%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM178^1 : 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 : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:51 AM UTC 2026
% Result : Theorem 2.57s 1.62s
% Output : Refutation 2.57s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM178^1 : TPTP v9.2.1. Released v7.0.0.
% 0.10/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.33 % Computer : n003.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Feb 24 04:15:55 EST 2026
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.43 ---- Original DTF file ---
% 0.15/0.43 thf(spec,logic,$$dhol).
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 % File : COM178^1 : TPTP v9.2.1. Released v7.0.0.
% 0.15/0.43 % Domain : Computing Theory
% 0.15/0.43 % Problem : Koenig's lemma 206
% 0.15/0.43 % Version : [Bla16] axioms : Especial.
% 0.15/0.43 % English :
% 0.15/0.43
% 0.15/0.43 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.15/0.43 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.15/0.43 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.15/0.43 % Source : [Bla16]
% 0.15/0.43 % Names : koenigslemma__206.p [Bla16]
% 0.15/0.43
% 0.15/0.43 % Status : Theorem
% 0.15/0.43 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.15/0.43 % Syntax : Number of formulae : 305 ( 140 unt; 44 typ; 0 def)
% 0.15/0.43 % Number of atoms : 673 ( 290 equ; 0 cnn)
% 0.15/0.43 % Maximal formula atoms : 8 ( 2 avg)
% 0.15/0.43 % Number of connectives : 4179 ( 161 ~; 53 |; 82 &;3599 @)
% 0.15/0.43 % ( 0 <=>; 284 =>; 0 <=; 0 <~>)
% 0.15/0.43 % Maximal formula depth : 28 ( 8 avg)
% 0.15/0.43 % Number of types : 2 ( 1 usr)
% 0.15/0.43 % Number of type conns : 287 ( 287 >; 0 *; 0 +; 0 <<)
% 0.15/0.43 % Number of symbols : 46 ( 43 usr; 4 con; 0-6 aty)
% 0.15/0.43 % Number of variables : 1150 ( 37 ^;1001 !; 68 ?;1150 :)
% 0.15/0.43 % ( 44 !>; 0 ?*; 0 @-; 0 @+)
% 0.15/0.43 % SPC : TH1_THM_EQU_NAR
% 0.15/0.43
% 0.15/0.43 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.15/0.43 % 2016-07-13 14:47:17.094
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 %----Could-be-implicit typings (5)
% 0.15/0.43 thf(ty_t_Coinductive__List_Ollist,type,
% 0.15/0.43 coinductive_llist: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_Product__Type_Oprod,type,
% 0.15/0.43 product_prod: $tType > $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_Set_Oset,type,
% 0.15/0.43 set: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_itself,type,
% 0.15/0.43 itself: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_tf_node,type,
% 0.15/0.43 node: $tType ).
% 0.15/0.43
% 0.15/0.43 %----Explicit typings (39)
% 0.15/0.43 thf(sy_cl_HOL_Otype,type,
% 0.15/0.43 type:
% 0.15/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Obot,type,
% 0.15/0.43 bot:
% 0.15/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 0.15/0.43 coindu328551480prefix:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 0.15/0.43 coinductive_gen_lset:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Olappend,type,
% 0.15/0.43 coinductive_lappend:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Oldistinct,type,
% 0.15/0.43 coindu351974385stinct:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_OldropWhile,type,
% 0.15/0.43 coindu218763757pWhile:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.15/0.43 coinductive_lfinite:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollast,type,
% 0.15/0.43 coinductive_llast:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollexord,type,
% 0.15/0.43 coinductive_llexord:
% 0.15/0.43 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.15/0.43 coinductive_LCons:
% 0.15/0.43 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.15/0.43 coinductive_LNil:
% 0.15/0.43 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 0.15/0.43 coindu1381640503_llist:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 0.15/0.43 coinductive_lhd:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.15/0.43 coinductive_lnull:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.15/0.43 coinductive_lset:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ollist_Oltl,type,
% 0.15/0.43 coinductive_ltl:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Olmember,type,
% 0.15/0.43 coinductive_lmember:
% 0.15/0.43 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Olprefix,type,
% 0.15/0.43 coinductive_lprefix:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 0.15/0.43 coindu1478340336prefix:
% 0.15/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 0.15/0.43 coindu501562517eWhile:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Olzip,type,
% 0.15/0.43 coinductive_lzip:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > ( coinductive_llist @ ( product_prod @ A @ B ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Oord_Olsorted,type,
% 0.15/0.43 coinductive_lsorted:
% 0.15/0.43 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.15/0.43 coindu1441602521_llist:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_If,type,
% 0.15/0.43 if:
% 0.15/0.43 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
% 0.15/0.43 koenig793108494nected:
% 0.15/0.43 !>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.15/0.43 bot_bot:
% 0.15/0.43 !>[A: $tType] : A ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_OPair,type,
% 0.15/0.43 product_Pair:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.15/0.43 produc2004651681e_prod:
% 0.15/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.15/0.43 product_rec_prod:
% 0.15/0.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Pure_Otype,type,
% 0.15/0.43 type2:
% 0.15/0.43 !>[A: $tType] : ( itself @ A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_Oinv__imagep,type,
% 0.15/0.43 inv_imagep:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_OCollect,type,
% 0.15/0.43 collect:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_Ois__empty,type,
% 0.15/0.43 is_empty:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_member,type,
% 0.15/0.43 member:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_f____,type,
% 0.15/0.43 f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_graph,type,
% 0.15/0.43 graph: node > node > $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_n,type,
% 0.15/0.43 n: node ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_ns____,type,
% 0.15/0.43 ns: set @ node ).
% 0.15/0.43
% 0.15/0.43 %----Relevant facts (254)
% 0.15/0.43 thf(fact_0__092_060open_062lhd_A_If_A_In_M_Ans_J_J_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
% 0.15/0.43 member @ node @ ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % \<open>lhd (f (n, ns)) \<in> lset (f (n, ns))\<close>
% 0.15/0.43 thf(fact_1_ns__def,axiom,
% 0.15/0.43 ( ns
% 0.15/0.43 = ( bot_bot @ ( set @ node ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ns_def
% 0.15/0.43 thf(fact_2_f__simps_I2_J,axiom,
% 0.15/0.43 ! [Na: node,Nsa: set @ node] :
% 0.15/0.43 ( ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) )
% 0.15/0.43 = Na ) ).
% 0.15/0.43
% 0.15/0.43 % f_simps(2)
% 0.15/0.43 thf(fact_3_f__simps_I1_J,axiom,
% 0.15/0.43 ! [Na: node,Nsa: set @ node] :
% 0.15/0.43 ~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % f_simps(1)
% 0.15/0.43 thf(fact_4_prod_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.15/0.43 = ( ( X1 = Y1 )
% 0.15/0.43 & ( X2 = Y2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod.inject
% 0.15/0.43 thf(fact_5_old_Oprod_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.15/0.43 = ( ( A2 = A3 )
% 0.15/0.43 & ( B2 = B3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.inject
% 0.15/0.43 thf(fact_6_connected,axiom,
% 0.15/0.43 koenig793108494nected @ node @ graph ).
% 0.15/0.43
% 0.15/0.43 % connected
% 0.15/0.43 thf(fact_7_surj__pair,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.15/0.43 ? [X: A,Y: B] :
% 0.15/0.43 ( P
% 0.15/0.43 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 0.15/0.43
% 0.15/0.43 % surj_pair
% 0.15/0.43 thf(fact_8_prod__cases,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.15/0.43 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.15/0.43 => ( P2 @ P ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases
% 0.15/0.43 thf(fact_9_Pair__inject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.15/0.43 => ~ ( ( A2 = A3 )
% 0.15/0.43 => ( B2 != B3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Pair_inject
% 0.15/0.43 thf(fact_10_prod__cases3,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.15/0.43 ~ ! [A4: A,B4: B,C2: C] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases3
% 0.15/0.43 thf(fact_11_prod__cases4,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.15/0.43 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases4
% 0.15/0.43 thf(fact_12_prod__cases5,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.15/0.43 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases5
% 0.15/0.43 thf(fact_13_prod__cases6,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.15/0.43 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases6
% 0.15/0.43 thf(fact_14_old_Oprod_Oinducts,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.15/0.43 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.15/0.43 => ( P2 @ Prod ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.inducts
% 0.15/0.43 thf(fact_15_old_Oprod_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.15/0.43 ~ ! [A4: A,B4: B] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.exhaust
% 0.15/0.43 thf(fact_16_prod__induct7,axiom,
% 0.15/0.43 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.15/0.43 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.15/0.43 => ( P2 @ X3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct7
% 0.15/0.43 thf(fact_17_prod__induct6,axiom,
% 0.15/0.43 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.15/0.43 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.15/0.43 => ( P2 @ X3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct6
% 0.15/0.43 thf(fact_18_prod__induct5,axiom,
% 0.15/0.43 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.15/0.43 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.15/0.43 => ( P2 @ X3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct5
% 0.15/0.43 thf(fact_19_prod__induct4,axiom,
% 0.15/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.15/0.43 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.15/0.43 => ( P2 @ X3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct4
% 0.15/0.43 thf(fact_20_prod__induct3,axiom,
% 0.15/0.43 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.15/0.43 ( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.15/0.43 => ( P2 @ X3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct3
% 0.15/0.43 thf(fact_21_prod__cases7,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.15/0.43 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases7
% 0.15/0.43 thf(fact_22_lset__eq__empty,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coinductive_lset @ A @ Xs )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_eq_empty
% 0.15/0.43 thf(fact_23_old_Oprod_Orec,axiom,
% 0.15/0.43 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.15/0.43 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.15/0.43 = ( F1 @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.rec
% 0.15/0.43 thf(fact_24_llist_Oset__sel_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,A2: coinductive_llist @ A] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ A2 )
% 0.15/0.43 => ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.set_sel(1)
% 0.15/0.43 thf(fact_25_lset__lnull,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( coinductive_lset @ A @ Xs )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_lnull
% 0.15/0.43 thf(fact_26_empty__iff,axiom,
% 0.15/0.43 ! [A: $tType,C3: A] :
% 0.15/0.43 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % empty_iff
% 0.15/0.43 thf(fact_27_all__not__in__conv,axiom,
% 0.15/0.43 ! [A: $tType,A5: set @ A] :
% 0.15/0.43 ( ( ! [X4: A] :
% 0.15/0.43 ~ ( member @ A @ X4 @ A5 ) )
% 0.15/0.43 = ( A5
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % all_not_in_conv
% 0.15/0.43 thf(fact_28_Collect__empty__eq,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o] :
% 0.15/0.43 ( ( ( collect @ A @ P2 )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( ! [X4: A] :
% 0.15/0.43 ~ ( P2 @ X4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_empty_eq
% 0.15/0.43 thf(fact_29_empty__Collect__eq,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o] :
% 0.15/0.43 ( ( ( bot_bot @ ( set @ A ) )
% 0.15/0.43 = ( collect @ A @ P2 ) )
% 0.15/0.43 = ( ! [X4: A] :
% 0.15/0.43 ~ ( P2 @ X4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % empty_Collect_eq
% 0.15/0.43 thf(fact_30_bot__apply,axiom,
% 0.15/0.43 ! [C: $tType,D: $tType] :
% 0.15/0.43 ( ( bot @ C @ ( type2 @ C ) )
% 0.15/0.43 => ( ( bot_bot @ ( D > C ) )
% 0.15/0.43 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_apply
% 0.15/0.43 thf(fact_31_ltakeWhile_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
% 0.15/0.43 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.exhaust
% 0.15/0.43 thf(fact_32_internal__case__prod__conv,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.15/0.43 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.15/0.43 = ( C3 @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % internal_case_prod_conv
% 0.15/0.43 thf(fact_33_bot__set__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( bot_bot @ ( set @ A ) )
% 0.15/0.43 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_set_def
% 0.15/0.43 thf(fact_34_bot__fun__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( bot @ B @ ( type2 @ B ) )
% 0.15/0.43 => ( ( bot_bot @ ( A > B ) )
% 0.15/0.43 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_fun_def
% 0.15/0.43 thf(fact_35_ex__in__conv,axiom,
% 0.15/0.43 ! [A: $tType,A5: set @ A] :
% 0.15/0.43 ( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
% 0.15/0.43 = ( A5
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ex_in_conv
% 0.15/0.43 thf(fact_36_equals0I,axiom,
% 0.15/0.43 ! [A: $tType,A5: set @ A] :
% 0.15/0.43 ( ! [Y: A] :
% 0.15/0.43 ~ ( member @ A @ Y @ A5 )
% 0.15/0.43 => ( A5
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % equals0I
% 0.15/0.43 thf(fact_37_equals0D,axiom,
% 0.15/0.43 ! [A: $tType,A5: set @ A,A2: A] :
% 0.15/0.43 ( ( A5
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 => ~ ( member @ A @ A2 @ A5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % equals0D
% 0.15/0.43 thf(fact_38_emptyE,axiom,
% 0.15/0.43 ! [A: $tType,A2: A] :
% 0.15/0.43 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % emptyE
% 0.15/0.43 thf(fact_39_lappend_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lappend.exhaust
% 0.15/0.43 thf(fact_40_lzip_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ( coinductive_lnull @ B @ Ys ) )
% 0.15/0.43 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.exhaust
% 0.15/0.43 thf(fact_41_lset__code,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lset @ A )
% 0.15/0.43 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_code
% 0.15/0.43 thf(fact_42_Set_Ois__empty__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( is_empty @ A )
% 0.15/0.43 = ( ^ [A6: set @ A] :
% 0.15/0.43 ( A6
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Set.is_empty_def
% 0.15/0.43 thf(fact_43_mem__Collect__eq,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,P2: A > $o] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.15/0.43 = ( P2 @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mem_Collect_eq
% 0.15/0.43 thf(fact_44_Collect__mem__eq,axiom,
% 0.15/0.43 ! [A: $tType,A5: set @ A] :
% 0.15/0.43 ( ( collect @ A
% 0.15/0.43 @ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
% 0.15/0.43 = A5 ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_mem_eq
% 0.15/0.43 thf(fact_45_Collect__cong,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.15/0.43 ( ! [X: A] :
% 0.15/0.43 ( ( P2 @ X )
% 0.15/0.43 = ( Q @ X ) )
% 0.15/0.43 => ( ( collect @ A @ P2 )
% 0.15/0.43 = ( collect @ A @ Q ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_cong
% 0.15/0.43 thf(fact_46_ext,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 0.15/0.43 ( ! [X: A] :
% 0.15/0.43 ( ( F3 @ X )
% 0.15/0.43 = ( G3 @ X ) )
% 0.15/0.43 => ( F3 = G3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % ext
% 0.15/0.43 thf(fact_47_lset__lmember,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 = ( coinductive_lmember @ A @ X3 @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_lmember
% 0.15/0.43 thf(fact_48_Collect__empty__eq__bot,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o] :
% 0.15/0.43 ( ( ( collect @ A @ P2 )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( P2
% 0.15/0.43 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_empty_eq_bot
% 0.15/0.43 thf(fact_49_bot__empty__eq,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( bot_bot @ ( A > $o ) )
% 0.15/0.43 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_empty_eq
% 0.15/0.43 thf(fact_50_ssubst__Pair__rhs,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 0.15/0.43 => ( ( S2 = S )
% 0.15/0.43 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ssubst_Pair_rhs
% 0.15/0.43 thf(fact_51_lhd__lzip,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ B @ Ys )
% 0.15/0.43 => ( ( coinductive_lhd @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 0.15/0.43 = ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ B @ Ys ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lhd_lzip
% 0.15/0.43 thf(fact_52_llist__set__induct,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: A > ( coinductive_llist @ A ) > $o] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( P2 @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A,Y: A] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( ( member @ A @ Y @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 0.15/0.43 => ( ( P2 @ Y @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.15/0.43 => ( P2 @ Y @ Xs2 ) ) ) )
% 0.15/0.43 => ( P2 @ X3 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist_set_induct
% 0.15/0.43 thf(fact_53_lnull__ldropWhile,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( ! [X4: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( P2 @ X4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_ldropWhile
% 0.15/0.43 thf(fact_54_lzip_Odisc__iff_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( ~ ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
% 0.15/0.43 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 & ~ ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.disc_iff(2)
% 0.15/0.43 thf(fact_55_lnull__lzip,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 0.15/0.43 = ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_lzip
% 0.15/0.43 thf(fact_56_ltl__lzip,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ B @ Ys )
% 0.15/0.43 => ( ( coinductive_ltl @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 0.15/0.43 = ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ B @ Ys ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltl_lzip
% 0.15/0.43 thf(fact_57_lnull__ltlI,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_ltlI
% 0.15/0.43 thf(fact_58_in__lset__ltlD,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
% 0.15/0.43 => ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % in_lset_ltlD
% 0.15/0.43 thf(fact_59_lzip_Odisc_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ B @ Ys )
% 0.15/0.43 => ~ ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.disc(2)
% 0.15/0.43 thf(fact_60_lzip_Odisc_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ( coinductive_lnull @ B @ Ys ) )
% 0.15/0.43 => ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.disc(1)
% 0.15/0.43 thf(fact_61_in__lset__ldropWhileD,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) )
% 0.15/0.43 => ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % in_lset_ldropWhileD
% 0.15/0.43 thf(fact_62_ldropWhile__cong,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P2: A > $o,Q: A > $o] :
% 0.15/0.43 ( ( Xs = Ys )
% 0.15/0.43 => ( ! [X: A] :
% 0.15/0.43 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.15/0.43 => ( ( P2 @ X )
% 0.15/0.43 = ( Q @ X ) ) )
% 0.15/0.43 => ( ( coindu218763757pWhile @ A @ P2 @ Xs )
% 0.15/0.43 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldropWhile_cong
% 0.15/0.43 thf(fact_63_llist_Oset__sel_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,A2: coinductive_llist @ A,X3: A] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ A2 )
% 0.15/0.43 => ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
% 0.15/0.43 => ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.set_sel(2)
% 0.15/0.43 thf(fact_64_lset__lzipD2,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X3: A,Y3: B,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( coinductive_lset @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
% 0.15/0.43 => ( member @ B @ Y3 @ ( coinductive_lset @ B @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_lzipD2
% 0.15/0.43 thf(fact_65_lset__lzipD1,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,X3: A,Y3: B,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( coinductive_lset @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
% 0.15/0.43 => ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_lzipD1
% 0.15/0.43 thf(fact_66_llist_Oexpand,axiom,
% 0.15/0.43 ! [A: $tType,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Llist )
% 0.15/0.43 = ( coinductive_lnull @ A @ Llist2 ) )
% 0.15/0.43 => ( ( ~ ( coinductive_lnull @ A @ Llist )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Llist2 )
% 0.15/0.43 => ( ( ( coinductive_lhd @ A @ Llist )
% 0.15/0.43 = ( coinductive_lhd @ A @ Llist2 ) )
% 0.15/0.43 & ( ( coinductive_ltl @ A @ Llist )
% 0.15/0.43 = ( coinductive_ltl @ A @ Llist2 ) ) ) ) )
% 0.15/0.43 => ( Llist = Llist2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.expand
% 0.15/0.43 thf(fact_67_llist_Ocoinduct,axiom,
% 0.15/0.43 ! [A: $tType,R2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 0.15/0.43 ( ( R2 @ Llist @ Llist2 )
% 0.15/0.43 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 0.15/0.43 ( ( R2 @ Llist3 @ Llist4 )
% 0.15/0.43 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 0.15/0.43 = ( coinductive_lnull @ A @ Llist4 ) )
% 0.15/0.43 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 0.15/0.43 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 0.15/0.43 = ( coinductive_lhd @ A @ Llist4 ) )
% 0.15/0.43 & ( R2 @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
% 0.15/0.43 => ( Llist = Llist2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.coinduct
% 0.15/0.43 thf(fact_68_llist_Ocoinduct__strong,axiom,
% 0.15/0.43 ! [A: $tType,R2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 0.15/0.43 ( ( R2 @ Llist @ Llist2 )
% 0.15/0.43 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 0.15/0.43 ( ( R2 @ Llist3 @ Llist4 )
% 0.15/0.43 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 0.15/0.43 = ( coinductive_lnull @ A @ Llist4 ) )
% 0.15/0.43 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 0.15/0.43 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 0.15/0.43 = ( coinductive_lhd @ A @ Llist4 ) )
% 0.15/0.43 & ( ( R2 @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
% 0.15/0.43 | ( ( coinductive_ltl @ A @ Llist3 )
% 0.15/0.43 = ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
% 0.15/0.43 => ( Llist = Llist2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.coinduct_strong
% 0.15/0.43 thf(fact_69_lhd__ldropWhile__in__lset,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ? [X5: A] :
% 0.15/0.43 ( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 & ~ ( P2 @ X5 ) )
% 0.15/0.43 => ( member @ A @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lhd_ldropWhile_in_lset
% 0.15/0.43 thf(fact_70_lhd__ldropWhile,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ? [X5: A] :
% 0.15/0.43 ( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 & ~ ( P2 @ X5 ) )
% 0.15/0.43 => ~ ( P2 @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lhd_ldropWhile
% 0.15/0.43 thf(fact_71_lzip_Octr_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ B @ Ys )
% 0.15/0.43 => ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 0.15/0.43 = ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ B @ Ys ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ B @ Ys ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.ctr(2)
% 0.15/0.43 thf(fact_72_unfold__llist__id,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
% 0.15/0.43 = Xs ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist_id
% 0.15/0.43 thf(fact_73_ldistinct__lhdD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_lhdD
% 0.15/0.43 thf(fact_74_ldistinct__coinduct,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ Xs )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ Xs2 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 0.15/0.43 & ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.15/0.43 | ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
% 0.15/0.43 => ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_coinduct
% 0.15/0.43 thf(fact_75_llist_Ocase__eq__if,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( coindu1381640503_llist @ B @ A )
% 0.15/0.43 = ( ^ [F12: B,F22: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F12 @ ( F22 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.case_eq_if
% 0.15/0.43 thf(fact_76_lhd__LCons__ltl,axiom,
% 0.15/0.43 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Llist )
% 0.15/0.43 => ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
% 0.15/0.43 = Llist ) ) ).
% 0.15/0.43
% 0.15/0.43 % lhd_LCons_ltl
% 0.15/0.43 thf(fact_77_ord_Olsorted__lhdD,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lsorted @ A @ Less_eq @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) )
% 0.15/0.43 => ( Less_eq @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_lhdD
% 0.15/0.43 thf(fact_78_ord_Olsorted__coinduct_H,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Less_eq: A > A > $o] :
% 0.15/0.43 ( ( X6 @ Xs )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ Xs2 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.15/0.43 => ( ( Less_eq @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 0.15/0.43 & ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.15/0.43 | ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) ) )
% 0.15/0.43 => ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_coinduct'
% 0.15/0.43 thf(fact_79_llist_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 0.15/0.43 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.15/0.43 = ( ( X21 = Y21 )
% 0.15/0.43 & ( X22 = Y22 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.inject
% 0.15/0.43 thf(fact_80_ldropWhile__LCons,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( P2 @ X3 )
% 0.15/0.43 => ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( coindu218763757pWhile @ A @ P2 @ Xs ) ) )
% 0.15/0.43 & ( ~ ( P2 @ X3 )
% 0.15/0.43 => ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldropWhile_LCons
% 0.15/0.43 thf(fact_81_unfold__llist__eq__LCons,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 0.15/0.43 = ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( ~ ( IS_LNIL @ B2 )
% 0.15/0.43 & ( X3
% 0.15/0.43 = ( LHD @ B2 ) )
% 0.15/0.43 & ( Xs
% 0.15/0.43 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist_eq_LCons
% 0.15/0.43 thf(fact_82_unfold__llist_Odisc__iff_I1_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
% 0.15/0.43 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
% 0.15/0.43 = ( P @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.disc_iff(1)
% 0.15/0.43 thf(fact_83_unfold__llist_Odisc__iff_I2_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
% 0.15/0.43 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) )
% 0.15/0.43 = ( ~ ( P @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.disc_iff(2)
% 0.15/0.43 thf(fact_84_ldistinct__LCons,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 & ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_LCons
% 0.15/0.43 thf(fact_85_lzip__simps_I3_J,axiom,
% 0.15/0.43 ! [C: $tType,B: $tType,X3: C,Xs: coinductive_llist @ C,Y3: B,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( coinductive_lzip @ C @ B @ ( coinductive_LCons @ C @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
% 0.15/0.43 = ( coinductive_LCons @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Y3 ) @ ( coinductive_lzip @ C @ B @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip_simps(3)
% 0.15/0.43 thf(fact_86_ldistinct__lzipI2,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ B] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ Ys )
% 0.15/0.43 => ( coindu351974385stinct @ ( product_prod @ B @ A ) @ ( coinductive_lzip @ B @ A @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_lzipI2
% 0.15/0.43 thf(fact_87_ldistinct__lzipI1,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ Xs )
% 0.15/0.43 => ( coindu351974385stinct @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_lzipI1
% 0.15/0.43 thf(fact_88_llist_Osimps_I5_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,F1: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.15/0.43 = ( F23 @ X21 @ X22 ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.simps(5)
% 0.15/0.43 thf(fact_89_unfold__llist_Octr_I2_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.43 ( ~ ( P @ A2 )
% 0.15/0.43 => ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
% 0.15/0.43 = ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.ctr(2)
% 0.15/0.43 thf(fact_90_ldistinct_OLCons,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( ( coindu351974385stinct @ A @ Xs )
% 0.15/0.43 => ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct.LCons
% 0.15/0.43 thf(fact_91_ord_OLCons__LCons,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( Less_eq @ X3 @ Y3 )
% 0.15/0.43 => ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs ) )
% 0.15/0.43 => ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.LCons_LCons
% 0.15/0.43 thf(fact_92_ord_Olsorted__LCons__LCons,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
% 0.15/0.43 = ( ( Less_eq @ X3 @ Y3 )
% 0.15/0.43 & ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_LCons_LCons
% 0.15/0.43 thf(fact_93_lzip__eq__LCons__conv,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Z: product_prod @ A @ B,Zs: coinductive_llist @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 0.15/0.43 = ( coinductive_LCons @ ( product_prod @ A @ B ) @ Z @ Zs ) )
% 0.15/0.43 = ( ? [X4: A,Xs3: coinductive_llist @ A,Y4: B,Ys2: coinductive_llist @ B] :
% 0.15/0.43 ( ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs3 ) )
% 0.15/0.43 & ( Ys
% 0.15/0.43 = ( coinductive_LCons @ B @ Y4 @ Ys2 ) )
% 0.15/0.43 & ( Z
% 0.15/0.43 = ( product_Pair @ A @ B @ X4 @ Y4 ) )
% 0.15/0.43 & ( Zs
% 0.15/0.43 = ( coinductive_lzip @ A @ B @ Xs3 @ Ys2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip_eq_LCons_conv
% 0.15/0.43 thf(fact_94_ord_Olsorted__LCons_H,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( Less_eq @ X3 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 & ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_LCons'
% 0.15/0.43 thf(fact_95_not__lnull__conv,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 0.15/0.43 = ( ? [X4: A,Xs3: coinductive_llist @ A] :
% 0.15/0.43 ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % not_lnull_conv
% 0.15/0.43 thf(fact_96_llist_OdiscI_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 0.15/0.43 ( ( Llist
% 0.15/0.43 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.15/0.43 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.discI(2)
% 0.15/0.43 thf(fact_97_llist_Odisc_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.15/0.43 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.disc(2)
% 0.15/0.43 thf(fact_98_lset__intros_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,X7: A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_intros(2)
% 0.15/0.43 thf(fact_99_lset__intros_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] : ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_intros(1)
% 0.15/0.43 thf(fact_100_llist_Oset__intros_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,A22: coinductive_llist @ A,A1: A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ A22 ) )
% 0.15/0.43 => ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.set_intros(2)
% 0.15/0.43 thf(fact_101_llist_Oset__intros_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.set_intros(1)
% 0.15/0.43 thf(fact_102_lset__cases,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( ! [Xs4: coinductive_llist @ A] :
% 0.15/0.43 ( Xs
% 0.15/0.43 != ( coinductive_LCons @ A @ X3 @ Xs4 ) )
% 0.15/0.43 => ~ ! [X8: A,Xs4: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ X8 @ Xs4 ) )
% 0.15/0.43 => ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_cases
% 0.15/0.43 thf(fact_103_lset__induct,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] : ( P2 @ ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 0.15/0.43 => ( ! [X8: A,Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
% 0.15/0.43 => ( ( X3 != X8 )
% 0.15/0.43 => ( ( P2 @ Xs2 )
% 0.15/0.43 => ( P2 @ ( coinductive_LCons @ A @ X8 @ Xs2 ) ) ) ) )
% 0.15/0.43 => ( P2 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_induct
% 0.15/0.43 thf(fact_104_lset__induct_H,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] : ( P2 @ ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 0.15/0.43 => ( ! [X8: A,Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
% 0.15/0.43 => ( ( P2 @ Xs2 )
% 0.15/0.43 => ( P2 @ ( coinductive_LCons @ A @ X8 @ Xs2 ) ) ) )
% 0.15/0.43 => ( P2 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_induct'
% 0.15/0.43 thf(fact_105_llist_Oset__cases,axiom,
% 0.15/0.43 ! [A: $tType,E3: A,A2: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ E3 @ ( coinductive_lset @ A @ A2 ) )
% 0.15/0.43 => ( ! [Z2: coinductive_llist @ A] :
% 0.15/0.43 ( A2
% 0.15/0.43 != ( coinductive_LCons @ A @ E3 @ Z2 ) )
% 0.15/0.43 => ~ ! [Z1: A,Z2: coinductive_llist @ A] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 0.15/0.43 => ~ ( member @ A @ E3 @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.set_cases
% 0.15/0.43 thf(fact_106_llist_Oset__induct,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,A2: coinductive_llist @ A,P2: A > ( coinductive_llist @ A ) > $o] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) )
% 0.15/0.43 => ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P2 @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 0.15/0.43 => ( ! [Z1: A,Z2: coinductive_llist @ A,Xa: A] :
% 0.15/0.43 ( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z2 ) )
% 0.15/0.43 => ( ( P2 @ Xa @ Z2 )
% 0.15/0.43 => ( P2 @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
% 0.15/0.43 => ( P2 @ X3 @ A2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.set_induct
% 0.15/0.43 thf(fact_107_ltl__simps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.15/0.43 = X22 ) ).
% 0.15/0.43
% 0.15/0.43 % ltl_simps(2)
% 0.15/0.43 thf(fact_108_lhd__LCons,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.15/0.43 = X21 ) ).
% 0.15/0.43
% 0.15/0.43 % lhd_LCons
% 0.15/0.43 thf(fact_109_ldistinct__ltlI,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ Xs )
% 0.15/0.43 => ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_ltlI
% 0.15/0.43 thf(fact_110_ord_Olsorted__ltlI,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lsorted @ A @ Less_eq @ Xs )
% 0.15/0.43 => ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_ltlI
% 0.15/0.43 thf(fact_111_unfold__llist_Odisc_I1_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.43 ( ( P @ A2 )
% 0.15/0.43 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.disc(1)
% 0.15/0.43 thf(fact_112_unfold__llist_Odisc_I2_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.43 ( ~ ( P @ A2 )
% 0.15/0.43 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.disc(2)
% 0.15/0.43 thf(fact_113_unfold__llist_Osimps_I4_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.43 ( ~ ( P @ A2 )
% 0.15/0.43 => ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
% 0.15/0.43 = ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.simps(4)
% 0.15/0.43 thf(fact_114_unfold__llist_Osimps_I3_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.43 ( ~ ( P @ A2 )
% 0.15/0.43 => ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
% 0.15/0.43 = ( G21 @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.simps(3)
% 0.15/0.43 thf(fact_115_lmember__code_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lmember @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( X3 = Y3 )
% 0.15/0.43 | ( coinductive_lmember @ A @ X3 @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lmember_code(2)
% 0.15/0.43 thf(fact_116_lzip_Ocode,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( coinductive_lzip @ A @ B )
% 0.15/0.43 = ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ B] :
% 0.15/0.43 ( if @ ( coinductive_llist @ ( product_prod @ A @ B ) )
% 0.15/0.43 @ ( ( coinductive_lnull @ A @ Xs5 )
% 0.15/0.43 | ( coinductive_lnull @ B @ Ys3 ) )
% 0.15/0.43 @ ( coinductive_LNil @ ( product_prod @ A @ B ) )
% 0.15/0.43 @ ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs5 ) @ ( coinductive_lhd @ B @ Ys3 ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs5 ) @ ( coinductive_ltl @ B @ Ys3 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.code
% 0.15/0.43 thf(fact_117_llist_Osplit__sel,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P2: B > $o,F1: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 0.15/0.43 ( ( P2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ Llist ) )
% 0.15/0.43 = ( ( ( Llist
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 => ( P2 @ F1 ) )
% 0.15/0.43 & ( ( Llist
% 0.15/0.43 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
% 0.15/0.43 => ( P2 @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.split_sel
% 0.15/0.43 thf(fact_118_llist_Osplit__sel__asm,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P2: B > $o,F1: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 0.15/0.43 ( ( P2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ Llist ) )
% 0.15/0.43 = ( ~ ( ( ( Llist
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 & ~ ( P2 @ F1 ) )
% 0.15/0.43 | ( ( Llist
% 0.15/0.43 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
% 0.15/0.43 & ~ ( P2 @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.split_sel_asm
% 0.15/0.43 thf(fact_119_ltakeWhile_Octr_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 0.15/0.43 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.ctr(2)
% 0.15/0.43 thf(fact_120_llexord__coinduct,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R: A > A > $o] :
% 0.15/0.43 ( ( X6 @ Xs @ Ys )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ Xs2 @ Ys4 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Ys4 )
% 0.15/0.43 & ( ~ ( coinductive_lnull @ A @ Ys4 )
% 0.15/0.43 => ( ( R @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ Ys4 ) )
% 0.15/0.43 | ( ( ( coinductive_lhd @ A @ Xs2 )
% 0.15/0.43 = ( coinductive_lhd @ A @ Ys4 ) )
% 0.15/0.43 & ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) )
% 0.15/0.43 | ( coinductive_llexord @ A @ R @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) ) ) ) ) ) ) ) )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_coinduct
% 0.15/0.43 thf(fact_121_ldistinct_Ocases,axiom,
% 0.15/0.43 ! [A: $tType,A2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ A2 )
% 0.15/0.43 => ( ( A2
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 => ~ ! [X: A,Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.15/0.43 => ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 0.15/0.43 => ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct.cases
% 0.15/0.43 thf(fact_122_llexord__refl,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ Xs ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_refl
% 0.15/0.43 thf(fact_123_ltakeWhile__LNil,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o] :
% 0.15/0.43 ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile_LNil
% 0.15/0.43 thf(fact_124_llexord__LCons__LCons,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( ( X3 = Y3 )
% 0.15/0.43 & ( coinductive_llexord @ A @ R @ Xs @ Ys ) )
% 0.15/0.43 | ( R @ X3 @ Y3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_LCons_LCons
% 0.15/0.43 thf(fact_125_ldistinct__LNil__code,axiom,
% 0.15/0.43 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_LNil_code
% 0.15/0.43 thf(fact_126_llexord__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_code(1)
% 0.15/0.43 thf(fact_127_llexord__LNil__right,axiom,
% 0.15/0.43 ! [A: $tType,Ys: coinductive_llist @ A,R: A > A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Ys )
% 0.15/0.43 => ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 0.15/0.43 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_LNil_right
% 0.15/0.43 thf(fact_128_ldropWhile__LNil,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o] :
% 0.15/0.43 ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldropWhile_LNil
% 0.15/0.43 thf(fact_129_ltakeWhile__LCons,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( P2 @ X3 )
% 0.15/0.43 => ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( coinductive_LCons @ A @ X3 @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) )
% 0.15/0.43 & ( ~ ( P2 @ X3 )
% 0.15/0.43 => ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile_LCons
% 0.15/0.43 thf(fact_130_ltakeWhile_Odisc__iff_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) )
% 0.15/0.43 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 & ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.disc_iff(2)
% 0.15/0.43 thf(fact_131_ltakeWhile_Odisc__iff_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.disc_iff(1)
% 0.15/0.43 thf(fact_132_lnull__ltakeWhile,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_ltakeWhile
% 0.15/0.43 thf(fact_133_lzip__simps_I2_J,axiom,
% 0.15/0.43 ! [D: $tType,C: $tType,Xs: coinductive_llist @ C] :
% 0.15/0.43 ( ( coinductive_lzip @ C @ D @ Xs @ ( coinductive_LNil @ D ) )
% 0.15/0.43 = ( coinductive_LNil @ ( product_prod @ C @ D ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip_simps(2)
% 0.15/0.43 thf(fact_134_lzip__simps_I1_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( coinductive_lzip @ A @ B @ ( coinductive_LNil @ A ) @ Ys )
% 0.15/0.43 = ( coinductive_LNil @ ( product_prod @ A @ B ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip_simps(1)
% 0.15/0.43 thf(fact_135_llexord__code_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ~ ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_code(2)
% 0.15/0.43 thf(fact_136_llexord_Ocases,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ A1 @ A22 )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A,X: A] :
% 0.15/0.43 ( ( A1
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.15/0.43 => ( ( A22
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 0.15/0.43 => ~ ( coinductive_llexord @ A @ R @ Xs2 @ Ys4 ) ) )
% 0.15/0.43 => ( ! [X: A] :
% 0.15/0.43 ( ? [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( A1
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.15/0.43 => ! [Y: A] :
% 0.15/0.43 ( ? [Ys4: coinductive_llist @ A] :
% 0.15/0.43 ( A22
% 0.15/0.43 = ( coinductive_LCons @ A @ Y @ Ys4 ) )
% 0.15/0.43 => ~ ( R @ X @ Y ) ) )
% 0.15/0.43 => ~ ( ( A1
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 => ! [Ys4: coinductive_llist @ A] : ( A22 != Ys4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord.cases
% 0.15/0.43 thf(fact_137_llexord_Osimps,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_llexord @ A )
% 0.15/0.43 = ( ^ [R3: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 0.15/0.43 ( ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X4: A] :
% 0.15/0.43 ( ( A12
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 0.15/0.43 & ( A23
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Ys3 ) )
% 0.15/0.43 & ( coinductive_llexord @ A @ R3 @ Xs5 @ Ys3 ) )
% 0.15/0.43 | ? [X4: A,Y4: A,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 0.15/0.43 ( ( A12
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 0.15/0.43 & ( A23
% 0.15/0.43 = ( coinductive_LCons @ A @ Y4 @ Ys3 ) )
% 0.15/0.43 & ( R3 @ X4 @ Y4 ) )
% 0.15/0.43 | ? [Ys3: coinductive_llist @ A] :
% 0.15/0.43 ( ( A12
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 & ( A23 = Ys3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord.simps
% 0.15/0.43 thf(fact_138_llexord_Ocoinduct,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa2: coinductive_llist @ A,R: A > A > $o] :
% 0.15/0.43 ( ( X6 @ X3 @ Xa2 )
% 0.15/0.43 => ( ! [X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ X @ Xa )
% 0.15/0.43 => ( ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LCons @ A @ Xb @ Xs6 ) )
% 0.15/0.43 & ( Xa
% 0.15/0.43 = ( coinductive_LCons @ A @ Xb @ Ys5 ) )
% 0.15/0.43 & ( ( X6 @ Xs6 @ Ys5 )
% 0.15/0.43 | ( coinductive_llexord @ A @ R @ Xs6 @ Ys5 ) ) )
% 0.15/0.43 | ? [Xb: A,Y5: A,Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LCons @ A @ Xb @ Xs6 ) )
% 0.15/0.43 & ( Xa
% 0.15/0.43 = ( coinductive_LCons @ A @ Y5 @ Ys5 ) )
% 0.15/0.43 & ( R @ Xb @ Y5 ) )
% 0.15/0.43 | ? [Ys5: coinductive_llist @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 & ( Xa = Ys5 ) ) ) )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ X3 @ Xa2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord.coinduct
% 0.15/0.43 thf(fact_139_llexord__LNil,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_LNil
% 0.15/0.43 thf(fact_140_llexord__antisym,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_llexord @ A @ R @ Ys @ Xs )
% 0.15/0.43 => ( ! [A4: A,B4: A] :
% 0.15/0.43 ( ( R @ A4 @ B4 )
% 0.15/0.43 => ~ ( R @ B4 @ A4 ) )
% 0.15/0.43 => ( Xs = Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_antisym
% 0.15/0.43 thf(fact_141_llexord__linear,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ! [X: A,Y: A] :
% 0.15/0.43 ( ( R @ X @ Y )
% 0.15/0.43 | ( X = Y )
% 0.15/0.43 | ( R @ Y @ X ) )
% 0.15/0.43 => ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 0.15/0.43 | ( coinductive_llexord @ A @ R @ Ys @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_linear
% 0.15/0.43 thf(fact_142_llexord__trans,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_llexord @ A @ R @ Ys @ Zs )
% 0.15/0.43 => ( ! [A4: A,B4: A,C2: A] :
% 0.15/0.43 ( ( R @ A4 @ B4 )
% 0.15/0.43 => ( ( R @ B4 @ C2 )
% 0.15/0.43 => ( R @ A4 @ C2 ) ) )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ Xs @ Zs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_trans
% 0.15/0.43 thf(fact_143_lzip__eq__LNil__conv,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 0.15/0.43 = ( coinductive_LNil @ ( product_prod @ A @ B ) ) )
% 0.15/0.43 = ( ( Xs
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ( Ys
% 0.15/0.43 = ( coinductive_LNil @ B ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip_eq_LNil_conv
% 0.15/0.43 thf(fact_144_ltakeWhile__eq__LNil__iff,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( ( Xs
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 => ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile_eq_LNil_iff
% 0.15/0.43 thf(fact_145_ltakeWhile_Octr_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
% 0.15/0.43 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.ctr(1)
% 0.15/0.43 thf(fact_146_ltl__ltakeWhile,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) )
% 0.15/0.43 & ( ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltl_ltakeWhile
% 0.15/0.43 thf(fact_147_lset__ltakeWhileD,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) )
% 0.15/0.43 => ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 & ( P2 @ X3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_ltakeWhileD
% 0.15/0.43 thf(fact_148_ltakeWhile__cong,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P2: A > $o,Q: A > $o] :
% 0.15/0.43 ( ( Xs = Ys )
% 0.15/0.43 => ( ! [X: A] :
% 0.15/0.43 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.15/0.43 => ( ( P2 @ X )
% 0.15/0.43 = ( Q @ X ) ) )
% 0.15/0.43 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 0.15/0.43 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile_cong
% 0.15/0.43 thf(fact_149_ltakeWhile__all,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ! [X: A] :
% 0.15/0.43 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( P2 @ X ) )
% 0.15/0.43 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 0.15/0.43 = Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile_all
% 0.15/0.43 thf(fact_150_neq__LNil__conv,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( ? [X4: A,Xs3: coinductive_llist @ A] :
% 0.15/0.43 ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % neq_LNil_conv
% 0.15/0.43 thf(fact_151_llist_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,Y3: coinductive_llist @ A] :
% 0.15/0.43 ( ( Y3
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 0.15/0.43 ( Y3
% 0.15/0.43 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.exhaust
% 0.15/0.43 thf(fact_152_llist_Odistinct_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_LNil @ A )
% 0.15/0.43 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.distinct(1)
% 0.15/0.43 thf(fact_153_llist_Odisc_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.disc(1)
% 0.15/0.43 thf(fact_154_llist_OdiscI_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.15/0.43 ( ( Llist
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 => ( coinductive_lnull @ A @ Llist ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.discI(1)
% 0.15/0.43 thf(fact_155_llist_Ocollapse_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Llist )
% 0.15/0.43 => ( Llist
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.collapse(1)
% 0.15/0.43 thf(fact_156_lnull__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lnull @ A )
% 0.15/0.43 = ( ^ [Llist5: coinductive_llist @ A] :
% 0.15/0.43 ( Llist5
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_def
% 0.15/0.43 thf(fact_157_ltl__simps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltl_simps(1)
% 0.15/0.43 thf(fact_158_llexord__LCons__less,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( R @ X3 @ Y3 )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_LCons_less
% 0.15/0.43 thf(fact_159_llexord__LCons__eq,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_LCons_eq
% 0.15/0.43 thf(fact_160_llexord__LCons__left,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
% 0.15/0.43 = ( ? [Y4: A,Ys2: coinductive_llist @ A] :
% 0.15/0.43 ( ( Ys
% 0.15/0.43 = ( coinductive_LCons @ A @ Y4 @ Ys2 ) )
% 0.15/0.43 & ( ( ( X3 = Y4 )
% 0.15/0.43 & ( coinductive_llexord @ A @ R @ Xs @ Ys2 ) )
% 0.15/0.43 | ( R @ X3 @ Y4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_LCons_left
% 0.15/0.43 thf(fact_161_llexord__code_I3_J,axiom,
% 0.15/0.43 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( R @ X3 @ Y3 )
% 0.15/0.43 | ( ( X3 = Y3 )
% 0.15/0.43 & ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_code(3)
% 0.15/0.43 thf(fact_162_lnull__llexord,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_llexord
% 0.15/0.43 thf(fact_163_ord_OLNil,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.LNil
% 0.15/0.43 thf(fact_164_ord_Olsorted__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_code(1)
% 0.15/0.43 thf(fact_165_llist_Osimps_I4_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,F1: B,F23: A > ( coinductive_llist @ A ) > B] :
% 0.15/0.43 ( ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = F1 ) ).
% 0.15/0.43
% 0.15/0.43 % llist.simps(4)
% 0.15/0.43 thf(fact_166_ldistinct_OLNil,axiom,
% 0.15/0.43 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct.LNil
% 0.15/0.43 thf(fact_167_unfold__llist_Octr_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.43 ( ( P @ A2 )
% 0.15/0.43 => ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
% 0.15/0.43 = ( coinductive_LNil @ B ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.ctr(1)
% 0.15/0.43 thf(fact_168_gen__lset__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,A5: set @ A] :
% 0.15/0.43 ( ( coinductive_gen_lset @ A @ A5 @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = A5 ) ).
% 0.15/0.43
% 0.15/0.43 % gen_lset_code(1)
% 0.15/0.43 thf(fact_169_lmember__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,X3: A] :
% 0.15/0.43 ~ ( coinductive_lmember @ A @ X3 @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % lmember_code(1)
% 0.15/0.43 thf(fact_170_ltakeWhile_Ocode,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coindu501562517eWhile @ A )
% 0.15/0.43 = ( ^ [P3: A > $o,Xs5: coinductive_llist @ A] :
% 0.15/0.43 ( if @ ( coinductive_llist @ A )
% 0.15/0.43 @ ( ( coinductive_lnull @ A @ Xs5 )
% 0.15/0.43 | ~ ( P3 @ ( coinductive_lhd @ A @ Xs5 ) ) )
% 0.15/0.43 @ ( coinductive_LNil @ A )
% 0.15/0.43 @ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs5 ) @ ( coindu501562517eWhile @ A @ P3 @ ( coinductive_ltl @ A @ Xs5 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.code
% 0.15/0.43 thf(fact_171_ltakeWhile_Odisc_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 => ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.disc(2)
% 0.15/0.43 thf(fact_172_ltakeWhile_Odisc_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
% 0.15/0.43 => ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.disc(1)
% 0.15/0.43 thf(fact_173_lhd__ltakeWhile,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 => ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lhd_ltakeWhile
% 0.15/0.43 thf(fact_174_lset__LNil,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lset @ A @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lset_LNil
% 0.15/0.43 thf(fact_175_ord_OSingleton,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.Singleton
% 0.15/0.43 thf(fact_176_ord_Olsorted_Ocoinduct,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Less_eq: A > A > $o] :
% 0.15/0.43 ( ( X6 @ X3 )
% 0.15/0.43 => ( ! [X: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ X )
% 0.15/0.43 => ( ( X
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [Xa3: A] :
% 0.15/0.43 ( X
% 0.15/0.43 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
% 0.15/0.43 | ? [Xa3: A,Y5: A,Xs6: coinductive_llist @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) ) )
% 0.15/0.43 & ( Less_eq @ Xa3 @ Y5 )
% 0.15/0.43 & ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) )
% 0.15/0.43 | ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) ) ) ) ) )
% 0.15/0.43 => ( coinductive_lsorted @ A @ Less_eq @ X3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted.coinduct
% 0.15/0.43 thf(fact_177_ord_Olsorted_Osimps,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lsorted @ A )
% 0.15/0.43 = ( ^ [Less_eq2: A > A > $o,A7: coinductive_llist @ A] :
% 0.15/0.43 ( ( A7
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [X4: A] :
% 0.15/0.43 ( A7
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ ( coinductive_LNil @ A ) ) )
% 0.15/0.43 | ? [X4: A,Y4: A,Xs5: coinductive_llist @ A] :
% 0.15/0.43 ( ( A7
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
% 0.15/0.43 & ( Less_eq2 @ X4 @ Y4 )
% 0.15/0.43 & ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted.simps
% 0.15/0.43 thf(fact_178_ord_Olsorted_Ocases,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,A2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lsorted @ A @ Less_eq @ A2 )
% 0.15/0.43 => ( ( A2
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 => ( ! [X: A] :
% 0.15/0.43 ( A2
% 0.15/0.43 != ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 0.15/0.43 => ~ ! [X: A,Y: A,Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs2 ) ) )
% 0.15/0.43 => ( ( Less_eq @ X @ Y )
% 0.15/0.43 => ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted.cases
% 0.15/0.43 thf(fact_179_ord_Olsorted__code_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_code(2)
% 0.15/0.43 thf(fact_180_lzip_Octr_I1_J,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 | ( coinductive_lnull @ B @ Ys ) )
% 0.15/0.43 => ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 0.15/0.43 = ( coinductive_LNil @ ( product_prod @ A @ B ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lzip.ctr(1)
% 0.15/0.43 thf(fact_181_ldropWhile__eq__LNil__iff,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coindu218763757pWhile @ A @ P2 @ Xs )
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( ! [X4: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 => ( P2 @ X4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldropWhile_eq_LNil_iff
% 0.15/0.43 thf(fact_182_unfold__llist_Ocode,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( coindu1441602521_llist @ A @ B )
% 0.15/0.43 = ( ^ [P4: A > $o,G212: A > B,G222: A > A,A7: A] : ( if @ ( coinductive_llist @ B ) @ ( P4 @ A7 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A7 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G212 @ G222 @ ( G222 @ A7 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % unfold_llist.code
% 0.15/0.43 thf(fact_183_ltl__unfold__llist,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
% 0.15/0.43 ( ( ( IS_LNIL @ A2 )
% 0.15/0.43 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 0.15/0.43 = ( coinductive_LNil @ A ) ) )
% 0.15/0.43 & ( ~ ( IS_LNIL @ A2 )
% 0.15/0.43 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 0.15/0.43 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltl_unfold_llist
% 0.15/0.43 thf(fact_184_ltakeWhile_Osimps_I4_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.15/0.43 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 0.15/0.43 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ltakeWhile.simps(4)
% 0.15/0.43 thf(fact_185_eq__LConsD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 => ( ( Xs
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 & ( ( coinductive_lhd @ A @ Xs )
% 0.15/0.43 = Y3 )
% 0.15/0.43 & ( ( coinductive_ltl @ A @ Xs )
% 0.15/0.43 = Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_LConsD
% 0.15/0.43 thf(fact_186_llist_Oexhaust__sel,axiom,
% 0.15/0.43 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.15/0.43 ( ( Llist
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 => ( Llist
% 0.15/0.43 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.exhaust_sel
% 0.15/0.43 thf(fact_187_ldistinct_Ocoinduct,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ X3 )
% 0.15/0.43 => ( ! [X: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ X )
% 0.15/0.43 => ( ( X
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [Xa3: A,Xs6: coinductive_llist @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LCons @ A @ Xa3 @ Xs6 ) )
% 0.15/0.43 & ~ ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Xs6 ) )
% 0.15/0.43 & ( ( X6 @ Xs6 )
% 0.15/0.43 | ( coindu351974385stinct @ A @ Xs6 ) ) ) ) )
% 0.15/0.43 => ( coindu351974385stinct @ A @ X3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct.coinduct
% 0.15/0.43 thf(fact_188_ldistinct_Osimps,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coindu351974385stinct @ A )
% 0.15/0.43 = ( ^ [A7: coinductive_llist @ A] :
% 0.15/0.43 ( ( A7
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [X4: A,Xs5: coinductive_llist @ A] :
% 0.15/0.43 ( ( A7
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 0.15/0.43 & ~ ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs5 ) )
% 0.15/0.43 & ( coindu351974385stinct @ A @ Xs5 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct.simps
% 0.15/0.43 thf(fact_189_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( Xs
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [Xs3: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ Y3 @ Xs3 ) )
% 0.15/0.43 & ( coindu328551480prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Coinductive_List.finite_lprefix_nitpick_simps(3)
% 0.15/0.43 thf(fact_190_lstrict__prefix__code_I3_J,axiom,
% 0.15/0.43 ! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
% 0.15/0.43 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 0.15/0.43
% 0.15/0.43 % lstrict_prefix_code(3)
% 0.15/0.43 thf(fact_191_lstrict__prefix__code_I2_J,axiom,
% 0.15/0.43 ! [B: $tType,Y3: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) ) ).
% 0.15/0.43
% 0.15/0.43 % lstrict_prefix_code(2)
% 0.15/0.43 thf(fact_192_llast__singleton,axiom,
% 0.15/0.43 ! [A: $tType,X3: A] :
% 0.15/0.43 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 0.15/0.43 = X3 ) ).
% 0.15/0.43
% 0.15/0.43 % llast_singleton
% 0.15/0.43 thf(fact_193_llast__LCons2,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Y3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
% 0.15/0.43 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llast_LCons2
% 0.15/0.43 thf(fact_194_lstrict__prefix__code_I4_J,axiom,
% 0.15/0.43 ! [B: $tType,X3: B,Xs: coinductive_llist @ B,Y3: B,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( X3 = Y3 )
% 0.15/0.43 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lstrict_prefix_code(4)
% 0.15/0.43 thf(fact_195_lstrict__prefix__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % lstrict_prefix_code(1)
% 0.15/0.43 thf(fact_196_llist__less__induct,axiom,
% 0.15/0.43 ! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ! [Ys5: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu1478340336prefix @ A @ Ys5 @ Xs2 )
% 0.15/0.43 => ( P2 @ Ys5 ) )
% 0.15/0.43 => ( P2 @ Xs2 ) )
% 0.15/0.43 => ( P2 @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist_less_induct
% 0.15/0.43 thf(fact_197_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.15/0.43
% 0.15/0.43 % Coinductive_List.finite_lprefix_nitpick_simps(2)
% 0.15/0.43 thf(fact_198_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( Xs
% 0.15/0.43 = ( coinductive_LNil @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Coinductive_List.finite_lprefix_nitpick_simps(1)
% 0.15/0.43 thf(fact_199_llast__LCons,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
% 0.15/0.43 ( ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = X3 ) )
% 0.15/0.43 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llast_LCons
% 0.15/0.43 thf(fact_200_lprefix__coinduct,axiom,
% 0.15/0.43 ! [A: $tType,P2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( P2 @ Xs @ Ys )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
% 0.15/0.43 ( ( P2 @ Xs2 @ Ys4 )
% 0.15/0.43 => ( ( ( coinductive_lnull @ A @ Ys4 )
% 0.15/0.43 => ( coinductive_lnull @ A @ Xs2 ) )
% 0.15/0.43 & ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Ys4 )
% 0.15/0.43 => ( ( ( coinductive_lhd @ A @ Xs2 )
% 0.15/0.43 = ( coinductive_lhd @ A @ Ys4 ) )
% 0.15/0.43 & ( ( P2 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) )
% 0.15/0.43 | ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) ) ) ) ) ) ) )
% 0.15/0.43 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_coinduct
% 0.15/0.43 thf(fact_201_lprefix__expand,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Ys )
% 0.15/0.43 & ( ( coinductive_lhd @ A @ Xs )
% 0.15/0.43 = ( coinductive_lhd @ A @ Ys ) )
% 0.15/0.43 & ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) )
% 0.15/0.43 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_expand
% 0.15/0.43 thf(fact_202_llist_Oleq__refl,axiom,
% 0.15/0.43 ! [A: $tType,X3: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X3 @ X3 ) ).
% 0.15/0.43
% 0.15/0.43 % llist.leq_refl
% 0.15/0.43 thf(fact_203_lprefix__refl,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_refl
% 0.15/0.43 thf(fact_204_LCons__lprefix__LCons,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( X3 = Y3 )
% 0.15/0.43 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % LCons_lprefix_LCons
% 0.15/0.43 thf(fact_205_lprefix__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_code(1)
% 0.15/0.43 thf(fact_206_lprefix__LNil,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.15/0.43 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_LNil
% 0.15/0.43 thf(fact_207_Coinductive__List_Ofinite__lprefix__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coindu328551480prefix @ A )
% 0.15/0.43 = ( coinductive_lprefix @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % Coinductive_List.finite_lprefix_def
% 0.15/0.43 thf(fact_208_lstrict__prefix__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coindu1478340336prefix @ A )
% 0.15/0.43 = ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs5 @ Ys3 )
% 0.15/0.43 & ( Xs5 != Ys3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lstrict_prefix_def
% 0.15/0.43 thf(fact_209_lprefix__LCons__conv,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.15/0.43 = ( ( Xs
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [Xs3: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs
% 0.15/0.43 = ( coinductive_LCons @ A @ Y3 @ Xs3 ) )
% 0.15/0.43 & ( coinductive_lprefix @ A @ Xs3 @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_LCons_conv
% 0.15/0.43 thf(fact_210_lprefix_Ocoinduct,axiom,
% 0.15/0.43 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ X3 @ Xa2 )
% 0.15/0.43 => ( ! [X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 0.15/0.43 ( ( X6 @ X @ Xa )
% 0.15/0.43 => ( ? [Xs6: coinductive_llist @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 & ( Xa = Xs6 ) )
% 0.15/0.43 | ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 = ( coinductive_LCons @ A @ Xb @ Xs6 ) )
% 0.15/0.43 & ( Xa
% 0.15/0.43 = ( coinductive_LCons @ A @ Xb @ Ys5 ) )
% 0.15/0.43 & ( ( X6 @ Xs6 @ Ys5 )
% 0.15/0.43 | ( coinductive_lprefix @ A @ Xs6 @ Ys5 ) ) ) ) )
% 0.15/0.43 => ( coinductive_lprefix @ A @ X3 @ Xa2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix.coinduct
% 0.15/0.43 thf(fact_211_lprefix_Osimps,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lprefix @ A )
% 0.15/0.43 = ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 0.15/0.43 ( ? [Xs5: coinductive_llist @ A] :
% 0.15/0.43 ( ( A12
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 & ( A23 = Xs5 ) )
% 0.15/0.43 | ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X4: A] :
% 0.15/0.43 ( ( A12
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 0.15/0.43 & ( A23
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Ys3 ) )
% 0.15/0.43 & ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix.simps
% 0.15/0.43 thf(fact_212_lprefix_Ocases,axiom,
% 0.15/0.43 ! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ A1 @ A22 )
% 0.15/0.43 => ( ( ( A1
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 => ! [Xs2: coinductive_llist @ A] : ( A22 != Xs2 ) )
% 0.15/0.43 => ~ ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A,X: A] :
% 0.15/0.43 ( ( A1
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.15/0.43 => ( ( A22
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 0.15/0.43 => ~ ( coinductive_lprefix @ A @ Xs2 @ Ys4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix.cases
% 0.15/0.43 thf(fact_213_lprefix__code_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_code(2)
% 0.15/0.43 thf(fact_214_lprefixI,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X6: set @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) )] :
% 0.15/0.43 ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs @ Ys ) @ X6 )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
% 0.15/0.43 ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs2 @ Ys4 ) @ X6 )
% 0.15/0.43 => ( ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 | ? [X5: A,Xs7: coinductive_llist @ A,Ys6: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs2
% 0.15/0.43 = ( coinductive_LCons @ A @ X5 @ Xs7 ) )
% 0.15/0.43 & ( Ys4
% 0.15/0.43 = ( coinductive_LCons @ A @ X5 @ Ys6 ) )
% 0.15/0.43 & ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs7 @ Ys6 ) @ X6 )
% 0.15/0.43 | ( coinductive_lprefix @ A @ Xs7 @ Ys6 ) ) ) ) )
% 0.15/0.43 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefixI
% 0.15/0.43 thf(fact_215_lprefix__lhdD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( ( coinductive_lhd @ A @ Xs )
% 0.15/0.43 = ( coinductive_lhd @ A @ Ys ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_lhdD
% 0.15/0.43 thf(fact_216_lprefix__down__linear,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_down_linear
% 0.15/0.43 thf(fact_217_llist_Oleq__antisym,axiom,
% 0.15/0.43 ! [A: $tType,X3: coinductive_llist @ A,Y3: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ X3 @ Y3 )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Y3 @ X3 )
% 0.15/0.43 => ( X3 = Y3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.leq_antisym
% 0.15/0.43 thf(fact_218_lprefix__antisym,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 0.15/0.43 => ( Xs = Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_antisym
% 0.15/0.43 thf(fact_219_llist_Oleq__trans,axiom,
% 0.15/0.43 ! [A: $tType,X3: coinductive_llist @ A,Y3: coinductive_llist @ A,Z: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ X3 @ Y3 )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Y3 @ Z )
% 0.15/0.43 => ( coinductive_lprefix @ A @ X3 @ Z ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llist.leq_trans
% 0.15/0.43 thf(fact_220_lprefix__trans,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.15/0.43 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_trans
% 0.15/0.43 thf(fact_221_ldistinct__lprefix,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu351974385stinct @ A @ Xs )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 0.15/0.43 => ( coindu351974385stinct @ A @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ldistinct_lprefix
% 0.15/0.43 thf(fact_222_ord_Olsorted__lprefixD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less_eq: A > A > $o] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_lsorted @ A @ Less_eq @ Ys )
% 0.15/0.43 => ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord.lsorted_lprefixD
% 0.15/0.43 thf(fact_223_lprefix__ltlI,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_ltlI
% 0.15/0.43 thf(fact_224_lprefix__not__lnullD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_not_lnullD
% 0.15/0.43 thf(fact_225_lprefix__lnullD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_lnull @ A @ Ys )
% 0.15/0.43 => ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_lnullD
% 0.15/0.43 thf(fact_226_lprefix__lnull,axiom,
% 0.15/0.43 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Ys )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_lnull
% 0.15/0.43 thf(fact_227_lnull__lprefix,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_lprefix
% 0.15/0.43 thf(fact_228_Le__LCons,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Le_LCons
% 0.15/0.43 thf(fact_229_LCons__lprefix__conv,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
% 0.15/0.43 = ( ? [Ys2: coinductive_llist @ A] :
% 0.15/0.43 ( ( Ys
% 0.15/0.43 = ( coinductive_LCons @ A @ X3 @ Ys2 ) )
% 0.15/0.43 & ( coinductive_lprefix @ A @ Xs @ Ys2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % LCons_lprefix_conv
% 0.15/0.43 thf(fact_230_lprefix__ltakeWhile,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) @ Xs ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_ltakeWhile
% 0.15/0.43 thf(fact_231_lprefix__imp__llexord,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R: A > A > $o] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_imp_llexord
% 0.15/0.43 thf(fact_232_LNil__lprefix,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.15/0.43
% 0.15/0.43 % LNil_lprefix
% 0.15/0.43 thf(fact_233_llimit__induct,axiom,
% 0.15/0.43 ! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( P2 @ ( coinductive_LNil @ A ) )
% 0.15/0.43 => ( ! [X: A,Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.15/0.43 => ( ( P2 @ Xs2 )
% 0.15/0.43 => ( P2 @ ( coinductive_LCons @ A @ X @ Xs2 ) ) ) )
% 0.15/0.43 => ( ( ! [Ys5: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu1478340336prefix @ A @ Ys5 @ Xs )
% 0.15/0.43 => ( P2 @ Ys5 ) )
% 0.15/0.43 => ( P2 @ Xs ) )
% 0.15/0.43 => ( P2 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llimit_induct
% 0.15/0.43 thf(fact_234_in__inv__imagep,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( inv_imagep @ A @ B )
% 0.15/0.43 = ( ^ [R3: A > A > $o,F4: B > A,X4: B,Y4: B] : ( R3 @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % in_inv_imagep
% 0.15/0.43 thf(fact_235_lfinite__LCons,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 0.15/0.43 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_LCons
% 0.15/0.43 thf(fact_236_lfinite__code_I2_J,axiom,
% 0.15/0.43 ! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
% 0.15/0.43 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) )
% 0.15/0.43 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_code(2)
% 0.15/0.43 thf(fact_237_lfinite__code_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_code(1)
% 0.15/0.43 thf(fact_238_lfinite__ltl,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
% 0.15/0.43 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_ltl
% 0.15/0.43 thf(fact_239_lfinite__lzip,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.15/0.43 ( ( coinductive_lfinite @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 0.15/0.43 = ( ( coinductive_lfinite @ A @ Xs )
% 0.15/0.43 | ( coinductive_lfinite @ B @ Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_lzip
% 0.15/0.43 thf(fact_240_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lprefix @ A )
% 0.15/0.43 = ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 0.15/0.43 ( ( ( coinductive_lfinite @ A @ Xs5 )
% 0.15/0.43 => ( coindu328551480prefix @ A @ Xs5 @ Ys3 ) )
% 0.15/0.43 & ( ~ ( coinductive_lfinite @ A @ Xs5 )
% 0.15/0.43 => ( Xs5 = Ys3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Coinductive_List.lprefix_nitpick_simps
% 0.15/0.43 thf(fact_241_not__lfinite__lprefix__conv__eq,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.15/0.43 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 = ( Xs = Ys ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % not_lfinite_lprefix_conv_eq
% 0.15/0.43 thf(fact_242_lprefix__lfiniteD,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( ( coinductive_lfinite @ A @ Ys )
% 0.15/0.43 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lprefix_lfiniteD
% 0.15/0.43 thf(fact_243_lfinite__LNil,axiom,
% 0.15/0.43 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_LNil
% 0.15/0.43 thf(fact_244_lfinite__LConsI,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ Xs )
% 0.15/0.43 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_LConsI
% 0.15/0.43 thf(fact_245_lnull__imp__lfinite,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Xs )
% 0.15/0.43 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnull_imp_lfinite
% 0.15/0.43 thf(fact_246_lfinite__ldropWhile,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( ? [X4: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 & ~ ( P2 @ X4 ) )
% 0.15/0.43 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_ldropWhile
% 0.15/0.43 thf(fact_247_lfinite__ltakeWhile,axiom,
% 0.15/0.43 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 0.15/0.43 = ( ( coinductive_lfinite @ A @ Xs )
% 0.15/0.43 | ? [X4: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.15/0.43 & ~ ( P2 @ X4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_ltakeWhile
% 0.15/0.43 thf(fact_248_lfinite__induct,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ Xs )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( P2 @ Xs2 ) )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.15/0.43 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.15/0.43 => ( ( P2 @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.15/0.43 => ( P2 @ Xs2 ) ) ) )
% 0.15/0.43 => ( P2 @ Xs ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite_induct
% 0.15/0.43 thf(fact_249_lfinite_Ocases,axiom,
% 0.15/0.43 ! [A: $tType,A2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ A2 )
% 0.15/0.43 => ( ( A2
% 0.15/0.43 != ( coinductive_LNil @ A ) )
% 0.15/0.43 => ~ ! [Xs2: coinductive_llist @ A] :
% 0.15/0.43 ( ? [X: A] :
% 0.15/0.43 ( A2
% 0.15/0.43 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.15/0.43 => ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite.cases
% 0.15/0.43 thf(fact_250_lfinite_Osimps,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_lfinite @ A )
% 0.15/0.43 = ( ^ [A7: coinductive_llist @ A] :
% 0.15/0.43 ( ( A7
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ? [Xs5: coinductive_llist @ A,X4: A] :
% 0.15/0.43 ( ( A7
% 0.15/0.43 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 0.15/0.43 & ( coinductive_lfinite @ A @ Xs5 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite.simps
% 0.15/0.43 thf(fact_251_lfinite_Oinducts,axiom,
% 0.15/0.43 ! [A: $tType,X3: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ X3 )
% 0.15/0.43 => ( ( P2 @ ( coinductive_LNil @ A ) )
% 0.15/0.43 => ( ! [Xs2: coinductive_llist @ A,X: A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.15/0.43 => ( ( P2 @ Xs2 )
% 0.15/0.43 => ( P2 @ ( coinductive_LCons @ A @ X @ Xs2 ) ) ) )
% 0.15/0.43 => ( P2 @ X3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lfinite.inducts
% 0.15/0.43 thf(fact_252_lstrict__prefix__lfinite1,axiom,
% 0.15/0.43 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.15/0.43 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 0.15/0.43 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.15/0.43
% 0.15/0.43 % lstrict_prefix_lfinite1
% 0.15/0.43 thf(fact_253_llexord__conv,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( coinductive_llexord @ A )
% 0.15/0.43 = ( ^ [R3: A > A > $o,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 0.15/0.43 ( ( Xs5 = Ys3 )
% 0.15/0.43 | ? [Zs2: coinductive_llist @ A,Xs3: coinductive_llist @ A,Y4: A,Ys2: coinductive_llist @ A] :
% 0.15/0.43 ( ( coinductive_lfinite @ A @ Zs2 )
% 0.15/0.43 & ( Xs5
% 0.15/0.43 = ( coinductive_lappend @ A @ Zs2 @ Xs3 ) )
% 0.15/0.43 & ( Ys3
% 0.15/0.43 = ( coinductive_lappend @ A @ Zs2 @ ( coinductive_LCons @ A @ Y4 @ Ys2 ) ) )
% 0.15/0.43 & ( ( Xs3
% 0.15/0.43 = ( coinductive_LNil @ A ) )
% 0.15/0.43 | ( R3 @ ( coinductive_lhd @ A @ Xs3 ) @ Y4 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % llexord_conv
% 0.15/0.43
% 0.15/0.43 %----Type constructors (3)
% 0.15/0.43 thf(tcon_fun___Orderings_Obot,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( bot @ A9 @ ( type2 @ A9 ) )
% 0.15/0.43 => ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Obot_1,axiom,
% 0.15/0.43 ! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Obot_2,axiom,
% 0.15/0.43 bot @ $o @ ( type2 @ $o ) ).
% 0.15/0.43
% 0.15/0.43 %----Helper facts (3)
% 0.15/0.43 thf(help_If_3_1_T,axiom,
% 0.15/0.43 ! [P2: $o] :
% 0.15/0.43 ( ( P2 = $true )
% 0.15/0.43 | ( P2 = $false ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(help_If_2_1_T,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Y3: A] :
% 0.15/0.43 ( ( if @ A @ $false @ X3 @ Y3 )
% 0.15/0.43 = Y3 ) ).
% 0.15/0.43
% 0.15/0.43 thf(help_If_1_1_T,axiom,
% 0.15/0.43 ! [A: $tType,X3: A,Y3: A] :
% 0.15/0.43 ( ( if @ A @ $true @ X3 @ Y3 )
% 0.15/0.43 = X3 ) ).
% 0.15/0.43
% 0.15/0.43 %----Conjectures (1)
% 0.15/0.43 thf(conj_0,conjecture,
% 0.15/0.43 member @ node @ n @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 0.15/0.43
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 ------------------------
% 2.38/1.46 ---- Embedded in THF ---
% 2.38/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.38/1.46 %%% Generated on Tue Feb 24 04:15:56 EST 2026
% 2.38/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.38/1.46 %%% Logic specification used:
% 2.38/1.46 %%% thf(spec, logic, $$dhol).
% 2.38/1.46
% 2.38/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.udRHUCu2u7/DTF2DTF19871.p
% 2.38/1.46 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.38/1.46 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.38/1.46 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.38/1.46 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.38/1.46 thf(ty_tf_node, type, node: $tType).
% 2.38/1.46 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.38/1.46 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ofinite__lprefix, type, coindu328551480prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ogen__lset, type, coinductive_gen_lset: (!> [A:$tType]: (((set @ A) > ((coinductive_llist @ A) > (set @ A)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Olappend, type, coinductive_lappend: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Oldistinct, type, coindu351974385stinct: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_OldropWhile, type, coindu218763757pWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollast, type, coinductive_llast: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollexord, type, coinductive_llexord: (!> [A:$tType]: (((A > (A > $o)) > ((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_Ocase__llist, type, coindu1381640503_llist: (!> [B:$tType,A:$tType]: ((B > ((A > ((coinductive_llist @ A) > B)) > ((coinductive_llist @ A) > B)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_Olhd, type, coinductive_lhd: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ollist_Oltl, type, coinductive_ltl: (!> [A:$tType]: (((coinductive_llist @ A) > (coinductive_llist @ A))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Olprefix, type, coinductive_lprefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Olstrict__prefix, type, coindu1478340336prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_OltakeWhile, type, coindu501562517eWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Olzip, type, coinductive_lzip: (!> [A:$tType,B:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ B) > (coinductive_llist @ ((product_prod @ A) @ B))))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Oord_Olsorted, type, coinductive_lsorted: (!> [A:$tType]: (((A > (A > $o)) > ((coinductive_llist @ A) > $o))))).
% 2.38/1.46 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 2.38/1.46 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.38/1.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected, type, koenig793108494nected: (!> [Node:$tType]: (((Node > (Node > $o)) > $o)))).
% 2.38/1.46 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.38/1.46 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.38/1.46 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.38/1.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.38/1.46 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.38/1.46 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.38/1.46 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.38/1.46 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.38/1.46 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.38/1.46 thf(sy_v_f____, type, f: (((product_prod @ node) @ (set @ node)) > (coinductive_llist @ node))).
% 2.38/1.46 thf(sy_v_graph, type, graph: (node > (node > $o))).
% 2.38/1.46 thf(sy_v_n, type, n: node).
% 2.38/1.46 thf(sy_v_ns____, type, ns: (set @ node)).
% 2.38/1.46 thf(fact_0__092_060open_062lhd_A_If_A_In_M_Ans_J_J_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062, axiom, (((member @ node) @ ((coinductive_lhd @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ n) @ ns)))) @ ((coinductive_lset @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ n) @ ns))))).
% 2.38/1.46 thf(fact_1_ns__def, axiom, (ns = (bot_bot @ (set @ node)))).
% 2.38/1.46 thf(fact_2_f__simps_I2_J, axiom, (! [Na:node,Nsa:(set @ node)]: ((((coinductive_lhd @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ Na) @ Nsa))) = Na)))).
% 2.38/1.46 thf(fact_3_f__simps_I1_J, axiom, (! [Na:node,Nsa:(set @ node)]: ((~ (((coinductive_lnull @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ Na) @ Nsa)))))))).
% 2.38/1.46 thf(fact_4_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.38/1.46 thf(fact_5_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.38/1.46 thf(fact_6_connected, axiom, ((koenig793108494nected @ node) @ graph)).
% 2.38/1.46 thf(fact_7_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X:A,Y:B]: ((P = ((((product_Pair @ A) @ B) @ X) @ Y))))))).
% 2.38/1.46 thf(fact_8_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P2 @ P))))).
% 2.38/1.46 thf(fact_9_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.38/1.46 thf(fact_10_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C2:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 2.38/1.46 thf(fact_11_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.38/1.46 thf(fact_12_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.38/1.46 thf(fact_13_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.38/1.46 thf(fact_14_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P2 @ Prod))))).
% 2.38/1.46 thf(fact_15_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 2.38/1.46 thf(fact_16_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P2 @ X3))))).
% 2.38/1.46 thf(fact_17_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P2 @ X3))))).
% 2.38/1.46 thf(fact_18_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X3))))).
% 2.38/1.46 thf(fact_19_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C2:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P2 @ X3))))).
% 2.38/1.46 thf(fact_20_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C2:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P2 @ X3))))).
% 2.38/1.46 thf(fact_21_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.38/1.46 thf(fact_22_lset__eq__empty, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lset @ A) @ Xs) = (bot_bot @ (set @ A))) = ((coinductive_lnull @ A) @ Xs))))).
% 2.38/1.46 thf(fact_23_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.38/1.46 thf(fact_24_llist_Oset__sel_I1_J, axiom, (! [A:$tType,A2:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ A2))) => (((member @ A) @ ((coinductive_lhd @ A) @ A2)) @ ((coinductive_lset @ A) @ A2)))))).
% 2.38/1.46 thf(fact_25_lset__lnull, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lset @ A) @ Xs) = (bot_bot @ (set @ A))))))).
% 2.38/1.46 thf(fact_26_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 2.38/1.46 thf(fact_27_all__not__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A5))))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.38/1.46 thf(fact_28_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P2 @ X4))))))))).
% 2.38/1.46 thf(fact_29_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X4:A]: ((~ ((P2 @ X4))))))))).
% 2.38/1.46 thf(fact_30_bot__apply, axiom, (! [C:$tType,D:$tType]: ((((bot @ C) @ (type2 @ C)) => ((bot_bot @ (D > C)) = (^ [X4:D]: ((bot_bot @ C)))))))).
% 2.38/1.46 thf(fact_31_ltakeWhile_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((~ ((((coinductive_lnull @ A) @ Xs) | (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))))))))).
% 2.38/1.46 thf(fact_32_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.38/1.46 thf(fact_33_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.38/1.46 thf(fact_34_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X4:A]: ((bot_bot @ B)))))))).
% 2.38/1.46 thf(fact_35_ex__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A5))) = (A5 != (bot_bot @ (set @ A))))))).
% 2.38/1.46 thf(fact_36_equals0I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [Y:A]: ((~ ((((member @ A) @ Y) @ A5))))) => (A5 = (bot_bot @ (set @ A))))))).
% 2.38/1.46 thf(fact_37_equals0D, axiom, (! [A:$tType,A5:(set @ A),A2:A]: (((A5 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A5))))))).
% 2.38/1.46 thf(fact_38_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.38/1.46 thf(fact_39_lappend_Oexhaust, axiom, (! [A:$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)))))))).
% 2.38/1.46 thf(fact_40_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)))))))).
% 2.38/1.46 thf(fact_41_lset__code, axiom, (! [A:$tType]: (((coinductive_lset @ A) = ((coinductive_gen_lset @ A) @ (bot_bot @ (set @ A))))))).
% 2.38/1.46 thf(fact_42_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A6:(set @ A)]: ((A6 = (bot_bot @ (set @ A))))))))).
% 2.38/1.46 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.38/1.46 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A5)))) = A5)))).
% 2.38/1.46 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P2 @ X) = (Q @ X)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.38/1.46 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G3:(A > B)]: (((! [X:A]: (((F3 @ X) = (G3 @ X)))) => (F3 = G3))))).
% 2.38/1.46 thf(fact_47_lset__lmember, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X3) @ Xs))))).
% 2.38/1.46 thf(fact_48_Collect__empty__eq__bot, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (P2 = (bot_bot @ (A > $o))))))).
% 2.38/1.46 thf(fact_49_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 2.38/1.46 thf(fact_50_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R2:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R2) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R2)))))).
% 2.38/1.46 thf(fact_51_lhd__lzip, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ B) @ Ys))) => (((coinductive_lhd @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)) = ((((product_Pair @ A) @ B) @ ((coinductive_lhd @ A) @ Xs)) @ ((coinductive_lhd @ B) @ Ys)))))))).
% 2.38/1.46 thf(fact_52_llist__set__induct, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A),P2:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs2))) => ((P2 @ ((coinductive_lhd @ A) @ Xs2)) @ Xs2)))) => ((! [Xs2:(coinductive_llist @ A),Y:A]: (((~ (((coinductive_lnull @ A) @ Xs2))) => ((((member @ A) @ Y) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs2))) => (((P2 @ Y) @ ((coinductive_ltl @ A) @ Xs2)) => ((P2 @ Y) @ Xs2)))))) => ((P2 @ X3) @ Xs))))))).
% 2.38/1.46 thf(fact_53_lnull__ldropWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu218763757pWhile @ A) @ P2) @ Xs)) = (! [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) => (P2 @ X4)))))))).
% 2.38/1.46 thf(fact_54_lzip_Odisc__iff_I2_J, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ (((coinductive_lnull @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)))) = ((~ (((coinductive_lnull @ A) @ Xs))) & (~ (((coinductive_lnull @ B) @ Ys)))))))).
% 2.38/1.46 thf(fact_55_lnull__lzip, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: ((((coinductive_lnull @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)) = (((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))))).
% 2.38/1.46 thf(fact_56_ltl__lzip, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ B) @ Ys))) => (((coinductive_ltl @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)) = ((((coinductive_lzip @ A) @ B) @ ((coinductive_ltl @ A) @ Xs)) @ ((coinductive_ltl @ B) @ Ys)))))))).
% 2.38/1.46 thf(fact_57_lnull__ltlI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lnull @ A) @ ((coinductive_ltl @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_58_in__lset__ltlD, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs))) => (((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_59_lzip_Odisc_I2_J, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ B) @ Ys))) => (~ (((coinductive_lnull @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys))))))))).
% 2.38/1.46 thf(fact_60_lzip_Odisc_I1_J, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)) => ((coinductive_lnull @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_61_in__lset__ldropWhileD, axiom, (! [A:$tType,X3:A,P2:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ (((coindu218763757pWhile @ A) @ P2) @ Xs))) => (((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_62_ldropWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P2:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)) => ((P2 @ X) = (Q @ X))))) => ((((coindu218763757pWhile @ A) @ P2) @ Xs) = (((coindu218763757pWhile @ A) @ Q) @ Ys))))))).
% 2.38/1.46 thf(fact_63_llist_Oset__sel_I2_J, axiom, (! [A:$tType,A2:(coinductive_llist @ A),X3:A]: (((~ (((coinductive_lnull @ A) @ A2))) => ((((member @ A) @ X3) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ A2))) => (((member @ A) @ X3) @ ((coinductive_lset @ A) @ A2))))))).
% 2.38/1.46 thf(fact_64_lset__lzipD2, axiom, (! [A:$tType,B:$tType,X3:A,Y3:B,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((coinductive_lset @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys))) => (((member @ B) @ Y3) @ ((coinductive_lset @ B) @ Ys)))))).
% 2.38/1.46 thf(fact_65_lset__lzipD1, axiom, (! [B:$tType,A:$tType,X3:A,Y3:B,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((coinductive_lset @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys))) => (((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_66_llist_Oexpand, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),Llist2:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Llist) = ((coinductive_lnull @ A) @ Llist2)) => (((~ (((coinductive_lnull @ A) @ Llist))) => ((~ (((coinductive_lnull @ A) @ Llist2))) => ((((coinductive_lhd @ A) @ Llist) = ((coinductive_lhd @ A) @ Llist2)) & (((coinductive_ltl @ A) @ Llist) = ((coinductive_ltl @ A) @ Llist2))))) => (Llist = Llist2)))))).
% 2.38/1.46 thf(fact_67_llist_Ocoinduct, axiom, (! [A:$tType,R2:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),Llist:(coinductive_llist @ A),Llist2:(coinductive_llist @ A)]: ((((R2 @ Llist) @ Llist2) => ((! [Llist3:(coinductive_llist @ A),Llist4:(coinductive_llist @ A)]: ((((R2 @ Llist3) @ Llist4) => ((((coinductive_lnull @ A) @ Llist3) = ((coinductive_lnull @ A) @ Llist4)) & ((~ (((coinductive_lnull @ A) @ Llist3))) => ((~ (((coinductive_lnull @ A) @ Llist4))) => ((((coinductive_lhd @ A) @ Llist3) = ((coinductive_lhd @ A) @ Llist4)) & ((R2 @ ((coinductive_ltl @ A) @ Llist3)) @ ((coinductive_ltl @ A) @ Llist4))))))))) => (Llist = Llist2)))))).
% 2.38/1.46 thf(fact_68_llist_Ocoinduct__strong, axiom, (! [A:$tType,R2:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),Llist:(coinductive_llist @ A),Llist2:(coinductive_llist @ A)]: ((((R2 @ Llist) @ Llist2) => ((! [Llist3:(coinductive_llist @ A),Llist4:(coinductive_llist @ A)]: ((((R2 @ Llist3) @ Llist4) => ((((coinductive_lnull @ A) @ Llist3) = ((coinductive_lnull @ A) @ Llist4)) & ((~ (((coinductive_lnull @ A) @ Llist3))) => ((~ (((coinductive_lnull @ A) @ Llist4))) => ((((coinductive_lhd @ A) @ Llist3) = ((coinductive_lhd @ A) @ Llist4)) & (((R2 @ ((coinductive_ltl @ A) @ Llist3)) @ ((coinductive_ltl @ A) @ Llist4)) | (((coinductive_ltl @ A) @ Llist3) = ((coinductive_ltl @ A) @ Llist4)))))))))) => (Llist = Llist2)))))).
% 2.38/1.46 thf(fact_69_lhd__ldropWhile__in__lset, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P2 @ X5)))))) => (((member @ A) @ ((coinductive_lhd @ A) @ (((coindu218763757pWhile @ A) @ P2) @ Xs))) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_70_lhd__ldropWhile, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P2 @ X5)))))) => (~ ((P2 @ ((coinductive_lhd @ A) @ (((coindu218763757pWhile @ A) @ P2) @ Xs))))))))).
% 2.38/1.46 thf(fact_71_lzip_Octr_I2_J, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ B) @ Ys))) => (((((coinductive_lzip @ A) @ B) @ Xs) @ Ys) = (((coinductive_LCons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ ((coinductive_lhd @ A) @ Xs)) @ ((coinductive_lhd @ B) @ Ys))) @ ((((coinductive_lzip @ A) @ B) @ ((coinductive_ltl @ A) @ Xs)) @ ((coinductive_ltl @ B) @ Ys))))))))).
% 2.38/1.46 thf(fact_72_unfold__llist__id, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((((((coindu1441602521_llist @ (coinductive_llist @ A)) @ A) @ (coinductive_lnull @ A)) @ (coinductive_lhd @ A)) @ (coinductive_ltl @ A)) @ Xs) = Xs)))).
% 2.38/1.46 thf(fact_73_ldistinct__lhdD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ Xs) => ((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((((member @ A) @ ((coinductive_lhd @ A) @ Xs)) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs)))))))))).
% 2.38/1.46 thf(fact_74_ldistinct__coinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((X6 @ Xs) => ((! [Xs2:(coinductive_llist @ A)]: (((X6 @ Xs2) => ((~ (((coinductive_lnull @ A) @ Xs2))) => ((~ ((((member @ A) @ ((coinductive_lhd @ A) @ Xs2)) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs2))))) & ((X6 @ ((coinductive_ltl @ A) @ Xs2)) | ((coindu351974385stinct @ A) @ ((coinductive_ltl @ A) @ Xs2)))))))) => ((coindu351974385stinct @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_75_llist_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((coindu1381640503_llist @ B) @ A) = (^ [F12:B,F22:(A > ((coinductive_llist @ A) > B)),Llist5:(coinductive_llist @ A)]: (((((if @ B) @ ((coinductive_lnull @ A) @ Llist5)) @ F12) @ ((F22 @ ((coinductive_lhd @ A) @ Llist5)) @ ((coinductive_ltl @ A) @ Llist5))))))))).
% 2.38/1.46 thf(fact_76_lhd__LCons__ltl, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Llist))) => ((((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist)) = Llist))))).
% 2.38/1.46 thf(fact_77_ord_Olsorted__lhdD, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Xs:(coinductive_llist @ A)]: (((((coinductive_lsorted @ A) @ Less_eq) @ Xs) => ((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ A) @ ((coinductive_ltl @ A) @ Xs)))) => ((Less_eq @ ((coinductive_lhd @ A) @ Xs)) @ ((coinductive_lhd @ A) @ ((coinductive_ltl @ A) @ Xs))))))))).
% 2.38/1.46 thf(fact_78_ord_Olsorted__coinduct_H, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A),Less_eq:(A > (A > $o))]: (((X6 @ Xs) => ((! [Xs2:(coinductive_llist @ A)]: (((X6 @ Xs2) => ((~ (((coinductive_lnull @ A) @ Xs2))) => ((~ (((coinductive_lnull @ A) @ ((coinductive_ltl @ A) @ Xs2)))) => (((Less_eq @ ((coinductive_lhd @ A) @ Xs2)) @ ((coinductive_lhd @ A) @ ((coinductive_ltl @ A) @ Xs2))) & ((X6 @ ((coinductive_ltl @ A) @ Xs2)) | (((coinductive_lsorted @ A) @ Less_eq) @ ((coinductive_ltl @ A) @ Xs2))))))))) => (((coinductive_lsorted @ A) @ Less_eq) @ Xs)))))).
% 2.38/1.46 thf(fact_79_llist_Oinject, axiom, (! [A:$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)))))).
% 2.38/1.46 thf(fact_80_ldropWhile__LCons, axiom, (! [A:$tType,P2:(A > $o),X3:A,Xs:(coinductive_llist @ A)]: ((((P2 @ X3) => ((((coindu218763757pWhile @ A) @ P2) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = (((coindu218763757pWhile @ A) @ P2) @ Xs))) & ((~ ((P2 @ X3))) => ((((coindu218763757pWhile @ A) @ P2) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = (((coinductive_LCons @ A) @ X3) @ Xs))))))).
% 2.38/1.46 thf(fact_81_unfold__llist__eq__LCons, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B2:B,X3:A,Xs:(coinductive_llist @ A)]: (((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ B2) = (((coinductive_LCons @ A) @ X3) @ Xs)) = ((~ ((IS_LNIL @ B2))) & ((X3 = (LHD @ B2)) & (Xs = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B2))))))))).
% 2.38/1.46 thf(fact_82_unfold__llist_Odisc__iff_I1_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),G21:(A > B),G22:(A > A),A2:A]: ((((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2)) = (P @ A2))))).
% 2.38/1.46 thf(fact_83_unfold__llist_Odisc__iff_I2_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),G21:(A > B),G22:(A > A),A2:A]: (((~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2)))) = (~ ((P @ A2))))))).
% 2.38/1.46 thf(fact_84_ldistinct__LCons, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = ((~ ((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)))) & ((coindu351974385stinct @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_85_lzip__simps_I3_J, axiom, (! [C:$tType,B:$tType,X3:C,Xs:(coinductive_llist @ C),Y3:B,Ys:(coinductive_llist @ B)]: ((((((coinductive_lzip @ C) @ B) @ (((coinductive_LCons @ C) @ X3) @ Xs)) @ (((coinductive_LCons @ B) @ Y3) @ Ys)) = (((coinductive_LCons @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ X3) @ Y3)) @ ((((coinductive_lzip @ C) @ B) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_86_ldistinct__lzipI2, axiom, (! [B:$tType,A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ B)]: ((((coindu351974385stinct @ A) @ Ys) => ((coindu351974385stinct @ ((product_prod @ B) @ A)) @ ((((coinductive_lzip @ B) @ A) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_87_ldistinct__lzipI1, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: ((((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_88_llist_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F23:(A > ((coinductive_llist @ A) > B)),X21:A,X22:(coinductive_llist @ A)]: (((((((coindu1381640503_llist @ B) @ A) @ F1) @ F23) @ (((coinductive_LCons @ A) @ X21) @ X22)) = ((F23 @ X21) @ X22))))).
% 2.38/1.46 thf(fact_89_unfold__llist_Octr_I2_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P @ A2))) => (((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2) = (((coinductive_LCons @ B) @ (G21 @ A2)) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ (G22 @ A2)))))))).
% 2.38/1.46 thf(fact_90_ldistinct_OLCons, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: (((~ ((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)))) => (((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs))))))).
% 2.38/1.46 thf(fact_91_ord_OLCons__LCons, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X3:A,Y3:A,Xs:(coinductive_llist @ A)]: ((((Less_eq @ X3) @ Y3) => ((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ Y3) @ Xs)) => (((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X3) @ (((coinductive_LCons @ A) @ Y3) @ Xs)))))))).
% 2.38/1.46 thf(fact_92_ord_Olsorted__LCons__LCons, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X3:A,Y3:A,Xs:(coinductive_llist @ A)]: (((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X3) @ (((coinductive_LCons @ A) @ Y3) @ Xs))) = (((Less_eq @ X3) @ Y3) & (((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ Y3) @ Xs))))))).
% 2.38/1.46 thf(fact_93_lzip__eq__LCons__conv, axiom, (! [B:$tType,A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),Z:((product_prod @ A) @ B),Zs:(coinductive_llist @ ((product_prod @ A) @ B))]: (((((((coinductive_lzip @ A) @ B) @ Xs) @ Ys) = (((coinductive_LCons @ ((product_prod @ A) @ B)) @ Z) @ Zs)) = (? [X4:A,Xs3:(coinductive_llist @ A),Y4:B,Ys2:(coinductive_llist @ B)]: (((Xs = (((coinductive_LCons @ A) @ X4) @ Xs3)) & ((Ys = (((coinductive_LCons @ B) @ Y4) @ Ys2)) & ((Z = ((((product_Pair @ A) @ B) @ X4) @ Y4)) & (Zs = ((((coinductive_lzip @ A) @ B) @ Xs3) @ Ys2))))))))))).
% 2.38/1.46 thf(fact_94_ord_Olsorted__LCons_H, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X3:A,Xs:(coinductive_llist @ A)]: (((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = ((~ (((coinductive_lnull @ A) @ Xs))) => (((Less_eq @ X3) @ ((coinductive_lhd @ A) @ Xs)) & (((coinductive_lsorted @ A) @ Less_eq) @ Xs))))))).
% 2.38/1.46 thf(fact_95_not__lnull__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) = (? [X4:A,Xs3:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X4) @ Xs3)))))))).
% 2.38/1.46 thf(fact_96_llist_OdiscI_I2_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),X21:A,X22:(coinductive_llist @ A)]: (((Llist = (((coinductive_LCons @ A) @ X21) @ X22)) => (~ (((coinductive_lnull @ A) @ Llist))))))).
% 2.38/1.46 thf(fact_97_llist_Odisc_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((~ (((coinductive_lnull @ A) @ (((coinductive_LCons @ A) @ X21) @ X22))))))).
% 2.38/1.46 thf(fact_98_lset__intros_I2_J, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A),X7:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (((member @ A) @ X3) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X7) @ Xs))))))).
% 2.38/1.46 thf(fact_99_lset__intros_I1_J, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: ((((member @ A) @ X3) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)))))).
% 2.38/1.46 thf(fact_100_llist_Oset__intros_I2_J, axiom, (! [A:$tType,X3:A,A22:(coinductive_llist @ A),A1:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ A22)) => (((member @ A) @ X3) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22))))))).
% 2.38/1.46 thf(fact_101_llist_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(coinductive_llist @ A)]: ((((member @ A) @ A1) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22)))))).
% 2.38/1.46 thf(fact_102_lset__cases, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs4:(coinductive_llist @ A)]: ((Xs != (((coinductive_LCons @ A) @ X3) @ Xs4)))) => (~ ((! [X8:A,Xs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X8) @ Xs4)) => (~ ((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs4)))))))))))))).
% 2.38/1.46 thf(fact_103_lset__induct, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A),P2:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((P2 @ (((coinductive_LCons @ A) @ X3) @ Xs2)))) => ((! [X8:A,Xs2:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs2)) => ((X3 != X8) => ((P2 @ Xs2) => (P2 @ (((coinductive_LCons @ A) @ X8) @ Xs2))))))) => (P2 @ Xs))))))).
% 2.38/1.46 thf(fact_104_lset__induct_H, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A),P2:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((P2 @ (((coinductive_LCons @ A) @ X3) @ Xs2)))) => ((! [X8:A,Xs2:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs2)) => ((P2 @ Xs2) => (P2 @ (((coinductive_LCons @ A) @ X8) @ Xs2)))))) => (P2 @ Xs))))))).
% 2.38/1.46 thf(fact_105_llist_Oset__cases, axiom, (! [A:$tType,E3:A,A2:(coinductive_llist @ A)]: (((((member @ A) @ E3) @ ((coinductive_lset @ A) @ A2)) => ((! [Z2:(coinductive_llist @ A)]: ((A2 != (((coinductive_LCons @ A) @ E3) @ Z2)))) => (~ ((! [Z1:A,Z2:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ Z1) @ Z2)) => (~ ((((member @ A) @ E3) @ ((coinductive_lset @ A) @ Z2)))))))))))))).
% 2.38/1.46 thf(fact_106_llist_Oset__induct, axiom, (! [A:$tType,X3:A,A2:(coinductive_llist @ A),P2:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ A2)) => ((! [Z1:A,Z2:(coinductive_llist @ A)]: (((P2 @ Z1) @ (((coinductive_LCons @ A) @ Z1) @ Z2)))) => ((! [Z1:A,Z2:(coinductive_llist @ A),Xa:A]: (((((member @ A) @ Xa) @ ((coinductive_lset @ A) @ Z2)) => (((P2 @ Xa) @ Z2) => ((P2 @ Xa) @ (((coinductive_LCons @ A) @ Z1) @ Z2)))))) => ((P2 @ X3) @ A2))))))).
% 2.38/1.46 thf(fact_107_ltl__simps_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((((coinductive_ltl @ A) @ (((coinductive_LCons @ A) @ X21) @ X22)) = X22)))).
% 2.38/1.46 thf(fact_108_lhd__LCons, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((((coinductive_lhd @ A) @ (((coinductive_LCons @ A) @ X21) @ X22)) = X21)))).
% 2.38/1.46 thf(fact_109_ldistinct__ltlI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ A) @ ((coinductive_ltl @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_110_ord_Olsorted__ltlI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Xs:(coinductive_llist @ A)]: (((((coinductive_lsorted @ A) @ Less_eq) @ Xs) => (((coinductive_lsorted @ A) @ Less_eq) @ ((coinductive_ltl @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_111_unfold__llist_Odisc_I1_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((P @ A2) => ((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2)))))).
% 2.38/1.46 thf(fact_112_unfold__llist_Odisc_I2_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P @ A2))) => (~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2)))))))).
% 2.38/1.46 thf(fact_113_unfold__llist_Osimps_I4_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P @ A2))) => (((coinductive_ltl @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2)) = ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ (G22 @ A2))))))).
% 2.38/1.46 thf(fact_114_unfold__llist_Osimps_I3_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P @ A2))) => (((coinductive_lhd @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2)) = (G21 @ A2)))))).
% 2.38/1.46 thf(fact_115_lmember__code_I2_J, axiom, (! [A:$tType,X3:A,Y3:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lmember @ A) @ X3) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = ((X3 = Y3) | (((coinductive_lmember @ A) @ X3) @ Ys)))))).
% 2.38/1.46 thf(fact_116_lzip_Ocode, axiom, (! [B:$tType,A:$tType]: ((((coinductive_lzip @ A) @ B) = (^ [Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ B)]: (((((if @ (coinductive_llist @ ((product_prod @ A) @ B))) @ (((coinductive_lnull @ A) @ Xs5) | ((coinductive_lnull @ B) @ Ys3))) @ (coinductive_LNil @ ((product_prod @ A) @ B))) @ (((coinductive_LCons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ ((coinductive_lhd @ A) @ Xs5)) @ ((coinductive_lhd @ B) @ Ys3))) @ ((((coinductive_lzip @ A) @ B) @ ((coinductive_ltl @ A) @ Xs5)) @ ((coinductive_ltl @ B) @ Ys3)))))))))).
% 2.38/1.46 thf(fact_117_llist_Osplit__sel, axiom, (! [B:$tType,A:$tType,P2:(B > $o),F1:B,F23:(A > ((coinductive_llist @ A) > B)),Llist:(coinductive_llist @ A)]: (((P2 @ (((((coindu1381640503_llist @ B) @ A) @ F1) @ F23) @ Llist)) = (((Llist = (coinductive_LNil @ A)) => (P2 @ F1)) & ((Llist = (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist))) => (P2 @ ((F23 @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist))))))))).
% 2.38/1.46 thf(fact_118_llist_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P2:(B > $o),F1:B,F23:(A > ((coinductive_llist @ A) > B)),Llist:(coinductive_llist @ A)]: (((P2 @ (((((coindu1381640503_llist @ B) @ A) @ F1) @ F23) @ Llist)) = (~ ((((Llist = (coinductive_LNil @ A)) & (~ ((P2 @ F1)))) | ((Llist = (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist))) & (~ ((P2 @ ((F23 @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist))))))))))))).
% 2.38/1.46 thf(fact_119_ltakeWhile_Octr_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P2 @ ((coinductive_lhd @ A) @ Xs)) => ((((coindu501562517eWhile @ A) @ P2) @ Xs) = (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Xs)) @ (((coindu501562517eWhile @ A) @ P2) @ ((coinductive_ltl @ A) @ Xs))))))))).
% 2.38/1.46 thf(fact_120_llexord__coinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),R:(A > (A > $o))]: ((((X6 @ Xs) @ Ys) => ((! [Xs2:(coinductive_llist @ A),Ys4:(coinductive_llist @ A)]: ((((X6 @ Xs2) @ Ys4) => ((~ (((coinductive_lnull @ A) @ Xs2))) => ((~ (((coinductive_lnull @ A) @ Ys4))) & ((~ (((coinductive_lnull @ A) @ Ys4))) => (((R @ ((coinductive_lhd @ A) @ Xs2)) @ ((coinductive_lhd @ A) @ Ys4)) | ((((coinductive_lhd @ A) @ Xs2) = ((coinductive_lhd @ A) @ Ys4)) & (((X6 @ ((coinductive_ltl @ A) @ Xs2)) @ ((coinductive_ltl @ A) @ Ys4)) | ((((coinductive_llexord @ A) @ R) @ ((coinductive_ltl @ A) @ Xs2)) @ ((coinductive_ltl @ A) @ Ys4))))))))))) => ((((coinductive_llexord @ A) @ R) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_121_ldistinct_Ocases, axiom, (! [A:$tType,A2:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ A2) => ((A2 != (coinductive_LNil @ A)) => (~ ((! [X:A,Xs2:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ X) @ Xs2)) => ((~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs2)))) => (~ (((coindu351974385stinct @ A) @ Xs2)))))))))))))).
% 2.38/1.46 thf(fact_122_llexord__refl, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(coinductive_llist @ A)]: (((((coinductive_llexord @ A) @ R) @ Xs) @ Xs)))).
% 2.38/1.46 thf(fact_123_ltakeWhile__LNil, axiom, (! [A:$tType,P2:(A > $o)]: (((((coindu501562517eWhile @ A) @ P2) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_124_llexord__LCons__LCons, axiom, (! [A:$tType,R:(A > (A > $o)),X3:A,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: ((((((coinductive_llexord @ A) @ R) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = (((X3 = Y3) & ((((coinductive_llexord @ A) @ R) @ Xs) @ Ys)) | ((R @ X3) @ Y3)))))).
% 2.38/1.46 thf(fact_125_ldistinct__LNil__code, axiom, (! [A:$tType]: (((coindu351974385stinct @ A) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_126_llexord__code_I1_J, axiom, (! [A:$tType,R:(A > (A > $o)),Ys:(coinductive_llist @ A)]: (((((coinductive_llexord @ A) @ R) @ (coinductive_LNil @ A)) @ Ys)))).
% 2.38/1.46 thf(fact_127_llexord__LNil__right, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),R:(A > (A > $o)),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => (((((coinductive_llexord @ A) @ R) @ Xs) @ Ys) = ((coinductive_lnull @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_128_ldropWhile__LNil, axiom, (! [A:$tType,P2:(A > $o)]: (((((coindu218763757pWhile @ A) @ P2) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_129_ltakeWhile__LCons, axiom, (! [A:$tType,P2:(A > $o),X3:A,Xs:(coinductive_llist @ A)]: ((((P2 @ X3) => ((((coindu501562517eWhile @ A) @ P2) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = (((coinductive_LCons @ A) @ X3) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)))) & ((~ ((P2 @ X3))) => ((((coindu501562517eWhile @ A) @ P2) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = (coinductive_LNil @ A))))))).
% 2.38/1.46 thf(fact_130_ltakeWhile_Odisc__iff_I2_J, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)))) = ((~ (((coinductive_lnull @ A) @ Xs))) & (P2 @ ((coinductive_lhd @ A) @ Xs))))))).
% 2.38/1.46 thf(fact_131_ltakeWhile_Odisc__iff_I1_J, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = (((coinductive_lnull @ A) @ Xs) | (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.38/1.46 thf(fact_132_lnull__ltakeWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = ((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.38/1.46 thf(fact_133_lzip__simps_I2_J, axiom, (! [D:$tType,C:$tType,Xs:(coinductive_llist @ C)]: ((((((coinductive_lzip @ C) @ D) @ Xs) @ (coinductive_LNil @ D)) = (coinductive_LNil @ ((product_prod @ C) @ D)))))).
% 2.38/1.46 thf(fact_134_lzip__simps_I1_J, axiom, (! [B:$tType,A:$tType,Ys:(coinductive_llist @ B)]: ((((((coinductive_lzip @ A) @ B) @ (coinductive_LNil @ A)) @ Ys) = (coinductive_LNil @ ((product_prod @ A) @ B)))))).
% 2.38/1.46 thf(fact_135_llexord__code_I2_J, axiom, (! [A:$tType,R:(A > (A > $o)),X3:A,Xs:(coinductive_llist @ A)]: ((~ (((((coinductive_llexord @ A) @ R) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (coinductive_LNil @ A))))))).
% 2.38/1.46 thf(fact_136_llexord_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A1:(coinductive_llist @ A),A22:(coinductive_llist @ A)]: ((((((coinductive_llexord @ A) @ R) @ A1) @ A22) => ((! [Xs2:(coinductive_llist @ A),Ys4:(coinductive_llist @ A),X:A]: (((A1 = (((coinductive_LCons @ A) @ X) @ Xs2)) => ((A22 = (((coinductive_LCons @ A) @ X) @ Ys4)) => (~ (((((coinductive_llexord @ A) @ R) @ Xs2) @ Ys4))))))) => ((! [X:A]: (((? [Xs2:(coinductive_llist @ A)]: ((A1 = (((coinductive_LCons @ A) @ X) @ Xs2)))) => (! [Y:A]: (((? [Ys4:(coinductive_llist @ A)]: ((A22 = (((coinductive_LCons @ A) @ Y) @ Ys4)))) => (~ (((R @ X) @ Y))))))))) => (~ (((A1 = (coinductive_LNil @ A)) => (! [Ys4:(coinductive_llist @ A)]: ((A22 != Ys4)))))))))))).
% 2.38/1.46 thf(fact_137_llexord_Osimps, axiom, (! [A:$tType]: (((coinductive_llexord @ A) = (^ [R3:(A > (A > $o)),A12:(coinductive_llist @ A),A23:(coinductive_llist @ A)]: (((? [Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A),X4:A]: (((A12 = (((coinductive_LCons @ A) @ X4) @ Xs5)) & ((A23 = (((coinductive_LCons @ A) @ X4) @ Ys3)) & ((((coinductive_llexord @ A) @ R3) @ Xs5) @ Ys3))))) | ((? [X4:A,Y4:A,Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A)]: (((A12 = (((coinductive_LCons @ A) @ X4) @ Xs5)) & ((A23 = (((coinductive_LCons @ A) @ Y4) @ Ys3)) & ((R3 @ X4) @ Y4))))) | (? [Ys3:(coinductive_llist @ A)]: (((A12 = (coinductive_LNil @ A)) & (A23 = Ys3)))))))))))).
% 2.38/1.46 thf(fact_138_llexord_Ocoinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),X3:(coinductive_llist @ A),Xa2:(coinductive_llist @ A),R:(A > (A > $o))]: ((((X6 @ X3) @ Xa2) => ((! [X:(coinductive_llist @ A),Xa:(coinductive_llist @ A)]: ((((X6 @ X) @ Xa) => ((? [Xs6:(coinductive_llist @ A),Ys5:(coinductive_llist @ A),Xb:A]: (((X = (((coinductive_LCons @ A) @ Xb) @ Xs6)) & ((Xa = (((coinductive_LCons @ A) @ Xb) @ Ys5)) & (((X6 @ Xs6) @ Ys5) | ((((coinductive_llexord @ A) @ R) @ Xs6) @ Ys5)))))) | ((? [Xb:A,Y5:A,Xs6:(coinductive_llist @ A),Ys5:(coinductive_llist @ A)]: (((X = (((coinductive_LCons @ A) @ Xb) @ Xs6)) & ((Xa = (((coinductive_LCons @ A) @ Y5) @ Ys5)) & ((R @ Xb) @ Y5))))) | (? [Ys5:(coinductive_llist @ A)]: (((X = (coinductive_LNil @ A)) & (Xa = Ys5))))))))) => ((((coinductive_llexord @ A) @ R) @ X3) @ Xa2)))))).
% 2.38/1.46 thf(fact_139_llexord__LNil, axiom, (! [A:$tType,R:(A > (A > $o)),Ys:(coinductive_llist @ A)]: (((((coinductive_llexord @ A) @ R) @ (coinductive_LNil @ A)) @ Ys)))).
% 2.38/1.46 thf(fact_140_llexord__antisym, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((((coinductive_llexord @ A) @ R) @ Xs) @ Ys) => (((((coinductive_llexord @ A) @ R) @ Ys) @ Xs) => ((! [A4:A,B4:A]: ((((R @ A4) @ B4) => (~ (((R @ B4) @ A4)))))) => (Xs = Ys))))))).
% 2.38/1.46 thf(fact_141_llexord__linear, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((! [X:A,Y:A]: ((((R @ X) @ Y) | ((X = Y) | ((R @ Y) @ X))))) => (((((coinductive_llexord @ A) @ R) @ Xs) @ Ys) | ((((coinductive_llexord @ A) @ R) @ Ys) @ Xs)))))).
% 2.38/1.46 thf(fact_142_llexord__trans, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: ((((((coinductive_llexord @ A) @ R) @ Xs) @ Ys) => (((((coinductive_llexord @ A) @ R) @ Ys) @ Zs) => ((! [A4:A,B4:A,C2:A]: ((((R @ A4) @ B4) => (((R @ B4) @ C2) => ((R @ A4) @ C2))))) => ((((coinductive_llexord @ A) @ R) @ Xs) @ Zs))))))).
% 2.38/1.46 thf(fact_143_lzip__eq__LNil__conv, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coinductive_lzip @ A) @ B) @ Xs) @ Ys) = (coinductive_LNil @ ((product_prod @ A) @ B))) = ((Xs = (coinductive_LNil @ A)) | (Ys = (coinductive_LNil @ B))))))).
% 2.38/1.46 thf(fact_144_ltakeWhile__eq__LNil__iff, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coindu501562517eWhile @ A) @ P2) @ Xs) = (coinductive_LNil @ A)) = ((Xs != (coinductive_LNil @ A)) => (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.38/1.46 thf(fact_145_ltakeWhile_Octr_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((((coinductive_lnull @ A) @ Xs) | (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))) => ((((coindu501562517eWhile @ A) @ P2) @ Xs) = (coinductive_LNil @ A)))))).
% 2.38/1.46 thf(fact_146_ltl__ltakeWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((P2 @ ((coinductive_lhd @ A) @ Xs)) => (((coinductive_ltl @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = (((coindu501562517eWhile @ A) @ P2) @ ((coinductive_ltl @ A) @ Xs)))) & ((~ ((P2 @ ((coinductive_lhd @ A) @ Xs)))) => (((coinductive_ltl @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = (coinductive_LNil @ A))))))).
% 2.38/1.46 thf(fact_147_lset__ltakeWhileD, axiom, (! [A:$tType,X3:A,P2:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs))) => ((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) & (P2 @ X3)))))).
% 2.38/1.46 thf(fact_148_ltakeWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P2:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)) => ((P2 @ X) = (Q @ X))))) => ((((coindu501562517eWhile @ A) @ P2) @ Xs) = (((coindu501562517eWhile @ A) @ Q) @ Ys))))))).
% 2.38/1.46 thf(fact_149_ltakeWhile__all, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (P2 @ X)))) => ((((coindu501562517eWhile @ A) @ P2) @ Xs) = Xs))))).
% 2.38/1.46 thf(fact_150_neq__LNil__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((Xs != (coinductive_LNil @ A)) = (? [X4:A,Xs3:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X4) @ Xs3)))))))).
% 2.38/1.46 thf(fact_151_llist_Oexhaust, axiom, (! [A:$tType,Y3:(coinductive_llist @ A)]: (((Y3 != (coinductive_LNil @ A)) => (~ ((! [X212:A,X222:(coinductive_llist @ A)]: ((Y3 != (((coinductive_LCons @ A) @ X212) @ X222)))))))))).
% 2.38/1.46 thf(fact_152_llist_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: (((coinductive_LNil @ A) != (((coinductive_LCons @ A) @ X21) @ X22))))).
% 2.38/1.46 thf(fact_153_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coinductive_lnull @ A) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_154_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((Llist = (coinductive_LNil @ A)) => ((coinductive_lnull @ A) @ Llist))))).
% 2.38/1.46 thf(fact_155_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Llist) => (Llist = (coinductive_LNil @ A)))))).
% 2.38/1.46 thf(fact_156_lnull__def, axiom, (! [A:$tType]: (((coinductive_lnull @ A) = (^ [Llist5:(coinductive_llist @ A)]: ((Llist5 = (coinductive_LNil @ A)))))))).
% 2.38/1.46 thf(fact_157_ltl__simps_I1_J, axiom, (! [A:$tType]: ((((coinductive_ltl @ A) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_158_llexord__LCons__less, axiom, (! [A:$tType,R:(A > (A > $o)),X3:A,Y3:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((R @ X3) @ Y3) => ((((coinductive_llexord @ A) @ R) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (((coinductive_LCons @ A) @ Y3) @ Ys)))))).
% 2.38/1.46 thf(fact_159_llexord__LCons__eq, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),X3:A]: ((((((coinductive_llexord @ A) @ R) @ Xs) @ Ys) => ((((coinductive_llexord @ A) @ R) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (((coinductive_LCons @ A) @ X3) @ Ys)))))).
% 2.38/1.46 thf(fact_160_llexord__LCons__left, axiom, (! [A:$tType,R:(A > (A > $o)),X3:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((((coinductive_llexord @ A) @ R) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ Ys) = (? [Y4:A,Ys2:(coinductive_llist @ A)]: (((Ys = (((coinductive_LCons @ A) @ Y4) @ Ys2)) & (((X3 = Y4) & ((((coinductive_llexord @ A) @ R) @ Xs) @ Ys2)) | ((R @ X3) @ Y4))))))))).
% 2.38/1.46 thf(fact_161_llexord__code_I3_J, axiom, (! [A:$tType,R:(A > (A > $o)),X3:A,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: ((((((coinductive_llexord @ A) @ R) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = (((R @ X3) @ Y3) | ((X3 = Y3) & ((((coinductive_llexord @ A) @ R) @ Xs) @ Ys))))))).
% 2.38/1.46 thf(fact_162_lnull__llexord, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),R:(A > (A > $o)),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((((coinductive_llexord @ A) @ R) @ Xs) @ Ys))))).
% 2.38/1.46 thf(fact_163_ord_OLNil, axiom, (! [A:$tType,Less_eq:(A > (A > $o))]: ((((coinductive_lsorted @ A) @ Less_eq) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_164_ord_Olsorted__code_I1_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o))]: ((((coinductive_lsorted @ A) @ Less_eq) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_165_llist_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F23:(A > ((coinductive_llist @ A) > B))]: (((((((coindu1381640503_llist @ B) @ A) @ F1) @ F23) @ (coinductive_LNil @ A)) = F1)))).
% 2.38/1.46 thf(fact_166_ldistinct_OLNil, axiom, (! [A:$tType]: (((coindu351974385stinct @ A) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_167_unfold__llist_Octr_I1_J, axiom, (! [A:$tType,B:$tType,P:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((P @ A2) => (((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G22) @ A2) = (coinductive_LNil @ B)))))).
% 2.38/1.46 thf(fact_168_gen__lset__code_I1_J, axiom, (! [A:$tType,A5:(set @ A)]: (((((coinductive_gen_lset @ A) @ A5) @ (coinductive_LNil @ A)) = A5)))).
% 2.38/1.46 thf(fact_169_lmember__code_I1_J, axiom, (! [A:$tType,X3:A]: ((~ ((((coinductive_lmember @ A) @ X3) @ (coinductive_LNil @ A))))))).
% 2.38/1.46 thf(fact_170_ltakeWhile_Ocode, axiom, (! [A:$tType]: (((coindu501562517eWhile @ A) = (^ [P3:(A > $o),Xs5:(coinductive_llist @ A)]: (((((if @ (coinductive_llist @ A)) @ (((coinductive_lnull @ A) @ Xs5) | (~ ((P3 @ ((coinductive_lhd @ A) @ Xs5)))))) @ (coinductive_LNil @ A)) @ (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Xs5)) @ (((coindu501562517eWhile @ A) @ P3) @ ((coinductive_ltl @ A) @ Xs5)))))))))).
% 2.38/1.46 thf(fact_171_ltakeWhile_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P2 @ ((coinductive_lhd @ A) @ Xs)) => (~ (((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs))))))))).
% 2.38/1.46 thf(fact_172_ltakeWhile_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((((coinductive_lnull @ A) @ Xs) | (~ ((P2 @ ((coinductive_lhd @ A) @ Xs))))) => ((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)))))).
% 2.38/1.46 thf(fact_173_lhd__ltakeWhile, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P2 @ ((coinductive_lhd @ A) @ Xs)) => (((coinductive_lhd @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = ((coinductive_lhd @ A) @ Xs))))))).
% 2.38/1.46 thf(fact_174_lset__LNil, axiom, (! [A:$tType]: ((((coinductive_lset @ A) @ (coinductive_LNil @ A)) = (bot_bot @ (set @ A)))))).
% 2.38/1.46 thf(fact_175_ord_OSingleton, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X3:A]: ((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A)))))).
% 2.38/1.46 thf(fact_176_ord_Olsorted_Ocoinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > $o),X3:(coinductive_llist @ A),Less_eq:(A > (A > $o))]: (((X6 @ X3) => ((! [X:(coinductive_llist @ A)]: (((X6 @ X) => ((X = (coinductive_LNil @ A)) | ((? [Xa3:A]: ((X = (((coinductive_LCons @ A) @ Xa3) @ (coinductive_LNil @ A))))) | (? [Xa3:A,Y5:A,Xs6:(coinductive_llist @ A)]: (((X = (((coinductive_LCons @ A) @ Xa3) @ (((coinductive_LCons @ A) @ Y5) @ Xs6))) & (((Less_eq @ Xa3) @ Y5) & ((X6 @ (((coinductive_LCons @ A) @ Y5) @ Xs6)) | (((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ Y5) @ Xs6)))))))))))) => (((coinductive_lsorted @ A) @ Less_eq) @ X3)))))).
% 2.38/1.46 thf(fact_177_ord_Olsorted_Osimps, axiom, (! [A:$tType]: (((coinductive_lsorted @ A) = (^ [Less_eq2:(A > (A > $o)),A7:(coinductive_llist @ A)]: (((A7 = (coinductive_LNil @ A)) | ((? [X4:A]: ((A7 = (((coinductive_LCons @ A) @ X4) @ (coinductive_LNil @ A))))) | (? [X4:A,Y4:A,Xs5:(coinductive_llist @ A)]: (((A7 = (((coinductive_LCons @ A) @ X4) @ (((coinductive_LCons @ A) @ Y4) @ Xs5))) & (((Less_eq2 @ X4) @ Y4) & (((coinductive_lsorted @ A) @ Less_eq2) @ (((coinductive_LCons @ A) @ Y4) @ Xs5)))))))))))))).
% 2.38/1.46 thf(fact_178_ord_Olsorted_Ocases, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),A2:(coinductive_llist @ A)]: (((((coinductive_lsorted @ A) @ Less_eq) @ A2) => ((A2 != (coinductive_LNil @ A)) => ((! [X:A]: ((A2 != (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))) => (~ ((! [X:A,Y:A,Xs2:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs2))) => (((Less_eq @ X) @ Y) => (~ ((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ Y) @ Xs2)))))))))))))))).
% 2.38/1.46 thf(fact_179_ord_Olsorted__code_I2_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X3:A]: ((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A)))))).
% 2.38/1.46 thf(fact_180_lzip_Octr_I1_J, axiom, (! [B:$tType,A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)) => (((((coinductive_lzip @ A) @ B) @ Xs) @ Ys) = (coinductive_LNil @ ((product_prod @ A) @ B))))))).
% 2.38/1.46 thf(fact_181_ldropWhile__eq__LNil__iff, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coindu218763757pWhile @ A) @ P2) @ Xs) = (coinductive_LNil @ A)) = (! [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) => (P2 @ X4)))))))).
% 2.38/1.46 thf(fact_182_unfold__llist_Ocode, axiom, (! [B:$tType,A:$tType]: ((((coindu1441602521_llist @ A) @ B) = (^ [P4:(A > $o),G212:(A > B),G222:(A > A),A7:A]: (((((if @ (coinductive_llist @ B)) @ (P4 @ A7)) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ (G212 @ A7)) @ ((((((coindu1441602521_llist @ A) @ B) @ P4) @ G212) @ G222) @ (G222 @ A7)))))))))).
% 2.38/1.46 thf(fact_183_ltl__unfold__llist, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),A2:B,LHD:(B > A),LTL:(B > B)]: ((((IS_LNIL @ A2) => (((coinductive_ltl @ A) @ ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ A2)) = (coinductive_LNil @ A))) & ((~ ((IS_LNIL @ A2))) => (((coinductive_ltl @ A) @ ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ A2)) = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ A2)))))))).
% 2.38/1.46 thf(fact_184_ltakeWhile_Osimps_I4_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P2 @ ((coinductive_lhd @ A) @ Xs)) => (((coinductive_ltl @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = (((coindu501562517eWhile @ A) @ P2) @ ((coinductive_ltl @ A) @ Xs)))))))).
% 2.38/1.46 thf(fact_185_eq__LConsD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y3) @ Ys)) => ((Xs != (coinductive_LNil @ A)) & ((((coinductive_lhd @ A) @ Xs) = Y3) & (((coinductive_ltl @ A) @ Xs) = Ys))))))).
% 2.38/1.46 thf(fact_186_llist_Oexhaust__sel, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((Llist != (coinductive_LNil @ A)) => (Llist = (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist))))))).
% 2.38/1.46 thf(fact_187_ldistinct_Ocoinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > $o),X3:(coinductive_llist @ A)]: (((X6 @ X3) => ((! [X:(coinductive_llist @ A)]: (((X6 @ X) => ((X = (coinductive_LNil @ A)) | (? [Xa3:A,Xs6:(coinductive_llist @ A)]: (((X = (((coinductive_LCons @ A) @ Xa3) @ Xs6)) & ((~ ((((member @ A) @ Xa3) @ ((coinductive_lset @ A) @ Xs6)))) & ((X6 @ Xs6) | ((coindu351974385stinct @ A) @ Xs6)))))))))) => ((coindu351974385stinct @ A) @ X3)))))).
% 2.38/1.46 thf(fact_188_ldistinct_Osimps, axiom, (! [A:$tType]: (((coindu351974385stinct @ A) = (^ [A7:(coinductive_llist @ A)]: (((A7 = (coinductive_LNil @ A)) | (? [X4:A,Xs5:(coinductive_llist @ A)]: (((A7 = (((coinductive_LCons @ A) @ X4) @ Xs5)) & ((~ ((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs5)))) & ((coindu351974385stinct @ A) @ Xs5)))))))))))).
% 2.38/1.46 thf(fact_189_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs3:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y3) @ Xs3)) & (((coindu328551480prefix @ A) @ Xs3) @ Ys))))))))).
% 2.38/1.46 thf(fact_190_lstrict__prefix__code_I3_J, axiom, (! [B:$tType,X3:B,Xs:(coinductive_llist @ B)]: ((~ ((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X3) @ Xs)) @ (coinductive_LNil @ B))))))).
% 2.38/1.46 thf(fact_191_lstrict__prefix__code_I2_J, axiom, (! [B:$tType,Y3:B,Ys:(coinductive_llist @ B)]: ((((coindu1478340336prefix @ B) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ Y3) @ Ys))))).
% 2.38/1.46 thf(fact_192_llast__singleton, axiom, (! [A:$tType,X3:A]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A))) = X3)))).
% 2.38/1.46 thf(fact_193_llast__LCons2, axiom, (! [A:$tType,X3:A,Y3:A,Xs:(coinductive_llist @ A)]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X3) @ (((coinductive_LCons @ A) @ Y3) @ Xs))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y3) @ Xs)))))).
% 2.38/1.46 thf(fact_194_lstrict__prefix__code_I4_J, axiom, (! [B:$tType,X3:B,Xs:(coinductive_llist @ B),Y3:B,Ys:(coinductive_llist @ B)]: (((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X3) @ Xs)) @ (((coinductive_LCons @ B) @ Y3) @ Ys)) = ((X3 = Y3) & (((coindu1478340336prefix @ B) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_195_lstrict__prefix__code_I1_J, axiom, (! [A:$tType]: ((~ ((((coindu1478340336prefix @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A))))))).
% 2.38/1.46 thf(fact_196_llist__less__induct, axiom, (! [A:$tType,P2:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((! [Xs2:(coinductive_llist @ A)]: (((! [Ys5:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys5) @ Xs2) => (P2 @ Ys5)))) => (P2 @ Xs2)))) => (P2 @ Xs))))).
% 2.38/1.46 thf(fact_197_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu328551480prefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.38/1.46 thf(fact_198_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (coinductive_LNil @ A)) = (Xs = (coinductive_LNil @ A)))))).
% 2.38/1.46 thf(fact_199_llast__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X3:A]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = X3)) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = ((coinductive_llast @ A) @ Xs))))))).
% 2.38/1.46 thf(fact_200_lprefix__coinduct, axiom, (! [A:$tType,P2:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((P2 @ Xs) @ Ys) => ((! [Xs2:(coinductive_llist @ A),Ys4:(coinductive_llist @ A)]: ((((P2 @ Xs2) @ Ys4) => ((((coinductive_lnull @ A) @ Ys4) => ((coinductive_lnull @ A) @ Xs2)) & ((~ (((coinductive_lnull @ A) @ Xs2))) => ((~ (((coinductive_lnull @ A) @ Ys4))) => ((((coinductive_lhd @ A) @ Xs2) = ((coinductive_lhd @ A) @ Ys4)) & (((P2 @ ((coinductive_ltl @ A) @ Xs2)) @ ((coinductive_ltl @ A) @ Ys4)) | (((coinductive_lprefix @ A) @ ((coinductive_ltl @ A) @ Xs2)) @ ((coinductive_ltl @ A) @ Ys4)))))))))) => (((coinductive_lprefix @ A) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_201_lprefix__expand, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ A) @ Ys))) & ((((coinductive_lhd @ A) @ Xs) = ((coinductive_lhd @ A) @ Ys)) & (((coinductive_lprefix @ A) @ ((coinductive_ltl @ A) @ Xs)) @ ((coinductive_ltl @ A) @ Ys))))) => (((coinductive_lprefix @ A) @ Xs) @ Ys))))).
% 2.38/1.46 thf(fact_202_llist_Oleq__refl, axiom, (! [A:$tType,X3:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ X3) @ X3)))).
% 2.38/1.46 thf(fact_203_lprefix__refl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ Xs)))).
% 2.38/1.46 thf(fact_204_LCons__lprefix__LCons, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = ((X3 = Y3) & (((coinductive_lprefix @ A) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_205_lprefix__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (coinductive_LNil @ A)) @ Ys)))).
% 2.38/1.46 thf(fact_206_lprefix__LNil, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (coinductive_LNil @ A)) = ((coinductive_lnull @ A) @ Xs))))).
% 2.38/1.46 thf(fact_207_Coinductive__List_Ofinite__lprefix__def, axiom, (! [A:$tType]: (((coindu328551480prefix @ A) = (coinductive_lprefix @ A))))).
% 2.38/1.46 thf(fact_208_lstrict__prefix__def, axiom, (! [A:$tType]: (((coindu1478340336prefix @ A) = (^ [Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs5) @ Ys3) & (Xs5 != Ys3)))))))).
% 2.38/1.46 thf(fact_209_lprefix__LCons__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs3:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y3) @ Xs3)) & (((coinductive_lprefix @ A) @ Xs3) @ Ys))))))))).
% 2.38/1.46 thf(fact_210_lprefix_Ocoinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),X3:(coinductive_llist @ A),Xa2:(coinductive_llist @ A)]: ((((X6 @ X3) @ Xa2) => ((! [X:(coinductive_llist @ A),Xa:(coinductive_llist @ A)]: ((((X6 @ X) @ Xa) => ((? [Xs6:(coinductive_llist @ A)]: (((X = (coinductive_LNil @ A)) & (Xa = Xs6)))) | (? [Xs6:(coinductive_llist @ A),Ys5:(coinductive_llist @ A),Xb:A]: (((X = (((coinductive_LCons @ A) @ Xb) @ Xs6)) & ((Xa = (((coinductive_LCons @ A) @ Xb) @ Ys5)) & (((X6 @ Xs6) @ Ys5) | (((coinductive_lprefix @ A) @ Xs6) @ Ys5)))))))))) => (((coinductive_lprefix @ A) @ X3) @ Xa2)))))).
% 2.38/1.46 thf(fact_211_lprefix_Osimps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [A12:(coinductive_llist @ A),A23:(coinductive_llist @ A)]: (((? [Xs5:(coinductive_llist @ A)]: (((A12 = (coinductive_LNil @ A)) & (A23 = Xs5)))) | (? [Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A),X4:A]: (((A12 = (((coinductive_LCons @ A) @ X4) @ Xs5)) & ((A23 = (((coinductive_LCons @ A) @ X4) @ Ys3)) & (((coinductive_lprefix @ A) @ Xs5) @ Ys3)))))))))))).
% 2.38/1.46 thf(fact_212_lprefix_Ocases, axiom, (! [A:$tType,A1:(coinductive_llist @ A),A22:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ A1) @ A22) => (((A1 = (coinductive_LNil @ A)) => (! [Xs2:(coinductive_llist @ A)]: ((A22 != Xs2)))) => (~ ((! [Xs2:(coinductive_llist @ A),Ys4:(coinductive_llist @ A),X:A]: (((A1 = (((coinductive_LCons @ A) @ X) @ Xs2)) => ((A22 = (((coinductive_LCons @ A) @ X) @ Ys4)) => (~ ((((coinductive_lprefix @ A) @ Xs2) @ Ys4)))))))))))))).
% 2.38/1.46 thf(fact_213_lprefix__code_I2_J, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: ((~ ((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (coinductive_LNil @ A))))))).
% 2.38/1.46 thf(fact_214_lprefixI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),X6:(set @ ((product_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A)))]: (((((member @ ((product_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A))) @ ((((product_Pair @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ Xs) @ Ys)) @ X6) => ((! [Xs2:(coinductive_llist @ A),Ys4:(coinductive_llist @ A)]: (((((member @ ((product_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A))) @ ((((product_Pair @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ Xs2) @ Ys4)) @ X6) => (((coinductive_lnull @ A) @ Xs2) | (? [X5:A,Xs7:(coinductive_llist @ A),Ys6:(coinductive_llist @ A)]: (((Xs2 = (((coinductive_LCons @ A) @ X5) @ Xs7)) & ((Ys4 = (((coinductive_LCons @ A) @ X5) @ Ys6)) & ((((member @ ((product_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A))) @ ((((product_Pair @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ Xs7) @ Ys6)) @ X6) | (((coinductive_lprefix @ A) @ Xs7) @ Ys6)))))))))) => (((coinductive_lprefix @ A) @ Xs) @ Ys)))))).
% 2.38/1.46 thf(fact_215_lprefix__lhdD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_lhd @ A) @ Xs) = ((coinductive_lhd @ A) @ Ys))))))).
% 2.38/1.46 thf(fact_216_lprefix__down__linear, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Zs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Zs) => ((((coinductive_lprefix @ A) @ Ys) @ Zs) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) | (((coinductive_lprefix @ A) @ Ys) @ Xs))))))).
% 2.38/1.46 thf(fact_217_llist_Oleq__antisym, axiom, (! [A:$tType,X3:(coinductive_llist @ A),Y3:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X3) @ Y3) => ((((coinductive_lprefix @ A) @ Y3) @ X3) => (X3 = Y3)))))).
% 2.38/1.46 thf(fact_218_lprefix__antisym, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Xs) => (Xs = Ys)))))).
% 2.38/1.46 thf(fact_219_llist_Oleq__trans, axiom, (! [A:$tType,X3:(coinductive_llist @ A),Y3:(coinductive_llist @ A),Z:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X3) @ Y3) => ((((coinductive_lprefix @ A) @ Y3) @ Z) => (((coinductive_lprefix @ A) @ X3) @ Z)))))).
% 2.38/1.46 thf(fact_220_lprefix__trans, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Zs) => (((coinductive_lprefix @ A) @ Xs) @ Zs)))))).
% 2.38/1.46 thf(fact_221_ldistinct__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ Xs) => ((((coinductive_lprefix @ A) @ Ys) @ Xs) => ((coindu351974385stinct @ A) @ Ys)))))).
% 2.38/1.46 thf(fact_222_ord_Olsorted__lprefixD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less_eq:(A > (A > $o))]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lsorted @ A) @ Less_eq) @ Ys) => (((coinductive_lsorted @ A) @ Less_eq) @ Xs)))))).
% 2.38/1.46 thf(fact_223_lprefix__ltlI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ ((coinductive_ltl @ A) @ Xs)) @ ((coinductive_ltl @ A) @ Ys)))))).
% 2.38/1.46 thf(fact_224_lprefix__not__lnullD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((~ (((coinductive_lnull @ A) @ Xs))) => (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.38/1.46 thf(fact_225_lprefix__lnullD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lnull @ A) @ Ys) => ((coinductive_lnull @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_226_lprefix__lnull, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = ((coinductive_lnull @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_227_lnull__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lprefix @ A) @ Xs) @ Ys))))).
% 2.38/1.46 thf(fact_228_Le__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),X3:A]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ (((coinductive_LCons @ A) @ X3) @ Ys)))))).
% 2.38/1.46 thf(fact_229_LCons__lprefix__conv, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) @ Ys) = (? [Ys2:(coinductive_llist @ A)]: (((Ys = (((coinductive_LCons @ A) @ X3) @ Ys2)) & (((coinductive_lprefix @ A) @ Xs) @ Ys2)))))))).
% 2.38/1.46 thf(fact_230_lprefix__ltakeWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) @ Xs)))).
% 2.38/1.46 thf(fact_231_lprefix__imp__llexord, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),R:(A > (A > $o))]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_llexord @ A) @ R) @ Xs) @ Ys))))).
% 2.38/1.46 thf(fact_232_LNil__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.38/1.46 thf(fact_233_llimit__induct, axiom, (! [A:$tType,P2:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((P2 @ (coinductive_LNil @ A)) => ((! [X:A,Xs2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs2) => ((P2 @ Xs2) => (P2 @ (((coinductive_LCons @ A) @ X) @ Xs2)))))) => (((! [Ys5:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys5) @ Xs) => (P2 @ Ys5)))) => (P2 @ Xs)) => (P2 @ Xs))))))).
% 2.38/1.46 thf(fact_234_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R3:(A > (A > $o)),F4:(B > A),X4:B,Y4:B]: (((R3 @ (F4 @ X4)) @ (F4 @ Y4)))))))).
% 2.38/1.46 thf(fact_235_lfinite__LCons, axiom, (! [A:$tType,X3:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.38/1.46 thf(fact_236_lfinite__code_I2_J, axiom, (! [B:$tType,X3:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X3) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 2.38/1.46 thf(fact_237_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_238_lfinite__ltl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ ((coinductive_ltl @ A) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.38/1.46 thf(fact_239_lfinite__lzip, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: ((((coinductive_lfinite @ ((product_prod @ A) @ B)) @ ((((coinductive_lzip @ A) @ B) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) | ((coinductive_lfinite @ B) @ Ys)))))).
% 2.38/1.46 thf(fact_240_Coinductive__List_Olprefix__nitpick__simps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A)]: (((((coinductive_lfinite @ A) @ Xs5) => (((coindu328551480prefix @ A) @ Xs5) @ Ys3)) & ((~ (((coinductive_lfinite @ A) @ Xs5))) => (Xs5 = Ys3))))))))).
% 2.38/1.46 thf(fact_241_not__lfinite__lprefix__conv__eq, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = (Xs = Ys)))))).
% 2.38/1.46 thf(fact_242_lprefix__lfiniteD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_243_lfinite__LNil, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.38/1.46 thf(fact_244_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X3:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X3) @ Xs)))))).
% 2.38/1.46 thf(fact_245_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.38/1.46 thf(fact_246_lfinite__ldropWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu218763757pWhile @ A) @ P2) @ Xs)) = ((? [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P2 @ X4)))))) => ((coinductive_lfinite @ A) @ Xs)))))).
% 2.38/1.46 thf(fact_247_lfinite__ltakeWhile, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu501562517eWhile @ A) @ P2) @ Xs)) = (((coinductive_lfinite @ A) @ Xs) | (? [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P2 @ X4))))))))))).
% 2.38/1.46 thf(fact_248_lfinite__induct, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((! [Xs2:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs2) => (P2 @ Xs2)))) => ((! [Xs2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs2) => ((~ (((coinductive_lnull @ A) @ Xs2))) => ((P2 @ ((coinductive_ltl @ A) @ Xs2)) => (P2 @ Xs2)))))) => (P2 @ Xs))))))).
% 2.38/1.46 thf(fact_249_lfinite_Ocases, axiom, (! [A:$tType,A2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ A2) => ((A2 != (coinductive_LNil @ A)) => (~ ((! [Xs2:(coinductive_llist @ A)]: (((? [X:A]: ((A2 = (((coinductive_LCons @ A) @ X) @ Xs2)))) => (~ (((coinductive_lfinite @ A) @ Xs2))))))))))))).
% 2.38/1.46 thf(fact_250_lfinite_Osimps, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) = (^ [A7:(coinductive_llist @ A)]: (((A7 = (coinductive_LNil @ A)) | (? [Xs5:(coinductive_llist @ A),X4:A]: (((A7 = (((coinductive_LCons @ A) @ X4) @ Xs5)) & ((coinductive_lfinite @ A) @ Xs5))))))))))).
% 2.38/1.46 thf(fact_251_lfinite_Oinducts, axiom, (! [A:$tType,X3:(coinductive_llist @ A),P2:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ X3) => ((P2 @ (coinductive_LNil @ A)) => ((! [Xs2:(coinductive_llist @ A),X:A]: ((((coinductive_lfinite @ A) @ Xs2) => ((P2 @ Xs2) => (P2 @ (((coinductive_LCons @ A) @ X) @ Xs2)))))) => (P2 @ X3))))))).
% 2.38/1.46 thf(fact_252_lstrict__prefix__lfinite1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ Ys) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.38/1.46 thf(fact_253_llexord__conv, axiom, (! [A:$tType]: (((coinductive_llexord @ A) = (^ [R3:(A > (A > $o)),Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A)]: (((Xs5 = Ys3) | (? [Zs2:(coinductive_llist @ A),Xs3:(coinductive_llist @ A),Y4:A,Ys2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Zs2) & ((Xs5 = (((coinductive_lappend @ A) @ Zs2) @ Xs3)) & ((Ys3 = (((coinductive_lappend @ A) @ Zs2) @ (((coinductive_LCons @ A) @ Y4) @ Ys2))) & ((Xs3 = (coinductive_LNil @ A)) | ((R3 @ ((coinductive_lhd @ A) @ Xs3)) @ Y4)))))))))))))).
% 2.38/1.46 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: ((((bot @ A9) @ (type2 @ A9)) => ((bot @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.38/1.46 thf(tcon_Set_Oset___Orderings_Obot_1, axiom, (! [A8:$tType]: (((bot @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.38/1.46 thf(tcon_HOL_Obool___Orderings_Obot_2, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.38/1.46 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 2.38/1.46 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y3:A]: ((((((if @ A) @ $false) @ X3) @ Y3) = Y3)))).
% 2.38/1.46 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y3:A]: ((((((if @ A) @ $true) @ X3) @ Y3) = X3)))).
% 2.38/1.46 thf(conj_0, conjecture, (((member @ node) @ n) @ ((coinductive_lset @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ n) @ ns))))).
% 2.38/1.46 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.udRHUCu2u7/DTF2DTF19871.p
% 2.38/1.46 ------------------------
% 2.52/1.50 ---- Cleaned THF ---
% 2.52/1.50 thf(ty_t_Coinductive__List_Ollist,type,
% 2.52/1.50 coinductive_llist: $tType > $tType ).
% 2.52/1.50
% 2.52/1.50 thf(ty_t_Product__Type_Oprod,type,
% 2.52/1.50 product_prod: $tType > $tType > $tType ).
% 2.52/1.50
% 2.52/1.50 thf(ty_t_Set_Oset,type,
% 2.52/1.50 set: $tType > $tType ).
% 2.52/1.50
% 2.52/1.50 thf(ty_t_itself,type,
% 2.52/1.50 itself: $tType > $tType ).
% 2.52/1.50
% 2.52/1.50 thf(ty_tf_node,type,
% 2.52/1.50 node: $tType ).
% 2.52/1.50
% 2.52/1.50 thf(sy_cl_HOL_Otype,type,
% 2.52/1.50 type:
% 2.52/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_cl_Orderings_Obot,type,
% 2.52/1.50 bot:
% 2.52/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 2.52/1.50 coindu328551480prefix:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 2.52/1.50 coinductive_gen_lset:
% 2.52/1.50 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Olappend,type,
% 2.52/1.50 coinductive_lappend:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Oldistinct,type,
% 2.52/1.50 coindu351974385stinct:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_OldropWhile,type,
% 2.52/1.50 coindu218763757pWhile:
% 2.52/1.50 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.52/1.50 coinductive_lfinite:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollast,type,
% 2.52/1.50 coinductive_llast:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollexord,type,
% 2.52/1.50 coinductive_llexord:
% 2.52/1.50 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 2.52/1.50 coinductive_LCons:
% 2.52/1.50 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 2.52/1.50 coinductive_LNil:
% 2.52/1.50 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 2.52/1.50 coindu1381640503_llist:
% 2.52/1.50 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 2.52/1.50 coinductive_lhd:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 2.52/1.50 coinductive_lnull:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 2.52/1.50 coinductive_lset:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ollist_Oltl,type,
% 2.52/1.50 coinductive_ltl:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Olmember,type,
% 2.52/1.50 coinductive_lmember:
% 2.52/1.50 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Olprefix,type,
% 2.52/1.50 coinductive_lprefix:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 2.52/1.50 coindu1478340336prefix:
% 2.52/1.50 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 2.52/1.50 coindu501562517eWhile:
% 2.52/1.50 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Olzip,type,
% 2.52/1.50 coinductive_lzip:
% 2.52/1.50 !>[A: $tType,B: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > ( coinductive_llist @ ( product_prod @ A @ B ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Oord_Olsorted,type,
% 2.52/1.50 coinductive_lsorted:
% 2.52/1.50 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 2.52/1.50 coindu1441602521_llist:
% 2.52/1.50 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_If,type,
% 2.52/1.50 if:
% 2.52/1.50 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
% 2.52/1.50 koenig793108494nected:
% 2.52/1.50 !>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.52/1.50 bot_bot:
% 2.52/1.50 !>[A: $tType] : A ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Product__Type_OPair,type,
% 2.52/1.50 product_Pair:
% 2.52/1.50 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.52/1.50 produc2004651681e_prod:
% 2.52/1.50 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.52/1.50 product_rec_prod:
% 2.52/1.50 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Pure_Otype,type,
% 2.52/1.50 type2:
% 2.52/1.50 !>[A: $tType] : ( itself @ A ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Relation_Oinv__imagep,type,
% 2.52/1.50 inv_imagep:
% 2.52/1.50 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Set_OCollect,type,
% 2.52/1.50 collect:
% 2.52/1.50 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_Set_Ois__empty,type,
% 2.52/1.50 is_empty:
% 2.52/1.50 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_c_member,type,
% 2.52/1.50 member:
% 2.52/1.50 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_v_f____,type,
% 2.52/1.50 f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
% 2.52/1.50
% 2.52/1.50 thf(sy_v_graph,type,
% 2.52/1.50 graph: node > node > $o ).
% 2.52/1.50
% 2.52/1.50 thf(sy_v_n,type,
% 2.52/1.50 n: node ).
% 2.52/1.50
% 2.52/1.50 thf(sy_v_ns____,type,
% 2.52/1.50 ns: set @ node ).
% 2.52/1.50
% 2.52/1.50 thf(fact_0__092_060open_062lhd_A_If_A_In_M_Ans_J_J_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
% 2.52/1.50 member @ node @ ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_1_ns__def,axiom,
% 2.52/1.50 ( ns
% 2.52/1.50 = ( bot_bot @ ( set @ node ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_2_f__simps_I2_J,axiom,
% 2.52/1.50 ! [Na: node,Nsa: set @ node] :
% 2.52/1.50 ( ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) )
% 2.52/1.50 = Na ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_3_f__simps_I1_J,axiom,
% 2.52/1.50 ! [Na: node,Nsa: set @ node] :
% 2.52/1.50 ~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_4_prod_Oinject,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.52/1.50 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.52/1.50 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.52/1.50 = ( ( X1 = Y1 )
% 2.52/1.50 & ( X2 = Y2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_5_old_Oprod_Oinject,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.52/1.50 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.52/1.50 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.52/1.50 = ( ( A2 = A3 )
% 2.52/1.50 & ( B2 = B3 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_6_connected,axiom,
% 2.52/1.50 koenig793108494nected @ node @ graph ).
% 2.52/1.50
% 2.52/1.50 thf(fact_7_surj__pair,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.52/1.50 ? [X: A,Y: B] :
% 2.52/1.50 ( P
% 2.52/1.50 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_8_prod__cases,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.52/1.50 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.52/1.50 => ( P2 @ P ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_9_Pair__inject,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.52/1.50 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.52/1.50 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.52/1.50 => ~ ( ( A2 = A3 )
% 2.52/1.50 => ( B2 != B3 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_10_prod__cases3,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.52/1.50 ~ ! [A4: A,B4: B,C2: C] :
% 2.52/1.50 ( Y3
% 2.52/1.50 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_11_prod__cases4,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.52/1.50 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 2.52/1.50 ( Y3
% 2.52/1.50 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_12_prod__cases5,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.52/1.50 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 2.52/1.50 ( Y3
% 2.52/1.50 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_13_prod__cases6,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.52/1.50 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 2.52/1.50 ( Y3
% 2.52/1.50 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_14_old_Oprod_Oinducts,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.52/1.50 ( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.52/1.50 => ( P2 @ Prod ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_15_old_Oprod_Oexhaust,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.52/1.50 ~ ! [A4: A,B4: B] :
% 2.52/1.50 ( Y3
% 2.52/1.50 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_16_prod__induct7,axiom,
% 2.52/1.50 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.52/1.50 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 2.52/1.50 => ( P2 @ X3 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_17_prod__induct6,axiom,
% 2.52/1.50 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.52/1.50 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.52/1.50 => ( P2 @ X3 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_18_prod__induct5,axiom,
% 2.52/1.50 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.52/1.50 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.52/1.50 => ( P2 @ X3 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_19_prod__induct4,axiom,
% 2.52/1.50 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.52/1.50 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.52/1.50 => ( P2 @ X3 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_20_prod__induct3,axiom,
% 2.52/1.50 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.52/1.50 ( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 2.52/1.50 => ( P2 @ X3 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_21_prod__cases7,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.52/1.50 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 2.52/1.50 ( Y3
% 2.52/1.50 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_22_lset__eq__empty,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( coinductive_lset @ A @ Xs )
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) )
% 2.52/1.50 = ( coinductive_lnull @ A @ Xs ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_23_old_Oprod_Orec,axiom,
% 2.52/1.50 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.52/1.50 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.52/1.50 = ( F1 @ A2 @ B2 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_24_llist_Oset__sel_I1_J,axiom,
% 2.52/1.50 ! [A: $tType,A2: coinductive_llist @ A] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ A2 )
% 2.52/1.50 => ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_25_lset__lnull,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ( coinductive_lset @ A @ Xs )
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_26_empty__iff,axiom,
% 2.52/1.50 ! [A: $tType,C3: A] :
% 2.52/1.50 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_27_all__not__in__conv,axiom,
% 2.52/1.50 ! [A: $tType,A5: set @ A] :
% 2.52/1.50 ( ( ! [X4: A] :
% 2.52/1.50 ~ ( member @ A @ X4 @ A5 ) )
% 2.52/1.50 = ( A5
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_28_Collect__empty__eq,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o] :
% 2.52/1.50 ( ( ( collect @ A @ P2 )
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) )
% 2.52/1.50 = ( ! [X4: A] :
% 2.52/1.50 ~ ( P2 @ X4 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_29_empty__Collect__eq,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o] :
% 2.52/1.50 ( ( ( bot_bot @ ( set @ A ) )
% 2.52/1.50 = ( collect @ A @ P2 ) )
% 2.52/1.50 = ( ! [X4: A] :
% 2.52/1.50 ~ ( P2 @ X4 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_30_bot__apply,axiom,
% 2.52/1.50 ! [C: $tType,D: $tType] :
% 2.52/1.50 ( ( bot @ C @ ( type2 @ C ) )
% 2.52/1.50 => ( ( bot_bot @ ( D > C ) )
% 2.52/1.50 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_31_ltakeWhile_Oexhaust,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.50 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
% 2.52/1.50 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_32_internal__case__prod__conv,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.52/1.50 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.52/1.50 = ( C3 @ A2 @ B2 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_33_bot__set__def,axiom,
% 2.52/1.50 ! [A: $tType] :
% 2.52/1.50 ( ( bot_bot @ ( set @ A ) )
% 2.52/1.50 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_34_bot__fun__def,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType] :
% 2.52/1.50 ( ( bot @ B @ ( type2 @ B ) )
% 2.52/1.50 => ( ( bot_bot @ ( A > B ) )
% 2.52/1.50 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_35_ex__in__conv,axiom,
% 2.52/1.50 ! [A: $tType,A5: set @ A] :
% 2.52/1.50 ( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
% 2.52/1.50 = ( A5
% 2.52/1.50 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_36_equals0I,axiom,
% 2.52/1.50 ! [A: $tType,A5: set @ A] :
% 2.52/1.50 ( ! [Y: A] :
% 2.52/1.50 ~ ( member @ A @ Y @ A5 )
% 2.52/1.50 => ( A5
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_37_equals0D,axiom,
% 2.52/1.50 ! [A: $tType,A5: set @ A,A2: A] :
% 2.52/1.50 ( ( A5
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) )
% 2.52/1.50 => ~ ( member @ A @ A2 @ A5 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_38_emptyE,axiom,
% 2.52/1.50 ! [A: $tType,A2: A] :
% 2.52/1.50 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_39_lappend_Oexhaust,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ~ ( coinductive_lnull @ A @ Ys ) )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_40_lzip_Oexhaust,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 | ( coinductive_lnull @ B @ Ys ) )
% 2.52/1.50 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_41_lset__code,axiom,
% 2.52/1.50 ! [A: $tType] :
% 2.52/1.50 ( ( coinductive_lset @ A )
% 2.52/1.50 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_42_Set_Ois__empty__def,axiom,
% 2.52/1.50 ! [A: $tType] :
% 2.52/1.50 ( ( is_empty @ A )
% 2.52/1.50 = ( ^ [A6: set @ A] :
% 2.52/1.50 ( A6
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_43_mem__Collect__eq,axiom,
% 2.52/1.50 ! [A: $tType,A2: A,P2: A > $o] :
% 2.52/1.50 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.52/1.50 = ( P2 @ A2 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_44_Collect__mem__eq,axiom,
% 2.52/1.50 ! [A: $tType,A5: set @ A] :
% 2.52/1.50 ( ( collect @ A
% 2.52/1.50 @ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
% 2.52/1.50 = A5 ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_45_Collect__cong,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.52/1.50 ( ! [X: A] :
% 2.52/1.50 ( ( P2 @ X )
% 2.52/1.50 = ( Q @ X ) )
% 2.52/1.50 => ( ( collect @ A @ P2 )
% 2.52/1.50 = ( collect @ A @ Q ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_46_ext,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 2.52/1.50 ( ! [X: A] :
% 2.52/1.50 ( ( F3 @ X )
% 2.52/1.50 = ( G3 @ X ) )
% 2.52/1.50 => ( F3 = G3 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_47_lset__lmember,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 = ( coinductive_lmember @ A @ X3 @ Xs ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_48_Collect__empty__eq__bot,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o] :
% 2.52/1.50 ( ( ( collect @ A @ P2 )
% 2.52/1.50 = ( bot_bot @ ( set @ A ) ) )
% 2.52/1.50 = ( P2
% 2.52/1.50 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_49_bot__empty__eq,axiom,
% 2.52/1.50 ! [A: $tType] :
% 2.52/1.50 ( ( bot_bot @ ( A > $o ) )
% 2.52/1.50 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_50_ssubst__Pair__rhs,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 2.52/1.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 2.52/1.50 => ( ( S2 = S )
% 2.52/1.50 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_51_lhd__lzip,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ B @ Ys )
% 2.52/1.50 => ( ( coinductive_lhd @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 2.52/1.50 = ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ B @ Ys ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_52_llist__set__induct,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: A > ( coinductive_llist @ A ) > $o] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.50 => ( P2 @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A,Y: A] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.50 => ( ( member @ A @ Y @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 2.52/1.50 => ( ( P2 @ Y @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.52/1.50 => ( P2 @ Y @ Xs2 ) ) ) )
% 2.52/1.50 => ( P2 @ X3 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_53_lnull__ldropWhile,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) )
% 2.52/1.50 = ( ! [X4: A] :
% 2.52/1.50 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( P2 @ X4 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_54_lzip_Odisc__iff_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( ~ ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
% 2.52/1.50 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 & ~ ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_55_lnull__lzip,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 2.52/1.50 = ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 | ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_56_ltl__lzip,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ B @ Ys )
% 2.52/1.50 => ( ( coinductive_ltl @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 2.52/1.50 = ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ B @ Ys ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_57_lnull__ltlI,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_58_in__lset__ltlD,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
% 2.52/1.50 => ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_59_lzip_Odisc_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ B @ Ys )
% 2.52/1.50 => ~ ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_60_lzip_Odisc_I1_J,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 | ( coinductive_lnull @ B @ Ys ) )
% 2.52/1.50 => ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_61_in__lset__ldropWhileD,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) )
% 2.52/1.50 => ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_62_ldropWhile__cong,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P2: A > $o,Q: A > $o] :
% 2.52/1.50 ( ( Xs = Ys )
% 2.52/1.50 => ( ! [X: A] :
% 2.52/1.50 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 2.52/1.50 => ( ( P2 @ X )
% 2.52/1.50 = ( Q @ X ) ) )
% 2.52/1.50 => ( ( coindu218763757pWhile @ A @ P2 @ Xs )
% 2.52/1.50 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_63_llist_Oset__sel_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,A2: coinductive_llist @ A,X3: A] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ A2 )
% 2.52/1.50 => ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
% 2.52/1.50 => ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_64_lset__lzipD2,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,X3: A,Y3: B,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( coinductive_lset @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
% 2.52/1.50 => ( member @ B @ Y3 @ ( coinductive_lset @ B @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_65_lset__lzipD1,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,X3: A,Y3: B,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( coinductive_lset @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
% 2.52/1.50 => ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_66_llist_Oexpand,axiom,
% 2.52/1.50 ! [A: $tType,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( coinductive_lnull @ A @ Llist )
% 2.52/1.50 = ( coinductive_lnull @ A @ Llist2 ) )
% 2.52/1.50 => ( ( ~ ( coinductive_lnull @ A @ Llist )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Llist2 )
% 2.52/1.50 => ( ( ( coinductive_lhd @ A @ Llist )
% 2.52/1.50 = ( coinductive_lhd @ A @ Llist2 ) )
% 2.52/1.50 & ( ( coinductive_ltl @ A @ Llist )
% 2.52/1.50 = ( coinductive_ltl @ A @ Llist2 ) ) ) ) )
% 2.52/1.50 => ( Llist = Llist2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_67_llist_Ocoinduct,axiom,
% 2.52/1.50 ! [A: $tType,R2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 2.52/1.50 ( ( R2 @ Llist @ Llist2 )
% 2.52/1.50 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 2.52/1.50 ( ( R2 @ Llist3 @ Llist4 )
% 2.52/1.50 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 2.52/1.50 = ( coinductive_lnull @ A @ Llist4 ) )
% 2.52/1.50 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 2.52/1.50 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 2.52/1.50 = ( coinductive_lhd @ A @ Llist4 ) )
% 2.52/1.50 & ( R2 @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
% 2.52/1.50 => ( Llist = Llist2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_68_llist_Ocoinduct__strong,axiom,
% 2.52/1.50 ! [A: $tType,R2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 2.52/1.50 ( ( R2 @ Llist @ Llist2 )
% 2.52/1.50 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 2.52/1.50 ( ( R2 @ Llist3 @ Llist4 )
% 2.52/1.50 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 2.52/1.50 = ( coinductive_lnull @ A @ Llist4 ) )
% 2.52/1.50 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 2.52/1.50 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 2.52/1.50 = ( coinductive_lhd @ A @ Llist4 ) )
% 2.52/1.50 & ( ( R2 @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
% 2.52/1.50 | ( ( coinductive_ltl @ A @ Llist3 )
% 2.52/1.50 = ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
% 2.52/1.50 => ( Llist = Llist2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_69_lhd__ldropWhile__in__lset,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.50 ( ? [X5: A] :
% 2.52/1.50 ( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 & ~ ( P2 @ X5 ) )
% 2.52/1.50 => ( member @ A @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_70_lhd__ldropWhile,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.50 ( ? [X5: A] :
% 2.52/1.50 ( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 & ~ ( P2 @ X5 ) )
% 2.52/1.50 => ~ ( P2 @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_71_lzip_Octr_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ B @ Ys )
% 2.52/1.50 => ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 2.52/1.50 = ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ B @ Ys ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ B @ Ys ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_72_unfold__llist__id,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
% 2.52/1.50 = Xs ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_73_ldistinct__lhdD,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coindu351974385stinct @ A @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_74_ldistinct__coinduct,axiom,
% 2.52/1.50 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( X6 @ Xs )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( ( X6 @ Xs2 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.50 => ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 2.52/1.50 & ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.52/1.50 | ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
% 2.52/1.50 => ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_75_llist_Ocase__eq__if,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType] :
% 2.52/1.50 ( ( coindu1381640503_llist @ B @ A )
% 2.52/1.50 = ( ^ [F12: B,F22: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F12 @ ( F22 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_76_lhd__LCons__ltl,axiom,
% 2.52/1.50 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Llist )
% 2.52/1.50 => ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
% 2.52/1.50 = Llist ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_77_ord_Olsorted__lhdD,axiom,
% 2.52/1.50 ! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lsorted @ A @ Less_eq @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) )
% 2.52/1.50 => ( Less_eq @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_78_ord_Olsorted__coinduct_H,axiom,
% 2.52/1.50 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Less_eq: A > A > $o] :
% 2.52/1.50 ( ( X6 @ Xs )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( ( X6 @ Xs2 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.52/1.50 => ( ( Less_eq @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 2.52/1.50 & ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.52/1.50 | ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) ) )
% 2.52/1.50 => ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_79_llist_Oinject,axiom,
% 2.52/1.50 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 2.52/1.50 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 2.52/1.50 = ( ( X21 = Y21 )
% 2.52/1.50 & ( X22 = Y22 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_80_ldropWhile__LCons,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( P2 @ X3 )
% 2.52/1.50 => ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( coindu218763757pWhile @ A @ P2 @ Xs ) ) )
% 2.52/1.50 & ( ~ ( P2 @ X3 )
% 2.52/1.50 => ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_81_unfold__llist__eq__LCons,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 2.52/1.50 = ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( ~ ( IS_LNIL @ B2 )
% 2.52/1.50 & ( X3
% 2.52/1.50 = ( LHD @ B2 ) )
% 2.52/1.50 & ( Xs
% 2.52/1.50 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_82_unfold__llist_Odisc__iff_I1_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
% 2.52/1.50 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
% 2.52/1.50 = ( P @ A2 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_83_unfold__llist_Odisc__iff_I2_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
% 2.52/1.50 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) )
% 2.52/1.50 = ( ~ ( P @ A2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_84_ldistinct__LCons,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 & ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_85_lzip__simps_I3_J,axiom,
% 2.52/1.50 ! [C: $tType,B: $tType,X3: C,Xs: coinductive_llist @ C,Y3: B,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( coinductive_lzip @ C @ B @ ( coinductive_LCons @ C @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
% 2.52/1.50 = ( coinductive_LCons @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Y3 ) @ ( coinductive_lzip @ C @ B @ Xs @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_86_ldistinct__lzipI2,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ B] :
% 2.52/1.50 ( ( coindu351974385stinct @ A @ Ys )
% 2.52/1.50 => ( coindu351974385stinct @ ( product_prod @ B @ A ) @ ( coinductive_lzip @ B @ A @ Xs @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_87_ldistinct__lzipI1,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( coindu351974385stinct @ A @ Xs )
% 2.52/1.50 => ( coindu351974385stinct @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_88_llist_Osimps_I5_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,F1: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
% 2.52/1.50 ( ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.52/1.50 = ( F23 @ X21 @ X22 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_89_unfold__llist_Octr_I2_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.52/1.50 ( ~ ( P @ A2 )
% 2.52/1.50 => ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
% 2.52/1.50 = ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_90_ldistinct_OLCons,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( ( coindu351974385stinct @ A @ Xs )
% 2.52/1.50 => ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_91_ord_OLCons__LCons,axiom,
% 2.52/1.50 ! [A: $tType,Less_eq: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( Less_eq @ X3 @ Y3 )
% 2.52/1.50 => ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs ) )
% 2.52/1.50 => ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_92_ord_Olsorted__LCons__LCons,axiom,
% 2.52/1.50 ! [A: $tType,Less_eq: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
% 2.52/1.50 = ( ( Less_eq @ X3 @ Y3 )
% 2.52/1.50 & ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_93_lzip__eq__LCons__conv,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Z: product_prod @ A @ B,Zs: coinductive_llist @ ( product_prod @ A @ B )] :
% 2.52/1.50 ( ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 2.52/1.50 = ( coinductive_LCons @ ( product_prod @ A @ B ) @ Z @ Zs ) )
% 2.52/1.50 = ( ? [X4: A,Xs3: coinductive_llist @ A,Y4: B,Ys2: coinductive_llist @ B] :
% 2.52/1.50 ( ( Xs
% 2.52/1.50 = ( coinductive_LCons @ A @ X4 @ Xs3 ) )
% 2.52/1.50 & ( Ys
% 2.52/1.50 = ( coinductive_LCons @ B @ Y4 @ Ys2 ) )
% 2.52/1.50 & ( Z
% 2.52/1.50 = ( product_Pair @ A @ B @ X4 @ Y4 ) )
% 2.52/1.50 & ( Zs
% 2.52/1.50 = ( coinductive_lzip @ A @ B @ Xs3 @ Ys2 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_94_ord_Olsorted__LCons_H,axiom,
% 2.52/1.50 ! [A: $tType,Less_eq: A > A > $o,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ( Less_eq @ X3 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.50 & ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_95_not__lnull__conv,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 2.52/1.50 = ( ? [X4: A,Xs3: coinductive_llist @ A] :
% 2.52/1.50 ( Xs
% 2.52/1.50 = ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_96_llist_OdiscI_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 2.52/1.50 ( ( Llist
% 2.52/1.50 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.52/1.50 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_97_llist_Odisc_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.52/1.50 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_98_lset__intros_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,X7: A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_99_lset__intros_I1_J,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] : ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_100_llist_Oset__intros_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,A22: coinductive_llist @ A,A1: A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ A22 ) )
% 2.52/1.50 => ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_101_llist_Oset__intros_I1_J,axiom,
% 2.52/1.50 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_102_lset__cases,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( ! [Xs4: coinductive_llist @ A] :
% 2.52/1.50 ( Xs
% 2.52/1.50 != ( coinductive_LCons @ A @ X3 @ Xs4 ) )
% 2.52/1.50 => ~ ! [X8: A,Xs4: coinductive_llist @ A] :
% 2.52/1.50 ( ( Xs
% 2.52/1.50 = ( coinductive_LCons @ A @ X8 @ Xs4 ) )
% 2.52/1.50 => ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_103_lset__induct,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A] : ( P2 @ ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 2.52/1.50 => ( ! [X8: A,Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
% 2.52/1.50 => ( ( X3 != X8 )
% 2.52/1.50 => ( ( P2 @ Xs2 )
% 2.52/1.50 => ( P2 @ ( coinductive_LCons @ A @ X8 @ Xs2 ) ) ) ) )
% 2.52/1.50 => ( P2 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_104_lset__induct_H,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A] : ( P2 @ ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 2.52/1.50 => ( ! [X8: A,Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
% 2.52/1.50 => ( ( P2 @ Xs2 )
% 2.52/1.50 => ( P2 @ ( coinductive_LCons @ A @ X8 @ Xs2 ) ) ) )
% 2.52/1.50 => ( P2 @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_105_llist_Oset__cases,axiom,
% 2.52/1.50 ! [A: $tType,E3: A,A2: coinductive_llist @ A] :
% 2.52/1.50 ( ( member @ A @ E3 @ ( coinductive_lset @ A @ A2 ) )
% 2.52/1.50 => ( ! [Z2: coinductive_llist @ A] :
% 2.52/1.50 ( A2
% 2.52/1.50 != ( coinductive_LCons @ A @ E3 @ Z2 ) )
% 2.52/1.50 => ~ ! [Z1: A,Z2: coinductive_llist @ A] :
% 2.52/1.50 ( ( A2
% 2.52/1.50 = ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 2.52/1.50 => ~ ( member @ A @ E3 @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_106_llist_Oset__induct,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,A2: coinductive_llist @ A,P2: A > ( coinductive_llist @ A ) > $o] :
% 2.52/1.50 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) )
% 2.52/1.50 => ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P2 @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 2.52/1.50 => ( ! [Z1: A,Z2: coinductive_llist @ A,Xa: A] :
% 2.52/1.50 ( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z2 ) )
% 2.52/1.50 => ( ( P2 @ Xa @ Z2 )
% 2.52/1.50 => ( P2 @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
% 2.52/1.50 => ( P2 @ X3 @ A2 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_107_ltl__simps_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.52/1.50 = X22 ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_108_lhd__LCons,axiom,
% 2.52/1.50 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.52/1.50 = X21 ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_109_ldistinct__ltlI,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coindu351974385stinct @ A @ Xs )
% 2.52/1.50 => ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_110_ord_Olsorted__ltlI,axiom,
% 2.52/1.50 ! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lsorted @ A @ Less_eq @ Xs )
% 2.52/1.50 => ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_111_unfold__llist_Odisc_I1_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.52/1.50 ( ( P @ A2 )
% 2.52/1.50 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_112_unfold__llist_Odisc_I2_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.52/1.50 ( ~ ( P @ A2 )
% 2.52/1.50 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_113_unfold__llist_Osimps_I4_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.52/1.50 ( ~ ( P @ A2 )
% 2.52/1.50 => ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
% 2.52/1.50 = ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_114_unfold__llist_Osimps_I3_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.52/1.50 ( ~ ( P @ A2 )
% 2.52/1.50 => ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
% 2.52/1.50 = ( G21 @ A2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_115_lmember__code_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,X3: A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lmember @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.50 = ( ( X3 = Y3 )
% 2.52/1.50 | ( coinductive_lmember @ A @ X3 @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_116_lzip_Ocode,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType] :
% 2.52/1.50 ( ( coinductive_lzip @ A @ B )
% 2.52/1.50 = ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ B] :
% 2.52/1.50 ( if @ ( coinductive_llist @ ( product_prod @ A @ B ) )
% 2.52/1.50 @ ( ( coinductive_lnull @ A @ Xs5 )
% 2.52/1.50 | ( coinductive_lnull @ B @ Ys3 ) )
% 2.52/1.50 @ ( coinductive_LNil @ ( product_prod @ A @ B ) )
% 2.52/1.50 @ ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs5 ) @ ( coinductive_lhd @ B @ Ys3 ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs5 ) @ ( coinductive_ltl @ B @ Ys3 ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_117_llist_Osplit__sel,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P2: B > $o,F1: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 2.52/1.50 ( ( P2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ Llist ) )
% 2.52/1.50 = ( ( ( Llist
% 2.52/1.50 = ( coinductive_LNil @ A ) )
% 2.52/1.50 => ( P2 @ F1 ) )
% 2.52/1.50 & ( ( Llist
% 2.52/1.50 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
% 2.52/1.50 => ( P2 @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_118_llist_Osplit__sel__asm,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,P2: B > $o,F1: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 2.52/1.50 ( ( P2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ Llist ) )
% 2.52/1.50 = ( ~ ( ( ( Llist
% 2.52/1.50 = ( coinductive_LNil @ A ) )
% 2.52/1.50 & ~ ( P2 @ F1 ) )
% 2.52/1.50 | ( ( Llist
% 2.52/1.50 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
% 2.52/1.50 & ~ ( P2 @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_119_ltakeWhile_Octr_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.50 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.50 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 2.52/1.50 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_120_llexord__coinduct,axiom,
% 2.52/1.50 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R: A > A > $o] :
% 2.52/1.50 ( ( X6 @ Xs @ Ys )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
% 2.52/1.50 ( ( X6 @ Xs2 @ Ys4 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.50 => ( ~ ( coinductive_lnull @ A @ Ys4 )
% 2.52/1.50 & ( ~ ( coinductive_lnull @ A @ Ys4 )
% 2.52/1.50 => ( ( R @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ Ys4 ) )
% 2.52/1.50 | ( ( ( coinductive_lhd @ A @ Xs2 )
% 2.52/1.50 = ( coinductive_lhd @ A @ Ys4 ) )
% 2.52/1.50 & ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) )
% 2.52/1.50 | ( coinductive_llexord @ A @ R @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) ) ) ) ) ) ) ) )
% 2.52/1.50 => ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_121_ldistinct_Ocases,axiom,
% 2.52/1.50 ! [A: $tType,A2: coinductive_llist @ A] :
% 2.52/1.50 ( ( coindu351974385stinct @ A @ A2 )
% 2.52/1.50 => ( ( A2
% 2.52/1.50 != ( coinductive_LNil @ A ) )
% 2.52/1.50 => ~ ! [X: A,Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( ( A2
% 2.52/1.50 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.52/1.50 => ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 2.52/1.50 => ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_122_llexord__refl,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ Xs ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_123_ltakeWhile__LNil,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o] :
% 2.52/1.50 ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LNil @ A ) )
% 2.52/1.50 = ( coinductive_LNil @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_124_llexord__LCons__LCons,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.50 = ( ( ( X3 = Y3 )
% 2.52/1.50 & ( coinductive_llexord @ A @ R @ Xs @ Ys ) )
% 2.52/1.50 | ( R @ X3 @ Y3 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_125_ldistinct__LNil__code,axiom,
% 2.52/1.50 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_126_llexord__code_I1_J,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_127_llexord__LNil__right,axiom,
% 2.52/1.50 ! [A: $tType,Ys: coinductive_llist @ A,R: A > A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lnull @ A @ Ys )
% 2.52/1.50 => ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 2.52/1.50 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_128_ldropWhile__LNil,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o] :
% 2.52/1.50 ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LNil @ A ) )
% 2.52/1.50 = ( coinductive_LNil @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_129_ltakeWhile__LCons,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( ( P2 @ X3 )
% 2.52/1.50 => ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( coinductive_LCons @ A @ X3 @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) )
% 2.52/1.50 & ( ~ ( P2 @ X3 )
% 2.52/1.50 => ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.50 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_130_ltakeWhile_Odisc__iff_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) )
% 2.52/1.50 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 & ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_131_ltakeWhile_Odisc__iff_I1_J,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.50 = ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_132_lnull__ltakeWhile,axiom,
% 2.52/1.50 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.50 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.50 => ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_133_lzip__simps_I2_J,axiom,
% 2.52/1.50 ! [D: $tType,C: $tType,Xs: coinductive_llist @ C] :
% 2.52/1.50 ( ( coinductive_lzip @ C @ D @ Xs @ ( coinductive_LNil @ D ) )
% 2.52/1.50 = ( coinductive_LNil @ ( product_prod @ C @ D ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_134_lzip__simps_I1_J,axiom,
% 2.52/1.50 ! [B: $tType,A: $tType,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( coinductive_lzip @ A @ B @ ( coinductive_LNil @ A ) @ Ys )
% 2.52/1.50 = ( coinductive_LNil @ ( product_prod @ A @ B ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_135_llexord__code_I2_J,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.50 ~ ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_136_llexord_Ocases,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_llexord @ A @ R @ A1 @ A22 )
% 2.52/1.50 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A,X: A] :
% 2.52/1.50 ( ( A1
% 2.52/1.50 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.52/1.50 => ( ( A22
% 2.52/1.50 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 2.52/1.50 => ~ ( coinductive_llexord @ A @ R @ Xs2 @ Ys4 ) ) )
% 2.52/1.50 => ( ! [X: A] :
% 2.52/1.50 ( ? [Xs2: coinductive_llist @ A] :
% 2.52/1.50 ( A1
% 2.52/1.50 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.52/1.50 => ! [Y: A] :
% 2.52/1.50 ( ? [Ys4: coinductive_llist @ A] :
% 2.52/1.50 ( A22
% 2.52/1.50 = ( coinductive_LCons @ A @ Y @ Ys4 ) )
% 2.52/1.50 => ~ ( R @ X @ Y ) ) )
% 2.52/1.50 => ~ ( ( A1
% 2.52/1.50 = ( coinductive_LNil @ A ) )
% 2.52/1.50 => ! [Ys4: coinductive_llist @ A] : ( A22 != Ys4 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_137_llexord_Osimps,axiom,
% 2.52/1.50 ! [A: $tType] :
% 2.52/1.50 ( ( coinductive_llexord @ A )
% 2.52/1.50 = ( ^ [R3: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 2.52/1.50 ( ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X4: A] :
% 2.52/1.50 ( ( A12
% 2.52/1.50 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 2.52/1.50 & ( A23
% 2.52/1.50 = ( coinductive_LCons @ A @ X4 @ Ys3 ) )
% 2.52/1.50 & ( coinductive_llexord @ A @ R3 @ Xs5 @ Ys3 ) )
% 2.52/1.50 | ? [X4: A,Y4: A,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 2.52/1.50 ( ( A12
% 2.52/1.50 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 2.52/1.50 & ( A23
% 2.52/1.50 = ( coinductive_LCons @ A @ Y4 @ Ys3 ) )
% 2.52/1.50 & ( R3 @ X4 @ Y4 ) )
% 2.52/1.50 | ? [Ys3: coinductive_llist @ A] :
% 2.52/1.50 ( ( A12
% 2.52/1.50 = ( coinductive_LNil @ A ) )
% 2.52/1.50 & ( A23 = Ys3 ) ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_138_llexord_Ocoinduct,axiom,
% 2.52/1.50 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa2: coinductive_llist @ A,R: A > A > $o] :
% 2.52/1.50 ( ( X6 @ X3 @ Xa2 )
% 2.52/1.50 => ( ! [X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 2.52/1.50 ( ( X6 @ X @ Xa )
% 2.52/1.50 => ( ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
% 2.52/1.50 ( ( X
% 2.52/1.50 = ( coinductive_LCons @ A @ Xb @ Xs6 ) )
% 2.52/1.50 & ( Xa
% 2.52/1.50 = ( coinductive_LCons @ A @ Xb @ Ys5 ) )
% 2.52/1.50 & ( ( X6 @ Xs6 @ Ys5 )
% 2.52/1.50 | ( coinductive_llexord @ A @ R @ Xs6 @ Ys5 ) ) )
% 2.52/1.50 | ? [Xb: A,Y5: A,Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 2.52/1.50 ( ( X
% 2.52/1.50 = ( coinductive_LCons @ A @ Xb @ Xs6 ) )
% 2.52/1.50 & ( Xa
% 2.52/1.50 = ( coinductive_LCons @ A @ Y5 @ Ys5 ) )
% 2.52/1.50 & ( R @ Xb @ Y5 ) )
% 2.52/1.50 | ? [Ys5: coinductive_llist @ A] :
% 2.52/1.50 ( ( X
% 2.52/1.50 = ( coinductive_LNil @ A ) )
% 2.52/1.50 & ( Xa = Ys5 ) ) ) )
% 2.52/1.50 => ( coinductive_llexord @ A @ R @ X3 @ Xa2 ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_139_llexord__LNil,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_140_llexord__antisym,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 2.52/1.50 => ( ( coinductive_llexord @ A @ R @ Ys @ Xs )
% 2.52/1.50 => ( ! [A4: A,B4: A] :
% 2.52/1.50 ( ( R @ A4 @ B4 )
% 2.52/1.50 => ~ ( R @ B4 @ A4 ) )
% 2.52/1.50 => ( Xs = Ys ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_141_llexord__linear,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.50 ( ! [X: A,Y: A] :
% 2.52/1.50 ( ( R @ X @ Y )
% 2.52/1.50 | ( X = Y )
% 2.52/1.50 | ( R @ Y @ X ) )
% 2.52/1.50 => ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 2.52/1.50 | ( coinductive_llexord @ A @ R @ Ys @ Xs ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_142_llexord__trans,axiom,
% 2.52/1.50 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.52/1.50 ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 2.52/1.50 => ( ( coinductive_llexord @ A @ R @ Ys @ Zs )
% 2.52/1.50 => ( ! [A4: A,B4: A,C2: A] :
% 2.52/1.50 ( ( R @ A4 @ B4 )
% 2.52/1.50 => ( ( R @ B4 @ C2 )
% 2.52/1.50 => ( R @ A4 @ C2 ) ) )
% 2.52/1.50 => ( coinductive_llexord @ A @ R @ Xs @ Zs ) ) ) ) ).
% 2.52/1.50
% 2.52/1.50 thf(fact_143_lzip__eq__LNil__conv,axiom,
% 2.52/1.50 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.50 ( ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 2.52/1.50 = ( coinductive_LNil @ ( product_prod @ A @ B ) ) )
% 2.52/1.50 = ( ( Xs
% 2.52/1.50 = ( coinductive_LNil @ A ) )
% 2.52/1.50 | ( Ys
% 2.52/1.50 = ( coinductive_LNil @ B ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_144_ltakeWhile__eq__LNil__iff,axiom,
% 2.52/1.51 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( ( Xs
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 => ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_145_ltakeWhile_Octr_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.51 ( ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
% 2.52/1.51 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 2.52/1.51 = ( coinductive_LNil @ A ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_146_ltl__ltakeWhile,axiom,
% 2.52/1.51 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.51 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.51 = ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) )
% 2.52/1.51 & ( ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.51 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.51 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_147_lset__ltakeWhileD,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) )
% 2.52/1.51 => ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.51 & ( P2 @ X3 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_148_ltakeWhile__cong,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P2: A > $o,Q: A > $o] :
% 2.52/1.51 ( ( Xs = Ys )
% 2.52/1.51 => ( ! [X: A] :
% 2.52/1.51 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 2.52/1.51 => ( ( P2 @ X )
% 2.52/1.51 = ( Q @ X ) ) )
% 2.52/1.51 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 2.52/1.51 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_149_ltakeWhile__all,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.51 ( ! [X: A] :
% 2.52/1.51 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.51 => ( P2 @ X ) )
% 2.52/1.51 => ( ( coindu501562517eWhile @ A @ P2 @ Xs )
% 2.52/1.51 = Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_150_neq__LNil__conv,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( Xs
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( ? [X4: A,Xs3: coinductive_llist @ A] :
% 2.52/1.51 ( Xs
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_151_llist_Oexhaust,axiom,
% 2.52/1.51 ! [A: $tType,Y3: coinductive_llist @ A] :
% 2.52/1.51 ( ( Y3
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 2.52/1.51 ( Y3
% 2.52/1.51 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_152_llist_Odistinct_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_LNil @ A )
% 2.52/1.51 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_153_llist_Odisc_I1_J,axiom,
% 2.52/1.51 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_154_llist_OdiscI_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.52/1.51 ( ( Llist
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 => ( coinductive_lnull @ A @ Llist ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_155_llist_Ocollapse_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lnull @ A @ Llist )
% 2.52/1.51 => ( Llist
% 2.52/1.51 = ( coinductive_LNil @ A ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_156_lnull__def,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_lnull @ A )
% 2.52/1.51 = ( ^ [Llist5: coinductive_llist @ A] :
% 2.52/1.51 ( Llist5
% 2.52/1.51 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_157_ltl__simps_I1_J,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_158_llexord__LCons__less,axiom,
% 2.52/1.51 ! [A: $tType,R: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( R @ X3 @ Y3 )
% 2.52/1.51 => ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_159_llexord__LCons__eq,axiom,
% 2.52/1.51 ! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
% 2.52/1.51 ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
% 2.52/1.51 => ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_160_llexord__LCons__left,axiom,
% 2.52/1.51 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
% 2.52/1.51 = ( ? [Y4: A,Ys2: coinductive_llist @ A] :
% 2.52/1.51 ( ( Ys
% 2.52/1.51 = ( coinductive_LCons @ A @ Y4 @ Ys2 ) )
% 2.52/1.51 & ( ( ( X3 = Y4 )
% 2.52/1.51 & ( coinductive_llexord @ A @ R @ Xs @ Ys2 ) )
% 2.52/1.51 | ( R @ X3 @ Y4 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_161_llexord__code_I3_J,axiom,
% 2.52/1.51 ! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.51 = ( ( R @ X3 @ Y3 )
% 2.52/1.51 | ( ( X3 = Y3 )
% 2.52/1.51 & ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_162_lnull__llexord,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_163_ord_OLNil,axiom,
% 2.52/1.51 ! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_164_ord_Olsorted__code_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_165_llist_Osimps_I4_J,axiom,
% 2.52/1.51 ! [A: $tType,B: $tType,F1: B,F23: A > ( coinductive_llist @ A ) > B] :
% 2.52/1.51 ( ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ ( coinductive_LNil @ A ) )
% 2.52/1.51 = F1 ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_166_ldistinct_OLNil,axiom,
% 2.52/1.51 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_167_unfold__llist_Octr_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.52/1.51 ( ( P @ A2 )
% 2.52/1.51 => ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
% 2.52/1.51 = ( coinductive_LNil @ B ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_168_gen__lset__code_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,A5: set @ A] :
% 2.52/1.51 ( ( coinductive_gen_lset @ A @ A5 @ ( coinductive_LNil @ A ) )
% 2.52/1.51 = A5 ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_169_lmember__code_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,X3: A] :
% 2.52/1.51 ~ ( coinductive_lmember @ A @ X3 @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_170_ltakeWhile_Ocode,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coindu501562517eWhile @ A )
% 2.52/1.51 = ( ^ [P3: A > $o,Xs5: coinductive_llist @ A] :
% 2.52/1.51 ( if @ ( coinductive_llist @ A )
% 2.52/1.51 @ ( ( coinductive_lnull @ A @ Xs5 )
% 2.52/1.51 | ~ ( P3 @ ( coinductive_lhd @ A @ Xs5 ) ) )
% 2.52/1.51 @ ( coinductive_LNil @ A )
% 2.52/1.51 @ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs5 ) @ ( coindu501562517eWhile @ A @ P3 @ ( coinductive_ltl @ A @ Xs5 ) ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_171_ltakeWhile_Odisc_I2_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.51 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.51 => ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_172_ltakeWhile_Odisc_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.51 ( ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 | ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
% 2.52/1.51 => ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_173_lhd__ltakeWhile,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.51 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.51 => ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.51 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_174_lset__LNil,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_lset @ A @ ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_175_ord_OSingleton,axiom,
% 2.52/1.51 ! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_176_ord_Olsorted_Ocoinduct,axiom,
% 2.52/1.51 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Less_eq: A > A > $o] :
% 2.52/1.51 ( ( X6 @ X3 )
% 2.52/1.51 => ( ! [X: coinductive_llist @ A] :
% 2.52/1.51 ( ( X6 @ X )
% 2.52/1.51 => ( ( X
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [Xa3: A] :
% 2.52/1.51 ( X
% 2.52/1.51 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
% 2.52/1.51 | ? [Xa3: A,Y5: A,Xs6: coinductive_llist @ A] :
% 2.52/1.51 ( ( X
% 2.52/1.51 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) ) )
% 2.52/1.51 & ( Less_eq @ Xa3 @ Y5 )
% 2.52/1.51 & ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) )
% 2.52/1.51 | ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) ) ) ) ) )
% 2.52/1.51 => ( coinductive_lsorted @ A @ Less_eq @ X3 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_177_ord_Olsorted_Osimps,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_lsorted @ A )
% 2.52/1.51 = ( ^ [Less_eq2: A > A > $o,A7: coinductive_llist @ A] :
% 2.52/1.51 ( ( A7
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [X4: A] :
% 2.52/1.51 ( A7
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ ( coinductive_LNil @ A ) ) )
% 2.52/1.51 | ? [X4: A,Y4: A,Xs5: coinductive_llist @ A] :
% 2.52/1.51 ( ( A7
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
% 2.52/1.51 & ( Less_eq2 @ X4 @ Y4 )
% 2.52/1.51 & ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_178_ord_Olsorted_Ocases,axiom,
% 2.52/1.51 ! [A: $tType,Less_eq: A > A > $o,A2: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lsorted @ A @ Less_eq @ A2 )
% 2.52/1.51 => ( ( A2
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 => ( ! [X: A] :
% 2.52/1.51 ( A2
% 2.52/1.51 != ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 2.52/1.51 => ~ ! [X: A,Y: A,Xs2: coinductive_llist @ A] :
% 2.52/1.51 ( ( A2
% 2.52/1.51 = ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs2 ) ) )
% 2.52/1.51 => ( ( Less_eq @ X @ Y )
% 2.52/1.51 => ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs2 ) ) ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_179_ord_Olsorted__code_I2_J,axiom,
% 2.52/1.51 ! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_180_lzip_Octr_I1_J,axiom,
% 2.52/1.51 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.51 ( ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 | ( coinductive_lnull @ B @ Ys ) )
% 2.52/1.51 => ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
% 2.52/1.51 = ( coinductive_LNil @ ( product_prod @ A @ B ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_181_ldropWhile__eq__LNil__iff,axiom,
% 2.52/1.51 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( ( coindu218763757pWhile @ A @ P2 @ Xs )
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( ! [X4: A] :
% 2.52/1.51 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.51 => ( P2 @ X4 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_182_unfold__llist_Ocode,axiom,
% 2.52/1.51 ! [B: $tType,A: $tType] :
% 2.52/1.51 ( ( coindu1441602521_llist @ A @ B )
% 2.52/1.51 = ( ^ [P4: A > $o,G212: A > B,G222: A > A,A7: A] : ( if @ ( coinductive_llist @ B ) @ ( P4 @ A7 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A7 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G212 @ G222 @ ( G222 @ A7 ) ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_183_ltl__unfold__llist,axiom,
% 2.52/1.51 ! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
% 2.52/1.51 ( ( ( IS_LNIL @ A2 )
% 2.52/1.51 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 2.52/1.51 = ( coinductive_LNil @ A ) ) )
% 2.52/1.51 & ( ~ ( IS_LNIL @ A2 )
% 2.52/1.51 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 2.52/1.51 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_184_ltakeWhile_Osimps_I4_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.52/1.51 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
% 2.52/1.51 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.51 = ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_185_eq__LConsD,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( Xs
% 2.52/1.51 = ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.51 => ( ( Xs
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 & ( ( coinductive_lhd @ A @ Xs )
% 2.52/1.51 = Y3 )
% 2.52/1.51 & ( ( coinductive_ltl @ A @ Xs )
% 2.52/1.51 = Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_186_llist_Oexhaust__sel,axiom,
% 2.52/1.51 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.52/1.51 ( ( Llist
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 => ( Llist
% 2.52/1.51 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_187_ldistinct_Ocoinduct,axiom,
% 2.52/1.51 ! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A] :
% 2.52/1.51 ( ( X6 @ X3 )
% 2.52/1.51 => ( ! [X: coinductive_llist @ A] :
% 2.52/1.51 ( ( X6 @ X )
% 2.52/1.51 => ( ( X
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [Xa3: A,Xs6: coinductive_llist @ A] :
% 2.52/1.51 ( ( X
% 2.52/1.51 = ( coinductive_LCons @ A @ Xa3 @ Xs6 ) )
% 2.52/1.51 & ~ ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Xs6 ) )
% 2.52/1.51 & ( ( X6 @ Xs6 )
% 2.52/1.51 | ( coindu351974385stinct @ A @ Xs6 ) ) ) ) )
% 2.52/1.51 => ( coindu351974385stinct @ A @ X3 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_188_ldistinct_Osimps,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coindu351974385stinct @ A )
% 2.52/1.51 = ( ^ [A7: coinductive_llist @ A] :
% 2.52/1.51 ( ( A7
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [X4: A,Xs5: coinductive_llist @ A] :
% 2.52/1.51 ( ( A7
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 2.52/1.51 & ~ ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs5 ) )
% 2.52/1.51 & ( coindu351974385stinct @ A @ Xs5 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_189_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.51 = ( ( Xs
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [Xs3: coinductive_llist @ A] :
% 2.52/1.51 ( ( Xs
% 2.52/1.51 = ( coinductive_LCons @ A @ Y3 @ Xs3 ) )
% 2.52/1.51 & ( coindu328551480prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_190_lstrict__prefix__code_I3_J,axiom,
% 2.52/1.51 ! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
% 2.52/1.51 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_191_lstrict__prefix__code_I2_J,axiom,
% 2.52/1.51 ! [B: $tType,Y3: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_192_llast__singleton,axiom,
% 2.52/1.51 ! [A: $tType,X3: A] :
% 2.52/1.51 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 2.52/1.51 = X3 ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_193_llast__LCons2,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Y3: A,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
% 2.52/1.51 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_194_lstrict__prefix__code_I4_J,axiom,
% 2.52/1.51 ! [B: $tType,X3: B,Xs: coinductive_llist @ B,Y3: B,Ys: coinductive_llist @ B] :
% 2.52/1.51 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
% 2.52/1.51 = ( ( X3 = Y3 )
% 2.52/1.51 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_195_lstrict__prefix__code_I1_J,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_196_llist__less__induct,axiom,
% 2.52/1.51 ! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ! [Xs2: coinductive_llist @ A] :
% 2.52/1.51 ( ! [Ys5: coinductive_llist @ A] :
% 2.52/1.51 ( ( coindu1478340336prefix @ A @ Ys5 @ Xs2 )
% 2.52/1.51 => ( P2 @ Ys5 ) )
% 2.52/1.51 => ( P2 @ Xs2 ) )
% 2.52/1.51 => ( P2 @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_197_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_198_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( Xs
% 2.52/1.51 = ( coinductive_LNil @ A ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_199_llast__LCons,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
% 2.52/1.51 ( ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.51 = X3 ) )
% 2.52/1.51 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.51 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_200_lprefix__coinduct,axiom,
% 2.52/1.51 ! [A: $tType,P2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( P2 @ Xs @ Ys )
% 2.52/1.51 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
% 2.52/1.51 ( ( P2 @ Xs2 @ Ys4 )
% 2.52/1.51 => ( ( ( coinductive_lnull @ A @ Ys4 )
% 2.52/1.51 => ( coinductive_lnull @ A @ Xs2 ) )
% 2.52/1.51 & ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.51 => ( ~ ( coinductive_lnull @ A @ Ys4 )
% 2.52/1.51 => ( ( ( coinductive_lhd @ A @ Xs2 )
% 2.52/1.51 = ( coinductive_lhd @ A @ Ys4 ) )
% 2.52/1.51 & ( ( P2 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) )
% 2.52/1.51 | ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) ) ) ) ) ) ) )
% 2.52/1.51 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_201_lprefix__expand,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ~ ( coinductive_lnull @ A @ Ys )
% 2.52/1.51 & ( ( coinductive_lhd @ A @ Xs )
% 2.52/1.51 = ( coinductive_lhd @ A @ Ys ) )
% 2.52/1.51 & ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) )
% 2.52/1.51 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_202_llist_Oleq__refl,axiom,
% 2.52/1.51 ! [A: $tType,X3: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X3 @ X3 ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_203_lprefix__refl,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_204_LCons__lprefix__LCons,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.51 = ( ( X3 = Y3 )
% 2.52/1.51 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_205_lprefix__code_I1_J,axiom,
% 2.52/1.51 ! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_206_lprefix__LNil,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.52/1.51 = ( coinductive_lnull @ A @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_207_Coinductive__List_Ofinite__lprefix__def,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coindu328551480prefix @ A )
% 2.52/1.51 = ( coinductive_lprefix @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_208_lstrict__prefix__def,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coindu1478340336prefix @ A )
% 2.52/1.51 = ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs5 @ Ys3 )
% 2.52/1.51 & ( Xs5 != Ys3 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_209_lprefix__LCons__conv,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 2.52/1.51 = ( ( Xs
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [Xs3: coinductive_llist @ A] :
% 2.52/1.51 ( ( Xs
% 2.52/1.51 = ( coinductive_LCons @ A @ Y3 @ Xs3 ) )
% 2.52/1.51 & ( coinductive_lprefix @ A @ Xs3 @ Ys ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_210_lprefix_Ocoinduct,axiom,
% 2.52/1.51 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
% 2.52/1.51 ( ( X6 @ X3 @ Xa2 )
% 2.52/1.51 => ( ! [X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 2.52/1.51 ( ( X6 @ X @ Xa )
% 2.52/1.51 => ( ? [Xs6: coinductive_llist @ A] :
% 2.52/1.51 ( ( X
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 & ( Xa = Xs6 ) )
% 2.52/1.51 | ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
% 2.52/1.51 ( ( X
% 2.52/1.51 = ( coinductive_LCons @ A @ Xb @ Xs6 ) )
% 2.52/1.51 & ( Xa
% 2.52/1.51 = ( coinductive_LCons @ A @ Xb @ Ys5 ) )
% 2.52/1.51 & ( ( X6 @ Xs6 @ Ys5 )
% 2.52/1.51 | ( coinductive_lprefix @ A @ Xs6 @ Ys5 ) ) ) ) )
% 2.52/1.51 => ( coinductive_lprefix @ A @ X3 @ Xa2 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_211_lprefix_Osimps,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_lprefix @ A )
% 2.52/1.51 = ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 2.52/1.51 ( ? [Xs5: coinductive_llist @ A] :
% 2.52/1.51 ( ( A12
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 & ( A23 = Xs5 ) )
% 2.52/1.51 | ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X4: A] :
% 2.52/1.51 ( ( A12
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 2.52/1.51 & ( A23
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ Ys3 ) )
% 2.52/1.51 & ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_212_lprefix_Ocases,axiom,
% 2.52/1.51 ! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ A1 @ A22 )
% 2.52/1.51 => ( ( ( A1
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 => ! [Xs2: coinductive_llist @ A] : ( A22 != Xs2 ) )
% 2.52/1.51 => ~ ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A,X: A] :
% 2.52/1.51 ( ( A1
% 2.52/1.51 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.52/1.51 => ( ( A22
% 2.52/1.51 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 2.52/1.51 => ~ ( coinductive_lprefix @ A @ Xs2 @ Ys4 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_213_lprefix__code_I2_J,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.51 ~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_214_lprefixI,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X6: set @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) )] :
% 2.52/1.51 ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs @ Ys ) @ X6 )
% 2.52/1.51 => ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
% 2.52/1.51 ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs2 @ Ys4 ) @ X6 )
% 2.52/1.51 => ( ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.51 | ? [X5: A,Xs7: coinductive_llist @ A,Ys6: coinductive_llist @ A] :
% 2.52/1.51 ( ( Xs2
% 2.52/1.51 = ( coinductive_LCons @ A @ X5 @ Xs7 ) )
% 2.52/1.51 & ( Ys4
% 2.52/1.51 = ( coinductive_LCons @ A @ X5 @ Ys6 ) )
% 2.52/1.51 & ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs7 @ Ys6 ) @ X6 )
% 2.52/1.51 | ( coinductive_lprefix @ A @ Xs7 @ Ys6 ) ) ) ) )
% 2.52/1.51 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_215_lprefix__lhdD,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( ( coinductive_lhd @ A @ Xs )
% 2.52/1.51 = ( coinductive_lhd @ A @ Ys ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_216_lprefix__down__linear,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_217_llist_Oleq__antisym,axiom,
% 2.52/1.51 ! [A: $tType,X3: coinductive_llist @ A,Y3: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ X3 @ Y3 )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Y3 @ X3 )
% 2.52/1.51 => ( X3 = Y3 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_218_lprefix__antisym,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 2.52/1.51 => ( Xs = Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_219_llist_Oleq__trans,axiom,
% 2.52/1.51 ! [A: $tType,X3: coinductive_llist @ A,Y3: coinductive_llist @ A,Z: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ X3 @ Y3 )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Y3 @ Z )
% 2.52/1.51 => ( coinductive_lprefix @ A @ X3 @ Z ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_220_lprefix__trans,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 2.52/1.51 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_221_ldistinct__lprefix,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coindu351974385stinct @ A @ Xs )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 2.52/1.51 => ( coindu351974385stinct @ A @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_222_ord_Olsorted__lprefixD,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less_eq: A > A > $o] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ( coinductive_lsorted @ A @ Less_eq @ Ys )
% 2.52/1.51 => ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_223_lprefix__ltlI,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_224_lprefix__not__lnullD,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_225_lprefix__lnullD,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ( coinductive_lnull @ A @ Ys )
% 2.52/1.51 => ( coinductive_lnull @ A @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_226_lprefix__lnull,axiom,
% 2.52/1.51 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lnull @ A @ Ys )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_227_lnull__lprefix,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_228_Le__LCons,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_229_LCons__lprefix__conv,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
% 2.52/1.51 = ( ? [Ys2: coinductive_llist @ A] :
% 2.52/1.51 ( ( Ys
% 2.52/1.51 = ( coinductive_LCons @ A @ X3 @ Ys2 ) )
% 2.52/1.51 & ( coinductive_lprefix @ A @ Xs @ Ys2 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_230_lprefix__ltakeWhile,axiom,
% 2.52/1.51 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) @ Xs ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_231_lprefix__imp__llexord,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R: A > A > $o] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_232_LNil__lprefix,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_233_llimit__induct,axiom,
% 2.52/1.51 ! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( P2 @ ( coinductive_LNil @ A ) )
% 2.52/1.51 => ( ! [X: A,Xs2: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.52/1.51 => ( ( P2 @ Xs2 )
% 2.52/1.51 => ( P2 @ ( coinductive_LCons @ A @ X @ Xs2 ) ) ) )
% 2.52/1.51 => ( ( ! [Ys5: coinductive_llist @ A] :
% 2.52/1.51 ( ( coindu1478340336prefix @ A @ Ys5 @ Xs )
% 2.52/1.51 => ( P2 @ Ys5 ) )
% 2.52/1.51 => ( P2 @ Xs ) )
% 2.52/1.51 => ( P2 @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_234_in__inv__imagep,axiom,
% 2.52/1.51 ! [B: $tType,A: $tType] :
% 2.52/1.51 ( ( inv_imagep @ A @ B )
% 2.52/1.51 = ( ^ [R3: A > A > $o,F4: B > A,X4: B,Y4: B] : ( R3 @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_235_lfinite__LCons,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
% 2.52/1.51 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_236_lfinite__code_I2_J,axiom,
% 2.52/1.51 ! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
% 2.52/1.51 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) )
% 2.52/1.51 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_237_lfinite__code_I1_J,axiom,
% 2.52/1.51 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_238_lfinite__ltl,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
% 2.52/1.51 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_239_lfinite__lzip,axiom,
% 2.52/1.51 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.52/1.51 ( ( coinductive_lfinite @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
% 2.52/1.51 = ( ( coinductive_lfinite @ A @ Xs )
% 2.52/1.51 | ( coinductive_lfinite @ B @ Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_240_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_lprefix @ A )
% 2.52/1.51 = ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 2.52/1.51 ( ( ( coinductive_lfinite @ A @ Xs5 )
% 2.52/1.51 => ( coindu328551480prefix @ A @ Xs5 @ Ys3 ) )
% 2.52/1.51 & ( ~ ( coinductive_lfinite @ A @ Xs5 )
% 2.52/1.51 => ( Xs5 = Ys3 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_241_not__lfinite__lprefix__conv__eq,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.52/1.51 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 = ( Xs = Ys ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_242_lprefix__lfiniteD,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( ( coinductive_lfinite @ A @ Ys )
% 2.52/1.51 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_243_lfinite__LNil,axiom,
% 2.52/1.51 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_244_lfinite__LConsI,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ Xs )
% 2.52/1.51 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_245_lnull__imp__lfinite,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lnull @ A @ Xs )
% 2.52/1.51 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_246_lfinite__ldropWhile,axiom,
% 2.52/1.51 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) )
% 2.52/1.51 = ( ? [X4: A] :
% 2.52/1.51 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.51 & ~ ( P2 @ X4 ) )
% 2.52/1.51 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_247_lfinite__ltakeWhile,axiom,
% 2.52/1.51 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
% 2.52/1.51 = ( ( coinductive_lfinite @ A @ Xs )
% 2.52/1.51 | ? [X4: A] :
% 2.52/1.51 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 2.52/1.51 & ~ ( P2 @ X4 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_248_lfinite__induct,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ Xs )
% 2.52/1.51 => ( ! [Xs2: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.51 => ( P2 @ Xs2 ) )
% 2.52/1.51 => ( ! [Xs2: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.52/1.51 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.52/1.51 => ( ( P2 @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.52/1.51 => ( P2 @ Xs2 ) ) ) )
% 2.52/1.51 => ( P2 @ Xs ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_249_lfinite_Ocases,axiom,
% 2.52/1.51 ! [A: $tType,A2: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ A2 )
% 2.52/1.51 => ( ( A2
% 2.52/1.51 != ( coinductive_LNil @ A ) )
% 2.52/1.51 => ~ ! [Xs2: coinductive_llist @ A] :
% 2.52/1.51 ( ? [X: A] :
% 2.52/1.51 ( A2
% 2.52/1.51 = ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.52/1.51 => ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_250_lfinite_Osimps,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_lfinite @ A )
% 2.52/1.51 = ( ^ [A7: coinductive_llist @ A] :
% 2.52/1.51 ( ( A7
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ? [Xs5: coinductive_llist @ A,X4: A] :
% 2.52/1.51 ( ( A7
% 2.52/1.51 = ( coinductive_LCons @ A @ X4 @ Xs5 ) )
% 2.52/1.51 & ( coinductive_lfinite @ A @ Xs5 ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_251_lfinite_Oinducts,axiom,
% 2.52/1.51 ! [A: $tType,X3: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ X3 )
% 2.52/1.51 => ( ( P2 @ ( coinductive_LNil @ A ) )
% 2.52/1.51 => ( ! [Xs2: coinductive_llist @ A,X: A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.52/1.51 => ( ( P2 @ Xs2 )
% 2.52/1.51 => ( P2 @ ( coinductive_LCons @ A @ X @ Xs2 ) ) ) )
% 2.52/1.51 => ( P2 @ X3 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_252_lstrict__prefix__lfinite1,axiom,
% 2.52/1.51 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.52/1.51 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 2.52/1.51 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(fact_253_llexord__conv,axiom,
% 2.52/1.51 ! [A: $tType] :
% 2.52/1.51 ( ( coinductive_llexord @ A )
% 2.52/1.51 = ( ^ [R3: A > A > $o,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 2.52/1.51 ( ( Xs5 = Ys3 )
% 2.52/1.51 | ? [Zs2: coinductive_llist @ A,Xs3: coinductive_llist @ A,Y4: A,Ys2: coinductive_llist @ A] :
% 2.52/1.51 ( ( coinductive_lfinite @ A @ Zs2 )
% 2.52/1.51 & ( Xs5
% 2.52/1.51 = ( coinductive_lappend @ A @ Zs2 @ Xs3 ) )
% 2.52/1.51 & ( Ys3
% 2.52/1.51 = ( coinductive_lappend @ A @ Zs2 @ ( coinductive_LCons @ A @ Y4 @ Ys2 ) ) )
% 2.52/1.51 & ( ( Xs3
% 2.52/1.51 = ( coinductive_LNil @ A ) )
% 2.52/1.51 | ( R3 @ ( coinductive_lhd @ A @ Xs3 ) @ Y4 ) ) ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(tcon_fun___Orderings_Obot,axiom,
% 2.52/1.51 ! [A8: $tType,A9: $tType] :
% 2.52/1.51 ( ( bot @ A9 @ ( type2 @ A9 ) )
% 2.52/1.51 => ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(tcon_Set_Oset___Orderings_Obot_1,axiom,
% 2.52/1.51 ! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(tcon_HOL_Obool___Orderings_Obot_2,axiom,
% 2.52/1.51 bot @ $o @ ( type2 @ $o ) ).
% 2.52/1.51
% 2.52/1.51 thf(help_If_3_1_T,axiom,
% 2.52/1.51 ! [P2: $o] :
% 2.52/1.51 ( ( P2 = $true )
% 2.52/1.51 | ( P2 = $false ) ) ).
% 2.52/1.51
% 2.52/1.51 thf(help_If_2_1_T,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Y3: A] :
% 2.52/1.51 ( ( if @ A @ $false @ X3 @ Y3 )
% 2.52/1.51 = Y3 ) ).
% 2.52/1.51
% 2.52/1.51 thf(help_If_1_1_T,axiom,
% 2.52/1.51 ! [A: $tType,X3: A,Y3: A] :
% 2.52/1.51 ( ( if @ A @ $true @ X3 @ Y3 )
% 2.52/1.51 = X3 ) ).
% 2.52/1.51
% 2.52/1.51 thf(conj_0,conjecture,
% 2.52/1.51 member @ node @ n @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 2.52/1.51 ------------------------
% 2.52/1.53 % (19977)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_19871 for (2999ds/2Mi)
% 2.52/1.53 % (19977)Instruction limit reached!
% 2.52/1.53 % (19977)------------------------------
% 2.52/1.53 % (19977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (19977)Termination reason: Unknown
% 2.52/1.53 % (19977)Termination phase: shuffling
% 2.52/1.53
% 2.52/1.53 % (19977)Memory used [KB]: 1407
% 2.52/1.53 % (19977)Time elapsed: 0.003 s
% 2.52/1.53 % (19977)Instructions burned: 5 (million)
% 2.52/1.53 % (19977)------------------------------
% 2.52/1.53 % (19977)------------------------------
% 2.52/1.53 % (19976)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/2Mi)
% 2.52/1.53 % (19974)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_19871 for (2999ds/4Mi)
% 2.52/1.53 % (19975)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_19871 for (2999ds/27Mi)
% 2.52/1.53 % (19973)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/183Mi)
% 2.52/1.53 % (19978)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_19871 for (2999ds/275Mi)
% 2.52/1.53 % (19980)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/18Mi)
% 2.52/1.53 % (19976)Instruction limit reached!
% 2.52/1.53 % (19976)------------------------------
% 2.52/1.53 % (19976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (19976)Termination reason: Unknown
% 2.52/1.53 % (19976)Termination phase: shuffling
% 2.52/1.53
% 2.52/1.53 % (19976)Memory used [KB]: 1407
% 2.52/1.53 % (19976)Time elapsed: 0.003 s
% 2.52/1.53 % (19976)Instructions burned: 3 (million)
% 2.52/1.53 % (19976)------------------------------
% 2.52/1.53 % (19976)------------------------------
% 2.52/1.53 % (19974)Instruction limit reached!
% 2.52/1.53 % (19974)------------------------------
% 2.52/1.53 % (19974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (19974)Termination reason: Unknown
% 2.52/1.53 % (19974)Termination phase: shuffling
% 2.52/1.53
% 2.52/1.53 % (19974)Memory used [KB]: 1407
% 2.52/1.53 % (19974)Time elapsed: 0.004 s
% 2.52/1.53 % (19974)Instructions burned: 5 (million)
% 2.52/1.53 % (19974)------------------------------
% 2.52/1.53 % (19974)------------------------------
% 2.52/1.54 % (19981)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/3Mi)
% 2.52/1.54 % (19981)Instruction limit reached!
% 2.52/1.54 % (19981)------------------------------
% 2.52/1.54 % (19981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (19981)Termination reason: Unknown
% 2.52/1.54 % (19981)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (19981)Memory used [KB]: 1407
% 2.52/1.54 % (19981)Time elapsed: 0.002 s
% 2.52/1.54 % (19981)Instructions burned: 3 (million)
% 2.52/1.54 % (19981)------------------------------
% 2.52/1.54 % (19981)------------------------------
% 2.52/1.54 % (19980)Instruction limit reached!
% 2.52/1.54 % (19980)------------------------------
% 2.52/1.54 % (19980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (19980)Termination reason: Unknown
% 2.52/1.54 % (19980)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (19980)Memory used [KB]: 1791
% 2.52/1.54 % (19980)Time elapsed: 0.011 s
% 2.52/1.54 % (19980)Instructions burned: 18 (million)
% 2.52/1.54 % (19980)------------------------------
% 2.52/1.54 % (19980)------------------------------
% 2.52/1.55 % (19975)Instruction limit reached!
% 2.52/1.55 % (19975)------------------------------
% 2.52/1.55 % (19975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (19975)Termination reason: Unknown
% 2.52/1.55 % (19975)Termination phase: Property scanning
% 2.52/1.55
% 2.52/1.55 % (19975)Memory used [KB]: 1918
% 2.52/1.55 % (19975)Time elapsed: 0.015 s
% 2.52/1.55 % (19975)Instructions burned: 27 (million)
% 2.52/1.55 % (19975)------------------------------
% 2.52/1.55 % (19975)------------------------------
% 2.52/1.55 % (19983)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_19871 for (2999ds/15Mi)
% 2.52/1.55 % (19982)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_19871 for (2999ds/37Mi)
% 2.52/1.55 % (19983)Instruction limit reached!
% 2.52/1.55 % (19983)------------------------------
% 2.52/1.55 % (19983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (19983)Termination reason: Unknown
% 2.52/1.55 % (19983)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (19983)Memory used [KB]: 1663
% 2.52/1.55 % (19983)Time elapsed: 0.006 s
% 2.52/1.55 % (19983)Instructions burned: 15 (million)
% 2.52/1.55 % (19983)------------------------------
% 2.52/1.55 % (19983)------------------------------
% 2.52/1.55 % (19984)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_19871 for (2999ds/3Mi)
% 2.52/1.56 % (19984)Instruction limit reached!
% 2.52/1.56 % (19984)------------------------------
% 2.52/1.56 % (19984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (19984)Termination reason: Unknown
% 2.52/1.56 % (19984)Termination phase: shuffling
% 2.52/1.56
% 2.52/1.56 % (19984)Memory used [KB]: 1407
% 2.52/1.56 % (19984)Time elapsed: 0.003 s
% 2.52/1.56 % (19984)Instructions burned: 3 (million)
% 2.52/1.56 % (19984)------------------------------
% 2.52/1.56 % (19984)------------------------------
% 2.52/1.56 % (19985)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_19871 for (2999ds/1041Mi)
% 2.52/1.56 % (19987)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/16Mi)
% 2.52/1.56 % (19986)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/7Mi)
% 2.52/1.57 % (19986)Instruction limit reached!
% 2.52/1.57 % (19986)------------------------------
% 2.52/1.57 % (19986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (19986)Termination reason: Unknown
% 2.52/1.57 % (19986)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (19986)Memory used [KB]: 1535
% 2.52/1.57 % (19986)Time elapsed: 0.005 s
% 2.52/1.57 % (19986)Instructions burned: 7 (million)
% 2.52/1.57 % (19986)------------------------------
% 2.52/1.57 % (19986)------------------------------
% 2.52/1.57 % (19982)Instruction limit reached!
% 2.52/1.57 % (19982)------------------------------
% 2.52/1.57 % (19982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (19982)Termination reason: Unknown
% 2.52/1.57 % (19982)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (19982)Memory used [KB]: 2174
% 2.52/1.57 % (19982)Time elapsed: 0.019 s
% 2.52/1.57 % (19982)Instructions burned: 38 (million)
% 2.52/1.57 % (19982)------------------------------
% 2.52/1.57 % (19982)------------------------------
% 2.52/1.57 % (19987)Instruction limit reached!
% 2.52/1.57 % (19987)------------------------------
% 2.52/1.57 % (19987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (19987)Termination reason: Unknown
% 2.52/1.57 % (19987)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (19987)Memory used [KB]: 1663
% 2.52/1.57 % (19987)Time elapsed: 0.008 s
% 2.52/1.57 % (19987)Instructions burned: 17 (million)
% 2.52/1.57 % (19987)------------------------------
% 2.52/1.57 % (19987)------------------------------
% 2.52/1.57 % (19978)Refutation not found, incomplete strategy
% 2.52/1.57 % (19978)------------------------------
% 2.52/1.57 % (19978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (19978)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.57
% 2.52/1.57
% 2.52/1.57 % (19978)Memory used [KB]: 6908
% 2.52/1.57 % (19978)Time elapsed: 0.038 s
% 2.52/1.57 % (19978)Instructions burned: 79 (million)
% 2.52/1.57 % (19978)------------------------------
% 2.52/1.57 % (19978)------------------------------
% 2.52/1.57 % (19988)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/3Mi)
% 2.52/1.57 % (19988)Instruction limit reached!
% 2.52/1.57 % (19988)------------------------------
% 2.52/1.57 % (19988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (19988)Termination reason: Unknown
% 2.52/1.57 % (19988)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (19988)Memory used [KB]: 1407
% 2.52/1.57 % (19988)Time elapsed: 0.003 s
% 2.52/1.57 % (19988)Instructions burned: 3 (million)
% 2.52/1.57 % (19988)------------------------------
% 2.52/1.57 % (19988)------------------------------
% 2.52/1.58 % (19990)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/7Mi)
% 2.52/1.58 % (19990)Instruction limit reached!
% 2.52/1.58 % (19990)------------------------------
% 2.52/1.58 % (19990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (19990)Termination reason: Unknown
% 2.52/1.58 % (19990)Termination phase: shuffling
% 2.52/1.58
% 2.52/1.58 % (19990)Memory used [KB]: 1535
% 2.52/1.58 % (19990)Time elapsed: 0.004 s
% 2.52/1.58 % (19990)Instructions burned: 9 (million)
% 2.52/1.58 % (19990)------------------------------
% 2.52/1.58 % (19990)------------------------------
% 2.52/1.58 % (19989)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/3Mi)
% 2.52/1.58 % (19989)Instruction limit reached!
% 2.52/1.58 % (19989)------------------------------
% 2.52/1.58 % (19989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (19989)Termination reason: Unknown
% 2.52/1.58 % (19989)Termination phase: shuffling
% 2.52/1.58
% 2.52/1.58 % (19989)Memory used [KB]: 1407
% 2.52/1.58 % (19989)Time elapsed: 0.003 s
% 2.52/1.58 % (19989)Instructions burned: 3 (million)
% 2.52/1.58 % (19989)------------------------------
% 2.52/1.58 % (19989)------------------------------
% 2.52/1.59 % (19991)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/3Mi)
% 2.52/1.59 % (19992)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/4Mi)
% 2.52/1.59 % (19991)Instruction limit reached!
% 2.52/1.59 % (19991)------------------------------
% 2.52/1.59 % (19991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (19991)Termination reason: Unknown
% 2.52/1.59 % (19991)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (19991)Memory used [KB]: 1407
% 2.52/1.59 % (19991)Time elapsed: 0.003 s
% 2.52/1.59 % (19991)Instructions burned: 4 (million)
% 2.52/1.59 % (19991)------------------------------
% 2.52/1.59 % (19991)------------------------------
% 2.52/1.59 % (19992)Instruction limit reached!
% 2.52/1.59 % (19992)------------------------------
% 2.52/1.59 % (19992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (19992)Termination reason: Unknown
% 2.52/1.59 % (19992)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (19992)Memory used [KB]: 1407
% 2.52/1.59 % (19992)Time elapsed: 0.004 s
% 2.52/1.59 % (19992)Instructions burned: 5 (million)
% 2.52/1.59 % (19992)------------------------------
% 2.52/1.59 % (19992)------------------------------
% 2.52/1.59 % (19993)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/18Mi)
% 2.52/1.59 % (19994)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/710Mi)
% 2.57/1.60 % (19993)Instruction limit reached!
% 2.57/1.60 % (19993)------------------------------
% 2.57/1.60 % (19993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.60 % (19993)Termination reason: Unknown
% 2.57/1.60 % (19993)Termination phase: shuffling
% 2.57/1.60
% 2.57/1.60 % (19993)Memory used [KB]: 1791
% 2.57/1.60 % (19993)Time elapsed: 0.011 s
% 2.57/1.60 % (19993)Instructions burned: 19 (million)
% 2.57/1.60 % (19993)------------------------------
% 2.57/1.60 % (19993)------------------------------
% 2.57/1.60 % (19995)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/6Mi)
% 2.57/1.60 % (19996)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/902Mi)
% 2.57/1.60 % (19997)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/21Mi)
% 2.57/1.61 % (19995)Instruction limit reached!
% 2.57/1.61 % (19995)------------------------------
% 2.57/1.61 % (19995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.61 % (19995)Termination reason: Unknown
% 2.57/1.61 % (19995)Termination phase: shuffling
% 2.57/1.61
% 2.57/1.61 % (19995)Memory used [KB]: 1535
% 2.57/1.61 % (19995)Time elapsed: 0.005 s
% 2.57/1.61 % (19995)Instructions burned: 7 (million)
% 2.57/1.61 % (19995)------------------------------
% 2.57/1.61 % (19995)------------------------------
% 2.57/1.61 % (19998)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_19871 for (2999ds/5Mi)
% 2.57/1.62 % (19997)Instruction limit reached!
% 2.57/1.62 % (19997)------------------------------
% 2.57/1.62 % (19997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.62 % (19997)Termination reason: Unknown
% 2.57/1.62 % (19997)Termination phase: shuffling
% 2.57/1.62
% 2.57/1.62 % (19997)Memory used [KB]: 1918
% 2.57/1.62 % (19997)Time elapsed: 0.012 s
% 2.57/1.62 % (19997)Instructions burned: 21 (million)
% 2.57/1.62 % (19997)------------------------------
% 2.57/1.62 % (19997)------------------------------
% 2.57/1.62 % (19973)Instruction limit reached!
% 2.57/1.62 % (19973)------------------------------
% 2.57/1.62 % (19973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.62 % (19973)Termination reason: Unknown
% 2.57/1.62 % (19973)Termination phase: Saturation
% 2.57/1.62
% 2.57/1.62 % (19973)Memory used [KB]: 7291
% 2.57/1.62 % (19973)Time elapsed: 0.086 s
% 2.57/1.62 % (19973)Instructions burned: 185 (million)
% 2.57/1.62 % (19973)------------------------------
% 2.57/1.62 % (19973)------------------------------
% 2.57/1.62 % (19998)Instruction limit reached!
% 2.57/1.62 % (19998)------------------------------
% 2.57/1.62 % (19998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.62 % (19998)Termination reason: Unknown
% 2.57/1.62 % (19998)Termination phase: shuffling
% 2.57/1.62
% 2.57/1.62 % (19998)Memory used [KB]: 1407
% 2.57/1.62 % (19998)Time elapsed: 0.004 s
% 2.57/1.62 % (19998)Instructions burned: 6 (million)
% 2.57/1.62 % (19998)------------------------------
% 2.57/1.62 % (19998)------------------------------
% 2.57/1.62 % (19994)First to succeed.
% 2.57/1.62 % (19994)Refutation found. Thanks to Tanya!
% 2.57/1.62 % SZS status Theorem for DTF2THF_19871
% 2.57/1.62 % SZS output start Proof for DTF2THF_19871
% 2.57/1.62 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.57/1.62 thf(type_def_7, type, coinductive_llist: ($tType) > $tType).
% 2.57/1.62 thf(type_def_8, type, set: ($tType) > $tType).
% 2.57/1.62 thf(type_def_9, type, product_prod: ($tType * $tType) > $tType).
% 2.57/1.62 thf(type_def_10, type, node: $tType).
% 2.57/1.62 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.57/1.62 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 2.57/1.62 thf(func_def_2, type, set: ($tType) > $tType).
% 2.57/1.62 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.57/1.62 thf(func_def_4, type, node: $tType).
% 2.57/1.62 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.57/1.62 thf(func_def_6, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.57/1.62 thf(func_def_7, type, coindu328551480prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_8, type, coinductive_gen_lset: !>[X0: $tType]:(set @ X0 > coinductive_llist @ X0 > set @ X0)).
% 2.57/1.62 thf(func_def_9, type, coinductive_lappend: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_10, type, coindu351974385stinct: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_11, type, coindu218763757pWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_12, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_13, type, coinductive_llast: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.57/1.62 thf(func_def_14, type, coinductive_llexord: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_15, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_16, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_17, type, coindu1381640503_llist: !>[X0: $tType, X1: $tType]:(X0 > (X1 > coinductive_llist @ X1 > X0) > coinductive_llist @ X1 > X0)).
% 2.57/1.62 thf(func_def_18, type, coinductive_lhd: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.57/1.62 thf(func_def_19, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_20, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 2.57/1.62 thf(func_def_21, type, coinductive_ltl: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_22, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_23, type, coinductive_lprefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_24, type, coindu1478340336prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_25, type, coindu501562517eWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_26, type, coinductive_lzip: !>[X0: $tType, X1: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X1 > coinductive_llist @ product_prod @ X0 @ X1)).
% 2.57/1.62 thf(func_def_27, type, coinductive_lsorted: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > $o)).
% 2.57/1.62 thf(func_def_28, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 2.57/1.62 thf(func_def_29, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.57/1.62 thf(func_def_30, type, koenig793108494nected: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 2.57/1.62 thf(func_def_31, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.57/1.62 thf(func_def_32, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.57/1.62 thf(func_def_33, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.57/1.62 thf(func_def_34, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.57/1.62 thf(func_def_35, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.57/1.62 thf(func_def_36, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 2.57/1.62 thf(func_def_37, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.57/1.62 thf(func_def_38, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.57/1.62 thf(func_def_39, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.57/1.62 thf(func_def_40, type, f: product_prod @ node @ set @ node > coinductive_llist @ node).
% 2.57/1.62 thf(func_def_41, type, graph: node > node > $o).
% 2.57/1.62 thf(func_def_42, type, n: node).
% 2.57/1.62 thf(func_def_43, type, ns: set @ node).
% 2.57/1.62 thf(func_def_61, type, sK0: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_62, type, sK1: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > X0)).
% 2.57/1.62 thf(func_def_63, type, sK2: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_64, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.57/1.62 thf(func_def_65, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.57/1.62 thf(func_def_66, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.57/1.62 thf(func_def_67, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.57/1.62 thf(func_def_68, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.57/1.62 thf(func_def_69, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.57/1.62 thf(func_def_70, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.57/1.62 thf(func_def_71, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.57/1.62 thf(func_def_72, type, sK11: !>[X0: $tType]:(set @ product_prod @ coinductive_llist @ X0 @ coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_73, type, sK12: !>[X0: $tType]:(set @ product_prod @ coinductive_llist @ X0 @ coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_74, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.57/1.62 thf(func_def_75, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.57/1.62 thf(func_def_76, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_77, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_78, type, sK17: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_79, type, sK18: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_80, type, sK19: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.57/1.62 thf(func_def_81, type, sK20: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.57/1.62 thf(func_def_82, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.57/1.62 thf(func_def_83, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.57/1.62 thf(func_def_84, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.57/1.62 thf(func_def_85, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.57/1.62 thf(func_def_86, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.57/1.62 thf(func_def_87, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.57/1.62 thf(func_def_88, type, sK27: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.57/1.62 thf(func_def_89, type, sK28: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.57/1.62 thf(func_def_90, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.57/1.62 thf(func_def_91, type, sK30: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_92, type, sK31: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_93, type, sK32: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_94, type, sK33: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_95, type, sK34: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_96, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.57/1.62 thf(func_def_97, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.57/1.62 thf(func_def_98, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.57/1.62 thf(func_def_99, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.57/1.62 thf(func_def_100, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_101, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_102, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.57/1.62 thf(func_def_103, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.57/1.62 thf(func_def_104, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.57/1.62 thf(func_def_105, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.57/1.62 thf(func_def_106, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.57/1.62 thf(func_def_107, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_108, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_109, type, sK48: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_110, type, sK49: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > X0)).
% 2.57/1.62 thf(func_def_111, type, sK50: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_112, type, sK51: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_113, type, sK52: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_114, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.57/1.62 thf(func_def_115, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.57/1.62 thf(func_def_116, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.57/1.62 thf(func_def_117, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.57/1.62 thf(func_def_118, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.57/1.62 thf(func_def_119, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.57/1.62 thf(func_def_120, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.57/1.62 thf(func_def_121, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.57/1.62 thf(func_def_122, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_123, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_124, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.57/1.62 thf(func_def_125, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.57/1.62 thf(func_def_126, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.57/1.62 thf(func_def_127, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.57/1.62 thf(func_def_128, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.57/1.62 thf(func_def_129, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.57/1.62 thf(func_def_130, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.57/1.62 thf(func_def_131, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.57/1.62 thf(func_def_132, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.57/1.62 thf(func_def_133, type, sK72: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_134, type, sK73: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > X0)).
% 2.57/1.62 thf(func_def_135, type, sK74: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_136, type, sK75: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(func_def_137, type, sK76: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > X0)).
% 2.57/1.62 thf(func_def_138, type, sK77: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.57/1.62 thf(f863,plain,(
% 2.57/1.62 $false),
% 2.57/1.62 inference(subsumption_resolution,[status(thm)],[f855,f862])).
% 2.57/1.62 thf(f862,plain,(
% 2.57/1.62 ((member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)))) = $true)),
% 2.57/1.62 inference(backward_demodulation,[status(thm)],[f854,f815])).
% 2.57/1.62 thf(f815,plain,(
% 2.57/1.62 ( ! [X0 : node,X1 : set @ node] : (((coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ X0 @ X1))) = X0)) )),
% 2.57/1.62 inference(cnf_transformation,[status(thm)],[f702])).
% 2.57/1.62 thf(f702,plain,(
% 2.57/1.62 ! [X0 : node,X1 : set @ node] : ((coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ X0 @ X1))) = X0)),
% 2.57/1.62 inference(rectify,[status(thm)],[f3])).
% 2.57/1.62 thf(f3,axiom,(
% 2.57/1.62 ! [X5 : node,X6 : set @ node] : ((coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ X5 @ X6))) = X5)),
% 2.57/1.62 file('/export/starexec/sandbox2/tmp/tmp.udRHUCu2u7/DTF2THF_19871.p',fact_2_f__simps_I2_J)).
% 2.57/1.62 thf(f854,plain,(
% 2.57/1.62 ((member @ node @ (coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node))) @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)))) = $true)),
% 2.57/1.62 inference(definition_unfolding,[status(thm)],[f810,f831,f831])).
% 2.57/1.62 thf(f831,plain,(
% 2.57/1.62 (ns = bot_bot @ set @ node)),
% 2.57/1.62 inference(cnf_transformation,[status(thm)],[f2])).
% 2.57/1.62 thf(f2,axiom,(
% 2.57/1.62 (ns = bot_bot @ set @ node)),
% 2.57/1.62 file('/export/starexec/sandbox2/tmp/tmp.udRHUCu2u7/DTF2THF_19871.p',fact_1_ns__def)).
% 2.57/1.62 thf(f810,plain,(
% 2.57/1.62 ((member @ node @ (coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))) @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns)))) = $true)),
% 2.57/1.62 inference(cnf_transformation,[status(thm)],[f543])).
% 2.57/1.62 thf(f543,plain,(
% 2.57/1.62 ((member @ node @ (coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))) @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns)))) = $true)),
% 2.57/1.62 inference(fool_elimination,[status(thm)],[f542])).
% 2.57/1.62 thf(f542,plain,(
% 2.57/1.62 (member @ node @ (coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))) @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))))),
% 2.57/1.62 inference(rectify,[status(thm)],[f1])).
% 2.57/1.62 thf(f1,axiom,(
% 2.57/1.62 (member @ node @ (coinductive_lhd @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))) @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))))),
% 2.57/1.62 file('/export/starexec/sandbox2/tmp/tmp.udRHUCu2u7/DTF2THF_19871.p',fact_0__092_060open_062lhd_A_If_A_In_M_Ans_J_J_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062)).
% 2.57/1.62 thf(f855,plain,(
% 2.57/1.62 ((member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)))) != $true)),
% 2.57/1.62 inference(definition_unfolding,[status(thm)],[f845,f831])).
% 2.57/1.62 thf(f845,plain,(
% 2.57/1.62 ((member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns)))) != $true)),
% 2.57/1.62 inference(cnf_transformation,[status(thm)],[f700])).
% 2.57/1.62 thf(f700,plain,(
% 2.57/1.62 ((member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns)))) != $true)),
% 2.57/1.62 inference(flattening,[status(thm)],[f441])).
% 2.57/1.62 thf(f441,plain,(
% 2.57/1.62 ~((member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns)))) = $true)),
% 2.57/1.62 inference(fool_elimination,[status(thm)],[f440])).
% 2.57/1.62 thf(f440,plain,(
% 2.57/1.62 ~(member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))))),
% 2.57/1.62 inference(rectify,[status(thm)],[f262])).
% 2.57/1.62 thf(f262,negated_conjecture,(
% 2.57/1.62 ~(member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))))),
% 2.57/1.62 inference(negated_conjecture,[status(cth)],[f261])).
% 2.57/1.62 thf(f261,conjecture,(
% 2.57/1.62 (member @ node @ n @ (coinductive_lset @ node @ (f @ (product_Pair @ node @ set @ node @ n @ ns))))),
% 2.57/1.62 file('/export/starexec/sandbox2/tmp/tmp.udRHUCu2u7/DTF2THF_19871.p',conj_0)).
% 2.57/1.62 % SZS output end Proof for DTF2THF_19871
% 2.57/1.62 % (19994)------------------------------
% 2.57/1.62 % (19994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.62 % (19994)Termination reason: Refutation
% 2.57/1.62
% 2.57/1.62 % (19994)Memory used [KB]: 7036
% 2.57/1.62 % (19994)Time elapsed: 0.029 s
% 2.57/1.62 % (19994)Instructions burned: 85 (million)
% 2.57/1.62 % (19994)------------------------------
% 2.57/1.62 % (19994)------------------------------
% 2.57/1.62 % (19972)Success in time 0.1 s
%------------------------------------------------------------------------------