%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP100^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n007.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 08:55:33 AM UTC 2026
% Result : Theorem 2.90s 1.62s
% Output : Refutation 2.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP100^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n007.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:25:53 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.46 ---- Original DTF file ---
% 0.26/0.46 thf(spec,logic,$$dhol).
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 % File : ITP100^2 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.46 % Domain : Interactive Theorem Proving
% 0.26/0.46 % Problem : Sledgehammer ListInf problem prob_52__5408414_1
% 0.26/0.46 % Version : Especial.
% 0.26/0.46 % English :
% 0.26/0.46
% 0.26/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.46 % Source : [Des21]
% 0.26/0.46 % Names : ListInf/prob_52__5408414_1 [Des21]
% 0.26/0.46
% 0.26/0.46 % Status : Theorem
% 0.26/0.46 % Rating : 0.00 v7.5.0
% 0.26/0.46 % Syntax : Number of formulae : 327 ( 132 unt; 54 typ; 0 def)
% 0.26/0.46 % Number of atoms : 768 ( 467 equ; 0 cnn)
% 0.26/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.26/0.46 % Number of connectives : 4717 ( 163 ~; 42 |; 91 &;4026 @)
% 0.26/0.46 % ( 0 <=>; 395 =>; 0 <=; 0 <~>)
% 0.26/0.46 % Maximal formula depth : 34 ( 8 avg)
% 0.26/0.46 % Number of types : 3 ( 2 usr)
% 0.26/0.46 % Number of type conns : 316 ( 316 >; 0 *; 0 +; 0 <<)
% 0.26/0.46 % Number of symbols : 55 ( 52 usr; 4 con; 0-6 aty)
% 0.26/0.46 % Number of variables : 1255 ( 27 ^;1100 !; 71 ?;1255 :)
% 0.26/0.46 % ( 57 !>; 0 ?*; 0 @-; 0 @+)
% 0.26/0.46 % SPC : TH1_THM_EQU_NAR
% 0.26/0.46
% 0.26/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:15:32.855
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 % Could-be-implicit typings (4)
% 0.26/0.46 thf(ty_t_List_Olist,type,
% 0.26/0.46 list: $tType > $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Set_Oset,type,
% 0.26/0.46 set: $tType > $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Nat_Onat,type,
% 0.26/0.46 nat: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_tf_a,type,
% 0.26/0.46 a: $tType ).
% 0.26/0.46
% 0.26/0.46 % Explicit typings (50)
% 0.26/0.46 thf(sy_cl_HOL_Otype,type,
% 0.26/0.46 type:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.26/0.46 ord:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.26/0.46 order:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.26/0.46 linorder:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.26/0.46 preorder:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
% 0.26/0.46 bNF_Greatest_Shift:
% 0.26/0.46 !>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
% 0.26/0.46 bNF_Greatest_Succ:
% 0.26/0.46 !>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
% 0.26/0.46 bNF_Greatest_shift:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_If,type,
% 0.26/0.46 if:
% 0.26/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List2_Olist__asc,type,
% 0.26/0.46 list_asc:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List2_Olist__ord,type,
% 0.26/0.46 list_ord:
% 0.26/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List2_Olist__strict__asc,type,
% 0.26/0.46 list_strict_asc:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List2_Omap2,type,
% 0.26/0.46 map2:
% 0.26/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
% 0.26/0.46 listIn521021761append:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oappend,type,
% 0.26/0.46 append:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oarg__min__list,type,
% 0.26/0.46 arg_min_list:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Obind,type,
% 0.26/0.46 bind:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Obutlast,type,
% 0.26/0.46 butlast:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oconcat,type,
% 0.26/0.46 concat:
% 0.26/0.46 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Odistinct__adj,type,
% 0.26/0.46 distinct_adj:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Ofoldr,type,
% 0.26/0.46 foldr:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oinsert,type,
% 0.26/0.46 insert:
% 0.26/0.46 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olast,type,
% 0.26/0.46 last:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olist_OCons,type,
% 0.26/0.46 cons:
% 0.26/0.46 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olist_ONil,type,
% 0.26/0.46 nil:
% 0.26/0.46 !>[A: $tType] : ( list @ A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olist_Ocase__list,type,
% 0.26/0.46 case_list:
% 0.26/0.46 !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olist_Ohd,type,
% 0.26/0.46 hd:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olist_Otl,type,
% 0.26/0.46 tl:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olist__ex1,type,
% 0.26/0.46 list_ex1:
% 0.26/0.46 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Olistrelp,type,
% 0.26/0.46 listrelp:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Omap__tailrec__rev,type,
% 0.26/0.46 map_tailrec_rev:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Omaps,type,
% 0.26/0.46 maps:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Omember,type,
% 0.26/0.46 member:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oord_Olexordp__eq,type,
% 0.26/0.46 lexordp_eq:
% 0.26/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oord__class_Olexordp,type,
% 0.26/0.46 ord_lexordp:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oord__class_Olexordp__eq,type,
% 0.26/0.46 ord_lexordp_eq:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oproduct__lists,type,
% 0.26/0.46 product_lists:
% 0.26/0.46 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oremdups__adj,type,
% 0.26/0.46 remdups_adj:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Orotate1,type,
% 0.26/0.46 rotate1:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oshuffles,type,
% 0.26/0.46 shuffles:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Osplice,type,
% 0.26/0.46 splice:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Osubseqs,type,
% 0.26/0.46 subseqs:
% 0.26/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Osuccessively,type,
% 0.26/0.46 successively:
% 0.26/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.26/0.46 size_size:
% 0.26/0.46 !>[A: $tType] : ( A > nat ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.26/0.46 ord_less:
% 0.26/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Set_OCollect,type,
% 0.26/0.46 collect:
% 0.26/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_member,type,
% 0.26/0.46 member2:
% 0.26/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_v_f,type,
% 0.26/0.46 f: nat > a ).
% 0.26/0.46
% 0.26/0.46 thf(sy_v_x,type,
% 0.26/0.46 x: a ).
% 0.26/0.46
% 0.26/0.46 thf(sy_v_xs,type,
% 0.26/0.46 xs: list @ a ).
% 0.26/0.46
% 0.26/0.46 % Relevant facts (255)
% 0.26/0.46 thf(fact_0_i__append__Nil,axiom,
% 0.26/0.46 ! [A: $tType,F: nat > A] :
% 0.26/0.46 ( ( listIn521021761append @ A @ ( nil @ A ) @ F )
% 0.26/0.46 = F ) ).
% 0.26/0.46
% 0.26/0.46 % i_append_Nil
% 0.26/0.46 thf(fact_1_list_Oinject,axiom,
% 0.26/0.46 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 0.26/0.46 ( ( ( cons @ A @ X21 @ X22 )
% 0.26/0.46 = ( cons @ A @ Y21 @ Y22 ) )
% 0.26/0.46 = ( ( X21 = Y21 )
% 0.26/0.46 & ( X22 = Y22 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.inject
% 0.26/0.46 thf(fact_2_list_Odistinct_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,X21: A,X22: list @ A] :
% 0.26/0.46 ( ( nil @ A )
% 0.26/0.46 != ( cons @ A @ X21 @ X22 ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.distinct(1)
% 0.26/0.46 thf(fact_3_list_OdiscI,axiom,
% 0.26/0.46 ! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
% 0.26/0.46 ( ( List
% 0.26/0.46 = ( cons @ A @ X21 @ X22 ) )
% 0.26/0.46 => ( List
% 0.26/0.46 != ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.discI
% 0.26/0.46 thf(fact_4_list_Oexhaust,axiom,
% 0.26/0.46 ! [A: $tType,Y: list @ A] :
% 0.26/0.46 ( ( Y
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ~ ! [X212: A,X222: list @ A] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( cons @ A @ X212 @ X222 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.exhaust
% 0.26/0.46 thf(fact_5_list_Oinducts,axiom,
% 0.26/0.46 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 0.26/0.46 ( ( P @ ( nil @ A ) )
% 0.26/0.46 => ( ! [X1: A,X2: list @ A] :
% 0.26/0.46 ( ( P @ X2 )
% 0.26/0.46 => ( P @ ( cons @ A @ X1 @ X2 ) ) )
% 0.26/0.46 => ( P @ List ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.inducts
% 0.26/0.46 thf(fact_6_neq__Nil__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 = ( ? [Y2: A,Ys: list @ A] :
% 0.26/0.46 ( Xs
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % neq_Nil_conv
% 0.26/0.46 thf(fact_7_list__induct2_H,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
% 0.26/0.46 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
% 0.26/0.46 => ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
% 0.26/0.46 ( ( P @ Xs2 @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
% 0.26/0.46 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_induct2'
% 0.26/0.46 thf(fact_8_splice_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.26/0.46 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ( P @ Ys3 @ Xs2 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 ) )
% 0.26/0.46 => ( P @ A0 @ A1 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % splice.induct
% 0.26/0.46 thf(fact_9_induct__list012,axiom,
% 0.26/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.26/0.46 ( ( P @ ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ( ! [X: A,Y3: A,Zs: list @ A] :
% 0.26/0.46 ( ( P @ Zs )
% 0.26/0.46 => ( ( P @ ( cons @ A @ Y3 @ Zs ) )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
% 0.26/0.46 => ( P @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % induct_list012
% 0.26/0.46 thf(fact_10_min__list_Ocases,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: list @ A] :
% 0.26/0.46 ( ! [X: A,Xs2: list @ A] :
% 0.26/0.46 ( X3
% 0.26/0.46 != ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ( X3
% 0.26/0.46 = ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % min_list.cases
% 0.26/0.46 thf(fact_11_min__list_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.26/0.46 ( ! [X: A,Xs2: list @ A] :
% 0.26/0.46 ( ! [X213: A,X223: list @ A] :
% 0.26/0.46 ( ( Xs2
% 0.26/0.46 = ( cons @ A @ X213 @ X223 ) )
% 0.26/0.46 => ( P @ Xs2 ) )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) ) )
% 0.26/0.46 => ( ( P @ ( nil @ A ) )
% 0.26/0.46 => ( P @ A0 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % min_list.induct
% 0.26/0.46 thf(fact_12_shuffles_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.26/0.46 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.26/0.46 => ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
% 0.26/0.46 ( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
% 0.26/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % shuffles.induct
% 0.26/0.46 thf(fact_13_not__Cons__self2,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.26/0.46 ( ( cons @ A @ X3 @ Xs )
% 0.26/0.46 != Xs ) ).
% 0.26/0.46
% 0.26/0.46 % not_Cons_self2
% 0.26/0.46 thf(fact_14_strict__sorted_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.26/0.46 ( ( P @ ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A,Ys3: list @ A] :
% 0.26/0.46 ( ( P @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Ys3 ) ) )
% 0.26/0.46 => ( P @ A0 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % strict_sorted.induct
% 0.26/0.46 thf(fact_15_strict__sorted_Ocases,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [X3: list @ A] :
% 0.26/0.46 ( ( X3
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ~ ! [X: A,Ys3: list @ A] :
% 0.26/0.46 ( X3
% 0.26/0.46 != ( cons @ A @ X @ Ys3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % strict_sorted.cases
% 0.26/0.46 thf(fact_16_map__tailrec__rev_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
% 0.26/0.46 ( ! [F2: A > B,X_1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X_1 )
% 0.26/0.46 => ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
% 0.26/0.46 ( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
% 0.26/0.46 => ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 0.26/0.46 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % map_tailrec_rev.induct
% 0.26/0.46 thf(fact_17_list__nonempty__induct,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A] :
% 0.26/0.46 ( ( Xs2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( P @ Xs2 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
% 0.26/0.46 => ( P @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_nonempty_induct
% 0.26/0.46 thf(fact_18_successively_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.26/0.46 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 0.26/0.46 => ( ! [P2: A > A > $o,X: A] : ( P @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ( ! [P2: A > A > $o,X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( ( P @ P2 @ ( cons @ A @ Y3 @ Xs2 ) )
% 0.26/0.46 => ( P @ P2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
% 0.26/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % successively.induct
% 0.26/0.46 thf(fact_19_arg__min__list_Oinduct,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( linorder @ B )
% 0.26/0.46 => ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
% 0.26/0.46 ( ! [F2: A > B,X: A] : ( P @ F2 @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ( ! [F2: A > B,X: A,Y3: A,Zs: list @ A] :
% 0.26/0.46 ( ( P @ F2 @ ( cons @ A @ Y3 @ Zs ) )
% 0.26/0.46 => ( P @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) )
% 0.26/0.46 => ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
% 0.26/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % arg_min_list.induct
% 0.26/0.46 thf(fact_20_remdups__adj_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 0.26/0.46 ( ( P @ ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( ( ( X = Y3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) ) )
% 0.26/0.46 => ( ( ( X != Y3 )
% 0.26/0.46 => ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
% 0.26/0.46 => ( P @ A0 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % remdups_adj.induct
% 0.26/0.46 thf(fact_21_sorted__wrt_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.26/0.46 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 0.26/0.46 => ( ! [P2: A > A > $o,X: A,Ys3: list @ A] :
% 0.26/0.46 ( ( P @ P2 @ Ys3 )
% 0.26/0.46 => ( P @ P2 @ ( cons @ A @ X @ Ys3 ) ) )
% 0.26/0.46 => ( P @ A0 @ A1 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % sorted_wrt.induct
% 0.26/0.46 thf(fact_22_remdups__adj_Ocases,axiom,
% 0.26/0.46 ! [A: $tType,X3: list @ A] :
% 0.26/0.46 ( ( X3
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] :
% 0.26/0.46 ( X3
% 0.26/0.46 != ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( X3
% 0.26/0.46 != ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % remdups_adj.cases
% 0.26/0.46 thf(fact_23_transpose_Ocases,axiom,
% 0.26/0.46 ! [A: $tType,X3: list @ ( list @ A )] :
% 0.26/0.46 ( ( X3
% 0.26/0.46 != ( nil @ ( list @ A ) ) )
% 0.26/0.46 => ( ! [Xss: list @ ( list @ A )] :
% 0.26/0.46 ( X3
% 0.26/0.46 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.26/0.46 => ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
% 0.26/0.46 ( X3
% 0.26/0.46 != ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % transpose.cases
% 0.26/0.46 thf(fact_24_list__ord_Oinduct,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.26/0.46 ( ! [Ord: A > A > $o,X1: A,X2: A,Xs2: list @ A] :
% 0.26/0.46 ( ( P @ Ord @ ( cons @ A @ X2 @ Xs2 ) )
% 0.26/0.46 => ( P @ Ord @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.26/0.46 => ( ! [Ord: A > A > $o] : ( P @ Ord @ ( nil @ A ) )
% 0.26/0.46 => ( ! [Ord: A > A > $o,V: A] : ( P @ Ord @ ( cons @ A @ V @ ( nil @ A ) ) )
% 0.26/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.induct
% 0.26/0.46 thf(fact_25_insert__Nil,axiom,
% 0.26/0.46 ! [A: $tType,X3: A] :
% 0.26/0.46 ( ( insert @ A @ X3 @ ( nil @ A ) )
% 0.26/0.46 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % insert_Nil
% 0.26/0.46 thf(fact_26_list__ex1__simps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o] :
% 0.26/0.46 ~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ex1_simps(1)
% 0.26/0.46 thf(fact_27_map__tailrec__rev_Oelims,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
% 0.26/0.46 ( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
% 0.26/0.46 = Y )
% 0.26/0.46 => ( ( ( Xa
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( Y != Xb ) )
% 0.26/0.46 => ~ ! [A3: A,As: list @ A] :
% 0.26/0.46 ( ( Xa
% 0.26/0.46 = ( cons @ A @ A3 @ As ) )
% 0.26/0.46 => ( Y
% 0.26/0.46 != ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % map_tailrec_rev.elims
% 0.26/0.46 thf(fact_28_arg__min__list_Osimps_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( linorder @ B )
% 0.26/0.46 => ! [F: A > B,X3: A] :
% 0.26/0.46 ( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.26/0.46 = X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % arg_min_list.simps(1)
% 0.26/0.46 thf(fact_29_ord_Olexordp__eq__simps_I3_J,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
% 0.26/0.46 ~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq_simps(3)
% 0.26/0.46 thf(fact_30_bind__simps_I1_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.26/0.46 ( ( bind @ B @ A @ ( nil @ B ) @ F )
% 0.26/0.46 = ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % bind_simps(1)
% 0.26/0.46 thf(fact_31_listrelp_Oinducts,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,R: A > B > $o,X12: list @ A,X23: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 0.26/0.46 ( ( listrelp @ A @ B @ R @ X12 @ X23 )
% 0.26/0.46 => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.26/0.46 => ( ! [X: A,Y3: B,Xs2: list @ A,Ys3: list @ B] :
% 0.26/0.46 ( ( R @ X @ Y3 )
% 0.26/0.46 => ( ( listrelp @ A @ B @ R @ Xs2 @ Ys3 )
% 0.26/0.46 => ( ( P @ Xs2 @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) ) )
% 0.26/0.46 => ( P @ X12 @ X23 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % listrelp.inducts
% 0.26/0.46 thf(fact_32_listrelp_Osimps,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( listrelp @ A @ B )
% 0.26/0.46 = ( ^ [R2: A > B > $o,A12: list @ A,A22: list @ B] :
% 0.26/0.46 ( ( ( A12
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( nil @ B ) ) )
% 0.26/0.46 | ? [X4: A,Y2: B,Xs3: list @ A,Ys: list @ B] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ B @ Y2 @ Ys ) )
% 0.26/0.46 & ( R2 @ X4 @ Y2 )
% 0.26/0.46 & ( listrelp @ A @ B @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % listrelp.simps
% 0.26/0.46 thf(fact_33_listrelp_Ocases,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,R: A > B > $o,A1: list @ A,A2: list @ B] :
% 0.26/0.46 ( ( listrelp @ A @ B @ R @ A1 @ A2 )
% 0.26/0.46 => ( ( ( A1
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( A2
% 0.26/0.46 != ( nil @ B ) ) )
% 0.26/0.46 => ~ ! [X: A,Y3: B,Xs2: list @ A] :
% 0.26/0.46 ( ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Ys3: list @ B] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( cons @ B @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ( R @ X @ Y3 )
% 0.26/0.46 => ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % listrelp.cases
% 0.26/0.46 thf(fact_34_lexordp__eq__simps_I3_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Xs: list @ A] :
% 0.26/0.46 ~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_simps(3)
% 0.26/0.46 thf(fact_35_lexordp__eq__simps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Xs: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
% 0.26/0.46 = ( Xs
% 0.26/0.46 = ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_simps(2)
% 0.26/0.46 thf(fact_36_lexordp__eq__simps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_simps(1)
% 0.26/0.46 thf(fact_37_ord_Olexordp__eq__simps_I4_J,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.26/0.46 ( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.26/0.46 = ( ( Less @ X3 @ Y )
% 0.26/0.46 | ( ~ ( Less @ Y @ X3 )
% 0.26/0.46 & ( lexordp_eq @ A @ Less @ Xs @ Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq_simps(4)
% 0.26/0.46 thf(fact_38_ord_Olexordp__eq__simps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,Xs: list @ A] :
% 0.26/0.46 ( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
% 0.26/0.46 = ( Xs
% 0.26/0.46 = ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq_simps(2)
% 0.26/0.46 thf(fact_39_ord_Olexordp__eq__simps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq_simps(1)
% 0.26/0.46 thf(fact_40_ord_Olexordp__eq_Ocong,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( lexordp_eq @ A )
% 0.26/0.46 = ( lexordp_eq @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.cong
% 0.26/0.46 thf(fact_41_ord_Olexordp__eq__refl,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq_refl
% 0.26/0.46 thf(fact_42_lexordp__eq__refl,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_refl
% 0.26/0.46 thf(fact_43_lexordp__eq__trans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ( ord_lexordp_eq @ A @ Ys2 @ Zs2 )
% 0.26/0.46 => ( ord_lexordp_eq @ A @ Xs @ Zs2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_trans
% 0.26/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.26/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.26/0.46 ( ( member2 @ A @ A4 @ ( collect @ A @ P ) )
% 0.26/0.46 = ( P @ A4 ) ) ).
% 0.26/0.46
% 0.26/0.46 % mem_Collect_eq
% 0.26/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.26/0.46 ! [A: $tType,A5: set @ A] :
% 0.26/0.46 ( ( collect @ A
% 0.26/0.46 @ ^ [X4: A] : ( member2 @ A @ X4 @ A5 ) )
% 0.26/0.46 = A5 ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_mem_eq
% 0.26/0.46 thf(fact_46_Collect__cong,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.26/0.46 ( ! [X: A] :
% 0.26/0.46 ( ( P @ X )
% 0.26/0.46 = ( Q @ X ) )
% 0.26/0.46 => ( ( collect @ A @ P )
% 0.26/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_cong
% 0.26/0.46 thf(fact_47_ext,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.26/0.46 ( ! [X: A] :
% 0.26/0.46 ( ( F @ X )
% 0.26/0.46 = ( G @ X ) )
% 0.26/0.46 => ( F = G ) ) ).
% 0.26/0.46
% 0.26/0.46 % ext
% 0.26/0.46 thf(fact_48_lexordp__eq__linear,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.26/0.46 | ( ord_lexordp_eq @ A @ Ys2 @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_linear
% 0.26/0.46 thf(fact_49_lexordp__eq__antisym,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ( ord_lexordp_eq @ A @ Ys2 @ Xs )
% 0.26/0.46 => ( Xs = Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_antisym
% 0.26/0.46 thf(fact_50_lexordp__eq_ONil,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq.Nil
% 0.26/0.46 thf(fact_51_ord_Olexordp__eq_OCons,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( Less @ X3 @ Y )
% 0.26/0.46 => ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.Cons
% 0.26/0.46 thf(fact_52_ord_Olexordp__eq_OCons__eq,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ~ ( Less @ X3 @ Y )
% 0.26/0.46 => ( ~ ( Less @ Y @ X3 )
% 0.26/0.46 => ( ( lexordp_eq @ A @ Less @ Xs @ Ys2 )
% 0.26/0.46 => ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.Cons_eq
% 0.26/0.46 thf(fact_53_ord_Olexordp__eq_ONil,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.Nil
% 0.26/0.46 thf(fact_54_listrelp_OCons,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,R: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 0.26/0.46 ( ( R @ X3 @ Y )
% 0.26/0.46 => ( ( listrelp @ A @ B @ R @ Xs @ Ys2 )
% 0.26/0.46 => ( listrelp @ A @ B @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % listrelp.Cons
% 0.26/0.46 thf(fact_55_listrelp_ONil,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% 0.26/0.46
% 0.26/0.46 % listrelp.Nil
% 0.26/0.46 thf(fact_56_map__tailrec__rev_Osimps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
% 0.26/0.46 ( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A4 @ As2 ) @ Bs2 )
% 0.26/0.46 = ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A4 ) @ Bs2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % map_tailrec_rev.simps(2)
% 0.26/0.46 thf(fact_57_map__tailrec__rev_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,Bs2: list @ B] :
% 0.26/0.46 ( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs2 )
% 0.26/0.46 = Bs2 ) ).
% 0.26/0.46
% 0.26/0.46 % map_tailrec_rev.simps(1)
% 0.26/0.46 thf(fact_58_ord_Olexordp__eq_Ocases,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,A1: list @ A,A2: list @ A] :
% 0.26/0.46 ( ( lexordp_eq @ A @ Less @ A1 @ A2 )
% 0.26/0.46 => ( ( A1
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] :
% 0.26/0.46 ( ? [Xs2: list @ A] :
% 0.26/0.46 ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Y3: A] :
% 0.26/0.46 ( ? [Ys3: list @ A] :
% 0.26/0.46 ( A2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ~ ( Less @ X @ Y3 ) ) )
% 0.26/0.46 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Ys3: list @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ~ ( Less @ X @ Y3 )
% 0.26/0.46 => ( ~ ( Less @ Y3 @ X )
% 0.26/0.46 => ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.cases
% 0.26/0.46 thf(fact_59_ord_Olexordp__eq_Osimps,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( lexordp_eq @ A )
% 0.26/0.46 = ( ^ [Less2: A > A > $o,A12: list @ A,A22: list @ A] :
% 0.26/0.46 ( ? [Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( A22 = Ys ) )
% 0.26/0.46 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) )
% 0.26/0.46 & ( Less2 @ X4 @ Y2 ) )
% 0.26/0.46 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) )
% 0.26/0.46 & ~ ( Less2 @ X4 @ Y2 )
% 0.26/0.46 & ~ ( Less2 @ Y2 @ X4 )
% 0.26/0.46 & ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.simps
% 0.26/0.46 thf(fact_60_ord_Olexordp__eq_Oinducts,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.26/0.46 ( ( lexordp_eq @ A @ Less @ X12 @ X23 )
% 0.26/0.46 => ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ( Less @ X @ Y3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ~ ( Less @ X @ Y3 )
% 0.26/0.46 => ( ~ ( Less @ Y3 @ X )
% 0.26/0.46 => ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 )
% 0.26/0.46 => ( ( P @ Xs2 @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
% 0.26/0.46 => ( P @ X12 @ X23 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq.inducts
% 0.26/0.46 thf(fact_61_product__lists_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 0.26/0.46 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % product_lists.simps(1)
% 0.26/0.46 thf(fact_62_subseqs_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( subseqs @ A @ ( nil @ A ) )
% 0.26/0.46 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % subseqs.simps(1)
% 0.26/0.46 thf(fact_63_lexordp__eq_Ocases,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [A1: list @ A,A2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ A1 @ A2 )
% 0.26/0.46 => ( ( A1
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] :
% 0.26/0.46 ( ? [Xs2: list @ A] :
% 0.26/0.46 ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Y3: A] :
% 0.26/0.46 ( ? [Ys3: list @ A] :
% 0.26/0.46 ( A2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ~ ( ord_less @ A @ X @ Y3 ) ) )
% 0.26/0.46 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Ys3: list @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.26/0.46 => ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq.cases
% 0.26/0.46 thf(fact_64_lexordp__eq_Osimps,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ( ( ord_lexordp_eq @ A )
% 0.26/0.46 = ( ^ [A12: list @ A,A22: list @ A] :
% 0.26/0.46 ( ? [Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( A22 = Ys ) )
% 0.26/0.46 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) )
% 0.26/0.46 & ( ord_less @ A @ X4 @ Y2 ) )
% 0.26/0.46 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) )
% 0.26/0.46 & ~ ( ord_less @ A @ X4 @ Y2 )
% 0.26/0.46 & ~ ( ord_less @ A @ Y2 @ X4 )
% 0.26/0.46 & ( ord_lexordp_eq @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq.simps
% 0.26/0.46 thf(fact_65_lexordp__eq_Oinducts,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ X12 @ X23 )
% 0.26/0.46 => ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.26/0.46 => ( ( ord_lexordp_eq @ A @ Xs2 @ Ys3 )
% 0.26/0.46 => ( ( P @ Xs2 @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
% 0.26/0.46 => ( P @ X12 @ X23 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq.inducts
% 0.26/0.46 thf(fact_66_member__rec_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,Y: A] :
% 0.26/0.46 ~ ( member @ A @ ( nil @ A ) @ Y ) ).
% 0.26/0.46
% 0.26/0.46 % member_rec(2)
% 0.26/0.46 thf(fact_67_member__rec_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A,Y: A] :
% 0.26/0.46 ( ( member @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
% 0.26/0.46 = ( ( X3 = Y )
% 0.26/0.46 | ( member @ A @ Xs @ Y ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % member_rec(1)
% 0.26/0.46 thf(fact_68_splice_Oelims,axiom,
% 0.26/0.46 ! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
% 0.26/0.46 ( ( ( splice @ A @ X3 @ Xa )
% 0.26/0.46 = Y )
% 0.26/0.46 => ( ( ( X3
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( Y != Xa ) )
% 0.26/0.46 => ~ ! [X: A,Xs2: list @ A] :
% 0.26/0.46 ( ( X3
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ( Y
% 0.26/0.46 != ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % splice.elims
% 0.26/0.46 thf(fact_69_bind__simps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X3: B,Xs: list @ B,F: B > ( list @ A )] :
% 0.26/0.46 ( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F )
% 0.26/0.46 = ( append @ A @ ( F @ X3 ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bind_simps(2)
% 0.26/0.46 thf(fact_70_same__append__eq,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.26/0.46 = ( append @ A @ Xs @ Zs2 ) )
% 0.26/0.46 = ( Ys2 = Zs2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % same_append_eq
% 0.26/0.46 thf(fact_71_append__same__eq,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Ys2 @ Xs )
% 0.26/0.46 = ( append @ A @ Zs2 @ Xs ) )
% 0.26/0.46 = ( Ys2 = Zs2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_same_eq
% 0.26/0.46 thf(fact_72_append__assoc,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs2 )
% 0.26/0.46 = ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_assoc
% 0.26/0.46 thf(fact_73_append_Oassoc,axiom,
% 0.26/0.46 ! [A: $tType,A4: list @ A,B2: list @ A,C2: list @ A] :
% 0.26/0.46 ( ( append @ A @ ( append @ A @ A4 @ B2 ) @ C2 )
% 0.26/0.46 = ( append @ A @ A4 @ ( append @ A @ B2 @ C2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append.assoc
% 0.26/0.46 thf(fact_74_append_Oright__neutral,axiom,
% 0.26/0.46 ! [A: $tType,A4: list @ A] :
% 0.26/0.46 ( ( append @ A @ A4 @ ( nil @ A ) )
% 0.26/0.46 = A4 ) ).
% 0.26/0.46
% 0.26/0.46 % append.right_neutral
% 0.26/0.46 thf(fact_75_append__is__Nil__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 = ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_is_Nil_conv
% 0.26/0.46 thf(fact_76_Nil__is__append__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( nil @ A )
% 0.26/0.46 = ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Nil_is_append_conv
% 0.26/0.46 thf(fact_77_self__append__conv2,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( Ys2
% 0.26/0.46 = ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( Xs
% 0.26/0.46 = ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % self_append_conv2
% 0.26/0.46 thf(fact_78_append__self__conv2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.26/0.46 = Ys2 )
% 0.26/0.46 = ( Xs
% 0.26/0.46 = ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_self_conv2
% 0.26/0.46 thf(fact_79_self__append__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 = ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( Ys2
% 0.26/0.46 = ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % self_append_conv
% 0.26/0.46 thf(fact_80_append__self__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.26/0.46 = Xs )
% 0.26/0.46 = ( Ys2
% 0.26/0.46 = ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_self_conv
% 0.26/0.46 thf(fact_81_append__Nil2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( append @ A @ Xs @ ( nil @ A ) )
% 0.26/0.46 = Xs ) ).
% 0.26/0.46
% 0.26/0.46 % append_Nil2
% 0.26/0.46 thf(fact_82_split__Nil__iff,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( splice @ A @ Xs @ Ys2 )
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 = ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % split_Nil_iff
% 0.26/0.46 thf(fact_83_splice__Nil2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( splice @ A @ Xs @ ( nil @ A ) )
% 0.26/0.46 = Xs ) ).
% 0.26/0.46
% 0.26/0.46 % splice_Nil2
% 0.26/0.46 thf(fact_84_i__append__assoc,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,F: nat > A] :
% 0.26/0.46 ( ( listIn521021761append @ A @ Xs @ ( listIn521021761append @ A @ Ys2 @ F ) )
% 0.26/0.46 = ( listIn521021761append @ A @ ( append @ A @ Xs @ Ys2 ) @ F ) ) ).
% 0.26/0.46
% 0.26/0.46 % i_append_assoc
% 0.26/0.46 thf(fact_85_append1__eq__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A,Y: A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.26/0.46 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 0.26/0.46 = ( ( Xs = Ys2 )
% 0.26/0.46 & ( X3 = Y ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append1_eq_conv
% 0.26/0.46 thf(fact_86_lexordp__eq__simps_I4_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.26/0.46 = ( ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 | ( ~ ( ord_less @ A @ Y @ X3 )
% 0.26/0.46 & ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_simps(4)
% 0.26/0.46 thf(fact_87_append__eq__append__conv2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A,Ts: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.26/0.46 = ( append @ A @ Zs2 @ Ts ) )
% 0.26/0.46 = ( ? [Us: list @ A] :
% 0.26/0.46 ( ( ( Xs
% 0.26/0.46 = ( append @ A @ Zs2 @ Us ) )
% 0.26/0.46 & ( ( append @ A @ Us @ Ys2 )
% 0.26/0.46 = Ts ) )
% 0.26/0.46 | ( ( ( append @ A @ Xs @ Us )
% 0.26/0.46 = Zs2 )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_eq_append_conv2
% 0.26/0.46 thf(fact_88_append__eq__appendI,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys2: list @ A,Us2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ Xs1 )
% 0.26/0.46 = Zs2 )
% 0.26/0.46 => ( ( Ys2
% 0.26/0.46 = ( append @ A @ Xs1 @ Us2 ) )
% 0.26/0.46 => ( ( append @ A @ Xs @ Ys2 )
% 0.26/0.46 = ( append @ A @ Zs2 @ Us2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_eq_appendI
% 0.26/0.46 thf(fact_89_append__Cons,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
% 0.26/0.46 = ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_Cons
% 0.26/0.46 thf(fact_90_Cons__eq__appendI,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ( cons @ A @ X3 @ Xs1 )
% 0.26/0.46 = Ys2 )
% 0.26/0.46 => ( ( Xs
% 0.26/0.46 = ( append @ A @ Xs1 @ Zs2 ) )
% 0.26/0.46 => ( ( cons @ A @ X3 @ Xs )
% 0.26/0.46 = ( append @ A @ Ys2 @ Zs2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Cons_eq_appendI
% 0.26/0.46 thf(fact_91_append_Oleft__neutral,axiom,
% 0.26/0.46 ! [A: $tType,A4: list @ A] :
% 0.26/0.46 ( ( append @ A @ ( nil @ A ) @ A4 )
% 0.26/0.46 = A4 ) ).
% 0.26/0.46
% 0.26/0.46 % append.left_neutral
% 0.26/0.46 thf(fact_92_append__Nil,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A] :
% 0.26/0.46 ( ( append @ A @ ( nil @ A ) @ Ys2 )
% 0.26/0.46 = Ys2 ) ).
% 0.26/0.46
% 0.26/0.46 % append_Nil
% 0.26/0.46 thf(fact_93_eq__Nil__appendI,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( Xs = Ys2 )
% 0.26/0.46 => ( Xs
% 0.26/0.46 = ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_Nil_appendI
% 0.26/0.46 thf(fact_94_lexordp__eq__pref,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [U: list @ A,V2: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_pref
% 0.26/0.46 thf(fact_95_ord_Olexordp__eq__pref,axiom,
% 0.26/0.46 ! [A: $tType,Less: A > A > $o,U: list @ A,V2: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % ord.lexordp_eq_pref
% 0.26/0.46 thf(fact_96_splice_Osimps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
% 0.26/0.46 = ( cons @ A @ X3 @ ( splice @ A @ Ys2 @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % splice.simps(2)
% 0.26/0.46 thf(fact_97_splice_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A] :
% 0.26/0.46 ( ( splice @ A @ ( nil @ A ) @ Ys2 )
% 0.26/0.46 = Ys2 ) ).
% 0.26/0.46
% 0.26/0.46 % splice.simps(1)
% 0.26/0.46 thf(fact_98_rev__nonempty__induct,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A] :
% 0.26/0.46 ( ( Xs2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( P @ Xs2 )
% 0.26/0.46 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
% 0.26/0.46 => ( P @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rev_nonempty_induct
% 0.26/0.46 thf(fact_99_append__eq__Cons__conv,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Zs2: list @ A,X3: A,Xs: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Ys2 @ Zs2 )
% 0.26/0.46 = ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( Zs2
% 0.26/0.46 = ( cons @ A @ X3 @ Xs ) ) )
% 0.26/0.46 | ? [Ys4: list @ A] :
% 0.26/0.46 ( ( Ys2
% 0.26/0.46 = ( cons @ A @ X3 @ Ys4 ) )
% 0.26/0.46 & ( ( append @ A @ Ys4 @ Zs2 )
% 0.26/0.46 = Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_eq_Cons_conv
% 0.26/0.46 thf(fact_100_Cons__eq__append__conv,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ( cons @ A @ X3 @ Xs )
% 0.26/0.46 = ( append @ A @ Ys2 @ Zs2 ) )
% 0.26/0.46 = ( ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( ( cons @ A @ X3 @ Xs )
% 0.26/0.46 = Zs2 ) )
% 0.26/0.46 | ? [Ys4: list @ A] :
% 0.26/0.46 ( ( ( cons @ A @ X3 @ Ys4 )
% 0.26/0.46 = Ys2 )
% 0.26/0.46 & ( Xs
% 0.26/0.46 = ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Cons_eq_append_conv
% 0.26/0.46 thf(fact_101_rev__exhaust,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ~ ! [Ys3: list @ A,Y3: A] :
% 0.26/0.46 ( Xs
% 0.26/0.46 != ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rev_exhaust
% 0.26/0.46 thf(fact_102_rev__induct,axiom,
% 0.26/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.26/0.46 ( ( P @ ( nil @ A ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A] :
% 0.26/0.46 ( ( P @ Xs2 )
% 0.26/0.46 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
% 0.26/0.46 => ( P @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rev_induct
% 0.26/0.46 thf(fact_103_append__eq__Cons,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.26/0.46 ( ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs )
% 0.26/0.46 = ( cons @ A @ X3 @ Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_eq_Cons
% 0.26/0.46 thf(fact_104_lexordp__eq_OCons,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 => ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq.Cons
% 0.26/0.46 thf(fact_105_lexordp__eq_OCons__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 => ( ~ ( ord_less @ A @ Y @ X3 )
% 0.26/0.46 => ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq.Cons_eq
% 0.26/0.46 thf(fact_106_maps__simps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: B > ( list @ A ),X3: B,Xs: list @ B] :
% 0.26/0.46 ( ( maps @ B @ A @ F @ ( cons @ B @ X3 @ Xs ) )
% 0.26/0.46 = ( append @ A @ ( F @ X3 ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % maps_simps(1)
% 0.26/0.46 thf(fact_107_concat__eq__append__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ( concat @ A @ Xss2 )
% 0.26/0.46 = ( append @ A @ Ys2 @ Zs2 ) )
% 0.26/0.46 = ( ( ( Xss2
% 0.26/0.46 = ( nil @ ( list @ A ) ) )
% 0.26/0.46 => ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( Zs2
% 0.26/0.46 = ( nil @ A ) ) ) )
% 0.26/0.46 & ( ( Xss2
% 0.26/0.46 != ( nil @ ( list @ A ) ) )
% 0.26/0.46 => ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
% 0.26/0.46 ( ( Xss2
% 0.26/0.46 = ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
% 0.26/0.46 & ( Zs2
% 0.26/0.46 = ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % concat_eq_append_conv
% 0.26/0.46 thf(fact_108_rotate1_Osimps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.26/0.46 ( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rotate1.simps(2)
% 0.26/0.46 thf(fact_109_butlast__snoc,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.26/0.46 = Xs ) ).
% 0.26/0.46
% 0.26/0.46 % butlast_snoc
% 0.26/0.46 thf(fact_110_last__snoc,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.26/0.46 = X3 ) ).
% 0.26/0.46
% 0.26/0.46 % last_snoc
% 0.26/0.46 thf(fact_111_SuccI,axiom,
% 0.26/0.46 ! [A: $tType,Kl: list @ A,K: A,Kl2: set @ ( list @ A )] :
% 0.26/0.46 ( ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 )
% 0.26/0.46 => ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % SuccI
% 0.26/0.46 thf(fact_112_SuccD,axiom,
% 0.26/0.46 ! [A: $tType,K: A,Kl2: set @ ( list @ A ),Kl: list @ A] :
% 0.26/0.46 ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) )
% 0.26/0.46 => ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % SuccD
% 0.26/0.46 thf(fact_113_rotate1__is__Nil__conv,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( ( rotate1 @ A @ Xs )
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 = ( Xs
% 0.26/0.46 = ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rotate1_is_Nil_conv
% 0.26/0.46 thf(fact_114_last__appendL,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( last @ A @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % last_appendL
% 0.26/0.46 thf(fact_115_last__appendR,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( Ys2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( last @ A @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % last_appendR
% 0.26/0.46 thf(fact_116_concat__append,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
% 0.26/0.46 ( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys2 ) )
% 0.26/0.46 = ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % concat_append
% 0.26/0.46 thf(fact_117_append__butlast__last__id,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
% 0.26/0.46 = Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % append_butlast_last_id
% 0.26/0.46 thf(fact_118_butlast_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( butlast @ A @ ( nil @ A ) )
% 0.26/0.46 = ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % butlast.simps(1)
% 0.26/0.46 thf(fact_119_snoc__eq__iff__butlast,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.26/0.46 = Ys2 )
% 0.26/0.46 = ( ( Ys2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 & ( ( butlast @ A @ Ys2 )
% 0.26/0.46 = Xs )
% 0.26/0.46 & ( ( last @ A @ Ys2 )
% 0.26/0.46 = X3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % snoc_eq_iff_butlast
% 0.26/0.46 thf(fact_120_rotate1_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( rotate1 @ A @ ( nil @ A ) )
% 0.26/0.46 = ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % rotate1.simps(1)
% 0.26/0.46 thf(fact_121_last_Osimps,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = X3 ) )
% 0.26/0.46 & ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( last @ A @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % last.simps
% 0.26/0.46 thf(fact_122_last__ConsL,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = X3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % last_ConsL
% 0.26/0.46 thf(fact_123_last__ConsR,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( last @ A @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % last_ConsR
% 0.26/0.46 thf(fact_124_last__append,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( last @ A @ Xs ) ) )
% 0.26/0.46 & ( ( Ys2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( last @ A @ Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % last_append
% 0.26/0.46 thf(fact_125_longest__common__suffix,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ? [Ss: list @ A,Xs5: list @ A,Ys5: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 = ( append @ A @ Xs5 @ Ss ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( append @ A @ Ys5 @ Ss ) )
% 0.26/0.46 & ( ( Xs5
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( Ys5
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( ( last @ A @ Xs5 )
% 0.26/0.46 != ( last @ A @ Ys5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % longest_common_suffix
% 0.26/0.46 thf(fact_126_butlast_Osimps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( nil @ A ) ) )
% 0.26/0.46 & ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( cons @ A @ X3 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % butlast.simps(2)
% 0.26/0.46 thf(fact_127_concat_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( concat @ A @ ( nil @ ( list @ A ) ) )
% 0.26/0.46 = ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % concat.simps(1)
% 0.26/0.46 thf(fact_128_concat_Osimps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] :
% 0.26/0.46 ( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) )
% 0.26/0.46 = ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % concat.simps(2)
% 0.26/0.46 thf(fact_129_butlast__append,axiom,
% 0.26/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( butlast @ A @ Xs ) ) )
% 0.26/0.46 & ( ( Ys2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( append @ A @ Xs @ ( butlast @ A @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % butlast_append
% 0.26/0.46 thf(fact_130_maps__simps_I2_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.26/0.46 ( ( maps @ B @ A @ F @ ( nil @ B ) )
% 0.26/0.46 = ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % maps_simps(2)
% 0.26/0.46 thf(fact_131_concat__eq__appendD,axiom,
% 0.26/0.46 ! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ( concat @ A @ Xss2 )
% 0.26/0.46 = ( append @ A @ Ys2 @ Zs2 ) )
% 0.26/0.46 => ( ( Xss2
% 0.26/0.46 != ( nil @ ( list @ A ) ) )
% 0.26/0.46 => ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
% 0.26/0.46 ( ( Xss2
% 0.26/0.46 = ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs5 ) @ Xss23 ) ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
% 0.26/0.46 & ( Zs2
% 0.26/0.46 = ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % concat_eq_appendD
% 0.26/0.46 thf(fact_132_shift__def,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( bNF_Greatest_shift @ A @ B )
% 0.26/0.46 = ( ^ [Lab: ( list @ A ) > B,K2: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % shift_def
% 0.26/0.46 thf(fact_133_empty__Shift,axiom,
% 0.26/0.46 ! [A: $tType,Kl2: set @ ( list @ A ),K: A] :
% 0.26/0.46 ( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl2 )
% 0.26/0.46 => ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ ( nil @ A ) ) )
% 0.26/0.46 => ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % empty_Shift
% 0.26/0.46 thf(fact_134_Succ__Shift,axiom,
% 0.26/0.46 ! [A: $tType,Kl2: set @ ( list @ A ),K: A,Kl: list @ A] :
% 0.26/0.46 ( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) @ Kl )
% 0.26/0.46 = ( bNF_Greatest_Succ @ A @ Kl2 @ ( cons @ A @ K @ Kl ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Succ_Shift
% 0.26/0.46 thf(fact_135_list__ord__snoc,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( list_ord @ A @ Ord2 @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.26/0.46 = ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( ( Ord2 @ ( last @ A @ Xs ) @ X3 )
% 0.26/0.46 & ( list_ord @ A @ Ord2 @ Xs ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_snoc
% 0.26/0.46 thf(fact_136_concat__conv__foldr,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( concat @ A )
% 0.26/0.46 = ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % concat_conv_foldr
% 0.26/0.46 thf(fact_137_foldr__append,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: B > A > A,Xs: list @ B,Ys2: list @ B,A4: A] :
% 0.26/0.46 ( ( foldr @ B @ A @ F @ ( append @ B @ Xs @ Ys2 ) @ A4 )
% 0.26/0.46 = ( foldr @ B @ A @ F @ Xs @ ( foldr @ B @ A @ F @ Ys2 @ A4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % foldr_append
% 0.26/0.46 thf(fact_138_list__ord_Oelims_I3_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A > A > $o,Xa: list @ A] :
% 0.26/0.46 ( ~ ( list_ord @ A @ X3 @ Xa )
% 0.26/0.46 => ~ ! [X1: A,X2: A,Xs2: list @ A] :
% 0.26/0.46 ( ( Xa
% 0.26/0.46 = ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.26/0.46 => ( ( X3 @ X1 @ X2 )
% 0.26/0.46 & ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.elims(3)
% 0.26/0.46 thf(fact_139_list__ord_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,X12: A,X23: A,Xs: list @ A] :
% 0.26/0.46 ( ( list_ord @ A @ Ord2 @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs ) ) )
% 0.26/0.46 = ( ( Ord2 @ X12 @ X23 )
% 0.26/0.46 & ( list_ord @ A @ Ord2 @ ( cons @ A @ X23 @ Xs ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.simps(1)
% 0.26/0.46 thf(fact_140_list__ord_Osimps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o] : ( list_ord @ A @ Ord2 @ ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.simps(2)
% 0.26/0.46 thf(fact_141_list__ord__Nil,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o] : ( list_ord @ A @ Ord2 @ ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_Nil
% 0.26/0.46 thf(fact_142_list__ord__imp,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,Ord3: A > A > $o,Xs: list @ A] :
% 0.26/0.46 ( ! [X: A,Y3: A] :
% 0.26/0.46 ( ( Ord2 @ X @ Y3 )
% 0.26/0.46 => ( Ord3 @ X @ Y3 ) )
% 0.26/0.46 => ( ( list_ord @ A @ Ord2 @ Xs )
% 0.26/0.46 => ( list_ord @ A @ Ord3 @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_imp
% 0.26/0.46 thf(fact_143_list__ord__one,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,X3: A] : ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_one
% 0.26/0.46 thf(fact_144_list__ord_Osimps_I3_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,V2: A] : ( list_ord @ A @ Ord2 @ ( cons @ A @ V2 @ ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.simps(3)
% 0.26/0.46 thf(fact_145_list__ord_Oelims_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A > A > $o,Xa: list @ A,Y: $o] :
% 0.26/0.46 ( ( ( list_ord @ A @ X3 @ Xa )
% 0.26/0.46 = Y )
% 0.26/0.46 => ( ! [X1: A,X2: A,Xs2: list @ A] :
% 0.26/0.46 ( ( Xa
% 0.26/0.46 = ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.26/0.46 => ( Y
% 0.26/0.46 = ( ~ ( ( X3 @ X1 @ X2 )
% 0.26/0.46 & ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) )
% 0.26/0.46 => ( ( ( Xa
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ~ Y )
% 0.26/0.46 => ~ ( ? [V: A] :
% 0.26/0.46 ( Xa
% 0.26/0.46 = ( cons @ A @ V @ ( nil @ A ) ) )
% 0.26/0.46 => ~ Y ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.elims(1)
% 0.26/0.46 thf(fact_146_list__ord_Oelims_I2_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A > A > $o,Xa: list @ A] :
% 0.26/0.46 ( ( list_ord @ A @ X3 @ Xa )
% 0.26/0.46 => ( ! [X1: A,X2: A,Xs2: list @ A] :
% 0.26/0.46 ( ( Xa
% 0.26/0.46 = ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.26/0.46 => ~ ( ( X3 @ X1 @ X2 )
% 0.26/0.46 & ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.26/0.46 => ( ( Xa
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ~ ! [V: A] :
% 0.26/0.46 ( Xa
% 0.26/0.46 != ( cons @ A @ V @ ( nil @ A ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord.elims(2)
% 0.26/0.46 thf(fact_147_ShiftD,axiom,
% 0.26/0.46 ! [A: $tType,Kl: list @ A,Kl2: set @ ( list @ A ),K: A] :
% 0.26/0.46 ( ( member2 @ ( list @ A ) @ Kl @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) )
% 0.26/0.46 => ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl ) @ Kl2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % ShiftD
% 0.26/0.46 thf(fact_148_list__strict__asc__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ( ( list_strict_asc @ A )
% 0.26/0.46 = ( list_ord @ A @ ( ord_less @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_strict_asc_def
% 0.26/0.46 thf(fact_149_list__ord__append,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( list_ord @ A @ Ord2 @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( ( list_ord @ A @ Ord2 @ Xs )
% 0.26/0.46 & ( ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( ( list_ord @ A @ Ord2 @ Ys2 )
% 0.26/0.46 & ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( Ord2 @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_append
% 0.26/0.46 thf(fact_150_lexordp__iff,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ( ( ord_lexordp @ A )
% 0.26/0.46 = ( ^ [Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ? [X4: A,Vs: list @ A] :
% 0.26/0.46 ( Ys
% 0.26/0.46 = ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs ) ) )
% 0.26/0.46 | ? [Us: list @ A,A6: A,B3: A,Vs: list @ A,Ws: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ A6 @ B3 )
% 0.26/0.46 & ( Xs3
% 0.26/0.46 = ( append @ A @ Us @ ( cons @ A @ A6 @ Vs ) ) )
% 0.26/0.46 & ( Ys
% 0.26/0.46 = ( append @ A @ Us @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_iff
% 0.26/0.46 thf(fact_151_lexordp__append__left__rightI,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Y: A,Us2: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 => ( ord_lexordp @ A @ ( append @ A @ Us2 @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_append_left_rightI
% 0.26/0.46 thf(fact_152_lexordp__simps_I2_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Xs: list @ A] :
% 0.26/0.46 ~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_simps(2)
% 0.26/0.46 thf(fact_153_lexordp__simps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ ( nil @ A ) @ Ys2 )
% 0.26/0.46 = ( Ys2
% 0.26/0.46 != ( nil @ A ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_simps(1)
% 0.26/0.46 thf(fact_154_lexordp__simps_I3_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.26/0.46 = ( ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 | ( ~ ( ord_less @ A @ Y @ X3 )
% 0.26/0.46 & ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_simps(3)
% 0.26/0.46 thf(fact_155_hd__append2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( hd @ A @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % hd_append2
% 0.26/0.46 thf(fact_156_lexordp__append__leftI,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Us2: list @ A,Vs2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Us2 @ Vs2 )
% 0.26/0.46 => ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_append_leftI
% 0.26/0.46 thf(fact_157_lexordp__antisym,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 => ~ ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_antisym
% 0.26/0.46 thf(fact_158_lexordp__trans,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ( ord_lexordp @ A @ Ys2 @ Zs2 )
% 0.26/0.46 => ( ord_lexordp @ A @ Xs @ Zs2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_trans
% 0.26/0.46 thf(fact_159_lexordp__linear,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 | ( Xs = Ys2 )
% 0.26/0.46 | ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_linear
% 0.26/0.46 thf(fact_160_lexordp__irreflexive_H,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( order @ A )
% 0.26/0.46 => ! [Xs: list @ A] :
% 0.26/0.46 ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_irreflexive'
% 0.26/0.46 thf(fact_161_lexordp__irreflexive,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Xs: list @ A] :
% 0.26/0.46 ( ! [X: A] :
% 0.26/0.46 ~ ( ord_less @ A @ X @ X )
% 0.26/0.46 => ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_irreflexive
% 0.26/0.46 thf(fact_162_list_Osel_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,X21: A,X22: list @ A] :
% 0.26/0.46 ( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
% 0.26/0.46 = X21 ) ).
% 0.26/0.46
% 0.26/0.46 % list.sel(1)
% 0.26/0.46 thf(fact_163_hd__concat,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ ( list @ A ) ) )
% 0.26/0.46 => ( ( ( hd @ ( list @ A ) @ Xs )
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( hd @ A @ ( concat @ A @ Xs ) )
% 0.26/0.46 = ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % hd_concat
% 0.26/0.46 thf(fact_164_lexordp__into__lexordp__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_into_lexordp_eq
% 0.26/0.46 thf(fact_165_lexordp__eq__conv__lexord,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ( ( ord_lexordp_eq @ A )
% 0.26/0.46 = ( ^ [Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( Xs3 = Ys )
% 0.26/0.46 | ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_eq_conv_lexord
% 0.26/0.46 thf(fact_166_lexordp__conv__lexordp__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ( ( ord_lexordp @ A )
% 0.26/0.46 = ( ^ [Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( ord_lexordp_eq @ A @ Xs3 @ Ys )
% 0.26/0.46 & ~ ( ord_lexordp_eq @ A @ Ys @ Xs3 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_conv_lexordp_eq
% 0.26/0.46 thf(fact_167_hd__append,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( hd @ A @ Ys2 ) ) )
% 0.26/0.46 & ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( hd @ A @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % hd_append
% 0.26/0.46 thf(fact_168_longest__common__prefix,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ? [Ps: list @ A,Xs5: list @ A,Ys5: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 = ( append @ A @ Ps @ Xs5 ) )
% 0.26/0.46 & ( Ys2
% 0.26/0.46 = ( append @ A @ Ps @ Ys5 ) )
% 0.26/0.46 & ( ( Xs5
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( Ys5
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( ( hd @ A @ Xs5 )
% 0.26/0.46 != ( hd @ A @ Ys5 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % longest_common_prefix
% 0.26/0.46 thf(fact_169_list__ord__Cons__imp,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,Xs: list @ A,X3: A] :
% 0.26/0.46 ( ( list_ord @ A @ Ord2 @ Xs )
% 0.26/0.46 => ( ( Ord2 @ X3 @ ( hd @ A @ Xs ) )
% 0.26/0.46 => ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ Xs ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_Cons_imp
% 0.26/0.46 thf(fact_170_lexordp_OCons,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 => ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp.Cons
% 0.26/0.46 thf(fact_171_lexordp_OCons__eq,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.26/0.46 => ( ~ ( ord_less @ A @ Y @ X3 )
% 0.26/0.46 => ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp.Cons_eq
% 0.26/0.46 thf(fact_172_lexordp__append__leftD,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Xs: list @ A,Us2: list @ A,Vs2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) )
% 0.26/0.46 => ( ! [A3: A] :
% 0.26/0.46 ~ ( ord_less @ A @ A3 @ A3 )
% 0.26/0.46 => ( ord_lexordp @ A @ Us2 @ Vs2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_append_leftD
% 0.26/0.46 thf(fact_173_lexordp_ONil,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Y: A,Ys2: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp.Nil
% 0.26/0.46 thf(fact_174_lexordp__append__rightI,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ys2: list @ A,Xs: list @ A] :
% 0.26/0.46 ( ( Ys2
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_append_rightI
% 0.26/0.46 thf(fact_175_list__ord__Cons,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [Ord2: A > A > $o,X3: A,Xs: list @ A] :
% 0.26/0.46 ( ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 = ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( ( Ord2 @ X3 @ ( hd @ A @ Xs ) )
% 0.26/0.46 & ( list_ord @ A @ Ord2 @ Xs ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % list_ord_Cons
% 0.26/0.46 thf(fact_176_lexordp_Ocases,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [A1: list @ A,A2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ A1 @ A2 )
% 0.26/0.46 => ( ( ( A1
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ! [Y3: A,Ys3: list @ A] :
% 0.26/0.46 ( A2
% 0.26/0.46 != ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.26/0.46 => ( ! [X: A] :
% 0.26/0.46 ( ? [Xs2: list @ A] :
% 0.26/0.46 ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Y3: A] :
% 0.26/0.46 ( ? [Ys3: list @ A] :
% 0.26/0.46 ( A2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ~ ( ord_less @ A @ X @ Y3 ) ) )
% 0.26/0.46 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( ( A1
% 0.26/0.46 = ( cons @ A @ X @ Xs2 ) )
% 0.26/0.46 => ! [Ys3: list @ A] :
% 0.26/0.46 ( ( A2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.26/0.46 => ~ ( ord_lexordp @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp.cases
% 0.26/0.46 thf(fact_177_lexordp_Osimps,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ( ( ord_lexordp @ A )
% 0.26/0.46 = ( ^ [A12: list @ A,A22: list @ A] :
% 0.26/0.46 ( ? [Y2: A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) ) )
% 0.26/0.46 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) )
% 0.26/0.46 & ( ord_less @ A @ X4 @ Y2 ) )
% 0.26/0.46 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.26/0.46 ( ( A12
% 0.26/0.46 = ( cons @ A @ X4 @ Xs3 ) )
% 0.26/0.46 & ( A22
% 0.26/0.46 = ( cons @ A @ Y2 @ Ys ) )
% 0.26/0.46 & ~ ( ord_less @ A @ X4 @ Y2 )
% 0.26/0.46 & ~ ( ord_less @ A @ Y2 @ X4 )
% 0.26/0.46 & ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp.simps
% 0.26/0.46 thf(fact_178_lexordp_Oinducts,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( ord @ A )
% 0.26/0.46 => ! [X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.26/0.46 ( ( ord_lexordp @ A @ X12 @ X23 )
% 0.26/0.46 => ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.26/0.46 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.26/0.46 => ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
% 0.26/0.46 => ( ( P @ Xs2 @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
% 0.26/0.46 => ( P @ X12 @ X23 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp.inducts
% 0.26/0.46 thf(fact_179_lexordp__cases,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 => ! [Y3: A,Ys5: list @ A] :
% 0.26/0.46 ( Ys2
% 0.26/0.46 != ( cons @ A @ Y3 @ Ys5 ) ) )
% 0.26/0.46 => ( ! [X: A] :
% 0.26/0.46 ( ? [Xs5: list @ A] :
% 0.26/0.46 ( Xs
% 0.26/0.46 = ( cons @ A @ X @ Xs5 ) )
% 0.26/0.46 => ! [Y3: A] :
% 0.26/0.46 ( ? [Ys5: list @ A] :
% 0.26/0.46 ( Ys2
% 0.26/0.46 = ( cons @ A @ Y3 @ Ys5 ) )
% 0.26/0.46 => ~ ( ord_less @ A @ X @ Y3 ) ) )
% 0.26/0.46 => ~ ! [X: A,Xs5: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 = ( cons @ A @ X @ Xs5 ) )
% 0.26/0.46 => ! [Ys5: list @ A] :
% 0.26/0.46 ( ( Ys2
% 0.26/0.46 = ( cons @ A @ X @ Ys5 ) )
% 0.26/0.46 => ~ ( ord_lexordp @ A @ Xs5 @ Ys5 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_cases
% 0.26/0.46 thf(fact_180_lexordp__induct,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( linorder @ A )
% 0.26/0.46 => ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.26/0.46 => ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
% 0.26/0.46 ( ( ord_less @ A @ X @ Y3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.26/0.46 => ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
% 0.26/0.46 ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
% 0.26/0.46 => ( ( P @ Xs2 @ Ys3 )
% 0.26/0.46 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys3 ) ) ) )
% 0.26/0.46 => ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lexordp_induct
% 0.26/0.46 thf(fact_181_distinct__adj__append__iff,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( ( distinct_adj @ A @ Xs )
% 0.26/0.46 & ( distinct_adj @ A @ Ys2 )
% 0.26/0.46 & ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( ( last @ A @ Xs )
% 0.26/0.46 != ( hd @ A @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % distinct_adj_append_iff
% 0.26/0.46 thf(fact_182_rotate1__hd__tl,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( rotate1 @ A @ Xs )
% 0.26/0.46 = ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rotate1_hd_tl
% 0.26/0.46 thf(fact_183_successively__append__iff,axiom,
% 0.26/0.46 ! [A: $tType,P: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( successively @ A @ P @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( ( successively @ A @ P @ Xs )
% 0.26/0.46 & ( successively @ A @ P @ Ys2 )
% 0.26/0.46 & ( ( Xs
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( Ys2
% 0.26/0.46 = ( nil @ A ) )
% 0.26/0.46 | ( P @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % successively_append_iff
% 0.26/0.46 thf(fact_184_distinct__adj__Cons__Cons,axiom,
% 0.26/0.46 ! [B: $tType,X3: B,Y: B,Xs: list @ B] :
% 0.26/0.46 ( ( distinct_adj @ B @ ( cons @ B @ X3 @ ( cons @ B @ Y @ Xs ) ) )
% 0.26/0.46 = ( ( X3 != Y )
% 0.26/0.46 & ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % distinct_adj_Cons_Cons
% 0.26/0.46 thf(fact_185_tl__append2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( tl @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 = ( append @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % tl_append2
% 0.26/0.46 thf(fact_186_hd__Cons__tl,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( Xs
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
% 0.26/0.46 = Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % hd_Cons_tl
% 0.26/0.46 thf(fact_187_list_Ocollapse,axiom,
% 0.26/0.46 ! [A: $tType,List: list @ A] :
% 0.26/0.46 ( ( List
% 0.26/0.46 != ( nil @ A ) )
% 0.26/0.46 => ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
% 0.26/0.46 = List ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.collapse
% 0.26/0.46 thf(fact_188_list_Osel_I3_J,axiom,
% 0.26/0.46 ! [A: $tType,X21: A,X22: list @ A] :
% 0.26/0.46 ( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
% 0.26/0.46 = X22 ) ).
% 0.26/0.46
% 0.26/0.46 % list.sel(3)
% 0.26/0.46 thf(fact_189_list_Osel_I2_J,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( tl @ A @ ( nil @ A ) )
% 0.26/0.46 = ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % list.sel(2)
% 0.26/0.46 thf(fact_190_successively_Oelims_I3_J,axiom,
% 0.26/0.46 ! [A: $tType,X3: A > A > $o,Xa: list @ A] :
% 0.26/0.46 ( ~ ( successively @ A @ X3 @ Xa )
% 0.26/0.46 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.46 ( ( Xa
% 0.26/0.46 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.26/0.46 => ( ( X3 @ X @ Y3 )
% 0.26/0.46 & ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % successively.elims(3)
% 0.26/0.46 thf(fact_191_successively_Osimps_I3_J,axiom,
% 0.26/0.46 ! [A: $tType,P: A > A > $o,X3: A,Y: A,Xs: list @ A] :
% 0.26/0.46 ( ( successively @ A @ P @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
% 0.26/0.46 = ( ( P @ X3 @ Y )
% 0.26/0.46 & ( successively @ A @ P @ ( cons @ A @ Y @ Xs ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % successively.simps(3)
% 0.26/0.46 thf(fact_192_successively_Osimps_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,P: A > A > $o] : ( successively @ A @ P @ ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % successively.simps(1)
% 0.26/0.46 thf(fact_193_distinct__adj__appendD1,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 => ( distinct_adj @ A @ Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % distinct_adj_appendD1
% 0.26/0.46 thf(fact_194_distinct__adj__appendD2,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.46 ( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.46 => ( distinct_adj @ A @ Ys2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % distinct_adj_appendD2
% 0.26/0.46 thf(fact_195_distinct__adj__Nil,axiom,
% 0.26/0.46 ! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % distinct_adj_Nil
% 0.26/0.46 thf(fact_196_distinct__adj__ConsD,axiom,
% 0.26/0.46 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.26/0.46 ( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.46 => ( distinct_adj @ A @ Xs ) ) ).
% 0.26/0.46
% 0.26/0.46 % distinct_adj_ConsD
% 0.26/0.46 thf(fact_197_butlast__tl,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.46 ( ( butlast @ A @ ( tl @ A @ Xs ) )
% 0.26/0.46 = ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % butlast_tl
% 0.26/0.46 thf(fact_198_tl__Nil,axiom,
% 0.26/0.46 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( ( tl @ A @ Xs )
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 = ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ? [X4: A] :
% 0.26/0.47 ( Xs
% 0.26/0.47 = ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % tl_Nil
% 0.26/0.47 thf(fact_199_Nil__tl,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( ( nil @ A )
% 0.26/0.47 = ( tl @ A @ Xs ) )
% 0.26/0.47 = ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ? [X4: A] :
% 0.26/0.47 ( Xs
% 0.26/0.47 = ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Nil_tl
% 0.26/0.47 thf(fact_200_list_Oexpand,axiom,
% 0.26/0.47 ! [A: $tType,List: list @ A,List2: list @ A] :
% 0.26/0.47 ( ( ( List
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 = ( List2
% 0.26/0.47 = ( nil @ A ) ) )
% 0.26/0.47 => ( ( ( List
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 => ( ( List2
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 => ( ( ( hd @ A @ List )
% 0.26/0.47 = ( hd @ A @ List2 ) )
% 0.26/0.47 & ( ( tl @ A @ List )
% 0.26/0.47 = ( tl @ A @ List2 ) ) ) ) )
% 0.26/0.47 => ( List = List2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list.expand
% 0.26/0.47 thf(fact_201_last__tl,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ( ( tl @ A @ Xs )
% 0.26/0.47 != ( nil @ A ) ) )
% 0.26/0.47 => ( ( last @ A @ ( tl @ A @ Xs ) )
% 0.26/0.47 = ( last @ A @ Xs ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % last_tl
% 0.26/0.47 thf(fact_202_successively_Osimps_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,P: A > A > $o,X3: A] : ( successively @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % successively.simps(2)
% 0.26/0.47 thf(fact_203_successively_Oelims_I1_J,axiom,
% 0.26/0.47 ! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
% 0.26/0.47 ( ( ( successively @ A @ X3 @ Xa )
% 0.26/0.47 = Y )
% 0.26/0.47 => ( ( ( Xa
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 => ~ Y )
% 0.26/0.47 => ( ( ? [X: A] :
% 0.26/0.47 ( Xa
% 0.26/0.47 = ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.47 => ~ Y )
% 0.26/0.47 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.47 ( ( Xa
% 0.26/0.47 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.26/0.47 => ( Y
% 0.26/0.47 = ( ~ ( ( X3 @ X @ Y3 )
% 0.26/0.47 & ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % successively.elims(1)
% 0.26/0.47 thf(fact_204_successively_Oelims_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,X3: A > A > $o,Xa: list @ A] :
% 0.26/0.47 ( ( successively @ A @ X3 @ Xa )
% 0.26/0.47 => ( ( Xa
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 => ( ! [X: A] :
% 0.26/0.47 ( Xa
% 0.26/0.47 != ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.47 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.47 ( ( Xa
% 0.26/0.47 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.26/0.47 => ~ ( ( X3 @ X @ Y3 )
% 0.26/0.47 & ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % successively.elims(2)
% 0.26/0.47 thf(fact_205_distinct__adj__singleton,axiom,
% 0.26/0.47 ! [B: $tType,X3: B] : ( distinct_adj @ B @ ( cons @ B @ X3 @ ( nil @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % distinct_adj_singleton
% 0.26/0.47 thf(fact_206_list_Oexhaust__sel,axiom,
% 0.26/0.47 ! [A: $tType,List: list @ A] :
% 0.26/0.47 ( ( List
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 => ( List
% 0.26/0.47 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list.exhaust_sel
% 0.26/0.47 thf(fact_207_successively__Cons,axiom,
% 0.26/0.47 ! [A: $tType,P: A > A > $o,X3: A,Xs: list @ A] :
% 0.26/0.47 ( ( successively @ A @ P @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.47 = ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ( ( P @ X3 @ ( hd @ A @ Xs ) )
% 0.26/0.47 & ( successively @ A @ P @ Xs ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % successively_Cons
% 0.26/0.47 thf(fact_208_distinct__adj__Cons,axiom,
% 0.26/0.47 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.26/0.47 ( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.26/0.47 = ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ( ( X3
% 0.26/0.47 != ( hd @ A @ Xs ) )
% 0.26/0.47 & ( distinct_adj @ A @ Xs ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % distinct_adj_Cons
% 0.26/0.47 thf(fact_209_list_Osplit__sel,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
% 0.26/0.47 ( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
% 0.26/0.47 = ( ( ( List
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 => ( P @ F1 ) )
% 0.26/0.47 & ( ( List
% 0.26/0.47 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
% 0.26/0.47 => ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list.split_sel
% 0.26/0.47 thf(fact_210_list_Osplit__sel__asm,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
% 0.26/0.47 ( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
% 0.26/0.47 = ( ~ ( ( ( List
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 & ~ ( P @ F1 ) )
% 0.26/0.47 | ( ( List
% 0.26/0.47 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
% 0.26/0.47 & ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list.split_sel_asm
% 0.26/0.47 thf(fact_211_list_Ocase__eq__if,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType] :
% 0.26/0.47 ( ( case_list @ B @ A )
% 0.26/0.47 = ( ^ [F12: B,F23: A > ( list @ A ) > B,List3: list @ A] :
% 0.26/0.47 ( if @ B
% 0.26/0.47 @ ( List3
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 @ F12
% 0.26/0.47 @ ( F23 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list.case_eq_if
% 0.26/0.47 thf(fact_212_list_Osimps_I5_J,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X22: list @ A] :
% 0.26/0.47 ( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X22 ) )
% 0.26/0.47 = ( F22 @ X21 @ X22 ) ) ).
% 0.26/0.47
% 0.26/0.47 % list.simps(5)
% 0.26/0.47 thf(fact_213_list_Osimps_I4_J,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] :
% 0.26/0.47 ( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) )
% 0.26/0.47 = F1 ) ).
% 0.26/0.47
% 0.26/0.47 % list.simps(4)
% 0.26/0.47 thf(fact_214_Cons__in__shuffles__iff,axiom,
% 0.26/0.47 ! [A: $tType,Z: A,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.47 ( ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.26/0.47 = ( ( ( Xs
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 & ( ( hd @ A @ Xs )
% 0.26/0.47 = Z )
% 0.26/0.47 & ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) )
% 0.26/0.47 | ( ( Ys2
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 & ( ( hd @ A @ Ys2 )
% 0.26/0.47 = Z )
% 0.26/0.47 & ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys2 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Cons_in_shuffles_iff
% 0.26/0.47 thf(fact_215_map2__Cons__not__empty,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,Xs: list @ A,F: A > C > B,Y: C,Ys2: list @ C] :
% 0.26/0.47 ( ( Xs
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 => ( ( map2 @ A @ C @ B @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
% 0.26/0.47 = ( cons @ B @ ( F @ ( hd @ A @ Xs ) @ Y ) @ ( map2 @ A @ C @ B @ F @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_Cons_not_empty
% 0.26/0.47 thf(fact_216_Nil__in__shuffles,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.47 ( ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.26/0.47 = ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 & ( Ys2
% 0.26/0.47 = ( nil @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Nil_in_shuffles
% 0.26/0.47 thf(fact_217_splice__in__shuffles,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A] : ( member2 @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % splice_in_shuffles
% 0.26/0.47 thf(fact_218_shufflesE,axiom,
% 0.26/0.47 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.47 ( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.26/0.47 => ( ( ( Zs2 = Xs )
% 0.26/0.47 => ( Ys2
% 0.26/0.47 != ( nil @ A ) ) )
% 0.26/0.47 => ( ( ( Zs2 = Ys2 )
% 0.26/0.47 => ( Xs
% 0.26/0.47 != ( nil @ A ) ) )
% 0.26/0.47 => ( ! [X: A,Xs5: list @ A] :
% 0.26/0.47 ( ( Xs
% 0.26/0.47 = ( cons @ A @ X @ Xs5 ) )
% 0.26/0.47 => ! [Z2: A,Zs3: list @ A] :
% 0.26/0.47 ( ( Zs2
% 0.26/0.47 = ( cons @ A @ Z2 @ Zs3 ) )
% 0.26/0.47 => ( ( X = Z2 )
% 0.26/0.47 => ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs5 @ Ys2 ) ) ) ) )
% 0.26/0.47 => ~ ! [Y3: A,Ys5: list @ A] :
% 0.26/0.47 ( ( Ys2
% 0.26/0.47 = ( cons @ A @ Y3 @ Ys5 ) )
% 0.26/0.47 => ! [Z2: A,Zs3: list @ A] :
% 0.26/0.47 ( ( Zs2
% 0.26/0.47 = ( cons @ A @ Z2 @ Zs3 ) )
% 0.26/0.47 => ( ( Y3 = Z2 )
% 0.26/0.47 => ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % shufflesE
% 0.26/0.47 thf(fact_219_map2__Cons__Cons,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,X3: B,Xs: list @ B,Y: C,Ys2: list @ C] :
% 0.26/0.47 ( ( map2 @ B @ C @ A @ F @ ( cons @ B @ X3 @ Xs ) @ ( cons @ C @ Y @ Ys2 ) )
% 0.26/0.47 = ( cons @ A @ ( F @ X3 @ Y ) @ ( map2 @ B @ C @ A @ F @ Xs @ Ys2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_Cons_Cons
% 0.26/0.47 thf(fact_220_map2_Osimps_I1_J,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,F: A > B > C,Ys2: list @ B] :
% 0.26/0.47 ( ( map2 @ A @ B @ C @ F @ ( nil @ A ) @ Ys2 )
% 0.26/0.47 = ( nil @ C ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2.simps(1)
% 0.26/0.47 thf(fact_221_map2__Nil,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: B > C > A,Ys2: list @ C] :
% 0.26/0.47 ( ( map2 @ B @ C @ A @ F @ ( nil @ B ) @ Ys2 )
% 0.26/0.47 = ( nil @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_Nil
% 0.26/0.47 thf(fact_222_map2__empty__conv,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType,F: B > C > A,Xs: list @ B,Ys2: list @ C] :
% 0.26/0.47 ( ( ( map2 @ B @ C @ A @ F @ Xs @ Ys2 )
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 = ( Xs
% 0.26/0.47 = ( nil @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_empty_conv
% 0.26/0.47 thf(fact_223_map2__not__empty__conv,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType,F: B > C > A,Xs: list @ B,Ys2: list @ C] :
% 0.26/0.47 ( ( ( map2 @ B @ C @ A @ F @ Xs @ Ys2 )
% 0.26/0.47 != ( nil @ A ) )
% 0.26/0.47 = ( Xs
% 0.26/0.47 != ( nil @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_not_empty_conv
% 0.26/0.47 thf(fact_224_Cons__in__shuffles__leftI,axiom,
% 0.26/0.47 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A,Z: A] :
% 0.26/0.47 ( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.26/0.47 => ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ ( cons @ A @ Z @ Xs ) @ Ys2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Cons_in_shuffles_leftI
% 0.26/0.47 thf(fact_225_Cons__in__shuffles__rightI,axiom,
% 0.26/0.47 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A,Z: A] :
% 0.26/0.47 ( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.26/0.47 => ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z @ Ys2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Cons_in_shuffles_rightI
% 0.26/0.47 thf(fact_226_Nil__in__shufflesI,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.47 ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 => ( ( Ys2
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 => ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Nil_in_shufflesI
% 0.26/0.47 thf(fact_227_shuffles__commutes,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( shuffles @ A )
% 0.26/0.47 = ( ^ [Xs3: list @ A,Ys: list @ A] : ( shuffles @ A @ Ys @ Xs3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % shuffles_commutes
% 0.26/0.47 thf(fact_228_map2_Osimps_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X3: A,Xs: list @ A,Ys2: list @ B] :
% 0.26/0.47 ( ( map2 @ A @ B @ C @ F @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
% 0.26/0.47 = ( cons @ C @ ( F @ X3 @ ( hd @ B @ Ys2 ) ) @ ( map2 @ A @ B @ C @ F @ Xs @ ( tl @ B @ Ys2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2.simps(2)
% 0.26/0.47 thf(fact_229_map2__Cons__if,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,Xs: list @ B,F: B > C > A,Y: C,Ys2: list @ C] :
% 0.26/0.47 ( ( ( Xs
% 0.26/0.47 = ( nil @ B ) )
% 0.26/0.47 => ( ( map2 @ B @ C @ A @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
% 0.26/0.47 = ( nil @ A ) ) )
% 0.26/0.47 & ( ( Xs
% 0.26/0.47 != ( nil @ B ) )
% 0.26/0.47 => ( ( map2 @ B @ C @ A @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
% 0.26/0.47 = ( cons @ A @ ( F @ ( hd @ B @ Xs ) @ Y ) @ ( map2 @ B @ C @ A @ F @ ( tl @ B @ Xs ) @ Ys2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_Cons_if
% 0.26/0.47 thf(fact_230_remdups__adj__append_H,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.47 ( ( ( Xs
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ( Ys2
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 | ( ( last @ A @ Xs )
% 0.26/0.47 != ( hd @ A @ Ys2 ) ) )
% 0.26/0.47 => ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.26/0.47 = ( append @ A @ ( remdups_adj @ A @ Xs ) @ ( remdups_adj @ A @ Ys2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj_append'
% 0.26/0.47 thf(fact_231_remdups__adj__append,axiom,
% 0.26/0.47 ! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] :
% 0.26/0.47 ( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) )
% 0.26/0.47 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj_append
% 0.26/0.47 thf(fact_232_remdups__adj__Nil__iff,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( ( remdups_adj @ A @ Xs )
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 = ( Xs
% 0.26/0.47 = ( nil @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj_Nil_iff
% 0.26/0.47 thf(fact_233_hd__remdups__adj,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
% 0.26/0.47 = ( hd @ A @ Xs ) ) ).
% 0.26/0.47
% 0.26/0.47 % hd_remdups_adj
% 0.26/0.47 thf(fact_234_last__remdups__adj,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( last @ A @ ( remdups_adj @ A @ Xs ) )
% 0.26/0.47 = ( last @ A @ Xs ) ) ).
% 0.26/0.47
% 0.26/0.47 % last_remdups_adj
% 0.26/0.47 thf(fact_235_remdups__adj__Cons__alt,axiom,
% 0.26/0.47 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.26/0.47 ( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
% 0.26/0.47 = ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj_Cons_alt
% 0.26/0.47 thf(fact_236_successively__remdups__adjI,axiom,
% 0.26/0.47 ! [A: $tType,P: A > A > $o,Xs: list @ A] :
% 0.26/0.47 ( ( successively @ A @ P @ Xs )
% 0.26/0.47 => ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % successively_remdups_adjI
% 0.26/0.47 thf(fact_237_distinct__adj__remdups__adj,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] : ( distinct_adj @ A @ ( remdups_adj @ A @ Xs ) ) ).
% 0.26/0.47
% 0.26/0.47 % distinct_adj_remdups_adj
% 0.26/0.47 thf(fact_238_distinct__adj__altdef,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( distinct_adj @ A )
% 0.26/0.47 = ( ^ [Xs3: list @ A] :
% 0.26/0.47 ( ( remdups_adj @ A @ Xs3 )
% 0.26/0.47 = Xs3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % distinct_adj_altdef
% 0.26/0.47 thf(fact_239_remdups__adj_Oelims,axiom,
% 0.26/0.47 ! [A: $tType,X3: list @ A,Y: list @ A] :
% 0.26/0.47 ( ( ( remdups_adj @ A @ X3 )
% 0.26/0.47 = Y )
% 0.26/0.47 => ( ( ( X3
% 0.26/0.47 = ( nil @ A ) )
% 0.26/0.47 => ( Y
% 0.26/0.47 != ( nil @ A ) ) )
% 0.26/0.47 => ( ! [X: A] :
% 0.26/0.47 ( ( X3
% 0.26/0.47 = ( cons @ A @ X @ ( nil @ A ) ) )
% 0.26/0.47 => ( Y
% 0.26/0.47 != ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.26/0.47 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.26/0.47 ( ( X3
% 0.26/0.47 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.26/0.47 => ~ ( ( ( X = Y3 )
% 0.26/0.47 => ( Y
% 0.26/0.47 = ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) ) ) )
% 0.26/0.47 & ( ( X != Y3 )
% 0.26/0.47 => ( Y
% 0.26/0.47 = ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj.elims
% 0.26/0.47 thf(fact_240_remdups__adj_Osimps_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,X3: A] :
% 0.26/0.47 ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.26/0.47 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj.simps(2)
% 0.26/0.47 thf(fact_241_remdups__adj_Osimps_I1_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( remdups_adj @ A @ ( nil @ A ) )
% 0.26/0.47 = ( nil @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj.simps(1)
% 0.26/0.47 thf(fact_242_remdups__adj_Osimps_I3_J,axiom,
% 0.26/0.47 ! [A: $tType,X3: A,Y: A,Xs: list @ A] :
% 0.26/0.47 ( ( ( X3 = Y )
% 0.26/0.47 => ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
% 0.26/0.47 = ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
% 0.26/0.47 & ( ( X3 != Y )
% 0.26/0.47 => ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
% 0.26/0.47 = ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj.simps(3)
% 0.26/0.47 thf(fact_243_remdups__adj__append__two,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,X3: A,Y: A] :
% 0.26/0.47 ( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
% 0.26/0.47 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % remdups_adj_append_two
% 0.26/0.47 thf(fact_244_map2__snoc__snoc,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,F: A > B > C,X3: A,Y: B] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.26/0.47 => ( ( map2 @ A @ B @ C @ F @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y @ ( nil @ B ) ) ) )
% 0.26/0.47 = ( append @ C @ ( map2 @ A @ B @ C @ F @ Xs @ Ys2 ) @ ( cons @ C @ ( F @ X3 @ Y ) @ ( nil @ C ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map2_snoc_snoc
% 0.26/0.47 thf(fact_245_list__strict__asc__imp__list__asc,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( preorder @ A )
% 0.26/0.47 => ! [Xs: list @ A] :
% 0.26/0.47 ( ( list_strict_asc @ A @ Xs )
% 0.26/0.47 => ( list_asc @ A @ Xs ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list_strict_asc_imp_list_asc
% 0.26/0.47 thf(fact_246_append__eq__append__conv,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Us2: list @ A,Vs2: list @ A] :
% 0.26/0.47 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.26/0.47 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.26/0.47 | ( ( size_size @ ( list @ A ) @ Us2 )
% 0.26/0.47 = ( size_size @ ( list @ A ) @ Vs2 ) ) )
% 0.26/0.47 => ( ( ( append @ A @ Xs @ Us2 )
% 0.26/0.47 = ( append @ A @ Ys2 @ Vs2 ) )
% 0.26/0.47 = ( ( Xs = Ys2 )
% 0.26/0.47 & ( Us2 = Vs2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % append_eq_append_conv
% 0.26/0.47 thf(fact_247_length__rotate1,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A] :
% 0.26/0.47 ( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
% 0.26/0.47 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.26/0.47
% 0.26/0.47 % length_rotate1
% 0.26/0.47 thf(fact_248_length__greater__imp__not__empty,axiom,
% 0.26/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.26/0.47 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.26/0.47 => ( Xs
% 0.26/0.47 != ( nil @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % length_greater_imp_not_empty
% 0.26/0.47 thf(fact_249_neq__if__length__neq,axiom,
% 0.26/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.26/0.47 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.26/0.47 => ( Xs != Ys2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % neq_if_length_neq
% 0.26/0.47 thf(fact_250_Ex__list__of__length,axiom,
% 0.26/0.47 ! [A: $tType,N: nat] :
% 0.26/0.47 ? [Xs2: list @ A] :
% 0.26/0.47 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.26/0.47 = N ) ).
% 0.26/0.47
% 0.26/0.47 % Ex_list_of_length
% 0.26/0.47 thf(fact_251_length__induct,axiom,
% 0.26/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.26/0.47 ( ! [Xs2: list @ A] :
% 0.26/0.47 ( ! [Ys6: list @ A] :
% 0.26/0.47 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.26/0.47 => ( P @ Ys6 ) )
% 0.26/0.47 => ( P @ Xs2 ) )
% 0.26/0.47 => ( P @ Xs ) ) ).
% 0.26/0.47
% 0.26/0.47 % length_induct
% 0.26/0.47 thf(fact_252_list__induct2,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.26/0.47 => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.26/0.47 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.26/0.47 => ( ( P @ Xs2 @ Ys3 )
% 0.26/0.47 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) )
% 0.26/0.47 => ( P @ Xs @ Ys2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list_induct2
% 0.26/0.47 thf(fact_253_list__induct3,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.26/0.47 => ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 0.26/0.47 = ( size_size @ ( list @ C ) @ Zs2 ) )
% 0.26/0.47 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
% 0.26/0.47 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z2: C,Zs: list @ C] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.26/0.47 => ( ( ( size_size @ ( list @ B ) @ Ys3 )
% 0.26/0.47 = ( size_size @ ( list @ C ) @ Zs ) )
% 0.26/0.47 => ( ( P @ Xs2 @ Ys3 @ Zs )
% 0.26/0.47 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C @ Z2 @ Zs ) ) ) ) )
% 0.26/0.47 => ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list_induct3
% 0.26/0.47 thf(fact_254_list__induct4,axiom,
% 0.26/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.26/0.47 => ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 0.26/0.47 = ( size_size @ ( list @ C ) @ Zs2 ) )
% 0.26/0.47 => ( ( ( size_size @ ( list @ C ) @ Zs2 )
% 0.26/0.47 = ( size_size @ ( list @ D ) @ Ws2 ) )
% 0.26/0.47 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
% 0.26/0.47 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z2: C,Zs: list @ C,W: D,Ws3: list @ D] :
% 0.26/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.26/0.47 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.26/0.47 => ( ( ( size_size @ ( list @ B ) @ Ys3 )
% 0.26/0.47 = ( size_size @ ( list @ C ) @ Zs ) )
% 0.26/0.47 => ( ( ( size_size @ ( list @ C ) @ Zs )
% 0.26/0.47 = ( size_size @ ( list @ D ) @ Ws3 ) )
% 0.26/0.47 => ( ( P @ Xs2 @ Ys3 @ Zs @ Ws3 )
% 0.26/0.47 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C @ Z2 @ Zs ) @ ( cons @ D @ W @ Ws3 ) ) ) ) ) )
% 0.26/0.47 => ( P @ Xs @ Ys2 @ Zs2 @ Ws2 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % list_induct4
% 0.26/0.47
% 0.26/0.47 % Type constructors (14)
% 0.26/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.26/0.47 ! [A7: $tType,A8: $tType] :
% 0.26/0.47 ( ( preorder @ A8 )
% 0.26/0.47 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.26/0.47 ! [A7: $tType,A8: $tType] :
% 0.26/0.47 ( ( order @ A8 )
% 0.26/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.26/0.47 ! [A7: $tType,A8: $tType] :
% 0.26/0.47 ( ( ord @ A8 )
% 0.26/0.47 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.26/0.47 preorder @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.26/0.47 linorder @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.26/0.47 order @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.26/0.47 ord @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.26/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.26/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.26/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.26/0.47 preorder @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.26/0.47 linorder @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.26/0.47 order @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.26/0.47 ord @ $o ).
% 0.26/0.47
% 0.26/0.47 % Helper facts (3)
% 0.26/0.47 thf(help_If_3_1_T,axiom,
% 0.26/0.47 ! [P: $o] :
% 0.26/0.47 ( ( P = $true )
% 0.26/0.47 | ( P = $false ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(help_If_2_1_T,axiom,
% 0.26/0.47 ! [A: $tType,X3: A,Y: A] :
% 0.26/0.47 ( ( if @ A @ $false @ X3 @ Y )
% 0.26/0.47 = Y ) ).
% 0.26/0.47
% 0.26/0.47 thf(help_If_1_1_T,axiom,
% 0.26/0.47 ! [A: $tType,X3: A,Y: A] :
% 0.26/0.47 ( ( if @ A @ $true @ X3 @ Y )
% 0.26/0.47 = X3 ) ).
% 0.26/0.47
% 0.26/0.47 % Conjectures (1)
% 0.26/0.47 thf(conj_0,conjecture,
% 0.26/0.47 ( ( listIn521021761append @ a @ ( cons @ a @ x @ xs ) @ f )
% 0.26/0.47 = ( listIn521021761append @ a @ ( cons @ a @ x @ ( nil @ a ) ) @ ( listIn521021761append @ a @ xs @ f ) ) ) ).
% 0.26/0.47
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 ------------------------
% 0.31/1.50 ---- Embedded in THF ---
% 0.31/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.50 %%% Generated on Tue Feb 24 08:25:54 EST 2026
% 0.31/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.50 %%% Logic specification used:
% 0.31/1.50 %%% thf(spec, logic, $$dhol).
% 0.31/1.50
% 0.31/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.nSD6Sk1FeO/DTF2DTF14040.p
% 0.31/1.50 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 0.31/1.50 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.31/1.50 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.31/1.50 thf(ty_tf_a, type, a: $tType).
% 0.31/1.50 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_c_BNF__Greatest__Fixpoint_OShift, type, bNF_Greatest_Shift: (!> [A:$tType]: (((set @ (list @ A)) > (A > (set @ (list @ A))))))).
% 0.31/1.50 thf(sy_c_BNF__Greatest__Fixpoint_OSucc, type, bNF_Greatest_Succ: (!> [A:$tType]: (((set @ (list @ A)) > ((list @ A) > (set @ A)))))).
% 0.31/1.50 thf(sy_c_BNF__Greatest__Fixpoint_Oshift, type, bNF_Greatest_shift: (!> [A:$tType,B:$tType]: ((((list @ A) > B) > (A > ((list @ A) > B)))))).
% 0.31/1.50 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 0.31/1.50 thf(sy_c_List2_Olist__asc, type, list_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 0.31/1.50 thf(sy_c_List2_Olist__ord, type, list_ord: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))).
% 0.31/1.50 thf(sy_c_List2_Olist__strict__asc, type, list_strict_asc: (!> [A:$tType]: (((list @ A) > $o)))).
% 0.31/1.50 thf(sy_c_List2_Omap2, type, map2: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > ((list @ A) > ((list @ B) > (list @ C))))))).
% 0.31/1.50 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append, type, listIn521021761append: (!> [A:$tType]: (((list @ A) > ((nat > A) > (nat > A)))))).
% 0.31/1.50 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 0.31/1.50 thf(sy_c_List_Oarg__min__list, type, arg_min_list: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > A))))).
% 0.31/1.50 thf(sy_c_List_Obind, type, bind: (!> [A:$tType,B:$tType]: (((list @ A) > ((A > (list @ B)) > (list @ B)))))).
% 0.31/1.50 thf(sy_c_List_Obutlast, type, butlast: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.31/1.50 thf(sy_c_List_Oconcat, type, concat: (!> [A:$tType]: (((list @ (list @ A)) > (list @ A))))).
% 0.31/1.50 thf(sy_c_List_Odistinct__adj, type, distinct_adj: (!> [A:$tType]: (((list @ A) > $o)))).
% 0.31/1.50 thf(sy_c_List_Ofoldr, type, foldr: (!> [A:$tType,B:$tType]: (((A > (B > B)) > ((list @ A) > (B > B)))))).
% 0.31/1.50 thf(sy_c_List_Oinsert, type, insert: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 0.31/1.50 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 0.31/1.50 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 0.31/1.50 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 0.31/1.50 thf(sy_c_List_Olist_Ocase__list, type, case_list: (!> [B:$tType,A:$tType]: ((B > ((A > ((list @ A) > B)) > ((list @ A) > B)))))).
% 0.31/1.50 thf(sy_c_List_Olist_Ohd, type, hd: (!> [A:$tType]: (((list @ A) > A)))).
% 0.31/1.50 thf(sy_c_List_Olist_Otl, type, tl: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.31/1.50 thf(sy_c_List_Olist__ex1, type, list_ex1: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 0.31/1.50 thf(sy_c_List_Olistrelp, type, listrelp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((list @ A) > ((list @ B) > $o)))))).
% 0.31/1.50 thf(sy_c_List_Omap__tailrec__rev, type, map_tailrec_rev: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > ((list @ B) > (list @ B))))))).
% 0.31/1.50 thf(sy_c_List_Omaps, type, maps: (!> [A:$tType,B:$tType]: (((A > (list @ B)) > ((list @ A) > (list @ B)))))).
% 0.31/1.50 thf(sy_c_List_Omember, type, member: (!> [A:$tType]: (((list @ A) > (A > $o))))).
% 0.31/1.50 thf(sy_c_List_Oord_Olexordp__eq, type, lexordp_eq: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))).
% 0.31/1.50 thf(sy_c_List_Oord__class_Olexordp, type, ord_lexordp: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 0.31/1.50 thf(sy_c_List_Oord__class_Olexordp__eq, type, ord_lexordp_eq: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 0.31/1.50 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 0.31/1.50 thf(sy_c_List_Oremdups__adj, type, remdups_adj: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.31/1.50 thf(sy_c_List_Orotate1, type, rotate1: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.31/1.50 thf(sy_c_List_Oshuffles, type, shuffles: (!> [A:$tType]: (((list @ A) > ((list @ A) > (set @ (list @ A))))))).
% 0.31/1.50 thf(sy_c_List_Osplice, type, splice: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 0.31/1.50 thf(sy_c_List_Osubseqs, type, subseqs: (!> [A:$tType]: (((list @ A) > (list @ (list @ A)))))).
% 0.31/1.50 thf(sy_c_List_Osuccessively, type, successively: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))).
% 0.31/1.50 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 0.31/1.50 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.31/1.50 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.31/1.50 thf(sy_c_member, type, member2: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_v_f, type, f: (nat > a)).
% 0.31/1.50 thf(sy_v_x, type, x: a).
% 0.31/1.50 thf(sy_v_xs, type, xs: (list @ a)).
% 0.31/1.50 thf(fact_0_i__append__Nil, axiom, (! [A:$tType,F:(nat > A)]: (((((listIn521021761append @ A) @ (nil @ A)) @ F) = F)))).
% 0.31/1.50 thf(fact_1_list_Oinject, axiom, (! [A:$tType,X21:A,X22:(list @ A),Y21:A,Y22:(list @ A)]: ((((((cons @ A) @ X21) @ X22) = (((cons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 0.31/1.50 thf(fact_2_list_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(list @ A)]: (((nil @ A) != (((cons @ A) @ X21) @ X22))))).
% 0.31/1.50 thf(fact_3_list_OdiscI, axiom, (! [A:$tType,List:(list @ A),X21:A,X22:(list @ A)]: (((List = (((cons @ A) @ X21) @ X22)) => (List != (nil @ A)))))).
% 0.31/1.50 thf(fact_4_list_Oexhaust, axiom, (! [A:$tType,Y:(list @ A)]: (((Y != (nil @ A)) => (~ ((! [X212:A,X222:(list @ A)]: ((Y != (((cons @ A) @ X212) @ X222)))))))))).
% 0.31/1.50 thf(fact_5_list_Oinducts, axiom, (! [A:$tType,P:((list @ A) > $o),List:(list @ A)]: (((P @ (nil @ A)) => ((! [X1:A,X2:(list @ A)]: (((P @ X2) => (P @ (((cons @ A) @ X1) @ X2))))) => (P @ List)))))).
% 0.31/1.50 thf(fact_6_neq__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) = (? [Y2:A,Ys:(list @ A)]: ((Xs = (((cons @ A) @ Y2) @ Ys)))))))).
% 0.31/1.50 thf(fact_7_list__induct2_H, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),Xs:(list @ A),Ys2:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [X:A,Xs2:(list @ A)]: (((P @ (((cons @ A) @ X) @ Xs2)) @ (nil @ B)))) => ((! [Y3:B,Ys3:(list @ B)]: (((P @ (nil @ A)) @ (((cons @ B) @ Y3) @ Ys3)))) => ((! [X:A,Xs2:(list @ A),Y3:B,Ys3:(list @ B)]: ((((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ B) @ Y3) @ Ys3))))) => ((P @ Xs) @ Ys2)))))))).
% 0.31/1.50 thf(fact_8_splice_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [X:A,Xs2:(list @ A),Ys3:(list @ A)]: ((((P @ Ys3) @ Xs2) => ((P @ (((cons @ A) @ X) @ Xs2)) @ Ys3)))) => ((P @ A0) @ A1)))))).
% 0.31/1.50 thf(fact_9_induct__list012, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X:A]: ((P @ (((cons @ A) @ X) @ (nil @ A))))) => ((! [X:A,Y3:A,Zs:(list @ A)]: (((P @ Zs) => ((P @ (((cons @ A) @ Y3) @ Zs)) => (P @ (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Zs))))))) => (P @ Xs))))))).
% 0.31/1.50 thf(fact_10_min__list_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(list @ A)]: (((! [X:A,Xs2:(list @ A)]: ((X3 != (((cons @ A) @ X) @ Xs2)))) => (X3 = (nil @ A))))))))).
% 0.31/1.50 thf(fact_11_min__list_Oinduct, axiom, (! [A:$tType]: (((ord @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((! [X:A,Xs2:(list @ A)]: (((! [X213:A,X223:(list @ A)]: (((Xs2 = (((cons @ A) @ X213) @ X223)) => (P @ Xs2)))) => (P @ (((cons @ A) @ X) @ Xs2))))) => ((P @ (nil @ A)) => (P @ A0))))))))).
% 0.31/1.50 thf(fact_12_shuffles_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [Xs2:(list @ A)]: (((P @ Xs2) @ (nil @ A)))) => ((! [X:A,Xs2:(list @ A),Y3:A,Ys3:(list @ A)]: ((((P @ Xs2) @ (((cons @ A) @ Y3) @ Ys3)) => (((P @ (((cons @ A) @ X) @ Xs2)) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3)))))) => ((P @ A0) @ A1))))))).
% 0.31/1.50 thf(fact_13_not__Cons__self2, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((cons @ A) @ X3) @ Xs) != Xs)))).
% 0.31/1.50 thf(fact_14_strict__sorted_Oinduct, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X:A,Ys3:(list @ A)]: (((P @ Ys3) => (P @ (((cons @ A) @ X) @ Ys3))))) => (P @ A0))))))))).
% 0.31/1.50 thf(fact_15_strict__sorted_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:(list @ A)]: (((X3 != (nil @ A)) => (~ ((! [X:A,Ys3:(list @ A)]: ((X3 != (((cons @ A) @ X) @ Ys3))))))))))))).
% 0.31/1.50 thf(fact_16_map__tailrec__rev_Oinduct, axiom, (! [A:$tType,B:$tType,P:((A > B) > ((list @ A) > ((list @ B) > $o))),A0:(A > B),A1:(list @ A),A2:(list @ B)]: (((! [F2:(A > B),X_1:(list @ B)]: ((((P @ F2) @ (nil @ A)) @ X_1))) => ((! [F2:(A > B),A3:A,As:(list @ A),Bs:(list @ B)]: (((((P @ F2) @ As) @ (((cons @ B) @ (F2 @ A3)) @ Bs)) => (((P @ F2) @ (((cons @ A) @ A3) @ As)) @ Bs)))) => (((P @ A0) @ A1) @ A2)))))).
% 0.31/1.50 thf(fact_17_list__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X:A]: ((P @ (((cons @ A) @ X) @ (nil @ A))))) => ((! [X:A,Xs2:(list @ A)]: (((Xs2 != (nil @ A)) => ((P @ Xs2) => (P @ (((cons @ A) @ X) @ Xs2)))))) => (P @ Xs))))))).
% 0.31/1.50 thf(fact_18_successively_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [P2:(A > (A > $o))]: (((P @ P2) @ (nil @ A)))) => ((! [P2:(A > (A > $o)),X:A]: (((P @ P2) @ (((cons @ A) @ X) @ (nil @ A))))) => ((! [P2:(A > (A > $o)),X:A,Y3:A,Xs2:(list @ A)]: ((((P @ P2) @ (((cons @ A) @ Y3) @ Xs2)) => ((P @ P2) @ (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2)))))) => ((P @ A0) @ A1))))))).
% 0.31/1.50 thf(fact_19_arg__min__list_Oinduct, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [P:((A > B) > ((list @ A) > $o)),A0:(A > B),A1:(list @ A)]: (((! [F2:(A > B),X:A]: (((P @ F2) @ (((cons @ A) @ X) @ (nil @ A))))) => ((! [F2:(A > B),X:A,Y3:A,Zs:(list @ A)]: ((((P @ F2) @ (((cons @ A) @ Y3) @ Zs)) => ((P @ F2) @ (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Zs)))))) => ((! [A3:(A > B)]: (((P @ A3) @ (nil @ A)))) => ((P @ A0) @ A1)))))))))).
% 0.31/1.50 thf(fact_20_remdups__adj_Oinduct, axiom, (! [A:$tType,P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X:A]: ((P @ (((cons @ A) @ X) @ (nil @ A))))) => ((! [X:A,Y3:A,Xs2:(list @ A)]: ((((X = Y3) => (P @ (((cons @ A) @ X) @ Xs2))) => (((X != Y3) => (P @ (((cons @ A) @ Y3) @ Xs2))) => (P @ (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2))))))) => (P @ A0))))))).
% 0.31/1.50 thf(fact_21_sorted__wrt_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [P2:(A > (A > $o))]: (((P @ P2) @ (nil @ A)))) => ((! [P2:(A > (A > $o)),X:A,Ys3:(list @ A)]: ((((P @ P2) @ Ys3) => ((P @ P2) @ (((cons @ A) @ X) @ Ys3))))) => ((P @ A0) @ A1)))))).
% 0.31/1.50 thf(fact_22_remdups__adj_Ocases, axiom, (! [A:$tType,X3:(list @ A)]: (((X3 != (nil @ A)) => ((! [X:A]: ((X3 != (((cons @ A) @ X) @ (nil @ A))))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: ((X3 != (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2)))))))))))).
% 0.31/1.50 thf(fact_23_transpose_Ocases, axiom, (! [A:$tType,X3:(list @ (list @ A))]: (((X3 != (nil @ (list @ A))) => ((! [Xss:(list @ (list @ A))]: ((X3 != (((cons @ (list @ A)) @ (nil @ A)) @ Xss)))) => (~ ((! [X:A,Xs2:(list @ A),Xss:(list @ (list @ A))]: ((X3 != (((cons @ (list @ A)) @ (((cons @ A) @ X) @ Xs2)) @ Xss))))))))))).
% 0.31/1.50 thf(fact_24_list__ord_Oinduct, axiom, (! [A:$tType]: (((ord @ A) => (! [P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [Ord:(A > (A > $o)),X1:A,X2:A,Xs2:(list @ A)]: ((((P @ Ord) @ (((cons @ A) @ X2) @ Xs2)) => ((P @ Ord) @ (((cons @ A) @ X1) @ (((cons @ A) @ X2) @ Xs2)))))) => ((! [Ord:(A > (A > $o))]: (((P @ Ord) @ (nil @ A)))) => ((! [Ord:(A > (A > $o)),V:A]: (((P @ Ord) @ (((cons @ A) @ V) @ (nil @ A))))) => ((P @ A0) @ A1)))))))))).
% 0.31/1.50 thf(fact_25_insert__Nil, axiom, (! [A:$tType,X3:A]: (((((insert @ A) @ X3) @ (nil @ A)) = (((cons @ A) @ X3) @ (nil @ A)))))).
% 0.31/1.50 thf(fact_26_list__ex1__simps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: ((~ ((((list_ex1 @ A) @ P) @ (nil @ A))))))).
% 0.31/1.50 thf(fact_27_map__tailrec__rev_Oelims, axiom, (! [A:$tType,B:$tType,X3:(A > B),Xa:(list @ A),Xb:(list @ B),Y:(list @ B)]: ((((((((map_tailrec_rev @ A) @ B) @ X3) @ Xa) @ Xb) = Y) => (((Xa = (nil @ A)) => (Y != Xb)) => (~ ((! [A3:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A3) @ As)) => (Y != (((((map_tailrec_rev @ A) @ B) @ X3) @ As) @ (((cons @ B) @ (X3 @ A3)) @ Xb))))))))))))).
% 0.31/1.50 thf(fact_28_arg__min__list_Osimps_I1_J, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [F:(A > B),X3:A]: ((((((arg_min_list @ A) @ B) @ F) @ (((cons @ A) @ X3) @ (nil @ A))) = X3))))))).
% 0.31/1.50 thf(fact_29_ord_Olexordp__eq__simps_I3_J, axiom, (! [A:$tType,Less:(A > (A > $o)),X3:A,Xs:(list @ A)]: ((~ (((((lexordp_eq @ A) @ Less) @ (((cons @ A) @ X3) @ Xs)) @ (nil @ A))))))).
% 0.31/1.50 thf(fact_30_bind__simps_I1_J, axiom, (! [B:$tType,A:$tType,F:(B > (list @ A))]: ((((((bind @ B) @ A) @ (nil @ B)) @ F) = (nil @ A))))).
% 0.31/1.50 thf(fact_31_listrelp_Oinducts, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X12:(list @ A),X23:(list @ B),P:((list @ A) > ((list @ B) > $o))]: (((((((listrelp @ A) @ B) @ R) @ X12) @ X23) => (((P @ (nil @ A)) @ (nil @ B)) => ((! [X:A,Y3:B,Xs2:(list @ A),Ys3:(list @ B)]: ((((R @ X) @ Y3) => ((((((listrelp @ A) @ B) @ R) @ Xs2) @ Ys3) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ B) @ Y3) @ Ys3))))))) => ((P @ X12) @ X23))))))).
% 0.31/1.50 thf(fact_32_listrelp_Osimps, axiom, (! [B:$tType,A:$tType]: ((((listrelp @ A) @ B) = (^ [R2:(A > (B > $o)),A12:(list @ A),A22:(list @ B)]: ((((A12 = (nil @ A)) & (A22 = (nil @ B))) | (? [X4:A,Y2:B,Xs3:(list @ A),Ys:(list @ B)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ B) @ Y2) @ Ys)) & (((R2 @ X4) @ Y2) & (((((listrelp @ A) @ B) @ R2) @ Xs3) @ Ys))))))))))))).
% 0.31/1.50 thf(fact_33_listrelp_Ocases, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),A1:(list @ A),A2:(list @ B)]: (((((((listrelp @ A) @ B) @ R) @ A1) @ A2) => (((A1 = (nil @ A)) => (A2 != (nil @ B))) => (~ ((! [X:A,Y3:B,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X) @ Xs2)) => (! [Ys3:(list @ B)]: (((A2 = (((cons @ B) @ Y3) @ Ys3)) => (((R @ X) @ Y3) => (~ ((((((listrelp @ A) @ B) @ R) @ Xs2) @ Ys3))))))))))))))))).
% 0.31/1.50 thf(fact_34_lexordp__eq__simps_I3_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Xs:(list @ A)]: ((~ ((((ord_lexordp_eq @ A) @ (((cons @ A) @ X3) @ Xs)) @ (nil @ A)))))))))).
% 0.31/1.50 thf(fact_35_lexordp__eq__simps_I2_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A)]: (((((ord_lexordp_eq @ A) @ Xs) @ (nil @ A)) = (Xs = (nil @ A))))))))).
% 0.31/1.50 thf(fact_36_lexordp__eq__simps_I1_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Ys2:(list @ A)]: ((((ord_lexordp_eq @ A) @ (nil @ A)) @ Ys2))))))).
% 0.31/1.50 thf(fact_37_ord_Olexordp__eq__simps_I4_J, axiom, (! [A:$tType,Less:(A > (A > $o)),X3:A,Xs:(list @ A),Y:A,Ys2:(list @ A)]: ((((((lexordp_eq @ A) @ Less) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)) = (((Less @ X3) @ Y) | ((~ (((Less @ Y) @ X3))) & ((((lexordp_eq @ A) @ Less) @ Xs) @ Ys2))))))).
% 0.31/1.50 thf(fact_38_ord_Olexordp__eq__simps_I2_J, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(list @ A)]: ((((((lexordp_eq @ A) @ Less) @ Xs) @ (nil @ A)) = (Xs = (nil @ A)))))).
% 0.31/1.50 thf(fact_39_ord_Olexordp__eq__simps_I1_J, axiom, (! [A:$tType,Less:(A > (A > $o)),Ys2:(list @ A)]: (((((lexordp_eq @ A) @ Less) @ (nil @ A)) @ Ys2)))).
% 0.31/1.50 thf(fact_40_ord_Olexordp__eq_Ocong, axiom, (! [A:$tType]: (((lexordp_eq @ A) = (lexordp_eq @ A))))).
% 0.31/1.50 thf(fact_41_ord_Olexordp__eq__refl, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(list @ A)]: (((((lexordp_eq @ A) @ Less) @ Xs) @ Xs)))).
% 0.31/1.50 thf(fact_42_lexordp__eq__refl, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A)]: ((((ord_lexordp_eq @ A) @ Xs) @ Xs))))))).
% 0.31/1.50 thf(fact_43_lexordp__eq__trans, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A),Zs2:(list @ A)]: (((((ord_lexordp_eq @ A) @ Xs) @ Ys2) => ((((ord_lexordp_eq @ A) @ Ys2) @ Zs2) => (((ord_lexordp_eq @ A) @ Xs) @ Zs2))))))))).
% 0.31/1.50 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member2 @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 0.31/1.50 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member2 @ A) @ X4) @ A5)))) = A5)))).
% 0.31/1.50 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.31/1.50 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 0.31/1.50 thf(fact_48_lexordp__eq__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp_eq @ A) @ Xs) @ Ys2) | (((ord_lexordp_eq @ A) @ Ys2) @ Xs)))))))).
% 0.31/1.50 thf(fact_49_lexordp__eq__antisym, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp_eq @ A) @ Xs) @ Ys2) => ((((ord_lexordp_eq @ A) @ Ys2) @ Xs) => (Xs = Ys2))))))))).
% 0.31/1.50 thf(fact_50_lexordp__eq_ONil, axiom, (! [A:$tType]: (((ord @ A) => (! [Ys2:(list @ A)]: ((((ord_lexordp_eq @ A) @ (nil @ A)) @ Ys2))))))).
% 0.31/1.50 thf(fact_51_ord_Olexordp__eq_OCons, axiom, (! [A:$tType,Less:(A > (A > $o)),X3:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: ((((Less @ X3) @ Y) => ((((lexordp_eq @ A) @ Less) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)))))).
% 0.31/1.50 thf(fact_52_ord_Olexordp__eq_OCons__eq, axiom, (! [A:$tType,Less:(A > (A > $o)),X3:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: (((~ (((Less @ X3) @ Y))) => ((~ (((Less @ Y) @ X3))) => (((((lexordp_eq @ A) @ Less) @ Xs) @ Ys2) => ((((lexordp_eq @ A) @ Less) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)))))))).
% 0.31/1.50 thf(fact_53_ord_Olexordp__eq_ONil, axiom, (! [A:$tType,Less:(A > (A > $o)),Ys2:(list @ A)]: (((((lexordp_eq @ A) @ Less) @ (nil @ A)) @ Ys2)))).
% 0.31/1.50 thf(fact_54_listrelp_OCons, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X3:A,Y:B,Xs:(list @ A),Ys2:(list @ B)]: ((((R @ X3) @ Y) => ((((((listrelp @ A) @ B) @ R) @ Xs) @ Ys2) => (((((listrelp @ A) @ B) @ R) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ B) @ Y) @ Ys2))))))).
% 0.31/1.50 thf(fact_55_listrelp_ONil, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: ((((((listrelp @ A) @ B) @ R) @ (nil @ A)) @ (nil @ B))))).
% 0.31/1.50 thf(fact_56_map__tailrec__rev_Osimps_I2_J, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:A,As2:(list @ A),Bs2:(list @ B)]: (((((((map_tailrec_rev @ A) @ B) @ F) @ (((cons @ A) @ A4) @ As2)) @ Bs2) = (((((map_tailrec_rev @ A) @ B) @ F) @ As2) @ (((cons @ B) @ (F @ A4)) @ Bs2)))))).
% 0.31/1.50 thf(fact_57_map__tailrec__rev_Osimps_I1_J, axiom, (! [A:$tType,B:$tType,F:(A > B),Bs2:(list @ B)]: (((((((map_tailrec_rev @ A) @ B) @ F) @ (nil @ A)) @ Bs2) = Bs2)))).
% 0.31/1.50 thf(fact_58_ord_Olexordp__eq_Ocases, axiom, (! [A:$tType,Less:(A > (A > $o)),A1:(list @ A),A2:(list @ A)]: ((((((lexordp_eq @ A) @ Less) @ A1) @ A2) => ((A1 != (nil @ A)) => ((! [X:A]: (((? [Xs2:(list @ A)]: ((A1 = (((cons @ A) @ X) @ Xs2)))) => (! [Y3:A]: (((? [Ys3:(list @ A)]: ((A2 = (((cons @ A) @ Y3) @ Ys3)))) => (~ (((Less @ X) @ Y3))))))))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X) @ Xs2)) => (! [Ys3:(list @ A)]: (((A2 = (((cons @ A) @ Y3) @ Ys3)) => ((~ (((Less @ X) @ Y3))) => ((~ (((Less @ Y3) @ X))) => (~ (((((lexordp_eq @ A) @ Less) @ Xs2) @ Ys3))))))))))))))))))).
% 0.31/1.50 thf(fact_59_ord_Olexordp__eq_Osimps, axiom, (! [A:$tType]: (((lexordp_eq @ A) = (^ [Less2:(A > (A > $o)),A12:(list @ A),A22:(list @ A)]: (((? [Ys:(list @ A)]: (((A12 = (nil @ A)) & (A22 = Ys)))) | ((? [X4:A,Y2:A,Xs3:(list @ A),Ys:(list @ A)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ A) @ Y2) @ Ys)) & ((Less2 @ X4) @ Y2))))) | (? [X4:A,Y2:A,Xs3:(list @ A),Ys:(list @ A)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ A) @ Y2) @ Ys)) & ((~ (((Less2 @ X4) @ Y2))) & ((~ (((Less2 @ Y2) @ X4))) & ((((lexordp_eq @ A) @ Less2) @ Xs3) @ Ys))))))))))))))).
% 0.31/1.50 thf(fact_60_ord_Olexordp__eq_Oinducts, axiom, (! [A:$tType,Less:(A > (A > $o)),X12:(list @ A),X23:(list @ A),P:((list @ A) > ((list @ A) > $o))]: ((((((lexordp_eq @ A) @ Less) @ X12) @ X23) => ((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [X:A,Y3:A,Xs2:(list @ A),Ys3:(list @ A)]: ((((Less @ X) @ Y3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3))))) => ((! [X:A,Y3:A,Xs2:(list @ A),Ys3:(list @ A)]: (((~ (((Less @ X) @ Y3))) => ((~ (((Less @ Y3) @ X))) => (((((lexordp_eq @ A) @ Less) @ Xs2) @ Ys3) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3)))))))) => ((P @ X12) @ X23)))))))).
% 0.31/1.50 thf(fact_61_product__lists_Osimps_I1_J, axiom, (! [A:$tType]: ((((product_lists @ A) @ (nil @ (list @ A))) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 0.31/1.50 thf(fact_62_subseqs_Osimps_I1_J, axiom, (! [A:$tType]: ((((subseqs @ A) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 0.31/1.50 thf(fact_63_lexordp__eq_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [A1:(list @ A),A2:(list @ A)]: (((((ord_lexordp_eq @ A) @ A1) @ A2) => ((A1 != (nil @ A)) => ((! [X:A]: (((? [Xs2:(list @ A)]: ((A1 = (((cons @ A) @ X) @ Xs2)))) => (! [Y3:A]: (((? [Ys3:(list @ A)]: ((A2 = (((cons @ A) @ Y3) @ Ys3)))) => (~ ((((ord_less @ A) @ X) @ Y3))))))))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X) @ Xs2)) => (! [Ys3:(list @ A)]: (((A2 = (((cons @ A) @ Y3) @ Ys3)) => ((~ ((((ord_less @ A) @ X) @ Y3))) => ((~ ((((ord_less @ A) @ Y3) @ X))) => (~ ((((ord_lexordp_eq @ A) @ Xs2) @ Ys3)))))))))))))))))))))).
% 0.31/1.50 thf(fact_64_lexordp__eq_Osimps, axiom, (! [A:$tType]: (((ord @ A) => ((ord_lexordp_eq @ A) = (^ [A12:(list @ A),A22:(list @ A)]: (((? [Ys:(list @ A)]: (((A12 = (nil @ A)) & (A22 = Ys)))) | ((? [X4:A,Y2:A,Xs3:(list @ A),Ys:(list @ A)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ A) @ Y2) @ Ys)) & (((ord_less @ A) @ X4) @ Y2))))) | (? [X4:A,Y2:A,Xs3:(list @ A),Ys:(list @ A)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ A) @ Y2) @ Ys)) & ((~ ((((ord_less @ A) @ X4) @ Y2))) & ((~ ((((ord_less @ A) @ Y2) @ X4))) & (((ord_lexordp_eq @ A) @ Xs3) @ Ys)))))))))))))))).
% 0.31/1.50 thf(fact_65_lexordp__eq_Oinducts, axiom, (! [A:$tType]: (((ord @ A) => (! [X12:(list @ A),X23:(list @ A),P:((list @ A) > ((list @ A) > $o))]: (((((ord_lexordp_eq @ A) @ X12) @ X23) => ((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [X:A,Y3:A,Xs2:(list @ A),Ys3:(list @ A)]: (((((ord_less @ A) @ X) @ Y3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3))))) => ((! [X:A,Y3:A,Xs2:(list @ A),Ys3:(list @ A)]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((~ ((((ord_less @ A) @ Y3) @ X))) => ((((ord_lexordp_eq @ A) @ Xs2) @ Ys3) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3)))))))) => ((P @ X12) @ X23))))))))))).
% 0.31/1.50 thf(fact_66_member__rec_I2_J, axiom, (! [A:$tType,Y:A]: ((~ ((((member @ A) @ (nil @ A)) @ Y)))))).
% 0.31/1.50 thf(fact_67_member__rec_I1_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:A]: (((((member @ A) @ (((cons @ A) @ X3) @ Xs)) @ Y) = ((X3 = Y) | (((member @ A) @ Xs) @ Y)))))).
% 0.31/1.50 thf(fact_68_splice_Oelims, axiom, (! [A:$tType,X3:(list @ A),Xa:(list @ A),Y:(list @ A)]: ((((((splice @ A) @ X3) @ Xa) = Y) => (((X3 = (nil @ A)) => (Y != Xa)) => (~ ((! [X:A,Xs2:(list @ A)]: (((X3 = (((cons @ A) @ X) @ Xs2)) => (Y != (((cons @ A) @ X) @ (((splice @ A) @ Xa) @ Xs2))))))))))))).
% 0.31/1.50 thf(fact_69_bind__simps_I2_J, axiom, (! [A:$tType,B:$tType,X3:B,Xs:(list @ B),F:(B > (list @ A))]: ((((((bind @ B) @ A) @ (((cons @ B) @ X3) @ Xs)) @ F) = (((append @ A) @ (F @ X3)) @ ((((bind @ B) @ A) @ Xs) @ F)))))).
% 0.31/1.50 thf(fact_70_same__append__eq, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Xs) @ Zs2)) = (Ys2 = Zs2))))).
% 0.31/1.50 thf(fact_71_append__same__eq, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Ys2) @ Xs) = (((append @ A) @ Zs2) @ Xs)) = (Ys2 = Zs2))))).
% 0.31/1.50 thf(fact_72_append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs2:(list @ A)]: (((((append @ A) @ (((append @ A) @ Xs) @ Ys2)) @ Zs2) = (((append @ A) @ Xs) @ (((append @ A) @ Ys2) @ Zs2)))))).
% 0.31/1.50 thf(fact_73_append_Oassoc, axiom, (! [A:$tType,A4:(list @ A),B2:(list @ A),C2:(list @ A)]: (((((append @ A) @ (((append @ A) @ A4) @ B2)) @ C2) = (((append @ A) @ A4) @ (((append @ A) @ B2) @ C2)))))).
% 0.31/1.50 thf(fact_74_append_Oright__neutral, axiom, (! [A:$tType,A4:(list @ A)]: (((((append @ A) @ A4) @ (nil @ A)) = A4)))).
% 0.31/1.50 thf(fact_75_append__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = (nil @ A)) = ((Xs = (nil @ A)) & (Ys2 = (nil @ A))))))).
% 0.31/1.50 thf(fact_76_Nil__is__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((nil @ A) = (((append @ A) @ Xs) @ Ys2)) = ((Xs = (nil @ A)) & (Ys2 = (nil @ A))))))).
% 0.31/1.50 thf(fact_77_self__append__conv2, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 = (((append @ A) @ Xs) @ Ys2)) = (Xs = (nil @ A)))))).
% 0.31/1.50 thf(fact_78_append__self__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = Ys2) = (Xs = (nil @ A)))))).
% 0.31/1.50 thf(fact_79_self__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs = (((append @ A) @ Xs) @ Ys2)) = (Ys2 = (nil @ A)))))).
% 0.31/1.50 thf(fact_80_append__self__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = Xs) = (Ys2 = (nil @ A)))))).
% 0.31/1.50 thf(fact_81_append__Nil2, axiom, (! [A:$tType,Xs:(list @ A)]: (((((append @ A) @ Xs) @ (nil @ A)) = Xs)))).
% 0.31/1.50 thf(fact_82_split__Nil__iff, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((splice @ A) @ Xs) @ Ys2) = (nil @ A)) = ((Xs = (nil @ A)) & (Ys2 = (nil @ A))))))).
% 0.31/1.50 thf(fact_83_splice__Nil2, axiom, (! [A:$tType,Xs:(list @ A)]: (((((splice @ A) @ Xs) @ (nil @ A)) = Xs)))).
% 0.31/1.50 thf(fact_84_i__append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),F:(nat > A)]: (((((listIn521021761append @ A) @ Xs) @ (((listIn521021761append @ A) @ Ys2) @ F)) = (((listIn521021761append @ A) @ (((append @ A) @ Xs) @ Ys2)) @ F))))).
% 0.31/1.50 thf(fact_85_append1__eq__conv, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys2:(list @ A),Y:A]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))) = (((append @ A) @ Ys2) @ (((cons @ A) @ Y) @ (nil @ A)))) = ((Xs = Ys2) & (X3 = Y)))))).
% 0.31/1.50 thf(fact_86_lexordp__eq__simps_I4_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Xs:(list @ A),Y:A,Ys2:(list @ A)]: (((((ord_lexordp_eq @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)) = ((((ord_less @ A) @ X3) @ Y) | ((~ ((((ord_less @ A) @ Y) @ X3))) & (((ord_lexordp_eq @ A) @ Xs) @ Ys2)))))))))).
% 0.31/1.50 thf(fact_87_append__eq__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs2:(list @ A),Ts:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Zs2) @ Ts)) = (? [Us:(list @ A)]: ((((Xs = (((append @ A) @ Zs2) @ Us)) & ((((append @ A) @ Us) @ Ys2) = Ts)) | (((((append @ A) @ Xs) @ Us) = Zs2) & (Ys2 = (((append @ A) @ Us) @ Ts)))))))))).
% 0.31/1.50 thf(fact_88_append__eq__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs1:(list @ A),Zs2:(list @ A),Ys2:(list @ A),Us2:(list @ A)]: ((((((append @ A) @ Xs) @ Xs1) = Zs2) => ((Ys2 = (((append @ A) @ Xs1) @ Us2)) => ((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Zs2) @ Us2))))))).
% 0.31/1.50 thf(fact_89_append__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys2:(list @ A)]: (((((append @ A) @ (((cons @ A) @ X3) @ Xs)) @ Ys2) = (((cons @ A) @ X3) @ (((append @ A) @ Xs) @ Ys2)))))).
% 0.31/1.50 thf(fact_90_Cons__eq__appendI, axiom, (! [A:$tType,X3:A,Xs1:(list @ A),Ys2:(list @ A),Xs:(list @ A),Zs2:(list @ A)]: ((((((cons @ A) @ X3) @ Xs1) = Ys2) => ((Xs = (((append @ A) @ Xs1) @ Zs2)) => ((((cons @ A) @ X3) @ Xs) = (((append @ A) @ Ys2) @ Zs2))))))).
% 0.31/1.50 thf(fact_91_append_Oleft__neutral, axiom, (! [A:$tType,A4:(list @ A)]: (((((append @ A) @ (nil @ A)) @ A4) = A4)))).
% 0.31/1.50 thf(fact_92_append__Nil, axiom, (! [A:$tType,Ys2:(list @ A)]: (((((append @ A) @ (nil @ A)) @ Ys2) = Ys2)))).
% 0.31/1.50 thf(fact_93_eq__Nil__appendI, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs = Ys2) => (Xs = (((append @ A) @ (nil @ A)) @ Ys2)))))).
% 0.31/1.50 thf(fact_94_lexordp__eq__pref, axiom, (! [A:$tType]: (((ord @ A) => (! [U:(list @ A),V2:(list @ A)]: ((((ord_lexordp_eq @ A) @ U) @ (((append @ A) @ U) @ V2)))))))).
% 0.31/1.50 thf(fact_95_ord_Olexordp__eq__pref, axiom, (! [A:$tType,Less:(A > (A > $o)),U:(list @ A),V2:(list @ A)]: (((((lexordp_eq @ A) @ Less) @ U) @ (((append @ A) @ U) @ V2))))).
% 0.31/1.50 thf(fact_96_splice_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys2:(list @ A)]: (((((splice @ A) @ (((cons @ A) @ X3) @ Xs)) @ Ys2) = (((cons @ A) @ X3) @ (((splice @ A) @ Ys2) @ Xs)))))).
% 0.31/1.50 thf(fact_97_splice_Osimps_I1_J, axiom, (! [A:$tType,Ys2:(list @ A)]: (((((splice @ A) @ (nil @ A)) @ Ys2) = Ys2)))).
% 0.31/1.50 thf(fact_98_rev__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X:A]: ((P @ (((cons @ A) @ X) @ (nil @ A))))) => ((! [X:A,Xs2:(list @ A)]: (((Xs2 != (nil @ A)) => ((P @ Xs2) => (P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X) @ (nil @ A)))))))) => (P @ Xs))))))).
% 0.31/1.50 thf(fact_99_append__eq__Cons__conv, axiom, (! [A:$tType,Ys2:(list @ A),Zs2:(list @ A),X3:A,Xs:(list @ A)]: ((((((append @ A) @ Ys2) @ Zs2) = (((cons @ A) @ X3) @ Xs)) = (((Ys2 = (nil @ A)) & (Zs2 = (((cons @ A) @ X3) @ Xs))) | (? [Ys4:(list @ A)]: (((Ys2 = (((cons @ A) @ X3) @ Ys4)) & ((((append @ A) @ Ys4) @ Zs2) = Xs))))))))).
% 0.31/1.50 thf(fact_100_Cons__eq__append__conv, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys2:(list @ A),Zs2:(list @ A)]: ((((((cons @ A) @ X3) @ Xs) = (((append @ A) @ Ys2) @ Zs2)) = (((Ys2 = (nil @ A)) & ((((cons @ A) @ X3) @ Xs) = Zs2)) | (? [Ys4:(list @ A)]: ((((((cons @ A) @ X3) @ Ys4) = Ys2) & (Xs = (((append @ A) @ Ys4) @ Zs2)))))))))).
% 0.31/1.50 thf(fact_101_rev__exhaust, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (~ ((! [Ys3:(list @ A),Y3:A]: ((Xs != (((append @ A) @ Ys3) @ (((cons @ A) @ Y3) @ (nil @ A)))))))))))).
% 0.31/1.50 thf(fact_102_rev__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X:A,Xs2:(list @ A)]: (((P @ Xs2) => (P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X) @ (nil @ A))))))) => (P @ Xs)))))).
% 0.31/1.50 thf(fact_103_append__eq__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((append @ A) @ (((cons @ A) @ X3) @ (nil @ A))) @ Xs) = (((cons @ A) @ X3) @ Xs))))).
% 0.31/1.50 thf(fact_104_lexordp__eq_OCons, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less @ A) @ X3) @ Y) => (((ord_lexordp_eq @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))))))))).
% 0.31/1.50 thf(fact_105_lexordp__eq_OCons__eq, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: (((~ ((((ord_less @ A) @ X3) @ Y))) => ((~ ((((ord_less @ A) @ Y) @ X3))) => ((((ord_lexordp_eq @ A) @ Xs) @ Ys2) => (((ord_lexordp_eq @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))))))))))).
% 0.31/1.50 thf(fact_106_maps__simps_I1_J, axiom, (! [A:$tType,B:$tType,F:(B > (list @ A)),X3:B,Xs:(list @ B)]: ((((((maps @ B) @ A) @ F) @ (((cons @ B) @ X3) @ Xs)) = (((append @ A) @ (F @ X3)) @ ((((maps @ B) @ A) @ F) @ Xs)))))).
% 0.31/1.50 thf(fact_107_concat__eq__append__conv, axiom, (! [A:$tType,Xss2:(list @ (list @ A)),Ys2:(list @ A),Zs2:(list @ A)]: (((((concat @ A) @ Xss2) = (((append @ A) @ Ys2) @ Zs2)) = (((Xss2 = (nil @ (list @ A))) => ((Ys2 = (nil @ A)) & (Zs2 = (nil @ A)))) & ((Xss2 != (nil @ (list @ A))) => (? [Xss1:(list @ (list @ A)),Xs3:(list @ A),Xs4:(list @ A),Xss22:(list @ (list @ A))]: (((Xss2 = (((append @ (list @ A)) @ Xss1) @ (((cons @ (list @ A)) @ (((append @ A) @ Xs3) @ Xs4)) @ Xss22))) & ((Ys2 = (((append @ A) @ ((concat @ A) @ Xss1)) @ Xs3)) & (Zs2 = (((append @ A) @ Xs4) @ ((concat @ A) @ Xss22))))))))))))).
% 0.31/1.50 thf(fact_108_rotate1_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((rotate1 @ A) @ (((cons @ A) @ X3) @ Xs)) = (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))))))).
% 0.31/1.50 thf(fact_109_butlast__snoc, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((butlast @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) = Xs)))).
% 0.31/1.50 thf(fact_110_last__snoc, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((last @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) = X3)))).
% 0.31/1.50 thf(fact_111_SuccI, axiom, (! [A:$tType,Kl:(list @ A),K:A,Kl2:(set @ (list @ A))]: (((((member2 @ (list @ A)) @ (((append @ A) @ Kl) @ (((cons @ A) @ K) @ (nil @ A)))) @ Kl2) => (((member2 @ A) @ K) @ (((bNF_Greatest_Succ @ A) @ Kl2) @ Kl)))))).
% 0.31/1.50 thf(fact_112_SuccD, axiom, (! [A:$tType,K:A,Kl2:(set @ (list @ A)),Kl:(list @ A)]: (((((member2 @ A) @ K) @ (((bNF_Greatest_Succ @ A) @ Kl2) @ Kl)) => (((member2 @ (list @ A)) @ (((append @ A) @ Kl) @ (((cons @ A) @ K) @ (nil @ A)))) @ Kl2))))).
% 0.31/1.50 thf(fact_113_rotate1__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((rotate1 @ A) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))).
% 0.31/1.50 thf(fact_114_last__appendL, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_115_last__appendR, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Ys2)))))).
% 0.31/1.50 thf(fact_116_concat__append, axiom, (! [A:$tType,Xs:(list @ (list @ A)),Ys2:(list @ (list @ A))]: ((((concat @ A) @ (((append @ (list @ A)) @ Xs) @ Ys2)) = (((append @ A) @ ((concat @ A) @ Xs)) @ ((concat @ A) @ Ys2)))))).
% 0.31/1.50 thf(fact_117_append__butlast__last__id, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((append @ A) @ ((butlast @ A) @ Xs)) @ (((cons @ A) @ ((last @ A) @ Xs)) @ (nil @ A))) = Xs))))).
% 0.31/1.50 thf(fact_118_butlast_Osimps_I1_J, axiom, (! [A:$tType]: ((((butlast @ A) @ (nil @ A)) = (nil @ A))))).
% 0.31/1.50 thf(fact_119_snoc__eq__iff__butlast, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))) = Ys2) = ((Ys2 != (nil @ A)) & ((((butlast @ A) @ Ys2) = Xs) & (((last @ A) @ Ys2) = X3))))))).
% 0.31/1.50 thf(fact_120_rotate1_Osimps_I1_J, axiom, (! [A:$tType]: ((((rotate1 @ A) @ (nil @ A)) = (nil @ A))))).
% 0.31/1.50 thf(fact_121_last_Osimps, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((Xs = (nil @ A)) => (((last @ A) @ (((cons @ A) @ X3) @ Xs)) = X3)) & ((Xs != (nil @ A)) => (((last @ A) @ (((cons @ A) @ X3) @ Xs)) = ((last @ A) @ Xs))))))).
% 0.31/1.50 thf(fact_122_last__ConsL, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: (((Xs = (nil @ A)) => (((last @ A) @ (((cons @ A) @ X3) @ Xs)) = X3))))).
% 0.31/1.50 thf(fact_123_last__ConsR, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: (((Xs != (nil @ A)) => (((last @ A) @ (((cons @ A) @ X3) @ Xs)) = ((last @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_124_last__append, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: ((((Ys2 = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Xs))) & ((Ys2 != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Ys2))))))).
% 0.31/1.50 thf(fact_125_longest__common__suffix, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((? [Ss:(list @ A),Xs5:(list @ A),Ys5:(list @ A)]: (((Xs = (((append @ A) @ Xs5) @ Ss)) & ((Ys2 = (((append @ A) @ Ys5) @ Ss)) & ((Xs5 = (nil @ A)) | ((Ys5 = (nil @ A)) | (((last @ A) @ Xs5) != ((last @ A) @ Ys5))))))))))).
% 0.31/1.50 thf(fact_126_butlast_Osimps_I2_J, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((Xs = (nil @ A)) => (((butlast @ A) @ (((cons @ A) @ X3) @ Xs)) = (nil @ A))) & ((Xs != (nil @ A)) => (((butlast @ A) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ ((butlast @ A) @ Xs)))))))).
% 0.31/1.50 thf(fact_127_concat_Osimps_I1_J, axiom, (! [A:$tType]: ((((concat @ A) @ (nil @ (list @ A))) = (nil @ A))))).
% 0.31/1.50 thf(fact_128_concat_Osimps_I2_J, axiom, (! [A:$tType,X3:(list @ A),Xs:(list @ (list @ A))]: ((((concat @ A) @ (((cons @ (list @ A)) @ X3) @ Xs)) = (((append @ A) @ X3) @ ((concat @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_129_butlast__append, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: ((((Ys2 = (nil @ A)) => (((butlast @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((butlast @ A) @ Xs))) & ((Ys2 != (nil @ A)) => (((butlast @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((append @ A) @ Xs) @ ((butlast @ A) @ Ys2)))))))).
% 0.31/1.50 thf(fact_130_maps__simps_I2_J, axiom, (! [B:$tType,A:$tType,F:(B > (list @ A))]: ((((((maps @ B) @ A) @ F) @ (nil @ B)) = (nil @ A))))).
% 0.31/1.50 thf(fact_131_concat__eq__appendD, axiom, (! [A:$tType,Xss2:(list @ (list @ A)),Ys2:(list @ A),Zs2:(list @ A)]: (((((concat @ A) @ Xss2) = (((append @ A) @ Ys2) @ Zs2)) => ((Xss2 != (nil @ (list @ A))) => (? [Xss12:(list @ (list @ A)),Xs2:(list @ A),Xs5:(list @ A),Xss23:(list @ (list @ A))]: (((Xss2 = (((append @ (list @ A)) @ Xss12) @ (((cons @ (list @ A)) @ (((append @ A) @ Xs2) @ Xs5)) @ Xss23))) & ((Ys2 = (((append @ A) @ ((concat @ A) @ Xss12)) @ Xs2)) & (Zs2 = (((append @ A) @ Xs5) @ ((concat @ A) @ Xss23)))))))))))).
% 0.31/1.50 thf(fact_132_shift__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Greatest_shift @ A) @ B) = (^ [Lab:((list @ A) > B),K2:A,Kl3:(list @ A)]: ((Lab @ (((cons @ A) @ K2) @ Kl3)))))))).
% 0.31/1.50 thf(fact_133_empty__Shift, axiom, (! [A:$tType,Kl2:(set @ (list @ A)),K:A]: (((((member2 @ (list @ A)) @ (nil @ A)) @ Kl2) => ((((member2 @ A) @ K) @ (((bNF_Greatest_Succ @ A) @ Kl2) @ (nil @ A))) => (((member2 @ (list @ A)) @ (nil @ A)) @ (((bNF_Greatest_Shift @ A) @ Kl2) @ K))))))).
% 0.31/1.50 thf(fact_134_Succ__Shift, axiom, (! [A:$tType,Kl2:(set @ (list @ A)),K:A,Kl:(list @ A)]: (((((bNF_Greatest_Succ @ A) @ (((bNF_Greatest_Shift @ A) @ Kl2) @ K)) @ Kl) = (((bNF_Greatest_Succ @ A) @ Kl2) @ (((cons @ A) @ K) @ Kl)))))).
% 0.31/1.50 thf(fact_135_list__ord__snoc, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),Xs:(list @ A),X3:A]: (((((list_ord @ A) @ Ord2) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) = ((Xs = (nil @ A)) | (((Ord2 @ ((last @ A) @ Xs)) @ X3) & (((list_ord @ A) @ Ord2) @ Xs)))))))))).
% 0.31/1.50 thf(fact_136_concat__conv__foldr, axiom, (! [A:$tType]: (((concat @ A) = (^ [Xss3:(list @ (list @ A))]: ((((((foldr @ (list @ A)) @ (list @ A)) @ (append @ A)) @ Xss3) @ (nil @ A)))))))).
% 0.31/1.50 thf(fact_137_foldr__append, axiom, (! [B:$tType,A:$tType,F:(B > (A > A)),Xs:(list @ B),Ys2:(list @ B),A4:A]: (((((((foldr @ B) @ A) @ F) @ (((append @ B) @ Xs) @ Ys2)) @ A4) = (((((foldr @ B) @ A) @ F) @ Xs) @ (((((foldr @ B) @ A) @ F) @ Ys2) @ A4)))))).
% 0.31/1.50 thf(fact_138_list__ord_Oelims_I3_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(A > (A > $o)),Xa:(list @ A)]: (((~ ((((list_ord @ A) @ X3) @ Xa))) => (~ ((! [X1:A,X2:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X1) @ (((cons @ A) @ X2) @ Xs2))) => (((X3 @ X1) @ X2) & (((list_ord @ A) @ X3) @ (((cons @ A) @ X2) @ Xs2))))))))))))))).
% 0.31/1.50 thf(fact_139_list__ord_Osimps_I1_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),X12:A,X23:A,Xs:(list @ A)]: (((((list_ord @ A) @ Ord2) @ (((cons @ A) @ X12) @ (((cons @ A) @ X23) @ Xs))) = (((Ord2 @ X12) @ X23) & (((list_ord @ A) @ Ord2) @ (((cons @ A) @ X23) @ Xs)))))))))).
% 0.31/1.50 thf(fact_140_list__ord_Osimps_I2_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o))]: ((((list_ord @ A) @ Ord2) @ (nil @ A)))))))).
% 0.31/1.50 thf(fact_141_list__ord__Nil, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o))]: ((((list_ord @ A) @ Ord2) @ (nil @ A)))))))).
% 0.31/1.50 thf(fact_142_list__ord__imp, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),Ord3:(A > (A > $o)),Xs:(list @ A)]: (((! [X:A,Y3:A]: ((((Ord2 @ X) @ Y3) => ((Ord3 @ X) @ Y3)))) => ((((list_ord @ A) @ Ord2) @ Xs) => (((list_ord @ A) @ Ord3) @ Xs))))))))).
% 0.31/1.50 thf(fact_143_list__ord__one, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),X3:A]: ((((list_ord @ A) @ Ord2) @ (((cons @ A) @ X3) @ (nil @ A))))))))).
% 0.31/1.50 thf(fact_144_list__ord_Osimps_I3_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),V2:A]: ((((list_ord @ A) @ Ord2) @ (((cons @ A) @ V2) @ (nil @ A))))))))).
% 0.31/1.50 thf(fact_145_list__ord_Oelims_I1_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(A > (A > $o)),Xa:(list @ A),Y:$o]: ((((((list_ord @ A) @ X3) @ Xa) = Y) => ((! [X1:A,X2:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X1) @ (((cons @ A) @ X2) @ Xs2))) => (Y = (~ ((((X3 @ X1) @ X2) & (((list_ord @ A) @ X3) @ (((cons @ A) @ X2) @ Xs2))))))))) => (((Xa = (nil @ A)) => (~ (Y))) => (~ (((? [V:A]: ((Xa = (((cons @ A) @ V) @ (nil @ A))))) => (~ (Y)))))))))))))).
% 0.31/1.50 thf(fact_146_list__ord_Oelims_I2_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(A > (A > $o)),Xa:(list @ A)]: (((((list_ord @ A) @ X3) @ Xa) => ((! [X1:A,X2:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X1) @ (((cons @ A) @ X2) @ Xs2))) => (~ ((((X3 @ X1) @ X2) & (((list_ord @ A) @ X3) @ (((cons @ A) @ X2) @ Xs2)))))))) => ((Xa != (nil @ A)) => (~ ((! [V:A]: ((Xa != (((cons @ A) @ V) @ (nil @ A)))))))))))))))).
% 0.31/1.50 thf(fact_147_ShiftD, axiom, (! [A:$tType,Kl:(list @ A),Kl2:(set @ (list @ A)),K:A]: (((((member2 @ (list @ A)) @ Kl) @ (((bNF_Greatest_Shift @ A) @ Kl2) @ K)) => (((member2 @ (list @ A)) @ (((cons @ A) @ K) @ Kl)) @ Kl2))))).
% 0.31/1.50 thf(fact_148_list__strict__asc__def, axiom, (! [A:$tType]: (((ord @ A) => ((list_strict_asc @ A) = ((list_ord @ A) @ (ord_less @ A))))))).
% 0.31/1.50 thf(fact_149_list__ord__append, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),Xs:(list @ A),Ys2:(list @ A)]: (((((list_ord @ A) @ Ord2) @ (((append @ A) @ Xs) @ Ys2)) = ((((list_ord @ A) @ Ord2) @ Xs) & ((Ys2 = (nil @ A)) | ((((list_ord @ A) @ Ord2) @ Ys2) & ((Xs = (nil @ A)) | ((Ord2 @ ((last @ A) @ Xs)) @ ((hd @ A) @ Ys2))))))))))))).
% 0.31/1.50 thf(fact_150_lexordp__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_lexordp @ A) = (^ [Xs3:(list @ A),Ys:(list @ A)]: (((? [X4:A,Vs:(list @ A)]: ((Ys = (((append @ A) @ Xs3) @ (((cons @ A) @ X4) @ Vs))))) | (? [Us:(list @ A),A6:A,B3:A,Vs:(list @ A),Ws:(list @ A)]: (((((ord_less @ A) @ A6) @ B3) & ((Xs3 = (((append @ A) @ Us) @ (((cons @ A) @ A6) @ Vs))) & (Ys = (((append @ A) @ Us) @ (((cons @ A) @ B3) @ Ws))))))))))))))).
% 0.31/1.50 thf(fact_151_lexordp__append__left__rightI, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A,Us2:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less @ A) @ X3) @ Y) => (((ord_lexordp @ A) @ (((append @ A) @ Us2) @ (((cons @ A) @ X3) @ Xs))) @ (((append @ A) @ Us2) @ (((cons @ A) @ Y) @ Ys2)))))))))).
% 0.31/1.50 thf(fact_152_lexordp__simps_I2_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A)]: ((~ ((((ord_lexordp @ A) @ Xs) @ (nil @ A)))))))))).
% 0.31/1.50 thf(fact_153_lexordp__simps_I1_J, axiom, (! [A:$tType]: (((ord @ A) => (! [Ys2:(list @ A)]: (((((ord_lexordp @ A) @ (nil @ A)) @ Ys2) = (Ys2 != (nil @ A))))))))).
% 0.31/1.50 thf(fact_154_lexordp__simps_I3_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Xs:(list @ A),Y:A,Ys2:(list @ A)]: (((((ord_lexordp @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)) = ((((ord_less @ A) @ X3) @ Y) | ((~ ((((ord_less @ A) @ Y) @ X3))) & (((ord_lexordp @ A) @ Xs) @ Ys2)))))))))).
% 0.31/1.50 thf(fact_155_hd__append2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs != (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((hd @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_156_lexordp__append__leftI, axiom, (! [A:$tType]: (((ord @ A) => (! [Us2:(list @ A),Vs2:(list @ A),Xs:(list @ A)]: (((((ord_lexordp @ A) @ Us2) @ Vs2) => (((ord_lexordp @ A) @ (((append @ A) @ Xs) @ Us2)) @ (((append @ A) @ Xs) @ Vs2))))))))).
% 0.31/1.50 thf(fact_157_lexordp__antisym, axiom, (! [A:$tType]: (((order @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => (~ ((((ord_lexordp @ A) @ Ys2) @ Xs)))))))))).
% 0.31/1.50 thf(fact_158_lexordp__trans, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A),Zs2:(list @ A)]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => ((((ord_lexordp @ A) @ Ys2) @ Zs2) => (((ord_lexordp @ A) @ Xs) @ Zs2))))))))).
% 0.31/1.50 thf(fact_159_lexordp__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp @ A) @ Xs) @ Ys2) | ((Xs = Ys2) | (((ord_lexordp @ A) @ Ys2) @ Xs))))))))).
% 0.31/1.50 thf(fact_160_lexordp__irreflexive_H, axiom, (! [A:$tType]: (((order @ A) => (! [Xs:(list @ A)]: ((~ ((((ord_lexordp @ A) @ Xs) @ Xs))))))))).
% 0.31/1.50 thf(fact_161_lexordp__irreflexive, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A)]: (((! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))) => (~ ((((ord_lexordp @ A) @ Xs) @ Xs)))))))))).
% 0.31/1.50 thf(fact_162_list_Osel_I1_J, axiom, (! [A:$tType,X21:A,X22:(list @ A)]: ((((hd @ A) @ (((cons @ A) @ X21) @ X22)) = X21)))).
% 0.31/1.50 thf(fact_163_hd__concat, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((Xs != (nil @ (list @ A))) => ((((hd @ (list @ A)) @ Xs) != (nil @ A)) => (((hd @ A) @ ((concat @ A) @ Xs)) = ((hd @ A) @ ((hd @ (list @ A)) @ Xs)))))))).
% 0.31/1.50 thf(fact_164_lexordp__into__lexordp__eq, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => (((ord_lexordp_eq @ A) @ Xs) @ Ys2)))))))).
% 0.31/1.50 thf(fact_165_lexordp__eq__conv__lexord, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_lexordp_eq @ A) = (^ [Xs3:(list @ A),Ys:(list @ A)]: (((Xs3 = Ys) | (((ord_lexordp @ A) @ Xs3) @ Ys))))))))).
% 0.31/1.50 thf(fact_166_lexordp__conv__lexordp__eq, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_lexordp @ A) = (^ [Xs3:(list @ A),Ys:(list @ A)]: (((((ord_lexordp_eq @ A) @ Xs3) @ Ys) & (~ ((((ord_lexordp_eq @ A) @ Ys) @ Xs3))))))))))).
% 0.31/1.50 thf(fact_167_hd__append, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((Xs = (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((hd @ A) @ Ys2))) & ((Xs != (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((hd @ A) @ Xs))))))).
% 0.31/1.50 thf(fact_168_longest__common__prefix, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((? [Ps:(list @ A),Xs5:(list @ A),Ys5:(list @ A)]: (((Xs = (((append @ A) @ Ps) @ Xs5)) & ((Ys2 = (((append @ A) @ Ps) @ Ys5)) & ((Xs5 = (nil @ A)) | ((Ys5 = (nil @ A)) | (((hd @ A) @ Xs5) != ((hd @ A) @ Ys5))))))))))).
% 0.31/1.50 thf(fact_169_list__ord__Cons__imp, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),Xs:(list @ A),X3:A]: (((((list_ord @ A) @ Ord2) @ Xs) => (((Ord2 @ X3) @ ((hd @ A) @ Xs)) => (((list_ord @ A) @ Ord2) @ (((cons @ A) @ X3) @ Xs)))))))))).
% 0.31/1.50 thf(fact_170_lexordp_OCons, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less @ A) @ X3) @ Y) => (((ord_lexordp @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))))))))).
% 0.31/1.50 thf(fact_171_lexordp_OCons__eq, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: (((~ ((((ord_less @ A) @ X3) @ Y))) => ((~ ((((ord_less @ A) @ Y) @ X3))) => ((((ord_lexordp @ A) @ Xs) @ Ys2) => (((ord_lexordp @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))))))))))).
% 0.31/1.50 thf(fact_172_lexordp__append__leftD, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A),Us2:(list @ A),Vs2:(list @ A)]: (((((ord_lexordp @ A) @ (((append @ A) @ Xs) @ Us2)) @ (((append @ A) @ Xs) @ Vs2)) => ((! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))) => (((ord_lexordp @ A) @ Us2) @ Vs2))))))))).
% 0.31/1.50 thf(fact_173_lexordp_ONil, axiom, (! [A:$tType]: (((ord @ A) => (! [Y:A,Ys2:(list @ A)]: ((((ord_lexordp @ A) @ (nil @ A)) @ (((cons @ A) @ Y) @ Ys2)))))))).
% 0.31/1.50 thf(fact_174_lexordp__append__rightI, axiom, (! [A:$tType]: (((ord @ A) => (! [Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 != (nil @ A)) => (((ord_lexordp @ A) @ Xs) @ (((append @ A) @ Xs) @ Ys2))))))))).
% 0.31/1.50 thf(fact_175_list__ord__Cons, axiom, (! [A:$tType]: (((ord @ A) => (! [Ord2:(A > (A > $o)),X3:A,Xs:(list @ A)]: (((((list_ord @ A) @ Ord2) @ (((cons @ A) @ X3) @ Xs)) = ((Xs = (nil @ A)) | (((Ord2 @ X3) @ ((hd @ A) @ Xs)) & (((list_ord @ A) @ Ord2) @ Xs)))))))))).
% 0.31/1.50 thf(fact_176_lexordp_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [A1:(list @ A),A2:(list @ A)]: (((((ord_lexordp @ A) @ A1) @ A2) => (((A1 = (nil @ A)) => (! [Y3:A,Ys3:(list @ A)]: ((A2 != (((cons @ A) @ Y3) @ Ys3))))) => ((! [X:A]: (((? [Xs2:(list @ A)]: ((A1 = (((cons @ A) @ X) @ Xs2)))) => (! [Y3:A]: (((? [Ys3:(list @ A)]: ((A2 = (((cons @ A) @ Y3) @ Ys3)))) => (~ ((((ord_less @ A) @ X) @ Y3))))))))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X) @ Xs2)) => (! [Ys3:(list @ A)]: (((A2 = (((cons @ A) @ Y3) @ Ys3)) => ((~ ((((ord_less @ A) @ X) @ Y3))) => ((~ ((((ord_less @ A) @ Y3) @ X))) => (~ ((((ord_lexordp @ A) @ Xs2) @ Ys3)))))))))))))))))))))).
% 0.31/1.50 thf(fact_177_lexordp_Osimps, axiom, (! [A:$tType]: (((ord @ A) => ((ord_lexordp @ A) = (^ [A12:(list @ A),A22:(list @ A)]: (((? [Y2:A,Ys:(list @ A)]: (((A12 = (nil @ A)) & (A22 = (((cons @ A) @ Y2) @ Ys))))) | ((? [X4:A,Y2:A,Xs3:(list @ A),Ys:(list @ A)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ A) @ Y2) @ Ys)) & (((ord_less @ A) @ X4) @ Y2))))) | (? [X4:A,Y2:A,Xs3:(list @ A),Ys:(list @ A)]: (((A12 = (((cons @ A) @ X4) @ Xs3)) & ((A22 = (((cons @ A) @ Y2) @ Ys)) & ((~ ((((ord_less @ A) @ X4) @ Y2))) & ((~ ((((ord_less @ A) @ Y2) @ X4))) & (((ord_lexordp @ A) @ Xs3) @ Ys)))))))))))))))).
% 0.31/1.50 thf(fact_178_lexordp_Oinducts, axiom, (! [A:$tType]: (((ord @ A) => (! [X12:(list @ A),X23:(list @ A),P:((list @ A) > ((list @ A) > $o))]: (((((ord_lexordp @ A) @ X12) @ X23) => ((! [Y3:A,Ys3:(list @ A)]: (((P @ (nil @ A)) @ (((cons @ A) @ Y3) @ Ys3)))) => ((! [X:A,Y3:A,Xs2:(list @ A),Ys3:(list @ A)]: (((((ord_less @ A) @ X) @ Y3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3))))) => ((! [X:A,Y3:A,Xs2:(list @ A),Ys3:(list @ A)]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((~ ((((ord_less @ A) @ Y3) @ X))) => ((((ord_lexordp @ A) @ Xs2) @ Ys3) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3)))))))) => ((P @ X12) @ X23))))))))))).
% 0.31/1.50 thf(fact_179_lexordp__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => (((Xs = (nil @ A)) => (! [Y3:A,Ys5:(list @ A)]: ((Ys2 != (((cons @ A) @ Y3) @ Ys5))))) => ((! [X:A]: (((? [Xs5:(list @ A)]: ((Xs = (((cons @ A) @ X) @ Xs5)))) => (! [Y3:A]: (((? [Ys5:(list @ A)]: ((Ys2 = (((cons @ A) @ Y3) @ Ys5)))) => (~ ((((ord_less @ A) @ X) @ Y3))))))))) => (~ ((! [X:A,Xs5:(list @ A)]: (((Xs = (((cons @ A) @ X) @ Xs5)) => (! [Ys5:(list @ A)]: (((Ys2 = (((cons @ A) @ X) @ Ys5)) => (~ ((((ord_lexordp @ A) @ Xs5) @ Ys5)))))))))))))))))))).
% 0.31/1.50 thf(fact_180_lexordp__induct, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A),P:((list @ A) > ((list @ A) > $o))]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => ((! [Y3:A,Ys3:(list @ A)]: (((P @ (nil @ A)) @ (((cons @ A) @ Y3) @ Ys3)))) => ((! [X:A,Xs2:(list @ A),Y3:A,Ys3:(list @ A)]: (((((ord_less @ A) @ X) @ Y3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ Y3) @ Ys3))))) => ((! [X:A,Xs2:(list @ A),Ys3:(list @ A)]: (((((ord_lexordp @ A) @ Xs2) @ Ys3) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ A) @ X) @ Ys3)))))) => ((P @ Xs) @ Ys2))))))))))).
% 0.31/1.50 thf(fact_181_distinct__adj__append__iff, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((distinct_adj @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((distinct_adj @ A) @ Xs) & (((distinct_adj @ A) @ Ys2) & ((Xs = (nil @ A)) | ((Ys2 = (nil @ A)) | (((last @ A) @ Xs) != ((hd @ A) @ Ys2)))))))))).
% 0.31/1.50 thf(fact_182_rotate1__hd__tl, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((rotate1 @ A) @ Xs) = (((append @ A) @ ((tl @ A) @ Xs)) @ (((cons @ A) @ ((hd @ A) @ Xs)) @ (nil @ A)))))))).
% 0.31/1.50 thf(fact_183_successively__append__iff, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A),Ys2:(list @ A)]: (((((successively @ A) @ P) @ (((append @ A) @ Xs) @ Ys2)) = ((((successively @ A) @ P) @ Xs) & ((((successively @ A) @ P) @ Ys2) & ((Xs = (nil @ A)) | ((Ys2 = (nil @ A)) | ((P @ ((last @ A) @ Xs)) @ ((hd @ A) @ Ys2)))))))))).
% 0.31/1.50 thf(fact_184_distinct__adj__Cons__Cons, axiom, (! [B:$tType,X3:B,Y:B,Xs:(list @ B)]: ((((distinct_adj @ B) @ (((cons @ B) @ X3) @ (((cons @ B) @ Y) @ Xs))) = ((X3 != Y) & ((distinct_adj @ B) @ (((cons @ B) @ Y) @ Xs))))))).
% 0.31/1.50 thf(fact_185_tl__append2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs != (nil @ A)) => (((tl @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((append @ A) @ ((tl @ A) @ Xs)) @ Ys2)))))).
% 0.31/1.50 thf(fact_186_hd__Cons__tl, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((cons @ A) @ ((hd @ A) @ Xs)) @ ((tl @ A) @ Xs)) = Xs))))).
% 0.31/1.50 thf(fact_187_list_Ocollapse, axiom, (! [A:$tType,List:(list @ A)]: (((List != (nil @ A)) => ((((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List)) = List))))).
% 0.31/1.50 thf(fact_188_list_Osel_I3_J, axiom, (! [A:$tType,X21:A,X22:(list @ A)]: ((((tl @ A) @ (((cons @ A) @ X21) @ X22)) = X22)))).
% 0.31/1.50 thf(fact_189_list_Osel_I2_J, axiom, (! [A:$tType]: ((((tl @ A) @ (nil @ A)) = (nil @ A))))).
% 0.31/1.50 thf(fact_190_successively_Oelims_I3_J, axiom, (! [A:$tType,X3:(A > (A > $o)),Xa:(list @ A)]: (((~ ((((successively @ A) @ X3) @ Xa))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2))) => (((X3 @ X) @ Y3) & (((successively @ A) @ X3) @ (((cons @ A) @ Y3) @ Xs2)))))))))))).
% 0.31/1.50 thf(fact_191_successively_Osimps_I3_J, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A,Y:A,Xs:(list @ A)]: (((((successively @ A) @ P) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Xs))) = (((P @ X3) @ Y) & (((successively @ A) @ P) @ (((cons @ A) @ Y) @ Xs))))))).
% 0.31/1.50 thf(fact_192_successively_Osimps_I1_J, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((successively @ A) @ P) @ (nil @ A))))).
% 0.31/1.50 thf(fact_193_distinct__adj__appendD1, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((distinct_adj @ A) @ (((append @ A) @ Xs) @ Ys2)) => ((distinct_adj @ A) @ Xs))))).
% 0.31/1.50 thf(fact_194_distinct__adj__appendD2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((distinct_adj @ A) @ (((append @ A) @ Xs) @ Ys2)) => ((distinct_adj @ A) @ Ys2))))).
% 0.31/1.50 thf(fact_195_distinct__adj__Nil, axiom, (! [A:$tType]: (((distinct_adj @ A) @ (nil @ A))))).
% 0.31/1.50 thf(fact_196_distinct__adj__ConsD, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((distinct_adj @ A) @ (((cons @ A) @ X3) @ Xs)) => ((distinct_adj @ A) @ Xs))))).
% 0.31/1.50 thf(fact_197_butlast__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((butlast @ A) @ ((tl @ A) @ Xs)) = ((tl @ A) @ ((butlast @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_198_tl__Nil, axiom, (! [A:$tType,Xs:(list @ A)]: (((((tl @ A) @ Xs) = (nil @ A)) = ((Xs = (nil @ A)) | (? [X4:A]: ((Xs = (((cons @ A) @ X4) @ (nil @ A)))))))))).
% 0.31/1.50 thf(fact_199_Nil__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((nil @ A) = ((tl @ A) @ Xs)) = ((Xs = (nil @ A)) | (? [X4:A]: ((Xs = (((cons @ A) @ X4) @ (nil @ A)))))))))).
% 0.31/1.50 thf(fact_200_list_Oexpand, axiom, (! [A:$tType,List:(list @ A),List2:(list @ A)]: ((((List = (nil @ A)) = (List2 = (nil @ A))) => (((List != (nil @ A)) => ((List2 != (nil @ A)) => ((((hd @ A) @ List) = ((hd @ A) @ List2)) & (((tl @ A) @ List) = ((tl @ A) @ List2))))) => (List = List2)))))).
% 0.31/1.50 thf(fact_201_last__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((Xs = (nil @ A)) | (((tl @ A) @ Xs) != (nil @ A))) => (((last @ A) @ ((tl @ A) @ Xs)) = ((last @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_202_successively_Osimps_I2_J, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A]: ((((successively @ A) @ P) @ (((cons @ A) @ X3) @ (nil @ A)))))).
% 0.31/1.50 thf(fact_203_successively_Oelims_I1_J, axiom, (! [A:$tType,X3:(A > (A > $o)),Xa:(list @ A),Y:$o]: ((((((successively @ A) @ X3) @ Xa) = Y) => (((Xa = (nil @ A)) => (~ (Y))) => (((? [X:A]: ((Xa = (((cons @ A) @ X) @ (nil @ A))))) => (~ (Y))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2))) => (Y = (~ ((((X3 @ X) @ Y3) & (((successively @ A) @ X3) @ (((cons @ A) @ Y3) @ Xs2))))))))))))))))).
% 0.31/1.50 thf(fact_204_successively_Oelims_I2_J, axiom, (! [A:$tType,X3:(A > (A > $o)),Xa:(list @ A)]: (((((successively @ A) @ X3) @ Xa) => ((Xa != (nil @ A)) => ((! [X:A]: ((Xa != (((cons @ A) @ X) @ (nil @ A))))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2))) => (~ ((((X3 @ X) @ Y3) & (((successively @ A) @ X3) @ (((cons @ A) @ Y3) @ Xs2)))))))))))))))).
% 0.31/1.50 thf(fact_205_distinct__adj__singleton, axiom, (! [B:$tType,X3:B]: (((distinct_adj @ B) @ (((cons @ B) @ X3) @ (nil @ B)))))).
% 0.31/1.50 thf(fact_206_list_Oexhaust__sel, axiom, (! [A:$tType,List:(list @ A)]: (((List != (nil @ A)) => (List = (((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List))))))).
% 0.31/1.50 thf(fact_207_successively__Cons, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A,Xs:(list @ A)]: (((((successively @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = ((Xs = (nil @ A)) | (((P @ X3) @ ((hd @ A) @ Xs)) & (((successively @ A) @ P) @ Xs))))))).
% 0.31/1.50 thf(fact_208_distinct__adj__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((distinct_adj @ A) @ (((cons @ A) @ X3) @ Xs)) = ((Xs = (nil @ A)) | ((X3 != ((hd @ A) @ Xs)) & ((distinct_adj @ A) @ Xs))))))).
% 0.31/1.50 thf(fact_209_list_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > ((list @ A) > B)),List:(list @ A)]: (((P @ (((((case_list @ B) @ A) @ F1) @ F22) @ List)) = (((List = (nil @ A)) => (P @ F1)) & ((List = (((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List))) => (P @ ((F22 @ ((hd @ A) @ List)) @ ((tl @ A) @ List))))))))).
% 0.31/1.50 thf(fact_210_list_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > ((list @ A) > B)),List:(list @ A)]: (((P @ (((((case_list @ B) @ A) @ F1) @ F22) @ List)) = (~ ((((List = (nil @ A)) & (~ ((P @ F1)))) | ((List = (((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List))) & (~ ((P @ ((F22 @ ((hd @ A) @ List)) @ ((tl @ A) @ List))))))))))))).
% 0.31/1.50 thf(fact_211_list_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((case_list @ B) @ A) = (^ [F12:B,F23:(A > ((list @ A) > B)),List3:(list @ A)]: (((((if @ B) @ (List3 = (nil @ A))) @ F12) @ ((F23 @ ((hd @ A) @ List3)) @ ((tl @ A) @ List3))))))))).
% 0.31/1.50 thf(fact_212_list_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F22:(A > ((list @ A) > B)),X21:A,X22:(list @ A)]: (((((((case_list @ B) @ A) @ F1) @ F22) @ (((cons @ A) @ X21) @ X22)) = ((F22 @ X21) @ X22))))).
% 0.31/1.50 thf(fact_213_list_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F22:(A > ((list @ A) > B))]: (((((((case_list @ B) @ A) @ F1) @ F22) @ (nil @ A)) = F1)))).
% 0.31/1.50 thf(fact_214_Cons__in__shuffles__iff, axiom, (! [A:$tType,Z:A,Zs2:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: (((((member2 @ (list @ A)) @ (((cons @ A) @ Z) @ Zs2)) @ (((shuffles @ A) @ Xs) @ Ys2)) = (((Xs != (nil @ A)) & ((((hd @ A) @ Xs) = Z) & (((member2 @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ ((tl @ A) @ Xs)) @ Ys2)))) | ((Ys2 != (nil @ A)) & ((((hd @ A) @ Ys2) = Z) & (((member2 @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ ((tl @ A) @ Ys2)))))))))).
% 0.31/1.50 thf(fact_215_map2__Cons__not__empty, axiom, (! [A:$tType,B:$tType,C:$tType,Xs:(list @ A),F:(A > (C > B)),Y:C,Ys2:(list @ C)]: (((Xs != (nil @ A)) => (((((((map2 @ A) @ C) @ B) @ F) @ Xs) @ (((cons @ C) @ Y) @ Ys2)) = (((cons @ B) @ ((F @ ((hd @ A) @ Xs)) @ Y)) @ ((((((map2 @ A) @ C) @ B) @ F) @ ((tl @ A) @ Xs)) @ Ys2))))))).
% 0.31/1.50 thf(fact_216_Nil__in__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((member2 @ (list @ A)) @ (nil @ A)) @ (((shuffles @ A) @ Xs) @ Ys2)) = ((Xs = (nil @ A)) & (Ys2 = (nil @ A))))))).
% 0.31/1.50 thf(fact_217_splice__in__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((member2 @ (list @ A)) @ (((splice @ A) @ Xs) @ Ys2)) @ (((shuffles @ A) @ Xs) @ Ys2))))).
% 0.31/1.50 thf(fact_218_shufflesE, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: (((((member2 @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys2)) => (((Zs2 = Xs) => (Ys2 != (nil @ A))) => (((Zs2 = Ys2) => (Xs != (nil @ A))) => ((! [X:A,Xs5:(list @ A)]: (((Xs = (((cons @ A) @ X) @ Xs5)) => (! [Z2:A,Zs3:(list @ A)]: (((Zs2 = (((cons @ A) @ Z2) @ Zs3)) => ((X = Z2) => (~ ((((member2 @ (list @ A)) @ Zs3) @ (((shuffles @ A) @ Xs5) @ Ys2))))))))))) => (~ ((! [Y3:A,Ys5:(list @ A)]: (((Ys2 = (((cons @ A) @ Y3) @ Ys5)) => (! [Z2:A,Zs3:(list @ A)]: (((Zs2 = (((cons @ A) @ Z2) @ Zs3)) => ((Y3 = Z2) => (~ ((((member2 @ (list @ A)) @ Zs3) @ (((shuffles @ A) @ Xs) @ Ys5)))))))))))))))))))).
% 0.31/1.50 thf(fact_219_map2__Cons__Cons, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),X3:B,Xs:(list @ B),Y:C,Ys2:(list @ C)]: ((((((((map2 @ B) @ C) @ A) @ F) @ (((cons @ B) @ X3) @ Xs)) @ (((cons @ C) @ Y) @ Ys2)) = (((cons @ A) @ ((F @ X3) @ Y)) @ ((((((map2 @ B) @ C) @ A) @ F) @ Xs) @ Ys2)))))).
% 0.31/1.50 thf(fact_220_map2_Osimps_I1_J, axiom, (! [B:$tType,A:$tType,C:$tType,F:(A > (B > C)),Ys2:(list @ B)]: ((((((((map2 @ A) @ B) @ C) @ F) @ (nil @ A)) @ Ys2) = (nil @ C))))).
% 0.31/1.50 thf(fact_221_map2__Nil, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > (C > A)),Ys2:(list @ C)]: ((((((((map2 @ B) @ C) @ A) @ F) @ (nil @ B)) @ Ys2) = (nil @ A))))).
% 0.31/1.50 thf(fact_222_map2__empty__conv, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > (C > A)),Xs:(list @ B),Ys2:(list @ C)]: (((((((((map2 @ B) @ C) @ A) @ F) @ Xs) @ Ys2) = (nil @ A)) = (Xs = (nil @ B)))))).
% 0.31/1.50 thf(fact_223_map2__not__empty__conv, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > (C > A)),Xs:(list @ B),Ys2:(list @ C)]: (((((((((map2 @ B) @ C) @ A) @ F) @ Xs) @ Ys2) != (nil @ A)) = (Xs != (nil @ B)))))).
% 0.31/1.50 thf(fact_224_Cons__in__shuffles__leftI, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys2:(list @ A),Z:A]: (((((member2 @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys2)) => (((member2 @ (list @ A)) @ (((cons @ A) @ Z) @ Zs2)) @ (((shuffles @ A) @ (((cons @ A) @ Z) @ Xs)) @ Ys2)))))).
% 0.31/1.50 thf(fact_225_Cons__in__shuffles__rightI, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys2:(list @ A),Z:A]: (((((member2 @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys2)) => (((member2 @ (list @ A)) @ (((cons @ A) @ Z) @ Zs2)) @ (((shuffles @ A) @ Xs) @ (((cons @ A) @ Z) @ Ys2))))))).
% 0.31/1.50 thf(fact_226_Nil__in__shufflesI, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs = (nil @ A)) => ((Ys2 = (nil @ A)) => (((member2 @ (list @ A)) @ (nil @ A)) @ (((shuffles @ A) @ Xs) @ Ys2))))))).
% 0.31/1.50 thf(fact_227_shuffles__commutes, axiom, (! [A:$tType]: (((shuffles @ A) = (^ [Xs3:(list @ A),Ys:(list @ A)]: ((((shuffles @ A) @ Ys) @ Xs3))))))).
% 0.31/1.50 thf(fact_228_map2_Osimps_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X3:A,Xs:(list @ A),Ys2:(list @ B)]: ((((((((map2 @ A) @ B) @ C) @ F) @ (((cons @ A) @ X3) @ Xs)) @ Ys2) = (((cons @ C) @ ((F @ X3) @ ((hd @ B) @ Ys2))) @ ((((((map2 @ A) @ B) @ C) @ F) @ Xs) @ ((tl @ B) @ Ys2))))))).
% 0.31/1.50 thf(fact_229_map2__Cons__if, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ B),F:(B > (C > A)),Y:C,Ys2:(list @ C)]: ((((Xs = (nil @ B)) => (((((((map2 @ B) @ C) @ A) @ F) @ Xs) @ (((cons @ C) @ Y) @ Ys2)) = (nil @ A))) & ((Xs != (nil @ B)) => (((((((map2 @ B) @ C) @ A) @ F) @ Xs) @ (((cons @ C) @ Y) @ Ys2)) = (((cons @ A) @ ((F @ ((hd @ B) @ Xs)) @ Y)) @ ((((((map2 @ B) @ C) @ A) @ F) @ ((tl @ B) @ Xs)) @ Ys2)))))))).
% 0.31/1.50 thf(fact_230_remdups__adj__append_H, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((Xs = (nil @ A)) | ((Ys2 = (nil @ A)) | (((last @ A) @ Xs) != ((hd @ A) @ Ys2)))) => (((remdups_adj @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((append @ A) @ ((remdups_adj @ A) @ Xs)) @ ((remdups_adj @ A) @ Ys2))))))).
% 0.31/1.50 thf(fact_231_remdups__adj__append, axiom, (! [A:$tType,Xs_1:(list @ A),X3:A,Xs_2:(list @ A)]: ((((remdups_adj @ A) @ (((append @ A) @ Xs_1) @ (((cons @ A) @ X3) @ Xs_2))) = (((append @ A) @ ((remdups_adj @ A) @ (((append @ A) @ Xs_1) @ (((cons @ A) @ X3) @ (nil @ A))))) @ ((tl @ A) @ ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs_2)))))))).
% 0.31/1.50 thf(fact_232_remdups__adj__Nil__iff, axiom, (! [A:$tType,Xs:(list @ A)]: (((((remdups_adj @ A) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))).
% 0.31/1.50 thf(fact_233_hd__remdups__adj, axiom, (! [A:$tType,Xs:(list @ A)]: ((((hd @ A) @ ((remdups_adj @ A) @ Xs)) = ((hd @ A) @ Xs))))).
% 0.31/1.50 thf(fact_234_last__remdups__adj, axiom, (! [A:$tType,Xs:(list @ A)]: ((((last @ A) @ ((remdups_adj @ A) @ Xs)) = ((last @ A) @ Xs))))).
% 0.31/1.50 thf(fact_235_remdups__adj__Cons__alt, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((cons @ A) @ X3) @ ((tl @ A) @ ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)))) = ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)))))).
% 0.31/1.50 thf(fact_236_successively__remdups__adjI, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A)]: (((((successively @ A) @ P) @ Xs) => (((successively @ A) @ P) @ ((remdups_adj @ A) @ Xs)))))).
% 0.31/1.50 thf(fact_237_distinct__adj__remdups__adj, axiom, (! [A:$tType,Xs:(list @ A)]: (((distinct_adj @ A) @ ((remdups_adj @ A) @ Xs))))).
% 0.31/1.50 thf(fact_238_distinct__adj__altdef, axiom, (! [A:$tType]: (((distinct_adj @ A) = (^ [Xs3:(list @ A)]: ((((remdups_adj @ A) @ Xs3) = Xs3))))))).
% 0.31/1.50 thf(fact_239_remdups__adj_Oelims, axiom, (! [A:$tType,X3:(list @ A),Y:(list @ A)]: (((((remdups_adj @ A) @ X3) = Y) => (((X3 = (nil @ A)) => (Y != (nil @ A))) => ((! [X:A]: (((X3 = (((cons @ A) @ X) @ (nil @ A))) => (Y != (((cons @ A) @ X) @ (nil @ A)))))) => (~ ((! [X:A,Y3:A,Xs2:(list @ A)]: (((X3 = (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Xs2))) => (~ ((((X = Y3) => (Y = ((remdups_adj @ A) @ (((cons @ A) @ X) @ Xs2)))) & ((X != Y3) => (Y = (((cons @ A) @ X) @ ((remdups_adj @ A) @ (((cons @ A) @ Y3) @ Xs2))))))))))))))))))).
% 0.31/1.50 thf(fact_240_remdups__adj_Osimps_I2_J, axiom, (! [A:$tType,X3:A]: ((((remdups_adj @ A) @ (((cons @ A) @ X3) @ (nil @ A))) = (((cons @ A) @ X3) @ (nil @ A)))))).
% 0.31/1.50 thf(fact_241_remdups__adj_Osimps_I1_J, axiom, (! [A:$tType]: ((((remdups_adj @ A) @ (nil @ A)) = (nil @ A))))).
% 0.31/1.50 thf(fact_242_remdups__adj_Osimps_I3_J, axiom, (! [A:$tType,X3:A,Y:A,Xs:(list @ A)]: ((((X3 = Y) => (((remdups_adj @ A) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Xs))) = ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)))) & ((X3 != Y) => (((remdups_adj @ A) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Xs))) = (((cons @ A) @ X3) @ ((remdups_adj @ A) @ (((cons @ A) @ Y) @ Xs))))))))).
% 0.31/1.50 thf(fact_243_remdups__adj__append__two, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Y:A]: ((((remdups_adj @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ (nil @ A))))) = (((append @ A) @ ((remdups_adj @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))))) @ ((((if @ (list @ A)) @ (X3 = Y)) @ (nil @ A)) @ (((cons @ A) @ Y) @ (nil @ A)))))))).
% 0.31/1.50 thf(fact_244_map2__snoc__snoc, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys2:(list @ B),F:(A > (B > C)),X3:A,Y:B]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => (((((((map2 @ A) @ B) @ C) @ F) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) @ (((append @ B) @ Ys2) @ (((cons @ B) @ Y) @ (nil @ B)))) = (((append @ C) @ ((((((map2 @ A) @ B) @ C) @ F) @ Xs) @ Ys2)) @ (((cons @ C) @ ((F @ X3) @ Y)) @ (nil @ C)))))))).
% 0.31/1.50 thf(fact_245_list__strict__asc__imp__list__asc, axiom, (! [A:$tType]: (((preorder @ A) => (! [Xs:(list @ A)]: ((((list_strict_asc @ A) @ Xs) => ((list_asc @ A) @ Xs)))))))).
% 0.31/1.50 thf(fact_246_append__eq__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Us2:(list @ A),Vs2:(list @ A)]: ((((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys2)) | (((size_size @ (list @ A)) @ Us2) = ((size_size @ (list @ A)) @ Vs2))) => (((((append @ A) @ Xs) @ Us2) = (((append @ A) @ Ys2) @ Vs2)) = ((Xs = Ys2) & (Us2 = Vs2))))))).
% 0.31/1.50 thf(fact_247_length__rotate1, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((rotate1 @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))).
% 0.31/1.50 thf(fact_248_length__greater__imp__not__empty, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (Xs != (nil @ A)))))).
% 0.31/1.50 thf(fact_249_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys2)) => (Xs != Ys2))))).
% 0.31/1.50 thf(fact_250_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 0.31/1.50 thf(fact_251_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys6:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys6)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys6)))) => (P @ Xs2)))) => (P @ Xs))))).
% 0.31/1.50 thf(fact_252_list__induct2, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),P:((list @ A) > ((list @ B) > $o))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => (((P @ (nil @ A)) @ (nil @ B)) => ((! [X:A,Xs2:(list @ A),Y3:B,Ys3:(list @ B)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ B)) @ Ys3)) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ B) @ Y3) @ Ys3)))))) => ((P @ Xs) @ Ys2))))))).
% 0.31/1.50 thf(fact_253_list__induct3, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys2:(list @ B),Zs2:(list @ C),P:((list @ A) > ((list @ B) > ((list @ C) > $o)))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((size_size @ (list @ B)) @ Ys2) = ((size_size @ (list @ C)) @ Zs2)) => ((((P @ (nil @ A)) @ (nil @ B)) @ (nil @ C)) => ((! [X:A,Xs2:(list @ A),Y3:B,Ys3:(list @ B),Z2:C,Zs:(list @ C)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ B)) @ Ys3)) => ((((size_size @ (list @ B)) @ Ys3) = ((size_size @ (list @ C)) @ Zs)) => ((((P @ Xs2) @ Ys3) @ Zs) => (((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ B) @ Y3) @ Ys3)) @ (((cons @ C) @ Z2) @ Zs))))))) => (((P @ Xs) @ Ys2) @ Zs2)))))))).
% 0.31/1.50 thf(fact_254_list__induct4, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,Xs:(list @ A),Ys2:(list @ B),Zs2:(list @ C),Ws2:(list @ D),P:((list @ A) > ((list @ B) > ((list @ C) > ((list @ D) > $o))))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((size_size @ (list @ B)) @ Ys2) = ((size_size @ (list @ C)) @ Zs2)) => ((((size_size @ (list @ C)) @ Zs2) = ((size_size @ (list @ D)) @ Ws2)) => (((((P @ (nil @ A)) @ (nil @ B)) @ (nil @ C)) @ (nil @ D)) => ((! [X:A,Xs2:(list @ A),Y3:B,Ys3:(list @ B),Z2:C,Zs:(list @ C),W:D,Ws3:(list @ D)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ B)) @ Ys3)) => ((((size_size @ (list @ B)) @ Ys3) = ((size_size @ (list @ C)) @ Zs)) => ((((size_size @ (list @ C)) @ Zs) = ((size_size @ (list @ D)) @ Ws3)) => (((((P @ Xs2) @ Ys3) @ Zs) @ Ws3) => ((((P @ (((cons @ A) @ X) @ Xs2)) @ (((cons @ B) @ Y3) @ Ys3)) @ (((cons @ C) @ Z2) @ Zs)) @ (((cons @ D) @ W) @ Ws3)))))))) => ((((P @ Xs) @ Ys2) @ Zs2) @ Ws2))))))))).
% 0.31/1.50 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 0.31/1.50 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 0.31/1.50 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 0.31/1.50 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 0.31/1.50 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 0.31/1.50 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 0.31/1.50 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 0.31/1.50 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, (preorder @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, (linorder @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, (order @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, (ord @ $o)).
% 0.31/1.50 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 0.31/1.50 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y:A]: ((((((if @ A) @ $false) @ X3) @ Y) = Y)))).
% 0.31/1.50 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y:A]: ((((((if @ A) @ $true) @ X3) @ Y) = X3)))).
% 0.31/1.50 thf(conj_0, conjecture, ((((listIn521021761append @ a) @ (((cons @ a) @ x) @ xs)) @ f) = (((listIn521021761append @ a) @ (((cons @ a) @ x) @ (nil @ a))) @ (((listIn521021761append @ a) @ xs) @ f)))).
% 0.31/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.nSD6Sk1FeO/DTF2DTF14040.p
% 0.31/1.50 ------------------------
% 0.31/1.55 ---- Cleaned THF ---
% 0.31/1.56 thf(ty_t_List_Olist,type,
% 0.31/1.56 list: $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Set_Oset,type,
% 0.31/1.56 set: $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Nat_Onat,type,
% 0.31/1.56 nat: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_tf_a,type,
% 0.31/1.56 a: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(sy_cl_HOL_Otype,type,
% 0.31/1.56 type:
% 0.31/1.56 !>[A: $tType] : $o ).
% 0.31/1.56
% 0.31/1.56 thf(sy_cl_Orderings_Oord,type,
% 0.31/1.56 ord:
% 0.31/1.56 !>[A: $tType] : $o ).
% 0.31/1.56
% 0.31/1.56 thf(sy_cl_Orderings_Oorder,type,
% 0.31/1.56 order:
% 0.31/1.56 !>[A: $tType] : $o ).
% 0.31/1.56
% 0.31/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 0.31/1.56 linorder:
% 0.31/1.56 !>[A: $tType] : $o ).
% 0.31/1.56
% 0.31/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 0.31/1.56 preorder:
% 0.31/1.56 !>[A: $tType] : $o ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
% 0.31/1.56 bNF_Greatest_Shift:
% 0.31/1.56 !>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
% 0.31/1.56 bNF_Greatest_Succ:
% 0.31/1.56 !>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
% 0.31/1.56 bNF_Greatest_shift:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_If,type,
% 0.31/1.56 if:
% 0.31/1.56 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List2_Olist__asc,type,
% 0.31/1.56 list_asc:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List2_Olist__ord,type,
% 0.31/1.56 list_ord:
% 0.31/1.56 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List2_Olist__strict__asc,type,
% 0.31/1.56 list_strict_asc:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List2_Omap2,type,
% 0.31/1.56 map2:
% 0.31/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
% 0.31/1.56 listIn521021761append:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oappend,type,
% 0.31/1.56 append:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oarg__min__list,type,
% 0.31/1.56 arg_min_list:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Obind,type,
% 0.31/1.56 bind:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Obutlast,type,
% 0.31/1.56 butlast:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oconcat,type,
% 0.31/1.56 concat:
% 0.31/1.56 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Odistinct__adj,type,
% 0.31/1.56 distinct_adj:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Ofoldr,type,
% 0.31/1.56 foldr:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oinsert,type,
% 0.31/1.56 insert:
% 0.31/1.56 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olast,type,
% 0.31/1.56 last:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olist_OCons,type,
% 0.31/1.56 cons:
% 0.31/1.56 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olist_ONil,type,
% 0.31/1.56 nil:
% 0.31/1.56 !>[A: $tType] : ( list @ A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olist_Ocase__list,type,
% 0.31/1.56 case_list:
% 0.31/1.56 !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olist_Ohd,type,
% 0.31/1.56 hd:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olist_Otl,type,
% 0.31/1.56 tl:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olist__ex1,type,
% 0.31/1.56 list_ex1:
% 0.31/1.56 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Olistrelp,type,
% 0.31/1.56 listrelp:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Omap__tailrec__rev,type,
% 0.31/1.56 map_tailrec_rev:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Omaps,type,
% 0.31/1.56 maps:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Omember,type,
% 0.31/1.56 member:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oord_Olexordp__eq,type,
% 0.31/1.56 lexordp_eq:
% 0.31/1.56 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oord__class_Olexordp,type,
% 0.31/1.56 ord_lexordp:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oord__class_Olexordp__eq,type,
% 0.31/1.56 ord_lexordp_eq:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oproduct__lists,type,
% 0.31/1.56 product_lists:
% 0.31/1.56 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oremdups__adj,type,
% 0.31/1.56 remdups_adj:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Orotate1,type,
% 0.31/1.56 rotate1:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Oshuffles,type,
% 0.31/1.56 shuffles:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Osplice,type,
% 0.31/1.56 splice:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Osubseqs,type,
% 0.31/1.56 subseqs:
% 0.31/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_List_Osuccessively,type,
% 0.31/1.56 successively:
% 0.31/1.56 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.31/1.56 size_size:
% 0.31/1.56 !>[A: $tType] : ( A > nat ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.31/1.56 ord_less:
% 0.31/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Set_OCollect,type,
% 0.31/1.56 collect:
% 0.31/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_member,type,
% 0.31/1.56 member2:
% 0.31/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_f,type,
% 0.31/1.56 f: nat > a ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_x,type,
% 0.31/1.56 x: a ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_xs,type,
% 0.31/1.56 xs: list @ a ).
% 0.31/1.56
% 0.31/1.56 thf(fact_0_i__append__Nil,axiom,
% 0.31/1.56 ! [A: $tType,F: nat > A] :
% 0.31/1.56 ( ( listIn521021761append @ A @ ( nil @ A ) @ F )
% 0.31/1.56 = F ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_1_list_Oinject,axiom,
% 0.31/1.56 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 0.31/1.56 ( ( ( cons @ A @ X21 @ X22 )
% 0.31/1.56 = ( cons @ A @ Y21 @ Y22 ) )
% 0.31/1.56 = ( ( X21 = Y21 )
% 0.31/1.56 & ( X22 = Y22 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_2_list_Odistinct_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,X21: A,X22: list @ A] :
% 0.31/1.56 ( ( nil @ A )
% 0.31/1.56 != ( cons @ A @ X21 @ X22 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_3_list_OdiscI,axiom,
% 0.31/1.56 ! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
% 0.31/1.56 ( ( List
% 0.31/1.56 = ( cons @ A @ X21 @ X22 ) )
% 0.31/1.56 => ( List
% 0.31/1.56 != ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_4_list_Oexhaust,axiom,
% 0.31/1.56 ! [A: $tType,Y: list @ A] :
% 0.31/1.56 ( ( Y
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ~ ! [X212: A,X222: list @ A] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( cons @ A @ X212 @ X222 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_5_list_Oinducts,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 0.31/1.56 ( ( P @ ( nil @ A ) )
% 0.31/1.56 => ( ! [X1: A,X2: list @ A] :
% 0.31/1.56 ( ( P @ X2 )
% 0.31/1.56 => ( P @ ( cons @ A @ X1 @ X2 ) ) )
% 0.31/1.56 => ( P @ List ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_6_neq__Nil__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 = ( ? [Y2: A,Ys: list @ A] :
% 0.31/1.56 ( Xs
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_7_list__induct2_H,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
% 0.31/1.56 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
% 0.31/1.56 => ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
% 0.31/1.56 ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
% 0.31/1.56 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_8_splice_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.31/1.56 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ( P @ Ys3 @ Xs2 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 ) )
% 0.31/1.56 => ( P @ A0 @ A1 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_9_induct__list012,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.31/1.56 ( ( P @ ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( ! [X: A,Y3: A,Zs: list @ A] :
% 0.31/1.56 ( ( P @ Zs )
% 0.31/1.56 => ( ( P @ ( cons @ A @ Y3 @ Zs ) )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
% 0.31/1.56 => ( P @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_10_min__list_Ocases,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: list @ A] :
% 0.31/1.56 ( ! [X: A,Xs2: list @ A] :
% 0.31/1.56 ( X3
% 0.31/1.56 != ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ( X3
% 0.31/1.56 = ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_11_min__list_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.31/1.56 ( ! [X: A,Xs2: list @ A] :
% 0.31/1.56 ( ! [X213: A,X223: list @ A] :
% 0.31/1.56 ( ( Xs2
% 0.31/1.56 = ( cons @ A @ X213 @ X223 ) )
% 0.31/1.56 => ( P @ Xs2 ) )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) ) )
% 0.31/1.56 => ( ( P @ ( nil @ A ) )
% 0.31/1.56 => ( P @ A0 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_12_shuffles_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.31/1.56 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.31/1.56 => ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
% 0.31/1.56 ( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
% 0.31/1.56 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_13_not__Cons__self2,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( cons @ A @ X3 @ Xs )
% 0.31/1.56 != Xs ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_14_strict__sorted_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.31/1.56 ( ( P @ ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A,Ys3: list @ A] :
% 0.31/1.56 ( ( P @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Ys3 ) ) )
% 0.31/1.56 => ( P @ A0 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_15_strict__sorted_Ocases,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [X3: list @ A] :
% 0.31/1.56 ( ( X3
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ~ ! [X: A,Ys3: list @ A] :
% 0.31/1.56 ( X3
% 0.31/1.56 != ( cons @ A @ X @ Ys3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_16_map__tailrec__rev_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
% 0.31/1.56 ( ! [F2: A > B,X_1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X_1 )
% 0.31/1.56 => ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
% 0.31/1.56 ( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
% 0.31/1.56 => ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 0.31/1.56 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_17_list__nonempty__induct,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xs2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( P @ Xs2 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
% 0.31/1.56 => ( P @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_18_successively_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.31/1.56 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 0.31/1.56 => ( ! [P2: A > A > $o,X: A] : ( P @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( ! [P2: A > A > $o,X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( P @ P2 @ ( cons @ A @ Y3 @ Xs2 ) )
% 0.31/1.56 => ( P @ P2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
% 0.31/1.56 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_19_arg__min__list_Oinduct,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType] :
% 0.31/1.56 ( ( linorder @ B )
% 0.31/1.56 => ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
% 0.31/1.56 ( ! [F2: A > B,X: A] : ( P @ F2 @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( ! [F2: A > B,X: A,Y3: A,Zs: list @ A] :
% 0.31/1.56 ( ( P @ F2 @ ( cons @ A @ Y3 @ Zs ) )
% 0.31/1.56 => ( P @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) )
% 0.31/1.56 => ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
% 0.31/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_20_remdups__adj_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 0.31/1.56 ( ( P @ ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( ( X = Y3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) ) )
% 0.31/1.56 => ( ( ( X != Y3 )
% 0.31/1.56 => ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
% 0.31/1.56 => ( P @ A0 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_21_sorted__wrt_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.31/1.56 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 0.31/1.56 => ( ! [P2: A > A > $o,X: A,Ys3: list @ A] :
% 0.31/1.56 ( ( P @ P2 @ Ys3 )
% 0.31/1.56 => ( P @ P2 @ ( cons @ A @ X @ Ys3 ) ) )
% 0.31/1.56 => ( P @ A0 @ A1 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_22_remdups__adj_Ocases,axiom,
% 0.31/1.56 ! [A: $tType,X3: list @ A] :
% 0.31/1.56 ( ( X3
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( X3
% 0.31/1.56 != ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( X3
% 0.31/1.56 != ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_23_transpose_Ocases,axiom,
% 0.31/1.56 ! [A: $tType,X3: list @ ( list @ A )] :
% 0.31/1.56 ( ( X3
% 0.31/1.56 != ( nil @ ( list @ A ) ) )
% 0.31/1.56 => ( ! [Xss: list @ ( list @ A )] :
% 0.31/1.56 ( X3
% 0.31/1.56 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.31/1.56 => ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
% 0.31/1.56 ( X3
% 0.31/1.56 != ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_24_list__ord_Oinduct,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.31/1.56 ( ! [Ord: A > A > $o,X1: A,X2: A,Xs2: list @ A] :
% 0.31/1.56 ( ( P @ Ord @ ( cons @ A @ X2 @ Xs2 ) )
% 0.31/1.56 => ( P @ Ord @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.31/1.56 => ( ! [Ord: A > A > $o] : ( P @ Ord @ ( nil @ A ) )
% 0.31/1.56 => ( ! [Ord: A > A > $o,V: A] : ( P @ Ord @ ( cons @ A @ V @ ( nil @ A ) ) )
% 0.31/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_25_insert__Nil,axiom,
% 0.31/1.56 ! [A: $tType,X3: A] :
% 0.31/1.56 ( ( insert @ A @ X3 @ ( nil @ A ) )
% 0.31/1.56 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_26_list__ex1__simps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,P: A > $o] :
% 0.31/1.56 ~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_27_map__tailrec__rev_Oelims,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
% 0.31/1.56 ( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
% 0.31/1.56 = Y )
% 0.31/1.56 => ( ( ( Xa
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( Y != Xb ) )
% 0.31/1.56 => ~ ! [A3: A,As: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ A3 @ As ) )
% 0.31/1.56 => ( Y
% 0.31/1.56 != ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_28_arg__min__list_Osimps_I1_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType] :
% 0.31/1.56 ( ( linorder @ B )
% 0.31/1.56 => ! [F: A > B,X3: A] :
% 0.31/1.56 ( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.31/1.56 = X3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_29_ord_Olexordp__eq__simps_I3_J,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
% 0.31/1.56 ~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_30_bind__simps_I1_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.31/1.56 ( ( bind @ B @ A @ ( nil @ B ) @ F )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_31_listrelp_Oinducts,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,R: A > B > $o,X12: list @ A,X23: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 0.31/1.56 ( ( listrelp @ A @ B @ R @ X12 @ X23 )
% 0.31/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.31/1.56 => ( ! [X: A,Y3: B,Xs2: list @ A,Ys3: list @ B] :
% 0.31/1.56 ( ( R @ X @ Y3 )
% 0.31/1.56 => ( ( listrelp @ A @ B @ R @ Xs2 @ Ys3 )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) ) )
% 0.31/1.56 => ( P @ X12 @ X23 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_32_listrelp_Osimps,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType] :
% 0.31/1.56 ( ( listrelp @ A @ B )
% 0.31/1.56 = ( ^ [R2: A > B > $o,A12: list @ A,A22: list @ B] :
% 0.31/1.56 ( ( ( A12
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( nil @ B ) ) )
% 0.31/1.56 | ? [X4: A,Y2: B,Xs3: list @ A,Ys: list @ B] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ B @ Y2 @ Ys ) )
% 0.31/1.56 & ( R2 @ X4 @ Y2 )
% 0.31/1.56 & ( listrelp @ A @ B @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_33_listrelp_Ocases,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,R: A > B > $o,A1: list @ A,A2: list @ B] :
% 0.31/1.56 ( ( listrelp @ A @ B @ R @ A1 @ A2 )
% 0.31/1.56 => ( ( ( A1
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( A2
% 0.31/1.56 != ( nil @ B ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: B,Xs2: list @ A] :
% 0.31/1.56 ( ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Ys3: list @ B] :
% 0.31/1.56 ( ( A2
% 0.31/1.56 = ( cons @ B @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ( R @ X @ Y3 )
% 0.31/1.56 => ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_34_lexordp__eq__simps_I3_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Xs: list @ A] :
% 0.31/1.56 ~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_35_lexordp__eq__simps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Xs: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_36_lexordp__eq__simps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_37_ord_Olexordp__eq__simps_I4_J,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.31/1.56 ( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.31/1.56 = ( ( Less @ X3 @ Y )
% 0.31/1.56 | ( ~ ( Less @ Y @ X3 )
% 0.31/1.56 & ( lexordp_eq @ A @ Less @ Xs @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_38_ord_Olexordp__eq__simps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,Xs: list @ A] :
% 0.31/1.56 ( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_39_ord_Olexordp__eq__simps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_40_ord_Olexordp__eq_Ocong,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( lexordp_eq @ A )
% 0.31/1.56 = ( lexordp_eq @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_41_ord_Olexordp__eq__refl,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_42_lexordp__eq__refl,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_43_lexordp__eq__trans,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ( ord_lexordp_eq @ A @ Ys2 @ Zs2 )
% 0.31/1.56 => ( ord_lexordp_eq @ A @ Xs @ Zs2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_44_mem__Collect__eq,axiom,
% 0.31/1.56 ! [A: $tType,A4: A,P: A > $o] :
% 0.31/1.56 ( ( member2 @ A @ A4 @ ( collect @ A @ P ) )
% 0.31/1.56 = ( P @ A4 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_45_Collect__mem__eq,axiom,
% 0.31/1.56 ! [A: $tType,A5: set @ A] :
% 0.31/1.56 ( ( collect @ A
% 0.31/1.56 @ ^ [X4: A] : ( member2 @ A @ X4 @ A5 ) )
% 0.31/1.56 = A5 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_46_Collect__cong,axiom,
% 0.31/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/1.56 ( ! [X: A] :
% 0.31/1.56 ( ( P @ X )
% 0.31/1.56 = ( Q @ X ) )
% 0.31/1.56 => ( ( collect @ A @ P )
% 0.31/1.56 = ( collect @ A @ Q ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_47_ext,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/1.56 ( ! [X: A] :
% 0.31/1.56 ( ( F @ X )
% 0.31/1.56 = ( G @ X ) )
% 0.31/1.56 => ( F = G ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_48_lexordp__eq__linear,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.31/1.56 | ( ord_lexordp_eq @ A @ Ys2 @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_49_lexordp__eq__antisym,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ( ord_lexordp_eq @ A @ Ys2 @ Xs )
% 0.31/1.56 => ( Xs = Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_50_lexordp__eq_ONil,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_51_ord_Olexordp__eq_OCons,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( Less @ X3 @ Y )
% 0.31/1.56 => ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_52_ord_Olexordp__eq_OCons__eq,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ~ ( Less @ X3 @ Y )
% 0.31/1.56 => ( ~ ( Less @ Y @ X3 )
% 0.31/1.56 => ( ( lexordp_eq @ A @ Less @ Xs @ Ys2 )
% 0.31/1.56 => ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_53_ord_Olexordp__eq_ONil,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_54_listrelp_OCons,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,R: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 0.31/1.56 ( ( R @ X3 @ Y )
% 0.31/1.56 => ( ( listrelp @ A @ B @ R @ Xs @ Ys2 )
% 0.31/1.56 => ( listrelp @ A @ B @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_55_listrelp_ONil,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_56_map__tailrec__rev_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,F: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
% 0.31/1.56 ( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A4 @ As2 ) @ Bs2 )
% 0.31/1.56 = ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A4 ) @ Bs2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_57_map__tailrec__rev_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,F: A > B,Bs2: list @ B] :
% 0.31/1.56 ( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs2 )
% 0.31/1.56 = Bs2 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_58_ord_Olexordp__eq_Ocases,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,A1: list @ A,A2: list @ A] :
% 0.31/1.56 ( ( lexordp_eq @ A @ Less @ A1 @ A2 )
% 0.31/1.56 => ( ( A1
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( ? [Xs2: list @ A] :
% 0.31/1.56 ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Y3: A] :
% 0.31/1.56 ( ? [Ys3: list @ A] :
% 0.31/1.56 ( A2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ~ ( Less @ X @ Y3 ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Ys3: list @ A] :
% 0.31/1.56 ( ( A2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ~ ( Less @ X @ Y3 )
% 0.31/1.56 => ( ~ ( Less @ Y3 @ X )
% 0.31/1.56 => ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_59_ord_Olexordp__eq_Osimps,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( lexordp_eq @ A )
% 0.31/1.56 = ( ^ [Less2: A > A > $o,A12: list @ A,A22: list @ A] :
% 0.31/1.56 ( ? [Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( A22 = Ys ) )
% 0.31/1.56 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) )
% 0.31/1.56 & ( Less2 @ X4 @ Y2 ) )
% 0.31/1.56 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) )
% 0.31/1.56 & ~ ( Less2 @ X4 @ Y2 )
% 0.31/1.56 & ~ ( Less2 @ Y2 @ X4 )
% 0.31/1.56 & ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_60_ord_Olexordp__eq_Oinducts,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.31/1.56 ( ( lexordp_eq @ A @ Less @ X12 @ X23 )
% 0.31/1.56 => ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ( Less @ X @ Y3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ~ ( Less @ X @ Y3 )
% 0.31/1.56 => ( ~ ( Less @ Y3 @ X )
% 0.31/1.56 => ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
% 0.31/1.56 => ( P @ X12 @ X23 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_61_product__lists_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 0.31/1.56 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_62_subseqs_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( subseqs @ A @ ( nil @ A ) )
% 0.31/1.56 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_63_lexordp__eq_Ocases,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [A1: list @ A,A2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ A1 @ A2 )
% 0.31/1.56 => ( ( A1
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( ? [Xs2: list @ A] :
% 0.31/1.56 ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Y3: A] :
% 0.31/1.56 ( ? [Ys3: list @ A] :
% 0.31/1.56 ( A2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ~ ( ord_less @ A @ X @ Y3 ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Ys3: list @ A] :
% 0.31/1.56 ( ( A2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.31/1.56 => ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_64_lexordp__eq_Osimps,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ( ( ord_lexordp_eq @ A )
% 0.31/1.56 = ( ^ [A12: list @ A,A22: list @ A] :
% 0.31/1.56 ( ? [Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( A22 = Ys ) )
% 0.31/1.56 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) )
% 0.31/1.56 & ( ord_less @ A @ X4 @ Y2 ) )
% 0.31/1.56 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) )
% 0.31/1.56 & ~ ( ord_less @ A @ X4 @ Y2 )
% 0.31/1.56 & ~ ( ord_less @ A @ Y2 @ X4 )
% 0.31/1.56 & ( ord_lexordp_eq @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_65_lexordp__eq_Oinducts,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ X12 @ X23 )
% 0.31/1.56 => ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.31/1.56 => ( ( ord_lexordp_eq @ A @ Xs2 @ Ys3 )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
% 0.31/1.56 => ( P @ X12 @ X23 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_66_member__rec_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,Y: A] :
% 0.31/1.56 ~ ( member @ A @ ( nil @ A ) @ Y ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_67_member__rec_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A,Y: A] :
% 0.31/1.56 ( ( member @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
% 0.31/1.56 = ( ( X3 = Y )
% 0.31/1.56 | ( member @ A @ Xs @ Y ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_68_splice_Oelims,axiom,
% 0.31/1.56 ! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
% 0.31/1.56 ( ( ( splice @ A @ X3 @ Xa )
% 0.31/1.56 = Y )
% 0.31/1.56 => ( ( ( X3
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( Y != Xa ) )
% 0.31/1.56 => ~ ! [X: A,Xs2: list @ A] :
% 0.31/1.56 ( ( X3
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ( Y
% 0.31/1.56 != ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_69_bind__simps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,X3: B,Xs: list @ B,F: B > ( list @ A )] :
% 0.31/1.56 ( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F )
% 0.31/1.56 = ( append @ A @ ( F @ X3 ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_70_same__append__eq,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ Ys2 )
% 0.31/1.56 = ( append @ A @ Xs @ Zs2 ) )
% 0.31/1.56 = ( Ys2 = Zs2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_71_append__same__eq,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Ys2 @ Xs )
% 0.31/1.56 = ( append @ A @ Zs2 @ Xs ) )
% 0.31/1.56 = ( Ys2 = Zs2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_72_append__assoc,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs2 )
% 0.31/1.56 = ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_73_append_Oassoc,axiom,
% 0.31/1.56 ! [A: $tType,A4: list @ A,B2: list @ A,C2: list @ A] :
% 0.31/1.56 ( ( append @ A @ ( append @ A @ A4 @ B2 ) @ C2 )
% 0.31/1.56 = ( append @ A @ A4 @ ( append @ A @ B2 @ C2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_74_append_Oright__neutral,axiom,
% 0.31/1.56 ! [A: $tType,A4: list @ A] :
% 0.31/1.56 ( ( append @ A @ A4 @ ( nil @ A ) )
% 0.31/1.56 = A4 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_75_append__is__Nil__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ Ys2 )
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_76_Nil__is__append__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( nil @ A )
% 0.31/1.56 = ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_77_self__append__conv2,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 = ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_78_append__self__conv2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ Ys2 )
% 0.31/1.56 = Ys2 )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_79_self__append__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( Ys2
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_80_append__self__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ Ys2 )
% 0.31/1.56 = Xs )
% 0.31/1.56 = ( Ys2
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_81_append__Nil2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( append @ A @ Xs @ ( nil @ A ) )
% 0.31/1.56 = Xs ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_82_split__Nil__iff,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( splice @ A @ Xs @ Ys2 )
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_83_splice__Nil2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( splice @ A @ Xs @ ( nil @ A ) )
% 0.31/1.56 = Xs ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_84_i__append__assoc,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,F: nat > A] :
% 0.31/1.56 ( ( listIn521021761append @ A @ Xs @ ( listIn521021761append @ A @ Ys2 @ F ) )
% 0.31/1.56 = ( listIn521021761append @ A @ ( append @ A @ Xs @ Ys2 ) @ F ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_85_append1__eq__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A,Y: A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.31/1.56 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 0.31/1.56 = ( ( Xs = Ys2 )
% 0.31/1.56 & ( X3 = Y ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_86_lexordp__eq__simps_I4_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.31/1.56 = ( ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 | ( ~ ( ord_less @ A @ Y @ X3 )
% 0.31/1.56 & ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_87_append__eq__append__conv2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A,Ts: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ Ys2 )
% 0.31/1.56 = ( append @ A @ Zs2 @ Ts ) )
% 0.31/1.56 = ( ? [Us: list @ A] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( append @ A @ Zs2 @ Us ) )
% 0.31/1.56 & ( ( append @ A @ Us @ Ys2 )
% 0.31/1.56 = Ts ) )
% 0.31/1.56 | ( ( ( append @ A @ Xs @ Us )
% 0.31/1.56 = Zs2 )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_88_append__eq__appendI,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys2: list @ A,Us2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ Xs1 )
% 0.31/1.56 = Zs2 )
% 0.31/1.56 => ( ( Ys2
% 0.31/1.56 = ( append @ A @ Xs1 @ Us2 ) )
% 0.31/1.56 => ( ( append @ A @ Xs @ Ys2 )
% 0.31/1.56 = ( append @ A @ Zs2 @ Us2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_89_append__Cons,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
% 0.31/1.56 = ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_90_Cons__eq__appendI,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ( cons @ A @ X3 @ Xs1 )
% 0.31/1.56 = Ys2 )
% 0.31/1.56 => ( ( Xs
% 0.31/1.56 = ( append @ A @ Xs1 @ Zs2 ) )
% 0.31/1.56 => ( ( cons @ A @ X3 @ Xs )
% 0.31/1.56 = ( append @ A @ Ys2 @ Zs2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_91_append_Oleft__neutral,axiom,
% 0.31/1.56 ! [A: $tType,A4: list @ A] :
% 0.31/1.56 ( ( append @ A @ ( nil @ A ) @ A4 )
% 0.31/1.56 = A4 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_92_append__Nil,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A] :
% 0.31/1.56 ( ( append @ A @ ( nil @ A ) @ Ys2 )
% 0.31/1.56 = Ys2 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_93_eq__Nil__appendI,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( Xs = Ys2 )
% 0.31/1.56 => ( Xs
% 0.31/1.56 = ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_94_lexordp__eq__pref,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [U: list @ A,V2: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_95_ord_Olexordp__eq__pref,axiom,
% 0.31/1.56 ! [A: $tType,Less: A > A > $o,U: list @ A,V2: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_96_splice_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
% 0.31/1.56 = ( cons @ A @ X3 @ ( splice @ A @ Ys2 @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_97_splice_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A] :
% 0.31/1.56 ( ( splice @ A @ ( nil @ A ) @ Ys2 )
% 0.31/1.56 = Ys2 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_98_rev__nonempty__induct,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xs2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( P @ Xs2 )
% 0.31/1.56 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
% 0.31/1.56 => ( P @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_99_append__eq__Cons__conv,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Zs2: list @ A,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Ys2 @ Zs2 )
% 0.31/1.56 = ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( Zs2
% 0.31/1.56 = ( cons @ A @ X3 @ Xs ) ) )
% 0.31/1.56 | ? [Ys4: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 = ( cons @ A @ X3 @ Ys4 ) )
% 0.31/1.56 & ( ( append @ A @ Ys4 @ Zs2 )
% 0.31/1.56 = Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_100_Cons__eq__append__conv,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ( cons @ A @ X3 @ Xs )
% 0.31/1.56 = ( append @ A @ Ys2 @ Zs2 ) )
% 0.31/1.56 = ( ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( ( cons @ A @ X3 @ Xs )
% 0.31/1.56 = Zs2 ) )
% 0.31/1.56 | ? [Ys4: list @ A] :
% 0.31/1.56 ( ( ( cons @ A @ X3 @ Ys4 )
% 0.31/1.56 = Ys2 )
% 0.31/1.56 & ( Xs
% 0.31/1.56 = ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_101_rev__exhaust,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ~ ! [Ys3: list @ A,Y3: A] :
% 0.31/1.56 ( Xs
% 0.31/1.56 != ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_102_rev__induct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.31/1.56 ( ( P @ ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A] :
% 0.31/1.56 ( ( P @ Xs2 )
% 0.31/1.56 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
% 0.31/1.56 => ( P @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_103_append__eq__Cons,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs )
% 0.31/1.56 = ( cons @ A @ X3 @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_104_lexordp__eq_OCons,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 => ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_105_lexordp__eq_OCons__eq,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 => ( ~ ( ord_less @ A @ Y @ X3 )
% 0.31/1.56 => ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_106_maps__simps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,F: B > ( list @ A ),X3: B,Xs: list @ B] :
% 0.31/1.56 ( ( maps @ B @ A @ F @ ( cons @ B @ X3 @ Xs ) )
% 0.31/1.56 = ( append @ A @ ( F @ X3 ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_107_concat__eq__append__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ( concat @ A @ Xss2 )
% 0.31/1.56 = ( append @ A @ Ys2 @ Zs2 ) )
% 0.31/1.56 = ( ( ( Xss2
% 0.31/1.56 = ( nil @ ( list @ A ) ) )
% 0.31/1.56 => ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( Zs2
% 0.31/1.56 = ( nil @ A ) ) ) )
% 0.31/1.56 & ( ( Xss2
% 0.31/1.56 != ( nil @ ( list @ A ) ) )
% 0.31/1.56 => ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
% 0.31/1.56 ( ( Xss2
% 0.31/1.56 = ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
% 0.31/1.56 & ( Zs2
% 0.31/1.56 = ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_108_rotate1_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_109_butlast__snoc,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.31/1.56 = Xs ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_110_last__snoc,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.31/1.56 = X3 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_111_SuccI,axiom,
% 0.31/1.56 ! [A: $tType,Kl: list @ A,K: A,Kl2: set @ ( list @ A )] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 )
% 0.31/1.56 => ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_112_SuccD,axiom,
% 0.31/1.56 ! [A: $tType,K: A,Kl2: set @ ( list @ A ),Kl: list @ A] :
% 0.31/1.56 ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) )
% 0.31/1.56 => ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_113_rotate1__is__Nil__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( ( rotate1 @ A @ Xs )
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_114_last__appendL,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( last @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_115_last__appendR,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( last @ A @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_116_concat__append,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
% 0.31/1.56 ( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys2 ) )
% 0.31/1.56 = ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_117_append__butlast__last__id,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
% 0.31/1.56 = Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_118_butlast_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( butlast @ A @ ( nil @ A ) )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_119_snoc__eq__iff__butlast,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.31/1.56 = Ys2 )
% 0.31/1.56 = ( ( Ys2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 & ( ( butlast @ A @ Ys2 )
% 0.31/1.56 = Xs )
% 0.31/1.56 & ( ( last @ A @ Ys2 )
% 0.31/1.56 = X3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_120_rotate1_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( rotate1 @ A @ ( nil @ A ) )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_121_last_Osimps,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = X3 ) )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( last @ A @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_122_last__ConsL,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = X3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_123_last__ConsR,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( last @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_124_last__append,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( last @ A @ Xs ) ) )
% 0.31/1.56 & ( ( Ys2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( last @ A @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_125_longest__common__suffix,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ? [Ss: list @ A,Xs5: list @ A,Ys5: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( append @ A @ Xs5 @ Ss ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( append @ A @ Ys5 @ Ss ) )
% 0.31/1.56 & ( ( Xs5
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( Ys5
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( last @ A @ Xs5 )
% 0.31/1.56 != ( last @ A @ Ys5 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_126_butlast_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( nil @ A ) ) )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( cons @ A @ X3 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_127_concat_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( concat @ A @ ( nil @ ( list @ A ) ) )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_128_concat_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] :
% 0.31/1.56 ( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) )
% 0.31/1.56 = ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_129_butlast__append,axiom,
% 0.31/1.56 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( butlast @ A @ Xs ) ) )
% 0.31/1.56 & ( ( Ys2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( append @ A @ Xs @ ( butlast @ A @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_130_maps__simps_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.31/1.56 ( ( maps @ B @ A @ F @ ( nil @ B ) )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_131_concat__eq__appendD,axiom,
% 0.31/1.56 ! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ( concat @ A @ Xss2 )
% 0.31/1.56 = ( append @ A @ Ys2 @ Zs2 ) )
% 0.31/1.56 => ( ( Xss2
% 0.31/1.56 != ( nil @ ( list @ A ) ) )
% 0.31/1.56 => ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
% 0.31/1.56 ( ( Xss2
% 0.31/1.56 = ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs5 ) @ Xss23 ) ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
% 0.31/1.56 & ( Zs2
% 0.31/1.56 = ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_132_shift__def,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType] :
% 0.31/1.56 ( ( bNF_Greatest_shift @ A @ B )
% 0.31/1.56 = ( ^ [Lab: ( list @ A ) > B,K2: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_133_empty__Shift,axiom,
% 0.31/1.56 ! [A: $tType,Kl2: set @ ( list @ A ),K: A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl2 )
% 0.31/1.56 => ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ ( nil @ A ) ) )
% 0.31/1.56 => ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_134_Succ__Shift,axiom,
% 0.31/1.56 ! [A: $tType,Kl2: set @ ( list @ A ),K: A,Kl: list @ A] :
% 0.31/1.56 ( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) @ Kl )
% 0.31/1.56 = ( bNF_Greatest_Succ @ A @ Kl2 @ ( cons @ A @ K @ Kl ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_135_list__ord__snoc,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( list_ord @ A @ Ord2 @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( Ord2 @ ( last @ A @ Xs ) @ X3 )
% 0.31/1.56 & ( list_ord @ A @ Ord2 @ Xs ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_136_concat__conv__foldr,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( concat @ A )
% 0.31/1.56 = ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_137_foldr__append,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,F: B > A > A,Xs: list @ B,Ys2: list @ B,A4: A] :
% 0.31/1.56 ( ( foldr @ B @ A @ F @ ( append @ B @ Xs @ Ys2 ) @ A4 )
% 0.31/1.56 = ( foldr @ B @ A @ F @ Xs @ ( foldr @ B @ A @ F @ Ys2 @ A4 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_138_list__ord_Oelims_I3_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A > A > $o,Xa: list @ A] :
% 0.31/1.56 ( ~ ( list_ord @ A @ X3 @ Xa )
% 0.31/1.56 => ~ ! [X1: A,X2: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.31/1.56 => ( ( X3 @ X1 @ X2 )
% 0.31/1.56 & ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_139_list__ord_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,X12: A,X23: A,Xs: list @ A] :
% 0.31/1.56 ( ( list_ord @ A @ Ord2 @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs ) ) )
% 0.31/1.56 = ( ( Ord2 @ X12 @ X23 )
% 0.31/1.56 & ( list_ord @ A @ Ord2 @ ( cons @ A @ X23 @ Xs ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_140_list__ord_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o] : ( list_ord @ A @ Ord2 @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_141_list__ord__Nil,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o] : ( list_ord @ A @ Ord2 @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_142_list__ord__imp,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,Ord3: A > A > $o,Xs: list @ A] :
% 0.31/1.56 ( ! [X: A,Y3: A] :
% 0.31/1.56 ( ( Ord2 @ X @ Y3 )
% 0.31/1.56 => ( Ord3 @ X @ Y3 ) )
% 0.31/1.56 => ( ( list_ord @ A @ Ord2 @ Xs )
% 0.31/1.56 => ( list_ord @ A @ Ord3 @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_143_list__ord__one,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,X3: A] : ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_144_list__ord_Osimps_I3_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,V2: A] : ( list_ord @ A @ Ord2 @ ( cons @ A @ V2 @ ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_145_list__ord_Oelims_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A > A > $o,Xa: list @ A,Y: $o] :
% 0.31/1.56 ( ( ( list_ord @ A @ X3 @ Xa )
% 0.31/1.56 = Y )
% 0.31/1.56 => ( ! [X1: A,X2: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.31/1.56 => ( Y
% 0.31/1.56 = ( ~ ( ( X3 @ X1 @ X2 )
% 0.31/1.56 & ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) )
% 0.31/1.56 => ( ( ( Xa
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ~ Y )
% 0.31/1.56 => ~ ( ? [V: A] :
% 0.31/1.56 ( Xa
% 0.31/1.56 = ( cons @ A @ V @ ( nil @ A ) ) )
% 0.31/1.56 => ~ Y ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_146_list__ord_Oelims_I2_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A > A > $o,Xa: list @ A] :
% 0.31/1.56 ( ( list_ord @ A @ X3 @ Xa )
% 0.31/1.56 => ( ! [X1: A,X2: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.31/1.56 => ~ ( ( X3 @ X1 @ X2 )
% 0.31/1.56 & ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.31/1.56 => ( ( Xa
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ~ ! [V: A] :
% 0.31/1.56 ( Xa
% 0.31/1.56 != ( cons @ A @ V @ ( nil @ A ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_147_ShiftD,axiom,
% 0.31/1.56 ! [A: $tType,Kl: list @ A,Kl2: set @ ( list @ A ),K: A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ Kl @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) )
% 0.31/1.56 => ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl ) @ Kl2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_148_list__strict__asc__def,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ( ( list_strict_asc @ A )
% 0.31/1.56 = ( list_ord @ A @ ( ord_less @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_149_list__ord__append,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( list_ord @ A @ Ord2 @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( ( list_ord @ A @ Ord2 @ Xs )
% 0.31/1.56 & ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( list_ord @ A @ Ord2 @ Ys2 )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( Ord2 @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_150_lexordp__iff,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ( ( ord_lexordp @ A )
% 0.31/1.56 = ( ^ [Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ? [X4: A,Vs: list @ A] :
% 0.31/1.56 ( Ys
% 0.31/1.56 = ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs ) ) )
% 0.31/1.56 | ? [Us: list @ A,A6: A,B3: A,Vs: list @ A,Ws: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ A6 @ B3 )
% 0.31/1.56 & ( Xs3
% 0.31/1.56 = ( append @ A @ Us @ ( cons @ A @ A6 @ Vs ) ) )
% 0.31/1.56 & ( Ys
% 0.31/1.56 = ( append @ A @ Us @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_151_lexordp__append__left__rightI,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Y: A,Us2: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 => ( ord_lexordp @ A @ ( append @ A @ Us2 @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_152_lexordp__simps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Xs: list @ A] :
% 0.31/1.56 ~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_153_lexordp__simps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ ( nil @ A ) @ Ys2 )
% 0.31/1.56 = ( Ys2
% 0.31/1.56 != ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_154_lexordp__simps_I3_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.31/1.56 = ( ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 | ( ~ ( ord_less @ A @ Y @ X3 )
% 0.31/1.56 & ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_155_hd__append2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( hd @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_156_lexordp__append__leftI,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Us2: list @ A,Vs2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Us2 @ Vs2 )
% 0.31/1.56 => ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_157_lexordp__antisym,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( order @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 => ~ ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_158_lexordp__trans,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ( ord_lexordp @ A @ Ys2 @ Zs2 )
% 0.31/1.56 => ( ord_lexordp @ A @ Xs @ Zs2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_159_lexordp__linear,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 | ( Xs = Ys2 )
% 0.31/1.56 | ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_160_lexordp__irreflexive_H,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( order @ A )
% 0.31/1.56 => ! [Xs: list @ A] :
% 0.31/1.56 ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_161_lexordp__irreflexive,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Xs: list @ A] :
% 0.31/1.56 ( ! [X: A] :
% 0.31/1.56 ~ ( ord_less @ A @ X @ X )
% 0.31/1.56 => ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_162_list_Osel_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,X21: A,X22: list @ A] :
% 0.31/1.56 ( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
% 0.31/1.56 = X21 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_163_hd__concat,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ ( list @ A ) ) )
% 0.31/1.56 => ( ( ( hd @ ( list @ A ) @ Xs )
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( hd @ A @ ( concat @ A @ Xs ) )
% 0.31/1.56 = ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_164_lexordp__into__lexordp__eq,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_165_lexordp__eq__conv__lexord,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ( ( ord_lexordp_eq @ A )
% 0.31/1.56 = ( ^ [Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( Xs3 = Ys )
% 0.31/1.56 | ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_166_lexordp__conv__lexordp__eq,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ( ( ord_lexordp @ A )
% 0.31/1.56 = ( ^ [Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( ord_lexordp_eq @ A @ Xs3 @ Ys )
% 0.31/1.56 & ~ ( ord_lexordp_eq @ A @ Ys @ Xs3 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_167_hd__append,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( hd @ A @ Ys2 ) ) )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( hd @ A @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_168_longest__common__prefix,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ? [Ps: list @ A,Xs5: list @ A,Ys5: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( append @ A @ Ps @ Xs5 ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( append @ A @ Ps @ Ys5 ) )
% 0.31/1.56 & ( ( Xs5
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( Ys5
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( hd @ A @ Xs5 )
% 0.31/1.56 != ( hd @ A @ Ys5 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_169_list__ord__Cons__imp,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,Xs: list @ A,X3: A] :
% 0.31/1.56 ( ( list_ord @ A @ Ord2 @ Xs )
% 0.31/1.56 => ( ( Ord2 @ X3 @ ( hd @ A @ Xs ) )
% 0.31/1.56 => ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ Xs ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_170_lexordp_OCons,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 => ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_171_lexordp_OCons__eq,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ~ ( ord_less @ A @ X3 @ Y )
% 0.31/1.56 => ( ~ ( ord_less @ A @ Y @ X3 )
% 0.31/1.56 => ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_172_lexordp__append__leftD,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Xs: list @ A,Us2: list @ A,Vs2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) )
% 0.31/1.56 => ( ! [A3: A] :
% 0.31/1.56 ~ ( ord_less @ A @ A3 @ A3 )
% 0.31/1.56 => ( ord_lexordp @ A @ Us2 @ Vs2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_173_lexordp_ONil,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Y: A,Ys2: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_174_lexordp__append__rightI,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ys2: list @ A,Xs: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_175_list__ord__Cons,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [Ord2: A > A > $o,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( Ord2 @ X3 @ ( hd @ A @ Xs ) )
% 0.31/1.56 & ( list_ord @ A @ Ord2 @ Xs ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_176_lexordp_Ocases,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [A1: list @ A,A2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ A1 @ A2 )
% 0.31/1.56 => ( ( ( A1
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ! [Y3: A,Ys3: list @ A] :
% 0.31/1.56 ( A2
% 0.31/1.56 != ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( ? [Xs2: list @ A] :
% 0.31/1.56 ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Y3: A] :
% 0.31/1.56 ( ? [Ys3: list @ A] :
% 0.31/1.56 ( A2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ~ ( ord_less @ A @ X @ Y3 ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( A1
% 0.31/1.56 = ( cons @ A @ X @ Xs2 ) )
% 0.31/1.56 => ! [Ys3: list @ A] :
% 0.31/1.56 ( ( A2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.31/1.56 => ~ ( ord_lexordp @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_177_lexordp_Osimps,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ( ( ord_lexordp @ A )
% 0.31/1.56 = ( ^ [A12: list @ A,A22: list @ A] :
% 0.31/1.56 ( ? [Y2: A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) ) )
% 0.31/1.56 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) )
% 0.31/1.56 & ( ord_less @ A @ X4 @ Y2 ) )
% 0.31/1.56 | ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
% 0.31/1.56 ( ( A12
% 0.31/1.56 = ( cons @ A @ X4 @ Xs3 ) )
% 0.31/1.56 & ( A22
% 0.31/1.56 = ( cons @ A @ Y2 @ Ys ) )
% 0.31/1.56 & ~ ( ord_less @ A @ X4 @ Y2 )
% 0.31/1.56 & ~ ( ord_less @ A @ Y2 @ X4 )
% 0.31/1.56 & ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_178_lexordp_Oinducts,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( ord @ A )
% 0.31/1.56 => ! [X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.31/1.56 ( ( ord_lexordp @ A @ X12 @ X23 )
% 0.31/1.56 => ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.31/1.56 => ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.31/1.56 => ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
% 0.31/1.56 => ( P @ X12 @ X23 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_179_lexordp__cases,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ! [Y3: A,Ys5: list @ A] :
% 0.31/1.56 ( Ys2
% 0.31/1.56 != ( cons @ A @ Y3 @ Ys5 ) ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( ? [Xs5: list @ A] :
% 0.31/1.56 ( Xs
% 0.31/1.56 = ( cons @ A @ X @ Xs5 ) )
% 0.31/1.56 => ! [Y3: A] :
% 0.31/1.56 ( ? [Ys5: list @ A] :
% 0.31/1.56 ( Ys2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys5 ) )
% 0.31/1.56 => ~ ( ord_less @ A @ X @ Y3 ) ) )
% 0.31/1.56 => ~ ! [X: A,Xs5: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( cons @ A @ X @ Xs5 ) )
% 0.31/1.56 => ! [Ys5: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 = ( cons @ A @ X @ Ys5 ) )
% 0.31/1.56 => ~ ( ord_lexordp @ A @ Xs5 @ Ys5 ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_180_lexordp__induct,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( linorder @ A )
% 0.31/1.56 => ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.31/1.56 => ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
% 0.31/1.56 ( ( ord_less @ A @ X @ Y3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
% 0.31/1.56 ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys3 ) ) ) )
% 0.31/1.56 => ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_181_distinct__adj__append__iff,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( ( distinct_adj @ A @ Xs )
% 0.31/1.56 & ( distinct_adj @ A @ Ys2 )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( last @ A @ Xs )
% 0.31/1.56 != ( hd @ A @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_182_rotate1__hd__tl,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( rotate1 @ A @ Xs )
% 0.31/1.56 = ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_183_successively__append__iff,axiom,
% 0.31/1.56 ! [A: $tType,P: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( successively @ A @ P @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( ( successively @ A @ P @ Xs )
% 0.31/1.56 & ( successively @ A @ P @ Ys2 )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( P @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_184_distinct__adj__Cons__Cons,axiom,
% 0.31/1.56 ! [B: $tType,X3: B,Y: B,Xs: list @ B] :
% 0.31/1.56 ( ( distinct_adj @ B @ ( cons @ B @ X3 @ ( cons @ B @ Y @ Xs ) ) )
% 0.31/1.56 = ( ( X3 != Y )
% 0.31/1.56 & ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_185_tl__append2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( tl @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( append @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_186_hd__Cons__tl,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
% 0.31/1.56 = Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_187_list_Ocollapse,axiom,
% 0.31/1.56 ! [A: $tType,List: list @ A] :
% 0.31/1.56 ( ( List
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
% 0.31/1.56 = List ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_188_list_Osel_I3_J,axiom,
% 0.31/1.56 ! [A: $tType,X21: A,X22: list @ A] :
% 0.31/1.56 ( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
% 0.31/1.56 = X22 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_189_list_Osel_I2_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( tl @ A @ ( nil @ A ) )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_190_successively_Oelims_I3_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A > A > $o,Xa: list @ A] :
% 0.31/1.56 ( ~ ( successively @ A @ X3 @ Xa )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.31/1.56 => ( ( X3 @ X @ Y3 )
% 0.31/1.56 & ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_191_successively_Osimps_I3_J,axiom,
% 0.31/1.56 ! [A: $tType,P: A > A > $o,X3: A,Y: A,Xs: list @ A] :
% 0.31/1.56 ( ( successively @ A @ P @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
% 0.31/1.56 = ( ( P @ X3 @ Y )
% 0.31/1.56 & ( successively @ A @ P @ ( cons @ A @ Y @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_192_successively_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,P: A > A > $o] : ( successively @ A @ P @ ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_193_distinct__adj__appendD1,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 => ( distinct_adj @ A @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_194_distinct__adj__appendD2,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 => ( distinct_adj @ A @ Ys2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_195_distinct__adj__Nil,axiom,
% 0.31/1.56 ! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_196_distinct__adj__ConsD,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 => ( distinct_adj @ A @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_197_butlast__tl,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( butlast @ A @ ( tl @ A @ Xs ) )
% 0.31/1.56 = ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_198_tl__Nil,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( ( tl @ A @ Xs )
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ? [X4: A] :
% 0.31/1.56 ( Xs
% 0.31/1.56 = ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_199_Nil__tl,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( ( nil @ A )
% 0.31/1.56 = ( tl @ A @ Xs ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ? [X4: A] :
% 0.31/1.56 ( Xs
% 0.31/1.56 = ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_200_list_Oexpand,axiom,
% 0.31/1.56 ! [A: $tType,List: list @ A,List2: list @ A] :
% 0.31/1.56 ( ( ( List
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( List2
% 0.31/1.56 = ( nil @ A ) ) )
% 0.31/1.56 => ( ( ( List
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( List2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( ( hd @ A @ List )
% 0.31/1.56 = ( hd @ A @ List2 ) )
% 0.31/1.56 & ( ( tl @ A @ List )
% 0.31/1.56 = ( tl @ A @ List2 ) ) ) ) )
% 0.31/1.56 => ( List = List2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_201_last__tl,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( tl @ A @ Xs )
% 0.31/1.56 != ( nil @ A ) ) )
% 0.31/1.56 => ( ( last @ A @ ( tl @ A @ Xs ) )
% 0.31/1.56 = ( last @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_202_successively_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,P: A > A > $o,X3: A] : ( successively @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_203_successively_Oelims_I1_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
% 0.31/1.56 ( ( ( successively @ A @ X3 @ Xa )
% 0.31/1.56 = Y )
% 0.31/1.56 => ( ( ( Xa
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ~ Y )
% 0.31/1.56 => ( ( ? [X: A] :
% 0.31/1.56 ( Xa
% 0.31/1.56 = ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ~ Y )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.31/1.56 => ( Y
% 0.31/1.56 = ( ~ ( ( X3 @ X @ Y3 )
% 0.31/1.56 & ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_204_successively_Oelims_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A > A > $o,Xa: list @ A] :
% 0.31/1.56 ( ( successively @ A @ X3 @ Xa )
% 0.31/1.56 => ( ( Xa
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( Xa
% 0.31/1.56 != ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( Xa
% 0.31/1.56 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.31/1.56 => ~ ( ( X3 @ X @ Y3 )
% 0.31/1.56 & ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_205_distinct__adj__singleton,axiom,
% 0.31/1.56 ! [B: $tType,X3: B] : ( distinct_adj @ B @ ( cons @ B @ X3 @ ( nil @ B ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_206_list_Oexhaust__sel,axiom,
% 0.31/1.56 ! [A: $tType,List: list @ A] :
% 0.31/1.56 ( ( List
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( List
% 0.31/1.56 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_207_successively__Cons,axiom,
% 0.31/1.56 ! [A: $tType,P: A > A > $o,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( successively @ A @ P @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( P @ X3 @ ( hd @ A @ Xs ) )
% 0.31/1.56 & ( successively @ A @ P @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_208_distinct__adj__Cons,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( X3
% 0.31/1.56 != ( hd @ A @ Xs ) )
% 0.31/1.56 & ( distinct_adj @ A @ Xs ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_209_list_Osplit__sel,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
% 0.31/1.56 ( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
% 0.31/1.56 = ( ( ( List
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( P @ F1 ) )
% 0.31/1.56 & ( ( List
% 0.31/1.56 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
% 0.31/1.56 => ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_210_list_Osplit__sel__asm,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
% 0.31/1.56 ( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
% 0.31/1.56 = ( ~ ( ( ( List
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ~ ( P @ F1 ) )
% 0.31/1.56 | ( ( List
% 0.31/1.56 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
% 0.31/1.56 & ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_211_list_Ocase__eq__if,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType] :
% 0.31/1.56 ( ( case_list @ B @ A )
% 0.31/1.56 = ( ^ [F12: B,F23: A > ( list @ A ) > B,List3: list @ A] :
% 0.31/1.56 ( if @ B
% 0.31/1.56 @ ( List3
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 @ F12
% 0.31/1.56 @ ( F23 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_212_list_Osimps_I5_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X22: list @ A] :
% 0.31/1.56 ( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X22 ) )
% 0.31/1.56 = ( F22 @ X21 @ X22 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_213_list_Osimps_I4_J,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] :
% 0.31/1.56 ( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) )
% 0.31/1.56 = F1 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_214_Cons__in__shuffles__iff,axiom,
% 0.31/1.56 ! [A: $tType,Z: A,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 & ( ( hd @ A @ Xs )
% 0.31/1.56 = Z )
% 0.31/1.56 & ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) )
% 0.31/1.56 | ( ( Ys2
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 & ( ( hd @ A @ Ys2 )
% 0.31/1.56 = Z )
% 0.31/1.56 & ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_215_map2__Cons__not__empty,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,C: $tType,Xs: list @ A,F: A > C > B,Y: C,Ys2: list @ C] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 => ( ( map2 @ A @ C @ B @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
% 0.31/1.56 = ( cons @ B @ ( F @ ( hd @ A @ Xs ) @ Y ) @ ( map2 @ A @ C @ B @ F @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_216_Nil__in__shuffles,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 & ( Ys2
% 0.31/1.56 = ( nil @ A ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_217_splice__in__shuffles,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] : ( member2 @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_218_shufflesE,axiom,
% 0.31/1.56 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.31/1.56 => ( ( ( Zs2 = Xs )
% 0.31/1.56 => ( Ys2
% 0.31/1.56 != ( nil @ A ) ) )
% 0.31/1.56 => ( ( ( Zs2 = Ys2 )
% 0.31/1.56 => ( Xs
% 0.31/1.56 != ( nil @ A ) ) )
% 0.31/1.56 => ( ! [X: A,Xs5: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( cons @ A @ X @ Xs5 ) )
% 0.31/1.56 => ! [Z2: A,Zs3: list @ A] :
% 0.31/1.56 ( ( Zs2
% 0.31/1.56 = ( cons @ A @ Z2 @ Zs3 ) )
% 0.31/1.56 => ( ( X = Z2 )
% 0.31/1.56 => ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs5 @ Ys2 ) ) ) ) )
% 0.31/1.56 => ~ ! [Y3: A,Ys5: list @ A] :
% 0.31/1.56 ( ( Ys2
% 0.31/1.56 = ( cons @ A @ Y3 @ Ys5 ) )
% 0.31/1.56 => ! [Z2: A,Zs3: list @ A] :
% 0.31/1.56 ( ( Zs2
% 0.31/1.56 = ( cons @ A @ Z2 @ Zs3 ) )
% 0.31/1.56 => ( ( Y3 = Z2 )
% 0.31/1.56 => ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_219_map2__Cons__Cons,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,X3: B,Xs: list @ B,Y: C,Ys2: list @ C] :
% 0.31/1.56 ( ( map2 @ B @ C @ A @ F @ ( cons @ B @ X3 @ Xs ) @ ( cons @ C @ Y @ Ys2 ) )
% 0.31/1.56 = ( cons @ A @ ( F @ X3 @ Y ) @ ( map2 @ B @ C @ A @ F @ Xs @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_220_map2_Osimps_I1_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,C: $tType,F: A > B > C,Ys2: list @ B] :
% 0.31/1.56 ( ( map2 @ A @ B @ C @ F @ ( nil @ A ) @ Ys2 )
% 0.31/1.56 = ( nil @ C ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_221_map2__Nil,axiom,
% 0.31/1.56 ! [C: $tType,B: $tType,A: $tType,F: B > C > A,Ys2: list @ C] :
% 0.31/1.56 ( ( map2 @ B @ C @ A @ F @ ( nil @ B ) @ Ys2 )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_222_map2__empty__conv,axiom,
% 0.31/1.56 ! [A: $tType,C: $tType,B: $tType,F: B > C > A,Xs: list @ B,Ys2: list @ C] :
% 0.31/1.56 ( ( ( map2 @ B @ C @ A @ F @ Xs @ Ys2 )
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ B ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_223_map2__not__empty__conv,axiom,
% 0.31/1.56 ! [A: $tType,C: $tType,B: $tType,F: B > C > A,Xs: list @ B,Ys2: list @ C] :
% 0.31/1.56 ( ( ( map2 @ B @ C @ A @ F @ Xs @ Ys2 )
% 0.31/1.56 != ( nil @ A ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 != ( nil @ B ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_224_Cons__in__shuffles__leftI,axiom,
% 0.31/1.56 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A,Z: A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.31/1.56 => ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ ( cons @ A @ Z @ Xs ) @ Ys2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_225_Cons__in__shuffles__rightI,axiom,
% 0.31/1.56 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A,Z: A] :
% 0.31/1.56 ( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.31/1.56 => ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_226_Nil__in__shufflesI,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_227_shuffles__commutes,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( shuffles @ A )
% 0.31/1.56 = ( ^ [Xs3: list @ A,Ys: list @ A] : ( shuffles @ A @ Ys @ Xs3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_228_map2_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X3: A,Xs: list @ A,Ys2: list @ B] :
% 0.31/1.56 ( ( map2 @ A @ B @ C @ F @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
% 0.31/1.56 = ( cons @ C @ ( F @ X3 @ ( hd @ B @ Ys2 ) ) @ ( map2 @ A @ B @ C @ F @ Xs @ ( tl @ B @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_229_map2__Cons__if,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,C: $tType,Xs: list @ B,F: B > C > A,Y: C,Ys2: list @ C] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( nil @ B ) )
% 0.31/1.56 => ( ( map2 @ B @ C @ A @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
% 0.31/1.56 = ( nil @ A ) ) )
% 0.31/1.56 & ( ( Xs
% 0.31/1.56 != ( nil @ B ) )
% 0.31/1.56 => ( ( map2 @ B @ C @ A @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
% 0.31/1.56 = ( cons @ A @ ( F @ ( hd @ B @ Xs ) @ Y ) @ ( map2 @ B @ C @ A @ F @ ( tl @ B @ Xs ) @ Ys2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_230_remdups__adj__append_H,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( Xs
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( Ys2
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 | ( ( last @ A @ Xs )
% 0.31/1.56 != ( hd @ A @ Ys2 ) ) )
% 0.31/1.56 => ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.31/1.56 = ( append @ A @ ( remdups_adj @ A @ Xs ) @ ( remdups_adj @ A @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_231_remdups__adj__append,axiom,
% 0.31/1.56 ! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] :
% 0.31/1.56 ( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) )
% 0.31/1.56 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_232_remdups__adj__Nil__iff,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( ( remdups_adj @ A @ Xs )
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 = ( Xs
% 0.31/1.56 = ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_233_hd__remdups__adj,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
% 0.31/1.56 = ( hd @ A @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_234_last__remdups__adj,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( last @ A @ ( remdups_adj @ A @ Xs ) )
% 0.31/1.56 = ( last @ A @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_235_remdups__adj__Cons__alt,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Xs: list @ A] :
% 0.31/1.56 ( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
% 0.31/1.56 = ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_236_successively__remdups__adjI,axiom,
% 0.31/1.56 ! [A: $tType,P: A > A > $o,Xs: list @ A] :
% 0.31/1.56 ( ( successively @ A @ P @ Xs )
% 0.31/1.56 => ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_237_distinct__adj__remdups__adj,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] : ( distinct_adj @ A @ ( remdups_adj @ A @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_238_distinct__adj__altdef,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( distinct_adj @ A )
% 0.31/1.56 = ( ^ [Xs3: list @ A] :
% 0.31/1.56 ( ( remdups_adj @ A @ Xs3 )
% 0.31/1.56 = Xs3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_239_remdups__adj_Oelims,axiom,
% 0.31/1.56 ! [A: $tType,X3: list @ A,Y: list @ A] :
% 0.31/1.56 ( ( ( remdups_adj @ A @ X3 )
% 0.31/1.56 = Y )
% 0.31/1.56 => ( ( ( X3
% 0.31/1.56 = ( nil @ A ) )
% 0.31/1.56 => ( Y
% 0.31/1.56 != ( nil @ A ) ) )
% 0.31/1.56 => ( ! [X: A] :
% 0.31/1.56 ( ( X3
% 0.31/1.56 = ( cons @ A @ X @ ( nil @ A ) ) )
% 0.31/1.56 => ( Y
% 0.31/1.56 != ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.31/1.56 => ~ ! [X: A,Y3: A,Xs2: list @ A] :
% 0.31/1.56 ( ( X3
% 0.31/1.56 = ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
% 0.31/1.56 => ~ ( ( ( X = Y3 )
% 0.31/1.56 => ( Y
% 0.31/1.56 = ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) ) ) )
% 0.31/1.56 & ( ( X != Y3 )
% 0.31/1.56 => ( Y
% 0.31/1.56 = ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_240_remdups__adj_Osimps_I2_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A] :
% 0.31/1.56 ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.31/1.56 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_241_remdups__adj_Osimps_I1_J,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( remdups_adj @ A @ ( nil @ A ) )
% 0.31/1.56 = ( nil @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_242_remdups__adj_Osimps_I3_J,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Y: A,Xs: list @ A] :
% 0.31/1.56 ( ( ( X3 = Y )
% 0.31/1.56 => ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
% 0.31/1.56 = ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
% 0.31/1.56 & ( ( X3 != Y )
% 0.31/1.56 => ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
% 0.31/1.56 = ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_243_remdups__adj__append__two,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,X3: A,Y: A] :
% 0.31/1.56 ( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
% 0.31/1.56 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_244_map2__snoc__snoc,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,F: A > B > C,X3: A,Y: B] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.31/1.56 => ( ( map2 @ A @ B @ C @ F @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y @ ( nil @ B ) ) ) )
% 0.31/1.56 = ( append @ C @ ( map2 @ A @ B @ C @ F @ Xs @ Ys2 ) @ ( cons @ C @ ( F @ X3 @ Y ) @ ( nil @ C ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_245_list__strict__asc__imp__list__asc,axiom,
% 0.31/1.56 ! [A: $tType] :
% 0.31/1.56 ( ( preorder @ A )
% 0.31/1.56 => ! [Xs: list @ A] :
% 0.31/1.56 ( ( list_strict_asc @ A @ Xs )
% 0.31/1.56 => ( list_asc @ A @ Xs ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_246_append__eq__append__conv,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Us2: list @ A,Vs2: list @ A] :
% 0.31/1.56 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/1.56 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.31/1.56 | ( ( size_size @ ( list @ A ) @ Us2 )
% 0.31/1.56 = ( size_size @ ( list @ A ) @ Vs2 ) ) )
% 0.31/1.56 => ( ( ( append @ A @ Xs @ Us2 )
% 0.31/1.56 = ( append @ A @ Ys2 @ Vs2 ) )
% 0.31/1.56 = ( ( Xs = Ys2 )
% 0.31/1.56 & ( Us2 = Vs2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_247_length__rotate1,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A] :
% 0.31/1.56 ( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
% 0.31/1.56 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_248_length__greater__imp__not__empty,axiom,
% 0.31/1.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.31/1.56 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.31/1.56 => ( Xs
% 0.31/1.56 != ( nil @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_249_neq__if__length__neq,axiom,
% 0.31/1.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/1.56 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.31/1.56 => ( Xs != Ys2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_250_Ex__list__of__length,axiom,
% 0.31/1.56 ! [A: $tType,N: nat] :
% 0.31/1.56 ? [Xs2: list @ A] :
% 0.31/1.56 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.31/1.56 = N ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_251_length__induct,axiom,
% 0.31/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.31/1.56 ( ! [Xs2: list @ A] :
% 0.31/1.56 ( ! [Ys6: list @ A] :
% 0.31/1.56 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.31/1.56 => ( P @ Ys6 ) )
% 0.31/1.56 => ( P @ Xs2 ) )
% 0.31/1.56 => ( P @ Xs ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_252_list__induct2,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.31/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) )
% 0.31/1.56 => ( P @ Xs @ Ys2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_253_list__induct3,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.31/1.56 => ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 0.31/1.56 = ( size_size @ ( list @ C ) @ Zs2 ) )
% 0.31/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z2: C,Zs: list @ C] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.31/1.56 => ( ( ( size_size @ ( list @ B ) @ Ys3 )
% 0.31/1.56 = ( size_size @ ( list @ C ) @ Zs ) )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 @ Zs )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C @ Z2 @ Zs ) ) ) ) )
% 0.31/1.56 => ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_254_list__induct4,axiom,
% 0.31/1.56 ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.31/1.56 => ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 0.31/1.56 = ( size_size @ ( list @ C ) @ Zs2 ) )
% 0.31/1.56 => ( ( ( size_size @ ( list @ C ) @ Zs2 )
% 0.31/1.56 = ( size_size @ ( list @ D ) @ Ws2 ) )
% 0.31/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
% 0.31/1.56 => ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z2: C,Zs: list @ C,W: D,Ws3: list @ D] :
% 0.31/1.56 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.31/1.56 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.31/1.56 => ( ( ( size_size @ ( list @ B ) @ Ys3 )
% 0.31/1.56 = ( size_size @ ( list @ C ) @ Zs ) )
% 0.31/1.56 => ( ( ( size_size @ ( list @ C ) @ Zs )
% 0.31/1.56 = ( size_size @ ( list @ D ) @ Ws3 ) )
% 0.31/1.56 => ( ( P @ Xs2 @ Ys3 @ Zs @ Ws3 )
% 0.31/1.56 => ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C @ Z2 @ Zs ) @ ( cons @ D @ W @ Ws3 ) ) ) ) ) )
% 0.31/1.56 => ( P @ Xs @ Ys2 @ Zs2 @ Ws2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/1.56 ! [A7: $tType,A8: $tType] :
% 0.31/1.56 ( ( preorder @ A8 )
% 0.31/1.56 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/1.56 ! [A7: $tType,A8: $tType] :
% 0.31/1.56 ( ( order @ A8 )
% 0.31/1.56 => ( order @ ( A7 > A8 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/1.56 ! [A7: $tType,A8: $tType] :
% 0.31/1.56 ( ( ord @ A8 )
% 0.31/1.56 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.31/1.56 preorder @ nat ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.31/1.56 linorder @ nat ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.31/1.56 order @ nat ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.31/1.56 ord @ nat ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.31/1.56 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.31/1.56 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.31/1.56 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.31/1.56 preorder @ $o ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.31/1.56 linorder @ $o ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.31/1.56 order @ $o ).
% 0.31/1.56
% 0.31/1.56 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.31/1.56 ord @ $o ).
% 0.31/1.56
% 0.31/1.56 thf(help_If_3_1_T,axiom,
% 0.31/1.56 ! [P: $o] :
% 0.31/1.56 ( ( P = $true )
% 0.31/1.56 | ( P = $false ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(help_If_2_1_T,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Y: A] :
% 0.31/1.56 ( ( if @ A @ $false @ X3 @ Y )
% 0.31/1.56 = Y ) ).
% 0.31/1.56
% 0.31/1.56 thf(help_If_1_1_T,axiom,
% 0.31/1.56 ! [A: $tType,X3: A,Y: A] :
% 0.31/1.56 ( ( if @ A @ $true @ X3 @ Y )
% 0.31/1.56 = X3 ) ).
% 0.31/1.56
% 0.31/1.56 thf(conj_0,conjecture,
% 0.31/1.56 ( ( listIn521021761append @ a @ ( cons @ a @ x @ xs ) @ f )
% 0.31/1.56 = ( listIn521021761append @ a @ ( cons @ a @ x @ ( nil @ a ) ) @ ( listIn521021761append @ a @ xs @ f ) ) ) ).
% 0.31/1.56 ------------------------
% 0.31/1.59 % (14306)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14040 for (2999ds/2Mi)
% 0.31/1.60 % (14305)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_14040 for (2999ds/27Mi)
% 0.31/1.60 % (14306)Instruction limit reached!
% 0.31/1.60 % (14306)------------------------------
% 0.31/1.60 % (14306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.60 % (14306)Termination reason: Unknown
% 0.31/1.60 % (14306)Termination phase: shuffling
% 0.31/1.60
% 0.31/1.60 % (14306)Memory used [KB]: 1407
% 0.31/1.60 % (14306)Time elapsed: 0.003 s
% 0.31/1.60 % (14306)Instructions burned: 4 (million)
% 0.31/1.60 % (14306)------------------------------
% 0.31/1.60 % (14306)------------------------------
% 0.31/1.60 % (14308)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_14040 for (2999ds/275Mi)
% 2.90/1.60 % (14303)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14040 for (2999ds/183Mi)
% 2.90/1.60 % (14304)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_14040 for (2999ds/4Mi)
% 2.90/1.60 % (14307)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_14040 for (2999ds/2Mi)
% 2.90/1.60 % (14309)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14040 for (2999ds/18Mi)
% 2.90/1.60 % (14307)Instruction limit reached!
% 2.90/1.60 % (14307)------------------------------
% 2.90/1.60 % (14307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (14307)Termination reason: Unknown
% 2.90/1.60 % (14307)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (14307)Memory used [KB]: 1407
% 2.90/1.60 % (14307)Time elapsed: 0.004 s
% 2.90/1.60 % (14307)Instructions burned: 3 (million)
% 2.90/1.60 % (14307)------------------------------
% 2.90/1.60 % (14307)------------------------------
% 2.90/1.60 % (14304)Instruction limit reached!
% 2.90/1.60 % (14304)------------------------------
% 2.90/1.60 % (14304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (14304)Termination reason: Unknown
% 2.90/1.60 % (14304)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (14304)Memory used [KB]: 1407
% 2.90/1.60 % (14304)Time elapsed: 0.005 s
% 2.90/1.60 % (14304)Instructions burned: 5 (million)
% 2.90/1.60 % (14304)------------------------------
% 2.90/1.60 % (14304)------------------------------
% 2.90/1.60 % (14310)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14040 for (2999ds/3Mi)
% 2.90/1.61 % (14305)Instruction limit reached!
% 2.90/1.61 % (14305)------------------------------
% 2.90/1.61 % (14305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (14305)Termination reason: Unknown
% 2.90/1.61 % (14305)Termination phase: Property scanning
% 2.90/1.61
% 2.90/1.61 % (14305)Memory used [KB]: 1918
% 2.90/1.61 % (14305)Time elapsed: 0.012 s
% 2.90/1.61 % (14305)Instructions burned: 28 (million)
% 2.90/1.61 % (14305)------------------------------
% 2.90/1.61 % (14305)------------------------------
% 2.90/1.61 % (14310)Instruction limit reached!
% 2.90/1.61 % (14310)------------------------------
% 2.90/1.61 % (14310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (14310)Termination reason: Unknown
% 2.90/1.61 % (14310)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (14310)Memory used [KB]: 1407
% 2.90/1.61 % (14310)Time elapsed: 0.002 s
% 2.90/1.61 % (14310)Instructions burned: 3 (million)
% 2.90/1.61 % (14310)------------------------------
% 2.90/1.61 % (14310)------------------------------
% 2.90/1.61 % (14309)Instruction limit reached!
% 2.90/1.61 % (14309)------------------------------
% 2.90/1.61 % (14309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (14309)Termination reason: Unknown
% 2.90/1.61 % (14309)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (14309)Memory used [KB]: 1791
% 2.90/1.61 % (14309)Time elapsed: 0.013 s
% 2.90/1.61 % (14309)Instructions burned: 18 (million)
% 2.90/1.61 % (14309)------------------------------
% 2.90/1.61 % (14309)------------------------------
% 2.90/1.62 % (14314)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_14040 for (2999ds/1041Mi)
% 2.90/1.62 % (14313)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_14040 for (2999ds/3Mi)
% 2.90/1.62 % (14308)First to succeed.
% 2.90/1.62 % (14313)Instruction limit reached!
% 2.90/1.62 % (14313)------------------------------
% 2.90/1.62 % (14313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (14313)Termination reason: Unknown
% 2.90/1.62 % (14313)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (14313)Memory used [KB]: 1407
% 2.90/1.62 % (14313)Time elapsed: 0.003 s
% 2.90/1.62 % (14313)Instructions burned: 3 (million)
% 2.90/1.62 % (14313)------------------------------
% 2.90/1.62 % (14313)------------------------------
% 2.90/1.62 % (14308)Refutation found. Thanks to Tanya!
% 2.90/1.62 % SZS status Theorem for DTF2THF_14040
% 2.90/1.62 % SZS output start Proof for DTF2THF_14040
% 2.90/1.62 thf(type_def_5, type, list: ($tType) > $tType).
% 2.90/1.62 thf(type_def_6, type, set: ($tType) > $tType).
% 2.90/1.62 thf(type_def_8, type, nat: $tType).
% 2.90/1.62 thf(type_def_9, type, a: $tType).
% 2.90/1.62 thf(func_def_0, type, list: ($tType) > $tType).
% 2.90/1.62 thf(func_def_1, type, set: ($tType) > $tType).
% 2.90/1.62 thf(func_def_2, type, nat: $tType).
% 2.90/1.62 thf(func_def_3, type, a: $tType).
% 2.90/1.62 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.90/1.62 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 2.90/1.62 thf(func_def_6, type, order: !>[X0: $tType]:($o)).
% 2.90/1.62 thf(func_def_7, type, linorder: !>[X0: $tType]:($o)).
% 2.90/1.62 thf(func_def_8, type, preorder: !>[X0: $tType]:($o)).
% 2.90/1.62 thf(func_def_9, type, bNF_Greatest_Shift: !>[X0: $tType]:(set @ list @ X0 > X0 > set @ list @ X0)).
% 2.90/1.62 thf(func_def_10, type, bNF_Greatest_Succ: !>[X0: $tType]:(set @ list @ X0 > list @ X0 > set @ X0)).
% 2.90/1.62 thf(func_def_11, type, bNF_Greatest_shift: !>[X0: $tType, X1: $tType]:((list @ X0 > X1) > X0 > list @ X0 > X1)).
% 2.90/1.62 thf(func_def_12, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.90/1.62 thf(func_def_13, type, list_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.90/1.62 thf(func_def_14, type, list_ord: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > $o)).
% 2.90/1.62 thf(func_def_15, type, list_strict_asc: !>[X0: $tType]:(list @ X0 > $o)).
% 2.90/1.62 thf(func_def_16, type, map2: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > list @ X0 > list @ X1 > list @ X2)).
% 2.90/1.62 thf(func_def_17, type, listIn521021761append: !>[X0: $tType]:(list @ X0 > (nat > X0) > nat > X0)).
% 2.90/1.62 thf(func_def_18, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_19, type, arg_min_list: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > X0)).
% 2.90/1.62 thf(func_def_20, type, bind: !>[X0: $tType, X1: $tType]:(list @ X0 > (X0 > list @ X1) > list @ X1)).
% 2.90/1.62 thf(func_def_21, type, butlast: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_22, type, concat: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_23, type, distinct_adj: !>[X0: $tType]:(list @ X0 > $o)).
% 2.90/1.62 thf(func_def_24, type, foldr: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > list @ X0 > X1 > X1)).
% 2.90/1.62 thf(func_def_25, type, insert: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_26, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_27, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_28, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.90/1.62 thf(func_def_29, type, case_list: !>[X0: $tType, X1: $tType]:(X0 > (X1 > list @ X1 > X0) > list @ X1 > X0)).
% 2.90/1.62 thf(func_def_30, type, hd: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_31, type, tl: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_32, type, list_ex1: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 2.90/1.62 thf(func_def_33, type, listrelp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > list @ X0 > list @ X1 > $o)).
% 2.90/1.62 thf(func_def_34, type, map_tailrec_rev: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X1 > list @ X1)).
% 2.90/1.62 thf(func_def_35, type, maps: !>[X0: $tType, X1: $tType]:((X0 > list @ X1) > list @ X0 > list @ X1)).
% 2.90/1.62 thf(func_def_36, type, member: !>[X0: $tType]:(list @ X0 > X0 > $o)).
% 2.90/1.62 thf(func_def_37, type, lexordp_eq: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > $o)).
% 2.90/1.62 thf(func_def_38, type, ord_lexordp: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 2.90/1.62 thf(func_def_39, type, ord_lexordp_eq: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 2.90/1.62 thf(func_def_40, type, product_lists: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.90/1.62 thf(func_def_41, type, remdups_adj: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_42, type, rotate1: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_43, type, shuffles: !>[X0: $tType]:(list @ X0 > list @ X0 > set @ list @ X0)).
% 2.90/1.62 thf(func_def_44, type, splice: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_45, type, subseqs: !>[X0: $tType]:(list @ X0 > list @ list @ X0)).
% 2.90/1.62 thf(func_def_46, type, successively: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > $o)).
% 2.90/1.62 thf(func_def_47, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.90/1.62 thf(func_def_48, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.90/1.62 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.90/1.62 thf(func_def_50, type, member2: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.90/1.62 thf(func_def_51, type, f: nat > a).
% 2.90/1.62 thf(func_def_52, type, x: a).
% 2.90/1.62 thf(func_def_53, type, xs: list @ a).
% 2.90/1.62 thf(func_def_70, type, sK0: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_71, type, sK1: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.90/1.62 thf(func_def_72, type, sK2: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0 > X0 > list @ X0)).
% 2.90/1.62 thf(func_def_73, type, sK3: !>[X0: $tType]:(list @ X0 > X0 > list @ X0 > list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_74, type, sK4: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_75, type, sK5: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_76, type, sK6: !>[X0: $tType]:(list @ list @ X0 > X0)).
% 2.90/1.62 thf(func_def_77, type, sK7: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_78, type, sK8: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.90/1.62 thf(func_def_79, type, sK9: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.90/1.62 thf(func_def_80, type, sK10: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0 > list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_81, type, sK11: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_82, type, sK12: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_83, type, sK13: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_84, type, sK14: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.90/1.62 thf(func_def_85, type, sK15: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.90/1.62 thf(func_def_86, type, sK16: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_87, type, sK17: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_88, type, sK18: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.90/1.62 thf(func_def_89, type, sK19: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_90, type, sK20: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.90/1.62 thf(func_def_91, type, sK21: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.90/1.62 thf(func_def_92, type, sK22: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_93, type, sK23: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_94, type, sK24: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > list @ X0)).
% 2.90/1.62 thf(func_def_95, type, sK25: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_96, type, sK26: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_97, type, sK27: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_98, type, sK28: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_99, type, sK29: !>[X0: $tType]:(list @ X0 > X0)).
% 2.90/1.62 thf(func_def_100, type, sK30: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.90/1.62 thf(func_def_101, type, sK31: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_102, type, sK32: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.90/1.62 thf(func_def_103, type, sK33: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.90/1.62 thf(func_def_105, type, ph35: !>[X0: $tType]:(X0)).
% 2.90/1.62 thf(func_def_106, type, sK36: nat).
% 2.90/1.62 thf(f855,plain,(
% 2.90/1.62 $false),
% 2.90/1.62 inference(trivial_inequality_removal,[status(thm)],[f854])).
% 2.90/1.62 thf(f854,plain,(
% 2.90/1.62 ((listIn521021761append @ a @ (cons @ a @ x @ xs) @ f) != (listIn521021761append @ a @ (cons @ a @ x @ xs) @ f))),
% 2.90/1.62 inference(forward_demodulation,[status(thm)],[f853,f802])).
% 2.90/1.62 thf(f802,plain,(
% 2.90/1.62 ( ! [X0 : $tType,X2 : list @ X0,X1 : X0] : (((append @ X0 @ (cons @ X0 @ X1 @ nil @ X0) @ X2) = (cons @ X0 @ X1 @ X2))) )),
% 2.90/1.62 inference(cnf_transformation,[status(thm)],[f655])).
% 2.90/1.62 thf(f655,plain,(
% 2.90/1.62 ! [X0 : $tType,X1 : X0,X2 : list @ X0] : ((append @ X0 @ (cons @ X0 @ X1 @ nil @ X0) @ X2) = (cons @ X0 @ X1 @ X2))),
% 2.90/1.62 inference(rectify,[status(thm)],[f104])).
% 2.90/1.62 thf(f104,axiom,(
% 2.90/1.62 ! [X0 : $tType,X27 : X0,X15 : list @ X0] : ((cons @ X0 @ X27 @ X15) = (append @ X0 @ (cons @ X0 @ X27 @ nil @ X0) @ X15))),
% 2.90/1.62 file('/export/starexec/sandbox2/tmp/tmp.nSD6Sk1FeO/DTF2THF_14040.p',fact_103_append__eq__Cons)).
% 2.90/1.62 thf(f853,plain,(
% 2.90/1.62 ((listIn521021761append @ a @ (cons @ a @ x @ xs) @ f) != (listIn521021761append @ a @ (append @ a @ (cons @ a @ x @ nil @ a) @ xs) @ f))),
% 2.90/1.62 inference(superposition,[status(thm)],[f770,f808])).
% 2.90/1.62 thf(f808,plain,(
% 2.90/1.62 ( ! [X0 : $tType,X2 : list @ X0,X3 : nat > X0,X1 : list @ X0] : (((listIn521021761append @ X0 @ (append @ X0 @ X1 @ X2) @ X3) = (listIn521021761append @ X0 @ X1 @ (listIn521021761append @ X0 @ X2 @ X3)))) )),
% 2.90/1.62 inference(cnf_transformation,[status(thm)],[f634])).
% 2.90/1.62 thf(f634,plain,(
% 2.90/1.62 ! [X0 : $tType,X1 : list @ X0,X2 : list @ X0,X3 : nat > X0] : ((listIn521021761append @ X0 @ (append @ X0 @ X1 @ X2) @ X3) = (listIn521021761append @ X0 @ X1 @ (listIn521021761append @ X0 @ X2 @ X3)))),
% 2.90/1.62 inference(rectify,[status(thm)],[f85])).
% 2.90/1.62 thf(f85,axiom,(
% 2.90/1.62 ! [X0 : $tType,X15 : list @ X0,X18 : list @ X0,X3 : nat > X0] : ((listIn521021761append @ X0 @ X15 @ (listIn521021761append @ X0 @ X18 @ X3)) = (listIn521021761append @ X0 @ (append @ X0 @ X15 @ X18) @ X3))),
% 2.90/1.62 file('/export/starexec/sandbox2/tmp/tmp.nSD6Sk1FeO/DTF2THF_14040.p',fact_84_i__append__assoc)).
% 2.90/1.62 thf(f770,plain,(
% 2.90/1.62 ((listIn521021761append @ a @ (cons @ a @ x @ xs) @ f) != (listIn521021761append @ a @ (cons @ a @ x @ nil @ a) @ (listIn521021761append @ a @ xs @ f)))),
% 2.90/1.62 inference(cnf_transformation,[status(thm)],[f632])).
% 2.90/1.62 thf(f632,plain,(
% 2.90/1.62 ((listIn521021761append @ a @ (cons @ a @ x @ xs) @ f) != (listIn521021761append @ a @ (cons @ a @ x @ nil @ a) @ (listIn521021761append @ a @ xs @ f)))),
% 2.90/1.62 inference(flattening,[status(thm)],[f274])).
% 2.90/1.62 thf(f274,negated_conjecture,(
% 2.90/1.62 ~((listIn521021761append @ a @ (cons @ a @ x @ xs) @ f) = (listIn521021761append @ a @ (cons @ a @ x @ nil @ a) @ (listIn521021761append @ a @ xs @ f)))),
% 2.90/1.62 inference(negated_conjecture,[status(cth)],[f273])).
% 2.90/1.62 thf(f273,conjecture,(
% 2.90/1.62 ((listIn521021761append @ a @ (cons @ a @ x @ xs) @ f) = (listIn521021761append @ a @ (cons @ a @ x @ nil @ a) @ (listIn521021761append @ a @ xs @ f)))),
% 2.90/1.62 file('/export/starexec/sandbox2/tmp/tmp.nSD6Sk1FeO/DTF2THF_14040.p',conj_0)).
% 2.90/1.62 % SZS output end Proof for DTF2THF_14040
% 2.90/1.62 % (14308)------------------------------
% 2.90/1.62 % (14308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (14308)Termination reason: Refutation
% 2.90/1.62
% 2.90/1.62 % (14308)Memory used [KB]: 6652
% 2.90/1.62 % (14308)Time elapsed: 0.023 s
% 2.90/1.62 % (14308)Instructions burned: 60 (million)
% 2.90/1.62 % (14308)------------------------------
% 2.90/1.62 % (14308)------------------------------
% 2.90/1.62 % (14302)Success in time 0.051 s
%------------------------------------------------------------------------------