%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM190^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:54 AM UTC 2026
% Result : Theorem 2.86s 1.64s
% Output : Refutation 2.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM190^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n029.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 04:16:29 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : COM190^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.46 % Domain : Computing Theory
% 0.30/0.46 % Problem : Grammars and languages 811
% 0.30/0.46 % Version : [Bla16] axioms : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.30/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Bla16]
% 0.30/0.46 % Names : gram_lang__811.p [Bla16]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.00 v7.1.0
% 0.30/0.46 % Syntax : Number of formulae : 312 ( 78 unt; 39 typ; 0 def)
% 0.30/0.46 % Number of atoms : 880 ( 279 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 13 ( 3 avg)
% 0.30/0.46 % Number of connectives : 3978 ( 125 ~; 26 |; 81 &;3265 @)
% 0.30/0.46 % ( 0 <=>; 481 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 27 ( 9 avg)
% 0.30/0.46 % Number of types : 4 ( 3 usr)
% 0.30/0.46 % Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 37 ( 36 usr; 2 con; 0-3 aty)
% 0.30/0.46 % Number of variables : 1124 ( 50 ^; 957 !; 88 ?;1124 :)
% 0.30/0.46 % ( 29 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.46 % 2016-07-13 14:44:57.308
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 %----Could-be-implicit typings (6)
% 0.30/0.46 thf(ty_t_DTree_Odtree,type,
% 0.30/0.46 dtree: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_List_Olist,type,
% 0.30/0.46 list: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Nat_Onat,type,
% 0.30/0.46 nat: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_DTree_ON,type,
% 0.30/0.46 n: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_itself,type,
% 0.30/0.46 itself: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 %----Explicit typings (33)
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.46 ord:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.46 no_bot:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/0.46 no_top:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/0.46 wellorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/0.46 dense_order:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/0.46 dense_linorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.30/0.46 condit1656338222tinuum:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.30/0.46 condit1037483654norder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Opath,type,
% 0.30/0.46 gram_L250615845e_path: ( n > dtree ) > ( list @ n ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Oappend,type,
% 0.30/0.46 append:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Odistinct,type,
% 0.30/0.46 distinct:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Olast,type,
% 0.30/0.46 last:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Olist_OCons,type,
% 0.30/0.46 cons:
% 0.30/0.46 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Olist_ONil,type,
% 0.30/0.46 nil:
% 0.30/0.46 !>[A: $tType] : ( list @ A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Olist_Ohd,type,
% 0.30/0.46 hd:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Olist_Oset,type,
% 0.30/0.46 set2:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_On__lists,type,
% 0.30/0.46 n_lists:
% 0.30/0.46 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Oproduct__lists,type,
% 0.30/0.46 product_lists:
% 0.30/0.46 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Osublists,type,
% 0.30/0.46 sublists:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_List_Ounion,type,
% 0.30/0.46 union:
% 0.30/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.30/0.46 size_size:
% 0.30/0.46 !>[A: $tType] : ( A > nat ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.46 ord_less:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Pure_Otype,type,
% 0.30/0.46 type2:
% 0.30/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_f,type,
% 0.30/0.46 f: n > dtree ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_nl,type,
% 0.30/0.46 nl: list @ n ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_nla____,type,
% 0.30/0.46 nla: list @ n ).
% 0.30/0.46
% 0.30/0.46 %----Relevant facts (255)
% 0.30/0.46 thf(fact_0_assms,axiom,
% 0.30/0.46 gram_L250615845e_path @ f @ nl ).
% 0.30/0.46
% 0.30/0.46 % assms
% 0.30/0.46 thf(fact_1__C1_Oprems_C,axiom,
% 0.30/0.46 gram_L250615845e_path @ f @ nla ).
% 0.30/0.46
% 0.30/0.46 % "1.prems"
% 0.30/0.46 thf(fact_2__C1_Ohyps_C,axiom,
% 0.30/0.46 ! [Ys: list @ n] :
% 0.30/0.46 ( ( ord_less @ nat @ ( size_size @ ( list @ n ) @ Ys ) @ ( size_size @ ( list @ n ) @ nla ) )
% 0.30/0.46 => ( ( gram_L250615845e_path @ f @ Ys )
% 0.30/0.46 => ? [Nl: list @ n] :
% 0.30/0.46 ( ( gram_L250615845e_path @ f @ Nl )
% 0.30/0.46 & ( ( hd @ n @ Nl )
% 0.30/0.46 = ( hd @ n @ Ys ) )
% 0.30/0.46 & ( ( last @ n @ Nl )
% 0.30/0.46 = ( last @ n @ Ys ) )
% 0.30/0.46 & ( ord_less_eq @ ( set @ n ) @ ( set2 @ n @ Nl ) @ ( set2 @ n @ Ys ) )
% 0.30/0.46 & ( distinct @ n @ Nl ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % "1.hyps"
% 0.30/0.46 thf(fact_3_path__NE,axiom,
% 0.30/0.46 ! [F: n > dtree,Nl2: list @ n] :
% 0.30/0.46 ( ( gram_L250615845e_path @ F @ Nl2 )
% 0.30/0.46 => ( Nl2
% 0.30/0.46 != ( nil @ n ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % path_NE
% 0.30/0.46 thf(fact_4_path_OBase,axiom,
% 0.30/0.46 ! [F: n > dtree,N: n] : ( gram_L250615845e_path @ F @ ( cons @ n @ N @ ( nil @ n ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % path.Base
% 0.30/0.46 thf(fact_5_path__post,axiom,
% 0.30/0.46 ! [F: n > dtree,N: n,Nl2: list @ n] :
% 0.30/0.46 ( ( gram_L250615845e_path @ F @ ( cons @ n @ N @ Nl2 ) )
% 0.30/0.46 => ( ( Nl2
% 0.30/0.46 != ( nil @ n ) )
% 0.30/0.46 => ( gram_L250615845e_path @ F @ Nl2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % path_post
% 0.30/0.46 thf(fact_6_path__post__concat,axiom,
% 0.30/0.46 ! [F: n > dtree,Nl1: list @ n,Nl22: list @ n] :
% 0.30/0.46 ( ( gram_L250615845e_path @ F @ ( append @ n @ Nl1 @ Nl22 ) )
% 0.30/0.46 => ( ( Nl22
% 0.30/0.46 != ( nil @ n ) )
% 0.30/0.46 => ( gram_L250615845e_path @ F @ Nl22 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % path_post_concat
% 0.30/0.46 thf(fact_7_path__concat,axiom,
% 0.30/0.46 ! [F: n > dtree,Nl1: list @ n,Nl22: list @ n] :
% 0.30/0.46 ( ( gram_L250615845e_path @ F @ Nl1 )
% 0.30/0.46 => ( ( gram_L250615845e_path @ F @ ( cons @ n @ ( last @ n @ Nl1 ) @ Nl22 ) )
% 0.30/0.46 => ( gram_L250615845e_path @ F @ ( append @ n @ Nl1 @ Nl22 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % path_concat
% 0.30/0.46 thf(fact_8_last__snoc,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,X: A] :
% 0.30/0.46 ( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % last_snoc
% 0.30/0.46 thf(fact_9_last__appendL,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.30/0.46 ( ( Ys2
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( last @ A @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % last_appendL
% 0.30/0.46 thf(fact_10_last__appendR,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.30/0.46 ( ( Ys2
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( last @ A @ Ys2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % last_appendR
% 0.30/0.46 thf(fact_11_hd__append2,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( hd @ A @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % hd_append2
% 0.30/0.46 thf(fact_12_append1__eq__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y: A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.30/0.46 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 0.30/0.46 = ( ( Xs = Ys2 )
% 0.30/0.46 & ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append1_eq_conv
% 0.30/0.46 thf(fact_13_append__eq__append__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Us: list @ A,Vs: list @ A] :
% 0.30/0.46 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.46 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.46 | ( ( size_size @ ( list @ A ) @ Us )
% 0.30/0.46 = ( size_size @ ( list @ A ) @ Vs ) ) )
% 0.30/0.46 => ( ( ( append @ A @ Xs @ Us )
% 0.30/0.46 = ( append @ A @ Ys2 @ Vs ) )
% 0.30/0.46 = ( ( Xs = Ys2 )
% 0.30/0.46 & ( Us = Vs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_eq_append_conv
% 0.30/0.46 thf(fact_14_append__Nil2,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A] :
% 0.30/0.46 ( ( append @ A @ Xs @ ( nil @ A ) )
% 0.30/0.46 = Xs ) ).
% 0.30/0.46
% 0.30/0.46 % append_Nil2
% 0.30/0.46 thf(fact_15_append__self__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.46 = Xs )
% 0.30/0.46 = ( Ys2
% 0.30/0.46 = ( nil @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_self_conv
% 0.30/0.46 thf(fact_16_self__append__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( Ys2
% 0.30/0.46 = ( nil @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % self_append_conv
% 0.30/0.46 thf(fact_17_append__self__conv2,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.46 = Ys2 )
% 0.30/0.46 = ( Xs
% 0.30/0.46 = ( nil @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_self_conv2
% 0.30/0.46 thf(fact_18_self__append__conv2,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.30/0.46 ( ( Ys2
% 0.30/0.46 = ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( Xs
% 0.30/0.46 = ( nil @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % self_append_conv2
% 0.30/0.46 thf(fact_19_list_Oinject,axiom,
% 0.30/0.46 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 0.30/0.46 ( ( ( cons @ A @ X21 @ X22 )
% 0.30/0.46 = ( cons @ A @ Y21 @ Y22 ) )
% 0.30/0.46 = ( ( X21 = Y21 )
% 0.30/0.46 & ( X22 = Y22 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.inject
% 0.30/0.46 thf(fact_20_same__append__eq,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.46 = ( append @ A @ Xs @ Zs ) )
% 0.30/0.46 = ( Ys2 = Zs ) ) ).
% 0.30/0.46
% 0.30/0.46 % same_append_eq
% 0.30/0.46 thf(fact_21_append__same__eq,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Ys2 @ Xs )
% 0.30/0.46 = ( append @ A @ Zs @ Xs ) )
% 0.30/0.46 = ( Ys2 = Zs ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_same_eq
% 0.30/0.46 thf(fact_22_append__assoc,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.30/0.46 ( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs )
% 0.30/0.46 = ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_assoc
% 0.30/0.46 thf(fact_23_append__is__Nil__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 = ( ( Xs
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 & ( Ys2
% 0.30/0.46 = ( nil @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_is_Nil_conv
% 0.30/0.46 thf(fact_24_Nil__is__append__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( ( nil @ A )
% 0.30/0.46 = ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( ( Xs
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 & ( Ys2
% 0.30/0.46 = ( nil @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Nil_is_append_conv
% 0.30/0.46 thf(fact_25_not__Cons__self2,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( cons @ A @ X @ Xs )
% 0.30/0.46 != Xs ) ).
% 0.30/0.46
% 0.30/0.46 % not_Cons_self2
% 0.30/0.46 thf(fact_26_neq__if__length__neq,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.46 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.46 => ( Xs != Ys2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % neq_if_length_neq
% 0.30/0.46 thf(fact_27_Ex__list__of__length,axiom,
% 0.30/0.46 ! [A: $tType,N: nat] :
% 0.30/0.46 ? [Xs2: list @ A] :
% 0.30/0.46 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.30/0.46 = N ) ).
% 0.30/0.46
% 0.30/0.46 % Ex_list_of_length
% 0.30/0.46 thf(fact_28_append__eq__append__conv2,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,Ts: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.46 = ( append @ A @ Zs @ Ts ) )
% 0.30/0.46 = ( ? [Us2: list @ A] :
% 0.30/0.46 ( ( ( Xs
% 0.30/0.46 = ( append @ A @ Zs @ Us2 ) )
% 0.30/0.46 & ( ( append @ A @ Us2 @ Ys2 )
% 0.30/0.46 = Ts ) )
% 0.30/0.46 | ( ( ( append @ A @ Xs @ Us2 )
% 0.30/0.46 = Zs )
% 0.30/0.46 & ( Ys2
% 0.30/0.46 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_eq_append_conv2
% 0.30/0.46 thf(fact_29_append__eq__appendI,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys2: list @ A,Us: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Xs @ Xs1 )
% 0.30/0.46 = Zs )
% 0.30/0.46 => ( ( Ys2
% 0.30/0.46 = ( append @ A @ Xs1 @ Us ) )
% 0.30/0.46 => ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.46 = ( append @ A @ Zs @ Us ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_eq_appendI
% 0.30/0.46 thf(fact_30_map__tailrec__rev_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
% 0.30/0.46 ( ! [F2: A > B,X1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X1 )
% 0.30/0.46 => ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
% 0.30/0.46 ( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
% 0.30/0.46 => ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 0.30/0.46 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_tailrec_rev.induct
% 0.30/0.46 thf(fact_31_list__nonempty__induct,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A] :
% 0.30/0.46 ( ( Xs2
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( P @ Xs2 )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.30/0.46 => ( P @ Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list_nonempty_induct
% 0.30/0.46 thf(fact_32_remdups__adj_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 0.30/0.46 ( ( P @ ( nil @ A ) )
% 0.30/0.46 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.30/0.46 => ( ! [X2: A,Y2: A,Xs2: list @ A] :
% 0.30/0.46 ( ( ( X2 = Y2 )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
% 0.30/0.46 => ( ( ( X2 != Y2 )
% 0.30/0.46 => ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
% 0.30/0.46 => ( P @ A0 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % remdups_adj.induct
% 0.30/0.46 thf(fact_33_remdups__adj_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,X: list @ A] :
% 0.30/0.46 ( ( X
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ! [X2: A] :
% 0.30/0.46 ( X
% 0.30/0.46 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.30/0.46 => ~ ! [X2: A,Y2: A,Xs2: list @ A] :
% 0.30/0.46 ( X
% 0.30/0.46 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % remdups_adj.cases
% 0.30/0.46 thf(fact_34_transpose_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,X: list @ ( list @ A )] :
% 0.30/0.46 ( ( X
% 0.30/0.46 != ( nil @ ( list @ A ) ) )
% 0.30/0.46 => ( ! [Xss: list @ ( list @ A )] :
% 0.30/0.46 ( X
% 0.30/0.46 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.30/0.46 => ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
% 0.30/0.46 ( X
% 0.30/0.46 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transpose.cases
% 0.30/0.46 thf(fact_35_splice_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.30/0.46 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 0.30/0.46 => ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A,Y2: A,Ys3: list @ A] :
% 0.30/0.46 ( ( P @ Xs2 @ Ys3 )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % splice.induct
% 0.30/0.46 thf(fact_36_list__induct2_H,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
% 0.30/0.46 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
% 0.30/0.46 => ( ! [Y2: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys3 ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B] :
% 0.30/0.46 ( ( P @ Xs2 @ Ys3 )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) ) )
% 0.30/0.46 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list_induct2'
% 0.30/0.46 thf(fact_37_neq__Nil__conv,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 = ( ? [Y3: A,Ys4: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neq_Nil_conv
% 0.30/0.46 thf(fact_38_list_Oinducts,axiom,
% 0.30/0.46 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 0.30/0.46 ( ( P @ ( nil @ A ) )
% 0.30/0.46 => ( ! [X1: A,X23: list @ A] :
% 0.30/0.46 ( ( P @ X23 )
% 0.30/0.46 => ( P @ ( cons @ A @ X1 @ X23 ) ) )
% 0.30/0.46 => ( P @ List ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.inducts
% 0.30/0.46 thf(fact_39_list_Oexhaust,axiom,
% 0.30/0.46 ! [A: $tType,Y: list @ A] :
% 0.30/0.46 ( ( Y
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ~ ! [X212: A,X222: list @ A] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( cons @ A @ X212 @ X222 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.exhaust
% 0.30/0.46 thf(fact_40_list_OdiscI,axiom,
% 0.30/0.46 ! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
% 0.30/0.46 ( ( List
% 0.30/0.46 = ( cons @ A @ X21 @ X22 ) )
% 0.30/0.46 => ( List
% 0.30/0.46 != ( nil @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.discI
% 0.30/0.46 thf(fact_41_list_Odistinct_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X21: A,X22: list @ A] :
% 0.30/0.46 ( ( nil @ A )
% 0.30/0.46 != ( cons @ A @ X21 @ X22 ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.distinct(1)
% 0.30/0.46 thf(fact_42_list_Oset__cases,axiom,
% 0.30/0.46 ! [A: $tType,E: A,A4: list @ A] :
% 0.30/0.46 ( ( member @ A @ E @ ( set2 @ A @ A4 ) )
% 0.30/0.46 => ( ! [Z2: list @ A] :
% 0.30/0.46 ( A4
% 0.30/0.46 != ( cons @ A @ E @ Z2 ) )
% 0.30/0.46 => ~ ! [Z1: A,Z2: list @ A] :
% 0.30/0.46 ( ( A4
% 0.30/0.46 = ( cons @ A @ Z1 @ Z2 ) )
% 0.30/0.46 => ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.set_cases
% 0.30/0.46 thf(fact_43_set__ConsD,axiom,
% 0.30/0.46 ! [A: $tType,Y: A,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
% 0.30/0.46 => ( ( Y = X )
% 0.30/0.46 | ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_ConsD
% 0.30/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.30/0.46 = ( P @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 0.30/0.46 = A5 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_46_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 = ( Q @ X2 ) )
% 0.30/0.46 => ( ( collect @ A @ P )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_47_list_Oset__intros_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,A1: A,A2: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.set_intros(1)
% 0.30/0.46 thf(fact_48_list_Oset__intros_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A2: list @ A,A1: A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ A2 ) )
% 0.30/0.46 => ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.set_intros(2)
% 0.30/0.46 thf(fact_49_subset__code_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
% 0.30/0.46 = ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 => ( member @ A @ X3 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_code(1)
% 0.30/0.46 thf(fact_50_length__induct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.30/0.46 ( ! [Xs2: list @ A] :
% 0.30/0.46 ( ! [Ys: list @ A] :
% 0.30/0.46 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.30/0.46 => ( P @ Ys ) )
% 0.30/0.46 => ( P @ Xs2 ) )
% 0.30/0.46 => ( P @ Xs ) ) ).
% 0.30/0.46
% 0.30/0.46 % length_induct
% 0.30/0.46 thf(fact_51_append__Cons,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 0.30/0.46 = ( cons @ A @ X @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_Cons
% 0.30/0.46 thf(fact_52_Cons__eq__appendI,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
% 0.30/0.46 ( ( ( cons @ A @ X @ Xs1 )
% 0.30/0.46 = Ys2 )
% 0.30/0.46 => ( ( Xs
% 0.30/0.46 = ( append @ A @ Xs1 @ Zs ) )
% 0.30/0.46 => ( ( cons @ A @ X @ Xs )
% 0.30/0.46 = ( append @ A @ Ys2 @ Zs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Cons_eq_appendI
% 0.30/0.46 thf(fact_53_append__Nil,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A] :
% 0.30/0.46 ( ( append @ A @ ( nil @ A ) @ Ys2 )
% 0.30/0.46 = Ys2 ) ).
% 0.30/0.46
% 0.30/0.46 % append_Nil
% 0.30/0.46 thf(fact_54_eq__Nil__appendI,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( Xs = Ys2 )
% 0.30/0.46 => ( Xs
% 0.30/0.46 = ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_Nil_appendI
% 0.30/0.46 thf(fact_55_distinct__length__2__or__more,axiom,
% 0.30/0.46 ! [A: $tType,A4: A,B3: A,Xs: list @ A] :
% 0.30/0.46 ( ( distinct @ A @ ( cons @ A @ A4 @ ( cons @ A @ B3 @ Xs ) ) )
% 0.30/0.46 = ( ( A4 != B3 )
% 0.30/0.46 & ( distinct @ A @ ( cons @ A @ A4 @ Xs ) )
% 0.30/0.46 & ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % distinct_length_2_or_more
% 0.30/0.46 thf(fact_56_distinct_Osimps_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % distinct.simps(1)
% 0.30/0.46 thf(fact_57_list_Osel_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X21: A,X22: list @ A] :
% 0.30/0.46 ( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
% 0.30/0.46 = X21 ) ).
% 0.30/0.46
% 0.30/0.46 % list.sel(1)
% 0.30/0.46 thf(fact_58_set__subset__Cons,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_subset_Cons
% 0.30/0.46 thf(fact_59_list__induct3,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
% 0.30/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.46 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.46 => ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 0.30/0.46 = ( size_size @ ( list @ C ) @ Zs ) )
% 0.30/0.46 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B,Z: C,Zs2: list @ C] :
% 0.30/0.46 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.30/0.46 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.30/0.46 => ( ( ( size_size @ ( list @ B ) @ Ys3 )
% 0.30/0.46 = ( size_size @ ( list @ C ) @ Zs2 ) )
% 0.30/0.46 => ( ( P @ Xs2 @ Ys3 @ Zs2 )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) @ ( cons @ C @ Z @ Zs2 ) ) ) ) )
% 0.30/0.46 => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list_induct3
% 0.30/0.46 thf(fact_60_list__induct2,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 0.30/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.46 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.46 => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B] :
% 0.30/0.46 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.30/0.46 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.30/0.46 => ( ( P @ Xs2 @ Ys3 )
% 0.30/0.46 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) ) ) )
% 0.30/0.46 => ( P @ Xs @ Ys2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list_induct2
% 0.30/0.46 thf(fact_61_rev__nonempty__induct,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A] :
% 0.30/0.46 ( ( Xs2
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( P @ Xs2 )
% 0.30/0.46 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 0.30/0.46 => ( P @ Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_nonempty_induct
% 0.30/0.46 thf(fact_62_append__eq__Cons__conv,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( ( append @ A @ Ys2 @ Zs )
% 0.30/0.46 = ( cons @ A @ X @ Xs ) )
% 0.30/0.46 = ( ( ( Ys2
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 & ( Zs
% 0.30/0.46 = ( cons @ A @ X @ Xs ) ) )
% 0.30/0.46 | ? [Ys5: list @ A] :
% 0.30/0.46 ( ( Ys2
% 0.30/0.46 = ( cons @ A @ X @ Ys5 ) )
% 0.30/0.46 & ( ( append @ A @ Ys5 @ Zs )
% 0.30/0.46 = Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % append_eq_Cons_conv
% 0.30/0.46 thf(fact_63_Cons__eq__append__conv,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.30/0.46 ( ( ( cons @ A @ X @ Xs )
% 0.30/0.46 = ( append @ A @ Ys2 @ Zs ) )
% 0.30/0.46 = ( ( ( Ys2
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 & ( ( cons @ A @ X @ Xs )
% 0.30/0.46 = Zs ) )
% 0.30/0.46 | ? [Ys5: list @ A] :
% 0.30/0.46 ( ( ( cons @ A @ X @ Ys5 )
% 0.30/0.46 = Ys2 )
% 0.30/0.46 & ( Xs
% 0.30/0.46 = ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Cons_eq_append_conv
% 0.30/0.46 thf(fact_64_rev__exhaust,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ~ ! [Ys3: list @ A,Y2: A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 != ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_exhaust
% 0.30/0.46 thf(fact_65_rev__induct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.30/0.46 ( ( P @ ( nil @ A ) )
% 0.30/0.46 => ( ! [X2: A,Xs2: list @ A] :
% 0.30/0.46 ( ( P @ Xs2 )
% 0.30/0.46 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 0.30/0.46 => ( P @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_induct
% 0.30/0.46 thf(fact_66_split__list__first__prop__iff,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ( ? [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X3 ) ) )
% 0.30/0.46 = ( ? [Ys4: list @ A,X3: A] :
% 0.30/0.46 ( ? [Zs3: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs3 ) ) )
% 0.30/0.46 & ( P @ X3 )
% 0.30/0.46 & ! [Y3: A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ ( set2 @ A @ Ys4 ) )
% 0.30/0.46 => ~ ( P @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_first_prop_iff
% 0.30/0.46 thf(fact_67_split__list__last__prop__iff,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ( ? [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X3 ) ) )
% 0.30/0.46 = ( ? [Ys4: list @ A,X3: A,Zs3: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs3 ) ) )
% 0.30/0.46 & ( P @ X3 )
% 0.30/0.46 & ! [Y3: A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ ( set2 @ A @ Zs3 ) )
% 0.30/0.46 => ~ ( P @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_last_prop_iff
% 0.30/0.46 thf(fact_68_in__set__conv__decomp__first,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 = ( ? [Ys4: list @ A,Zs3: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs3 ) ) )
% 0.30/0.46 & ~ ( member @ A @ X @ ( set2 @ A @ Ys4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_set_conv_decomp_first
% 0.30/0.46 thf(fact_69_in__set__conv__decomp__last,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 = ( ? [Ys4: list @ A,Zs3: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs3 ) ) )
% 0.30/0.46 & ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_set_conv_decomp_last
% 0.30/0.46 thf(fact_70_split__list__first__propE,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X4 ) )
% 0.30/0.46 => ~ ! [Ys3: list @ A,X2: A] :
% 0.30/0.46 ( ? [Zs2: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 0.30/0.46 => ( ( P @ X2 )
% 0.30/0.46 => ~ ! [Xa: A] :
% 0.30/0.46 ( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
% 0.30/0.46 => ~ ( P @ Xa ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_first_propE
% 0.30/0.46 thf(fact_71_split__list__last__propE,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X4 ) )
% 0.30/0.46 => ~ ! [Ys3: list @ A,X2: A,Zs2: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 0.30/0.46 => ( ( P @ X2 )
% 0.30/0.46 => ~ ! [Xa: A] :
% 0.30/0.46 ( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
% 0.30/0.46 => ~ ( P @ Xa ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_last_propE
% 0.30/0.46 thf(fact_72_split__list__first__prop,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X4 ) )
% 0.30/0.46 => ? [Ys3: list @ A,X2: A] :
% 0.30/0.46 ( ? [Zs2: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 0.30/0.46 & ( P @ X2 )
% 0.30/0.46 & ! [Xa: A] :
% 0.30/0.46 ( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
% 0.30/0.46 => ~ ( P @ Xa ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_first_prop
% 0.30/0.46 thf(fact_73_split__list__last__prop,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X4 ) )
% 0.30/0.46 => ? [Ys3: list @ A,X2: A,Zs2: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 0.30/0.46 & ( P @ X2 )
% 0.30/0.46 & ! [Xa: A] :
% 0.30/0.46 ( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
% 0.30/0.46 => ~ ( P @ Xa ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_last_prop
% 0.30/0.46 thf(fact_74_in__set__conv__decomp,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 = ( ? [Ys4: list @ A,Zs3: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_set_conv_decomp
% 0.30/0.46 thf(fact_75_split__list__propE,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X4 ) )
% 0.30/0.46 => ~ ! [Ys3: list @ A,X2: A] :
% 0.30/0.46 ( ? [Zs2: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 0.30/0.46 => ~ ( P @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_propE
% 0.30/0.46 thf(fact_76_split__list__first,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 => ? [Ys3: list @ A,Zs2: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
% 0.30/0.46 & ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_first
% 0.30/0.46 thf(fact_77_split__list__prop,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( P @ X4 ) )
% 0.30/0.46 => ? [Ys3: list @ A,X2: A] :
% 0.30/0.46 ( ? [Zs2: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 0.30/0.46 & ( P @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_prop
% 0.30/0.46 thf(fact_78_split__list__last,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 => ? [Ys3: list @ A,Zs2: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
% 0.30/0.46 & ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list_last
% 0.30/0.46 thf(fact_79_split__list,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 => ? [Ys3: list @ A,Zs2: list @ A] :
% 0.30/0.46 ( Xs
% 0.30/0.46 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_list
% 0.30/0.46 thf(fact_80_distinct__singleton,axiom,
% 0.30/0.46 ! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % distinct_singleton
% 0.30/0.46 thf(fact_81_distinct_Osimps_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.46 ( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
% 0.30/0.46 = ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.46 & ( distinct @ A @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % distinct.simps(2)
% 0.30/0.46 thf(fact_82_hd__in__set,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % hd_in_set
% 0.30/0.46 thf(fact_83_list_Oset__sel_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,A4: list @ A] :
% 0.30/0.46 ( ( A4
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( member @ A @ ( hd @ A @ A4 ) @ ( set2 @ A @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % list.set_sel(1)
% 0.30/0.46 thf(fact_84_last__ConsR,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,X: A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.30/0.46 = ( last @ A @ Xs ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % last_ConsR
% 0.30/0.46 thf(fact_85_last__ConsL,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,X: A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % last_ConsL
% 0.30/0.46 thf(fact_86_last_Osimps,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,X: A] :
% 0.30/0.46 ( ( ( Xs
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.30/0.46 = X ) )
% 0.30/0.46 & ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.30/0.46 = ( last @ A @ Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % last.simps
% 0.30/0.46 thf(fact_87_last__in__set,axiom,
% 0.30/0.46 ! [A: $tType,As2: list @ A] :
% 0.30/0.46 ( ( As2
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % last_in_set
% 0.30/0.46 thf(fact_88_longest__common__prefix,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ? [Ps: list @ A,Xs3: list @ A,Ys6: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Ps @ Xs3 ) )
% 0.30/0.46 & ( Ys2
% 0.30/0.46 = ( append @ A @ Ps @ Ys6 ) )
% 0.30/0.46 & ( ( Xs3
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 | ( Ys6
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 | ( ( hd @ A @ Xs3 )
% 0.30/0.46 != ( hd @ A @ Ys6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % longest_common_prefix
% 0.30/0.46 thf(fact_89_hd__append,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( ( Xs
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( hd @ A @ Ys2 ) ) )
% 0.30/0.46 & ( ( Xs
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( hd @ A @ Xs ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % hd_append
% 0.30/0.46 thf(fact_90_longest__common__suffix,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ? [Ss: list @ A,Xs3: list @ A,Ys6: list @ A] :
% 0.30/0.46 ( ( Xs
% 0.30/0.46 = ( append @ A @ Xs3 @ Ss ) )
% 0.30/0.46 & ( Ys2
% 0.30/0.46 = ( append @ A @ Ys6 @ Ss ) )
% 0.30/0.46 & ( ( Xs3
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 | ( Ys6
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 | ( ( last @ A @ Xs3 )
% 0.30/0.46 != ( last @ A @ Ys6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % longest_common_suffix
% 0.30/0.46 thf(fact_91_last__append,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.30/0.46 ( ( ( Ys2
% 0.30/0.46 = ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( last @ A @ Xs ) ) )
% 0.30/0.46 & ( ( Ys2
% 0.30/0.46 != ( nil @ A ) )
% 0.30/0.46 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( last @ A @ Ys2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % last_append
% 0.30/0.46 thf(fact_92_not__distinct__decomp,axiom,
% 0.30/0.46 ! [A: $tType,Ws: list @ A] :
% 0.30/0.46 ( ~ ( distinct @ A @ Ws )
% 0.30/0.46 => ? [Xs2: list @ A,Ys3: list @ A,Zs2: list @ A,Y2: A] :
% 0.30/0.46 ( Ws
% 0.30/0.46 = ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_distinct_decomp
% 0.30/0.46 thf(fact_93_not__distinct__conv__prefix,axiom,
% 0.30/0.46 ! [A: $tType,As2: list @ A] :
% 0.30/0.46 ( ( ~ ( distinct @ A @ As2 ) )
% 0.30/0.46 = ( ? [Xs4: list @ A,Y3: A,Ys4: list @ A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ ( set2 @ A @ Xs4 ) )
% 0.30/0.46 & ( distinct @ A @ Xs4 )
% 0.30/0.46 & ( As2
% 0.30/0.46 = ( append @ A @ Xs4 @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_distinct_conv_prefix
% 0.30/0.46 thf(fact_94_subset__antisym,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A5 )
% 0.30/0.46 => ( A5 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_antisym
% 0.30/0.46 thf(fact_95_subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( member @ A @ X2 @ A5 )
% 0.30/0.46 => ( member @ A @ X2 @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetI
% 0.30/0.46 thf(fact_96_distinct__union,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.46 ( ( distinct @ A @ ( union @ A @ Xs @ Ys2 ) )
% 0.30/0.46 = ( distinct @ A @ Ys2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % distinct_union
% 0.30/0.46 thf(fact_97_order__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_98_set__n__lists,axiom,
% 0.30/0.46 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.46 ( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
% 0.30/0.46 = ( collect @ ( list @ A )
% 0.30/0.46 @ ^ [Ys4: list @ A] :
% 0.30/0.46 ( ( ( size_size @ ( list @ A ) @ Ys4 )
% 0.30/0.46 = N )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_n_lists
% 0.30/0.46 thf(fact_99_complete__interval,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,P: A > $o] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( P @ A4 )
% 0.30/0.46 => ( ~ ( P @ B3 )
% 0.30/0.46 => ? [C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ C2 )
% 0.30/0.46 & ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/0.46 & ! [X4: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ A4 @ X4 )
% 0.30/0.46 & ( ord_less @ A @ X4 @ C2 ) )
% 0.30/0.46 => ( P @ X4 ) )
% 0.30/0.46 & ! [D: A] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ A4 @ X2 )
% 0.30/0.46 & ( ord_less @ A @ X2 @ D ) )
% 0.30/0.46 => ( P @ X2 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ D @ C2 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % complete_interval
% 0.30/0.46 thf(fact_100_order_Onot__eq__order__implies__strict,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A] :
% 0.30/0.46 ( ( A4 != B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.not_eq_order_implies_strict
% 0.30/0.46 thf(fact_101_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A4: A] :
% 0.30/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.30/0.46 => ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.strict_implies_order
% 0.30/0.46 thf(fact_102_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( ord_less @ A )
% 0.30/0.46 = ( ^ [B4: A,A6: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B4 @ A6 )
% 0.30/0.46 & ( A6 != B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.strict_iff_order
% 0.30/0.46 thf(fact_103_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( ord_less_eq @ A )
% 0.30/0.46 = ( ^ [B4: A,A6: A] :
% 0.30/0.46 ( ( ord_less @ A @ B4 @ A6 )
% 0.30/0.46 | ( A6 = B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.order_iff_strict
% 0.30/0.46 thf(fact_104_psubsetI,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( A5 != B2 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetI
% 0.30/0.46 thf(fact_105_psubsetE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetE
% 0.30/0.46 thf(fact_106_psubset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 0.30/0.46 & ( A7 != B5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_eq
% 0.30/0.46 thf(fact_107_psubset__imp__subset,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_imp_subset
% 0.30/0.46 thf(fact_108_psubset__subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_subset_trans
% 0.30/0.46 thf(fact_109_subset__not__subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 0.30/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_not_subset_eq
% 0.30/0.46 thf(fact_110_subset__psubset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_psubset_trans
% 0.30/0.46 thf(fact_111_subset__iff__psubset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B5 )
% 0.30/0.46 | ( A7 = B5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff_psubset_eq
% 0.30/0.46 thf(fact_112_length__n__lists__elem,axiom,
% 0.30/0.46 ! [A: $tType,Ys2: list @ A,N: nat,Xs: list @ A] :
% 0.30/0.46 ( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
% 0.30/0.46 => ( ( size_size @ ( list @ A ) @ Ys2 )
% 0.30/0.46 = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % length_n_lists_elem
% 0.30/0.46 thf(fact_113_less__eq__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ( ord_less_eq @ ( A > $o )
% 0.30/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A7 )
% 0.30/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_set_def
% 0.30/0.46 thf(fact_114_impossible__Cons,axiom,
% 0.30/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.46 => ( Xs
% 0.30/0.46 != ( cons @ A @ X @ Ys2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % impossible_Cons
% 0.30/0.46 thf(fact_115_le__funD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funD
% 0.30/0.46 thf(fact_116_le__funE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funE
% 0.30/0.46 thf(fact_117_le__funI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ! [F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funI
% 0.30/0.46 thf(fact_118_le__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.46 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/0.46 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_fun_def
% 0.30/0.46 thf(fact_119_order__subst1,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B3 @ C4 )
% 0.30/0.46 => ( ! [X2: B,Y2: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst1
% 0.30/0.46 thf(fact_120_order__subst2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.30/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C4 )
% 0.30/0.46 => ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst2
% 0.30/0.46 thf(fact_121_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 0.30/0.46 ( ( A4
% 0.30/0.46 = ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B3 @ C4 )
% 0.30/0.46 => ( ! [X2: B,Y2: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_122_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,B3: A,F: A > B,C4: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ( F @ B3 )
% 0.30/0.46 = C4 )
% 0.30/0.46 => ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_123_eq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.30/0.46 = ( ^ [X3: A,Y3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.46 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_124_antisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_125_linear,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linear
% 0.30/0.46 thf(fact_126_eq__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( X = Y )
% 0.30/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_127_le__cases,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases
% 0.30/0.46 thf(fact_128_order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ C4 )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_129_le__cases3,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A,Z4: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y @ Z4 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X @ Z4 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ X @ Z4 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z4 @ Y ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Z4 @ Y )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z4 @ X ) )
% 0.30/0.46 => ~ ( ( ord_less_eq @ A @ Z4 @ X )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases3
% 0.30/0.46 thf(fact_130_antisym__conv,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [Y: A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 = ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_131_ord__eq__le__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.46 ( ( A4 = B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ C4 )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_trans
% 0.30/0.46 thf(fact_132_ord__le__eq__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( B3 = C4 )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_trans
% 0.30/0.46 thf(fact_133_order__class_Oorder_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.46 => ( A4 = B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.antisym
% 0.30/0.46 thf(fact_134_order__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A,Z4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.30/0.46 => ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_trans
% 0.30/0.46 thf(fact_135_dual__order_Orefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.refl
% 0.30/0.46 thf(fact_136_linorder__wlog,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [P: A > A > $o,A4: A,B3: A] :
% 0.30/0.46 ( ! [A3: A,B6: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B6 )
% 0.30/0.46 => ( P @ A3 @ B6 ) )
% 0.30/0.46 => ( ! [A3: A,B6: A] :
% 0.30/0.46 ( ( P @ B6 @ A3 )
% 0.30/0.46 => ( P @ A3 @ B6 ) )
% 0.30/0.46 => ( P @ A4 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linorder_wlog
% 0.30/0.46 thf(fact_137_dual__order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A4: A,C4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ C4 @ B3 )
% 0.30/0.46 => ( ord_less_eq @ A @ C4 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.trans
% 0.30/0.46 thf(fact_138_dual__order_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( A4 = B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.antisym
% 0.30/0.46 thf(fact_139_ord__eq__less__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 0.30/0.46 ( ( A4
% 0.30/0.46 = ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less @ B @ B3 @ C4 )
% 0.30/0.46 => ( ! [X2: B,Y2: B] :
% 0.30/0.46 ( ( ord_less @ B @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_less_subst
% 0.30/0.46 thf(fact_140_ord__less__eq__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,B3: A,F: A > B,C4: B] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ( F @ B3 )
% 0.30/0.46 = C4 )
% 0.30/0.46 => ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less @ B @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_less_eq_subst
% 0.30/0.46 thf(fact_141_order__less__subst1,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less @ B @ B3 @ C4 )
% 0.30/0.46 => ( ! [X2: B,Y2: B] :
% 0.30/0.46 ( ( ord_less @ B @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_less_subst1
% 0.30/0.46 thf(fact_142_order__less__subst2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.30/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less @ C @ ( F @ B3 ) @ C4 )
% 0.30/0.46 => ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_less_subst2
% 0.30/0.46 thf(fact_143_lt__ex,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % lt_ex
% 0.30/0.46 thf(fact_144_gt__ex,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( no_top @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % gt_ex
% 0.30/0.46 thf(fact_145_neqE,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( X != Y )
% 0.30/0.46 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neqE
% 0.30/0.46 thf(fact_146_neq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( X != Y )
% 0.30/0.46 = ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neq_iff
% 0.30/0.46 thf(fact_147_order_Oasym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.asym
% 0.30/0.46 thf(fact_148_dense,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ? [Z: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Z )
% 0.30/0.46 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dense
% 0.30/0.46 thf(fact_149_less__imp__neq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( X != Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_imp_neq
% 0.30/0.46 thf(fact_150_less__asym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_asym
% 0.30/0.46 thf(fact_151_less__asym_H,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_asym'
% 0.30/0.46 thf(fact_152_less__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A,Z4: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( ( ord_less @ A @ Y @ Z4 )
% 0.30/0.46 => ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_trans
% 0.30/0.46 thf(fact_153_less__linear,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 | ( X = Y )
% 0.30/0.46 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_linear
% 0.30/0.46 thf(fact_154_less__irrefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_irrefl
% 0.30/0.46 thf(fact_155_ord__eq__less__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.46 ( ( A4 = B3 )
% 0.30/0.46 => ( ( ord_less @ A @ B3 @ C4 )
% 0.30/0.46 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_less_trans
% 0.30/0.46 thf(fact_156_ord__less__eq__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( B3 = C4 )
% 0.30/0.46 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_less_eq_trans
% 0.30/0.46 thf(fact_157_dual__order_Oasym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A4: A] :
% 0.30/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.30/0.46 => ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.asym
% 0.30/0.46 thf(fact_158_less__imp__not__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( X != Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_imp_not_eq
% 0.30/0.46 thf(fact_159_less__not__sym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_not_sym
% 0.30/0.46 thf(fact_160_less__induct,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [P: A > $o,A4: A] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ! [Y5: A] :
% 0.30/0.46 ( ( ord_less @ A @ Y5 @ X2 )
% 0.30/0.46 => ( P @ Y5 ) )
% 0.30/0.46 => ( P @ X2 ) )
% 0.30/0.46 => ( P @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_induct
% 0.30/0.46 thf(fact_161_antisym__conv3,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [Y: A,X: A] :
% 0.30/0.46 ( ~ ( ord_less @ A @ Y @ X )
% 0.30/0.46 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv3
% 0.30/0.46 thf(fact_162_less__imp__not__eq2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( Y != X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_imp_not_eq2
% 0.30/0.46 thf(fact_163_less__imp__triv,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A,P: $o] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( ( ord_less @ A @ Y @ X )
% 0.30/0.46 => P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_imp_triv
% 0.30/0.46 thf(fact_164_linorder__cases,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ( ( X != Y )
% 0.30/0.46 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linorder_cases
% 0.30/0.46 thf(fact_165_dual__order_Oirrefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.irrefl
% 0.30/0.46 thf(fact_166_order_Ostrict__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less @ A @ B3 @ C4 )
% 0.30/0.46 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.strict_trans
% 0.30/0.46 thf(fact_167_less__imp__not__less,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ Y )
% 0.30/0.46 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_imp_not_less
% 0.30/0.46 thf(fact_168_dual__order_Ostrict__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A4: A,C4: A] :
% 0.30/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.30/0.46 => ( ( ord_less @ A @ C4 @ B3 )
% 0.30/0.46 => ( ord_less @ A @ C4 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.strict_trans
% 0.30/0.46 thf(fact_169_not__less__iff__gr__or__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.46 = ( ( ord_less @ A @ Y @ X )
% 0.30/0.46 | ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_less_iff_gr_or_eq
% 0.30/0.46 thf(fact_170_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( A4 != B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.strict_implies_not_eq
% 0.30/0.46 thf(fact_171_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A4: A] :
% 0.30/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.30/0.46 => ( A4 != B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.strict_implies_not_eq
% 0.30/0.46 thf(fact_172_ex__gt__or__lt,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ? [B6: A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B6 )
% 0.30/0.46 | ( ord_less @ A @ B6 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ex_gt_or_lt
% 0.30/0.46 thf(fact_173_set__mp,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_mp
% 0.30/0.46 thf(fact_174_in__mono,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_mono
% 0.30/0.46 thf(fact_175_subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ C4 @ A5 )
% 0.30/0.46 => ( member @ A @ C4 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetD
% 0.30/0.46 thf(fact_176_subsetCE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ C4 @ A5 )
% 0.30/0.46 => ( member @ A @ C4 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetCE
% 0.30/0.46 thf(fact_177_equalityE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityE
% 0.30/0.46 thf(fact_178_subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A7 )
% 0.30/0.46 => ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_eq
% 0.30/0.46 thf(fact_179_equalityD1,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD1
% 0.30/0.46 thf(fact_180_equalityD2,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD2
% 0.30/0.46 thf(fact_181_set__rev__mp,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_rev_mp
% 0.30/0.46 thf(fact_182_subset__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ! [T: A] :
% 0.30/0.46 ( ( member @ A @ T @ A7 )
% 0.30/0.46 => ( member @ A @ T @ B5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff
% 0.30/0.46 thf(fact_183_rev__subsetD,axiom,
% 0.30/0.46 ! [A: $tType,C4: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C4 @ A5 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( member @ A @ C4 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_subsetD
% 0.30/0.46 thf(fact_184_subset__refl,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.30/0.46
% 0.30/0.46 % subset_refl
% 0.30/0.46 thf(fact_185_Collect__mono,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 => ( Q @ X2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono
% 0.30/0.46 thf(fact_186_subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_trans
% 0.30/0.46 thf(fact_187_set__eq__subset,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ^ [Y4: set @ A,Z3: set @ A] : ( Y4 = Z3 ) )
% 0.30/0.46 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_eq_subset
% 0.30/0.46 thf(fact_188_contra__subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ~ ( member @ A @ C4 @ B2 )
% 0.30/0.46 => ~ ( member @ A @ C4 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % contra_subsetD
% 0.30/0.46 thf(fact_189_Collect__mono__iff,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.46 = ( ! [X3: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono_iff
% 0.30/0.46 thf(fact_190_leD,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [Y: A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % leD
% 0.30/0.46 thf(fact_191_leI,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % leI
% 0.30/0.47 thf(fact_192_le__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 | ( X3 = Y3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_less
% 0.30/0.47 thf(fact_193_less__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 & ( X3 != Y3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_le
% 0.30/0.47 thf(fact_194_order__le__less__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B3 @ C4 )
% 0.30/0.47 => ( ! [X2: B,Y2: B] :
% 0.30/0.47 ( ( ord_less @ B @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_le_less_subst1
% 0.30/0.47 thf(fact_195_order__le__less__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C @ ( type2 @ C ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C4 )
% 0.30/0.47 => ( ! [X2: A,Y2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_le_less_subst2
% 0.30/0.47 thf(fact_196_order__less__le__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B3 @ C4 )
% 0.30/0.47 => ( ! [X2: B,Y2: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_le_subst1
% 0.30/0.47 thf(fact_197_order__less__le__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C @ ( type2 @ C ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C4 )
% 0.30/0.47 => ( ! [X2: A,Y2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_le_subst2
% 0.30/0.47 thf(fact_198_not__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.30/0.47 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_le
% 0.30/0.47 thf(fact_199_not__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less
% 0.30/0.47 thf(fact_200_le__neq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A4: A,B3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.47 => ( ( A4 != B3 )
% 0.30/0.47 => ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_neq_trans
% 0.30/0.47 thf(fact_201_less__imp__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_le
% 0.30/0.47 thf(fact_202_antisym__conv1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv1
% 0.30/0.47 thf(fact_203_antisym__conv2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv2
% 0.30/0.47 thf(fact_204_le__less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ Y @ Z4 )
% 0.30/0.47 => ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_less_trans
% 0.30/0.47 thf(fact_205_less__le__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ Z4 )
% 0.30/0.47 => ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_le_trans
% 0.30/0.47 thf(fact_206_dense__ge,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Z4: A,Y: A] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_ge
% 0.30/0.47 thf(fact_207_dense__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Y: A,Z4: A] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ X2 @ Z4 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_le
% 0.30/0.47 thf(fact_208_le__less__linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_less_linear
% 0.30/0.47 thf(fact_209_le__imp__less__or__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 | ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_imp_less_or_eq
% 0.30/0.47 thf(fact_210_less__le__not__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_le_not_le
% 0.30/0.47 thf(fact_211_not__le__imp__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_le_imp_less
% 0.30/0.47 thf(fact_212_order_Ostrict__trans1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.47 => ( ( ord_less @ A @ B3 @ C4 )
% 0.30/0.47 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans1
% 0.30/0.47 thf(fact_213_order_Ostrict__trans2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A4: A,B3: A,C4: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B3 @ C4 )
% 0.30/0.47 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans2
% 0.30/0.47 thf(fact_214_order_Oorder__iff__strict,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [A6: A,B4: A] :
% 0.30/0.47 ( ( ord_less @ A @ A6 @ B4 )
% 0.30/0.47 | ( A6 = B4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.order_iff_strict
% 0.30/0.47 thf(fact_215_order_Ostrict__iff__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [A6: A,B4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A6 @ B4 )
% 0.30/0.47 & ( A6 != B4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_iff_order
% 0.30/0.47 thf(fact_216_dual__order_Ostrict__trans1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A4: A,C4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.47 => ( ( ord_less @ A @ C4 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C4 @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans1
% 0.30/0.47 thf(fact_217_dual__order_Ostrict__trans2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B3: A,A4: A,C4: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C4 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C4 @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans2
% 0.30/0.47 thf(fact_218_dense__ge__bounded,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Z4: A,X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X )
% 0.30/0.47 => ( ! [W: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ W )
% 0.30/0.47 => ( ( ord_less @ A @ W @ X )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_ge_bounded
% 0.30/0.47 thf(fact_219_dense__le__bounded,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ! [W: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ W )
% 0.30/0.47 => ( ( ord_less @ A @ W @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ W @ Z4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_le_bounded
% 0.30/0.47 thf(fact_220_order_Ostrict__implies__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A4: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_order
% 0.30/0.47 thf(fact_221_distinct__n__lists,axiom,
% 0.30/0.47 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.47 ( ( distinct @ A @ Xs )
% 0.30/0.47 => ( distinct @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % distinct_n_lists
% 0.30/0.47 thf(fact_222_sublists_Osimps_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( sublists @ A @ ( nil @ A ) )
% 0.30/0.47 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sublists.simps(1)
% 0.30/0.47 thf(fact_223_minf_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(8)
% 0.30/0.47 thf(fact_224_minf_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(6)
% 0.30/0.47 thf(fact_225_pinf_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(8)
% 0.30/0.47 thf(fact_226_predicate1I,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( P @ X2 )
% 0.30/0.47 => ( Q @ X2 ) )
% 0.30/0.47 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.30/0.47
% 0.30/0.47 % predicate1I
% 0.30/0.47 thf(fact_227_rev__predicate1D,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.30/0.47 ( ( P @ X )
% 0.30/0.47 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/0.47 => ( Q @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_predicate1D
% 0.30/0.47 thf(fact_228_less__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 => ( ( ord_less @ ( A > B ) )
% 0.30/0.47 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.30/0.47 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_fun_def
% 0.30/0.47 thf(fact_229_predicate1D,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/0.47 => ( ( P @ X )
% 0.30/0.47 => ( Q @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % predicate1D
% 0.30/0.47 thf(fact_230_psubset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.47 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_trans
% 0.30/0.47 thf(fact_231_less__set__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.30/0.47 ( ord_less @ ( A > $o )
% 0.30/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A7 )
% 0.30/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_set_def
% 0.30/0.47 thf(fact_232_psubsetD,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.47 => ( ( member @ A @ C4 @ A5 )
% 0.30/0.47 => ( member @ A @ C4 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetD
% 0.30/0.47 thf(fact_233_pinf_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z5 @ X2 )
% 0.30/0.47 => ( ( P @ X2 )
% 0.30/0.47 = ( P2 @ X2 ) ) )
% 0.30/0.47 => ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z5 @ X2 )
% 0.30/0.47 => ( ( Q @ X2 )
% 0.30/0.47 = ( Q2 @ X2 ) ) )
% 0.30/0.47 => ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ( ( ( P @ X4 )
% 0.30/0.47 & ( Q @ X4 ) )
% 0.30/0.47 = ( ( P2 @ X4 )
% 0.30/0.47 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(1)
% 0.30/0.47 thf(fact_234_pinf_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z5 @ X2 )
% 0.30/0.47 => ( ( P @ X2 )
% 0.30/0.47 = ( P2 @ X2 ) ) )
% 0.30/0.47 => ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z5 @ X2 )
% 0.30/0.47 => ( ( Q @ X2 )
% 0.30/0.47 = ( Q2 @ X2 ) ) )
% 0.30/0.47 => ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ( ( ( P @ X4 )
% 0.30/0.47 | ( Q @ X4 ) )
% 0.30/0.47 = ( ( P2 @ X4 )
% 0.30/0.47 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(2)
% 0.30/0.47 thf(fact_235_pinf_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ( X4 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(3)
% 0.30/0.47 thf(fact_236_pinf_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ( X4 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(4)
% 0.30/0.47 thf(fact_237_pinf_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(5)
% 0.30/0.47 thf(fact_238_pinf_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(7)
% 0.30/0.47 thf(fact_239_pinf_I11_J,axiom,
% 0.30/0.47 ! [C: $tType,D2: $tType] :
% 0.30/0.47 ( ( ord @ C @ ( type2 @ C ) )
% 0.30/0.47 => ! [F4: D2] :
% 0.30/0.47 ? [Z: C] :
% 0.30/0.47 ! [X4: C] :
% 0.30/0.47 ( ( ord_less @ C @ Z @ X4 )
% 0.30/0.47 => ( F4 = F4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(11)
% 0.30/0.47 thf(fact_240_minf_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Z5 )
% 0.30/0.47 => ( ( P @ X2 )
% 0.30/0.47 = ( P2 @ X2 ) ) )
% 0.30/0.47 => ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Z5 )
% 0.30/0.47 => ( ( Q @ X2 )
% 0.30/0.47 = ( Q2 @ X2 ) ) )
% 0.30/0.47 => ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ( ( ( P @ X4 )
% 0.30/0.47 & ( Q @ X4 ) )
% 0.30/0.47 = ( ( P2 @ X4 )
% 0.30/0.47 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(1)
% 0.30/0.47 thf(fact_241_minf_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Z5 )
% 0.30/0.47 => ( ( P @ X2 )
% 0.30/0.47 = ( P2 @ X2 ) ) )
% 0.30/0.47 => ( ? [Z5: A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Z5 )
% 0.30/0.47 => ( ( Q @ X2 )
% 0.30/0.47 = ( Q2 @ X2 ) ) )
% 0.30/0.47 => ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ( ( ( P @ X4 )
% 0.30/0.47 | ( Q @ X4 ) )
% 0.30/0.47 = ( ( P2 @ X4 )
% 0.30/0.47 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(2)
% 0.30/0.47 thf(fact_242_minf_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ( X4 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(3)
% 0.30/0.47 thf(fact_243_minf_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ( X4 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(4)
% 0.30/0.47 thf(fact_244_minf_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(5)
% 0.30/0.47 thf(fact_245_minf_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Z )
% 0.30/0.47 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(7)
% 0.30/0.47 thf(fact_246_minf_I11_J,axiom,
% 0.30/0.47 ! [C: $tType,D2: $tType] :
% 0.30/0.47 ( ( ord @ C @ ( type2 @ C ) )
% 0.30/0.47 => ! [F4: D2] :
% 0.30/0.47 ? [Z: C] :
% 0.30/0.47 ! [X4: C] :
% 0.30/0.47 ( ( ord_less @ C @ X4 @ Z )
% 0.30/0.47 => ( F4 = F4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(11)
% 0.30/0.47 thf(fact_247_pinf_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z: A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(6)
% 0.30/0.47 thf(fact_248_pred__subset__eq,axiom,
% 0.30/0.47 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.30/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.30/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_subset_eq
% 0.30/0.47 thf(fact_249_distinct__product__lists,axiom,
% 0.30/0.47 ! [A: $tType,Xss2: list @ ( list @ A )] :
% 0.30/0.47 ( ! [X2: list @ A] :
% 0.30/0.47 ( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss2 ) )
% 0.30/0.47 => ( distinct @ A @ X2 ) )
% 0.30/0.47 => ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % distinct_product_lists
% 0.30/0.47 thf(fact_250_in__set__product__lists__length,axiom,
% 0.30/0.47 ! [A: $tType,Xs: list @ A,Xss2: list @ ( list @ A )] :
% 0.30/0.47 ( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) )
% 0.30/0.47 => ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.47 = ( size_size @ ( list @ ( list @ A ) ) @ Xss2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_set_product_lists_length
% 0.30/0.47 thf(fact_251_product__lists_Osimps_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 0.30/0.47 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % product_lists.simps(1)
% 0.30/0.47 thf(fact_252_nat__less__le,axiom,
% 0.30/0.47 ( ( ord_less @ nat )
% 0.30/0.47 = ( ^ [M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.30/0.47 & ( M != N2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_less_le
% 0.30/0.47 thf(fact_253_less__imp__le__nat,axiom,
% 0.30/0.47 ! [M2: nat,N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ M2 @ N )
% 0.30/0.47 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_le_nat
% 0.30/0.47 thf(fact_254_nat__neq__iff,axiom,
% 0.30/0.47 ! [M2: nat,N: nat] :
% 0.30/0.47 ( ( M2 != N )
% 0.30/0.47 = ( ( ord_less @ nat @ M2 @ N )
% 0.30/0.47 | ( ord_less @ nat @ N @ M2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_neq_iff
% 0.30/0.47
% 0.30/0.47 %----Type constructors (17)
% 0.30/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 0.30/0.47 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( order @ A9 @ ( type2 @ A9 ) )
% 0.30/0.47 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.47 ! [A8: $tType,A9: $tType] :
% 0.30/0.47 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 0.30/0.47 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.30/0.47 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.30/0.47 wellorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.30/0.47 preorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.47 linorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.30/0.47 no_top @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.30/0.47 order @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.30/0.47 ord @ nat @ ( type2 @ nat ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.30/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.30/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.30/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.30/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.30/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.30/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.30/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.30/0.47
% 0.30/0.47 %----Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 gram_L250615845e_path @ f @ nla ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.76/1.54 ---- Embedded in THF ---
% 2.76/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.76/1.54 %%% Generated on Tue Feb 24 04:16:30 EST 2026
% 2.76/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.76/1.54 %%% Logic specification used:
% 2.76/1.54 %%% thf(spec, logic, $$dhol).
% 2.76/1.54
% 2.76/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.vbi8mpeqj5/DTF2DTF23649.p
% 2.76/1.54 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 2.76/1.54 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.76/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.76/1.54 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.76/1.54 thf(ty_t_DTree_ON, type, n: $tType).
% 2.76/1.54 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.76/1.54 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.76/1.54 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Opath, type, gram_L250615845e_path: ((n > dtree) > ((list @ n) > $o))).
% 2.76/1.54 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.76/1.54 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.76/1.54 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 2.76/1.54 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.76/1.54 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.76/1.54 thf(sy_c_List_Olist_Ohd, type, hd: (!> [A:$tType]: (((list @ A) > A)))).
% 2.76/1.54 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.76/1.54 thf(sy_c_List_On__lists, type, n_lists: (!> [A:$tType]: ((nat > ((list @ A) > (list @ (list @ A))))))).
% 2.76/1.54 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 2.76/1.54 thf(sy_c_List_Osublists, type, sublists: (!> [A:$tType]: (((list @ A) > (list @ (list @ A)))))).
% 2.76/1.54 thf(sy_c_List_Ounion, type, union: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.76/1.54 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.76/1.54 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.76/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.76/1.54 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.76/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.76/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.76/1.54 thf(sy_v_f, type, f: (n > dtree)).
% 2.76/1.54 thf(sy_v_nl, type, nl: (list @ n)).
% 2.76/1.54 thf(sy_v_nla____, type, nla: (list @ n)).
% 2.76/1.54 thf(fact_0_assms, axiom, ((gram_L250615845e_path @ f) @ nl)).
% 2.76/1.54 thf(fact_1__C1_Oprems_C, axiom, ((gram_L250615845e_path @ f) @ nla)).
% 2.76/1.54 thf(fact_2__C1_Ohyps_C, axiom, (! [Ys:(list @ n)]: (((((ord_less @ nat) @ ((size_size @ (list @ n)) @ Ys)) @ ((size_size @ (list @ n)) @ nla)) => (((gram_L250615845e_path @ f) @ Ys) => (? [Nl:(list @ n)]: ((((gram_L250615845e_path @ f) @ Nl) & ((((hd @ n) @ Nl) = ((hd @ n) @ Ys)) & ((((last @ n) @ Nl) = ((last @ n) @ Ys)) & ((((ord_less_eq @ (set @ n)) @ ((set2 @ n) @ Nl)) @ ((set2 @ n) @ Ys)) & ((distinct @ n) @ Nl)))))))))))).
% 2.76/1.54 thf(fact_3_path__NE, axiom, (! [F:(n > dtree),Nl2:(list @ n)]: ((((gram_L250615845e_path @ F) @ Nl2) => (Nl2 != (nil @ n)))))).
% 2.76/1.54 thf(fact_4_path_OBase, axiom, (! [F:(n > dtree),N:n]: (((gram_L250615845e_path @ F) @ (((cons @ n) @ N) @ (nil @ n)))))).
% 2.76/1.54 thf(fact_5_path__post, axiom, (! [F:(n > dtree),N:n,Nl2:(list @ n)]: ((((gram_L250615845e_path @ F) @ (((cons @ n) @ N) @ Nl2)) => ((Nl2 != (nil @ n)) => ((gram_L250615845e_path @ F) @ Nl2)))))).
% 2.76/1.54 thf(fact_6_path__post__concat, axiom, (! [F:(n > dtree),Nl1:(list @ n),Nl22:(list @ n)]: ((((gram_L250615845e_path @ F) @ (((append @ n) @ Nl1) @ Nl22)) => ((Nl22 != (nil @ n)) => ((gram_L250615845e_path @ F) @ Nl22)))))).
% 2.76/1.54 thf(fact_7_path__concat, axiom, (! [F:(n > dtree),Nl1:(list @ n),Nl22:(list @ n)]: ((((gram_L250615845e_path @ F) @ Nl1) => (((gram_L250615845e_path @ F) @ (((cons @ n) @ ((last @ n) @ Nl1)) @ Nl22)) => ((gram_L250615845e_path @ F) @ (((append @ n) @ Nl1) @ Nl22))))))).
% 2.76/1.54 thf(fact_8_last__snoc, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((last @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) = X)))).
% 2.76/1.54 thf(fact_9_last__appendL, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Xs)))))).
% 2.76/1.54 thf(fact_10_last__appendR, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Ys2)))))).
% 2.76/1.54 thf(fact_11_hd__append2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs != (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((hd @ A) @ Xs)))))).
% 2.76/1.54 thf(fact_12_append1__eq__conv, axiom, (! [A:$tType,Xs:(list @ A),X:A,Ys2:(list @ A),Y:A]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A))) = (((append @ A) @ Ys2) @ (((cons @ A) @ Y) @ (nil @ A)))) = ((Xs = Ys2) & (X = Y)))))).
% 2.76/1.54 thf(fact_13_append__eq__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Us:(list @ A),Vs:(list @ A)]: ((((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys2)) | (((size_size @ (list @ A)) @ Us) = ((size_size @ (list @ A)) @ Vs))) => (((((append @ A) @ Xs) @ Us) = (((append @ A) @ Ys2) @ Vs)) = ((Xs = Ys2) & (Us = Vs))))))).
% 2.76/1.54 thf(fact_14_append__Nil2, axiom, (! [A:$tType,Xs:(list @ A)]: (((((append @ A) @ Xs) @ (nil @ A)) = Xs)))).
% 2.76/1.54 thf(fact_15_append__self__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = Xs) = (Ys2 = (nil @ A)))))).
% 2.76/1.54 thf(fact_16_self__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs = (((append @ A) @ Xs) @ Ys2)) = (Ys2 = (nil @ A)))))).
% 2.76/1.54 thf(fact_17_append__self__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = Ys2) = (Xs = (nil @ A)))))).
% 2.76/1.54 thf(fact_18_self__append__conv2, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: (((Ys2 = (((append @ A) @ Xs) @ Ys2)) = (Xs = (nil @ A)))))).
% 2.76/1.54 thf(fact_19_list_Oinject, axiom, (! [A:$tType,X21:A,X22:(list @ A),Y21:A,Y22:(list @ A)]: ((((((cons @ A) @ X21) @ X22) = (((cons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.76/1.54 thf(fact_20_same__append__eq, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Xs) @ Zs)) = (Ys2 = Zs))))).
% 2.76/1.54 thf(fact_21_append__same__eq, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A),Zs:(list @ A)]: ((((((append @ A) @ Ys2) @ Xs) = (((append @ A) @ Zs) @ Xs)) = (Ys2 = Zs))))).
% 2.76/1.54 thf(fact_22_append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: (((((append @ A) @ (((append @ A) @ Xs) @ Ys2)) @ Zs) = (((append @ A) @ Xs) @ (((append @ A) @ Ys2) @ Zs)))))).
% 2.76/1.54 thf(fact_23_append__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = (nil @ A)) = ((Xs = (nil @ A)) & (Ys2 = (nil @ A))))))).
% 2.76/1.54 thf(fact_24_Nil__is__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((nil @ A) = (((append @ A) @ Xs) @ Ys2)) = ((Xs = (nil @ A)) & (Ys2 = (nil @ A))))))).
% 2.76/1.54 thf(fact_25_not__Cons__self2, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((cons @ A) @ X) @ Xs) != Xs)))).
% 2.76/1.54 thf(fact_26_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys2)) => (Xs != Ys2))))).
% 2.76/1.54 thf(fact_27_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 2.76/1.54 thf(fact_28_append__eq__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A),Ts:(list @ A)]: ((((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Zs) @ Ts)) = (? [Us2:(list @ A)]: ((((Xs = (((append @ A) @ Zs) @ Us2)) & ((((append @ A) @ Us2) @ Ys2) = Ts)) | (((((append @ A) @ Xs) @ Us2) = Zs) & (Ys2 = (((append @ A) @ Us2) @ Ts)))))))))).
% 2.76/1.54 thf(fact_29_append__eq__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs1:(list @ A),Zs:(list @ A),Ys2:(list @ A),Us:(list @ A)]: ((((((append @ A) @ Xs) @ Xs1) = Zs) => ((Ys2 = (((append @ A) @ Xs1) @ Us)) => ((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Zs) @ Us))))))).
% 2.76/1.54 thf(fact_30_map__tailrec__rev_Oinduct, axiom, (! [A:$tType,B:$tType,P:((A > B) > ((list @ A) > ((list @ B) > $o))),A0:(A > B),A1:(list @ A),A2:(list @ B)]: (((! [F2:(A > B),X1:(list @ B)]: ((((P @ F2) @ (nil @ A)) @ X1))) => ((! [F2:(A > B),A3:A,As:(list @ A),Bs:(list @ B)]: (((((P @ F2) @ As) @ (((cons @ B) @ (F2 @ A3)) @ Bs)) => (((P @ F2) @ (((cons @ A) @ A3) @ As)) @ Bs)))) => (((P @ A0) @ A1) @ A2)))))).
% 2.76/1.54 thf(fact_31_list__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs2:(list @ A)]: (((Xs2 != (nil @ A)) => ((P @ Xs2) => (P @ (((cons @ A) @ X2) @ Xs2)))))) => (P @ Xs))))))).
% 2.76/1.54 thf(fact_32_remdups__adj_Oinduct, axiom, (! [A:$tType,P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Xs2:(list @ A)]: ((((X2 = Y2) => (P @ (((cons @ A) @ X2) @ Xs2))) => (((X2 != Y2) => (P @ (((cons @ A) @ Y2) @ Xs2))) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs2))))))) => (P @ A0))))))).
% 2.76/1.54 thf(fact_33_remdups__adj_Ocases, axiom, (! [A:$tType,X:(list @ A)]: (((X != (nil @ A)) => ((! [X2:A]: ((X != (((cons @ A) @ X2) @ (nil @ A))))) => (~ ((! [X2:A,Y2:A,Xs2:(list @ A)]: ((X != (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs2)))))))))))).
% 2.76/1.54 thf(fact_34_transpose_Ocases, axiom, (! [A:$tType,X:(list @ (list @ A))]: (((X != (nil @ (list @ A))) => ((! [Xss:(list @ (list @ A))]: ((X != (((cons @ (list @ A)) @ (nil @ A)) @ Xss)))) => (~ ((! [X2:A,Xs2:(list @ A),Xss:(list @ (list @ A))]: ((X != (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xss))))))))))).
% 2.76/1.54 thf(fact_35_splice_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X1:(list @ A)]: (((P @ (nil @ A)) @ X1))) => ((! [V:A,Va:(list @ A)]: (((P @ (((cons @ A) @ V) @ Va)) @ (nil @ A)))) => ((! [X2:A,Xs2:(list @ A),Y2:A,Ys3:(list @ A)]: ((((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y2) @ Ys3))))) => ((P @ A0) @ A1))))))).
% 2.76/1.54 thf(fact_36_list__induct2_H, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),Xs:(list @ A),Ys2:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [X2:A,Xs2:(list @ A)]: (((P @ (((cons @ A) @ X2) @ Xs2)) @ (nil @ B)))) => ((! [Y2:B,Ys3:(list @ B)]: (((P @ (nil @ A)) @ (((cons @ B) @ Y2) @ Ys3)))) => ((! [X2:A,Xs2:(list @ A),Y2:B,Ys3:(list @ B)]: ((((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ B) @ Y2) @ Ys3))))) => ((P @ Xs) @ Ys2)))))))).
% 2.76/1.54 thf(fact_37_neq__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) = (? [Y3:A,Ys4:(list @ A)]: ((Xs = (((cons @ A) @ Y3) @ Ys4)))))))).
% 2.76/1.54 thf(fact_38_list_Oinducts, axiom, (! [A:$tType,P:((list @ A) > $o),List:(list @ A)]: (((P @ (nil @ A)) => ((! [X1:A,X23:(list @ A)]: (((P @ X23) => (P @ (((cons @ A) @ X1) @ X23))))) => (P @ List)))))).
% 2.76/1.54 thf(fact_39_list_Oexhaust, axiom, (! [A:$tType,Y:(list @ A)]: (((Y != (nil @ A)) => (~ ((! [X212:A,X222:(list @ A)]: ((Y != (((cons @ A) @ X212) @ X222)))))))))).
% 2.76/1.54 thf(fact_40_list_OdiscI, axiom, (! [A:$tType,List:(list @ A),X21:A,X22:(list @ A)]: (((List = (((cons @ A) @ X21) @ X22)) => (List != (nil @ A)))))).
% 2.76/1.54 thf(fact_41_list_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(list @ A)]: (((nil @ A) != (((cons @ A) @ X21) @ X22))))).
% 2.76/1.54 thf(fact_42_list_Oset__cases, axiom, (! [A:$tType,E:A,A4:(list @ A)]: (((((member @ A) @ E) @ ((set2 @ A) @ A4)) => ((! [Z2:(list @ A)]: ((A4 != (((cons @ A) @ E) @ Z2)))) => (~ ((! [Z1:A,Z2:(list @ A)]: (((A4 = (((cons @ A) @ Z1) @ Z2)) => (~ ((((member @ A) @ E) @ ((set2 @ A) @ Z2)))))))))))))).
% 2.76/1.54 thf(fact_43_set__ConsD, axiom, (! [A:$tType,Y:A,X:A,Xs:(list @ A)]: (((((member @ A) @ Y) @ ((set2 @ A) @ (((cons @ A) @ X) @ Xs))) => ((Y = X) | (((member @ A) @ Y) @ ((set2 @ A) @ Xs))))))).
% 2.76/1.54 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.76/1.54 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) = A5)))).
% 2.76/1.54 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.76/1.54 thf(fact_47_list_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A2:(list @ A)]: ((((member @ A) @ A1) @ ((set2 @ A) @ (((cons @ A) @ A1) @ A2)))))).
% 2.76/1.54 thf(fact_48_list_Oset__intros_I2_J, axiom, (! [A:$tType,X:A,A2:(list @ A),A1:A]: (((((member @ A) @ X) @ ((set2 @ A) @ A2)) => (((member @ A) @ X) @ ((set2 @ A) @ (((cons @ A) @ A1) @ A2))))))).
% 2.76/1.54 thf(fact_49_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B2) = (! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X3) @ B2)))))))).
% 2.76/1.54 thf(fact_50_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys)))) => (P @ Xs2)))) => (P @ Xs))))).
% 2.76/1.54 thf(fact_51_append__Cons, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys2:(list @ A)]: (((((append @ A) @ (((cons @ A) @ X) @ Xs)) @ Ys2) = (((cons @ A) @ X) @ (((append @ A) @ Xs) @ Ys2)))))).
% 2.76/1.54 thf(fact_52_Cons__eq__appendI, axiom, (! [A:$tType,X:A,Xs1:(list @ A),Ys2:(list @ A),Xs:(list @ A),Zs:(list @ A)]: ((((((cons @ A) @ X) @ Xs1) = Ys2) => ((Xs = (((append @ A) @ Xs1) @ Zs)) => ((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys2) @ Zs))))))).
% 2.76/1.54 thf(fact_53_append__Nil, axiom, (! [A:$tType,Ys2:(list @ A)]: (((((append @ A) @ (nil @ A)) @ Ys2) = Ys2)))).
% 2.76/1.54 thf(fact_54_eq__Nil__appendI, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs = Ys2) => (Xs = (((append @ A) @ (nil @ A)) @ Ys2)))))).
% 2.76/1.54 thf(fact_55_distinct__length__2__or__more, axiom, (! [A:$tType,A4:A,B3:A,Xs:(list @ A)]: ((((distinct @ A) @ (((cons @ A) @ A4) @ (((cons @ A) @ B3) @ Xs))) = ((A4 != B3) & (((distinct @ A) @ (((cons @ A) @ A4) @ Xs)) & ((distinct @ A) @ (((cons @ A) @ B3) @ Xs)))))))).
% 2.76/1.54 thf(fact_56_distinct_Osimps_I1_J, axiom, (! [A:$tType]: (((distinct @ A) @ (nil @ A))))).
% 2.76/1.54 thf(fact_57_list_Osel_I1_J, axiom, (! [A:$tType,X21:A,X22:(list @ A)]: ((((hd @ A) @ (((cons @ A) @ X21) @ X22)) = X21)))).
% 2.76/1.54 thf(fact_58_set__subset__Cons, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ (((cons @ A) @ X) @ Xs)))))).
% 2.76/1.54 thf(fact_59_list__induct3, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys2:(list @ B),Zs:(list @ C),P:((list @ A) > ((list @ B) > ((list @ C) > $o)))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((size_size @ (list @ B)) @ Ys2) = ((size_size @ (list @ C)) @ Zs)) => ((((P @ (nil @ A)) @ (nil @ B)) @ (nil @ C)) => ((! [X2:A,Xs2:(list @ A),Y2:B,Ys3:(list @ B),Z:C,Zs2:(list @ C)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ B)) @ Ys3)) => ((((size_size @ (list @ B)) @ Ys3) = ((size_size @ (list @ C)) @ Zs2)) => ((((P @ Xs2) @ Ys3) @ Zs2) => (((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ B) @ Y2) @ Ys3)) @ (((cons @ C) @ Z) @ Zs2))))))) => (((P @ Xs) @ Ys2) @ Zs)))))))).
% 2.76/1.54 thf(fact_60_list__induct2, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),P:((list @ A) > ((list @ B) > $o))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => (((P @ (nil @ A)) @ (nil @ B)) => ((! [X2:A,Xs2:(list @ A),Y2:B,Ys3:(list @ B)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ B)) @ Ys3)) => (((P @ Xs2) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ B) @ Y2) @ Ys3)))))) => ((P @ Xs) @ Ys2))))))).
% 2.76/1.54 thf(fact_61_rev__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs2:(list @ A)]: (((Xs2 != (nil @ A)) => ((P @ Xs2) => (P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X2) @ (nil @ A)))))))) => (P @ Xs))))))).
% 2.76/1.54 thf(fact_62_append__eq__Cons__conv, axiom, (! [A:$tType,Ys2:(list @ A),Zs:(list @ A),X:A,Xs:(list @ A)]: ((((((append @ A) @ Ys2) @ Zs) = (((cons @ A) @ X) @ Xs)) = (((Ys2 = (nil @ A)) & (Zs = (((cons @ A) @ X) @ Xs))) | (? [Ys5:(list @ A)]: (((Ys2 = (((cons @ A) @ X) @ Ys5)) & ((((append @ A) @ Ys5) @ Zs) = Xs))))))))).
% 2.76/1.54 thf(fact_63_Cons__eq__append__conv, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: ((((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys2) @ Zs)) = (((Ys2 = (nil @ A)) & ((((cons @ A) @ X) @ Xs) = Zs)) | (? [Ys5:(list @ A)]: ((((((cons @ A) @ X) @ Ys5) = Ys2) & (Xs = (((append @ A) @ Ys5) @ Zs)))))))))).
% 2.76/1.54 thf(fact_64_rev__exhaust, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (~ ((! [Ys3:(list @ A),Y2:A]: ((Xs != (((append @ A) @ Ys3) @ (((cons @ A) @ Y2) @ (nil @ A)))))))))))).
% 2.76/1.54 thf(fact_65_rev__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A,Xs2:(list @ A)]: (((P @ Xs2) => (P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X2) @ (nil @ A))))))) => (P @ Xs)))))).
% 2.76/1.54 thf(fact_66_split__list__first__prop__iff, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (P @ X3)))) = (? [Ys4:(list @ A),X3:A]: (((? [Zs3:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X3) @ Zs3))))) & ((P @ X3) & (! [Y3:A]: (((((member @ A) @ Y3) @ ((set2 @ A) @ Ys4)) => (~ ((P @ Y3)))))))))))))).
% 2.76/1.54 thf(fact_67_split__list__last__prop__iff, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (P @ X3)))) = (? [Ys4:(list @ A),X3:A,Zs3:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X3) @ Zs3))) & ((P @ X3) & (! [Y3:A]: (((((member @ A) @ Y3) @ ((set2 @ A) @ Zs3)) => (~ ((P @ Y3)))))))))))))).
% 2.76/1.54 thf(fact_68_in__set__conv__decomp__first, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) = (? [Ys4:(list @ A),Zs3:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X) @ Zs3))) & (~ ((((member @ A) @ X) @ ((set2 @ A) @ Ys4))))))))))).
% 2.76/1.54 thf(fact_69_in__set__conv__decomp__last, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) = (? [Ys4:(list @ A),Zs3:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X) @ Zs3))) & (~ ((((member @ A) @ X) @ ((set2 @ A) @ Zs3))))))))))).
% 2.76/1.54 thf(fact_70_split__list__first__propE, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => (~ ((! [Ys3:(list @ A),X2:A]: (((? [Zs2:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X2) @ Zs2))))) => ((P @ X2) => (~ ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Ys3)) => (~ ((P @ Xa)))))))))))))))))).
% 2.76/1.54 thf(fact_71_split__list__last__propE, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => (~ ((! [Ys3:(list @ A),X2:A,Zs2:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X2) @ Zs2))) => ((P @ X2) => (~ ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Zs2)) => (~ ((P @ Xa)))))))))))))))))).
% 2.76/1.54 thf(fact_72_split__list__first__prop, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => (? [Ys3:(list @ A),X2:A]: (((? [Zs2:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X2) @ Zs2))))) & ((P @ X2) & (! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Ys3)) => (~ ((P @ Xa)))))))))))))).
% 2.76/1.54 thf(fact_73_split__list__last__prop, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => (? [Ys3:(list @ A),X2:A,Zs2:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X2) @ Zs2))) & ((P @ X2) & (! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Zs2)) => (~ ((P @ Xa)))))))))))))).
% 2.76/1.54 thf(fact_74_in__set__conv__decomp, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) = (? [Ys4:(list @ A),Zs3:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X) @ Zs3))))))))).
% 2.76/1.54 thf(fact_75_split__list__propE, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => (~ ((! [Ys3:(list @ A),X2:A]: (((? [Zs2:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X2) @ Zs2))))) => (~ ((P @ X2)))))))))))).
% 2.76/1.54 thf(fact_76_split__list__first, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (? [Ys3:(list @ A),Zs2:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X) @ Zs2))) & (~ ((((member @ A) @ X) @ ((set2 @ A) @ Ys3))))))))))).
% 2.76/1.54 thf(fact_77_split__list__prop, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => (? [Ys3:(list @ A),X2:A]: (((? [Zs2:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X2) @ Zs2))))) & (P @ X2)))))))).
% 2.76/1.54 thf(fact_78_split__list__last, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (? [Ys3:(list @ A),Zs2:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X) @ Zs2))) & (~ ((((member @ A) @ X) @ ((set2 @ A) @ Zs2))))))))))).
% 2.76/1.54 thf(fact_79_split__list, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (? [Ys3:(list @ A),Zs2:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X) @ Zs2))))))))).
% 2.76/1.54 thf(fact_80_distinct__singleton, axiom, (! [A:$tType,X:A]: (((distinct @ A) @ (((cons @ A) @ X) @ (nil @ A)))))).
% 2.76/1.54 thf(fact_81_distinct_Osimps_I2_J, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: ((((distinct @ A) @ (((cons @ A) @ X) @ Xs)) = ((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs)))) & ((distinct @ A) @ Xs)))))).
% 2.76/1.54 thf(fact_82_hd__in__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((member @ A) @ ((hd @ A) @ Xs)) @ ((set2 @ A) @ Xs)))))).
% 2.76/1.54 thf(fact_83_list_Oset__sel_I1_J, axiom, (! [A:$tType,A4:(list @ A)]: (((A4 != (nil @ A)) => (((member @ A) @ ((hd @ A) @ A4)) @ ((set2 @ A) @ A4)))))).
% 2.76/1.54 thf(fact_84_last__ConsR, axiom, (! [A:$tType,Xs:(list @ A),X:A]: (((Xs != (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = ((last @ A) @ Xs)))))).
% 2.76/1.54 thf(fact_85_last__ConsL, axiom, (! [A:$tType,Xs:(list @ A),X:A]: (((Xs = (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = X))))).
% 2.76/1.54 thf(fact_86_last_Osimps, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((Xs = (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = X)) & ((Xs != (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = ((last @ A) @ Xs))))))).
% 2.76/1.54 thf(fact_87_last__in__set, axiom, (! [A:$tType,As2:(list @ A)]: (((As2 != (nil @ A)) => (((member @ A) @ ((last @ A) @ As2)) @ ((set2 @ A) @ As2)))))).
% 2.76/1.54 thf(fact_88_longest__common__prefix, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((? [Ps:(list @ A),Xs3:(list @ A),Ys6:(list @ A)]: (((Xs = (((append @ A) @ Ps) @ Xs3)) & ((Ys2 = (((append @ A) @ Ps) @ Ys6)) & ((Xs3 = (nil @ A)) | ((Ys6 = (nil @ A)) | (((hd @ A) @ Xs3) != ((hd @ A) @ Ys6))))))))))).
% 2.76/1.54 thf(fact_89_hd__append, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((Xs = (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((hd @ A) @ Ys2))) & ((Xs != (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((hd @ A) @ Xs))))))).
% 2.76/1.54 thf(fact_90_longest__common__suffix, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((? [Ss:(list @ A),Xs3:(list @ A),Ys6:(list @ A)]: (((Xs = (((append @ A) @ Xs3) @ Ss)) & ((Ys2 = (((append @ A) @ Ys6) @ Ss)) & ((Xs3 = (nil @ A)) | ((Ys6 = (nil @ A)) | (((last @ A) @ Xs3) != ((last @ A) @ Ys6))))))))))).
% 2.76/1.54 thf(fact_91_last__append, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: ((((Ys2 = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Xs))) & ((Ys2 != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys2)) = ((last @ A) @ Ys2))))))).
% 2.76/1.54 thf(fact_92_not__distinct__decomp, axiom, (! [A:$tType,Ws:(list @ A)]: (((~ (((distinct @ A) @ Ws))) => (? [Xs2:(list @ A),Ys3:(list @ A),Zs2:(list @ A),Y2:A]: ((Ws = (((append @ A) @ Xs2) @ (((append @ A) @ (((cons @ A) @ Y2) @ (nil @ A))) @ (((append @ A) @ Ys3) @ (((append @ A) @ (((cons @ A) @ Y2) @ (nil @ A))) @ Zs2))))))))))).
% 2.76/1.54 thf(fact_93_not__distinct__conv__prefix, axiom, (! [A:$tType,As2:(list @ A)]: (((~ (((distinct @ A) @ As2))) = (? [Xs4:(list @ A),Y3:A,Ys4:(list @ A)]: (((((member @ A) @ Y3) @ ((set2 @ A) @ Xs4)) & (((distinct @ A) @ Xs4) & (As2 = (((append @ A) @ Xs4) @ (((cons @ A) @ Y3) @ Ys4))))))))))).
% 2.76/1.54 thf(fact_94_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A5) => (A5 = B2)))))).
% 2.76/1.54 thf(fact_95_subsetI, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B2))))).
% 2.76/1.54 thf(fact_96_distinct__union, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((distinct @ A) @ (((union @ A) @ Xs) @ Ys2)) = ((distinct @ A) @ Ys2))))).
% 2.76/1.54 thf(fact_97_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.76/1.54 thf(fact_98_set__n__lists, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((set2 @ (list @ A)) @ (((n_lists @ A) @ N) @ Xs)) = ((collect @ (list @ A)) @ (^ [Ys4:(list @ A)]: (((((size_size @ (list @ A)) @ Ys4) = N) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Ys4)) @ ((set2 @ A) @ Xs)))))))))).
% 2.76/1.54 thf(fact_99_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A4:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A4) @ B3) => ((P @ A4) => ((~ ((P @ B3))) => (? [C2:A]: (((((ord_less_eq @ A) @ A4) @ C2) & ((((ord_less_eq @ A) @ C2) @ B3) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A4) @ X4) & (((ord_less @ A) @ X4) @ C2)) => (P @ X4)))) & (! [D:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A4) @ X2) & (((ord_less @ A) @ X2) @ D)) => (P @ X2)))) => (((ord_less_eq @ A) @ D) @ C2)))))))))))))))))).
% 2.76/1.54 thf(fact_100_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((A4 != B3) => ((((ord_less_eq @ A) @ A4) @ B3) => (((ord_less @ A) @ A4) @ B3))))))))).
% 2.76/1.54 thf(fact_101_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) => (((ord_less_eq @ A) @ B3) @ A4)))))))).
% 2.76/1.54 thf(fact_102_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B4:A,A6:A]: (((((ord_less_eq @ A) @ B4) @ A6) & (A6 != B4))))))))).
% 2.76/1.54 thf(fact_103_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B4:A,A6:A]: (((((ord_less @ A) @ B4) @ A6) | (A6 = B4))))))))).
% 2.76/1.54 thf(fact_104_psubsetI, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((A5 != B2) => (((ord_less @ (set @ A)) @ A5) @ B2)))))).
% 2.76/1.54 thf(fact_105_psubsetE, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B2) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A5)))))))).
% 2.76/1.54 thf(fact_106_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B5) & (A7 != B5)))))))).
% 2.76/1.54 thf(fact_107_psubset__imp__subset, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B2) => (((ord_less_eq @ (set @ A)) @ A5) @ B2))))).
% 2.76/1.54 thf(fact_108_psubset__subset__trans, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A5) @ C3)))))).
% 2.76/1.54 thf(fact_109_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B5) & (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A7)))))))))).
% 2.76/1.54 thf(fact_110_subset__psubset__trans, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A5) @ C3)))))).
% 2.76/1.54 thf(fact_111_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B5) | (A7 = B5)))))))).
% 2.76/1.54 thf(fact_112_length__n__lists__elem, axiom, (! [A:$tType,Ys2:(list @ A),N:nat,Xs:(list @ A)]: (((((member @ (list @ A)) @ Ys2) @ ((set2 @ (list @ A)) @ (((n_lists @ A) @ N) @ Xs))) => (((size_size @ (list @ A)) @ Ys2) = N))))).
% 2.76/1.54 thf(fact_113_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A7)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B5)))))))))).
% 2.76/1.54 thf(fact_114_impossible__Cons, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),X:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys2)) => (Xs != (((cons @ A) @ X) @ Ys2)))))).
% 2.76/1.54 thf(fact_115_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.76/1.54 thf(fact_116_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.76/1.54 thf(fact_117_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.76/1.54 thf(fact_118_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F3 @ X3)) @ (G2 @ X3))))))))))).
% 2.76/1.54 thf(fact_119_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B3:B,C4:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C4) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C4))))))))))).
% 2.76/1.54 thf(fact_120_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B3:A,F:(A > C),C4:C]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C4) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C4)))))))))).
% 2.76/1.54 thf(fact_121_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B3:B,C4:B]: (((A4 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C4) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C4))))))))))).
% 2.76/1.54 thf(fact_122_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B3:A,F:(A > B),C4:B]: (((((ord_less_eq @ A) @ A4) @ B3) => (((F @ B3) = C4) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C4)))))))))).
% 2.76/1.54 thf(fact_123_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 2.76/1.54 thf(fact_124_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.76/1.54 thf(fact_125_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.76/1.54 thf(fact_126_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.76/1.54 thf(fact_127_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.76/1.54 thf(fact_128_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ C4) => (((ord_less_eq @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_129_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z4:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z4)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z4)))) => (((((ord_less_eq @ A) @ X) @ Z4) => (~ ((((ord_less_eq @ A) @ Z4) @ Y)))) => (((((ord_less_eq @ A) @ Z4) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z4) => (~ ((((ord_less_eq @ A) @ Z4) @ X)))) => (~ (((((ord_less_eq @ A) @ Z4) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.76/1.54 thf(fact_130_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.76/1.54 thf(fact_131_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((A4 = B3) => ((((ord_less_eq @ A) @ B3) @ C4) => (((ord_less_eq @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_132_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((B3 = C4) => (((ord_less_eq @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_133_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ A4) => (A4 = B3))))))))).
% 2.76/1.54 thf(fact_134_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z4:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z4) => (((ord_less_eq @ A) @ X) @ Z4))))))))).
% 2.76/1.54 thf(fact_135_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.76/1.54 thf(fact_136_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A4:A,B3:A]: (((! [A3:A,B6:A]: (((((ord_less_eq @ A) @ A3) @ B6) => ((P @ A3) @ B6)))) => ((! [A3:A,B6:A]: ((((P @ B6) @ A3) => ((P @ A3) @ B6)))) => ((P @ A4) @ B3))))))))).
% 2.76/1.54 thf(fact_137_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A,C4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ C4) @ B3) => (((ord_less_eq @ A) @ C4) @ A4))))))))).
% 2.76/1.54 thf(fact_138_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ A4) @ B3) => (A4 = B3))))))))).
% 2.76/1.54 thf(fact_139_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B3:B,C4:B]: (((A4 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C4) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C4))))))))))).
% 2.76/1.54 thf(fact_140_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B3:A,F:(A > B),C4:B]: (((((ord_less @ A) @ A4) @ B3) => (((F @ B3) = C4) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ B) @ (F @ A4)) @ C4)))))))))).
% 2.76/1.54 thf(fact_141_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B3:B,C4:B]: (((((ord_less @ A) @ A4) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C4) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C4))))))))))).
% 2.76/1.54 thf(fact_142_order__less__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B3:A,F:(A > C),C4:C]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C4) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ C) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C4)))))))))).
% 2.76/1.54 thf(fact_143_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X))))))))).
% 2.76/1.54 thf(fact_144_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.76/1.54 thf(fact_145_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.76/1.54 thf(fact_146_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.76/1.54 thf(fact_147_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (~ ((((ord_less @ A) @ B3) @ A4)))))))))).
% 2.76/1.54 thf(fact_148_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z:A]: (((((ord_less @ A) @ X) @ Z) & (((ord_less @ A) @ Z) @ Y))))))))))).
% 2.76/1.54 thf(fact_149_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.76/1.54 thf(fact_150_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.76/1.54 thf(fact_151_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (~ ((((ord_less @ A) @ B3) @ A4)))))))))).
% 2.76/1.54 thf(fact_152_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z4:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z4) => (((ord_less @ A) @ X) @ Z4))))))))).
% 2.76/1.54 thf(fact_153_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.76/1.54 thf(fact_154_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.76/1.54 thf(fact_155_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((A4 = B3) => ((((ord_less @ A) @ B3) @ C4) => (((ord_less @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_156_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((((ord_less @ A) @ A4) @ B3) => ((B3 = C4) => (((ord_less @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_157_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) => (~ ((((ord_less @ A) @ A4) @ B3)))))))))).
% 2.76/1.54 thf(fact_158_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.76/1.54 thf(fact_159_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.76/1.54 thf(fact_160_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A4:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (P @ Y5)))) => (P @ X2)))) => (P @ A4)))))))).
% 2.76/1.54 thf(fact_161_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.76/1.54 thf(fact_162_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 2.76/1.54 thf(fact_163_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 2.76/1.54 thf(fact_164_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.76/1.54 thf(fact_165_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((~ ((((ord_less @ A) @ A4) @ A4))))))))).
% 2.76/1.54 thf(fact_166_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less @ A) @ B3) @ C4) => (((ord_less @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_167_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.76/1.54 thf(fact_168_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A,C4:A]: (((((ord_less @ A) @ B3) @ A4) => ((((ord_less @ A) @ C4) @ B3) => (((ord_less @ A) @ C4) @ A4))))))))).
% 2.76/1.54 thf(fact_169_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 2.76/1.54 thf(fact_170_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (A4 != B3)))))))).
% 2.76/1.54 thf(fact_171_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) => (A4 != B3)))))))).
% 2.76/1.54 thf(fact_172_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A4:A]: ((? [B6:A]: (((((ord_less @ A) @ A4) @ B6) | (((ord_less @ A) @ B6) @ A4)))))))))).
% 2.76/1.54 thf(fact_173_set__mp, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ X) @ A5) => (((member @ A) @ X) @ B2)))))).
% 2.76/1.54 thf(fact_174_in__mono, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ X) @ A5) => (((member @ A) @ X) @ B2)))))).
% 2.76/1.54 thf(fact_175_subsetD, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ C4) @ A5) => (((member @ A) @ C4) @ B2)))))).
% 2.76/1.54 thf(fact_176_subsetCE, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ C4) @ A5) => (((member @ A) @ C4) @ B2)))))).
% 2.76/1.54 thf(fact_177_equalityE, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((A5 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A5)))))))))).
% 2.76/1.54 thf(fact_178_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A7) => (((member @ A) @ X3) @ B5)))))))))).
% 2.76/1.54 thf(fact_179_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((A5 = B2) => (((ord_less_eq @ (set @ A)) @ A5) @ B2))))).
% 2.76/1.54 thf(fact_180_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((A5 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A5))))).
% 2.76/1.54 thf(fact_181_set__rev__mp, axiom, (! [A:$tType,X:A,A5:(set @ A),B2:(set @ A)]: (((((member @ A) @ X) @ A5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((member @ A) @ X) @ B2)))))).
% 2.76/1.54 thf(fact_182_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A7) => (((member @ A) @ T) @ B5)))))))))).
% 2.76/1.54 thf(fact_183_rev__subsetD, axiom, (! [A:$tType,C4:A,A5:(set @ A),B2:(set @ A)]: (((((member @ A) @ C4) @ A5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((member @ A) @ C4) @ B2)))))).
% 2.76/1.54 thf(fact_184_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 2.76/1.54 thf(fact_185_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.76/1.54 thf(fact_186_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ A5) @ C3)))))).
% 2.76/1.54 thf(fact_187_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z3:(set @ A)]: ((Y4 = Z3))) = (^ [A7:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A7)))))))).
% 2.76/1.54 thf(fact_188_contra__subsetD, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((~ ((((member @ A) @ C4) @ B2))) => (~ ((((member @ A) @ C4) @ A5)))))))).
% 2.76/1.54 thf(fact_189_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.76/1.54 thf(fact_190_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 2.76/1.54 thf(fact_191_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.76/1.54 thf(fact_192_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 2.76/1.54 thf(fact_193_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (X3 != Y3))))))))).
% 2.76/1.54 thf(fact_194_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B3:B,C4:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C4) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C4))))))))))).
% 2.76/1.54 thf(fact_195_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B3:A,F:(A > C),C4:C]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C4) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C4)))))))))).
% 2.76/1.54 thf(fact_196_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B3:B,C4:B]: (((((ord_less @ A) @ A4) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C4) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C4))))))))))).
% 2.76/1.54 thf(fact_197_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B3:A,F:(A > C),C4:C]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C4) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ C) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C4)))))))))).
% 2.76/1.54 thf(fact_198_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 2.76/1.54 thf(fact_199_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.76/1.54 thf(fact_200_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((A4 != B3) => (((ord_less @ A) @ A4) @ B3))))))))).
% 2.76/1.54 thf(fact_201_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.76/1.54 thf(fact_202_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.76/1.54 thf(fact_203_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.76/1.54 thf(fact_204_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z4:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z4) => (((ord_less @ A) @ X) @ Z4))))))))).
% 2.76/1.54 thf(fact_205_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z4:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z4) => (((ord_less @ A) @ X) @ Z4))))))))).
% 2.76/1.54 thf(fact_206_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z4:A,Y:A]: (((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => (((ord_less_eq @ A) @ Y) @ X2)))) => (((ord_less_eq @ A) @ Y) @ Z4)))))))).
% 2.76/1.54 thf(fact_207_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z4:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Z4)))) => (((ord_less_eq @ A) @ Y) @ Z4)))))))).
% 2.76/1.54 thf(fact_208_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 2.76/1.54 thf(fact_209_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 2.76/1.54 thf(fact_210_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X3))))))))))).
% 2.76/1.54 thf(fact_211_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 2.76/1.54 thf(fact_212_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less @ A) @ B3) @ C4) => (((ord_less @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_213_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A,C4:A]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ C4) => (((ord_less @ A) @ A4) @ C4))))))))).
% 2.76/1.54 thf(fact_214_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A6:A,B4:A]: (((((ord_less @ A) @ A6) @ B4) | (A6 = B4))))))))).
% 2.76/1.54 thf(fact_215_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B4:A]: (((((ord_less_eq @ A) @ A6) @ B4) & (A6 != B4))))))))).
% 2.76/1.54 thf(fact_216_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A,C4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less @ A) @ C4) @ B3) => (((ord_less @ A) @ C4) @ A4))))))))).
% 2.76/1.54 thf(fact_217_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A4:A,C4:A]: (((((ord_less @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ C4) @ B3) => (((ord_less @ A) @ C4) @ A4))))))))).
% 2.76/1.54 thf(fact_218_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z4:A,X:A,Y:A]: (((((ord_less @ A) @ Z4) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z4) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z4))))))))).
% 2.76/1.54 thf(fact_219_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z4:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z4))))) => (((ord_less_eq @ A) @ Y) @ Z4))))))))).
% 2.76/1.54 thf(fact_220_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (((ord_less_eq @ A) @ A4) @ B3)))))))).
% 2.76/1.54 thf(fact_221_distinct__n__lists, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: ((((distinct @ A) @ Xs) => ((distinct @ (list @ A)) @ (((n_lists @ A) @ N) @ Xs)))))).
% 2.76/1.54 thf(fact_222_sublists_Osimps_I1_J, axiom, (! [A:$tType]: ((((sublists @ A) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.76/1.54 thf(fact_223_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (~ ((((ord_less_eq @ A) @ T2) @ X4)))))))))))))).
% 2.76/1.54 thf(fact_224_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((ord_less_eq @ A) @ X4) @ T2)))))))))))).
% 2.76/1.54 thf(fact_225_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((ord_less_eq @ A) @ T2) @ X4)))))))))))).
% 2.76/1.54 thf(fact_226_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 2.76/1.54 thf(fact_227_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X)))))).
% 2.76/1.54 thf(fact_228_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F3))))))))))).
% 2.76/1.54 thf(fact_229_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X) => (Q @ X)))))).
% 2.76/1.54 thf(fact_230_psubset__trans, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A5) @ C3)))))).
% 2.76/1.54 thf(fact_231_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A7)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B5)))))))))).
% 2.76/1.54 thf(fact_232_psubsetD, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C4:A]: (((((ord_less @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ C4) @ A5) => (((member @ A) @ C4) @ B2)))))).
% 2.76/1.54 thf(fact_233_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ Z5) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ Z5) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 2.76/1.54 thf(fact_234_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ Z5) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ Z5) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 2.76/1.54 thf(fact_235_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (X4 != T2)))))))))))).
% 2.76/1.54 thf(fact_236_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (X4 != T2)))))))))))).
% 2.76/1.54 thf(fact_237_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (~ ((((ord_less @ A) @ X4) @ T2)))))))))))))).
% 2.76/1.54 thf(fact_238_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((ord_less @ A) @ T2) @ X4)))))))))))).
% 2.76/1.54 thf(fact_239_pinf_I11_J, axiom, (! [C:$tType,D2:$tType]: ((((ord @ C) @ (type2 @ C)) => (! [F4:D2]: ((? [Z:C]: ((! [X4:C]: (((((ord_less @ C) @ Z) @ X4) => (F4 = F4)))))))))))).
% 2.76/1.54 thf(fact_240_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z5) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z5) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 2.76/1.54 thf(fact_241_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z5) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z5:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z5) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 2.76/1.54 thf(fact_242_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (X4 != T2)))))))))))).
% 2.76/1.54 thf(fact_243_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (X4 != T2)))))))))))).
% 2.76/1.54 thf(fact_244_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((ord_less @ A) @ X4) @ T2)))))))))))).
% 2.76/1.54 thf(fact_245_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (~ ((((ord_less @ A) @ T2) @ X4)))))))))))))).
% 2.76/1.54 thf(fact_246_minf_I11_J, axiom, (! [C:$tType,D2:$tType]: ((((ord @ C) @ (type2 @ C)) => (! [F4:D2]: ((? [Z:C]: ((! [X4:C]: (((((ord_less @ C) @ X4) @ Z) => (F4 = F4)))))))))))).
% 2.76/1.54 thf(fact_247_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ T2)))))))))))))).
% 2.76/1.54 thf(fact_248_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 2.76/1.54 thf(fact_249_distinct__product__lists, axiom, (! [A:$tType,Xss2:(list @ (list @ A))]: (((! [X2:(list @ A)]: (((((member @ (list @ A)) @ X2) @ ((set2 @ (list @ A)) @ Xss2)) => ((distinct @ A) @ X2)))) => ((distinct @ (list @ A)) @ ((product_lists @ A) @ Xss2)))))).
% 2.76/1.54 thf(fact_250_in__set__product__lists__length, axiom, (! [A:$tType,Xs:(list @ A),Xss2:(list @ (list @ A))]: (((((member @ (list @ A)) @ Xs) @ ((set2 @ (list @ A)) @ ((product_lists @ A) @ Xss2))) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ (list @ A))) @ Xss2)))))).
% 2.76/1.54 thf(fact_251_product__lists_Osimps_I1_J, axiom, (! [A:$tType]: ((((product_lists @ A) @ (nil @ (list @ A))) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.76/1.54 thf(fact_252_nat__less__le, axiom, ((ord_less @ nat) = (^ [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) & (M != N2)))))).
% 2.76/1.54 thf(fact_253_less__imp__le__nat, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.76/1.54 thf(fact_254_nat__neq__iff, axiom, (! [M2:nat,N:nat]: (((M2 != N) = ((((ord_less @ nat) @ M2) @ N) | (((ord_less @ nat) @ N) @ M2)))))).
% 2.76/1.54 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A9) @ (type2 @ A9)) => ((preorder @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.76/1.54 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A9) @ (type2 @ A9)) => ((order @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.76/1.54 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A9) @ (type2 @ A9)) => ((ord @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.76/1.54 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.76/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A8:$tType]: (((preorder @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.76/1.54 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A8:$tType]: (((order @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.76/1.54 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A8:$tType]: (((ord @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.76/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.76/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.76/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.76/1.54 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.76/1.54 thf(conj_0, conjecture, ((gram_L250615845e_path @ f) @ nla)).
% 2.76/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.vbi8mpeqj5/DTF2DTF23649.p
% 2.76/1.54 ------------------------
% 2.76/1.57 ---- Cleaned THF ---
% 2.76/1.57 thf(ty_t_DTree_Odtree,type,
% 2.76/1.57 dtree: $tType ).
% 2.76/1.57
% 2.76/1.57 thf(ty_t_List_Olist,type,
% 2.76/1.57 list: $tType > $tType ).
% 2.76/1.57
% 2.76/1.57 thf(ty_t_Set_Oset,type,
% 2.76/1.57 set: $tType > $tType ).
% 2.76/1.57
% 2.76/1.57 thf(ty_t_Nat_Onat,type,
% 2.76/1.57 nat: $tType ).
% 2.76/1.57
% 2.76/1.57 thf(ty_t_DTree_ON,type,
% 2.76/1.57 n: $tType ).
% 2.76/1.57
% 2.76/1.57 thf(ty_t_itself,type,
% 2.76/1.57 itself: $tType > $tType ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_HOL_Otype,type,
% 2.76/1.57 type:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Oord,type,
% 2.76/1.57 ord:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Oorder,type,
% 2.76/1.57 order:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Ono__bot,type,
% 2.76/1.57 no_bot:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Ono__top,type,
% 2.76/1.57 no_top:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.76/1.57 linorder:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Opreorder,type,
% 2.76/1.57 preorder:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Owellorder,type,
% 2.76/1.57 wellorder:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Odense__order,type,
% 2.76/1.57 dense_order:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.76/1.57 dense_linorder:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.76/1.57 condit1656338222tinuum:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.76/1.57 condit1037483654norder:
% 2.76/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Opath,type,
% 2.76/1.57 gram_L250615845e_path: ( n > dtree ) > ( list @ n ) > $o ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Oappend,type,
% 2.76/1.57 append:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Odistinct,type,
% 2.76/1.57 distinct:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Olast,type,
% 2.76/1.57 last:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Olist_OCons,type,
% 2.76/1.57 cons:
% 2.76/1.57 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Olist_ONil,type,
% 2.76/1.57 nil:
% 2.76/1.57 !>[A: $tType] : ( list @ A ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Olist_Ohd,type,
% 2.76/1.57 hd:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Olist_Oset,type,
% 2.76/1.57 set2:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_On__lists,type,
% 2.76/1.57 n_lists:
% 2.76/1.57 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Oproduct__lists,type,
% 2.76/1.57 product_lists:
% 2.76/1.57 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Osublists,type,
% 2.76/1.57 sublists:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_List_Ounion,type,
% 2.76/1.57 union:
% 2.76/1.57 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.76/1.57 size_size:
% 2.76/1.57 !>[A: $tType] : ( A > nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.76/1.57 ord_less:
% 2.76/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.76/1.57 ord_less_eq:
% 2.76/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_Pure_Otype,type,
% 2.76/1.57 type2:
% 2.76/1.57 !>[A: $tType] : ( itself @ A ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_Set_OCollect,type,
% 2.76/1.57 collect:
% 2.76/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_c_member,type,
% 2.76/1.57 member:
% 2.76/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(sy_v_f,type,
% 2.76/1.57 f: n > dtree ).
% 2.76/1.57
% 2.76/1.57 thf(sy_v_nl,type,
% 2.76/1.57 nl: list @ n ).
% 2.76/1.57
% 2.76/1.57 thf(sy_v_nla____,type,
% 2.76/1.57 nla: list @ n ).
% 2.76/1.57
% 2.76/1.57 thf(fact_0_assms,axiom,
% 2.76/1.57 gram_L250615845e_path @ f @ nl ).
% 2.76/1.57
% 2.76/1.57 thf(fact_1__C1_Oprems_C,axiom,
% 2.76/1.57 gram_L250615845e_path @ f @ nla ).
% 2.76/1.57
% 2.76/1.57 thf(fact_2__C1_Ohyps_C,axiom,
% 2.76/1.57 ! [Ys: list @ n] :
% 2.76/1.57 ( ( ord_less @ nat @ ( size_size @ ( list @ n ) @ Ys ) @ ( size_size @ ( list @ n ) @ nla ) )
% 2.76/1.57 => ( ( gram_L250615845e_path @ f @ Ys )
% 2.76/1.57 => ? [Nl: list @ n] :
% 2.76/1.57 ( ( gram_L250615845e_path @ f @ Nl )
% 2.76/1.57 & ( ( hd @ n @ Nl )
% 2.76/1.57 = ( hd @ n @ Ys ) )
% 2.76/1.57 & ( ( last @ n @ Nl )
% 2.76/1.57 = ( last @ n @ Ys ) )
% 2.76/1.57 & ( ord_less_eq @ ( set @ n ) @ ( set2 @ n @ Nl ) @ ( set2 @ n @ Ys ) )
% 2.76/1.57 & ( distinct @ n @ Nl ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_3_path__NE,axiom,
% 2.76/1.57 ! [F: n > dtree,Nl2: list @ n] :
% 2.76/1.57 ( ( gram_L250615845e_path @ F @ Nl2 )
% 2.76/1.57 => ( Nl2
% 2.76/1.57 != ( nil @ n ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_4_path_OBase,axiom,
% 2.76/1.57 ! [F: n > dtree,N: n] : ( gram_L250615845e_path @ F @ ( cons @ n @ N @ ( nil @ n ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_5_path__post,axiom,
% 2.76/1.57 ! [F: n > dtree,N: n,Nl2: list @ n] :
% 2.76/1.57 ( ( gram_L250615845e_path @ F @ ( cons @ n @ N @ Nl2 ) )
% 2.76/1.57 => ( ( Nl2
% 2.76/1.57 != ( nil @ n ) )
% 2.76/1.57 => ( gram_L250615845e_path @ F @ Nl2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_6_path__post__concat,axiom,
% 2.76/1.57 ! [F: n > dtree,Nl1: list @ n,Nl22: list @ n] :
% 2.76/1.57 ( ( gram_L250615845e_path @ F @ ( append @ n @ Nl1 @ Nl22 ) )
% 2.76/1.57 => ( ( Nl22
% 2.76/1.57 != ( nil @ n ) )
% 2.76/1.57 => ( gram_L250615845e_path @ F @ Nl22 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_7_path__concat,axiom,
% 2.76/1.57 ! [F: n > dtree,Nl1: list @ n,Nl22: list @ n] :
% 2.76/1.57 ( ( gram_L250615845e_path @ F @ Nl1 )
% 2.76/1.57 => ( ( gram_L250615845e_path @ F @ ( cons @ n @ ( last @ n @ Nl1 ) @ Nl22 ) )
% 2.76/1.57 => ( gram_L250615845e_path @ F @ ( append @ n @ Nl1 @ Nl22 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_8_last__snoc,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,X: A] :
% 2.76/1.57 ( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 2.76/1.57 = X ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_9_last__appendL,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 2.76/1.57 ( ( Ys2
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( last @ A @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_10_last__appendR,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 2.76/1.57 ( ( Ys2
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( last @ A @ Ys2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_11_hd__append2,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( hd @ A @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_12_append1__eq__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y: A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 2.76/1.57 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 2.76/1.57 = ( ( Xs = Ys2 )
% 2.76/1.57 & ( X = Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_13_append__eq__append__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Us: list @ A,Vs: list @ A] :
% 2.76/1.57 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.76/1.57 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.76/1.57 | ( ( size_size @ ( list @ A ) @ Us )
% 2.76/1.57 = ( size_size @ ( list @ A ) @ Vs ) ) )
% 2.76/1.57 => ( ( ( append @ A @ Xs @ Us )
% 2.76/1.57 = ( append @ A @ Ys2 @ Vs ) )
% 2.76/1.57 = ( ( Xs = Ys2 )
% 2.76/1.57 & ( Us = Vs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_14_append__Nil2,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A] :
% 2.76/1.57 ( ( append @ A @ Xs @ ( nil @ A ) )
% 2.76/1.57 = Xs ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_15_append__self__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ Ys2 )
% 2.76/1.57 = Xs )
% 2.76/1.57 = ( Ys2
% 2.76/1.57 = ( nil @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_16_self__append__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( Ys2
% 2.76/1.57 = ( nil @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_17_append__self__conv2,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ Ys2 )
% 2.76/1.57 = Ys2 )
% 2.76/1.57 = ( Xs
% 2.76/1.57 = ( nil @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_18_self__append__conv2,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 2.76/1.57 ( ( Ys2
% 2.76/1.57 = ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( Xs
% 2.76/1.57 = ( nil @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_19_list_Oinject,axiom,
% 2.76/1.57 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 2.76/1.57 ( ( ( cons @ A @ X21 @ X22 )
% 2.76/1.57 = ( cons @ A @ Y21 @ Y22 ) )
% 2.76/1.57 = ( ( X21 = Y21 )
% 2.76/1.57 & ( X22 = Y22 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_20_same__append__eq,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ Ys2 )
% 2.76/1.57 = ( append @ A @ Xs @ Zs ) )
% 2.76/1.57 = ( Ys2 = Zs ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_21_append__same__eq,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Ys2 @ Xs )
% 2.76/1.57 = ( append @ A @ Zs @ Xs ) )
% 2.76/1.57 = ( Ys2 = Zs ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_22_append__assoc,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 2.76/1.57 ( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs )
% 2.76/1.57 = ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_23_append__is__Nil__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ Ys2 )
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 = ( ( Xs
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 & ( Ys2
% 2.76/1.57 = ( nil @ A ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_24_Nil__is__append__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( ( nil @ A )
% 2.76/1.57 = ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( ( Xs
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 & ( Ys2
% 2.76/1.57 = ( nil @ A ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_25_not__Cons__self2,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( cons @ A @ X @ Xs )
% 2.76/1.57 != Xs ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_26_neq__if__length__neq,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.76/1.57 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.76/1.57 => ( Xs != Ys2 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_27_Ex__list__of__length,axiom,
% 2.76/1.57 ! [A: $tType,N: nat] :
% 2.76/1.57 ? [Xs2: list @ A] :
% 2.76/1.57 ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.76/1.57 = N ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_28_append__eq__append__conv2,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,Ts: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ Ys2 )
% 2.76/1.57 = ( append @ A @ Zs @ Ts ) )
% 2.76/1.57 = ( ? [Us2: list @ A] :
% 2.76/1.57 ( ( ( Xs
% 2.76/1.57 = ( append @ A @ Zs @ Us2 ) )
% 2.76/1.57 & ( ( append @ A @ Us2 @ Ys2 )
% 2.76/1.57 = Ts ) )
% 2.76/1.57 | ( ( ( append @ A @ Xs @ Us2 )
% 2.76/1.57 = Zs )
% 2.76/1.57 & ( Ys2
% 2.76/1.57 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_29_append__eq__appendI,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys2: list @ A,Us: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Xs @ Xs1 )
% 2.76/1.57 = Zs )
% 2.76/1.57 => ( ( Ys2
% 2.76/1.57 = ( append @ A @ Xs1 @ Us ) )
% 2.76/1.57 => ( ( append @ A @ Xs @ Ys2 )
% 2.76/1.57 = ( append @ A @ Zs @ Us ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_30_map__tailrec__rev_Oinduct,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
% 2.76/1.57 ( ! [F2: A > B,X1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X1 )
% 2.76/1.57 => ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
% 2.76/1.57 ( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
% 2.76/1.57 => ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 2.76/1.57 => ( P @ A0 @ A1 @ A2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_31_list__nonempty__induct,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A] :
% 2.76/1.57 ( ( Xs2
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( P @ Xs2 )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 2.76/1.57 => ( P @ Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_32_remdups__adj_Oinduct,axiom,
% 2.76/1.57 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 2.76/1.57 ( ( P @ ( nil @ A ) )
% 2.76/1.57 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.76/1.57 => ( ! [X2: A,Y2: A,Xs2: list @ A] :
% 2.76/1.57 ( ( ( X2 = Y2 )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
% 2.76/1.57 => ( ( ( X2 != Y2 )
% 2.76/1.57 => ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
% 2.76/1.57 => ( P @ A0 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_33_remdups__adj_Ocases,axiom,
% 2.76/1.57 ! [A: $tType,X: list @ A] :
% 2.76/1.57 ( ( X
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ! [X2: A] :
% 2.76/1.57 ( X
% 2.76/1.57 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.76/1.57 => ~ ! [X2: A,Y2: A,Xs2: list @ A] :
% 2.76/1.57 ( X
% 2.76/1.57 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_34_transpose_Ocases,axiom,
% 2.76/1.57 ! [A: $tType,X: list @ ( list @ A )] :
% 2.76/1.57 ( ( X
% 2.76/1.57 != ( nil @ ( list @ A ) ) )
% 2.76/1.57 => ( ! [Xss: list @ ( list @ A )] :
% 2.76/1.57 ( X
% 2.76/1.57 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 2.76/1.57 => ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
% 2.76/1.57 ( X
% 2.76/1.57 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_35_splice_Oinduct,axiom,
% 2.76/1.57 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.76/1.57 ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
% 2.76/1.57 => ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A,Y2: A,Ys3: list @ A] :
% 2.76/1.57 ( ( P @ Xs2 @ Ys3 )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 2.76/1.57 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_36_list__induct2_H,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
% 2.76/1.57 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
% 2.76/1.57 => ( ! [Y2: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys3 ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B] :
% 2.76/1.57 ( ( P @ Xs2 @ Ys3 )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) ) )
% 2.76/1.57 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_37_neq__Nil__conv,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 = ( ? [Y3: A,Ys4: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_38_list_Oinducts,axiom,
% 2.76/1.57 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 2.76/1.57 ( ( P @ ( nil @ A ) )
% 2.76/1.57 => ( ! [X1: A,X23: list @ A] :
% 2.76/1.57 ( ( P @ X23 )
% 2.76/1.57 => ( P @ ( cons @ A @ X1 @ X23 ) ) )
% 2.76/1.57 => ( P @ List ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_39_list_Oexhaust,axiom,
% 2.76/1.57 ! [A: $tType,Y: list @ A] :
% 2.76/1.57 ( ( Y
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ~ ! [X212: A,X222: list @ A] :
% 2.76/1.57 ( Y
% 2.76/1.57 != ( cons @ A @ X212 @ X222 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_40_list_OdiscI,axiom,
% 2.76/1.57 ! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
% 2.76/1.57 ( ( List
% 2.76/1.57 = ( cons @ A @ X21 @ X22 ) )
% 2.76/1.57 => ( List
% 2.76/1.57 != ( nil @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_41_list_Odistinct_I1_J,axiom,
% 2.76/1.57 ! [A: $tType,X21: A,X22: list @ A] :
% 2.76/1.57 ( ( nil @ A )
% 2.76/1.57 != ( cons @ A @ X21 @ X22 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_42_list_Oset__cases,axiom,
% 2.76/1.57 ! [A: $tType,E: A,A4: list @ A] :
% 2.76/1.57 ( ( member @ A @ E @ ( set2 @ A @ A4 ) )
% 2.76/1.57 => ( ! [Z2: list @ A] :
% 2.76/1.57 ( A4
% 2.76/1.57 != ( cons @ A @ E @ Z2 ) )
% 2.76/1.57 => ~ ! [Z1: A,Z2: list @ A] :
% 2.76/1.57 ( ( A4
% 2.76/1.57 = ( cons @ A @ Z1 @ Z2 ) )
% 2.76/1.57 => ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_43_set__ConsD,axiom,
% 2.76/1.57 ! [A: $tType,Y: A,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
% 2.76/1.57 => ( ( Y = X )
% 2.76/1.57 | ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_44_mem__Collect__eq,axiom,
% 2.76/1.57 ! [A: $tType,A4: A,P: A > $o] :
% 2.76/1.57 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.76/1.57 = ( P @ A4 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_45_Collect__mem__eq,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A] :
% 2.76/1.57 ( ( collect @ A
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 2.76/1.57 = A5 ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_46_Collect__cong,axiom,
% 2.76/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( P @ X2 )
% 2.76/1.57 = ( Q @ X2 ) )
% 2.76/1.57 => ( ( collect @ A @ P )
% 2.76/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_47_list_Oset__intros_I1_J,axiom,
% 2.76/1.57 ! [A: $tType,A1: A,A2: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_48_list_Oset__intros_I2_J,axiom,
% 2.76/1.57 ! [A: $tType,X: A,A2: list @ A,A1: A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ A2 ) )
% 2.76/1.57 => ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_49_subset__code_I1_J,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,B2: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
% 2.76/1.57 = ( ! [X3: A] :
% 2.76/1.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 => ( member @ A @ X3 @ B2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_50_length__induct,axiom,
% 2.76/1.57 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.76/1.57 ( ! [Xs2: list @ A] :
% 2.76/1.57 ( ! [Ys: list @ A] :
% 2.76/1.57 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.76/1.57 => ( P @ Ys ) )
% 2.76/1.57 => ( P @ Xs2 ) )
% 2.76/1.57 => ( P @ Xs ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_51_append__Cons,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 2.76/1.57 = ( cons @ A @ X @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_52_Cons__eq__appendI,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
% 2.76/1.57 ( ( ( cons @ A @ X @ Xs1 )
% 2.76/1.57 = Ys2 )
% 2.76/1.57 => ( ( Xs
% 2.76/1.57 = ( append @ A @ Xs1 @ Zs ) )
% 2.76/1.57 => ( ( cons @ A @ X @ Xs )
% 2.76/1.57 = ( append @ A @ Ys2 @ Zs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_53_append__Nil,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A] :
% 2.76/1.57 ( ( append @ A @ ( nil @ A ) @ Ys2 )
% 2.76/1.57 = Ys2 ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_54_eq__Nil__appendI,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( Xs = Ys2 )
% 2.76/1.57 => ( Xs
% 2.76/1.57 = ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_55_distinct__length__2__or__more,axiom,
% 2.76/1.57 ! [A: $tType,A4: A,B3: A,Xs: list @ A] :
% 2.76/1.57 ( ( distinct @ A @ ( cons @ A @ A4 @ ( cons @ A @ B3 @ Xs ) ) )
% 2.76/1.57 = ( ( A4 != B3 )
% 2.76/1.57 & ( distinct @ A @ ( cons @ A @ A4 @ Xs ) )
% 2.76/1.57 & ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_56_distinct_Osimps_I1_J,axiom,
% 2.76/1.57 ! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_57_list_Osel_I1_J,axiom,
% 2.76/1.57 ! [A: $tType,X21: A,X22: list @ A] :
% 2.76/1.57 ( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
% 2.76/1.57 = X21 ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_58_set__subset__Cons,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_59_list__induct3,axiom,
% 2.76/1.57 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
% 2.76/1.57 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.76/1.57 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 2.76/1.57 => ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 2.76/1.57 = ( size_size @ ( list @ C ) @ Zs ) )
% 2.76/1.57 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B,Z: C,Zs2: list @ C] :
% 2.76/1.57 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.76/1.57 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 2.76/1.57 => ( ( ( size_size @ ( list @ B ) @ Ys3 )
% 2.76/1.57 = ( size_size @ ( list @ C ) @ Zs2 ) )
% 2.76/1.57 => ( ( P @ Xs2 @ Ys3 @ Zs2 )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) @ ( cons @ C @ Z @ Zs2 ) ) ) ) )
% 2.76/1.57 => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_60_list__induct2,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 2.76/1.57 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.76/1.57 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 2.76/1.57 => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B] :
% 2.76/1.57 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.76/1.57 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 2.76/1.57 => ( ( P @ Xs2 @ Ys3 )
% 2.76/1.57 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) ) ) )
% 2.76/1.57 => ( P @ Xs @ Ys2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_61_rev__nonempty__induct,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A] :
% 2.76/1.57 ( ( Xs2
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( P @ Xs2 )
% 2.76/1.57 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 2.76/1.57 => ( P @ Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_62_append__eq__Cons__conv,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( ( append @ A @ Ys2 @ Zs )
% 2.76/1.57 = ( cons @ A @ X @ Xs ) )
% 2.76/1.57 = ( ( ( Ys2
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 & ( Zs
% 2.76/1.57 = ( cons @ A @ X @ Xs ) ) )
% 2.76/1.57 | ? [Ys5: list @ A] :
% 2.76/1.57 ( ( Ys2
% 2.76/1.57 = ( cons @ A @ X @ Ys5 ) )
% 2.76/1.57 & ( ( append @ A @ Ys5 @ Zs )
% 2.76/1.57 = Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_63_Cons__eq__append__conv,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 2.76/1.57 ( ( ( cons @ A @ X @ Xs )
% 2.76/1.57 = ( append @ A @ Ys2 @ Zs ) )
% 2.76/1.57 = ( ( ( Ys2
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 & ( ( cons @ A @ X @ Xs )
% 2.76/1.57 = Zs ) )
% 2.76/1.57 | ? [Ys5: list @ A] :
% 2.76/1.57 ( ( ( cons @ A @ X @ Ys5 )
% 2.76/1.57 = Ys2 )
% 2.76/1.57 & ( Xs
% 2.76/1.57 = ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_64_rev__exhaust,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ~ ! [Ys3: list @ A,Y2: A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 != ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_65_rev__induct,axiom,
% 2.76/1.57 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.76/1.57 ( ( P @ ( nil @ A ) )
% 2.76/1.57 => ( ! [X2: A,Xs2: list @ A] :
% 2.76/1.57 ( ( P @ Xs2 )
% 2.76/1.57 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 2.76/1.57 => ( P @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_66_split__list__first__prop__iff,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ( ? [X3: A] :
% 2.76/1.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X3 ) ) )
% 2.76/1.57 = ( ? [Ys4: list @ A,X3: A] :
% 2.76/1.57 ( ? [Zs3: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs3 ) ) )
% 2.76/1.57 & ( P @ X3 )
% 2.76/1.57 & ! [Y3: A] :
% 2.76/1.57 ( ( member @ A @ Y3 @ ( set2 @ A @ Ys4 ) )
% 2.76/1.57 => ~ ( P @ Y3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_67_split__list__last__prop__iff,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ( ? [X3: A] :
% 2.76/1.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X3 ) ) )
% 2.76/1.57 = ( ? [Ys4: list @ A,X3: A,Zs3: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs3 ) ) )
% 2.76/1.57 & ( P @ X3 )
% 2.76/1.57 & ! [Y3: A] :
% 2.76/1.57 ( ( member @ A @ Y3 @ ( set2 @ A @ Zs3 ) )
% 2.76/1.57 => ~ ( P @ Y3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_68_in__set__conv__decomp__first,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 = ( ? [Ys4: list @ A,Zs3: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs3 ) ) )
% 2.76/1.57 & ~ ( member @ A @ X @ ( set2 @ A @ Ys4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_69_in__set__conv__decomp__last,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 = ( ? [Ys4: list @ A,Zs3: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs3 ) ) )
% 2.76/1.57 & ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_70_split__list__first__propE,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ? [X4: A] :
% 2.76/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X4 ) )
% 2.76/1.57 => ~ ! [Ys3: list @ A,X2: A] :
% 2.76/1.57 ( ? [Zs2: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 2.76/1.57 => ( ( P @ X2 )
% 2.76/1.57 => ~ ! [Xa: A] :
% 2.76/1.57 ( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
% 2.76/1.57 => ~ ( P @ Xa ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_71_split__list__last__propE,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ? [X4: A] :
% 2.76/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X4 ) )
% 2.76/1.57 => ~ ! [Ys3: list @ A,X2: A,Zs2: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 2.76/1.57 => ( ( P @ X2 )
% 2.76/1.57 => ~ ! [Xa: A] :
% 2.76/1.57 ( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
% 2.76/1.57 => ~ ( P @ Xa ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_72_split__list__first__prop,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ? [X4: A] :
% 2.76/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X4 ) )
% 2.76/1.57 => ? [Ys3: list @ A,X2: A] :
% 2.76/1.57 ( ? [Zs2: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 2.76/1.57 & ( P @ X2 )
% 2.76/1.57 & ! [Xa: A] :
% 2.76/1.57 ( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
% 2.76/1.57 => ~ ( P @ Xa ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_73_split__list__last__prop,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ? [X4: A] :
% 2.76/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X4 ) )
% 2.76/1.57 => ? [Ys3: list @ A,X2: A,Zs2: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 2.76/1.57 & ( P @ X2 )
% 2.76/1.57 & ! [Xa: A] :
% 2.76/1.57 ( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
% 2.76/1.57 => ~ ( P @ Xa ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_74_in__set__conv__decomp,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 = ( ? [Ys4: list @ A,Zs3: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_75_split__list__propE,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ? [X4: A] :
% 2.76/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X4 ) )
% 2.76/1.57 => ~ ! [Ys3: list @ A,X2: A] :
% 2.76/1.57 ( ? [Zs2: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 2.76/1.57 => ~ ( P @ X2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_76_split__list__first,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 => ? [Ys3: list @ A,Zs2: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
% 2.76/1.57 & ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_77_split__list__prop,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 2.76/1.57 ( ? [X4: A] :
% 2.76/1.57 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( P @ X4 ) )
% 2.76/1.57 => ? [Ys3: list @ A,X2: A] :
% 2.76/1.57 ( ? [Zs2: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
% 2.76/1.57 & ( P @ X2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_78_split__list__last,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 => ? [Ys3: list @ A,Zs2: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
% 2.76/1.57 & ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_79_split__list,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 => ? [Ys3: list @ A,Zs2: list @ A] :
% 2.76/1.57 ( Xs
% 2.76/1.57 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_80_distinct__singleton,axiom,
% 2.76/1.57 ! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_81_distinct_Osimps_I2_J,axiom,
% 2.76/1.57 ! [A: $tType,X: A,Xs: list @ A] :
% 2.76/1.57 ( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
% 2.76/1.57 = ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 2.76/1.57 & ( distinct @ A @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_82_hd__in__set,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_83_list_Oset__sel_I1_J,axiom,
% 2.76/1.57 ! [A: $tType,A4: list @ A] :
% 2.76/1.57 ( ( A4
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( member @ A @ ( hd @ A @ A4 ) @ ( set2 @ A @ A4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_84_last__ConsR,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,X: A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.76/1.57 = ( last @ A @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_85_last__ConsL,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,X: A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.76/1.57 = X ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_86_last_Osimps,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,X: A] :
% 2.76/1.57 ( ( ( Xs
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.76/1.57 = X ) )
% 2.76/1.57 & ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.76/1.57 = ( last @ A @ Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_87_last__in__set,axiom,
% 2.76/1.57 ! [A: $tType,As2: list @ A] :
% 2.76/1.57 ( ( As2
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_88_longest__common__prefix,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ? [Ps: list @ A,Xs3: list @ A,Ys6: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Ps @ Xs3 ) )
% 2.76/1.57 & ( Ys2
% 2.76/1.57 = ( append @ A @ Ps @ Ys6 ) )
% 2.76/1.57 & ( ( Xs3
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 | ( Ys6
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 | ( ( hd @ A @ Xs3 )
% 2.76/1.57 != ( hd @ A @ Ys6 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_89_hd__append,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( ( Xs
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( hd @ A @ Ys2 ) ) )
% 2.76/1.57 & ( ( Xs
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( hd @ A @ Xs ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_90_longest__common__suffix,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ? [Ss: list @ A,Xs3: list @ A,Ys6: list @ A] :
% 2.76/1.57 ( ( Xs
% 2.76/1.57 = ( append @ A @ Xs3 @ Ss ) )
% 2.76/1.57 & ( Ys2
% 2.76/1.57 = ( append @ A @ Ys6 @ Ss ) )
% 2.76/1.57 & ( ( Xs3
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 | ( Ys6
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 | ( ( last @ A @ Xs3 )
% 2.76/1.57 != ( last @ A @ Ys6 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_91_last__append,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 2.76/1.57 ( ( ( Ys2
% 2.76/1.57 = ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( last @ A @ Xs ) ) )
% 2.76/1.57 & ( ( Ys2
% 2.76/1.57 != ( nil @ A ) )
% 2.76/1.57 => ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( last @ A @ Ys2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_92_not__distinct__decomp,axiom,
% 2.76/1.57 ! [A: $tType,Ws: list @ A] :
% 2.76/1.57 ( ~ ( distinct @ A @ Ws )
% 2.76/1.57 => ? [Xs2: list @ A,Ys3: list @ A,Zs2: list @ A,Y2: A] :
% 2.76/1.57 ( Ws
% 2.76/1.57 = ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_93_not__distinct__conv__prefix,axiom,
% 2.76/1.57 ! [A: $tType,As2: list @ A] :
% 2.76/1.57 ( ( ~ ( distinct @ A @ As2 ) )
% 2.76/1.57 = ( ? [Xs4: list @ A,Y3: A,Ys4: list @ A] :
% 2.76/1.57 ( ( member @ A @ Y3 @ ( set2 @ A @ Xs4 ) )
% 2.76/1.57 & ( distinct @ A @ Xs4 )
% 2.76/1.57 & ( As2
% 2.76/1.57 = ( append @ A @ Xs4 @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_94_subset__antisym,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A5 )
% 2.76/1.57 => ( A5 = B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_95_subsetI,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( member @ A @ X2 @ A5 )
% 2.76/1.57 => ( member @ A @ X2 @ B2 ) )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_96_distinct__union,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.76/1.57 ( ( distinct @ A @ ( union @ A @ Xs @ Ys2 ) )
% 2.76/1.57 = ( distinct @ A @ Ys2 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_97_order__refl,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_98_set__n__lists,axiom,
% 2.76/1.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.76/1.57 ( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
% 2.76/1.57 = ( collect @ ( list @ A )
% 2.76/1.57 @ ^ [Ys4: list @ A] :
% 2.76/1.57 ( ( ( size_size @ ( list @ A ) @ Ys4 )
% 2.76/1.57 = N )
% 2.76/1.57 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_99_complete__interval,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,P: A > $o] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( P @ A4 )
% 2.76/1.57 => ( ~ ( P @ B3 )
% 2.76/1.57 => ? [C2: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ C2 )
% 2.76/1.57 & ( ord_less_eq @ A @ C2 @ B3 )
% 2.76/1.57 & ! [X4: A] :
% 2.76/1.57 ( ( ( ord_less_eq @ A @ A4 @ X4 )
% 2.76/1.57 & ( ord_less @ A @ X4 @ C2 ) )
% 2.76/1.57 => ( P @ X4 ) )
% 2.76/1.57 & ! [D: A] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( ( ord_less_eq @ A @ A4 @ X2 )
% 2.76/1.57 & ( ord_less @ A @ X2 @ D ) )
% 2.76/1.57 => ( P @ X2 ) )
% 2.76/1.57 => ( ord_less_eq @ A @ D @ C2 ) ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_100_order_Onot__eq__order__implies__strict,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( A4 != B3 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_101_dual__order_Ostrict__implies__order,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A] :
% 2.76/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.76/1.57 => ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_102_dual__order_Ostrict__iff__order,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less @ A )
% 2.76/1.57 = ( ^ [B4: A,A6: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ B4 @ A6 )
% 2.76/1.57 & ( A6 != B4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_103_dual__order_Oorder__iff__strict,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less_eq @ A )
% 2.76/1.57 = ( ^ [B4: A,A6: A] :
% 2.76/1.57 ( ( ord_less @ A @ B4 @ A6 )
% 2.76/1.57 | ( A6 = B4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_104_psubsetI,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( A5 != B2 )
% 2.76/1.57 => ( ord_less @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_105_psubsetE,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_106_psubset__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 2.76/1.57 & ( A7 != B5 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_107_psubset__imp__subset,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_108_psubset__subset__trans,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.76/1.57 => ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_109_subset__not__subset__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 2.76/1.57 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_110_subset__psubset__trans,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.76/1.57 => ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_111_subset__iff__psubset__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B5 )
% 2.76/1.57 | ( A7 = B5 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_112_length__n__lists__elem,axiom,
% 2.76/1.57 ! [A: $tType,Ys2: list @ A,N: nat,Xs: list @ A] :
% 2.76/1.57 ( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
% 2.76/1.57 => ( ( size_size @ ( list @ A ) @ Ys2 )
% 2.76/1.57 = N ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_113_less__eq__set__def,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ( ord_less_eq @ ( A > $o )
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A7 )
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_114_impossible__Cons,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.76/1.57 => ( Xs
% 2.76/1.57 != ( cons @ A @ X @ Ys2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_115_le__funD,axiom,
% 2.76/1.57 ! [B: $tType,A: $tType] :
% 2.76/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 => ! [F: A > B,G: A > B,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.76/1.57 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_116_le__funE,axiom,
% 2.76/1.57 ! [B: $tType,A: $tType] :
% 2.76/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 => ! [F: A > B,G: A > B,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.76/1.57 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_117_le__funI,axiom,
% 2.76/1.57 ! [B: $tType,A: $tType] :
% 2.76/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 => ! [F: A > B,G: A > B] :
% 2.76/1.57 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.76/1.57 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_118_le__fun__def,axiom,
% 2.76/1.57 ! [B: $tType,A: $tType] :
% 2.76/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 => ( ( ord_less_eq @ ( A > B ) )
% 2.76/1.57 = ( ^ [F3: A > B,G2: A > B] :
% 2.76/1.57 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_119_order__subst1,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 2.76/1.57 => ( ( ord_less_eq @ B @ B3 @ C4 )
% 2.76/1.57 => ( ! [X2: B,Y2: B] :
% 2.76/1.57 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less_eq @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_120_order__subst2,axiom,
% 2.76/1.57 ! [A: $tType,C: $tType] :
% 2.76/1.57 ( ( ( order @ C @ ( type2 @ C ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C4 )
% 2.76/1.57 => ( ! [X2: A,Y2: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less_eq @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_121_ord__eq__le__subst,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 2.76/1.57 ( ( A4
% 2.76/1.57 = ( F @ B3 ) )
% 2.76/1.57 => ( ( ord_less_eq @ B @ B3 @ C4 )
% 2.76/1.57 => ( ! [X2: B,Y2: B] :
% 2.76/1.57 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less_eq @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_122_ord__le__eq__subst,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,B3: A,F: A > B,C4: B] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ( F @ B3 )
% 2.76/1.57 = C4 )
% 2.76/1.57 => ( ! [X2: A,Y2: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less_eq @ B @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_123_eq__iff,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.76/1.57 = ( ^ [X3: A,Y3: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.76/1.57 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_124_antisym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less_eq @ A @ Y @ X )
% 2.76/1.57 => ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_125_linear,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_126_eq__refl,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( X = Y )
% 2.76/1.57 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_127_le__cases,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_128_order_Otrans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ B3 @ C4 )
% 2.76/1.57 => ( ord_less_eq @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_129_le__cases3,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,Z4: A] :
% 2.76/1.57 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ Y @ Z4 ) )
% 2.76/1.57 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ X @ Z4 ) )
% 2.76/1.57 => ( ( ( ord_less_eq @ A @ X @ Z4 )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ Z4 @ Y ) )
% 2.76/1.57 => ( ( ( ord_less_eq @ A @ Z4 @ Y )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.76/1.57 => ( ( ( ord_less_eq @ A @ Y @ Z4 )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ Z4 @ X ) )
% 2.76/1.57 => ~ ( ( ord_less_eq @ A @ Z4 @ X )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_130_antisym__conv,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Y: A,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ Y @ X )
% 2.76/1.57 => ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 = ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_131_ord__eq__le__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( A4 = B3 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ B3 @ C4 )
% 2.76/1.57 => ( ord_less_eq @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_132_ord__le__eq__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( B3 = C4 )
% 2.76/1.57 => ( ord_less_eq @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_133_order__class_Oorder_Oantisym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.76/1.57 => ( A4 = B3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_134_order__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,Z4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less_eq @ A @ Y @ Z4 )
% 2.76/1.57 => ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_135_dual__order_Orefl,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_136_linorder__wlog,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [P: A > A > $o,A4: A,B3: A] :
% 2.76/1.57 ( ! [A3: A,B6: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A3 @ B6 )
% 2.76/1.57 => ( P @ A3 @ B6 ) )
% 2.76/1.57 => ( ! [A3: A,B6: A] :
% 2.76/1.57 ( ( P @ B6 @ A3 )
% 2.76/1.57 => ( P @ A3 @ B6 ) )
% 2.76/1.57 => ( P @ A4 @ B3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_137_dual__order_Otrans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ C4 @ B3 )
% 2.76/1.57 => ( ord_less_eq @ A @ C4 @ A4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_138_dual__order_Oantisym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( A4 = B3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_139_ord__eq__less__subst,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 2.76/1.57 ( ( A4
% 2.76/1.57 = ( F @ B3 ) )
% 2.76/1.57 => ( ( ord_less @ B @ B3 @ C4 )
% 2.76/1.57 => ( ! [X2: B,Y2: B] :
% 2.76/1.57 ( ( ord_less @ B @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_140_ord__less__eq__subst,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,B3: A,F: A > B,C4: B] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ( F @ B3 )
% 2.76/1.57 = C4 )
% 2.76/1.57 => ( ! [X2: A,Y2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ B @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_141_order__less__subst1,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ ( F @ B3 ) )
% 2.76/1.57 => ( ( ord_less @ B @ B3 @ C4 )
% 2.76/1.57 => ( ! [X2: B,Y2: B] :
% 2.76/1.57 ( ( ord_less @ B @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_142_order__less__subst2,axiom,
% 2.76/1.57 ! [A: $tType,C: $tType] :
% 2.76/1.57 ( ( ( order @ C @ ( type2 @ C ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less @ C @ ( F @ B3 ) @ C4 )
% 2.76/1.57 => ( ! [X2: A,Y2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_143_lt__ex,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A] :
% 2.76/1.57 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_144_gt__ex,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( no_top @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A] :
% 2.76/1.57 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_145_neqE,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( X != Y )
% 2.76/1.57 => ( ~ ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_146_neq__iff,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( X != Y )
% 2.76/1.57 = ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_147_order_Oasym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_148_dense,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ? [Z: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Z )
% 2.76/1.57 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_149_less__imp__neq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( X != Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_150_less__asym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_151_less__asym_H,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_152_less__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,Z4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less @ A @ Y @ Z4 )
% 2.76/1.57 => ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_153_less__linear,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 | ( X = Y )
% 2.76/1.57 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_154_less__irrefl,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A] :
% 2.76/1.57 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_155_ord__eq__less__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( A4 = B3 )
% 2.76/1.57 => ( ( ord_less @ A @ B3 @ C4 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_156_ord__less__eq__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( B3 = C4 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_157_dual__order_Oasym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A] :
% 2.76/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.76/1.57 => ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_158_less__imp__not__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( X != Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_159_less__not__sym,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_160_less__induct,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [P: A > $o,A4: A] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ! [Y5: A] :
% 2.76/1.57 ( ( ord_less @ A @ Y5 @ X2 )
% 2.76/1.57 => ( P @ Y5 ) )
% 2.76/1.57 => ( P @ X2 ) )
% 2.76/1.57 => ( P @ A4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_161_antisym__conv3,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Y: A,X: A] :
% 2.76/1.57 ( ~ ( ord_less @ A @ Y @ X )
% 2.76/1.57 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.76/1.57 = ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_162_less__imp__not__eq2,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( Y != X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_163_less__imp__triv,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,P: $o] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less @ A @ Y @ X )
% 2.76/1.57 => P ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_164_linorder__cases,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ~ ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ( X != Y )
% 2.76/1.57 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_165_dual__order_Oirrefl,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A] :
% 2.76/1.57 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_166_order_Ostrict__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less @ A @ B3 @ C4 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_167_less__imp__not__less,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_168_dual__order_Ostrict__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A,C4: A] :
% 2.76/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.76/1.57 => ( ( ord_less @ A @ C4 @ B3 )
% 2.76/1.57 => ( ord_less @ A @ C4 @ A4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_169_not__less__iff__gr__or__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.76/1.57 = ( ( ord_less @ A @ Y @ X )
% 2.76/1.57 | ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_170_order_Ostrict__implies__not__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( A4 != B3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_171_dual__order_Ostrict__implies__not__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A] :
% 2.76/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.76/1.57 => ( A4 != B3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_172_ex__gt__or__lt,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A] :
% 2.76/1.57 ? [B6: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B6 )
% 2.76/1.57 | ( ord_less @ A @ B6 @ A4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_173_set__mp,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( member @ A @ X @ A5 )
% 2.76/1.57 => ( member @ A @ X @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_174_in__mono,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( member @ A @ X @ A5 )
% 2.76/1.57 => ( member @ A @ X @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_175_subsetD,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( member @ A @ C4 @ A5 )
% 2.76/1.57 => ( member @ A @ C4 @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_176_subsetCE,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( member @ A @ C4 @ A5 )
% 2.76/1.57 => ( member @ A @ C4 @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_177_equalityE,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( A5 = B2 )
% 2.76/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_178_subset__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ! [X3: A] :
% 2.76/1.57 ( ( member @ A @ X3 @ A7 )
% 2.76/1.57 => ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_179_equalityD1,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( A5 = B2 )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_180_equalityD2,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( A5 = B2 )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_181_set__rev__mp,axiom,
% 2.76/1.57 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( member @ A @ X @ A5 )
% 2.76/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( member @ A @ X @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_182_subset__iff,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ! [T: A] :
% 2.76/1.57 ( ( member @ A @ T @ A7 )
% 2.76/1.57 => ( member @ A @ T @ B5 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_183_rev__subsetD,axiom,
% 2.76/1.57 ! [A: $tType,C4: A,A5: set @ A,B2: set @ A] :
% 2.76/1.57 ( ( member @ A @ C4 @ A5 )
% 2.76/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( member @ A @ C4 @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_184_subset__refl,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_185_Collect__mono,axiom,
% 2.76/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( P @ X2 )
% 2.76/1.57 => ( Q @ X2 ) )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_186_subset__trans,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.76/1.57 => ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_187_set__eq__subset,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ^ [Y4: set @ A,Z3: set @ A] : ( Y4 = Z3 ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 2.76/1.57 & ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_188_contra__subsetD,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ~ ( member @ A @ C4 @ B2 )
% 2.76/1.57 => ~ ( member @ A @ C4 @ A5 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_189_Collect__mono__iff,axiom,
% 2.76/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.76/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.76/1.57 = ( ! [X3: A] :
% 2.76/1.57 ( ( P @ X3 )
% 2.76/1.57 => ( Q @ X3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_190_leD,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Y: A,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ Y @ X )
% 2.76/1.57 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_191_leI,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ~ ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_192_le__less,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less_eq @ A )
% 2.76/1.57 = ( ^ [X3: A,Y3: A] :
% 2.76/1.57 ( ( ord_less @ A @ X3 @ Y3 )
% 2.76/1.57 | ( X3 = Y3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_193_less__le,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less @ A )
% 2.76/1.57 = ( ^ [X3: A,Y3: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.76/1.57 & ( X3 != Y3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_194_order__le__less__subst1,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 2.76/1.57 => ( ( ord_less @ B @ B3 @ C4 )
% 2.76/1.57 => ( ! [X2: B,Y2: B] :
% 2.76/1.57 ( ( ord_less @ B @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_195_order__le__less__subst2,axiom,
% 2.76/1.57 ! [A: $tType,C: $tType] :
% 2.76/1.57 ( ( ( order @ C @ ( type2 @ C ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less @ C @ ( F @ B3 ) @ C4 )
% 2.76/1.57 => ( ! [X2: A,Y2: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_196_order__less__le__subst1,axiom,
% 2.76/1.57 ! [A: $tType,B: $tType] :
% 2.76/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,F: B > A,B3: B,C4: B] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ ( F @ B3 ) )
% 2.76/1.57 => ( ( ord_less_eq @ B @ B3 @ C4 )
% 2.76/1.57 => ( ! [X2: B,Y2: B] :
% 2.76/1.57 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ A @ A4 @ ( F @ C4 ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_197_order__less__le__subst2,axiom,
% 2.76/1.57 ! [A: $tType,C: $tType] :
% 2.76/1.57 ( ( ( order @ C @ ( type2 @ C ) )
% 2.76/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.76/1.57 => ! [A4: A,B3: A,F: A > C,C4: C] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C4 )
% 2.76/1.57 => ( ! [X2: A,Y2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.76/1.57 => ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.76/1.57 => ( ord_less @ C @ ( F @ A4 ) @ C4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_198_not__le,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 2.76/1.57 = ( ord_less @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_199_not__less,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.76/1.57 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_200_le__neq__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( A4 != B3 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_201_less__imp__le,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_202_antisym__conv1,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ~ ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 = ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_203_antisym__conv2,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.76/1.57 = ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_204_le__less__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,Z4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less @ A @ Y @ Z4 )
% 2.76/1.57 => ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_205_less__le__trans,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,Z4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less_eq @ A @ Y @ Z4 )
% 2.76/1.57 => ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_206_dense__ge,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Z4: A,Y: A] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z4 @ X2 )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ X2 ) )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_207_dense__le,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Y: A,Z4: A] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.76/1.57 => ( ord_less_eq @ A @ X2 @ Z4 ) )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_208_le__less__linear,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_209_le__imp__less__or__eq,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.76/1.57 => ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 | ( X = Y ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_210_less__le__not__le,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less @ A )
% 2.76/1.57 = ( ^ [X3: A,Y3: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.76/1.57 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_211_not__le__imp__less,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Y: A,X: A] :
% 2.76/1.57 ( ~ ( ord_less_eq @ A @ Y @ X )
% 2.76/1.57 => ( ord_less @ A @ X @ Y ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_212_order_Ostrict__trans1,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less @ A @ B3 @ C4 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_213_order_Ostrict__trans2,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A,C4: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ B3 @ C4 )
% 2.76/1.57 => ( ord_less @ A @ A4 @ C4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_214_order_Oorder__iff__strict,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less_eq @ A )
% 2.76/1.57 = ( ^ [A6: A,B4: A] :
% 2.76/1.57 ( ( ord_less @ A @ A6 @ B4 )
% 2.76/1.57 | ( A6 = B4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_215_order_Ostrict__iff__order,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ( ( ord_less @ A )
% 2.76/1.57 = ( ^ [A6: A,B4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ A6 @ B4 )
% 2.76/1.57 & ( A6 != B4 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_216_dual__order_Ostrict__trans1,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A,C4: A] :
% 2.76/1.57 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.76/1.57 => ( ( ord_less @ A @ C4 @ B3 )
% 2.76/1.57 => ( ord_less @ A @ C4 @ A4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_217_dual__order_Ostrict__trans2,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [B3: A,A4: A,C4: A] :
% 2.76/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.76/1.57 => ( ( ord_less_eq @ A @ C4 @ B3 )
% 2.76/1.57 => ( ord_less @ A @ C4 @ A4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_218_dense__ge__bounded,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [Z4: A,X: A,Y: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z4 @ X )
% 2.76/1.57 => ( ! [W: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z4 @ W )
% 2.76/1.57 => ( ( ord_less @ A @ W @ X )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_219_dense__le__bounded,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [X: A,Y: A,Z4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ Y )
% 2.76/1.57 => ( ! [W: A] :
% 2.76/1.57 ( ( ord_less @ A @ X @ W )
% 2.76/1.57 => ( ( ord_less @ A @ W @ Y )
% 2.76/1.57 => ( ord_less_eq @ A @ W @ Z4 ) ) )
% 2.76/1.57 => ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_220_order_Ostrict__implies__order,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [A4: A,B3: A] :
% 2.76/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.76/1.57 => ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_221_distinct__n__lists,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,N: nat] :
% 2.76/1.57 ( ( distinct @ A @ Xs )
% 2.76/1.57 => ( distinct @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_222_sublists_Osimps_I1_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( sublists @ A @ ( nil @ A ) )
% 2.76/1.57 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_223_minf_I8_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_224_minf_I6_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_225_pinf_I8_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_226_predicate1I,axiom,
% 2.76/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.76/1.57 ( ! [X2: A] :
% 2.76/1.57 ( ( P @ X2 )
% 2.76/1.57 => ( Q @ X2 ) )
% 2.76/1.57 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_227_rev__predicate1D,axiom,
% 2.76/1.57 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 2.76/1.57 ( ( P @ X )
% 2.76/1.57 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.76/1.57 => ( Q @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_228_less__fun__def,axiom,
% 2.76/1.57 ! [B: $tType,A: $tType] :
% 2.76/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.76/1.57 => ( ( ord_less @ ( A > B ) )
% 2.76/1.57 = ( ^ [F3: A > B,G2: A > B] :
% 2.76/1.57 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 2.76/1.57 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_229_predicate1D,axiom,
% 2.76/1.57 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 2.76/1.57 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.76/1.57 => ( ( P @ X )
% 2.76/1.57 => ( Q @ X ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_230_psubset__trans,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C3: set @ A] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.76/1.57 => ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_231_less__set__def,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) )
% 2.76/1.57 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.76/1.57 ( ord_less @ ( A > $o )
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A7 )
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_232_psubsetD,axiom,
% 2.76/1.57 ! [A: $tType,A5: set @ A,B2: set @ A,C4: A] :
% 2.76/1.57 ( ( ord_less @ ( set @ A ) @ A5 @ B2 )
% 2.76/1.57 => ( ( member @ A @ C4 @ A5 )
% 2.76/1.57 => ( member @ A @ C4 @ B2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_233_pinf_I1_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.76/1.57 ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z5 @ X2 )
% 2.76/1.57 => ( ( P @ X2 )
% 2.76/1.57 = ( P2 @ X2 ) ) )
% 2.76/1.57 => ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z5 @ X2 )
% 2.76/1.57 => ( ( Q @ X2 )
% 2.76/1.57 = ( Q2 @ X2 ) ) )
% 2.76/1.57 => ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ( ( ( P @ X4 )
% 2.76/1.57 & ( Q @ X4 ) )
% 2.76/1.57 = ( ( P2 @ X4 )
% 2.76/1.57 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_234_pinf_I2_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.76/1.57 ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z5 @ X2 )
% 2.76/1.57 => ( ( P @ X2 )
% 2.76/1.57 = ( P2 @ X2 ) ) )
% 2.76/1.57 => ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z5 @ X2 )
% 2.76/1.57 => ( ( Q @ X2 )
% 2.76/1.57 = ( Q2 @ X2 ) ) )
% 2.76/1.57 => ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ( ( ( P @ X4 )
% 2.76/1.57 | ( Q @ X4 ) )
% 2.76/1.57 = ( ( P2 @ X4 )
% 2.76/1.57 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_235_pinf_I3_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ( X4 != T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_236_pinf_I4_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ( X4 != T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_237_pinf_I5_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_238_pinf_I7_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_239_pinf_I11_J,axiom,
% 2.76/1.57 ! [C: $tType,D2: $tType] :
% 2.76/1.57 ( ( ord @ C @ ( type2 @ C ) )
% 2.76/1.57 => ! [F4: D2] :
% 2.76/1.57 ? [Z: C] :
% 2.76/1.57 ! [X4: C] :
% 2.76/1.57 ( ( ord_less @ C @ Z @ X4 )
% 2.76/1.57 => ( F4 = F4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_240_minf_I1_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.76/1.57 ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Z5 )
% 2.76/1.57 => ( ( P @ X2 )
% 2.76/1.57 = ( P2 @ X2 ) ) )
% 2.76/1.57 => ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Z5 )
% 2.76/1.57 => ( ( Q @ X2 )
% 2.76/1.57 = ( Q2 @ X2 ) ) )
% 2.76/1.57 => ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ( ( ( P @ X4 )
% 2.76/1.57 & ( Q @ X4 ) )
% 2.76/1.57 = ( ( P2 @ X4 )
% 2.76/1.57 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_241_minf_I2_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.76/1.57 ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Z5 )
% 2.76/1.57 => ( ( P @ X2 )
% 2.76/1.57 = ( P2 @ X2 ) ) )
% 2.76/1.57 => ( ? [Z5: A] :
% 2.76/1.57 ! [X2: A] :
% 2.76/1.57 ( ( ord_less @ A @ X2 @ Z5 )
% 2.76/1.57 => ( ( Q @ X2 )
% 2.76/1.57 = ( Q2 @ X2 ) ) )
% 2.76/1.57 => ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ( ( ( P @ X4 )
% 2.76/1.57 | ( Q @ X4 ) )
% 2.76/1.57 = ( ( P2 @ X4 )
% 2.76/1.57 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_242_minf_I3_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ( X4 != T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_243_minf_I4_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ( X4 != T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_244_minf_I5_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_245_minf_I7_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ X4 @ Z )
% 2.76/1.57 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_246_minf_I11_J,axiom,
% 2.76/1.57 ! [C: $tType,D2: $tType] :
% 2.76/1.57 ( ( ord @ C @ ( type2 @ C ) )
% 2.76/1.57 => ! [F4: D2] :
% 2.76/1.57 ? [Z: C] :
% 2.76/1.57 ! [X4: C] :
% 2.76/1.57 ( ( ord_less @ C @ X4 @ Z )
% 2.76/1.57 => ( F4 = F4 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_247_pinf_I6_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.76/1.57 => ! [T2: A] :
% 2.76/1.57 ? [Z: A] :
% 2.76/1.57 ! [X4: A] :
% 2.76/1.57 ( ( ord_less @ A @ Z @ X4 )
% 2.76/1.57 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_248_pred__subset__eq,axiom,
% 2.76/1.57 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.76/1.57 ( ( ord_less_eq @ ( A > $o )
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 2.76/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 2.76/1.57 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_249_distinct__product__lists,axiom,
% 2.76/1.57 ! [A: $tType,Xss2: list @ ( list @ A )] :
% 2.76/1.57 ( ! [X2: list @ A] :
% 2.76/1.57 ( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss2 ) )
% 2.76/1.57 => ( distinct @ A @ X2 ) )
% 2.76/1.57 => ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_250_in__set__product__lists__length,axiom,
% 2.76/1.57 ! [A: $tType,Xs: list @ A,Xss2: list @ ( list @ A )] :
% 2.76/1.57 ( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) )
% 2.76/1.57 => ( ( size_size @ ( list @ A ) @ Xs )
% 2.76/1.57 = ( size_size @ ( list @ ( list @ A ) ) @ Xss2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_251_product__lists_Osimps_I1_J,axiom,
% 2.76/1.57 ! [A: $tType] :
% 2.76/1.57 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 2.76/1.57 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_252_nat__less__le,axiom,
% 2.76/1.57 ( ( ord_less @ nat )
% 2.76/1.57 = ( ^ [M: nat,N2: nat] :
% 2.76/1.57 ( ( ord_less_eq @ nat @ M @ N2 )
% 2.76/1.57 & ( M != N2 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_253_less__imp__le__nat,axiom,
% 2.76/1.57 ! [M2: nat,N: nat] :
% 2.76/1.57 ( ( ord_less @ nat @ M2 @ N )
% 2.76/1.57 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(fact_254_nat__neq__iff,axiom,
% 2.76/1.57 ! [M2: nat,N: nat] :
% 2.76/1.57 ( ( M2 != N )
% 2.76/1.57 = ( ( ord_less @ nat @ M2 @ N )
% 2.76/1.57 | ( ord_less @ nat @ N @ M2 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.76/1.57 ! [A8: $tType,A9: $tType] :
% 2.76/1.57 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 2.76/1.57 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.76/1.57 ! [A8: $tType,A9: $tType] :
% 2.76/1.57 ( ( order @ A9 @ ( type2 @ A9 ) )
% 2.76/1.57 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.76/1.57 ! [A8: $tType,A9: $tType] :
% 2.76/1.57 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 2.76/1.57 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.76/1.57 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.76/1.57 wellorder @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.76/1.57 preorder @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.76/1.57 linorder @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.76/1.57 no_top @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.76/1.57 order @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.76/1.57 ord @ nat @ ( type2 @ nat ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.76/1.57 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.76/1.57 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.76/1.57 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.76/1.57 preorder @ $o @ ( type2 @ $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 2.76/1.57 linorder @ $o @ ( type2 @ $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.76/1.57 order @ $o @ ( type2 @ $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.76/1.57 ord @ $o @ ( type2 @ $o ) ).
% 2.76/1.57
% 2.76/1.57 thf(conj_0,conjecture,
% 2.76/1.57 gram_L250615845e_path @ f @ nla ).
% 2.76/1.57 ------------------------
% 2.76/1.59 % (23851)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_23649 for (2999ds/2Mi)
% 2.76/1.59 % (23850)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/2Mi)
% 2.76/1.59 % (23847)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/183Mi)
% 2.76/1.59 % (23851)Instruction limit reached!
% 2.76/1.59 % (23851)------------------------------
% 2.76/1.59 % (23851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.59 % (23848)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_23649 for (2999ds/4Mi)
% 2.76/1.59 % (23851)Termination reason: Unknown
% 2.76/1.59 % (23851)Termination phase: shuffling
% 2.76/1.59
% 2.76/1.59 % (23851)Memory used [KB]: 1279
% 2.76/1.59 % (23851)Time elapsed: 0.002 s
% 2.76/1.59 % (23851)Instructions burned: 2 (million)
% 2.76/1.59 % (23851)------------------------------
% 2.76/1.59 % (23851)------------------------------
% 2.76/1.59 % (23852)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_23649 for (2999ds/275Mi)
% 2.76/1.59 % (23850)Instruction limit reached!
% 2.76/1.59 % (23850)------------------------------
% 2.76/1.59 % (23850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.59 % (23850)Termination reason: Unknown
% 2.76/1.59 % (23850)Termination phase: shuffling
% 2.76/1.59
% 2.76/1.59 % (23850)Memory used [KB]: 1407
% 2.76/1.59 % (23850)Time elapsed: 0.003 s
% 2.76/1.59 % (23850)Instructions burned: 4 (million)
% 2.76/1.59 % (23850)------------------------------
% 2.76/1.59 % (23850)------------------------------
% 2.76/1.59 % (23848)Instruction limit reached!
% 2.76/1.59 % (23848)------------------------------
% 2.76/1.59 % (23848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.59 % (23848)Termination reason: Unknown
% 2.76/1.59 % (23848)Termination phase: shuffling
% 2.76/1.59
% 2.76/1.59 % (23848)Memory used [KB]: 1407
% 2.76/1.59 % (23848)Time elapsed: 0.003 s
% 2.76/1.59 % (23848)Instructions burned: 6 (million)
% 2.76/1.59 % (23848)------------------------------
% 2.76/1.59 % (23848)------------------------------
% 2.76/1.59 % (23853)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/18Mi)
% 2.76/1.59 % (23849)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_23649 for (2999ds/27Mi)
% 2.86/1.60 % (23853)Instruction limit reached!
% 2.86/1.60 % (23853)------------------------------
% 2.86/1.60 % (23853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.60 % (23853)Termination reason: Unknown
% 2.86/1.60 % (23853)Termination phase: shuffling
% 2.86/1.60
% 2.86/1.60 % (23853)Memory used [KB]: 1663
% 2.86/1.60 % (23853)Time elapsed: 0.008 s
% 2.86/1.60 % (23853)Instructions burned: 18 (million)
% 2.86/1.60 % (23853)------------------------------
% 2.86/1.60 % (23853)------------------------------
% 2.86/1.60 % (23857)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/37Mi)
% 2.86/1.60 % (23858)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/15Mi)
% 2.86/1.61 % (23849)Instruction limit reached!
% 2.86/1.61 % (23849)------------------------------
% 2.86/1.61 % (23849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23849)Termination reason: Unknown
% 2.86/1.61 % (23849)Termination phase: Property scanning
% 2.86/1.61
% 2.86/1.61 % (23849)Memory used [KB]: 1791
% 2.86/1.61 % (23849)Time elapsed: 0.013 s
% 2.86/1.61 % (23849)Instructions burned: 27 (million)
% 2.86/1.61 % (23849)------------------------------
% 2.86/1.61 % (23849)------------------------------
% 2.86/1.61 % (23856)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/3Mi)
% 2.86/1.61 % (23859)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_23649 for (2999ds/3Mi)
% 2.86/1.61 % (23856)Instruction limit reached!
% 2.86/1.61 % (23856)------------------------------
% 2.86/1.61 % (23856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23856)Termination reason: Unknown
% 2.86/1.61 % (23856)Termination phase: shuffling
% 2.86/1.61
% 2.86/1.61 % (23856)Memory used [KB]: 1407
% 2.86/1.61 % (23856)Time elapsed: 0.003 s
% 2.86/1.61 % (23856)Instructions burned: 4 (million)
% 2.86/1.61 % (23856)------------------------------
% 2.86/1.61 % (23856)------------------------------
% 2.86/1.61 % (23859)Instruction limit reached!
% 2.86/1.61 % (23859)------------------------------
% 2.86/1.61 % (23859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23859)Termination reason: Unknown
% 2.86/1.61 % (23859)Termination phase: shuffling
% 2.86/1.61
% 2.86/1.61 % (23859)Memory used [KB]: 1407
% 2.86/1.61 % (23859)Time elapsed: 0.003 s
% 2.86/1.61 % (23859)Instructions burned: 3 (million)
% 2.86/1.61 % (23859)------------------------------
% 2.86/1.61 % (23859)------------------------------
% 2.86/1.61 % (23858)Instruction limit reached!
% 2.86/1.61 % (23858)------------------------------
% 2.86/1.61 % (23858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23858)Termination reason: Unknown
% 2.86/1.61 % (23858)Termination phase: shuffling
% 2.86/1.61
% 2.86/1.61 % (23858)Memory used [KB]: 1663
% 2.86/1.61 % (23858)Time elapsed: 0.010 s
% 2.86/1.61 % (23858)Instructions burned: 15 (million)
% 2.86/1.61 % (23858)------------------------------
% 2.86/1.61 % (23858)------------------------------
% 2.86/1.62 % (23857)Instruction limit reached!
% 2.86/1.62 % (23857)------------------------------
% 2.86/1.62 % (23857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.62 % (23857)Termination reason: Unknown
% 2.86/1.62 % (23857)Termination phase: shuffling
% 2.86/1.62
% 2.86/1.62 % (23857)Memory used [KB]: 2046
% 2.86/1.62 % (23857)Time elapsed: 0.018 s
% 2.86/1.62 % (23857)Instructions burned: 37 (million)
% 2.86/1.62 % (23857)------------------------------
% 2.86/1.62 % (23857)------------------------------
% 2.86/1.62 % (23869)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/16Mi)
% 2.86/1.63 % (23868)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/7Mi)
% 2.86/1.63 % (23867)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_23649 for (2999ds/1041Mi)
% 2.86/1.63 % (23870)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/3Mi)
% 2.86/1.63 % (23870)Instruction limit reached!
% 2.86/1.63 % (23870)------------------------------
% 2.86/1.63 % (23870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.63 % (23870)Termination reason: Unknown
% 2.86/1.63 % (23870)Termination phase: shuffling
% 2.86/1.63
% 2.86/1.63 % (23870)Memory used [KB]: 1407
% 2.86/1.63 % (23870)Time elapsed: 0.003 s
% 2.86/1.63 % (23870)Instructions burned: 3 (million)
% 2.86/1.63 % (23870)------------------------------
% 2.86/1.63 % (23870)------------------------------
% 2.86/1.63 % (23868)Instruction limit reached!
% 2.86/1.63 % (23868)------------------------------
% 2.86/1.63 % (23868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.63 % (23868)Termination reason: Unknown
% 2.86/1.63 % (23868)Termination phase: shuffling
% 2.86/1.63
% 2.86/1.63 % (23868)Memory used [KB]: 1535
% 2.86/1.63 % (23868)Time elapsed: 0.005 s
% 2.86/1.63 % (23868)Instructions burned: 7 (million)
% 2.86/1.63 % (23868)------------------------------
% 2.86/1.63 % (23868)------------------------------
% 2.86/1.63 % (23869)Instruction limit reached!
% 2.86/1.63 % (23869)------------------------------
% 2.86/1.63 % (23869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.63 % (23869)Termination reason: Unknown
% 2.86/1.63 % (23869)Termination phase: shuffling
% 2.86/1.63
% 2.86/1.63 % (23869)Memory used [KB]: 1663
% 2.86/1.63 % (23869)Time elapsed: 0.011 s
% 2.86/1.63 % (23869)Instructions burned: 16 (million)
% 2.86/1.63 % (23869)------------------------------
% 2.86/1.63 % (23869)------------------------------
% 2.86/1.63 % (23872)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_23649 for (2999ds/3Mi)
% 2.86/1.64 % (23872)Instruction limit reached!
% 2.86/1.64 % (23872)------------------------------
% 2.86/1.64 % (23872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.64 % (23872)Termination reason: Unknown
% 2.86/1.64 % (23872)Termination phase: shuffling
% 2.86/1.64
% 2.86/1.64 % (23872)Memory used [KB]: 1407
% 2.86/1.64 % (23872)Time elapsed: 0.003 s
% 2.86/1.64 % (23872)Instructions burned: 3 (million)
% 2.86/1.64 % (23872)------------------------------
% 2.86/1.64 % (23872)------------------------------
% 2.86/1.64 % (23852)First to succeed.
% 2.86/1.64 % (23852)Refutation found. Thanks to Tanya!
% 2.86/1.64 % SZS status Theorem for DTF2THF_23649
% 2.86/1.64 % SZS output start Proof for DTF2THF_23649
% 2.86/1.64 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.86/1.64 thf(type_def_7, type, n: $tType).
% 2.86/1.64 thf(type_def_8, type, dtree: $tType).
% 2.86/1.64 thf(type_def_9, type, list: ($tType) > $tType).
% 2.86/1.64 thf(type_def_10, type, set: ($tType) > $tType).
% 2.86/1.64 thf(type_def_11, type, nat: $tType).
% 2.86/1.64 thf(func_def_0, type, dtree: $tType).
% 2.86/1.64 thf(func_def_1, type, list: ($tType) > $tType).
% 2.86/1.64 thf(func_def_2, type, set: ($tType) > $tType).
% 2.86/1.64 thf(func_def_3, type, nat: $tType).
% 2.86/1.64 thf(func_def_4, type, n: $tType).
% 2.86/1.64 thf(func_def_5, type, itself: ($tType) > $tType).
% 2.86/1.64 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_13, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_14, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_15, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_16, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_17, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.86/1.64 thf(func_def_18, type, gram_L250615845e_path: (n > dtree) > list @ n > $o).
% 2.86/1.64 thf(func_def_19, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_20, type, distinct: !>[X0: $tType]:(list @ X0 > $o)).
% 2.86/1.64 thf(func_def_21, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_22, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_23, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.86/1.64 thf(func_def_24, type, hd: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_25, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.86/1.64 thf(func_def_26, type, n_lists: !>[X0: $tType]:(nat > list @ X0 > list @ list @ X0)).
% 2.86/1.64 thf(func_def_27, type, product_lists: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.86/1.64 thf(func_def_28, type, sublists: !>[X0: $tType]:(list @ X0 > list @ list @ X0)).
% 2.86/1.64 thf(func_def_29, type, union: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_30, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.86/1.64 thf(func_def_31, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.86/1.64 thf(func_def_32, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.86/1.64 thf(func_def_33, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.86/1.64 thf(func_def_34, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.86/1.64 thf(func_def_35, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.86/1.64 thf(func_def_36, type, f: n > dtree).
% 2.86/1.64 thf(func_def_37, type, nl: list @ n).
% 2.86/1.64 thf(func_def_38, type, nla: list @ n).
% 2.86/1.64 thf(func_def_52, type, sK0: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_53, type, sK1: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_54, type, sK2: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_55, type, sK3: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_56, type, sK4: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_57, type, sK5: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_58, type, sK6: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_59, type, sK7: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_60, type, sK8: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_61, type, sK9: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_62, type, sK10: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_63, type, sK11: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_64, type, sK12: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_65, type, sK13: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_66, type, sK14: !>[X0: $tType]:(list @ X0 > (X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_67, type, sK15: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_68, type, sK16: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_69, type, sK17: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_70, type, sK18: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_71, type, sK19: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_72, type, sK20: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_73, type, sK21: list @ n > list @ n).
% 2.86/1.64 thf(func_def_74, type, sK22: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > X0 > X1)).
% 2.86/1.64 thf(func_def_75, type, sK23: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > X0)).
% 2.86/1.64 thf(func_def_76, type, sK24: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_77, type, sK25: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.86/1.64 thf(func_def_78, type, sK26: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > X0 > X1)).
% 2.86/1.64 thf(func_def_79, type, sK27: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.86/1.64 thf(func_def_80, type, sK28: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_81, type, sK29: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_82, type, sK30: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_83, type, sK31: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_84, type, sK32: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_85, type, sK33: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_86, type, sK34: !>[X0: $tType]:(nat > list @ X0)).
% 2.86/1.64 thf(func_def_87, type, sK35: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_88, type, sK36: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_89, type, sK37: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_90, type, sK38: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_91, type, sK39: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_92, type, sK40: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_93, type, sK41: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_94, type, sK42: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_95, type, sK43: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_96, type, sK44: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_97, type, sK45: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_98, type, sK46: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_99, type, sK47: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_100, type, sK48: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_101, type, sK49: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_102, type, sK50: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_103, type, sK51: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_104, type, sK52: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_105, type, sK53: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X0)).
% 2.86/1.64 thf(func_def_106, type, sK54: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_107, type, sK55: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X1)).
% 2.86/1.64 thf(func_def_108, type, sK56: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.86/1.64 thf(func_def_109, type, sK57: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_110, type, sK58: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_111, type, sK59: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_112, type, sK60: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_113, type, sK61: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_114, type, sK62: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_115, type, sK63: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_116, type, sK64: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_117, type, sK65: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_118, type, sK66: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_119, type, sK67: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_120, type, sK68: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_121, type, sK69: !>[X0: $tType]:(list @ X0 > X0 > X0)).
% 2.86/1.64 thf(func_def_122, type, sK70: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_123, type, sK71: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_124, type, sK72: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_125, type, sK73: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_126, type, sK74: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.86/1.64 thf(func_def_127, type, sK75: !>[X0: $tType]:(list @ list @ X0 > X0)).
% 2.86/1.64 thf(func_def_128, type, sK76: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_129, type, sK77: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.86/1.64 thf(func_def_130, type, sK78: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_131, type, sK79: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_132, type, sK80: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_133, type, sK81: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > X1)).
% 2.86/1.64 thf(func_def_134, type, sK82: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > list @ X1)).
% 2.86/1.64 thf(func_def_135, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > X0)).
% 2.86/1.64 thf(func_def_136, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_137, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > X2)).
% 2.86/1.64 thf(func_def_138, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > list @ X2)).
% 2.86/1.64 thf(func_def_139, type, sK87: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_140, type, sK88: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_141, type, sK89: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_142, type, sK90: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_143, type, sK91: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_144, type, sK92: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_145, type, sK93: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_146, type, sK94: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_147, type, sK95: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_148, type, sK96: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_149, type, sK97: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_150, type, sK98: !>[X0: $tType]:(list @ X0 > X0 > list @ X0 > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_151, type, sK99: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X0)).
% 2.86/1.64 thf(func_def_152, type, sK100: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_153, type, sK101: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X1)).
% 2.86/1.64 thf(func_def_154, type, sK102: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.86/1.64 thf(func_def_155, type, sK103: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X0)).
% 2.86/1.64 thf(func_def_156, type, sK104: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_157, type, sK105: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X1)).
% 2.86/1.64 thf(func_def_158, type, sK106: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.86/1.64 thf(func_def_159, type, sK107: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_160, type, sK108: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_161, type, sK109: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_162, type, sK110: !>[X0: $tType]:((X0 > $o) > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_163, type, sK111: !>[X0: $tType]:(list @ X0 > (X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_164, type, sK112: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_165, type, sK113: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.86/1.64 thf(func_def_166, type, sK114: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_167, type, sK115: !>[X0: $tType]:(set @ X0 > list @ X0 > X0)).
% 2.86/1.64 thf(func_def_168, type, sK116: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_169, type, sK117: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_170, type, sK118: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.86/1.64 thf(func_def_171, type, sK119: !>[X0: $tType]:(list @ X0 > X0)).
% 2.86/1.64 thf(func_def_172, type, sK120: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.86/1.64 thf(func_def_173, type, sK121: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_174, type, sK122: !>[X0: $tType]:(list @ X0 > X0 > list @ X0)).
% 2.86/1.64 thf(func_def_176, type, ph124: !>[X0: $tType]:(X0)).
% 2.86/1.64 thf(f1344,plain,(
% 2.86/1.64 $false),
% 2.86/1.64 inference(subsumption_resolution,[status(thm)],[f1234,f1274])).
% 2.86/1.64 thf(f1274,plain,(
% 2.86/1.64 ((gram_L250615845e_path @ f @ nla) = $true)),
% 2.86/1.64 inference(cnf_transformation,[status(thm)],[f497])).
% 2.86/1.64 thf(f497,plain,(
% 2.86/1.64 ((gram_L250615845e_path @ f @ nla) = $true)),
% 2.86/1.64 inference(fool_elimination,[status(thm)],[f496])).
% 2.86/1.64 thf(f496,plain,(
% 2.86/1.64 (gram_L250615845e_path @ f @ nla)),
% 2.86/1.64 inference(rectify,[status(thm)],[f2])).
% 2.86/1.64 thf(f2,axiom,(
% 2.86/1.64 (gram_L250615845e_path @ f @ nla)),
% 2.86/1.64 file('/export/starexec/sandbox/tmp/tmp.vbi8mpeqj5/DTF2THF_23649.p',fact_1__C1_Oprems_C)).
% 2.86/1.64 thf(f1234,plain,(
% 2.86/1.64 ((gram_L250615845e_path @ f @ nla) != $true)),
% 2.86/1.64 inference(cnf_transformation,[status(thm)],[f756])).
% 2.86/1.64 thf(f756,plain,(
% 2.86/1.64 ((gram_L250615845e_path @ f @ nla) != $true)),
% 2.86/1.64 inference(flattening,[status(thm)],[f379])).
% 2.86/1.64 thf(f379,plain,(
% 2.86/1.64 ~((gram_L250615845e_path @ f @ nla) = $true)),
% 2.86/1.64 inference(fool_elimination,[status(thm)],[f378])).
% 2.86/1.64 thf(f378,plain,(
% 2.86/1.64 ~(gram_L250615845e_path @ f @ nla)),
% 2.86/1.64 inference(rectify,[status(thm)],[f274])).
% 2.86/1.64 thf(f274,negated_conjecture,(
% 2.86/1.64 ~(gram_L250615845e_path @ f @ nla)),
% 2.86/1.64 inference(negated_conjecture,[status(cth)],[f273])).
% 2.86/1.64 thf(f273,conjecture,(
% 2.86/1.64 (gram_L250615845e_path @ f @ nla)),
% 2.86/1.64 file('/export/starexec/sandbox/tmp/tmp.vbi8mpeqj5/DTF2THF_23649.p',conj_0)).
% 2.86/1.64 % SZS output end Proof for DTF2THF_23649
% 2.86/1.64 % (23852)------------------------------
% 2.86/1.64 % (23852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.64 % (23852)Termination reason: Refutation
% 2.86/1.64
% 2.86/1.64 % (23852)Memory used [KB]: 7036
% 2.86/1.64 % (23852)Time elapsed: 0.049 s
% 2.86/1.64 % (23852)Instructions burned: 95 (million)
% 2.86/1.64 % (23852)------------------------------
% 2.86/1.64 % (23852)------------------------------
% 2.86/1.64 % (23846)Success in time 0.053 s
%------------------------------------------------------------------------------