%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT172^1 : 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 : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:45:01 AM UTC 2026
% Result : Theorem 2.79s 1.66s
% Output : Refutation 2.79s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : DAT172^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.06 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.06/0.25 % Computer : n024.cluster.edu
% 0.06/0.25 % Model : x86_64 x86_64
% 0.06/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.25 % Memory : 8042.1875MB
% 0.06/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.25 % CPULimit : 300
% 0.06/0.25 % WCLimit : 300
% 0.06/0.25 % DateTime : Tue Feb 24 01:09:10 EST 2026
% 0.06/0.25 % CPUTime :
% 0.06/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.06/0.33 ---- Original DTF file ---
% 0.06/0.33 thf(spec,logic,$$dhol).
% 0.06/0.33 %------------------------------------------------------------------------------
% 0.06/0.33 % File : DAT172^1 : TPTP v9.2.1. Released v7.0.0.
% 0.06/0.33 % Domain : Data Structures
% 0.06/0.33 % Problem : Huffman 2088
% 0.06/0.33 % Version : [Bla16] axioms : Especial.
% 0.06/0.33 % English :
% 0.06/0.33
% 0.06/0.33 % Refs : [Bla08] Blanchette (2008), The Textbook Proof of Huffman's Alg
% 0.06/0.33 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.06/0.33 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.06/0.33 % Source : [Bla16]
% 0.06/0.33 % Names : huffman__2088.p [Bla16]
% 0.06/0.33
% 0.06/0.33 % Status : Theorem
% 0.06/0.33 % Rating : 0.33 v9.1.0, 0.00 v7.5.0, 0.67 v7.2.0, 0.50 v7.1.0
% 0.06/0.33 % Syntax : Number of formulae : 347 ( 92 unt; 59 typ; 0 def)
% 0.06/0.33 % Number of atoms : 813 ( 243 equ; 0 cnn)
% 0.06/0.33 % Maximal formula atoms : 13 ( 2 avg)
% 0.06/0.33 % Number of connectives : 3694 ( 79 ~; 7 |; 38 &;3170 @)
% 0.06/0.33 % ( 0 <=>; 400 =>; 0 <=; 0 <~>)
% 0.06/0.33 % Maximal formula depth : 23 ( 8 avg)
% 0.06/0.33 % Number of types : 3 ( 2 usr)
% 0.06/0.33 % Number of type conns : 122 ( 122 >; 0 *; 0 +; 0 <<)
% 0.06/0.33 % Number of symbols : 60 ( 57 usr; 9 con; 0-6 aty)
% 0.06/0.33 % Number of variables : 887 ( 33 ^; 789 !; 18 ?; 887 :)
% 0.06/0.33 % ( 47 !>; 0 ?*; 0 @-; 0 @+)
% 0.06/0.33 % SPC : TH1_THM_EQU_NAR
% 0.06/0.33
% 0.06/0.33 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.06/0.33 % 2016-07-13 14:43:44.792
% 0.06/0.33 %------------------------------------------------------------------------------
% 0.06/0.33 %----Could-be-implicit typings (5)
% 0.06/0.33 thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree,type,
% 0.06/0.33 huffma16452318e_tree: $tType > $tType ).
% 0.06/0.33
% 0.06/0.33 thf(ty_t_Set_Oset,type,
% 0.06/0.33 set: $tType > $tType ).
% 0.06/0.33
% 0.06/0.33 thf(ty_t_Nat_Onat,type,
% 0.06/0.33 nat: $tType ).
% 0.06/0.33
% 0.06/0.33 thf(ty_t_itself,type,
% 0.06/0.33 itself: $tType > $tType ).
% 0.06/0.33
% 0.06/0.33 thf(ty_tf_a,type,
% 0.06/0.33 a: $tType ).
% 0.06/0.33
% 0.06/0.33 %----Explicit typings (54)
% 0.06/0.33 thf(sy_cl_HOL_Otype,type,
% 0.06/0.33 type:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ozero,type,
% 0.06/0.33 zero:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Orderings_Oord,type,
% 0.06/0.33 ord:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Orderings_Oorder,type,
% 0.06/0.33 order:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ogroup__add,type,
% 0.06/0.33 group_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Omonoid__add,type,
% 0.06/0.33 monoid_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Orderings_Olinorder,type,
% 0.06/0.33 linorder:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Orderings_Opreorder,type,
% 0.06/0.33 preorder:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.06/0.33 semigroup_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.06/0.33 comm_monoid_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Fields_Olinordered__field,type,
% 0.06/0.33 linordered_field:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.06/0.33 ab_semigroup_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.06/0.33 cancel_semigroup_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.06/0.33 cancel1352612707id_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.06/0.33 linord219039673up_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.06/0.33 ordere216010020id_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.06/0.33 ordere779506340up_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.06/0.33 canoni770627133id_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 0.06/0.33 ordere1818651114id_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 0.06/0.33 strict797366125id_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.06/0.33 ordere236663937imp_le:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.06/0.33 ordere223160158up_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.06/0.33 strict2144017051up_add:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.06/0.33 ordere516151231imp_le:
% 0.06/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.06/0.33 plus_plus:
% 0.06/0.33 !>[A: $tType] : ( A > A > A ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.06/0.33 zero_zero:
% 0.06/0.33 !>[A: $tType] : A ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oalphabet,type,
% 0.06/0.33 huffma505251170phabet:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( set @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oconsistent,type,
% 0.06/0.33 huffma1050891809istent:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ocost,type,
% 0.06/0.33 huffma636208924e_cost:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Odepth,type,
% 0.06/0.33 huffma223349076_depth:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq,type,
% 0.06/0.33 huffma854352999e_freq:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight,type,
% 0.06/0.33 huffma1554076246height:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OmergeSibling,type,
% 0.06/0.33 huffma1954420889ibling:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > ( huffma16452318e_tree @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ominima,type,
% 0.06/0.33 huffma1154738298minima:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ooptimum,type,
% 0.06/0.33 huffma936049440ptimum:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Osibling,type,
% 0.06/0.33 huffma943100115ibling:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OsplitLeaf,type,
% 0.06/0.33 huffma454997449itLeaf:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapFourSyms,type,
% 0.06/0.33 huffma304375860urSyms:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > A > A > ( huffma16452318e_tree @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapLeaves,type,
% 0.06/0.33 huffma2094459102Leaves:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapSyms,type,
% 0.06/0.33 huffma469337550apSyms:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > ( huffma16452318e_tree @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oweight,type,
% 0.06/0.33 huffma691733767weight:
% 0.06/0.33 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_If,type,
% 0.06/0.33 if:
% 0.06/0.33 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.06/0.33 ord_less:
% 0.06/0.33 !>[A: $tType] : ( A > A > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.06/0.33 ord_less_eq:
% 0.06/0.33 !>[A: $tType] : ( A > A > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Pure_Otype,type,
% 0.06/0.33 type2:
% 0.06/0.33 !>[A: $tType] : ( itself @ A ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_Set_OCollect,type,
% 0.06/0.33 collect:
% 0.06/0.33 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_c_member,type,
% 0.06/0.33 member:
% 0.06/0.33 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_a,type,
% 0.06/0.33 a2: a ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_b,type,
% 0.06/0.33 b: a ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_c____,type,
% 0.06/0.33 c: a ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_t,type,
% 0.06/0.33 t: huffma16452318e_tree @ a ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_u____,type,
% 0.06/0.33 u: huffma16452318e_tree @ a ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_w_092_060_094sub_062a,type,
% 0.06/0.33 w_a: nat ).
% 0.06/0.33
% 0.06/0.33 thf(sy_v_w_092_060_094sub_062b,type,
% 0.06/0.33 w_b: nat ).
% 0.06/0.33
% 0.06/0.33 %----Relevant facts (254)
% 0.06/0.33 thf(fact_0_c_092_060_094sub_062u,axiom,
% 0.06/0.33 huffma1050891809istent @ a @ u ).
% 0.06/0.33
% 0.06/0.33 % c\<^sub>u
% 0.06/0.33 thf(fact_1_ab,axiom,
% 0.06/0.33 a2 != b ).
% 0.06/0.33
% 0.06/0.33 % ab
% 0.06/0.33 thf(fact_2_assms_I1_J,axiom,
% 0.06/0.33 huffma1050891809istent @ a @ t ).
% 0.06/0.33
% 0.06/0.33 % assms(1)
% 0.06/0.33 thf(fact_3_assms_I2_J,axiom,
% 0.06/0.33 huffma936049440ptimum @ a @ t ).
% 0.06/0.33
% 0.06/0.33 % assms(2)
% 0.06/0.33 thf(fact_4_dc,axiom,
% 0.06/0.33 ( ( huffma943100115ibling @ a @ u @ c )
% 0.06/0.33 != c ) ).
% 0.06/0.33
% 0.06/0.33 % dc
% 0.06/0.33 thf(fact_5_sibling__sibling__id,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_sibling_id
% 0.06/0.33 thf(fact_6_a_092_060_094sub_062c,axiom,
% 0.06/0.33 member @ a @ c @ ( huffma505251170phabet @ a @ u ) ).
% 0.06/0.33
% 0.06/0.33 % a\<^sub>c
% 0.06/0.33 thf(fact_7_sibling__reciprocal,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.33 = B )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ T @ B )
% 0.06/0.33 = A2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_reciprocal
% 0.06/0.33 thf(fact_8_consistent__swapFourSyms,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( huffma1050891809istent @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % consistent_swapFourSyms
% 0.06/0.33 thf(fact_9_a_092_060_094sub_062d,axiom,
% 0.06/0.33 member @ a @ ( huffma943100115ibling @ a @ u @ c ) @ ( huffma505251170phabet @ a @ u ) ).
% 0.06/0.33
% 0.06/0.33 % a\<^sub>d
% 0.06/0.33 thf(fact_10_a_092_060_094sub_062b,axiom,
% 0.06/0.33 member @ a @ b @ ( huffma505251170phabet @ a @ u ) ).
% 0.06/0.33
% 0.06/0.33 % a\<^sub>b
% 0.06/0.33 thf(fact_11_a_092_060_094sub_062a,axiom,
% 0.06/0.33 member @ a @ a2 @ ( huffma505251170phabet @ a @ u ) ).
% 0.06/0.33
% 0.06/0.33 % a\<^sub>a
% 0.06/0.33 thf(fact_12_d_092_060_094sub_062c,axiom,
% 0.06/0.33 ( ( huffma223349076_depth @ a @ u @ c )
% 0.06/0.33 = ( huffma1554076246height @ a @ u ) ) ).
% 0.06/0.33
% 0.06/0.33 % d\<^sub>c
% 0.06/0.33 thf(fact_13_sibling__swapSyms__sibling,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 0.06/0.33 != B )
% 0.06/0.33 => ( ( A2 != B )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ ( huffma943100115ibling @ A @ T @ B ) ) @ A2 )
% 0.06/0.33 = B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_swapSyms_sibling
% 0.06/0.33 thf(fact_14_sibling__swapSyms__other__sibling,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 0.06/0.33 != A2 )
% 0.06/0.33 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 0.06/0.33 != B )
% 0.06/0.33 => ( ( A2 != B )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) @ ( huffma943100115ibling @ A @ T @ B ) )
% 0.06/0.33 = A2 ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_swapSyms_other_sibling
% 0.06/0.33 thf(fact_15_d_092_060_094sub_062d,axiom,
% 0.06/0.33 ( ( huffma223349076_depth @ a @ u @ ( huffma943100115ibling @ a @ u @ c ) )
% 0.06/0.33 = ( huffma1554076246height @ a @ u ) ) ).
% 0.06/0.33
% 0.06/0.33 % d\<^sub>d
% 0.06/0.33 thf(fact_16_sibling__swapLeaves__sibling,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A,W_a: nat,W_s: nat] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 0.06/0.33 != B )
% 0.06/0.33 => ( ( A2 != B )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_s @ ( huffma943100115ibling @ A @ T @ B ) ) @ A2 )
% 0.06/0.33 = B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_swapLeaves_sibling
% 0.06/0.33 thf(fact_17_depth__sibling,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( huffma223349076_depth @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) )
% 0.06/0.33 = ( huffma223349076_depth @ A @ T @ A2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % depth_sibling
% 0.06/0.33 thf(fact_18_sibling__swapFourSyms__when__4th__is__sibling,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( A2 != B )
% 0.06/0.33 => ( ( ( huffma943100115ibling @ A @ T @ C )
% 0.06/0.33 != C )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ ( huffma943100115ibling @ A @ T @ C ) ) @ A2 )
% 0.06/0.33 = B ) ) ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_swapFourSyms_when_4th_is_sibling
% 0.06/0.33 thf(fact_19_assms_I4_J,axiom,
% 0.06/0.33 ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) ) ).
% 0.06/0.33
% 0.06/0.33 % assms(4)
% 0.06/0.33 thf(fact_20_assms_I3_J,axiom,
% 0.06/0.33 member @ a @ a2 @ ( huffma505251170phabet @ a @ t ) ).
% 0.06/0.33
% 0.06/0.33 % assms(3)
% 0.06/0.33 thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060lbrakk_062c_A_092_060in_062_Aalphabet_Au_059_Adepth_Au_Ac_A_061_Aheight_Au_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.06/0.33 ~ ! [C2: a] :
% 0.06/0.33 ( ( member @ a @ C2 @ ( huffma505251170phabet @ a @ u ) )
% 0.06/0.33 => ( ( huffma223349076_depth @ a @ u @ C2 )
% 0.06/0.33 != ( huffma1554076246height @ a @ u ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % \<open>\<And>thesis. (\<And>c. \<lbrakk>c \<in> alphabet u; depth u c = height u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.06/0.33 thf(fact_22__092_060open_062_092_060exists_062a_092_060in_062alphabet_Au_O_Adepth_Au_Aa_A_061_Aheight_Au_092_060close_062,axiom,
% 0.06/0.33 ? [X: a] :
% 0.06/0.33 ( ( member @ a @ X @ ( huffma505251170phabet @ a @ u ) )
% 0.06/0.33 & ( ( huffma223349076_depth @ a @ u @ X )
% 0.06/0.33 = ( huffma1554076246height @ a @ u ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % \<open>\<exists>a\<in>alphabet u. depth u a = height u\<close>
% 0.06/0.33 thf(fact_23_notin__alphabet__imp__sibling__id,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 0.06/0.33 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % notin_alphabet_imp_sibling_id
% 0.06/0.33 thf(fact_24_swapLeaves__id__when__notin__alphabet,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W: nat,W2: nat] :
% 0.06/0.33 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma2094459102Leaves @ A @ T @ W @ A2 @ W2 @ A2 )
% 0.06/0.33 = T ) ) ).
% 0.06/0.33
% 0.06/0.33 % swapLeaves_id_when_notin_alphabet
% 0.06/0.33 thf(fact_25_alphabet__swapSyms,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,B: A] :
% 0.06/0.33 ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma505251170phabet @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) )
% 0.06/0.33 = ( huffma505251170phabet @ A @ T ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % alphabet_swapSyms
% 0.06/0.33 thf(fact_26_height__swapLeaves,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
% 0.06/0.33 ( ( huffma1554076246height @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 0.06/0.33 = ( huffma1554076246height @ A @ T ) ) ).
% 0.06/0.33
% 0.06/0.33 % height_swapLeaves
% 0.06/0.33 thf(fact_27_swapSyms__id,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( huffma469337550apSyms @ A @ T @ A2 @ A2 )
% 0.06/0.33 = T ) ) ).
% 0.06/0.33
% 0.06/0.33 % swapSyms_id
% 0.06/0.33 thf(fact_28_alphabet__swapFourSyms,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma505251170phabet @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) )
% 0.06/0.33 = ( huffma505251170phabet @ A @ T ) ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % alphabet_swapFourSyms
% 0.06/0.33 thf(fact_29_depth__swapLeaves__neither,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B: A,W_a: nat,W_b: nat] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( C != A2 )
% 0.06/0.33 => ( ( C != B )
% 0.06/0.33 => ( ( huffma223349076_depth @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) @ C )
% 0.06/0.33 = ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % depth_swapLeaves_neither
% 0.06/0.33 thf(fact_30_depth__swapSyms__neither,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( C != A2 )
% 0.06/0.33 => ( ( C != B )
% 0.06/0.33 => ( ( huffma223349076_depth @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) @ C )
% 0.06/0.33 = ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % depth_swapSyms_neither
% 0.06/0.33 thf(fact_31_exists__at__height,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ? [X: A] :
% 0.06/0.33 ( ( member @ A @ X @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 & ( ( huffma223349076_depth @ A @ T @ X )
% 0.06/0.33 = ( huffma1554076246height @ A @ T ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % exists_at_height
% 0.06/0.33 thf(fact_32_exists__in__alphabet,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A] :
% 0.06/0.33 ? [A3: A] : ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) ) ).
% 0.06/0.33
% 0.06/0.33 % exists_in_alphabet
% 0.06/0.33 thf(fact_33_in__alphabet__imp__sibling__in__alphabet,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 0.06/0.33 ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( member @ A @ ( huffma943100115ibling @ A @ T @ A2 ) @ ( huffma505251170phabet @ A @ T ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % in_alphabet_imp_sibling_in_alphabet
% 0.06/0.33 thf(fact_34_sibling__ne__imp__sibling__in__alphabet,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.33 != A2 )
% 0.06/0.33 => ( member @ A @ ( huffma943100115ibling @ A @ T @ A2 ) @ ( huffma505251170phabet @ A @ T ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % sibling_ne_imp_sibling_in_alphabet
% 0.06/0.33 thf(fact_35_consistent__swapLeaves,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( huffma1050891809istent @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % consistent_swapLeaves
% 0.06/0.33 thf(fact_36_consistent__swapSyms,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( huffma1050891809istent @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % consistent_swapSyms
% 0.06/0.33 thf(fact_37_swapFourSyms__def,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( huffma304375860urSyms @ A )
% 0.06/0.33 = ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A,C3: A,D2: A] : ( if @ ( huffma16452318e_tree @ A ) @ ( A4 = D2 ) @ ( huffma469337550apSyms @ A @ T2 @ B2 @ C3 ) @ ( if @ ( huffma16452318e_tree @ A ) @ ( B2 = C3 ) @ ( huffma469337550apSyms @ A @ T2 @ A4 @ D2 ) @ ( huffma469337550apSyms @ A @ ( huffma469337550apSyms @ A @ T2 @ A4 @ C3 ) @ B2 @ D2 ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % swapFourSyms_def
% 0.06/0.33 thf(fact_38_a_092_060_094sub_062u,axiom,
% 0.06/0.33 ( ( huffma505251170phabet @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
% 0.06/0.33 = ( huffma505251170phabet @ a @ u ) ) ).
% 0.06/0.33
% 0.06/0.33 % a\<^sub>u
% 0.06/0.33 thf(fact_39_freq__swapFourSyms,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma854352999e_freq @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) )
% 0.06/0.33 = ( huffma854352999e_freq @ A @ T ) ) ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % freq_swapFourSyms
% 0.06/0.33 thf(fact_40_False,axiom,
% 0.06/0.33 ( ( huffma1554076246height @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
% 0.06/0.33 != ( zero_zero @ nat ) ) ).
% 0.06/0.33
% 0.06/0.33 % False
% 0.06/0.33 thf(fact_41_f_092_060_094sub_062u,axiom,
% 0.06/0.33 ( ( huffma854352999e_freq @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
% 0.06/0.33 = ( huffma854352999e_freq @ a @ u ) ) ).
% 0.06/0.33
% 0.06/0.33 % f\<^sub>u
% 0.06/0.33 thf(fact_42_freq__swapSyms,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma854352999e_freq @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) )
% 0.06/0.33 = ( huffma854352999e_freq @ A @ T ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % freq_swapSyms
% 0.06/0.33 thf(fact_43_consistent__splitLeaf,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,W_a: nat,A2: A,W_b: nat] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( huffma1050891809istent @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % consistent_splitLeaf
% 0.06/0.33 thf(fact_44_height__0__imp__sibling__id,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( ( huffma1554076246height @ A @ T )
% 0.06/0.33 = ( zero_zero @ nat ) )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % height_0_imp_sibling_id
% 0.06/0.33 thf(fact_45_mem__Collect__eq,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,P: A > $o] :
% 0.06/0.33 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.06/0.33 = ( P @ A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % mem_Collect_eq
% 0.06/0.33 thf(fact_46_Collect__mem__eq,axiom,
% 0.06/0.33 ! [A: $tType,A5: set @ A] :
% 0.06/0.33 ( ( collect @ A
% 0.06/0.33 @ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
% 0.06/0.33 = A5 ) ).
% 0.06/0.33
% 0.06/0.33 % Collect_mem_eq
% 0.06/0.33 thf(fact_47_Collect__cong,axiom,
% 0.06/0.33 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.06/0.33 ( ! [X: A] :
% 0.06/0.33 ( ( P @ X )
% 0.06/0.33 = ( Q @ X ) )
% 0.06/0.33 => ( ( collect @ A @ P )
% 0.06/0.33 = ( collect @ A @ Q ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % Collect_cong
% 0.06/0.33 thf(fact_48_ext,axiom,
% 0.06/0.33 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 0.06/0.33 ( ! [X: A] :
% 0.06/0.33 ( ( F @ X )
% 0.06/0.33 = ( G @ X ) )
% 0.06/0.33 => ( F = G ) ) ).
% 0.06/0.33
% 0.06/0.33 % ext
% 0.06/0.33 thf(fact_49_notin__alphabet__imp__splitLeaf__id,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W_a: nat,W_b: nat,B: A] :
% 0.06/0.33 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B )
% 0.06/0.33 = T ) ) ).
% 0.06/0.33
% 0.06/0.33 % notin_alphabet_imp_splitLeaf_id
% 0.06/0.33 thf(fact_50_swapLeaves__id,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( huffma2094459102Leaves @ A @ T @ ( huffma854352999e_freq @ A @ T @ A2 ) @ A2 @ ( huffma854352999e_freq @ A @ T @ A2 ) @ A2 )
% 0.06/0.33 = T ) ) ).
% 0.06/0.33
% 0.06/0.33 % swapLeaves_id
% 0.06/0.33 thf(fact_51_depth__height__imp__sibling__ne,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( ( huffma223349076_depth @ A @ T @ A2 )
% 0.06/0.33 = ( huffma1554076246height @ A @ T ) )
% 0.06/0.33 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T ) )
% 0.06/0.33 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.33 != A2 ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % depth_height_imp_sibling_ne
% 0.06/0.33 thf(fact_52_h_092_060_094sub_062u,axiom,
% 0.06/0.33 ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ a @ u ) ).
% 0.06/0.33
% 0.06/0.33 % h\<^sub>u
% 0.06/0.33 thf(fact_53_notin__alphabet__imp__freq__0,axiom,
% 0.06/0.33 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 0.06/0.33 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.33 => ( ( huffma854352999e_freq @ A @ T @ A2 )
% 0.06/0.33 = ( zero_zero @ nat ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % notin_alphabet_imp_freq_0
% 0.06/0.33 thf(fact_54_assms_I7_J,axiom,
% 0.06/0.33 ord_less_eq @ nat @ w_a @ w_b ).
% 0.06/0.33
% 0.06/0.33 % assms(7)
% 0.06/0.33 thf(fact_55_freq__swapLeaves,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,W_a: nat,W_b: nat] :
% 0.06/0.33 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( A2 != B )
% 0.06/0.33 => ( ( huffma854352999e_freq @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 0.06/0.33 = ( ^ [C3: A] : ( if @ nat @ ( C3 = A2 ) @ ( if @ nat @ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) ) @ W_a @ ( zero_zero @ nat ) ) @ ( if @ nat @ ( C3 = B ) @ ( if @ nat @ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) ) @ W_b @ ( zero_zero @ nat ) ) @ ( huffma854352999e_freq @ A @ T @ C3 ) ) ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % freq_swapLeaves
% 0.06/0.33 thf(fact_56_assms_I6_J,axiom,
% 0.06/0.33 ! [X3: a] :
% 0.06/0.33 ( ( member @ a @ X3 @ ( huffma505251170phabet @ a @ t ) )
% 0.06/0.33 => ( ( ord_less_eq @ nat @ w_a @ ( huffma854352999e_freq @ a @ t @ X3 ) )
% 0.06/0.33 & ( ord_less_eq @ nat @ w_b @ ( huffma854352999e_freq @ a @ t @ X3 ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % assms(6)
% 0.06/0.33 thf(fact_57_assms_I5_J,axiom,
% 0.06/0.33 ( ( huffma854352999e_freq @ a @ t @ a2 )
% 0.06/0.33 = ( plus_plus @ nat @ w_a @ w_b ) ) ).
% 0.06/0.33
% 0.06/0.33 % assms(5)
% 0.06/0.33 thf(fact_58_height__gt__0__alphabet__eq__imp__height__gt__0,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,U: huffma16452318e_tree @ A] :
% 0.06/0.33 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T ) )
% 0.06/0.33 => ( ( huffma1050891809istent @ A @ T )
% 0.06/0.33 => ( ( ( huffma505251170phabet @ A @ T )
% 0.06/0.33 = ( huffma505251170phabet @ A @ U ) )
% 0.06/0.33 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ U ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % height_gt_0_alphabet_eq_imp_height_gt_0
% 0.06/0.33 thf(fact_59_swapSyms__def,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( huffma469337550apSyms @ A )
% 0.06/0.33 = ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A] : ( huffma2094459102Leaves @ A @ T2 @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ A4 @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ B2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % swapSyms_def
% 0.06/0.33 thf(fact_60_neq0__conv,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ( ( N
% 0.06/0.33 != ( zero_zero @ nat ) )
% 0.06/0.33 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % neq0_conv
% 0.06/0.33 thf(fact_61_not__gr__zero,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [N: A] :
% 0.06/0.33 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.06/0.33 = ( N
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % not_gr_zero
% 0.06/0.33 thf(fact_62_infinite__descent0__measure,axiom,
% 0.06/0.33 ! [A: $tType,V: A > nat,P: A > $o,X4: A] :
% 0.06/0.33 ( ! [X: A] :
% 0.06/0.33 ( ( ( V @ X )
% 0.06/0.33 = ( zero_zero @ nat ) )
% 0.06/0.33 => ( P @ X ) )
% 0.06/0.33 => ( ! [X: A] :
% 0.06/0.33 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
% 0.06/0.33 => ( ~ ( P @ X )
% 0.06/0.33 => ? [Y: A] :
% 0.06/0.33 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X ) )
% 0.06/0.33 & ~ ( P @ Y ) ) ) )
% 0.06/0.33 => ( P @ X4 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % infinite_descent0_measure
% 0.06/0.33 thf(fact_63_less__nat__zero__code,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_nat_zero_code
% 0.06/0.33 thf(fact_64_infinite__descent0,axiom,
% 0.06/0.33 ! [P: nat > $o,N: nat] :
% 0.06/0.33 ( ( P @ ( zero_zero @ nat ) )
% 0.06/0.33 => ( ! [N2: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.06/0.33 => ( ~ ( P @ N2 )
% 0.06/0.33 => ? [M: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ M @ N2 )
% 0.06/0.33 & ~ ( P @ M ) ) ) )
% 0.06/0.33 => ( P @ N ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % infinite_descent0
% 0.06/0.33 thf(fact_65_gr__implies__not0,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ M2 @ N )
% 0.06/0.33 => ( N
% 0.06/0.33 != ( zero_zero @ nat ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % gr_implies_not0
% 0.06/0.33 thf(fact_66_less__zeroE,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_zeroE
% 0.06/0.33 thf(fact_67_add__left__cancel,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ A2 @ B )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ C ) )
% 0.06/0.33 = ( B = C ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_left_cancel
% 0.06/0.33 thf(fact_68_add__right__cancel,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A,C: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ B @ A2 )
% 0.06/0.33 = ( plus_plus @ A @ C @ A2 ) )
% 0.06/0.33 = ( B = C ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_right_cancel
% 0.06/0.33 thf(fact_69_le__zero__eq,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [N: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.06/0.33 = ( N
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_zero_eq
% 0.06/0.33 thf(fact_70_add__le__cancel__left,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [C: A,A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.06/0.33 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_cancel_left
% 0.06/0.33 thf(fact_71_add__le__cancel__right,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,C: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.06/0.33 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_cancel_right
% 0.06/0.33 thf(fact_72_add__cancel__right__right,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( A2
% 0.06/0.33 = ( plus_plus @ A @ A2 @ B ) )
% 0.06/0.33 = ( B
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_cancel_right_right
% 0.06/0.33 thf(fact_73_add__cancel__right__left,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( A2
% 0.06/0.33 = ( plus_plus @ A @ B @ A2 ) )
% 0.06/0.33 = ( B
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_cancel_right_left
% 0.06/0.33 thf(fact_74_add__cancel__left__right,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ A2 @ B )
% 0.06/0.33 = A2 )
% 0.06/0.33 = ( B
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_cancel_left_right
% 0.06/0.33 thf(fact_75_add__cancel__left__left,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ B @ A2 )
% 0.06/0.33 = A2 )
% 0.06/0.33 = ( B
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_cancel_left_left
% 0.06/0.33 thf(fact_76_double__zero__sym,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( ( zero_zero @ A )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.06/0.33 = ( A2
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % double_zero_sym
% 0.06/0.33 thf(fact_77_double__zero,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 = ( A2
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % double_zero
% 0.06/0.33 thf(fact_78_add_Oright__neutral,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( monoid_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.right_neutral
% 0.06/0.33 thf(fact_79_add_Oleft__neutral,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( monoid_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.left_neutral
% 0.06/0.33 thf(fact_80_add__less__cancel__right,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,C: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.06/0.33 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_cancel_right
% 0.06/0.33 thf(fact_81_add__less__cancel__left,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [C: A,A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.06/0.33 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_cancel_left
% 0.06/0.33 thf(fact_82_Nat_Oadd__0__right,axiom,
% 0.06/0.33 ! [M2: nat] :
% 0.06/0.33 ( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
% 0.06/0.33 = M2 ) ).
% 0.06/0.33
% 0.06/0.33 % Nat.add_0_right
% 0.06/0.33 thf(fact_83_add__is__0,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ( plus_plus @ nat @ M2 @ N )
% 0.06/0.33 = ( zero_zero @ nat ) )
% 0.06/0.33 = ( ( M2
% 0.06/0.33 = ( zero_zero @ nat ) )
% 0.06/0.33 & ( N
% 0.06/0.33 = ( zero_zero @ nat ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_is_0
% 0.06/0.33 thf(fact_84_le0,axiom,
% 0.06/0.33 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.06/0.33
% 0.06/0.33 % le0
% 0.06/0.33 thf(fact_85_nat__add__left__cancel__less,axiom,
% 0.06/0.33 ! [K: nat,M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 0.06/0.33 = ( ord_less @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_add_left_cancel_less
% 0.06/0.33 thf(fact_86_nat__add__left__cancel__le,axiom,
% 0.06/0.33 ! [K: nat,M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 0.06/0.33 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_add_left_cancel_le
% 0.06/0.33 thf(fact_87_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.06/0.33 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % zero_le_double_add_iff_zero_le_single_add
% 0.06/0.33 thf(fact_88_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.06/0.33 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % double_add_le_zero_iff_single_add_le_zero
% 0.06/0.33 thf(fact_89_le__add__same__cancel2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.06/0.33 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_add_same_cancel2
% 0.06/0.33 thf(fact_90_le__add__same__cancel1,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.06/0.33 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_add_same_cancel1
% 0.06/0.33 thf(fact_91_add__le__same__cancel2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.06/0.33 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_same_cancel2
% 0.06/0.33 thf(fact_92_add__le__same__cancel1,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.06/0.33 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_same_cancel1
% 0.06/0.33 thf(fact_93_add__less__same__cancel1,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.06/0.33 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_same_cancel1
% 0.06/0.33 thf(fact_94_add__less__same__cancel2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.06/0.33 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_same_cancel2
% 0.06/0.33 thf(fact_95_less__add__same__cancel1,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.06/0.33 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_add_same_cancel1
% 0.06/0.33 thf(fact_96_less__add__same__cancel2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.06/0.33 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_add_same_cancel2
% 0.06/0.33 thf(fact_97_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.06/0.33 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % double_add_less_zero_iff_single_add_less_zero
% 0.06/0.33 thf(fact_98_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.06/0.33 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % zero_less_double_add_iff_zero_less_single_add
% 0.06/0.33 thf(fact_99_add__gr__0,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
% 0.06/0.33 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.06/0.33 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_gr_0
% 0.06/0.33 thf(fact_100_add__less__imp__less__right,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,C: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.06/0.33 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_imp_less_right
% 0.06/0.33 thf(fact_101_add__less__imp__less__left,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [C: A,A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.06/0.33 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_imp_less_left
% 0.06/0.33 thf(fact_102_add__strict__increasing2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere1818651114id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less @ A @ B @ C )
% 0.06/0.33 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_strict_increasing2
% 0.06/0.33 thf(fact_103_add__strict__right__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ B )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_strict_right_mono
% 0.06/0.33 thf(fact_104_add__eq__0__iff__both__eq__0,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [X4: A,Y2: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ X4 @ Y2 )
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 = ( ( X4
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 & ( Y2
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_eq_0_iff_both_eq_0
% 0.06/0.33 thf(fact_105_add__strict__increasing,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere1818651114id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less_eq @ A @ B @ C )
% 0.06/0.33 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_strict_increasing
% 0.06/0.33 thf(fact_106_add__strict__left__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ B )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_strict_left_mono
% 0.06/0.33 thf(fact_107_add__less__le__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ B )
% 0.06/0.33 => ( ( ord_less_eq @ A @ C @ D )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_le_mono
% 0.06/0.33 thf(fact_108_add__le__less__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.33 => ( ( ord_less @ A @ C @ D )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_less_mono
% 0.06/0.33 thf(fact_109_add__strict__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( strict2144017051up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ B )
% 0.06/0.33 => ( ( ord_less @ A @ C @ D )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_strict_mono
% 0.06/0.33 thf(fact_110_add__nonpos__eq__0__iff,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [X4: A,Y2: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ( plus_plus @ A @ X4 @ Y2 )
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 = ( ( X4
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 & ( Y2
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_nonpos_eq_0_iff
% 0.06/0.33 thf(fact_111_add__nonneg__eq__0__iff,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [X4: A,Y2: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
% 0.06/0.33 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 0.06/0.33 => ( ( ( plus_plus @ A @ X4 @ Y2 )
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 = ( ( X4
% 0.06/0.33 = ( zero_zero @ A ) )
% 0.06/0.33 & ( Y2
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_nonneg_eq_0_iff
% 0.06/0.33 thf(fact_112_add__nonpos__nonpos,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_nonpos_nonpos
% 0.06/0.33 thf(fact_113_add__nonneg__nonneg,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.06/0.33 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_nonneg_nonneg
% 0.06/0.33 thf(fact_114_add__increasing2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [C: A,B: A,A2: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.06/0.33 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.06/0.33 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_increasing2
% 0.06/0.33 thf(fact_115_add__decreasing2,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [C: A,A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_decreasing2
% 0.06/0.33 thf(fact_116_add__pos__nonneg,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.06/0.33 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_pos_nonneg
% 0.06/0.33 thf(fact_117_add__nonpos__neg,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_nonpos_neg
% 0.06/0.33 thf(fact_118_add__nonneg__pos,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.06/0.33 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_nonneg_pos
% 0.06/0.33 thf(fact_119_add__neg__nonpos,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_neg_nonpos
% 0.06/0.33 thf(fact_120_add__increasing,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less_eq @ A @ B @ C )
% 0.06/0.33 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_increasing
% 0.06/0.33 thf(fact_121_add__decreasing,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,C: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less_eq @ A @ C @ B )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_decreasing
% 0.06/0.33 thf(fact_122_zero__le,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [X4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) ) ).
% 0.06/0.33
% 0.06/0.33 % zero_le
% 0.06/0.33 thf(fact_123_add_Ogroup__left__neutral,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( group_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.group_left_neutral
% 0.06/0.33 thf(fact_124_add_Ocomm__neutral,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.comm_neutral
% 0.06/0.33 thf(fact_125_comm__monoid__add__class_Oadd__0,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A] :
% 0.06/0.33 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 = A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % comm_monoid_add_class.add_0
% 0.06/0.33 thf(fact_126_less__mono__imp__le__mono,axiom,
% 0.06/0.33 ! [F: nat > nat,I: nat,J: nat] :
% 0.06/0.33 ( ! [I2: nat,J2: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I2 @ J2 )
% 0.06/0.33 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.06/0.33 => ( ( ord_less_eq @ nat @ I @ J )
% 0.06/0.33 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_mono_imp_le_mono
% 0.06/0.33 thf(fact_127_le__neq__implies__less,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.06/0.33 => ( ( M2 != N )
% 0.06/0.33 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_neq_implies_less
% 0.06/0.33 thf(fact_128_mono__nat__linear__lb,axiom,
% 0.06/0.33 ! [F: nat > nat,M2: nat,K: nat] :
% 0.06/0.33 ( ! [M3: nat,N2: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ M3 @ N2 )
% 0.06/0.33 => ( ord_less @ nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
% 0.06/0.33 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % mono_nat_linear_lb
% 0.06/0.33 thf(fact_129_less__or__eq__imp__le,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ( ord_less @ nat @ M2 @ N )
% 0.06/0.33 | ( M2 = N ) )
% 0.06/0.33 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_or_eq_imp_le
% 0.06/0.33 thf(fact_130_less__add__eq__less,axiom,
% 0.06/0.33 ! [K: nat,L: nat,M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ K @ L )
% 0.06/0.33 => ( ( ( plus_plus @ nat @ M2 @ L )
% 0.06/0.33 = ( plus_plus @ nat @ K @ N ) )
% 0.06/0.33 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_add_eq_less
% 0.06/0.33 thf(fact_131_le__eq__less__or__eq,axiom,
% 0.06/0.33 ( ( ord_less_eq @ nat )
% 0.06/0.33 = ( ^ [M4: nat,N3: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ M4 @ N3 )
% 0.06/0.33 | ( M4 = N3 ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_eq_less_or_eq
% 0.06/0.33 thf(fact_132_add__eq__self__zero,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ( plus_plus @ nat @ M2 @ N )
% 0.06/0.33 = M2 )
% 0.06/0.33 => ( N
% 0.06/0.33 = ( zero_zero @ nat ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_eq_self_zero
% 0.06/0.33 thf(fact_133_trans__less__add2,axiom,
% 0.06/0.33 ! [I: nat,J: nat,M2: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I @ J )
% 0.06/0.33 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % trans_less_add2
% 0.06/0.33 thf(fact_134_trans__less__add1,axiom,
% 0.06/0.33 ! [I: nat,J: nat,M2: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I @ J )
% 0.06/0.33 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % trans_less_add1
% 0.06/0.33 thf(fact_135_less__imp__le__nat,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ M2 @ N )
% 0.06/0.33 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_imp_le_nat
% 0.06/0.33 thf(fact_136_plus__nat_Oadd__0,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.06/0.33 = N ) ).
% 0.06/0.33
% 0.06/0.33 % plus_nat.add_0
% 0.06/0.33 thf(fact_137_add__less__mono1,axiom,
% 0.06/0.33 ! [I: nat,J: nat,K: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I @ J )
% 0.06/0.33 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_mono1
% 0.06/0.33 thf(fact_138_not__add__less2,axiom,
% 0.06/0.33 ! [J: nat,I: nat] :
% 0.06/0.33 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 0.06/0.33
% 0.06/0.33 % not_add_less2
% 0.06/0.33 thf(fact_139_not__add__less1,axiom,
% 0.06/0.33 ! [I: nat,J: nat] :
% 0.06/0.33 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 0.06/0.33
% 0.06/0.33 % not_add_less1
% 0.06/0.33 thf(fact_140_add__less__mono,axiom,
% 0.06/0.33 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I @ J )
% 0.06/0.33 => ( ( ord_less @ nat @ K @ L )
% 0.06/0.33 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_less_mono
% 0.06/0.33 thf(fact_141_nat__less__le,axiom,
% 0.06/0.33 ( ( ord_less @ nat )
% 0.06/0.33 = ( ^ [M4: nat,N3: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 0.06/0.33 & ( M4 != N3 ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_less_le
% 0.06/0.33 thf(fact_142_add__lessD1,axiom,
% 0.06/0.33 ! [I: nat,J: nat,K: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.06/0.33 => ( ord_less @ nat @ I @ K ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_lessD1
% 0.06/0.33 thf(fact_143_le__0__eq,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.06/0.33 = ( N
% 0.06/0.33 = ( zero_zero @ nat ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_0_eq
% 0.06/0.33 thf(fact_144_add__mono__thms__linordered__field_I1_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( ord_less @ A @ I @ J )
% 0.06/0.33 & ( K = L ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_field(1)
% 0.06/0.33 thf(fact_145_add__mono__thms__linordered__field_I2_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( I = J )
% 0.06/0.33 & ( ord_less @ A @ K @ L ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_field(2)
% 0.06/0.33 thf(fact_146_add__mono__thms__linordered__field_I3_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( ord_less @ A @ I @ J )
% 0.06/0.33 & ( ord_less_eq @ A @ K @ L ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_field(3)
% 0.06/0.33 thf(fact_147_add__mono__thms__linordered__field_I4_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( ord_less_eq @ A @ I @ J )
% 0.06/0.33 & ( ord_less @ A @ K @ L ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_field(4)
% 0.06/0.33 thf(fact_148_add__mono__thms__linordered__field_I5_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( ord_less @ A @ I @ J )
% 0.06/0.33 & ( ord_less @ A @ K @ L ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_field(5)
% 0.06/0.33 thf(fact_149_less__eq__nat_Osimps_I1_J,axiom,
% 0.06/0.33 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.06/0.33
% 0.06/0.33 % less_eq_nat.simps(1)
% 0.06/0.33 thf(fact_150_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % ab_semigroup_add_class.add_ac(1)
% 0.06/0.33 thf(fact_151_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( I = J )
% 0.06/0.33 & ( K = L ) )
% 0.06/0.33 => ( ( plus_plus @ A @ I @ K )
% 0.06/0.33 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_semiring(4)
% 0.06/0.33 thf(fact_152_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( ord_less_eq @ A @ I @ J )
% 0.06/0.33 & ( K = L ) )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_semiring(3)
% 0.06/0.33 thf(fact_153_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( I = J )
% 0.06/0.33 & ( ord_less_eq @ A @ K @ L ) )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_semiring(2)
% 0.06/0.33 thf(fact_154_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [I: A,J: A,K: A,L: A] :
% 0.06/0.33 ( ( ( ord_less_eq @ A @ I @ J )
% 0.06/0.33 & ( ord_less_eq @ A @ K @ L ) )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono_thms_linordered_semiring(1)
% 0.06/0.33 thf(fact_155_add__leE,axiom,
% 0.06/0.33 ! [M2: nat,K: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 0.06/0.33 => ~ ( ( ord_less_eq @ nat @ M2 @ N )
% 0.06/0.33 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_leE
% 0.06/0.33 thf(fact_156_le__add1,axiom,
% 0.06/0.33 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_add1
% 0.06/0.33 thf(fact_157_le__add2,axiom,
% 0.06/0.33 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_add2
% 0.06/0.33 thf(fact_158_le__refl,axiom,
% 0.06/0.33 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.06/0.33
% 0.06/0.33 % le_refl
% 0.06/0.33 thf(fact_159_add__leD1,axiom,
% 0.06/0.33 ! [M2: nat,K: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 0.06/0.33 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_leD1
% 0.06/0.33 thf(fact_160_add__leD2,axiom,
% 0.06/0.33 ! [M2: nat,K: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 0.06/0.33 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_leD2
% 0.06/0.33 thf(fact_161_le__trans,axiom,
% 0.06/0.33 ! [I: nat,J: nat,K: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ I @ J )
% 0.06/0.33 => ( ( ord_less_eq @ nat @ J @ K )
% 0.06/0.33 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_trans
% 0.06/0.33 thf(fact_162_eq__imp__le,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( M2 = N )
% 0.06/0.33 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.06/0.33
% 0.06/0.33 % eq_imp_le
% 0.06/0.33 thf(fact_163_le__Suc__ex,axiom,
% 0.06/0.33 ! [K: nat,L: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ K @ L )
% 0.06/0.33 => ? [N2: nat] :
% 0.06/0.33 ( L
% 0.06/0.33 = ( plus_plus @ nat @ K @ N2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_Suc_ex
% 0.06/0.33 thf(fact_164_le__antisym,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.06/0.33 => ( ( ord_less_eq @ nat @ N @ M2 )
% 0.06/0.33 => ( M2 = N ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_antisym
% 0.06/0.33 thf(fact_165_add__le__mono,axiom,
% 0.06/0.33 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ I @ J )
% 0.06/0.33 => ( ( ord_less_eq @ nat @ K @ L )
% 0.06/0.33 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_mono
% 0.06/0.33 thf(fact_166_add__le__mono1,axiom,
% 0.06/0.33 ! [I: nat,J: nat,K: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ I @ J )
% 0.06/0.33 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_mono1
% 0.06/0.33 thf(fact_167_nat__le__linear,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.06/0.33 | ( ord_less_eq @ nat @ N @ M2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_le_linear
% 0.06/0.33 thf(fact_168_trans__le__add1,axiom,
% 0.06/0.33 ! [I: nat,J: nat,M2: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ I @ J )
% 0.06/0.33 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % trans_le_add1
% 0.06/0.33 thf(fact_169_trans__le__add2,axiom,
% 0.06/0.33 ! [I: nat,J: nat,M2: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ I @ J )
% 0.06/0.33 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % trans_le_add2
% 0.06/0.33 thf(fact_170_nat__le__iff__add,axiom,
% 0.06/0.33 ( ( ord_less_eq @ nat )
% 0.06/0.33 = ( ^ [M4: nat,N3: nat] :
% 0.06/0.33 ? [K2: nat] :
% 0.06/0.33 ( N3
% 0.06/0.33 = ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_le_iff_add
% 0.06/0.33 thf(fact_171_nat__add__left__cancel,axiom,
% 0.06/0.33 ! [K: nat,M2: nat,N: nat] :
% 0.06/0.33 ( ( ( plus_plus @ nat @ K @ M2 )
% 0.06/0.33 = ( plus_plus @ nat @ K @ N ) )
% 0.06/0.33 = ( M2 = N ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_add_left_cancel
% 0.06/0.33 thf(fact_172_nat__add__right__cancel,axiom,
% 0.06/0.33 ! [M2: nat,K: nat,N: nat] :
% 0.06/0.33 ( ( ( plus_plus @ nat @ M2 @ K )
% 0.06/0.33 = ( plus_plus @ nat @ N @ K ) )
% 0.06/0.33 = ( M2 = N ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_add_right_cancel
% 0.06/0.33 thf(fact_173_add_Oassoc,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.assoc
% 0.06/0.33 thf(fact_174_add_Oleft__cancel,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( group_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ A2 @ B )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ C ) )
% 0.06/0.33 = ( B = C ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.left_cancel
% 0.06/0.33 thf(fact_175_add_Oright__cancel,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( group_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A,C: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ B @ A2 )
% 0.06/0.33 = ( plus_plus @ A @ C @ A2 ) )
% 0.06/0.33 = ( B = C ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.right_cancel
% 0.06/0.33 thf(fact_176_add_Ocommute,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ( ( plus_plus @ A )
% 0.06/0.33 = ( ^ [A4: A,B2: A] : ( plus_plus @ A @ B2 @ A4 ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.commute
% 0.06/0.33 thf(fact_177_add_Oleft__commute,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A,C: A] :
% 0.06/0.33 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add.left_commute
% 0.06/0.33 thf(fact_178_add__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A,D: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.33 => ( ( ord_less_eq @ A @ C @ D )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_mono
% 0.06/0.33 thf(fact_179_add__left__imp__eq,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ A2 @ B )
% 0.06/0.33 = ( plus_plus @ A @ A2 @ C ) )
% 0.06/0.33 => ( B = C ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_left_imp_eq
% 0.06/0.33 thf(fact_180_add__right__imp__eq,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [B: A,A2: A,C: A] :
% 0.06/0.33 ( ( ( plus_plus @ A @ B @ A2 )
% 0.06/0.33 = ( plus_plus @ A @ C @ A2 ) )
% 0.06/0.33 => ( B = C ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_right_imp_eq
% 0.06/0.33 thf(fact_181_add__left__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_left_mono
% 0.06/0.33 thf(fact_182_add__right__mono,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.33 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_right_mono
% 0.06/0.33 thf(fact_183_le__iff__add,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ( ( ord_less_eq @ A )
% 0.06/0.33 = ( ^ [A4: A,B2: A] :
% 0.06/0.33 ? [C3: A] :
% 0.06/0.33 ( B2
% 0.06/0.33 = ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % le_iff_add
% 0.06/0.33 thf(fact_184_add__le__imp__le__left,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [C: A,A2: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.06/0.33 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_imp_le_left
% 0.06/0.33 thf(fact_185_add__le__imp__le__right,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,C: A,B: A] :
% 0.06/0.33 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.06/0.33 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_le_imp_le_right
% 0.06/0.33 thf(fact_186_add__neg__neg,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 0.06/0.33 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_neg_neg
% 0.06/0.33 thf(fact_187_add__pos__pos,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.06/0.33 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % add_pos_pos
% 0.06/0.33 thf(fact_188_pos__add__strict,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( strict797366125id_add @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [A2: A,B: A,C: A] :
% 0.06/0.33 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.06/0.33 => ( ( ord_less @ A @ B @ C )
% 0.06/0.33 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % pos_add_strict
% 0.06/0.33 thf(fact_189_less__imp__add__positive,axiom,
% 0.06/0.33 ! [I: nat,J: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I @ J )
% 0.06/0.33 => ? [K3: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
% 0.06/0.33 & ( ( plus_plus @ nat @ I @ K3 )
% 0.06/0.33 = J ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_imp_add_positive
% 0.06/0.33 thf(fact_190_ex__least__nat__le,axiom,
% 0.06/0.33 ! [P: nat > $o,N: nat] :
% 0.06/0.33 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.06/0.33 => ( ( P @ N )
% 0.06/0.33 => ? [K3: nat] :
% 0.06/0.33 ( ( ord_less_eq @ nat @ K3 @ N )
% 0.06/0.33 & ! [I3: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ I3 @ K3 )
% 0.06/0.33 => ~ ( P @ I3 ) )
% 0.06/0.33 & ( P @ K3 ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % ex_least_nat_le
% 0.06/0.33 thf(fact_191_depth__le__height,axiom,
% 0.06/0.33 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] : ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma1554076246height @ A @ T ) ) ).
% 0.06/0.33
% 0.06/0.33 % depth_le_height
% 0.06/0.33 thf(fact_192_zero__reorient,axiom,
% 0.06/0.33 ! [A: $tType] :
% 0.06/0.33 ( ( zero @ A @ ( type2 @ A ) )
% 0.06/0.33 => ! [X4: A] :
% 0.06/0.33 ( ( ( zero_zero @ A )
% 0.06/0.33 = X4 )
% 0.06/0.33 = ( X4
% 0.06/0.33 = ( zero_zero @ A ) ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % zero_reorient
% 0.06/0.33 thf(fact_193_nat__neq__iff,axiom,
% 0.06/0.33 ! [M2: nat,N: nat] :
% 0.06/0.33 ( ( M2 != N )
% 0.06/0.33 = ( ( ord_less @ nat @ M2 @ N )
% 0.06/0.33 | ( ord_less @ nat @ N @ M2 ) ) ) ).
% 0.06/0.33
% 0.06/0.33 % nat_neq_iff
% 0.06/0.33 thf(fact_194_less__not__refl,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ~ ( ord_less @ nat @ N @ N ) ).
% 0.06/0.33
% 0.06/0.33 % less_not_refl
% 0.06/0.33 thf(fact_195_less__not__refl2,axiom,
% 0.06/0.33 ! [N: nat,M2: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ N @ M2 )
% 0.06/0.33 => ( M2 != N ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_not_refl2
% 0.06/0.33 thf(fact_196_less__not__refl3,axiom,
% 0.06/0.33 ! [S: nat,T: nat] :
% 0.06/0.33 ( ( ord_less @ nat @ S @ T )
% 0.06/0.33 => ( S != T ) ) ).
% 0.06/0.33
% 0.06/0.33 % less_not_refl3
% 0.06/0.33 thf(fact_197_measure__induct,axiom,
% 0.06/0.33 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.06/0.33 ( ! [X: A] :
% 0.06/0.33 ( ! [Y: A] :
% 0.06/0.33 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X ) )
% 0.06/0.33 => ( P @ Y ) )
% 0.06/0.33 => ( P @ X ) )
% 0.06/0.33 => ( P @ A2 ) ) ).
% 0.06/0.33
% 0.06/0.33 % measure_induct
% 0.06/0.33 thf(fact_198_less__irrefl__nat,axiom,
% 0.06/0.33 ! [N: nat] :
% 0.06/0.33 ~ ( ord_less @ nat @ N @ N ) ).
% 0.06/0.33
% 0.06/0.33 % less_irrefl_nat
% 0.06/0.33 thf(fact_199_nat__less__induct,axiom,
% 0.06/0.34 ! [P: nat > $o,N: nat] :
% 0.06/0.34 ( ! [N2: nat] :
% 0.06/0.34 ( ! [M: nat] :
% 0.06/0.34 ( ( ord_less @ nat @ M @ N2 )
% 0.06/0.34 => ( P @ M ) )
% 0.06/0.34 => ( P @ N2 ) )
% 0.06/0.34 => ( P @ N ) ) ).
% 0.06/0.34
% 0.06/0.34 % nat_less_induct
% 0.06/0.34 thf(fact_200_infinite__descent,axiom,
% 0.06/0.34 ! [P: nat > $o,N: nat] :
% 0.06/0.34 ( ! [N2: nat] :
% 0.06/0.34 ( ~ ( P @ N2 )
% 0.06/0.34 => ? [M: nat] :
% 0.06/0.34 ( ( ord_less @ nat @ M @ N2 )
% 0.06/0.34 & ~ ( P @ M ) ) )
% 0.06/0.34 => ( P @ N ) ) ).
% 0.06/0.34
% 0.06/0.34 % infinite_descent
% 0.06/0.34 thf(fact_201_linorder__neqE__nat,axiom,
% 0.06/0.34 ! [X4: nat,Y2: nat] :
% 0.06/0.34 ( ( X4 != Y2 )
% 0.06/0.34 => ( ~ ( ord_less @ nat @ X4 @ Y2 )
% 0.06/0.34 => ( ord_less @ nat @ Y2 @ X4 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % linorder_neqE_nat
% 0.06/0.34 thf(fact_202_measure__induct__rule,axiom,
% 0.06/0.34 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.06/0.34 ( ! [X: A] :
% 0.06/0.34 ( ! [Y: A] :
% 0.06/0.34 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X ) )
% 0.06/0.34 => ( P @ Y ) )
% 0.06/0.34 => ( P @ X ) )
% 0.06/0.34 => ( P @ A2 ) ) ).
% 0.06/0.34
% 0.06/0.34 % measure_induct_rule
% 0.06/0.34 thf(fact_203_infinite__descent__measure,axiom,
% 0.06/0.34 ! [A: $tType,P: A > $o,V: A > nat,X4: A] :
% 0.06/0.34 ( ! [X: A] :
% 0.06/0.34 ( ~ ( P @ X )
% 0.06/0.34 => ? [Y: A] :
% 0.06/0.34 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X ) )
% 0.06/0.34 & ~ ( P @ Y ) ) )
% 0.06/0.34 => ( P @ X4 ) ) ).
% 0.06/0.34
% 0.06/0.34 % infinite_descent_measure
% 0.06/0.34 thf(fact_204_gr__zeroI,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [N: A] :
% 0.06/0.34 ( ( N
% 0.06/0.34 != ( zero_zero @ A ) )
% 0.06/0.34 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % gr_zeroI
% 0.06/0.34 thf(fact_205_not__less__zero,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [N: A] :
% 0.06/0.34 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % not_less_zero
% 0.06/0.34 thf(fact_206_gr__implies__not__zero,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [M2: A,N: A] :
% 0.06/0.34 ( ( ord_less @ A @ M2 @ N )
% 0.06/0.34 => ( N
% 0.06/0.34 != ( zero_zero @ A ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % gr_implies_not_zero
% 0.06/0.34 thf(fact_207_zero__less__iff__neq__zero,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [N: A] :
% 0.06/0.34 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.06/0.34 = ( N
% 0.06/0.34 != ( zero_zero @ A ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % zero_less_iff_neq_zero
% 0.06/0.34 thf(fact_208_gr0I,axiom,
% 0.06/0.34 ! [N: nat] :
% 0.06/0.34 ( ( N
% 0.06/0.34 != ( zero_zero @ nat ) )
% 0.06/0.34 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.06/0.34
% 0.06/0.34 % gr0I
% 0.06/0.34 thf(fact_209_not__gr0,axiom,
% 0.06/0.34 ! [N: nat] :
% 0.06/0.34 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.06/0.34 = ( N
% 0.06/0.34 = ( zero_zero @ nat ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % not_gr0
% 0.06/0.34 thf(fact_210_not__less0,axiom,
% 0.06/0.34 ! [N: nat] :
% 0.06/0.34 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.06/0.34
% 0.06/0.34 % not_less0
% 0.06/0.34 thf(fact_211_field__le__epsilon,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A] :
% 0.06/0.34 ( ! [E: A] :
% 0.06/0.34 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.06/0.34 => ( ord_less_eq @ A @ X4 @ ( plus_plus @ A @ Y2 @ E ) ) )
% 0.06/0.34 => ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % field_le_epsilon
% 0.06/0.34 thf(fact_212_ex__has__greatest__nat__lemma,axiom,
% 0.06/0.34 ! [A: $tType,P: A > $o,K: A,M2: A > nat,N: nat] :
% 0.06/0.34 ( ( P @ K )
% 0.06/0.34 => ( ! [X: A] :
% 0.06/0.34 ( ( P @ X )
% 0.06/0.34 => ? [Y: A] :
% 0.06/0.34 ( ( P @ Y )
% 0.06/0.34 & ~ ( ord_less_eq @ nat @ ( M2 @ Y ) @ ( M2 @ X ) ) ) )
% 0.06/0.34 => ? [Y3: A] :
% 0.06/0.34 ( ( P @ Y3 )
% 0.06/0.34 & ~ ( ord_less @ nat @ ( M2 @ Y3 ) @ ( plus_plus @ nat @ ( M2 @ K ) @ N ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ex_has_greatest_nat_lemma
% 0.06/0.34 thf(fact_213_minima__def,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( huffma1154738298minima @ A )
% 0.06/0.34 = ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A] :
% 0.06/0.34 ( ( member @ A @ A4 @ ( huffma505251170phabet @ A @ T2 ) )
% 0.06/0.34 & ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T2 ) )
% 0.06/0.34 & ( A4 != B2 )
% 0.06/0.34 & ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ ( huffma854352999e_freq @ A @ T2 @ B2 ) )
% 0.06/0.34 & ! [X2: A] :
% 0.06/0.34 ( ( member @ A @ X2 @ ( huffma505251170phabet @ A @ T2 ) )
% 0.06/0.34 => ( ( X2 != A4 )
% 0.06/0.34 => ( ( X2 != B2 )
% 0.06/0.34 => ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ ( huffma854352999e_freq @ A @ T2 @ X2 ) )
% 0.06/0.34 & ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ ( huffma854352999e_freq @ A @ T2 @ X2 ) ) ) ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % minima_def
% 0.06/0.34 thf(fact_214_cost__splitLeaf,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,W_a: nat,W_b: nat,B: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( ( huffma854352999e_freq @ A @ T @ A2 )
% 0.06/0.34 = ( plus_plus @ nat @ W_a @ W_b ) )
% 0.06/0.34 => ( ( huffma636208924e_cost @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 0.06/0.34 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ W_a ) @ W_b ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % cost_splitLeaf
% 0.06/0.34 thf(fact_215_cost__swapSyms__le,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma854352999e_freq @ A @ T @ B ) )
% 0.06/0.34 => ( ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ B ) )
% 0.06/0.34 => ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) @ ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % cost_swapSyms_le
% 0.06/0.34 thf(fact_216_freq__mergeSibling,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.34 != A2 )
% 0.06/0.34 => ( ( huffma854352999e_freq @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) )
% 0.06/0.34 = ( ^ [C3: A] :
% 0.06/0.34 ( if @ nat @ ( C3 = A2 ) @ ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma854352999e_freq @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) ) )
% 0.06/0.34 @ ( if @ nat
% 0.06/0.34 @ ( C3
% 0.06/0.34 = ( huffma943100115ibling @ A @ T @ A2 ) )
% 0.06/0.34 @ ( zero_zero @ nat )
% 0.06/0.34 @ ( huffma854352999e_freq @ A @ T @ C3 ) ) ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % freq_mergeSibling
% 0.06/0.34 thf(fact_217_optimum__def,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( huffma936049440ptimum @ A )
% 0.06/0.34 = ( ^ [T2: huffma16452318e_tree @ A] :
% 0.06/0.34 ! [U2: huffma16452318e_tree @ A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ U2 )
% 0.06/0.34 => ( ( ( huffma505251170phabet @ A @ T2 )
% 0.06/0.34 = ( huffma505251170phabet @ A @ U2 ) )
% 0.06/0.34 => ( ( ( huffma854352999e_freq @ A @ T2 )
% 0.06/0.34 = ( huffma854352999e_freq @ A @ U2 ) )
% 0.06/0.34 => ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ T2 ) @ ( huffma636208924e_cost @ A @ U2 ) ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % optimum_def
% 0.06/0.34 thf(fact_218_notin__alphabet__imp__mergeSibling__id,axiom,
% 0.06/0.34 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 0.06/0.34 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( huffma1954420889ibling @ A @ T @ A2 )
% 0.06/0.34 = T ) ) ).
% 0.06/0.34
% 0.06/0.34 % notin_alphabet_imp_mergeSibling_id
% 0.06/0.34 thf(fact_219_height__0__imp__cost__0,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A] :
% 0.06/0.34 ( ( ( huffma1554076246height @ A @ T )
% 0.06/0.34 = ( zero_zero @ nat ) )
% 0.06/0.34 => ( ( huffma636208924e_cost @ A @ T )
% 0.06/0.34 = ( zero_zero @ nat ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % height_0_imp_cost_0
% 0.06/0.34 thf(fact_220_consistent__mergeSibling,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( huffma1050891809istent @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % consistent_mergeSibling
% 0.06/0.34 thf(fact_221_cost__mergeSibling,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 0.06/0.34 != A2 )
% 0.06/0.34 => ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) ) )
% 0.06/0.34 = ( huffma636208924e_cost @ A @ T ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % cost_mergeSibling
% 0.06/0.34 thf(fact_222_linordered__field__no__ub,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X3: A] :
% 0.06/0.34 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.06/0.34
% 0.06/0.34 % linordered_field_no_ub
% 0.06/0.34 thf(fact_223_linordered__field__no__lb,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X3: A] :
% 0.06/0.34 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 0.06/0.34
% 0.06/0.34 % linordered_field_no_lb
% 0.06/0.34 thf(fact_224_linordered__field__class_Osign__simps_I26_J,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [A2: A,B: A,C: A] :
% 0.06/0.34 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.06/0.34 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % linordered_field_class.sign_simps(26)
% 0.06/0.34 thf(fact_225_linordered__field__class_Osign__simps_I27_J,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.06/0.34 => ( ( plus_plus @ A )
% 0.06/0.34 = ( ^ [A4: A,B2: A] : ( plus_plus @ A @ B2 @ A4 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % linordered_field_class.sign_simps(27)
% 0.06/0.34 thf(fact_226_linordered__field__class_Osign__simps_I28_J,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [B: A,A2: A,C: A] :
% 0.06/0.34 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.06/0.34 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % linordered_field_class.sign_simps(28)
% 0.06/0.34 thf(fact_227_ex__has__least__nat,axiom,
% 0.06/0.34 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.06/0.34 ( ( P @ K )
% 0.06/0.34 => ? [X: A] :
% 0.06/0.34 ( ( P @ X )
% 0.06/0.34 & ! [Y: A] :
% 0.06/0.34 ( ( P @ Y )
% 0.06/0.34 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ Y ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ex_has_least_nat
% 0.06/0.34 thf(fact_228_cost__swapFourSyms__le,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( ( huffma1154738298minima @ A @ T @ A2 @ B )
% 0.06/0.34 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( ( huffma223349076_depth @ A @ T @ C )
% 0.06/0.34 = ( huffma1554076246height @ A @ T ) )
% 0.06/0.34 => ( ( ( huffma223349076_depth @ A @ T @ D )
% 0.06/0.34 = ( huffma1554076246height @ A @ T ) )
% 0.06/0.34 => ( ( C != D )
% 0.06/0.34 => ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) ) @ ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % cost_swapFourSyms_le
% 0.06/0.34 thf(fact_229_ex__has__greatest__nat,axiom,
% 0.06/0.34 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B: nat] :
% 0.06/0.34 ( ( P @ K )
% 0.06/0.34 => ( ! [Y3: A] :
% 0.06/0.34 ( ( P @ Y3 )
% 0.06/0.34 => ( ord_less @ nat @ ( M2 @ Y3 ) @ B ) )
% 0.06/0.34 => ? [X: A] :
% 0.06/0.34 ( ( P @ X )
% 0.06/0.34 & ! [Y: A] :
% 0.06/0.34 ( ( P @ Y )
% 0.06/0.34 => ( ord_less_eq @ nat @ ( M2 @ Y ) @ ( M2 @ X ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ex_has_greatest_nat
% 0.06/0.34 thf(fact_230_weight__splitLeaf,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,W_a: nat,W_b: nat,B: A] :
% 0.06/0.34 ( ( huffma1050891809istent @ A @ T )
% 0.06/0.34 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 0.06/0.34 => ( ( ( huffma854352999e_freq @ A @ T @ A2 )
% 0.06/0.34 = ( plus_plus @ nat @ W_a @ W_b ) )
% 0.06/0.34 => ( ( huffma691733767weight @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 0.06/0.34 = ( huffma691733767weight @ A @ T ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % weight_splitLeaf
% 0.06/0.34 thf(fact_231_order__refl,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( preorder @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.06/0.34
% 0.06/0.34 % order_refl
% 0.06/0.34 thf(fact_232_weight__mergeSibling,axiom,
% 0.06/0.34 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 0.06/0.34 ( ( huffma691733767weight @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) )
% 0.06/0.34 = ( huffma691733767weight @ A @ T ) ) ).
% 0.06/0.34
% 0.06/0.34 % weight_mergeSibling
% 0.06/0.34 thf(fact_233_le__funD,axiom,
% 0.06/0.34 ! [B3: $tType,A: $tType] :
% 0.06/0.34 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 => ! [F: A > B3,G: A > B3,X4: A] :
% 0.06/0.34 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.06/0.34 => ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % le_funD
% 0.06/0.34 thf(fact_234_le__funE,axiom,
% 0.06/0.34 ! [B3: $tType,A: $tType] :
% 0.06/0.34 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 => ! [F: A > B3,G: A > B3,X4: A] :
% 0.06/0.34 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.06/0.34 => ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % le_funE
% 0.06/0.34 thf(fact_235_le__funI,axiom,
% 0.06/0.34 ! [B3: $tType,A: $tType] :
% 0.06/0.34 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 => ! [F: A > B3,G: A > B3] :
% 0.06/0.34 ( ! [X: A] : ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) )
% 0.06/0.34 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % le_funI
% 0.06/0.34 thf(fact_236_le__fun__def,axiom,
% 0.06/0.34 ! [B3: $tType,A: $tType] :
% 0.06/0.34 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 => ( ( ord_less_eq @ ( A > B3 ) )
% 0.06/0.34 = ( ^ [F2: A > B3,G2: A > B3] :
% 0.06/0.34 ! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % le_fun_def
% 0.06/0.34 thf(fact_237_order__subst1,axiom,
% 0.06/0.34 ! [A: $tType,B3: $tType] :
% 0.06/0.34 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 & ( order @ A @ ( type2 @ A ) ) )
% 0.06/0.34 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.06/0.34 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.06/0.34 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.06/0.34 => ( ! [X: B3,Y3: B3] :
% 0.06/0.34 ( ( ord_less_eq @ B3 @ X @ Y3 )
% 0.06/0.34 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.06/0.34 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % order_subst1
% 0.06/0.34 thf(fact_238_order__subst2,axiom,
% 0.06/0.34 ! [A: $tType,C4: $tType] :
% 0.06/0.34 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 0.06/0.34 & ( order @ A @ ( type2 @ A ) ) )
% 0.06/0.34 => ! [A2: A,B: A,F: A > C4,C: C4] :
% 0.06/0.34 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.34 => ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
% 0.06/0.34 => ( ! [X: A,Y3: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.06/0.34 => ( ord_less_eq @ C4 @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.06/0.34 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % order_subst2
% 0.06/0.34 thf(fact_239_ord__eq__le__subst,axiom,
% 0.06/0.34 ! [A: $tType,B3: $tType] :
% 0.06/0.34 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 & ( ord @ A @ ( type2 @ A ) ) )
% 0.06/0.34 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.06/0.34 ( ( A2
% 0.06/0.34 = ( F @ B ) )
% 0.06/0.34 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.06/0.34 => ( ! [X: B3,Y3: B3] :
% 0.06/0.34 ( ( ord_less_eq @ B3 @ X @ Y3 )
% 0.06/0.34 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.06/0.34 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ord_eq_le_subst
% 0.06/0.34 thf(fact_240_ord__le__eq__subst,axiom,
% 0.06/0.34 ! [A: $tType,B3: $tType] :
% 0.06/0.34 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.06/0.34 & ( ord @ A @ ( type2 @ A ) ) )
% 0.06/0.34 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 0.06/0.34 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.34 => ( ( ( F @ B )
% 0.06/0.34 = C )
% 0.06/0.34 => ( ! [X: A,Y3: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.06/0.34 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.06/0.34 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ord_le_eq_subst
% 0.06/0.34 thf(fact_241_eq__iff,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( order @ A @ ( type2 @ A ) )
% 0.06/0.34 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.06/0.34 = ( ^ [X2: A,Y5: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.06/0.34 & ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % eq_iff
% 0.06/0.34 thf(fact_242_antisym,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( order @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.06/0.34 => ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.06/0.34 => ( X4 = Y2 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % antisym
% 0.06/0.34 thf(fact_243_linear,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linorder @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.06/0.34 | ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % linear
% 0.06/0.34 thf(fact_244_eq__refl,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( preorder @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A] :
% 0.06/0.34 ( ( X4 = Y2 )
% 0.06/0.34 => ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % eq_refl
% 0.06/0.34 thf(fact_245_le__cases,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linorder @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A] :
% 0.06/0.34 ( ~ ( ord_less_eq @ A @ X4 @ Y2 )
% 0.06/0.34 => ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % le_cases
% 0.06/0.34 thf(fact_246_order_Otrans,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( order @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [A2: A,B: A,C: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.34 => ( ( ord_less_eq @ A @ B @ C )
% 0.06/0.34 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % order.trans
% 0.06/0.34 thf(fact_247_le__cases3,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( linorder @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A,Z2: A] :
% 0.06/0.34 ( ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.06/0.34 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.06/0.34 => ( ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.06/0.34 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.06/0.34 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 0.06/0.34 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.06/0.34 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.06/0.34 => ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
% 0.06/0.34 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.06/0.34 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 0.06/0.34 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 0.06/0.34 => ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % le_cases3
% 0.06/0.34 thf(fact_248_antisym__conv,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( order @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [Y2: A,X4: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.06/0.34 => ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.06/0.34 = ( X4 = Y2 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % antisym_conv
% 0.06/0.34 thf(fact_249_ord__eq__le__trans,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( ord @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [A2: A,B: A,C: A] :
% 0.06/0.34 ( ( A2 = B )
% 0.06/0.34 => ( ( ord_less_eq @ A @ B @ C )
% 0.06/0.34 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ord_eq_le_trans
% 0.06/0.34 thf(fact_250_ord__le__eq__trans,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( ord @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [A2: A,B: A,C: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.34 => ( ( B = C )
% 0.06/0.34 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % ord_le_eq_trans
% 0.06/0.34 thf(fact_251_order__class_Oorder_Oantisym,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( order @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [A2: A,B: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ A2 @ B )
% 0.06/0.34 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.06/0.34 => ( A2 = B ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % order_class.order.antisym
% 0.06/0.34 thf(fact_252_order__trans,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( preorder @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [X4: A,Y2: A,Z2: A] :
% 0.06/0.34 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.06/0.34 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.06/0.34 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 % order_trans
% 0.06/0.34 thf(fact_253_dual__order_Orefl,axiom,
% 0.06/0.34 ! [A: $tType] :
% 0.06/0.34 ( ( order @ A @ ( type2 @ A ) )
% 0.06/0.34 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.06/0.34
% 0.06/0.34 % dual_order.refl
% 0.06/0.34
% 0.06/0.34 %----Type constructors (30)
% 0.06/0.34 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.06/0.34 ! [A6: $tType,A7: $tType] :
% 0.06/0.34 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.06/0.34 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.06/0.34 ! [A6: $tType,A7: $tType] :
% 0.06/0.34 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.06/0.34 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_fun___Orderings_Oord,axiom,
% 0.06/0.34 ! [A6: $tType,A7: $tType] :
% 0.06/0.34 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.06/0.34 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.06/0.34 ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.06/0.34 strict2144017051up_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.06/0.34 ordere223160158up_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.06/0.34 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.06/0.34 strict797366125id_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.06/0.34 ordere1818651114id_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.06/0.34 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.06/0.34 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 0.06/0.34 ordere216010020id_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.06/0.34 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.06/0.34 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.06/0.34 ab_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.06/0.34 comm_monoid_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.06/0.34 semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.06/0.34 preorder @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.06/0.34 linorder @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.06/0.34 monoid_add @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.06/0.34 order @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.06/0.34 ord @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.06/0.34 zero @ nat @ ( type2 @ nat ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.06/0.34 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.06/0.34 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.06/0.34 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.06/0.34 preorder @ $o @ ( type2 @ $o ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.06/0.34 linorder @ $o @ ( type2 @ $o ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.06/0.34 order @ $o @ ( type2 @ $o ) ).
% 0.06/0.34
% 0.06/0.34 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.06/0.34 ord @ $o @ ( type2 @ $o ) ).
% 0.06/0.34
% 0.06/0.34 %----Helper facts (3)
% 0.06/0.34 thf(help_If_3_1_T,axiom,
% 0.06/0.34 ! [P: $o] :
% 0.06/0.34 ( ( P = $true )
% 0.06/0.34 | ( P = $false ) ) ).
% 0.06/0.34
% 0.06/0.34 thf(help_If_2_1_T,axiom,
% 0.06/0.34 ! [A: $tType,X4: A,Y2: A] :
% 0.06/0.34 ( ( if @ A @ $false @ X4 @ Y2 )
% 0.06/0.34 = Y2 ) ).
% 0.06/0.34
% 0.06/0.34 thf(help_If_1_1_T,axiom,
% 0.06/0.34 ! [A: $tType,X4: A,Y2: A] :
% 0.06/0.34 ( ( if @ A @ $true @ X4 @ Y2 )
% 0.06/0.34 = X4 ) ).
% 0.06/0.34
% 0.06/0.34 %----Conjectures (1)
% 0.06/0.34 thf(conj_0,conjecture,
% 0.06/0.34 huffma1050891809istent @ a @ ( huffma304375860urSyms @ a @ u @ a2 @ b @ c @ ( huffma943100115ibling @ a @ u @ c ) ) ).
% 0.06/0.34
% 0.06/0.34 %------------------------------------------------------------------------------
% 0.06/0.34 ------------------------
% 2.67/1.54 ---- Embedded in THF ---
% 2.67/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.67/1.54 %%% Generated on Tue Feb 24 01:09:12 EST 2026
% 2.67/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.67/1.54 %%% Logic specification used:
% 2.67/1.54 %%% thf(spec, logic, $$dhol).
% 2.67/1.54
% 2.67/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.UCTj6kXiKr/DTF2DTF13488.p
% 2.67/1.54 thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree, type, huffma16452318e_tree: ($tType > $tType)).
% 2.67/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.67/1.54 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.67/1.54 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.67/1.54 thf(ty_tf_a, type, a: $tType).
% 2.67/1.54 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add, type, ordere1818651114id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict797366125id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.54 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.67/1.54 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oalphabet, type, huffma505251170phabet: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (set @ A))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oconsistent, type, huffma1050891809istent: (!> [A:$tType]: (((huffma16452318e_tree @ A) > $o)))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ocost, type, huffma636208924e_cost: (!> [A:$tType]: (((huffma16452318e_tree @ A) > nat)))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Odepth, type, huffma223349076_depth: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > nat))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq, type, huffma854352999e_freq: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > nat))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight, type, huffma1554076246height: (!> [A:$tType]: (((huffma16452318e_tree @ A) > nat)))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OmergeSibling, type, huffma1954420889ibling: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > (huffma16452318e_tree @ A)))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ominima, type, huffma1154738298minima: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > (A > $o)))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ooptimum, type, huffma936049440ptimum: (!> [A:$tType]: (((huffma16452318e_tree @ A) > $o)))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Osibling, type, huffma943100115ibling: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > A))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OsplitLeaf, type, huffma454997449itLeaf: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (nat > (A > (nat > (A > (huffma16452318e_tree @ A))))))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapFourSyms, type, huffma304375860urSyms: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > (A > (A > (A > (huffma16452318e_tree @ A))))))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapLeaves, type, huffma2094459102Leaves: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (nat > (A > (nat > (A > (huffma16452318e_tree @ A))))))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapSyms, type, huffma469337550apSyms: (!> [A:$tType]: (((huffma16452318e_tree @ A) > (A > (A > (huffma16452318e_tree @ A))))))).
% 2.67/1.54 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oweight, type, huffma691733767weight: (!> [A:$tType]: (((huffma16452318e_tree @ A) > nat)))).
% 2.67/1.54 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.67/1.54 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.67/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.67/1.54 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.67/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.67/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.67/1.54 thf(sy_v_a, type, a2: a).
% 2.67/1.54 thf(sy_v_b, type, b: a).
% 2.67/1.54 thf(sy_v_c____, type, c: a).
% 2.67/1.54 thf(sy_v_t, type, t: (huffma16452318e_tree @ a)).
% 2.67/1.54 thf(sy_v_u____, type, u: (huffma16452318e_tree @ a)).
% 2.67/1.54 thf(sy_v_w_092_060_094sub_062a, type, w_a: nat).
% 2.67/1.54 thf(sy_v_w_092_060_094sub_062b, type, w_b: nat).
% 2.67/1.54 thf(fact_0_c_092_060_094sub_062u, axiom, ((huffma1050891809istent @ a) @ u)).
% 2.67/1.54 thf(fact_1_ab, axiom, (a2 != b)).
% 2.67/1.54 thf(fact_2_assms_I1_J, axiom, ((huffma1050891809istent @ a) @ t)).
% 2.67/1.54 thf(fact_3_assms_I2_J, axiom, ((huffma936049440ptimum @ a) @ t)).
% 2.67/1.54 thf(fact_4_dc, axiom, ((((huffma943100115ibling @ a) @ u) @ c) != c)).
% 2.67/1.54 thf(fact_5_sibling__sibling__id, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => ((((huffma943100115ibling @ A) @ T) @ (((huffma943100115ibling @ A) @ T) @ A2)) = A2))))).
% 2.67/1.54 thf(fact_6_a_092_060_094sub_062c, axiom, (((member @ a) @ c) @ ((huffma505251170phabet @ a) @ u))).
% 2.67/1.54 thf(fact_7_sibling__reciprocal, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma943100115ibling @ A) @ T) @ A2) = B) => ((((huffma943100115ibling @ A) @ T) @ B) = A2)))))).
% 2.67/1.54 thf(fact_8_consistent__swapFourSyms, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A,C:A,D:A]: ((((huffma1050891809istent @ A) @ T) => ((huffma1050891809istent @ A) @ ((((((huffma304375860urSyms @ A) @ T) @ A2) @ B) @ C) @ D)))))).
% 2.67/1.54 thf(fact_9_a_092_060_094sub_062d, axiom, (((member @ a) @ (((huffma943100115ibling @ a) @ u) @ c)) @ ((huffma505251170phabet @ a) @ u))).
% 2.67/1.54 thf(fact_10_a_092_060_094sub_062b, axiom, (((member @ a) @ b) @ ((huffma505251170phabet @ a) @ u))).
% 2.67/1.54 thf(fact_11_a_092_060_094sub_062a, axiom, (((member @ a) @ a2) @ ((huffma505251170phabet @ a) @ u))).
% 2.67/1.54 thf(fact_12_d_092_060_094sub_062c, axiom, ((((huffma223349076_depth @ a) @ u) @ c) = ((huffma1554076246height @ a) @ u))).
% 2.67/1.54 thf(fact_13_sibling__swapSyms__sibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),B:A,A2:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma943100115ibling @ A) @ T) @ B) != B) => ((A2 != B) => ((((huffma943100115ibling @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ (((huffma943100115ibling @ A) @ T) @ B))) @ A2) = B))))))).
% 2.67/1.54 thf(fact_14_sibling__swapSyms__other__sibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),B:A,A2:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma943100115ibling @ A) @ T) @ B) != A2) => (((((huffma943100115ibling @ A) @ T) @ B) != B) => ((A2 != B) => ((((huffma943100115ibling @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ B)) @ (((huffma943100115ibling @ A) @ T) @ B)) = A2)))))))).
% 2.67/1.54 thf(fact_15_d_092_060_094sub_062d, axiom, ((((huffma223349076_depth @ a) @ u) @ (((huffma943100115ibling @ a) @ u) @ c)) = ((huffma1554076246height @ a) @ u))).
% 2.67/1.54 thf(fact_16_sibling__swapLeaves__sibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),B:A,A2:A,W_a:nat,W_s:nat]: ((((huffma1050891809istent @ A) @ T) => (((((huffma943100115ibling @ A) @ T) @ B) != B) => ((A2 != B) => ((((huffma943100115ibling @ A) @ ((((((huffma2094459102Leaves @ A) @ T) @ W_a) @ A2) @ W_s) @ (((huffma943100115ibling @ A) @ T) @ B))) @ A2) = B))))))).
% 2.67/1.54 thf(fact_17_depth__sibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => ((((huffma223349076_depth @ A) @ T) @ (((huffma943100115ibling @ A) @ T) @ A2)) = (((huffma223349076_depth @ A) @ T) @ A2)))))).
% 2.67/1.54 thf(fact_18_sibling__swapFourSyms__when__4th__is__sibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A,C:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ C) @ ((huffma505251170phabet @ A) @ T)) => ((A2 != B) => (((((huffma943100115ibling @ A) @ T) @ C) != C) => ((((huffma943100115ibling @ A) @ ((((((huffma304375860urSyms @ A) @ T) @ A2) @ B) @ C) @ (((huffma943100115ibling @ A) @ T) @ C))) @ A2) = B)))))))))).
% 2.67/1.54 thf(fact_19_assms_I4_J, axiom, (~ ((((member @ a) @ b) @ ((huffma505251170phabet @ a) @ t))))).
% 2.67/1.54 thf(fact_20_assms_I3_J, axiom, (((member @ a) @ a2) @ ((huffma505251170phabet @ a) @ t))).
% 2.67/1.54 thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060lbrakk_062c_A_092_060in_062_Aalphabet_Au_059_Adepth_Au_Ac_A_061_Aheight_Au_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [C2:a]: (((((member @ a) @ C2) @ ((huffma505251170phabet @ a) @ u)) => ((((huffma223349076_depth @ a) @ u) @ C2) != ((huffma1554076246height @ a) @ u)))))))).
% 2.67/1.54 thf(fact_22__092_060open_062_092_060exists_062a_092_060in_062alphabet_Au_O_Adepth_Au_Aa_A_061_Aheight_Au_092_060close_062, axiom, (? [X:a]: (((((member @ a) @ X) @ ((huffma505251170phabet @ a) @ u)) & ((((huffma223349076_depth @ a) @ u) @ X) = ((huffma1554076246height @ a) @ u)))))).
% 2.67/1.54 thf(fact_23_notin__alphabet__imp__sibling__id, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A)]: (((~ ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)))) => ((((huffma943100115ibling @ A) @ T) @ A2) = A2))))).
% 2.67/1.54 thf(fact_24_swapLeaves__id__when__notin__alphabet, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A),W:nat,W2:nat]: (((~ ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)))) => (((((((huffma2094459102Leaves @ A) @ T) @ W) @ A2) @ W2) @ A2) = T))))).
% 2.67/1.54 thf(fact_25_alphabet__swapSyms, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A),B:A]: (((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)) => (((huffma505251170phabet @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ B)) = ((huffma505251170phabet @ A) @ T))))))).
% 2.67/1.54 thf(fact_26_height__swapLeaves, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),W_a:nat,A2:A,W_b:nat,B:A]: ((((huffma1554076246height @ A) @ ((((((huffma2094459102Leaves @ A) @ T) @ W_a) @ A2) @ W_b) @ B)) = ((huffma1554076246height @ A) @ T))))).
% 2.67/1.54 thf(fact_27_swapSyms__id, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma469337550apSyms @ A) @ T) @ A2) @ A2) = T))))).
% 2.67/1.54 thf(fact_28_alphabet__swapFourSyms, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A),B:A,C:A,D:A]: (((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ C) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ D) @ ((huffma505251170phabet @ A) @ T)) => (((huffma505251170phabet @ A) @ ((((((huffma304375860urSyms @ A) @ T) @ A2) @ B) @ C) @ D)) = ((huffma505251170phabet @ A) @ T))))))))).
% 2.67/1.54 thf(fact_29_depth__swapLeaves__neither, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),C:A,A2:A,B:A,W_a:nat,W_b:nat]: ((((huffma1050891809istent @ A) @ T) => ((C != A2) => ((C != B) => ((((huffma223349076_depth @ A) @ ((((((huffma2094459102Leaves @ A) @ T) @ W_a) @ A2) @ W_b) @ B)) @ C) = (((huffma223349076_depth @ A) @ T) @ C)))))))).
% 2.67/1.54 thf(fact_30_depth__swapSyms__neither, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),C:A,A2:A,B:A]: ((((huffma1050891809istent @ A) @ T) => ((C != A2) => ((C != B) => ((((huffma223349076_depth @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ B)) @ C) = (((huffma223349076_depth @ A) @ T) @ C)))))))).
% 2.67/1.54 thf(fact_31_exists__at__height, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A)]: ((((huffma1050891809istent @ A) @ T) => (? [X:A]: (((((member @ A) @ X) @ ((huffma505251170phabet @ A) @ T)) & ((((huffma223349076_depth @ A) @ T) @ X) = ((huffma1554076246height @ A) @ T))))))))).
% 2.67/1.54 thf(fact_32_exists__in__alphabet, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A)]: ((? [A3:A]: ((((member @ A) @ A3) @ ((huffma505251170phabet @ A) @ T))))))).
% 2.67/1.54 thf(fact_33_in__alphabet__imp__sibling__in__alphabet, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A)]: (((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => (((member @ A) @ (((huffma943100115ibling @ A) @ T) @ A2)) @ ((huffma505251170phabet @ A) @ T)))))).
% 2.67/1.54 thf(fact_34_sibling__ne__imp__sibling__in__alphabet, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((((huffma943100115ibling @ A) @ T) @ A2) != A2) => (((member @ A) @ (((huffma943100115ibling @ A) @ T) @ A2)) @ ((huffma505251170phabet @ A) @ T)))))).
% 2.67/1.54 thf(fact_35_consistent__swapLeaves, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),W_a:nat,A2:A,W_b:nat,B:A]: ((((huffma1050891809istent @ A) @ T) => ((huffma1050891809istent @ A) @ ((((((huffma2094459102Leaves @ A) @ T) @ W_a) @ A2) @ W_b) @ B)))))).
% 2.67/1.54 thf(fact_36_consistent__swapSyms, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A]: ((((huffma1050891809istent @ A) @ T) => ((huffma1050891809istent @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ B)))))).
% 2.67/1.54 thf(fact_37_swapFourSyms__def, axiom, (! [A:$tType]: (((huffma304375860urSyms @ A) = (^ [T2:(huffma16452318e_tree @ A),A4:A,B2:A,C3:A,D2:A]: (((((if @ (huffma16452318e_tree @ A)) @ (A4 = D2)) @ ((((huffma469337550apSyms @ A) @ T2) @ B2) @ C3)) @ ((((if @ (huffma16452318e_tree @ A)) @ (B2 = C3)) @ ((((huffma469337550apSyms @ A) @ T2) @ A4) @ D2)) @ ((((huffma469337550apSyms @ A) @ ((((huffma469337550apSyms @ A) @ T2) @ A4) @ C3)) @ B2) @ D2))))))))).
% 2.67/1.54 thf(fact_38_a_092_060_094sub_062u, axiom, (((huffma505251170phabet @ a) @ ((((((huffma454997449itLeaf @ a) @ t) @ w_a) @ a2) @ w_b) @ b)) = ((huffma505251170phabet @ a) @ u))).
% 2.67/1.54 thf(fact_39_freq__swapFourSyms, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A,C:A,D:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ C) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ D) @ ((huffma505251170phabet @ A) @ T)) => (((huffma854352999e_freq @ A) @ ((((((huffma304375860urSyms @ A) @ T) @ A2) @ B) @ C) @ D)) = ((huffma854352999e_freq @ A) @ T)))))))))).
% 2.67/1.54 thf(fact_40_False, axiom, (((huffma1554076246height @ a) @ ((((((huffma454997449itLeaf @ a) @ t) @ w_a) @ a2) @ w_b) @ b)) != (zero_zero @ nat))).
% 2.67/1.54 thf(fact_41_f_092_060_094sub_062u, axiom, (((huffma854352999e_freq @ a) @ ((((((huffma454997449itLeaf @ a) @ t) @ w_a) @ a2) @ w_b) @ b)) = ((huffma854352999e_freq @ a) @ u))).
% 2.67/1.54 thf(fact_42_freq__swapSyms, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)) => (((huffma854352999e_freq @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ B)) = ((huffma854352999e_freq @ A) @ T)))))))).
% 2.67/1.54 thf(fact_43_consistent__splitLeaf, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),B:A,W_a:nat,A2:A,W_b:nat]: ((((huffma1050891809istent @ A) @ T) => ((~ ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)))) => ((huffma1050891809istent @ A) @ ((((((huffma454997449itLeaf @ A) @ T) @ W_a) @ A2) @ W_b) @ B))))))).
% 2.67/1.54 thf(fact_44_height__0__imp__sibling__id, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: (((((huffma1554076246height @ A) @ T) = (zero_zero @ nat)) => ((((huffma943100115ibling @ A) @ T) @ A2) = A2))))).
% 2.67/1.54 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.67/1.54 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A5)))) = A5)))).
% 2.67/1.54 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.67/1.54 thf(fact_48_ext, axiom, (! [B3:$tType,A:$tType,F:(A > B3),G:(A > B3)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 2.67/1.54 thf(fact_49_notin__alphabet__imp__splitLeaf__id, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A),W_a:nat,W_b:nat,B:A]: (((~ ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)))) => (((((((huffma454997449itLeaf @ A) @ T) @ W_a) @ A2) @ W_b) @ B) = T))))).
% 2.67/1.54 thf(fact_50_swapLeaves__id, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => (((((((huffma2094459102Leaves @ A) @ T) @ (((huffma854352999e_freq @ A) @ T) @ A2)) @ A2) @ (((huffma854352999e_freq @ A) @ T) @ A2)) @ A2) = T))))).
% 2.67/1.54 thf(fact_51_depth__height__imp__sibling__ne, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma223349076_depth @ A) @ T) @ A2) = ((huffma1554076246height @ A) @ T)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((huffma1554076246height @ A) @ T)) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((huffma943100115ibling @ A) @ T) @ A2) != A2)))))))).
% 2.67/1.54 thf(fact_52_h_092_060_094sub_062u, axiom, (((ord_less @ nat) @ (zero_zero @ nat)) @ ((huffma1554076246height @ a) @ u))).
% 2.67/1.54 thf(fact_53_notin__alphabet__imp__freq__0, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A)]: (((~ ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)))) => ((((huffma854352999e_freq @ A) @ T) @ A2) = (zero_zero @ nat)))))).
% 2.67/1.54 thf(fact_54_assms_I7_J, axiom, (((ord_less_eq @ nat) @ w_a) @ w_b)).
% 2.67/1.54 thf(fact_55_freq__swapLeaves, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A,W_a:nat,W_b:nat]: ((((huffma1050891809istent @ A) @ T) => ((A2 != B) => (((huffma854352999e_freq @ A) @ ((((((huffma2094459102Leaves @ A) @ T) @ W_a) @ A2) @ W_b) @ B)) = (^ [C3:A]: (((((if @ nat) @ (C3 = A2)) @ ((((if @ nat) @ (((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T))) @ W_a) @ (zero_zero @ nat))) @ ((((if @ nat) @ (C3 = B)) @ ((((if @ nat) @ (((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T))) @ W_b) @ (zero_zero @ nat))) @ (((huffma854352999e_freq @ A) @ T) @ C3))))))))))).
% 2.67/1.54 thf(fact_56_assms_I6_J, axiom, (! [X3:a]: (((((member @ a) @ X3) @ ((huffma505251170phabet @ a) @ t)) => ((((ord_less_eq @ nat) @ w_a) @ (((huffma854352999e_freq @ a) @ t) @ X3)) & (((ord_less_eq @ nat) @ w_b) @ (((huffma854352999e_freq @ a) @ t) @ X3))))))).
% 2.67/1.54 thf(fact_57_assms_I5_J, axiom, ((((huffma854352999e_freq @ a) @ t) @ a2) = (((plus_plus @ nat) @ w_a) @ w_b))).
% 2.67/1.54 thf(fact_58_height__gt__0__alphabet__eq__imp__height__gt__0, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),U:(huffma16452318e_tree @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((huffma1554076246height @ A) @ T)) => (((huffma1050891809istent @ A) @ T) => ((((huffma505251170phabet @ A) @ T) = ((huffma505251170phabet @ A) @ U)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((huffma1554076246height @ A) @ U)))))))).
% 2.67/1.54 thf(fact_59_swapSyms__def, axiom, (! [A:$tType]: (((huffma469337550apSyms @ A) = (^ [T2:(huffma16452318e_tree @ A),A4:A,B2:A]: (((((((huffma2094459102Leaves @ A) @ T2) @ (((huffma854352999e_freq @ A) @ T2) @ A4)) @ A4) @ (((huffma854352999e_freq @ A) @ T2) @ B2)) @ B2))))))).
% 2.67/1.54 thf(fact_60_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.67/1.54 thf(fact_61_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_62_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X4:A]: (((! [X:A]: ((((V @ X) = (zero_zero @ nat)) => (P @ X)))) => ((! [X:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X)) => ((~ ((P @ X))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X)) & (~ ((P @ Y)))))))))) => (P @ X4)))))).
% 2.67/1.54 thf(fact_63_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.67/1.54 thf(fact_64_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M:nat]: (((((ord_less @ nat) @ M) @ N2) & (~ ((P @ M)))))))))) => (P @ N)))))).
% 2.67/1.54 thf(fact_65_gr__implies__not0, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (N != (zero_zero @ nat)))))).
% 2.67/1.54 thf(fact_66_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.67/1.54 thf(fact_67_add__left__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 2.67/1.54 thf(fact_68_add__right__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 2.67/1.54 thf(fact_69_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_70_add__le__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_71_add__le__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_72_add__cancel__right__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((A2 = (((plus_plus @ A) @ A2) @ B)) = (B = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_73_add__cancel__right__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((A2 = (((plus_plus @ A) @ B) @ A2)) = (B = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_74_add__cancel__left__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: ((((((plus_plus @ A) @ A2) @ B) = A2) = (B = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_75_add__cancel__left__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [B:A,A2:A]: ((((((plus_plus @ A) @ B) @ A2) = A2) = (B = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_76_double__zero__sym, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_77_double__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_78_add_Oright__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.67/1.54 thf(fact_79_add_Oleft__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.67/1.54 thf(fact_80_add__less__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_81_add__less__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_82_Nat_Oadd__0__right, axiom, (! [M2:nat]: (((((plus_plus @ nat) @ M2) @ (zero_zero @ nat)) = M2)))).
% 2.67/1.54 thf(fact_83_add__is__0, axiom, (! [M2:nat,N:nat]: ((((((plus_plus @ nat) @ M2) @ N) = (zero_zero @ nat)) = ((M2 = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 2.67/1.54 thf(fact_84_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.67/1.54 thf(fact_85_nat__add__left__cancel__less, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_86_nat__add__left__cancel__le, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_87_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.54 thf(fact_88_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_89_le__add__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.67/1.54 thf(fact_90_le__add__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.67/1.54 thf(fact_91_add__le__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_92_add__le__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_93_add__less__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_94_add__less__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_95_less__add__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B)) = (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.67/1.54 thf(fact_96_less__add__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.67/1.54 thf(fact_97_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_98_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.54 thf(fact_99_add__gr__0, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M2) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 2.67/1.54 thf(fact_100_add__less__imp__less__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) => (((ord_less @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_101_add__less__imp__less__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) => (((ord_less @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_102_add__strict__increasing2, axiom, (! [A:$tType]: ((((ordere1818651114id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.67/1.54 thf(fact_103_add__strict__right__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_104_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: ((((((plus_plus @ A) @ X4) @ Y2) = (zero_zero @ A)) = ((X4 = (zero_zero @ A)) & (Y2 = (zero_zero @ A)))))))))).
% 2.67/1.54 thf(fact_105_add__strict__increasing, axiom, (! [A:$tType]: ((((ordere1818651114id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.67/1.54 thf(fact_106_add__strict__left__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B))))))))).
% 2.67/1.54 thf(fact_107_add__less__le__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.67/1.54 thf(fact_108_add__le__less__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.67/1.54 thf(fact_109_add__strict__mono, axiom, (! [A:$tType]: ((((strict2144017051up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.67/1.54 thf(fact_110_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y2) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X4) @ Y2) = (zero_zero @ A)) = ((X4 = (zero_zero @ A)) & (Y2 = (zero_zero @ A)))))))))))).
% 2.67/1.54 thf(fact_111_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y2) => (((((plus_plus @ A) @ X4) @ Y2) = (zero_zero @ A)) = ((X4 = (zero_zero @ A)) & (Y2 = (zero_zero @ A)))))))))))).
% 2.67/1.54 thf(fact_112_add__nonpos__nonpos, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.67/1.54 thf(fact_113_add__nonneg__nonneg, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.67/1.54 thf(fact_114_add__increasing2, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [C:A,B:A,A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => ((((ord_less_eq @ A) @ B) @ A2) => (((ord_less_eq @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.67/1.54 thf(fact_115_add__decreasing2, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B))))))))).
% 2.67/1.54 thf(fact_116_add__pos__nonneg, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.67/1.54 thf(fact_117_add__nonpos__neg, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.67/1.54 thf(fact_118_add__nonneg__pos, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.67/1.54 thf(fact_119_add__neg__nonpos, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.67/1.54 thf(fact_120_add__increasing, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.67/1.54 thf(fact_121_add__decreasing, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B))))))))).
% 2.67/1.54 thf(fact_122_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X4:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X4))))))).
% 2.67/1.54 thf(fact_123_add_Ogroup__left__neutral, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.67/1.54 thf(fact_124_add_Ocomm__neutral, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.67/1.54 thf(fact_125_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.67/1.54 thf(fact_126_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 2.67/1.54 thf(fact_127_le__neq__implies__less, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((M2 != N) => (((ord_less @ nat) @ M2) @ N)))))).
% 2.67/1.54 thf(fact_128_mono__nat__linear__lb, axiom, (! [F:(nat > nat),M2:nat,K:nat]: (((! [M3:nat,N2:nat]: (((((ord_less @ nat) @ M3) @ N2) => (((ord_less @ nat) @ (F @ M3)) @ (F @ N2))))) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (F @ M2)) @ K)) @ (F @ (((plus_plus @ nat) @ M2) @ K))))))).
% 2.67/1.54 thf(fact_129_less__or__eq__imp__le, axiom, (! [M2:nat,N:nat]: ((((((ord_less @ nat) @ M2) @ N) | (M2 = N)) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_130_less__add__eq__less, axiom, (! [K:nat,L:nat,M2:nat,N:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M2) @ L) = (((plus_plus @ nat) @ K) @ N)) => (((ord_less @ nat) @ M2) @ N)))))).
% 2.67/1.54 thf(fact_131_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M4:nat,N3:nat]: (((((ord_less @ nat) @ M4) @ N3) | (M4 = N3)))))).
% 2.67/1.54 thf(fact_132_add__eq__self__zero, axiom, (! [M2:nat,N:nat]: ((((((plus_plus @ nat) @ M2) @ N) = M2) => (N = (zero_zero @ nat)))))).
% 2.67/1.54 thf(fact_133_trans__less__add2, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ M2) @ J)))))).
% 2.67/1.54 thf(fact_134_trans__less__add1, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M2)))))).
% 2.67/1.54 thf(fact_135_less__imp__le__nat, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_136_plus__nat_Oadd__0, axiom, (! [N:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N) = N)))).
% 2.67/1.54 thf(fact_137_add__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.67/1.54 thf(fact_138_not__add__less2, axiom, (! [J:nat,I:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ I)))))).
% 2.67/1.54 thf(fact_139_not__add__less1, axiom, (! [I:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ I)))))).
% 2.67/1.54 thf(fact_140_add__less__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ K) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.67/1.54 thf(fact_141_nat__less__le, axiom, ((ord_less @ nat) = (^ [M4:nat,N3:nat]: (((((ord_less_eq @ nat) @ M4) @ N3) & (M4 != N3)))))).
% 2.67/1.54 thf(fact_142_add__lessD1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) => (((ord_less @ nat) @ I) @ K))))).
% 2.67/1.54 thf(fact_143_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.67/1.54 thf(fact_144_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_145_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_146_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_147_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_148_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_149_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.67/1.54 thf(fact_150_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_151_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_152_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_153_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_154_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.67/1.54 thf(fact_155_add__leE, axiom, (! [M2:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ N) => (~ (((((ord_less_eq @ nat) @ M2) @ N) => (~ ((((ord_less_eq @ nat) @ K) @ N)))))))))).
% 2.67/1.54 thf(fact_156_le__add1, axiom, (! [N:nat,M2:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M2))))).
% 2.67/1.54 thf(fact_157_le__add2, axiom, (! [N:nat,M2:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_158_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.67/1.54 thf(fact_159_add__leD1, axiom, (! [M2:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_160_add__leD2, axiom, (! [M2:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ N) => (((ord_less_eq @ nat) @ K) @ N))))).
% 2.67/1.54 thf(fact_161_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.67/1.54 thf(fact_162_eq__imp__le, axiom, (! [M2:nat,N:nat]: (((M2 = N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 2.67/1.54 thf(fact_163_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N2:nat]: ((L = (((plus_plus @ nat) @ K) @ N2)))))))).
% 2.67/1.54 thf(fact_164_le__antisym, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ N) @ M2) => (M2 = N)))))).
% 2.67/1.54 thf(fact_165_add__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.67/1.54 thf(fact_166_add__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.67/1.54 thf(fact_167_nat__le__linear, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) | (((ord_less_eq @ nat) @ N) @ M2))))).
% 2.67/1.54 thf(fact_168_trans__le__add1, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M2)))))).
% 2.67/1.54 thf(fact_169_trans__le__add2, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M2) @ J)))))).
% 2.67/1.54 thf(fact_170_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M4:nat,N3:nat]: ((? [K2:nat]: ((N3 = (((plus_plus @ nat) @ M4) @ K2)))))))).
% 2.67/1.54 thf(fact_171_nat__add__left__cancel, axiom, (! [K:nat,M2:nat,N:nat]: ((((((plus_plus @ nat) @ K) @ M2) = (((plus_plus @ nat) @ K) @ N)) = (M2 = N))))).
% 2.67/1.54 thf(fact_172_nat__add__right__cancel, axiom, (! [M2:nat,K:nat,N:nat]: ((((((plus_plus @ nat) @ M2) @ K) = (((plus_plus @ nat) @ N) @ K)) = (M2 = N))))).
% 2.67/1.54 thf(fact_173_add_Oassoc, axiom, (! [A:$tType]: ((((semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_174_add_Oleft__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 2.67/1.54 thf(fact_175_add_Oright__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 2.67/1.54 thf(fact_176_add_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => ((plus_plus @ A) = (^ [A4:A,B2:A]: ((((plus_plus @ A) @ B2) @ A4)))))))).
% 2.67/1.54 thf(fact_177_add_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: (((((plus_plus @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_178_add__mono, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.67/1.54 thf(fact_179_add__left__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) => (B = C)))))))).
% 2.67/1.54 thf(fact_180_add__right__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B = C)))))))).
% 2.67/1.54 thf(fact_181_add__left__mono, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B))))))))).
% 2.67/1.54 thf(fact_182_add__right__mono, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_183_le__iff__add, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B2:A]: ((? [C3:A]: ((B2 = (((plus_plus @ A) @ A4) @ C3))))))))))).
% 2.67/1.54 thf(fact_184_add__le__imp__le__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_185_add__le__imp__le__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.67/1.54 thf(fact_186_add__neg__neg, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.67/1.54 thf(fact_187_add__pos__pos, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.67/1.54 thf(fact_188_pos__add__strict, axiom, (! [A:$tType]: ((((strict797366125id_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.67/1.54 thf(fact_189_less__imp__add__positive, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (? [K3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K3) & ((((plus_plus @ nat) @ I) @ K3) = J)))))))).
% 2.67/1.54 thf(fact_190_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((~ ((P @ (zero_zero @ nat)))) => ((P @ N) => (? [K3:nat]: (((((ord_less_eq @ nat) @ K3) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K3) => (~ ((P @ I3)))))) & (P @ K3)))))))))).
% 2.67/1.54 thf(fact_191_depth__le__height, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((ord_less_eq @ nat) @ (((huffma223349076_depth @ A) @ T) @ A2)) @ ((huffma1554076246height @ A) @ T))))).
% 2.67/1.54 thf(fact_192_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type2 @ A)) => (! [X4:A]: ((((zero_zero @ A) = X4) = (X4 = (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_193_nat__neq__iff, axiom, (! [M2:nat,N:nat]: (((M2 != N) = ((((ord_less @ nat) @ M2) @ N) | (((ord_less @ nat) @ N) @ M2)))))).
% 2.67/1.54 thf(fact_194_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.67/1.54 thf(fact_195_less__not__refl2, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ N) @ M2) => (M2 != N))))).
% 2.67/1.54 thf(fact_196_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.67/1.54 thf(fact_197_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X:A]: (((! [Y:A]: (((((ord_less @ nat) @ (F @ Y)) @ (F @ X)) => (P @ Y)))) => (P @ X)))) => (P @ A2))))).
% 2.67/1.54 thf(fact_198_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.67/1.54 thf(fact_199_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M:nat]: (((((ord_less @ nat) @ M) @ N2) => (P @ M)))) => (P @ N2)))) => (P @ N))))).
% 2.67/1.54 thf(fact_200_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M:nat]: (((((ord_less @ nat) @ M) @ N2) & (~ ((P @ M))))))))) => (P @ N))))).
% 2.67/1.54 thf(fact_201_linorder__neqE__nat, axiom, (! [X4:nat,Y2:nat]: (((X4 != Y2) => ((~ ((((ord_less @ nat) @ X4) @ Y2))) => (((ord_less @ nat) @ Y2) @ X4)))))).
% 2.67/1.54 thf(fact_202_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X:A]: (((! [Y:A]: (((((ord_less @ nat) @ (F @ Y)) @ (F @ X)) => (P @ Y)))) => (P @ X)))) => (P @ A2))))).
% 2.67/1.54 thf(fact_203_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X4:A]: (((! [X:A]: (((~ ((P @ X))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X)) & (~ ((P @ Y))))))))) => (P @ X4))))).
% 2.67/1.54 thf(fact_204_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.67/1.54 thf(fact_205_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.67/1.54 thf(fact_206_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [M2:A,N:A]: (((((ord_less @ A) @ M2) @ N) => (N != (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_207_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.67/1.54 thf(fact_208_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.67/1.54 thf(fact_209_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.67/1.54 thf(fact_210_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.67/1.54 thf(fact_211_field__le__epsilon, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((! [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) => (((ord_less_eq @ A) @ X4) @ (((plus_plus @ A) @ Y2) @ E))))) => (((ord_less_eq @ A) @ X4) @ Y2)))))))).
% 2.67/1.54 thf(fact_212_ex__has__greatest__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),N:nat]: (((P @ K) => ((! [X:A]: (((P @ X) => (? [Y:A]: (((P @ Y) & (~ ((((ord_less_eq @ nat) @ (M2 @ Y)) @ (M2 @ X)))))))))) => (? [Y3:A]: (((P @ Y3) & (~ ((((ord_less @ nat) @ (M2 @ Y3)) @ (((plus_plus @ nat) @ (M2 @ K)) @ N)))))))))))).
% 2.67/1.54 thf(fact_213_minima__def, axiom, (! [A:$tType]: (((huffma1154738298minima @ A) = (^ [T2:(huffma16452318e_tree @ A),A4:A,B2:A]: (((((member @ A) @ A4) @ ((huffma505251170phabet @ A) @ T2)) & ((((member @ A) @ B2) @ ((huffma505251170phabet @ A) @ T2)) & ((A4 != B2) & ((((ord_less_eq @ nat) @ (((huffma854352999e_freq @ A) @ T2) @ A4)) @ (((huffma854352999e_freq @ A) @ T2) @ B2)) & (! [X2:A]: (((((member @ A) @ X2) @ ((huffma505251170phabet @ A) @ T2)) => ((X2 != A4) => ((X2 != B2) => ((((ord_less_eq @ nat) @ (((huffma854352999e_freq @ A) @ T2) @ A4)) @ (((huffma854352999e_freq @ A) @ T2) @ X2)) & (((ord_less_eq @ nat) @ (((huffma854352999e_freq @ A) @ T2) @ B2)) @ (((huffma854352999e_freq @ A) @ T2) @ X2)))))))))))))))))).
% 2.67/1.54 thf(fact_214_cost__splitLeaf, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,W_a:nat,W_b:nat,B:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => (((((huffma854352999e_freq @ A) @ T) @ A2) = (((plus_plus @ nat) @ W_a) @ W_b)) => (((huffma636208924e_cost @ A) @ ((((((huffma454997449itLeaf @ A) @ T) @ W_a) @ A2) @ W_b) @ B)) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ ((huffma636208924e_cost @ A) @ T)) @ W_a)) @ W_b)))))))).
% 2.67/1.54 thf(fact_215_cost__swapSyms__le, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ B) @ ((huffma505251170phabet @ A) @ T)) => ((((ord_less_eq @ nat) @ (((huffma854352999e_freq @ A) @ T) @ A2)) @ (((huffma854352999e_freq @ A) @ T) @ B)) => ((((ord_less_eq @ nat) @ (((huffma223349076_depth @ A) @ T) @ A2)) @ (((huffma223349076_depth @ A) @ T) @ B)) => (((ord_less_eq @ nat) @ ((huffma636208924e_cost @ A) @ ((((huffma469337550apSyms @ A) @ T) @ A2) @ B))) @ ((huffma636208924e_cost @ A) @ T)))))))))).
% 2.67/1.54 thf(fact_216_freq__mergeSibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => (((((huffma943100115ibling @ A) @ T) @ A2) != A2) => (((huffma854352999e_freq @ A) @ (((huffma1954420889ibling @ A) @ T) @ A2)) = (^ [C3:A]: (((((if @ nat) @ (C3 = A2)) @ (((plus_plus @ nat) @ (((huffma854352999e_freq @ A) @ T) @ A2)) @ (((huffma854352999e_freq @ A) @ T) @ (((huffma943100115ibling @ A) @ T) @ A2)))) @ ((((if @ nat) @ (C3 = (((huffma943100115ibling @ A) @ T) @ A2))) @ (zero_zero @ nat)) @ (((huffma854352999e_freq @ A) @ T) @ C3)))))))))))).
% 2.67/1.54 thf(fact_217_optimum__def, axiom, (! [A:$tType]: (((huffma936049440ptimum @ A) = (^ [T2:(huffma16452318e_tree @ A)]: ((! [U2:(huffma16452318e_tree @ A)]: ((((huffma1050891809istent @ A) @ U2) => ((((huffma505251170phabet @ A) @ T2) = ((huffma505251170phabet @ A) @ U2)) => ((((huffma854352999e_freq @ A) @ T2) = ((huffma854352999e_freq @ A) @ U2)) => (((ord_less_eq @ nat) @ ((huffma636208924e_cost @ A) @ T2)) @ ((huffma636208924e_cost @ A) @ U2))))))))))))).
% 2.67/1.54 thf(fact_218_notin__alphabet__imp__mergeSibling__id, axiom, (! [A:$tType,A2:A,T:(huffma16452318e_tree @ A)]: (((~ ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)))) => ((((huffma1954420889ibling @ A) @ T) @ A2) = T))))).
% 2.67/1.54 thf(fact_219_height__0__imp__cost__0, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A)]: (((((huffma1554076246height @ A) @ T) = (zero_zero @ nat)) => (((huffma636208924e_cost @ A) @ T) = (zero_zero @ nat)))))).
% 2.67/1.54 thf(fact_220_consistent__mergeSibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => ((huffma1050891809istent @ A) @ (((huffma1954420889ibling @ A) @ T) @ A2)))))).
% 2.67/1.54 thf(fact_221_cost__mergeSibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma943100115ibling @ A) @ T) @ A2) != A2) => ((((plus_plus @ nat) @ (((plus_plus @ nat) @ ((huffma636208924e_cost @ A) @ (((huffma1954420889ibling @ A) @ T) @ A2))) @ (((huffma854352999e_freq @ A) @ T) @ A2))) @ (((huffma854352999e_freq @ A) @ T) @ (((huffma943100115ibling @ A) @ T) @ A2))) = ((huffma636208924e_cost @ A) @ T))))))).
% 2.67/1.54 thf(fact_222_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X3:A]: ((? [X1:A]: ((((ord_less @ A) @ X3) @ X1))))))))).
% 2.67/1.54 thf(fact_223_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X3:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X3))))))))).
% 2.67/1.54 thf(fact_224_linordered__field__class_Osign__simps_I26_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_225_linordered__field__class_Osign__simps_I27_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => ((plus_plus @ A) = (^ [A4:A,B2:A]: ((((plus_plus @ A) @ B2) @ A4)))))))).
% 2.67/1.54 thf(fact_226_linordered__field__class_Osign__simps_I28_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: (((((plus_plus @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.67/1.54 thf(fact_227_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (? [X:A]: (((P @ X) & (! [Y:A]: (((P @ Y) => (((ord_less_eq @ nat) @ (M2 @ X)) @ (M2 @ Y)))))))))))).
% 2.67/1.54 thf(fact_228_cost__swapFourSyms__le, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,B:A,C:A,D:A]: ((((huffma1050891809istent @ A) @ T) => (((((huffma1154738298minima @ A) @ T) @ A2) @ B) => ((((member @ A) @ C) @ ((huffma505251170phabet @ A) @ T)) => ((((member @ A) @ D) @ ((huffma505251170phabet @ A) @ T)) => (((((huffma223349076_depth @ A) @ T) @ C) = ((huffma1554076246height @ A) @ T)) => (((((huffma223349076_depth @ A) @ T) @ D) = ((huffma1554076246height @ A) @ T)) => ((C != D) => (((ord_less_eq @ nat) @ ((huffma636208924e_cost @ A) @ ((((((huffma304375860urSyms @ A) @ T) @ A2) @ B) @ C) @ D))) @ ((huffma636208924e_cost @ A) @ T)))))))))))).
% 2.67/1.54 thf(fact_229_ex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),B:nat]: (((P @ K) => ((! [Y3:A]: (((P @ Y3) => (((ord_less @ nat) @ (M2 @ Y3)) @ B)))) => (? [X:A]: (((P @ X) & (! [Y:A]: (((P @ Y) => (((ord_less_eq @ nat) @ (M2 @ Y)) @ (M2 @ X))))))))))))).
% 2.67/1.54 thf(fact_230_weight__splitLeaf, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A,W_a:nat,W_b:nat,B:A]: ((((huffma1050891809istent @ A) @ T) => ((((member @ A) @ A2) @ ((huffma505251170phabet @ A) @ T)) => (((((huffma854352999e_freq @ A) @ T) @ A2) = (((plus_plus @ nat) @ W_a) @ W_b)) => (((huffma691733767weight @ A) @ ((((((huffma454997449itLeaf @ A) @ T) @ W_a) @ A2) @ W_b) @ B)) = ((huffma691733767weight @ A) @ T)))))))).
% 2.67/1.54 thf(fact_231_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 2.67/1.54 thf(fact_232_weight__mergeSibling, axiom, (! [A:$tType,T:(huffma16452318e_tree @ A),A2:A]: ((((huffma691733767weight @ A) @ (((huffma1954420889ibling @ A) @ T) @ A2)) = ((huffma691733767weight @ A) @ T))))).
% 2.67/1.54 thf(fact_233_le__funD, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G:(A > B3),X4:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X4)) @ (G @ X4))))))))).
% 2.67/1.54 thf(fact_234_le__funE, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G:(A > B3),X4:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X4)) @ (G @ X4))))))))).
% 2.67/1.54 thf(fact_235_le__funI, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G:(A > B3)]: (((! [X:A]: ((((ord_less_eq @ B3) @ (F @ X)) @ (G @ X)))) => (((ord_less_eq @ (A > B3)) @ F) @ G)))))))).
% 2.67/1.54 thf(fact_236_le__fun__def, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => ((ord_less_eq @ (A > B3)) = (^ [F2:(A > B3),G2:(A > B3)]: ((! [X2:A]: ((((ord_less_eq @ B3) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 2.67/1.54 thf(fact_237_order__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X:B3,Y3:B3]: (((((ord_less_eq @ B3) @ X) @ Y3) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.54 thf(fact_238_order__subst2, axiom, (! [A:$tType,C4:$tType]: (((((order @ C4) @ (type2 @ C4)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C4) @ (F @ B)) @ C) => ((! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ C4) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ C4) @ (F @ A2)) @ C)))))))))).
% 2.67/1.54 thf(fact_239_ord__eq__le__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((A2 = (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X:B3,Y3:B3]: (((((ord_less_eq @ B3) @ X) @ Y3) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.54 thf(fact_240_ord__le__eq__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > B3),C:B3]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ B3) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ B3) @ (F @ A2)) @ C)))))))))).
% 2.67/1.54 thf(fact_241_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X2:A,Y5:A]: (((((ord_less_eq @ A) @ X2) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X2))))))))).
% 2.67/1.54 thf(fact_242_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X4) => (X4 = Y2))))))))).
% 2.67/1.54 thf(fact_243_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X4)))))))).
% 2.67/1.54 thf(fact_244_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((X4 = Y2) => (((ord_less_eq @ A) @ X4) @ Y2)))))))).
% 2.67/1.54 thf(fact_245_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X4:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X4)))))))).
% 2.67/1.54 thf(fact_246_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.67/1.54 thf(fact_247_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X4:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X4) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z2)))) => (((((ord_less_eq @ A) @ X4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X4)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y2))))))))))))))))).
% 2.67/1.54 thf(fact_248_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y2:A,X4:A]: (((((ord_less_eq @ A) @ Y2) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y2) = (X4 = Y2))))))))).
% 2.67/1.54 thf(fact_249_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.67/1.54 thf(fact_250_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.67/1.54 thf(fact_251_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 2.67/1.54 thf(fact_252_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X4:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X4) @ Z2))))))))).
% 2.67/1.54 thf(fact_253_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.67/1.54 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.67/1.54 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.67/1.54 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, ((ordere516151231imp_le @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, axiom, ((strict2144017051up_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, ((ordere223160158up_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, ((ordere236663937imp_le @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add, axiom, ((strict797366125id_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add, axiom, ((ordere1818651114id_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, ((ordere779506340up_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, axiom, ((ordere216010020id_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, ((cancel1352612707id_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, ((cancel_semigroup_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, ((ab_semigroup_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, ((comm_monoid_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, ((semigroup_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Omonoid__add, axiom, ((monoid_add @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Nat_Onat___Groups_Ozero, axiom, ((zero @ nat) @ (type2 @ nat))).
% 2.67/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.67/1.54 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.67/1.54 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.67/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.67/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.67/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.67/1.54 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.67/1.54 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.67/1.54 thf(help_If_2_1_T, axiom, (! [A:$tType,X4:A,Y2:A]: ((((((if @ A) @ $false) @ X4) @ Y2) = Y2)))).
% 2.67/1.54 thf(help_If_1_1_T, axiom, (! [A:$tType,X4:A,Y2:A]: ((((((if @ A) @ $true) @ X4) @ Y2) = X4)))).
% 2.67/1.54 thf(conj_0, conjecture, ((huffma1050891809istent @ a) @ ((((((huffma304375860urSyms @ a) @ u) @ a2) @ b) @ c) @ (((huffma943100115ibling @ a) @ u) @ c)))).
% 2.67/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.UCTj6kXiKr/DTF2DTF13488.p
% 2.67/1.54 ------------------------
% 2.67/1.59 ---- Cleaned THF ---
% 2.67/1.59 thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree,type,
% 2.67/1.59 huffma16452318e_tree: $tType > $tType ).
% 2.67/1.59
% 2.67/1.59 thf(ty_t_Set_Oset,type,
% 2.67/1.59 set: $tType > $tType ).
% 2.67/1.59
% 2.67/1.59 thf(ty_t_Nat_Onat,type,
% 2.67/1.59 nat: $tType ).
% 2.67/1.59
% 2.67/1.59 thf(ty_t_itself,type,
% 2.67/1.59 itself: $tType > $tType ).
% 2.67/1.59
% 2.67/1.59 thf(ty_tf_a,type,
% 2.67/1.59 a: $tType ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_HOL_Otype,type,
% 2.67/1.59 type:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ozero,type,
% 2.67/1.59 zero:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Orderings_Oord,type,
% 2.67/1.59 ord:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Orderings_Oorder,type,
% 2.67/1.59 order:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ogroup__add,type,
% 2.67/1.59 group_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Omonoid__add,type,
% 2.67/1.59 monoid_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Orderings_Olinorder,type,
% 2.67/1.59 linorder:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Orderings_Opreorder,type,
% 2.67/1.59 preorder:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.67/1.59 semigroup_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.67/1.59 comm_monoid_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Fields_Olinordered__field,type,
% 2.67/1.59 linordered_field:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.67/1.59 ab_semigroup_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.67/1.59 cancel_semigroup_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.67/1.59 cancel1352612707id_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.67/1.59 linord219039673up_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 2.67/1.59 ordere216010020id_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.67/1.59 ordere779506340up_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.67/1.59 canoni770627133id_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 2.67/1.59 ordere1818651114id_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 2.67/1.59 strict797366125id_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.67/1.59 ordere236663937imp_le:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.67/1.59 ordere223160158up_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.67/1.59 strict2144017051up_add:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.67/1.59 ordere516151231imp_le:
% 2.67/1.59 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.67/1.59 plus_plus:
% 2.67/1.59 !>[A: $tType] : ( A > A > A ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.67/1.59 zero_zero:
% 2.67/1.59 !>[A: $tType] : A ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oalphabet,type,
% 2.67/1.59 huffma505251170phabet:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( set @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oconsistent,type,
% 2.67/1.59 huffma1050891809istent:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ocost,type,
% 2.67/1.59 huffma636208924e_cost:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Odepth,type,
% 2.67/1.59 huffma223349076_depth:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq,type,
% 2.67/1.59 huffma854352999e_freq:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight,type,
% 2.67/1.59 huffma1554076246height:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OmergeSibling,type,
% 2.67/1.59 huffma1954420889ibling:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > ( huffma16452318e_tree @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ominima,type,
% 2.67/1.59 huffma1154738298minima:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ooptimum,type,
% 2.67/1.59 huffma936049440ptimum:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Osibling,type,
% 2.67/1.59 huffma943100115ibling:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OsplitLeaf,type,
% 2.67/1.59 huffma454997449itLeaf:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapFourSyms,type,
% 2.67/1.59 huffma304375860urSyms:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > A > A > ( huffma16452318e_tree @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapLeaves,type,
% 2.67/1.59 huffma2094459102Leaves:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapSyms,type,
% 2.67/1.59 huffma469337550apSyms:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > ( huffma16452318e_tree @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oweight,type,
% 2.67/1.59 huffma691733767weight:
% 2.67/1.59 !>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_If,type,
% 2.67/1.59 if:
% 2.67/1.59 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.67/1.59 ord_less:
% 2.67/1.59 !>[A: $tType] : ( A > A > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.67/1.59 ord_less_eq:
% 2.67/1.59 !>[A: $tType] : ( A > A > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Pure_Otype,type,
% 2.67/1.59 type2:
% 2.67/1.59 !>[A: $tType] : ( itself @ A ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_Set_OCollect,type,
% 2.67/1.59 collect:
% 2.67/1.59 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_c_member,type,
% 2.67/1.59 member:
% 2.67/1.59 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_a,type,
% 2.67/1.59 a2: a ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_b,type,
% 2.67/1.59 b: a ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_c____,type,
% 2.67/1.59 c: a ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_t,type,
% 2.67/1.59 t: huffma16452318e_tree @ a ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_u____,type,
% 2.67/1.59 u: huffma16452318e_tree @ a ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_w_092_060_094sub_062a,type,
% 2.67/1.59 w_a: nat ).
% 2.67/1.59
% 2.67/1.59 thf(sy_v_w_092_060_094sub_062b,type,
% 2.67/1.59 w_b: nat ).
% 2.67/1.59
% 2.67/1.59 thf(fact_0_c_092_060_094sub_062u,axiom,
% 2.67/1.59 huffma1050891809istent @ a @ u ).
% 2.67/1.59
% 2.67/1.59 thf(fact_1_ab,axiom,
% 2.67/1.59 a2 != b ).
% 2.67/1.59
% 2.67/1.59 thf(fact_2_assms_I1_J,axiom,
% 2.67/1.59 huffma1050891809istent @ a @ t ).
% 2.67/1.59
% 2.67/1.59 thf(fact_3_assms_I2_J,axiom,
% 2.67/1.59 huffma936049440ptimum @ a @ t ).
% 2.67/1.59
% 2.67/1.59 thf(fact_4_dc,axiom,
% 2.67/1.59 ( ( huffma943100115ibling @ a @ u @ c )
% 2.67/1.59 != c ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_5_sibling__sibling__id,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_6_a_092_060_094sub_062c,axiom,
% 2.67/1.59 member @ a @ c @ ( huffma505251170phabet @ a @ u ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_7_sibling__reciprocal,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 = B )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ T @ B )
% 2.67/1.59 = A2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_8_consistent__swapFourSyms,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( huffma1050891809istent @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_9_a_092_060_094sub_062d,axiom,
% 2.67/1.59 member @ a @ ( huffma943100115ibling @ a @ u @ c ) @ ( huffma505251170phabet @ a @ u ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_10_a_092_060_094sub_062b,axiom,
% 2.67/1.59 member @ a @ b @ ( huffma505251170phabet @ a @ u ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_11_a_092_060_094sub_062a,axiom,
% 2.67/1.59 member @ a @ a2 @ ( huffma505251170phabet @ a @ u ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_12_d_092_060_094sub_062c,axiom,
% 2.67/1.59 ( ( huffma223349076_depth @ a @ u @ c )
% 2.67/1.59 = ( huffma1554076246height @ a @ u ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_13_sibling__swapSyms__sibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 2.67/1.59 != B )
% 2.67/1.59 => ( ( A2 != B )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ ( huffma943100115ibling @ A @ T @ B ) ) @ A2 )
% 2.67/1.59 = B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_14_sibling__swapSyms__other__sibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 2.67/1.59 != A2 )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 2.67/1.59 != B )
% 2.67/1.59 => ( ( A2 != B )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) @ ( huffma943100115ibling @ A @ T @ B ) )
% 2.67/1.59 = A2 ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_15_d_092_060_094sub_062d,axiom,
% 2.67/1.59 ( ( huffma223349076_depth @ a @ u @ ( huffma943100115ibling @ a @ u @ c ) )
% 2.67/1.59 = ( huffma1554076246height @ a @ u ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_16_sibling__swapLeaves__sibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A,W_a: nat,W_s: nat] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ B )
% 2.67/1.59 != B )
% 2.67/1.59 => ( ( A2 != B )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_s @ ( huffma943100115ibling @ A @ T @ B ) ) @ A2 )
% 2.67/1.59 = B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_17_depth__sibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( huffma223349076_depth @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) )
% 2.67/1.59 = ( huffma223349076_depth @ A @ T @ A2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_18_sibling__swapFourSyms__when__4th__is__sibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( A2 != B )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ C )
% 2.67/1.59 != C )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ ( huffma943100115ibling @ A @ T @ C ) ) @ A2 )
% 2.67/1.59 = B ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_19_assms_I4_J,axiom,
% 2.67/1.59 ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_20_assms_I3_J,axiom,
% 2.67/1.59 member @ a @ a2 @ ( huffma505251170phabet @ a @ t ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060lbrakk_062c_A_092_060in_062_Aalphabet_Au_059_Adepth_Au_Ac_A_061_Aheight_Au_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.67/1.59 ~ ! [C2: a] :
% 2.67/1.59 ( ( member @ a @ C2 @ ( huffma505251170phabet @ a @ u ) )
% 2.67/1.59 => ( ( huffma223349076_depth @ a @ u @ C2 )
% 2.67/1.59 != ( huffma1554076246height @ a @ u ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_22__092_060open_062_092_060exists_062a_092_060in_062alphabet_Au_O_Adepth_Au_Aa_A_061_Aheight_Au_092_060close_062,axiom,
% 2.67/1.59 ? [X: a] :
% 2.67/1.59 ( ( member @ a @ X @ ( huffma505251170phabet @ a @ u ) )
% 2.67/1.59 & ( ( huffma223349076_depth @ a @ u @ X )
% 2.67/1.59 = ( huffma1554076246height @ a @ u ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_23_notin__alphabet__imp__sibling__id,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_24_swapLeaves__id__when__notin__alphabet,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W: nat,W2: nat] :
% 2.67/1.59 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma2094459102Leaves @ A @ T @ W @ A2 @ W2 @ A2 )
% 2.67/1.59 = T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_25_alphabet__swapSyms,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,B: A] :
% 2.67/1.59 ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma505251170phabet @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) )
% 2.67/1.59 = ( huffma505251170phabet @ A @ T ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_26_height__swapLeaves,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
% 2.67/1.59 ( ( huffma1554076246height @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 2.67/1.59 = ( huffma1554076246height @ A @ T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_27_swapSyms__id,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( huffma469337550apSyms @ A @ T @ A2 @ A2 )
% 2.67/1.59 = T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_28_alphabet__swapFourSyms,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma505251170phabet @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) )
% 2.67/1.59 = ( huffma505251170phabet @ A @ T ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_29_depth__swapLeaves__neither,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B: A,W_a: nat,W_b: nat] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( C != A2 )
% 2.67/1.59 => ( ( C != B )
% 2.67/1.59 => ( ( huffma223349076_depth @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) @ C )
% 2.67/1.59 = ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_30_depth__swapSyms__neither,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( C != A2 )
% 2.67/1.59 => ( ( C != B )
% 2.67/1.59 => ( ( huffma223349076_depth @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) @ C )
% 2.67/1.59 = ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_31_exists__at__height,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ? [X: A] :
% 2.67/1.59 ( ( member @ A @ X @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 & ( ( huffma223349076_depth @ A @ T @ X )
% 2.67/1.59 = ( huffma1554076246height @ A @ T ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_32_exists__in__alphabet,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ? [A3: A] : ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_33_in__alphabet__imp__sibling__in__alphabet,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( member @ A @ ( huffma943100115ibling @ A @ T @ A2 ) @ ( huffma505251170phabet @ A @ T ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_34_sibling__ne__imp__sibling__in__alphabet,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 != A2 )
% 2.67/1.59 => ( member @ A @ ( huffma943100115ibling @ A @ T @ A2 ) @ ( huffma505251170phabet @ A @ T ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_35_consistent__swapLeaves,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( huffma1050891809istent @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_36_consistent__swapSyms,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( huffma1050891809istent @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_37_swapFourSyms__def,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( huffma304375860urSyms @ A )
% 2.67/1.59 = ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A,C3: A,D2: A] : ( if @ ( huffma16452318e_tree @ A ) @ ( A4 = D2 ) @ ( huffma469337550apSyms @ A @ T2 @ B2 @ C3 ) @ ( if @ ( huffma16452318e_tree @ A ) @ ( B2 = C3 ) @ ( huffma469337550apSyms @ A @ T2 @ A4 @ D2 ) @ ( huffma469337550apSyms @ A @ ( huffma469337550apSyms @ A @ T2 @ A4 @ C3 ) @ B2 @ D2 ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_38_a_092_060_094sub_062u,axiom,
% 2.67/1.59 ( ( huffma505251170phabet @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
% 2.67/1.59 = ( huffma505251170phabet @ a @ u ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_39_freq__swapFourSyms,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma854352999e_freq @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) )
% 2.67/1.59 = ( huffma854352999e_freq @ A @ T ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_40_False,axiom,
% 2.67/1.59 ( ( huffma1554076246height @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
% 2.67/1.59 != ( zero_zero @ nat ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_41_f_092_060_094sub_062u,axiom,
% 2.67/1.59 ( ( huffma854352999e_freq @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
% 2.67/1.59 = ( huffma854352999e_freq @ a @ u ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_42_freq__swapSyms,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma854352999e_freq @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) )
% 2.67/1.59 = ( huffma854352999e_freq @ A @ T ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_43_consistent__splitLeaf,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,B: A,W_a: nat,A2: A,W_b: nat] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( huffma1050891809istent @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_44_height__0__imp__sibling__id,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( ( huffma1554076246height @ A @ T )
% 2.67/1.59 = ( zero_zero @ nat ) )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,P: A > $o] :
% 2.67/1.59 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.67/1.59 = ( P @ A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.67/1.59 ! [A: $tType,A5: set @ A] :
% 2.67/1.59 ( ( collect @ A
% 2.67/1.59 @ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
% 2.67/1.59 = A5 ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_47_Collect__cong,axiom,
% 2.67/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.67/1.59 ( ! [X: A] :
% 2.67/1.59 ( ( P @ X )
% 2.67/1.59 = ( Q @ X ) )
% 2.67/1.59 => ( ( collect @ A @ P )
% 2.67/1.59 = ( collect @ A @ Q ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_48_ext,axiom,
% 2.67/1.59 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 2.67/1.59 ( ! [X: A] :
% 2.67/1.59 ( ( F @ X )
% 2.67/1.59 = ( G @ X ) )
% 2.67/1.59 => ( F = G ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_49_notin__alphabet__imp__splitLeaf__id,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W_a: nat,W_b: nat,B: A] :
% 2.67/1.59 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B )
% 2.67/1.59 = T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_50_swapLeaves__id,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( huffma2094459102Leaves @ A @ T @ ( huffma854352999e_freq @ A @ T @ A2 ) @ A2 @ ( huffma854352999e_freq @ A @ T @ A2 ) @ A2 )
% 2.67/1.59 = T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_51_depth__height__imp__sibling__ne,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma223349076_depth @ A @ T @ A2 )
% 2.67/1.59 = ( huffma1554076246height @ A @ T ) )
% 2.67/1.59 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 != A2 ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_52_h_092_060_094sub_062u,axiom,
% 2.67/1.59 ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ a @ u ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_53_notin__alphabet__imp__freq__0,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma854352999e_freq @ A @ T @ A2 )
% 2.67/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_54_assms_I7_J,axiom,
% 2.67/1.59 ord_less_eq @ nat @ w_a @ w_b ).
% 2.67/1.59
% 2.67/1.59 thf(fact_55_freq__swapLeaves,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,W_a: nat,W_b: nat] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( A2 != B )
% 2.67/1.59 => ( ( huffma854352999e_freq @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 2.67/1.59 = ( ^ [C3: A] : ( if @ nat @ ( C3 = A2 ) @ ( if @ nat @ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) ) @ W_a @ ( zero_zero @ nat ) ) @ ( if @ nat @ ( C3 = B ) @ ( if @ nat @ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) ) @ W_b @ ( zero_zero @ nat ) ) @ ( huffma854352999e_freq @ A @ T @ C3 ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_56_assms_I6_J,axiom,
% 2.67/1.59 ! [X3: a] :
% 2.67/1.59 ( ( member @ a @ X3 @ ( huffma505251170phabet @ a @ t ) )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ w_a @ ( huffma854352999e_freq @ a @ t @ X3 ) )
% 2.67/1.59 & ( ord_less_eq @ nat @ w_b @ ( huffma854352999e_freq @ a @ t @ X3 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_57_assms_I5_J,axiom,
% 2.67/1.59 ( ( huffma854352999e_freq @ a @ t @ a2 )
% 2.67/1.59 = ( plus_plus @ nat @ w_a @ w_b ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_58_height__gt__0__alphabet__eq__imp__height__gt__0,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,U: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T ) )
% 2.67/1.59 => ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma505251170phabet @ A @ T )
% 2.67/1.59 = ( huffma505251170phabet @ A @ U ) )
% 2.67/1.59 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ U ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_59_swapSyms__def,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( huffma469337550apSyms @ A )
% 2.67/1.59 = ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A] : ( huffma2094459102Leaves @ A @ T2 @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ A4 @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ B2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_60_neq0__conv,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ( ( N
% 2.67/1.59 != ( zero_zero @ nat ) )
% 2.67/1.59 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_61_not__gr__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [N: A] :
% 2.67/1.59 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.67/1.59 = ( N
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_62_infinite__descent0__measure,axiom,
% 2.67/1.59 ! [A: $tType,V: A > nat,P: A > $o,X4: A] :
% 2.67/1.59 ( ! [X: A] :
% 2.67/1.59 ( ( ( V @ X )
% 2.67/1.59 = ( zero_zero @ nat ) )
% 2.67/1.59 => ( P @ X ) )
% 2.67/1.59 => ( ! [X: A] :
% 2.67/1.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
% 2.67/1.59 => ( ~ ( P @ X )
% 2.67/1.59 => ? [Y: A] :
% 2.67/1.59 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X ) )
% 2.67/1.59 & ~ ( P @ Y ) ) ) )
% 2.67/1.59 => ( P @ X4 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_63_less__nat__zero__code,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_64_infinite__descent0,axiom,
% 2.67/1.59 ! [P: nat > $o,N: nat] :
% 2.67/1.59 ( ( P @ ( zero_zero @ nat ) )
% 2.67/1.59 => ( ! [N2: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.67/1.59 => ( ~ ( P @ N2 )
% 2.67/1.59 => ? [M: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M @ N2 )
% 2.67/1.59 & ~ ( P @ M ) ) ) )
% 2.67/1.59 => ( P @ N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_65_gr__implies__not0,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M2 @ N )
% 2.67/1.59 => ( N
% 2.67/1.59 != ( zero_zero @ nat ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_66_less__zeroE,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_67_add__left__cancel,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ A2 @ B )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ C ) )
% 2.67/1.59 = ( B = C ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_68_add__right__cancel,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A,C: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ B @ A2 )
% 2.67/1.59 = ( plus_plus @ A @ C @ A2 ) )
% 2.67/1.59 = ( B = C ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_69_le__zero__eq,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [N: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.67/1.59 = ( N
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_70_add__le__cancel__left,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [C: A,A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.67/1.59 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_71_add__le__cancel__right,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,C: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.67/1.59 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_72_add__cancel__right__right,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( A2
% 2.67/1.59 = ( plus_plus @ A @ A2 @ B ) )
% 2.67/1.59 = ( B
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_73_add__cancel__right__left,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( A2
% 2.67/1.59 = ( plus_plus @ A @ B @ A2 ) )
% 2.67/1.59 = ( B
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_74_add__cancel__left__right,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ A2 @ B )
% 2.67/1.59 = A2 )
% 2.67/1.59 = ( B
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_75_add__cancel__left__left,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ B @ A2 )
% 2.67/1.59 = A2 )
% 2.67/1.59 = ( B
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_76_double__zero__sym,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( ( zero_zero @ A )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ A2 ) )
% 2.67/1.59 = ( A2
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_77_double__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ A2 @ A2 )
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 = ( A2
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_78_add_Oright__neutral,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( monoid_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_79_add_Oleft__neutral,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( monoid_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_80_add__less__cancel__right,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,C: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.67/1.59 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_81_add__less__cancel__left,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [C: A,A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.67/1.59 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_82_Nat_Oadd__0__right,axiom,
% 2.67/1.59 ! [M2: nat] :
% 2.67/1.59 ( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
% 2.67/1.59 = M2 ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_83_add__is__0,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ( plus_plus @ nat @ M2 @ N )
% 2.67/1.59 = ( zero_zero @ nat ) )
% 2.67/1.59 = ( ( M2
% 2.67/1.59 = ( zero_zero @ nat ) )
% 2.67/1.59 & ( N
% 2.67/1.59 = ( zero_zero @ nat ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_84_le0,axiom,
% 2.67/1.59 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_85_nat__add__left__cancel__less,axiom,
% 2.67/1.59 ! [K: nat,M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 2.67/1.59 = ( ord_less @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_86_nat__add__left__cancel__le,axiom,
% 2.67/1.59 ! [K: nat,M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 2.67/1.59 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_87_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.67/1.59 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_88_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.67/1.59 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_89_le__add__same__cancel2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.67/1.59 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_90_le__add__same__cancel1,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.67/1.59 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_91_add__le__same__cancel2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.67/1.59 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_92_add__le__same__cancel1,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.67/1.59 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_93_add__less__same__cancel1,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.67/1.59 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_94_add__less__same__cancel2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.67/1.59 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_95_less__add__same__cancel1,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.67/1.59 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_96_less__add__same__cancel2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.67/1.59 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_97_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.67/1.59 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_98_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.67/1.59 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_99_add__gr__0,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
% 2.67/1.59 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 2.67/1.59 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_100_add__less__imp__less__right,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,C: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.67/1.59 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_101_add__less__imp__less__left,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [C: A,A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.67/1.59 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_102_add__strict__increasing2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere1818651114id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less @ A @ B @ C )
% 2.67/1.59 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_103_add__strict__right__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ B )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_104_add__eq__0__iff__both__eq__0,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ X4 @ Y2 )
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 = ( ( X4
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 & ( Y2
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_105_add__strict__increasing,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere1818651114id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ C )
% 2.67/1.59 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_106_add__strict__left__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ B )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_107_add__less__le__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less_eq @ A @ C @ D )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_108_add__le__less__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less @ A @ C @ D )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_109_add__strict__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( strict2144017051up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less @ A @ C @ D )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_110_add__nonpos__eq__0__iff,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ( plus_plus @ A @ X4 @ Y2 )
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 = ( ( X4
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 & ( Y2
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_111_add__nonneg__eq__0__iff,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
% 2.67/1.59 => ( ( ( plus_plus @ A @ X4 @ Y2 )
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 = ( ( X4
% 2.67/1.59 = ( zero_zero @ A ) )
% 2.67/1.59 & ( Y2
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_112_add__nonpos__nonpos,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_113_add__nonneg__nonneg,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.67/1.59 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_114_add__increasing2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [C: A,B: A,A2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.67/1.59 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_115_add__decreasing2,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [C: A,A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_116_add__pos__nonneg,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.67/1.59 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_117_add__nonpos__neg,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_118_add__nonneg__pos,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.67/1.59 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_119_add__neg__nonpos,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_120_add__increasing,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ C )
% 2.67/1.59 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_121_add__decreasing,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,C: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less_eq @ A @ C @ B )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_122_zero__le,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_123_add_Ogroup__left__neutral,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( group_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_124_add_Ocomm__neutral,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_125_comm__monoid__add__class_Oadd__0,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] :
% 2.67/1.59 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 = A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_126_less__mono__imp__le__mono,axiom,
% 2.67/1.59 ! [F: nat > nat,I: nat,J: nat] :
% 2.67/1.59 ( ! [I2: nat,J2: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I2 @ J2 )
% 2.67/1.59 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ I @ J )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_127_le__neq__implies__less,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.67/1.59 => ( ( M2 != N )
% 2.67/1.59 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_128_mono__nat__linear__lb,axiom,
% 2.67/1.59 ! [F: nat > nat,M2: nat,K: nat] :
% 2.67/1.59 ( ! [M3: nat,N2: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M3 @ N2 )
% 2.67/1.59 => ( ord_less @ nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_129_less__or__eq__imp__le,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ( ord_less @ nat @ M2 @ N )
% 2.67/1.59 | ( M2 = N ) )
% 2.67/1.59 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_130_less__add__eq__less,axiom,
% 2.67/1.59 ! [K: nat,L: nat,M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ K @ L )
% 2.67/1.59 => ( ( ( plus_plus @ nat @ M2 @ L )
% 2.67/1.59 = ( plus_plus @ nat @ K @ N ) )
% 2.67/1.59 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_131_le__eq__less__or__eq,axiom,
% 2.67/1.59 ( ( ord_less_eq @ nat )
% 2.67/1.59 = ( ^ [M4: nat,N3: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M4 @ N3 )
% 2.67/1.59 | ( M4 = N3 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_132_add__eq__self__zero,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ( plus_plus @ nat @ M2 @ N )
% 2.67/1.59 = M2 )
% 2.67/1.59 => ( N
% 2.67/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_133_trans__less__add2,axiom,
% 2.67/1.59 ! [I: nat,J: nat,M2: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I @ J )
% 2.67/1.59 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_134_trans__less__add1,axiom,
% 2.67/1.59 ! [I: nat,J: nat,M2: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I @ J )
% 2.67/1.59 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_135_less__imp__le__nat,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M2 @ N )
% 2.67/1.59 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_136_plus__nat_Oadd__0,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 2.67/1.59 = N ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_137_add__less__mono1,axiom,
% 2.67/1.59 ! [I: nat,J: nat,K: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I @ J )
% 2.67/1.59 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_138_not__add__less2,axiom,
% 2.67/1.59 ! [J: nat,I: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_139_not__add__less1,axiom,
% 2.67/1.59 ! [I: nat,J: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_140_add__less__mono,axiom,
% 2.67/1.59 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I @ J )
% 2.67/1.59 => ( ( ord_less @ nat @ K @ L )
% 2.67/1.59 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_141_nat__less__le,axiom,
% 2.67/1.59 ( ( ord_less @ nat )
% 2.67/1.59 = ( ^ [M4: nat,N3: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 2.67/1.59 & ( M4 != N3 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_142_add__lessD1,axiom,
% 2.67/1.59 ! [I: nat,J: nat,K: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 2.67/1.59 => ( ord_less @ nat @ I @ K ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_143_le__0__eq,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.67/1.59 = ( N
% 2.67/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_144_add__mono__thms__linordered__field_I1_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( ord_less @ A @ I @ J )
% 2.67/1.59 & ( K = L ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_145_add__mono__thms__linordered__field_I2_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( I = J )
% 2.67/1.59 & ( ord_less @ A @ K @ L ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_146_add__mono__thms__linordered__field_I3_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( ord_less @ A @ I @ J )
% 2.67/1.59 & ( ord_less_eq @ A @ K @ L ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_147_add__mono__thms__linordered__field_I4_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( ord_less_eq @ A @ I @ J )
% 2.67/1.59 & ( ord_less @ A @ K @ L ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_148_add__mono__thms__linordered__field_I5_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( ord_less @ A @ I @ J )
% 2.67/1.59 & ( ord_less @ A @ K @ L ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_149_less__eq__nat_Osimps_I1_J,axiom,
% 2.67/1.59 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_150_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_151_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( I = J )
% 2.67/1.59 & ( K = L ) )
% 2.67/1.59 => ( ( plus_plus @ A @ I @ K )
% 2.67/1.59 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_152_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( ord_less_eq @ A @ I @ J )
% 2.67/1.59 & ( K = L ) )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_153_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( I = J )
% 2.67/1.59 & ( ord_less_eq @ A @ K @ L ) )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_154_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.67/1.59 ( ( ( ord_less_eq @ A @ I @ J )
% 2.67/1.59 & ( ord_less_eq @ A @ K @ L ) )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_155_add__leE,axiom,
% 2.67/1.59 ! [M2: nat,K: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 2.67/1.59 => ~ ( ( ord_less_eq @ nat @ M2 @ N )
% 2.67/1.59 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_156_le__add1,axiom,
% 2.67/1.59 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_157_le__add2,axiom,
% 2.67/1.59 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_158_le__refl,axiom,
% 2.67/1.59 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_159_add__leD1,axiom,
% 2.67/1.59 ! [M2: nat,K: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 2.67/1.59 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_160_add__leD2,axiom,
% 2.67/1.59 ! [M2: nat,K: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 2.67/1.59 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_161_le__trans,axiom,
% 2.67/1.59 ! [I: nat,J: nat,K: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ J @ K )
% 2.67/1.59 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_162_eq__imp__le,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( M2 = N )
% 2.67/1.59 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_163_le__Suc__ex,axiom,
% 2.67/1.59 ! [K: nat,L: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ K @ L )
% 2.67/1.59 => ? [N2: nat] :
% 2.67/1.59 ( L
% 2.67/1.59 = ( plus_plus @ nat @ K @ N2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_164_le__antisym,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ N @ M2 )
% 2.67/1.59 => ( M2 = N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_165_add__le__mono,axiom,
% 2.67/1.59 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ K @ L )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_166_add__le__mono1,axiom,
% 2.67/1.59 ! [I: nat,J: nat,K: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_167_nat__le__linear,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ M2 @ N )
% 2.67/1.59 | ( ord_less_eq @ nat @ N @ M2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_168_trans__le__add1,axiom,
% 2.67/1.59 ! [I: nat,J: nat,M2: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.67/1.59 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_169_trans__le__add2,axiom,
% 2.67/1.59 ! [I: nat,J: nat,M2: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.67/1.59 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_170_nat__le__iff__add,axiom,
% 2.67/1.59 ( ( ord_less_eq @ nat )
% 2.67/1.59 = ( ^ [M4: nat,N3: nat] :
% 2.67/1.59 ? [K2: nat] :
% 2.67/1.59 ( N3
% 2.67/1.59 = ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_171_nat__add__left__cancel,axiom,
% 2.67/1.59 ! [K: nat,M2: nat,N: nat] :
% 2.67/1.59 ( ( ( plus_plus @ nat @ K @ M2 )
% 2.67/1.59 = ( plus_plus @ nat @ K @ N ) )
% 2.67/1.59 = ( M2 = N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_172_nat__add__right__cancel,axiom,
% 2.67/1.59 ! [M2: nat,K: nat,N: nat] :
% 2.67/1.59 ( ( ( plus_plus @ nat @ M2 @ K )
% 2.67/1.59 = ( plus_plus @ nat @ N @ K ) )
% 2.67/1.59 = ( M2 = N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_173_add_Oassoc,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_174_add_Oleft__cancel,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( group_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ A2 @ B )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ C ) )
% 2.67/1.59 = ( B = C ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_175_add_Oright__cancel,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( group_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A,C: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ B @ A2 )
% 2.67/1.59 = ( plus_plus @ A @ C @ A2 ) )
% 2.67/1.59 = ( B = C ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_176_add_Ocommute,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ( ( plus_plus @ A )
% 2.67/1.59 = ( ^ [A4: A,B2: A] : ( plus_plus @ A @ B2 @ A4 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_177_add_Oleft__commute,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A,C: A] :
% 2.67/1.59 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_178_add__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less_eq @ A @ C @ D )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_179_add__left__imp__eq,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ A2 @ B )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ C ) )
% 2.67/1.59 => ( B = C ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_180_add__right__imp__eq,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A,C: A] :
% 2.67/1.59 ( ( ( plus_plus @ A @ B @ A2 )
% 2.67/1.59 = ( plus_plus @ A @ C @ A2 ) )
% 2.67/1.59 => ( B = C ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_181_add__left__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_182_add__right__mono,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_183_le__iff__add,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ( ( ord_less_eq @ A )
% 2.67/1.59 = ( ^ [A4: A,B2: A] :
% 2.67/1.59 ? [C3: A] :
% 2.67/1.59 ( B2
% 2.67/1.59 = ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_184_add__le__imp__le__left,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [C: A,A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_185_add__le__imp__le__right,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,C: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_186_add__neg__neg,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 2.67/1.59 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_187_add__pos__pos,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.67/1.59 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_188_pos__add__strict,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( strict797366125id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.67/1.59 => ( ( ord_less @ A @ B @ C )
% 2.67/1.59 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_189_less__imp__add__positive,axiom,
% 2.67/1.59 ! [I: nat,J: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I @ J )
% 2.67/1.59 => ? [K3: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
% 2.67/1.59 & ( ( plus_plus @ nat @ I @ K3 )
% 2.67/1.59 = J ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_190_ex__least__nat__le,axiom,
% 2.67/1.59 ! [P: nat > $o,N: nat] :
% 2.67/1.59 ( ~ ( P @ ( zero_zero @ nat ) )
% 2.67/1.59 => ( ( P @ N )
% 2.67/1.59 => ? [K3: nat] :
% 2.67/1.59 ( ( ord_less_eq @ nat @ K3 @ N )
% 2.67/1.59 & ! [I3: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ I3 @ K3 )
% 2.67/1.59 => ~ ( P @ I3 ) )
% 2.67/1.59 & ( P @ K3 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_191_depth__le__height,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] : ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma1554076246height @ A @ T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_192_zero__reorient,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( zero @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A] :
% 2.67/1.59 ( ( ( zero_zero @ A )
% 2.67/1.59 = X4 )
% 2.67/1.59 = ( X4
% 2.67/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_193_nat__neq__iff,axiom,
% 2.67/1.59 ! [M2: nat,N: nat] :
% 2.67/1.59 ( ( M2 != N )
% 2.67/1.59 = ( ( ord_less @ nat @ M2 @ N )
% 2.67/1.59 | ( ord_less @ nat @ N @ M2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_194_less__not__refl,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ N @ N ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_195_less__not__refl2,axiom,
% 2.67/1.59 ! [N: nat,M2: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ N @ M2 )
% 2.67/1.59 => ( M2 != N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_196_less__not__refl3,axiom,
% 2.67/1.59 ! [S: nat,T: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ S @ T )
% 2.67/1.59 => ( S != T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_197_measure__induct,axiom,
% 2.67/1.59 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.67/1.59 ( ! [X: A] :
% 2.67/1.59 ( ! [Y: A] :
% 2.67/1.59 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X ) )
% 2.67/1.59 => ( P @ Y ) )
% 2.67/1.59 => ( P @ X ) )
% 2.67/1.59 => ( P @ A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_198_less__irrefl__nat,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ N @ N ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_199_nat__less__induct,axiom,
% 2.67/1.59 ! [P: nat > $o,N: nat] :
% 2.67/1.59 ( ! [N2: nat] :
% 2.67/1.59 ( ! [M: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M @ N2 )
% 2.67/1.59 => ( P @ M ) )
% 2.67/1.59 => ( P @ N2 ) )
% 2.67/1.59 => ( P @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_200_infinite__descent,axiom,
% 2.67/1.59 ! [P: nat > $o,N: nat] :
% 2.67/1.59 ( ! [N2: nat] :
% 2.67/1.59 ( ~ ( P @ N2 )
% 2.67/1.59 => ? [M: nat] :
% 2.67/1.59 ( ( ord_less @ nat @ M @ N2 )
% 2.67/1.59 & ~ ( P @ M ) ) )
% 2.67/1.59 => ( P @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_201_linorder__neqE__nat,axiom,
% 2.67/1.59 ! [X4: nat,Y2: nat] :
% 2.67/1.59 ( ( X4 != Y2 )
% 2.67/1.59 => ( ~ ( ord_less @ nat @ X4 @ Y2 )
% 2.67/1.59 => ( ord_less @ nat @ Y2 @ X4 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_202_measure__induct__rule,axiom,
% 2.67/1.59 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.67/1.59 ( ! [X: A] :
% 2.67/1.59 ( ! [Y: A] :
% 2.67/1.59 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X ) )
% 2.67/1.59 => ( P @ Y ) )
% 2.67/1.59 => ( P @ X ) )
% 2.67/1.59 => ( P @ A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_203_infinite__descent__measure,axiom,
% 2.67/1.59 ! [A: $tType,P: A > $o,V: A > nat,X4: A] :
% 2.67/1.59 ( ! [X: A] :
% 2.67/1.59 ( ~ ( P @ X )
% 2.67/1.59 => ? [Y: A] :
% 2.67/1.59 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X ) )
% 2.67/1.59 & ~ ( P @ Y ) ) )
% 2.67/1.59 => ( P @ X4 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_204_gr__zeroI,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [N: A] :
% 2.67/1.59 ( ( N
% 2.67/1.59 != ( zero_zero @ A ) )
% 2.67/1.59 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_205_not__less__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [N: A] :
% 2.67/1.59 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_206_gr__implies__not__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [M2: A,N: A] :
% 2.67/1.59 ( ( ord_less @ A @ M2 @ N )
% 2.67/1.59 => ( N
% 2.67/1.59 != ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_207_zero__less__iff__neq__zero,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [N: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.67/1.59 = ( N
% 2.67/1.59 != ( zero_zero @ A ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_208_gr0I,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ( ( N
% 2.67/1.59 != ( zero_zero @ nat ) )
% 2.67/1.59 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_209_not__gr0,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.67/1.59 = ( N
% 2.67/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_210_not__less0,axiom,
% 2.67/1.59 ! [N: nat] :
% 2.67/1.59 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_211_field__le__epsilon,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ! [E: A] :
% 2.67/1.59 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 2.67/1.59 => ( ord_less_eq @ A @ X4 @ ( plus_plus @ A @ Y2 @ E ) ) )
% 2.67/1.59 => ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_212_ex__has__greatest__nat__lemma,axiom,
% 2.67/1.59 ! [A: $tType,P: A > $o,K: A,M2: A > nat,N: nat] :
% 2.67/1.59 ( ( P @ K )
% 2.67/1.59 => ( ! [X: A] :
% 2.67/1.59 ( ( P @ X )
% 2.67/1.59 => ? [Y: A] :
% 2.67/1.59 ( ( P @ Y )
% 2.67/1.59 & ~ ( ord_less_eq @ nat @ ( M2 @ Y ) @ ( M2 @ X ) ) ) )
% 2.67/1.59 => ? [Y3: A] :
% 2.67/1.59 ( ( P @ Y3 )
% 2.67/1.59 & ~ ( ord_less @ nat @ ( M2 @ Y3 ) @ ( plus_plus @ nat @ ( M2 @ K ) @ N ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_213_minima__def,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( huffma1154738298minima @ A )
% 2.67/1.59 = ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A] :
% 2.67/1.59 ( ( member @ A @ A4 @ ( huffma505251170phabet @ A @ T2 ) )
% 2.67/1.59 & ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T2 ) )
% 2.67/1.59 & ( A4 != B2 )
% 2.67/1.59 & ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ ( huffma854352999e_freq @ A @ T2 @ B2 ) )
% 2.67/1.59 & ! [X2: A] :
% 2.67/1.59 ( ( member @ A @ X2 @ ( huffma505251170phabet @ A @ T2 ) )
% 2.67/1.59 => ( ( X2 != A4 )
% 2.67/1.59 => ( ( X2 != B2 )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ ( huffma854352999e_freq @ A @ T2 @ X2 ) )
% 2.67/1.59 & ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ ( huffma854352999e_freq @ A @ T2 @ X2 ) ) ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_214_cost__splitLeaf,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,W_a: nat,W_b: nat,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( ( huffma854352999e_freq @ A @ T @ A2 )
% 2.67/1.59 = ( plus_plus @ nat @ W_a @ W_b ) )
% 2.67/1.59 => ( ( huffma636208924e_cost @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 2.67/1.59 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ W_a ) @ W_b ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_215_cost__swapSyms__le,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma854352999e_freq @ A @ T @ B ) )
% 2.67/1.59 => ( ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ B ) )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) @ ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_216_freq__mergeSibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 != A2 )
% 2.67/1.59 => ( ( huffma854352999e_freq @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) )
% 2.67/1.59 = ( ^ [C3: A] :
% 2.67/1.59 ( if @ nat @ ( C3 = A2 ) @ ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma854352999e_freq @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) ) )
% 2.67/1.59 @ ( if @ nat
% 2.67/1.59 @ ( C3
% 2.67/1.59 = ( huffma943100115ibling @ A @ T @ A2 ) )
% 2.67/1.59 @ ( zero_zero @ nat )
% 2.67/1.59 @ ( huffma854352999e_freq @ A @ T @ C3 ) ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_217_optimum__def,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( huffma936049440ptimum @ A )
% 2.67/1.59 = ( ^ [T2: huffma16452318e_tree @ A] :
% 2.67/1.59 ! [U2: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ U2 )
% 2.67/1.59 => ( ( ( huffma505251170phabet @ A @ T2 )
% 2.67/1.59 = ( huffma505251170phabet @ A @ U2 ) )
% 2.67/1.59 => ( ( ( huffma854352999e_freq @ A @ T2 )
% 2.67/1.59 = ( huffma854352999e_freq @ A @ U2 ) )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ T2 ) @ ( huffma636208924e_cost @ A @ U2 ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_218_notin__alphabet__imp__mergeSibling__id,axiom,
% 2.67/1.59 ! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( huffma1954420889ibling @ A @ T @ A2 )
% 2.67/1.59 = T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_219_height__0__imp__cost__0,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A] :
% 2.67/1.59 ( ( ( huffma1554076246height @ A @ T )
% 2.67/1.59 = ( zero_zero @ nat ) )
% 2.67/1.59 => ( ( huffma636208924e_cost @ A @ T )
% 2.67/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_220_consistent__mergeSibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( huffma1050891809istent @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_221_cost__mergeSibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( ( huffma943100115ibling @ A @ T @ A2 )
% 2.67/1.59 != A2 )
% 2.67/1.59 => ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) ) )
% 2.67/1.59 = ( huffma636208924e_cost @ A @ T ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_222_linordered__field__no__ub,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X3: A] :
% 2.67/1.59 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_223_linordered__field__no__lb,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X3: A] :
% 2.67/1.59 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_224_linordered__field__class_Osign__simps_I26_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_225_linordered__field__class_Osign__simps_I27_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.67/1.59 => ( ( plus_plus @ A )
% 2.67/1.59 = ( ^ [A4: A,B2: A] : ( plus_plus @ A @ B2 @ A4 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_226_linordered__field__class_Osign__simps_I28_J,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [B: A,A2: A,C: A] :
% 2.67/1.59 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 2.67/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_227_ex__has__least__nat,axiom,
% 2.67/1.59 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 2.67/1.59 ( ( P @ K )
% 2.67/1.59 => ? [X: A] :
% 2.67/1.59 ( ( P @ X )
% 2.67/1.59 & ! [Y: A] :
% 2.67/1.59 ( ( P @ Y )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ Y ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_228_cost__swapFourSyms__le,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( huffma1154738298minima @ A @ T @ A2 @ B )
% 2.67/1.59 => ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( ( huffma223349076_depth @ A @ T @ C )
% 2.67/1.59 = ( huffma1554076246height @ A @ T ) )
% 2.67/1.59 => ( ( ( huffma223349076_depth @ A @ T @ D )
% 2.67/1.59 = ( huffma1554076246height @ A @ T ) )
% 2.67/1.59 => ( ( C != D )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) ) @ ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_229_ex__has__greatest__nat,axiom,
% 2.67/1.59 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B: nat] :
% 2.67/1.59 ( ( P @ K )
% 2.67/1.59 => ( ! [Y3: A] :
% 2.67/1.59 ( ( P @ Y3 )
% 2.67/1.59 => ( ord_less @ nat @ ( M2 @ Y3 ) @ B ) )
% 2.67/1.59 => ? [X: A] :
% 2.67/1.59 ( ( P @ X )
% 2.67/1.59 & ! [Y: A] :
% 2.67/1.59 ( ( P @ Y )
% 2.67/1.59 => ( ord_less_eq @ nat @ ( M2 @ Y ) @ ( M2 @ X ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_230_weight__splitLeaf,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A,W_a: nat,W_b: nat,B: A] :
% 2.67/1.59 ( ( huffma1050891809istent @ A @ T )
% 2.67/1.59 => ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
% 2.67/1.59 => ( ( ( huffma854352999e_freq @ A @ T @ A2 )
% 2.67/1.59 = ( plus_plus @ nat @ W_a @ W_b ) )
% 2.67/1.59 => ( ( huffma691733767weight @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
% 2.67/1.59 = ( huffma691733767weight @ A @ T ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_231_order__refl,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( preorder @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_232_weight__mergeSibling,axiom,
% 2.67/1.59 ! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
% 2.67/1.59 ( ( huffma691733767weight @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) )
% 2.67/1.59 = ( huffma691733767weight @ A @ T ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_233_le__funD,axiom,
% 2.67/1.59 ! [B3: $tType,A: $tType] :
% 2.67/1.59 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 => ! [F: A > B3,G: A > B3,X4: A] :
% 2.67/1.59 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 2.67/1.59 => ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_234_le__funE,axiom,
% 2.67/1.59 ! [B3: $tType,A: $tType] :
% 2.67/1.59 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 => ! [F: A > B3,G: A > B3,X4: A] :
% 2.67/1.59 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 2.67/1.59 => ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_235_le__funI,axiom,
% 2.67/1.59 ! [B3: $tType,A: $tType] :
% 2.67/1.59 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 => ! [F: A > B3,G: A > B3] :
% 2.67/1.59 ( ! [X: A] : ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) )
% 2.67/1.59 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_236_le__fun__def,axiom,
% 2.67/1.59 ! [B3: $tType,A: $tType] :
% 2.67/1.59 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 => ( ( ord_less_eq @ ( A > B3 ) )
% 2.67/1.59 = ( ^ [F2: A > B3,G2: A > B3] :
% 2.67/1.59 ! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_237_order__subst1,axiom,
% 2.67/1.59 ! [A: $tType,B3: $tType] :
% 2.67/1.59 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 & ( order @ A @ ( type2 @ A ) ) )
% 2.67/1.59 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.67/1.59 => ( ( ord_less_eq @ B3 @ B @ C )
% 2.67/1.59 => ( ! [X: B3,Y3: B3] :
% 2.67/1.59 ( ( ord_less_eq @ B3 @ X @ Y3 )
% 2.67/1.59 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_238_order__subst2,axiom,
% 2.67/1.59 ! [A: $tType,C4: $tType] :
% 2.67/1.59 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 2.67/1.59 & ( order @ A @ ( type2 @ A ) ) )
% 2.67/1.59 => ! [A2: A,B: A,F: A > C4,C: C4] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
% 2.67/1.59 => ( ! [X: A,Y3: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.67/1.59 => ( ord_less_eq @ C4 @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.67/1.59 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_239_ord__eq__le__subst,axiom,
% 2.67/1.59 ! [A: $tType,B3: $tType] :
% 2.67/1.59 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 & ( ord @ A @ ( type2 @ A ) ) )
% 2.67/1.59 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 2.67/1.59 ( ( A2
% 2.67/1.59 = ( F @ B ) )
% 2.67/1.59 => ( ( ord_less_eq @ B3 @ B @ C )
% 2.67/1.59 => ( ! [X: B3,Y3: B3] :
% 2.67/1.59 ( ( ord_less_eq @ B3 @ X @ Y3 )
% 2.67/1.59 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_240_ord__le__eq__subst,axiom,
% 2.67/1.59 ! [A: $tType,B3: $tType] :
% 2.67/1.59 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.67/1.59 & ( ord @ A @ ( type2 @ A ) ) )
% 2.67/1.59 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( ( F @ B )
% 2.67/1.59 = C )
% 2.67/1.59 => ( ! [X: A,Y3: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.67/1.59 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.67/1.59 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_241_eq__iff,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( order @ A @ ( type2 @ A ) )
% 2.67/1.59 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.67/1.59 = ( ^ [X2: A,Y5: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 2.67/1.59 & ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_242_antisym,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( order @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ Y2 @ X4 )
% 2.67/1.59 => ( X4 = Y2 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_243_linear,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linorder @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.67/1.59 | ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_244_eq__refl,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( preorder @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ( X4 = Y2 )
% 2.67/1.59 => ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_245_le__cases,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linorder @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A] :
% 2.67/1.59 ( ~ ( ord_less_eq @ A @ X4 @ Y2 )
% 2.67/1.59 => ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_246_order_Otrans,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( order @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ C )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_247_le__cases3,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( linorder @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A,Z2: A] :
% 2.67/1.59 ( ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.67/1.59 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.67/1.59 => ( ( ( ord_less_eq @ A @ Y2 @ X4 )
% 2.67/1.59 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.67/1.59 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 2.67/1.59 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.67/1.59 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.67/1.59 => ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
% 2.67/1.59 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.67/1.59 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 2.67/1.59 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 2.67/1.59 => ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_248_antisym__conv,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( order @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [Y2: A,X4: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ Y2 @ X4 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.67/1.59 = ( X4 = Y2 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_249_ord__eq__le__trans,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ord @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( A2 = B )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ C )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_250_ord__le__eq__trans,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( ord @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A,C: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( B = C )
% 2.67/1.59 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_251_order__class_Oorder_Oantisym,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( order @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A,B: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.67/1.59 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.67/1.59 => ( A2 = B ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_252_order__trans,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( preorder @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [X4: A,Y2: A,Z2: A] :
% 2.67/1.59 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.67/1.59 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.67/1.59 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(fact_253_dual__order_Orefl,axiom,
% 2.67/1.59 ! [A: $tType] :
% 2.67/1.59 ( ( order @ A @ ( type2 @ A ) )
% 2.67/1.59 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.67/1.59 ! [A6: $tType,A7: $tType] :
% 2.67/1.59 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 2.67/1.59 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.67/1.59 ! [A6: $tType,A7: $tType] :
% 2.67/1.59 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.67/1.59 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_fun___Orderings_Oord,axiom,
% 2.67/1.59 ! [A6: $tType,A7: $tType] :
% 2.67/1.59 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 2.67/1.59 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.67/1.59 ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.67/1.59 strict2144017051up_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.67/1.59 ordere223160158up_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.67/1.59 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 2.67/1.59 strict797366125id_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 2.67/1.59 ordere1818651114id_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.67/1.59 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.67/1.59 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 2.67/1.59 ordere216010020id_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.67/1.59 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.67/1.59 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.67/1.59 ab_semigroup_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.67/1.59 comm_monoid_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.67/1.59 semigroup_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.67/1.59 preorder @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.67/1.59 linorder @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 2.67/1.59 monoid_add @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.67/1.59 order @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.67/1.59 ord @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.67/1.59 zero @ nat @ ( type2 @ nat ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.67/1.59 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.67/1.59 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.67/1.59 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.67/1.59 preorder @ $o @ ( type2 @ $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 2.67/1.59 linorder @ $o @ ( type2 @ $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.67/1.59 order @ $o @ ( type2 @ $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.67/1.59 ord @ $o @ ( type2 @ $o ) ).
% 2.67/1.59
% 2.67/1.59 thf(help_If_3_1_T,axiom,
% 2.67/1.59 ! [P: $o] :
% 2.67/1.59 ( ( P = $true )
% 2.67/1.59 | ( P = $false ) ) ).
% 2.67/1.59
% 2.67/1.59 thf(help_If_2_1_T,axiom,
% 2.67/1.59 ! [A: $tType,X4: A,Y2: A] :
% 2.67/1.59 ( ( if @ A @ $false @ X4 @ Y2 )
% 2.67/1.59 = Y2 ) ).
% 2.67/1.59
% 2.67/1.59 thf(help_If_1_1_T,axiom,
% 2.67/1.59 ! [A: $tType,X4: A,Y2: A] :
% 2.67/1.59 ( ( if @ A @ $true @ X4 @ Y2 )
% 2.67/1.59 = X4 ) ).
% 2.67/1.59
% 2.67/1.59 thf(conj_0,conjecture,
% 2.67/1.59 huffma1050891809istent @ a @ ( huffma304375860urSyms @ a @ u @ a2 @ b @ c @ ( huffma943100115ibling @ a @ u @ c ) ) ).
% 2.67/1.59 ------------------------
% 2.79/1.63 % (13602)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13488 for (2999ds/2Mi)
% 2.79/1.63 % (13601)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_13488 for (2999ds/27Mi)
% 2.79/1.63 % (13600)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_13488 for (2999ds/4Mi)
% 2.79/1.63 % (13599)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13488 for (2999ds/183Mi)
% 2.79/1.63 % (13603)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_13488 for (2999ds/2Mi)
% 2.79/1.63 % (13604)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_13488 for (2999ds/275Mi)
% 2.79/1.63 % (13605)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13488 for (2999ds/18Mi)
% 2.79/1.63 % (13602)Instruction limit reached!
% 2.79/1.63 % (13602)------------------------------
% 2.79/1.63 % (13602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.63 % (13602)Termination reason: Unknown
% 2.79/1.63 % (13602)Termination phase: shuffling
% 2.79/1.63
% 2.79/1.63 % (13602)Memory used [KB]: 1279
% 2.79/1.63 % (13602)Time elapsed: 0.003 s
% 2.79/1.63 % (13602)Instructions burned: 3 (million)
% 2.79/1.63 % (13602)------------------------------
% 2.79/1.63 % (13602)------------------------------
% 2.79/1.63 % (13603)Instruction limit reached!
% 2.79/1.63 % (13603)------------------------------
% 2.79/1.63 % (13603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.63 % (13603)Termination reason: Unknown
% 2.79/1.63 % (13603)Termination phase: shuffling
% 2.79/1.63
% 2.79/1.63 % (13603)Memory used [KB]: 1407
% 2.79/1.63 % (13603)Time elapsed: 0.003 s
% 2.79/1.63 % (13603)Instructions burned: 3 (million)
% 2.79/1.63 % (13603)------------------------------
% 2.79/1.63 % (13603)------------------------------
% 2.79/1.63 % (13600)Instruction limit reached!
% 2.79/1.63 % (13600)------------------------------
% 2.79/1.63 % (13600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.63 % (13600)Termination reason: Unknown
% 2.79/1.63 % (13600)Termination phase: shuffling
% 2.79/1.63
% 2.79/1.63 % (13600)Memory used [KB]: 1407
% 2.79/1.63 % (13600)Time elapsed: 0.004 s
% 2.79/1.63 % (13600)Instructions burned: 5 (million)
% 2.79/1.63 % (13600)------------------------------
% 2.79/1.63 % (13600)------------------------------
% 2.79/1.63 % (13605)Instruction limit reached!
% 2.79/1.63 % (13605)------------------------------
% 2.79/1.63 % (13605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.63 % (13605)Termination reason: Unknown
% 2.79/1.63 % (13605)Termination phase: shuffling
% 2.79/1.63
% 2.79/1.63 % (13605)Memory used [KB]: 1663
% 2.79/1.63 % (13605)Time elapsed: 0.011 s
% 2.79/1.63 % (13605)Instructions burned: 19 (million)
% 2.79/1.63 % (13605)------------------------------
% 2.79/1.63 % (13605)------------------------------
% 2.79/1.64 % (13601)Instruction limit reached!
% 2.79/1.64 % (13601)------------------------------
% 2.79/1.64 % (13601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.64 % (13601)Termination reason: Unknown
% 2.79/1.64 % (13601)Termination phase: Property scanning
% 2.79/1.64
% 2.79/1.64 % (13601)Memory used [KB]: 1663
% 2.79/1.64 % (13601)Time elapsed: 0.015 s
% 2.79/1.64 % (13601)Instructions burned: 27 (million)
% 2.79/1.64 % (13601)------------------------------
% 2.79/1.64 % (13601)------------------------------
% 2.79/1.64 % (13606)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13488 for (2999ds/3Mi)
% 2.79/1.64 % (13607)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_13488 for (2999ds/37Mi)
% 2.79/1.64 % (13608)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_13488 for (2999ds/15Mi)
% 2.79/1.64 % (13606)Instruction limit reached!
% 2.79/1.64 % (13606)------------------------------
% 2.79/1.64 % (13606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.64 % (13606)Termination reason: Unknown
% 2.79/1.64 % (13606)Termination phase: shuffling
% 2.79/1.64
% 2.79/1.64 % (13606)Memory used [KB]: 1407
% 2.79/1.64 % (13606)Time elapsed: 0.003 s
% 2.79/1.64 % (13606)Instructions burned: 3 (million)
% 2.79/1.64 % (13606)------------------------------
% 2.79/1.64 % (13606)------------------------------
% 2.79/1.65 % (13608)Instruction limit reached!
% 2.79/1.65 % (13608)------------------------------
% 2.79/1.65 % (13608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.65 % (13608)Termination reason: Unknown
% 2.79/1.65 % (13608)Termination phase: shuffling
% 2.79/1.65
% 2.79/1.65 % (13609)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_13488 for (2999ds/3Mi)
% 2.79/1.65 % (13608)Memory used [KB]: 1535
% 2.79/1.65 % (13608)Time elapsed: 0.009 s
% 2.79/1.65 % (13608)Instructions burned: 15 (million)
% 2.79/1.65 % (13608)------------------------------
% 2.79/1.65 % (13608)------------------------------
% 2.79/1.65 % (13609)Instruction limit reached!
% 2.79/1.65 % (13609)------------------------------
% 2.79/1.65 % (13609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.65 % (13609)Termination reason: Unknown
% 2.79/1.65 % (13609)Termination phase: shuffling
% 2.79/1.65
% 2.79/1.65 % (13609)Memory used [KB]: 1407
% 2.79/1.65 % (13609)Time elapsed: 0.003 s
% 2.79/1.65 % (13609)Instructions burned: 3 (million)
% 2.79/1.65 % (13609)------------------------------
% 2.79/1.65 % (13609)------------------------------
% 2.79/1.65 % (13604)First to succeed.
% 2.79/1.65 % (13610)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_13488 for (2999ds/1041Mi)
% 2.79/1.66 % (13604)Refutation found. Thanks to Tanya!
% 2.79/1.66 % SZS status Theorem for DTF2THF_13488
% 2.79/1.66 % SZS output start Proof for DTF2THF_13488
% 2.79/1.66 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.79/1.66 thf(type_def_7, type, huffma16452318e_tree: ($tType) > $tType).
% 2.79/1.66 thf(type_def_8, type, set: ($tType) > $tType).
% 2.79/1.66 thf(type_def_9, type, nat: $tType).
% 2.79/1.66 thf(type_def_10, type, a: $tType).
% 2.79/1.66 thf(func_def_0, type, huffma16452318e_tree: ($tType) > $tType).
% 2.79/1.66 thf(func_def_1, type, set: ($tType) > $tType).
% 2.79/1.66 thf(func_def_2, type, nat: $tType).
% 2.79/1.66 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.79/1.66 thf(func_def_4, type, a: $tType).
% 2.79/1.66 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_6, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_9, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_10, type, monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_13, type, semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_14, type, comm_monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_15, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_16, type, ab_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_17, type, cancel_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_18, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_19, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_20, type, ordere216010020id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_21, type, ordere779506340up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_22, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_23, type, ordere1818651114id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_24, type, strict797366125id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_25, type, ordere236663937imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_26, type, ordere223160158up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_27, type, strict2144017051up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_28, type, ordere516151231imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.79/1.66 thf(func_def_29, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.79/1.66 thf(func_def_30, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.79/1.66 thf(func_def_31, type, huffma505251170phabet: !>[X0: $tType]:(huffma16452318e_tree @ X0 > set @ X0)).
% 2.79/1.66 thf(func_def_32, type, huffma1050891809istent: !>[X0: $tType]:(huffma16452318e_tree @ X0 > $o)).
% 2.79/1.66 thf(func_def_33, type, huffma636208924e_cost: !>[X0: $tType]:(huffma16452318e_tree @ X0 > nat)).
% 2.79/1.66 thf(func_def_34, type, huffma223349076_depth: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > nat)).
% 2.79/1.66 thf(func_def_35, type, huffma854352999e_freq: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > nat)).
% 2.79/1.66 thf(func_def_36, type, huffma1554076246height: !>[X0: $tType]:(huffma16452318e_tree @ X0 > nat)).
% 2.79/1.66 thf(func_def_37, type, huffma1954420889ibling: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > huffma16452318e_tree @ X0)).
% 2.79/1.66 thf(func_def_38, type, huffma1154738298minima: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > X0 > $o)).
% 2.79/1.66 thf(func_def_39, type, huffma936049440ptimum: !>[X0: $tType]:(huffma16452318e_tree @ X0 > $o)).
% 2.79/1.66 thf(func_def_40, type, huffma943100115ibling: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > X0)).
% 2.79/1.66 thf(func_def_41, type, huffma454997449itLeaf: !>[X0: $tType]:(huffma16452318e_tree @ X0 > nat > X0 > nat > X0 > huffma16452318e_tree @ X0)).
% 2.79/1.66 thf(func_def_42, type, huffma304375860urSyms: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > X0 > X0 > X0 > huffma16452318e_tree @ X0)).
% 2.79/1.66 thf(func_def_43, type, huffma2094459102Leaves: !>[X0: $tType]:(huffma16452318e_tree @ X0 > nat > X0 > nat > X0 > huffma16452318e_tree @ X0)).
% 2.79/1.66 thf(func_def_44, type, huffma469337550apSyms: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0 > X0 > huffma16452318e_tree @ X0)).
% 2.79/1.66 thf(func_def_45, type, huffma691733767weight: !>[X0: $tType]:(huffma16452318e_tree @ X0 > nat)).
% 2.79/1.66 thf(func_def_46, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.79/1.66 thf(func_def_47, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.79/1.66 thf(func_def_48, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.79/1.66 thf(func_def_49, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.79/1.66 thf(func_def_50, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.79/1.66 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.79/1.66 thf(func_def_52, type, a2: a).
% 2.79/1.66 thf(func_def_53, type, b: a).
% 2.79/1.66 thf(func_def_54, type, c: a).
% 2.79/1.66 thf(func_def_55, type, t: huffma16452318e_tree @ a).
% 2.79/1.66 thf(func_def_56, type, u: huffma16452318e_tree @ a).
% 2.79/1.66 thf(func_def_57, type, w_a: nat).
% 2.79/1.66 thf(func_def_58, type, w_b: nat).
% 2.79/1.66 thf(func_def_75, type, sK0: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.79/1.66 thf(func_def_76, type, sK1: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.79/1.66 thf(func_def_77, type, sK2: (nat > $o) > nat).
% 2.79/1.66 thf(func_def_78, type, sK3: (nat > $o) > nat).
% 2.79/1.66 thf(func_def_79, type, sK4: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.79/1.66 thf(func_def_80, type, sK5: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.79/1.66 thf(func_def_81, type, sK6: (nat > $o) > nat).
% 2.79/1.66 thf(func_def_82, type, sK7: (nat > nat) > nat).
% 2.79/1.66 thf(func_def_83, type, sK8: (nat > nat) > nat).
% 2.79/1.66 thf(func_def_84, type, sK9: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0)).
% 2.79/1.66 thf(func_def_85, type, sK10: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.79/1.66 thf(func_def_86, type, sK11: nat > (nat > $o) > nat).
% 2.79/1.66 thf(func_def_87, type, sK12: nat > nat > nat).
% 2.79/1.66 thf(func_def_88, type, sK13: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.79/1.66 thf(func_def_89, type, sK14: a).
% 2.79/1.66 thf(func_def_90, type, sK15: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.79/1.66 thf(func_def_91, type, sK16: !>[X0: $tType]:(huffma16452318e_tree @ X0 > X0)).
% 2.79/1.66 thf(func_def_92, type, sK17: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.79/1.66 thf(func_def_93, type, sK18: !>[X0: $tType]:(nat > (X0 > nat) > X0 > (X0 > $o) > X0)).
% 2.79/1.66 thf(func_def_94, type, sK19: a).
% 2.79/1.66 thf(func_def_95, type, sK20: (nat > nat) > nat).
% 2.79/1.66 thf(func_def_96, type, sK21: (nat > nat) > nat).
% 2.79/1.66 thf(func_def_97, type, sK22: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.79/1.66 thf(func_def_99, type, ph24: !>[X0: $tType]:(X0)).
% 2.79/1.66 thf(f1106,plain,(
% 2.79/1.66 $false),
% 2.79/1.66 inference(subsumption_resolution,[status(thm)],[f1096,f1050])).
% 2.79/1.66 thf(f1050,plain,(
% 2.79/1.66 ((huffma1050891809istent @ a @ u) = $true)),
% 2.79/1.66 inference(cnf_transformation,[status(thm)],[f344])).
% 2.79/1.66 thf(f344,plain,(
% 2.79/1.66 ((huffma1050891809istent @ a @ u) = $true)),
% 2.79/1.66 inference(fool_elimination,[status(thm)],[f343])).
% 2.79/1.66 thf(f343,plain,(
% 2.79/1.66 (huffma1050891809istent @ a @ u)),
% 2.79/1.66 inference(rectify,[status(thm)],[f1])).
% 2.79/1.66 thf(f1,axiom,(
% 2.79/1.66 (huffma1050891809istent @ a @ u)),
% 2.79/1.66 file('/export/starexec/sandbox/tmp/tmp.UCTj6kXiKr/DTF2THF_13488.p',fact_0_c_092_060_094sub_062u)).
% 2.79/1.66 thf(f1096,plain,(
% 2.79/1.66 ((huffma1050891809istent @ a @ u) != $true)),
% 2.79/1.66 inference(trivial_inequality_removal,[status(thm)],[f1095])).
% 2.79/1.66 thf(f1095,plain,(
% 2.79/1.66 ($true != $true) | ((huffma1050891809istent @ a @ u) != $true)),
% 2.79/1.66 inference(superposition,[status(thm)],[f985,f964])).
% 2.79/1.66 thf(f964,plain,(
% 2.79/1.66 ( ! [X0 : $tType,X2 : X0,X3 : X0,X1 : huffma16452318e_tree @ X0,X4 : X0,X5 : X0] : (((huffma1050891809istent @ X0 @ (huffma304375860urSyms @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) = $true) | ((huffma1050891809istent @ X0 @ X1) != $true)) )),
% 2.79/1.66 inference(cnf_transformation,[status(thm)],[f903])).
% 2.79/1.66 thf(f903,plain,(
% 2.79/1.66 ! [X0 : $tType,X1 : huffma16452318e_tree @ X0,X2 : X0,X3 : X0,X4 : X0,X5 : X0] : (((huffma1050891809istent @ X0 @ (huffma304375860urSyms @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) = $true) | ((huffma1050891809istent @ X0 @ X1) != $true))),
% 2.79/1.66 inference(ennf_transformation,[status(thm)],[f398])).
% 2.79/1.66 thf(f398,plain,(
% 2.79/1.66 ! [X0 : $tType,X1 : huffma16452318e_tree @ X0,X2 : X0,X3 : X0,X4 : X0,X5 : X0] : (((huffma1050891809istent @ X0 @ X1) = $true) => ((huffma1050891809istent @ X0 @ (huffma304375860urSyms @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) = $true))),
% 2.79/1.66 inference(fool_elimination,[status(thm)],[f397])).
% 2.79/1.66 thf(f397,plain,(
% 2.79/1.66 ! [X0 : $tType,X1 : huffma16452318e_tree @ X0,X2 : X0,X3 : X0,X4 : X0,X5 : X0] : ((huffma1050891809istent @ X0 @ X1) => (huffma1050891809istent @ X0 @ (huffma304375860urSyms @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 2.79/1.66 inference(rectify,[status(thm)],[f9])).
% 2.79/1.66 thf(f9,axiom,(
% 2.79/1.66 ! [X0 : $tType,X1 : huffma16452318e_tree @ X0,X2 : X0,X3 : X0,X4 : X0,X5 : X0] : ((huffma1050891809istent @ X0 @ X1) => (huffma1050891809istent @ X0 @ (huffma304375860urSyms @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 2.79/1.66 file('/export/starexec/sandbox/tmp/tmp.UCTj6kXiKr/DTF2THF_13488.p',fact_8_consistent__swapFourSyms)).
% 2.79/1.66 thf(f985,plain,(
% 2.79/1.66 ((huffma1050891809istent @ a @ (huffma304375860urSyms @ a @ u @ a2 @ b @ c @ (huffma943100115ibling @ a @ u @ c))) != $true)),
% 2.79/1.66 inference(cnf_transformation,[status(thm)],[f827])).
% 2.79/1.66 thf(f827,plain,(
% 2.79/1.66 ((huffma1050891809istent @ a @ (huffma304375860urSyms @ a @ u @ a2 @ b @ c @ (huffma943100115ibling @ a @ u @ c))) != $true)),
% 2.79/1.66 inference(flattening,[status(thm)],[f480])).
% 2.79/1.66 thf(f480,plain,(
% 2.79/1.66 ~((huffma1050891809istent @ a @ (huffma304375860urSyms @ a @ u @ a2 @ b @ c @ (huffma943100115ibling @ a @ u @ c))) = $true)),
% 2.79/1.66 inference(fool_elimination,[status(thm)],[f479])).
% 2.79/1.66 thf(f479,plain,(
% 2.79/1.66 ~(huffma1050891809istent @ a @ (huffma304375860urSyms @ a @ u @ a2 @ b @ c @ (huffma943100115ibling @ a @ u @ c)))),
% 2.79/1.66 inference(rectify,[status(thm)],[f289])).
% 2.79/1.66 thf(f289,negated_conjecture,(
% 2.79/1.66 ~(huffma1050891809istent @ a @ (huffma304375860urSyms @ a @ u @ a2 @ b @ c @ (huffma943100115ibling @ a @ u @ c)))),
% 2.79/1.66 inference(negated_conjecture,[status(cth)],[f288])).
% 2.79/1.66 thf(f288,conjecture,(
% 2.79/1.66 (huffma1050891809istent @ a @ (huffma304375860urSyms @ a @ u @ a2 @ b @ c @ (huffma943100115ibling @ a @ u @ c)))),
% 2.79/1.66 file('/export/starexec/sandbox/tmp/tmp.UCTj6kXiKr/DTF2THF_13488.p',conj_0)).
% 2.79/1.66 % SZS output end Proof for DTF2THF_13488
% 2.79/1.66 % (13604)------------------------------
% 2.79/1.66 % (13604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.66 % (13604)Termination reason: Refutation
% 2.79/1.66
% 2.79/1.66 % (13604)Memory used [KB]: 6652
% 2.79/1.66 % (13604)Time elapsed: 0.031 s
% 2.79/1.66 % (13604)Instructions burned: 56 (million)
% 2.79/1.66 % (13604)------------------------------
% 2.79/1.66 % (13604)------------------------------
% 2.79/1.66 % (13598)Success in time 0.058 s
%------------------------------------------------------------------------------