TPTP Problem File: DAT166^1.p
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%------------------------------------------------------------------------------
% File : DAT166^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Huffman 1018
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Bla08] Blanchette (2008), The Textbook Proof of Huffman's Alg
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : huffman__1018.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 373 ( 117 unt; 83 typ; 0 def)
% Number of atoms : 675 ( 298 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 3875 ( 105 ~; 7 |; 34 &;3406 @)
% ( 0 <=>; 323 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 168 ( 168 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 81 usr; 4 con; 0-5 aty)
% Number of variables : 985 ( 36 ^; 858 !; 16 ?; 985 :)
% ( 75 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:17.528
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree,type,
huffma16452318e_tree: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (77)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oalphabet,type,
huffma505251170phabet:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( set @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oalphabet_092_060_094sub_062F,type,
huffma279473244abet_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OcachedWeight,type,
huffma787811817Weight:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oconsistent,type,
huffma1050891809istent:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oconsistent_092_060_094sub_062F,type,
huffma2111480347tent_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ocost,type,
huffma636208924e_cost:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Odepth,type,
huffma223349076_depth:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq,type,
huffma854352999e_freq:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq_092_060_094sub_062F,type,
huffma2047054433freq_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight,type,
huffma1554076246height:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight_092_060_094sub_062F,type,
huffma279770448ight_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ohuffman,type,
huffma149336734uffman:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ohuffman__rel,type,
huffma316836827an_rel:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > ( list @ ( huffma16452318e_tree @ A ) ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OinsortTree,type,
huffma725507568rtTree:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( list @ ( huffma16452318e_tree @ A ) ) > ( list @ ( huffma16452318e_tree @ A ) ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Osibling,type,
huffma943100115ibling:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_OInnerNode,type,
huffma1759677307erNode:
!>[A: $tType] : ( nat > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_OLeaf,type,
huffma1554276827e_Leaf:
!>[A: $tType] : ( nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OuniteTrees,type,
huffma453905539eTrees:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oweight,type,
huffma691733767weight:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo292404081st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_t,type,
t: huffma16452318e_tree @ a ).
%----Relevant facts (252)
thf(fact_0_exists__in__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
? [A2: A] : ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) ) ).
% exists_in_alphabet
thf(fact_1_sibling_Osimps_I4_J,axiom,
! [A: $tType,A3: A,T_1: huffma16452318e_tree @ A,W: nat,V: nat,Va: huffma16452318e_tree @ A,Vb: huffma16452318e_tree @ A] :
( ( ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) @ A3 )
= ( huffma943100115ibling @ A @ T_1 @ A3 ) ) )
& ( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( ( member @ A @ A3 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) @ A3 )
= ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ A3 ) ) )
& ( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) @ A3 )
= A3 ) ) ) ) ) ).
% sibling.simps(4)
thf(fact_2_sibling_Osimps_I3_J,axiom,
! [A: $tType,A3: A,V: nat,Va: huffma16452318e_tree @ A,Vb: huffma16452318e_tree @ A,W: nat,T_2: huffma16452318e_tree @ A] :
( ( ( member @ A @ A3 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ T_2 ) @ A3 )
= ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ A3 ) ) )
& ( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_2 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ T_2 ) @ A3 )
= ( huffma943100115ibling @ A @ T_2 @ A3 ) ) )
& ( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_2 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ T_2 ) @ A3 )
= A3 ) ) ) ) ) ).
% sibling.simps(3)
thf(fact_3_sibling_Osimps_I1_J,axiom,
! [A: $tType,W_b: nat,B: A,A3: A] :
( ( huffma943100115ibling @ A @ ( huffma1554276827e_Leaf @ A @ W_b @ B ) @ A3 )
= A3 ) ).
% sibling.simps(1)
thf(fact_4_finite__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] : ( finite_finite2 @ A @ ( huffma505251170phabet @ A @ T ) ) ).
% finite_alphabet
thf(fact_5_alphabet__uniteTrees,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma505251170phabet @ A @ ( huffma453905539eTrees @ A @ T_1 @ T_2 ) )
= ( sup_sup @ ( set @ A ) @ ( huffma505251170phabet @ A @ T_1 ) @ ( huffma505251170phabet @ A @ T_2 ) ) ) ).
% alphabet_uniteTrees
thf(fact_6_notin__alphabet__imp__freq__0,axiom,
! [A: $tType,A3: A,T: huffma16452318e_tree @ A] :
( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma854352999e_freq @ A @ T @ A3 )
= ( zero_zero @ nat ) ) ) ).
% notin_alphabet_imp_freq_0
thf(fact_7_sibling_Osimps_I2_J,axiom,
! [A: $tType,A3: A,B: A,W: nat,W_b: nat,W_c: nat,C: A] :
( ( ( A3 = B )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1554276827e_Leaf @ A @ W_b @ B ) @ ( huffma1554276827e_Leaf @ A @ W_c @ C ) ) @ A3 )
= C ) )
& ( ( A3 != B )
=> ( ( ( A3 = C )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1554276827e_Leaf @ A @ W_b @ B ) @ ( huffma1554276827e_Leaf @ A @ W_c @ C ) ) @ A3 )
= B ) )
& ( ( A3 != C )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1554276827e_Leaf @ A @ W_b @ B ) @ ( huffma1554276827e_Leaf @ A @ W_c @ C ) ) @ A3 )
= A3 ) ) ) ) ) ).
% sibling.simps(2)
thf(fact_8_sibling_Oinduct,axiom,
! [A: $tType,P: ( huffma16452318e_tree @ A ) > A > $o,A0: huffma16452318e_tree @ A,A1: A] :
( ! [W_b2: nat,B2: A,X1: A] : ( P @ ( huffma1554276827e_Leaf @ A @ W_b2 @ B2 ) @ X1 )
=> ( ! [W2: nat,W_b2: nat,B2: A,W_c2: nat,C2: A,X1: A] : ( P @ ( huffma1759677307erNode @ A @ W2 @ ( huffma1554276827e_Leaf @ A @ W_b2 @ B2 ) @ ( huffma1554276827e_Leaf @ A @ W_c2 @ C2 ) ) @ X1 )
=> ( ! [W2: nat,V2: nat,Va2: huffma16452318e_tree @ A,Vb2: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,A2: A] :
( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) ) )
=> ( P @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) @ A2 ) )
=> ( ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) ) )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_22 ) )
=> ( P @ T_22 @ A2 ) ) )
=> ( P @ ( huffma1759677307erNode @ A @ W2 @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) @ T_22 ) @ A2 ) ) )
=> ( ! [W2: nat,T_12: huffma16452318e_tree @ A,V2: nat,Va2: huffma16452318e_tree @ A,Vb2: huffma16452318e_tree @ A,A2: A] :
( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_12 ) )
=> ( P @ T_12 @ A2 ) )
=> ( ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_12 ) )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) ) )
=> ( P @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) @ A2 ) ) )
=> ( P @ ( huffma1759677307erNode @ A @ W2 @ T_12 @ ( huffma1759677307erNode @ A @ V2 @ Va2 @ Vb2 ) ) @ A2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% sibling.induct
thf(fact_9_alphabet__huffman,axiom,
! [A: $tType,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( Ts
!= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( ( huffma505251170phabet @ A @ ( huffma149336734uffman @ A @ Ts ) )
= ( huffma279473244abet_F @ A @ Ts ) ) ) ).
% alphabet_huffman
thf(fact_10_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 ) ) ) ) ).
% exists_at_height
thf(fact_11_alphabet_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( sup_sup @ ( set @ A ) @ ( huffma505251170phabet @ A @ T_1 ) @ ( huffma505251170phabet @ A @ T_2 ) ) ) ).
% alphabet.simps(2)
thf(fact_12_alphabet_092_060_094sub_062F__insortTree,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma279473244abet_F @ A @ ( huffma725507568rtTree @ A @ T @ Ts ) )
= ( sup_sup @ ( set @ A ) @ ( huffma505251170phabet @ A @ T ) @ ( huffma279473244abet_F @ A @ Ts ) ) ) ).
% alphabet\<^sub>F_insortTree
thf(fact_13_tree_Oinject_I2_J,axiom,
! [A: $tType,X21: nat,X22: huffma16452318e_tree @ A,X23: huffma16452318e_tree @ A,Y21: nat,Y22: huffma16452318e_tree @ A,Y23: huffma16452318e_tree @ A] :
( ( ( huffma1759677307erNode @ A @ X21 @ X22 @ X23 )
= ( huffma1759677307erNode @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% tree.inject(2)
thf(fact_14_tree_Oinject_I1_J,axiom,
! [A: $tType,X11: nat,X12: A,Y11: nat,Y12: A] :
( ( ( huffma1554276827e_Leaf @ A @ X11 @ X12 )
= ( huffma1554276827e_Leaf @ A @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tree.inject(1)
thf(fact_15_freq_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A3: A] :
( ( huffma854352999e_freq @ A @ ( huffma1554276827e_Leaf @ A @ W @ A3 ) )
= ( ^ [B3: A] : ( if @ nat @ ( B3 = A3 ) @ W @ ( zero_zero @ nat ) ) ) ) ).
% freq.simps(1)
thf(fact_16_depth_Osimps_I1_J,axiom,
! [A: $tType,W: nat,B: A,A3: A] :
( ( huffma223349076_depth @ A @ ( huffma1554276827e_Leaf @ A @ W @ B ) @ A3 )
= ( zero_zero @ nat ) ) ).
% depth.simps(1)
thf(fact_17_height_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A3: A] :
( ( huffma1554076246height @ A @ ( huffma1554276827e_Leaf @ A @ W @ A3 ) )
= ( zero_zero @ nat ) ) ).
% height.simps(1)
thf(fact_18_tree_Odistinct_I1_J,axiom,
! [A: $tType,X11: nat,X12: A,X21: nat,X22: huffma16452318e_tree @ A,X23: huffma16452318e_tree @ A] :
( ( huffma1554276827e_Leaf @ A @ X11 @ X12 )
!= ( huffma1759677307erNode @ A @ X21 @ X22 @ X23 ) ) ).
% tree.distinct(1)
thf(fact_19_consistent_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A3: A] : ( huffma1050891809istent @ A @ ( huffma1554276827e_Leaf @ A @ W @ A3 ) ) ).
% consistent.simps(1)
thf(fact_20_tree_Oinduct,axiom,
! [A: $tType,P: ( huffma16452318e_tree @ A ) > $o,Tree: huffma16452318e_tree @ A] :
( ! [X1: nat,X2: A] : ( P @ ( huffma1554276827e_Leaf @ A @ X1 @ X2 ) )
=> ( ! [X1: nat,X2: huffma16452318e_tree @ A,X3: huffma16452318e_tree @ A] :
( ( P @ X2 )
=> ( ( P @ X3 )
=> ( P @ ( huffma1759677307erNode @ A @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% tree.induct
thf(fact_21_tree_Oexhaust,axiom,
! [A: $tType,Y: huffma16452318e_tree @ A] :
( ! [X112: nat,X122: A] :
( Y
!= ( huffma1554276827e_Leaf @ A @ X112 @ X122 ) )
=> ~ ! [X212: nat,X222: huffma16452318e_tree @ A,X232: huffma16452318e_tree @ A] :
( Y
!= ( huffma1759677307erNode @ A @ X212 @ X222 @ X232 ) ) ) ).
% tree.exhaust
thf(fact_22_insortTree__ne__Nil,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma725507568rtTree @ A @ T @ Ts )
!= ( nil @ ( huffma16452318e_tree @ A ) ) ) ).
% insortTree_ne_Nil
thf(fact_23_finite__Un,axiom,
! [A: $tType,F: set @ A,G: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F @ G ) )
= ( ( finite_finite2 @ A @ F )
& ( finite_finite2 @ A @ G ) ) ) ).
% finite_Un
thf(fact_24_freq__huffman,axiom,
! [A: $tType,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( Ts
!= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( ( huffma854352999e_freq @ A @ ( huffma149336734uffman @ A @ Ts ) )
= ( huffma2047054433freq_F @ A @ Ts ) ) ) ).
% freq_huffman
thf(fact_25_UnCI,axiom,
! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C @ B4 )
=> ( member @ A @ C @ A4 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_26_Un__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C @ A4 )
| ( member @ A @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_27_sup__apply,axiom,
! [B5: $tType,A: $tType] :
( ( semilattice_sup @ B5 @ ( type2 @ B5 ) )
=> ( ( sup_sup @ ( A > B5 ) )
= ( ^ [F2: A > B5,G2: A > B5,X4: A] : ( sup_sup @ B5 @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% sup_apply
thf(fact_28_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_29_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( sup_sup @ A @ X5 @ X5 )
= X5 ) ) ).
% sup_idem
thf(fact_30_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B ) )
= ( sup_sup @ A @ A3 @ B ) ) ) ).
% sup.left_idem
thf(fact_31_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( sup_sup @ A @ X5 @ ( sup_sup @ A @ X5 @ Y ) )
= ( sup_sup @ A @ X5 @ Y ) ) ) ).
% sup_left_idem
thf(fact_32_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B ) @ B )
= ( sup_sup @ A @ A3 @ B ) ) ) ).
% sup.right_idem
thf(fact_33_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_34_notin__alphabet_092_060_094sub_062F__imp__freq_092_060_094sub_062F__0,axiom,
! [A: $tType,A3: A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ~ ( member @ A @ A3 @ ( huffma279473244abet_F @ A @ Ts ) )
=> ( ( huffma2047054433freq_F @ A @ Ts @ A3 )
= ( zero_zero @ nat ) ) ) ).
% notin_alphabet\<^sub>F_imp_freq\<^sub>F_0
thf(fact_35_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_36_finite__set__choice,axiom,
! [B5: $tType,A: $tType,A4: set @ A,P: A > B5 > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ? [X13: B5] : ( P @ X @ X13 ) )
=> ? [F3: A > B5] :
! [X6: A] :
( ( member @ A @ X6 @ A4 )
=> ( P @ X6 @ ( F3 @ X6 ) ) ) ) ) ).
% finite_set_choice
thf(fact_37_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A,Z: A] :
( ( sup_sup @ A @ X5 @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X5 @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_38_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A,C: A] :
( ( sup_sup @ A @ B @ ( sup_sup @ A @ A3 @ C ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% sup.left_commute
thf(fact_39_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y2: A] : ( sup_sup @ A @ Y2 @ X4 ) ) ) ) ).
% sup_commute
thf(fact_40_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B3: A] : ( sup_sup @ A @ B3 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_41_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X5 @ Y ) @ Z )
= ( sup_sup @ A @ X5 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_42_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B ) @ C )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% sup.assoc
thf(fact_43_sup__fun__def,axiom,
! [B5: $tType,A: $tType] :
( ( semilattice_sup @ B5 @ ( type2 @ B5 ) )
=> ( ( sup_sup @ ( A > B5 ) )
= ( ^ [F2: A > B5,G2: A > B5,X4: A] : ( sup_sup @ B5 @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% sup_fun_def
thf(fact_44_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y2: A] : ( sup_sup @ A @ Y2 @ X4 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
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 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B5: $tType,A: $tType,F4: A > B5,G3: A > B5] :
( ! [X: A] :
( ( F4 @ X )
= ( G3 @ X ) )
=> ( F4 = G3 ) ) ).
% ext
thf(fact_49_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X5 @ Y ) @ Z )
= ( sup_sup @ A @ X5 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_50_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A,Z: A] :
( ( sup_sup @ A @ X5 @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X5 @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_51_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( sup_sup @ A @ X5 @ ( sup_sup @ A @ X5 @ Y ) )
= ( sup_sup @ A @ X5 @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_52_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_53_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_54_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_55_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_56_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_assoc
thf(fact_57_ball__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_58_bex__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_59_UnI2,axiom,
! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_60_UnI1,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_61_UnE,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( ~ ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% UnE
thf(fact_62_freq_092_060_094sub_062F_Osimps_I1_J,axiom,
! [A: $tType] :
( ( huffma2047054433freq_F @ A @ ( nil @ ( huffma16452318e_tree @ A ) ) )
= ( ^ [B3: A] : ( zero_zero @ nat ) ) ) ).
% freq\<^sub>F.simps(1)
thf(fact_63_infinite__Un,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_Un
thf(fact_64_Un__infinite,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_65_finite__UnI,axiom,
! [A: $tType,F: set @ A,G: set @ A] :
( ( finite_finite2 @ A @ F )
=> ( ( finite_finite2 @ A @ G )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_66_consistent__huffman,axiom,
! [A: $tType,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma2111480347tent_F @ A @ Ts )
=> ( ( Ts
!= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( huffma1050891809istent @ A @ ( huffma149336734uffman @ A @ Ts ) ) ) ) ).
% consistent_huffman
thf(fact_67_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 ) ) ) ).
% height_0_imp_cost_0
thf(fact_68_height_092_060_094sub_062F_Osimps_I1_J,axiom,
! [A: $tType] :
( ( huffma279770448ight_F @ A @ ( nil @ ( huffma16452318e_tree @ A ) ) )
= ( zero_zero @ nat ) ) ).
% height\<^sub>F.simps(1)
thf(fact_69_freq_092_060_094sub_062F__insortTree,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma2047054433freq_F @ A @ ( huffma725507568rtTree @ A @ T @ Ts ) )
= ( ^ [A6: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T @ A6 ) @ ( huffma2047054433freq_F @ A @ Ts @ A6 ) ) ) ) ).
% freq\<^sub>F_insortTree
thf(fact_70_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 ) ) ) ) ) ).
% height_gt_0_alphabet_eq_imp_height_gt_0
thf(fact_71_cost_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A3: A] :
( ( huffma636208924e_cost @ A @ ( huffma1554276827e_Leaf @ A @ W @ A3 ) )
= ( zero_zero @ nat ) ) ).
% cost.simps(1)
thf(fact_72_huffman_Osimps_I2_J,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma149336734uffman @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T_1 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_2 @ Ts ) ) )
= ( huffma149336734uffman @ A @ ( huffma725507568rtTree @ A @ ( huffma453905539eTrees @ A @ T_1 @ T_2 ) @ Ts ) ) ) ).
% huffman.simps(2)
thf(fact_73_huffman_Oinduct,axiom,
! [A: $tType,P: ( list @ ( huffma16452318e_tree @ A ) ) > $o,A0: list @ ( huffma16452318e_tree @ A )] :
( ! [T3: huffma16452318e_tree @ A] : ( P @ ( cons @ ( huffma16452318e_tree @ A ) @ T3 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( ( P @ ( huffma725507568rtTree @ A @ ( huffma453905539eTrees @ A @ T_12 @ T_22 ) @ Ts2 ) )
=> ( P @ ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) ) )
=> ( ( P @ ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( P @ A0 ) ) ) ) ).
% huffman.induct
thf(fact_74_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( ( plus_plus @ A @ A3 @ B )
= ( plus_plus @ A @ A3 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_75_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A,C: A] :
( ( ( plus_plus @ A @ B @ A3 )
= ( plus_plus @ A @ C @ A3 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_76_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_77_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.left_neutral
thf(fact_78_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_79_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_80_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_81_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A] :
( ( ( plus_plus @ A @ B @ A3 )
= A3 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_82_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( ( plus_plus @ A @ A3 @ B )
= A3 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_83_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( A3
= ( plus_plus @ A @ B @ A3 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_84_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_85_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A3: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A3 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_86_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A3: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A3 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_87_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_88_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_89_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel2
thf(fact_90_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel1
thf(fact_91_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B ) @ B )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_92_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B @ A3 ) @ B )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_93_freq__uniteTrees,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma854352999e_freq @ A @ ( huffma453905539eTrees @ A @ T_1 @ T_2 ) )
= ( ^ [A6: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T_1 @ A6 ) @ ( huffma854352999e_freq @ A @ T_2 @ A6 ) ) ) ) ).
% freq_uniteTrees
thf(fact_94_consistent_092_060_094sub_062F__insortTree,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma2111480347tent_F @ A @ ( huffma725507568rtTree @ A @ T @ Ts ) )
= ( huffma2111480347tent_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) ) ) ).
% consistent\<^sub>F_insortTree
thf(fact_95_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ( plus_plus @ A @ X5 @ Y )
= ( zero_zero @ A ) )
= ( ( X5
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_96_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_97_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_98_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_99_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 ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_100_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_101_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_102_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_103_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B ) @ C )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_104_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 ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_105_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 ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_106_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 ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_107_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 ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_108_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B ) @ C )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.assoc
thf(fact_109_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( ( plus_plus @ A @ A3 @ B )
= ( plus_plus @ A @ A3 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_110_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A,C: A] :
( ( ( plus_plus @ A @ B @ A3 )
= ( plus_plus @ A @ C @ A3 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_111_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B3: A] : ( plus_plus @ A @ B3 @ A6 ) ) ) ) ).
% add.commute
thf(fact_112_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A3 @ C ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.left_commute
thf(fact_113_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( ( plus_plus @ A @ A3 @ B )
= ( plus_plus @ A @ A3 @ C ) )
=> ( B = C ) ) ) ).
% add_left_imp_eq
thf(fact_114_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A,C: A] :
( ( ( plus_plus @ A @ B @ A3 )
= ( plus_plus @ A @ C @ A3 ) )
=> ( B = C ) ) ) ).
% add_right_imp_eq
thf(fact_115_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A3 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_116_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( ord_less @ A @ A3 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_strict_left_mono
thf(fact_117_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( ord_less @ A @ A3 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_118_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A3: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less @ A @ A3 @ B ) ) ) ).
% add_less_imp_less_left
thf(fact_119_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A3: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less @ A @ A3 @ B ) ) ) ).
% add_less_imp_less_right
thf(fact_120_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A3 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_121_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B ) ) ) ) ) ).
% add_pos_pos
thf(fact_122_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_123_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X5: A,A3: A,B: A] :
( ( ord_less @ A @ X5 @ A3 )
=> ( ord_less @ A @ X5 @ ( sup_sup @ A @ A3 @ B ) ) ) ) ).
% less_supI1
thf(fact_124_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X5: A,B: A,A3: A] :
( ( ord_less @ A @ X5 @ B )
=> ( ord_less @ A @ X5 @ ( sup_sup @ A @ A3 @ B ) ) ) ) ).
% less_supI2
thf(fact_125_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,C: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B @ C ) @ A3 )
=> ~ ( ( ord_less @ A @ B @ A3 )
=> ~ ( ord_less @ A @ C @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_126_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A6: A] :
( ( A6
= ( sup_sup @ A @ A6 @ B3 ) )
& ( A6 != B3 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_127_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A3: A,B: A] :
( ( ord_less @ A @ C @ A3 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A3 @ B ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_128_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,B: A,A3: A] :
( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A3 @ B ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_129_freq_092_060_094sub_062F_Osimps_I2_J,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma2047054433freq_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) )
= ( ^ [B3: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T @ B3 ) @ ( huffma2047054433freq_F @ A @ Ts @ B3 ) ) ) ) ).
% freq\<^sub>F.simps(2)
thf(fact_130_huffman_Ocases,axiom,
! [A: $tType,X5: list @ ( huffma16452318e_tree @ A )] :
( ! [T3: huffma16452318e_tree @ A] :
( X5
!= ( cons @ ( huffma16452318e_tree @ A ) @ T3 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( X5
!= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) )
=> ( X5
= ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ) ).
% huffman.cases
thf(fact_131_freq_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma854352999e_freq @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( ^ [B3: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T_1 @ B3 ) @ ( huffma854352999e_freq @ A @ T_2 @ B3 ) ) ) ) ).
% freq.simps(2)
thf(fact_132_insortTree_Osimps_I1_J,axiom,
! [A: $tType,U: huffma16452318e_tree @ A] :
( ( huffma725507568rtTree @ A @ U @ ( nil @ ( huffma16452318e_tree @ A ) ) )
= ( cons @ ( huffma16452318e_tree @ A ) @ U @ ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ).
% insortTree.simps(1)
thf(fact_133_huffman_Osimps_I1_J,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( huffma149336734uffman @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
= T ) ).
% huffman.simps(1)
thf(fact_134_consistent_092_060_094sub_062F_Osimps_I1_J,axiom,
! [A: $tType] : ( huffma2111480347tent_F @ A @ ( nil @ ( huffma16452318e_tree @ A ) ) ) ).
% consistent\<^sub>F.simps(1)
thf(fact_135_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( ( zero_zero @ A )
= X5 )
= ( X5
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_136_alphabet_092_060_094sub_062F_Osimps_I2_J,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma279473244abet_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) )
= ( sup_sup @ ( set @ A ) @ ( huffma505251170phabet @ A @ T ) @ ( huffma279473244abet_F @ A @ Ts ) ) ) ).
% alphabet\<^sub>F.simps(2)
thf(fact_137_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_138_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_139_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_140_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_141_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_142_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_143_finite__psubset__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [B7: set @ A] :
( ( ord_less @ ( set @ A ) @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_144_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V3: A > nat,X5: A] :
( ! [X: A] :
( ~ ( P @ X )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V3 @ Y3 ) @ ( V3 @ X ) )
& ~ ( P @ Y3 ) ) )
=> ( P @ X5 ) ) ).
% infinite_descent_measure
thf(fact_145_measure__induct__rule,axiom,
! [A: $tType,F4: A > nat,P: A > $o,A3: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F4 @ Y3 ) @ ( F4 @ X ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A3 ) ) ).
% measure_induct_rule
thf(fact_146_linorder__neqE__nat,axiom,
! [X5: nat,Y: nat] :
( ( X5 != Y )
=> ( ~ ( ord_less @ nat @ X5 @ Y )
=> ( ord_less @ nat @ Y @ X5 ) ) ) ).
% linorder_neqE_nat
thf(fact_147_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_148_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_149_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_150_measure__induct,axiom,
! [A: $tType,F4: A > nat,P: A > $o,A3: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F4 @ Y3 ) @ ( F4 @ X ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A3 ) ) ).
% measure_induct
thf(fact_151_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less @ nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_152_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_153_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_154_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_155_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N @ K ) )
= ( M = N ) ) ).
% nat_add_right_cancel
thf(fact_156_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N ) )
= ( M = N ) ) ).
% nat_add_left_cancel
thf(fact_157_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_158_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_159_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_160_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_161_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_162_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_163_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_164_infinite__descent0__measure,axiom,
! [A: $tType,V3: A > nat,P: A > $o,X5: A] :
( ! [X: A] :
( ( ( V3 @ X )
= ( zero_zero @ nat ) )
=> ( P @ X ) )
=> ( ! [X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V3 @ X ) )
=> ( ~ ( P @ X )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V3 @ Y3 ) @ ( V3 @ X ) )
& ~ ( P @ Y3 ) ) ) )
=> ( P @ X5 ) ) ) ).
% infinite_descent0_measure
thf(fact_165_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_166_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_167_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_168_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_169_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_170_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 ) ) ) ).
% add_less_mono1
thf(fact_171_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_172_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_173_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 ) ) ) ) ).
% add_less_mono
thf(fact_174_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_175_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_176_height_092_060_094sub_062F__0__imp__Leaf__freq_092_060_094sub_062F__in__set,axiom,
! [A: $tType,Ts: list @ ( huffma16452318e_tree @ A ),A3: A] :
( ( huffma2111480347tent_F @ A @ Ts )
=> ( ( ( huffma279770448ight_F @ A @ Ts )
= ( zero_zero @ nat ) )
=> ( ( member @ A @ A3 @ ( huffma279473244abet_F @ A @ Ts ) )
=> ( member @ ( huffma16452318e_tree @ A ) @ ( huffma1554276827e_Leaf @ A @ ( huffma2047054433freq_F @ A @ Ts @ A3 ) @ A3 ) @ ( set2 @ ( huffma16452318e_tree @ A ) @ Ts ) ) ) ) ) ).
% height\<^sub>F_0_imp_Leaf_freq\<^sub>F_in_set
thf(fact_177_huffman_Oelims,axiom,
! [A: $tType,X5: list @ ( huffma16452318e_tree @ A ),Y: huffma16452318e_tree @ A] :
( ( ( huffma149336734uffman @ A @ X5 )
= Y )
=> ( ! [T3: huffma16452318e_tree @ A] :
( ( X5
= ( cons @ ( huffma16452318e_tree @ A ) @ T3 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( Y != T3 ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( ( X5
= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) )
=> ( Y
!= ( huffma149336734uffman @ A @ ( huffma725507568rtTree @ A @ ( huffma453905539eTrees @ A @ T_12 @ T_22 ) @ Ts2 ) ) ) )
=> ~ ( ( X5
= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( Y
!= ( undefined @ ( huffma16452318e_tree @ A ) ) ) ) ) ) ) ).
% huffman.elims
thf(fact_178_uniteTrees__def,axiom,
! [A: $tType] :
( ( huffma453905539eTrees @ A )
= ( ^ [T_13: huffma16452318e_tree @ A,T_23: huffma16452318e_tree @ A] : ( huffma1759677307erNode @ A @ ( plus_plus @ nat @ ( huffma787811817Weight @ A @ T_13 ) @ ( huffma787811817Weight @ A @ T_23 ) ) @ T_13 @ T_23 ) ) ) ).
% uniteTrees_def
thf(fact_179_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_180_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_181_psubsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_182_finite__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_183_list_Oset__cases,axiom,
! [A: $tType,E: A,A3: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A3 ) )
=> ( ! [Z2: list @ A] :
( A3
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A3
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_184_set__ConsD,axiom,
! [A: $tType,Y: A,X5: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X5 @ Xs ) ) )
=> ( ( Y = X5 )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_185_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_186_list_Oset__intros_I2_J,axiom,
! [A: $tType,X5: A,A22: list @ A,A1: A] :
( ( member @ A @ X5 @ ( set2 @ A @ A22 ) )
=> ( member @ A @ X5 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_187_cachedWeight_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma787811817Weight @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= W ) ).
% cachedWeight.simps(2)
thf(fact_188_cachedWeight_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A3: A] :
( ( huffma787811817Weight @ A @ ( huffma1554276827e_Leaf @ A @ W @ A3 ) )
= W ) ).
% cachedWeight.simps(1)
thf(fact_189_not__Cons__self2,axiom,
! [A: $tType,X5: A,Xs: list @ A] :
( ( cons @ A @ X5 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_190_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B5: $tType,P: ( A > B5 ) > ( list @ A ) > ( list @ B5 ) > $o,A0: A > B5,A1: list @ A,A22: list @ B5] :
( ! [F3: A > B5,X1: list @ B5] : ( P @ F3 @ ( nil @ A ) @ X1 )
=> ( ! [F3: A > B5,A2: A,As: list @ A,Bs: list @ B5] :
( ( P @ F3 @ As @ ( cons @ B5 @ ( F3 @ A2 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_191_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_192_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y4: A,Xs2: list @ A] :
( ( ( X = Y4 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_193_remdups__adj_Ocases,axiom,
! [A: $tType,X5: list @ A] :
( ( X5
!= ( nil @ A ) )
=> ( ! [X: A] :
( X5
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y4: A,Xs2: list @ A] :
( X5
!= ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_194_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [V2: A,Va2: list @ A] : ( P @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y4: A,Ys: list @ A] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y4 @ Ys ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_195_list__induct2_H,axiom,
! [A: $tType,B5: $tType,P: ( list @ A ) > ( list @ B5 ) > $o,Xs: list @ A,Ys2: list @ B5] :
( ( P @ ( nil @ A ) @ ( nil @ B5 ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B5 ) )
=> ( ! [Y4: B5,Ys: list @ B5] : ( P @ ( nil @ A ) @ ( cons @ B5 @ Y4 @ Ys ) )
=> ( ! [X: A,Xs2: list @ A,Y4: B5,Ys: list @ B5] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B5 @ Y4 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_196_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_197_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X2: list @ A] :
( ( P @ X2 )
=> ( P @ ( cons @ A @ X1 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_198_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_199_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_200_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_201_transpose_Ocases,axiom,
! [A: $tType,X5: list @ ( list @ A )] :
( ( X5
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X5
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X5
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_202_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% set_union
thf(fact_203_height__0__imp__cachedWeight__eq__weight,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( ( huffma1554076246height @ A @ T )
= ( zero_zero @ nat ) )
=> ( ( huffma787811817Weight @ A @ T )
= ( huffma691733767weight @ A @ T ) ) ) ).
% height_0_imp_cachedWeight_eq_weight
thf(fact_204_the__elem__set,axiom,
! [A: $tType,X5: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
= X5 ) ).
% the_elem_set
thf(fact_205_weight_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A3: A] :
( ( huffma691733767weight @ A @ ( huffma1554276827e_Leaf @ A @ W @ A3 ) )
= W ) ).
% weight.simps(1)
thf(fact_206_weight_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma691733767weight @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ A @ T_1 ) @ ( huffma691733767weight @ A @ T_2 ) ) ) ).
% weight.simps(2)
thf(fact_207_cost_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma636208924e_cost @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ A @ T_1 ) @ ( huffma636208924e_cost @ A @ T_1 ) ) @ ( huffma691733767weight @ A @ T_2 ) ) @ ( huffma636208924e_cost @ A @ T_2 ) ) ) ).
% cost.simps(2)
thf(fact_208_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_209_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_210_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_211_huffman_Opelims,axiom,
! [A: $tType,X5: list @ ( huffma16452318e_tree @ A ),Y: huffma16452318e_tree @ A] :
( ( ( huffma149336734uffman @ A @ X5 )
= Y )
=> ( ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ X5 )
=> ( ! [T3: huffma16452318e_tree @ A] :
( ( X5
= ( cons @ ( huffma16452318e_tree @ A ) @ T3 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( ( Y = T3 )
=> ~ ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ ( cons @ ( huffma16452318e_tree @ A ) @ T3 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( ( X5
= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) )
=> ( ( Y
= ( huffma149336734uffman @ A @ ( huffma725507568rtTree @ A @ ( huffma453905539eTrees @ A @ T_12 @ T_22 ) @ Ts2 ) ) )
=> ~ ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) ) ) )
=> ~ ( ( X5
= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( ( Y
= ( undefined @ ( huffma16452318e_tree @ A ) ) )
=> ~ ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ) ) ) ) ) ).
% huffman.pelims
thf(fact_212_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% semiring_normalization_rules(6)
thf(fact_213_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% semiring_normalization_rules(5)
thf(fact_214_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A,C: A,D: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_215_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,C4: A] : ( plus_plus @ A @ C4 @ A6 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_216_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B ) @ C )
= ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C ) @ B ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_217_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A,C: A,D: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ C @ ( plus_plus @ A @ A3 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_218_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B ) @ C )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_219_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B ) @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_220_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B: A,A3: A] :
( ( B
= ( plus_plus @ A @ B @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_221_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_222_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_223_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A3: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B ) @ C )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_224_depth_Osimps_I2_J,axiom,
! [A: $tType,A3: A,T_1: huffma16452318e_tree @ A,W: nat,T_2: huffma16452318e_tree @ A] :
( ( ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma223349076_depth @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A3 )
= ( plus_plus @ nat @ ( huffma223349076_depth @ A @ T_1 @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_2 ) )
=> ( ( huffma223349076_depth @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A3 )
= ( plus_plus @ nat @ ( huffma223349076_depth @ A @ T_2 @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T_2 ) )
=> ( ( huffma223349076_depth @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% depth.simps(2)
thf(fact_225_count__notin,axiom,
! [A: $tType,X5: A,Xs: list @ A] :
( ~ ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X5 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_226_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_227_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_228_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_229_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_230_count__list_Osimps_I2_J,axiom,
! [A: $tType,X5: A,Y: A,Xs: list @ A] :
( ( ( X5 = Y )
=> ( ( count_list @ A @ ( cons @ A @ X5 @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X5 != Y )
=> ( ( count_list @ A @ ( cons @ A @ X5 @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_231_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( ( one_one @ A )
= X5 )
= ( X5
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_232_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_233_zero__less__two,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_234_fold__atLeastAtMost__nat_Oinduct,axiom,
! [A: $tType,P: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
( ! [F3: nat > A > A,A2: nat,B2: nat,Acc: A] :
( ( ~ ( ord_less @ nat @ B2 @ A2 )
=> ( P @ F3 @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F3 @ A2 @ Acc ) ) )
=> ( P @ F3 @ A2 @ B2 @ Acc ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ).
% fold_atLeastAtMost_nat.induct
thf(fact_235_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( X5 != Y )
=> ( ~ ( ord_less @ A @ X5 @ Y )
=> ( ord_less @ A @ Y @ X5 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_236_bounded__nat__set__is__finite,axiom,
! [N3: set @ nat,N: nat] :
( ! [X: nat] :
( ( member @ nat @ X @ N3 )
=> ( ord_less @ nat @ X @ N ) )
=> ( finite_finite2 @ nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_237_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N4: set @ nat] :
? [M3: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N4 )
=> ( ord_less @ nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_238_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_239_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_240_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_241_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_242_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_243_sublist__singleton,axiom,
! [A: $tType,A4: set @ nat,X5: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( sublist @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ A4 )
= ( cons @ A @ X5 @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( sublist @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ A4 )
= ( nil @ A ) ) ) ) ).
% sublist_singleton
thf(fact_244_sublist__nil,axiom,
! [A: $tType,A4: set @ nat] :
( ( sublist @ A @ ( nil @ A ) @ A4 )
= ( nil @ A ) ) ).
% sublist_nil
thf(fact_245_in__set__sublistD,axiom,
! [A: $tType,X5: A,Xs: list @ A,I2: set @ nat] :
( ( member @ A @ X5 @ ( set2 @ A @ ( sublist @ A @ Xs @ I2 ) ) )
=> ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_sublistD
thf(fact_246_notin__set__sublistI,axiom,
! [A: $tType,X5: A,Xs: list @ A,I2: set @ nat] :
( ~ ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X5 @ ( set2 @ A @ ( sublist @ A @ Xs @ I2 ) ) ) ) ).
% notin_set_sublistI
thf(fact_247_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_248_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_249_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo292404081st_nat @ A )
= ( ^ [F2: nat > A > A,A6: nat,B3: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A6 ) @ Acc2 @ ( set_fo292404081st_nat @ A @ F2 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A6 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_250_sgn0,axiom,
! [A: $tType] :
( ( sgn_if @ A @ ( type2 @ A ) )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn0
thf(fact_251_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
%----Type constructors (33)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_1,axiom,
semilattice_sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_4,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_6,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_7,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
lattice @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X5: A,Y: A] :
( ( if @ A @ $false @ X5 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X5: A,Y: A] :
( ( if @ A @ $true @ X5 @ Y )
= X5 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t ) ) ).
thf(conj_1,conjecture,
( ( huffma943100115ibling @ a @ t @ a2 )
= a2 ) ).
%------------------------------------------------------------------------------