TPTP Problem File: DAT173^1.p
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%------------------------------------------------------------------------------
% File : DAT173^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Huffman 2222
% 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__2222.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 359 ( 98 unt; 69 typ; 0 def)
% Number of atoms : 781 ( 253 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 4689 ( 85 ~; 5 |; 32 &;4140 @)
% ( 0 <=>; 427 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 162 ( 162 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 9 con; 0-6 aty)
% Number of variables : 1089 ( 62 ^; 961 !; 10 ?;1089 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:44:17.483
%------------------------------------------------------------------------------
%----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 (63)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[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_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[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_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > 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_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_Ominima,type,
huffma1154738298minima:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ooptimum,type,
huffma936049440ptimum:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OsplitLeaf,type,
huffma454997449itLeaf:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OsplitLeaf_092_060_094sub_062F,type,
huffma130202051Leaf_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > nat > A > nat > A > ( list @ ( huffma16452318e_tree @ A ) ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapFourSyms,type,
huffma304375860urSyms:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > A > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapLeaves,type,
huffma2094459102Leaves:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapSyms,type,
huffma469337550apSyms:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > 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_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
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__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[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_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_b,type,
b: a ).
thf(sy_v_t_092_060_094sub_0621____,type,
t_1: huffma16452318e_tree @ a ).
thf(sy_v_t_092_060_094sub_0622____,type,
t_2: huffma16452318e_tree @ a ).
thf(sy_v_tsa____,type,
tsa: list @ ( huffma16452318e_tree @ a ) ).
thf(sy_v_w_092_060_094sub_062a,type,
w_a: nat ).
thf(sy_v_w_092_060_094sub_062b,type,
w_b: nat ).
%----Relevant facts (255)
thf(fact_0__092_060open_062a_A_092_060notin_062_Aalphabet_092_060_094sub_062F_Ats_092_060close_062,axiom,
~ ( member @ a @ a2 @ ( huffma279473244abet_F @ a @ tsa ) ) ).
% \<open>a \<notin> alphabet\<^sub>F ts\<close>
thf(fact_1__092_060open_062a_A_092_060notin_062_Aalphabet_At_092_060_094sub_0622_092_060close_062,axiom,
~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_2 ) ) ).
% \<open>a \<notin> alphabet t\<^sub>2\<close>
thf(fact_2_True,axiom,
member @ a @ a2 @ ( huffma505251170phabet @ a @ t_1 ) ).
% True
thf(fact_3_hyps_I2_J,axiom,
huffma2111480347tent_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t_1 @ ( cons @ ( huffma16452318e_tree @ a ) @ t_2 @ tsa ) ) ).
% hyps(2)
thf(fact_4_hyps_I3_J,axiom,
( ( cons @ ( huffma16452318e_tree @ a ) @ t_1 @ ( cons @ ( huffma16452318e_tree @ a ) @ t_2 @ tsa ) )
!= ( nil @ ( huffma16452318e_tree @ a ) ) ) ).
% hyps(3)
thf(fact_5_hyps_I4_J,axiom,
member @ a @ a2 @ ( huffma279473244abet_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t_1 @ ( cons @ ( huffma16452318e_tree @ a ) @ t_2 @ tsa ) ) ) ).
% hyps(4)
thf(fact_6_hyps_I5_J,axiom,
( ( huffma2047054433freq_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t_1 @ ( cons @ ( huffma16452318e_tree @ a ) @ t_2 @ tsa ) ) @ a2 )
= ( plus_plus @ nat @ w_a @ w_b ) ) ).
% hyps(5)
thf(fact_7__C2_Ohyps_C,axiom,
( ( huffma2111480347tent_F @ a @ ( huffma725507568rtTree @ a @ ( huffma453905539eTrees @ a @ t_1 @ t_2 ) @ tsa ) )
=> ( ( ( huffma725507568rtTree @ a @ ( huffma453905539eTrees @ a @ t_1 @ t_2 ) @ tsa )
!= ( nil @ ( huffma16452318e_tree @ a ) ) )
=> ( ( member @ a @ a2 @ ( huffma279473244abet_F @ a @ ( huffma725507568rtTree @ a @ ( huffma453905539eTrees @ a @ t_1 @ t_2 ) @ tsa ) ) )
=> ( ( ( huffma2047054433freq_F @ a @ ( huffma725507568rtTree @ a @ ( huffma453905539eTrees @ a @ t_1 @ t_2 ) @ tsa ) @ a2 )
= ( plus_plus @ nat @ w_a @ w_b ) )
=> ( ( huffma454997449itLeaf @ a @ ( huffma149336734uffman @ a @ ( huffma725507568rtTree @ a @ ( huffma453905539eTrees @ a @ t_1 @ t_2 ) @ tsa ) ) @ w_a @ a2 @ w_b @ b )
= ( huffma149336734uffman @ a @ ( huffma130202051Leaf_F @ a @ ( huffma725507568rtTree @ a @ ( huffma453905539eTrees @ a @ t_1 @ t_2 ) @ tsa ) @ w_a @ a2 @ w_b @ b ) ) ) ) ) ) ) ).
% "2.hyps"
thf(fact_8_notin__alphabet_092_060_094sub_062F__imp__splitLeaf_092_060_094sub_062F__id,axiom,
! [A: $tType,A2: A,Ts: list @ ( huffma16452318e_tree @ A ),W_a: nat,W_b: nat,B: A] :
( ~ ( member @ A @ A2 @ ( huffma279473244abet_F @ A @ Ts ) )
=> ( ( huffma130202051Leaf_F @ A @ Ts @ W_a @ A2 @ W_b @ B )
= Ts ) ) ).
% notin_alphabet\<^sub>F_imp_splitLeaf\<^sub>F_id
thf(fact_9_splitLeaf_092_060_094sub_062F_Osimps_I2_J,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A ),W_a: nat,A2: A,W_b: nat,B: A] :
( ( huffma130202051Leaf_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) @ W_a @ A2 @ W_b @ B )
= ( cons @ ( huffma16452318e_tree @ A ) @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) @ ( huffma130202051Leaf_F @ A @ Ts @ W_a @ A2 @ W_b @ B ) ) ) ).
% splitLeaf\<^sub>F.simps(2)
thf(fact_10_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 ) ) ).
% notin_alphabet_imp_splitLeaf_id
thf(fact_11_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_12_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_13_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_14_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_15_splitLeaf_092_060_094sub_062F_Osimps_I1_J,axiom,
! [A: $tType,W_a: nat,A2: A,W_b: nat,B: A] :
( ( huffma130202051Leaf_F @ A @ ( nil @ ( huffma16452318e_tree @ A ) ) @ W_a @ A2 @ W_b @ B )
= ( nil @ ( huffma16452318e_tree @ A ) ) ) ).
% splitLeaf\<^sub>F.simps(1)
thf(fact_16_huffman_Ocases,axiom,
! [A: $tType,X: list @ ( huffma16452318e_tree @ A )] :
( ! [T2: huffma16452318e_tree @ A] :
( X
!= ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( X
!= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) )
=> ( X
= ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ) ).
% huffman.cases
thf(fact_17_sortedByWeight_Ocases,axiom,
! [A: $tType,X: list @ ( huffma16452318e_tree @ A )] :
( ( X
!= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( ! [T2: huffma16452318e_tree @ A] :
( X
!= ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ~ ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( X
!= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) ) ) ) ).
% sortedByWeight.cases
thf(fact_18_sortedByWeight_Oinduct,axiom,
! [A: $tType,P: ( list @ ( huffma16452318e_tree @ A ) ) > $o,A0: list @ ( huffma16452318e_tree @ A )] :
( ( P @ ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( ! [T2: huffma16452318e_tree @ A] : ( P @ ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( ( P @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) )
=> ( P @ ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% sortedByWeight.induct
thf(fact_19_cachedWeight__splitLeaf,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
( ( huffma787811817Weight @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( huffma787811817Weight @ A @ T ) ) ).
% cachedWeight_splitLeaf
thf(fact_20_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_21_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_22_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_23_splitLeaf_092_060_094sub_062F__insortTree__when__in__alphabet_092_060_094sub_062F__tail,axiom,
! [A: $tType,A2: A,Ts: list @ ( huffma16452318e_tree @ A ),T: huffma16452318e_tree @ A,W_a: nat,W_b: nat,B: A] :
( ( member @ A @ A2 @ ( huffma279473244abet_F @ A @ Ts ) )
=> ( ( huffma2111480347tent_F @ A @ Ts )
=> ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( huffma2047054433freq_F @ A @ Ts @ A2 )
= ( plus_plus @ nat @ W_a @ W_b ) )
=> ( ( huffma130202051Leaf_F @ A @ ( huffma725507568rtTree @ A @ T @ Ts ) @ W_a @ A2 @ W_b @ B )
= ( huffma725507568rtTree @ A @ T @ ( huffma130202051Leaf_F @ A @ Ts @ W_a @ A2 @ W_b @ B ) ) ) ) ) ) ) ).
% splitLeaf\<^sub>F_insortTree_when_in_alphabet\<^sub>F_tail
thf(fact_24_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_25_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_26_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_27_exists__in__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
? [A3: A] : ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) ) ).
% exists_in_alphabet
thf(fact_28_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B2: $tType,P: ( A > B2 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2,A1: list @ A,A22: list @ B2] :
( ! [F: A > B2,X1: list @ B2] : ( P @ F @ ( nil @ A ) @ X1 )
=> ( ! [F: A > B2,A3: A,As: list @ A,Bs: list @ B2] :
( ( P @ F @ As @ ( cons @ B2 @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_29_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_30_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y: A,Xs2: list @ A] :
( ( ( X2 = Y )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons @ A @ Y @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_31_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_32_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y: A,Ys: list @ A] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_33_list__induct2_H,axiom,
! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,Xs: list @ A,Ys2: list @ B2] :
( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B2 ) )
=> ( ! [Y: B2,Ys: list @ B2] : ( P @ ( nil @ A ) @ ( cons @ B2 @ Y @ Ys ) )
=> ( ! [X2: A,Xs2: list @ A,Y: B2,Ys: list @ B2] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B2 @ Y @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_34_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_35_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_36_list_Oexhaust,axiom,
! [A: $tType,Y3: list @ A] :
( ( Y3
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y3
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_37_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_38_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_39_huffman_Oinduct,axiom,
! [A: $tType,P: ( list @ ( huffma16452318e_tree @ A ) ) > $o,A0: list @ ( huffma16452318e_tree @ A )] :
( ! [T2: huffma16452318e_tree @ A] : ( P @ ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( 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_40_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 ) ) ) ).
% add_left_cancel
thf(fact_41_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 ) ) ) ).
% add_right_cancel
thf(fact_42_huffman_Oelims,axiom,
! [A: $tType,X: list @ ( huffma16452318e_tree @ A ),Y3: huffma16452318e_tree @ A] :
( ( ( huffma149336734uffman @ A @ X )
= Y3 )
=> ( ! [T2: huffma16452318e_tree @ A] :
( ( X
= ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( Y3 != T2 ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( ( X
= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) )
=> ( Y3
!= ( huffma149336734uffman @ A @ ( huffma725507568rtTree @ A @ ( huffma453905539eTrees @ A @ T_12 @ T_22 ) @ Ts2 ) ) ) )
=> ~ ( ( X
= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( Y3
!= ( undefined @ ( huffma16452318e_tree @ A ) ) ) ) ) ) ) ).
% huffman.elims
thf(fact_43_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B2: $tType,A: $tType,F2: A > B2,G: A > B2] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_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_49_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 ) ) ) ) ) ) ).
% alphabet_swapFourSyms
thf(fact_50_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_51_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 ) )
= ( ^ [A5: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T @ A5 ) @ ( huffma2047054433freq_F @ A @ Ts @ A5 ) ) ) ) ).
% freq\<^sub>F_insortTree
thf(fact_52_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_53_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_54_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_55_freq__uniteTrees,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma854352999e_freq @ A @ ( huffma453905539eTrees @ A @ T_1 @ T_2 ) )
= ( ^ [A5: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T_1 @ A5 ) @ ( huffma854352999e_freq @ A @ T_2 @ A5 ) ) ) ) ).
% freq_uniteTrees
thf(fact_56_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 ) ) ) ) ) ) ) ).
% freq_swapFourSyms
thf(fact_57_splitLeaf_092_060_094sub_062F__insortTree__when__in__alphabet__left,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A ),W_a: nat,W_b: nat,B: A] :
( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma1050891809istent @ A @ T )
=> ( ~ ( member @ A @ A2 @ ( huffma279473244abet_F @ A @ Ts ) )
=> ( ( ( huffma854352999e_freq @ A @ T @ A2 )
= ( plus_plus @ nat @ W_a @ W_b ) )
=> ( ( huffma130202051Leaf_F @ A @ ( huffma725507568rtTree @ A @ T @ Ts ) @ W_a @ A2 @ W_b @ B )
= ( huffma725507568rtTree @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) @ Ts ) ) ) ) ) ) ).
% splitLeaf\<^sub>F_insortTree_when_in_alphabet_left
thf(fact_58_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 ) ) ) ).
% consistent_swapFourSyms
thf(fact_59_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 ) ) ) ) ).
% consistent_splitLeaf
thf(fact_60_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 ) ) ) ).
% add_right_imp_eq
thf(fact_61_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 ) ) ) ).
% add_left_imp_eq
thf(fact_62_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 ) ) ) ) ).
% add.left_commute
thf(fact_63_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% add.commute
thf(fact_64_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 ) ) ) ).
% add.right_cancel
thf(fact_65_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 ) ) ) ).
% add.left_cancel
thf(fact_66_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 ) ) ) ) ).
% add.assoc
thf(fact_67_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_68_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 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_69_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_70_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_71_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 ) ) ) ) ) ).
% cost_splitLeaf
thf(fact_72_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_73_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_74_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ B )
= ( sup_sup @ A @ A2 @ B ) ) ) ).
% sup.right_idem
thf(fact_75_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
= ( sup_sup @ A @ X @ Y3 ) ) ) ).
% sup_left_idem
thf(fact_76_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) )
= ( sup_sup @ A @ A2 @ B ) ) ) ).
% sup.left_idem
thf(fact_77_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_78_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_79_sup__apply,axiom,
! [B2: $tType,A: $tType] :
( ( semilattice_sup @ B2 @ ( type2 @ B2 ) )
=> ( ( sup_sup @ ( A > B2 ) )
= ( ^ [F3: A > B2,G2: A > B2,X3: A] : ( sup_sup @ B2 @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_80_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
= ( sup_sup @ A @ X @ Y3 ) ) ) ).
% inf_sup_aci(8)
thf(fact_81_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
= ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_82_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_83_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_84_sup__fun__def,axiom,
! [B2: $tType,A: $tType] :
( ( semilattice_sup @ B2 @ ( type2 @ B2 ) )
=> ( ( sup_sup @ ( A > B2 ) )
= ( ^ [F3: A > B2,G2: A > B2,X3: A] : ( sup_sup @ B2 @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_85_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ C )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% sup.assoc
thf(fact_86_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% sup_assoc
thf(fact_87_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B3: A] : ( sup_sup @ A @ B3 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_88_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_89_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( sup_sup @ A @ B @ ( sup_sup @ A @ A2 @ C ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% sup.left_commute
thf(fact_90_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
= ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_91_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_92_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_93_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_94_bex__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_95_ball__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_96_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_97_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_98_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Un_commute
thf(fact_99_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_100_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_101_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 ) ) ) ) ) ).
% weight_splitLeaf
thf(fact_102_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_103_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 ) ) ) ) ) ).
% freq_swapSyms
thf(fact_104_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_105_huffman_Opelims,axiom,
! [A: $tType,X: list @ ( huffma16452318e_tree @ A ),Y3: huffma16452318e_tree @ A] :
( ( ( huffma149336734uffman @ A @ X )
= Y3 )
=> ( ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ X )
=> ( ! [T2: huffma16452318e_tree @ A] :
( ( X
= ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) )
=> ( ( Y3 = T2 )
=> ~ ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ) )
=> ( ! [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A,Ts2: list @ ( huffma16452318e_tree @ A )] :
( ( X
= ( cons @ ( huffma16452318e_tree @ A ) @ T_12 @ ( cons @ ( huffma16452318e_tree @ A ) @ T_22 @ Ts2 ) ) )
=> ( ( Y3
= ( 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 ) ) ) ) )
=> ~ ( ( X
= ( nil @ ( huffma16452318e_tree @ A ) ) )
=> ( ( Y3
= ( undefined @ ( huffma16452318e_tree @ A ) ) )
=> ~ ( accp @ ( list @ ( huffma16452318e_tree @ A ) ) @ ( huffma316836827an_rel @ A ) @ ( nil @ ( huffma16452318e_tree @ A ) ) ) ) ) ) ) ) ) ).
% huffman.pelims
thf(fact_106_optimum__splitLeaf,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma936049440ptimum @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( huffma854352999e_freq @ A @ T @ A2 )
= ( plus_plus @ nat @ W_a @ W_b ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ord_less_eq @ nat @ W_a @ ( huffma854352999e_freq @ A @ T @ X2 ) )
& ( ord_less_eq @ nat @ W_b @ ( huffma854352999e_freq @ A @ T @ X2 ) ) ) )
=> ( ( ord_less_eq @ nat @ W_a @ W_b )
=> ( huffma936049440ptimum @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ) ) ) ) ) ) ).
% optimum_splitLeaf
thf(fact_107_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 ) ) ) ).
% add_le_cancel_right
thf(fact_108_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 ) ) ) ).
% add_le_cancel_left
thf(fact_109_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_110_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_111_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 ) ) ) ) ).
% alphabet_swapSyms
thf(fact_112_swapSyms__id,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma469337550apSyms @ A @ T @ A2 @ A2 )
= T ) ) ).
% swapSyms_id
thf(fact_113_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 ) ) ) ).
% add_le_imp_le_right
thf(fact_114_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 ) ) ) ).
% add_le_imp_le_left
thf(fact_115_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
? [C3: A] :
( B3
= ( plus_plus @ A @ A5 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_116_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 ) ) ) ) ).
% add_right_mono
thf(fact_117_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 ) ) ) ) ).
% add_left_mono
thf(fact_118_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 ) ) ) ) ) ).
% add_mono
thf(fact_119_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 ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_120_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 ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_121_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 ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_122_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% inf_sup_ord(4)
thf(fact_123_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% inf_sup_ord(3)
thf(fact_124_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B @ X ) ) ) ) ).
% le_supE
thf(fact_125_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ X ) ) ) ) ).
% le_supI
thf(fact_126_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% sup_ge1
thf(fact_127_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% sup_ge2
thf(fact_128_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% le_supI1
thf(fact_129_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B: A,A2: A] :
( ( ord_less_eq @ A @ X @ B )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% le_supI2
thf(fact_130_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,D: A,B: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ( ord_less_eq @ A @ D @ B )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D ) @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% sup.mono
thf(fact_131_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B @ D )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ ( sup_sup @ A @ C @ D ) ) ) ) ) ).
% sup_mono
thf(fact_132_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_133_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_134_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% sup.orderE
thf(fact_135_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B ) )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% sup.orderI
thf(fact_136_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F2: A > A > A,X: A,Y3: A] :
( ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( F2 @ X2 @ Y ) )
=> ( ! [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ ( F2 @ X2 @ Y ) )
=> ( ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ ( F2 @ Y @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X @ Y3 )
= ( F2 @ X @ Y3 ) ) ) ) ) ) ).
% sup_unique
thf(fact_137_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( sup_sup @ A @ A2 @ B )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_138_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( sup_sup @ A @ A2 @ B )
= B ) ) ) ).
% sup.absorb2
thf(fact_139_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( sup_sup @ A @ X @ Y3 )
= X ) ) ) ).
% sup_absorb1
thf(fact_140_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( sup_sup @ A @ X @ Y3 )
= Y3 ) ) ) ).
% sup_absorb2
thf(fact_141_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B @ A2 )
=> ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_142_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_143_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B3 ) ) ) ) ) ).
% sup.order_iff
thf(fact_144_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) ) ) ).
% sup.cobounded1
thf(fact_145_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] : ( ord_less_eq @ A @ B @ ( sup_sup @ A @ A2 @ B ) ) ) ).
% sup.cobounded2
thf(fact_146_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( sup_sup @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_147_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( sup_sup @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_148_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% sup.coboundedI1
thf(fact_149_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,B: A,A2: A] :
( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% sup.coboundedI2
thf(fact_150_consistent__swapSyms,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( huffma1050891809istent @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) ) ).
% consistent_swapSyms
thf(fact_151_swapFourSyms__def,axiom,
! [A: $tType] :
( ( huffma304375860urSyms @ A )
= ( ^ [T3: huffma16452318e_tree @ A,A5: A,B3: A,C3: A,D2: A] : ( if @ ( huffma16452318e_tree @ A ) @ ( A5 = D2 ) @ ( huffma469337550apSyms @ A @ T3 @ B3 @ C3 ) @ ( if @ ( huffma16452318e_tree @ A ) @ ( B3 = C3 ) @ ( huffma469337550apSyms @ A @ T3 @ A5 @ D2 ) @ ( huffma469337550apSyms @ A @ ( huffma469337550apSyms @ A @ T3 @ A5 @ C3 ) @ B3 @ D2 ) ) ) ) ) ).
% swapFourSyms_def
thf(fact_152_optimum__def,axiom,
! [A: $tType] :
( ( huffma936049440ptimum @ A )
= ( ^ [T3: huffma16452318e_tree @ A] :
! [U2: huffma16452318e_tree @ A] :
( ( huffma1050891809istent @ A @ U2 )
=> ( ( ( huffma505251170phabet @ A @ T3 )
= ( huffma505251170phabet @ A @ U2 ) )
=> ( ( ( huffma854352999e_freq @ A @ T3 )
= ( huffma854352999e_freq @ A @ U2 ) )
=> ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ T3 ) @ ( huffma636208924e_cost @ A @ U2 ) ) ) ) ) ) ) ).
% optimum_def
thf(fact_153_insortTree_Osimps_I2_J,axiom,
! [A: $tType,U: huffma16452318e_tree @ A,T: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( ( ord_less_eq @ nat @ ( huffma787811817Weight @ A @ U ) @ ( huffma787811817Weight @ A @ T ) )
=> ( ( huffma725507568rtTree @ A @ U @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) )
= ( cons @ ( huffma16452318e_tree @ A ) @ U @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( huffma787811817Weight @ A @ U ) @ ( huffma787811817Weight @ A @ T ) )
=> ( ( huffma725507568rtTree @ A @ U @ ( cons @ ( huffma16452318e_tree @ A ) @ T @ Ts ) )
= ( cons @ ( huffma16452318e_tree @ A ) @ T @ ( huffma725507568rtTree @ A @ U @ Ts ) ) ) ) ) ).
% insortTree.simps(2)
thf(fact_154_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_155_minima__def,axiom,
! [A: $tType] :
( ( huffma1154738298minima @ A )
= ( ^ [T3: huffma16452318e_tree @ A,A5: A,B3: A] :
( ( member @ A @ A5 @ ( huffma505251170phabet @ A @ T3 ) )
& ( member @ A @ B3 @ ( huffma505251170phabet @ A @ T3 ) )
& ( A5 != B3 )
& ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T3 @ A5 ) @ ( huffma854352999e_freq @ A @ T3 @ B3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( huffma505251170phabet @ A @ T3 ) )
=> ( ( X3 != A5 )
=> ( ( X3 != B3 )
=> ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T3 @ A5 ) @ ( huffma854352999e_freq @ A @ T3 @ X3 ) )
& ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T3 @ B3 ) @ ( huffma854352999e_freq @ A @ T3 @ X3 ) ) ) ) ) ) ) ) ) ).
% minima_def
thf(fact_156_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_157_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 ) ) ) ) ) ) ) ).
% cost_swapSyms_le
thf(fact_158_weight__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( A2 != B )
=> ( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ B ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ A @ T ) @ W_a ) @ W_b ) ) )
& ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ A @ T ) @ W_b ) ) ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) @ ( huffma854352999e_freq @ A @ T @ B ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ A @ T ) @ W_a ) ) )
& ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( huffma691733767weight @ A @ T ) ) ) ) ) ) ) ) ).
% weight_swapLeaves
thf(fact_159_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C2 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_160_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 ) ) ).
% swapLeaves_id_when_notin_alphabet
thf(fact_161_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 ) ) ).
% swapLeaves_id
thf(fact_162_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 ) ) ) ) ) ).
% depth_swapLeaves_neither
thf(fact_163_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 ) ) ) ) ) ).
% depth_swapSyms_neither
thf(fact_164_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_165_Un__absorb2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_166_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_167_Un__upper2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_168_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_169_Un__least,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_170_Un__mono,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B4: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_171_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 ) ) ) ).
% consistent_swapLeaves
thf(fact_172_le__funD,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ! [F2: A > B2,G: A > B2,X: A] :
( ( ord_less_eq @ ( A > B2 ) @ F2 @ G )
=> ( ord_less_eq @ B2 @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_173_le__funE,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ! [F2: A > B2,G: A > B2,X: A] :
( ( ord_less_eq @ ( A > B2 ) @ F2 @ G )
=> ( ord_less_eq @ B2 @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_174_le__funI,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ! [F2: A > B2,G: A > B2] :
( ! [X2: A] : ( ord_less_eq @ B2 @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B2 ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_175_le__fun__def,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ( ( ord_less_eq @ ( A > B2 ) )
= ( ^ [F3: A > B2,G2: A > B2] :
! [X3: A] : ( ord_less_eq @ B2 @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_176_order__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 @ ( type2 @ B2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B2 > A,B: B2,C: B2] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq @ B2 @ B @ C )
=> ( ! [X2: B2,Y: B2] :
( ( ord_less_eq @ B2 @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_177_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C4 @ ( F2 @ B ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C4 @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ C4 @ ( F2 @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_178_ord__eq__le__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 @ ( type2 @ B2 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B2 > A,B: B2,C: B2] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq @ B2 @ B @ C )
=> ( ! [X2: B2,Y: B2] :
( ( ord_less_eq @ B2 @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_179_ord__le__eq__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 @ ( type2 @ B2 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > B2,C: B2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B2 @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq @ B2 @ ( F2 @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_180_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_181_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X )
=> ( X = Y3 ) ) ) ) ).
% antisym
thf(fact_182_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% linear
thf(fact_183_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( X = Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% eq_refl
thf(fact_184_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% le_cases
thf(fact_185_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 ) ) ) ) ).
% order.trans
thf(fact_186_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_187_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_less_eq @ A @ X @ Y3 )
= ( X = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_188_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 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_189_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 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_190_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 ) ) ) ) ).
% order_class.order.antisym
thf(fact_191_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_192_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_193_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B: A] :
( ! [A3: A,B6: A] :
( ( ord_less_eq @ A @ A3 @ B6 )
=> ( P @ A3 @ B6 ) )
=> ( ! [A3: A,B6: A] :
( ( P @ B6 @ A3 )
=> ( P @ A3 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_wlog
thf(fact_194_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_195_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( A2 = B ) ) ) ) ).
% dual_order.antisym
thf(fact_196_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_197_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_198_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_199_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 ) ) ) ).
% le_trans
thf(fact_200_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_201_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_202_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_203_swapSyms__def,axiom,
! [A: $tType] :
( ( huffma469337550apSyms @ A )
= ( ^ [T3: huffma16452318e_tree @ A,A5: A,B3: A] : ( huffma2094459102Leaves @ A @ T3 @ ( huffma854352999e_freq @ A @ T3 @ A5 ) @ A5 @ ( huffma854352999e_freq @ A @ T3 @ B3 ) @ B3 ) ) ) ).
% swapSyms_def
thf(fact_204_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_205_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_206_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_207_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 ) ) ) ).
% add_le_mono1
thf(fact_208_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 ) ) ) ) ).
% add_le_mono
thf(fact_209_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_210_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_211_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_212_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_213_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_214_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_215_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 ) ) ) ) ) ) ) ) ) ).
% cost_swapFourSyms_le
thf(fact_216_cost__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 ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ A2 ) ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ B ) @ ( huffma223349076_depth @ A @ T @ B ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ B ) ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ B ) @ ( huffma223349076_depth @ A @ T @ A2 ) ) ) ) ) ) ) ).
% cost_swapSyms
thf(fact_217_cost__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( A2 != B )
=> ( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ A2 ) ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ B ) @ ( huffma223349076_depth @ A @ T @ B ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ ( times_times @ nat @ W_a @ ( huffma223349076_depth @ A @ T @ B ) ) ) @ ( times_times @ nat @ W_b @ ( huffma223349076_depth @ A @ T @ A2 ) ) ) ) )
& ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ A2 ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ ( times_times @ nat @ W_b @ ( huffma223349076_depth @ A @ T @ A2 ) ) ) ) ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ A @ T @ B ) @ ( huffma223349076_depth @ A @ T @ B ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ ( times_times @ nat @ W_a @ ( huffma223349076_depth @ A @ T @ B ) ) ) ) )
& ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma636208924e_cost @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ) ).
% cost_swapLeaves
thf(fact_218_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_219_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_220_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 ) ) ).
% height_swapLeaves
thf(fact_221_set__mp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_222_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_223_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetD
thf(fact_224_subsetCE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetCE
thf(fact_225_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_226_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_227_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_228_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_229_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_230_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_231_rev__subsetD,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% rev_subsetD
thf(fact_232_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_233_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_234_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_235_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_236_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ~ ( member @ A @ C @ B4 )
=> ~ ( member @ A @ C @ A4 ) ) ) ).
% contra_subsetD
thf(fact_237_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_238_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_239_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_240_mult__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 @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_241_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_242_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_243_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_244_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_245_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_246_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.assoc
thf(fact_247_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ B3 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_248_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.left_commute
thf(fact_249_le__le__imp__sum__mult__le__sum__mult,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ N ) @ ( times_times @ nat @ J @ M ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ I @ M ) @ ( times_times @ nat @ J @ N ) ) ) ) ) ).
% le_le_imp_sum_mult_le_sum_mult
thf(fact_250_depth__le__height,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] : ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma1554076246height @ A @ T ) ) ).
% depth_le_height
thf(fact_251_exists__at__height,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( huffma1050891809istent @ A @ T )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( huffma505251170phabet @ A @ T ) )
& ( ( huffma223349076_depth @ A @ T @ X2 )
= ( huffma1554076246height @ A @ T ) ) ) ) ).
% exists_at_height
thf(fact_252_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_253_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_254_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,E: A,B: A,C: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B @ E ) @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ E ) @ C ) ) ) ).
% combine_common_factor
%----Type constructors (31)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
=> ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 @ ( type2 @ A8 ) )
=> ( lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_1,axiom,
semilattice_sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_3,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
semiring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_6,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_14,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_16,axiom,
ord @ $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,X: A,Y3: A] :
( ( if @ A @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $true @ X @ Y3 )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( huffma454997449itLeaf @ a @ ( huffma149336734uffman @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t_1 @ ( cons @ ( huffma16452318e_tree @ a ) @ t_2 @ tsa ) ) ) @ w_a @ a2 @ w_b @ b )
= ( huffma149336734uffman @ a @ ( huffma130202051Leaf_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t_1 @ ( cons @ ( huffma16452318e_tree @ a ) @ t_2 @ tsa ) ) @ w_a @ a2 @ w_b @ b ) ) ) ).
%------------------------------------------------------------------------------