TPTP Problem File: SWW550_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWW550_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Huffman's Algorithm line 1274
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla09] Blanchette (2009), Proof Pearl: Mechanizing the Textbo
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : huff_1274 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 180 ( 52 unt; 50 typ; 0 def)
% Number of atoms : 280 ( 124 equ)
% Maximal formula atoms : 15 ( 1 avg)
% Number of connectives : 193 ( 43 ~; 5 |; 24 &)
% ( 18 <=>; 103 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 40 ( 22 >; 18 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-2 aty)
% Number of functors : 33 ( 33 usr; 10 con; 0-6 aty)
% Number of variables : 404 ( 359 !; 2 ?; 404 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:53
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_t_a,type,
a: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Huffman__Mirabelle__lalbadcutu_Otree,type,
huffma1450048681e_tree: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (45)
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oalphabet,type,
huffma675207370phabet:
!>[A: $tType] : ( huffma1450048681e_tree(A) > fun(A,bool) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_OcachedWeight,type,
huffma854194513Weight:
!>[A: $tType] : ( huffma1450048681e_tree(A) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oconsistent,type,
huffma1518433673istent:
!>[A: $tType] : ( huffma1450048681e_tree(A) > $o ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Ocost,type,
huffma1134658180e_cost:
!>[A: $tType] : ( huffma1450048681e_tree(A) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Ofreq,type,
huffma1352802255e_freq:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * A ) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Osibling,type,
huffma1401021291ibling:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * A ) > A ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_OswapLeaves,type,
huffma414517318Leaves:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * nat * A * nat * A ) > huffma1450048681e_tree(A) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_OInnerNode,type,
huffma1146269203erNode:
!>[A: $tType] : ( ( nat * huffma1450048681e_tree(A) * huffma1450048681e_tree(A) ) > huffma1450048681e_tree(A) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_Otree__case,type,
huffma107959123e_case:
!>[A: $tType,T1: $tType] : ( ( fun(nat,fun(A,T1)) * fun(nat,fun(huffma1450048681e_tree(A),fun(huffma1450048681e_tree(A),T1))) * huffma1450048681e_tree(A) ) > T1 ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_OuniteTrees,type,
huffma921447403eTrees:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * huffma1450048681e_tree(A) ) > huffma1450048681e_tree(A) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oweight,type,
huffma83463279weight:
!>[A: $tType] : ( huffma1450048681e_tree(A) > nat ) ).
tff(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa1:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_aa____,type,
aa: a ).
tff(sy_v_b,type,
b: a ).
tff(sy_v_t_092_060_094isub_0621____,type,
t_1: huffma1450048681e_tree(a) ).
tff(sy_v_t_092_060_094isub_0622____,type,
t_2: huffma1450048681e_tree(a) ).
tff(sy_v_w_092_060_094isub_062a,type,
w_a: nat ).
tff(sy_v_w_092_060_094isub_062b,type,
w_b: nat ).
tff(sy_v_w____,type,
w: nat ).
%----Relevant facts (99)
tff(fact_0_step_092_060_094isub_0621_Oprems,axiom,
aa != b ).
tff(fact_1_step_092_060_094isub_0621_I5_J,axiom,
~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,t_2))) ).
tff(fact_2_step_092_060_094isub_0621_I4_J,axiom,
pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,t_1))) ).
tff(fact_3_step_092_060_094isub_0621_I1_J,axiom,
huffma1518433673istent(a,t_1) ).
tff(fact_4_step_092_060_094isub_0621_I2_J,axiom,
huffma1518433673istent(a,t_2) ).
tff(fact_5_tree_Osimps_I2_J,axiom,
! [B: $tType,Tree21: huffma1450048681e_tree(B),Tree11: huffma1450048681e_tree(B),Nat1: nat,Tree2: huffma1450048681e_tree(B),Tree1: huffma1450048681e_tree(B),Nat: nat] :
( ( huffma1146269203erNode(B,Nat,Tree1,Tree2) = huffma1146269203erNode(B,Nat1,Tree11,Tree21) )
<=> ( ( Nat = Nat1 )
& ( Tree1 = Tree11 )
& ( Tree2 = Tree21 ) ) ) ).
tff(fact_6_step_092_060_094isub_0621_I6_J,axiom,
( ( aa != b )
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,t_1)))
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_1)))
=> ( plus_plus(nat,plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,t_1,w_a,aa,w_b,b)),huffma1352802255e_freq(a,t_1,aa)),huffma1352802255e_freq(a,t_1,b)) = plus_plus(nat,plus_plus(nat,huffma83463279weight(a,t_1),w_a),w_b) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_1)))
=> ( plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,t_1,w_a,aa,w_b,b)),huffma1352802255e_freq(a,t_1,aa)) = plus_plus(nat,huffma83463279weight(a,t_1),w_b) ) ) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,t_1)))
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_1)))
=> ( plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,t_1,w_a,aa,w_b,b)),huffma1352802255e_freq(a,t_1,b)) = plus_plus(nat,huffma83463279weight(a,t_1),w_a) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_1)))
=> ( huffma83463279weight(a,huffma414517318Leaves(a,t_1,w_a,aa,w_b,b)) = huffma83463279weight(a,t_1) ) ) ) ) ) ) ).
tff(fact_7_step_092_060_094isub_0621_I7_J,axiom,
( ( aa != b )
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,t_2)))
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_2)))
=> ( plus_plus(nat,plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,t_2,w_a,aa,w_b,b)),huffma1352802255e_freq(a,t_2,aa)),huffma1352802255e_freq(a,t_2,b)) = plus_plus(nat,plus_plus(nat,huffma83463279weight(a,t_2),w_a),w_b) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_2)))
=> ( plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,t_2,w_a,aa,w_b,b)),huffma1352802255e_freq(a,t_2,aa)) = plus_plus(nat,huffma83463279weight(a,t_2),w_b) ) ) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,t_2)))
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_2)))
=> ( plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,t_2,w_a,aa,w_b,b)),huffma1352802255e_freq(a,t_2,b)) = plus_plus(nat,huffma83463279weight(a,t_2),w_a) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,t_2)))
=> ( huffma83463279weight(a,huffma414517318Leaves(a,t_2,w_a,aa,w_b,b)) = huffma83463279weight(a,t_2) ) ) ) ) ) ) ).
tff(fact_8_swapLeaves_Osimps_I2_J,axiom,
! [A: $tType,B2: A,W_b1: nat,A1: A,W_a1: nat,T_21: huffma1450048681e_tree(A),T_11: huffma1450048681e_tree(A),W: nat] : huffma414517318Leaves(A,huffma1146269203erNode(A,W,T_11,T_21),W_a1,A1,W_b1,B2) = huffma1146269203erNode(A,W,huffma414517318Leaves(A,T_11,W_a1,A1,W_b1,B2),huffma414517318Leaves(A,T_21,W_a1,A1,W_b1,B2)) ).
tff(fact_9_freq_Osimps_I2_J,axiom,
! [A: $tType,T_21: huffma1450048681e_tree(A),T_11: huffma1450048681e_tree(A),W: nat,X: A] : huffma1352802255e_freq(A,huffma1146269203erNode(A,W,T_11,T_21),X) = plus_plus(nat,huffma1352802255e_freq(A,T_11,X),huffma1352802255e_freq(A,T_21,X)) ).
tff(fact_10_weight_Osimps_I2_J,axiom,
! [A: $tType,T_21: huffma1450048681e_tree(A),T_11: huffma1450048681e_tree(A),W: nat] : huffma83463279weight(A,huffma1146269203erNode(A,W,T_11,T_21)) = plus_plus(nat,huffma83463279weight(A,T_11),huffma83463279weight(A,T_21)) ).
tff(fact_11_swapLeaves__id__when__notin__alphabet,axiom,
! [B: $tType,W1: nat,Wa: nat,Ta: huffma1450048681e_tree(B),Ab: B] :
( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,Ta)))
=> ( huffma414517318Leaves(B,Ta,Wa,Ab,W1,Ab) = Ta ) ) ).
tff(fact_12_cost_Osimps_I2_J,axiom,
! [A: $tType,T_21: huffma1450048681e_tree(A),T_11: huffma1450048681e_tree(A),W: nat] : huffma1134658180e_cost(A,huffma1146269203erNode(A,W,T_11,T_21)) = plus_plus(nat,plus_plus(nat,plus_plus(nat,huffma83463279weight(A,T_11),huffma1134658180e_cost(A,T_11)),huffma83463279weight(A,T_21)),huffma1134658180e_cost(A,T_21)) ).
tff(fact_13_freq__uniteTrees,axiom,
! [A: $tType,T_21: huffma1450048681e_tree(A),T_11: huffma1450048681e_tree(A),X: A] : huffma1352802255e_freq(A,huffma921447403eTrees(A,T_11,T_21),X) = plus_plus(nat,huffma1352802255e_freq(A,T_11,X),huffma1352802255e_freq(A,T_21,X)) ).
tff(fact_14_nat__add__left__cancel,axiom,
! [N: nat,M: nat,K1: nat] :
( ( plus_plus(nat,K1,M) = plus_plus(nat,K1,N) )
<=> ( M = N ) ) ).
tff(fact_15_nat__add__right__cancel,axiom,
! [N: nat,K1: nat,M: nat] :
( ( plus_plus(nat,M,K1) = plus_plus(nat,N,K1) )
<=> ( M = N ) ) ).
tff(fact_16_add__left__cancel,axiom,
! [B: $tType] :
( cancel_semigroup_add(B)
=> ! [C2: B,Ba: B,Ab: B] :
( ( plus_plus(B,Ab,Ba) = plus_plus(B,Ab,C2) )
<=> ( Ba = C2 ) ) ) ).
tff(fact_17_add__right__cancel,axiom,
! [B: $tType] :
( cancel_semigroup_add(B)
=> ! [C2: B,Ab: B,Ba: B] :
( ( plus_plus(B,Ba,Ab) = plus_plus(B,C2,Ab) )
<=> ( Ba = C2 ) ) ) ).
tff(fact_18_step_092_060_094isub_0621_I3_J,axiom,
inf_inf(fun(a,bool),huffma675207370phabet(a,t_1),huffma675207370phabet(a,t_2)) = bot_bot(fun(a,bool)) ).
tff(fact_19_tree_Osimps_I6_J,axiom,
! [B: $tType,C: $tType,Tree2: huffma1450048681e_tree(C),Tree1: huffma1450048681e_tree(C),Nat: nat,F2: fun(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B))),F1: fun(nat,fun(C,B))] : huffma107959123e_case(C,B,F1,F2,huffma1146269203erNode(C,Nat,Tree1,Tree2)) = aa1(huffma1450048681e_tree(C),B,aa1(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B),aa1(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B)),F2,Nat),Tree1),Tree2) ).
tff(fact_20_exists__in__alphabet,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B)] :
? [A3: B] : pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),A3),huffma675207370phabet(B,Ta))) ).
tff(fact_21_sibling_Osimps_I4_J,axiom,
! [B: $tType,Vb: huffma1450048681e_tree(B),Va: huffma1450048681e_tree(B),V: nat,Wa: nat,T_12: huffma1450048681e_tree(B),Ab: B] :
( ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,T_12)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_12,huffma1146269203erNode(B,V,Va,Vb)),Ab) = huffma1401021291ibling(B,T_12,Ab) ) )
& ( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,T_12)))
=> ( ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb))))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_12,huffma1146269203erNode(B,V,Va,Vb)),Ab) = huffma1401021291ibling(B,huffma1146269203erNode(B,V,Va,Vb),Ab) ) )
& ( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb))))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_12,huffma1146269203erNode(B,V,Va,Vb)),Ab) = Ab ) ) ) ) ) ).
tff(fact_22_sibling_Osimps_I3_J,axiom,
! [B: $tType,T_22: huffma1450048681e_tree(B),Wa: nat,Vb: huffma1450048681e_tree(B),Va: huffma1450048681e_tree(B),V: nat,Ab: B] :
( ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb))))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,huffma1146269203erNode(B,V,Va,Vb),T_22),Ab) = huffma1401021291ibling(B,huffma1146269203erNode(B,V,Va,Vb),Ab) ) )
& ( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb))))
=> ( ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,T_22)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,huffma1146269203erNode(B,V,Va,Vb),T_22),Ab) = huffma1401021291ibling(B,T_22,Ab) ) )
& ( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,T_22)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,huffma1146269203erNode(B,V,Va,Vb),T_22),Ab) = Ab ) ) ) ) ) ).
tff(fact_23_swapLeaves__id,axiom,
! [A: $tType,A1: A,T: huffma1450048681e_tree(A)] :
( huffma1518433673istent(A,T)
=> ( huffma414517318Leaves(A,T,huffma1352802255e_freq(A,T,A1),A1,huffma1352802255e_freq(A,T,A1),A1) = T ) ) ).
tff(fact_24_consistent_Osimps_I2_J,axiom,
! [B: $tType,T_22: huffma1450048681e_tree(B),T_12: huffma1450048681e_tree(B),Wa: nat] :
( huffma1518433673istent(B,huffma1146269203erNode(B,Wa,T_12,T_22))
<=> ( huffma1518433673istent(B,T_12)
& huffma1518433673istent(B,T_22)
& ( inf_inf(fun(B,bool),huffma675207370phabet(B,T_12),huffma675207370phabet(B,T_22)) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_25_sibling__sibling__id,axiom,
! [A: $tType,A1: A,T: huffma1450048681e_tree(A)] :
( huffma1518433673istent(A,T)
=> ( huffma1401021291ibling(A,T,huffma1401021291ibling(A,T,A1)) = A1 ) ) ).
tff(fact_26_consistent__uniteTrees,axiom,
! [B: $tType,T_22: huffma1450048681e_tree(B),T_12: huffma1450048681e_tree(B)] :
( huffma1518433673istent(B,T_12)
=> ( huffma1518433673istent(B,T_22)
=> ( ( inf_inf(fun(B,bool),huffma675207370phabet(B,T_12),huffma675207370phabet(B,T_22)) = bot_bot(fun(B,bool)) )
=> huffma1518433673istent(B,huffma921447403eTrees(B,T_12,T_22)) ) ) ) ).
tff(fact_27_sibling__reciprocal,axiom,
! [A: $tType,B2: A,A1: A,T: huffma1450048681e_tree(A)] :
( huffma1518433673istent(A,T)
=> ( ( huffma1401021291ibling(A,T,A1) = B2 )
=> ( huffma1401021291ibling(A,T,B2) = A1 ) ) ) ).
tff(fact_28_in__alphabet__imp__sibling__in__alphabet,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B),Ab: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,Ta)))
=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),huffma1401021291ibling(B,Ta,Ab)),huffma675207370phabet(B,Ta))) ) ).
tff(fact_29_sibling__ne__imp__sibling__in__alphabet,axiom,
! [B: $tType,Ab: B,Ta: huffma1450048681e_tree(B)] :
( ( huffma1401021291ibling(B,Ta,Ab) != Ab )
=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),huffma1401021291ibling(B,Ta,Ab)),huffma675207370phabet(B,Ta))) ) ).
tff(fact_30_notin__alphabet__imp__sibling__id,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B),Ab: B] :
( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,Ta)))
=> ( huffma1401021291ibling(B,Ta,Ab) = Ab ) ) ).
tff(fact_31_consistent__swapLeaves,axiom,
! [A: $tType,B2: A,W_b1: nat,A1: A,W_a1: nat,T: huffma1450048681e_tree(A)] :
( huffma1518433673istent(A,T)
=> huffma1518433673istent(A,huffma414517318Leaves(A,T,W_a1,A1,W_b1,B2)) ) ).
tff(fact_32_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C3: A,A1: A,B2: A] :
( ( plus_plus(A,B2,A1) = plus_plus(A,C3,A1) )
=> ( B2 = C3 ) ) ) ).
tff(fact_33_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C3: A,B2: A,A1: A] :
( ( plus_plus(A,A1,B2) = plus_plus(A,A1,C3) )
=> ( B2 = C3 ) ) ) ).
tff(fact_34_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C3: A,B2: A,A1: A] :
( ( plus_plus(A,A1,B2) = plus_plus(A,A1,C3) )
=> ( B2 = C3 ) ) ) ).
tff(fact_35_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C3: A,B2: A,A1: A] : plus_plus(A,plus_plus(A,A1,B2),C3) = plus_plus(A,A1,plus_plus(A,B2,C3)) ) ).
tff(fact_36_nat__add__assoc,axiom,
! [K: nat,N1: nat,M1: nat] : plus_plus(nat,plus_plus(nat,M1,N1),K) = plus_plus(nat,M1,plus_plus(nat,N1,K)) ).
tff(fact_37_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X4: nat] : plus_plus(nat,X4,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X4,Z)) ).
tff(fact_38_nat__add__commute,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,M1,N1) = plus_plus(nat,N1,M1) ).
tff(fact_39_inf__bot__left,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [X4: A] : inf_inf(A,bot_bot(A),X4) = bot_bot(A) ) ).
tff(fact_40_inf__bot__right,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [X4: A] : inf_inf(A,X4,bot_bot(A)) = bot_bot(A) ) ).
tff(fact_41_Int__iff,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool),C2: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),inf_inf(fun(B,bool),A2,B1)))
<=> ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),A2))
& pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),B1)) ) ) ).
tff(fact_42_inf1I,axiom,
! [B: $tType,B1: fun(B,bool),X1: B,A2: fun(B,bool)] :
( pp(aa1(B,bool,A2,X1))
=> ( pp(aa1(B,bool,B1,X1))
=> pp(aa1(B,bool,inf_inf(fun(B,bool),A2,B1),X1)) ) ) ).
tff(fact_43_IntI,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool),C2: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),A2))
=> ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),B1))
=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),inf_inf(fun(B,bool),A2,B1))) ) ) ).
tff(fact_44_IntE,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool),C2: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),inf_inf(fun(B,bool),A2,B1)))
=> ~ ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),A2))
=> ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),B1)) ) ) ).
tff(fact_45_inf1E,axiom,
! [B: $tType,X1: B,B1: fun(B,bool),A2: fun(B,bool)] :
( pp(aa1(B,bool,inf_inf(fun(B,bool),A2,B1),X1))
=> ~ ( pp(aa1(B,bool,A2,X1))
=> ~ pp(aa1(B,bool,B1,X1)) ) ) ).
tff(fact_46_emptyE,axiom,
! [B: $tType,Ab: B] : ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),bot_bot(fun(B,bool)))) ).
tff(fact_47_Collect__empty__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( collect(B,P1) = bot_bot(fun(B,bool)) )
<=> ! [X2: B] : ~ pp(aa1(B,bool,P1,X2)) ) ).
tff(fact_48_empty__iff,axiom,
! [B: $tType,C2: B] : ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),bot_bot(fun(B,bool)))) ).
tff(fact_49_empty__Collect__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,P1) )
<=> ! [X2: B] : ~ pp(aa1(B,bool,P1,X2)) ) ).
tff(fact_50_all__not__in__conv,axiom,
! [B: $tType,A2: fun(B,bool)] :
( ! [X2: B] : ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X2),A2))
<=> ( A2 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_51_inf__left__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Y: A,X4: A] : inf_inf(A,X4,inf_inf(A,X4,Y)) = inf_inf(A,X4,Y) ) ).
tff(fact_52_inf_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A1: A] : inf_inf(A,A1,inf_inf(A,A1,B2)) = inf_inf(A,A1,B2) ) ).
tff(fact_53_equals0D,axiom,
! [B: $tType,Ab: B,A2: fun(B,bool)] :
( ( A2 = bot_bot(fun(B,bool)) )
=> ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),A2)) ) ).
tff(fact_54_ex__in__conv,axiom,
! [B: $tType,A2: fun(B,bool)] :
( ? [X2: B] : pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X2),A2))
<=> ( A2 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_55_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_56_bot__empty__eq,axiom,
! [B: $tType,X: B] :
( pp(aa1(B,bool,bot_bot(fun(B,bool)),X))
<=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X),bot_bot(fun(B,bool)))) ) ).
tff(fact_57_inf__assoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Z: A,Y: A,X4: A] : inf_inf(A,inf_inf(A,X4,Y),Z) = inf_inf(A,X4,inf_inf(A,Y,Z)) ) ).
tff(fact_58_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z: A,Y: A,X4: A] : inf_inf(A,inf_inf(A,X4,Y),Z) = inf_inf(A,X4,inf_inf(A,Y,Z)) ) ).
tff(fact_59_inf_Oassoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [C3: A,B2: A,A1: A] : inf_inf(A,inf_inf(A,A1,B2),C3) = inf_inf(A,A1,inf_inf(A,B2,C3)) ) ).
tff(fact_60_inf__left__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Z: A,Y: A,X4: A] : inf_inf(A,X4,inf_inf(A,Y,Z)) = inf_inf(A,Y,inf_inf(A,X4,Z)) ) ).
tff(fact_61_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z: A,Y: A,X4: A] : inf_inf(A,X4,inf_inf(A,Y,Z)) = inf_inf(A,Y,inf_inf(A,X4,Z)) ) ).
tff(fact_62_inf_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [C3: A,A1: A,B2: A] : inf_inf(A,B2,inf_inf(A,A1,C3)) = inf_inf(A,A1,inf_inf(A,B2,C3)) ) ).
tff(fact_63_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X4: A] : inf_inf(A,X4,inf_inf(A,X4,Y)) = inf_inf(A,X4,Y) ) ).
tff(fact_64_inf__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Y: A,X4: A] : inf_inf(A,X4,Y) = inf_inf(A,Y,X4) ) ).
tff(fact_65_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X4: A] : inf_inf(A,X4,Y) = inf_inf(A,Y,X4) ) ).
tff(fact_66_inf_Ocommute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A1: A] : inf_inf(A,A1,B2) = inf_inf(A,B2,A1) ) ).
tff(fact_67_inf__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X4: A] : inf_inf(A,X4,X4) = X4 ) ).
tff(fact_68_inf_Oidem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A1: A] : inf_inf(A,A1,A1) = A1 ) ).
tff(fact_69_inf1D2,axiom,
! [B: $tType,X1: B,B1: fun(B,bool),A2: fun(B,bool)] :
( pp(aa1(B,bool,inf_inf(fun(B,bool),A2,B1),X1))
=> pp(aa1(B,bool,B1,X1)) ) ).
tff(fact_70_inf1D1,axiom,
! [B: $tType,X1: B,B1: fun(B,bool),A2: fun(B,bool)] :
( pp(aa1(B,bool,inf_inf(fun(B,bool),A2,B1),X1))
=> pp(aa1(B,bool,A2,X1)) ) ).
tff(fact_71_IntD2,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool),C2: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),inf_inf(fun(B,bool),A2,B1)))
=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),B1)) ) ).
tff(fact_72_IntD1,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool),C2: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),inf_inf(fun(B,bool),A2,B1)))
=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),C2),A2)) ) ).
tff(fact_73_Int__assoc,axiom,
! [B: $tType,C1: fun(B,bool),B1: fun(B,bool),A2: fun(B,bool)] : inf_inf(fun(B,bool),inf_inf(fun(B,bool),A2,B1),C1) = inf_inf(fun(B,bool),A2,inf_inf(fun(B,bool),B1,C1)) ).
tff(fact_74_Int__left__commute,axiom,
! [B: $tType,C1: fun(B,bool),B1: fun(B,bool),A2: fun(B,bool)] : inf_inf(fun(B,bool),A2,inf_inf(fun(B,bool),B1,C1)) = inf_inf(fun(B,bool),B1,inf_inf(fun(B,bool),A2,C1)) ).
tff(fact_75_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X3: B] : aa1(B,C,F,X3) = aa1(B,C,G,X3)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [B: $tType,A2: fun(B,bool),X1: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X1),A2))
<=> pp(aa1(B,bool,A2,X1)) ) ).
tff(fact_77_Collect__def,axiom,
! [B: $tType,P1: fun(B,bool)] : collect(B,P1) = P1 ).
tff(fact_78_Int__left__absorb,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool)] : inf_inf(fun(B,bool),A2,inf_inf(fun(B,bool),A2,B1)) = inf_inf(fun(B,bool),A2,B1) ).
tff(fact_79_Int__commute,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool)] : inf_inf(fun(B,bool),A2,B1) = inf_inf(fun(B,bool),B1,A2) ).
tff(fact_80_Int__def,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool)] : inf_inf(fun(B,bool),A2,B1) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,combc(B,fun(B,bool),bool,member(B),A2)),combc(B,fun(B,bool),bool,member(B),B1))) ).
tff(fact_81_Int__absorb,axiom,
! [B: $tType,A2: fun(B,bool)] : inf_inf(fun(B,bool),A2,A2) = A2 ).
tff(fact_82_inf__Int__eq,axiom,
! [B: $tType,S: fun(B,bool),R1: fun(B,bool),X: B] :
( pp(aa1(B,bool,inf_inf(fun(B,bool),combc(B,fun(B,bool),bool,member(B),R1),combc(B,fun(B,bool),bool,member(B),S)),X))
<=> pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X),inf_inf(fun(B,bool),R1,S))) ) ).
tff(fact_83_Int__Collect,axiom,
! [B: $tType,P1: fun(B,bool),A2: fun(B,bool),X1: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X1),inf_inf(fun(B,bool),A2,collect(B,P1))))
<=> ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X1),A2))
& pp(aa1(B,bool,P1,X1)) ) ) ).
tff(fact_84_Collect__conj__eq,axiom,
! [B: $tType,Q1: fun(B,bool),P1: fun(B,bool)] : collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,P1),Q1)) = inf_inf(fun(B,bool),collect(B,P1),collect(B,Q1)) ).
tff(fact_85_disjoint__iff__not__equal,axiom,
! [B: $tType,B1: fun(B,bool),A2: fun(B,bool)] :
( ( inf_inf(fun(B,bool),A2,B1) = bot_bot(fun(B,bool)) )
<=> ! [X2: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),X2),A2))
=> ! [Xa: B] :
( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Xa),B1))
=> ( X2 != Xa ) ) ) ) ).
tff(fact_86_Int__empty__right,axiom,
! [B: $tType,A2: fun(B,bool)] : inf_inf(fun(B,bool),A2,bot_bot(fun(B,bool))) = bot_bot(fun(B,bool)) ).
tff(fact_87_Int__empty__left,axiom,
! [B: $tType,B1: fun(B,bool)] : inf_inf(fun(B,bool),bot_bot(fun(B,bool)),B1) = bot_bot(fun(B,bool)) ).
tff(fact_88_inf__apply,axiom,
! [B: $tType,C: $tType] :
( lattice(B)
=> ! [X1: C,G: fun(C,B),F: fun(C,B)] : aa1(C,B,inf_inf(fun(C,B),F,G),X1) = inf_inf(B,aa1(C,B,F,X1),aa1(C,B,G,X1)) ) ).
tff(fact_89_inf__fun__def,axiom,
! [C: $tType,B: $tType] :
( lattice(C)
=> ! [G: fun(B,C),F: fun(B,C),X: B] : aa1(B,C,inf_inf(fun(B,C),F,G),X) = inf_inf(C,aa1(B,C,F,X),aa1(B,C,G,X)) ) ).
tff(fact_90_uniteTrees__def,axiom,
! [A: $tType,T_21: huffma1450048681e_tree(A),T_11: huffma1450048681e_tree(A)] : huffma921447403eTrees(A,T_11,T_21) = huffma1146269203erNode(A,plus_plus(nat,huffma854194513Weight(A,T_11),huffma854194513Weight(A,T_21)),T_11,T_21) ).
tff(fact_91_bot__apply,axiom,
! [C: $tType,B: $tType] :
( bot(B)
=> ! [X1: C] : aa1(C,B,bot_bot(fun(C,B)),X1) = bot_bot(B) ) ).
tff(fact_92_bot__fun__def,axiom,
! [B: $tType,C: $tType] :
( bot(C)
=> ! [X: B] : aa1(B,C,bot_bot(fun(B,C)),X) = bot_bot(C) ) ).
tff(fact_93_freq__swapLeaves,axiom,
! [B: $tType,W_b: nat,W_a: nat,Ba: B,Ab: B,Ta: huffma1450048681e_tree(B)] :
( huffma1518433673istent(B,Ta)
=> ( ( Ab != Ba )
=> ! [X: B] :
( ( ( X = Ab )
=> ( ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ba),huffma675207370phabet(B,Ta)))
=> ( huffma1352802255e_freq(B,huffma414517318Leaves(B,Ta,W_a,Ab,W_b,Ba),X) = W_a ) )
& ( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ba),huffma675207370phabet(B,Ta)))
=> ( huffma1352802255e_freq(B,huffma414517318Leaves(B,Ta,W_a,Ab,W_b,Ba),X) = zero_zero(nat) ) ) ) )
& ( ( X != Ab )
=> ( ( ( X = Ba )
=> ( ( pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,Ta)))
=> ( huffma1352802255e_freq(B,huffma414517318Leaves(B,Ta,W_a,Ab,W_b,Ba),X) = W_b ) )
& ( ~ pp(aa1(fun(B,bool),bool,aa1(B,fun(fun(B,bool),bool),member(B),Ab),huffma675207370phabet(B,Ta)))
=> ( huffma1352802255e_freq(B,huffma414517318Leaves(B,Ta,W_a,Ab,W_b,Ba),X) = zero_zero(nat) ) ) ) )
& ( ( X != Ba )
=> ( huffma1352802255e_freq(B,huffma414517318Leaves(B,Ta,W_a,Ab,W_b,Ba),X) = huffma1352802255e_freq(B,Ta,X) ) ) ) ) ) ) ) ).
tff(fact_94_double__zero__sym,axiom,
! [B: $tType] :
( linord219039673up_add(B)
=> ! [Ab: B] :
( ( zero_zero(B) = plus_plus(B,Ab,Ab) )
<=> ( Ab = zero_zero(B) ) ) ) ).
tff(fact_95_add__is__0,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
tff(fact_96_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
tff(fact_97_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_98_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
%----Arities (19)
tff(arity_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice(bool) ).
tff(arity_fun___Lattices_Obounded__lattice,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Obounded__lattice__bot,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice_bot(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Osemilattice__inf,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> semilattice_inf(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Olattice,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> lattice(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Lattices_Osemilattice__inf,axiom,
semilattice_inf(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Lattices_Olattice,axiom,
lattice(nat) ).
tff(arity_Nat_Onat___Orderings_Obot,axiom,
bot(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Lattices_Obounded__lattice__bot,axiom,
bounded_lattice_bot(bool) ).
tff(arity_HOL_Obool___Lattices_Osemilattice__inf,axiom,
semilattice_inf(bool) ).
tff(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
%----Helper facts (11)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa1(A,C,combb(B,C,A,P,Q),R) = aa1(B,C,P,aa1(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa1(A,C,combc(A,B,C,P,Q),R) = aa1(B,C,aa1(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa1(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa1(A,C,combs(A,B,C,P,Q),R) = aa1(B,C,aa1(A,fun(B,C),P,R),aa1(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa1(bool,bool,aa1(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa1(bool,bool,aa1(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa1(bool,bool,aa1(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,huffma1146269203erNode(a,w,t_1,t_2))))
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,huffma1146269203erNode(a,w,t_1,t_2))))
=> ( plus_plus(nat,plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,huffma1146269203erNode(a,w,t_1,t_2),w_a,aa,w_b,b)),huffma1352802255e_freq(a,huffma1146269203erNode(a,w,t_1,t_2),aa)),huffma1352802255e_freq(a,huffma1146269203erNode(a,w,t_1,t_2),b)) = plus_plus(nat,plus_plus(nat,huffma83463279weight(a,huffma1146269203erNode(a,w,t_1,t_2)),w_a),w_b) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,huffma1146269203erNode(a,w,t_1,t_2))))
=> ( plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,huffma1146269203erNode(a,w,t_1,t_2),w_a,aa,w_b,b)),huffma1352802255e_freq(a,huffma1146269203erNode(a,w,t_1,t_2),aa)) = plus_plus(nat,huffma83463279weight(a,huffma1146269203erNode(a,w,t_1,t_2)),w_b) ) ) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),aa),huffma675207370phabet(a,huffma1146269203erNode(a,w,t_1,t_2))))
=> ( ( pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,huffma1146269203erNode(a,w,t_1,t_2))))
=> ( plus_plus(nat,huffma83463279weight(a,huffma414517318Leaves(a,huffma1146269203erNode(a,w,t_1,t_2),w_a,aa,w_b,b)),huffma1352802255e_freq(a,huffma1146269203erNode(a,w,t_1,t_2),b)) = plus_plus(nat,huffma83463279weight(a,huffma1146269203erNode(a,w,t_1,t_2)),w_a) ) )
& ( ~ pp(aa1(fun(a,bool),bool,aa1(a,fun(fun(a,bool),bool),member(a),b),huffma675207370phabet(a,huffma1146269203erNode(a,w,t_1,t_2))))
=> ( huffma83463279weight(a,huffma414517318Leaves(a,huffma1146269203erNode(a,w,t_1,t_2),w_a,aa,w_b,b)) = huffma83463279weight(a,huffma1146269203erNode(a,w,t_1,t_2)) ) ) ) ) ) ).
%------------------------------------------------------------------------------