TPTP Problem File: SWW542_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW542_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Huffman's Algorithm line 1048
% 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_1048 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 165 ( 49 unt; 40 typ; 0 def)
% Number of atoms : 271 ( 94 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 189 ( 43 ~; 7 |; 20 &)
% ( 27 <=>; 92 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 10 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 38 ( 22 >; 16 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 6 con; 0-5 aty)
% Number of variables : 324 ( 286 !; 8 ?; 324 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:49
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_t_a,type,
a1: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Huffman__Mirabelle__lalbadcutu_Otree,type,
huffma1450048681e_tree: $tType > $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (34)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
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_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_Odepth,type,
huffma410068972_depth:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * A ) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Ofreq,type,
huffma1352802255e_freq:
!>[A: $tType] : ( huffma1450048681e_tree(A) > fun(A,nat) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oheight,type,
huffma945805758height:
!>[A: $tType] : ( huffma1450048681e_tree(A) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Ooptimum,type,
huffma1393970616ptimum:
!>[A: $tType] : ( huffma1450048681e_tree(A) > $o ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Osibling,type,
huffma1401021291ibling:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * A ) > 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_OLeaf,type,
huffma2021818691e_Leaf:
!>[A: $tType] : ( ( nat * A ) > huffma1450048681e_tree(A) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_Otree__case,type,
huffma107959123e_case:
!>[A: $tType,T2: $tType] : ( ( fun(nat,fun(A,T2)) * fun(nat,fun(huffma1450048681e_tree(A),fun(huffma1450048681e_tree(A),T2))) * huffma1450048681e_tree(A) ) > T2 ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_Otree__rec,type,
huffma1280178957ee_rec:
!>[A: $tType,T2: $tType] : ( ( fun(nat,fun(A,T2)) * fun(nat,fun(huffma1450048681e_tree(A),fun(huffma1450048681e_tree(A),fun(T2,fun(T2,T2))))) * huffma1450048681e_tree(A) ) > T2 ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_aa,type,
aa:
!>[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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_a,type,
a: a1 ).
tff(sy_v_t_092_060_094isub_0621,type,
t_1: huffma1450048681e_tree(a1) ).
tff(sy_v_t_092_060_094isub_0622,type,
t_2: huffma1450048681e_tree(a1) ).
tff(sy_v_w,type,
w: nat ).
%----Relevant facts (99)
tff(fact_0_tree_Osimps_I2_J,axiom,
! [B: $tType,Tree23: huffma1450048681e_tree(B),Tree13: huffma1450048681e_tree(B),Nat4: nat,Tree2: huffma1450048681e_tree(B),Tree1: huffma1450048681e_tree(B),Nat: nat] :
( ( huffma1146269203erNode(B,Nat,Tree1,Tree2) = huffma1146269203erNode(B,Nat4,Tree13,Tree23) )
<=> ( ( Nat = Nat4 )
& ( Tree1 = Tree13 )
& ( Tree2 = Tree23 ) ) ) ).
tff(fact_1_sibling_Osimps_I4_J,axiom,
! [B: $tType,Vb: huffma1450048681e_tree(B),Va: huffma1450048681e_tree(B),V: nat,Wa: nat,T_11: huffma1450048681e_tree(B),Aa: B] :
( ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,huffma1146269203erNode(B,V,Va,Vb)),Aa) = huffma1401021291ibling(B,T_11,Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( ( member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,huffma1146269203erNode(B,V,Va,Vb)),Aa) = huffma1401021291ibling(B,huffma1146269203erNode(B,V,Va,Vb),Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,huffma1146269203erNode(B,V,Va,Vb)),Aa) = Aa ) ) ) ) ) ).
tff(fact_2_sibling_Osimps_I3_J,axiom,
! [B: $tType,T_21: huffma1450048681e_tree(B),Wa: nat,Vb: huffma1450048681e_tree(B),Va: huffma1450048681e_tree(B),V: nat,Aa: B] :
( ( member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,huffma1146269203erNode(B,V,Va,Vb),T_21),Aa) = huffma1401021291ibling(B,huffma1146269203erNode(B,V,Va,Vb),Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( ( member(B,Aa,huffma675207370phabet(B,T_21))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,huffma1146269203erNode(B,V,Va,Vb),T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,T_21))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,huffma1146269203erNode(B,V,Va,Vb),T_21),Aa) = Aa ) ) ) ) ) ).
tff(fact_3_notin__alphabet__imp__sibling__id,axiom,
! [B: $tType,T1: huffma1450048681e_tree(B),Aa: B] :
( ~ member(B,Aa,huffma675207370phabet(B,T1))
=> ( huffma1401021291ibling(B,T1,Aa) = Aa ) ) ).
tff(fact_4_height__gt__0__notin__alphabet__imp__sibling__left,axiom,
! [B: $tType,T_21: huffma1450048681e_tree(B),Wa: nat,Aa: B,T_11: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_11))
=> ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) ) ) ).
tff(fact_5_height__gt__0__notin__alphabet__imp__sibling__right,axiom,
! [B: $tType,Wa: nat,T_11: huffma1450048681e_tree(B),Aa: B,T_21: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_21))
=> ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) ) ) ).
tff(fact_6_height__gt__0__in__alphabet__imp__sibling__left,axiom,
! [B: $tType,T_21: huffma1450048681e_tree(B),Wa: nat,Aa: B,T_11: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_11))
=> ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,T_21),Aa) = huffma1401021291ibling(B,T_11,Aa) ) ) ) ).
tff(fact_7_height__gt__0__in__alphabet__imp__sibling__right,axiom,
! [B: $tType,Wa: nat,T_11: huffma1450048681e_tree(B),Aa: B,T_21: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_21))
=> ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,Wa,T_11,T_21),Aa) = huffma1401021291ibling(B,T_11,Aa) ) ) ) ).
tff(fact_8_height__0__imp__sibling__id,axiom,
! [A: $tType,A2: A,T: huffma1450048681e_tree(A)] :
( ( huffma945805758height(A,T) = zero_zero(nat) )
=> ( huffma1401021291ibling(A,T,A2) = A2 ) ) ).
tff(fact_9_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_10_less__nat__zero__code,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_11_less__zeroE,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_12_height__gt__0__alphabet__eq__imp__height__gt__0,axiom,
! [B: $tType,U1: huffma1450048681e_tree(B),T1: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T1))
=> ( huffma1518433673istent(B,T1)
=> ( ( huffma675207370phabet(B,T1) = huffma675207370phabet(B,U1) )
=> ord_less(nat,zero_zero(nat),huffma945805758height(B,U1)) ) ) ) ).
tff(fact_13_not__less0,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_14_gr__implies__not0,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ( N2 != zero_zero(nat) ) ) ).
tff(fact_15_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N2) ) ).
tff(fact_16_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)) = aa(huffma1450048681e_tree(C),B,aa(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B),aa(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B)),F2,Nat),Tree1),Tree2) ).
tff(fact_17_exists__in__alphabet,axiom,
! [B: $tType,T1: huffma1450048681e_tree(B)] :
? [A3: B] : member(B,A3,huffma675207370phabet(B,T1)) ).
tff(fact_18_sibling_Osimps_I2_J,axiom,
! [A: $tType,C1: A,W_c: nat,W_b: nat,W: nat,B1: A,A2: A] :
( ( ( A2 = B1 )
=> ( huffma1401021291ibling(A,huffma1146269203erNode(A,W,huffma2021818691e_Leaf(A,W_b,B1),huffma2021818691e_Leaf(A,W_c,C1)),A2) = C1 ) )
& ( ( A2 != B1 )
=> ( ( ( A2 = C1 )
=> ( huffma1401021291ibling(A,huffma1146269203erNode(A,W,huffma2021818691e_Leaf(A,W_b,B1),huffma2021818691e_Leaf(A,W_c,C1)),A2) = B1 ) )
& ( ( A2 != C1 )
=> ( huffma1401021291ibling(A,huffma1146269203erNode(A,W,huffma2021818691e_Leaf(A,W_b,B1),huffma2021818691e_Leaf(A,W_c,C1)),A2) = A2 ) ) ) ) ) ).
tff(fact_19_height__0__imp__cost__0,axiom,
! [A: $tType,T: huffma1450048681e_tree(A)] :
( ( huffma945805758height(A,T) = zero_zero(nat) )
=> ( huffma1134658180e_cost(A,T) = zero_zero(nat) ) ) ).
tff(fact_20_tree_Osimps_I1_J,axiom,
! [B: $tType,A4: B,Nat4: nat,Aa: B,Nat: nat] :
( ( huffma2021818691e_Leaf(B,Nat,Aa) = huffma2021818691e_Leaf(B,Nat4,A4) )
<=> ( ( Nat = Nat4 )
& ( Aa = A4 ) ) ) ).
tff(fact_21_consistent_Osimps_I1_J,axiom,
! [A: $tType,A2: A,W: nat] : huffma1518433673istent(A,huffma2021818691e_Leaf(A,W,A2)) ).
tff(fact_22_height_Osimps_I1_J,axiom,
! [A: $tType,A2: A,W: nat] : huffma945805758height(A,huffma2021818691e_Leaf(A,W,A2)) = zero_zero(nat) ).
tff(fact_23_cost_Osimps_I1_J,axiom,
! [A: $tType,A2: A,W: nat] : huffma1134658180e_cost(A,huffma2021818691e_Leaf(A,W,A2)) = zero_zero(nat) ).
tff(fact_24_tree_Osimps_I5_J,axiom,
! [B: $tType,C: $tType,Aa: 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,huffma2021818691e_Leaf(C,Nat,Aa)) = aa(C,B,aa(nat,fun(C,B),F1,Nat),Aa) ).
tff(fact_25_tree_Osimps_I4_J,axiom,
! [A: $tType,A2: A,Nat2: nat,Tree22: huffma1450048681e_tree(A),Tree12: huffma1450048681e_tree(A),Nat3: nat] : huffma1146269203erNode(A,Nat3,Tree12,Tree22) != huffma2021818691e_Leaf(A,Nat2,A2) ).
tff(fact_26_tree_Osimps_I3_J,axiom,
! [A: $tType,Tree22: huffma1450048681e_tree(A),Tree12: huffma1450048681e_tree(A),Nat3: nat,A2: A,Nat2: nat] : huffma2021818691e_Leaf(A,Nat2,A2) != huffma1146269203erNode(A,Nat3,Tree12,Tree22) ).
tff(fact_27_sibling_Osimps_I1_J,axiom,
! [A: $tType,A2: A,B1: A,W_b: nat] : huffma1401021291ibling(A,huffma2021818691e_Leaf(A,W_b,B1),A2) = A2 ).
tff(fact_28_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X2: B] :
( ( zero_zero(B) = X2 )
<=> ( X2 = zero_zero(B) ) ) ) ).
tff(fact_29_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N1: nat,M1: nat] :
( ( ord_less(nat,M1,N1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ( M1 = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ord_less(nat,N1,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) ) ) ) ).
tff(fact_30_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_31_less__not__refl2,axiom,
! [M2: nat,N2: nat] :
( ord_less(nat,N2,M2)
=> ( M2 != N2 ) ) ).
tff(fact_32_less__irrefl__nat,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_33_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_34_nat__neq__iff,axiom,
! [N1: nat,M1: nat] :
( ( M1 != N1 )
<=> ( ord_less(nat,M1,N1)
| ord_less(nat,N1,M1) ) ) ).
tff(fact_35_less__not__refl,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_36_tree_Osize_I3_J,axiom,
! [A: $tType,A2: A,Nat2: nat] : size_size(huffma1450048681e_tree(A),huffma2021818691e_Leaf(A,Nat2,A2)) = zero_zero(nat) ).
tff(fact_37_depth_Osimps_I1_J,axiom,
! [A: $tType,A2: A,B1: A,W: nat] : huffma410068972_depth(A,huffma2021818691e_Leaf(A,W,B1),A2) = zero_zero(nat) ).
tff(fact_38_of__nat__0__less__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [N1: nat] :
( ord_less(B,zero_zero(B),semiring_1_of_nat(B,N1))
<=> ord_less(nat,zero_zero(nat),N1) ) ) ).
tff(fact_39_tree_Oexhaust,axiom,
! [A: $tType,Y: huffma1450048681e_tree(A)] :
( ! [Nat1: nat,A3: A] : Y != huffma2021818691e_Leaf(A,Nat1,A3)
=> ~ ! [Nat1: nat,Tree11: huffma1450048681e_tree(A),Tree21: huffma1450048681e_tree(A)] : Y != huffma1146269203erNode(A,Nat1,Tree11,Tree21) ) ).
tff(fact_40_freq_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A2: A,X4: A] :
( ( ( X4 = A2 )
=> ( aa(A,nat,huffma1352802255e_freq(A,huffma2021818691e_Leaf(A,W,A2)),X4) = W ) )
& ( ( X4 != A2 )
=> ( aa(A,nat,huffma1352802255e_freq(A,huffma2021818691e_Leaf(A,W,A2)),X4) = zero_zero(nat) ) ) ) ).
tff(fact_41_exists__at__height,axiom,
! [B: $tType,T1: huffma1450048681e_tree(B)] :
( huffma1518433673istent(B,T1)
=> ? [X3: B] :
( member(B,X3,huffma675207370phabet(B,T1))
& ( huffma410068972_depth(B,T1,X3) = huffma945805758height(B,T1) ) ) ) ).
tff(fact_42_infinite__descent0,axiom,
! [N1: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N: nat] :
( ord_less(nat,zero_zero(nat),N)
=> ( ~ pp(aa(nat,bool,P1,N))
=> ? [M3: nat] :
( ord_less(nat,M3,N)
& ~ pp(aa(nat,bool,P1,M3)) ) ) )
=> pp(aa(nat,bool,P1,N1)) ) ) ).
tff(fact_43_of__nat__eq__iff,axiom,
! [B: $tType] :
( semiring_char_0(B)
=> ! [N1: nat,M1: nat] :
( ( semiring_1_of_nat(B,M1) = semiring_1_of_nat(B,N1) )
<=> ( M1 = N1 ) ) ) ).
tff(fact_44_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_45_of__nat__less__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [N1: nat,M1: nat] :
( ord_less(B,semiring_1_of_nat(B,M1),semiring_1_of_nat(B,N1))
<=> ord_less(nat,M1,N1) ) ) ).
tff(fact_46_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M2: nat] : ~ ord_less(A,semiring_1_of_nat(A,M2),zero_zero(A)) ) ).
tff(fact_47_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ord_less(A,semiring_1_of_nat(A,M2),semiring_1_of_nat(A,N2)) ) ) ).
tff(fact_48_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,M2: nat] :
( ord_less(A,semiring_1_of_nat(A,M2),semiring_1_of_nat(A,N2))
=> ord_less(nat,M2,N2) ) ) ).
tff(fact_49_notin__alphabet__imp__freq__0,axiom,
! [B: $tType,T1: huffma1450048681e_tree(B),Aa: B] :
( ~ member(B,Aa,huffma675207370phabet(B,T1))
=> ( aa(B,nat,huffma1352802255e_freq(B,T1),Aa) = zero_zero(nat) ) ) ).
tff(fact_50_zero__less__int__conv,axiom,
! [N1: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,N1))
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_51_optimum__def,axiom,
! [B: $tType,T1: huffma1450048681e_tree(B)] :
( huffma1393970616ptimum(B,T1)
<=> ! [U: huffma1450048681e_tree(B)] :
( huffma1518433673istent(B,U)
=> ( ( huffma675207370phabet(B,T1) = huffma675207370phabet(B,U) )
=> ( ( huffma1352802255e_freq(B,T1) = huffma1352802255e_freq(B,U) )
=> ord_less_eq(nat,huffma1134658180e_cost(B,T1),huffma1134658180e_cost(B,U)) ) ) ) ) ).
tff(fact_52_tree_Orecs_I1_J,axiom,
! [B: $tType,C: $tType,Aa: C,Nat: nat,F2: fun(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B))))),F1: fun(nat,fun(C,B))] : huffma1280178957ee_rec(C,B,F1,F2,huffma2021818691e_Leaf(C,Nat,Aa)) = aa(C,B,aa(nat,fun(C,B),F1,Nat),Aa) ).
tff(fact_53_tree_Orecs_I2_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),fun(B,fun(B,B))))),F1: fun(nat,fun(C,B))] : huffma1280178957ee_rec(C,B,F1,F2,huffma1146269203erNode(C,Nat,Tree1,Tree2)) = aa(B,B,aa(B,fun(B,B),aa(huffma1450048681e_tree(C),fun(B,fun(B,B)),aa(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B))),aa(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B)))),F2,Nat),Tree1),Tree2),huffma1280178957ee_rec(C,B,F1,F2,Tree1)),huffma1280178957ee_rec(C,B,F1,F2,Tree2)) ).
tff(fact_54_le__0__eq,axiom,
! [N1: nat] :
( ord_less_eq(nat,N1,zero_zero(nat))
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_55_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ord_less_eq(nat,zero_zero(nat),N2) ).
tff(fact_56_le0,axiom,
! [N2: nat] : ord_less_eq(nat,zero_zero(nat),N2) ).
tff(fact_57_of__nat__le__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [N1: nat,M1: nat] :
( ord_less_eq(B,semiring_1_of_nat(B,M1),semiring_1_of_nat(B,N1))
<=> ord_less_eq(nat,M1,N1) ) ) ).
tff(fact_58_int__less__0__conv,axiom,
! [K: nat] : ~ ord_less(int,semiring_1_of_nat(int,K),zero_zero(int)) ).
tff(fact_59_int__int__eq,axiom,
! [N1: nat,M1: nat] :
( ( semiring_1_of_nat(int,M1) = semiring_1_of_nat(int,N1) )
<=> ( M1 = N1 ) ) ).
tff(fact_60_nat__less__le,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,M1,N1)
<=> ( ord_less_eq(nat,M1,N1)
& ( M1 != N1 ) ) ) ).
tff(fact_61_le__eq__less__or__eq,axiom,
! [N1: nat,M1: nat] :
( ord_less_eq(nat,M1,N1)
<=> ( ord_less(nat,M1,N1)
| ( M1 = N1 ) ) ) ).
tff(fact_62_less__imp__le__nat,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_63_le__neq__implies__less,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
=> ( ( M2 != N2 )
=> ord_less(nat,M2,N2) ) ) ).
tff(fact_64_less__or__eq__imp__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less(nat,M2,N2)
| ( M2 = N2 ) )
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_65_le__antisym,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
=> ( ord_less_eq(nat,N2,M2)
=> ( M2 = N2 ) ) ) ).
tff(fact_66_le__trans,axiom,
! [K: nat,J2: nat,I3: nat] :
( ord_less_eq(nat,I3,J2)
=> ( ord_less_eq(nat,J2,K)
=> ord_less_eq(nat,I3,K) ) ) ).
tff(fact_67_eq__imp__le,axiom,
! [N2: nat,M2: nat] :
( ( M2 = N2 )
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_68_nat__le__linear,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
| ord_less_eq(nat,N2,M2) ) ).
tff(fact_69_le__refl,axiom,
! [N2: nat] : ord_less_eq(nat,N2,N2) ).
tff(fact_70_of__nat__0__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,N2)) ) ).
tff(fact_71_zero__le__imp__of__nat,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M2: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,M2)) ) ).
tff(fact_72_depth__le__height,axiom,
! [A: $tType,A2: A,T: huffma1450048681e_tree(A)] : ord_less_eq(nat,huffma410068972_depth(A,T,A2),huffma945805758height(A,T)) ).
tff(fact_73_int__eq__0__conv,axiom,
! [N1: nat] :
( ( semiring_1_of_nat(int,N1) = zero_zero(int) )
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_74_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X3: B] : aa(B,C,F,X3) = aa(B,C,G,X3)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [B: $tType,A1: fun(B,bool),X2: B] :
( member(B,X2,A1)
<=> pp(aa(B,bool,A1,X2)) ) ).
tff(fact_76_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_77_zless__int,axiom,
! [N1: nat,M1: nat] :
( ord_less(int,semiring_1_of_nat(int,M1),semiring_1_of_nat(int,N1))
<=> ord_less(nat,M1,N1) ) ).
tff(fact_78_zero__less__imp__eq__int,axiom,
! [K: int] :
( ord_less(int,zero_zero(int),K)
=> ? [N: nat] :
( ord_less(nat,zero_zero(nat),N)
& ( K = semiring_1_of_nat(int,N) ) ) ) ).
tff(fact_79_ex__least__nat__le,axiom,
! [N1: nat,P1: fun(nat,bool)] :
( ~ pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( pp(aa(nat,bool,P1,N1))
=> ? [K1: nat] :
( ord_less_eq(nat,K1,N1)
& ! [I2: nat] :
( ord_less(nat,I2,K1)
=> ~ pp(aa(nat,bool,P1,I2)) )
& pp(aa(nat,bool,P1,K1)) ) ) ) ).
tff(fact_80_transfer__int__nat__relations_I2_J,axiom,
! [Y1: nat,X2: nat] :
( ord_less(int,semiring_1_of_nat(int,X2),semiring_1_of_nat(int,Y1))
<=> ord_less(nat,X2,Y1) ) ).
tff(fact_81_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = semiring_1_of_nat(int,zero_zero(nat)) ).
tff(fact_82_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,Z)) ).
tff(fact_83_transfer__int__nat__quantifiers_I2_J,axiom,
! [P1: fun(int,bool)] :
( ? [X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
& pp(aa(int,bool,P1,X1)) )
<=> ? [X1: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X1))) ) ).
tff(fact_84_transfer__int__nat__quantifiers_I1_J,axiom,
! [P1: fun(int,bool)] :
( ! [X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
=> pp(aa(int,bool,P1,X1)) )
<=> ! [X1: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X1))) ) ).
tff(fact_85_transfer__int__nat__relations_I3_J,axiom,
! [Y1: nat,X2: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,X2),semiring_1_of_nat(int,Y1))
<=> ord_less_eq(nat,X2,Y1) ) ).
tff(fact_86_less__int__def,axiom,
! [Wa: int,Z1: int] :
( ord_less(int,Z1,Wa)
<=> ( ord_less_eq(int,Z1,Wa)
& ( Z1 != Wa ) ) ) ).
tff(fact_87_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_88_zero__zle__int,axiom,
! [N2: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,N2)) ).
tff(fact_89_zle__int,axiom,
! [N1: nat,M1: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,M1),semiring_1_of_nat(int,N1))
<=> ord_less_eq(nat,M1,N1) ) ).
tff(fact_90_int__le__0__conv,axiom,
! [N1: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,N1),zero_zero(int))
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_91_transfer__int__nat__relations_I1_J,axiom,
! [Y1: nat,X2: nat] :
( ( semiring_1_of_nat(int,X2) = semiring_1_of_nat(int,Y1) )
<=> ( X2 = Y1 ) ) ).
tff(fact_92_int__if__cong,axiom,
! [Y1: nat,X2: nat,P1: bool] :
( ( pp(P1)
=> ( semiring_1_of_nat(int,X2) = semiring_1_of_nat(int,if(nat,P1,X2,Y1)) ) )
& ( ~ pp(P1)
=> ( semiring_1_of_nat(int,Y1) = semiring_1_of_nat(int,if(nat,P1,X2,Y1)) ) ) ) ).
tff(fact_93_nonneg__eq__int,axiom,
! [Z: int] :
( ord_less_eq(int,zero_zero(int),Z)
=> ~ ! [M: nat] : Z != semiring_1_of_nat(int,M) ) ).
tff(fact_94_zero__le__imp__eq__int,axiom,
! [K: int] :
( ord_less_eq(int,zero_zero(int),K)
=> ? [N: nat] : K = semiring_1_of_nat(int,N) ) ).
tff(fact_95_less__mono__imp__le__mono,axiom,
! [J: nat,I: nat,F: fun(nat,nat)] :
( ! [I1: nat,J1: nat] :
( ord_less(nat,I1,J1)
=> ord_less(nat,aa(nat,nat,F,I1),aa(nat,nat,F,J1)) )
=> ( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,aa(nat,nat,F,I),aa(nat,nat,F,J)) ) ) ).
tff(fact_96_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_97_le__fun__def,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [G: fun(B,C),F: fun(B,C)] :
( ord_less_eq(fun(B,C),F,G)
<=> ! [X1: B] : ord_less_eq(C,aa(B,C,F,X1),aa(B,C,G,X1)) ) ) ).
tff(fact_98_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less_eq(A,Y,X) ) ) ).
%----Arities (19)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (5)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
( ord_less(nat,zero_zero(nat),huffma945805758height(a1,t_1))
| ord_less(nat,zero_zero(nat),huffma945805758height(a1,t_2)) ) ).
tff(conj_1,conjecture,
( ( member(a1,a,huffma675207370phabet(a1,t_1))
=> ( huffma1401021291ibling(a1,huffma1146269203erNode(a1,w,t_1,t_2),a) = huffma1401021291ibling(a1,t_1,a) ) )
& ( ~ member(a1,a,huffma675207370phabet(a1,t_1))
=> ( huffma1401021291ibling(a1,huffma1146269203erNode(a1,w,t_1,t_2),a) = huffma1401021291ibling(a1,t_2,a) ) ) ) ).
%------------------------------------------------------------------------------