TPTP Problem File: SWW532_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW532_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Huffman's Algorithm line 625
% 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_625 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 188 ( 68 unt; 60 typ; 0 def)
% Number of atoms : 236 ( 92 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 154 ( 46 ~; 6 |; 4 &)
% ( 22 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 63 ( 38 >; 25 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-6 aty)
% Number of functors : 36 ( 36 usr; 4 con; 0-5 aty)
% Number of variables : 289 ( 247 !; 1 ?; 289 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:59
%------------------------------------------------------------------------------
%----Should-be-implicit typings (11)
tff(ty_t_a,type,
a1: $tType ).
tff(ty_tc_Code__Numeral_Ocode__numeral,type,
code_code_numeral: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Huffman__Mirabelle__lalbadcutu_Otree,type,
huffma1450048681e_tree: $tType > $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_String_Ochar,type,
char1: $tType ).
tff(ty_tc_String_Oliteral,type,
literal: $tType ).
tff(ty_tc_String_Onibble,type,
nibble: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (49)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[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_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_Code__Numeral_OSuc__code__numeral,type,
code_S1047413653umeral: code_code_numeral > code_code_numeral ).
tff(sy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__case,type,
code_c600343653l_case:
!>[T: $tType] : ( ( T * fun(code_code_numeral,T) * code_code_numeral ) > T ) ).
tff(sy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec,type,
code_c1430087035al_rec:
!>[T: $tType] : ( ( T * fun(code_code_numeral,fun(T,T)) * code_code_numeral ) > T ) ).
tff(sy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__size,type,
code_c271388182l_size: code_code_numeral > nat ).
tff(sy_c_Code__Numeral_Oof__nat,type,
code_of_nat: nat > code_code_numeral ).
tff(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_HOL_Obool_Obool__size,type,
bool_size: bool > nat ).
tff(sy_c_HOL_Oequal__class_Oequal,type,
equal_equal:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oalphabet_092_060_094isub_062F,type,
huffma1516701463abet_F:
!>[A: $tType] : ( list(huffma1450048681e_tree(A)) > fun(A,bool) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oconsistent_092_060_094isub_062F,type,
huffma594769176tent_F:
!>[A: $tType] : ( list(huffma1450048681e_tree(A)) > $o ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Ofreq_092_060_094isub_062F,type,
huffma409467474freq_F:
!>[A: $tType] : ( ( list(huffma1450048681e_tree(A)) * A ) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oheight_092_060_094isub_062F,type,
huffma632063779ight_F:
!>[A: $tType] : ( list(huffma1450048681e_tree(A)) > nat ) ).
tff(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( fun(product_prod(A,A),bool) * nat ) > fun(product_prod(list(A),list(A)),bool) ) ).
tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord467138063of_set:
!>[A: $tType] : ( fun(A,bool) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Olist_Olist__rec,type,
list_rec:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),fun(T,T))) * list(A) ) > T ) ).
tff(sy_c_List_Olist_Olist__size,type,
list_size:
!>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $o ) ).
tff(sy_c_Quickcheck_Obeyond,type,
beyond: ( code_code_numeral * code_code_numeral ) > code_code_numeral ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_String_Ochar_OChar,type,
char: ( nibble * nibble ) > char1 ).
tff(sy_c_String_Ochar_Ochar__case,type,
char_case:
!>[T: $tType] : ( ( fun(nibble,fun(nibble,T)) * char1 ) > T ) ).
tff(sy_c_String_Ochar_Ochar__rec,type,
char_rec:
!>[T: $tType] : ( ( fun(nibble,fun(nibble,T)) * char1 ) > T ) ).
tff(sy_c_String_Ochar_Ochar__size,type,
char_size: char1 > nat ).
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_ts,type,
ts: list(huffma1450048681e_tree(a1)) ).
%----Relevant facts (99)
tff(fact_0_freq_092_060_094isub_062F_Osimps_I1_J,axiom,
! [A: $tType,X3: A] : huffma409467474freq_F(A,nil(huffma1450048681e_tree(A)),X3) = zero_zero(nat) ).
tff(fact_1_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X2: B] :
( ( zero_zero(B) = X2 )
<=> ( X2 = zero_zero(B) ) ) ) ).
tff(fact_2_bool_Osize_I1_J,axiom,
bool_size(fTrue) = zero_zero(nat) ).
tff(fact_3_bool_Osize_I2_J,axiom,
bool_size(fFalse) = zero_zero(nat) ).
tff(fact_4_char__size,axiom,
! [C2: char1] : char_size(C2) = zero_zero(nat) ).
tff(fact_5_nat_Osize_I1_J,axiom,
nat_size(zero_zero(nat)) = zero_zero(nat) ).
tff(fact_6_size__char,axiom,
! [C2: char1] : size_size(char1,C2) = zero_zero(nat) ).
tff(fact_7_bool_Osize_I3_J,axiom,
size_size(bool,fTrue) = zero_zero(nat) ).
tff(fact_8_bool_Osize_I4_J,axiom,
size_size(bool,fFalse) = zero_zero(nat) ).
tff(fact_9_size__literal__def,axiom,
! [S: literal] : size_size(literal,S) = zero_zero(nat) ).
tff(fact_10_nat_Osize_I3_J,axiom,
size_size(nat,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_11_nat__size,axiom,
! [N1: nat] : size_size(nat,N1) = N1 ).
tff(fact_12_size__bool,axiom,
! [B1: bool] : size_size(bool,B1) = zero_zero(nat) ).
tff(fact_13_height_092_060_094isub_062F_Osimps_I1_J,axiom,
! [A: $tType] : huffma632063779ight_F(A,nil(huffma1450048681e_tree(A))) = zero_zero(nat) ).
tff(fact_14_consistent_092_060_094isub_062F_Osimps_I1_J,axiom,
! [A: $tType] : huffma594769176tent_F(A,nil(huffma1450048681e_tree(A))) ).
tff(fact_15_alphabet_092_060_094isub_062F_Osimps_I1_J,axiom,
! [B: $tType] : huffma1516701463abet_F(B,nil(huffma1450048681e_tree(B))) = bot_bot(fun(B,bool)) ).
tff(fact_16_char_Osize_I2_J,axiom,
! [Nibble23: nibble,Nibble13: nibble] : size_size(char1,char(Nibble13,Nibble23)) = zero_zero(nat) ).
tff(fact_17_char_Osize_I1_J,axiom,
! [Nibble23: nibble,Nibble13: nibble] : char_size(char(Nibble13,Nibble23)) = zero_zero(nat) ).
tff(fact_18_list_Osize_I1_J,axiom,
! [B: $tType,Fa: fun(B,nat)] : list_size(B,Fa,nil(B)) = zero_zero(nat) ).
tff(fact_19_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_20_semigroup_Oassoc,axiom,
! [B: $tType,C1: B,B1: B,Aa: B,F: fun(B,fun(B,B))] :
( semigroup(B,F)
=> ( aa(B,B,aa(B,fun(B,B),F,aa(B,B,aa(B,fun(B,B),F,Aa),B1)),C1) = aa(B,B,aa(B,fun(B,B),F,Aa),aa(B,B,aa(B,fun(B,B),F,B1),C1)) ) ) ).
tff(fact_21_code__numeral_Osize_I1_J,axiom,
code_c271388182l_size(zero_zero(code_code_numeral)) = zero_zero(nat) ).
tff(fact_22_char_Oinject,axiom,
! [Nibble22: nibble,Nibble12: nibble,Nibble2: nibble,Nibble1: nibble] :
( ( char(Nibble1,Nibble2) = char(Nibble12,Nibble22) )
<=> ( ( Nibble1 = Nibble12 )
& ( Nibble2 = Nibble22 ) ) ) ).
tff(fact_23_code__numeral_Osize_I3_J,axiom,
size_size(code_code_numeral,zero_zero(code_code_numeral)) = zero_zero(nat) ).
tff(fact_24_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).
tff(fact_25_emptyE,axiom,
! [B: $tType,Aa: B] : ~ member(B,Aa,bot_bot(fun(B,bool))) ).
tff(fact_26_Collect__empty__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( collect(B,P1) = bot_bot(fun(B,bool)) )
<=> ! [X4: B] : ~ pp(aa(B,bool,P1,X4)) ) ).
tff(fact_27_empty__iff,axiom,
! [B: $tType,C1: B] : ~ member(B,C1,bot_bot(fun(B,bool))) ).
tff(fact_28_empty__Collect__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,P1) )
<=> ! [X4: B] : ~ pp(aa(B,bool,P1,X4)) ) ).
tff(fact_29_all__not__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [X4: B] : ~ member(B,X4,A1)
<=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_30_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_31_ex__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ? [X4: B] : member(B,X4,A1)
<=> ( A1 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_32_equals0D,axiom,
! [B: $tType,Aa: B,A1: fun(B,bool)] :
( ( A1 = bot_bot(fun(B,bool)) )
=> ~ member(B,Aa,A1) ) ).
tff(fact_33_sgn__0__0,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Aa: B] :
( ( sgn_sgn(B,Aa) = zero_zero(B) )
<=> ( Aa = zero_zero(B) ) ) ) ).
tff(fact_34_bot__fun__def,axiom,
! [B: $tType,C: $tType] :
( bot(C)
=> ! [X3: B] : aa(B,C,bot_bot(fun(B,C)),X3) = bot_bot(C) ) ).
tff(fact_35_bot__apply,axiom,
! [C: $tType,B: $tType] :
( bot(B)
=> ! [X2: C] : aa(C,B,bot_bot(fun(C,B)),X2) = bot_bot(B) ) ).
tff(fact_36_sorted__list__of__set__empty,axiom,
! [B: $tType] :
( linorder(B)
=> ( linord467138063of_set(B,bot_bot(fun(B,bool))) = nil(B) ) ) ).
tff(fact_37_code__numeral_Osimps_I4_J,axiom,
! [B: $tType,F2: fun(code_code_numeral,B),F1: B] : code_c600343653l_case(B,F1,F2,zero_zero(code_code_numeral)) = F1 ).
tff(fact_38_char_Oexhaust,axiom,
! [Y1: char1] :
~ ! [Nibble11: nibble,Nibble21: nibble] : Y1 != char(Nibble11,Nibble21) ).
tff(fact_39_char_Osimps_I2_J,axiom,
! [B: $tType,Nibble2: nibble,Nibble1: nibble,F1: fun(nibble,fun(nibble,B))] : char_case(B,F1,char(Nibble1,Nibble2)) = aa(nibble,B,aa(nibble,fun(nibble,B),F1,Nibble1),Nibble2) ).
tff(fact_40_char_Orecs,axiom,
! [B: $tType,Nibble2: nibble,Nibble1: nibble,F1: fun(nibble,fun(nibble,B))] : char_rec(B,F1,char(Nibble1,Nibble2)) = aa(nibble,B,aa(nibble,fun(nibble,B),F1,Nibble1),Nibble2) ).
tff(fact_41_list_Osimps_I4_J,axiom,
! [C: $tType,B: $tType,F2: fun(C,fun(list(C),B)),F1: B] : list_case(B,C,F1,F2,nil(C)) = F1 ).
tff(fact_42_equals0I,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [Y: B] : ~ member(B,Y,A1)
=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_43_list__ex1__simps_I1_J,axiom,
! [B: $tType,P1: fun(B,bool)] : ~ list_ex1(B,P1,nil(B)) ).
tff(fact_44_lexn_Osimps_I1_J,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] : lexn(B,R,zero_zero(nat)) = bot_bot(fun(product_prod(list(B),list(B)),bool)) ).
tff(fact_45_code__numeral_Orecs_I1_J,axiom,
! [B: $tType,F2: fun(code_code_numeral,fun(B,B)),F1: B] : code_c1430087035al_rec(B,F1,F2,zero_zero(code_code_numeral)) = F1 ).
tff(fact_46_list_Orecs_I1_J,axiom,
! [C: $tType,B: $tType,F2: fun(C,fun(list(C),fun(B,B))),F1: B] : list_rec(B,C,F1,F2,nil(C)) = F1 ).
tff(fact_47_code__numeral_Osimps_I5_J,axiom,
! [B: $tType,Code_numeral3: code_code_numeral,F2: fun(code_code_numeral,B),F1: B] : code_c600343653l_case(B,F1,F2,code_S1047413653umeral(Code_numeral3)) = aa(code_code_numeral,B,F2,Code_numeral3) ).
tff(fact_48_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
tff(fact_49_code__numeral_Oinject,axiom,
! [Code_numeral4: code_code_numeral,Code_numeral3: code_code_numeral] :
( ( code_S1047413653umeral(Code_numeral3) = code_S1047413653umeral(Code_numeral4) )
<=> ( Code_numeral3 = Code_numeral4 ) ) ).
tff(fact_50_code__numeral_Orecs_I2_J,axiom,
! [B: $tType,Code_numeral3: code_code_numeral,F2: fun(code_code_numeral,fun(B,B)),F1: B] : code_c1430087035al_rec(B,F1,F2,code_S1047413653umeral(Code_numeral3)) = aa(B,B,aa(code_code_numeral,fun(B,B),F2,Code_numeral3),code_c1430087035al_rec(B,F1,F2,Code_numeral3)) ).
tff(fact_51_code__numeral_Osimps_I2_J,axiom,
! [Code_numeral2: code_code_numeral] : zero_zero(code_code_numeral) != code_S1047413653umeral(Code_numeral2) ).
tff(fact_52_code__numeral_Osimps_I3_J,axiom,
! [Code_numeral1: code_code_numeral] : code_S1047413653umeral(Code_numeral1) != zero_zero(code_code_numeral) ).
tff(fact_53_code__numeral_Oexhaust,axiom,
! [Y1: code_code_numeral] :
( ( Y1 != zero_zero(code_code_numeral) )
=> ~ ! [Code_numeral: code_code_numeral] : Y1 != code_S1047413653umeral(Code_numeral) ) ).
tff(fact_54_,axiom,
! [B: $tType,K1: code_code_numeral,Rhs: fun(code_code_numeral,B),Random_aux: fun(code_code_numeral,B)] :
( ( aa(code_code_numeral,B,Random_aux,zero_zero(code_code_numeral)) = aa(code_code_numeral,B,Rhs,zero_zero(code_code_numeral)) )
=> ( ! [K2: code_code_numeral] : aa(code_code_numeral,B,Random_aux,code_S1047413653umeral(K2)) = aa(code_code_numeral,B,Rhs,code_S1047413653umeral(K2))
=> ( aa(code_code_numeral,B,Random_aux,K1) = aa(code_code_numeral,B,Rhs,K1) ) ) ) ).
tff(fact_55_bot__empty__eq,axiom,
! [B: $tType,X3: B] :
( pp(aa(B,bool,bot_bot(fun(B,bool)),X3))
<=> member(B,X3,bot_bot(fun(B,bool))) ) ).
tff(fact_56_zero__code__numeral__def,axiom,
zero_zero(code_code_numeral) = code_of_nat(zero_zero(nat)) ).
tff(fact_57_Code__Numeral_Oof__nat__inject,axiom,
! [M: nat,N: nat] :
( ( code_of_nat(N) = code_of_nat(M) )
<=> ( N = M ) ) ).
tff(fact_58_,axiom,
! [K: code_code_numeral] : beyond(K,zero_zero(code_code_numeral)) = zero_zero(code_code_numeral) ).
tff(fact_59_code__numeral__case,axiom,
! [K: code_code_numeral] :
~ ! [N2: nat] : K != code_of_nat(N2) ).
tff(fact_60_code__numeral__induct__raw,axiom,
! [K1: code_code_numeral,P1: fun(code_code_numeral,bool)] :
( ! [N2: nat] : pp(aa(code_code_numeral,bool,P1,code_of_nat(N2)))
=> pp(aa(code_code_numeral,bool,P1,K1)) ) ).
tff(fact_61_,axiom,
! [L: code_code_numeral,K: code_code_numeral] :
( ( ord_less(code_code_numeral,K,L)
=> ( beyond(K,L) = L ) )
& ( ~ ord_less(code_code_numeral,K,L)
=> ( beyond(K,L) = zero_zero(code_code_numeral) ) ) ) ).
tff(fact_62_in__inv__imagep,axiom,
! [B: $tType,C: $tType,Y2: C,X2: C,F: fun(C,B),R: fun(B,fun(B,bool))] :
( inv_imagep(B,C,R,F,X2,Y2)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R,aa(C,B,F,X2)),aa(C,B,F,Y2))) ) ).
tff(fact_63_eq__Nil__null,axiom,
! [B: $tType,Xs: list(B)] :
( ( Xs = nil(B) )
<=> null(B,Xs) ) ).
tff(fact_64_sgn__less,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Aa: B] :
( ord_less(B,sgn_sgn(B,Aa),zero_zero(B))
<=> ord_less(B,Aa,zero_zero(B)) ) ) ).
tff(fact_65_sgn__greater,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Aa: B] :
( ord_less(B,zero_zero(B),sgn_sgn(B,Aa))
<=> ord_less(B,zero_zero(B),Aa) ) ) ).
tff(fact_66_not__less__bot,axiom,
! [A: $tType] :
( bot(A)
=> ! [A2: A] : ~ ord_less(A,A2,bot_bot(A)) ) ).
tff(fact_67_bot__less,axiom,
! [B: $tType] :
( bot(B)
=> ! [Aa: B] :
( ( Aa != bot_bot(B) )
<=> ord_less(B,bot_bot(B),Aa) ) ) ).
tff(fact_68_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ( ( X1 != Y1 )
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_69_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_70_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_71_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ( B2 = C2 )
=> ord_less(A,A2,C2) ) ) ) ).
tff(fact_72_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less(A,B2,C2)
=> ord_less(A,A2,C2) ) ) ) ).
tff(fact_73_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ~ ord_less(A,B2,A2) ) ) ).
tff(fact_74_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X: B] : aa(B,C,F,X) = aa(B,C,G,X)
=> ( 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_Collect__def,axiom,
! [B: $tType,P1: fun(B,bool)] : collect(B,P1) = P1 ).
tff(fact_77_order__less__imp__triv,axiom,
! [B: $tType] :
( preorder(B)
=> ! [P1: bool,Y2: B,X2: B] :
( ord_less(B,X2,Y2)
=> ( ord_less(B,Y2,X2)
=> pp(P1) ) ) ) ).
tff(fact_78_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( Y1 != X1 ) ) ) ).
tff(fact_79_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( X1 != Y1 ) ) ) ).
tff(fact_80_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_81_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_82_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( X1 != Y1 ) ) ) ).
tff(fact_83_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
=> ( ~ ord_less(A,X1,Y1)
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_84_linorder__antisym__conv3,axiom,
! [B: $tType] :
( linorder(B)
=> ! [X2: B,Y2: B] :
( ~ ord_less(B,Y2,X2)
=> ( ~ ord_less(B,X2,Y2)
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_85_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
| ( X1 = Y1 )
| ord_less(A,Y1,X1) ) ) ).
tff(fact_86_not__less__iff__gr__or__eq,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y2: B,X2: B] :
( ~ ord_less(B,X2,Y2)
<=> ( ord_less(B,Y2,X2)
| ( X2 = Y2 ) ) ) ) ).
tff(fact_87_linorder__neq__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y2: B,X2: B] :
( ( X2 != Y2 )
<=> ( ord_less(B,X2,Y2)
| ord_less(B,Y2,X2) ) ) ) ).
tff(fact_88_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ~ ord_less(A,X1,X1) ) ).
tff(fact_89_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
=> ( ~ ord_less(A,X1,Y1)
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_90_null__rec_I2_J,axiom,
! [B: $tType] : null(B,nil(B)) ).
tff(fact_91_List_Onull__def,axiom,
! [B: $tType,Xs: list(B)] :
( null(B,Xs)
<=> ( Xs = nil(B) ) ) ).
tff(fact_92_equal__Nil__null,axiom,
! [B: $tType,Xs: list(B)] :
( equal_equal(list(B),Xs,nil(B))
<=> null(B,Xs) ) ).
tff(fact_93_order__less__subst1,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,B1: C,F: fun(C,B),Aa: B] :
( ord_less(B,Aa,aa(C,B,F,B1))
=> ( ord_less(C,B1,C1)
=> ( ! [X: C,Y: C] :
( ord_less(C,X,Y)
=> ord_less(B,aa(C,B,F,X),aa(C,B,F,Y)) )
=> ord_less(B,Aa,aa(C,B,F,C1)) ) ) ) ) ).
tff(fact_94_ord__eq__less__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,B1: C,F: fun(C,B),Aa: B] :
( ( Aa = aa(C,B,F,B1) )
=> ( ord_less(C,B1,C1)
=> ( ! [X: C,Y: C] :
( ord_less(C,X,Y)
=> ord_less(B,aa(C,B,F,X),aa(C,B,F,Y)) )
=> ord_less(B,Aa,aa(C,B,F,C1)) ) ) ) ) ).
tff(fact_95_less__zeroE,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_96_less__nat__zero__code,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_97_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_98_not__psubset__empty,axiom,
! [B: $tType,A1: fun(B,bool)] : ~ ord_less(fun(B,bool),A1,bot_bot(fun(B,bool))) ).
%----Arities (20)
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_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(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_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Orderings_Obot,axiom,
bot(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_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
tff(arity_Code__Numeral_Ocode__numeral___Orderings_Opreorder,axiom,
preorder(code_code_numeral) ).
tff(arity_Code__Numeral_Ocode__numeral___Orderings_Olinorder,axiom,
linorder(code_code_numeral) ).
tff(arity_Code__Numeral_Ocode__numeral___Orderings_Oorder,axiom,
order(code_code_numeral) ).
tff(arity_Code__Numeral_Ocode__numeral___Orderings_Oord,axiom,
ord(code_code_numeral) ).
tff(arity_Code__Numeral_Ocode__numeral___Groups_Ozero,axiom,
zero(code_code_numeral) ).
%----Helper facts (7)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
~ member(a1,a,huffma1516701463abet_F(a1,ts)) ).
tff(conj_1,conjecture,
huffma409467474freq_F(a1,ts,a) = zero_zero(nat) ).
%------------------------------------------------------------------------------