TPTP Problem File: DAT165^1.p
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%------------------------------------------------------------------------------
% File : DAT165^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Huffman 651
% 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__651.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 388 ( 138 unt; 83 typ; 0 def)
% Number of atoms : 671 ( 286 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 3378 ( 81 ~; 9 |; 35 &;2986 @)
% ( 0 <=>; 267 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 188 ( 188 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 79 usr; 5 con; 0-5 aty)
% Number of variables : 913 ( 34 ^; 791 !; 16 ?; 913 :)
% ( 72 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:41.820
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree,type,
huffma16452318e_tree: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Num_Onum,type,
num: $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 (75)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[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_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_Odepth,type,
huffma223349076_depth:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq,type,
huffma854352999e_freq:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq_092_060_094sub_062F,type,
huffma2047054433freq_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight,type,
huffma1554076246height:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight_092_060_094sub_062F,type,
huffma279770448ight_F:
!>[A: $tType] : ( ( list @ ( huffma16452318e_tree @ A ) ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_OLeaf,type,
huffma1554276827e_Leaf:
!>[A: $tType] : ( nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_Ocase__tree,type,
huffma570615019e_tree:
!>[A: $tType,B: $tType] : ( ( nat > A > B ) > ( nat > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > B ) > ( huffma16452318e_tree @ A ) > B ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_Opred__tree,type,
huffma1727904060d_tree:
!>[A: $tType] : ( ( A > $o ) > ( huffma16452318e_tree @ A ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_Orec__tree,type,
huffma577982705c_tree:
!>[A: $tType,C: $tType] : ( ( nat > A > C ) > ( nat > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > C > C > C ) > ( huffma16452318e_tree @ A ) > C ) ).
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_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
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_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_t____,type,
t: huffma16452318e_tree @ a ).
thf(sy_v_tsa____,type,
tsa: list @ ( huffma16452318e_tree @ a ) ).
%----Relevant facts (253)
thf(fact_0_Cons_Oprems_I1_J,axiom,
huffma2111480347tent_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t @ tsa ) ).
% Cons.prems(1)
thf(fact_1_Cons_Oprems_I2_J,axiom,
( ( huffma279770448ight_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t @ tsa ) )
= ( zero_zero @ nat ) ) ).
% Cons.prems(2)
thf(fact_2_Cons_Oprems_I3_J,axiom,
member2 @ a @ a2 @ ( huffma279473244abet_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t @ tsa ) ) ).
% Cons.prems(3)
thf(fact_3_tree_Oinject_I1_J,axiom,
! [A: $tType,X11: nat,X12: A,Y11: nat,Y12: A] :
( ( ( huffma1554276827e_Leaf @ A @ X11 @ X12 )
= ( huffma1554276827e_Leaf @ A @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tree.inject(1)
thf(fact_4_Cons_Ohyps,axiom,
( ( huffma2111480347tent_F @ a @ tsa )
=> ( ( ( huffma279770448ight_F @ a @ tsa )
= ( zero_zero @ nat ) )
=> ( ( member2 @ a @ a2 @ ( huffma279473244abet_F @ a @ tsa ) )
=> ( member2 @ ( huffma16452318e_tree @ a ) @ ( huffma1554276827e_Leaf @ a @ ( huffma2047054433freq_F @ a @ tsa @ a2 ) @ a2 ) @ ( set2 @ ( huffma16452318e_tree @ a ) @ tsa ) ) ) ) ) ).
% Cons.hyps
thf(fact_5_notin__alphabet_092_060_094sub_062F__imp__freq_092_060_094sub_062F__0,axiom,
! [A: $tType,A2: A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ~ ( member2 @ A @ A2 @ ( huffma279473244abet_F @ A @ Ts ) )
=> ( ( huffma2047054433freq_F @ A @ Ts @ A2 )
= ( zero_zero @ nat ) ) ) ).
% notin_alphabet\<^sub>F_imp_freq\<^sub>F_0
thf(fact_6_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_7_list_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: list @ A,A1: A] :
( ( member2 @ A @ X @ ( set2 @ A @ A22 ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_8_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member2 @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_9_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member2 @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member2 @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_10_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member2 @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z2: list @ A] :
( A2
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member2 @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_11_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_12_tree_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: nat > A > B,F2: nat > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > B,X11: nat,X12: A] :
( ( huffma570615019e_tree @ A @ B @ F1 @ F2 @ ( huffma1554276827e_Leaf @ A @ X11 @ X12 ) )
= ( F1 @ X11 @ X12 ) ) ).
% tree.simps(5)
thf(fact_13_consistent_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A2: A] : ( huffma1050891809istent @ A @ ( huffma1554276827e_Leaf @ A @ W @ A2 ) ) ).
% consistent.simps(1)
thf(fact_14_tree_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: nat > A > C,F2: nat > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > C > C > C,X11: nat,X12: A] :
( ( huffma577982705c_tree @ A @ C @ F1 @ F2 @ ( huffma1554276827e_Leaf @ A @ X11 @ X12 ) )
= ( F1 @ X11 @ X12 ) ) ).
% tree.simps(7)
thf(fact_15_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_16_tree_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o,A2: nat,Aa: A] :
( ( huffma1727904060d_tree @ A @ P @ ( huffma1554276827e_Leaf @ A @ A2 @ Aa ) )
= ( P @ Aa ) ) ).
% tree.pred_inject(1)
thf(fact_17_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_18_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X2: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( member2 @ A @ X2 @ ( set2 @ A @ Xs2 ) ) @ Xs2 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_19_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_20_freq_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A2: A] :
( ( huffma854352999e_freq @ A @ ( huffma1554276827e_Leaf @ A @ W @ A2 ) )
= ( ^ [B2: A] : ( if @ nat @ ( B2 = A2 ) @ W @ ( zero_zero @ nat ) ) ) ) ).
% freq.simps(1)
thf(fact_21_depth_Osimps_I1_J,axiom,
! [A: $tType,W: nat,B3: A,A2: A] :
( ( huffma223349076_depth @ A @ ( huffma1554276827e_Leaf @ A @ W @ B3 ) @ A2 )
= ( zero_zero @ nat ) ) ).
% depth.simps(1)
thf(fact_22_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_23_in__set__member,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( member @ A @ Xs @ X ) ) ).
% in_set_member
thf(fact_24_height_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A2: A] :
( ( huffma1554076246height @ A @ ( huffma1554276827e_Leaf @ A @ W @ A2 ) )
= ( zero_zero @ nat ) ) ).
% height.simps(1)
thf(fact_25_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_26_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_27_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_28_exists__at__height,axiom,
! [A: $tType,T2: huffma16452318e_tree @ A] :
( ( huffma1050891809istent @ A @ T2 )
=> ? [X3: A] :
( ( member2 @ A @ X3 @ ( huffma505251170phabet @ A @ T2 ) )
& ( ( huffma223349076_depth @ A @ T2 @ X3 )
= ( huffma1554076246height @ A @ T2 ) ) ) ) ).
% exists_at_height
thf(fact_29_freq_092_060_094sub_062F_Osimps_I2_J,axiom,
! [A: $tType,T2: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma2047054433freq_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ Ts ) )
= ( ^ [B2: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ ( huffma2047054433freq_F @ A @ Ts @ B2 ) ) ) ) ).
% freq\<^sub>F.simps(2)
thf(fact_30_notin__alphabet__imp__freq__0,axiom,
! [A: $tType,A2: A,T2: huffma16452318e_tree @ A] :
( ~ ( member2 @ A @ A2 @ ( huffma505251170phabet @ A @ T2 ) )
=> ( ( huffma854352999e_freq @ A @ T2 @ A2 )
= ( zero_zero @ nat ) ) ) ).
% notin_alphabet_imp_freq_0
thf(fact_31_height_092_060_094sub_062F_Osimps_I2_J,axiom,
! [A: $tType,T2: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma279770448ight_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ Ts ) )
= ( ord_max @ nat @ ( huffma1554076246height @ A @ T2 ) @ ( huffma279770448ight_F @ A @ Ts ) ) ) ).
% height\<^sub>F.simps(2)
thf(fact_32_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_33_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [Inc: A > A,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_34_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_35_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( insert @ A @ X @ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% List.set_insert
thf(fact_36_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B3 = C2 ) ) ) ).
% add_right_cancel
thf(fact_37_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add_left_cancel
thf(fact_38_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_39_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_40_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_41_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_42_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= A2 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member2 @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= A2 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_48_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( plus_plus @ A @ B3 @ A2 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_49_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_50_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_51_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_52_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_53_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_54_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X22 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X22 ) ) ) ).
% list.simps(15)
thf(fact_55_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_56_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_57_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_58_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_59_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_60_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B2: A] : ( plus_plus @ A @ B2 @ A4 ) ) ) ) ).
% add.commute
thf(fact_61_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B3 = C2 ) ) ) ).
% add.right_cancel
thf(fact_62_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add.left_cancel
thf(fact_63_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_64_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_65_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_66_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_67_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q2 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q2 ) @ ( plus_plus @ nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_68_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_69_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_70_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_71_exists__in__alphabet,axiom,
! [A: $tType,T2: huffma16452318e_tree @ A] :
? [A5: A] : ( member2 @ A @ A5 @ ( huffma505251170phabet @ A @ T2 ) ) ).
% exists_in_alphabet
thf(fact_72_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_73_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_74_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_75_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_76_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_77_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_78_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F3: A > B,X1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X1 )
=> ( ! [F3: A > B,A5: A,As: list @ A,Bs: list @ B] :
( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A5 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A5 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_79_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_80_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y2: A,Xs3: list @ A] :
( ( ( X3 = Y2 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) ) )
=> ( ( ( X3 != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_81_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y2: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_82_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 ) )
=> ( ! [X3: A,Xs3: list @ A,Y2: A,Ys: list @ A] :
( ( P @ Xs3 @ Ys )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_83_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A] : ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys ) )
=> ( ! [X3: A,Xs3: list @ A,Y2: B,Ys: list @ B] :
( ( P @ Xs3 @ Ys )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_84_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_85_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_86_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_87_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_88_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_89_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_90_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_91_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_92_height_092_060_094sub_062F_Osimps_I1_J,axiom,
! [A: $tType] :
( ( huffma279770448ight_F @ A @ ( nil @ ( huffma16452318e_tree @ A ) ) )
= ( zero_zero @ nat ) ) ).
% height\<^sub>F.simps(1)
thf(fact_93_freq_092_060_094sub_062F_Osimps_I1_J,axiom,
! [A: $tType] :
( ( huffma2047054433freq_F @ A @ ( nil @ ( huffma16452318e_tree @ A ) ) )
= ( ^ [B2: A] : ( zero_zero @ nat ) ) ) ).
% freq\<^sub>F.simps(1)
thf(fact_94_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_max @ A @ A2 @ A2 )
= A2 ) ) ).
% max.idem
thf(fact_95_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_max @ A @ A2 @ ( ord_max @ A @ A2 @ B3 ) )
= ( ord_max @ A @ A2 @ B3 ) ) ) ).
% max.left_idem
thf(fact_96_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A2 @ B3 ) @ B3 )
= ( ord_max @ A @ A2 @ B3 ) ) ) ).
% max.right_idem
thf(fact_97_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( ~ ( member2 @ A @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member2 @ A @ A2 @ ( insert2 @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_98_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member2 @ A @ A2 @ ( insert2 @ A @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member2 @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_99_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ X @ A3 ) )
= ( insert2 @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_100_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F2 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_101_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_102_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( B3
= ( plus_plus @ A @ B3 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_103_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_104_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_105_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ C2 @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_106_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B3 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_107_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,C3: A] : ( plus_plus @ A @ C3 @ A4 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_108_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_109_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member2 @ A @ A2 @ A3 )
=> ? [B5: set @ A] :
( ( A3
= ( insert2 @ A @ A2 @ B5 ) )
& ~ ( member2 @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_110_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ Y @ A3 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_111_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: A,B4: set @ A] :
( ~ ( member2 @ A @ A2 @ A3 )
=> ( ~ ( member2 @ A @ B3 @ B4 )
=> ( ( ( insert2 @ A @ A2 @ A3 )
= ( insert2 @ A @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A3 = B4 ) )
& ( ( A2 != B3 )
=> ? [C4: set @ A] :
( ( A3
= ( insert2 @ A @ B3 @ C4 ) )
& ~ ( member2 @ A @ B3 @ C4 )
& ( B4
= ( insert2 @ A @ A2 @ C4 ) )
& ~ ( member2 @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_112_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member2 @ A @ A2 @ A3 )
=> ( ( insert2 @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_113_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B4: set @ A] :
( ~ ( member2 @ A @ X @ A3 )
=> ( ~ ( member2 @ A @ X @ B4 )
=> ( ( ( insert2 @ A @ X @ A3 )
= ( insert2 @ A @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_114_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member2 @ A @ X @ A3 )
=> ~ ! [B5: set @ A] :
( ( A3
= ( insert2 @ A @ X @ B5 ) )
=> ( member2 @ A @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_115_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( member2 @ A @ A2 @ B4 )
=> ( member2 @ A @ A2 @ ( insert2 @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_116_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member2 @ A @ A2 @ ( insert2 @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_117_insertE,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member2 @ A @ A2 @ ( insert2 @ A @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member2 @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_118_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_max @ A @ B3 @ ( ord_max @ A @ A2 @ C2 ) )
= ( ord_max @ A @ A2 @ ( ord_max @ A @ B3 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_119_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B2: A] : ( ord_max @ A @ B2 @ A4 ) ) ) ) ).
% max.commute
thf(fact_120_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A2 @ B3 ) @ C2 )
= ( ord_max @ A @ A2 @ ( ord_max @ A @ B3 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_121_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(6)
thf(fact_122_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(5)
thf(fact_123_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_124_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_125_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_126_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_127_sublist__singleton,axiom,
! [A: $tType,A3: set @ nat,X: A] :
( ( ( member2 @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( sublist @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A3 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member2 @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( sublist @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A3 )
= ( nil @ A ) ) ) ) ).
% sublist_singleton
thf(fact_128_sublist__nil,axiom,
! [A: $tType,A3: set @ nat] :
( ( sublist @ A @ ( nil @ A ) @ A3 )
= ( nil @ A ) ) ).
% sublist_nil
thf(fact_129_notin__set__sublistI,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: set @ nat] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member2 @ A @ X @ ( set2 @ A @ ( sublist @ A @ Xs @ I2 ) ) ) ) ).
% notin_set_sublistI
thf(fact_130_in__set__sublistD,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: set @ nat] :
( ( member2 @ A @ X @ ( set2 @ A @ ( sublist @ A @ Xs @ I2 ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_sublistD
thf(fact_131_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F2 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_132_height__gt__0__alphabet__eq__imp__height__gt__0,axiom,
! [A: $tType,T2: huffma16452318e_tree @ A,U: huffma16452318e_tree @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T2 ) )
=> ( ( huffma1050891809istent @ A @ T2 )
=> ( ( ( huffma505251170phabet @ A @ T2 )
= ( huffma505251170phabet @ A @ U ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ U ) ) ) ) ) ).
% height_gt_0_alphabet_eq_imp_height_gt_0
thf(fact_133_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_134_alphabet_092_060_094sub_062F_Osimps_I2_J,axiom,
! [A: $tType,T2: huffma16452318e_tree @ A,Ts: list @ ( huffma16452318e_tree @ A )] :
( ( huffma279473244abet_F @ A @ ( cons @ ( huffma16452318e_tree @ A ) @ T2 @ Ts ) )
= ( sup_sup @ ( set @ A ) @ ( huffma505251170phabet @ A @ T2 ) @ ( huffma279473244abet_F @ A @ Ts ) ) ) ).
% alphabet\<^sub>F.simps(2)
thf(fact_135_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F4 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_136_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_137_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_138_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_139_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_140_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_141_UnCI,axiom,
! [A: $tType,C2: A,B4: set @ A,A3: set @ A] :
( ( ~ ( member2 @ A @ C2 @ B4 )
=> ( member2 @ A @ C2 @ A3 ) )
=> ( member2 @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_142_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member2 @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member2 @ A @ C2 @ A3 )
| ( member2 @ A @ C2 @ B4 ) ) ) ).
% Un_iff
thf(fact_143_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_144_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_145_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_146_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_147_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_148_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Y @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_149_Un__insert__left,axiom,
! [A: $tType,A2: A,B4: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A2 @ B4 ) @ C5 )
= ( insert2 @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_150_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ A2 @ B4 ) )
= ( insert2 @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_151_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_152_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_153_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B3 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_154_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_155_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B3 ) @ B3 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_156_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A2 ) @ B3 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_157_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_158_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_159_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_160_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_161_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_162_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_163_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_164_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_165_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_166_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
| ( ord_less @ A @ Z @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_167_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B3 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_168_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A4: A] :
( ( A4
= ( ord_max @ A @ A4 @ B2 ) )
& ( A4 != B2 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_169_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B3 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_170_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [C2: A,B3: A,A2: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B3 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_171_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_172_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_173_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_174_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_175_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_176_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_177_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_178_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_179_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_180_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_181_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V2: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_182_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_183_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_184_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_185_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_186_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_187_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_188_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_189_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V2 @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_190_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_191_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_192_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_193_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_194_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_195_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_196_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_197_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_198_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_199_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_200_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_201_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_202_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_203_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_204_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_205_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_206_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member2 @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
=> ( ~ ( member2 @ A @ C2 @ A3 )
=> ( member2 @ A @ C2 @ B4 ) ) ) ).
% UnE
thf(fact_207_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member2 @ A @ C2 @ A3 )
=> ( member2 @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_208_UnI2,axiom,
! [A: $tType,C2: A,B4: set @ A,A3: set @ A] :
( ( member2 @ A @ C2 @ B4 )
=> ( member2 @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_209_bex__Un,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member2 @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member2 @ A @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member2 @ A @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_210_ball__Un,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member2 @ A @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member2 @ A @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_211_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) @ C5 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B4 @ C5 ) ) ) ).
% Un_assoc
thf(fact_212_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_213_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_214_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_215_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B4 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% Un_left_commute
thf(fact_216_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F4 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_217_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_218_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_219_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B3: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_220_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A2: A] :
( ( ord_less @ A @ X @ B3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_221_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B2: A] : ( sup_sup @ A @ B2 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_222_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_223_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_224_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_225_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_226_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A4: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B2 ) )
& ( A4 != B2 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_227_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_228_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,B3: A,A2: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_229_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A @ ( type2 @ A ) )
& ( linorder @ A @ ( type2 @ A ) ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_230_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_231_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_232_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_233_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_234_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
? [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_235_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% set_union
thf(fact_236_can__select__set__list__ex1,axiom,
! [A: $tType,P: A > $o,A3: list @ A] :
( ( can_select @ A @ P @ ( set2 @ A @ A3 ) )
= ( list_ex1 @ A @ P @ A3 ) ) ).
% can_select_set_list_ex1
thf(fact_237_complete__linorder__sup__max,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% complete_linorder_sup_max
thf(fact_238_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_239_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_240_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P2: A > $o,A6: set @ A] :
? [X2: A] :
( ( member2 @ A @ X2 @ A6 )
& ( P2 @ X2 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ A6 )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_241_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_242_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_243_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_244_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_245_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_246_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_247_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_248_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_249_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_250_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_251_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_252_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
%----Type constructors (48)
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___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 @ ( type2 @ A8 ) )
=> ( lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_1,axiom,
semilattice_sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Num_Onum___Orderings_Olinorder_3,axiom,
linorder @ num @ ( type2 @ num ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_6,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_7,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_9,axiom,
ordere516151231imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_10,axiom,
strict2144017051up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_11,axiom,
ordere223160158up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_12,axiom,
ordere236663937imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_13,axiom,
strict797366125id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_14,axiom,
ordere779506340up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_15,axiom,
ordere216010020id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_16,axiom,
cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_17,axiom,
cancel_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_18,axiom,
ab_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_19,axiom,
comm_monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_20,axiom,
comm_semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_21,axiom,
semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_22,axiom,
linorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_23,axiom,
monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_24,axiom,
semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_25,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
%----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,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member2 @ ( huffma16452318e_tree @ a ) @ ( huffma1554276827e_Leaf @ a @ ( huffma2047054433freq_F @ a @ ( cons @ ( huffma16452318e_tree @ a ) @ t @ tsa ) @ a2 ) @ a2 ) @ ( set2 @ ( huffma16452318e_tree @ a ) @ ( cons @ ( huffma16452318e_tree @ a ) @ t @ tsa ) ) ).
%------------------------------------------------------------------------------