TPTP Problem File: DAT172^1.p
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%------------------------------------------------------------------------------
% File : DAT172^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Huffman 2088
% 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__2088.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.67 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 347 ( 92 unt; 59 typ; 0 def)
% Number of atoms : 813 ( 243 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3694 ( 79 ~; 7 |; 38 &;3170 @)
% ( 0 <=>; 400 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 122 ( 122 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 9 con; 0-6 aty)
% Number of variables : 887 ( 33 ^; 789 !; 18 ?; 887 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:43:44.792
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree,type,
huffma16452318e_tree: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (54)
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_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_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_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_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_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere1818651114id_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_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_Oconsistent,type,
huffma1050891809istent:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ocost,type,
huffma636208924e_cost:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Odepth,type,
huffma223349076_depth:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ofreq,type,
huffma854352999e_freq:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oheight,type,
huffma1554076246height:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OmergeSibling,type,
huffma1954420889ibling:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ominima,type,
huffma1154738298minima:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Ooptimum,type,
huffma936049440ptimum:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > $o ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Osibling,type,
huffma943100115ibling:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OsplitLeaf,type,
huffma454997449itLeaf:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapFourSyms,type,
huffma304375860urSyms:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > A > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapLeaves,type,
huffma2094459102Leaves:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapSyms,type,
huffma469337550apSyms:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > A > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oweight,type,
huffma691733767weight:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_c____,type,
c: a ).
thf(sy_v_t,type,
t: huffma16452318e_tree @ a ).
thf(sy_v_u____,type,
u: huffma16452318e_tree @ a ).
thf(sy_v_w_092_060_094sub_062a,type,
w_a: nat ).
thf(sy_v_w_092_060_094sub_062b,type,
w_b: nat ).
%----Relevant facts (254)
thf(fact_0_c_092_060_094sub_062u,axiom,
huffma1050891809istent @ a @ u ).
% c\<^sub>u
thf(fact_1_ab,axiom,
a2 != b ).
% ab
thf(fact_2_assms_I1_J,axiom,
huffma1050891809istent @ a @ t ).
% assms(1)
thf(fact_3_assms_I2_J,axiom,
huffma936049440ptimum @ a @ t ).
% assms(2)
thf(fact_4_dc,axiom,
( ( huffma943100115ibling @ a @ u @ c )
!= c ) ).
% dc
thf(fact_5_sibling__sibling__id,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma943100115ibling @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) )
= A2 ) ) ).
% sibling_sibling_id
thf(fact_6_a_092_060_094sub_062c,axiom,
member @ a @ c @ ( huffma505251170phabet @ a @ u ) ).
% a\<^sub>c
thf(fact_7_sibling__reciprocal,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma943100115ibling @ A @ T @ A2 )
= B )
=> ( ( huffma943100115ibling @ A @ T @ B )
= A2 ) ) ) ).
% sibling_reciprocal
thf(fact_8_consistent__swapFourSyms,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
( ( huffma1050891809istent @ A @ T )
=> ( huffma1050891809istent @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) ) ) ).
% consistent_swapFourSyms
thf(fact_9_a_092_060_094sub_062d,axiom,
member @ a @ ( huffma943100115ibling @ a @ u @ c ) @ ( huffma505251170phabet @ a @ u ) ).
% a\<^sub>d
thf(fact_10_a_092_060_094sub_062b,axiom,
member @ a @ b @ ( huffma505251170phabet @ a @ u ) ).
% a\<^sub>b
thf(fact_11_a_092_060_094sub_062a,axiom,
member @ a @ a2 @ ( huffma505251170phabet @ a @ u ) ).
% a\<^sub>a
thf(fact_12_d_092_060_094sub_062c,axiom,
( ( huffma223349076_depth @ a @ u @ c )
= ( huffma1554076246height @ a @ u ) ) ).
% d\<^sub>c
thf(fact_13_sibling__swapSyms__sibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma943100115ibling @ A @ T @ B )
!= B )
=> ( ( A2 != B )
=> ( ( huffma943100115ibling @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ ( huffma943100115ibling @ A @ T @ B ) ) @ A2 )
= B ) ) ) ) ).
% sibling_swapSyms_sibling
thf(fact_14_sibling__swapSyms__other__sibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma943100115ibling @ A @ T @ B )
!= A2 )
=> ( ( ( huffma943100115ibling @ A @ T @ B )
!= B )
=> ( ( A2 != B )
=> ( ( huffma943100115ibling @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) @ ( huffma943100115ibling @ A @ T @ B ) )
= A2 ) ) ) ) ) ).
% sibling_swapSyms_other_sibling
thf(fact_15_d_092_060_094sub_062d,axiom,
( ( huffma223349076_depth @ a @ u @ ( huffma943100115ibling @ a @ u @ c ) )
= ( huffma1554076246height @ a @ u ) ) ).
% d\<^sub>d
thf(fact_16_sibling__swapLeaves__sibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,B: A,A2: A,W_a: nat,W_s: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma943100115ibling @ A @ T @ B )
!= B )
=> ( ( A2 != B )
=> ( ( huffma943100115ibling @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_s @ ( huffma943100115ibling @ A @ T @ B ) ) @ A2 )
= B ) ) ) ) ).
% sibling_swapLeaves_sibling
thf(fact_17_depth__sibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma223349076_depth @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) )
= ( huffma223349076_depth @ A @ T @ A2 ) ) ) ).
% depth_sibling
thf(fact_18_sibling__swapFourSyms__when__4th__is__sibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
=> ( ( A2 != B )
=> ( ( ( huffma943100115ibling @ A @ T @ C )
!= C )
=> ( ( huffma943100115ibling @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ ( huffma943100115ibling @ A @ T @ C ) ) @ A2 )
= B ) ) ) ) ) ) ) ).
% sibling_swapFourSyms_when_4th_is_sibling
thf(fact_19_assms_I4_J,axiom,
~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) ) ).
% assms(4)
thf(fact_20_assms_I3_J,axiom,
member @ a @ a2 @ ( huffma505251170phabet @ a @ t ) ).
% assms(3)
thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060lbrakk_062c_A_092_060in_062_Aalphabet_Au_059_Adepth_Au_Ac_A_061_Aheight_Au_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [C2: a] :
( ( member @ a @ C2 @ ( huffma505251170phabet @ a @ u ) )
=> ( ( huffma223349076_depth @ a @ u @ C2 )
!= ( huffma1554076246height @ a @ u ) ) ) ).
% \<open>\<And>thesis. (\<And>c. \<lbrakk>c \<in> alphabet u; depth u c = height u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_22__092_060open_062_092_060exists_062a_092_060in_062alphabet_Au_O_Adepth_Au_Aa_A_061_Aheight_Au_092_060close_062,axiom,
? [X: a] :
( ( member @ a @ X @ ( huffma505251170phabet @ a @ u ) )
& ( ( huffma223349076_depth @ a @ u @ X )
= ( huffma1554076246height @ a @ u ) ) ) ).
% \<open>\<exists>a\<in>alphabet u. depth u a = height u\<close>
thf(fact_23_notin__alphabet__imp__sibling__id,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma943100115ibling @ A @ T @ A2 )
= A2 ) ) ).
% notin_alphabet_imp_sibling_id
thf(fact_24_swapLeaves__id__when__notin__alphabet,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W: nat,W2: nat] :
( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma2094459102Leaves @ A @ T @ W @ A2 @ W2 @ A2 )
= T ) ) ).
% swapLeaves_id_when_notin_alphabet
thf(fact_25_alphabet__swapSyms,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A,B: A] :
( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma505251170phabet @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) )
= ( huffma505251170phabet @ A @ T ) ) ) ) ).
% alphabet_swapSyms
thf(fact_26_height__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
( ( huffma1554076246height @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( huffma1554076246height @ A @ T ) ) ).
% height_swapLeaves
thf(fact_27_swapSyms__id,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma469337550apSyms @ A @ T @ A2 @ A2 )
= T ) ) ).
% swapSyms_id
thf(fact_28_alphabet__swapFourSyms,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A,B: A,C: A,D: A] :
( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma505251170phabet @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) )
= ( huffma505251170phabet @ A @ T ) ) ) ) ) ) ).
% alphabet_swapFourSyms
thf(fact_29_depth__swapLeaves__neither,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( C != A2 )
=> ( ( C != B )
=> ( ( huffma223349076_depth @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) @ C )
= ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% depth_swapLeaves_neither
thf(fact_30_depth__swapSyms__neither,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( C != A2 )
=> ( ( C != B )
=> ( ( huffma223349076_depth @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) @ C )
= ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% depth_swapSyms_neither
thf(fact_31_exists__at__height,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( huffma1050891809istent @ A @ T )
=> ? [X: A] :
( ( member @ A @ X @ ( huffma505251170phabet @ A @ T ) )
& ( ( huffma223349076_depth @ A @ T @ X )
= ( huffma1554076246height @ A @ T ) ) ) ) ).
% exists_at_height
thf(fact_32_exists__in__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
? [A3: A] : ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) ) ).
% exists_in_alphabet
thf(fact_33_in__alphabet__imp__sibling__in__alphabet,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( member @ A @ ( huffma943100115ibling @ A @ T @ A2 ) @ ( huffma505251170phabet @ A @ T ) ) ) ).
% in_alphabet_imp_sibling_in_alphabet
thf(fact_34_sibling__ne__imp__sibling__in__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( ( huffma943100115ibling @ A @ T @ A2 )
!= A2 )
=> ( member @ A @ ( huffma943100115ibling @ A @ T @ A2 ) @ ( huffma505251170phabet @ A @ T ) ) ) ).
% sibling_ne_imp_sibling_in_alphabet
thf(fact_35_consistent__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( huffma1050891809istent @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ).
% consistent_swapLeaves
thf(fact_36_consistent__swapSyms,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( huffma1050891809istent @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) ) ).
% consistent_swapSyms
thf(fact_37_swapFourSyms__def,axiom,
! [A: $tType] :
( ( huffma304375860urSyms @ A )
= ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A,C3: A,D2: A] : ( if @ ( huffma16452318e_tree @ A ) @ ( A4 = D2 ) @ ( huffma469337550apSyms @ A @ T2 @ B2 @ C3 ) @ ( if @ ( huffma16452318e_tree @ A ) @ ( B2 = C3 ) @ ( huffma469337550apSyms @ A @ T2 @ A4 @ D2 ) @ ( huffma469337550apSyms @ A @ ( huffma469337550apSyms @ A @ T2 @ A4 @ C3 ) @ B2 @ D2 ) ) ) ) ) ).
% swapFourSyms_def
thf(fact_38_a_092_060_094sub_062u,axiom,
( ( huffma505251170phabet @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
= ( huffma505251170phabet @ a @ u ) ) ).
% a\<^sub>u
thf(fact_39_freq__swapFourSyms,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma854352999e_freq @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) )
= ( huffma854352999e_freq @ A @ T ) ) ) ) ) ) ) ).
% freq_swapFourSyms
thf(fact_40_False,axiom,
( ( huffma1554076246height @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
!= ( zero_zero @ nat ) ) ).
% False
thf(fact_41_f_092_060_094sub_062u,axiom,
( ( huffma854352999e_freq @ a @ ( huffma454997449itLeaf @ a @ t @ w_a @ a2 @ w_b @ b ) )
= ( huffma854352999e_freq @ a @ u ) ) ).
% f\<^sub>u
thf(fact_42_freq__swapSyms,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma854352999e_freq @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) )
= ( huffma854352999e_freq @ A @ T ) ) ) ) ) ).
% freq_swapSyms
thf(fact_43_consistent__splitLeaf,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,B: A,W_a: nat,A2: A,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ~ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( huffma1050891809istent @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) ) ) ) ).
% consistent_splitLeaf
thf(fact_44_height__0__imp__sibling__id,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( ( huffma1554076246height @ A @ T )
= ( zero_zero @ nat ) )
=> ( ( huffma943100115ibling @ A @ T @ A2 )
= A2 ) ) ).
% height_0_imp_sibling_id
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_notin__alphabet__imp__splitLeaf__id,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W_a: nat,W_b: nat,B: A] :
( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B )
= T ) ) ).
% notin_alphabet_imp_splitLeaf_id
thf(fact_50_swapLeaves__id,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma2094459102Leaves @ A @ T @ ( huffma854352999e_freq @ A @ T @ A2 ) @ A2 @ ( huffma854352999e_freq @ A @ T @ A2 ) @ A2 )
= T ) ) ).
% swapLeaves_id
thf(fact_51_depth__height__imp__sibling__ne,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma223349076_depth @ A @ T @ A2 )
= ( huffma1554076246height @ A @ T ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T ) )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma943100115ibling @ A @ T @ A2 )
!= A2 ) ) ) ) ) ).
% depth_height_imp_sibling_ne
thf(fact_52_h_092_060_094sub_062u,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ a @ u ) ).
% h\<^sub>u
thf(fact_53_notin__alphabet__imp__freq__0,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma854352999e_freq @ A @ T @ A2 )
= ( zero_zero @ nat ) ) ) ).
% notin_alphabet_imp_freq_0
thf(fact_54_assms_I7_J,axiom,
ord_less_eq @ nat @ w_a @ w_b ).
% assms(7)
thf(fact_55_freq__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( A2 != B )
=> ( ( huffma854352999e_freq @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( ^ [C3: A] : ( if @ nat @ ( C3 = A2 ) @ ( if @ nat @ ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) ) @ W_a @ ( zero_zero @ nat ) ) @ ( if @ nat @ ( C3 = B ) @ ( if @ nat @ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) ) @ W_b @ ( zero_zero @ nat ) ) @ ( huffma854352999e_freq @ A @ T @ C3 ) ) ) ) ) ) ) ).
% freq_swapLeaves
thf(fact_56_assms_I6_J,axiom,
! [X3: a] :
( ( member @ a @ X3 @ ( huffma505251170phabet @ a @ t ) )
=> ( ( ord_less_eq @ nat @ w_a @ ( huffma854352999e_freq @ a @ t @ X3 ) )
& ( ord_less_eq @ nat @ w_b @ ( huffma854352999e_freq @ a @ t @ X3 ) ) ) ) ).
% assms(6)
thf(fact_57_assms_I5_J,axiom,
( ( huffma854352999e_freq @ a @ t @ a2 )
= ( plus_plus @ nat @ w_a @ w_b ) ) ).
% assms(5)
thf(fact_58_height__gt__0__alphabet__eq__imp__height__gt__0,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,U: huffma16452318e_tree @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T ) )
=> ( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma505251170phabet @ A @ T )
= ( huffma505251170phabet @ A @ U ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ U ) ) ) ) ) ).
% height_gt_0_alphabet_eq_imp_height_gt_0
thf(fact_59_swapSyms__def,axiom,
! [A: $tType] :
( ( huffma469337550apSyms @ A )
= ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A] : ( huffma2094459102Leaves @ A @ T2 @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ A4 @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ B2 ) ) ) ).
% swapSyms_def
thf(fact_60_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_61_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_62_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X4: A] :
( ! [X: A] :
( ( ( V @ X )
= ( zero_zero @ nat ) )
=> ( P @ X ) )
=> ( ! [X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
=> ( ~ ( P @ X )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X ) )
& ~ ( P @ Y ) ) ) )
=> ( P @ X4 ) ) ) ).
% infinite_descent0_measure
thf(fact_63_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_64_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less @ nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_65_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_66_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_67_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_68_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_69_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_70_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_left
thf(fact_71_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_right
thf(fact_72_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_73_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ B @ A2 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_74_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ( plus_plus @ A @ A2 @ B )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_75_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ( plus_plus @ A @ B @ A2 )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_76_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_77_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_78_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_79_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_80_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_81_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_82_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_83_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_84_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_85_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_86_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_87_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_88_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_89_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% le_add_same_cancel2
thf(fact_90_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% le_add_same_cancel1
thf(fact_91_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_92_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_93_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_94_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_95_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel1
thf(fact_96_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel2
thf(fact_97_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_98_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_99_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_100_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_right
thf(fact_101_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_left
thf(fact_102_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_103_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_104_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ( plus_plus @ A @ X4 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_105_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing
thf(fact_106_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_strict_left_mono
thf(fact_107_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_108_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_109_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_110_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X4 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_111_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ( plus_plus @ A @ X4 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_112_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_113_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_114_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C: A,B: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_115_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% add_decreasing2
thf(fact_116_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_117_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_118_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_119_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_120_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_121_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% add_decreasing
thf(fact_122_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) ) ).
% zero_le
thf(fact_123_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_124_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_125_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_126_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_127_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_128_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_129_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_130_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_131_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_132_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= M2 )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_133_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_134_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_135_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_136_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_137_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_138_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_139_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_140_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_141_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_142_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_143_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_144_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_145_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_146_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_147_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ 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(4)
thf(fact_148_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_149_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_150_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_151_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_152_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_153_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_154_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_155_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_156_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% le_add1
thf(fact_157_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% le_add2
thf(fact_158_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_159_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_160_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_161_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_162_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_163_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_164_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_165_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_166_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_167_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
| ( ord_less_eq @ nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_168_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_169_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_170_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_171_nat__add__left__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( plus_plus @ nat @ K @ M2 )
= ( plus_plus @ nat @ K @ N ) )
= ( M2 = N ) ) ).
% nat_add_left_cancel
thf(fact_172_nat__add__right__cancel,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ K )
= ( plus_plus @ nat @ N @ K ) )
= ( M2 = N ) ) ).
% nat_add_right_cancel
thf(fact_173_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.assoc
thf(fact_174_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_175_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_176_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_177_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.left_commute
thf(fact_178_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_mono
thf(fact_179_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B = C ) ) ) ).
% add_left_imp_eq
thf(fact_180_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B = C ) ) ) ).
% add_right_imp_eq
thf(fact_181_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_left_mono
thf(fact_182_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_right_mono
thf(fact_183_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B2: A] :
? [C3: A] :
( B2
= ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_184_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_left
thf(fact_185_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_right
thf(fact_186_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_187_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_pos_pos
thf(fact_188_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_189_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ( plus_plus @ nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_190_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( P @ N )
=> ? [K3: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_191_depth__le__height,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] : ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma1554076246height @ A @ T ) ) ).
% depth_le_height
thf(fact_192_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X4: A] :
( ( ( zero_zero @ A )
= X4 )
= ( X4
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_193_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less @ nat @ M2 @ N )
| ( ord_less @ nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_194_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_195_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_196_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_197_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y: A] :
( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X ) )
=> ( P @ Y ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_198_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_199_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_200_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less @ nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_201_linorder__neqE__nat,axiom,
! [X4: nat,Y2: nat] :
( ( X4 != Y2 )
=> ( ~ ( ord_less @ nat @ X4 @ Y2 )
=> ( ord_less @ nat @ Y2 @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_202_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y: A] :
( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X ) )
=> ( P @ Y ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_203_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X4: A] :
( ! [X: A] :
( ~ ( P @ X )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X ) )
& ~ ( P @ Y ) ) )
=> ( P @ X4 ) ) ).
% infinite_descent_measure
thf(fact_204_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_205_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_206_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_207_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_208_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_209_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_210_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_211_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X4 @ ( plus_plus @ A @ Y2 @ E ) ) )
=> ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% field_le_epsilon
thf(fact_212_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X: A] :
( ( P @ X )
=> ? [Y: A] :
( ( P @ Y )
& ~ ( ord_less_eq @ nat @ ( M2 @ Y ) @ ( M2 @ X ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( M2 @ Y3 ) @ ( plus_plus @ nat @ ( M2 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_213_minima__def,axiom,
! [A: $tType] :
( ( huffma1154738298minima @ A )
= ( ^ [T2: huffma16452318e_tree @ A,A4: A,B2: A] :
( ( member @ A @ A4 @ ( huffma505251170phabet @ A @ T2 ) )
& ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T2 ) )
& ( A4 != B2 )
& ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ ( huffma854352999e_freq @ A @ T2 @ B2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( huffma505251170phabet @ A @ T2 ) )
=> ( ( X2 != A4 )
=> ( ( X2 != B2 )
=> ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ A4 ) @ ( huffma854352999e_freq @ A @ T2 @ X2 ) )
& ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T2 @ B2 ) @ ( huffma854352999e_freq @ A @ T2 @ X2 ) ) ) ) ) ) ) ) ) ).
% minima_def
thf(fact_214_cost__splitLeaf,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,W_a: nat,W_b: nat,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( huffma854352999e_freq @ A @ T @ A2 )
= ( plus_plus @ nat @ W_a @ W_b ) )
=> ( ( huffma636208924e_cost @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ T ) @ W_a ) @ W_b ) ) ) ) ) ).
% cost_splitLeaf
thf(fact_215_cost__swapSyms__le,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ B @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ord_less_eq @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma854352999e_freq @ A @ T @ B ) )
=> ( ( ord_less_eq @ nat @ ( huffma223349076_depth @ A @ T @ A2 ) @ ( huffma223349076_depth @ A @ T @ B ) )
=> ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ ( huffma469337550apSyms @ A @ T @ A2 @ B ) ) @ ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ).
% cost_swapSyms_le
thf(fact_216_freq__mergeSibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( huffma943100115ibling @ A @ T @ A2 )
!= A2 )
=> ( ( huffma854352999e_freq @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) )
= ( ^ [C3: A] :
( if @ nat @ ( C3 = A2 ) @ ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T @ A2 ) @ ( huffma854352999e_freq @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) ) )
@ ( if @ nat
@ ( C3
= ( huffma943100115ibling @ A @ T @ A2 ) )
@ ( zero_zero @ nat )
@ ( huffma854352999e_freq @ A @ T @ C3 ) ) ) ) ) ) ) ) ).
% freq_mergeSibling
thf(fact_217_optimum__def,axiom,
! [A: $tType] :
( ( huffma936049440ptimum @ A )
= ( ^ [T2: huffma16452318e_tree @ A] :
! [U2: huffma16452318e_tree @ A] :
( ( huffma1050891809istent @ A @ U2 )
=> ( ( ( huffma505251170phabet @ A @ T2 )
= ( huffma505251170phabet @ A @ U2 ) )
=> ( ( ( huffma854352999e_freq @ A @ T2 )
= ( huffma854352999e_freq @ A @ U2 ) )
=> ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ T2 ) @ ( huffma636208924e_cost @ A @ U2 ) ) ) ) ) ) ) ).
% optimum_def
thf(fact_218_notin__alphabet__imp__mergeSibling__id,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A] :
( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma1954420889ibling @ A @ T @ A2 )
= T ) ) ).
% notin_alphabet_imp_mergeSibling_id
thf(fact_219_height__0__imp__cost__0,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( ( huffma1554076246height @ A @ T )
= ( zero_zero @ nat ) )
=> ( ( huffma636208924e_cost @ A @ T )
= ( zero_zero @ nat ) ) ) ).
% height_0_imp_cost_0
thf(fact_220_consistent__mergeSibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( huffma1050891809istent @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) ) ) ).
% consistent_mergeSibling
thf(fact_221_cost__mergeSibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma943100115ibling @ A @ T @ A2 )
!= A2 )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ ( huffma943100115ibling @ A @ T @ A2 ) ) )
= ( huffma636208924e_cost @ A @ T ) ) ) ) ).
% cost_mergeSibling
thf(fact_222_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_223_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% linordered_field_no_lb
thf(fact_224_linordered__field__class_Osign__simps_I26_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% linordered_field_class.sign_simps(26)
thf(fact_225_linordered__field__class_Osign__simps_I27_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B2: A] : ( plus_plus @ A @ B2 @ A4 ) ) ) ) ).
% linordered_field_class.sign_simps(27)
thf(fact_226_linordered__field__class_Osign__simps_I28_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% linordered_field_class.sign_simps(28)
thf(fact_227_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X: A] :
( ( P @ X )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ Y ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_228_cost__swapFourSyms__le,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B: A,C: A,D: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( huffma1154738298minima @ A @ T @ A2 @ B )
=> ( ( member @ A @ C @ ( huffma505251170phabet @ A @ T ) )
=> ( ( member @ A @ D @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( huffma223349076_depth @ A @ T @ C )
= ( huffma1554076246height @ A @ T ) )
=> ( ( ( huffma223349076_depth @ A @ T @ D )
= ( huffma1554076246height @ A @ T ) )
=> ( ( C != D )
=> ( ord_less_eq @ nat @ ( huffma636208924e_cost @ A @ ( huffma304375860urSyms @ A @ T @ A2 @ B @ C @ D ) ) @ ( huffma636208924e_cost @ A @ T ) ) ) ) ) ) ) ) ) ).
% cost_swapFourSyms_le
thf(fact_229_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M2 @ Y3 ) @ B ) )
=> ? [X: A] :
( ( P @ X )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( M2 @ Y ) @ ( M2 @ X ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_230_weight__splitLeaf,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,W_a: nat,W_b: nat,B: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( huffma854352999e_freq @ A @ T @ A2 )
= ( plus_plus @ nat @ W_a @ W_b ) )
=> ( ( huffma691733767weight @ A @ ( huffma454997449itLeaf @ A @ T @ W_a @ A2 @ W_b @ B ) )
= ( huffma691733767weight @ A @ T ) ) ) ) ) ).
% weight_splitLeaf
thf(fact_231_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_232_weight__mergeSibling,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A] :
( ( huffma691733767weight @ A @ ( huffma1954420889ibling @ A @ T @ A2 ) )
= ( huffma691733767weight @ A @ T ) ) ).
% weight_mergeSibling
thf(fact_233_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X4: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_234_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X4: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_235_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3] :
( ! [X: A] : ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_236_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_237_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_238_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
=> ( ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ C4 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_239_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_240_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_241_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_242_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ) ).
% antisym
thf(fact_243_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% linear
thf(fact_244_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( X4 = Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% eq_refl
thf(fact_245_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% le_cases
thf(fact_246_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_247_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X4 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_248_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X4: A] :
( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_249_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_250_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_251_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( A2 = B ) ) ) ) ).
% order_class.order.antisym
thf(fact_252_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_253_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
%----Type constructors (30)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
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_Oordered__cancel__comm__monoid__add,axiom,
ordere1818651114id_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___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___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ 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___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X4: A,Y2: A] :
( ( if @ A @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y2: A] :
( ( if @ A @ $true @ X4 @ Y2 )
= X4 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
huffma1050891809istent @ a @ ( huffma304375860urSyms @ a @ u @ a2 @ b @ c @ ( huffma943100115ibling @ a @ u @ c ) ) ).
%------------------------------------------------------------------------------