TPTP Problem File: DAT168^1.p
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%------------------------------------------------------------------------------
% File : DAT168^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Huffman 1359
% 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__1359.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 362 ( 86 unt; 70 typ; 0 def)
% Number of atoms : 889 ( 335 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 4871 ( 66 ~; 20 |; 56 &;4362 @)
% ( 0 <=>; 367 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 90 ( 90 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 68 usr; 9 con; 0-6 aty)
% Number of variables : 916 ( 29 ^; 824 !; 5 ?; 916 :)
% ( 58 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:29.450
%------------------------------------------------------------------------------
%----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 (65)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[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_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[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_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere1490568538miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[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_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[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__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[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_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[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_OcachedWeight,type,
huffma787811817Weight:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat ) ).
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_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_OswapLeaves,type,
huffma2094459102Leaves:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_OInnerNode,type,
huffma1759677307erNode:
!>[A: $tType] : ( nat > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OuniteTrees,type,
huffma453905539eTrees:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ 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_t_092_060_094sub_0621____,type,
t_1: huffma16452318e_tree @ a ).
thf(sy_v_t_092_060_094sub_0622____,type,
t_2: 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 ).
thf(sy_v_w____,type,
w: nat ).
%----Relevant facts (254)
thf(fact_0_hyps_I4_J,axiom,
a2 != b ).
% hyps(4)
thf(fact_1__092_060open_062b_A_092_060notin_062_Aalphabet_At_092_060_094sub_0622_092_060close_062,axiom,
~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) ) ).
% \<open>b \<notin> alphabet t\<^sub>2\<close>
thf(fact_2_b_092_060_094sub_0621,axiom,
member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) ).
% b\<^sub>1
thf(fact_3_a_092_060_094sub_0621,axiom,
member @ a @ a2 @ ( huffma505251170phabet @ a @ t_1 ) ).
% a\<^sub>1
thf(fact_4_a_092_060_094sub_0622,axiom,
~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_2 ) ) ).
% a\<^sub>2
thf(fact_5_c,axiom,
huffma1050891809istent @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ).
% c
thf(fact_6_tree_Oinject_I2_J,axiom,
! [A: $tType,X21: nat,X22: huffma16452318e_tree @ A,X23: huffma16452318e_tree @ A,Y21: nat,Y22: huffma16452318e_tree @ A,Y23: huffma16452318e_tree @ A] :
( ( ( huffma1759677307erNode @ A @ X21 @ X22 @ X23 )
= ( huffma1759677307erNode @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% tree.inject(2)
thf(fact_7_w_092_060_094sub_0621,axiom,
( ( ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) ) @ ( huffma854352999e_freq @ a @ t_1 @ a2 ) ) @ ( huffma854352999e_freq @ a @ t_1 @ b ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ a @ t_1 ) @ w_a ) @ w_b ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) ) @ ( huffma854352999e_freq @ a @ t_1 @ a2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ a @ t_1 ) @ w_b ) ) ) ) )
& ( ~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) ) @ ( huffma854352999e_freq @ a @ t_1 @ b ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ a @ t_1 ) @ w_a ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) )
= ( huffma691733767weight @ a @ t_1 ) ) ) ) ) ) ).
% w\<^sub>1
thf(fact_8_w_092_060_094sub_0622,axiom,
( ( ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) ) @ ( huffma854352999e_freq @ a @ t_2 @ a2 ) ) @ ( huffma854352999e_freq @ a @ t_2 @ b ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ a @ t_2 ) @ w_a ) @ w_b ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) ) @ ( huffma854352999e_freq @ a @ t_2 @ a2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ a @ t_2 ) @ w_b ) ) ) ) )
& ( ~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) ) @ ( huffma854352999e_freq @ a @ t_2 @ b ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ a @ t_2 ) @ w_a ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( huffma691733767weight @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) )
= ( huffma691733767weight @ a @ t_2 ) ) ) ) ) ) ).
% w\<^sub>2
thf(fact_9_hyps_I2_J,axiom,
( ( huffma1050891809istent @ a @ t_2 )
=> ( ( a2 != b )
=> ( ( ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_2 @ a2 ) @ ( huffma223349076_depth @ a @ t_2 @ a2 ) ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_2 @ b ) @ ( huffma223349076_depth @ a @ t_2 @ b ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ t_2 ) @ ( times_times @ nat @ w_a @ ( huffma223349076_depth @ a @ t_2 @ b ) ) ) @ ( times_times @ nat @ w_b @ ( huffma223349076_depth @ a @ t_2 @ a2 ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_2 @ a2 ) @ ( huffma223349076_depth @ a @ t_2 @ a2 ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ t_2 ) @ ( times_times @ nat @ w_b @ ( huffma223349076_depth @ a @ t_2 @ a2 ) ) ) ) ) ) )
& ( ~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_2 @ b ) @ ( huffma223349076_depth @ a @ t_2 @ b ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ t_2 ) @ ( times_times @ nat @ w_a @ ( huffma223349076_depth @ a @ t_2 @ b ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_2 ) )
=> ( ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_2 @ w_a @ a2 @ w_b @ b ) )
= ( huffma636208924e_cost @ a @ t_2 ) ) ) ) ) ) ) ) ).
% hyps(2)
thf(fact_10_hyps_I1_J,axiom,
( ( huffma1050891809istent @ a @ t_1 )
=> ( ( a2 != b )
=> ( ( ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_1 @ a2 ) @ ( huffma223349076_depth @ a @ t_1 @ a2 ) ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_1 @ b ) @ ( huffma223349076_depth @ a @ t_1 @ b ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ t_1 ) @ ( times_times @ nat @ w_a @ ( huffma223349076_depth @ a @ t_1 @ b ) ) ) @ ( times_times @ nat @ w_b @ ( huffma223349076_depth @ a @ t_1 @ a2 ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_1 @ a2 ) @ ( huffma223349076_depth @ a @ t_1 @ a2 ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ t_1 ) @ ( times_times @ nat @ w_b @ ( huffma223349076_depth @ a @ t_1 @ a2 ) ) ) ) ) ) )
& ( ~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ t_1 @ b ) @ ( huffma223349076_depth @ a @ t_1 @ b ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ t_1 ) @ ( times_times @ nat @ w_a @ ( huffma223349076_depth @ a @ t_1 @ b ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t_1 ) )
=> ( ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ t_1 @ w_a @ a2 @ w_b @ b ) )
= ( huffma636208924e_cost @ a @ t_1 ) ) ) ) ) ) ) ) ).
% hyps(1)
thf(fact_11_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_12_freq_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma854352999e_freq @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( ^ [B: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T_1 @ B ) @ ( huffma854352999e_freq @ A @ T_2 @ B ) ) ) ) ).
% freq.simps(2)
thf(fact_13_swapLeaves_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B2: A] :
( ( huffma2094459102Leaves @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ W_a @ A2 @ W_b @ B2 )
= ( huffma1759677307erNode @ A @ W @ ( huffma2094459102Leaves @ A @ T_1 @ W_a @ A2 @ W_b @ B2 ) @ ( huffma2094459102Leaves @ A @ T_2 @ W_a @ A2 @ W_b @ B2 ) ) ) ).
% swapLeaves.simps(2)
thf(fact_14_exists__in__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
? [A3: A] : ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) ) ).
% exists_in_alphabet
thf(fact_15_depth__swapLeaves__neither,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,C: A,A2: A,B2: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( C != A2 )
=> ( ( C != B2 )
=> ( ( huffma223349076_depth @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) @ C )
= ( huffma223349076_depth @ A @ T @ C ) ) ) ) ) ).
% depth_swapLeaves_neither
thf(fact_16_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_17_cost_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma636208924e_cost @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ A @ T_1 ) @ ( huffma636208924e_cost @ A @ T_1 ) ) @ ( huffma691733767weight @ A @ T_2 ) ) @ ( huffma636208924e_cost @ A @ T_2 ) ) ) ).
% cost.simps(2)
thf(fact_18_weight_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma691733767weight @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ A @ T_1 ) @ ( huffma691733767weight @ A @ T_2 ) ) ) ).
% weight.simps(2)
thf(fact_19_weight__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B2: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( A2 != B2 )
=> ( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) ) @ ( huffma854352999e_freq @ A @ T @ B2 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma691733767weight @ A @ T ) @ W_a ) @ W_b ) ) )
& ( ~ ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) ) @ ( huffma854352999e_freq @ A @ T @ A2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ A @ T ) @ W_b ) ) ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( plus_plus @ nat @ ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) ) @ ( huffma854352999e_freq @ A @ T @ B2 ) )
= ( plus_plus @ nat @ ( huffma691733767weight @ A @ T ) @ W_a ) ) )
& ( ~ ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma691733767weight @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) )
= ( huffma691733767weight @ A @ T ) ) ) ) ) ) ) ) ).
% weight_swapLeaves
thf(fact_20_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_21_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_22_consistent__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( huffma1050891809istent @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) ) ) ).
% consistent_swapLeaves
thf(fact_23_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_24_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_25_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_26_freq__uniteTrees,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma854352999e_freq @ A @ ( huffma453905539eTrees @ A @ T_1 @ T_2 ) )
= ( ^ [A4: A] : ( plus_plus @ nat @ ( huffma854352999e_freq @ A @ T_1 @ A4 ) @ ( huffma854352999e_freq @ A @ T_2 @ A4 ) ) ) ) ).
% freq_uniteTrees
thf(fact_27_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_28_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_29_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_30_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B: A] : ( plus_plus @ A @ B @ A4 ) ) ) ) ).
% add.commute
thf(fact_31_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_32_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_33_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_34_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_35_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_36_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% mult.left_commute
thf(fact_37_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A4: A,B: A] : ( times_times @ A @ B @ A4 ) ) ) ) ).
% mult.commute
thf(fact_38_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% mult.assoc
thf(fact_39_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_40_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_41_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_42_uniteTrees__def,axiom,
! [A: $tType] :
( ( huffma453905539eTrees @ A )
= ( ^ [T_12: huffma16452318e_tree @ A,T_22: huffma16452318e_tree @ A] : ( huffma1759677307erNode @ A @ ( plus_plus @ nat @ ( huffma787811817Weight @ A @ T_12 ) @ ( huffma787811817Weight @ A @ T_22 ) ) @ T_12 @ T_22 ) ) ) ).
% uniteTrees_def
thf(fact_43_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ D ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D ) @ ( times_times @ A @ B2 @ C ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_44_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [W: A,Y: A,X: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ) ).
% crossproduct_eq
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] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,E: A,B2: A,C: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C ) ) ) ).
% combine_common_factor
thf(fact_50_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% distrib_right
thf(fact_51_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% distrib_left
thf(fact_52_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_53_semiring__normalization__rules_I1_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,M: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ M ) @ ( times_times @ A @ B2 @ M ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ M ) ) ) ).
% semiring_normalization_rules(1)
thf(fact_54_semiring__normalization__rules_I8_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% semiring_normalization_rules(8)
thf(fact_55_cachedWeight_Osimps_I2_J,axiom,
! [A: $tType,W: nat,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma787811817Weight @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) )
= W ) ).
% cachedWeight.simps(2)
thf(fact_56_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_57_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,C2: A] : ( plus_plus @ A @ C2 @ A4 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_58_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_59_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_60_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_61_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_62_semiring__normalization__rules_I19_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Rx: A,Ry: A] :
( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ Rx @ ( times_times @ A @ Lx @ Ry ) ) ) ) ).
% semiring_normalization_rules(19)
thf(fact_63_semiring__normalization__rules_I18_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Rx: A,Ry: A] :
( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ry ) ) ) ).
% semiring_normalization_rules(18)
thf(fact_64_semiring__normalization__rules_I17_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
= ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ Rx ) ) ) ) ).
% semiring_normalization_rules(17)
thf(fact_65_semiring__normalization__rules_I16_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
= ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ly ) ) ) ).
% semiring_normalization_rules(16)
thf(fact_66_semiring__normalization__rules_I15_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A,Ry: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ Rx @ ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Ry ) ) ) ) ).
% semiring_normalization_rules(15)
thf(fact_67_semiring__normalization__rules_I14_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A,Ry: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ ( times_times @ A @ Rx @ Ry ) ) ) ) ) ).
% semiring_normalization_rules(14)
thf(fact_68_semiring__normalization__rules_I13_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A,Ry: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ ( times_times @ A @ Ly @ Ry ) ) ) ) ).
% semiring_normalization_rules(13)
thf(fact_69_semiring__normalization__rules_I7_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A4: A,B: A] : ( times_times @ A @ B @ A4 ) ) ) ) ).
% semiring_normalization_rules(7)
thf(fact_70_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_71_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_72_semiring__normalization__rules_I34_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( times_times @ A @ X @ ( plus_plus @ A @ Y @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ X @ Z ) ) ) ) ).
% semiring_normalization_rules(34)
thf(fact_73_linordered__field__class_Osign__simps_I36_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(36)
thf(fact_74_linordered__field__class_Osign__simps_I35_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(35)
thf(fact_75_freq__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B2: A,W_a: nat,W_b: nat] :
( ( huffma1050891809istent @ A @ T )
=> ( ( A2 != B2 )
=> ( ( huffma854352999e_freq @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) )
= ( ^ [C2: A] : ( if @ nat @ ( C2 = A2 ) @ ( if @ nat @ ( member @ A @ B2 @ ( huffma505251170phabet @ A @ T ) ) @ W_a @ ( zero_zero @ nat ) ) @ ( if @ nat @ ( C2 = B2 ) @ ( if @ nat @ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) ) @ W_b @ ( zero_zero @ nat ) ) @ ( huffma854352999e_freq @ A @ T @ C2 ) ) ) ) ) ) ) ).
% freq_swapLeaves
thf(fact_76_exists__at__height,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( huffma1050891809istent @ A @ T )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( huffma505251170phabet @ A @ T ) )
& ( ( huffma223349076_depth @ A @ T @ X3 )
= ( huffma1554076246height @ A @ T ) ) ) ) ).
% exists_at_height
thf(fact_77_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_78_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_79_linordered__field__class_Osign__simps_I25_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(25)
thf(fact_80_linordered__field__class_Osign__simps_I24_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A4: A,B: A] : ( times_times @ A @ B @ A4 ) ) ) ) ).
% linordered_field_class.sign_simps(24)
thf(fact_81_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_82_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_83_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_84_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_85_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_86_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_87_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_88_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_89_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_90_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_91_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_92_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B2 @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_93_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B2 ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_94_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_95_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_96_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_97_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_98_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_99_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_100_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_101_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_102_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_103_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_104_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_105_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_106_height__swapLeaves,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,W_a: nat,A2: A,W_b: nat,B2: A] :
( ( huffma1554076246height @ A @ ( huffma2094459102Leaves @ A @ T @ W_a @ A2 @ W_b @ B2 ) )
= ( huffma1554076246height @ A @ T ) ) ).
% height_swapLeaves
thf(fact_107_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_108_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_109_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_110_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_111_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_112_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_113_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_114_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_115_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_116_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_117_height__0__imp__cachedWeight__eq__weight,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
( ( ( huffma1554076246height @ A @ T )
= ( zero_zero @ nat ) )
=> ( ( huffma787811817Weight @ A @ T )
= ( huffma691733767weight @ A @ T ) ) ) ).
% height_0_imp_cachedWeight_eq_weight
thf(fact_118_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere1490568538miring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_119_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_120_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_121_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_122_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_123_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_124_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_125_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_126_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ) ).
% mult_right_mono
thf(fact_127_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ C ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_128_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_129_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_130_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_131_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_132_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_133_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_134_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_135_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_136_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_137_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_138_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_139_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_140_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_141_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_142_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_143_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_144_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_145_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_146_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_147_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_148_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_149_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_150_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_151_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_152_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_153_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_154_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_155_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_156_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B: A] :
? [C2: A] :
( B
= ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_157_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_right_mono
thf(fact_158_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_159_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_160_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_161_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_162_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_163_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_164_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_165_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
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_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_168_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_169_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_170_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_171_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_172_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_173_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_174_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_175_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_176_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_177_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_178_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_179_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_180_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_181_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B2 @ C ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_182_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_183_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_184_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_185_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_186_semiring__normalization__rules_I9_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(9)
thf(fact_187_semiring__normalization__rules_I10_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(10)
thf(fact_188_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_189_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_190_mult__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 @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_191_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_192_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_193_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_194_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_195_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_196_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_197_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_198_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_199_sibling__reciprocal,axiom,
! [A: $tType,T: huffma16452318e_tree @ A,A2: A,B2: A] :
( ( huffma1050891809istent @ A @ T )
=> ( ( ( huffma943100115ibling @ A @ T @ A2 )
= B2 )
=> ( ( huffma943100115ibling @ A @ T @ B2 )
= A2 ) ) ) ).
% sibling_reciprocal
thf(fact_200_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [R: A,A2: A,B2: A,C: A,D: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R @ C ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_201_sibling_Osimps_I3_J,axiom,
! [A: $tType,A2: A,V: nat,Va: huffma16452318e_tree @ A,Vb: huffma16452318e_tree @ A,W: nat,T_2: huffma16452318e_tree @ A] :
( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ A2 ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_2 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_2 @ A2 ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_2 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ T_2 ) @ A2 )
= A2 ) ) ) ) ) ).
% sibling.simps(3)
thf(fact_202_sibling_Osimps_I4_J,axiom,
! [A: $tType,A2: A,T_1: huffma16452318e_tree @ A,W: nat,V: nat,Va: huffma16452318e_tree @ A,Vb: huffma16452318e_tree @ A] :
( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) @ A2 )
= ( huffma943100115ibling @ A @ T_1 @ A2 ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) @ A2 )
= ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) @ A2 ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ ( huffma1759677307erNode @ A @ V @ Va @ Vb ) ) @ A2 )
= A2 ) ) ) ) ) ).
% sibling.simps(4)
thf(fact_203_linordered__field__class_Osign__simps_I26_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(26)
thf(fact_204_linordered__field__class_Osign__simps_I27_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B: A] : ( plus_plus @ A @ B @ A4 ) ) ) ) ).
% linordered_field_class.sign_simps(27)
thf(fact_205_linordered__field__class_Osign__simps_I28_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(28)
thf(fact_206_linordered__field__class_Osign__simps_I23_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(23)
thf(fact_207_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_208_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_209_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_210_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_211_either__height__gt__0__imp__sibling,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A,A2: A,W: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T_1 ) )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T_2 ) ) )
=> ( ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_1 @ A2 ) ) )
& ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_2 @ A2 ) ) ) ) ) ).
% either_height_gt_0_imp_sibling
thf(fact_212_height__gt__0__in__alphabet__imp__sibling__left,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,A2: A,W: nat,T_2: huffma16452318e_tree @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T_1 ) )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_1 @ A2 ) ) ) ) ).
% height_gt_0_in_alphabet_imp_sibling_left
thf(fact_213_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_214_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_215_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_216_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_217_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_218_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_219_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_220_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_221_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_222_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_223_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_224_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_225_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_226_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_227_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_228_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_229_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_230_height__gt__0__notin__alphabet__imp__sibling__right,axiom,
! [A: $tType,T_2: huffma16452318e_tree @ A,A2: A,T_1: huffma16452318e_tree @ A,W: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T_2 ) )
=> ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_2 @ A2 ) ) ) ) ).
% height_gt_0_notin_alphabet_imp_sibling_right
thf(fact_231_height__gt__0__notin__alphabet__imp__sibling__left,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,A2: A,W: nat,T_2: huffma16452318e_tree @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T_1 ) )
=> ( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_2 @ A2 ) ) ) ) ).
% height_gt_0_notin_alphabet_imp_sibling_left
thf(fact_232_height__gt__0__in__alphabet__imp__sibling__right,axiom,
! [A: $tType,T_2: huffma16452318e_tree @ A,A2: A,T_1: huffma16452318e_tree @ A,W: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma1554076246height @ A @ T_2 ) )
=> ( ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T_1 ) )
=> ( ( huffma943100115ibling @ A @ ( huffma1759677307erNode @ A @ W @ T_1 @ T_2 ) @ A2 )
= ( huffma943100115ibling @ A @ T_1 @ A2 ) ) ) ) ).
% height_gt_0_in_alphabet_imp_sibling_right
thf(fact_233_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_234_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_235_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_236_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_237_order__le__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y3: B3] :
( ( ord_less @ B3 @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_238_order__le__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_239_order__less__le__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_240_order__less__le__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_241_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_242_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_243_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_244_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_245_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_246_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_247_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_248_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_249_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_250_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_251_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_252_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_253_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% less_le_not_le
%----Type constructors (34)
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_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___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ 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_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ 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___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring @ 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__mult,axiom,
semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring @ 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___Rings_Omult__zero,axiom,
mult_zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
semiring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
order @ $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,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,
( ( ( member @ a @ a2 @ ( huffma505251170phabet @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) )
=> ( ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ a2 ) @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ a2 ) ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ b ) @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ b ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) @ ( times_times @ nat @ w_a @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ b ) ) ) @ ( times_times @ nat @ w_b @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ a2 ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ a2 ) @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ a2 ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) @ ( times_times @ nat @ w_b @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ a2 ) ) ) ) ) ) )
& ( ~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) )
=> ( ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ w_a @ a2 @ w_b @ b ) ) @ ( times_times @ nat @ ( huffma854352999e_freq @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ b ) @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ b ) ) )
= ( plus_plus @ nat @ ( huffma636208924e_cost @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) @ ( times_times @ nat @ w_a @ ( huffma223349076_depth @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ b ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) )
=> ( ( huffma636208924e_cost @ a @ ( huffma2094459102Leaves @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) @ w_a @ a2 @ w_b @ b ) )
= ( huffma636208924e_cost @ a @ ( huffma1759677307erNode @ a @ w @ t_1 @ t_2 ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------