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standard_form_mult_poly.pvs
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standard_form_mult_poly % Welcome
: THEORY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%| Defines Multivariate Polynomials |%
%| and their standard form, with |%
%| properties of standard form |%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Written By: LW, JTS, AD
% Based off of standard_form_poly
% *** This corresponds to the much of section
% 2.1 in the paper, but ends at evalutation,
% partial evaluation, and showing the
% standard form is unique, which is shown
% later ****
%----- %
BEGIN
% -----%
%-------------------------------------------
%%Importing array definition of polynomials,
% and array2list to be able to relate the
% array and list definition of polynomials.
% Importing list_lemmas to have necisary
% properties of lists, and standard_form_poly
% from poly safe to have standard form of
% single variate polynomials
%-------------------------------------------
IMPORTING
structures@array2list[real],
structures@array2list[nat],
more_list_props[nat],
list_lemmas,
standard_form_poly
%-------------------------------------------
%%Define monomials and multivariate
% polynomial over reals
%-------------------------------------------
monomial: TYPE = [# C: real , alpha: list[nat] #]
MultPoly: TYPE = list[monomial]
%-------------------------------------------
%%Importing more_list_props for monomials
%-------------------------------------------
IMPORTING
more_list_props[monomial]
%--------------------------------------------
%%Making a subtype of Index that has
% uniform length
%-------------------------------------------
UnifInd: TYPE = {ll:list[monomial] | FORALL (i,j: below(length(ll))): length(nth(ll,i)`alpha) = length(nth(ll,j)`alpha)}
Unifind_is_an_Ind: Judgement UnifInd SUBTYPE_OF MultPoly
Unif?(p:MultPoly): bool =
null?(p) OR (FORALL (i,j: below(length(p))): length(nth(p,i)`alpha) = length(nth(p,j)`alpha))
%--------------------------------------------
%%Property of cons and Unif?
%--------------------------------------------
Unif_cons: LEMMA
FORALL(m:monomial, p:MultPoly | cons?(p) AND Unif?(p)): length(m`alpha) = length(car(p)`alpha) IMPLIES Unif?(cons(m,p))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Some useful functions of list[nat] and MultPoly
%% sum of a list, list of term degrees,
%% largest length of the alpha terms, lifting a MultPoly
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
caretzero_every: LEMMA
FORALL (n:nat):
every[number]
(LAMBDA (x: number):
number_field_pred(x) AND
real_pred(x) AND rational_pred(x)
AND integer_pred(x) AND (x >= 0))
((: 0 :)^n)
%---------------------------------------------
%%Define the sum of a list of natural numbers
%---------------------------------------------
sum(l:list[nat]): RECURSIVE real =
IF null?(l)
THEN 0
ELSE
car(l) + sum(cdr(l))
ENDIF
MEASURE l by <<
%---------------------------------------------
%%Last nth prop
%---------------------------------------------
last_last_nth: LEMMA
FORALL(l:list[nat]):
cons?(l) IMPLIES
(
last(l) = nth(l,length(l)-1))
%---------------------------------------------
%%Sum of a list of natural numbers is a
% natural number
%---------------------------------------------
sum_nat: LEMMA
FORALL (l:list[nat]):
rational_pred(sum(l)) AND
integer_pred(sum(l)) AND
sum(l) >= 0
%---------------------------------------------
%%Properties of sum
%---------------------------------------------
sum_append: LEMMA
FORALL (l1,l2:list[nat]):
sum(append(l1,l2)) = sum(l1) + sum(l2)
sum_0: LEMMA
FORALL(n:nat):
sum((: 0 :) ^ n ) = 0
sum_append_0: LEMMA
FORALL(l:list[nat],n:nat):
sum(append(l, (: 0 :)^n)) = sum(l)
sum_0_con: LEMMA
FORALL(l:list[nat]):
sum(l) = 0 IMPLIES EXISTS(n:nat): l = (: 0 :) ^ n
%---------------------------------------------
%%Add two monomials together that have the
% same alpha
%---------------------------------------------
add_mono(m1:monomial,
m2:{mm:monomial | mm`alpha = m1`alpha}):
monomial =
(# C:= m1`C + m2`C, alpha := m1`alpha #)
add_mono_commutative: LEMMA
FORALL(m1:monomial,
m2:{mm:monomial | mm`alpha = m1`alpha}):
add_mono(m1,m2) = add_mono(m2,m1)
%---------------------------------------------
%%Defining a list of the degree of each
% monomial in a MultPoly
%---------------------------------------------
term_deg(p:MultPoly): RECURSIVE list[nat] =
IF length(p) < 1
THEN null
ELSE
cons(sum(car(p)`alpha), term_deg(cdr(p)))
ENDIF
MEASURE length(p)
term_deg_example: LEMMA
term_deg((: (# C:= 1, alpha := (: 0,0 :) #),
(# C:= 2, alpha := (: 3,4 :) #),
(# C:= 1, alpha := (: 2,5 :) #) :)) = (: 0,7,7 :)
%---------------------------------------------
%%Homogeneous check
%---------------------------------------------
homogen?(p:MultPoly): bool =
IF p = null
THEN TRUE
ELSIF FORALL(j:below(length(term_deg(p))-1)):
nth(term_deg(p),j) = nth(term_deg(p),j+1)
THEN TRUE
ELSE
FALSE
ENDIF
%---------------------------------------------
%%Largest degree in the list of terms
%---------------------------------------------
max_deg(p:MultPoly): RECURSIVE
nat =
IF null?(p)
THEN 0
ELSE
max(car(term_deg(p)), max_deg(cdr(p)))
ENDIF
MEASURE length(p)
%---------------------------------------------
%%A function that finds the largest
% length of the alphas in a MultPoly
%---------------------------------------------
max_length(p:MultPoly): RECURSIVE
nat =
IF p=null
THEN 0
ELSE
max(length(car(p)`alpha), max_length(cdr(p)))
ENDIF
MEASURE length(p)
%---------------------------------------------
%%Examples of max_length
%---------------------------------------------
max_length_ex1: LEMMA
max_length((: (# C:= 0, alpha:= (: :) #) :)) = 0
max_length_ex2: LEMMA
max_length((: (# C:= 1, alpha := (: 0,0 :) #),
(# C:= 2, alpha := (: 3,4 :) #), (# C:= -1, alpha := (: 2,5 :) #) :)) = 2
%---------------------------------------------
%%Properties of max_length
%---------------------------------------------
max_length_is_max: LEMMA
FORALL (p:{q:MultPoly | cons?(q)}, j:below(length(p))):
length(nth(p,j)`alpha) <= max_length(p)
max_length_is_nth: LEMMA
FORALL (p:{q:MultPoly | cons?(q)}): EXISTS(j:below(length(p))):
length(nth(p,j)`alpha) = max_length(p)
max_length_Unif: LEMMA
FORALL (p:{q:UnifInd | cons?(q)}, j:below(length(p))):
length(nth(p,j)`alpha) = max_length(p)
max_length_Unif_cdr: LEMMA
FORALL (p:{q:UnifInd | cons?(q) AND cons?(cdr(q))}):
max_length(cdr(p)) = max_length(p)
%---------------------------------------------
%%Lift a MultPoly in R^k to R^n by making larger
% index variables to zero power
% define length_a to have a measure in the
% recursive definition of lft
%---------------------------------------------
length_a(p:MultPoly): RECURSIVE nat =
IF p=null THEN 0
ELSE length(car(p)`alpha) + length_a(cdr(p))
ENDIF
MEASURE length(p)
max_length_a: LEMMA
FORALL(p:MultPoly):
max_length(p)*length(p) >= length_a(p)
lft(p:MultPoly)(n:{nn:nat | nn >= max_length(p)}):
RECURSIVE MultPoly =
IF length(p) < 1
THEN null
ELSIF 0 <= length(car(p)`alpha) < n
THEN cons((# C:= car(p)`C, alpha :=
append(car(p)`alpha, (:0:)^(n-length(car(p)`alpha))) #),lft(cdr(p))(n))
ELSE
cons(car(p), lft(cdr(p))(n))
ENDIF
MEASURE length(p)+n*length(p) - length_a(p)
%---------------------------------------------
%% Establish alternate definition of lft
% to help prove later properties
%---------------------------------------------
lft_one(n: nat)((m:monomial | n >= length(m`alpha))):
monomial =
(# C:= m`C, alpha := append(m`alpha,
(:0:)^(n-length(m`alpha))) #)
lft_one_length: LEMMA
FORALL(m:monomial, n:nat | n >= length(m`alpha)):
length(lft_one(n)(m)`alpha)=n
lft_alt(p:MultPoly)((n:nat | n >= max_length(p))):
MultPoly = map(lft_one(n))(p)
lft_is_alt: LEMMA
FORALL (p:MultPoly, (n: nat | n >= max_length(p))):
lft(p)(n) = lft_alt(p)(n)
%---------------------------------------------
%% Example of lft
%---------------------------------------------
lft_example1: LEMMA
LET mp:MultPoly = (: (# C:= 1, alpha := (: 0,0 :) #),
(# C:= 2, alpha := (: 3,4 :) #), (# C:= -1, alpha := (: 2,5 :) #) :)
IN lft(mp)(3) = (: (# C:= 1, alpha := (: 0,0,0 :) #),
(# C:= 2, alpha := (: 3,4,0 :) #), (# C:= -1, alpha := (: 2,5,0 :) #) :)
%---------------------------------------------
%% Properties of lft
%---------------------------------------------
Lft_length: LEMMA
FORALL (p:MultPoly, n:{nn:nat | nn >= max_length(p)}):
length(p) = length(lft(p)(n))
max_length_lft: LEMMA
FORALL (p:MultPoly, n:{nn:nat | nn >= max_length(p)}):
max_length(p) <= max_length(lft(p)(n))
lft_length_nth: LEMMA
FORALL (p:{q:MultPoly | cons?(q)},
n:{nn:nat | nn >= max_length(p)}):
FORALL(j:below(length(p))):
length(nth(lft(p)(n),j)`alpha) = n
lft_nth_C: LEMMA
FORALL(p:MultPoly,n:{nn:nat | nn >= max_length(p)},
i:below(length(p))):
nth(p,i)`C = nth(lft(p)(n),i)`C
lft_nth_alpha: LEMMA
FORALL(p:MultPoly,n:{nn:nat | nn >= max_length(p)},
i:below(length(p))):
nth(lft(p)(n),i)`alpha = append( nth(p,i)`alpha,
(: 0 :)^(n - length(nth(p,i)`alpha)))
lft_Unif: LEMMA
FORALL (p:MultPoly): Unif?(lft(p)(max_length(p))) = TRUE
lft_n_last: LEMMA
FORALL(p:MultPoly, n:{nn:nat | nn >= max_length(p)},
j:below(length(p))):
n /= 0 AND length(nth(p,j)`alpha) = n
IMPLIES last(nth(p,j)`alpha) = last(nth(lft(p)(n),j)`alpha)
lft_n_max_length: LEMMA
FORALL(p:MultPoly,n:{nn:nat | nn >= max_length(p)}):
cons?(p) IMPLIES
max_length(lft(p)(n)) = n
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Standard form of a Multivariate Polynomial
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%---------------------------------------------
%% Sort a multivariate polynomial into standard
% order. Define leq that will define ordering
% this is graded lexicographical order
%---------------------------------------------
leq(l1,l2:monomial): RECURSIVE bool =
(length(l1`alpha) < length(l2`alpha)) OR
(null?(l1`alpha) AND null?(l2`alpha))
OR (length(l1`alpha) = length(l2`alpha) AND (sum(l1`alpha) < sum(l2`alpha)))
OR (length(l1`alpha) = length(l2`alpha) AND (sum(l1`alpha) = sum(l2`alpha) AND cons?(l1`alpha) AND cons?(l2`alpha) AND car(l1`alpha) < car(l2`alpha)))
OR (length(l1`alpha) = length(l2`alpha) AND (sum(l1`alpha) = sum(l2`alpha) AND cons?(l1`alpha) AND cons?(l2`alpha) AND car(l1`alpha) = car(l2`alpha) AND leq((# C:= l1`C, alpha:= cdr(l1`alpha) #), (# C:= l2`C, alpha:= cdr(l2`alpha) #) )))
MEASURE length(l1`alpha)
%---------------------------------------------
%%Properties of leq
%---------------------------------------------
leq_reflexive: LEMMA
reflexive?[monomial](leq)
leq_transitive: LEMMA
transitive?[monomial](leq)
leq_dichotomous: LEMMA
dichotomous?[monomial](leq)
almost_total_order: LEMMA
FORALL(m1,m2:monomial): leq(m1,m2) AND leq(m2,m1) IMPLIES m1`alpha = m2`alpha
%---------------------------------------------
%%Importing bubblesort with leq
%---------------------------------------------
IMPORTING sorting@bubblesort[monomial, leq],
sorting@sorting[monomial, leq]
%---------------------------------------------
%%leq is total preorder
%---------------------------------------------
leq_total_preorder: LEMMA
total_preorder?[monomial](leq)
%---------------------------------------------
%% Sort a polynomial
%---------------------------------------------
mv_sort(p:MultPoly): MultPoly =
IF null?(p)
THEN null
ELSE bubblesort(lft(p)(max_length(p)))
ENDIF
%---------------------------------------------
%% Example of mv_sort
%---------------------------------------------
mv_sort_ex1: LEMMA
mv_sort((: (# C:= 1, alpha := (: 0,0 :) #),
(# C:= 2, alpha := (: 3,4 :) #), (# C:= -1, alpha := (: 2,5 :) #) :))
= (: (# C:= 1, alpha := (: 0,0 :) #),
(# C:= -1, alpha := (: 2,5 :) #), (# C:= 2, alpha := (: 3,4 :) #) :)
%---------------------------------------------
%% Properties of mv_sort
%---------------------------------------------
sort_of_unif: LEMMA
FORALL (p:MultPoly):
Unif?(mv_sort(p))
mv_sort_max_length: LEMMA
FORALL(p:MultPoly):
max_length(mv_sort(p)) = max_length(p)
mv_is_sorted?(p:MultPoly):
bool =
Unif?(p) AND is_sorted?(p)
mv_sort_is_sorted: LEMMA
FORALL (p:MultPoly): mv_is_sorted?(mv_sort(p)) = TRUE
cons_smaller_to_sorted_is_sorted: LEMMA
FORALL (p:MultPoly, m:monomial):
cons?(p) AND is_sorted?(p) AND
leq(m,car(p)) IMPLIES is_sorted?(cons(m, p))
add_mono_sorted: LEMMA
FORALL (p:MultPoly,
m1:monomial,m2:{mm:monomial | m1`alpha = mm`alpha}):
cons?(p) AND is_sorted?(p)
AND leq(add_mono(m1,m2),car(p))
IMPLIES is_sorted?(cons(add_mono(m1,m2), p))
%---------------------------------------------
%% More properties of leq, that help with
% sort
%---------------------------------------------
constant_doesnt_matter: LEMMA
FORALL (m1,m2,m3:monomial):
(leq(m1,m3) AND m1`alpha = m2`alpha) IMPLIES leq(m2,m3)
constant_doesnt_matter_last: LEMMA
FORALL (m1,m2,m3:monomial):
(leq(m1,m2) AND m2`alpha = m3`alpha) IMPLIES leq(m1,m3)
append_leq: LEMMA
FORALL(l:list[nat],m1:monomial,
m2:{mm:monomial | length(m1`alpha) = length(mm`alpha)}):
LET M1 = (# C := m1`C, alpha := append(m1`alpha,l) #),
M2 = (# C := m2`C, alpha := append(m2`alpha,l) #)
IN leq(m1,m2) IMPLIES leq(M1,M2)
%---------------------------------------------
%%Check if polynomial is simplified, no two
% alphas are the same
%---------------------------------------------
simplified?(p:MultPoly): bool =
IF p=null or length(p) = 1
THEN TRUE
ELSIF FORALL(i,j:below(length(p))): i /= j IMPLIES nth(p,i)`alpha /= nth(p,j)`alpha
THEN TRUE
ELSE
FALSE
ENDIF
%---------------------------------------------
%%Combine terms that have the same alpha
%---------------------------------------------
sortedsimplify(p:{q:MultPoly | is_sorted?(q)}):
RECURSIVE MultPoly =
IF p = null
THEN null
ELSIF cdr(p) = null
THEN p
ELSIF car(p)`alpha = car(cdr(p))`alpha
THEN sortedsimplify(cons(add_mono(car(p),car(cdr(p))), cdr(cdr(p))))
ELSE
cons(car(p), sortedsimplify(cdr(p)))
ENDIF
MEASURE length(p)
simplify(p:MultPoly): MultPoly =
IF is_sorted?(p)
THEN sortedsimplify(p)
ELSE
sortedsimplify(mv_sort(p))
ENDIF
%---------------------------------------------
%%Properties of simplify
%---------------------------------------------
simplify_cdr: LEMMA
FORALL(p:MultPoly |cons?(p) AND simplified?(p)):
simplified?(cdr(p))
simplify_cons: LEMMA
FORALL(p:MultPoly |is_sorted?(p)):
cons?(p) IMPLIES cons?(sortedsimplify(p))
leq_simplify: LEMMA
FORALL(p:MultPoly |cons?(p) AND is_sorted?(p),
m:monomial):
leq(m,car(p)) IMPLIES leq(m,car(sortedsimplify(p)))
sorted_simp_is_sorted: LEMMA
FORALL(p:{q:MultPoly | is_sorted?(q)}):
is_sorted?(sortedsimplify(p))
cons_smaller_simplify: LEMMA
FORALL(p:MultPoly |cons?(p) AND is_sorted?(p)
AND simplified?(p), m:monomial):
leq(m,car(p)) AND m`alpha /= car(p)`alpha
IMPLIES simplified?(cons(m,p))
cons_simplify: LEMMA
FORALL(p:MultPoly |cons?(p) AND simplified?(p),
m:monomial):
(FORALL(j:below(length(p))): m`alpha /= nth(p,j)`alpha)
IMPLIES simplified?(cons(m,p))
car_equal_simplify: LEMMA
FORALL(p:MultPoly |cons?(p) AND is_sorted?(p)):
car(p)`alpha = car(sortedsimplify(p))`alpha
length_sortedsimplify_car_eq: LEMMA
FORALL (p: MultPoly,
m1, m2: {m:monomial | is_sorted?(cons(m, p))}):
m1`alpha = m2`alpha
IMPLIES
length(sortedsimplify(cons(m1,p))) = length(sortedsimplify(cons(m2,p)))
car_const_nth_sortedsimplify: LEMMA
FORALL (p: MultPoly,
m1, m2: {m:monomial | is_sorted?(cons(m, p))}):
m1`alpha = m2`alpha
IMPLIES
FORALL (i:below(length(sortedsimplify(cons(m1,p))))):
nth(sortedsimplify(cons(m1,p)),i)`alpha =
nth(sortedsimplify(cons(m2,p)),i)`alpha
not_equal_simplify: LEMMA
FORALL(p:MultPoly |cons?(p) AND is_sorted?(p), m:monomial):
m`alpha /= car(p)`alpha IMPLIES
m`alpha /= car(sortedsimplify(p))`alpha
simplify_alpha: LEMMA
FORALL(p:MultPoly |cons?(p) AND is_sorted?(p)):
FORALL(j:below(length(sortedsimplify(p)))):
EXISTS(i:below(length(p))): nth(p,i)`alpha
= nth(sortedsimplify(p),j)`alpha
simplify_alpha_reverse: LEMMA
FORALL(p:MultPoly |cons?(p) AND is_sorted?(p)):
FORALL(i:below(length(p))):
EXISTS(j:below(length(sortedsimplify(p)))):
nth(p,i)`alpha = nth(sortedsimplify(p),j)`alpha
simplified: LEMMA
FORALL(p:MultPoly | is_sorted?(p)):
simplified?(sortedsimplify(p)) = TRUE
simplify_Unif: LEMMA
FORALL(p:MultPoly | is_sorted?(p) AND Unif?(p)):
Unif?(sortedsimplify(p))
mv_sorted_simp_is_sorted: LEMMA
FORALL(p:{q:MultPoly | mv_is_sorted?(q)}):
mv_is_sorted?(sortedsimplify(p))
max_length_sorted_simplify: LEMMA
FORALL(p:MultPoly | is_sorted?(p)):
max_length(p) = max_length(sortedsimplify(p))
%---------------------------------------------
%%Sorted alternative definition
%---------------------------------------------
is_sorted_nth: LEMMA
FORALL(p:MultPoly):
is_sorted?(p)
IFF
(FORALL(i,j:below(length(p))):
i < j IMPLIES
leq(nth(p,i),nth(p,j)))
%---------------------------------------------
%%Check if any constants in polynomial are
% zero
%---------------------------------------------
allnonzero?(p:MultPoly): bool =
IF null = p
THEN TRUE
ELSIF EXISTS (i: below(length(p))): nth(p,i)`C = 0
THEN FALSE
ELSE
TRUE
ENDIF
%---------------------------------------------
%%Get rid of all zero terms in polynomial
%---------------------------------------------
allnonzero(p:MultPoly): RECURSIVE MultPoly =
IF null = p
THEN p
ELSIF car(p)`C = 0
THEN allnonzero(cdr(p))
ELSE
cons(car(p), allnonzero(cdr(p)))
ENDIF
MEASURE length(p)
%---------------------------------------------
%%Properties of allnonzero
%---------------------------------------------
allnonzero_cdr: LEMMA
FORALL(p:MultPoly): cons?(p) AND allnonzero?(p)
IMPLIES allnonzero?(cdr(p))
all_nonzero: LEMMA
FORALL(p:MultPoly): allnonzero?(allnonzero(p))
allnonzero_alpha: LEMMA
FORALL(p:MultPoly | cons?(p)):
FORALL(j:below(length(p))): nth(p,j)`C /= 0
IMPLIES EXISTS(k:below(length(allnonzero(p)))):
k <= j AND nth(p,j)`alpha = nth(allnonzero(p),k)`alpha
allnonzero_alpha_reverse: LEMMA
FORALL(p:MultPoly | cons?(p)): cons?(allnonzero(p))
IMPLIES FORALL(k:below(length(allnonzero(p)))):
EXISTS(j:below(length(p))): nth(p,j)`C /= 0
AND nth(p,j)`alpha = nth(allnonzero(p),k)`alpha
allnonzero_length: LEMMA
FORALL(p:MultPoly):
length(allnonzero(p)) <= length(p)
allnonzero_sorted: LEMMA
FORALL(p:MultPoly): mv_is_sorted?(p)
IMPLIES mv_is_sorted?(allnonzero(p))
allnonzero_simp: LEMMA
FORALL(p:MultPoly): simplified?(p)
IMPLIES simplified?(allnonzero(p))
allnonzero_lft: LEMMA
FORALL (p:MultPoly, n:nat):
allnonzero?(p) AND n >= max_length(p) IMPLIES
allnonzero?(lft(p)(n))
allnonzero_max_length: LEMMA
FORALL(p:MultPoly):
max_length(allnonzero(p)) <= max_length(p)
allnonzero_id: LEMMA
FORALL(p:MultPoly):
allnonzero?(p) IMPLIES p = allnonzero(p)
%---------------------------------------------
%% Make sure there are no unnecisary zeros
% at end of alphas in polynomials
%---------------------------------------------
minlength?(p:MultPoly): bool =
IF null?(p)
THEN TRUE
ELSIF FORALL (i: below(length(p))):
null?(nth(p,i)`alpha)
THEN TRUE
ELSIF Unif?(p) AND
(EXISTS (j: below(length(p))):
cons?(nth(p,j)`alpha) AND last(nth(p,j)`alpha) /= 0)
THEN TRUE
ELSE
FALSE
ENDIF
%---------------------------------------------
%% If something is minlength then it is Unif
%---------------------------------------------
minlength_unif: LEMMA
FORALL (p:MultPoly):
minlength?(p) IMPLIES Unif?(p)
%---------------------------------------------
%% Cut zeros of end of list
%---------------------------------------------
cutting(l:list[nat]): RECURSIVE list[nat] =
IF l = null
THEN null
ELSIF last(l) = 0
THEN cutting(rdc(l))
ELSE
l
ENDIF
MEASURE length(l)
%---------------------------------------------
%% Cut all entries of a polynomial
%---------------------------------------------
mv_cut(p:MultPoly): RECURSIVE MultPoly =
IF null = p
THEN p
ELSIF cons?(car(p)`alpha) AND
EXISTS (n:posnat): member(n,car(p)`alpha)
THEN cons( (# C:= car(p)`C,
alpha:= cutting(car(p)`alpha) #), mv_cut(cdr(p)))
ELSE
cons((# C := car(p)`C,
alpha:= (: :) #), mv_cut(cdr(p)))
ENDIF
MEASURE length(p) + length_a(p)
%--------------------------------------------
%% Alternative definition of cutting, to help
% with proving later properties
%--------------------------------------------
cut_monom(m:monomial):
monomial =
(# C:= m`C, alpha:= cutting (m`alpha) #)
%--------------------------------------------
%%Last nth property
%--------------------------------------------
last_is_nth: LEMMA
FORALL (l: (cons?[nat]) ):
last(l) = nth(l, length(l) - 1)
%--------------------------------------------
%%Properties of cut_monom
%--------------------------------------------
cut_monom_null: LEMMA
FORALL (m:monomial):
cut_monom(m)`alpha = null
IFF
NOT (cons?(m`alpha) AND
EXISTS (n:posnat): member(n,m`alpha))
mv_cut_alt(p:MultPoly):
MultPoly =
map(cut_monom)(p)
mv_cut_is_alt: LEMMA
FORALL (p: MultPoly):
mv_cut(p) = mv_cut_alt(p)
%--------------------------------------------
%% Properties of cutting
%--------------------------------------------
cutting_length: LEMMA
FORALL (l:list[nat]): length(l) >= length(cutting(l))
cutting_nth: LEMMA
FORALL(l:list[nat],i:below(length(cutting(l)))):
nth(cutting(l),i) = nth(l,i)
cutting_last: LEMMA
FORALL(l:list[nat]): cutting(l) = null
OR last(cutting(l)) /= 0
cutting_alt: LEMMA
FORALL (l: list[nat]):
l = append(cutting(l), (: 0 :)^(length(l) - length(cutting(l))))
cutting_sum: LEMMA
FORALL(l:list[nat]):
sum(l) = sum(cutting(l))
%--------------------------------------------
%%Properties of mv_cut
%--------------------------------------------
mv_cut_length: Lemma
FORALL (p:MultPoly):
length(p)=length(mv_cut(p))
mv_cut_C: LEMMA
FORALL(p:MultPoly,i:below(length(p))):
nth(p,i)`C = nth(mv_cut(p),i)`C
mv_cut_alpha: LEMMA
FORALL(p:MultPoly,i:below(length(p))):
nth(mv_cut(p),i)`alpha = cutting(nth(p,i)`alpha)
null_cut_cdr: LEMMA
FORALL(l:list[nat]):
cons?(l) AND null?(cutting(l))
IMPLIES
null?(cutting(cdr(l)))
zero_n_nth: LEMMA
FORALL(n:nat,i:below(n)):
nth( (: 0 :) ^ n , i) = 0
cutting_cdr: LEMMA
FORALL(l1:list[nat]):
(cons?(cutting(l1)) AND
cons?(l1)) IMPLIES
cutting(cdr(l1)) = cdr(cutting(l1))
cutting_append_length: LEMMA
FORALL(l1,l2:list[nat], n,m:nat):
(length(l1) = length(l2) AND
append(cutting(l1), (: 0 :) ^ n ) =
append(cutting(l2), (: 0 :) ^ m))
IMPLIES
length(cutting(l1)) = length(cutting(l2))
max_length_cut: LEMMA
FORALL (p:MultPoly):
max_length(p) >= max_length(mv_cut(p))
mv_cut_last_nonzero: LEMMA
FORALL(p:MultPoly | cons?(p)):
LET P = mv_cut(p)
IN
FORALL(j: below(length(P))):
null?(nth(P,j)`alpha) OR last(nth(P,j)`alpha) /= 0
%--------------------------------------------
%%Showing that lft of mv_cut is minlength
%--------------------------------------------
lft_max_0: LEMMA
FORALL(p:MultPoly |cons?(p)): max_length(p)<= 0
IMPLIES LET P = lft(p)(0)
IN
FORALL(j: below(length(P))): null?(nth(P,j)`alpha)
lft_minlength: LEMMA
FORALL(p:MultPoly |cons?(p)):
LET P = lft(p)(max_length(p))
IN
(EXISTS(j: below(length(p))): cons?(nth(p,j)`alpha) AND
length(nth(p,j)`alpha) = max_length(p)
AND last(nth(p,j)`alpha) /= 0) IMPLIES minlength?(P)
lft_all_null: LEMMA
FORALL(p:MultPoly |cons?(p)):
LET P = lft(p)(max_length(p))
IN (FORALL(j: below(length(P))):
null?(nth(p,j)`alpha)) IMPLIES minlength?(P)
lft_mv_cut: LEMMA
FORALL(p:MultPoly |cons?(p)):
LET P = lft(mv_cut(p))(max_length(mv_cut(p)))
IN minlength?(P)
%--------------------------------------------
%%Properties of lft of mv_cut
%--------------------------------------------
minlength_cdr: LEMMA
FORALL(p:MultPoly): cons?(p) AND minlength?(p)
IMPLIES
minlength?(lft(mv_cut(cdr(p)))(max_length(mv_cut(cdr(p)))))
minlength_simplify: LEMMA
FORALL(p:MultPoly): cons?(p) AND is_sorted?(p)
AND minlength?(p)
IMPLIES minlength?(sortedsimplify(p))
%--------------------------------------------
%%Show lft is sorted and allnonzero
%--------------------------------------------
lft_sorted: LEMMA
FORALL(p:MultPoly |cons?(p) AND Unif?(p)
AND is_sorted?(p),
n:{nn:nat | nn >= max_length(p)}):
LET P = lft(p)(n)
IN is_sorted?(P)
lft_allnonzero: LEMMA
FORALL(p:MultPoly | allnonzero?(p),
n:{nn:nat | nn >= max_length(p)}):
allnonzero?(lft(p)(n))
%--------------------------------------------
%%Show allnonzero of mv_cut and lft
%--------------------------------------------
cut_allnonzero: LEMMA
FORALL(p:MultPoly | allnonzero?(p)):
allnonzero?(mv_cut(p))
lft_cut_allnonzero: LEMMA
FORALL(p:MultPoly | allnonzero?(p),
n:{nn:nat | nn >= max_length(mv_cut(p))}):
allnonzero?(lft(mv_cut(p))(n))
lft_cut_is_lft: LEMMA
FORALL (m:monomial, n:nat):
n >= length(m`alpha) IMPLIES
lft_one(n)(cut_monom(m)) = lft_one(n)(m)
%--------------------------------------------
%%lft alpha nth property
%--------------------------------------------
lft_alpha: LEMMA
FORALL(p:MultPoly | cons?(p),
n:{nn:nat | nn >= max_length(p)}):
FORALL(i:below(length(p))):
append(nth(p,i)`alpha,(:0:)^(n-length(nth(p,i)`alpha)))
= nth(lft(p)(n),i)`alpha OR
nth(p,i)`alpha
= nth(lft(p)(n),i)`alpha
%--------------------------------------------
%%lft of simplified is simpified and lft of
% mv_cut is simplified
%--------------------------------------------
lft_simplify: LEMMA
FORALL((p:MultPoly | simplified?(p) AND Unif?(p)),
n:{nn:nat | nn >= max_length(p)}):
simplified?(lft(p)(n))
lft_cut_alpha: LEMMA
FORALL ((p:MultPoly | simplified?(p) AND Unif?(p)),
n:{nn:nat | nn >= max_length(p)}):
simplified?(lft(mv_cut(p))(n))
lft_cut_alpha_mv_cut: LEMMA
FORALL ((p:MultPoly | simplified?(p) AND Unif?(p)),
n:{nn:nat | nn >= max_length(mv_cut(p))}):
simplified?(lft(mv_cut(p))(n))
%--------------------------------------------
%% mv_sort of minlength is minlength
%--------------------------------------------
minlength_sorted: LEMMA
FORALL(p:MultPoly | cons?(p) AND minlength?(p)):
minlength?(mv_sort(p))
%--------------------------------------------
%% nth property of lft of mv_cut of
% polynomial
%--------------------------------------------
mv_cut_lft_alpha: LEMMA
FORALL(p:MultPoly,i:below(length(p)),
n:nat | n >= max_length(mv_cut(p))):
nth(lft(mv_cut(p))(n),i)`alpha =
append(cutting(nth(p,i)`alpha),
(: 0 :)^(n - length(cutting(nth(p,i)`alpha))))
%--------------------------------------------
%% more properties of leq
%--------------------------------------------
leq_Os: LEMMA
FORALL(n,m:nat,m1,m2:monomial):
(n <= m AND ( m1`alpha = (: 0 :)^n AND m2`alpha = (: 0 :)^m)) IMPLIES
leq(m1,m2)
leq_0_all: LEMMA
FORALL(m1,m2:monomial,n:nat):
(length(m1`alpha)= length(m2`alpha) AND m1`alpha = (: 0 :) ^n )
IMPLIES
leq(m1,m2)
leq_lft_monom: LEMMA
FORALL(m1,m2:monomial,
n:nat | n >= max(length(cut_monom(m1)`alpha),
length(cut_monom(m2)`alpha))):
(leq(m1,m2) AND length(m1`alpha) = length(m2`alpha))
IMPLIES