```-- Copyright (c) David Amos, 2011. All rights reserved.

{-# LANGUAGE TupleSections, NoMonomorphismRestriction #-}

-- |A module providing an efficient implementation of the Buchberger algorithm for calculating the (reduced) Groebner basis for an ideal,
-- together with some straightforward applications.
module Math.CommutativeAlgebra.GroebnerBasis where

import Data.List as L
import qualified Data.IntMap as IM
import qualified Data.Set as S

import Math.Core.Utils
import Math.Core.Field
import Math.Algebras.VectorSpace
import Math.Algebras.Structures
import Math.CommutativeAlgebra.Polynomial

-- Sources:
-- Cox, Little, O'Shea: Ideals, Varieties and Algorithms
-- Giovini, Mora, Niesi, Robbiano, Traverso, "One sugar cube please, or Selection strategies in the Buchberger algorithm"

sPoly f g = let d = tgcd (lt f) (lt g)
in (lt g `tdiv` d) *-> f - (lt f `tdiv` d) *-> g
-- The point about the s-poly is that it cancels out the leading terms of the two polys, exposing their second terms

isGB fs = all (\h -> h %% fs == 0) (pairWith sPoly fs)

-- Initial, naive version
-- Cox p87
gb1 fs = gb' fs (pairWith sPoly fs) where
gb' gs (h:hs) = let h' = h %% gs in
if h' == 0 then gb' gs hs else gb' (h':gs) (hs ++ map (sPoly h') gs)
gb' gs [] = gs

-- [f xi xj | xi <- xs, xj <- xs, i < j]
pairWith f (x:xs) = map (f x) xs ++ pairWith f xs
pairWith _ [] = []

-- Cox p89-90
reduce gs = reduce' gs [] where
reduce' (l:ls) rs = let l' = l %% (rs ++ ls) in
if l' == 0 then reduce' ls rs else reduce' ls (toMonic l':rs)
reduce' [] rs = L.sort rs
-- when using an elimination order, the elimination ideal will be at the end

-- Cox et al p106-7
-- No need to calculate an spoly fi fj if
-- 1. the lm fi and lm fj are coprime, or
-- 2. there exists some fk, with (i,k) (j,k) already considered, and lm fk divides lcm (lm fi) (lm fj)
-- some slight inefficiencies from looking up fi, fj repeatedly
gb2 fs = reduce \$ gb' fs (pairs [1..s]) s where
s = length fs
gb' gs ((i,j):ps) t =
let fi = gs!i; fj = gs!j in
if mcoprime (lm fi) (lm fj) || criterion fi fj
then gb' gs ps t
else let h = sPoly fi fj %% gs in
if h == 0 then gb' gs ps t else gb' (gs++[h]) (ps ++ [ (i,t+1) | i <- [1..t] ]) (t+1)
where
criterion fi fj = let l = mlcm (lm fi) (lm fj) in any (test l) [1..t]
test l k = k `notElem` [i,j]
&& ordpair i k `notElem` ps
&& ordpair j k `notElem` ps
&& lm (gs!k) `mdivides` l
gb' gs [] _ = gs

xs ! i = xs !! (i-1) -- in other words, index the list from 1 not 0

-- Doesn't result in any speedup
gb2a fs = reduce \$ gb' fs' (pairs [1..s]) s
where fs' = IM.fromList \$ zip [1..] \$ filter (/= 0) fs
s = IM.size fs'
gb' gs ((i,j):ps) t =
let fi = gs IM.! i; fj = gs IM.! j in
if mcoprime (lm fi) (lm fj) || criterion fi fj
then gb' gs ps t
else let h = sPoly fi fj %% IM.elems gs in
if h == 0
then gb' gs ps t
else let t' = t+1
gs' = IM.insert t' h gs
ps' = ps ++ map (,t') [1..t]
in gb' gs' ps' t'
where criterion fi fj = let l = mlcm (lm fi) (lm fj) in any (test l) [1..t]
test l k = k `notElem` [i,j]
&& ordpair i k `notElem` ps
&& ordpair j k `notElem` ps
&& lm (gs IM.! k) `mdivides` l
gb' gs [] _ = IM.elems gs

-- version of gb2 where we eliminate pairs as they're created, rather than as they're processed
gb3 fs = reduce \$ gb1' [] fs [] 0
where
gb1' gs (f:fs) ps t = let ps' = updatePairs gs ps f t
in gb1' (gs ++ [f]) fs ps' (t+1)
gb1' ls [] ps t = gb2' ls ps t
gb2' gs ((i,j):ps) t =
let h = sPoly (gs!i) (gs!j) %% gs in
if h == 0
then gb2' gs ps t
else let ps' = updatePairs gs ((i,j):ps) h t in gb2' (gs++[h]) ps' (t+1)
gb2' gs [] _ = gs
updatePairs gs ps f t =
[p | p@(i,j) <- ps,
not (lm f `mdivides` mlcm (lm (gs!i)) (lm (gs!j)))]
++ [ (i,t+1) | (gi,i) <- zip gs [1..t],
not (mcoprime (lm gi) (lm f)),
not (criterion (mlcm (lm gi) (lm f)) i) ]
where criterion l i = any (`mdivides` l) [lm gk | (gk,k) <- zip gs [1..t], k /= i, ordpair i k `notElem` ps]

-- Cox et al 108
-- 1. list smallest fs first, as more likely to reduce
-- 2. order the pairs with smallest lcm fi fj first ("normal selection strategy")
gb4 fs = reduce \$ gb1' [] fs' [] 0
where fs' = reverse \$ L.sort \$ filter (/=0) fs
gb1' gs (f:fs) ps t = gb1' (gs ++ [f]) fs ps' (t+1)
where ps' = updatePairs gs ps f t
gb1' ls [] ps t = gb2' ls ps t
gb2' gs ((l,(i,j)):ps) t =
let h = sPoly (gs!i) (gs!j) %% gs in
if h == 0
then gb2' gs ps t
else let ps' = updatePairs gs ((l,(i,j)):ps) h t in gb2' (gs++[h]) ps' (t+1)
gb2' gs [] _ = gs
updatePairs gs ps f t =
let oldps = [p | p@(l,(i,j)) <- ps, not (lm f `mdivides` l)]
newps = sortBy (flip cmpfst)
[ (l,(i,t+1)) | (gi,i) <- zip gs [1..t], let l = mlcm (lm gi) (lm f),
not (mcoprime (lm gi) (lm f)),
not (criterion l i) ]
in mergeBy (flip cmpfst) oldps newps
where criterion l i = any (`mdivides` l) [lm gk | (gk,k) <- zip gs [1..t], k /= i, ordpair i k `notElem` map snd ps]

mergeBy cmp (t:ts) (u:us) =
case cmp t u of
LT -> t : mergeBy cmp ts (u:us)
EQ -> t : mergeBy cmp ts (u:us)
GT -> u : mergeBy cmp (t:ts) us
mergeBy _ ts us = ts ++ us -- one of them is null

-- Giovini et al
-- The point of sugar is, given fi, fj, to give an upper bound on the degree of sPoly fi fj without having to calculate it
-- We can then select by preference pairs with lower sugar, expecting therefore that the s-polys will have lower degree

-- It is only for Lex ordering that sugar seems to give a substantial improvement
-- gb4 is fine for Grevlex

-- !! can probably get rid of the Ord k requirement in the following

-- |Given a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials.
gb :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m]
gb fs =
let fs' = reverse \$ sort \$ map toMonic \$ filter (/=0) fs
in reduce \$ gb1' [] fs' [] 0 where
gb1' gs (f:fs) ps t = gb1' (gs ++ [f]) fs ps' (t+1)
where ps' = updatePairs gs ps f (t+1)
gb1' ls [] ps t = gb2' ls ps t
gb2' gs (p@(_,(i,j)):ps) t =
if h == 0
then gb2' gs ps t
else gb2' (gs++[h]) ps' (t+1)
where h = toMonic \$ sPoly (gs!i) (gs!j) %% gs
ps' = updatePairs gs (p:ps) h (t+1)
gb2' gs [] _ = gs
updatePairs gs ps gk k =
let newps = [let l = mlcm (lm gi) (lm gk) in ((sugar gi gk l, l), (i,k)) | (gi,i) <- zip gs [1..k-1] ]
ps' = [p | p@((sij,tij),(i,j)) <- ps,
let ((sik,tik),_) = newps ! i, let ((sjk,tjk),_) = newps ! j,
not ( (tik `mproperlydivides` tij) && (tjk `mproperlydivides` tij) ) ] -- sloppy variant
newps'' = sortBy (flip cmpSug) \$ discard2 [] \$ sortBy (flip cmpNormal) newps'
in mergeBy (flip cmpSug) ps' newps''
where
if lm (gs!i) `mcoprime` lm gk
-- then discard [l | l@((_,tjk),_) <- ls, tjk /= tik] [r | r@((_,tjk),_) <- ls, tjk /= tik]
then discard1 (filter (\((_,tjk),_) -> tjk /= tik) ls) (filter (\((_,tjk),_) -> tjk /= tik) rs)
discard2 ls (r@((_sik,tik),(i,k)):rs) = discard2 (r:ls) \$ filter (\((_sjk,tjk),(j,k')) -> not (k == k' && tik `mdivides` tjk)) rs
-- The two calls to toMonic are designed to prevent coefficient explosion, but it is unproven that they are effective

-- sugar of sPoly f g, where h = lcm (lt f) (lt g)
-- this is an upper bound on deg (sPoly f g)
sugar f g h = mdeg h + max (deg f - mdeg (lm f)) (deg g - mdeg (lm g))

cmpNormal ((s1,t1),(i1,j1)) ((s2,t2),(i2,j2)) = compare (t1,j1) (t2,j2)

cmpSug ((s1,t1),(i1,j1)) ((s2,t2),(i2,j2)) = compare (-s1,t1,j1) (-s2,t2,j2)

{-
merge (t:ts) (u:us) =
if t <= u
then t : merge ts (u:us)
else u : merge (t:ts) us
merge ts us = ts ++ us -- one of them is null
-}

-- OPERATIONS ON IDEALS

memberGB f gs = f %% gs == 0

-- |@memberI f gs@ returns whether f is in the ideal generated by gs
memberI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
Vect k m -> [Vect k m] -> Bool
memberI f gs = memberGB f (gb gs)

-- Cox et al, p181
-- |Given ideals I and J, their sum is defined as I+J = {f+g | f \<- I, g \<- J}.
--
-- If fs and gs are generators for I and J, then @sumI fs gs@ returns generators for I+J.
--
-- The geometric interpretation is that the variety of the sum is the intersection of the varieties,
-- ie V(I+J) = V(I) intersect V(J)
sumI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> [Vect k m]
sumI fs gs = gb (fs ++ gs)

-- Cox et al, p183
-- |Given ideals I and J, their product I.J is the ideal generated by all products {f.g | f \<- I, g \<- J}.
--
-- If fs and gs are generators for I and J, then @productI fs gs@ returns generators for I.J.
--
-- The geometric interpretation is that the variety of the product is the union of the varieties,
-- ie V(I.J) = V(I) union V(J)
productI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> [Vect k m]
productI fs gs = gb [f * g | f <- fs, g <- gs]

-- Cox et al, p185-6
-- |The intersection of ideals I and J is the set of all polynomials which belong to both I and J.
--
-- If fs and gs are generators for I and J, then @intersectI fs gs@ returns generators for the intersection of I and J
--
-- The geometric interpretation is that the variety of the intersection is the union of the varieties,
-- ie V(I intersect J) = V(I) union V(J).
--
-- The reason for prefering the intersection over the product is that the intersection of radical ideals is radical,
-- whereas the product need not be.
intersectI :: (Fractional k, Ord k, Monomial m, Ord m) =>
[Vect k m] -> [Vect k m] -> [Vect k m]
intersectI fs gs =
let t = toElimFst \$ return \$ (mvar "t" :: Glex String)
hs = map ((t *) . toElimSnd) fs ++ map (((1-t) *) . toElimSnd) gs
in eliminateFst hs

toElimFst = fmap (\m -> Elim2 m munit)
toElimSnd = fmap (\m -> Elim2 munit m)
isElimFst = (/= munit) . (\(Elim2 m _) -> m) . lm
fromElimSnd = fmap (\(Elim2 _ m) -> m)
eliminateFst = map fromElimSnd . dropWhile isElimFst . gb

-- Cox et al, p193-4
-- |Given ideals I and J, their quotient is defined as I:J = {f | f \<- R, f.g is in I for all g in J}.
--
-- If fs and gs are generators for I and J, then @quotientI fs gs@ returns generators for I:J.
--
-- The ideal quotient is the algebraic analogue of the Zariski closure of a difference of varieties.
-- V(I:J) contains the Zariski closure of V(I)-V(J), with equality if k is algebraically closed and I is a radical ideal.
quotientI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> [Vect k m]
quotientI _ [] = [1]
quotientI fs gs = foldl1 intersectI \$ map (quotientP fs) gs
-- quotientI fs gs = foldl intersectI [1] \$ map (quotientP fs) gs

quotientP fs g = map ( // g ) \$ intersectI fs [g]
where h // g = let ([u],_) = quotRemMP h [g] in u

-- |@eliminate vs gs@ returns the elimination ideal obtained from the ideal generated by gs by eliminating the variables vs.
eliminate :: (Eq k, Fractional k, Ord k, MonomialConstructor m, Monomial (m v), Ord (m v)) =>
[Vect k (m v)] -> [Vect k (m v)] -> [Vect k (m v)]
eliminate vs gs = let subs = subFst vs in eliminateFst [g `bind` subs | g <- gs]
where subFst :: (Eq k, Num k, MonomialConstructor m, Eq (m v), Mon (m v)) =>
[Vect k (m v)] -> v -> Vect k (Elim2 (m v) (m v))
subFst vs = (\v -> let v' = var v in if v' `elem` vs then toElimFst v' else toElimSnd v')

{-
-- !! NOT WORKING
-- |@elimExcept vs gs@ returns the elimination ideal obtained from the ideal generated by gs by eliminating all variables except vs.
elimExcept :: (Fractional k, Ord k, MonomialConstructor m, Monomial (m v), Ord (m v)) =>
[Vect k (m v)] -> [Vect k (m v)] -> [Vect k (m v)]
elimExcept vs gs = let subs = subSnd vs in eliminateFst [g `bind` subs | g <- gs]
where subSnd :: (Num k, MonomialConstructor m, Eq (m v), Mon (m v)) =>
[Vect k (m v)] -> v -> Vect k (Elim2 (m v) (m v))
subSnd vs = (\v -> let v' = var v in if v' `elem` vs then toElimSnd v' else toElimFst v')
-}

-- MONOMIAL BASES FOR QUOTIENT ALGEBRAS

-- basis for the polynomial ring in variables vs
mbasis vs = mbasis' [1]
where mbasis' ms = ms ++ mbasis' (toSet [v*m | v <- vs, m <- ms])

-- |Given variables vs, and a Groebner basis gs, @mbasisQA vs gs@ returns a monomial basis for the quotient algebra k[vs]/\<gs\>.
-- For example, @mbasisQA [x,y] [x^2+y^2-1]@ returns a monomial basis for k[x,y]/\<x^2+y^2-1\>.
-- In general, the monomial basis is likely to be infinite.
mbasisQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> [Vect k m]
mbasisQA vs gs = mbasisQA' [1]
where mbasisQA' [] = []  -- the quotient algebra is finite-dimensional
mbasisQA' ms = ms ++ mbasisQA' (toSet [f | v <- vs, m <- ms, let f = v*m, f %% gs == f])

-- |Given an ideal I, the leading term ideal lt(I) consists of the leading terms of all elements of I.
-- If I is generated by gs, then @ltIdeal gs@ returns generators for lt(I).
ltIdeal :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m]
ltIdeal gs = map (return . lm) \$ gb gs

-- number of monomials of degree i in n variables
numMonomials n i = toInteger (i+n-1) `choose` toInteger (n-1)

-- |Given variables vs, and a homogeneous ideal gs, @hilbertFunQA vs gs@ returns the Hilbert function for the quotient algebra k[vs]/\<gs\>.
-- Given an integer i, the Hilbert function returns the number of degree i monomials in a basis for k[vs]/\<gs\>.
-- For a homogeneous ideal, this number is independent of the monomial ordering used
-- (even though the elements of the monomial basis themselves are dependent on the ordering).
--
-- If the ideal I is not homogeneous, then R/I is not graded, and the Hilbert function is not well-defined.
-- Specifically, the number of degree i monomials in a basis is likely to depend on which monomial ordering you use.
hilbertFunQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> Int -> Integer
hilbertFunQA vs gs i = hilbertFunQA' (ltIdeal gs) i
where n = length vs
hilbertFunQA' _ i | i < 0 = 0
hilbertFunQA' (m:ms) i = hilbertFunQA' ms i - hilbertFunQA' (ms `quotientP` m) (i - deg m)
hilbertFunQA' [] i = numMonomials n i
-- For example, consider k[x,y]/<x-y^2>
-- Under Lex ordering, the monomial basis is 1,y,y^2,y^3,...
-- Under Glex ordering, the monomial basis is 1,x,y,x^2,xy,x^3,x^2y,...
-- So the Hilbert function is not well-defined.
-- Note though that this function does still correctly return the number of degree i monomials for the given monomial ordering

-- naive implementation which simply counts monomials
hilbertSeriesQA1 vs gs = hilbertSeriesQA1' [1]
where hilbertSeriesQA1' [] = [] -- repeat 0
hilbertSeriesQA1' ms = length ms : hilbertSeriesQA1' (toSet [f | v <- vs, m <- ms, let f = v*m, f %% gs == f])

-- Eisenbud p325, p357 / Schenck p56
-- This can be made more efficient by choosing which m to recurse on
-- |Given variables vs, and a homogeneous ideal gs, @hilbertSeriesQA vs gs@ returns the Hilbert series for the quotient algebra k[vs]/\<gs\>.
-- The Hilbert series should be interpreted as a formal power series where the coefficient of t^i is the Hilbert function evaluated at i.
-- That is, the i'th element in the series is the number of degree i monomials in a basis for k[vs]/\<gs\>.
hilbertSeriesQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> [Integer]
hilbertSeriesQA vs gs = hilbertSeriesQA' \$ ltIdeal gs
where hilbertSeriesQA' (m:ms) = hilbertSeriesQA' ms <-> (replicate (deg m) 0 ++ hilbertSeriesQA' (ms `quotientI` [m]))
hilbertSeriesQA' [] = [numMonomials n i | i <- [0..] ]
n = length vs
(a:as) <-> (b:bs) = (a-b) : (as <-> bs)
as <-> [] = as
[] <-> bs = map negate bs

-- |In the case where every variable v occurs in some generator g of the homogeneous ideal (the usual case),
-- then the vs can be inferred from the gs.
-- @hilbertSeriesQA' gs@ returns the Hilbert series for the quotient algebra k[vs]/\<gs\>.
hilbertSeriesQA' :: (Fractional k, Ord k, MonomialConstructor m, Ord (m v), Monomial (m v), Algebra k (m v)) =>
[Vect k (m v)] -> [Integer]
hilbertSeriesQA' gs = hilbertSeriesQA vs gs where vs = toSet (concatMap vars gs)

-- |For i \>\> 0, the Hilbert function becomes a polynomial in i, called the Hilbert polynomial.
hilbertPolyQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) =>
[Vect k m] -> [Vect k m] -> GlexPoly Q String
hilbertPolyQA vs gs = hilbertPolyQA' (ltIdeal gs) i
where n = toInteger \$ length vs
i = glexvar "i"
hilbertPolyQA' [] x = product [ x + fromInteger j | j <- [1..n-1] ] / (fromInteger \$ product [1..n-1])
hilbertPolyQA' (m:ms) x = hilbertPolyQA' ms x - hilbertPolyQA' (ms `quotientP` m) (x - fromIntegral (deg m))

hilbertPolyQA' :: (Fractional k, Ord k, MonomialConstructor m, Ord (m v), Monomial (m v), Algebra k (m v)) =>
[Vect k (m v)] -> GlexPoly Q String
hilbertPolyQA' gs = hilbertPolyQA vs gs where vs = toSet (concatMap vars gs)

-- The dimension of a variety
dim vs gs = 1 + deg (hilbertPolyQA vs gs)

dim' gs = 1 + deg (hilbertPolyQA' gs)
```