\import Algebra.Field
\import Algebra.Group
\import Algebra.Group.Category
\import Algebra.Meta
\import Algebra.Module(*c)
\import Algebra.Monoid
\import Algebra.Monoid.Category
\import Algebra.Monoid.PermSet
\import Algebra.Ordered
\import Algebra.Pointed
\import Algebra.Pointed.Category
\import Algebra.Ring
\import Algebra.Ring.Category
\import Algebra.Ring.Integral
\import Algebra.Ring.MonoidRing
\import Algebra.Ring.MPoly
\import Algebra.Ring.Poly
\import Algebra.Ring.Reduced
\import Algebra.Ring.RingHom
\import Algebra.Ring.Sub
\import Algebra.Semiring
\import Arith.Fin
\import Arith.Nat
\import Category
\import Data.Array
\import Function \hiding (id, o)
\import Function.Meta
\import Logic
\import Logic.Meta
\import Meta
\import Order.LinearOrder
\import Order.PartialOrder
\import Order.StrictOrder
\import Paths
\import Paths.Meta
\import Relation.Equivalence
\import Set.Fin
\open Monoid \hiding (equals)

{- | If f : K -> R is integral and {K} is a field, then {R} is 0-dimensional.
-
-   It is actually enough to assume that {K} is 0-dimensional, but this is harder to prove.
-}
\func integralExt-zeroDim {K : DiscreteField} {R : CRing} {f : RingHom K R} (fi : isIntegralExt f) : R.IsZeroDimensional
=> \lam a => \case fi a \with {
| inP (p, inP pm, pe) => poly-zeroDim p pm.1 pm.3 a 0 (ide-right *> pe)
}
\where {
\lemma poly-zeroDim (p : Poly K) (i : Nat) (pi : polyCoef p i = 1) (a : R) (k : Nat) (pe : polyMapEval f p a * pow a k = 0) : ∃ (b : R) (n : Nat) (pow a n = pow a (suc n) * b) \elim p
| pzero => absurd (zro/=ide pi)
| padd p e => \case eitherZeroOrInv e \with {
| byLeft e=0 => \case \elim i, \elim pi \with {
| 0, pi => absurd $zro/=ide (inv e=0 *> pi) | suc i, pi => poly-zeroDim p i pi a (suc k) (unfold polyMapEval (rewrite (e=0,f.func-zro) equation) *> pe) } | byRight (x,q,_) => inP (negative (f x * polyMapEval f p a), k, equation {using (inv func-* *> pmap f q *> func-ide)}) } } \func polyImage_field {K : DiscreteField} {E : ImpotentCRing} (f : RingHom K E) (a : E) (ai : isIntegral f a) : DiscreteField { | CRing => SubRing.cStruct (polyImage f a) } => ImpotentCRing.zeroDim->field {ImpotentCRing.subring {SubRing.cStruct (polyImage f a)} SubRing.embed (\lam p => ext p)} (integralExt-zeroDim (polyImage_integralExt a ai)) \lemma integral_dom-zeroDim {R S : CRing} (f : RingHom R S) (inj : isInj f) (fi : isIntegralExt f) (Sd : S.IsZeroDimensional) : R.IsZeroDimensional => \lam x => \case Sd (f x) \with { | inP (y,k,s) => \case fi y \with { | inP (p, inP (d,pd,pm), pe) => TruncP.map (aux pd pm s 0$ rewrite (pe,f.func-zro) simplify) \lam s => (s.1, k, inj $f.func-pow *> s.2 *> pmap (* _) (pmap (* _) (inv f.func-pow) *> inv f.func-*) *> inv f.func-*) } } \where { \private \func aux {p : Poly R} {k d : Nat} (pd : degree<= p d) (pm : polyCoef p d = 1) {x : R} {y : S} (c : pow (f x) k = pow (f x) (suc k) * y) (a : R) (pe : pow (f x) (k Nat.+ d) * polyMapEval f p y = pow (f x) (suc k) * f a) : ∃ (b : R) (pow (f x) k = pow (f x) (suc k) * f b) \elim p, d | pzero, _ => inP (0, \have t {z : S} => inv ide-right *> pmap (z *) (inv f.func-ide *> pmap f (inv pm) *> f.func-zro) *> S.zro_*-right \in t *> inv t) | padd p e, 0 => inP (a, inv ide-right *> pmap (_ *) (simplify (inv f.func-ide)) *> (rewrite (pd,pm) in pe)) | padd p e, suc d => \have | t1 : pow (f x) (k Nat.+ d) * f x * y = pow (f x) (k Nat.+ d) => rewrite S.pow_+ equation | t2 : pow (f x) (k Nat.+ d) * f e = pow (f x) k * (f (pow x d) * f e) => rewrite S.pow_+$ *-assoc *> pmap (\lam x => _ * (x * _)) (inv f.func-pow)
\in aux pd pm c (a - e * pow x d) $rewrite (f.func-+,f.func-negative,f.func-*) equation } \func mPoly_integral {K : DiscreteField} {S : CRing} (f : RingHom K S) {a : S} {l : Array S} (p : MPoly (Fin (suc l.len)) K) (p/=0 : p /= 0) (pe : mPolyMapEval f p (a :: l) = 0) : \Sigma (E : CRing) (g : RingHom K E) (l' : Array E l.len) (RingHom.isAlgebraGenerated g l') (h : RingHom E S) (\Pi (j : Fin l.len) -> \Sigma (q : Poly E) (l j = polyMapEval h q a)) (isInj h) (isIntegral h a) (h ∘ g = f) => \let | d j => pow (suc (mdegree p)) (suc j) | l' => mkArray \lam j => l j - pow a (d j) | subring : SubRing S => mPolyImage f l' | t => mPolyImage-finGen f l' \in (SubRing.cStruct subring, mPolyImageHom f l', t.1, t.2, subring.embed, \lam j => (padd pzero (later (l' j, mPolyImage-element f l' j)) + pow (padd 1 0) (d j), inv$ func-+ {polyMapEvalRingHom _ _} *> pmap (+ _) (pmap (+ _) S.zro_*-left *> zro-left) *> +-assoc *> pmap (l j +) (+-comm *> S.toZero (MonoidHom.func-pow {polyMapEvalRingHom _ _} *> pmap (pow __ _) equation)) *> zro-right),
\lam q => ext q,
\case MPoly_poly-field (mPolyImageHom f l') p/=0 ((zro, transport (mPolyImage f l') f.func-zro (mPolyImage-ext f l' zro)) :: \lam j => (l' j, mPolyImage-element f l' j)) \with {
| inP s => isIntegral.fromInv s.2 s.3 $pmap (polyEval __ a) change-vars_map *> change-vars_eval *> pmap2 (mPolyEval __ __) (ext \case \elim __ \with { | 0 => simplify | suc j => simplify }) (inv (mPoly-map-comp {_} {_} {_} {_} {mPolyImageHom f l'} {subring.embed})) *> pe }, idp) \where { \func change-vars {R : CRing} {n : Nat} (p : MPoly (Fin n) R) (d : Nat) (l : Array R n) : Poly R => mPolyMapEval polyHom p \lam j => padd pzero (l j) + pow (padd 1 0) (pow d j) \lemma change-vars_map {R S : CRing} {f : RingHom R S} {n : Nat} {p : MPoly (Fin n) R} {d : Nat} {l : Array R n} : polyMap f (change-vars p d l) = change-vars (mPoly-map f p) d (map f l) => mPolyRingHom-unique (polyMapRingHom f RingCat.∘ mPolyMapEvalRingHom polyHom _) (mPolyMapEvalRingHom polyHom _ RingCat.∘ mPoly-mapHom f) (\lam r => simplify) (\lam j => simplify$ func-+ {polyMapRingHom f} *> pmap2 (padd pzero __ + __) f.func-zro (polyMapRingHom.polyMap_*c *> pmap (_ PolyRing.*c) (func-+ {polyMapRingHom f} *>
pmap (_ +) (MonoidHom.func-pow {polyMapRingHom f} *> pmap2 (\lam x y => pow (padd (padd pzero x) y) _) f.func-ide f.func-zro)))) p

\lemma change-vars_eval {R : CRing} {n : Nat} {p : MPoly (Fin n) R} {d : Nat} {l : Array R n} {a : R}
: polyEval (change-vars p d l) a = mPolyEval (\lam j => l j + pow a (pow d j)) p
=> mPolyRingHom-unique (polyEvalRingHom a RingCat.∘ mPolyMapEvalRingHom polyHom _) (mPolyEval _) (\lam r => simplify)
(\lam j => simplify $func-+ {polyEvalRingHom a} *> simplify (pmap (polyEvalRingHom a) PolyRing.ide_*c *> func-+ *> pmap2 (+) simplify (MonoidHom.func-pow *> simplify (simplify simplify)))) p \lemma change-vars_BigSum {R : CRing} {n : Nat} {l : Array (MPoly (Fin n) R)} {d : Nat} {a : Array R n} : change-vars (BigSum l) d a = BigSum \lam j => change-vars (l j) d a => AddMonoidHom.func-BigSum {mPolyMapEvalRingHom _ _} \lemma change-vars_*c {R : CRing} {n : Nat} {x : R} {p : MPoly (Fin n) R} {d : Nat} {a : Array R n} : change-vars (x *c p) d a = x PolyRing.*c change-vars p d a => func-* {mPolyMapEvalRingHom polyHom \lam j => padd pzero (a j) + pow (padd 1 0) (pow d j)} *> simplify (pmap (+ _) (path peq)) \lemma change-vars_monomial {R : CRing} {n : Nat} {c : PermSet (Fin n)} {d : Nat} {a : Array R n} : change-vars (msMonomial 1 c) d a = permSet-univ (\lam j => padd pzero (a j) + pow (padd 1 0) (pow d j)) c => simplify$ pmap (+ _) peq *> PolyRing.ide_*c

\lemma permSet-univ_monic {R : CRing} {ps : Array (Poly R)} {d : Array Nat ps.len} {c : PermSet (Fin ps.len)}
(p : \Pi (j : Fin ps.len) -> \Sigma (degree<= (ps j) (d j)) (polyCoef (ps j) (d j) = 1))
: \Sigma (degree<= (permSet-univ ps c) (permSet-univ {_} {AbMonoid.toCMonoid NatSemiring} d c))
(polyCoef (permSet-univ ps c) (permSet-univ {_} {AbMonoid.toCMonoid NatSemiring} d c) = 1) \elim c
| in~ l => (degree<=_BigProd {_} {map ps l} (map d l) \lam j => (p (l j)).1, leadCoef_BigProd {_} {map ps l} (map d l) (\lam j => (p (l j)).1) *> BigProd_ide \lam j => later $(p (l j)).2) \lemma MPoly_poly {R : CRing} {n : Nat} (pl : Array (PermSet (Fin n))) (pli : isInj pl) (pc : Array R pl.len) (pci : \Pi (j : Fin pl.len) -> Inv (pc j)) {d : Nat} (pd : \Pi (j : Fin pl.len) -> permSet-length (pl j) < d) (a : Array R n) (pl/=0 : pl.len /= 0) {p : MPoly (Fin n) R} (pp : p = BigSum \lam j => msMonomial (pc j) (pl j)) : ∃ (m : Nat) (degree<= (change-vars p d a) m) (Inv (polyCoef (change-vars p d a) m)) \elim pp | idp => \let | pow>0 {n} : 0 < pow d n => NatSemiring.pow>0$ zero<=_ <∘r pd (transport Fin (suc_pred pl/=0) 0)
| (m,rd,ri) => poly-degrees {_} {\lam j => change-vars (msMonomial 1 (pl j)) d a} pc
(\lam j => permSet-univ {_} {AbMonoid.toCMonoid NatSemiring} (later \lam i => pow d i) (pl j))
(\lam j => rewrite (change-vars_monomial {_} {_} {pl j}) $permSet-univ_monic {_} {\lam j => padd pzero (a j) + pow (padd 1 0) (pow d j)} {\lam j => pow d j} \lam j => (degree<=_+ (degree<=-trans ()$ suc_<_<= pow>0) $later$ transport (degree<= _) ide-left $degree<=_pow idp, polyCoef_+ *> pmap2 (+) (polyCoef=0 pow>0) (pmap (polyCoef _) (inv ide-left) *> leadCoef_pow idp *> pow_ide) *> zro-left)) pci (\lam {i} {j} q => pli$ cases (pl i arg addPath, pl j arg addPath, q) \with {
| in~ l1, l1p, in~ l2, l2p, q => PermSet.~-psequiv $EPerm.count_EPerm \lam m => later$
base-digit-sum {\lam j => count l1 j} (\lam k => later (rewrite l1p count<=) <∘r pd i) *> pmap (base-digit __ d m) (inv (groupSum (\lam j => pow d j)) *> q *> groupSum (\lam j => pow d j)) *> inv (base-digit-sum {\lam j => count l2 j} (\lam k => later (rewrite l2p count<=) <∘r pd j))
})
pl/=0 idp
| q : BigSum (\lam j => pc j PolyRing.*c change-vars (msMonomial R.ide (pl j)) d a) = change-vars p d a
=> pmap BigSum (exts \lam j => later $inv (change-vars_*c {_} {_} {_} {msMonomial 1 (pl j)}) *> simplify simplify) *> inv change-vars_BigSum \in inP (m, transport (degree<= __ m) q rd, transport (\lam x => Inv (polyCoef x m)) q ri) \where { \func group {n : Nat} (l : Array (Fin n)) : EPerm l (Big (++) nil \lam j => replicate (count l j) j) => EPerm.count_EPerm \lam j => inv$ count_Big++ {_} {\new Array _ _ _} *> BigSum-unique j (\lam k j/=k => count-none \lam _ => /=-sym j/=k) *> count-all \lam _ => idp

\lemma groupSum {R : CSemiring} {n : Nat} {l : Array (Fin n)} (f : Fin n -> R) : BigSum (map f l) = BigSum \lam j => natCoef (count l j) * f j
=> R.BigSum_EPerm (EPerm.EPerm_map f (group l)) *> pmap BigSum (inv (Big++_map f {\new Array _ _ _})) *> BigSum_Big++ *> pmap BigSum (exts \lam j => R.BigSum_replicate)

\func leadCoef_BigSum {R : CRing} {n : Nat} {l : Array (Poly R) (suc n)} (pd : Array Nat (suc n))
(pc : \Pi (j : Fin (suc n)) -> degree<= (l j) (pd j)) (inj : isInj pd)
: \Sigma (m : Nat) (degree<= (BigSum l) m) (j : Fin (suc n)) (polyCoef (BigSum l) m = polyCoef (l j) (pd j))
=> (NatSemiring.Big_∨ pd, degree<=_BigSum \lam j => degree<=-trans (pc j) $NatSemiring.Big_join-cond, (LinearOrder.findMax pd).1, rewrite (LinearOrder.findMax pd).2$ polyCoef_BigSum {_} {l} *> BigSum-unique {_} {map (polyCoef __ _) l} _ \lam k j/=k =>
degree<=.toCoefs (l k) (pd k) (pc k) $LinearOrder.<=_/= (transport (_ <=) (LinearOrder.findMax pd).2$ NatSemiring.Big_join-cond) \lam p => j/=k $inv (inj p) ) \func leadCoef_BigSum_/=0 {R : CRing} {l : Array (Poly R)} (pd : Array Nat l.len) (pc : \Pi (j : Fin l.len) -> degree<= (l j) (pd j)) (inj : isInj pd) (l/=0 : l.len /= 0) : \Sigma (m : Nat) (degree<= (BigSum l) m) (j : Fin l.len) (polyCoef (BigSum l) m = polyCoef (l j) (pd j)) \elim l | nil => absurd (l/=0 idp) | a :: l => leadCoef_BigSum pd pc inj \func poly-degrees {R : CRing} {l : Array (Poly R)} (pc : Array R l.len) (pd : Array Nat l.len) (pdp : \Pi (j : Fin l.len) -> \Sigma (degree<= (l j) (pd j)) (polyCoef (l j) (pd j) = 1)) (pcp : \Pi (j : Fin l.len) -> Monoid.Inv (pc j)) (inj : isInj pd) (l/=0 : l.len /= 0) {p : Poly R} (pp : p = AddMonoid.BigSum \lam j => pc j PolyRing.*c l j) : \Sigma (m : Nat) (degree<= p m) (Monoid.Inv (polyCoef p m)) \elim pp | idp => \let t => leadCoef_BigSum_/=0 {R} {\lam j => pc j PolyRing.*c l j} pd (\lam j => degree<=_*c (pdp j).1) inj l/=0 \in (t.1, t.2, rewrite (t.4 *> polyCoef_*c *> pmap (_ *) (pdp t.3).2 *> ide-right) (pcp t.3)) } \lemma MPoly_poly-field {K : DiscreteField} {E : CRing} (f : RingHom K E) {n : Nat} {p : MPoly (Fin n) K} (p/=0 : p /= 0) (a : Array E n) : ∃ (m : Nat) (degree<= (change-vars (mPoly-map f p) (suc (mdegree p)) a) m) (Inv (polyCoef (change-vars (mPoly-map f p) (suc (mdegree p)) a) m)) => unfold mdegree$ cases (monoidSetDec-equiv.ret p arg addPath) \with {
| in~ s, sp =>
\have sp' => inv (monoidSetDec-equiv.f_ret p) *> pmap decToMonoidSet sp
\in MPoly_poly (map __.2 s.1) s.2 (map (\lam t => f t.1) s.1) (\lam j => f.func-Inv $K.nonZero-Inv$ s.3 j)
(\lam j => NatSemiring.Big_<=_join1 {map (\lam t => permSet-length t.2) s.1} j <∘r id<suc) a
(\lam q => p/=0 $sp' *> rewrite (len=0 {_} {s.1} q) idp)$ pmap (mPoly-map f) sp' *>
inv (msMonomial_BigSum-split {_} {_} {map (\lam t => (f t.1, t.2)) s.1})
}
}

\func mPoly_integralExt {K : DiscreteField} {S : CRing} (f : RingHom K S) {n : Nat} {l : Array S (suc n)} (gen : RingHom.isAlgebraGenerated f l) (p : MPoly (Fin (suc n)) K) (p/=0 : p /= 0) (pe : mPolyMapEval f p l = 0)
: \Sigma (E : CRing) (g : RingHom K E) (l' : Array E n) (RingHom.isAlgebraGenerated g l') (h : RingHom E S) (isInj h) (isIntegralExt h) (h ∘ g = f)
=> \case mPoly_integral f p p/=0 pe \with {
| (E,g,l',gen',h,hg,hinj,hint,hg=f) => (E,g,l',gen',h,hinj, integralExt-finite-char (l 0 :: nil) (RingHom.isAlgebraGenerated-poly \lam x =>
TruncP.map (gen x) \lam s => (mPolyEval (\lam j => (hg j).1) $MPoly_Fin-suc' (mPoly-map g s.1), MPoly_Fin-suc'.mapEval-comm h {_} {_} {\lam j => (hg j).1} *> pmap2 (mPolyEval __ __) (exts \case \elim __ \with { | 0 => idp | suc j => inv (hg j).2 }) (inv mPoly-map-comp *> rewrite hg=f idp) *> inv s.2)) 1 0 \lam (0) => hint, hg=f) } {- | If {K} is a field and f : K -> E is finitely generated as an algebra, then {f} is integral if and only if {E} is 0-dimensional. It is actually enough to assume that {K} is reduced and 0-dimensional. Also, if {E} is local, then these equivalent conditions hold. -} \lemma finGenExt-char {K : DiscreteField} {E : CRing} {f : RingHom K E} (gen : RingHom.isFiniteAlgebra f) : isIntegralExt f <-> E.IsZeroDimensional \elim gen | inP s => (integralExt-zeroDim, aux __ f s.2) \where { \private \lemma aux {K : DiscreteField} {S : CRing} (Szd : S.IsZeroDimensional) (f : RingHom K S) {n : Nat} {l : Array S n} (gen : RingHom.isAlgebraGenerated f l) : isIntegralExt f \elim n, l | 0, nil => integralExt-finite-char nil gen 1 0 \case __ | suc n, a :: l => \let | (inP (b,m,q)) => Szd a | (inP (p,r)) => gen b | h : RingHom (MPoly (Fin (suc n)) K) S => mPolyMapEvalRingHom f (a :: l) | h0 : h (mVar zro) = a => simplify$ pmap (* _) f.func-ide *> ide-left
| (E,g,l',gen',h,hinj,hint,hg=f) => mPoly_integralExt f {n} {a :: l} gen (pow (mVar zro) m - pow (mVar zro) (suc m) * p)
(\lam q => K.zro/=ide $inv$ inv K.pow_ide *> inv msCoef_monomial *> pmap (msCoef __ _) (inv msMonomial_pow *> AddGroup.fromZero q) *> msCoef_mVar=0 id<suc)
(h.func-minus *> S.toZero (h.func-pow *> pmap (pow __ m) h0 *> q *> pmap (_ *) r *> inv (h.func-* *> pmap (* _) (h.func-* *> pmap2 (*) (h.func-pow *> pmap (pow __ m) h0) h0))))
\in rewriteI hg=f \$ integralExt-comp g h (aux (integral_dom-zeroDim h hinj hint Szd) g {n} gen') hint
}