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Division.lean
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Division.lean
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import Mathlib.Data.MvPolynomial.Basic
import Mathlib.Data.MvPolynomial.Division
import Mathlib.Data.MvPolynomial.CommRing
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.LibrarySearch
import Mathlib.Data.List.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Basic
import TermOrder
import Multideg
open BigOperators
open Classical
lemma List.filter_eq_nil' {α : Type _} {p : α→Prop} {l : List α} :
l.filter (fun x => decide (p x)) = [] ↔ ∀ a ∈ l , ¬p a := by
conv =>
enter [2, a, ha]
rw [Iff.intro (decide_eq_true (p:=p a)) of_decide_eq_true]
exact List.filter_eq_nil
lemma List.filter_ne_nil' {α : Type _} {p : α→Prop} {l : List α} :
l.filter (fun x => decide (p x)) ≠ [] ↔ ∃ a ∈ l , p a := by
have := (List.filter_eq_nil' (α:=α) (p:=p) (l:=l)).not
push_neg at this
exact this
lemma List.of_mem_filter' {α : Type _} {p : α→Prop} {a : α} {l : List α} :
a ∈ l.filter (fun x => decide (p x)) → p a := by
rw [Iff.intro (decide_eq_true (p:=p a)) of_decide_eq_true]
exact of_mem_filter
namespace MvPolynomial
section MvDiv
variable {σ : Type _} {s : σ →₀ ℕ} {k : Type _} [Field k]
variable [term_order_class: TermOrderClass (TermOrder (σ→₀ℕ))]
variable (p: MvPolynomial σ k) (G: List (MvPolynomial σ k))
set_option synthInstance.maxHeartbeats 40000
noncomputable def step:
(G.toFinset →₀ MvPolynomial σ k) × MvPolynomial σ k :=
if p = 0 then ⟨0, 0⟩ else
if hgs: ∃ g ∈ G, leading_term g∣ leading_term p
then
let gs := G.filter (leading_term ·∣ leading_term p)
let g := gs.head!
have gnonempty: gs ≠ [] := List.filter_ne_nil'.mpr hgs
have hG: g ∈ G := List.mem_of_mem_filter (gs.head!_mem_self gnonempty)
have hg: leading_term g ∣ leading_term p :=
List.of_mem_filter' (a:=g) (gs.head!_mem_self gnonempty) -- a is needed
⟨Finsupp.single ⟨g, List.mem_toFinset.mpr hG⟩ hg.choose, p - g * hg.choose⟩
else ⟨0, p⟩
def step_apply := by
have key: p.step = p.step := rfl
conv at key => rhs; unfold step
exact key
@[simp]
lemma step_multideg_le:
(p.step G).2.multideg ≤ p.multideg := by
unfold step
by_cases hp : p = 0
· -- simp? [hp]
simp only [hp, zero_sub, ite_true, multideg_zero, le_refl]
·
by_cases hgs: ∃ g ∈ G, leading_term g∣ leading_term p
·-- simp? [hp, hgs]
simp only [hp, ne_eq, hgs, dite_true, ite_false]
rw [sub_eq_add_neg, neg_eq_neg_one_mul]
apply multideg_add_le_left
rw [←multideg_leading_term, leading_term_mul, leading_term_mul'_right]
generalize_proofs _ h
rw [←Exists.choose_spec h]
-- simp?
simp only [leading_term_neg, leading_term_1, leading_term_leading_term,
neg_mul, one_mul, multideg_neg, multideg_leading_term, le_refl]
-- simp only [lt_neg, leading_term_1, leading_term_leading_term, one_mul, multideg_leading_term, ne_eq, le_refl]
·-- simp? [hgs, hp]
simp only [hp, ne_eq, hgs, dite_false, ite_false, le_refl]
@[simp]
lemma step_sub_multideg''_lt
{p: MvPolynomial σ k} (G: List (MvPolynomial σ k)) (hp: p ≠ 0):
multideg'' ((p.step G).2 - coeff p.multideg (p.step G).2•lm p) < multideg'' p
:= sub_multideg''_lt hp (step_multideg_le p G)
@[simp]
lemma step_zero: (0: MvPolynomial σ k).step G = ⟨0, 0⟩ := by
rw [step_apply]
simp only [zero_sub, ite_true]
lemma ne_zero_of_step_quo_ne_zero (hp : (p.step G).2 ≠ 0) : p ≠ 0 := by
by_contra h
simp [h] at hp
@[simp]
lemma step_sub_multideg_le
{p: MvPolynomial σ k} (G: List (MvPolynomial σ k)):
multideg ((p.step G).2- coeff p.multideg (p.step G).2 • lm p) ≤ multideg p
:= sub_multideg_le (step_multideg_le p G)
noncomputable def mv_div
(p: MvPolynomial σ k)
(G: List (MvPolynomial σ k))
: (G.toFinset →₀ MvPolynomial σ k) × MvPolynomial σ k
:=
if hp: p = 0 then ⟨0, p⟩ else
let xs_r := step p G
let xs'_r' := mv_div (xs_r.2 - coeff p.multideg xs_r.2 • lm p) G
⟨xs_r.1 + xs'_r'.1, xs'_r'.2 + coeff p.multideg xs_r.2 • lm p⟩
termination_by _ => multideg'' p
decreasing_by exact step_sub_multideg''_lt G hp
@[reducible]
noncomputable def quo: G.toFinset →₀ MvPolynomial σ k :=
(mv_div p G).1
@[reducible]
noncomputable def rem: MvPolynomial σ k :=
(mv_div p G).2
def mv_div_apply := by
have key: p.mv_div G = p.mv_div G := rfl
nth_rewrite 2 [mv_div] at key
exact key
lemma quo_apply: quo p G = (mv_div p G).1 := rfl
lemma rem_apply: rem p G = (mv_div p G).2 := rfl
@[simp]
lemma step_empty: step p [] = ⟨0, p⟩ := by
rw [step_apply]
-- simp?
simp only [ne_eq, List.find?, List.not_mem_nil, false_and, exists_false,
List.filter_nil, dite_false,
ite_eq_right_iff, Prod.mk.injEq, true_and]
intro hp
exact hp.symm
@[simp]
lemma mv_div_empty (p: MvPolynomial σ k): p.mv_div [] = ⟨0, p⟩ := by
rw [mv_div_apply]
-- simp?
simp only [ne_eq, step_empty, zero_add, dite_eq_ite,
ite_eq_left_iff, Prod.mk.injEq]
intro hp
rw [mv_div_empty (p - coeff (multideg p) p • lm p)]
-- simp?
simp only [ne_eq, sub_add_cancel, and_self]
termination_by mv_div_empty p => multideg'' p
decreasing_by
-- ↓↓↓ Workaround to make lean remember the instance ↓↓↓
let term_order_class := term_order_class -- Why should I do that??????
exact sub_multideg''_lt hp le_rfl
@[simp]
lemma rem_empty (p: MvPolynomial σ k): rem p [] = p := by
-- simp? [rem_apply]
simp only [rem_apply, mv_div_empty]
@[simp]
lemma quo_empty (p: MvPolynomial σ k): quo p [] = 0 := by
simp only [quo_apply, mv_div_empty]
@[simp]
lemma mv_div_zero: (0: MvPolynomial σ k).mv_div G = ⟨0, 0⟩ := by
rw [mv_div_apply]
simp only [step_zero, multideg_zero, coeff_zero, zero_smul, sub_self,
zero_add, add_zero, Prod.mk.eta, dite_eq_ite, ite_true]
@[simp]
lemma quo_zero: quo (0: MvPolynomial σ k) G = 0 := by
simp only [quo_apply, mv_div_zero]
@[simp]
lemma rem_zero: rem (0: MvPolynomial σ k) G = 0 := by
simp only [rem_apply, mv_div_zero]
lemma step_quo_sum_add_rem: (p.step G).1.sum (·*·) + (p.step G).2 = p := by
unfold step
by_cases hp: p = 0
· --simp? [hp]
simp only [hp, zero_sub, ite_true, add_zero]
rw [Finsupp.sum_zero_index]
-- simp only [hp, zero_sub, ite_true, Finsupp.sum_zero_index, add_zero]
·
by_cases hg: ∃ g, g ∈ G ∧ leading_term g ∣ leading_term p
·
-- simp? [hp, hg]
simp only [hp, ne_eq, hg, dite_true, ite_false, mul_zero,
Finsupp.sum_single_index, add_sub_cancel'_right]
·
--simp? [hp, hg]
simp only [hp, ne_eq, hg, dite_false, ite_false, zero_add]
rw [Finsupp.sum_zero_index, zero_add]
@[simp]
lemma step_quo_sum_eq_sub_rem: (p.step G).1.sum (·*·) = p - (p.step G).2 := by
rw [sub_eq_of_eq_add]
exact (step_quo_sum_add_rem p G).symm
@[simp]
lemma quo_sum_eq_sub_rem (p: MvPolynomial σ k) (G: List (MvPolynomial σ k)):
(p.quo G).sum (·*·) = p - p.rem G := by
rw [quo_apply, rem_apply, mv_div_apply]
by_cases hp: p = 0
·-- simp? [hp]
simp only [hp, multideg_zero, dite_eq_ite, ite_true,
Finsupp.sum_zero_index, sub_self]
·-- simp? [hp]
simp only [hp, ne_eq, dite_eq_ite, ite_false]
rw [Finsupp.sum_add_index]
·
-- rw [←quo_apply]
rw [quo_sum_eq_sub_rem
((step p G).snd - coeff (multideg p) (step p G).snd • lm p)
G,
step_quo_sum_eq_sub_rem]
ring
·-- simp?
simp only [ne_eq, Finset.mem_union, Finsupp.mem_support_iff, mul_zero,
implies_true, Subtype.forall, forall_const]
· intros a _ b₁ b₂
exact mul_add (↑a: MvPolynomial σ k) b₁ b₂
termination_by _ => multideg'' p
decreasing_by exact step_sub_multideg''_lt G hp
theorem quo_quo_sum_add_rem: (p.quo G).sum (·*·) + p.rem G = p := by
-- simp?
simp only [quo_sum_eq_sub_rem, sub_add_cancel]
@[simp]
lemma step_multideg''_quo_mul_le (hq: q ∈ G) :
(q * ((p.step G).1 ⟨q, List.mem_toFinset.mpr hq⟩)).multideg'' ≤ p.multideg'' :=
by
rw [step_apply]
by_cases hp: p = 0
·simp [hp]
·
-- simp? [hp]
simp only [hp, ne_eq, ite_false, mul_eq_zero]
by_cases hg: ∃ g, g ∈ G ∧ leading_term g ∣ leading_term p
·-- simp? [hg, Finsupp.single_apply]
simp only [hg, dite_true, ne_eq, Subtype.mk.injEq, Finsupp.single_apply,
mul_ite, mul_zero, ite_eq_right_iff, mul_eq_zero]
let g := List.head!
(G.filter (fun x => decide (leading_term x ∣ leading_term p)))
by_cases hg : g = q
·
-- simp? [hg]
simp only [hg, ite_true, mul_eq_zero, ne_eq, ge_iff_le]
generalize_proofs _ h
rw [←multideg''_leading_term, leading_term_mul'_right,
multideg''_leading_term, ←h.choose_spec, multideg''_leading_term]
·simp [hg]
·simp [hg]
theorem multideg''_quo_mul_le (p : MvPolynomial σ k) (G : List (MvPolynomial σ k))
(q: MvPolynomial σ k) (hq: q ∈ G):
(q * (p.quo G ⟨q, List.mem_toFinset.mpr hq⟩)).multideg'' ≤ p.multideg'' :=
by
rw [quo_apply]
unfold mv_div
by_cases hp: p = 0
·
-- simp? [hp]
simp only [hp, multideg''_zero, dite_eq_ite, ite_true,
Finsupp.coe_zero, Pi.zero_apply, mul_zero, le_refl]
·
-- simp? [hp]
simp only [hp, ne_eq, dite_eq_ite, ite_false, Finsupp.coe_add,
Pi.add_apply, mul_eq_zero]
rw [mul_add]
apply le_trans multideg''_add_le
-- simp? [step_multideg''_quo_mul_le p G hq]
simp only [mul_eq_zero, ne_eq, ge_iff_le, max_le_iff,
step_multideg''_quo_mul_le p G hq, true_and]
-- Why step_multideg_quo_mul_le not work?
rw [←quo_apply]
apply le_trans (multideg''_quo_mul_le
((p.step G).2 - (p.step G).2.coeff (multideg p) • lm p)
G q hq)
exact le_of_lt (step_sub_multideg''_lt G hp)
termination_by _ => multideg'' p
decreasing_by exact step_sub_multideg''_lt G hp
lemma step_eq_iff (hp : p ≠ 0):
(p.step G).2 = p ↔
∀ g ∈ G, g ≠ 0 → ¬ LE.le (α:=σ→₀ℕ) g.multideg p.multideg := by
constructor
·
intros h g hg hg'
-- simp? [step_apply, hp] at h
simp only [step_apply, ne_eq, multideg_eq_zero_iff,
not_exists, hp, ite_false] at h
by_cases hg's : ∃ g, g ∈ G ∧ leading_term g ∣ leading_term p
·
simp [hg's] at h
-- simp [step_quo_ne_zero'' p G hp hg's] at h
-- simp [step_quo_dvd_choose p G hp hg's] at h
-- simp only [hg's, dite_true, sub_eq_self, mul_eq_zero,
-- ne_eq, multideg_eq_zero_iff, not_exists] at h
let g's := G.filter (leading_term ·∣ leading_term p)
have gnonempty: g's ≠ [] := List.filter_ne_nil'.mpr (hg's)
have := List.of_mem_filter (List.head!_mem_self gnonempty)
have key := of_decide_eq_true this
cases' h with h h
· exfalso
rw [h] at key
simp [hp, monomial_dvd_monomial, leading_term_def] at key
· exfalso
have key := h ▸ key.choose_spec
simp [hp] at key
·
push_neg at hg's
specialize hg's g hg
simp [leading_term_def, hg', hp, monomial_dvd_monomial] at hg's
rw [imp_iff_not_or] at hg's
cases' hg's with hg's hg's
·exact hg's
· exfalso
rw [dvd_iff_exists_eq_mul_left, not_exists] at hg's
apply hg's (p.leading_coeff / g.leading_coeff)
rw [div_mul_cancel _ (g.leading_coeff_eq_zero_iff.not.mpr hg')]
·
intros hg
simp [step_apply, hp]
by_cases hg's : ∃ g, g ∈ G ∧ leading_term g ∣ leading_term p
· exfalso
rcases hg's with ⟨g',hg',hg'p⟩
simp [monomial_dvd_monomial, leading_term_def, hp] at hg'p
simp [(not_imp_not.mp (hg g' hg')) hg'p.1, hp] at hg'p
·
simp [hg's, hg]
lemma step_multideg_eq_iff_eq (h : (p.step G).2 ≠ 0):
(p.step G).2.multideg = p.multideg ↔ (p.step G).2 = p := by
have hp := ne_zero_of_step_quo_ne_zero p G h
simp [step, hp, h]
by_cases hg : ∃ g, g ∈ G ∧ leading_term g ∣ leading_term p
·
simp [hg, hp, step_apply] at h
simp [hg]
generalize_proofs _ hdvd
have hp' : p.leading_term ≠ 0 := p.leading_term_eq_zero_iff.not.mpr hp
rw [hdvd.choose_spec] at hp'
simp at hp'
simp [hp']
let g := List.head! (G.filter (leading_term · ∣ leading_term p))
have : p.leading_term = leading_term (g * hdvd.choose) := by
rw [leading_term_mul'_right, ←hdvd.choose_spec]
rw [leading_term_leading_term]
refine ne_of_lt ((multideg_sub_lt_left_iff ?_ h).mpr this)
apply le_of_eq
nth_rewrite 1 [leading_term_def] at this -- Why rw not work?
nth_rewrite 1 [leading_term_def] at this
rw [monomial_eq_monomial_iff] at this
simp [hp] at this
exact this.1.symm
·
simp [hg]
lemma step_multideg''_eq_iff :
(p.step G).2.multideg'' = p.multideg'' ↔ (p.step G).2 = p := by
by_cases hstep : (p.step G).2 = 0
·
by_cases hp : p = 0
· simp [multideg''_def, hp, hstep]
· simp [multideg''_def, hp, hstep]
rw [(⟨symm, symm⟩ : Iff (p=0) (0=p))] at hp
exact hp
·
simp [multideg''_def, hstep, ne_zero_of_step_quo_ne_zero p G hstep]
exact step_multideg_eq_iff_eq p G hstep
lemma step_multideg_eq_iff (h : (p.step G).2 ≠ 0):
(p.step G).2.multideg = p.multideg ↔
∀ g ∈ G, g ≠ 0 → ¬ LE.le (α:=σ→₀ℕ) g.multideg p.multideg :=
Trans.trans
(step_multideg_eq_iff_eq p G h)
(step_eq_iff p G (ne_zero_of_step_quo_ne_zero p G h))
theorem rem_support (p : MvPolynomial σ k) (G : List (MvPolynomial σ k))
{g : MvPolynomial σ k} (h : g ∈ G) (h' : g ≠ 0):
∀ s ∈ (p.rem G).support, ¬ LE.le (α:=σ→₀ℕ) g.multideg s := by
intros s hs
-- rw [rem_apply, mv_div_apply] at hs'
by_cases hp : p = 0
·
-- why simp [hp] at hs not work??
rw [hp, rem_zero, support_zero] at hs
exfalso
exact Finset.not_mem_empty s hs
·
-- workaround
rw [rem_apply, mv_div_apply] at hs
-- simp? only [hp] at hs
-- simp? only [←rem_apply] at hs
-- simp? at hs
-- simp? [hp, ←rem_apply, -coeff_smul] at hs
simp only [ne_eq, multideg_eq_zero_iff, not_exists, hp, rem_apply,
dite_eq_ite, ite_false, mem_support_iff, coeff_add] at hs
-- simp? only [coeff_smul] at hs
-- set_option synthInstance.etaExperiment true in
rw [coeff_smul] at hs
-- simp? at hs
rw [smul_eq_mul] at hs
have hnext := rem_support
((step p G).snd - (p.step G).2.coeff p.multideg • p.lm)
G h h' s
-- simp? at hnext
simp only [ne_eq, mem_support_iff, not_imp_not] at hnext
by_contra tmp
simp only [ne_eq, hp, hnext tmp , zero_add, mul_eq_zero] at hs
push_neg at hs
cases' hs with hs₁ hs₂
-- simp? [hp, lm, multideg'_eq_multideg hp] at hs₂
simp only [lm, ne_eq, hp, not_false_eq_true, multideg'_eq_multideg hp,
dite_eq_ite, ite_true, coeff_monomial, ite_eq_right_iff,
one_ne_zero, not_forall, exists_prop, and_true] at hs₂
have : (step p G).snd ≠ 0 := by
by_contra h
simp [h] at hs₁
apply not_not.mpr tmp
refine hs₂ ▸ (step_multideg_eq_iff p G this).mp ?_ g h h'
exact eq_of_le_of_not_lt
(step_multideg_le p G)
(not_lt_of_le (le_multideg_of_coeff_ne_zero hs₁))
termination_by _ => multideg'' p
decreasing_by exact step_sub_multideg''_lt G hp
variable (G': Finset (MvPolynomial σ k))
def is_rem
(r: MvPolynomial σ k)
:= (∀ g ∈ G', g ≠ 0 → ∀ s ∈ r.support, ¬LE.le (α:=σ→₀ℕ) g.multideg s) ∧
∃(q : G' →₀ MvPolynomial σ k),
(∀(g: MvPolynomial σ k) (hg : g ∈ G'),
(g * q ⟨g, hg⟩).multideg'' ≤ p.multideg'' )∧
p = q.sum (·*·) + r
theorem rem_is_rem : is_rem p G.toFinset (p.rem G) := by
constructor
·
intro g hg
exact rem_support p G (List.mem_toFinset.mp hg)
·
use p.quo G
constructor
·
intros g hg
exact multideg''_quo_mul_le p G g (List.mem_toFinset.mp hg)
·
exact (quo_quo_sum_add_rem p G).symm
lemma rem_is_rem' : is_rem p G' (p.rem G'.toList) := by
have := rem_is_rem p G'.toList
simp at this
exact this
theorem exists_rem : ∃ r : MvPolynomial σ k, is_rem p G' r := by
use p.rem G'.toList
simp only [rem_is_rem']
end MvDiv
end MvPolynomial