feat: port Data.Polynomial.Expand (#4083)

Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com>

Mathlib.lean

@@ -1197,6 +1197,7 @@ import Mathlib.Data.Polynomial.Derivative
 import Mathlib.Data.Polynomial.Div
 import Mathlib.Data.Polynomial.EraseLead
 import Mathlib.Data.Polynomial.Eval
+import Mathlib.Data.Polynomial.Expand
 import Mathlib.Data.Polynomial.FieldDivision
 import Mathlib.Data.Polynomial.HasseDeriv
 import Mathlib.Data.Polynomial.Identities

Mathlib/Data/Polynomial/Expand.lean

@@ -0,0 +1,299 @@
+/-
+Copyright (c) 2020 Kenny Lau. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kenny Lau
+
+! This file was ported from Lean 3 source module data.polynomial.expand
+! leanprover-community/mathlib commit bbeb185db4ccee8ed07dc48449414ebfa39cb821
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.RingTheory.Polynomial.Basic
+import Mathlib.RingTheory.Ideal.LocalRing
+
+/-!
+# Expand a polynomial by a factor of p, so `∑ aₙ xⁿ` becomes `∑ aₙ xⁿᵖ`.
+
+## Main definitions
+
+* `Polynomial.expand R p f`: expand the polynomial `f` with coefficients in a
+  commutative semiring `R` by a factor of p, so `expand R p (∑ aₙ xⁿ)` is `∑ aₙ xⁿᵖ`.
+* `Polynomial.contract p f`: the opposite of `expand`, so it sends `∑ aₙ xⁿᵖ` to `∑ aₙ xⁿ`.
+
+-/
+
+
+universe u v w
+
+open Classical BigOperators Polynomial
+
+open Finset
+
+namespace Polynomial
+
+section CommSemiring
+
+variable (R : Type u) [CommSemiring R] {S : Type v} [CommSemiring S] (p q : ℕ)
+
+/-- Expand the polynomial by a factor of p, so `∑ aₙ xⁿ` becomes `∑ aₙ xⁿᵖ`. -/
+noncomputable def expand : R[X] →ₐ[R] R[X] :=
+  { (eval₂RingHom C (X ^ p) : R[X] →+* R[X]) with commutes' := fun _ => eval₂_C _ _ }
+#align polynomial.expand Polynomial.expand
+
+theorem coe_expand : (expand R p : R[X] → R[X]) = eval₂ C (X ^ p) :=
+  rfl
+#align polynomial.coe_expand Polynomial.coe_expand
+
+variable {R}
+
+theorem expand_eq_sum {f : R[X]} : expand R p f = f.sum fun e a => C a * (X ^ p) ^ e := by
+  simp [expand, eval₂]
+#align polynomial.expand_eq_sum Polynomial.expand_eq_sum
+
+@[simp]
+theorem expand_C (r : R) : expand R p (C r) = C r :=
+  eval₂_C _ _
+set_option linter.uppercaseLean3 false in
+#align polynomial.expand_C Polynomial.expand_C
+
+@[simp]
+theorem expand_X : expand R p X = X ^ p :=
+  eval₂_X _ _
+set_option linter.uppercaseLean3 false in
+#align polynomial.expand_X Polynomial.expand_X
+
+@[simp]
+theorem expand_monomial (r : R) : expand R p (monomial q r) = monomial (q * p) r := by
+  simp_rw [← smul_X_eq_monomial, AlgHom.map_smul, AlgHom.map_pow, expand_X, mul_comm, pow_mul]
+#align polynomial.expand_monomial Polynomial.expand_monomial
+
+theorem expand_expand (f : R[X]) : expand R p (expand R q f) = expand R (p * q) f :=
+  Polynomial.induction_on f (fun r => by simp_rw [expand_C])
+    (fun f g ihf ihg => by simp_rw [AlgHom.map_add, ihf, ihg]) fun n r _ => by
+    simp_rw [AlgHom.map_mul, expand_C, AlgHom.map_pow, expand_X, AlgHom.map_pow, expand_X, pow_mul]
+#align polynomial.expand_expand Polynomial.expand_expand
+
+theorem expand_mul (f : R[X]) : expand R (p * q) f = expand R p (expand R q f) :=
+  (expand_expand p q f).symm
+#align polynomial.expand_mul Polynomial.expand_mul
+
+@[simp]
+theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by simp [expand]
+#align polynomial.expand_zero Polynomial.expand_zero
+
+@[simp]
+theorem expand_one (f : R[X]) : expand R 1 f = f :=
+  Polynomial.induction_on f (fun r => by rw [expand_C])
+    (fun f g ihf ihg => by rw [AlgHom.map_add, ihf, ihg]) fun n r _ => by
+    rw [AlgHom.map_mul, expand_C, AlgHom.map_pow, expand_X, pow_one]
+#align polynomial.expand_one Polynomial.expand_one
+
+theorem expand_pow (f : R[X]) : expand R (p ^ q) f = (expand R p^[q]) f :=
+  Nat.recOn q (by rw [pow_zero, expand_one, Function.iterate_zero, id]) fun n ih => by
+    rw [Function.iterate_succ_apply', pow_succ, expand_mul, ih]
+#align polynomial.expand_pow Polynomial.expand_pow
+
+theorem derivative_expand (f : R[X]) : Polynomial.derivative (expand R p f) =
+    expand R p (Polynomial.derivative f) * (p * (X ^ (p - 1) : R[X])) := by
+  rw [coe_expand, derivative_eval₂_C, derivative_pow, C_eq_nat_cast, derivative_X, mul_one]
+#align polynomial.derivative_expand Polynomial.derivative_expand
+
+theorem coeff_expand {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
+    (expand R p f).coeff n = if p ∣ n then f.coeff (n / p) else 0 := by
+  simp only [expand_eq_sum]
+  simp_rw [coeff_sum, ← pow_mul, C_mul_X_pow_eq_monomial, coeff_monomial, sum]
+  split_ifs with h
+  · rw [Finset.sum_eq_single (n / p), Nat.mul_div_cancel' h, if_pos rfl]
+    · intro b _ hb2
+      rw [if_neg]
+      intro hb3
+      apply hb2
+      rw [← hb3, Nat.mul_div_cancel_left b hp]
+    · intro hn
+      rw [not_mem_support_iff.1 hn]
+      split_ifs <;> rfl
+  · rw [Finset.sum_eq_zero]
+    intro k _
+    rw [if_neg]
+    exact fun hkn => h ⟨k, hkn.symm⟩
+#align polynomial.coeff_expand Polynomial.coeff_expand
+
+@[simp]
+theorem coeff_expand_mul {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
+    (expand R p f).coeff (n * p) = f.coeff n := by
+  rw [coeff_expand hp, if_pos (dvd_mul_left _ _), Nat.mul_div_cancel _ hp]
+#align polynomial.coeff_expand_mul Polynomial.coeff_expand_mul
+
+@[simp]
+theorem coeff_expand_mul' {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
+    (expand R p f).coeff (p * n) = f.coeff n := by rw [mul_comm, coeff_expand_mul hp]
+#align polynomial.coeff_expand_mul' Polynomial.coeff_expand_mul'
+
+/-- Expansion is injective. -/
+theorem expand_injective {n : ℕ} (hn : 0 < n) : Function.Injective (expand R n) := fun g g' H =>
+  ext fun k => by rw [← coeff_expand_mul hn, H, coeff_expand_mul hn]
+#align polynomial.expand_injective Polynomial.expand_injective
+
+theorem expand_inj {p : ℕ} (hp : 0 < p) {f g : R[X]} : expand R p f = expand R p g ↔ f = g :=
+  (expand_injective hp).eq_iff
+#align polynomial.expand_inj Polynomial.expand_inj
+
+theorem expand_eq_zero {p : ℕ} (hp : 0 < p) {f : R[X]} : expand R p f = 0 ↔ f = 0 :=
+  (expand_injective hp).eq_iff' (map_zero _)
+#align polynomial.expand_eq_zero Polynomial.expand_eq_zero
+
+theorem expand_ne_zero {p : ℕ} (hp : 0 < p) {f : R[X]} : expand R p f ≠ 0 ↔ f ≠ 0 :=
+  (expand_eq_zero hp).not
+#align polynomial.expand_ne_zero Polynomial.expand_ne_zero
+
+theorem expand_eq_C {p : ℕ} (hp : 0 < p) {f : R[X]} {r : R} : expand R p f = C r ↔ f = C r := by
+  rw [← expand_C, expand_inj hp, expand_C]
+set_option linter.uppercaseLean3 false in
+#align polynomial.expand_eq_C Polynomial.expand_eq_C
+
+theorem natDegree_expand (p : ℕ) (f : R[X]) : (expand R p f).natDegree = f.natDegree * p := by
+  cases' p.eq_zero_or_pos with hp hp
+  · rw [hp, coe_expand, pow_zero, MulZeroClass.mul_zero, ← C_1, eval₂_hom, natDegree_C]
+  by_cases hf : f = 0
+  · rw [hf, AlgHom.map_zero, natDegree_zero, MulZeroClass.zero_mul]
+  have hf1 : expand R p f ≠ 0 := mt (expand_eq_zero hp).1 hf
+  rw [← WithBot.coe_eq_coe]
+  convert (degree_eq_natDegree hf1).symm -- Porting note: was `rw [degree_eq_natDegree hf1]`
+  symm
+  refine' le_antisymm ((degree_le_iff_coeff_zero _ _).2 fun n hn => _) _
+  · rw [coeff_expand hp]
+    split_ifs with hpn
+    · rw [coeff_eq_zero_of_natDegree_lt]
+      contrapose! hn
+      erw [WithBot.coe_le_coe, ← Nat.div_mul_cancel hpn]
+      exact Nat.mul_le_mul_right p hn
+    · rfl
+  · refine' le_degree_of_ne_zero _
+    erw [coeff_expand_mul hp, ← leadingCoeff]
+    exact mt leadingCoeff_eq_zero.1 hf
+#align polynomial.nat_degree_expand Polynomial.natDegree_expand
+
+theorem Monic.expand {p : ℕ} {f : R[X]} (hp : 0 < p) (h : f.Monic) : (expand R p f).Monic := by
+  rw [Monic.def, Polynomial.leadingCoeff, natDegree_expand, coeff_expand hp]
+  simp [hp, h]
+#align polynomial.monic.expand Polynomial.Monic.expand
+
+theorem map_expand {p : ℕ} {f : R →+* S} {q : R[X]} :
+    map f (expand R p q) = expand S p (map f q) := by
+  by_cases hp : p = 0
+  · simp [hp]
+  ext
+  rw [coeff_map, coeff_expand (Nat.pos_of_ne_zero hp), coeff_expand (Nat.pos_of_ne_zero hp)]
+  split_ifs <;> simp_all
+#align polynomial.map_expand Polynomial.map_expand
+
+@[simp]
+theorem expand_eval (p : ℕ) (P : R[X]) (r : R) : eval r (expand R p P) = eval (r ^ p) P := by
+  refine' Polynomial.induction_on P (fun a => by simp) (fun f g hf hg => _) fun n a _ => by simp
+  rw [AlgHom.map_add, eval_add, eval_add, hf, hg]
+#align polynomial.expand_eval Polynomial.expand_eval
+
+@[simp]
+theorem expand_aeval {A : Type _} [Semiring A] [Algebra R A] (p : ℕ) (P : R[X]) (r : A) :
+    aeval r (expand R p P) = aeval (r ^ p) P := by
+  refine' Polynomial.induction_on P (fun a => by simp) (fun f g hf hg => _) fun n a _ => by simp
+  rw [AlgHom.map_add, aeval_add, aeval_add, hf, hg]
+#align polynomial.expand_aeval Polynomial.expand_aeval
+
+/-- The opposite of `expand`: sends `∑ aₙ xⁿᵖ` to `∑ aₙ xⁿ`. -/
+noncomputable def contract (p : ℕ) (f : R[X]) : R[X] :=
+  ∑ n in range (f.natDegree + 1), monomial n (f.coeff (n * p))
+#align polynomial.contract Polynomial.contract
+
+theorem coeff_contract {p : ℕ} (hp : p ≠ 0) (f : R[X]) (n : ℕ) :
+    (contract p f).coeff n = f.coeff (n * p) := by
+  simp only [contract, coeff_monomial, sum_ite_eq', finset_sum_coeff, mem_range, not_lt,
+    ite_eq_left_iff]
+  intro hn
+  apply (coeff_eq_zero_of_natDegree_lt _).symm
+  calc
+    f.natDegree < f.natDegree + 1 := Nat.lt_succ_self _
+    _ ≤ n * 1 := by simpa only [mul_one] using hn
+    _ ≤ n * p := mul_le_mul_of_nonneg_left (show 1 ≤ p from hp.bot_lt) (zero_le n)
+#align polynomial.coeff_contract Polynomial.coeff_contract
+
+theorem contract_expand {f : R[X]} (hp : p ≠ 0) : contract p (expand R p f) = f := by
+  ext
+  simp [coeff_contract hp, coeff_expand hp.bot_lt, Nat.mul_div_cancel _ hp.bot_lt]
+#align polynomial.contract_expand Polynomial.contract_expand
+
+section CharP
+
+variable [CharP R p]
+
+theorem expand_contract [NoZeroDivisors R] {f : R[X]} (hf : Polynomial.derivative f = 0)
+    (hp : p ≠ 0) : expand R p (contract p f) = f := by
+  ext n
+  rw [coeff_expand hp.bot_lt, coeff_contract hp]
+  split_ifs with h
+  · rw [Nat.div_mul_cancel h]
+  · cases' n with n
+    · exact absurd (dvd_zero p) h
+    have := coeff_derivative f n
+    rw [hf, coeff_zero, zero_eq_mul] at this
+    cases' this with h'
+    · rw [h']
+    rename_i _ _ _ _ h'
+    rw [← Nat.cast_succ, CharP.cast_eq_zero_iff R p] at h'
+    exact absurd h' h
+#align polynomial.expand_contract Polynomial.expand_contract
+
+variable [hp : Fact p.Prime]
+
+theorem expand_char (f : R[X]) : map (frobenius R p) (expand R p f) = f ^ p := by
+  refine' f.induction_on' (fun a b ha hb => _) fun n a => _
+  · rw [AlgHom.map_add, Polynomial.map_add, ha, hb, add_pow_char]
+  · rw [expand_monomial, map_monomial, ← C_mul_X_pow_eq_monomial, ← C_mul_X_pow_eq_monomial,
+      mul_pow, ← C.map_pow, frobenius_def]
+    ring
+#align polynomial.expand_char Polynomial.expand_char
+
+theorem map_expand_pow_char (f : R[X]) (n : ℕ) :
+    map (frobenius R p ^ n) (expand R (p ^ n) f) = f ^ p ^ n := by
+  induction' n with _ n_ih
+  · simp [RingHom.one_def]
+  symm
+  rw [pow_succ', pow_mul, ← n_ih, ← expand_char, pow_succ, RingHom.mul_def, ← map_map, mul_comm,
+    expand_mul, ← map_expand]
+#align polynomial.map_expand_pow_char Polynomial.map_expand_pow_char
+
+end CharP
+
+end CommSemiring
+
+section IsDomain
+
+variable (R : Type u) [CommRing R] [IsDomain R]
+
+theorem isLocalRingHom_expand {p : ℕ} (hp : 0 < p) :
+    IsLocalRingHom (↑(expand R p) : R[X] →+* R[X]) := by
+  refine' ⟨fun f hf1 => _⟩; norm_cast at hf1
+  have hf2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit hf1)
+  rw [coeff_expand hp, if_pos (dvd_zero _), p.zero_div] at hf2
+  rw [hf2, isUnit_C] at hf1; rw [expand_eq_C hp] at hf2; rwa [hf2, isUnit_C]
+#align polynomial.is_local_ring_hom_expand Polynomial.isLocalRingHom_expand
+
+variable {R}
+
+theorem of_irreducible_expand {p : ℕ} (hp : p ≠ 0) {f : R[X]} (hf : Irreducible (expand R p f)) :
+    Irreducible f :=
+  let _ := isLocalRingHom_expand R hp.bot_lt
+  of_irreducible_map (↑(expand R p)) hf
+#align polynomial.of_irreducible_expand Polynomial.of_irreducible_expand
+
+theorem of_irreducible_expand_pow {p : ℕ} (hp : p ≠ 0) {f : R[X]} {n : ℕ} :
+    Irreducible (expand R (p ^ n) f) → Irreducible f :=
+  Nat.recOn n (fun hf => by rwa [pow_zero, expand_one] at hf) fun n ih hf =>
+    ih <| of_irreducible_expand hp <| by
+      rw [pow_succ] at hf
+      rwa [expand_expand]
+#align polynomial.of_irreducible_expand_pow Polynomial.of_irreducible_expand_pow
+
+end IsDomain
+
+end Polynomial