[Merged by Bors] - feat: port Data.MvPolynomial.Derivation

Mathlib.lean

@@ -1275,6 +1275,7 @@ import Mathlib.Data.MvPolynomial.Cardinal
 import Mathlib.Data.MvPolynomial.Comap
 import Mathlib.Data.MvPolynomial.CommRing
 import Mathlib.Data.MvPolynomial.Counit
+import Mathlib.Data.MvPolynomial.Derivation
 import Mathlib.Data.MvPolynomial.Division
 import Mathlib.Data.MvPolynomial.Equiv
 import Mathlib.Data.MvPolynomial.Expand

Mathlib/Data/MvPolynomial/Derivation.lean

@@ -0,0 +1,173 @@
+/-
+Copyright (c) 2021 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+
+! This file was ported from Lean 3 source module data.mv_polynomial.derivation
+! leanprover-community/mathlib commit b608348ffaeb7f557f2fd46876037abafd326ff3
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.MvPolynomial.Supported
+import Mathlib.RingTheory.Derivation.Basic
+
+/-!
+# Derivations of multivariate polynomials
+
+In this file we prove that a derivation of `MvPolynomial σ R` is determined by its values on all
+monomials `MvPolynomial.X i`. We also provide a constructor `MvPolynomial.mkDerivation` that
+builds a derivation from its values on `X i`s and a linear equivalence
+`MvPolynomial.equivDerivation` between `σ → A` and `Derivation (MvPolynomial σ R) A`.
+-/
+
+
+namespace MvPolynomial
+
+open scoped BigOperators
+
+noncomputable section
+
+variable {σ R A : Type _} [CommSemiring R] [AddCommMonoid A] [Module R A]
+  [Module (MvPolynomial σ R) A]
+
+section
+
+variable (R)
+
+/-- The derivation on `MvPolynomial σ R` that takes value `f i` on `X i`, as a linear map.
+Use `MvPolynomial.mkDerivation` instead. -/
+def mkDerivationₗ (f : σ → A) : MvPolynomial σ R →ₗ[R] A :=
+  Finsupp.lsum R fun xs : σ →₀ ℕ =>
+    (LinearMap.ringLmapEquivSelf R R A).symm <|
+      xs.sum fun i k => monomial (xs - Finsupp.single i 1) (k : R) • f i
+#align mv_polynomial.mk_derivationₗ MvPolynomial.mkDerivationₗ
+
+end
+
+theorem mkDerivationₗ_monomial (f : σ → A) (s : σ →₀ ℕ) (r : R) :
+    mkDerivationₗ R f (monomial s r) =
+      r • s.sum fun i k => monomial (s - Finsupp.single i 1) (k : R) • f i :=
+  sum_monomial_eq <| LinearMap.map_zero _
+#align mv_polynomial.mk_derivationₗ_monomial MvPolynomial.mkDerivationₗ_monomial
+
+theorem mkDerivationₗ_C (f : σ → A) (r : R) : mkDerivationₗ R f (C r) = 0 :=
+  (mkDerivationₗ_monomial f _ _).trans (smul_zero _)
+set_option linter.uppercaseLean3 false in
+#align mv_polynomial.mk_derivationₗ_C MvPolynomial.mkDerivationₗ_C
+
+theorem mkDerivationₗ_X (f : σ → A) (i : σ) : mkDerivationₗ R f (X i) = f i :=
+  (mkDerivationₗ_monomial f _ _).trans <| by simp
+set_option linter.uppercaseLean3 false in
+#align mv_polynomial.mk_derivationₗ_X MvPolynomial.mkDerivationₗ_X
+
+@[simp]
+theorem derivation_C (D : Derivation R (MvPolynomial σ R) A) (a : R) : D (C a) = 0 :=
+  D.map_algebraMap a
+set_option linter.uppercaseLean3 false in
+#align mv_polynomial.derivation_C MvPolynomial.derivation_C
+
+-- @[simp] -- Porting note: simp normal form is `derivation_C_mul'`
+theorem derivation_C_mul (D : Derivation R (MvPolynomial σ R) A) (a : R) (f : MvPolynomial σ R) :
+    D (C a * f) = a • D f := by rw [C_mul', D.map_smul]
+set_option linter.uppercaseLean3 false in
+#align mv_polynomial.derivation_C_mul MvPolynomial.derivation_C_mul
+
+@[simp]
+theorem derivation_C_mul' (D : Derivation R (MvPolynomial σ R) A) (a : R) (f : MvPolynomial σ R) :
+    C (σ := σ) a • D f = a • D f := by
+  have : C (σ := σ) a • D f = D (C a * f) := by simp
+  rw [this, derivation_C_mul]
+
+/-- If two derivations agree on `X i`, `i ∈ s`, then they agree on all polynomials from
+`MvPolynomial.supported R s`. -/
+theorem derivation_eqOn_supported {D₁ D₂ : Derivation R (MvPolynomial σ R) A} {s : Set σ}
+    (h : Set.EqOn (D₁ ∘ X) (D₂ ∘ X) s) {f : MvPolynomial σ R} (hf : f ∈ supported R s) :
+    D₁ f = D₂ f :=
+  Derivation.eqOn_adjoin (Set.ball_image_iff.2 h) hf
+#align mv_polynomial.derivation_eq_on_supported MvPolynomial.derivation_eqOn_supported
+
+theorem derivation_eq_of_forall_mem_vars {D₁ D₂ : Derivation R (MvPolynomial σ R) A}
+    {f : MvPolynomial σ R} (h : ∀ i ∈ f.vars, D₁ (X i) = D₂ (X i)) : D₁ f = D₂ f :=
+  derivation_eqOn_supported h f.mem_supported_vars
+#align mv_polynomial.derivation_eq_of_forall_mem_vars MvPolynomial.derivation_eq_of_forall_mem_vars
+
+theorem derivation_eq_zero_of_forall_mem_vars {D : Derivation R (MvPolynomial σ R) A}
+    {f : MvPolynomial σ R} (h : ∀ i ∈ f.vars, D (X i) = 0) : D f = 0 :=
+  show D f = (0 : Derivation R (MvPolynomial σ R) A) f from derivation_eq_of_forall_mem_vars h
+#align mv_polynomial.derivation_eq_zero_of_forall_mem_vars MvPolynomial.derivation_eq_zero_of_forall_mem_vars
+
+@[ext]
+theorem derivation_ext {D₁ D₂ : Derivation R (MvPolynomial σ R) A} (h : ∀ i, D₁ (X i) = D₂ (X i)) :
+    D₁ = D₂ :=
+  Derivation.ext fun _ => derivation_eq_of_forall_mem_vars fun i _ => h i
+#align mv_polynomial.derivation_ext MvPolynomial.derivation_ext
+
+variable [IsScalarTower R (MvPolynomial σ R) A]
+
+theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
+    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
+    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
+  refine' ⟨fun H p i => H _ _, fun H => _⟩
+  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
+  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
+    intro p i
+    induction' p using MvPolynomial.induction_on' with s r p q hp hq
+    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
+        smul_add, D.map_smul, smul_comm r (X i)]
+    · rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
+  intro p q
+  induction q using MvPolynomial.induction_on with
+  | h_C c =>
+    rw [mul_comm, C_mul', hC, smul_zero, zero_add, D.map_smul, C_eq_smul_one, smul_one_smul]
+  | h_add q₁ q₂ h₁ h₂ => simp only [mul_add, map_add, h₁, h₂, smul_add, add_smul]; abel
+  | h_X q i hq =>
+    simp only [this, ← mul_assoc, hq, mul_smul, smul_add, add_assoc]
+    rw [smul_comm (X i), smul_comm (X i)]
+set_option linter.uppercaseLean3 false in
+#align mv_polynomial.leibniz_iff_X MvPolynomial.leibniz_iff_X
+
+variable (R)
+
+/-- The derivation on `MvPolynomial σ R` that takes value `f i` on `X i`. -/
+def mkDerivation (f : σ → A) : Derivation R (MvPolynomial σ R) A where
+  toLinearMap := mkDerivationₗ R f
+  map_one_eq_zero' := mkDerivationₗ_C _ 1
+  leibniz' :=
+    (leibniz_iff_X (mkDerivationₗ R f) (mkDerivationₗ_C _ 1)).2 fun s i => by
+      simp only [mkDerivationₗ_monomial, X, monomial_mul, one_smul, one_mul]
+      rw [Finsupp.sum_add_index'] <;>
+        [skip; simp; (intros; simp only [Nat.cast_add, (monomial _).map_add, add_smul])]
+      rw [Finsupp.sum_single_index, Finsupp.sum_single_index] <;> [skip; simp; simp]
+      rw [tsub_self, add_tsub_cancel_right, Nat.cast_one, ← C_apply, C_1, one_smul, add_comm,
+        Finsupp.smul_sum]
+      refine' congr_arg₂ (· + ·) rfl (Finset.sum_congr rfl fun j hj => _); dsimp only
+      rw [smul_smul, monomial_mul, one_mul, add_comm s, add_tsub_assoc_of_le]
+      rwa [Finsupp.single_le_iff, Nat.succ_le_iff, pos_iff_ne_zero, ← Finsupp.mem_support_iff]
+#align mv_polynomial.mk_derivation MvPolynomial.mkDerivation
+
+@[simp]
+theorem mkDerivation_X (f : σ → A) (i : σ) : mkDerivation R f (X i) = f i :=
+  mkDerivationₗ_X f i
+set_option linter.uppercaseLean3 false in
+#align mv_polynomial.mk_derivation_X MvPolynomial.mkDerivation_X
+
+theorem mkDerivation_monomial (f : σ → A) (s : σ →₀ ℕ) (r : R) :
+    mkDerivation R f (monomial s r) =
+      r • s.sum fun i k => monomial (s - Finsupp.single i 1) (k : R) • f i :=
+  mkDerivationₗ_monomial f s r
+#align mv_polynomial.mk_derivation_monomial MvPolynomial.mkDerivation_monomial
+
+/-- `MvPolynomial.mkDerivation` as a linear equivalence. -/
+def mkDerivationEquiv : (σ → A) ≃ₗ[R] Derivation R (MvPolynomial σ R) A :=
+  LinearEquiv.symm <|
+    { invFun := mkDerivation R
+      toFun := fun D i => D (X i)
+      map_add' := fun _ _ => rfl
+      map_smul' := fun _ _ => rfl
+      left_inv := fun _ => derivation_ext <| mkDerivation_X _ _
+      right_inv := fun _ => funext <| mkDerivation_X _ _ }
+#align mv_polynomial.mk_derivation_equiv MvPolynomial.mkDerivationEquiv
+
+end
+
+end MvPolynomial