state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hx :
x ∈
{ toAddSubmonoid := src✝.toAddSubmonoid,
smul_mem' :=
(_ :
∀ (c : R) {x : A}, x ∈ src✝.carrier → c • x ∈ sr... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | change ⁅x, y⁆ ∈ A' | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
| Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hx :
x ∈
{ toAddSubmonoid := src✝.toAddSubmonoid,
smul_mem' :=
(_ :
∀ (c : R) {x : A}, x ∈ src✝.carrier → c • x ∈ sr... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | change x ∈ A' at hx | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
cha... | Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hy :
y ∈
{ toAddSubmonoid := src✝.toAddSubmonoid,
smul_mem' :=
(_ :
∀ (c : R) {x : A}, x ∈ src✝.carrier → c • x ∈ sr... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | change y ∈ A' at hy | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
cha... | Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hx : x ∈ A'
hy : y ∈ A'
⊢ ⁅x, y⁆ ∈ A' | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | rw [LieRing.of_associative_ring_bracket] | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
cha... | Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hx : x ∈ A'
hy : y ∈ A'
⊢ x * y - y * x ∈ A' | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | have hxy := A'.mul_mem hx hy | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
cha... | Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hx : x ∈ A'
hy : y ∈ A'
hxy : x * y ∈ A'
⊢ x * y - y * x ∈ A' | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | have hyx := A'.mul_mem hy hx | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
cha... | Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
inst✝² : CommRing R
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
A' : Subalgebra R A
src✝ : Submodule R A := Subalgebra.toSubmodule A'
x y : A
hx : x ∈ A'
hy : y ∈ A'
hxy : x * y ∈ A'
hyx : y * x ∈ A'
⊢ x * y - y * x ∈ A' | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | exact Submodule.sub_mem (Subalgebra.toSubmodule A') hxy hyx | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A :=
{ Subalgebra.toSubmodule A' with
lie_mem' := fun {x y} hx hy => by
cha... | Mathlib.Algebra.Lie.OfAssociative.322_0.ll51mLev4p7Z1wP | /-- A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. -/
def lieSubalgebraOfSubalgebra (R : Type u) [CommRing R] (A : Type v) [Ring A] [Algebra R A]
(A' : Subalgebra R A) : LieSubalgebra R A | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
M₁ : Type v
M₂ : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M₁
inst✝² : Module R M₁
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
e : M₁ ≃ₗ[R] M₂
src✝ : Module.End R M₁ ≃ₗ[R] Module.End R M₂ := conj e
f g : Module.End R M₁
⊢ (conj e) ⁅f, g⁆ = ⁅(conj e) f, (conj e) g⁆ | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp only [LieRing.of_associative_ring_bracket, LinearMap.mul_eq_comp, e.conj_comp,
LinearEquiv.map_sub] | /-- A linear equivalence of two modules induces a Lie algebra equivalence of their endomorphisms. -/
def lieConj : Module.End R M₁ ≃ₗ⁅R⁆ Module.End R M₂ :=
{ e.conj with
map_lie' := fun {f g} =>
show e.conj ⁅f, g⁆ = ⁅e.conj f, e.conj g⁆ by
| Mathlib.Algebra.Lie.OfAssociative.342_0.ll51mLev4p7Z1wP | /-- A linear equivalence of two modules induces a Lie algebra equivalence of their endomorphisms. -/
def lieConj : Module.End R M₁ ≃ₗ⁅R⁆ Module.End R M₂ | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
A₁ : Type v
A₂ : Type w
inst✝⁴ : CommRing R
inst✝³ : Ring A₁
inst✝² : Ring A₂
inst✝¹ : Algebra R A₁
inst✝ : Algebra R A₂
e : A₁ ≃ₐ[R] A₂
src✝ : A₁ ≃ₗ[R] A₂ := toLinearEquiv e
x y : A₁
⊢ AddHom.toFun
{
toAddHom :=
{ toFun := e.toFun,
map_add' :=
(_ :
... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | have : e.toEquiv.toFun = e := rfl | /-- An equivalence of associative algebras is an equivalence of associated Lie algebras. -/
def toLieEquiv : A₁ ≃ₗ⁅R⁆ A₂ :=
{ e.toLinearEquiv with
toFun := e.toFun
map_lie' := fun {x y} => by
| Mathlib.Algebra.Lie.OfAssociative.371_0.ll51mLev4p7Z1wP | /-- An equivalence of associative algebras is an equivalence of associated Lie algebras. -/
def toLieEquiv : A₁ ≃ₗ⁅R⁆ A₂ | Mathlib_Algebra_Lie_OfAssociative |
R : Type u
A₁ : Type v
A₂ : Type w
inst✝⁴ : CommRing R
inst✝³ : Ring A₁
inst✝² : Ring A₂
inst✝¹ : Algebra R A₁
inst✝ : Algebra R A₂
e : A₁ ≃ₐ[R] A₂
src✝ : A₁ ≃ₗ[R] A₂ := toLinearEquiv e
x y : A₁
this : e.toFun = ⇑e
⊢ AddHom.toFun
{
toAddHom :=
{ toFun := e.toFun,
map_add' :=
... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | simp_rw [LieRing.of_associative_ring_bracket, this, map_sub, map_mul] | /-- An equivalence of associative algebras is an equivalence of associated Lie algebras. -/
def toLieEquiv : A₁ ≃ₗ⁅R⁆ A₂ :=
{ e.toLinearEquiv with
toFun := e.toFun
map_lie' := fun {x y} => by
have : e.toEquiv.toFun = e := rfl
| Mathlib.Algebra.Lie.OfAssociative.371_0.ll51mLev4p7Z1wP | /-- An equivalence of associative algebras is an equivalence of associated Lie algebras. -/
def toLieEquiv : A₁ ≃ₗ⁅R⁆ A₂ | Mathlib_Algebra_Lie_OfAssociative |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ T R (n + 2) = 2 * X * T R (n + 1) - T R n | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T] | @[simp]
theorem T_add_two (n : ℕ) : T R (n + 2) = 2 * X * T R (n + 1) - T R n := by | Mathlib.RingTheory.Polynomial.Chebyshev.81_0.SRy1jgYRAFbFJky | @[simp]
theorem T_add_two (n : ℕ) : T R (n + 2) = 2 * X * T R (n + 1) - T R n | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ T R 2 = 2 * X ^ 2 - 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T, sub_left_inj, sq, mul_assoc] | theorem T_two : T R 2 = 2 * X ^ 2 - 1 := by | Mathlib.RingTheory.Polynomial.Chebyshev.85_0.SRy1jgYRAFbFJky | theorem T_two : T R 2 = 2 * X ^ 2 - 1 | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h : 2 ≤ n
⊢ T R n = 2 * X * T R (n - 1) - T R (n - 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h | theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.88_0.SRy1jgYRAFbFJky | theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
case intro
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h : 2 ≤ 2 + n
⊢ T R (2 + n) = 2 * X * T R (2 + n - 1) - T R (2 + n - 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [add_comm] | theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
| Mathlib.RingTheory.Polynomial.Chebyshev.88_0.SRy1jgYRAFbFJky | theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
case intro
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h : 2 ≤ 2 + n
⊢ T R (n + 2) = 2 * X * T R (n + 2 - 1) - T R (n + 2 - 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | exact T_add_two R n | theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
| Mathlib.RingTheory.Polynomial.Chebyshev.88_0.SRy1jgYRAFbFJky | theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ U R (n + 2) = 2 * X * U R (n + 1) - U R n | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [U] | @[simp]
theorem U_add_two (n : ℕ) : U R (n + 2) = 2 * X * U R (n + 1) - U R n := by | Mathlib.RingTheory.Polynomial.Chebyshev.109_0.SRy1jgYRAFbFJky | @[simp]
theorem U_add_two (n : ℕ) : U R (n + 2) = 2 * X * U R (n + 1) - U R n | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ U R 2 = 4 * X ^ 2 - 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [U] | theorem U_two : U R 2 = 4 * X ^ 2 - 1 := by
| Mathlib.RingTheory.Polynomial.Chebyshev.113_0.SRy1jgYRAFbFJky | theorem U_two : U R 2 = 4 * X ^ 2 - 1 | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ 2 * X * (2 * X) - 1 = 4 * X ^ 2 - 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem U_two : U R 2 = 4 * X ^ 2 - 1 := by
simp only [U]
| Mathlib.RingTheory.Polynomial.Chebyshev.113_0.SRy1jgYRAFbFJky | theorem U_two : U R 2 = 4 * X ^ 2 - 1 | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h : 2 ≤ n
⊢ U R n = 2 * X * U R (n - 1) - U R (n - 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h | theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.118_0.SRy1jgYRAFbFJky | theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
case intro
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h : 2 ≤ 2 + n
⊢ U R (2 + n) = 2 * X * U R (2 + n - 1) - U R (2 + n - 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [add_comm] | theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
| Mathlib.RingTheory.Polynomial.Chebyshev.118_0.SRy1jgYRAFbFJky | theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
case intro
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h : 2 ≤ 2 + n
⊢ U R (n + 2) = 2 * X * U R (n + 2 - 1) - U R (n + 2 - 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | exact U_add_two R n | theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
| Mathlib.RingTheory.Polynomial.Chebyshev.118_0.SRy1jgYRAFbFJky | theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ U R (0 + 1) = X * U R 0 + T R (0 + 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T, U, two_mul, mul_one] | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.124_0.SRy1jgYRAFbFJky | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ U R (1 + 1) = X * U R 1 + T R (1 + 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T, U] | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by | Mathlib.RingTheory.Polynomial.Chebyshev.124_0.SRy1jgYRAFbFJky | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ 2 * X * (2 * X) - 1 = X * (2 * X) + (2 * X * X - 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; | Mathlib.RingTheory.Polynomial.Chebyshev.124_0.SRy1jgYRAFbFJky | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [U_add_two, U_eq_X_mul_U_add_T n, U_eq_X_mul_U_add_T (n + 1), U_eq_X_mul_U_add_T n] | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.124_0.SRy1jgYRAFbFJky | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) =
X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) := by
rw [U_add_two, U_eq_X_mul_U_add_T ... | Mathlib.RingTheory.Polynomial.Chebyshev.124_0.SRy1jgYRAFbFJky | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) = X * U R (n + 2) + T R (n + 2 + 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [U_add_two, T_add_two] | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) := by
rw [U_add_two, U_eq_X_mul_U_add_T ... | Mathlib.RingTheory.Polynomial.Chebyshev.124_0.SRy1jgYRAFbFJky | theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ T R (n + 1) = U R (n + 1) - X * U R n | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [U_eq_X_mul_U_add_T, add_comm (X * U R n), add_sub_cancel] | theorem T_eq_U_sub_X_mul_U (n : ℕ) : T R (n + 1) = U R (n + 1) - X * U R n := by
| Mathlib.RingTheory.Polynomial.Chebyshev.135_0.SRy1jgYRAFbFJky | theorem T_eq_U_sub_X_mul_U (n : ℕ) : T R (n + 1) = U R (n + 1) - X * U R n | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ T R (0 + 2) = X * T R (0 + 1) - (1 - X ^ 2) * U R 0 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T, U] | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ 2 * X * X - 1 = X * X - (1 - X ^ 2) * 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ T R (1 + 2) = X * T R (1 + 1) - (1 - X ^ 2) * U R 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T, U] | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ 2 * X * (2 * X * X - 1) - X = X * (2 * X * X - 1) - (1 - X ^ 2) * (2 * X) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ 2 * X * T R (n + 2 + 1) - T R (n + 2) =
2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) - (X * T R (n + 1) - (1 - X ^ 2) * U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T_eq_X_mul_T_sub_pol_U] | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) := T_add_two _ _
_ = 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R ... | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) - (X * T R (n + 1) - (1 - X ^ 2) * U R n) =
X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) := T_add_two _ _
_ = 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R ... | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n) =
X * T R (n + 2 + 1) - (1 - X ^ 2) * U R (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_add_two _ (n + 1), U_add_two] | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) := T_add_two _ _
_ = 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R ... | Mathlib.RingTheory.Polynomial.Chebyshev.139_0.SRy1jgYRAFbFJky | theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_eq_X_mul_T_sub_pol_U, ← sub_add, sub_self, zero_add] | theorem one_sub_X_sq_mul_U_eq_pol_in_T (n : ℕ) :
(1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.153_0.SRy1jgYRAFbFJky | theorem one_sub_X_sq_mul_U_eq_pol_in_T (n : ℕ) :
(1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
⊢ map f (T R 0) = T S 0 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T_zero, Polynomial.map_one] | @[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.160_0.SRy1jgYRAFbFJky | @[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by simp only [T_one, map_X]
| n + 2 => by
simp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, Polynomial.map_ofNat, map_T ... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
⊢ map f (T R 1) = T S 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T_one, map_X] | @[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by | Mathlib.RingTheory.Polynomial.Chebyshev.160_0.SRy1jgYRAFbFJky | @[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by simp only [T_one, map_X]
| n + 2 => by
simp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, Polynomial.map_ofNat, map_T ... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
n : ℕ
⊢ map f (T R (n + 2)) = T S (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, Polynomial.map_ofNat, map_T f (n + 1), map_T f n] | @[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by simp only [T_one, map_X]
| n + 2 => by
| Mathlib.RingTheory.Polynomial.Chebyshev.160_0.SRy1jgYRAFbFJky | @[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by simp only [T_one, map_X]
| n + 2 => by
simp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, Polynomial.map_ofNat, map_T ... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
⊢ map f (U R 0) = U S 0 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [U_zero, Polynomial.map_one] | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.169_0.SRy1jgYRAFbFJky | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
⊢ map f (U R 1) = U S 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one] | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
| Mathlib.RingTheory.Polynomial.Chebyshev.169_0.SRy1jgYRAFbFJky | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
n : ℕ
⊢ map f (U R (n + 2)) = U S (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, map_U f (n + 1), map_U f n] | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
| Mathlib.RingTheory.Polynomial.Chebyshev.169_0.SRy1jgYRAFbFJky | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
n : ℕ
⊢ map f 2 * X * U S (n + 1) - U S n = 2 * X * U S (n + 1) - U S n | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | norm_num | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_... | Mathlib.RingTheory.Polynomial.Chebyshev.169_0.SRy1jgYRAFbFJky | @[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ derivative (T R (0 + 1)) = (↑0 + 1) * U R 0 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one] | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ derivative (T R (1 + 1)) = (↑1 + 1) * U R 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul] | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
| Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ derivative (T R (n + 2 + 1)) = 2 * T R (n + 2) + 2 * X * derivative (T R (n + 1 + 1)) - derivative (T R (n + 1)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_add_two _ (n + 1), derivative_sub, derivative_mul, derivative_mul, derivative_X,
derivative_ofNat] | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (0 * X + 2 * 1) * T R (n + 1 + 1) + 2 * X * derivative (T R (n + 1 + 1)) - derivative (T R (n + 1)) =
2 * T R (n + 2) + 2 * X * derivative (T R (n + 1 + 1)) - derivative (T R (n + 1)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring_nf | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ 2 * T R (n + 2) + 2 * X * derivative (T R (n + 1 + 1)) - derivative (T R (n + 1)) =
2 * (U R (n + 1 + 1) - X * U R (n + 1)) + 2 * X * ((↑n + 1 + 1) * U R (n + 1)) - (↑n + 1) * U R n | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw_mod_cast [T_derivative_eq_U (n + 1), T_derivative_eq_U n, T_eq_U_sub_X_mul_U _ (n + 1)] | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ 2 * (U R (n + 1 + 1) - X * U R (n + 1)) + 2 * X * ((↑n + 1 + 1) * U R (n + 1)) - (↑n + 1) * U R n =
(↑n + 1) * (2 * X * U R (n + 1) - U R n) + 2 * U R (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (↑n + 1) * (2 * X * U R (n + 1) - U R n) + 2 * U R (n + 2) = (↑n + 1) * U R (n + 2) + 2 * U R (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [U_add_two] | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (↑n + 1) * U R (n + 2) + 2 * U R (n + 2) = (↑n + 2 + 1) * U R (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (↑n + 2 + 1) * U R (n + 2) = (↑(n + 2) + 1) * U R (n + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | norm_cast | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib.RingTheory.Polynomial.Chebyshev.180_0.SRy1jgYRAFbFJky | theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((↑n + 1) * U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_derivative_eq_U] | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) :=
calc
(1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((n + 1 : R[X]) * U R n) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.200_0.SRy1jgYRAFbFJky | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (1 - X ^ 2) * ((↑n + 1) * U R n) = (↑n + 1) * ((1 - X ^ 2) * U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) :=
calc
(1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((n + 1 : R[X]) * U R n) := by
rw [T_derivative_eq_U]
_ = (n + 1 : R[X]) * ((1 - X ^ 2) * U ... | Mathlib.RingTheory.Polynomial.Chebyshev.200_0.SRy1jgYRAFbFJky | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (↑n + 1) * ((1 - X ^ 2) * U R n) = (↑n + 1) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [one_sub_X_sq_mul_U_eq_pol_in_T, T_add_two] | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) :=
calc
(1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((n + 1 : R[X]) * U R n) := by
rw [T_derivative_eq_U]
_ = (n + 1 : R[X]) * ((1 - X ^ 2) * U ... | Mathlib.RingTheory.Polynomial.Chebyshev.200_0.SRy1jgYRAFbFJky | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (↑n + 1) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) = (↑n + 1) * (T R n - X * T R (n + 1)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) :=
calc
(1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((n + 1 : R[X]) * U R n) := by
rw [T_derivative_eq_U]
_ = (n + 1 : R[X]) * ((1 - X ^ 2) * U ... | Mathlib.RingTheory.Polynomial.Chebyshev.200_0.SRy1jgYRAFbFJky | theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ (↑n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_T_sub_pol_U]
simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,
one_mul, T_derivative... | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ derivative (T R (n + 2)) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | conv_lhs => rw [T_eq_X_mul_T_sub_pol_U] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
| derivative (T R (n + 2)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_eq_X_mul_T_sub_pol_U] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
| derivative (T R (n + 2)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_eq_X_mul_T_sub_pol_U] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
| derivative (T R (n + 2)) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_eq_X_mul_T_sub_pol_U] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ derivative (X * T R (n + 1) - (1 - X ^ 2) * U R n) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,
one_mul, T_derivative_eq_U] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ T R (n + 1) + X * ((↑n + 1) * U R n) - ((0 - C ↑2 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * derivative (U R n)) =
U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [T_eq_U_sub_X_mul_U, C_eq_nat_cast] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
⊢ U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) -
((0 - ↑2 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * derivative (U R n)) =
U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h :
derivative (T R (n + 2)) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n)
⊢ (↑n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | calc
((n : R[X]) + 1) * T R (n + 1) =
((n : R[X]) + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((n + 1 : R[X]) * U R n) -
(X * U R n + T R (n + 1)) :=
by ring
_ = derivative (T R (n + 2)) - X * derivative (T R (n + 1)) - U R (n + 1) := by
rw [← U_eq_X_mul_U_add_T, ← T_derivative_eq_U... | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h :
derivative (T R (n + 2)) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n)
⊢ (↑n + 1) * T R (n + 1) = (↑n + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((↑n + 1) * U R n) - (X * U R n ... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h :
derivative (T R (n + 2)) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n)
⊢ (↑n + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((↑n + 1) * U R n) - (X * U R n + T R (n + 1)) =
deri... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [← U_eq_X_mul_U_add_T, ← T_derivative_eq_U, ← Nat.cast_one, ← Nat.cast_add, Nat.cast_one, ←
T_derivative_eq_U (n + 1)] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h :
derivative (T R (n + 2)) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n)
⊢ derivative (T R (n + 2)) - X * derivative (T R (n + 1)) - U R (n + 1) =
U R (n + 1) - X * U R n + X ... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | rw [h] | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
h :
derivative (T R (n + 2)) =
U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n)
⊢ U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_... | Mathlib.RingTheory.Polynomial.Chebyshev.211_0.SRy1jgYRAFbFJky | theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ ∀ (k : ℕ), 2 * T R 0 * T R (0 + k) = T R (2 * 0 + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [two_mul, add_mul] | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ ∀ (k : ℕ), 2 * T R 1 * T R (1 + k) = T R (2 * 1 + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [add_comm] | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m : ℕ
⊢ ∀ (k : ℕ), 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | intro k | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
| Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | suffices 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k by
have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring
have h_nat₂ : m + 2 + k = m + k + 2 := by ring
simpa [h_nat₁, h_nat₂] using this | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
| Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
this : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k
⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
this : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k
⊢ 2 * (m + 2) + k = 2 * m + k + 4 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
this : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k
h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4
⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h_nat₂ : m + 2 + k = m + k + 2 := by ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
this : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k
h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4
⊢ m + 2 + k = m + k + 2 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
this : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k
h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4
h_nat₂ : m + 2 + k = m + k + 2
⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simpa [h_nat₁, h_nat₂] using this | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) := by
have h_nat₁ : m + 1 + (k + 1) = m + k + 2 := by ring
have h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring
simpa [h_nat₁, h_nat₂] using mul_T (m + 1) (k + 1) | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h_nat₁ : m + 1 + (k + 1) = m + k + 2 := by ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
⊢ m + 1 + (k + 1) = m + k + 2 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
h_nat₁ : m + 1 + (k + 1) = m + k + 2
⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
h_nat₁ : m + 1 + (k + 1) = m + k + 2
⊢ 2 * (m + 1) + (k + 1) = 2 * m + k + 3 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
h_nat₁ : m + 1 + (k + 1) = m + k + 2
h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3
⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simpa [h_nat₁, h_nat₂] using mul_T (m + 1) (k + 1) | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) := by
have h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]
have h_nat₂ : m + (k + 2) = m + k + 2 := by simp [add_assoc]
simpa [h_nat₁, h_nat₂] using mul_T m (k + 2) | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc] | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
⊢ 2 * m + (k + 2) = 2 * m + k + 2 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [add_assoc] | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2
⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h_nat₂ : m + (k + 2) = m + k + 2 := by simp [add_assoc] | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2
⊢ m + (k + 2) = m + k + 2 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp [add_assoc] | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2
h_nat₂ : m + (k + 2) = m + k + 2
⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simpa [h_nat₁, h_nat₂] using mul_T m (k + 2) | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)
⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h₁ := T_add_two R m | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)
h₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m
⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2) :=
T_add_two R (2 * m + k + 2) | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)
h₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m
h₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) -... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have h₃ := T_add_two R k | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m k : ℕ
H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)
H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)
h₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m
h₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) -... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | linear_combination 2 * T R (m + k + 2) * h₁ + 2 * (X : R[X]) * H₁ - H₂ - h₂ - h₃ | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib.RingTheory.Polynomial.Chebyshev.238_0.SRy1jgYRAFbFJky | /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices... | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ ∀ (n : ℕ), T R (0 * n) = comp (T R 0) (T R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by | Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ ∀ (n : ℕ), T R (1 * n) = comp (T R 1) (T R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | simp | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by | Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m : ℕ
⊢ ∀ (n : ℕ), T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | intro n | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
| Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
⊢ T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n) using 1 <;> ring_nf | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
| Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
⊢ 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | convert mul_T R n (m * n) using 1 | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
| Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
case h.e'_2
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
⊢ 2 * T R n * T R ((m + 1) * n) = 2 * T R n * T R (n + m * n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev... | ring_nf | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n... | Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
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