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 |
|---|---|---|---|---|---|---|
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
a : R
ha : a = 0
⊢ mirror ((monomial n) a) = (monomial n) a | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [ha, monomial_zero_right, mirror_zero] | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a := by
classical
by_cases ha : a = 0
· | Mathlib.Data.Polynomial.Mirror.47_0.jRYkh9xLrkM32QX | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
a : R
ha : ¬a = 0
⊢ mirror ((monomial n) a) = (monomial n) a | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mirror, reverse, natDegree_monomial n a, if_neg ha, natTrailingDegree_monomial ha, ←
C_mul_X_pow_eq_monomial, reflect_C_mul_X_pow, revAt_le (le_refl n), tsub_self, pow_zero,
mul_one] | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a := by
classical
by_cases ha : a = 0
· rw [ha, monomial_zero_right, mirror_zero]
· | Mathlib.Data.Polynomial.Mirror.47_0.jRYkh9xLrkM32QX | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ natDegree (mirror p) = natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases hp : p = 0 | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
| Mathlib.Data.Polynomial.Mirror.66_0.jRYkh9xLrkM32QX | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : p = 0
⊢ natDegree (mirror p) = natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [hp, mirror_zero] | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
by_cases hp : p = 0
· | Mathlib.Data.Polynomial.Mirror.66_0.jRYkh9xLrkM32QX | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : ¬p = 0
⊢ natDegree (mirror p) = natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | nontriviality R | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
by_cases hp : p = 0
· rw [hp, mirror_zero]
| Mathlib.Data.Polynomial.Mirror.66_0.jRYkh9xLrkM32QX | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : ¬p = 0
✝ : Nontrivial R
⊢ natDegree (mirror p) = natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree] | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
by_cases hp : p = 0
· rw [hp, mirror_zero]
nontriviality R
| Mathlib.Data.Polynomial.Mirror.66_0.jRYkh9xLrkM32QX | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : ¬p = 0
✝ : Nontrivial R
⊢ leadingCoeff (reverse p) * leadingCoeff (X ^ natTrailingDegree p) ≠ 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero] | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
by_cases hp : p = 0
· rw [hp, mirror_zero]
nontriviality R
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]
| Mathlib.Data.Polynomial.Mirror.66_0.jRYkh9xLrkM32QX | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ natTrailingDegree (mirror p) = natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases hp : p = 0 | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by
| Mathlib.Data.Polynomial.Mirror.75_0.jRYkh9xLrkM32QX | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : p = 0
⊢ natTrailingDegree (mirror p) = natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [hp, mirror_zero] | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by
by_cases hp : p = 0
· | Mathlib.Data.Polynomial.Mirror.75_0.jRYkh9xLrkM32QX | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : ¬p = 0
⊢ natTrailingDegree (mirror p) = natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mirror, natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp),
reverse_natTrailingDegree, zero_add] | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by
by_cases hp : p = 0
· rw [hp, mirror_zero]
· | Mathlib.Data.Polynomial.Mirror.75_0.jRYkh9xLrkM32QX | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
⊢ coeff (mirror p) n = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases h2 : p.natDegree < n | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
| Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : natDegree p < n
⊢ coeff (mirror p) n = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : natDegree p < n
⊢ natDegree (mirror p) < n | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [mirror_natDegree] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : natDegree p < n
⊢ 0 = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
| Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : natDegree p < n
h1 : n ≤ natDegree p + natTrailingDegree p
⊢ 0 = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : natDegree p < n
h1 : n ≤ natDegree p + natTrailingDegree p
⊢ natDegree p + natTrailingDegree p - n < natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _) | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : natDegree p < n
h1 : ¬n ≤ natDegree p + natTrailingDegree p
⊢ 0 = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [← revAtFun_eq, revAtFun, if_neg h1, coeff_eq_zero_of_natDegree_lt h2] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : ¬natDegree p < n
⊢ coeff (mirror p) n = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [not_lt] at h2 | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
⊢ coeff (mirror p) n = coeff p ((revAt (natDegree p + natTrailingDegree p)) n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [revAt_le (h2.trans (Nat.le_add_right _ _))] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases h3 : p.natTrailingDegree ≤ n | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
h3 : natTrailingDegree p ≤ n
⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [← tsub_add_eq_add_tsub h2, ← tsub_tsub_assoc h2 h3, mirror, coeff_mul_X_pow', if_pos h3,
coeff_reverse, revAt_le (tsub_le_self.trans h2)] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
h3 : ¬natTrailingDegree p ≤ n
⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [not_le] at h3 | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
h3 : n < natTrailingDegree p
⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [coeff_eq_zero_of_natDegree_lt (lt_tsub_iff_right.mpr (Nat.add_lt_add_left h3 _))] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
h3 : n < natTrailingDegree p
⊢ coeff (mirror p) n = 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact coeff_eq_zero_of_lt_natTrailingDegree (by rwa [mirror_natTrailingDegree]) | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
h2 : n ≤ natDegree p
h3 : n < natTrailingDegree p
⊢ n < natTrailingDegree (mirror p) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [mirror_natTrailingDegree] | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrail... | Mathlib.Data.Polynomial.Mirror.82_0.jRYkh9xLrkM32QX | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ eval 1 (mirror p) = eval 1 p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
| Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ (Finset.sum (Finset.range (natDegree p + 1)) fun x => coeff (mirror p) x) =
Finset.sum (Finset.range (natDegree p + 1)) fun x => coeff p x | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | refine' Finset.sum_bij_ne_zero _ _ _ _ _ | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
| Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_1
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ (a : ℕ) → a ∈ Finset.range (natDegree p + 1) → coeff (mirror p) a ≠ 0 → ℕ | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ ∀ a ∈ Finset.range (natDegree p + 1),
coeff (mirror p) a ≠ 0 → (revAt (natDegree p + natTrailingDegree p)) a ∈ Finset.range (natDegree p + 1) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | intro n hn hp | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (natDegree p + 1)
hp : coeff (mirror p) n ≠ 0
⊢ (revAt (natDegree p + natTrailingDegree p)) n ∈ Finset.range (natDegree p + 1) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [Finset.mem_range_succ_iff] at * | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
| Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ≤ natDegree p
hp : coeff (mirror p) n ≠ 0
⊢ (revAt (natDegree p + natTrailingDegree p)) n ≤ natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [revAt_le (hn.trans (Nat.le_add_right _ _))] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
| Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ≤ natDegree p
hp : coeff (mirror p) n ≠ 0
⊢ natDegree p + natTrailingDegree p - n ≤ natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [tsub_le_iff_tsub_le, add_comm, add_tsub_cancel_right, ← mirror_natTrailingDegree] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ≤ natDegree p
hp : coeff (mirror p) n ≠ 0
⊢ natTrailingDegree (mirror p) ≤ n | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact natTrailingDegree_le_of_ne_zero hp | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_3
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ ∀ (a₁ a₂ : ℕ),
a₁ ∈ Finset.range (natDegree p + 1) →
coeff (mirror p) a₁ ≠ 0 →
a₂ ∈ Finset.range (natDegree p + 1) →
coeff (mirror p) a₂ ≠ 0 →
(revAt (natDegree p + natTrailingDegree p)) a₁ = (revAt (natDegree p + natTra... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact fun n₁ n₂ _ _ _ _ h => by rw [← @revAt_invol _ n₁, h, revAt_invol] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n₁ n₂ : ℕ
x✝³ : n₁ ∈ Finset.range (natDegree p + 1)
x✝² : coeff (mirror p) n₁ ≠ 0
x✝¹ : n₂ ∈ Finset.range (natDegree p + 1)
x✝ : coeff (mirror p) n₂ ≠ 0
h : (revAt (natDegree p + natTrailingDegree p)) n₁ = (revAt (natDegree p + natTrailingDegree p)) n₂
⊢ n₁ = n₂ | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [← @revAt_invol _ n₁, h, revAt_invol] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_4
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ ∀ b ∈ Finset.range (natDegree p + 1),
coeff p b ≠ 0 →
∃ a,
∃ (_ : a ∈ Finset.range (natDegree p + 1)) (_ : coeff (mirror p) a ≠ 0),
b = (revAt (natDegree p + natTrailingDegree p)) a | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | intro n hn hp | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_4
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (natDegree p + 1)
hp : coeff p n ≠ 0
⊢ ∃ a,
∃ (_ : a ∈ Finset.range (natDegree p + 1)) (_ : coeff (mirror p) a ≠ 0),
n = (revAt (natDegree p + natTrailingDegree p)) a | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | use revAt (p.natDegree + p.natTrailingDegree) n | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (natDegree p + 1)
hp : coeff p n ≠ 0
⊢ ∃ (_ : (revAt (natDegree p + natTrailingDegree p)) n ∈ Finset.range (natDegree p + 1)) (_ :
coeff (mirror p) ((revAt (natDegree p + natTrailingDegree p)) n) ≠ 0),
n = (revAt (natDegree p + natTra... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | refine' ⟨_, _, revAt_invol.symm⟩ | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h.refine'_1
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (natDegree p + 1)
hp : coeff p n ≠ 0
⊢ (revAt (natDegree p + natTrailingDegree p)) n ∈ Finset.range (natDegree p + 1) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [Finset.mem_range_succ_iff] at * | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h.refine'_1
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ≤ natDegree p
hp : coeff p n ≠ 0
⊢ (revAt (natDegree p + natTrailingDegree p)) n ≤ natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [revAt_le (hn.trans (Nat.le_add_right _ _))] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h.refine'_1
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ≤ natDegree p
hp : coeff p n ≠ 0
⊢ natDegree p + natTrailingDegree p - n ≤ natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [tsub_le_iff_tsub_le, add_comm, add_tsub_cancel_right] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h.refine'_1
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ≤ natDegree p
hp : coeff p n ≠ 0
⊢ natTrailingDegree p ≤ n | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact natTrailingDegree_le_of_ne_zero hp | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h.refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (natDegree p + 1)
hp : coeff p n ≠ 0
⊢ coeff (mirror p) ((revAt (natDegree p + natTrailingDegree p)) n) ≠ 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | change p.mirror.coeff _ ≠ 0 | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case h.refine'_2
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (natDegree p + 1)
hp : coeff p n ≠ 0
⊢ coeff (mirror p) ((revAt (natDegree p + natTrailingDegree p)) n) ≠ 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [coeff_mirror, revAt_invol] | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
case refine'_5
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ ∀ a ∈ Finset.range (natDegree p + 1),
coeff (mirror p) a ≠ 0 → coeff (mirror p) a = coeff p ((revAt (natDegree p + natTrailingDegree p)) a) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact fun n _ _ => p.coeff_mirror n | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine' Finset.sum_bij_ne_zero _ _ _ _ _
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans ... | Mathlib.Data.Polynomial.Mirror.101_0.jRYkh9xLrkM32QX | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
⊢ coeff (mirror (mirror p)) n = coeff p n | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [coeff_mirror, coeff_mirror, mirror_natDegree, mirror_natTrailingDegree, revAt_invol] | theorem mirror_mirror : p.mirror.mirror = p :=
Polynomial.ext fun n => by
| Mathlib.Data.Polynomial.Mirror.123_0.jRYkh9xLrkM32QX | theorem mirror_mirror : p.mirror.mirror = p | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
h : mirror p = 0
⊢ p = 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [← p.mirror_mirror, h, mirror_zero] | @[simp]
theorem mirror_eq_zero : p.mirror = 0 ↔ p = 0 :=
⟨fun h => by | Mathlib.Data.Polynomial.Mirror.143_0.jRYkh9xLrkM32QX | @[simp]
theorem mirror_eq_zero : p.mirror = 0 ↔ p = 0 | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
h : p = 0
⊢ mirror p = 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [h, mirror_zero] | @[simp]
theorem mirror_eq_zero : p.mirror = 0 ↔ p = 0 :=
⟨fun h => by rw [← p.mirror_mirror, h, mirror_zero], fun h => by | Mathlib.Data.Polynomial.Mirror.143_0.jRYkh9xLrkM32QX | @[simp]
theorem mirror_eq_zero : p.mirror = 0 ↔ p = 0 | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ trailingCoeff (mirror p) = leadingCoeff p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [leadingCoeff, trailingCoeff, mirror_natTrailingDegree, coeff_mirror,
revAt_le (Nat.le_add_left _ _), add_tsub_cancel_right] | @[simp]
theorem mirror_trailingCoeff : p.mirror.trailingCoeff = p.leadingCoeff := by
| Mathlib.Data.Polynomial.Mirror.150_0.jRYkh9xLrkM32QX | @[simp]
theorem mirror_trailingCoeff : p.mirror.trailingCoeff = p.leadingCoeff | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ leadingCoeff (mirror p) = trailingCoeff p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [← p.mirror_mirror, mirror_trailingCoeff, p.mirror_mirror] | @[simp]
theorem mirror_leadingCoeff : p.mirror.leadingCoeff = p.trailingCoeff := by
| Mathlib.Data.Polynomial.Mirror.156_0.jRYkh9xLrkM32QX | @[simp]
theorem mirror_leadingCoeff : p.mirror.leadingCoeff = p.trailingCoeff | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ coeff (p * mirror p) (natDegree p + natTrailingDegree p) = sum p fun n x => x ^ 2 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) := by
| Mathlib.Data.Polynomial.Mirror.161_0.jRYkh9xLrkM32QX | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ (Finset.sum (Finset.range (Nat.succ (natDegree p + natTrailingDegree p))) fun k =>
coeff p (k, natDegree p + natTrailingDegree p - k).1 *
coeff (mirror p) (k, natDegree p + natTrailingDegree p - k).2) =
sum p fun n x => x ^ 2 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | refine'
(Finset.sum_congr rfl fun n hn => _).trans
(p.sum_eq_of_subset (fun _ => (· ^ 2)) (fun _ => zero_pow zero_lt_two) fun n hn =>
Finset.mem_range_succ_iff.mpr
((le_natDegree_of_mem_supp n hn).trans (Nat.le_add_right _ _))).symm | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) := by
rw [coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk]
| Mathlib.Data.Polynomial.Mirror.161_0.jRYkh9xLrkM32QX | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
hn : n ∈ Finset.range (Nat.succ (natDegree p + natTrailingDegree p))
⊢ coeff p (n, natDegree p + natTrailingDegree p - n).1 * coeff (mirror p) (n, natDegree p + natTrailingDegree p - n).2 =
coeff p n ^ 2 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [coeff_mirror, ← revAt_le (Finset.mem_range_succ_iff.mp hn), revAt_invol, ← sq] | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) := by
rw [coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk]
refine'
(Finset.sum_congr rfl fun n hn => _).trans
(p.sum_eq_of_subset (fun _ => (· ^ 2)) (fun _ => zero_pow zero_lt_two) ... | Mathlib.Data.Polynomial.Mirror.161_0.jRYkh9xLrkM32QX | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
⊢ natDegree (p * mirror p) = 2 * natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases hp : p = 0 | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree := by
| Mathlib.Data.Polynomial.Mirror.174_0.jRYkh9xLrkM32QX | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : p = 0
⊢ natDegree (p * mirror p) = 2 * natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [hp, zero_mul, natDegree_zero, mul_zero] | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree := by
by_cases hp : p = 0
· | Mathlib.Data.Polynomial.Mirror.174_0.jRYkh9xLrkM32QX | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
⊢ natDegree (p * mirror p) = 2 * natDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [natDegree_mul hp (mt mirror_eq_zero.mp hp), mirror_natDegree, two_mul] | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree := by
by_cases hp : p = 0
· rw [hp, zero_mul, natDegree_zero, mul_zero]
| Mathlib.Data.Polynomial.Mirror.174_0.jRYkh9xLrkM32QX | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases hp : p = 0 | theorem natTrailingDegree_mul_mirror :
(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree := by
| Mathlib.Data.Polynomial.Mirror.180_0.jRYkh9xLrkM32QX | theorem natTrailingDegree_mul_mirror :
(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : p = 0
⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [hp, zero_mul, natTrailingDegree_zero, mul_zero] | theorem natTrailingDegree_mul_mirror :
(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree := by
by_cases hp : p = 0
· | Mathlib.Data.Polynomial.Mirror.180_0.jRYkh9xLrkM32QX | theorem natTrailingDegree_mul_mirror :
(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [natTrailingDegree_mul hp (mt mirror_eq_zero.mp hp), mirror_natTrailingDegree, two_mul] | theorem natTrailingDegree_mul_mirror :
(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree := by
by_cases hp : p = 0
· rw [hp, zero_mul, natTrailingDegree_zero, mul_zero]
| Mathlib.Data.Polynomial.Mirror.180_0.jRYkh9xLrkM32QX | theorem natTrailingDegree_mul_mirror :
(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Ring R
p q : R[X]
⊢ mirror (-p) = -mirror p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mirror, mirror, reverse_neg, natTrailingDegree_neg, neg_mul_eq_neg_mul] | theorem mirror_neg : (-p).mirror = -p.mirror := by
| Mathlib.Data.Polynomial.Mirror.193_0.jRYkh9xLrkM32QX | theorem mirror_neg : (-p).mirror = -p.mirror | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
⊢ mirror (p * q) = mirror p * mirror q | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases hp : p = 0 | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
| Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : p = 0
⊢ mirror (p * q) = mirror p * mirror q | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [hp, zero_mul, mirror_zero, zero_mul] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· | Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
⊢ mirror (p * q) = mirror p * mirror q | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases hq : q = 0 | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
| Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case pos
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
hq : q = 0
⊢ mirror (p * q) = mirror p * mirror q | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [hq, mul_zero, mirror_zero, mul_zero] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· | Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
hq : ¬q = 0
⊢ mirror (p * q) = mirror p * mirror q | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· rw [hq, mul_zero, mirror_zero, mul_zero]
| Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
hq : ¬q = 0
⊢ reverse p * reverse q * (X ^ natTrailingDegree p * X ^ natTrailingDegree q) =
reverse p * X ^ natTrailingDegree p * (reverse q * X ^ natTrailingDegree q) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mul_assoc, ← mul_assoc q.reverse, ← X_pow_mul (p := reverse q)] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· rw [hq, mul_zero, mirror_zero, mul_zero]
rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add]
| Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
hq : ¬q = 0
⊢ reverse p * (X ^ natTrailingDegree p * reverse q * X ^ natTrailingDegree q) =
reverse p * X ^ natTrailingDegree p * (reverse q * X ^ natTrailingDegree q) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | repeat' rw [mul_assoc] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· rw [hq, mul_zero, mirror_zero, mul_zero]
rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add]
rw [mul_assoc, ← mu... | Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
hq : ¬q = 0
⊢ reverse p * (X ^ natTrailingDegree p * reverse q * X ^ natTrailingDegree q) =
reverse p * X ^ natTrailingDegree p * (reverse q * X ^ natTrailingDegree q) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mul_assoc] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· rw [hq, mul_zero, mirror_zero, mul_zero]
rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add]
rw [mul_assoc, ← mu... | Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
case neg
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
hp : ¬p = 0
hq : ¬q = 0
⊢ reverse p * (X ^ natTrailingDegree p * (reverse q * X ^ natTrailingDegree q)) =
reverse p * X ^ natTrailingDegree p * (reverse q * X ^ natTrailingDegree q) | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mul_assoc] | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· rw [hq, mul_zero, mirror_zero, mul_zero]
rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add]
rw [mul_assoc, ← mu... | Mathlib.Data.Polynomial.Mirror.199_0.jRYkh9xLrkM32QX | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : Ring R
p q : R[X]
inst✝ : NoZeroDivisors R
a : R
⊢ mirror (a • p) = a • mirror p | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [← C_mul', ← C_mul', mirror_mul_of_domain, mirror_C] | theorem mirror_smul (a : R) : (a • p).mirror = a • p.mirror := by
| Mathlib.Data.Polynomial.Mirror.209_0.jRYkh9xLrkM32QX | theorem mirror_smul (a : R) : (a • p).mirror = a • p.mirror | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
⊢ Irreducible f | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | constructor | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
| Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case not_unit
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
⊢ ¬IsUnit f | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact h1 | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
⊢ ∀ (a b : R[X]), f = a * b → IsUnit a ∨ IsUnit b | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | intro g h fgh | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
⊢ IsUnit g ∨ IsUnit h | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | let k := g * h.mirror | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
| Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
⊢ IsUnit g ∨ IsUn... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | have key : f * f.mirror = k * k.mirror := by
rw [fgh, mirror_mul_of_domain, mirror_mul_of_domain, mirror_mirror, mul_assoc, mul_comm h,
mul_comm g.mirror, mul_assoc, ← mul_assoc] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
| Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
⊢ f * mirror f = k * mirror k | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [fgh, mirror_mul_of_domain, mirror_mul_of_domain, mirror_mirror, mul_assoc, mul_comm h,
mul_comm g.mirror, mul_assoc, ← mul_assoc] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | have g_dvd_f : g ∣ f := by
rw [fgh]
exact dvd_mul_right g h | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
⊢ g ∣ ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [fgh] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
⊢ g ∣ ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact dvd_mul_right g h | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | have h_dvd_f : h ∣ f := by
rw [fgh]
exact dvd_mul_left h g | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [fgh] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact dvd_mul_left h g | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | have g_dvd_k : g ∣ k := dvd_mul_right g h.mirror | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | have h_dvd_k_rev : h ∣ k.mirror := by
rw [mirror_mul_of_domain, mirror_mirror]
exact dvd_mul_left h g.mirror | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rw [mirror_mul_of_domain, mirror_mirror] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact dvd_mul_left h g.mirror | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | have hk := h2 k key | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror ... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rcases hk with (hk | hk | hk | hk) | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mir... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact Or.inr (h3 h h_dvd_f (by rwa [← hk])) | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [← hk] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'.inr.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f *... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact Or.inr (h3 h h_dvd_f (by rwa [← neg_eq_iff_eq_neg.mpr hk, mirror_neg, dvd_neg])) | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [← neg_eq_iff_eq_neg.mpr hk, mirror_neg, dvd_neg] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'.inr.inr.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key :... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact Or.inl (h3 g g_dvd_f (by rwa [← hk])) | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [← hk] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
case isUnit_or_isUnit'.inr.inr.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key :... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | exact Or.inl (h3 g g_dvd_f (by rwa [← neg_eq_iff_eq_neg.mpr hk, dvd_neg])) | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
f : R[X]
h1 : ¬IsUnit f
h2 : ∀ (k : R[X]), f * mirror f = k * mirror k → k = f ∨ k = -f ∨ k = mirror f ∨ k = -mirror f
h3 : ∀ (g : R[X]), g ∣ f → g ∣ mirror f → IsUnit g
g h : R[X]
fgh : f = g * h
k : R[X] := g * mirror h
key : f * mirror f = k * mirror k
g_dvd_... | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | rwa [← neg_eq_iff_eq_neg.mpr hk, dvd_neg] | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f := by
constructor
· exact h1
· intro g h fgh
let k := g * h.mirror
have key : f * f.mirror = k * k.mirr... | Mathlib.Data.Polynomial.Mirror.219_0.jRYkh9xLrkM32QX | theorem irreducible_of_mirror (h1 : ¬IsUnit f)
(h2 : ∀ k, f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror)
(h3 : ∀ g, g ∣ f → g ∣ f.mirror → IsUnit g) : Irreducible f | Mathlib_Data_Polynomial_Mirror |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
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.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [AffineMap.continuous_linear_iff] | /-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.58_0.bJ3guivW1IqbUMZ | /-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
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.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | exact f.cont | /-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.58_0.bJ3guivW1IqbUMZ | /-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
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.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | @[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.70_0.bJ3guivW1IqbUMZ | @[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst... | /-
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.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | @[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.70_0.bJ3guivW1IqbUMZ | @[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
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