-- Mathlib/Algebra/Polynomial/Mirror.lean
theorem coeff_mirror (n : ℕ) :
    p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n)

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ p.natDegree + p.natTrailingDegree - n < p.natTrailingDegree

Tactic:
exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _)

Result:
no goals

Proof state:
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : n < p.natTrailingDegree
⊢ n < p.mirror.natTrailingDegree

Tactic:
rwa [mirror_natTrailingDegree]

Result:
no goals

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : ¬p.natTrailingDegree ≤ n
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

Tactic:
rw [not_le] at h3

Result:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : n < p.natTrailingDegree
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : p.natTrailingDegree ≤ n
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

Tactic:
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)]

Result:
no goals

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]

Result:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : ¬n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
rw [← revAtFun_eq, revAtFun, if_neg h1, coeff_eq_zero_of_natDegree_lt h2]

Result:
no goals

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : ¬p.natDegree < n
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
rw [not_lt] at h2

Result:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree]

Result:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ p.natDegree + p.natTrailingDegree - n < p.natTrailingDegree

Proof state:
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ p.mirror.natDegree < n

Tactic:
rwa [mirror_natDegree]

Result:
no goals

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree

Result:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
h1 : ¬n ≤ p.natDegree + p.natTrailingDegree
⊢ 0 = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : n < p.natTrailingDegree
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

Tactic:
rw [coeff_eq_zero_of_natDegree_lt (lt_tsub_iff_right.mpr (Nat.add_lt_add_left h3 _))]

Result:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : n < p.natTrailingDegree
⊢ p.mirror.coeff n = 0

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : n < p.natTrailingDegree
⊢ p.mirror.coeff n = 0

Tactic:
exact coeff_eq_zero_of_lt_natTrailingDegree (by rwa [mirror_natTrailingDegree])

Result:
no goals

Proof state:
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
by_cases h2 : p.natDegree < n

Result:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : p.natDegree < n
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : ¬p.natDegree < n
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
⊢ p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)

Tactic:
rw [revAt_le (h2.trans (Nat.le_add_right _ _))]

Result:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

Tactic:
by_cases h3 : p.natTrailingDegree ≤ n

Result:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : p.natTrailingDegree ≤ n
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)

case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h2 : n ≤ p.natDegree
h3 : ¬p.natTrailingDegree ≤ n
⊢ p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)