Theorem: constantCoeff_xInTermsOfW
File: Mathlib/RingTheory/WittVector/WittPolynomial.lean

Proof steps:
state_before: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
⊢ constantCoeff (xInTermsOfW p R n) = 0
state_after: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
⊢ ∀ (n : ℕ), (∀ m < n, constantCoeff (xInTermsOfW p R m) = 0) → constantCoeff (xInTermsOfW p R n) = 0
tactic: apply Nat.strongInductionOn n
state_before: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
⊢ ∀ (n : ℕ), (∀ m < n, constantCoeff (xInTermsOfW p R m) = 0) → constantCoeff (xInTermsOfW p R n) = 0
state_after: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
⊢ ∀ (n : ℕ), (∀ m < n, constantCoeff (xInTermsOfW p R m) = 0) → constantCoeff (xInTermsOfW p R n) = 0
tactic: clear n
state_before: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
⊢ ∀ (n : ℕ), (∀ m < n, constantCoeff (xInTermsOfW p R m) = 0) → constantCoeff (xInTermsOfW p R n) = 0
state_after: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ constantCoeff (xInTermsOfW p R n) = 0
tactic: intro n IH
state_before: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ constantCoeff (xInTermsOfW p R n) = 0
state_after: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ⅟↑p ^ n * ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0
tactic: rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C,
    constantCoeff_X, zero_sub, mul_neg, neg_eq_zero]
state_before: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ⅟↑p ^ n * ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0
state_after: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ⅟↑p ^ n * ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = ⅟↑p ^ n * 0
tactic: refine Eq.trans (?_ : _ = ((⅟↑p : R) ^ n)* 0) (mul_zero _)
state_before: p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ⅟↑p ^ n * ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = ⅟↑p ^ n * 0
state_after: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0
tactic: congr 1
state_before: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0
state_after: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ∀ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0
tactic: rw [sum_eq_zero]
state_before: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
⊢ ∀ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0
state_after: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m ∈ range n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0
tactic: intro m H
state_before: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m ∈ range n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0
state_after: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0
tactic: rw [mem_range] at H
state_before: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0
state_after: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ ↑p ^ m * 0 ^ p ^ (n - m) = 0
tactic: simp only [RingHom.map_mul, RingHom.map_pow, map_natCast, IH m H]
state_before: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ ↑p ^ m * 0 ^ p ^ (n - m) = 0
state_after: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ p ^ (n - m) ≠ 0
tactic: rw [zero_pow, mul_zero]
state_before: case e_a
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ p ^ (n - m) ≠ 0
state_after: no goals
tactic: exact pow_ne_zero _ hp.1.ne_zero