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

tactic: apply Nat.strongInductionOn n
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
⊢ n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ∀ (n : ℕ),
    (∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)) →
      n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)

tactic: clear n
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ∀ (n : ℕ),
    (∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)) →
      n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
⊢ ∀ (n : ℕ),
    (∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)) →
      n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)

tactic: intro n ih
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
⊢ ∀ (n : ℕ),
    (∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)) →
      n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)

tactic: rw [xInTermsOfW_eq, mul_comm, vars_C_mul _ (nonzero_of_invertible _),
  vars_sub_of_disjoint, vars_X, range_succ, insert_eq]
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1)
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∈ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ∧
    {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ {n} ∪ range n

case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ Disjoint (X n).vars (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars

tactic: on_goal 1 =>
  simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]
  intro i
  rw [mem_union, mem_union]
  apply Or.imp id
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∈ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ∧
    {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ {n} ∪ range n

case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ Disjoint (X n).vars (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ range n

case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ Disjoint (X n).vars (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars

tactic: on_goal 2 => rw [vars_X, disjoint_singleton_left]
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ range n

case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ Disjoint (X n).vars (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ range n

case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∉ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars

tactic: simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∈ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ∧
    {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ {n} ∪ range n
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ {n} ∪ range n

tactic: intro i
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ {n} ∪ range n
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ {n} ∪ range n

tactic: rw [mem_union, mem_union]
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ {n} ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ {n} ∪ range n
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ {n} ∨ i ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ {n} ∨ i ∈ range n

tactic: apply Or.imp id
state_before: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ {n} ∨ i ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ {n} ∨ i ∈ range n
state_after: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i : ℕ
⊢ i ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars → i ∈ range n

tactic: rw [vars_X, disjoint_singleton_left]
state_before: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ Disjoint (X n).vars (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
state_after: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∉ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars

tactic: intro H
state_before: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∉ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
state_after: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
H : n ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
⊢ False

tactic: replace H := vars_sum_subset _ _ H
state_before: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
H : n ∈ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
⊢ False
state_after: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
H : n ∈ (range n).biUnion fun i => (C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
⊢ False

tactic: rw [mem_biUnion] at H
state_before: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
H : n ∈ (range n).biUnion fun i => (C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars
⊢ False
state_after: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
H : ∃ a ∈ range n, n ∈ (C (↑p ^ a) * xInTermsOfW p ℚ a ^ p ^ (n - a)).vars
⊢ False

tactic: rcases H with ⟨j, hj, H⟩
state_before: case hpq
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
H : ∃ a ∈ range n, n ∈ (C (↑p ^ a) * xInTermsOfW p ℚ a ^ p ^ (n - a)).vars
⊢ False
state_after: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False

tactic: rw [vars_C_mul] at H
state_before: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False
state_after: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False

case hpq.intro.intro.ha
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ ↑p ^ j ≠ 0

tactic: swap
state_before: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False

case hpq.intro.intro.ha
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ ↑p ^ j ≠ 0
state_after: case hpq.intro.intro.ha
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ ↑p ^ j ≠ 0

case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False

tactic: rw [mem_range] at hj
state_before: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False
state_after: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False

tactic: replace H := (ih j hj).2 (vars_pow _ _ H)
state_before: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False
state_after: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n ∈ range (j + 1)
⊢ False

tactic: rw [mem_range] at H
state_before: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n ∈ range (j + 1)
⊢ False
state_after: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n < j + 1
⊢ False

tactic: apply pow_ne_zero
state_before: case hpq.intro.intro.ha
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ ↑p ^ j ≠ 0
state_after: case hpq.intro.intro.ha.h
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ ↑p ≠ 0

tactic: exact mod_cast hp.1.ne_zero
state_before: case hpq.intro.intro.ha.h
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (C (↑p ^ j) * xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ ↑p ≠ 0
state_after: no goals

tactic: rw [mem_range]
state_before: case intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i < j + 1
⊢ i ∈ range n
state_after: case intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i < j + 1
⊢ i < n

tactic: omega
state_before: case intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i < j + 1
⊢ i < n
state_after: no goals

tactic: omega
state_before: case hpq.intro.intro
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : DecidableEq R
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n < j + 1
⊢ False
state_after: no goals