Theorem: _private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_succ_dist_weak
File: Mathlib/NumberTheory/Padics/Hensel.lean

Proof steps:
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
⊢ 2 ≤ 2 ^ (n + 1)
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ^ 1 ≤ 2 ^ (n + 1)
⊢ 2 ≤ 2 ^ (n + 1)
tactic: have := pow_le_pow_right (by norm_num : 1 ≤ 2) (Nat.le_add_left _ _ : 1 ≤ n + 1)
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ^ 1 ≤ 2 ^ (n + 1)
⊢ 2 ≤ 2 ^ (n + 1)
state_after: no goals
tactic: simpa using this
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ 1 < 2
state_after: no goals
tactic: norm_num
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 1 =
    ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ * ‖Polynomial.eval a F‖ /
      (‖↑(Polynomial.eval a (Polynomial.derivative F))‖ * ‖↑(Polynomial.eval a (Polynomial.derivative F))‖) =
    ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖
tactic: rw [T_gen, sq, pow_one, norm_div, ← mul_div_assoc, PadicInt.padic_norm_e_of_padicInt,
        PadicInt.coe_mul, padicNormE.mul]
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ * ‖Polynomial.eval a F‖ /
      (‖↑(Polynomial.eval a (Polynomial.derivative F))‖ * ‖↑(Polynomial.eval a (Polynomial.derivative F))‖) =
    ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖
state_after: case hc
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ ≠ 0
tactic: apply mul_div_mul_left
state_before: case hc
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ ≠ 0
state_after: case hc.hnorm
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
tactic: apply deriv_norm_ne_zero
state_before: case hc.hnorm
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
n : ℕ
this : 2 ≤ 2 ^ (n + 1)
⊢ ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
state_after: no goals
tactic: assumption