Theorem: _private.Mathlib.NumberTheory.Padics.PadicNumbers.0.PadicSeq.norm_eq_of_equiv_aux
File: Mathlib/NumberTheory/Padics/PadicNumbers.lean

Proof steps:
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
⊢ False
state_after: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
⊢ False
tactic: have hpn : 0 < padicNorm p (f (stationaryPoint hf)) - padicNorm p (g (stationaryPoint hg)) :=
    sub_pos_of_lt hlt
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
⊢ False
tactic: cases' hfg _ hpn with N hN
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
⊢ False
tactic: let i := max N (max (stationaryPoint hf) (stationaryPoint hg))
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
⊢ False
tactic: have hi : N ≤ i := le_max_left _ _
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
⊢ False
tactic: have hN' := hN _ hi
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑(f - g) i) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
⊢ False
tactic: rw [lift_index_left hf N (stationaryPoint hg), lift_index_right hg N (stationaryPoint hf)]
    at hN' h hlt
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑(f - g) i) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑(f - g) i) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False
tactic: have hpne : padicNorm p (f i) ≠ padicNorm p (-g i) := by rwa [← padicNorm.neg (g i)] at h
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑(f - g) i) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False
tactic: rw [CauSeq.sub_apply, sub_eq_add_neg, add_eq_max_of_ne hpne, padicNorm.neg, max_eq_left_of_lt hlt]
    at hN'
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
this : padicNorm p (↑f i) < padicNorm p (↑f i)
⊢ False
tactic: have : padicNorm p (f i) < padicNorm p (f i) := by
    apply lt_of_lt_of_le hN'
    apply sub_le_self
    apply padicNorm.nonneg
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
this : padicNorm p (↑f i) < padicNorm p (↑f i)
⊢ False
state_after: no goals
tactic: exact lt_irrefl _ this