l3lab/ntp-mathlib huggingface dataset:
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  

Proof state:
case neg.intro.intro.intro
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
⊢ ↑(d { val := c₁ 1, property := (_ : c₁ 1 ∣ m) }) ^ s = ↑(d { val := c₁ 1 ^ s, property := hs' })

Next tactic:
have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  

Proof state:
case neg.intro.intro.intro
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
⊢ ↑(d { val := c₁ 1, property := (_ : c₁ 1 ∣ m) }) ^ s = ↑(d { val := c₁ 1 ^ s, property := hs' })

Next tactic:
rw [← c₂_def]
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  

Proof state:
case neg.intro.intro.intro
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
⊢ c₂ 1 ^ s = ↑(d { val := c₁ 1 ^ s, property := hs' })

Next tactic:
refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · 

Proof state:
case neg.intro.intro.intro.refine'_1
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
t u : Fin (s + 1)
h : t < u
⊢ c₂ t < c₂ u

Next tactic:
rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    

Proof state:
case neg.intro.intro.intro.refine'_1
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
t u : Fin (s + 1)
h : t < u
⊢ t < u

Next tactic:
exact h
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · 

Proof state:
case neg.intro.intro.intro.refine'_2
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
⊢ ∃ i, r = c₂ i

Next tactic:
have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    

Proof state:
case neg.intro.intro.intro.refine'_2
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this : r ≤ n
⊢ ∃ i, r = c₂ i

Next tactic:
suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      use i
      simp only [c₂_def, ← hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      

Proof state:
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this✝ : r ≤ n
this : (OrderIso.symm d) { val := r, property := this✝ } ≤ { val := c₁ 1 ^ s, property := hs' }
⊢ ∃ i, r = c₂ i

Next tactic:
obtain ⟨i, hi⟩ := hc₁''.1 this
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      

Proof state:
case intro
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this✝ : r ≤ n
this : (OrderIso.symm d) { val := r, property := this✝ } ≤ { val := c₁ 1 ^ s, property := hs' }
i : Fin (s + 1)
hi : ↑((OrderIso.symm d) { val := r, property := this✝ }) = c₁ i
⊢ ∃ i, r = c₂ i

Next tactic:
use i
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      use i
      

Proof state:
case h
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this✝ : r ≤ n
this : (OrderIso.symm d) { val := r, property := this✝ } ≤ { val := c₁ 1 ^ s, property := hs' }
i : Fin (s + 1)
hi : ↑((OrderIso.symm d) { val := r, property := this✝ }) = c₁ i
⊢ r = c₂ i

Next tactic:
simp only [c₂_def, ← hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      use i
      simp only [c₂_def, ← hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]
    

Proof state:
case neg.intro.intro.intro.refine'_2
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this : r ≤ n
⊢ (OrderIso.symm d) { val := r, property := this } ≤ { val := c₁ 1 ^ s, property := hs' }

Next tactic:
conv_rhs => rw [← d.symm_apply_apply ⟨c₁ 1 ^ s, hs'⟩]
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      use i
      simp only [c₂_def, ← hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]
    conv_rhs => 

Proof state:
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this : r ≤ n
| { val := c₁ 1 ^ s, property := hs' }

Next tactic:
rw [← d.symm_apply_apply ⟨c₁ 1 ^ s, hs'⟩]
theorem pow_image_of_prime_by_factor_orderIso_dvd
    [DecidableEq (Associates M)] {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
    (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
    (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s ≤ n := by
  by_cases hs : s = 0
  · simp [hs]
  suffices (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) ^ s =
      (d ⟨p ^ s, hs'⟩) by
    rw [this]
    apply Subtype.prop (d ⟨p ^ s, hs'⟩)
  obtain ⟨c₁, rfl, hc₁', hc₁''⟩ := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)
  let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
  have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
  rw [← c₂_def]
  refine'
    (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => ⟨@fun hr => _, _⟩) _).symm
  · rw [c₂_def, c₂_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc₁'.lt_iff_lt]
    exact h
  · have : r ≤ n := hr.trans (d ⟨c₁ 1 ^ s, _⟩).2
    suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      use i
      simp only [c₂_def, ← hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]
    conv_rhs => rw [← d.symm_apply_apply ⟨c₁ 1 ^ s, hs'⟩]
    

Proof state:
case neg.intro.intro.intro.refine'_2
M : Type u_1
inst✝⁴ : CancelCommMonoidWithZero M
N : Type u_2
inst✝³ : CancelCommMonoidWithZero N
inst✝² : UniqueFactorizationMonoid N
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : DecidableEq (Associates M)
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t => ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d { val := c₁ t, property := (_ : c₁ t ≤ m) })
r : Associates N
hr : r ≤ ↑(d { val := c₁ 1 ^ s, property := hs' })
this : r ≤ n
⊢ (OrderIso.symm d) { val := r, property := this } ≤ (OrderIso.symm d) (d { val := c₁ 1 ^ s, property := hs' })

Next tactic:
rw [d.symm.le_iff_le]