-- Mathlib/NumberTheory/MaricaSchoenheim.lean
/-- The special case of Graham's conjecture where all numbers are squarefree. -/
lemma grahamConjecture_of_squarefree {n : ℕ} (f : ℕ → ℕ) (hf' : ∀ k < n, Squarefree (f k)) :
    GrahamConjecture n f

Proof state:
case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
i : ℕ
hi : i < n
j : ℕ
hj : j < n
⊢ 0 < ∏ p ∈ (f i).primeFactors \ (f j).primeFactors, p ∧ ∏ p ∈ (f i).primeFactors \ (f j).primeFactors, p < n

Tactic:
rw [← primeFactors_div_gcd (hf' _ hi) (hf' _ hj).ne_zero,
  prod_primeFactors_of_squarefree <| hf'' _ hi _]

Result:
case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
i : ℕ
hi : i < n
j : ℕ
hj : j < n
⊢ 0 < f i / (f i).gcd (f j) ∧ f i / (f i).gcd (f j) < n

Proof state:
case calc_1
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ Set.InjOn (fun n => (f n).primeFactors) ↑(Iio n)

Tactic:
simpa using prod_primeFactors_invOn_squarefree.2.injOn.comp hf.injOn hf'

Result:
no goals

Proof state:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ n < n

Tactic:
calc
  n = #𝒜 := ?_
  _ ≤ #(𝒜 \\ 𝒜) := 𝒜.card_le_card_diffs
  _ ≤ #(Ioo 0 n) := card_le_card_of_injOn (fun s ↦ ∏ p ∈ s, p) ?_ ?_
  _ = n - 1 := by rw [card_Ioo, tsub_zero]
  _ < n := tsub_lt_self hn.bot_lt zero_lt_one

Result:
case calc_1
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ n = #𝒜

case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ ∀ a ∈ 𝒜 \\ 𝒜, (fun s => ∏ p ∈ s, p) a ∈ Ioo 0 n

case calc_3
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ Set.InjOn (fun s => ∏ p ∈ s, p) ↑(𝒜 \\ 𝒜)

Proof state:
case calc_3
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
a : ℕ
ha : a < n
b : ℕ
hb : b < n
c : ℕ
hc : c < n
d : ℕ
hd : d < n
⊢ f a / (f a).gcd (f b) = f c / (f c).gcd (f d) →
    (f a / (f a).gcd (f b)).primeFactors = (f c / (f c).gcd (f d)).primeFactors

Tactic:
rintro h

Result:
case calc_3
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
a : ℕ
ha : a < n
b : ℕ
hb : b < n
c : ℕ
hc : c < n
d : ℕ
hd : d < n
h : f a / (f a).gcd (f b) = f c / (f c).gcd (f d)
⊢ (f a / (f a).gcd (f b)).primeFactors = (f c / (f c).gcd (f d)).primeFactors

Proof state:
case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
i : ℕ
hi : i < n
j : ℕ
hj : j < n
⊢ 0 < f i / (f i).gcd (f j) ∧ f i / (f i).gcd (f j) < n

Tactic:
exact ⟨Nat.div_pos (gcd_le_left _ (hf' _ hi).ne_zero.bot_lt) <|
  Nat.gcd_pos_of_pos_left _ (hf' _ hi).ne_zero.bot_lt, Nat.div_lt_of_lt_mul <| this _ hi _ hj⟩

Result:
no goals

Proof state:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
⊢ n.GrahamConjecture f

Tactic:
rintro hn hf

Result:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
⊢ ∃ i < n, ∃ j < n, (f i).gcd (f j) * n ≤ f i

Proof state:
case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ ∀ a ∈ 𝒜 \\ 𝒜, (fun s => ∏ p ∈ s, p) a ∈ Ioo 0 n

Tactic:
simp only [𝒜, forall_mem_diffs, forall_mem_image, mem_Ioo, mem_Iio]

Result:
case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ ∀ ⦃x : ℕ⦄,
    x < n →
      ∀ ⦃x_1 : ℕ⦄,
        x_1 < n →
          0 < ∏ p ∈ (f x).primeFactors \ (f x_1).primeFactors, p ∧
            ∏ p ∈ (f x).primeFactors \ (f x_1).primeFactors, p < n

Proof state:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ False

Tactic:
refine lt_irrefl n ?_

Result:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ n < n

Proof state:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
⊢ False

Tactic:
have hf'' : ∀ i < n, ∀ j, Squarefree (f i / (f i).gcd (f j)) :=
  fun i hi j ↦ (hf' _ hi).squarefree_of_dvd <| div_dvd_of_dvd <| gcd_dvd_left _ _

Result:
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ False

Proof state:
case calc_3
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
a : ℕ
ha : a < n
b : ℕ
hb : b < n
c : ℕ
hc : c < n
d : ℕ
hd : d < n
h : f a / (f a).gcd (f b) = f c / (f c).gcd (f d)
⊢ (f a / (f a).gcd (f b)).primeFactors = (f c / (f c).gcd (f d)).primeFactors

Tactic:
rw [h]

Result:
no goals