internlm/Lean-Github huggingface dataset:
Theorem: infinity_of_primes₃
Proof state before:
case a
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
⊢ ↑two ^ p = 1
Tactic:
rw [two_desc]
Proof state after:
case a
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
⊢ 2 ^ p = 1
Theorem: infinity_of_primes₃
Proof state before:
case a
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
⊢ 2 ^ p = 1
Tactic:
exact h_mod_q'
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
⊢ two ≠ 1
Tactic:
by_contra h
Proof state after:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : two = 1
⊢ False
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : two = 1
⊢ False
Tactic:
rw [Units.ext_iff] at h
Proof state after:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : ↑two = ↑1
⊢ False
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : ↑two = ↑1
⊢ False
Tactic:
push_cast at h
Proof state after:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : ↑two = 1
⊢ False
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : ↑two = 1
⊢ False
Tactic:
rw [two_desc] at h
Proof state after:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : 2 = 1
⊢ False
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
h : 2 = 1
⊢ False
Tactic:
exact (ZMod.two_ne_one q) h
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ p ∣ q - 1
Tactic:
convert Subgroup.card_subgroup_dvd_card (Subgroup.zpowers (two))
Proof state after:
case h.e'_3
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ p = Fintype.card ↥(Subgroup.zpowers two)

case h.e'_4
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ q - 1 = Fintype.card (ZMod q)ˣ
Theorem: infinity_of_primes₃
Proof state before:
case h.e'_3
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ p = Fintype.card ↥(Subgroup.zpowers two)
Tactic:
rw [← orderOf_eq_prime h_two two_ne_one]
Proof state after:
case h.e'_3
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ orderOf two = Fintype.card ↥(Subgroup.zpowers two)
Theorem: infinity_of_primes₃
Proof state before:
case h.e'_3
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ orderOf two = Fintype.card ↥(Subgroup.zpowers two)
Tactic:
exact Fintype.card_zpowers.symm
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
case h.e'_4
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
⊢ q - 1 = Fintype.card (ZMod q)ˣ
Tactic:
exact ((prime_iff_card_units q).mp hq).symm
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
case h.left
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
⊢ q.Prime
Tactic:
refine' minFac_prime <| Nat.ne_of_gt _
Proof state after:
case h.left
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
⊢ 1 < m
Theorem: infinity_of_primes₃
Proof state before:
case h.left
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
⊢ 1 < m
Tactic:
calc 1 < 2^2 - 1 := by norm_num
    _  ≤ 2^p - 1 := sub_one_le_sub_one <| pow_le_pow_of_le_right (succ_pos 1) (Prime.two_le hp)
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
⊢ 1 < 2 ^ 2 - 1
Tactic:
norm_num
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
case h.right
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
⊢ p < q
Tactic:
have h2q : 2 ≤ q := by
  exact Prime.two_le <| minFac_prime <| Nat.ne_of_gt <| lt_of_succ_lt <|
      Nat.sub_le_sub_right ((pow_le_pow_of_le_right (succ_pos 1) (Prime.two_le hp))) 1
Proof state after:
case h.right
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
h2q : 2 ≤ q
⊢ p < q
Theorem: infinity_of_primes₃
Proof state before:
case h.right
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
h2q : 2 ≤ q
⊢ p < q
Tactic:
exact lt_of_le_of_lt (Nat.le_of_dvd  (Nat.sub_pos_of_lt <| h2q) h_piv_div_q_sub_one)
    <| pred_lt <| Nat.ne_of_gt <| Nat.le_of_lt h2q
Proof state after:
no goals
Theorem: infinity_of_primes₃
Proof state before:
p : ℕ
hp : p.Prime
this✝¹ : Fact p.Prime
m : ℕ := mersenne p
q : ℕ := m.minFac
hq : q.Prime
this✝ : Fact q.Prime
h_mod_q : q.ModEq (2 ^ p) 1
h_mod_q' : 2 ^ p = 1
this : 2 * 2 ^ (p - 1) = 1
two : (ZMod q)ˣ := Units.mkOfMulEqOne 2 (2 ^ (p - 1)) this
two_desc : ↑two = 2
h_two : two ^ p = 1
two_ne_one : two ≠ 1
h_piv_div_q_sub_one : p ∣ q - 1
⊢ 2 ≤ q
Tactic:
exact Prime.two_le <| minFac_prime <| Nat.ne_of_gt <| lt_of_succ_lt <|
    Nat.sub_le_sub_right ((pow_le_pow_of_le_right (succ_pos 1) (Prime.two_le hp))) 1
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
⊢ Tendsto π atTop atTop
Tactic:
have H_log_le_primeCountingReal_add_one (n : ℕ) (x : ℝ) (hxge : x ≥ n) (hxlt : x < n + 1) :
    Real.log x ≤ primeCountingReal x + 1 :=
  calc
    Real.log x ≤ ∑ k in Icc 1 n, (k : ℝ)⁻¹ := by sorry
    _ ≤ (∑' m : (S₁ x), (m : ℝ)⁻¹) := by sorry
    _ ≤ (∏ p in primesBelow ⌊x⌋.natAbs, (∑' k : ℕ, (p ^ k : ℝ)⁻¹)) := by sorry
    _ ≤ (∏ k in Icc 1 (primeCountingReal x), (nth Nat.Prime k) / ((nth Nat.Prime k) - 1)) := by sorry
    _ ≤ (∏ k in Icc 1 (primeCountingReal x), (k + 1) / k) := by sorry
    _ ≤ primeCountingReal x + 1 := by sorry
Proof state after:
H_log_le_primeCountingReal_add_one : ∀ (n : ℕ), ∀ x ≥ ↑n, x < ↑n + 1 → x.log ≤ ↑(primeCountingReal x) + 1
⊢ Tendsto π atTop atTop
Theorem: infinity_of_primes₄
Proof state before:
H_log_le_primeCountingReal_add_one : ∀ (n : ℕ), ∀ x ≥ ↑n, x < ↑n + 1 → x.log ≤ ↑(primeCountingReal x) + 1
⊢ Tendsto π atTop atTop
Tactic:
sorry
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
n : ℕ
x : ℝ
hxge : x ≥ ↑n
hxlt : x < ↑n + 1
⊢ x.log ≤ ∑ k ∈ Icc 1 n, (↑k)⁻¹
Tactic:
sorry
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
n : ℕ
x : ℝ
hxge : x ≥ ↑n
hxlt : x < ↑n + 1
⊢ ∑ k ∈ Icc 1 n, (↑k)⁻¹ ≤ ∑' (m : ↑(S₁ x)), (↑↑m)⁻¹
Tactic:
sorry
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
n : ℕ
x : ℝ
hxge : x ≥ ↑n
hxlt : x < ↑n + 1
⊢ ∑' (m : ↑(S₁ x)), (↑↑m)⁻¹ ≤ ∏ p ∈ ⌊x⌋.natAbs.primesBelow, ∑' (k : ℕ), (↑p ^ k)⁻¹
Tactic:
sorry
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
n : ℕ
x : ℝ
hxge : x ≥ ↑n
hxlt : x < ↑n + 1
⊢ ∏ p ∈ ⌊x⌋.natAbs.primesBelow, ∑' (k : ℕ), (↑p ^ k)⁻¹ ≤
    ↑(∏ k ∈ Icc 1 (primeCountingReal x), nth Nat.Prime k / (nth Nat.Prime k - 1))
Tactic:
sorry
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
n : ℕ
x : ℝ
hxge : x ≥ ↑n
hxlt : x < ↑n + 1
⊢ ↑(∏ k ∈ Icc 1 (primeCountingReal x), nth Nat.Prime k / (nth Nat.Prime k - 1)) ≤
    ↑(∏ k ∈ Icc 1 (primeCountingReal x), (k + 1) / k)
Tactic:
sorry
Proof state after:
no goals
Theorem: infinity_of_primes₄
Proof state before:
n : ℕ
x : ℝ
hxge : x ≥ ↑n
hxlt : x < ↑n + 1
⊢ ↑(∏ k ∈ Icc 1 (primeCountingReal x), (k + 1) / k) ≤ ↑(primeCountingReal x) + 1
Tactic:
sorry
Proof state after:
no goals
Theorem: monotone_primeCountingReal
Proof state before:
⊢ Monotone primeCountingReal
Tactic:
intro a b hab
Proof state after:
a b : ℝ
hab : a ≤ b
⊢ primeCountingReal a ≤ primeCountingReal b
Theorem: monotone_primeCountingReal
Proof state before:
a b : ℝ
hab : a ≤ b
⊢ primeCountingReal a ≤ primeCountingReal b
Tactic:
unfold primeCountingReal
Proof state after:
a b : ℝ
hab : a ≤ b
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Theorem: monotone_primeCountingReal
Proof state before:
a b : ℝ
hab : a ≤ b
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
by_cases ha : a ≤ 0
Proof state after:
case pos
a b : ℝ
hab : a ≤ b
ha : a ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting

case neg
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Theorem: monotone_primeCountingReal
Proof state before:
case pos
a b : ℝ
hab : a ≤ b
ha : a ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
by_cases hb : b ≤ 0
Proof state after:
case pos
a b : ℝ
hab : a ≤ b
ha : a ≤ 0
hb : b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting

case neg
a b : ℝ
hab : a ≤ b
ha : a ≤ 0
hb : ¬b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Theorem: monotone_primeCountingReal
Proof state before:
case pos
a b : ℝ
hab : a ≤ b
ha : a ≤ 0
hb : b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
simp [ha, hb]
Proof state after:
no goals
Theorem: monotone_primeCountingReal
Proof state before:
case neg
a b : ℝ
hab : a ≤ b
ha : a ≤ 0
hb : ¬b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
simp [ha, hb]
Proof state after:
no goals
Theorem: monotone_primeCountingReal
Proof state before:
case neg
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
by_cases hb : b ≤ 0
Proof state after:
case pos
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
hb : b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting

case neg
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
hb : ¬b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Theorem: monotone_primeCountingReal
Proof state before:
case pos
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
hb : b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
linarith
Proof state after:
no goals
Theorem: monotone_primeCountingReal
Proof state before:
case neg
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
hb : ¬b ≤ 0
⊢ (if a ≤ 0 then 0 else ⌊a⌋₊.primeCounting) ≤ if b ≤ 0 then 0 else ⌊b⌋₊.primeCounting
Tactic:
simp [ha, hb]
Proof state after:
case neg
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
hb : ¬b ≤ 0
⊢ ⌊a⌋₊.primeCounting ≤ ⌊b⌋₊.primeCounting
Theorem: monotone_primeCountingReal
Proof state before:
case neg
a b : ℝ
hab : a ≤ b
ha : ¬a ≤ 0
hb : ¬b ≤ 0
⊢ ⌊a⌋₊.primeCounting ≤ ⌊b⌋₊.primeCounting
Tactic:
exact monotone_primeCounting <| Nat.floor_mono hab
Proof state after:
no goals
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
⊢ p / (p - 1) ≤ (k + 1) / k
Tactic:
have h_k_nonzero: k ≠ 0 := ne_iff_lt_or_gt.mpr (Or.inr hk)
Proof state after:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
⊢ p / (p - 1) ≤ (k + 1) / k
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
⊢ p / (p - 1) ≤ (k + 1) / k
Tactic:
have h_p_pred_pos: p - 1 > 0 := by linarith
Proof state after:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
⊢ p / (p - 1) ≤ (k + 1) / k
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
⊢ p / (p - 1) ≤ (k + 1) / k
Tactic:
have h_p_pred_nonzero: p - 1 ≠ 0 := ne_iff_lt_or_gt.mpr (Or.inr h_p_pred_pos)
Proof state after:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
⊢ p / (p - 1) ≤ (k + 1) / k
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
⊢ p / (p - 1) ≤ (k + 1) / k
Tactic:
have h₁: p / (p - 1) = 1 + 1 / (p - 1) := by
  rw [one_add_div h_p_pred_nonzero, sub_add_cancel]
Proof state after:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
h₁ : p / (p - 1) = 1 + 1 / (p - 1)
⊢ p / (p - 1) ≤ (k + 1) / k
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
h₁ : p / (p - 1) = 1 + 1 / (p - 1)
⊢ p / (p - 1) ≤ (k + 1) / k
Tactic:
rw [← one_add_div h_k_nonzero, h₁, add_le_add_iff_left,
  one_div_le_one_div h_p_pred_pos hk, @le_sub_iff_add_le]
Proof state after:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
h₁ : p / (p - 1) = 1 + 1 / (p - 1)
⊢ k + 1 ≤ p
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
h₁ : p / (p - 1) = 1 + 1 / (p - 1)
⊢ k + 1 ≤ p
Tactic:
exact hp
Proof state after:
no goals
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
⊢ p - 1 > 0
Tactic:
linarith
Proof state after:
no goals
Theorem: H_P4_1
Proof state before:
k p : ℝ
hk : k > 0
hp : p ≥ k + 1
h_k_nonzero : k ≠ 0
h_p_pred_pos : p - 1 > 0
h_p_pred_nonzero : p - 1 ≠ 0
⊢ p / (p - 1) = 1 + 1 / (p - 1)
Tactic:
rw [one_add_div h_p_pred_nonzero, sub_add_cancel]
Proof state after:
no goals
Theorem: prod_Icc_succ_div
Proof state before:
n : ℕ
hn : 2 ≤ n
⊢ ∏ x ∈ Icc 1 n, (↑x + 1) / ↑x = ↑n + 1
Tactic:
rw [← Nat.Ico_succ_right]
Proof state after:
n : ℕ
hn : 2 ≤ n
⊢ ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
Theorem: prod_Icc_succ_div
Proof state before:
n : ℕ
hn : 2 ≤ n
⊢ ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
Tactic:
induction' n with n h
Proof state after:
case zero
hn : 2 ≤ 0
⊢ ∏ x ∈ Ico 1 (succ 0), (↑x + 1) / ↑x = ↑0 + 1

case succ
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
⊢ ∏ x ∈ Ico 1 (n + 1).succ, (↑x + 1) / ↑x = ↑(n + 1) + 1
Theorem: prod_Icc_succ_div
Proof state before:
case zero
hn : 2 ≤ 0
⊢ ∏ x ∈ Ico 1 (succ 0), (↑x + 1) / ↑x = ↑0 + 1
Tactic:
simp
Proof state after:
no goals
Theorem: prod_Icc_succ_div
Proof state before:
case succ
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
⊢ ∏ x ∈ Ico 1 (n + 1).succ, (↑x + 1) / ↑x = ↑(n + 1) + 1
Tactic:
rw [Finset.prod_Ico_succ_top <| Nat.le_add_left 1 n]
Proof state after:
case succ
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
⊢ (∏ k ∈ Ico 1 (n + 1), (↑k + 1) / ↑k) * ((↑(n + 1) + 1) / ↑(n + 1)) = ↑(n + 1) + 1
Theorem: prod_Icc_succ_div
Proof state before:
case succ
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
⊢ (∏ k ∈ Ico 1 (n + 1), (↑k + 1) / ↑k) * ((↑(n + 1) + 1) / ↑(n + 1)) = ↑(n + 1) + 1
Tactic:
norm_num
Proof state after:
case succ
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Theorem: prod_Icc_succ_div
Proof state before:
case succ
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Tactic:
cases' lt_or_ge n 2 with _ h2
Proof state after:
case succ.inl
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
h✝ : n < 2
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1

case succ.inr
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
h2 : n ≥ 2
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Theorem: prod_Icc_succ_div
Proof state before:
case succ.inr
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
h2 : n ≥ 2
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Tactic:
specialize h h2
Proof state after:
case succ.inr
n : ℕ
hn : 2 ≤ n + 1
h2 : n ≥ 2
h : ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Theorem: prod_Icc_succ_div
Proof state before:
case succ.inr
n : ℕ
hn : 2 ≤ n + 1
h2 : n ≥ 2
h : ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Tactic:
field_simp [Finset.prod_eq_zero_iff] at h ⊢
Proof state after:
case succ.inr
n : ℕ
hn : 2 ≤ n + 1
h2 : n ≥ 2
h : ∏ x ∈ Ico 1 (n + 1), (↑x + 1) = (↑n + 1) * ∏ x ∈ Ico 1 (n + 1), ↑x
⊢ (∏ x ∈ Ico 1 (n + 1), (↑x + 1)) * (↑n + 1 + 1) = (↑n + 1 + 1) * ((∏ x ∈ Ico 1 (n + 1), ↑x) * (↑n + 1))
Theorem: prod_Icc_succ_div
Proof state before:
case succ.inr
n : ℕ
hn : 2 ≤ n + 1
h2 : n ≥ 2
h : ∏ x ∈ Ico 1 (n + 1), (↑x + 1) = (↑n + 1) * ∏ x ∈ Ico 1 (n + 1), ↑x
⊢ (∏ x ∈ Ico 1 (n + 1), (↑x + 1)) * (↑n + 1 + 1) = (↑n + 1 + 1) * ((∏ x ∈ Ico 1 (n + 1), ↑x) * (↑n + 1))
Tactic:
rw [h]
Proof state after:
case succ.inr
n : ℕ
hn : 2 ≤ n + 1
h2 : n ≥ 2
h : ∏ x ∈ Ico 1 (n + 1), (↑x + 1) = (↑n + 1) * ∏ x ∈ Ico 1 (n + 1), ↑x
⊢ ((↑n + 1) * ∏ x ∈ Ico 1 (n + 1), ↑x) * (↑n + 1 + 1) = (↑n + 1 + 1) * ((∏ x ∈ Ico 1 (n + 1), ↑x) * (↑n + 1))
Theorem: prod_Icc_succ_div
Proof state before:
case succ.inr
n : ℕ
hn : 2 ≤ n + 1
h2 : n ≥ 2
h : ∏ x ∈ Ico 1 (n + 1), (↑x + 1) = (↑n + 1) * ∏ x ∈ Ico 1 (n + 1), ↑x
⊢ ((↑n + 1) * ∏ x ∈ Ico 1 (n + 1), ↑x) * (↑n + 1 + 1) = (↑n + 1 + 1) * ((∏ x ∈ Ico 1 (n + 1), ↑x) * (↑n + 1))
Tactic:
ring
Proof state after:
no goals
Theorem: prod_Icc_succ_div
Proof state before:
case succ.inl
n : ℕ
h : 2 ≤ n → ∏ x ∈ Ico 1 n.succ, (↑x + 1) / ↑x = ↑n + 1
hn : 2 ≤ n + 1
h✝ : n < 2
⊢ ((∏ x ∈ Ico 1 (n + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (n + 1), ↑x) * ((↑n + 1 + 1) / (↑n + 1)) = ↑n + 1 + 1
Tactic:
interval_cases n
Proof state after:
case succ.inl.«0»
n : ℕ
h : 2 ≤ 0 → ∏ x ∈ Ico 1 (succ 0), (↑x + 1) / ↑x = ↑0 + 1
hn : 2 ≤ 0 + 1
h✝ : 0 < 2
⊢ ((∏ x ∈ Ico 1 (0 + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (0 + 1), ↑x) * ((↑0 + 1 + 1) / (↑0 + 1)) = ↑0 + 1 + 1

case succ.inl.«1»
n : ℕ
h : 2 ≤ 1 → ∏ x ∈ Ico 1 (succ 1), (↑x + 1) / ↑x = ↑1 + 1
hn : 2 ≤ 1 + 1
h✝ : 1 < 2
⊢ ((∏ x ∈ Ico 1 (1 + 1), (↑x + 1)) / ∏ x ∈ Ico 1 (1 + 1), ↑x) * ((↑1 + 1 + 1) / (↑1 + 1)) = ↑1 + 1 + 1