internlm/Lean-Github huggingface dataset:
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ ∃ a, ((a ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑a ^ 2 ≤ s.level) ∧ (m ∣ a ^ ⌊s.level⌋₊ ∧ a ^ ⌊s.level⌋₊ ≠ 0) ∧ a ∣ m
Tactic:
rw [Finset.mem_Icc, Nat.le_floor_iff] at hm
Proof state after:
case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ ∃ a, ((a ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑a ^ 2 ≤ s.level) ∧ (m ∣ a ^ ⌊s.level⌋₊ ∧ a ^ ⌊s.level⌋₊ ≠ 0) ∧ a ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ ∃ a, ((a ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑a ^ 2 ≤ s.level) ∧ (m ∣ a ^ ⌊s.level⌋₊ ∧ a ^ ⌊s.level⌋₊ ≠ 0) ∧ a ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Tactic:
have hm_ne_zero : m ≠ 0 := by
  exact ne_of_gt $ Nat.succ_le.mp hm.1
Proof state after:
case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∃ a, ((a ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑a ^ 2 ≤ s.level) ∧ (m ∣ a ^ ⌊s.level⌋₊ ∧ a ^ ⌊s.level⌋₊ ≠ 0) ∧ a ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∃ a, ((a ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑a ^ 2 ≤ s.level) ∧ (m ∣ a ^ ⌊s.level⌋₊ ∧ a ^ ⌊s.level⌋₊ ≠ 0) ∧ a ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Tactic:
use ∏ p in m.primeFactors, p
Proof state after:
case h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ((∏ p ∈ m.primeFactors, p ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑(∏ p ∈ m.primeFactors, p) ^ 2 ≤ s.level) ∧
    (m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0) ∧
      ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ((∏ p ∈ m.primeFactors, p ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑(∏ p ∈ m.primeFactors, p) ^ 2 ≤ s.level) ∧
    (m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0) ∧
      ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Tactic:
constructor
Proof state after:
case h.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (∏ p ∈ m.primeFactors, p ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑(∏ p ∈ m.primeFactors, p) ^ 2 ≤ s.level

case h.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0) ∧
    ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (∏ p ∈ m.primeFactors, p ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0) ∧ ↑(∏ p ∈ m.primeFactors, p) ^ 2 ≤ s.level

case h.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0) ∧
    ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Tactic:
constructor
Proof state after:
case h.left.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∏ p ∈ m.primeFactors, p ∣ s.prodPrimes ∧ s.prodPrimes ≠ 0

case h.left.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ↑(∏ p ∈ m.primeFactors, p) ^ 2 ≤ s.level

case h.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0) ∧
    ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0) ∧
    ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Tactic:
constructor
Proof state after:
case h.right.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0

case h.right.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.right.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ∧ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0

case h.right.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Tactic:
constructor
Proof state after:
case h.right.left.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ m ∣ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊

case h.right.left.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0

case h.right.right
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∏ p ∈ m.primeFactors, p ∣ m

case calc_2.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ m ≤ ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ 0 ≤ √s.level
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
⊢ 0 < ∏ p ∈ m.primeFactors, p
Tactic:
apply Finset.prod_pos
Proof state after:
case h0
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
⊢ ∀ i ∈ m.primeFactors, 0 < i
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h0
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
⊢ ∀ i ∈ m.primeFactors, 0 < i
Tactic:
intro p hp
Proof state after:
case h0
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
p : ℕ
hp : p ∈ m.primeFactors
⊢ 0 < p
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h0
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
p : ℕ
hp : p ∈ m.primeFactors
⊢ 0 < p
Tactic:
exact Nat.pos_of_mem_factors $ List.mem_toFinset.mp hp
Proof state after:
no goals
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
⊢ (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
Tactic:
apply pow_ne_zero
Proof state after:
case h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
⊢ ∏ p ∈ m.primeFactors, p ≠ 0
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
⊢ ∏ p ∈ m.primeFactors, p ≠ 0
Tactic:
apply ne_of_gt
Proof state after:
case h.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
⊢ 0 < ∏ p ∈ m.primeFactors, p
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : m ∈ Finset.Icc 1 ⌊√s.level⌋₊
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
⊢ 0 < ∏ p ∈ m.primeFactors, p
Tactic:
apply hprod_pos
Proof state after:
no goals
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
⊢ m ≠ 0
Tactic:
exact ne_of_gt $ Nat.succ_le.mp hm.1
Proof state after:
no goals
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.left.left.left
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
⊢ ∏ p ∈ m.primeFactors, p ∣ s.prodPrimes
Tactic:
apply prod_primes_dvd_of_dvd <;> intro p hp
Proof state after:
case h.left.left.left.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
p : ℕ
hp : p ∈ m.primeFactors
⊢ p ∣ s.prodPrimes

case h.left.left.left.h'
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
p : ℕ
hp : p ∈ m.primeFactors
⊢ p.Prime
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.left.left.left.h'
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
p : ℕ
hp : p ∈ m.primeFactors
⊢ p.Prime
Tactic:
exact Nat.prime_of_mem_primeFactors hp
Proof state after:
no goals
Theorem: Sieve.selbergBoundingSum_ge_sum_div
Proof state before:
case h.left.left.left.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
p : ℕ
hp : p ∈ m.primeFactors
⊢ p ∣ s.prodPrimes
Tactic:
apply hP p $ Nat.prime_of_mem_primeFactors hp
Proof state after:
case h.left.left.left.h
s : SelbergSieve
hP : ∀ (p : ℕ), p.Prime → ↑p ≤ s.level → p ∣ s.prodPrimes
hnu : CompletelyMultiplicative s.nu
hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n
hnu_lt : ∀ (p : ℕ), p.Prime → p ∣ s.prodPrimes → s.nu p < 1
m : ℕ
hm : 1 ≤ m ∧ ↑m ≤ √s.level
hprod_pos : 0 < ∏ p ∈ m.primeFactors, p
hprod_ne_zero : (∏ p ∈ m.primeFactors, p) ^ ⌊s.level⌋₊ ≠ 0
hm_ne_zero : m ≠ 0
p : ℕ
hp : p ∈ m.primeFactors
⊢ ↑p ≤ s.level