More simplifications at the end of the proof

Changes

4 changed files (+24/-27)

• Sum.lean
  • @@ -188,19 +188,16 @@ theorem double_sum : converges_to (fun N : ℕ => ∑ a ∈ Ico 2 N, ∑ b ∈ I
    rw [telescope_sum n h₁] ring_nf field_simp - have h₂ : 0 < (1 : ℝ) / (-1 + n) := by + have : 0 < (1 : ℝ) / (-1 + n) := by field_simp omega - have h₃ : ∀ b ∈ Ico 2 n, 0 < (1 : ℝ) / (b ^ (n - 1) * (b - 1)) := by + have h₂ : ∀ b ∈ Ico 2 n, 0 < (1 : ℝ) / (b ^ (n - 1) * (b - 1)) := by intro b bico field_simp have : 1 < (b : ℝ) := Nat.one_lt_cast.mpr (List.left_le_of_mem_range' bico) - have h₄ : 0 < (b : ℝ) ^ (n - 1) := by + have h₃ : 0 < (b : ℝ) ^ (n - 1) := by apply pow_pos (by linarith) - exact mul_pos h₄ (by linarith) - have h₄ : 0 ≤ ∑ b ∈ Ico 2 n, (1 : ℝ) / (b ^ (n - 1) * (b - 1)) := sum_nonneg (fun b a => le_of_lt (h₃ b a)) - ring_nf at h₄ - field_simp at h₄ - suffices -(1 : ℝ) / (-1 + n) < 0 by linarith - nth_rw 1 [← mul_neg_one, mul_comm, mul_div_assoc] - linarith + exact mul_pos h₃ (by linarith) + have h₃ : 0 ≤ ∑ b ∈ Ico 2 n, (1 : ℝ) / (b ^ (n - 1) * (b - 1)) := sum_nonneg (fun b a => le_of_lt (h₂ b a)) + field_simp at h₃ + grind
• lake-manifest.json
  • @@ -5,20 +5,20 @@
    "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "79e94a093aff4a60fb1b1f92d9681e407124c2ca", + "rev": "90c0e186fd333cbdddbe0e5148aa3cd2d24dcf9a", "name": "mathlib", "manifestFile": "lake-manifest.json", - "inputRev": "v4.22.0", + "inputRev": "v4.23.0-rc2", "inherited": false, "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover-community/plausible", "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "b100ad4c5d74a464f497aaa8e7c74d86bf39a56f", + "rev": "240eddc1bb31420fbbc57fe5cc579435c2522493", "name": "plausible", "manifestFile": "lake-manifest.json", - "inputRev": "v4.22.0", + "inputRev": "main", "inherited": true, "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/LeanSearchClient",
  • @@ -35,57 +35,57 @@
    "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "eb164a46de87078f27640ee71e6c3841defc2484", + "rev": "dba7fbc707774d1ba830fd44d7f92a717e9bf57f", "name": "importGraph", "manifestFile": "lake-manifest.json", - "inputRev": "v4.22.0", + "inputRev": "main", "inherited": true, "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/ProofWidgets4", "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "1253a071e6939b0faf5c09d2b30b0bfc79dae407", + "rev": "6e47cc88cfbf1601ab364e9a4de5f33f13401ff8", "name": "proofwidgets", "manifestFile": "lake-manifest.json", - "inputRev": "v0.0.68", + "inputRev": "v0.0.71", "inherited": true, "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover-community/aesop", "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "1256a18522728c2eeed6109b02dd2b8f207a2a3c", + "rev": "523c8ee53f7057447fc62ec14e506fda4cf63dfa", "name": "aesop", "manifestFile": "lake-manifest.json", - "inputRev": "v4.22.0", + "inputRev": "master", "inherited": true, "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/quote4", "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "917bfa5064b812b7fbd7112d018ea0b4def25ab3", + "rev": "f85ad59c9b60647ef736719c23edd4578f723806", "name": "Qq", "manifestFile": "lake-manifest.json", - "inputRev": "v4.22.0", + "inputRev": "master", "inherited": true, "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/batteries", "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "240676e9568c254a69be94801889d4b13f3b249f", + "rev": "6e89c7370ca3a91b7d1f29ef7d727a9d027d7b0d", "name": "batteries", "manifestFile": "lake-manifest.json", - "inputRev": "v4.22.0", + "inputRev": "main", "inherited": true, "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover/lean4-cli", "type": "git", "subDir": null, "scope": "leanprover", - "rev": "c682c91d2d4dd59a7187e2ab977ac25bd1f87329", + "rev": "cacb481a1eaa4d7d4530a27b606c60923da21caf", "name": "Cli", "manifestFile": "lake-manifest.json", "inputRev": "main",
• lakefile.toml
  • @@ -13,4 +13,4 @@ root = "Gcd"
    [[require]] name = "mathlib" scope = "leanprover-community" - rev = "v4.22.0" + rev = "v4.23.0-rc2"
• lean-toolchain
  • @@ -1,1 +1,1 @@
    - leanprover/lean4:v4.22.0 + leanprover/lean4:v4.23.0-rc2