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This commit is contained in:
Stefan Kebekus
parent
4320db0533
commit
084841c35a
@ -219,6 +219,7 @@ theorem laplace_compCLMAt
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obtain ⟨u, hu₁, hu₂⟩ : ∃ u ∈ nhds x, ContDiffOn ℝ 2 f u := by
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apply ContDiffAt.contDiffOn h
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rfl
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simp
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obtain ⟨v, hv₁, hv₂, hv₃⟩ := mem_nhds_iff.1 hu₁
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use v
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constructor
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@ -281,7 +281,11 @@ lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
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unfold partialDeriv
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rw [fderiv_clm_apply]
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· simp
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· exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
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· apply Differentiable.differentiableAt
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rw [← one_add_one_eq_two] at hf
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rw [contDiff_succ_iff_fderiv] at hf
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apply hf.2.2.differentiable
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simp
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· simp
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@ -298,7 +302,9 @@ lemma partialDeriv_fderivOn
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· simp
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· convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
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apply ContDiffOn.differentiableOn _ (Preorder.le_refl 1)
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exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 hf).2
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rw [← one_add_one_eq_two] at hf
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rw [contDiffOn_succ_iff_fderiv_of_isOpen hs] at hf
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exact hf.2.2
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· simp
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@ -407,7 +413,8 @@ theorem partialDeriv_comm
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let f'' := (fderiv ℝ f' z)
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have h₁ : HasFDerivAt f' f'' z := by
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apply DifferentiableAt.hasFDerivAt
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apply (contDiff_succ_iff_fderiv.1 h).right.differentiable (Preorder.le_refl 1)
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rw [← one_add_one_eq_two] at h
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apply (contDiff_succ_iff_fderiv.1 h).2.2.differentiable (Preorder.le_refl 1)
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apply second_derivative_symmetric h₀ h₁ v₁ v₂
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@ -442,7 +449,8 @@ theorem partialDeriv_commOn
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apply DifferentiableAt.hasFDerivAt
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apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
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apply ContDiffOn.differentiableOn _ (Preorder.le_refl 1)
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exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 h).2
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rw [← one_add_one_eq_two] at h
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exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 h).2.2
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have h₀' : ∀ᶠ (y : E) in nhds z, HasFDerivAt f (f' y) y := by
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apply eventually_nhds_iff.mpr
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@ -463,7 +471,9 @@ theorem partialDeriv_commAt
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(h : ContDiffAt ℝ 2 f z) :
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∀ v₁ v₂ : E, partialDeriv ℝ v₁ (partialDeriv ℝ v₂ f) z = partialDeriv ℝ v₂ (partialDeriv ℝ v₁ f) z := by
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obtain ⟨u, hu₁, hu₂⟩ := h.contDiffOn le_rfl
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let A := h.contDiffOn le_rfl
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simp at A
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obtain ⟨u, hu₁, hu₂⟩ := A
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obtain ⟨v, hv₁, hv₂, hv₃⟩ := mem_nhds_iff.1 hu₁
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intro v₁ v₂
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@ -1,64 +1,24 @@
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{"version": "1.1.0",
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"packagesDir": ".lake/packages",
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"packages":
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[{"url": "https://github.com/leanprover-community/batteries",
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[{"url": "https://github.com/leanprover-community/mathlib4.git",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "c933dd9b00271d869e22b802a015092d1e8e454a",
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"name": "batteries",
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"scope": "",
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"rev": "7cf807751deab8d4943d867898dc1b31d61b746a",
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"name": "mathlib",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/quote4",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "303b23fbcea94ac4f96e590c1cad6618fd4f5f41",
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"name": "Qq",
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"manifestFile": "lake-manifest.json",
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"inputRev": "master",
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"inherited": true,
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"inputRev": null,
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"inherited": false,
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"configFile": "lakefile.lean"},
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{"url": "https://github.com/leanprover-community/aesop",
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{"url": "https://github.com/leanprover-community/plausible",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "de91b59101763419997026c35a41432ac8691f15",
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"name": "aesop",
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"rev": "8e5cb8d424df462f84997dd68af6f40e347c3e35",
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"name": "plausible",
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"manifestFile": "lake-manifest.json",
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"inputRev": "master",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/ProofWidgets4",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "1383e72b40dd62a566896a6e348ffe868801b172",
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"name": "proofwidgets",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v0.0.46",
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"inherited": true,
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"configFile": "lakefile.lean"},
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{"url": "https://github.com/leanprover/lean4-cli",
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"type": "git",
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"subDir": null,
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"scope": "leanprover",
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"rev": "726b3c9ad13acca724d4651f14afc4804a7b0e4d",
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"name": "Cli",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/import-graph",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "119b022b3ea88ec810a677888528e50f8144a26e",
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"name": "importGraph",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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"inputRev": "v4.15.0-rc1",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/LeanSearchClient",
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@ -71,25 +31,65 @@
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"inputRev": "main",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/plausible",
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{"url": "https://github.com/leanprover-community/import-graph",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "42dc02bdbc5d0c2f395718462a76c3d87318f7fa",
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"name": "plausible",
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"rev": "ed3b856bd8893ade75cafe13e8544d4c2660f377",
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"name": "importGraph",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v4.15.0-rc1",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/ProofWidgets4",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "2b000e02d50394af68cfb4770a291113d94801b5",
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"name": "proofwidgets",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v0.0.48",
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"inherited": true,
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"configFile": "lakefile.lean"},
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{"url": "https://github.com/leanprover-community/aesop",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "8b6048aa0a4a4b6bcf83597802d8dee734e64b7e",
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"name": "aesop",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v4.15.0-rc1",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/quote4",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "ad942fdf0b15c38bface6acbb01d63855a2519ac",
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"name": "Qq",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v4.14.0",
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"inherited": true,
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"configFile": "lakefile.lean"},
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{"url": "https://github.com/leanprover-community/batteries",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "7805acf1864ba1a2e359f86a8f092ccf1438ad83",
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"name": "batteries",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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"inherited": true,
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/mathlib4.git",
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{"url": "https://github.com/leanprover/lean4-cli",
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"type": "git",
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"subDir": null,
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"scope": "",
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"rev": "ab780650a6c8db2942d137b91ede7c94c47a4950",
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"name": "mathlib",
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"scope": "leanprover",
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"rev": "0c8ea32a15a4f74143e4e1e107ba2c412adb90fd",
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"name": "Cli",
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"manifestFile": "lake-manifest.json",
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"inputRev": null,
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"inherited": false,
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"configFile": "lakefile.lean"}],
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"inputRev": "main",
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"inherited": true,
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"configFile": "lakefile.toml"}],
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"name": "nevanlinna",
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"lakeDir": ".lake"}
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@ -1 +1 @@
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leanprover/lean4:v4.14.0-rc3
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leanprover/lean4:v4.15.0-rc1
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