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Stefan Kebekus
parent
c59e12a468
commit
6412671bc6
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@ -1,51 +1,60 @@
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import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Analysis.InnerProductSpace.Dual
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import Mathlib.Analysis.InnerProductSpace.PiL2
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import Mathlib.Algebra.BigOperators.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Bounds
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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import Mathlib.LinearAlgebra.Basis
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import Mathlib.LinearAlgebra.Contraction
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open BigOperators
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open Finset
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lemma OrthonormalBasis.sum_repr'
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{𝕜 : Type*} [RCLike 𝕜]
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
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[Fintype ι]
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(b : OrthonormalBasis ι 𝕜 E)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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simp_rw [b.repr_apply_apply v]
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noncomputable def realInnerAsElementOfDualTensorprod
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
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: TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
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noncomputable def dual'
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
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: (TensorProduct ℝ E E →ₗ[ℝ] ℝ) ≃ₗ[ℝ] TensorProduct ℝ E E := by
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let d := InnerProductSpace.toDual ℝ E
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let e := d.toLinearEquiv
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let a := TensorProduct.congr e e
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let b := homTensorHomEquiv ℝ E E ℝ ℝ
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sorry
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variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
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lemma vectorPresentation'
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theorem InvariantTensor
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[Fintype ι]
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(b : OrthonormalBasis ι ℝ E)
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--(hb : Orthonormal ℝ b)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
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(v : Basis ι ℝ E)
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(c : ι → ℝ)
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(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
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L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
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let A := b.sum_repr v
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let i : ι := by sorry
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let B := b.repr v i
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rw [L.map_sum]
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simp_rw [L.map_smul_univ]
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sorry
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nth_rw 1 [← (b.sum_repr v)]
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apply Fintype.sum_congr
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intro i
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--let A := b.repr v
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--have : (b.repr v) = ((OrthonormalBasis.toBasis b).repr v) := by tauto
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rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
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simp
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lemma vectorPresentation
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[Fintype ι]
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(b : Basis ι ℝ E)
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(hb : Orthonormal ℝ b)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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apply Fintype.sum_congr
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intro i
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rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
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simp
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theorem BilinearCalc
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[Fintype ι]
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@ -55,14 +64,10 @@ theorem BilinearCalc
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L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
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rw [L.map_sum]
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conv =>
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left
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arg 2
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intro r
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rw [L.map_smul_univ]
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simp
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simp_rw [L.map_smul_univ]
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sorry
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/-
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lemma c2
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[Fintype ι]
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(b : Basis ι ℝ E)
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@ -79,7 +84,7 @@ lemma c2
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intro i'
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rw [Orthonormal.inner_left_fintype hb]
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rw [Orthonormal.inner_left_fintype hb]
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-/
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lemma fin_sum
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[Fintype ι]
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@ -94,10 +99,8 @@ lemma fin_sum
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theorem LaplaceIndep
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[Fintype ι] [DecidableEq ι]
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(v₁ : Basis ι ℝ E)
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(hv₁ : Orthonormal ℝ v₁)
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(v₂ : Basis ι ℝ E)
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(hv₂ : Orthonormal ℝ v₂)
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(v₁ : OrthonormalBasis ι ℝ E)
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(v₂ : OrthonormalBasis ι ℝ E)
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(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
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∑ i, L (fun _ ↦ v₁ i) = ∑ i, L (fun _ => v₂ i) := by
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@ -116,7 +119,7 @@ theorem LaplaceIndep
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right
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arg 2
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intro i
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rw [vectorPresentation v₁ hv₁ (v₂ i)]
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rw [v₁.sum_repr' (v₂ i)]
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rw [BilinearCalc]
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rw [Finset.sum_comm]
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conv =>
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@ -160,10 +163,8 @@ noncomputable def Laplace_wrt_basis
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theorem LaplaceIndep'
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[Fintype ι] [DecidableEq ι]
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(v₁ : Basis ι ℝ E)
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(hv₁ : Orthonormal ℝ v₁)
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(v₂ : Basis ι ℝ E)
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(hv₂ : Orthonormal ℝ v₂)
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(v₁ : OrthonormalBasis ι ℝ E)
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(v₂ : OrthonormalBasis ι ℝ E)
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(f : E → F)
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: (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
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@ -1,11 +1,11 @@
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{"version": 7,
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{"version": "1.0.0",
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"packagesDir": ".lake/packages",
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"packages":
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[{"url": "https://github.com/leanprover/std4",
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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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"rev": "e840c18f7334c751efbd4cfe531476e10c943cdb",
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"name": "std",
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"rev": "555ec79bc6effe7009036184a5f73f7d8d4850ed",
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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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@ -13,7 +13,7 @@
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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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"rev": "64365c656d5e1bffa127d2a1795f471529ee0178",
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"rev": "a7bfa63f5dddbcab2d4e0569c4cac74b2585e2c6",
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"name": "Qq",
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"manifestFile": "lake-manifest.json",
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"inputRev": "master",
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@ -22,25 +22,25 @@
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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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"rev": "5fefb40a7c9038a7150e7edd92e43b1b94c49e79",
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"rev": "30619d94ce4a3d69cdb87bb1771562ca2e687cfa",
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"name": "aesop",
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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.lean"},
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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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"rev": "fb65c476595a453a9b8ffc4a1cea2db3a89b9cd8",
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"rev": "e6b6247c61280c77ade6bbf0bc3c66a44fe2e0c5",
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"name": "proofwidgets",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v0.0.30",
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"inputRev": "v0.0.36",
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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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"rev": "be8fa79a28b8b6897dce0713ef50e89c4a0f6ef5",
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"rev": "a11566029bd9ec4f68a65394e8c3ff1af74c1a29",
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"name": "Cli",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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@ -49,16 +49,16 @@
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{"url": "https://github.com/leanprover-community/import-graph.git",
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"type": "git",
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"subDir": null,
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"rev": "61a79185b6582573d23bf7e17f2137cd49e7e662",
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"rev": "1588be870b9c76fe62286e8f42f0b4dafa154c96",
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"name": "importGraph",
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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.lean"},
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"configFile": "lakefile.toml"},
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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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"rev": "e408da18380ef42acf7b8337c6b0536bfac09105",
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"rev": "9c4c6f78c9a1b7beba018504e284d497a3761af2",
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"name": "mathlib",
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"manifestFile": "lake-manifest.json",
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"inputRev": null,
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@ -1 +1 @@
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leanprover/lean4:v4.7.0
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leanprover/lean4:v4.9.0-rc3
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