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{
"id": "mathlibmpr_003",
"pr": 27229,
"formal_main_result": "DoubleCoset.finite_quotient_iff_exists_finset_iUnion_eq_univ",
"NL_main_result": "For subgroups `H, K ≤ G`, the double-coset quotient `H \\ G / K` is finite if and only if there exists a finite collection of cosets in this quotient whose double-coset preimages cover `G`.",
"formal_statement": "import Mathlib.GroupTheory.DoubleCoset\n\nopen DoubleCoset\n\nvariable {G : Type*} [Group G]\n\n/-- **Main result.** The double-coset quotient `H \\ G / K` is finite if and\nonly if there exists a finite collection of cosets in this quotient whose\ndouble-coset preimages cover `G`. -/\nlemma finite_quotient_iff_exists_finset_iUnion_eq_univ (H K : Subgroup G) :\n Finite (Quotient (H : Set G) K) ↔\n ∃ I : Finset (Quotient (H : Set G) K), ⋃ i ∈ I, quotToDoubleCoset H K i = .univ := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"DoubleCoset.union_quotToDoubleCoset",
"DoubleCoset.mk_out_eq_mul"
]
}
]
},
{
"id": "mathlibmpr_005",
"pr": 29235,
"formal_main_result": "ClosedSubmodule.closure_map_eq_mapEquiv_closure",
"NL_main_result": "For a continuous `R`-linear equivalence `f : M ≃L[R] N` and any submodule `s : Submodule R M`, the closure of the image `s.map f` equals the image under `mapEquiv f` of the closure of `s`. In other words, \"closure of image = image of closure\" through a continuous linear equivalence.",
"formal_statement": "import Mathlib\n\nnamespace ClosedSubmodule\n\nvariable {R M N : Type*} [Ring R]\n [TopologicalSpace M] [AddCommGroup M] [Module R M]\n [TopologicalSpace N] [AddCommGroup N] [Module R N]\n (f : M ≃L[R] N)\n\n/-- Action of a continuous `R`-linear equivalence on closed submodules. -/\ndef mapEquiv : ClosedSubmodule R M ≃ ClosedSubmodule R N where\n toFun s := ⟨s.toSubmodule.map f.toLinearMap, by simpa using s.isClosed⟩\n invFun t := ⟨t.toSubmodule.map f.symm.toLinearMap, by simpa using t.isClosed⟩\n left_inv := by intro _; ext _; simp\n right_inv := by intro _; ext _; simp\n\nvariable [ContinuousAdd M] [ContinuousConstSMul R M] [ContinuousAdd N] [ContinuousConstSMul R N]\n\n/-- **Main result.** For a continuous `R`-linear equivalence `f : M ≃L[R] N`\nand any submodule `s : Submodule R M`, closure of image = image of closure:\n`(s.map f).closure = s.closure.mapEquiv f`. -/\n@[simp]\ntheorem closure_map_eq_mapEquiv_closure (s : Submodule R M) :\n (s.map f.toLinearMap).closure = (s.closure).mapEquiv f := sorry\n\nend ClosedSubmodule",
"premise_group": [
{
"kind": "original",
"docs": [
"ContinuousLinearEquiv.image_closure",
"IsClosedMap.closure_image_eq_of_continuous"
]
},
{
"kind": "alternative",
"docs": [
"Homeomorph.image_closure",
"IsClosedMap.closure_image_eq_of_continuous"
]
}
]
},
{
"id": "mathlibmpr_006",
"pr": 30112,
"formal_main_result": "ProbabilityTheory.Kernel.exists_measurable_map_eq_unitInterval",
"NL_main_result": "Let `Y` be a standard Borel space and `κ : Kernel X Y` a Markov kernel. Then there exists a jointly measurable map `f : X → I → Y` such that for every `a : X`, the pushforward of the uniform measure `volume` on the unit interval `I` by `f a` equals `κ a`.",
"formal_statement": "import Mathlib\n\nopen MeasureTheory unitInterval\n\nnamespace ProbabilityTheory.Kernel\n\nvariable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] [StandardBorelSpace Y]\n [Nonempty Y]\n\n/-- **Main result.** Kallenberg's \"isolation of randomness\": for any Markov\nkernel `κ : Kernel X Y` with `Y` a standard Borel space, there exists a jointly\nmeasurable map `f : X → I → Y` such that for every `a : X` the pushforward of\nthe uniform measure on `I` by `f a` equals `κ a`. -/\ntheorem exists_measurable_map_eq_unitInterval (κ : Kernel X Y) [IsMarkovKernel κ] :\n ∃ f : X → unitInterval → Y,\n Measurable (Function.uncurry f) ∧\n ∀ a, Measure.map (f a) (volume : Measure unitInterval) = κ a := sorry\n\nend ProbabilityTheory.Kernel",
"premise_group": [
{
"kind": "original",
"docs": [
"measurableEmbedding_sigmoid_comp_embeddingReal",
"MeasurableEmbedding.comp"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.ext_of_Iic"
]
},
{
"kind": "alternative",
"docs": [
"MeasureTheory.exists_subset_real_measurableEquiv"
]
}
]
},
{
"id": "mathlibmpr_008",
"pr": 31121,
"formal_main_result": "GradedObject.eulerChar_eq_sum_finSet_of_finrankSupport_subset",
"NL_main_result": "Let `c` be a complex shape on an index type `ι` with an `EulerCharSigns` instance, and let `X` be a graded `R`-module object with `finrank R (X i) = 0` for every `i` outside a given finite set `s`. Then the `finsum` Euler characteristic equals the explicit finite sum: `eulerChar c X = ∑ i ∈ s, (χ i : ℤ) · finrank R (X i)`.",
"formal_statement": "import Mathlib\n\nopen CategoryTheory\n\nnamespace ComplexShape\n\nvariable {ι : Type*}\n\n/-- Alternating-sign data on the index type of a complex shape. -/\nclass EulerCharSigns (c : ComplexShape ι) where\n χ : ι → ℤˣ\n χ_neg_of_rel : ∀ ⦃i j⦄, c.Rel i j → χ j = -χ i\n\nend ComplexShape\n\nnamespace GradedObject\n\nvariable {ι R : Type*} [Ring R] (c : ComplexShape ι) [c.EulerCharSigns]\n {C : Type*} [CategoryTheory.Category C]\n\nvariable (X : ι → ModuleCat.{0} R)\n\n/-- The set of indices `i` with `finrank R (X i) ≠ 0`. -/\ndef finrankSupport : Set ι := Function.support (fun i => Module.finrank R (X i))\n\n/-- Generalized Euler characteristic of an `ι`-graded module. -/\nnoncomputable def eulerChar : ℤ :=\n ∑ᶠ i, (ComplexShape.EulerCharSigns.χ (c := c) i : ℤ) * Module.finrank R (X i)\n\n/-- **Main result.** If `finrank R (X i) = 0` for every `i ∉ s`, then the\n`finsum` Euler characteristic equals the explicit finite sum\n`∑ i ∈ s, (χ i : ℤ) · finrank R (X i)`. -/\ntheorem eulerChar_eq_sum_finSet_of_finrankSupport_subset\n (s : Finset ι) (hs : finrankSupport X ⊆ s) :\n eulerChar c X\n = ∑ i ∈ s, (ComplexShape.EulerCharSigns.χ (c := c) i : ℤ) * Module.finrank R (X i) :=\n sorry\n\nend GradedObject",
"premise_group": [
{
"kind": "original",
"docs": [
"finsum_eq_sum_of_support_subset",
"finsum_eq_finset_sum_of_support_subset",
"finsum_eq_sum_of_support_subset_of_finite"
]
}
]
},
{
"id": "mathlibmpr_009",
"pr": 31219,
"formal_main_result": "IsBaseChange.of_left_exact",
"NL_main_result": "Let `S` be a flat `R`-algebra and consider a commutative diagram with exact rows ``` M₁ --f--> M₂ --g--> M₃ | | | i₁ i₂ i₃ v v v N₁ --f'-> N₂ --g'-> N₃ ``` where `f, f'` are injective. If `i₂` and `i₃` exhibit `Nₖ` as the base change of `Mₖ` from `R` to `S`, then so does `i₁`.",
"formal_statement": "import Mathlib.RingTheory.IsTensorProduct\nimport Mathlib.RingTheory.Flat.Basic\nimport Mathlib.LinearAlgebra.TensorProduct.Tower\n\nvariable {R : Type*} [CommRing R] (S : Type*) [CommRing S] [Algebra R S]\n\nvariable {M₁ M₂ M₃ N₁ N₂ N₃ : Type*}\n [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃]\n [AddCommGroup N₁] [AddCommGroup N₂] [AddCommGroup N₃]\n [Module R M₁] [Module R M₂] [Module R M₃]\n [Module R N₁] [Module R N₂] [Module R N₃]\n [Module S N₁] [Module S N₂] [Module S N₃]\n [IsScalarTower R S N₁] [IsScalarTower R S N₂] [IsScalarTower R S N₃]\n (h₁ : M₁ →ₗ[R] N₁) (h₂ : M₂ →ₗ[R] N₂) (h₃ : M₃ →ₗ[R] N₃)\n {f : M₁ →ₗ[R] M₂} {g : M₂ →ₗ[R] M₃} {f' : N₁ →ₗ[S] N₂} {g' : N₂ →ₗ[S] N₃}\n\n/-- **Main result.** Given a commutative diagram of two left-exact\nsequences linked by `R`-linear maps `i₁, i₂, i₃` (with `f, f'` injective),\nif `S` is a flat `R`-algebra and `i₂, i₃` exhibit base change by `S`,\nthen so does `i₁`. -/\nlemma IsBaseChange.of_left_exact\n (comm₁ : h₂.comp f = (f'.restrictScalars R).comp h₁)\n (comm₂ : h₃.comp g = (g'.restrictScalars R).comp h₂) [Module.Flat R S]\n (isb₂ : IsBaseChange S h₂) (isb₃ : IsBaseChange S h₃)\n (exact₁ : Function.Exact f g) (inj₁ : Function.Injective f)\n (exact₂ : Function.Exact f' g') (inj₂ : Function.Injective f') :\n IsBaseChange S h₁ := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"Module.Flat.lTensor_exact",
"Module.Flat.iff_lTensor_exact",
"Module.Flat.iff_lTensor_exact'",
"Module.Flat.rTensor_exact"
]
},
{
"kind": "original",
"docs": [
"Module.Flat.lTensor_preserves_injective_linearMap",
"Module.Flat.rTensor_preserves_injective_linearMap",
"Module.Flat.iff_lTensor_preserves_injective_linearMap"
]
},
{
"kind": "original",
"docs": [
"MonoidHom.bijective_of_bijective_of_injective_of_left_exact"
]
}
]
},
{
"id": "mathlibmpr_010",
"pr": 31388,
"formal_main_result": "exists_isExtrOn_uIoo_of_tendsto",
"NL_main_result": "Let `α` be a densely-ordered linearly-ordered topological space and `β` a `T₂` regular space. If `f : α → β` is continuous on the unordered open interval `uIoo a b` (where `a ≠ b`) and tends, within `uIoo a b`, to the same limit `l` as the argument approaches either endpoint, then `f` attains a global extremum on `uIoo a b` at some interior point `c ∈ uIoo a b`.",
"formal_statement": "import Mathlib\n\nopen Set Filter Topology\n\nvariable {α β : Type*}\n [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [DenselyOrdered α] [OrderTopology α]\n [LinearOrder β] [TopologicalSpace β] [OrderTopology β]\n {f : α → β} {a b : α} {l : β}\n\n/-- **Main result.** If `f` is continuous on the unordered open\ninterval `uIoo a b` (with `a ≠ b`) and tends to the same limit `l`\nalong the neighbourhoods of both endpoints within the interval,\nthen `f` attains a global extremum on `uIoo a b` at some interior\npoint `c ∈ uIoo a b`. -/\nlemma exists_isExtrOn_uIoo_of_tendsto (hab : a ≠ b) (hfc : ContinuousOn f (uIoo a b))\n (ha : Tendsto f (𝓝[uIoo a b] a) (𝓝 l)) (hb : Tendsto f (𝓝[uIoo a b] b) (𝓝 l)) :\n ∃ c ∈ uIoo a b, IsExtrOn f (uIoo a b) c := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"exists_Ioo_extr_on_Icc",
"exists_isLocalExtr_Ioo"
]
}
]
},
{
"id": "mathlibmpr_011",
"pr": 31411,
"formal_main_result": "ConvexOn.sSup_affine_eq",
"NL_main_result": "Let `E` be a real locally convex topological vector space (with `𝕜 = ℝ` or `ℂ` providing the affine functionals), `s ⊆ E` a closed convex subset, and `φ : E → ℝ` a function that is convex and lower-semicontinuous on `s`. Then the family ``` 𝓕 := { f : s → ℝ | f ≤ φ|_s ∧ f = (Re ∘ l)|_s + c for some l ∈ E →L[𝕜] 𝕜, c ∈ ℝ } ``` of affine minorants of `φ|_s` has supremum equal to `φ|_s`.",
"formal_statement": "import Mathlib\n\nopen Function Set RCLike ContinuousLinearMap\n\nnamespace ConvexOn\n\nvariable {𝕜 E : Type*} {s : Set E} {φ : E → ℝ} [RCLike 𝕜]\n [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [AddGroup E]\n [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] [Module 𝕜 E] [ContinuousSMul 𝕜 E]\n\n/-- **Main result.** Let `E` be a real locally convex topological\nvector space, `s ⊆ E` a closed convex subset, and `φ : E → ℝ` convex\nand lower-semicontinuous on `s`. Then `φ|_s` is the supremum of all\nits continuous affine minorants of the form `(Re ∘ l)|_s + c`,\n`l ∈ E →L[𝕜] 𝕜`, `c ∈ ℝ`. -/\ntheorem sSup_affine_eq (hsc : IsClosed s)\n (hφc : LowerSemicontinuousOn φ s) (hφcv : ConvexOn ℝ s φ) :\n sSup {f : s → ℝ | f ≤ s.restrict φ ∧\n ∃ (l : E →L[𝕜] 𝕜) (c : ℝ), f = s.restrict (re ∘ l) + Function.const s c} =\n s.restrict φ := sorry\n\nend ConvexOn",
"premise_group": [
{
"kind": "original",
"docs": [
"geometric_hahn_banach_point_closed",
"geometric_hahn_banach_closed_point",
"RCLike.geometric_hahn_banach_point_closed"
]
},
{
"kind": "original",
"docs": [
"Convex.linear_preimage",
"Convex.is_linear_preimage"
]
},
{
"kind": "original",
"docs": [
"lowerSemicontinuousOn_iff_isClosed_epigraph"
]
},
{
"kind": "original",
"docs": [
"ConvexOn.convex_epigraph",
"convexOn_iff_convex_epigraph"
]
},
{
"kind": "alternative",
"docs": [
"iInter_halfSpaces_eq",
"RCLike.iInter_halfSpaces_eq"
]
}
]
},
{
"id": "mathlibmpr_012",
"pr": 31719,
"formal_main_result": "Complex.borelCaratheodory_zero",
"NL_main_result": "Let `f : ℂ → ℂ` be holomorphic on the open ball `‖z‖ < R` with `f 0 = 0`, and assume `Re (f z) < M` for every `z` in the ball (with `0 < M`, `0 < R`). Then for every `z` with `‖z‖ < R`, ``` ‖f z‖ ≤ 2 M ‖z‖ / (R − ‖z‖). ``` (The general Borel–Carathéodory bound `borelCaratheodory` follows by applying this to `f − f 0` after enlarging `M` to `M + ‖f 0‖`; see field 8.)",
"formal_statement": "import Mathlib.Analysis.Complex.Schwarz\n\nopen Metric Set\n\nnamespace Complex\n\nvariable {f : ℂ → ℂ} {M R : ℝ} {z : ℂ}\n\n/-- **Main result.** Borel–Carathéodory (case `f 0 = 0`). If `f` is\nholomorphic on the open ball `‖z‖ < R` with `f 0 = 0`, and\n`Re (f z) < M` for every `z` in the ball (with `0 < M`, `0 < R`), then\nfor every `z` with `‖z‖ < R`,\n`‖f z‖ ≤ 2 * M * ‖z‖ / (R - ‖z‖)`. -/\ntheorem borelCaratheodory_zero (hM : 0 < M) (hf : DifferentiableOn ℂ f (ball 0 R))\n (hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}) (hR : 0 < R) (hz : z ∈ ball 0 R)\n (hf₂ : f 0 = 0) : ‖f z‖ ≤ 2 * M * ‖z‖ / (R - ‖z‖) := sorry\n\nend Complex",
"premise_group": [
{
"kind": "original",
"docs": [
"Complex.dist_le_div_mul_dist_of_mapsTo_ball"
]
}
]
},
{
"id": "mathlibmpr_014",
"pr": 32186,
"formal_main_result": "norm_sub_le_integral_of_norm_deriv_le_of_le",
"NL_main_result": "Let `f : ℝ → E` be continuous on `[a, b]` and differentiable on `(a, b)`, with `‖f'(t)‖ ≤ B(t)` for almost every `t ∈ (a, b)` and `B` interval-integrable on `[a, b]`. Then",
"formal_statement": "import Mathlib\n\nopen MeasureTheory intervalIntegral\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]\n {f : ℝ → E} {B : ℝ → ℝ} {a b : ℝ}\n\n/-- **Main result.** If `f : ℝ → E` is continuous on `[a,b]` and differentiable\non `(a,b)` with `‖deriv f t‖ ≤ B t` a.e., and `B` is interval-integrable on\n`[a,b]`, then `‖f b − f a‖ ≤ ∫ t in a..b, B t`. -/\ntheorem norm_sub_le_integral_of_norm_deriv_le_of_le\n (hab : a ≤ b)\n (hfc : ContinuousOn f (Set.Icc a b))\n (hfd : DifferentiableOn ℝ f (Set.Ioo a b))\n (hfB : ∀ᵐ t, t ∈ Set.Ioo a b → ‖deriv f t‖ ≤ B t)\n (hBi : IntervalIntegrable B MeasureTheory.volume a b) :\n ‖f b - f a‖ ≤ ∫ t in a..b, B t := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le",
"intervalIntegral.integral_eq_sub_of_hasDeriv_right"
]
},
{
"kind": "original",
"docs": [
"intervalIntegral.norm_integral_le_integral_norm",
"MeasureTheory.norm_integral_le_integral_norm"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.integral_mono_ae"
]
}
]
},
{
"id": "mathlibmpr_015",
"pr": 32260,
"formal_main_result": "EuclideanGeometry.angle_eq_angle_div_two_of_oangle_eq_of_sSameSide",
"NL_main_result": "Let `p₁, p₂, p₃, p₄` be points in a real Euclidean affine space with `p₁ ≠ p₂`. If `p₃` bisects the oriented angle `∡p₁p₂p₄` (i.e. `∡p₁p₂p₃ = ∡p₃p₂p₄`) and `p₃, p₄` lie on the same (strictly defined) side of the line through `p₁` and `p₂`, then the unoriented angle satisfies",
"formal_statement": "import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine\n\nnamespace EuclideanGeometry\n\nvariable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]\n [NormedAddTorsor V P] [Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)]\n\n/-- **Main result.** Let `p₁ p₂ p₃ p₄` be points with `p₁ ≠ p₂`. If `p₃` bisects\nthe oriented angle `∡p₁p₂p₄` (i.e. `∡p₁p₂p₃ = ∡p₃p₂p₄`) and `p₃, p₄` lie on the\nsame strict side of the line through `p₁` and `p₂`, then\n`∠p₁p₂p₃ = ∠p₁p₂p₄ / 2`. -/\ntheorem angle_eq_angle_div_two_of_oangle_eq_of_sSameSide\n {p₁ p₂ p₃ p₄ : P} (hp₁p₂ : p₁ ≠ p₂)\n (h : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄)\n (hside : (line[ℝ, p₁, p₂] : AffineSubspace ℝ P).SSameSide p₃ p₄) :\n ∠ p₁ p₂ p₃ = ∠ p₁ p₂ p₄ / 2 := sorry\n\nend EuclideanGeometry",
"premise_group": [
{
"kind": "original",
"docs": [
"AffineSubspace.SSameSide.oangle_sign_eq"
]
},
{
"kind": "original",
"docs": [
"EuclideanGeometry.angle_eq_abs_oangle_toReal",
"Orientation.angle_eq_abs_oangle_toReal"
]
}
]
},
{
"id": "mathlibmpr_016",
"pr": 32744,
"formal_main_result": "Set.Definable.preimage_map",
"NL_main_result": "Let `L` be a first-order language, `M` an `L`-structure, `A ⊆ M` a parameter set, `β` a finite type, and `F : (α → M) → (β → M)` a *definable map* over `A` (each coordinate `v ↦ F(v)(i)` is a definable function over `A`). For every `A`-definable set `S ⊆ (β → M)`, the preimage `F⁻¹(S)` is `A`-definable.",
"formal_statement": "import Mathlib.ModelTheory.Definability\n\nnamespace Set\n\nopen FirstOrder Language\n\nvariable {M : Type*} {L : Language} [L.Structure M]\n {α β : Type*} (A : Set M)\n\n/-- A function whose tuple-graph is definable over `A`. -/\ndef DefinableFun (f : (α → M) → M) : Prop :=\n Definable A L { vw : α ⊕ Unit → M | vw (Sum.inr ()) = f (vw ∘ Sum.inl) }\n\n/-- A tuple-valued map is `A`-definable iff each coordinate is. -/\ndef DefinableMap (F : (α → M) → (β → M)) : Prop :=\n ∀ i : β, DefinableFun (L := L) A (fun v => F v i)\n\n/-- **Main result.** For a finite type `β` and an `A`-definable map\n`F : (α → M) → (β → M)`, the preimage of any `A`-definable set\n`S ⊆ (β → M)` under `F` is `A`-definable. -/\ntheorem Definable.preimage_map [Finite β] {F : (α → M) → (β → M)}\n (hF : DefinableMap (L := L) A F) {S : Set (β → M)} (hS : Definable A L S) :\n Definable A L (F ⁻¹' S) := sorry\n\nend Set",
"premise_group": [
{
"kind": "original",
"docs": [
"Set.Definable.preimage_comp"
]
},
{
"kind": "original",
"docs": [
"Set.definable_finset_inf",
"Set.definable_biInter_finset",
"Set.definable_finset_biInter"
]
},
{
"kind": "original",
"docs": [
"FirstOrder.Language.Formula.realize_iExs"
]
}
]
},
{
"id": "mathlibmpr_017",
"pr": 33189,
"formal_main_result": "ModelWithCorners.isOpen_interior",
"NL_main_result": "Let `M` be a C¹ manifold modelled on `(E, H, I : ModelWithCorners 𝕜 E H)` with `n ≠ 0`. Then `I.interior M = { x : M | extChartAt I x x ∈ interior (range I) }` is open.",
"formal_statement": "import Mathlib\n\nnamespace ModelWithCorners\n\nvariable {𝕜 : Type*} [NontriviallyNormedField 𝕜]\n {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]\n {H : Type*} [TopologicalSpace H]\n (I : ModelWithCorners 𝕜 E H)\n (M : Type*) [TopologicalSpace M] [ChartedSpace H M]\n {n : WithTop ℕ∞} [IsManifold I n M]\n\n/-- **Main result.** For a C¹ manifold (`n ≥ 1`), the interior\n`I.interior M = { x : M | extChartAt I x x ∈ interior (range I) }` is open. -/\nprotected theorem isOpen_interior (hn : n ≠ 0) : IsOpen (I.interior M) := sorry\n\nend ModelWithCorners",
"premise_group": [
{
"kind": "original",
"docs": [
"geometric_hahn_banach_point_open",
"geometric_hahn_banach_open_point",
"RCLike.geometric_hahn_banach_point_open",
"RCLike.geometric_hahn_banach_open_point"
]
},
{
"kind": "original",
"docs": [
"IsLocalMax.fderiv_eq_zero",
"IsLocalMax.hasFDerivAt_eq_zero",
"IsLocalExtr.fderiv_eq_zero",
"IsLocalExtr.hasFDerivAt_eq_zero"
]
},
{
"kind": "alternative",
"docs": [
"HasStrictFDerivAt.to_localInverse",
"ContDiffAt.to_localInverse"
]
}
]
},
{
"id": "mathlibmpr_018",
"pr": 33275,
"formal_main_result": "Polynomial.Chebyshev.eval_iterate_derivative_le_of_forall_abs_le_one",
"NL_main_result": "Let `P : ℝ[X]` have `deg P ≤ n` and satisfy `|P(x)| ≤ 1` for all `x ∈ [-1, 1]`. Then for every `k ∈ ℕ` and every `x ≥ 1`, `(P^{(k)})(x) ≤ (T_n^{(k)})(x)`, where `T_n` is the `n`-th Chebyshev polynomial of the first kind.",
"formal_statement": "import Mathlib\n\nnamespace Polynomial.Chebyshev\n\nvariable {n k : ℕ} {x : ℝ}\n\n/-- **Main result.** Let `P : ℝ[X]` have `degree P ≤ n` and satisfy\n`|P(x)| ≤ 1` for all `x ∈ [-1, 1]`. Then for every `k ∈ ℕ` and every `x ≥ 1`,\n`(P^{(k)})(x) ≤ (T_n^{(k)})(x)`. -/\ntheorem eval_iterate_derivative_le_of_forall_abs_le_one\n {P : Polynomial ℝ} (hdeg : P.degree ≤ n)\n (hbound : ∀ x ∈ Set.Icc (-1 : ℝ) 1, |P.eval x| ≤ 1)\n (hx : 1 ≤ x) :\n ((Polynomial.derivative^[k] P).eval x)\n ≤ ((Polynomial.derivative^[k] (Polynomial.Chebyshev.T ℝ n)).eval x) := sorry\n\nend Polynomial.Chebyshev",
"premise_group": [
{
"kind": "original",
"docs": [
"Lagrange.eval_iterate_derivative_eq_sum",
"Lagrange.iterate_derivative_interpolate"
]
},
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.sumNodes_le_sumNodes_T",
"Polynomial.Chebyshev.sumNodes_eq_sumNodes_T_iff"
]
}
]
},
{
"id": "mathlibmpr_019",
"pr": 33372,
"formal_main_result": "MeasureTheory.IsStoppingTime.iInf",
"NL_main_result": "Let `F = (F_t)` be a right-continuous filtration on `ι` (a conditionally complete linearly-ordered first-countable densely-ordered space with no maximum), and let `(τ_n)_{n : κ}` be a countable family of stopping times for `F` with values in `WithTop ι`. Then `ω ↦ ⨅_n τ_n(ω)` is again a stopping time.",
"formal_statement": "import Mathlib.Probability.Process.Stopping\n\nnamespace MeasureTheory\n\nopen ProbabilityTheory\n\nvariable {Ω ι κ : Type*} [Countable κ]\n [ConditionallyCompleteLinearOrderBot ι] [TopologicalSpace ι] [OrderTopology ι]\n [DenselyOrdered ι] [NoMaxOrder ι] [FirstCountableTopology ι]\n {m : MeasurableSpace Ω} {F : Filtration ι m}\n [F.IsRightContinuous]\n\n/-- **Main result.** Countable infimum of stopping times for a right-continuous\nfiltration is again a stopping time. -/\ntheorem IsStoppingTime.iInf\n {τ : κ → Ω → WithTop ι} (hτ : ∀ k, IsStoppingTime F (τ k)) :\n IsStoppingTime F (fun ω => ⨅ k, τ k ω) := sorry\n\nend MeasureTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"MeasureTheory.Filtration.IsRightContinuous"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.IsStoppingTime.measurableSet_ge'"
]
}
]
},
{
"id": "mathlibmpr_020",
"pr": 33449,
"formal_main_result": "ProbabilityTheory.tendsto_choose_mul_pow_of_tendsto_mul_atTop",
"NL_main_result": "Fix `k : ℕ`. Let `p : ℕ → ℝ` be a sequence with `n · p_n → r`. Then `(n choose k) · p_n^k · (1 − p_n)^{n − k} → e^{−r} · r^k / k!`.",
"formal_statement": "import Mathlib.Probability.Distributions.Poisson\nimport Mathlib.Analysis.SpecialFunctions.Exp\n\nopen Filter\n\nnamespace ProbabilityTheory\n\n/-- **Main result.** Poisson limit theorem: if `n · p_n → r`, then for each\nfixed `k`, `(n choose k) · p_n^k · (1 − p_n)^{n − k} → e^{−r} · r^k / k!`. -/\ntheorem tendsto_choose_mul_pow_of_tendsto_mul_atTop\n {p : ℕ → ℝ} {r : ℝ} (k : ℕ)\n (hp : Tendsto (fun n : ℕ => (n : ℝ) * p n) atTop (nhds r)) :\n Tendsto (fun n => (n.choose k : ℝ) * p n ^ k * (1 - p n) ^ (n - k))\n atTop (nhds (Real.exp (-r) * r ^ k / k.factorial)) := sorry\n\nend ProbabilityTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"Real.tendsto_one_add_pow_exp_of_tendsto",
"Complex.tendsto_one_add_pow_exp_of_tendsto",
"Real.tendsto_one_add_rpow_exp_of_tendsto"
]
},
{
"kind": "original",
"docs": [
"isEquivalent_choose"
]
}
]
},
{
"id": "mathlibmpr_021",
"pr": 33463,
"formal_main_result": "Polynomial.norm_coeff_le_choose_mul_mapMahlerMeasure",
"NL_main_result": "Let `A` be a normed ring and `v : A →+* ℂ` an isometry. For every polynomial `p ∈ A[X]` and every `n : ℕ`, `‖p.coeff n‖ ≤ binomial(deg p, n) · M_v(p)`, where `M_v(p) := mahlerMeasure (p.map v)` is the generalised Mahler measure.",
"formal_statement": "import Mathlib.Analysis.Polynomial.MahlerMeasure\n\nnamespace Polynomial\n\nvariable {A : Type*} [NormedRing A]\n\n/-- The Mahler measure of `p ∈ A[X]` computed via a ring hom `v : A →+* ℂ`. -/\nnoncomputable def mapMahlerMeasure (p : Polynomial A) (v : A →+* ℂ) : ℝ :=\n (p.map v).mahlerMeasure\n\n/-- **Main result.** Let `v : A →+* ℂ` be an isometry. For every polynomial\n`p ∈ A[X]` and every `n : ℕ`,\n`‖p.coeff n‖ ≤ Nat.choose (p.natDegree) n · mapMahlerMeasure p v`. -/\ntheorem norm_coeff_le_choose_mul_mapMahlerMeasure\n (p : Polynomial A) (v : A →+* ℂ) (hv : Isometry v) (n : ℕ) :\n ‖p.coeff n‖ ≤ (p.natDegree.choose n : ℝ) * p.mapMahlerMeasure v := sorry\n\nend Polynomial",
"premise_group": [
{
"kind": "original",
"docs": [
"Polynomial.norm_coeff_le_choose_mul_mahlerMeasure"
]
}
]
},
{
"id": "mathlibmpr_022",
"pr": 33479,
"formal_main_result": "EuclideanGeometry.cospherical_of_mul_dist_eq_mul_dist_of_angle_eq_pi",
"NL_main_result": "Let `p₁, p₂, p₃, p₄, p` be points in a real inner-product affine space `P`. Suppose `p` lies strictly between `p₁` and `p₂` and strictly between `p₃` and `p₄` (so `∠ p₁ p p₂ = ∠ p₃ p p₄ = π`), and that `p₁, p, p₃` are not collinear. If `dist p₁ p · dist p₂ p = dist p₃ p · dist p₄ p`, then `{p₁, p₂, p₃, p₄}` is cospherical.",
"formal_statement": "import Mathlib\n\nnamespace EuclideanGeometry\n\nvariable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]\n [MetricSpace P] [NormedAddTorsor V P]\n\n/-- **Main result.** (Converse of intersecting-chords theorem.) If\n`∠ p₁ p p₂ = π` and `∠ p₃ p p₄ = π` (i.e. `p` lies on lines `p₁p₂` and `p₃p₄`),\n`p₁, p, p₃` are not collinear, and\n`dist p₁ p · dist p₂ p = dist p₃ p · dist p₄ p`, then `{p₁, p₂, p₃, p₄}` is\ncospherical. -/\ntheorem cospherical_of_mul_dist_eq_mul_dist_of_angle_eq_pi\n {p₁ p₂ p₃ p₄ p : P}\n (h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p)\n (hp₁p₂ : ∠ p₁ p p₂ = π) (hp₃p₄ : ∠ p₃ p p₄ = π)\n (hn : ¬ Collinear ℝ ({p₁, p, p₃} : Set P)) :\n Cospherical ({p₁, p₂, p₃, p₄} : Set P) := sorry\n\nend EuclideanGeometry",
"premise_group": [
{
"kind": "original",
"docs": [
"EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear",
"EuclideanGeometry.concyclic_of_two_zsmul_oangle_eq_of_not_collinear",
"EuclideanGeometry.cospherical_or_collinear_of_two_zsmul_oangle_eq",
"EuclideanGeometry.concyclic_or_collinear_of_two_zsmul_oangle_eq"
]
},
{
"kind": "original",
"docs": [
"EuclideanGeometry.similar_of_side_angle_side"
]
},
{
"kind": "original",
"docs": [
"Similar.angle_eq_all"
]
},
{
"kind": "alternative",
"docs": [
"EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical"
]
},
{
"kind": "alternative",
"docs": [
"AffineIndependent.existsUnique_dist_eq",
"Affine.Simplex.circumsphere_unique_dist_eq"
]
},
{
"kind": "alternative",
"docs": [
"AffineIndependent.existsUnique_dist_eq",
"Affine.Simplex.circumsphere_unique_dist_eq"
]
},
{
"kind": "alternative",
"docs": [
"EuclideanGeometry.Sphere.secondInter_eq_lineMap"
]
}
]
},
{
"id": "mathlibmpr_023",
"pr": 33507,
"formal_main_result": "Polynomial.sum_sq_norm_coeff_eq_circleAverage",
"NL_main_result": "For any complex polynomial `p ∈ ℂ[X]`, `Σ_{i ∈ supp p} ‖p.coeff i‖² = (1 / 2π) ∫₀^{2π} ‖p(e^{iθ})‖² dθ`. That is, Parseval's identity on the unit circle, read coefficient-wise.",
"formal_statement": "import Mathlib\n\nopen Complex MeasureTheory Set AddCircle\nopen scoped Real\n\nnamespace Polynomial\n\nvariable (p : ℂ[X])\n\nlocal instance instTwoPiPos : Fact (0 < 2 * Real.pi) := Fact.mk Real.two_pi_pos\n\n@[fun_prop, continuity]\n theorem continuous_toCircle {T : ℝ} : Continuous (@toCircle T) :=\n continuous_coinduced_dom.mpr (Circle.exp.continuous.comp <| by fun_prop)\n\n/-- Algebra map from `ℂ[X]` to `C(AddCircle (2 * π), ℂ)` that evaluates polynomials on\nthe unit circle. -/\nnoncomputable def toAddCircle : ℂ[X] →ₐ[ℂ] C(AddCircle (2 * Real.pi), ℂ) :=\n Polynomial.aeval { toFun c := c.toCircle }\n\n/-- **Main result.** Parseval's identity for polynomials on the unit\ncircle: the sum of the squared norms of the coefficients of `p ∈ ℂ[X]`\nequals the average of `‖p(e^{iθ})‖²` over the unit circle. -/\ntheorem sum_sq_norm_coeff_eq_circleAverage : ∑ i ∈ p.support, ‖p.coeff i‖ ^ 2 =\n Real.circleAverage (fun θ ↦ ‖p.eval θ‖ ^ 2) 0 1 := sorry\n\nend Polynomial",
"premise_group": [
{
"kind": "original",
"docs": [
"tsum_sq_fourierCoeff",
"hasSum_sq_fourierCoeff"
]
},
{
"kind": "original",
"docs": [
"HilbertBasis.repr_apply_apply",
"OrthonormalBasis.repr_apply_apply"
]
},
{
"kind": "alternative",
"docs": [
"orthonormal_fourier"
]
},
{
"kind": "alternative",
"docs": [
"hasFPowerSeriesOn_cauchy_integral",
"sum_cauchyPowerSeries_eq_integral",
"DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul",
"DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul"
]
}
]
},
{
"id": "mathlibmpr_024",
"pr": 33520,
"formal_main_result": "FirstOrder.Language.MeetsDefinable.isElementary_closure",
"NL_main_result": "Let `M` be a structure for a first-order language `L`, and let `A ⊆ M` be a subset that meets every nonempty `L`-definable subset of `M`. Then the substructure of `M` generated by `A` is an elementary substructure.",
"formal_statement": "import Mathlib.ModelTheory.ElementarySubstructures\nimport Mathlib.ModelTheory.Definability\n\nnamespace FirstOrder.Language\n\nvariable {L : Language} {M : Type*} [L.Structure M]\n\n/-- A set meets definable sets if it meets every nonempty definable subset. -/\ndef MeetsDefinable (A : Set M) : Prop :=\n ∀ (D : Set M), D.Nonempty → A.Definable₁ L D → (D ∩ A).Nonempty\n\nnamespace MeetsDefinable\n\nvariable {A : Set M}\n\n/-- **Main result.** The closure of a set meeting every nonempty\n`L`-definable subset is an elementary substructure of `M`\n(set-input form of the Tarski–Vaught test). -/\ntheorem isElementary_closure (hA : L.MeetsDefinable A) :\n (Substructure.closure L A).IsElementary := sorry\n\nend MeetsDefinable\n\nend FirstOrder.Language",
"premise_group": [
{
"kind": "original",
"docs": [
"FirstOrder.Language.Substructure.isElementary_of_exists",
"FirstOrder.Language.Substructure.toElementarySubstructure"
]
},
{
"kind": "original",
"docs": [
"FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize",
"FirstOrder.Language.Substructure.mem_closure_iff_exists_term"
]
}
]
},
{
"id": "mathlibmpr_025",
"pr": 33676,
"formal_main_result": "Polynomial.Chebyshev.integral_eval_T_real_mul_eval_T_real_measureT_of_ne",
"NL_main_result": "For `n, m : ℕ` with `n ≠ m`, `∫ (T_n(x) · T_m(x)) / √(1 − x²) dx = 0`, where the integral is taken on `(−1, 1]` against the weighted Lebesgue measure `μ_T = 1_{(−1,1]} · (1 / √(1 − x²)) dx`.",
"formal_statement": "import Mathlib\n\nnamespace Polynomial.Chebyshev\n\nopen Real intervalIntegral MeasureTheory\n\n/-- Lebesgue measure scaled by `√(1 - x²)⁻¹`, restricted to `(-1, 1]`. -/\nnoncomputable def measureT : Measure ℝ :=\n (volume.withDensity\n fun x ↦ ENNReal.ofNNReal ⟨Real.sqrt (1 - x ^ 2)⁻¹, by positivity⟩).restrict (Set.Ioc (-1) 1)\n\n/-- **Main result.** Chebyshev `T` polynomials are orthogonal on `(-1, 1]`\nwith respect to the weighted measure `1/√(1 − x²) dx`: for `n ≠ m`,\n`∫ T_n(x) · T_m(x) / √(1 − x²) dx = 0`. -/\ntheorem integral_eval_T_real_mul_eval_T_real_measureT_of_ne {n m : ℕ} (h : n ≠ m) :\n ∫ x, (T ℝ n).eval x * (T ℝ m).eval x ∂measureT = 0 := sorry\n\nend Polynomial.Chebyshev",
"premise_group": [
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.T_mul_T",
"Polynomial.Chebyshev.C_mul_C"
]
},
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.T_real_cos",
"Polynomial.Chebyshev.T_complex_cos",
"Polynomial.Chebyshev.C_two_mul_real_cos"
]
},
{
"kind": "original",
"docs": [
"intervalIntegral.integral_comp_mul_deriv'''",
"intervalIntegral.integral_comp_mul_deriv'",
"intervalIntegral.integral_comp_mul_deriv",
"intervalIntegral.integral_comp_smul_deriv",
"intervalIntegral.integral_comp_smul_deriv'",
"intervalIntegral.integral_deriv_comp_mul_deriv"
]
}
]
},
{
"id": "mathlibmpr_027",
"pr": 33736,
"formal_main_result": "Polynomial.gaussNorm_mul",
"NL_main_result": "Let `R` be a ring, `v : R → ℝ` a non-archimedean absolute value, and `c > 0` a real number. Then for any `p, q ∈ R[X]`, `gaussNorm v c (p · q) = gaussNorm v c p · gaussNorm v c q`.",
"formal_statement": "import Mathlib.RingTheory.Polynomial.GaussNorm\nimport Mathlib.Algebra.Order.AbsoluteValue.Basic\nimport Mathlib.Algebra.Order.Ring.IsNonarchimedean\n\nnamespace Polynomial\n\nvariable {R : Type*} [Ring R] {v : AbsoluteValue R ℝ} {c : ℝ}\n (hna : IsNonarchimedean v) (hc : 0 < c)\n\ninclude hna hc in\n/-- **Main result.** If `v` is a non-archimedean absolute value on `R` and\n`c > 0`, then the Gauss norm `gaussNorm v c` is multiplicative on `R[X]`. -/\ntheorem gaussNorm_mul (p q : R[X]) :\n (p * q).gaussNorm v c = p.gaussNorm v c * q.gaussNorm v c := sorry\n\nend Polynomial",
"premise_group": [
{
"kind": "original",
"docs": [
"IsNonarchimedean",
"Valuation.map_add"
]
},
{
"kind": "original",
"docs": [
"Polynomial.le_gaussNorm",
"PowerSeries.le_gaussNorm"
]
},
{
"kind": "original",
"docs": [
"Polynomial.gaussNorm_nonneg",
"PowerSeries.gaussNorm_nonneg"
]
}
]
},
{
"id": "mathlibmpr_028",
"pr": 33802,
"formal_main_result": "Representation.IsIrreducible.bijective_or_eq_zero",
"NL_main_result": "Let `ρ : Representation k G V` and `σ : Representation k G W` be irreducible monoid representations of `G` on `k`-vector spaces. Then any intertwining map `f : ρ → σ` is either the zero map or a bijection.",
"formal_statement": "import Mathlib.RepresentationTheory.Irreducible\nimport Mathlib.RepresentationTheory.Intertwining\n\nnamespace Representation.IsIrreducible\n\nvariable {G k V W : Type*}\n [Monoid G] [Field k]\n [AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W]\n {ρ : Representation k G V} {σ : Representation k G W}\n [IsIrreducible ρ] [IsIrreducible σ]\n\n/-- **Main result.** Schur's lemma for irreducible monoid\nrepresentations: any intertwining map is bijective or zero. -/\ntheorem bijective_or_eq_zero (f : IntertwiningMap ρ σ) :\n Function.Bijective f ∨ f = 0 := sorry\n\nend Representation.IsIrreducible",
"premise_group": [
{
"kind": "original",
"docs": [
"LinearMap.bijective_or_eq_zero",
"LinearMap.bijective_of_ne_zero",
"CategoryTheory.isIso_of_hom_simple"
]
},
{
"kind": "original",
"docs": [
"Representation.irreducible_iff_isSimpleModule_asModule",
"Representation.is_simple_module_iff_irreducible_ofModule"
]
}
]
},
{
"id": "mathlibmpr_029",
"pr": 33860,
"formal_main_result": "Probability.integral_cauchyPDFReal",
"NL_main_result": "For every location `x₀ : ℝ` and positive scale `γ > 0`, the Cauchy probability density `x ↦ (πγ)⁻¹ · (1 + ((x − x₀)/γ)²)⁻¹` integrates to `1` over `ℝ`.",
"formal_statement": "import Mathlib\n\nopen MeasureTheory Measure Real\nopen scoped NNReal\n\nnamespace Probability\n\n/-- The pdf of the Cauchy distribution depending on its location `x₀` and scale `γ`. -/\nnoncomputable def cauchyPDFReal (x₀ : ℝ) (γ : ℝ≥0) (x : ℝ) : ℝ :=\n π⁻¹ * γ * ((x - x₀) ^ 2 + γ ^ 2)⁻¹\n\n/-- **Main result.** For positive scale `γ > 0`, the Cauchy probability\ndensity on `ℝ` integrates to `1`. -/\nlemma integral_cauchyPDFReal (x₀ : ℝ) {γ : ℝ≥0} (hγ : γ ≠ 0) :\n ∫ x, cauchyPDFReal x₀ γ x = 1 := sorry\n\nend Probability",
"premise_group": [
{
"kind": "original",
"docs": [
"integral_one_div_one_add_sq",
"integral_univ_inv_one_add_sq",
"integral_inv_one_add_sq"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.integral_sub_right_eq_self"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.integral_comp_div"
]
}
]
},
{
"id": "mathlibmpr_030",
"pr": 33941,
"formal_main_result": "ScottContinuousOn.comp",
"NL_main_result": "Let `f : α → β` and `g : β → γ` be Scott-continuous on a domain family `D` of subsets of `α` (resp. `D'` of `β`). Suppose `D` is closed under two-element subsets `{a,b}` (with `a ≤ b`) and that `f` sends `D`-sets to `D'`-sets after image. Then `g ∘ f` is Scott continuous on `D`.",
"formal_statement": "import Mathlib.Order.ScottContinuity\n\nopen Set\n\nvariable {α β γ : Type*}\n\nsection ScottContinuous\nvariable [Preorder α] [Preorder β] [Preorder γ] {D : Set (Set α)}\n {f : α → β}\n\n/-- **Main result.** Composition of Scott-continuous-on-a-domain functions\nis Scott continuous (on the doubleton-closed family `D`, given that `f`\nmaps `D` into `D'`). -/\ntheorem ScottContinuousOn.comp {g : β → γ} {D'}\n (hD : ∀ a b : α, a ≤ b → ({a, b} : Set α) ∈ D) (hD' : Set.MapsTo (f '' ·) D D')\n (hg : ScottContinuousOn D' g) (hf : ScottContinuousOn D f) :\n ScottContinuousOn D (g ∘ f) := sorry\n\nend ScottContinuous",
"premise_group": [
{
"kind": "original",
"docs": [
"ScottContinuousOn.monotone",
"ScottContinuous.monotone"
]
},
{
"kind": "alternative",
"docs": [
"Topology.IsScott.scottContinuousOn_iff_continuous"
]
}
]
},
{
"id": "mathlibmpr_031",
"pr": 33972,
"formal_main_result": "Polynomial.zero_lt_eval_of_roots_lt_of_leadingCoeff_nonneg",
"NL_main_result": "Let `P ∈ ℝ[X]` and `x ∈ ℝ`. If every (real) root of `P` is strictly less than `x` and `P` has a non-negative leading coefficient, then `P(x) > 0`.",
"formal_statement": "import Mathlib\n\nopen Real Polynomial\n\nnamespace Polynomial\n\nvariable {P : ℝ[X]} {x : ℝ}\n\n/-- **Main result.** A real polynomial is strictly positive at `x` if all\nits (real) roots are strictly less than `x` and its leading coefficient is\nnon-negative. -/\ntheorem zero_lt_eval_of_roots_lt_of_leadingCoeff_nonneg\n (hroots : ∀ y, P.IsRoot y → y < x) (hlc : 0 ≤ P.leadingCoeff) : 0 < P.eval x := sorry\n\nend Polynomial",
"premise_group": [
{
"kind": "original",
"docs": [
"intermediate_value_Icc",
"intermediate_value_uIcc",
"intermediate_value_Icc'"
]
},
{
"kind": "original",
"docs": [
"Polynomial.tendsto_atTop_of_leadingCoeff_nonneg"
]
}
]
},
{
"id": "mathlibmpr_032",
"pr": 34006,
"formal_main_result": "Submodule.CoFG.inf",
"NL_main_result": "Over a Noetherian ring `R`, if `S` and `T` are CoFG submodules of an `R`-module `M`, then so is their intersection `S ⊓ T`.",
"formal_statement": "import Mathlib.RingTheory.Noetherian.Basic\nimport Mathlib.RingTheory.Finiteness.Basic\n\nnamespace Submodule\n\nvariable {R : Type*} [Ring R] {M : Type*} [AddCommGroup M] [Module R M]\n\n/-- A submodule `S` of a module `M` is co-finitely generated (CoFG) if the quotient\n space `M ⧸ S` is finitely generated. -/\nabbrev CoFG (S : Submodule R M) : Prop := Module.Finite R (M ⧸ S)\n\nvariable [IsNoetherianRing R]\n\n/-- **Main result.** Over a Noetherian ring, the intersection of two\nco-finitely-generated submodules is co-finitely generated. -/\ntheorem CoFG.inf {S T : Submodule R M} (hS : S.CoFG) (hT : T.CoFG) :\n (S ⊓ T).CoFG := sorry\n\nend Submodule",
"premise_group": [
{
"kind": "original",
"docs": [
"LinearMap.ker_prod",
"ContinuousLinearMap.ker_prod"
]
},
{
"kind": "original",
"docs": [
"LinearMap.quotKerEquivRange"
]
}
]
},
{
"id": "mathlibmpr_033",
"pr": 34109,
"formal_main_result": "Nat.infinite_odd_abundant",
"NL_main_result": "The set of odd abundant natural numbers is infinite.",
"formal_statement": "import Mathlib\n\nnamespace Nat\n\n/-- **Main result.** The set of odd abundant natural numbers is infinite. -/\ntheorem infinite_odd_abundant : {n : ℕ | Odd n ∧ n.Abundant}.Infinite := sorry\n\nend Nat",
"premise_group": [
{
"kind": "original",
"docs": [
"Set.infinite_iff_exists_gt"
]
},
{
"kind": "original",
"docs": [
"Finset.sum_image"
]
},
{
"kind": "original",
"docs": [
"Finset.sum_le_sum_of_subset"
]
}
]
},
{
"id": "mathlibmpr_034",
"pr": 34255,
"formal_main_result": "Submodule.exists_eq_colon_of_mem_minimalPrimes",
"NL_main_result": "Let `R` be a Noetherian (commutative) ring, `M` an `R`-module, `N ⊆ M` a submodule, and `x ∈ M`. If `I` is a minimal prime over the annihilator-style ideal `N : {x}` (`= colon N {x}`), then there exists `x' ∈ M` with `I = N : {x'}`.",
"formal_statement": "import Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian\nimport Mathlib.RingTheory.Finiteness.Ideal\nimport Mathlib.RingTheory.Noetherian.Basic\n\nnamespace Submodule\n\nvariable {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M}\n {I : Ideal R} {x : M}\n\n/-- **Main result.** In a Noetherian ring, every minimal prime over a\ncolon ideal `N : {x}` is itself the colon ideal `N : {x'}` for some\n`x' ∈ M`. -/\ntheorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]\n (hI : I ∈ (N.colon {x}).minimalPrimes) : ∃ x' : M, I = N.colon {x'} := sorry\n\nend Submodule",
"premise_group": [
{
"kind": "original",
"docs": [
"Ideal.exists_radical_pow_le_of_fg",
"Ideal.exists_pow_le_of_le_radical_of_fg",
"Ideal.exists_pow_le_of_le_radical_of_fg_radical"
]
},
{
"kind": "original",
"docs": [
"Ideal.sInf_minimalPrimes",
"Ideal.radical_eq_sInf"
]
},
{
"kind": "original",
"docs": [
"Ideal.IsPrime.pow_le_iff"
]
},
{
"kind": "original",
"docs": [
"Ideal.IsPrime.inf_le'"
]
},
{
"kind": "original",
"docs": [
"Ideal.IsPrime.prod_mem_iff"
]
},
{
"kind": "original",
"docs": [
"Submodule.mem_colon_singleton"
]
}
]
},
{
"id": "mathlibmpr_035",
"pr": 34260,
"formal_main_result": "IsCompact.powerset_vietoris",
"NL_main_result": "For every compact subset `K` of a topological space `α`, its powerset `K.powerset = {t : Set α | t ⊆ K}`, equipped with the (subspace of the) Vietoris topology on `Set α`, is compact.",
"formal_statement": "import Mathlib.Topology.Sets.VietorisTopology\n\nnamespace IsCompact\n\nvariable {α : Type*} [TopologicalSpace α] {K : Set α}\n\nattribute [local instance] TopologicalSpace.vietoris\n\n/-- **Main result.** For every compact subset `K` of a topological space `α`,\nits powerset `K.powerset = {t : Set α | t ⊆ K}`, equipped with the (subspace\nof the) Vietoris topology on `Set α`, is compact. -/\ntheorem powerset_vietoris (hK : IsCompact K) :\n IsCompact (K.powerset : Set (Set α)) := sorry\n\nend IsCompact",
"premise_group": [
{
"kind": "original",
"docs": [
"isCompact_generateFrom",
"isCompact_generateFrom'",
"compactSpace_generateFrom",
"compactSpace_generateFrom'"
]
},
{
"kind": "original",
"docs": [
"IsCompact.elim_finite_subcover_image",
"isCompact_iff_finite_subcover"
]
},
{
"kind": "alternative",
"docs": [
"Pi.compactSpace",
"isCompact_univ_pi",
"Function.compactSpace"
]
},
{
"kind": "alternative",
"docs": [
"Pi.compactSpace",
"Function.compactSpace"
]
},
{
"kind": "alternative",
"docs": [
"isCompact_iff_ultrafilter_le_nhds",
"isCompact_iff_ultrafilter_le_nhds'"
]
}
]
},
{
"id": "mathlibmpr_036",
"pr": 34263,
"formal_main_result": "HasMFDerivAt.smul",
"NL_main_result": "Let `M` be a smooth manifold, `𝕜` a non-trivially normed field, and `V` a `𝕜`-normed space. Suppose `f : M → 𝕜` and `g : M → V` are maps having `HasMFDerivAt`-derivatives `f' : T_x M → 𝕜` and `g' : T_x M → V` at `x` (with both targets canonically identified with the corresponding tangent spaces via `NormedSpace.fromTangentSpace`). Then `f • g` has `HasMFDerivAt`-derivative `f x • g' + toSpanSingleton 𝕜 (g x) ∘L f'` at `x` (the standard product rule, after canonically identifying the tangent space to `V` at `(f • g) x` with `V`).",
"formal_statement": "import Mathlib.Geometry.Manifold.MFDeriv.Basic\nimport Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions\n\nopen Manifold\n\nvariable {𝕜 : Type*} [NontriviallyNormedField 𝕜]\n {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]\n {H : Type*} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H}\n {M : Type*} [TopologicalSpace M] [ChartedSpace H M]\n {V : Type*} [NormedAddCommGroup V] [NormedSpace 𝕜 V]\n\n/-- Identifying the tangent space at a normed space with the normed space itself.\nThis canonical identification (which, in mathlib, is implemented using an abuse of definitional\nequality) is very prevalent in a number of places: this device allows making it explicit. -/\ndef NormedSpace.fromTangentSpace (v : E) : TangentSpace 𝓘(𝕜, E) v ≃L[𝕜] E where\n toFun v := v\n invFun v := v\n map_add' := by simp\n map_smul' := by simp\n\n/-- **Main result.** Leibniz rule for `mfderiv`: if `f : M → 𝕜` and `g : M → V`\nhave `HasMFDerivAt`-derivatives `f'` and `g'` at `x`, then `f • g` has\n`HasMFDerivAt`-derivative `f x • g' + toSpanSingleton 𝕜 (g x) ∘L f'` at `x`\n(after canonically identifying tangent spaces of `V`). -/\ntheorem HasMFDerivAt.smul {f : M → 𝕜} {g : M → V} {x : M}\n {f' : TangentSpace I x →L[𝕜] 𝕜}\n {g' : TangentSpace I x →L[𝕜] V}\n (hf : HasMFDerivAt I (𝓘(𝕜, 𝕜)) f x f')\n (hg : HasMFDerivAt I (𝓘(𝕜, V)) g x g') :\n HasMFDerivAt I (𝓘(𝕜, V)) (fun y => f y • g y) x\n ((NormedSpace.fromTangentSpace (f x • g x)).symm.toContinuousLinearMap.comp\n (f x • (NormedSpace.fromTangentSpace (g x)).toContinuousLinearMap.comp g'\n + (ContinuousLinearMap.toSpanSingleton 𝕜 (g x)).comp f')) := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"HasFDerivWithinAt.smul",
"HasFDerivAt.smul",
"fderivWithin_smul",
"fderiv_smul"
]
},
{
"kind": "original",
"docs": [
"ContinuousLinearMap.toSpanSingleton"
]
},
{
"kind": "alternative",
"docs": [
"contMDiff_smul",
"contDiff_smul"
]
},
{
"kind": "alternative",
"docs": [
"mfderiv_comp",
"mfderiv_comp_apply"
]
}
]
},
{
"id": "mathlibmpr_037",
"pr": 34297,
"formal_main_result": "exteriorPower.ιMulti_family_span_fixedDegree_of_span",
"NL_main_result": "Let `M` be an `R`-module, `n : ℕ`, and `v : I → M` an indexed family with `I` linearly ordered. If the range of `v` spans `M`, then the family `(v(i₁) ∧ ⋯ ∧ v(iₙ))` indexed by ordered subsets `{i₁ < ⋯ < iₙ} ⊆ I` of size `n` spans the `n`th exterior power `⋀ⁿ M ⊆ ⋀M` (viewed as a submodule of the exterior algebra).",
"formal_statement": "import Mathlib.LinearAlgebra.ExteriorPower.Basic\n\nnamespace exteriorPower\n\nvariable {R M I : Type*} [CommRing R] [AddCommGroup M] [Module R M]\n [LinearOrder I] (n : ℕ) (v : I → M)\n\n/-- **Main result.** If the range of `v : I → M` spans `M`, then the family of\nordered wedges `(v i₁) ∧ ⋯ ∧ (v iₙ)` indexed by ordered `n`-subsets of `I`\nspans the `n`-th exterior power `⋀ⁿ M`. -/\nlemma ιMulti_family_span_fixedDegree_of_span {I : Type*} [LinearOrder I] {v : I → M}\n (hv : Submodule.span R (Set.range v) = ⊤) :\n Submodule.span R (Set.range (ExteriorAlgebra.ιMulti_family R n v)) = ⋀[R]^n M := sorry\n\nend exteriorPower",
"premise_group": [
{
"kind": "original",
"docs": [
"Submodule.span_pow",
"Submodule.pow_eq_span_pow_set"
]
},
{
"kind": "original",
"docs": [
"AlternatingMap.map_perm",
"AlternatingMap.map_congr_perm"
]
},
{
"kind": "alternative",
"docs": [
"exteriorPower.alternatingMapLinearEquiv"
]
}
]
},
{
"id": "mathlibmpr_038",
"pr": 34298,
"formal_main_result": "MeasureTheory.tendsto_integral_Ioi_zero",
"NL_main_result": "Let `f : ℝ → E` be a function and `μ` a measure on `ℝ`. For a function `b : ι → ℝ` and a filter `l` on `ι`, if `b → +∞` along `l`, then `∫_{Ioi (b i)} f dμ → 0` along `l`.",
"formal_statement": "import Mathlib\n\nnamespace MeasureTheory\n\nvariable {ι E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]\n {f : ℝ → E} {μ : Measure ℝ}\n\n/-- **Main result.** For `b : ι → ℝ` tending to `+∞` along filter `l`,\n`∫_{Ioi (b i)} f dμ → 0`. -/\ntheorem tendsto_integral_Ioi_zero\n {l : Filter ι} {b : ι → ℝ} (hb : Filter.Tendsto b l Filter.atTop) :\n Filter.Tendsto (fun i => ∫ x in Set.Ioi (b i), f x ∂μ) l (nhds 0) := sorry\n\nend MeasureTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"MeasureTheory.intervalIntegral_tendsto_integral_Ioi"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.setIntegral_union",
"MeasureTheory.integral_union_ae"
]
},
{
"kind": "alternative",
"docs": [
"MeasureTheory.tendsto_integral_of_dominated_convergence"
]
}
]
},
{
"id": "mathlibmpr_039",
"pr": 34364,
"formal_main_result": "Polynomial.Chebyshev.abs_iterate_derivative_T_real_le",
"NL_main_result": "For every integer `n`, every `k ∈ ℕ`, and every real `x` with `|x| ≤ 1`, `|T_n^{(k)}(x)| ≤ T_n^{(k)}(1)` (where `T_n` is the `n`th Chebyshev polynomial of the first kind, real-valued).",
"formal_statement": "import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev\n\nnamespace Polynomial.Chebyshev\n\n/-- **Main result.** For every integer `n`, every `k ∈ ℕ`, and every real `x`\nwith `|x| ≤ 1`, `|T_n^{(k)}(x)| ≤ T_n^{(k)}(1)`. -/\ntheorem abs_iterate_derivative_T_real_le (n : ℤ) (k : ℕ) {x : ℝ} (hx : |x| ≤ 1) :\n |((Polynomial.derivative^[k] (Polynomial.Chebyshev.T ℝ n)).eval x)|\n ≤ ((Polynomial.derivative^[k] (Polynomial.Chebyshev.T ℝ n)).eval 1) := sorry\n\nend Polynomial.Chebyshev",
"premise_group": [
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.abs_eval_T_real_le_one"
]
},
{
"kind": "alternative",
"docs": [
"Polynomial.Chebyshev.T_real_cos",
"Polynomial.Chebyshev.T_complex_cos"
]
},
{
"kind": "alternative",
"docs": [
"DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul",
"DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul"
]
}
]
},
{
"id": "mathlibmpr_040",
"pr": 34594,
"formal_main_result": "AddTorsor.convexCombination_assoc",
"NL_main_result": "Let `P` be an affine space over a strictly ordered ring `R` with model module `V`, and let `f : StdSimplex R (StdSimplex R P)` be a \"simplex of simplices\" (a probability distribution over probability distributions over `P`). Then the convex combination of the inner combinations equals the convex combination of the joint distribution: `convexCombination (f.map convexCombination) = convexCombination f.join`.",
"formal_statement": "import Mathlib\n\nnamespace AddTorsor\n\nvariable {R V P : Type*} [Ring R] [PartialOrder R] [IsStrictOrderedRing R]\n [AddCommGroup V] [Module R V] [AddTorsor V P]\n\n/-- The convex combination of points in an affine space, given a probability distribution. -/\nnoncomputable def convexCombination (s : StdSimplex R P) : P :=\n s.weights.support.affineCombination R id s.weights\n\n/-- **Main result.** Associativity of convex combinations in an affine space:\nfor `f : StdSimplex R (StdSimplex R P)`,\n`convexCombination (f.map convexCombination) = convexCombination f.join`. -/\ntheorem convexCombination_assoc (f : StdSimplex R (StdSimplex R P)) :\n convexCombination (f.map convexCombination) = convexCombination f.join := sorry\n\nend AddTorsor",
"premise_group": [
{
"kind": "original",
"docs": [
"Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one",
"Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one"
]
}
]
},
{
"id": "mathlibmpr_042",
"pr": 34632,
"formal_main_result": "SimpleGraph.completeBipartiteGraph_isContained_bipartiteDoubleCover",
"NL_main_result": "Let `G : SimpleGraph V` and `α, β` non-empty finite types. Then `K_{α,β}` (the complete bipartite graph) is contained as a subgraph of the bipartite double cover `G.bipartiteDoubleCover` iff it is contained in `G`.",
"formal_statement": "import Mathlib\n\nnamespace SimpleGraph\n\nvariable {V : Type*} (G : SimpleGraph V)\n\n/-- The bipartite double cover of `G`: a graph on `V ⊕ V` with `inl v ∼ inr w`\niff `v ∼ w` in `G`. -/\ndef bipartiteDoubleCover : SimpleGraph (V ⊕ V) where\n Adj\n | Sum.inl v, Sum.inr w => G.Adj v w\n | Sum.inr v, Sum.inl w => G.Adj v w\n | _, _ => False\n symm := by rintro (a|a) (b|b) h <;> simp_all [SimpleGraph.adj_comm]\n loopless := by rintro (a|a) h <;> simp_all\n\nvariable {α β : Type*} [Finite α] [Finite β] [Nonempty α] [Nonempty β]\n\n/-- **Main result.** `K_{α,β}` (the complete bipartite graph) is contained in\nthe bipartite double cover of `G` iff it is contained in `G`. -/\ntheorem completeBipartiteGraph_isContained_bipartiteDoubleCover :\n (completeBipartiteGraph α β).IsContained G.bipartiteDoubleCover ↔\n (completeBipartiteGraph α β).IsContained G := sorry\n\nend SimpleGraph",
"premise_group": [
{
"kind": "original",
"docs": [
"SimpleGraph.completeBipartiteGraph_isContained_iff"
]
}
]
},
{
"id": "mathlibmpr_043",
"pr": 34636,
"formal_main_result": "integral_preimage",
"NL_main_result": "For any Banach space `E` and any function `f : UnitAddTorus d → E`, and for any base point `a : d → ℝ`, `∫ x : UnitAddTorus d, f x = ∫ x in {x | ∀ i, x i ∈ (a i, a i + 1]}, f (fun i ↦ ↑(x i))`.",
"formal_statement": "import Mathlib.Analysis.Fourier.AddCircleMulti\n\nopen MeasureTheory\n\nvariable {d : Type*} [Fintype d] {E : Type*}\n [NormedAddCommGroup E] [NormedSpace ℝ E]\n\n/-- **Main result.** For any function `f : UnitAddTorus d → E` and any base\npoint `a : d → ℝ`,\n`∫ x : UnitAddTorus d, f x = ∫ x in {x | ∀ i, x i ∈ Ioc (a i) (a i + 1)}, f (fun i => ↑(x i))`. -/\ntheorem UnitAddTorus.integral_preimage (a : d → ℝ) (f : UnitAddTorus d → E) :\n ∫ x : UnitAddTorus d, f x\n = ∫ x in {x : d → ℝ | ∀ i, x i ∈ Set.Ioc (a i) (a i + 1)},\n f (fun i => ((x i : ℝ) : AddCircle (1 : ℝ))) := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"MeasureTheory.integral_map_equiv",
"MeasureTheory.lintegral_map_equiv",
"MeasureTheory.setIntegral_map_equiv"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.integral_subtype_comap"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.measurePreserving_pi"
]
},
{
"kind": "original",
"docs": [
"AddCircle.measurePreserving_mk"
]
}
]
},
{
"id": "mathlibmpr_044",
"pr": 34709,
"formal_main_result": "discrete_gronwall",
"NL_main_result": "Let `u, b, c : ℕ → ℝ` with `0 ≤ u n₀`, `0 ≤ b n`, `0 ≤ c n` for all `n ≥ n₀`, and assume the recurrence inequality `u (n+1) ≤ (1 + c n) · u n + b n` for `n ≥ n₀`. Then for every `n ≥ n₀`, `u n ≤ (u n₀ + ∑_{k = n₀}^{n−1} b k) · exp (∑_{i = n₀}^{n−1} c i)`.",
"formal_statement": "import Mathlib\n\n/-- **Main result.** Discrete Grönwall inequality: if `u (n+1) ≤ (1 + c n)·u n + b n`\nfor all `n ≥ n₀`, with `0 ≤ u n₀`, `0 ≤ b n`, `0 ≤ c n` for `n ≥ n₀`, then for\nevery `n ≥ n₀`,\n`u n ≤ (u n₀ + ∑_{k = n₀}^{n−1} b k) · exp (∑_{i = n₀}^{n−1} c i)`. -/\ntheorem discrete_gronwall {u b c : ℕ → ℝ} {n₀ : ℕ}\n (hu₀ : 0 ≤ u n₀)\n (hb : ∀ n, n₀ ≤ n → 0 ≤ b n)\n (hc : ∀ n, n₀ ≤ n → 0 ≤ c n)\n (hrec : ∀ n, n₀ ≤ n → u (n + 1) ≤ (1 + c n) * u n + b n) :\n ∀ n, n₀ ≤ n →\n u n ≤ (u n₀ + ∑ k ∈ Finset.Ico n₀ n, b k) *\n Real.exp (∑ i ∈ Finset.Ico n₀ n, c i) := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"Real.add_one_le_exp",
"Real.add_one_lt_exp"
]
}
]
},
{
"id": "mathlibmpr_045",
"pr": 34735,
"formal_main_result": "IsRelPrime.moebius_mul",
"NL_main_result": "Let `α` be a commutative monoid with zero that is a unique factorization monoid. For any `a, b : α` with `a` and `b` coprime (`IsRelPrime a b`), `μ(a · b) = μ(a) · μ(b)`.",
"formal_statement": "import Mathlib\n\nnoncomputable section\n\nnamespace UniqueFactorizationMonoid\n\nvariable {α : Type*} [CommMonoidWithZero α] [UniqueFactorizationMonoid α]\n [DecidableEq α] [NormalizedGCDMonoid α]\n\nopen Classical\n/-- The Möbius function on a unique factorization monoid:\n`μ(a) = (−1)^(factors a).card` if `a` is squarefree, and `0` otherwise. -/\nnoncomputable def moebius (a : α) : ℤ :=\n if Squarefree a then (-1 : ℤ) ^ (UniqueFactorizationMonoid.factors a).card else 0\n\nend UniqueFactorizationMonoid\n\nnamespace IsRelPrime\n\nvariable {α : Type*} [CommMonoidWithZero α] [UniqueFactorizationMonoid α]\n [DecidableEq α] [NormalizedGCDMonoid α]\n\n/-- **Main result.** The Möbius function on a UFM is multiplicative on coprime\narguments: `μ(a·b) = μ(a) · μ(b)` whenever `IsRelPrime a b`. -/\ntheorem moebius_mul {a b : α} (h : IsRelPrime a b) :\n UniqueFactorizationMonoid.moebius (a * b)\n = UniqueFactorizationMonoid.moebius a * UniqueFactorizationMonoid.moebius b := sorry\n\nend IsRelPrime",
"premise_group": [
{
"kind": "original",
"docs": [
"squarefree_mul_iff",
"Nat.squarefree_mul_iff"
]
},
{
"kind": "original",
"docs": [
"UniqueFactorizationMonoid.factors_mul"
]
},
{
"kind": "original",
"docs": [
"Multiset.card_eq_card_of_rel"
]
}
]
},
{
"id": "mathlibmpr_046",
"pr": 34858,
"formal_main_result": "Affine.Simplex.setInterior_map",
"NL_main_result": "Let `f : P →ᵃ[k] P₂` be an injective affine map, `s` an affine simplex in `P`, and `I ⊆ k` a weight-set defining the \"interior\" of a simplex. Then the interior of the image simplex `s.map f hf` (with weights in `I`) equals the image under `f` of the interior of `s`: `(s.map f hf).setInterior I = f '' s.setInterior I`.",
"formal_statement": "import Mathlib\n\nnamespace Affine.Simplex\n\nvariable {k V P V₂ P₂ : Type*}\n [Ring k]\n [AddCommGroup V] [Module k V] [AddTorsor V P]\n [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂]\n {n : ℕ} (s : Simplex k P n)\n (f : P →ᵃ[k] P₂) (hf : Function.Injective f)\n (I : Set k)\n\n/-- **Main result.** The interior of a simplex commutes with an injective\naffine map: `(s.map f hf).setInterior I = f '' s.setInterior I`. -/\ntheorem setInterior_map :\n (s.map f hf).setInterior I = f '' s.setInterior I := sorry\n\nend Affine.Simplex",
"premise_group": [
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_interior_iff",
"Affine.Simplex.affineCombination_mem_setInterior_iff"
]
},
{
"kind": "original",
"docs": [
"Finset.affineCombination_map",
"Finset.map_affineCombination"
]
},
{
"kind": "original",
"docs": [
"AffineSubspace.map_span"
]
}
]
},
{
"id": "mathlibmpr_048",
"pr": 34915,
"formal_main_result": "RestrictedProduct.isUnit_iff",
"NL_main_result": "Let `R = (R i)` be a family of monoids and `B = (B i)` a family of submonoids (one in each `R i`), and let `𝓕` be a filter on the index set. An element `x : Πʳ i, [R i, B i]_[𝓕]` is a unit if and only if (i) `x i` is a unit in `R i` for every `i`, and (ii) eventually along `𝓕`, the value `x i` lies in `B i` and is a unit in the submonoid `B i`.",
"formal_statement": "import Mathlib\n\nnamespace RestrictedProduct\n\nvariable {ι : Type*} {R : ι → Type*} [(i : ι) → Monoid (R i)]\n{S : ι → Type*} [(i : ι) → SetLike (S i) (R i)] [∀ (i : ι), SubmonoidClass (S i) (R i)]\n{B : (i : ι) → S i} {𝓕 : Filter ι}\n\n/-- **Main result.** An element `x` of the restricted product is a unit iff\neach `x i` is a unit in `R i`, and eventually along `𝓕`, `x i` lies in\n`B i` and is a unit there. -/\ntheorem isUnit_iff (x : Πʳ i, [R i, B i]_[𝓕]) :\n IsUnit x ↔\n (∀ i, IsUnit (x i)) ∧\n (∀ᶠ i in 𝓕, ∃ h : x i ∈ B i, IsUnit (⟨x i, h⟩ : B i)) := sorry\n\nend RestrictedProduct",
"premise_group": [
{
"kind": "original",
"docs": [
"MulEquiv.piUnits",
"ContinuousMulEquiv.piUnits"
]
}
]
},
{
"id": "mathlibmpr_049",
"pr": 34969,
"formal_main_result": "HopfAlgebra.antipode_mul",
"NL_main_result": "Let `R` be a commutative semiring and `A` an `R`-Hopf algebra with antipode `S` and multiplication `μ`. Then for all `a, b ∈ A`, `S(a · b) = S(b) · S(a)`.",
"formal_statement": "import Mathlib.RingTheory.HopfAlgebra.Basic\n\nnamespace HopfAlgebra\n\nvariable {R A : Type*} [CommSemiring R] [Semiring A] [HopfAlgebra R A]\n\n/-- **Main result.** The antipode of a Hopf algebra is an antihomomorphism:\n`S(a · b) = S(b) · S(a)`. -/\ntheorem antipode_mul (a b : A) :\n HopfAlgebra.antipode (R := R) (a * b)\n = HopfAlgebra.antipode (R := R) b * HopfAlgebra.antipode (R := R) a := sorry\n\nend HopfAlgebra",
"premise_group": [
{
"kind": "original",
"docs": [
"HopfAlgebra.mul_antipode_rTensor_comul_apply",
"HopfAlgebra.mul_antipode_rTensor_comul",
"HopfAlgebra.sum_mul_antipode_eq_smul",
"HopfAlgebra.sum_mul_antipode_eq_algebraMap_counit",
"HopfAlgebra.mul_antipode_lTensor_comul_apply"
]
},
{
"kind": "original",
"docs": [
"HopfAlgebra.sum_mul_antipode_eq_smul",
"HopfAlgebra.sum_mul_antipode_eq_algebraMap_counit",
"HopfAlgebra.mul_antipode_rTensor_comul_apply",
"HopfAlgebra.sum_antipode_mul_eq_smul"
]
},
{
"kind": "original",
"docs": [
"Bialgebra.comul_mul",
"Bialgebra.mul_compr₂_comul"
]
},
{
"kind": "original",
"docs": [
"LinearMap.convMul_apply"
]
}
]
},
{
"id": "mathlibmpr_050",
"pr": 34977,
"formal_main_result": "CategoryTheory.isContinuous_toGrothendieck_of_pullbacksPreservedBy",
"NL_main_result": "Let `F : C ⥤ D` be a functor between sites equipped with precoverages `J` on `C` and `K` on `D` that are stable under base change and have pullbacks. If `F` preserves the (pairwise) pullbacks of every `J`-covering and the image of every `J`-covering under `F` is a `K`-covering, then `F` is continuous for the Grothendieck topologies generated by `J` and `K` — i.e. the pullback `F.op ⋙ G` of any sheaf `G` on `(D, K_top)` is a sheaf on `(C, J_top)`.",
"formal_statement": "import Mathlib\n\nnamespace CategoryTheory\n\nopen CategoryTheory CategoryTheory.Limits\n\nvariable {C D : Type*} [Category C] [Category D]\n (F : C ⥤ D) (J : Precoverage C) (K : Precoverage D)\n\n/-- A functor `F` preserves pairwise pullbacks of a presieve `R` if for every pair\nof morphisms `f` and `g` in `R`, the pullback of `f` and `g` is preserved by `F`. -/\nclass Functor.PreservesPairwisePullbacks (F : C ⥤ D) {X : C} (R : Presieve X) : Prop where\n preservesLimit (R) ⦃Y Z : C⦄ ⦃f : Y ⟶ X⦄ ⦃g : Z ⟶ X⦄ : R f → R g →\n PreservesLimit (cospan f g) F := by infer_instance\n\n/-- Pullbacks are preserved by a functor `F : C ⥤ D` for the precoverage `J` on `C` if\n`F` preserves all pairwise pullbacks of presieves in `J`. -/\nclass Precoverage.PullbacksPreservedBy (J : Precoverage C) (F : C ⥤ D) : Prop where\n preservesPairwisePullbacks_of_mem ⦃X : C⦄ ⦃R : Presieve X⦄ :\n R ∈ J X → F.PreservesPairwisePullbacks R := by infer_instance\n\n/-- **Main result.** A pullback-preserving functor between sites is continuous\nfor the topologies generated by `J` and `K` whenever the image of every\n`J`-covering is a `K`-covering. -/\ntheorem isContinuous_toGrothendieck_of_pullbacksPreservedBy\n [J.IsStableUnderBaseChange] [J.HasPullbacks]\n [K.IsStableUnderBaseChange] [K.HasPullbacks]\n [J.PullbacksPreservedBy F]\n (h : J ≤ K.comap F) :\n Functor.IsContinuous F J.toGrothendieck K.toGrothendieck := sorry\n\nend CategoryTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"CategoryTheory.Presieve.isSheaf_coverage",
"CategoryTheory.Presieve.isSheaf_pretopology"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Presieve.isSheafFor_arrows_iff_pullbacks"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.pullbackComparison_comp_fst",
"CategoryTheory.Limits.pullbackComparison_comp_snd"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.hasPullback_of_preservesPullback"
]
}
]
},
{
"id": "mathlibmpr_051",
"pr": 35014,
"formal_main_result": "ProbabilityTheory.rnDeriv_compProd",
"NL_main_result": "For finite measures `μ, ν` on `α`, finite kernels `κ, η : Kernel α β`, with `μ ⊗ₘ κ ≪ μ ⊗ₘ η`, the Radon–Nikodym derivative of `μ ⊗ₘ κ` with respect to `ν ⊗ₘ η` (as measures on `α × β`) is, almost everywhere with respect to `ν ⊗ₘ η`, equal to the product `p ↦ (dμ/dν)(p.1) · (d(μ ⊗ₘ κ)/d(μ ⊗ₘ η))(p)`.",
"formal_statement": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\n/-- **Main result.** For finite measures `μ, ν` on `α`, finite kernels\n`κ, η : Kernel α β`, with `μ ⊗ₘ κ ≪ μ ⊗ₘ η`, the Radon–Nikodym derivative of\n`μ ⊗ₘ κ` with respect to `ν ⊗ₘ η` is `(ν ⊗ₘ η)`-a.e. equal to\n`p ↦ (dμ/dν)(p.1) · (d(μ ⊗ₘ κ)/d(μ ⊗ₘ η))(p)`. -/\ntheorem ProbabilityTheory.rnDeriv_compProd\n {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n {μ ν : Measure α} [IsFiniteMeasure μ] [IsFiniteMeasure ν]\n {κ η : Kernel α β} [IsFiniteKernel κ] [IsFiniteKernel η]\n (hac : μ ⊗ₘ κ ≪ μ ⊗ₘ η) :\n (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η)\n =ᵐ[ν ⊗ₘ η]\n fun p => μ.rnDeriv ν p.1 * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) p := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.rnDeriv_mul_rnDeriv",
"MeasureTheory.Measure.rnDeriv_mul_rnDeriv'"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.haveLebesgueDecomposition_add",
"MeasureTheory.Measure.singularPart_add_rnDeriv"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.MutuallySingular.compProd_of_left"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.rnDeriv_eq_zero_of_mutuallySingular",
"MeasureTheory.Measure.MutuallySingular.rnDeriv_ae_eq_zero"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.compProd_add_left"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.rnDeriv_withDensity",
"MeasureTheory.Measure.rnDeriv_withDensity₀"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.setLIntegral_rnDeriv_mul",
"MeasureTheory.lintegral_rnDeriv_mul"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.compProd_apply_prod"
]
}
]
},
{
"id": "mathlibmpr_052",
"pr": 35041,
"formal_main_result": "Finset.rado_selection",
"NL_main_result": "Let `α` be any type and `β : α → Type*` with each `β a` finite. Suppose for every finite set `s : Finset α` we have a function `g s : (a : α) → β a`. Then there is a single function `χ : (a : α) → β a` such that for every finite set `s`, there exists `t ⊇ s` with `χ x = g t x` for every `x ∈ s`. Informally: the family of local guesses `(g s)` can be stitched into a global function `χ` that, on each finite set `s`, agrees with `g t` for some super-finite-set `t`.",
"formal_statement": "import Mathlib.Topology.Compactness.Compact\nimport Mathlib.Data.Set.Finite.Basic\n\nvariable {α : Type*} {β : α → Type*} [∀ a, Finite (β a)]\n\n/-- **Main result.** Rado selection lemma: a family of \"local\napproximations\" (one per finite set) can be stitched into a global\nfunction that agrees with some member of the family on every finite\nset. -/\ntheorem Finset.rado_selection (g : Finset α → (a : α) → β a) :\n ∃ χ : (a : α) → β a,\n ∀ s : Finset α, ∃ t : Finset α, s ⊆ t ∧ ∀ x ∈ s, χ x = g t x := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"CompactSpace.iInter_nonempty",
"IsCompact.inter_iInter_nonempty"
]
},
{
"kind": "alternative",
"docs": [
"exists_orderEmbedding_covby_of_forall_covby_finite_of_bot"
]
}
]
},
{
"id": "mathlibmpr_053",
"pr": 35084,
"formal_main_result": "SimpleGraph.mulCayley_gc",
"NL_main_result": "For a magma `M`, the assignment `s ↦ mulCayley s` from `Set M` to `SimpleGraph M` is a left adjoint (a Galois connection): for every `S ⊆ M` and every simple graph `G` on `M`, `mulCayley S ≤ G ↔ S ⊆ {g ∈ M | ∀ a, a * g ≠ a → G.Adj (a * g) a}`. Equivalently, `mulCayley` is the smallest simple graph in which every element of `S` produces an edge from `a` to `a * g` (whenever distinct).",
"formal_statement": "import Mathlib\n\nnamespace SimpleGraph\n\nvariable {M : Type*} [Mul M]\n\n/-- The Cayley graph on `M` with generator set `s`: distinct vertices `u, v`\nare adjacent iff some `g ∈ s` satisfies `u * g = v` or `v * g = u`. -/\ndef mulCayley (s : Set M) : SimpleGraph M :=\n SimpleGraph.fromRel (fun u v => ∃ g ∈ s, u * g = v)\n\n/-- **Main result.** `s ↦ mulCayley s` is the left adjoint of a Galois\nconnection `Set M ⇆ SimpleGraph M`. -/\ntheorem mulCayley_gc :\n GaloisConnection (mulCayley (M := M))\n (fun G : SimpleGraph M => {g | ∀ a, a * g ≠ a → G.Adj (a * g) a}) := sorry\n\nend SimpleGraph",
"premise_group": [
{
"kind": "original",
"docs": [
"SimpleGraph.le_iff_adj"
]
}
]
},
{
"id": "mathlibmpr_055",
"pr": 35257,
"formal_main_result": "eq_of_zeroHypercover_target",
"NL_main_result": "Let `J` be a precoverage on `C` such that the morphism property \"is an isomorphism\" is local at the target for `J`. Let `f, g : X ⟶ Y` be two morphisms over `S` and let `𝒰 = {Uᵢ → S}` be a `J`-cover of `S`. If for every `i` the base-changed maps `f ×_S Uᵢ = g ×_S Uᵢ : X ×_S Uᵢ → Y ×_S Uᵢ` agree, then `f = g`.",
"formal_statement": "import Mathlib\n\nopen CategoryTheory CategoryTheory.Limits\n\nnamespace CategoryTheory\n\nuniverse v u\nvariable {C : Type u} [Category.{v} C]\n\n/-- **Main result.** Let `J` be a precoverage on `C` for which the\nproperty \"is an isomorphism\" is local at the target. Let `f, g : X ⟶ Y`\nbe two morphisms over `S`, and let `𝒰` be a `J`-cover of `S`. If for\nevery `i` the base-changed maps `f ×_S Uᵢ` and `g ×_S Uᵢ` agree, then\n`f = g`. -/\nlemma eq_of_zeroHypercover_target [HasEqualizers C] [HasPullbacks C]\n {X Y S : C} {f g : X ⟶ Y}\n {s : X ⟶ S} {t : Y ⟶ S} (hf : f ≫ t = s) (hg : g ≫ t = s)\n {J : Precoverage C}\n (𝒰 : Precoverage.ZeroHypercover.{v} J S) [J.IsStableUnderBaseChange]\n [(MorphismProperty.isomorphisms C).IsLocalAtTarget J]\n (H : ∀ i,\n pullback.map s (𝒰.f i) t (𝒰.f i) f (𝟙 (𝒰.X i)) (𝟙 S)\n (by simp [hf]) (by simp) =\n pullback.map s (𝒰.f i) t (𝒰.f i) g (𝟙 (𝒰.X i)) (𝟙 S)\n (by simp [hg]) (by simp)) :\n f = g := sorry\n\nend CategoryTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"CategoryTheory.MorphismProperty.IsLocalAtTarget.iff_of_zeroHypercover"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.isIso_limit_cone_parallelPair_of_epi"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.MorphismProperty.of_isPullback_of_descendsAlong"
]
}
]
},
{
"id": "mathlibmpr_056",
"pr": 35365,
"formal_main_result": "Affine.Simplex.closedInterior_eq_interior_union",
"NL_main_result": "For an `n`-simplex `s` in an affine space over a linearly ordered field (with `n ≥ 1`), the closed interior of `s` is the union of its open interior and the closed interiors of its `n + 1` opposite faces: `s.closedInterior = s.interior ∪ ⋃ i, (s.faceOpposite i).closedInterior`.",
"formal_statement": "import Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic\n\nnamespace Affine.Simplex\n\nvariable {k V P : Type*} [Field k] [LinearOrder k] [IsStrictOrderedRing k]\n [AddCommGroup V] [Module k V] [AddTorsor V P]\n\n/-- **Main result.** For an `n`-simplex `s` (with `n ≥ 1`), the closed\ninterior decomposes as the union of the open interior and the closed\ninteriors of the `n + 1` opposite faces. -/\ntheorem closedInterior_eq_interior_union {n : ℕ} [NeZero n]\n (s : Simplex k P n) :\n s.closedInterior =\n s.interior ∪ ⋃ i : Fin (n + 1), (s.faceOpposite i).closedInterior := sorry\n\nend Affine.Simplex",
"premise_group": [
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_closedInterior_iff"
]
},
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_interior_iff"
]
},
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_closedInterior_face_iff_mem_Icc"
]
},
{
"kind": "original",
"docs": [
"mem_affineSpan_iff_eq_affineCombination"
]
}
]
},
{
"id": "mathlibmpr_057",
"pr": 35497,
"formal_main_result": "CategoryTheory.Triangulated.extensionProduct_assoc",
"NL_main_result": "In a triangulated category `C`, for any three object properties `P, Q, R` we have `extensionProduct (extensionProduct P Q) R = extensionProduct P (extensionProduct Q R)`, i.e. the extension product is associative as an operation on `ObjectProperty C`.",
"formal_statement": "import Mathlib\n\nnamespace CategoryTheory.Triangulated\n\nopen CategoryTheory CategoryTheory.Pretriangulated CategoryTheory.ObjectProperty\n\nuniverse v u\nvariable {C : Type u} [Category.{v} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C]\n [∀ n : ℤ, (CategoryTheory.shiftFunctor C n).Additive] [Pretriangulated C]\n\nvariable (P Q R : ObjectProperty C)\n\n/-- An object `X` satisfies `extensionProduct P Q` if there exists a\ndistinguished triangle `Y ⟶ X ⟶ Z ⟶ Y⟦1⟧` with `P Y` and `Q Z`. -/\ndef extensionProduct : ObjectProperty C :=\n fun X => ∃ (Y Z : C) (f : Y ⟶ X) (g : X ⟶ Z) (h : Z ⟶ Y⟦(1 : ℤ)⟧),\n Triangle.mk f g h ∈ distTriang C ∧ P Y ∧ Q Z\n\n/-- **Main result.** The extension product of object properties is\nassociative in a triangulated category. -/\nlemma extensionProduct_assoc [IsTriangulated C] :\n extensionProduct (extensionProduct P Q) R =\n extensionProduct P (extensionProduct Q R) := sorry\n\nend CategoryTheory.Triangulated",
"premise_group": [
{
"kind": "original",
"docs": [
"CategoryTheory.Triangulated.someOctahedron",
"CategoryTheory.Triangulated.someOctahedron'",
"CategoryTheory.IsTriangulated.octahedron_axiom"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Pretriangulated.distinguished_cocone_triangle"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Pretriangulated.isomorphic_distinguished"
]
}
]
},
{
"id": "mathlibmpr_058",
"pr": 35508,
"formal_main_result": "AlgebraicGeometry.preservesLimitsOfShape_discrete_of_isSheaf_zariskiTopology",
"NL_main_result": "Let `F : Schemeᵒᵖ ⥤ Type v` be a Zariski sheaf. Then for any small index type `ι`, `F` preserves limits of shape `Discrete ι` — i.e. `F` carries arbitrary coproducts of schemes to products of types.",
"formal_statement": "import Mathlib.AlgebraicGeometry.Sites.BigZariski\nimport Mathlib.CategoryTheory.Sites.Preserves\n\nopen CategoryTheory CategoryTheory.Limits Opposite\n\nnamespace AlgebraicGeometry\n\nuniverse v u\n\n/-- **Main result.** Any Zariski sheaf `F : Schemeᵒᵖ ⥤ Type v`\npreserves arbitrary (small) products: for every small index type `ι`,\n`F` preserves limits of shape `Discrete ι`. -/\nlemma preservesLimitsOfShape_discrete_of_isSheaf_zariskiTopology\n {F : Scheme.{u}ᵒᵖ ⥤ Type v} {ι : Type*} [Small.{u} ι] [Small.{v} ι]\n (hF : Presieve.IsSheaf Scheme.zariskiTopology F) :\n PreservesLimitsOfShape (Discrete ι) F := sorry\n\nend AlgebraicGeometry",
"premise_group": [
{
"kind": "original",
"docs": [
"CategoryTheory.Presieve.preservesProduct_of_isSheafFor",
"CategoryTheory.Presieve.isSheafFor_iff_preservesProduct"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.preservesLimitsOfShape_of_discrete"
]
},
{
"kind": "original",
"docs": [
"AlgebraicGeometry.sigmaOpenCover"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.coproductIsCoproduct'",
"CategoryTheory.Limits.coproductIsCoproduct"
]
}
]
},
{
"id": "mathlibmpr_060",
"pr": 35582,
"formal_main_result": "ContinuousLinearMap.ker_adjoint_comp_self",
"NL_main_result": "For a continuous `𝕜`-linear map `T : E →L[𝕜] F` between Hilbert spaces, `ker (T† ∘L T) = ker T`. (Axler 7.64(b).)",
"formal_statement": "import Mathlib\n\nnamespace ContinuousLinearMap\n\nvariable {𝕜 E F : Type*} [RCLike 𝕜]\n [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]\n [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]\n\n/-- **Main result.** For a continuous `𝕜`-linear map `T : E →L[𝕜] F` between\nHilbert spaces, `ker (T† ∘L T) = ker T`. -/\ntheorem ker_adjoint_comp_self (T : E →L[𝕜] F) :\n (ContinuousLinearMap.adjoint T ∘L T).ker = T.ker := sorry\n\nend ContinuousLinearMap",
"premise_group": [
{
"kind": "original",
"docs": [
"ContinuousLinearMap.adjoint_inner_left",
"ContinuousLinearMap.adjoint_inner_right"
]
},
{
"kind": "alternative",
"docs": [
"ContinuousLinearMap.orthogonal_range"
]
},
{
"kind": "alternative",
"docs": [
"Submodule.orthogonal_orthogonal"
]
}
]
},
{
"id": "mathlibmpr_062",
"pr": 35745,
"formal_main_result": "Asymptotics.IsBigOWith.sum_congr",
"NL_main_result": "Let `(Aᵢ)_{i ∈ s}` and `(Bᵢ)_{i ∈ s}` be finite families of functions on a filter `l`. If for each `i` we have `Aᵢ =O_{Cᵢ}[l] Bᵢ` (i.e. eventually `‖Aᵢ x‖ ≤ Cᵢ · ‖Bᵢ x‖`), then the sum `∑ᵢ Aᵢ` is `O_{sup Cᵢ}[l]` of the function `∑ᵢ ‖Bᵢ‖`. The constant `sup Cᵢ` is the supremum of the individual constants over the index set.",
"formal_statement": "import Mathlib\n\nnamespace Asymptotics\n\nvariable {α ι E F : Type*}\n [NormedAddCommGroup E] [NormedAddCommGroup F]\n {l : Filter α} {s : Finset ι} {C : ι → ℝ} {A : ι → α → E} {B : ι → α → F}\n\n/-- **Main result.** Termwise big-O for finite sums: if `A i =O_{C i}[l] B i`\nfor each `i ∈ s`, then `∑ᵢ A i =O_{s.sup' _ C}[l] (∑ᵢ ‖B i‖)`. -/\ntheorem IsBigOWith.sum_congr (hs : s.Nonempty)\n (h : ∀ i ∈ s, IsBigOWith (C i) l (A i) (B i)) :\n IsBigOWith (s.sup' hs C) l (fun x => ∑ i ∈ s, A i x)\n (fun x => ∑ i ∈ s, ‖B i x‖) := sorry\n\nend Asymptotics",
"premise_group": [
{
"kind": "original",
"docs": [
"Finset.sup'_eq_csSup_image"
]
}
]
},
{
"id": "mathlibmpr_063",
"pr": 35767,
"formal_main_result": "ValueDistribution.logCounting_isBigO_one_iff_analyticOnNhd",
"NL_main_result": "Let `f : 𝕜 → E` be meromorphic. Then the pole counting function `r ↦ N(r, f, ⊤)` is `O(1)` (i.e. uniformly bounded for large `r`) if and only if the normal-form `toMeromorphicNFOn f univ` is analytic on all of `𝕜` — equivalently, every singularity of `f` is removable.",
"formal_statement": "import Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic\n\nopen Asymptotics Filter Function Real Set\n\nnamespace ValueDistribution\n\nvariable\n {𝕜 : Type*} [NontriviallyNormedField 𝕜] [ProperSpace 𝕜]\n {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]\n\n/-- **Main result.** A meromorphic function has only removable\nsingularities iff its pole-divisor `logCounting` is bounded at infinity. -/\ntheorem logCounting_isBigO_one_iff_analyticOnNhd {f : 𝕜 → E} (h : Meromorphic f) :\n logCounting f ⊤ =O[atTop] (1 : ℝ → ℝ) ↔\n AnalyticOnNhd 𝕜 (toMeromorphicNFOn f univ) univ := sorry\n\nend ValueDistribution",
"premise_group": [
{
"kind": "original",
"docs": [
"MeromorphicNFAt.meromorphicOrderAt_nonneg_iff_analyticAt"
]
},
{
"kind": "original",
"docs": [
"MeromorphicOn.divisor_apply"
]
},
{
"kind": "original",
"docs": [
"MeromorphicOn.divisor_of_toMeromorphicNFOn"
]
},
{
"kind": "original",
"docs": [
"Function.locallyFinsuppWithin.logCounting_nonneg"
]
},
{
"kind": "original",
"docs": [
"Function.locallyFinsuppWithin.logCounting_mono"
]
},
{
"kind": "original",
"docs": [
"Function.locallyFinsuppWithin.logCounting_le"
]
},
{
"kind": "alternative",
"docs": [
"Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt"
]
}
]
},
{
"id": "mathlibmpr_064",
"pr": 35810,
"formal_main_result": "ArithmeticFunction.isUnit_iff_isUnit_apply_one",
"NL_main_result": "Let `f : ArithmeticFunction R` over a commutative ring `R`. Then `f` is a unit in the Dirichlet convolution ring iff `f 1` is a unit in `R`.",
"formal_statement": "import Mathlib.NumberTheory.ArithmeticFunction\n\nnamespace ArithmeticFunction\n\nvariable {R : Type*} [CommRing R]\n\n/-- **Main result.** An arithmetic function over a commutative ring is\na unit (under Dirichlet convolution) iff its value at `1` is a unit. -/\ntheorem isUnit_iff_isUnit_apply_one {f : ArithmeticFunction R} :\n IsUnit f ↔ IsUnit (f 1) := sorry\n\nend ArithmeticFunction",
"premise_group": [
{
"kind": "original",
"docs": [
"ArithmeticFunction.mul_apply"
]
},
{
"kind": "original",
"docs": [
"ArithmeticFunction.mul_apply_one"
]
},
{
"kind": "original",
"docs": [
"Nat.cons_self_properDivisors",
"Nat.insert_self_properDivisors"
]
}
]
},
{
"id": "mathlibmpr_065",
"pr": 35918,
"formal_main_result": "Group.fg_of_descent",
"NL_main_result": "Let `G` be a group, `f : G →* G` an endomorphism such that `f(U) ⊆ U` for every subgroup `U`, and `h : G → ℝ` a height function. Suppose: a finite set `s` surjects onto `G ⧸ f(G)`; constants `0 ≤ a < b` and `c ≥ 0` satisfy `h x ≤ a · h(g·x) + c` for `g ∈ s`, `x ∈ G`, and `b · h x − c ≤ h(f x)` for all `x`; and every height-bounded set is finite. Then `G` is finitely generated.",
"formal_statement": "import Mathlib\n\nopen scoped Pointwise\n\n/-- **Main result.** Abstract descent theorem: if `G` carries a\nsubgroup-preserving endomorphism `f` and a height function `h` with\nquasi-multiplicativity bounds, a finite quotient generating set, and\nNorthcott-finiteness of bounded sublevel sets, then `G` is finitely\ngenerated. -/\ntheorem Group.fg_of_descent {G : Type*} [Group G] {f : G →* G}\n (hf : ∀ U : Subgroup G, U.map f ≤ U)\n {s : Set G} {h : G → ℝ} {a b c : ℝ}\n (ha : 0 ≤ a) (H₀ : a < b) (hs : s.Finite)\n (H₁ : s * f.range = .univ)\n (H₂ : ∀ g ∈ s, ∀ x, h x ≤ a * h (g * x) + c)\n (H₃ : ∀ x, b * h x - c ≤ h (f x))\n (H₄ : ∀ B, {x : G | h x ≤ B}.Finite) :\n Group.FG G := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"Set.exists_min_image",
"Finset.exists_min_image",
"Finite.exists_min"
]
},
{
"kind": "original",
"docs": [
"Subgroup.mem_closure_of_mem",
"Subgroup.subset_closure"
]
},
{
"kind": "original",
"docs": [
"Subgroup.notMem_of_notMem_closure"
]
}
]
},
{
"id": "mathlibmpr_066",
"pr": 35923,
"formal_main_result": "Height.mulHeight_mul_le",
"NL_main_result": "For tuples `x, y : ι → K` over an admissible-absolute-value field `K`, the multiplicative height of the pointwise product satisfies `H(x · y) ≤ H(x) · H(y)`.",
"formal_statement": "import Mathlib.NumberTheory.Height.Basic\n\nnamespace Height\n\nopen AdmissibleAbsValues Real Function\n\nvariable {K : Type*} [Field K] [AdmissibleAbsValues K] {ι : Type*} [Finite ι]\n\n/-- **Main result.** Submultiplicativity of the multiplicative height\non tuples: `H(x · y) ≤ H(x) · H(y)`. -/\nlemma mulHeight_mul_le (x y : ι → K) :\n mulHeight (x * y) ≤ mulHeight x * mulHeight y := sorry\n\nend Height",
"premise_group": [
{
"kind": "original",
"docs": [
"Real.iSup_fun_mul_eq_iSup_mul_iSup_of_nonneg",
"Cardinal.ciSup_mul_ciSup"
]
},
{
"kind": "original",
"docs": [
"Finite.le_ciSup",
"le_iSup"
]
},
{
"kind": "original",
"docs": [
"finprod_le_finprod",
"single_le_finprod"
]
},
{
"kind": "original",
"docs": [
"finprod_mul_distrib",
"finprod_mem_mul_distrib",
"Finset.prod_mul_distrib"
]
}
]
},
{
"id": "mathlibmpr_068",
"pr": 36091,
"formal_main_result": "geometric_hahn_banach_of_nonempty_interior",
"NL_main_result": "Let `E` be a real topological vector space, and let `s`, `t ⊆ E` be convex sets with `interior s` nonempty and disjoint from `t`, and `t` itself nonempty. Then there exist a *nonzero* continuous linear functional `f : E → ℝ` and a real number `u` such that `f a ≤ u` for all `a ∈ s` and `u ≤ f b` for all `b ∈ t`.",
"formal_statement": "import Mathlib.Analysis.LocallyConvex.Separation\nimport Mathlib.Analysis.Convex.Topology\n\nopen Set\n\nvariable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E]\n [IsTopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E}\n\n/-- **Main result.** Eidelheit's separation theorem: two convex subsets\n`s, t` of a real topological vector space, with `interior s` nonempty\nand disjoint from `t` and `t` nonempty, are weakly separated by a\nnonzero continuous linear functional. -/\ntheorem geometric_hahn_banach_of_nonempty_interior\n (hs : Convex ℝ s) (ht : Convex ℝ t) (hst : Disjoint (interior s) t)\n (hsint : (interior s).Nonempty) (htne : t.Nonempty) :\n ∃ (f : StrongDual ℝ E) (u : ℝ),\n f ≠ 0 ∧ (∀ a ∈ s, f a ≤ u) ∧ ∀ b ∈ t, u ≤ f b := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"geometric_hahn_banach_open",
"geometric_hahn_banach_open_open",
"RCLike.geometric_hahn_banach_open"
]
},
{
"kind": "original",
"docs": [
"Convex.closure_interior_eq_closure_of_nonempty_interior",
"Convex.interior_closure_eq_interior_of_nonempty_interior"
]
},
{
"kind": "alternative",
"docs": [
"continuous_gauge",
"Convex.uniformContinuous_gauge"
]
},
{
"kind": "alternative",
"docs": [
"exists_extension_of_le_sublinear"
]
}
]
},
{
"id": "mathlibmpr_069",
"pr": 36192,
"formal_main_result": "ArithmeticFunction.tendsTo_eulerProduct_of_tendsTo",
"NL_main_result": "Let `R` be a commutative semiring. Equip `ArithmeticFunction R` with the topology of pointwise convergence (the discrete topology on each coefficient slot). Let `f : ι → ArithmeticFunction R` be a family such that for every `n`, `(f i) n = (1 : ArithmeticFunction R) n` for all but finitely many `i`. Then for every `n`, the partial products `∏ i ∈ s, f i` evaluated at `n` are eventually equal to `(eulerProduct f) n` as `s : Finset ι` grows along `atTop`.",
"formal_statement": "import Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Algebra.InfiniteSum.Basic\nimport Mathlib.Topology.UniformSpace.Cauchy\n\nnamespace ArithmeticFunction\n\nopen Filter\n\nvariable {ι R : Type*} [CommSemiring R]\n\n/-- A private uniform space on `ArithmeticFunction R` (discrete on `R`,\nproduct on `ℕ → R`) used to define the Euler product as a `tprod`. -/\nlocal instance uniformSpace : UniformSpace (ArithmeticFunction R) :=\n letI : UniformSpace R := ⊥\n .comap ((↑) : ArithmeticFunction R → (ℕ → R)) inferInstance\n\n/-- The Euler product of a family of arithmetic functions. -/\nnoncomputable def eulerProduct (f : ι → ArithmeticFunction R) :\n ArithmeticFunction R := ∏' i, f i\n\n/-- **Main result.** If `f i n = 1 n` for all but finitely many `i` at\nevery `n`, then the partial products of the Euler-product family\nconverge pointwise to `eulerProduct f`. -/\ntheorem tendsTo_eulerProduct_of_tendsTo (f : ι → ArithmeticFunction R)\n (hf : ∀ n, ∀ᶠ i in cofinite, f i n = (1 : ArithmeticFunction R) n) :\n ∀ n, ∀ᶠ s in atTop, (∏ i ∈ s, f i) n = eulerProduct f n := sorry\n\nend ArithmeticFunction",
"premise_group": [
{
"kind": "original",
"docs": [
"multipliable_iff_cauchySeq_finset"
]
},
{
"kind": "original",
"docs": [
"isClosed_setOf_map_zero"
]
}
]
},
{
"id": "mathlibmpr_070",
"pr": 36320,
"formal_main_result": "Quiver.SchreierGraph.labelling_isCovering",
"NL_main_result": "Let a group `M` act on a set `V`, let `S` be a generator-index type with map `ι : S → M`, and form the Schreier graph `Γ := SchreierGraph V ι`. The labelling prefunctor `labelling : Γ ⥤q SingleObj S` is a *covering of quivers*: at every vertex `x : Γ`, the induced maps on stars and costars `Star_x → Star_{∗}` and `Costar_x → Costar_{∗}` are bijections.",
"formal_statement": "import Mathlib.Algebra.Group.Action.Defs\nimport Mathlib.Algebra.Group.Basic\nimport Mathlib.Combinatorics.Quiver.Covering\nimport Mathlib.Combinatorics.Quiver.SingleObj\n\nnamespace Quiver\n\n/-- A Schreier graph for a monoid `M` acting on `V` with generators\n`ι : S → M`: vertices `V`, with an edge `x → ι(s) • x` for each `s : S`. -/\n@[nolint unusedArguments, ext]\nstructure SchreierGraph (V : Type*) {M : Type*} [SMul M V] {S : Type*} (_ι : S → M) where\n ofVertex ::\n toVertex : V\n\nnamespace SchreierGraph\n\nsection Basic\nvariable (V : Type*) {M : Type*} [SMul M V] {S : Type*} (ι : S → M)\n\ninstance schreierGraphSMul : SMul M (SchreierGraph V ι) where\n smul x y := ⟨x • y.toVertex⟩\n\ninstance schreierGraphQuiver : Quiver (SchreierGraph V ι) where\n Hom x y := { s : S // (ι s) • x = y }\n\n/-- The labelling prefunctor sending each edge to its label in `S`. -/\n@[simps]\ndef labelling : SchreierGraph V ι ⥤q SingleObj S where\n obj _ := SingleObj.star S\n map e := e.val\n\nend Basic\n\nsection GroupAction\nvariable {V : Type*} {M : Type*} [Group M] [MulAction M V] {S : Type*} (ι : S → M)\n\n/-- **Main result.** For a group action, the labelling prefunctor of a\nSchreier graph is a covering of quivers (bijective on stars and costars\nat every vertex). -/\ntheorem labelling_isCovering : (labelling V ι).IsCovering := sorry\n\nend GroupAction\n\nend SchreierGraph\nend Quiver",
"premise_group": [
{
"kind": "original",
"docs": [
"Prefunctor.IsCovering",
"Prefunctor.isCovering_of_bijective_star",
"Prefunctor.isCovering_of_bijective_costar"
]
}
]
},
{
"id": "mathlibmpr_071",
"pr": 36405,
"formal_main_result": "Derivative.normalizedDerivOfComplex_mul",
"NL_main_result": "The normalised derivative `D` satisfies the Leibniz rule on the upper half-plane: for `MDifferentiable` functions `F, G : ℍ → ℂ`, `D (F · G) = (D F) · G + F · (D G)`.",
"formal_statement": "import Mathlib\n\nopen UpperHalfPlane hiding I\nopen Real Complex\nopen scoped Manifold\n\nnamespace Derivative\n\nnoncomputable section\n\n/-- The normalized derivative `D = (2π i)⁻¹ · d/dz` on functions on `ℍ`. -/\ndef normalizedDerivOfComplex (F : ℍ → ℂ) (z : ℍ) : ℂ :=\n (2 * π * I)⁻¹ * deriv (F ∘ ofComplex) z\n\nscoped notation \"D\" => normalizedDerivOfComplex\n\n/-- **Main result.** Leibniz rule for the normalised derivative `D` on\n`MDifferentiable` functions `F, G : ℍ → ℂ`:\n`D (F · G) = (D F) · G + F · (D G)`. -/\n@[simp]\ntheorem normalizedDerivOfComplex_mul (F G : ℍ → ℂ) (hF : MDiff F) (hG : MDiff G) :\n D (F * G) = D F * G + F * D G := sorry\n\nend\n\nend Derivative",
"premise_group": [
{
"kind": "original",
"docs": [
"deriv_mul"
]
},
{
"kind": "original",
"docs": [
"UpperHalfPlane.mdifferentiableAt_iff",
"deriv_mul"
]
},
{
"kind": "alternative",
"docs": [
"HasMFDerivAt.mul"
]
},
{
"kind": "alternative",
"docs": [
"mfderiv_comp",
"mfderiv_comp_apply"
]
}
]
},
{
"id": "mathlibmpr_072",
"pr": 36408,
"formal_main_result": "mem_unitary_iff_isStarNormal_and_realPart_sq_add_imaginaryPart_sq_eq_one",
"NL_main_result": "For a complex unital `*`-algebra `A` (with a `StarModule ℂ` and the obvious tower / commutation typeclasses), an element `x : A` is unitary iff it is star-normal and `(ℜ x)² + (ℑ x)² = 1`. (When one writes `x = a + i·b` with `a := ℜ x`, `b := ℑ x`, this is the familiar \"`x` is unitary iff `a` and `b` commute and `a² + b² = 1`\".)",
"formal_statement": "import Mathlib\n\nvariable {A : Type*}\n\n/-- **Main result.** For a complex unital `*`-algebra `A` (a `StarRing`\nwith a `StarModule ℂ` and the obvious tower / commutation typeclasses),\nan element `x : A` is unitary iff it is star-normal and\n`(ℜ x)² + (ℑ x)² = 1`. -/\ntheorem mem_unitary_iff_isStarNormal_and_realPart_sq_add_imaginaryPart_sq_eq_one\n [Ring A] [StarRing A] [Module ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A]\n [StarModule ℂ A] {x : A} :\n x ∈ unitary A ↔ IsStarNormal x ∧ (realPart x : A) ^ 2 + (imaginaryPart x : A) ^ 2 = (1 : A) :=\n sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"isStarNormal_iff"
]
},
{
"kind": "original",
"docs": [
"Unitary.mem_iff",
"Unitary.mem_iff_self_mul_star",
"Unitary.mem_iff_star_mul_self"
]
}
]
},
{
"id": "mathlibmpr_073",
"pr": 36447,
"formal_main_result": "FiniteDimensional.of_totallyBounded_nhds_zero",
"NL_main_result": "Let `E` be a Hausdorff topological vector space over a complete non-trivial normed field `𝕜`. If `E` admits a totally bounded neighbourhood of `0`, then `E` is finite-dimensional.",
"formal_statement": "import Mathlib.Analysis.Normed.Module.FiniteDimension\n\nopen Filter Topology\n\nvariable (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]\n {E : Type*} [AddCommGroup E] [UniformSpace E] [T2Space E] [IsUniformAddGroup E]\n [Module 𝕜 E] [ContinuousSMul 𝕜 E]\n\n/-- **Main result.** Riesz's theorem for general TVS: a Hausdorff\ntopological vector space over a complete non-trivial normed field that\nadmits a totally bounded neighbourhood of `0` is finite-dimensional. -/\ntheorem FiniteDimensional.of_totallyBounded_nhds_zero {U : Set E}\n (hU_nhds : U ∈ 𝓝 (0 : E)) (hU_tb : TotallyBounded U) :\n FiniteDimensional 𝕜 E := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"totallyBounded_iff_subset_finite_iUnion_nhds_zero",
"totallyBounded_iff_subset_finite_iUnion_nhds_one"
]
},
{
"kind": "original",
"docs": [
"TotallyBounded.isVonNBounded"
]
},
{
"kind": "original",
"docs": [
"Bornology.IsVonNBounded.tendsto_smallSets_nhds",
"Bornology.isVonNBounded_iff_tendsto_smallSets_nhds"
]
},
{
"kind": "original",
"docs": [
"Submodule.closed_of_finiteDimensional"
]
},
{
"kind": "original",
"docs": [
"Absorbent.submodule_eq_top"
]
},
{
"kind": "alternative",
"docs": [
"exists_seq_norm_le_one_le_norm_sub",
"exists_seq_norm_le_one_le_norm_sub'"
]
},
{
"kind": "alternative",
"docs": [
"Submodule.closed_of_finiteDimensional"
]
},
{
"kind": "alternative",
"docs": [
"TotallyBounded.image"
]
}
]
},
{
"id": "mathlibmpr_075",
"pr": 36490,
"formal_main_result": "CategoryTheory.Adjunction.isCocontinuous_iff_coverPreserving",
"NL_main_result": "Let `F ⊣ G : C ⥤ D` be an adjunction equipped with Grothendieck topologies `J` on `C` and `K` on `D`. Then `F` is cocontinuous from `J` to `K` (every `K`-cover of `F U` pulls back to a `J`-cover of `U`) if and only if `G` preserves covers from `K` to `J` (every `K`-cover of `X` is sent by `G` to a `J`-cover of `G X`).",
"formal_statement": "import Mathlib.CategoryTheory.Sites.CoverLifting\nimport Mathlib.CategoryTheory.Sites.CoverPreserving\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D]\n (J : GrothendieckTopology C) (K : GrothendieckTopology D)\n {F : C ⥤ D} {G : D ⥤ C}\n\n/-- **Main result.** For an adjunction `F ⊣ G` between sites, `F` is\ncocontinuous (every `K`-cover of `F U` pulls back to a `J`-cover of\n`U`) iff `G` preserves covers (every `K`-cover of `X` is sent by `G`\nto a `J`-cover of `G X`). -/\nlemma Adjunction.isCocontinuous_iff_coverPreserving (adj : F ⊣ G) :\n F.IsCocontinuous J K ↔ CoverPreserving K J G := sorry\n\nend CategoryTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"CategoryTheory.GrothendieckTopology.pullback_stable",
"CategoryTheory.GrothendieckTopology.pullback_stable'"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.GrothendieckTopology.superset_covering"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Functor.IsCocontinuous.cover_lift",
"CategoryTheory.Functor.cover_lift"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Adjunction.homEquiv_naturality_left_symm",
"CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left_symm"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Adjunction.homEquiv_naturality_right_symm",
"CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Functor.isContinuous_of_coverPreserving"
]
}
]
},
{
"id": "mathlibmpr_076",
"pr": 36520,
"formal_main_result": "ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_bot",
"NL_main_result": "Let `T : E →L[𝕜] E` be a compact self-adjoint operator on a complete inner product space `E`. Then the orthogonal complement of `⨆ μ, eigenspace T μ` (the closed span of all eigenspaces) is `⊥`, i.e. the eigenspaces of `T` together span a dense subspace of `E`.",
"formal_statement": "import Mathlib.Analysis.InnerProductSpace.Spectrum\nimport Mathlib.Analysis.Normed.Operator.Compact\n\nnamespace ContinuousLinearMap\n\nopen Module.End Submodule\n\nvariable {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]\n [CompleteSpace E] {T : E →L[𝕜] E}\n\n/-- **Main result.** The spectral theorem for compact self-adjoint\noperators on a complete inner product space: the orthogonal complement\nof `⨆ μ, eigenspace T μ` (the closed span of all eigenspaces) is `⊥`,\ni.e. the eigenspaces of `T` together span a dense subspace. -/\ntheorem orthogonalComplement_iSup_eigenspaces_eq_bot\n (hT : IsCompactOperator T) (hT' : T.IsSymmetric) :\n (⨆ μ, eigenspace (T : Module.End 𝕜 E) μ)ᗮ = ⊥ := sorry\n\nend ContinuousLinearMap",
"premise_group": [
{
"kind": "original",
"docs": [
"IsSelfAdjoint.spectralRadius_eq_nnnorm",
"IsSelfAdjoint.toReal_spectralRadius_eq_norm",
"IsStarNormal.spectralRadius_eq_nnnorm",
"IsometricContinuousFunctionalCalculus.isGreatest_spectrum"
]
},
{
"kind": "original",
"docs": [
"Module.End.hasEigenvalue_iff_mem_spectrum"
]
},
{
"kind": "original",
"docs": [
"LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_invariant"
]
},
{
"kind": "original",
"docs": [
"Submodule.inf_orthogonal_eq_bot",
"Submodule.orthogonal_disjoint",
"ClosedSubmodule.inf_orthogonal_eq_bot"
]
}
]
},
{
"id": "mathlibmpr_077",
"pr": 36946,
"formal_main_result": "PointedCone.DualFG.dual_dual_flip",
"NL_main_result": "Let `p : M →ₗ N →ₗ R` be a bilinear pairing between two `R`-modules and let `C : PointedCone R N` be a dually finitely generated cone (i.e. the dual w.r.t. `p` of some finite subset of `M`). Then `C` is recovered from its dual: `dual p (dual p.flip C) = C`.",
"formal_statement": "import Mathlib.Geometry.Convex.Cone.Dual\n\nnamespace PointedCone\n\nvariable {R M N : Type*}\n [CommRing R] [PartialOrder R] [IsOrderedRing R]\n [AddCommGroup M] [Module R M]\n [AddCommGroup N] [Module R N]\n (p : M →ₗ[R] N →ₗ[R] R)\n\n/-- A cone is dually finitely generated (`DualFG`) if it is the dual of\na finite set. -/\ndef DualFG (C : PointedCone R N) : Prop := ∃ s : Finset M, dual p s = C\n\nvariable {p}\n\n/-- **Main result.** The double dual of a dually finitely generated\ncone is the cone itself: for `C : PointedCone R N` with `C.DualFG p`,\nwe have `dual p (dual p.flip C) = C`. -/\nlemma DualFG.dual_dual_flip {C : PointedCone R N} (hC : C.DualFG p) :\n dual p (dual p.flip C) = C := sorry\n\nend PointedCone",
"premise_group": [
{
"kind": "original",
"docs": [
"PointedCone.dual_dual_flip_dual"
]
}
]
},
{
"id": "mathlibmpr_078",
"pr": 36950,
"formal_main_result": "AnalyticOnNhd.sum_divisor_le",
"NL_main_result": "Let `f : ℂ → ℂ` be analytic on the closed ball `B(c, R)`, satisfy `f c ≠ 0`, and `‖f z‖ ≤ M` on the sphere `∂B(c, R)` for some `M ≥ 1`. For any `r` with `0 < |r| < |R|`: the sum (with multiplicity) of the orders of zeros of `f` in `B(c, r)` is at most `log(M / ‖f c‖) / log(R / r)`.",
"formal_statement": "import Mathlib.Analysis.Complex.JensenFormula\n\nopen Real MeromorphicOn\n\nvariable {c : ℂ} {r R M : ℝ} {f : ℂ → ℂ}\n\n/-- **Main result.** Jensen's inequality bounding the number of zeros\nin `B(c, r)` by `log(M / ‖f c‖) / log(R / r)`, given `f` analytic on\n`B(c, |R|)`, `f c ≠ 0`, and `‖f z‖ ≤ M` on the boundary sphere. -/\ntheorem AnalyticOnNhd.sum_divisor_le\n (r_pos : 0 < |r|) (r_lt_R : |r| < |R|) (hM : 1 ≤ M)\n (h₁f : AnalyticOnNhd ℂ f (Metric.closedBall c |R|))\n (h₂f : f c ≠ 0)\n (f_bound : ∀ z ∈ Metric.sphere c |R|, ‖f z‖ ≤ M) :\n ∑ᶠ u, divisor f (Metric.closedBall c |r|) u\n ≤ Real.log (M / ‖f c‖) / Real.log (R / r) := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"MeromorphicOn.circleAverage_log_norm"
]
},
{
"kind": "original",
"docs": [
"Real.circleAverage_mono_on_of_le_circle",
"Real.circleAverage_mono"
]
}
]
},
{
"id": "mathlibmpr_079",
"pr": 37296,
"formal_main_result": "ClusterPt.limsSup",
"NL_main_result": "Let `α` be a conditionally complete linear order with the order topology and let `f` be a non-trivial filter on `α`. If `f` is bounded and cobounded above, then `f.limsSup` is a cluster point of `f`.",
"formal_statement": "import Mathlib.Topology.Order.LiminfLimsup\n\nopen Filter Topology\n\nvariable {α : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]\n\n/-- **Main result.** Let `f` be a non-trivial filter on a conditionally\ncomplete linear order `α` with the order topology. If `f` is bounded\nand cobounded above, then `f.limsSup` is a cluster point of `f`. -/\ntheorem ClusterPt.limsSup {f : Filter α} [NeBot f]\n (hc : f.IsCobounded (· ≤ ·) := by isBoundedDefault)\n (hb : f.IsBounded (· ≤ ·) := by isBoundedDefault) :\n ClusterPt f.limsSup f := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"Filter.frequently_lt_of_lt_limsSup"
]
},
{
"kind": "original",
"docs": [
"Filter.lt_mem_sets_of_limsSup_lt"
]
},
{
"kind": "original",
"docs": [
"clusterPt_iff_frequently",
"Filter.HasBasis.clusterPt_iff_frequently"
]
},
{
"kind": "alternative",
"docs": [
"Filter.limsSup_eq_iInf_sSup"
]
}
]
},
{
"id": "mathlibmpr_080",
"pr": 37297,
"formal_main_result": "Matrix.PosSemidef.hadamard",
"NL_main_result": "For matrices `A, B : Matrix ι ι 𝕜` over an RCLike-scalar field, if `A` and `B` are positive semidefinite then so is their entrywise (Hadamard) product `A ⊙ B`.",
"formal_statement": "import Mathlib.Analysis.Matrix.Order\n\nopen Matrix\nopen scoped ComplexOrder\nopen scoped MatrixOrder\n\ntheorem Matrix.PosSemidef.hadamard {ι 𝕜 : Type*} [RCLike 𝕜]\n {A B : Matrix ι ι 𝕜} (hA : A.PosSemidef) (hB : B.PosSemidef) :\n (A ⊙ B).PosSemidef := sorry",
"premise_group": [
{
"kind": "original",
"docs": [
"Matrix.PosSemidef.kronecker",
"Matrix.PosDef.kronecker"
]
},
{
"kind": "original",
"docs": [
"Matrix.PosSemidef.submatrix",
"Matrix.posSemidef_submatrix_equiv"
]
},
{
"kind": "original",
"docs": [
"Matrix.posSemidef_iff_dotProduct_mulVec"
]
},
{
"kind": "alternative",
"docs": [
"Matrix.posSemidef_vecMulVec_self_star",
"Matrix.posSemidef_vecMulVec_star_self"
]
}
]
},
{
"id": "mathlibmpr_081",
"pr": 37612,
"formal_main_result": "Real.exists_measure_rpow_eq_integral_rpowIntegrand₁₂",
"NL_main_result": "For each `p ∈ (1,2)` there exists a positive scalar multiple `μ` of Lebesgue measure on `(0,∞)` such that for every `x ≥ 0`, the function `t ↦ rpowIntegrand₁₂(p,t,x) = t^{p−1}(x/t + t/(t+x) − 1)` is integrable for `μ` and `x^p = ∫₀^∞ t^{p−1}(x/t + t/(t+x) − 1) dμ(t)`.",
"formal_statement": "import Mathlib.Analysis.SpecialFunctions.Pow.NNReal\nimport Mathlib.MeasureTheory.Integral.Bochner.Basic\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Basic\n\nnamespace Real\n\nopen Set MeasureTheory\n\nvariable {p : ℝ}\n\n/-- The integrand `t ^ (p - 1) * (t⁻¹ * x + t * (t + x)⁻¹ - 1)` whose\nintegral against a suitable measure on `(0, ∞)` recovers `x ^ p` for\n`p ∈ (1, 2)`. -/\nnoncomputable def rpowIntegrand₁₂ (p t x : ℝ) : ℝ :=\n t ^ (p - 1) * (t⁻¹ * x + t * (t + x)⁻¹ - 1)\n\n/-- **Main result.** For each `p ∈ (1, 2)` there exists a measure `μ`\non `ℝ` (a positive scalar multiple of Lebesgue measure on `(0, ∞)`)\nsuch that for every `x ≥ 0`, the function\n`t ↦ rpowIntegrand₁₂ p t x` is integrable on `(0, ∞)` and\n`x ^ p = ∫ t in (0, ∞), rpowIntegrand₁₂ p t x ∂μ`. -/\nlemma exists_measure_rpow_eq_integral_rpowIntegrand₁₂ (hp : p ∈ Ioo 1 2) :\n ∃ μ : Measure ℝ, ∀ x ∈ Ici (0 : ℝ),\n (IntegrableOn (fun t => rpowIntegrand₁₂ p t x) (Ioi 0) μ)\n ∧ x ^ p = ∫ t in Ioi 0, rpowIntegrand₁₂ p t x ∂μ := sorry\n\nend Real",
"premise_group": [
{
"kind": "original",
"docs": [
"Real.rpow_eq_const_mul_integral",
"Real.integral_rpowIntegrand₀₁_eq_rpow_mul_const"
]
}
]
},
{
"id": "mathlibmpr_082",
"pr": 38267,
"formal_main_result": "MeasureTheory.IsPredictable.predictablePart_eq",
"NL_main_result": "Let `f : ℕ → Ω → E` be predictable with respect to a filtration `ℱ` and integrable at every step. Then its predictable part equals (almost everywhere) the increment process from time `0`: `predictablePart f ℱ μ n =ᵐ[μ] f n − f 0`.",
"formal_statement": "import Mathlib.Probability.Martingale.Centering\n\nopen scoped MeasureTheory ProbabilityTheory\n\nnamespace MeasureTheory\n\nvariable {Ω E : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω} [NormedAddCommGroup E]\n [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : Filtration ℕ m0}\n\n/-- **Main result.** A predictable, integrable process `f : ℕ → Ω → E` has\npredictable part equal almost everywhere to the increment `f n − f 0`. -/\nlemma IsPredictable.predictablePart_eq [SecondCountableTopology E] [MeasurableSpace E]\n [BorelSpace E] [SigmaFiniteFiltration μ ℱ] (hf : IsPredictable ℱ f)\n (hfint : ∀ n, Integrable (f n) μ) (n : ℕ) :\n predictablePart f ℱ μ n =ᵐ[μ] f n - f 0 := sorry\n\nend MeasureTheory",
"premise_group": [
{
"kind": "original",
"docs": [
"Finset.sum_range_sub",
"Finset.sum_Ico_sub",
"List.sum_range_sub"
]
}
]
}
] |