| {"name":"SciLean.sub_restrict'","declaration":"theorem SciLean.sub_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : T.restrict A - S.restrict A = (T - S).restrict A"} | |
| {"name":"SciLean.Distribution.toMeasure","declaration":"def SciLean.Distribution.toMeasure {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] (f' : 𝒟'(X, R)) : MeasureTheory.Measure X"} | |
| {"name":"SciLean.Distribution.postComp.arg_T.IsSmoothLinarMap_rule","declaration":"theorem SciLean.Distribution.postComp.arg_T.IsSmoothLinarMap_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_5} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (T : W → 𝒟'(X, Y)) (f : Y ⊸[R] Z) (hT : SciLean.IsSmoothLinearMap R T) : SciLean.IsSmoothLinearMap R fun w => SciLean.Distribution.postComp (T w) f"} | |
| {"name":"SciLean.Distribution.smul_extAction","declaration":"theorem SciLean.Distribution.smul_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (r : R) (T : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (r • T).extAction φ L = r • T.extAction φ L"} | |
| {"name":"SciLean.Function.toDistribution_extAction","declaration":"theorem SciLean.Function.toDistribution_extAction {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_1} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (φ : X → R) : (↑f).extAction φ (fun y ⊸[R] fun r ⊸[R] r • y) = ∫' x, φ x • f x ∂MeasureTheory.volume"} | |
| {"name":"SciLean.Distribution.restrict","declaration":"def SciLean.Distribution.restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : 𝒟'(X, Y)"} | |
| {"name":"SciLean.«term𝒟'(_,_)»","declaration":"def SciLean.«term𝒟'(_,_)» : Lean.ParserDescr"} | |
| {"name":"SciLean.restrict_univ","declaration":"theorem SciLean.restrict_univ {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) : T.restrict Set.univ = T"} | |
| {"name":"SciLean.Distribution.extAction_iteD'","declaration":"theorem SciLean.Distribution.extAction_iteD' {R : Type u_2} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (A : Set X) (B : Set X) (t : 𝒟'(X, U)) (e : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : ((ifD A then\n t\n else\n e).restrict\n B).extAction\n φ L =\n (t.restrict B).extAction (fun x => if x ∈ A then φ x else 0) L +\n (e.restrict B).extAction (fun x => if x ∉ A then φ x else 0) L"} | |
| {"name":"SciLean.Distribution.prod'_extAction","declaration":"theorem SciLean.Distribution.prod'_extAction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_7} [SciLean.Vec R X] {Z : Type u_6} [SciLean.Vec R Z] {X₁ : Type u_2} [SciLean.Vec R X₁] {X₂ : Type u_4} [SciLean.Vec R X₂] {Y₁ : Type u_3} [SciLean.Vec R Y₁] {Y₂ : Type u_5} [SciLean.Vec R Y₂] (p : X₁ → X₂ → X) (T : 𝒟'(X₁, Y₁)) (S : X₁ → 𝒟'(X₂, Y₂)) (L : Y₁ ⊸[R] Y₂ ⊸[R] Z) (K : Z ⊸[R] R ⊸[R] Z) (φ : X → R) : (SciLean.Distribution.prod p T S L).extAction φ K =\n T.extAction (fun x₁ => (S x₁).extAction (fun x₂ => φ (p x₁ x₂)) (fun y₂ ⊸[R] fun r ⊸[R] r • y₂))\n (fun y₁ ⊸[R] fun y₂ ⊸[R] (K ((L y₁) y₂)) 1)"} | |
| {"name":"SciLean.Distribution.ext","declaration":"theorem SciLean.Distribution.ext {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (x : 𝒟'(X, Y)) (y : 𝒟'(X, Y)) : (∀ (φ : 𝒟 X), x φ = y φ) → x = y"} | |
| {"name":"Function.toDistribution","declaration":"def Function.toDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) : 𝒟'(X, Y)"} | |
| {"name":"SciLean.Distribution.extAction.arg_φ.IsSmoothLinearMap","declaration":"theorem SciLean.Distribution.extAction.arg_φ.IsSmoothLinearMap {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (φ : W → X → V) (L : U ⊸[R] V ⊸[R] Z) (hφ : SciLean.IsSmoothLinearMap R φ) : SciLean.IsSmoothLinearMap R fun w => T.extAction (φ w) L"} | |
| {"name":"SciLean.postComp_id","declaration":"theorem SciLean.postComp_id {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (u : 𝒟'(X, Y)) : SciLean.Distribution.postComp u (fun y ⊸[R] y) = u"} | |
| {"name":"SciLean.instInnerDistribution","declaration":"instance SciLean.instInnerDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [MeasureTheory.MeasureSpace X] [SciLean.SemiInnerProductSpace R Y] [Module ℝ Y] : Inner R (𝒟'(X, Y))"} | |
| {"name":"SciLean.HSub.hSub.arg_a0a1.toDistribution_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : (↑fun x => f x - g x) = ↑f - ↑g"} | |
| {"name":"SciLean.iteD.arg_te.IsSmoothLinearMap_rule","declaration":"theorem SciLean.iteD.arg_te.IsSmoothLinearMap_rule {R : Type u_2} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (t : W → 𝒟'(X, Y)) (e : W → 𝒟'(X, Y)) (ht : SciLean.IsSmoothLinearMap R t) (he : SciLean.IsSmoothLinearMap R e) : SciLean.IsSmoothLinearMap R fun w =>\n ifD A then\n t w\n else\n e w"} | |
| {"name":"SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule'","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (r : R) (f : X → Y) : r • ↑f = ↑fun x => r • f x"} | |
| {"name":"SciLean.Distribution.add_extAction","declaration":"theorem SciLean.Distribution.add_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (T' : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (T + T').extAction φ L = T.extAction φ L + T'.extAction φ L"} | |
| {"name":"SciLean.unexpandIteD","declaration":"def SciLean.unexpandIteD : Lean.PrettyPrinter.Unexpander"} | |
| {"name":"SciLean.Function.toDistribution_action","declaration":"theorem SciLean.Function.toDistribution_action {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (φ : 𝒟 X) : ↑f φ = ∫' x, φ x • f x ∂MeasureTheory.volume"} | |
| {"name":"SciLean.Distribution.mk_extAction","declaration":"theorem SciLean.Distribution.mk_extAction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : (X → R) → Y) (hT : SciLean.IsSmoothLinearMap R fun φ => T ⇑φ) (φ : X → R) : (fun φ ⊸[R] T ⇑φ).extAction φ (fun y ⊸[R] fun r ⊸[R] r • y) = T φ"} | |
| {"name":"SciLean.Distribution.neg_extAction","declaration":"theorem SciLean.Distribution.neg_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (-T).extAction φ L = -T.extAction φ L"} | |
| {"name":"SciLean.Distribution","declaration":"def SciLean.Distribution (R : Type u_1) [SciLean.RealScalar R] (X : Type u_2) [SciLean.Vec R X] (Y : Type u_3) [SciLean.Vec R Y] : Type (max (max u_2 u_1) u_3)"} | |
| {"name":"SciLean.TestFunction.apply_IsSmoothLinearMap","declaration":"theorem SciLean.TestFunction.apply_IsSmoothLinearMap {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] : SciLean.IsSmoothLinearMap R fun φ => ⇑φ"} | |
| {"name":"SciLean.unexpandDistribution","declaration":"def SciLean.unexpandDistribution : Lean.PrettyPrinter.Unexpander"} | |
| {"name":"SciLean.instModuleRealDistributionSemiringToAddCommMonoidToAddCommGroupToRCLikeToScalarInstVecSmoothLinearMapTestFunctionSpaceToTopologicalSpaceToUniformSpaceInstTCOrVecDiscreteTopologyInstVecToRCLikeToScalarTestFunctionSpace","declaration":"instance SciLean.instModuleRealDistributionSemiringToAddCommMonoidToAddCommGroupToRCLikeToScalarInstVecSmoothLinearMapTestFunctionSpaceToTopologicalSpaceToUniformSpaceInstTCOrVecDiscreteTopologyInstVecToRCLikeToScalarTestFunctionSpace {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] : Module ℝ (𝒟'(X, Y))"} | |
| {"name":"SciLean.postComp_comp","declaration":"theorem SciLean.postComp_comp {R : Type u_1} [SciLean.RealScalar R] {W : Type u_5} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (x : 𝒟'(X, U)) (g : U ⊸[R] V) (f : V ⊸[R] W) : SciLean.Distribution.postComp (SciLean.Distribution.postComp x g) f =\n SciLean.Distribution.postComp x (fun u ⊸[R] f (g u))"} | |
| {"name":"SciLean.postComp_restrict_extAction","declaration":"theorem SciLean.postComp_restrict_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_5} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (x : 𝒟'(X, U)) (f : U ⊸[R] V) (A : Set X) (φ : X → W) (L : V ⊸[R] W ⊸[R] Z) : ((SciLean.Distribution.postComp x f).restrict A).extAction φ L =\n (x.restrict A).extAction φ (fun u ⊸[R] fun w ⊸[R] (L (f u)) w)"} | |
| {"name":"SciLean.Distribution.extAction'","declaration":"def SciLean.Distribution.extAction' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (φ : X → R) : Y"} | |
| {"name":"SciLean.indextype_sum_restrict'","declaration":"theorem SciLean.indextype_sum_restrict' {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {I : Type u_1} [SciLean.IndexType I] (T : I → 𝒟'(X, R)) (A : Set X) : ∑ i, (T i).restrict A = ( ∑ i, T i).restrict A"} | |
| {"name":"SciLean.instSemiInnerProductSpaceToRCLikeToScalarDistribution","declaration":"instance SciLean.instSemiInnerProductSpaceToRCLikeToScalarDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [MeasureTheory.MeasureSpace X] [SciLean.SemiInnerProductSpace R Y] [Module ℝ Y] : SciLean.SemiInnerProductSpace R (𝒟'(X, Y))"} | |
| {"name":"SciLean.Distribution.IsFunction","declaration":"def SciLean.Distribution.IsFunction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (T : 𝒟'(X, Y)) : Prop"} | |
| {"name":"SciLean.Distribution.extAction","declaration":"def SciLean.Distribution.extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_5} [SciLean.Vec R Z] (T : 𝒟'(X, Y)) (φ : X → Z) (L : Y ⊸[R] Z ⊸[R] W) : W"} | |
| {"name":"SciLean.add_restrict'","declaration":"theorem SciLean.add_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : T.restrict A + S.restrict A = (T + S).restrict A"} | |
| {"name":"SciLean.instTestFunctionsDistribution","declaration":"instance SciLean.instTestFunctionsDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [MeasureTheory.MeasureSpace X] [SciLean.SemiInnerProductSpace R Y] [Module ℝ Y] : SciLean.TestFunctions (𝒟'(X, Y))"} | |
| {"name":"SciLean.Distribution.postComp","declaration":"def SciLean.Distribution.postComp {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (T : 𝒟'(X, Y)) (f : Y ⊸[R] Z) : 𝒟'(X, Z)"} | |
| {"name":"SciLean.postComp_assoc","declaration":"theorem SciLean.postComp_assoc {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_5} [SciLean.Vec R Y] {U : Type u_3} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (x : 𝒟'(X, U)) (y : U ⊸[R] 𝒟'(Y, V)) (f : V ⊸[R] W) (φ : Y → R) : SciLean.Distribution.postComp (SciLean.Distribution.postComp x y) (fun T ⊸[R] SciLean.Distribution.postComp T f) =\n SciLean.Distribution.postComp x (fun u ⊸[R] SciLean.Distribution.postComp (y u) f)"} | |
| {"name":"SciLean.finset_sum_restrict","declaration":"theorem SciLean.finset_sum_restrict {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {I : Type u_1} [Fintype I] (T : I → 𝒟'(X, Y)) (A : Set X) : (Finset.sum Finset.univ fun i => T i).restrict A = Finset.sum Finset.univ fun i => (T i).restrict A"} | |
| {"name":"SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule'","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : ↑f + ↑g = ↑fun x => f x + g x"} | |
| {"name":"SciLean.Distribution.zero_extAction","declaration":"theorem SciLean.Distribution.zero_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_3} [SciLean.Vec R W] {X : Type u_5} [SciLean.Vec R X] {U : Type u_2} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : 0.extAction φ L = 0"} | |
| {"name":"SciLean.action_dirac","declaration":"theorem SciLean.action_dirac {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] (x : X) (φ : 𝒟 X) : (SciLean.dirac x) φ = φ x"} | |
| {"name":"SciLean.Function.toDistribution_zero","declaration":"theorem SciLean.Function.toDistribution_zero {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] : (↑fun x => 0) = 0"} | |
| {"name":"SciLean.term𝒟'_","declaration":"def SciLean.term𝒟'_ : Lean.ParserDescr"} | |
| {"name":"SciLean.Distribution.prod_restrict","declaration":"theorem SciLean.Distribution.prod_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_7} [SciLean.Vec R X] {Z : Type u_6} [SciLean.Vec R Z] {X₁ : Type u_2} [SciLean.Vec R X₁] {X₂ : Type u_4} [SciLean.Vec R X₂] {Y₁ : Type u_3} [SciLean.Vec R Y₁] {Y₂ : Type u_5} [SciLean.Vec R Y₂] (p : X₁ → X₂ → X) (T : 𝒟'(X₁, Y₁)) (S : X₁ → 𝒟'(X₂, Y₂)) (L : Y₁ ⊸[R] Y₂ ⊸[R] Z) (A : Set X) : (SciLean.Distribution.prod p T S L).restrict A =\n SciLean.Distribution.prod p (T.restrict (Set.preimage1 p A)) (fun x₁ => (S x₁).restrict (p x₁ ⁻¹' A)) L"} | |
| {"name":"SciLean.Distribution.IsMeasure","declaration":"def SciLean.Distribution.IsMeasure {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] (f : 𝒟'(X, R)) : Prop"} | |
| {"name":"SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : (↑fun x => f x + g x) = ↑f + ↑g"} | |
| {"name":"SciLean.HSub.hSub.arg_a0a1.toDistribution_rule'","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : ↑f - ↑g = ↑fun x => f x - g x"} | |
| {"name":"SciLean.termIfD_Then_____Else____","declaration":"def SciLean.termIfD_Then_____Else____ : Lean.ParserDescr"} | |
| {"name":"SciLean.finset_sum_restrict'","declaration":"theorem SciLean.finset_sum_restrict' {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {I : Type u_1} [Fintype I] (T : I → 𝒟'(X, Y)) (A : Set X) : (Finset.sum Finset.univ fun i => (T i).restrict A) = (Finset.sum Finset.univ fun i => T i).restrict A"} | |
| {"name":"SciLean.HMul.hMul.arg_a0.toDistribution_rule","declaration":"theorem SciLean.HMul.hMul.arg_a0.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : (↑fun x => f x * r) = r • ↑f"} | |
| {"name":"SciLean.Distribution.toFunction","declaration":"def SciLean.Distribution.toFunction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (T : 𝒟'(X, Y)) : X → Y"} | |
| {"name":"SciLean.Distribution.extAction.arg_T.IsSmoothLinearMap","declaration":"theorem SciLean.Distribution.extAction.arg_T.IsSmoothLinearMap {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : W → 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] Z) (hT : SciLean.IsSmoothLinearMap R T) : SciLean.IsSmoothLinearMap R fun w => (T w).extAction φ L"} | |
| {"name":"SciLean.iteD_restrict","declaration":"theorem SciLean.iteD_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : (ifD A then\n T\n else\n 0) =\n T.restrict A"} | |
| {"name":"MeasureTheory.Measure.toDistribution","declaration":"def MeasureTheory.Measure.toDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] (μ : MeasureTheory.Measure X) : 𝒟'(X, R)"} | |
| {"name":"SciLean.Distribution.iteD_same","declaration":"theorem SciLean.Distribution.iteD_same {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (u : 𝒟'(X, Y)) : (ifD A then\n u\n else\n u) =\n u"} | |
| {"name":"SciLean.HMul.hMul.arg_a1.toDistribution_rule'","declaration":"theorem SciLean.HMul.hMul.arg_a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : r • ↑f = ↑fun x => r • f x"} | |
| {"name":"SciLean.neg_restrict","declaration":"theorem SciLean.neg_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : (-T).restrict A = -T.restrict A"} | |
| {"name":"SciLean.iteD_restrict'","declaration":"theorem SciLean.iteD_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : (ifD A then\n 0\n else\n T) =\n T.restrict Aᶜ"} | |
| {"name":"SciLean.sub_restrict","declaration":"theorem SciLean.sub_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : (T - S).restrict A = T.restrict A - S.restrict A"} | |
| {"name":"SciLean.smul_restrict","declaration":"theorem SciLean.smul_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (r : R) (T : 𝒟'(X, Y)) (A : Set X) : (r • T).restrict A = r • T.restrict A"} | |
| {"name":"SciLean.smul_restrict'","declaration":"theorem SciLean.smul_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (r : R) (T : 𝒟'(X, Y)) (A : Set X) : r • T.restrict A = (r • T).restrict A"} | |
| {"name":"SciLean.Distribution.indextype_sum_extAction","declaration":"theorem SciLean.Distribution.indextype_sum_extAction {R : Type u_2} [SciLean.RealScalar R] {W : Type (max (max u_2 u_3) u_4)} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {U : Type u_4} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] {I : Type u_1} [SciLean.IndexType I] (T : I → 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : ( ∑ i, T i).extAction φ L = ∑ i, (T i).extAction φ L"} | |
| {"name":"SciLean.HMul.hMul.arg_a0.toDistribution_rule'","declaration":"theorem SciLean.HMul.hMul.arg_a0.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : r • ↑f = ↑fun x => f x * r"} | |
| {"name":"SciLean.zero_restrict","declaration":"theorem SciLean.zero_restrict {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) : 0.restrict A = 0"} | |
| {"name":"SciLean.Distribution.prod","declaration":"def SciLean.Distribution.prod {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_3} [SciLean.Vec R Z] {X₁ : Type u_4} [SciLean.Vec R X₁] {X₂ : Type u_5} [SciLean.Vec R X₂] {Y₁ : Type u_6} [SciLean.Vec R Y₁] {Y₂ : Type u_7} [SciLean.Vec R Y₂] (p : X₁ → X₂ → X) (T : 𝒟'(X₁, Y₁)) (S : X₁ → 𝒟'(X₂, Y₂)) (L : Y₁ ⊸[R] Y₂ ⊸[R] Z) : 𝒟'(X, Z)"} | |
| {"name":"SciLean.Distribution.«term⟪_,_⟫[_]»","declaration":"def SciLean.Distribution.«term⟪_,_⟫[_]» : Lean.ParserDescr"} | |
| {"name":"SciLean.Distribution.extAction_iteD","declaration":"theorem SciLean.Distribution.extAction_iteD {R : Type u_2} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (A : Set X) (t : 𝒟'(X, U)) (e : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (ifD A then\n t\n else\n e).extAction\n φ L =\n t.extAction (fun x => if x ∈ A then φ x else 0) L + e.extAction (fun x => if x ∉ A then φ x else 0) L"} | |
| {"name":"SciLean.add_restrict","declaration":"theorem SciLean.add_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : (T + S).restrict A = T.restrict A + S.restrict A"} | |
| {"name":"SciLean.postComp_extAction","declaration":"theorem SciLean.postComp_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] {Z : Type u_5} [SciLean.Vec R Z] {U : Type u_2} [SciLean.Vec R U] {V : Type u_3} [SciLean.Vec R V] {y : U ⊸[R] V} (x : 𝒟'(X, U)) (f : U ⊸[R] V) (φ : X → W) (L : V ⊸[R] W ⊸[R] Z) : (SciLean.Distribution.postComp x y).extAction φ L = x.extAction φ (fun u ⊸[R] fun w ⊸[R] (L (f u)) w)"} | |
| {"name":"SciLean.iteD.arg_cte.toDistribution_rule","declaration":"theorem SciLean.iteD.arg_cte.toDistribution_rule {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (s : Set X) (t : X → Y) (e : X → Y) : (ifD s then\n ↑t\n else\n ↑e) =\n ↑fun x => if x ∈ s then t x else e x"} | |
| {"name":"SciLean.instSMulRealDistribution","declaration":"instance SciLean.instSMulRealDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] : SMul ℝ (𝒟'(X, Y))"} | |
| {"name":"SciLean.dirac","declaration":"def SciLean.dirac {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] (x : X) : 𝒟'(X, R)"} | |
| {"name":"SciLean.indextype_sum_restrict","declaration":"theorem SciLean.indextype_sum_restrict {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {I : Type u_1} [SciLean.IndexType I] (T : I → 𝒟'(X, R)) (A : Set X) : ( ∑ i, T i).restrict A = ∑ i, (T i).restrict A"} | |
| {"name":"SciLean.Distribution.bind","declaration":"def SciLean.Distribution.bind {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {U : Type u_5} [SciLean.Vec R U] {V : Type u_6} [SciLean.Vec R V] (x' : 𝒟'(X, U)) (f : X → 𝒟'(Y, V)) (L : U ⊸[R] V ⊸[R] W) : 𝒟'(Y, W)"} | |
| {"name":"SciLean.Distribution.integrate","declaration":"def SciLean.Distribution.integrate {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) : Y"} | |
| {"name":"SciLean.instCoeMeasureDistributionToVecToRCLikeToScalarInstSemiInnerProductSpace","declaration":"instance SciLean.instCoeMeasureDistributionToVecToRCLikeToScalarInstSemiInnerProductSpace {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] : Coe (MeasureTheory.Measure X) (𝒟'(X, R))"} | |
| {"name":"SciLean.action_bind","declaration":"theorem SciLean.action_bind {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (x : 𝒟'(X, U)) (f : X → 𝒟'(Y, V)) (L : U ⊸[R] V ⊸[R] W) (φ : 𝒟 Y) : (SciLean.Distribution.bind x f L) φ = x.extAction (fun x' => SciLean.Distribution.extAction' (f x') ⇑φ) L"} | |
| {"name":"SciLean.iteD","declaration":"def SciLean.iteD {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (t : 𝒟'(X, Y)) (e : 𝒟'(X, Y)) : 𝒟'(X, Y)"} | |
| {"name":"SciLean.Distribution.fintype_sum_extAction","declaration":"theorem SciLean.Distribution.fintype_sum_extAction {R : Type u_2} [SciLean.RealScalar R] {W : Type u_5} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {U : Type u_4} [SciLean.Vec R U] {V : Type u_6} [SciLean.Vec R V] {I : Type u_1} [Fintype I] (T : I → 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (Finset.sum Finset.univ fun i => T i).extAction φ L = Finset.sum Finset.univ fun i => (T i).extAction φ L"} | |
| {"name":"SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (r : R) (f : X → Y) : (↑fun x => r • f x) = r • ↑f"} | |
| {"name":"SciLean.Distribution.sub_extAction","declaration":"theorem SciLean.Distribution.sub_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (T' : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (T - T').extAction φ L = T.extAction φ L - T'.extAction φ L"} | |
| {"name":"SciLean.Distribution.action_iteD","declaration":"theorem SciLean.Distribution.action_iteD {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (t : 𝒟'(X, Y)) (e : 𝒟'(X, Y)) (φ : 𝒟 X) : (ifD A then\n t\n else\n e)\n φ =\n t.extAction (fun x => if x ∈ A then φ x else 0) (fun y ⊸[R] fun r ⊸[R] r • y) +\n e.extAction (fun x => if x ∉ A then φ x else 0) (fun y ⊸[R] fun r ⊸[R] r • y)"} | |
| {"name":"SciLean.HMul.hMul.arg_a1.toDistribution_rule","declaration":"theorem SciLean.HMul.hMul.arg_a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : (↑fun x => r • f x) = r • ↑f"} | |
| {"name":"SciLean.neg_restrict'","declaration":"theorem SciLean.neg_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : -T.restrict A = (-T).restrict A"} | |