Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated • 3
fact stringlengths 7 18.7k | type stringclasses 10
values | library stringclasses 5
values | imports listlengths 0 13 | filename stringclasses 441
values | symbolic_name stringlengths 1 65 | docstring stringclasses 503
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|---|---|---|---|---|---|---|
linkArgs := if System.Platform.isWindows then #[] else if System.Platform.isOSX then #["-L/opt/homebrew/opt/openblas/lib", "-L/usr/local/opt/openblas/lib", "-lblas"] else -- assuming linux #["-L/usr/lib/x86_64-linux-gnu/", "-lblas", "-lm"] | def | root | [
"import Lake"
] | lakefile.lean | linkArgs | null |
inclArgs := if System.Platform.isWindows then #[] else if System.Platform.isOSX then #["-I/opt/homebrew/opt/openblas/include", "-I/usr/local/opt/openblas/include"] else -- assuming linux #[] package scilean { moreLinkArgs := linkArgs } -- -- require mathlib from git "https://github.com/leanprover-community/mathlib4" @ ... | def | root | [
"import Lake"
] | lakefile.lean | inclArgs | null |
List.toArrayType (l : List α) {n : USize} (h : l.length = n.toNat) [PlainDataType α] : α^[n] := ⊞ i => l.get ⟨i.1.toNat,sorry_proof⟩ open Lean Meta Elab Term Qq macro_rules | `(⊞[ $x:term, $xs:term,* ]) => do let n := Syntax.mkNumLit (toString (xs.getElems.size + 1)) `(List.toArrayType [$x,$xs,*] (n:=$n) sorry_proof) -... | def | examples | [
"import SciLean",
"import SciLean.Core.Functions.ArgMinMax",
"import SciLean.Modules.SolversAndOptimizers.GradientDescent"
] | examples/Ballistic.lean | List.toArrayType | null |
g := ⊞[0.0, -9.81] | def | examples | [
"import SciLean",
"import SciLean.Core.Functions.ArgMinMax",
"import SciLean.Modules.SolversAndOptimizers.GradientDescent"
] | examples/Ballistic.lean | g | null |
ballisticMotion (x v : Float^[2]) := (v, g - (5.0 + ‖v‖₂) • v) #generate_revDeriv ballisticMotion x v prop_by unfold ballisticMotion; fprop trans_by unfold ballisticMotion; ftrans noncomputable approx aimToTarget (v₀ : Float^[2]) (optimizationRate : Float) := λ (T : Float) (target : Float^[2]) => let shoot := λ (v : Fl... | def | examples | [
"import SciLean",
"import SciLean.Core.Functions.ArgMinMax",
"import SciLean.Modules.SolversAndOptimizers.GradientDescent"
] | examples/Ballistic.lean | ballisticMotion | null |
shotTrajectoryPoints (n : ℕ) (T : ℝ) (v : ℝ×ℝ) : Array ((ℝ×ℝ)×(ℝ×ℝ)) := odeSolve_fixed_dt_array (λ (t : ℝ) (x,v) => ballisticMotion x v) midpoint_step n 0 (0,v) T /-- Do one step of optimization -/ | def | examples | [
"import SciLean",
"import SciLean.Core.Functions.ArgMinMax",
"import SciLean.Modules.SolversAndOptimizers.GradientDescent"
] | examples/Ballistic.lean | shotTrajectoryPoints | /-- Generate `n` trajectory points in the interval [0,T] -/ |
aimStep (target : ℝ×ℝ) (v₀ : ℝ×ℝ) := aimToTarget v₀ (5.0:ℝ) (1:ℝ) target | def | examples | [
"import SciLean",
"import SciLean.Core.Functions.ArgMinMax",
"import SciLean.Modules.SolversAndOptimizers.GradientDescent"
] | examples/Ballistic.lean | aimStep | /-- Do one step of optimization -/ |
frame : Frame where xmin := -1 ymin := -1 xSize := 2 width := 400 height := 400 | def | examples | [
"import ProofWidgets.Component.InteractiveSvg",
"import ProofWidgets.Component.HtmlDisplay",
"import examples.Ballistic"
] | examples/BallisticWidget.lean | frame | null |
State where v : ℝ×ℝ deriving ToJson, FromJson local instance {frame : Frame} : CoeHead (Point frame) (ℝ×ℝ) where coe p := (⟨p.toAbsolute.1⟩, ⟨p.toAbsolute.2⟩) local instance {frame : Frame} : CoeTail (ℝ×ℝ) (Point frame) where coe x := .abs x.1.toFloat x.2.toFloat | structure | examples | [
"import ProofWidgets.Component.InteractiveSvg",
"import ProofWidgets.Component.HtmlDisplay",
"import examples.Ballistic"
] | examples/BallisticWidget.lean | State | null |
isvg : InteractiveSvg State where init := { v := 0 } frame := frame update time Δt action mouseStart mouseEnd selected getData state := if let .some mouseEnd := mouseEnd then let target : ℝ×ℝ := mouseEnd let newVel := aimStep target state.v { v := newVel } else state render time mouseStart mouseEnd state := { elements ... | def | examples | [
"import ProofWidgets.Component.InteractiveSvg",
"import ProofWidgets.Component.HtmlDisplay",
"import examples.Ballistic"
] | examples/BallisticWidget.lean | isvg | null |
updateSvg (params : UpdateParams State) : RequestM (RequestTask (UpdateResult State)) := isvg.serverRpcMethod params -- TODO: the tsx file is pretty broken @[widget_module] | def | examples | [
"import ProofWidgets.Component.InteractiveSvg",
"import ProofWidgets.Component.HtmlDisplay",
"import examples.Ballistic"
] | examples/BallisticWidget.lean | updateSvg | null |
SvgWidget : Component (UpdateResult State) where javascript := include_str ".." / "lake-packages" / "proofwidgets" / "build" / "js" / "interactiveSvg.js" | def | examples | [
"import ProofWidgets.Component.InteractiveSvg",
"import ProofWidgets.Component.HtmlDisplay",
"import examples.Ballistic"
] | examples/BallisticWidget.lean | SvgWidget | null |
init : UpdateResult State := { html := Html.ofTHtml <div>Init!!!</div>, state := { state := isvg.init time := 0 selected := none mousePos := none idToData := isvg.render 0 none none isvg.init |>.idToDataList} } #html <SvgWidget html={init.html} state={init.state}/> | def | examples | [
"import ProofWidgets.Component.InteractiveSvg",
"import ProofWidgets.Component.HtmlDisplay",
"import examples.Ballistic"
] | examples/BallisticWidget.lean | init | null |
main : IO Unit := do IO.println "fix this test!" #exit let v := DenseVector.ofFn (Array:=FloatArray) (vstrg := .normal) (fun (i : Fin 3) => i.1.toFloat) IO.println s!"v := {v}" IO.println s!"0.1•v := {v.scal 0.1}" IO.println s!"v+v := {v.axpy 1 v}" IO.println s!"‖v‖₂² := {v.nrm2}" | def | examples | [
"import LeanBLAS"
] | examples/BlasTest.lean | main | null |
A (x : ℝ^{2}) : ℝ^{2} := ⊞ i, if i = 0 then x[0] + x[1] else x[0] - x[1] -- We can turn on debugging logging with -- set_option trace.Meta.Tactic.fun_trans.rewrite true in function_properties A (x : ℝ^{2}) argument x IsLin, HasAdjoint, IsSmooth, HasAdjDiff, abbrev ∂ := λ dx => A dx by unfold A; fun_trans, def † := λ x'... | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | A | /-- choose a particular A: (x, y) ↦ (x + y, x - y) -/ |
loss (x : ℝ^{2}) : ℝ := ⟪x, A x⟫ function_properties loss (x : ℝ^{2}) argument x IsSmooth, HasAdjDiff, abbrev ∂ := λ dx => (2 : ℝ) * ⟪dx, A x⟫ by { simp[loss]; fun_trans; -- 'trans' makes me think of 'transitive'. simp; funext dx; -- ⊢ ⟪dx, A x⟫ + ⟪x, A dx⟫ = 2 * ⟪dx, A x⟫ -- ⟪dx, A x⟫ + ⟪x, A dx⟫ -- = ⟪dx, A x⟫ + ⟪A† ... | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | loss | null |
gradLoss (x : ℝ^{2}) : ℝ^{2} := ∇ loss x /-- make gradLoss computable by transforming abstract gradient defn into computable definition. -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | gradLoss | /-- noncomputable version of gradient of loss -/ |
gradLoss' (x : ℝ^{2}) : ℝ^{2} := (∇ loss x) rewrite_by { unfold gradient; unfold loss; fun_trans } /-- Data at each training iteration -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | gradLoss' | /-- make gradLoss computable by transforming abstract gradient defn into computable
definition. -/ |
TrainIter where iteration : Nat -- iteration count. point : ℝ^{2} -- current point on the circle.. loss_at_point : ℝ -- loss value at current point. gradAmbient : ℝ^{2} -- gradient at current point in ambient space. gradSubmanifold : ℝ^{2} -- gradient at current point on the tagent space. | structure | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | TrainIter | /-- Data at each training iteration -/ |
normalize (p : ℝ^{2}) : ℝ^{2} := p / ‖p‖ /-- Project 'v' along 'direction'. -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | normalize | /-- Normalize a vector. -/ |
project (v : ℝ^{2}) (direction: ℝ^{2}) : ℝ^{2} := ⟪v, direction⟫ • direction /-- Project point 'x' to lie on the circle. This is a retraction of (ℝ^2 - origin) onto the embedded S¹. -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | project | /-- Project 'v' along 'direction'. -/ |
circle_project (x : ℝ^{2}) := normalize x /-- project a vector to the tangent space at `p` by deleting the normal component -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | circle_project | /-- Project point 'x' to lie on the circle.
This is a retraction of (ℝ^2 - origin) onto the embedded S¹. -/ |
projectToCircleTangentSpace (p : ℝ^{2}) (vec : ℝ^{2}) : ℝ^{2} := vec - project (direction := p) vec /-- Calculate gradient at current point -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | projectToCircleTangentSpace | /-- project a vector to the tangent space at `p`
by deleting the normal component -/ |
TrainIter.calcAtPoint (i : Nat) (p : ℝ^{2}) : TrainIter where iteration := i point := p loss_at_point := loss p gradAmbient := gradLoss' p gradSubmanifold := projectToCircleTangentSpace (p := p) (gradLoss' p) | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | TrainIter.calcAtPoint | /-- Calculate gradient at current point -/ |
HyperParams where learningRate : ℝ := 0.01 | structure | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | HyperParams | /-- Calculate gradient at current point -/ |
HyperParams.default : HyperParams := {} | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | HyperParams.default | /-- Calculate gradient at current point -/ |
TrainIter.nextPoint (hp : HyperParams) (step : TrainIter) : ℝ^{2} := circle_project <| step.point - hp.learningRate • step.gradSubmanifold /-- run the gradient descent algorithm -/ | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | TrainIter.nextPoint | /-- Step to the next point by gradient descent -/ |
gradientDescend (init: ℝ^{2}) (nsteps : ℕ) (hp : HyperParams := HyperParams.default) : Array TrainIter := Id.run do let mut cur : ℝ^{2} := circle_project init let mut out := #[] for i in List.range nsteps do let step := TrainIter.calcAtPoint i cur out := out.push <| step cur := step.nextPoint hp return out | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | gradientDescend | /-- run the gradient descent algorithm -/ |
main : IO Unit := do let init : ℝ^{2} := ⊞ i, if i = 0 then 1 else 0 let losses := gradientDescend init 1000 for loss in losses do IO.println s!"{loss}" | def | examples | [
"import SciLean"
] | examples/CircleOptimisation.lean | main | /-- run the gradient descent algorithm -/ |
g : ℝ := 9.81 open SciLean variable (f : Fin n → ℝ × ℝ) #check λ [i] => f i | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | g | null |
ParticleLagrangian {n} (ϕ : ℝ^{d} → ℝ) (m : ℝ^{n}) (x v : ℝ^{d}^{n}) : ℝ := ∑ i, (0.5:ℝ) * m[i] * ∥v[i]∥² - ∑ i, m[i] * ϕ x[i] | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | ParticleLagrangian | null |
chainConstraint {n} (l : ℝ^{n}) (x : ℝ^{2}^{n+1}) : Prop := ∀ i : Fin n, ∥x[i.succ] - x[⟨i, sorry_proof⟩]∥ = l[i] | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | chainConstraint | null |
chainConstraintFun {n} (l : ℝ^{n}) (x : ℝ^{2}^{n+1}) : ℝ^{n} := λ [i] => l[i]*l[i] - ∥x[i.succ] - x[⟨i, sorry_proof⟩]∥² | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | chainConstraintFun | null |
chainConstraintLhs {n} (l : ℝ^{n}) (x : ℝ^{2}^{n+1}) := λ i : Fin n => ∥x[i.succ] - x[⟨i, sorry_proof⟩]∥ | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | chainConstraintLhs | null |
chainConstraintRhs {n} (l : ℝ^{n}) (x : ℝ^{2}^{n+1}) := λ i : Fin n => l[i] | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | chainConstraintRhs | null |
endPoints {n n' : Nat} (l r : ℝ^{2}) (x : ℝ^{2}^{n+1}) (x' : ℝ^{2}^{n'+1}) : ℝ^{2} × ℝ^{2} × ℝ^{2} × ℝ^{2} := (-- top chain constraints x[0] - l, x[⟨n,sorry⟩] - r, -- bottow chain constraints x'[0] - x[⟨(n+1)/4, sorry⟩], x'[⟨n',sorry⟩] - x[⟨3*(n+1)/4, sorry⟩]) argument x isSmooth, diff, hasAdjDiff, adjDiff argument x' ... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | endPoints | null |
L' (m : ℝ×ℝ) (x v : ℝ^{2}×ℝ^{2}) : ℝ := (m.1/2) * ∥v.1∥² + (m.2/2) * ∥v.2∥² - m.1 * g * x.1[1] - m.2 * g * x.2[1] argument x isSmooth, diff, hasAdjDiff, adjDiff argument v isSmooth, diff, hasAdjDiff, adjDiff | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | L' | null |
parm (l : ℝ × ℝ) (θ : ℝ^{2}) : ℝ^{2}×ℝ^{2} := let d₁ : ℝ^{2} := ⟨Math.sin θ[0], Math.cos θ[0]⟩ let d₂ : ℝ^{2} := ⟨Math.sin θ[1], Math.cos θ[1]⟩ -- (l.1 * d₁, l.1 * d₁ + l.2 * d₂) (d₁, d₂) argument θ isSmooth | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | parm | null |
parm' {n} (l : ℝ^{n}) (θ : ℝ^{n}) (x₀ : ℝ^{2}) : (ℝ^{2})^{n+1} := Id.run do let mut x : (ℝ^{2})^{n+1} := 0 let mut y := x₀ for h : i in [0:n] do let i : Fin n := ⟨i, h.2⟩ let next := i.succ let dir : ℝ^{2} := ⟨Math.cos (θ[i]), Math.sin (θ[i])⟩ y += l[i] * dir x[next] := y x -- This impelementation is paying for zero in... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | parm' | null |
parm'' {n} (l : ℝ^{n}) (θ : ℝ^{n}) (x₀ : ℝ^{2}) : (ℝ^{2})^{n+1} := Id.run do let mut x : (ℝ^{2})^{n+1} := 0 let mut y := x₀ for h : i in [0:n] do let i : Fin n := ⟨i, h.2⟩ let next := i.succ let dir : ℝ^{2} := ⟨Math.cos (θ[i]), Math.sin (θ[i])⟩ y := y + l[i] * dir x[next] := y x -- There is no way the index access `θ[⟨... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | parm'' | null |
parm''' {n} (l : ℝ^{n}) (θ : ℝ^{n}) (x₀ : ℝ^{2}) : (ℝ^{2})^{n+1} := funRec (n+1) 0 step (reserveElem (n+1) 0) where step : (n' : Nat) → ((ℝ^{2})^{n'}) → (ℝ^{2})^{n'+1} | 0, x => (λ [i] => x₀) | n'+1, x => let dir : ℝ^{2} := ⟨Math.cos (θ[⟨n',sorry_proof⟩]), Math.sin (θ[⟨n',sorry_proof⟩])⟩ let y := x[⟨n', by simp⟩] + l[⟨... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | parm''' | null |
funRec.arg_x.diff_simp {α : Nat → Type} [Vec X] [∀ n, Vec (α n)] (f : (n : Nat) → α n → α (n + 1)) [∀ n, IsSmooth (f n)] (n m : Nat) : ∂ (λ x => funRec n m f x) = λ x dx => (funRec (α:=λ n' => α n' × α n') n m (λ n' (x',dx') => (f n' x', ∂ (f n') x' dx')) (x,dx)).2 := sorry_proof @[simp ↓] | theorem | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | funRec.arg_x.diff_simp | null |
funRec.arg_f.diff_simp {α : Nat → Type} [Vec X] [∀ n, Vec (α n)] (f : X → (n : Nat) → α n → α (n + 1)) [∀ n, IsSmooth (λ x => f x n)] [∀ x n, IsSmooth (f x n)] (n : Nat) : ∂ (λ x => funRec n m (f x)) = λ x dx y => (∂ (funRec (α:=λ n' => X × α n') n m (λ n' xy' => (xy'.1, f xy'.1 n' xy'.2))) (x,y) (dx,0)).2 := sorry_pro... | theorem | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | funRec.arg_f.diff_simp | null |
parm'''' {n} (l : ℝ^{n}) (θ : ℝ^{n}) (x₀ : ℝ) : ℝ^{n+1} := let step (n' : Nat) (x : ℝ^{n'}) : ℝ^{n'+1} := pushElem (Cont:=(ℝ^{·})) 1 (θ[⟨n',sorry⟩]) x funRec (n+1) 0 step 0 argument θ -- isSmooth := by simp[parm'''']; infer_instance, diff by simp[parm''''] enter [θ,x,a] simp only [funRec.arg_x.diff_simp (α:= λ n' => ℝ^... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | parm'''' | null |
parm'''' {n} (l : ℝ^{n}) (θ : ℝ^{n}) (x₀ : ℝ^{2}) : (ℝ^{2})^{n+1} := step 1 n (λ [i] => x₀) where step : (n m : Nat) → ((ℝ^{2})^{n+1-m}) → (ℝ^{2})^{n+1} | n, 0, x => x | n, m+1, x => pushElem 1 sorry (step n m x) -- | n'+1, x => -- let dir : ℝ^{2} := ⟨Math.cos (θ[⟨n',sorry_proof⟩]), Math.sin (θ[⟨n',sorry_proof⟩])⟩ -- l... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | parm'''' | null |
L (m l : ℝ×ℝ) (θ ω : ℝ^{2}) := (L' m (parm l θ) (ⅆ (t:=0), parm l (θ + (t:ℝ) * ω))) rewrite_by simp | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | L | null |
solver (m l : ℝ×ℝ) (steps : Nat) : Impl (ode_solve (LagrangianSystem (L m l))) := by simp [L, pure_impl, PureImpl] conv in (L _ _) => simp [L, L', parm] pattern (differential _); rmlamlet; enter [θ, ω]; simp -- autodiff simp -- Lagrangian is in a nice form now conv => pattern (LagrangianSystem _); whnf admit | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | solver | null |
solver_0 (m k l : ℝ) (steps : Nat) : Impl (λ x v : Q => (L m k l) x v) := by simp [LagrangianSystem, L, L', parm, gradient] conv => pattern (differential _); rmlamlet; enter [x, dx]; simp -- autodiff simp finish_impl | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | solver_0 | null |
solver_1 (m k l : ℝ) (steps : Nat) : Impl (λ x v : Q => ∇(swap (L m k l) v) x) := by conv in (L _ _ _) => enter [θ, ω] simp [L, L', parm] rmlamlet simp simp [gradient] conv => pattern (differential _); rmlamlet; enter [x, dx]; simp -- autodiff conv in (dual _) => pattern (dual _); rmlamlet; simp -- autodual . finish_im... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | solver_1 | null |
solver_2 (m k l : ℝ) (steps : Nat) : Impl (λ x v : Q => ∇(∂ (L m k l) x v) v) := by conv in (L _ _ _) => enter [θ, ω] simp [L, L', parm] rmlamlet simp conv => pattern (differential _); enter [x, dx, y]; rmlamlet; simp -- autodiff (we need to introduce all arguments!) conv => -- autograd - part1 pattern (gradient _) ent... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | solver_2 | null |
solver_3 (m k l : ℝ) (steps : Nat) : Impl (λ x v : Q => ∂(∇((L m k l) x)) v) := by conv in (L _ _ _) => simp [L, L', parm] { pattern (differential _); rmlamlet; enter [θ, ω]; simp } -- autodiff simp -- Lagrangian is in a nice form now conv => -- autograd - part1 pattern (gradient _) enter [x] simp [gradient] pattern (d... | def | examples | [
"import SciLean.Core",
"import SciLean.Mechanics",
"import SciLean.Operators.ODE",
"import SciLean.Solver.Solver",
"import SciLean.Data.DataArray",
"import SciLean.Core.Extra",
"import SciLean.Functions.Trigonometric",
"import SciLean.Data.FunRec"
] | examples/DoublePendulum.lean | solver_3 | null |
ε : ℝ := 0.001 | def | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | ε | null |
pair_add {X Y} [Add X] [Add Y] (x x' : X) (y y' : Y) : (x, y) + (x', y') = (x + x', y + y') := by simp[HAdd.hAdd, Add.add] done @[simp] | theorem | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | pair_add | null |
inner_zero {X} [Hilbert X] (x : X) : ⟪0,x⟫ = 0 := sorry @[simp] | theorem | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | inner_zero | null |
zero_inner {X} [Hilbert X] (x : X) : ⟪x,0⟫ = 0 := sorry | theorem | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | zero_inner | null |
parm (l₁ l₂ : ℝ) (Ω : ℝ×ℝ) : (ℝ^(2:ℕ)) × (ℝ^(2:ℕ)) := let e : ℝ^(2:ℕ) := ⟨0,-1⟩ let Δx₁ := rotate2d Ω.1 (l₁ * e) let Δx₂ := rotate2d Ω.2 (l₂ * e) (Δx₁, Δx₁ + Δx₂) argument Ω isSmooth, diff, hasAdjDiff, adjDiff? set_option trace.Meta.Tactic.simp true function_properties parm.arg_Ω.diff (l₁ l₂ : ℝ) (Ω dΩ : ℝ×ℝ) : (ℝ^(2:ℕ... | def | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | parm | null |
g : ℝ := 9.81 -- instance (priority := high) {X Y} [Vec X] [Vec Y] (f : X → Y) : IsSmooth f := sorry -- instance (priority := high) {X Y} [SemiHilbert X] [SemiHilbert Y] (f : X → Y) : HasAdjDiff f := sorry /- | def | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | g | null |
L (m₁ m₂ : ℝ) (l₁ l₂ : ℝ) (Ω ω : ℝ×ℝ) : ℝ := let e : ℝ^(2:ℕ) := ⟨0,-1⟩ let x := parm l₁ l₂ Ω let v := δ (λ t : ℝ => parm l₁ l₂ (Ω + t * ω)) 0 1 rewrite_by (simp) m₁/2 * ∥v.1∥² - m₁ * g * x.1.y + m₂/2 * ∥v.2∥² - m₂ * g * x.2.y argument Ω isSmooth, diff, hasAdjDiff, adjDiff argument ω isSmooth, diff, hasAdjDiff, adjDiff | def | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | L | null |
momentum (m₁ m₂ : ℝ) (l₁ l₂ : ℝ) (Ω ω : ℝ×ℝ) : ℝ×ℝ := (∂† (λ ω' => L m₁ m₂ l₁ l₂ Ω ω') ω 1) rewrite_by simp #print L.arg_Ω.diff | def | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | momentum | null |
velocity (m₁ m₂ : ℝ) (l₁ l₂ : ℝ) (Ω ω' : ℝ×ℝ) : ℝ×ℝ := ((λ ω => momentum m₁ m₂ l₁ l₂ Ω ω)⁻¹ ω') rewrite_by simp[momentum] simp[L.arg_ω.adjDiff] simp only [!?(∀ {X} [Hilbert X] (x y : X) (a : ℝ), ⟪a * x, y⟫ = a * ⟪x,y⟫)] simp only [!?(∀ (x y : ℝ^(2:ℕ)) (θ : ℝ), ⟪rotate2d θ x, rotate2d θ y⟫ = ⟪x,y⟫)] simp only [!?(∀ {X} ... | def | examples | [
"import SciLean"
] | examples/DoublePendulum2d.lean | velocity | null |
Array.intro {n α} (f : Fin n → α) : Array α := Id.run do let mut a : Array α := Array.mkEmpty n for i in [0:n] do a := a.push (f ⟨i, sorry⟩) a | def | examples | [
"import Lean",
"import SciLean.Functions.FFT"
] | examples/FFTWidget.lean | Array.intro | null |
generateData := Id.run do let n := 6 let steps := 2^n let mut data : Array (Array (Vec2 Float)) := Array.mkEmpty steps for waveNumber in [0:2^n] do let wave : ℝ^{2^n, 2} := introElem λ (i,j) => let θ := 2 * Math.pi * waveNumber * (i.1.toReal / 2^n) if j = 0 then Math.cos θ else Math.sin θ let wave' := FFT wave data := ... | def | examples | [
"import Lean",
"import SciLean.Functions.FFT"
] | examples/FFTWidget.lean | generateData | null |
helloWidget : UserWidgetDefinition where name := "Hello" javascript := include_str "." / "plot.js" #widget helloWidget (Json.mkObj [ ("data", toJson generateData), ("useTimer", true), ("yDomain", toJson [0, 1]) ]) | def | examples | [
"import Lean",
"import SciLean.Functions.FFT"
] | examples/FFTWidget.lean | helloWidget | null |
test1 : IO Unit := do let v : FloatVector (Fin 3) := VectorType.fromVec ![1,2,3] let v' : FloatVector' (.subvector (offset := 3) (inc := 2)) (Fin 3) := VectorType.fromVec ![1,2,3] let A : FloatMatrix' .RowMajor .normal (Fin 3) (Fin 3) := MatrixType.fromMatrix !![1,1,1; 0,-1,0; -1, 0, 42] IO.println v.data IO.println (1... | def | examples | [
"import SciLean"
] | examples/FloatMatrixTest.lean | test1 | null |
main : IO Unit := do test1 | def | examples | [
"import SciLean"
] | examples/FloatMatrixTest.lean | main | null |
main : IO Unit := do IO.println (Float.tgamma 3.1415) IO.println (Float.lgamma 3.1415) IO.println (FloatArray.mk #[1.0,2.0,3.0,4.0,3.0]) IO.println (FloatArray.mk #[1.0,2.0,3.0,4.0,3.0] |>.toByteArray) IO.println (FloatArray.mk #[1.0,2.0,3.0,4.0,3.0] |>.toByteArray |>.toFloatArray sorry) IO.println (FloatArray.mk #[2.0... | def | examples | [
"import SciLean"
] | examples/FloatTest.lean | main | null |
IndexType.sum_const {X} [AddCommGroup X] (x : X) : (∑ (i : I), x) = Size.size I • x := by sorry_proof | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | IndexType.sum_const | null |
likelihood (x : R^[D]^[N]) (w : R^[K]) (μ : R^[D]^[K]) (σ : R^[D,D]^[K]) : R := ∏ i, ∑ k, w[k] * gaussianS (μ[k]) (σ[k]) (x[i]) namespace Param | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | likelihood | null |
Q (q : R^[D]) (l : R^[((D-1)*D)/2]) : R^[D,D] := q.exp.diag + l.lowerTriangular D 1 def_rev_deriv Q in q l by unfold Q data_synth => lsimp def_rev_deriv' Q in q l by unfold Q data_synth => lsimp variable (q : R^[D]) (l : R^[((D-1)*D)/2]) -- properties of parametrization | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | Q | null |
det_Q : (Q q l).det = exp q.sum := sorry_proof | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | det_Q | null |
det_QTQ : ((Q q l)ᵀ * (Q q l)).det = exp (2 * q.sum) := by simp[DataArrayN.det_mul,det_Q,exp_pull]; ring_nf | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | det_QTQ | null |
Q_invertible : (Q q l).Invertible := sorry_proof | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | Q_invertible | null |
QTQ_invertible : ((Q q l)ᵀ * (Q q l)).Invertible := sorry_proof | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | QTQ_invertible | null |
trace_QTQ : ((Q q l)ᵀ * Q q l).trace = ‖q.exp‖₂² + ‖l‖₂² := sorry_proof attribute [simp, simp_core] det_Q det_QTQ Q_invertible trace_QTQ | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | trace_QTQ | null |
likelihood' (x : R^[D]^[N]) (α : R^[K]) (μ : R^[D]^[K]) (q : R^[D]^[K]) (l : R^[((D-1)*D)/2]^[K]) : R := likelihood x (α.softmax) μ (⊞ k => ((Q q[k] l[k])ᵀ * Q q[k] l[k])⁻¹) rewrite_by unfold likelihood simp (disch:=aesop) [gaussianS_ATA] | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | likelihood' | null |
prior (m : R) (σ : R^[D,D]^[K]) := ∏ k, /- C(D,m) -/ (σ[k].det)^m * exp (-((σ[k])⁻¹).trace/2) open Lean Meta /-- Take expression full of multiplications and divitions and split it into lists of multiplication and division factors. For example: ``` e = x₁ * x₂ ==> [x₁,x₂] [] e = x₁ * x₂ / y₁ ==> [x₁,x₂] [y₁] e = x₁ / (y... | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | prior | null |
parse (e : Expr) : Option (List Expr × List Expr × Bool) := match e.getAppFnArgs with | (``HMul.hMul, #[_, _, _, _, e₁, e₂]) => do match parse e₁, parse e₂ with | .none, .none => return ([e₁,e₂], [], false) | .some (m₁,d₁,neg₁), .none => return (m₁ ++ [e₂], d₁, neg₁) | .none, .some (m₂,d₂,neg₂) => return ([e₁] ++ m₂, d... | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | parse | /-- Take expression full of multiplications and divitions and split it into lists of
multiplication and division factors.
For example:
```
e = x₁ * x₂ ==> [x₁,x₂] []
e = x₁ * x₂ / y₁ ==> [x₁,x₂] [y₁]
e = x₁ / (y₁/x₂) ==> [x₁,x₂] [y₁]
e = x₁ * x₂ / y₁ * (x₃ * x₄) / (y₂/x₅)`
==> [x₁,x... |
neg_add_rev' {G : Type*} [SubtractionCommMonoid G] (a b : G) : -(a + b) = -a + -b := by simp[add_comm] @[rsimp] | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | neg_add_rev' | null |
sum_normalize (x : R^[I]) : ∑ i, x[i] = x.sum := rfl -- TODO: this theorem replaces (∑ i, ‖x[i]‖₂²) with (∑ i, ‖x[i]‖₂²) instead of ‖x‖₂² -- there must be some issue with transparency omit [PlainDataType R] in @[rsimp] | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | sum_normalize | null |
norm_normalize {X} [PlainDataType X] [NormedAddCommGroup X] [AdjointSpace R X] (x : X^[I]) : (∑ i, ‖x[i]‖₂²) = ‖x‖₂² := rfl -- theorem sum_over_prod {R} [AddCommMonoid R] {I J : Type*} [IndexType I] [IndexType J] -- {f : I → J → R} : ∑ i j, f i j = ∑ (i : I×J), f i.1 i.2 := sorry @[rsimp] | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | norm_normalize | null |
isum_sum (x : R^[I]^[J]) : ∑ i, x[i].sum = x.uncurry.sum := by simp[DataArrayN.uncurry_def,DataArrayN.sum,Function.HasUncurry.uncurry] rw[sum_sum_eq_sum_prod] @[rsimp] | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | isum_sum | null |
isum_norm_exp (x : R^[I]^[J]) : ∑ j, ‖x[j].exp‖₂² = ‖x.uncurry.exp‖₂² := by simp[DataArrayN.uncurry_def,DataArrayN.sum,Function.HasUncurry.uncurry, DataArrayN.norm2_def,DataArrayN.exp] rw[sum_sum_eq_sum_prod] @[rsimp] | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | isum_norm_exp | null |
isum_norm (x : R^[I]^[J]) : (∑ j, ‖x[j]‖₂²) = ‖x.uncurry‖₂² := by simp[DataArrayN.uncurry_def,DataArrayN.sum,Function.HasUncurry.uncurry, DataArrayN.norm2_def] rw[sum_sum_eq_sum_prod] open Param Scalar in | theorem | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | isum_norm | null |
loss (m : R) (x : R^[D]^[N]) (α : R^[K]) (μ : R^[D]^[K]) (q : R^[D]^[K]) (l : R^[((D-1)*D)/2]^[K]) : R := (let S := ⊞ k => ((Q q[k] l[k])ᵀ * Q q[k] l[k])⁻¹ (- log (likelihood x (α.softmax) μ S * prior m S))) rewrite_by unfold likelihood simp (disch:=aesop) [gaussianS_ATA] simp (disch:=aesop) [simp_core, likelihood, pri... | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | loss | null |
f {D N : Nat} (K : Nat) (m : Float) (x : Float^[D]^[N]) := { f := fun (α,μ,q,l) => loss (K:=K) m x α μ q l hf := by simp apply HasRevFDeriv_from_HasRevFDerivUpdate data_synth => skip f' := _ : ObjectiveFunction _ _} #check Nat set_option profiler true | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | f | null |
main : IO Unit := do let K := 2 let m := 0.0 let xs := ⊞[⊞[1.01,5.0], ⊞[1.03,5.1],⊞[1.1,4.8], ⊞[-1.03,5.0], ⊞[-0.9,4.7],⊞[-1.1,5.1]] let r ← optimize (f K m xs) {show_trace:=true : LBFGS Float K} (0,⊞[⊞[1,0],⊞[-1,2]],0,0) r.print #check Nat #eval! main #exit open Lean Parser Term in syntax "argmin" funBinder* ", " term... | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | main | null |
hoh {D N K} (m : R) (x : R^[D]^[N]) := (argmin (α : R^[K]) μ q l, loss m x α μ q l) rewrite_by skip #check solveFun #print hoh #exit variable (x : R → R^[n]) (x' xi') (hx : HasRevFDerivUpdate R x x') (hxi : ∀ i, HasRevFDerivUpdate R (fun w => (x w)[i]) xi') (i : Fin n) def_rev_deriv loss in x by unfold loss lsimp data_... | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | hoh | null |
foo (x y : R^[n]) := ‖⊞ i => x[i] + y[i]‖₂² def_rev_deriv foo in x y by unfold foo data_synth => lsimp | def | examples | [
"import SciLean",
"import SciLean.Tactic.DataSynth.DefRevDeriv",
"import SciLean.Data.DataArray.Operations.GaussianS"
] | examples/GaussianMixtureModel'.lean | foo | null |
asdf (e : Expr) : Bool := Id.run do match e with | .lam _ t b _ => if t.isAppOfArity' ``Prod 2 then return true else return asdf b | .mdata _ e => asdf e | _ => return false open Lean Meta in dsimproc_decl prodFunSimproc (_) := fun e => do -- unless asdf e do return .continue unless e.isLambda do return .continue lambd... | def | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | asdf | null |
ite.arg_te.Differentiable_rule {c : Prop} [h : Decidable c] (t e : X → Y) (ht : Differentiable R t) (he : Differentiable R e) : Differentiable R (fun x => ite c (t x) (e x)) := by split_ifs <;> assumption @[fun_prop] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | ite.arg_te.Differentiable_rule | null |
dite.arg_te.Differentiable_rule {c : Prop} [h : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) (ht : ∀ h, Differentiable R (t h)) (he : ∀ h, Differentiable R (e h)) : Differentiable R (fun x => dite c (t · x) (e · x)) := by split_ifs <;> aesop @[simp, simp_core] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | dite.arg_te.Differentiable_rule | null |
oneHot_unit [Zero α] (i : Unit) (x : α) : oneHot i x = x := rfl | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | oneHot_unit | null |
inner_QQT (x y : R^[I]) (Q : R^[I,J]) : ⟪x, Q*Qᵀ*y⟫ = ⟪Qᵀ*x,Qᵀ*y⟫ := by sorry @[scoped simp, scoped simp_core] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | inner_QQT | null |
inner_QQT' (x y : R^[I]) (Q : R^[I,J]) : ⟪x, Q*(Qᵀ*y)⟫ = ⟪Qᵀ*x,Qᵀ*y⟫ := sorry @[scoped simp, scoped simp_core] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | inner_QQT' | null |
inner_QTQ (x y : R^[J]) (Q : R^[I,J]) : ⟪x, Qᵀ*Q*y⟫ = ⟪Q*x,Q*y⟫ := sorry @[scoped simp, scoped simp_core] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | inner_QTQ | null |
inner_QTQ' (x y : R^[J]) (Q : R^[I,J]) : ⟪x, Qᵀ*(Q*y)⟫ = ⟪Q*x,Q*y⟫ := sorry @[scoped simp, scoped simp_core] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | inner_QTQ' | null |
gaussian_normalization_invQQT {d : ℕ} (Q : R^[d,d]) : (((2 * π) • (Q*Qᵀ)⁻¹).det) ^ (-(1:R) / 2) = (2 * π)^(-(d:R)/2) * Q.det := sorry @[scoped simp, scoped simp_core] | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | gaussian_normalization_invQQT | null |
gaussian_normalization_invQTQ {d : ℕ} (Q : R^[d,d]) : (((2 * π) • (Qᵀ*Q)⁻¹).det) ^ (-(1:R) / 2) = (2 * π)^(-(d:R)/2) * Q.det := sorry | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | gaussian_normalization_invQTQ | null |
log_sum_exp (x : I → R) : log (∑ i, exp (x i)) = (⊞ i => x i).logsumexp := sorry | theorem | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | log_sum_exp | null |
likelihood (x : R^[D]^[N]) (w : R^[K]) (μ : R^[D]^[K]) (σ : R^[D,D]^[K]) : R := ∏ i, ∑ k, w[k] * (((2*π) • σ[k]).det)^(-(1:R)/2) * exp (-(1:R)/2 * ⟪x[i] - μ[k], (σ[k]⁻¹ * (x[i] - μ[k]) : R^[D])⟫) namespace Param | def | examples | [
"import SciLean",
"import SciLean.Data.DataArray.Operations.Simps",
"import SciLean.Lean.Meta.Basic",
"import SciLean.Tactic.StructureDecomposition"
] | examples/GaussianMixtureModel.lean | likelihood | null |
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Lean4-SciLean
Structured dataset from SciLean — Scientific computing.
4,454 declarations extracted from Lean 4 source files.
Applications
- Training language models on formal proofs
- Fine-tuning theorem provers
- Retrieval-augmented generation for proof assistants
- Learning proof embeddings and representations
Source
- Repository: https://github.com/lecopivo/SciLean
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Declaration body |
| type | string | theorem, def, lemma, etc. |
| library | string | Source module |
| imports | list | Required imports |
| filename | string | Source file path |
| symbolic_name | string | Identifier |
| docstring | string | Documentation (if present) |
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