| name | lean-hit-development |
| description | Guides adding new computational-path constructions to the ComputationalPaths library. Use when defining new spaces, π₁ calculations, encode-decode proofs, or adding new topological constructions. |
Computational Path Construction Development
Add new computational-path constructions to the ComputationalPaths library.
File Location
ComputationalPaths/Path/CompPath/YourSpace.lean
Required Structure
import ComputationalPaths.Path.Basic
import ComputationalPaths.Path.Rewrite.SimpleEquiv
import ComputationalPaths.Path.Homotopy.FundamentalGroup
namespace ComputationalPaths.Path.CompPath
/-! ## Point type and basepoint -/
inductive YourSpace : Type u
| base : YourSpace
@[simp] def yourSpaceBase : YourSpace := YourSpace.base
/-! ## Path expressions with generators -/
inductive YourSpaceExpr : YourSpace → YourSpace → Type u
| loop : YourSpaceExpr yourSpaceBase yourSpaceBase
| refl (a : YourSpace) : YourSpaceExpr a a
| symm {a b} (p : YourSpaceExpr a b) : YourSpaceExpr b a
| trans {a b c} (p : YourSpaceExpr a b) (q : YourSpaceExpr b c) : YourSpaceExpr a c
/-! ## Relation, quotient, and encode/decode -/
def yourSpaceRel : YourSpaceExpr yourSpaceBase yourSpaceBase →
YourSpaceExpr yourSpaceBase yourSpaceBase → Prop := ...
def yourSpaceSetoid : Setoid (YourSpaceExpr yourSpaceBase yourSpaceBase) := ...
abbrev yourSpacePiOne : Type u := Quot yourSpaceSetoid.r
noncomputable def encode : yourSpacePiOne → Presentation := ...
noncomputable def decode : Presentation → yourSpacePiOne := ...
theorem decode_encode : decode (encode x) = x := by
-- quotient induction
...
theorem encode_decode : encode (decode y) = y := by
-- quotient induction
...
noncomputable def piOneEquiv : SimpleEquiv yourSpacePiOne Presentation := ...
end ComputationalPaths.Path.CompPath
Checklist
- Define the point type and basepoint
- Define path expressions and generators
- Provide a relation/normal form and a quotient for π₁
- Implement encode/decode over the quotient
- Prove round-trip properties
- Add to imports in
ComputationalPaths/Path.lean
- Update README
Common Constructions
| Space | Computational-path model |
|---|
| Circle (S¹) | One basepoint, one loop generator |
| Torus (T²) | Product of circle constructions |
| Sphere (S²) | Suspension/pushout-based loop triviality |
| Wedge (A ∨ B) | Pushout of basepoints, free product on π₁ |
