Lean 4 fork for HoTT-compatible kernel extensions (Path types, transport, HITs). Maintained against upstream leanprover/lean4.
751947482f
This PR changes `Fin.reverseInduction` from using well-founded recursion
to using `let rec`, which makes it have better definitional equality.
Co-authored by @digama0. See the test below:
```lean
namespace Fin
/-- The new one. -/
@[elab_as_elim] def reverseInduction' {motive : Fin (n + 1) → Sort _} (last : motive (Fin.last n))
(cast : ∀ i : Fin n, motive i.succ → motive (castSucc i)) (i : Fin (n + 1)) : motive i :=
let rec go (j : Nat) (h) (h2 : i ≤ j) (x : motive ⟨j, h⟩) : motive i :=
if hi : i.1 = j then (show i = ⟨j, h⟩ by simp [← hi]) ▸ x
else match j with
| 0 => by omega
| j+1 => go j (by omega) (by omega) (cast ⟨j, by omega⟩ x)
go _ _ (by omega) last
/-- Same code but using reverseInduction'. -/
@[elab_as_elim] def lastCases' {n : Nat} {motive : Fin (n + 1) → Sort _} (last : motive (Fin.last n))
(cast : ∀ i : Fin n, motive (castSucc i)) (i : Fin (n + 1)) : motive i :=
reverseInduction' last (fun i _ => cast i) i
end Fin
theorem foo : (Fin.lastCases (-4) (fun i ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int) = -4 := rfl
#eval (Fin.lastCases (-4) (fun i ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int)
theorem foo' : (Fin.lastCases' (-4) (fun i ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int) = -4 := rfl
#eval (Fin.lastCases' (-4) (fun i ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int)
theorem bar : (Fin.reverseInduction (n := 2) (motive := fun _ ↦ Int)
(-4) (fun i _ ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int) = -4 := rfl
#eval (Fin.reverseInduction (n := 2) (motive := fun _ ↦ Int)
(-4) (fun i _ ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int)
theorem bar' : (Fin.reverseInduction' (n := 2) (motive := fun _ ↦ Int)
(-4) (fun i _ ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int) = -4 := rfl
#eval (Fin.reverseInduction' (n := 2) (motive := fun _ ↦ Int)
(-4) (fun i _ ↦ (i : Int) * 2 + 1) (2 : Fin 3) : Int)
```
[Link to Lean 4
Web](https://live.lean-lang.org/#project=lean-nightly&codez=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)
Notice how `rfl` fails for the 1st and 5th tests that use the original
`Fin.reverseInduction`, but the 3rd and 7th tests that use the new code
in this PR succeed.
Closes #9141.
---------
Co-authored-by: Markus Himmel <markus@lean-fro.org>
|
||
|---|---|---|
| .github | ||
| doc | ||
| images | ||
| nix | ||
| releases_drafts | ||
| script | ||
| src | ||
| stage0 | ||
| tests | ||
| .gitattributes | ||
| .gitignore | ||
| .gitpod.Dockerfile | ||
| .gitpod.yml | ||
| .ignore | ||
| CMakeLists.txt | ||
| CMakePresets.json | ||
| CODEOWNERS | ||
| CONTRIBUTING.md | ||
| flake.lock | ||
| flake.nix | ||
| lean-toolchain | ||
| lean.code-workspace | ||
| LICENSE | ||
| LICENSES | ||
| README.md | ||
| RELEASES.md | ||
This is the repository for Lean 4.
About
- Quickstart
- Homepage
- Theorem Proving Tutorial
- Functional Programming in Lean
- Documentation Overview
- Language Reference
- Release notes starting at v4.0.0-m3
- Examples
- External Contribution Guidelines
Installation
See Setting Up Lean.
Contributing
Please read our Contribution Guidelines first.
Building from Source
See Building Lean.