Ecosyste.ms: Awesome
An open API service indexing awesome lists of open source software.
https://github.com/dmxlarchey/kruskal-fan
The Fan theorem for inductive bars and a constructive variant of König's lemma
https://github.com/dmxlarchey/kruskal-fan
Last synced: 8 days ago
JSON representation
The Fan theorem for inductive bars and a constructive variant of König's lemma
- Host: GitHub
- URL: https://github.com/dmxlarchey/kruskal-fan
- Owner: DmxLarchey
- License: mpl-2.0
- Created: 2024-05-21T09:45:48.000Z (6 months ago)
- Default Branch: main
- Last Pushed: 2024-08-28T10:05:49.000Z (3 months ago)
- Last Synced: 2024-10-11T21:49:24.925Z (about 1 month ago)
- Language: Coq
- Size: 20.5 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
```
(**************************************************************)
(* Copyright Dominique Larchey-Wendling [*] *)
(* *)
(* [*] Affiliation LORIA -- CNRS *)
(**************************************************************)
(* This file is distributed under the terms of the *)
(* Mozilla Public License Version 2.0, MPL-2.0 *)
(**************************************************************)
```
# Kruskal-Fan
This project is based on [`Kruskal-Trees`](https://github.com/DmxLarchey/Kruskal-Trees),
[`Kruskal-Finite`](https://github.com/DmxLarchey/Kruskal-Finite)
and [`Kruskal-AlmostFull`](https://github.com/DmxLarchey/Kruskal-AlmostFull), and
it implements two additional results related to monotone inductive bars.
## The Fan theorem for inductive bars
It states that if `P` is a _monotone_ predicate on lists
(ie `P l → P (x::l)`) and is bound to be reached by successive extensions starting from `[]`,
then `P` is bound to be reached uniformly over choice sequences of finitary fans:
```coq
Theorem FAN_theorem X (P : rel₁ (list X)) :
monotone P
→ bar P []
→ bar (λ lw, Forall2 (λ x l, x ∈ l) ∙ lw ⊆₁ P) [].
```
This Fan theorem can be used to justify [`Quasi-Morphisms`](https://github.com/DmxLarchey/Quasi-Morphisms) as used in the proof
of [Higman's lemma](https://github.com/DmxLarchey/Kruskal-Higman) and [Veldman's version of Kruskal's tree theorem](https://github.com/DmxLarchey/Kruskal-Veldman).
It is also the base for the termination of algorithms that explore a search-tree that
avoid good pairs in the search branches, see below.
## A constructive variant of König's lemma
If `R` is an almost full relation and `P` is a sequence of finitary choices, then there is a
bound `m` (computable if `af` is `Type`-bounded) such that any choice vector of length `m`
(and hence also above), contains a `R`-good pair:
```coq
Theorem af_konig X (R : rel₂ X) (P : nat → rel₁ X) :
af R
→ (∀ n : nat, fin (P n))
→ ∃ₜ m, ∀v : vec X m, (∀i, P (idx2nat i) v⦃i⦄) → ∃ i j, idx2nat i < idx2nat j ∧ R v⦃i⦄ v⦃j⦄
```
A `Type`-bounded variant of this lemma is used in the [constructive proof of decidability for implicational relevance logic](https://github.com/DmxLarchey/Relevant-decidability/tree/v2.0) and the `Prop`-bounded instance is used to establish the
termination of the computation of the [Friedman `TREE(n)`](https://github.com/DmxLarchey/Friedman-TREE) fast growing function.