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https://github.com/dmxlarchey/kruskal-theorems

Kruskal and Higman type tree theorems for the Kruskal-AlmostFull library
https://github.com/dmxlarchey/kruskal-theorems

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Kruskal and Higman type tree theorems for the Kruskal-AlmostFull library

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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         *)

(**************************************************************)

```

# What is this library?



This sub-project is part of the larger project `Coq-Kruskal` described here: https://github.com/DmxLarchey/Coq-Kruskal.



The library is build on top of [`Kruskal-Trees`](https://github.com/DmxLarchey/Kruskal-Trees), [`Kruskal-Finite`](https://github.com/DmxLarchey/Kruskal-Finite), [`Kruskal-AlmostFull`](https://github.com/DmxLarchey/Kruskal-AlmostFull), [`Kruskal-Higman`](https://github.com/DmxLarchey/Kruskal-Higman) and [`Kruskal-Veldman`](https://github.com/DmxLarchey/Kruskal-Veldman), and derives several instances of [Kruskal's tree theorem](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) for the homeomorphic embedding on rose trees, and of _Higman's theorem_ for the product embedding on trees of bounded breadth (the terminology/name for "Higman's theorem" is Wim Veldman's).



If your wish is to __understand the internals of the proof technique__ to establish Kruskal's tree theorem in inductive type theory,

and its beauty btw, _the place you want to look at is rather [`Kruskal-Veldman`](https://github.com/DmxLarchey/Kruskal-Veldman)_ which contains most 

of the details of the proof arguments, largely inspired by, but at the same time improving on the pen&paper proof of Wim Veldman.



In here you will just find easy consequences of that result. The proofs of those derived theorems are much simpler, 

ie. all the complexity is hidden in the [`Kruskal-Veldman`](https://github.com/DmxLarchey/Kruskal-Veldman) main result.

Those easy proofs proceed via surjective relational morphisms, or, as a degenerate case of morphism, simple inclusion 

between relations.



# How to install `Kruskal-Theorems`



It can be installed via `opam` since release `v1.0` which is now included 

in [`coq-opam`](https://github.com/coq/opam).

```console

opam repo add coq-released https://coq.inria.fr/opam/released

opam update

opam install coq-kruskal-theorems

```



Notice that to import it in a development, as with [`Kruskal-AlmostFull`](https://github.com/DmxLarchey/Kruskal-AlmostFull), one should

consistently choose between:

- the `Prop`-bounded version accessed via eg `From KruskalThmProp Require ...`;

- or the `Type`-bounded version via eg `From KruskalThmType Require ...`.



Mixing both versions is possible but hard and not recommended due 

to the total overlap of the namespaces except for the prefixes `KruskalThm{Prop,Type}`.



# What are the main results



There are several formulations of [_Kruskal's tree theorem_](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem#:~:text=In%20mathematics%2C%20Kruskal's%20tree%20theorem,quasi%2Dordered%20under%20homeomorphic%20embedding.)

available depending on the exact representation of indexed rose trees, either 

as vectors of rose trees,... or list of rose trees which is the

simplest to present:

```coq

Inductive ltree (X : Type) : Type :=

  | in_ltree : X → list (ltree X) → ltree X

where "⟨x|l⟩ₗ" := (@in_ltree _ x l).



Inductive list_embed {X Y} (R : X → Y → Prop) : list X → list Y → Prop :=

  | list_embed_nil :           [] ≤ₗ []

  | list_embed_head x l y m :  R x y → l ≤ₗ m → x::l ≤ₗ y::m

  | list_embed_skip l y m :    l ≤ₗ m → l ≤ₗ y::m

where "l ≤ₗ m" := (list_embed R l m).



Inductive ltree_homeo_embed {X} (R : rel₂ X) : ltree X → ltree X → Prop :=

  | homeo_embed_subt s t x l : t ∈ l → s ≤ₕ t → s ≤ₕ ⟨x|l⟩ₗ

  | homeo_embed_root x l y m : R x y → list_embed ≤ₕ l m → ⟨x|l⟩ₗ ≤ₕ ⟨y|m⟩ₗ

where "s ≤ₕ t" := (ltree_homeo_embed R s t).



Theorem af_ltree_homeo_embed X (R : rel₂ X) : af R → af (ltree_homeo_embed R).

```

and herein proved in [`kruskal_theorems.v`](theories/kruskal_theorems.v). The predicate `af`

characterizes almost full relations inductivelly and is defined and described in 

[`Kruskal-AlmostFull`](https://github.com/DmxLarchey/Kruskal-AlmostFull). So this

is a more abstract formulation than that of Kruskal's tree theorem (because the

`af` predicate abstracts the notion of [_Well Quasi Ordering_](https://en.wikipedia.org/wiki/Well-quasi-ordering) 

in a constructive way.



From this theorem formulated abstractly, we derive a proof of

[_Vazsonyi's conjecture_](https://en.wikipedia.org/wiki/Andrew_V%C3%A1zsonyi)

(which turned out to be a theorem) and which is proved here in [`vazsonyi_theorems`](theories/vazsonyi_theorems.v):

```coq

Theorem vazsonyi_theorem X (R : rel₂ (ltree X)) :

      (∀ s t x l, t ∈ l → R s t → R r ⟨x|l⟩ₗ)

    → (∀ x l y m, list_embed R l m → R ⟨x|l⟩ₗ ⟨y|m⟩ₗ)

    → ∀f : nat → ltree X, ∃ i j, i < j < n ∧ R (f i) (f j).

```

There, `R` abstract _any relation_ closed for the two rules of the homeomorphic embedding.