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(*   Copyright Dominique Larchey-Wendling [*]                 *)
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(*                             [*] Affiliation LORIA -- CNRS  *)
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(*      This file is distributed under the terms of the       *)
(*        Mozilla Public License Version 2.0, MPL-2.0         *)
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What is this library?

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

This library is an extension of Kruskal-AlmostFull. It contains a detailed and expanded constructive/inductive proofs of:

Some tools are also included, as required for the above proofs:

This README itself contains the outline of the proof with the critical steps. The terminology and proof structure are largely inspired from Wim Veldman's [1] which is (IMHO) the reference pen-and-paper work on this proof technique.

The main proof in Kruskal-Higman, that of af_utree_embed below, is a written as a downgrade of the proof of Kruskal-Veldman, on the much simpler case of unary trees, whereas the main proof in Kruskal-Veldman is concerned with the case of rose trees, aka finitely branching trees.

[1]. An intuitionistic proof of Kruskal's theorem, Wim Veldman, 2004

How to install Kruskal-Higman

It can be installed via opam since release v1.0 is now include into coq-opam.

opam repo add coq-released https://coq.inria.fr/opam/released
opam update
opam install coq-kruskal-higman

Notice that to import it in a development, as with Kruskal-AlmostFull, one should consistently choose between:

Mixing both versions is possible but hard and not recommended due to the total overlap of the namespaces except for the prefixes KruskalHigman{Prop,Type}.

The proof of Higman theorem for unary trees (sketch)

We define (decorated) unary trees and the (product) embedding between those:

Inductive utree X Y := ⟨ _ : X ⟩₀ | ⟨ _ : Y | _ : utree X Y ⟩₁.
  
Inductive utree_embed {X Y} R T : utree X Y → utree X Y → Prop :=
  | utree_embed_leaf x₁ x₂ : R x₁ x₂ → ⟨x₁⟩₀ ≤ₑ ⟨x₂⟩₀
  | utree_embed_subt s y t : s ≤ₑ t → s ≤ₑ ⟨y|t⟩₁
  | utree_embed_root y₁ t₁ y₂ t₂ : T y₁ y₂ → t₁ ≤ₑ t₂ → ⟨y₁|t₁⟩₁ ≤ₑ ⟨y₂|t₂⟩₁
where "s ≤ₑ t" := (utree_embed _ _ s t).

Higman's theorem for utree X Y is stated and proved as following:

Theorem af_utree_embed X Y (R : rel₂ X) (T : rel₂ Y) : af R → af T → af (utree_embed R T).

The proof proceeds as following (sketch):

  1. first a lexicographic induction on a kind of measure of the AF-complexity of (af T,af R). How this is implemented here is a bit complicated because it is a downgrade for the case of a list [af Rₙ;...;af R₁] of almost full predicates ordered using the easier ordering of [1] (see af/af_lex.v);
    • alternatively for utree X Y where n=2, this lexicographic ordering could just be implemented by nested induction, first on af T, then on af R;
  2. apply the second constructor of af. One needs to prove af (utree_embed R T)↑t for any t : utree X Y. We proceed by structural induction on t.
    • here we consider only the more complicated case where t = ⟨α|τ⟩₁ where α : Y and τ : utree X Y;
  3. the following propositions hold (see af/af_utree_embed_fun.v):
    • af (utree_embed R T)↑τ (by induction on t)
    • hence af R' where R' := R + T ⨉ (utree_embed R T)↑τ (by Coquand's af_product,af_sum)
    • af T' where T' := T↑α (because T ⊆ T' holds)
    • hence af (utree_embed R' T') (because T' = T↑α is smaller than T and the lexicographic product)
  4. finally we transfer af through af (utree_embed R' T') → af (utree_embed R T)↑⟨α|τ⟩₁ using a quasi-morphism of which the construction composes most of contents of the file af/af_utree_embed_fun.v;
  5. there is also version of that proof with a relational quasi morphism to illustrate the differences with the functional version above.
    • indeed Kruskal's tree theorem is only reasonably implementable with a relational version and the current project is an introduction to this involved proof, but with a comparable outline, see the project Kruskal-Veldman.

The quasi-morphism

We describe the quasi-morphism that implements the following transfer af (utree_embed R' T') → af (utree_embed R T)↑⟨α|τ⟩₁. Recall that we are in the case where the induction on t above is a unary tree ⟨α|τ⟩₁ and also when ∀y, af T↑y holds. In particular we have af T↑α. Recall the following definitions:

Arity 0                            |  Arity 1
-----------------------------------+------------
X' := X + Y ⨉ utree X Y            |  Y' := Y
R' := R + T ⨉ (utree_embed R T)↑τ  |  T' := T↑α

As argued above, af (utree_embed R' T') can be establish using (the consequences of) Ramsey's theorem and the induction hypotheses available.

We build an evaluation/analysis pair as a single binary relation between the types utree X' Y' (the analyses) and utree X Y (the evaluations). In the simple case of utree(s), that can be implemented directly as evaluation function. In the more complicated case of Kruskal's theorem (for roses trees), we will really need to view the evaluation/analysis as a relation between analyses and evaluations. So here we write ⤇ for this evaluation relation.

Terms in the type X' are either of the form

Evaluation consists in (recursively) replacing a leaf by a sub-tree, if it is of shape ⦗y,t⦘₂. Hence, we get the following rules for the evaluation relation:

                                                 t' ⤇ t
 -----------------   --------------------   ------------------
  ⟨⦗x⦘₁⟩₀ ⤇ ⟨x⟩₀     ⟨⦗y,t⦘₂⟩₀ ⤇ ⟨y|t⟩₁    ⟨y|t'⟩₁ ⤇ ⟨y|t⟩₁

but in the simple case of utree, these can also be implemented as the following fixpoint equations:

ev ⟨⦗x⦘₁⟩₀ = ⟨x⟩₀
ev ⟨⦗y,t⦘₂⟩₀ = ⟨y|t⟩₁
ev ⟨y|t'⟩₁ = ⟨y|ev t⟩₁

hence in this simple case we have: t' ⤇ t ↔ ev t' = t.

One can understand the analyses as ways to displace information in an evaluation. Nothing can be done at leaves but at a node of arity 1, it is possible to cut the utree there, and hide the sub-tree into a new leaf. For instance, the ⟨1|⟨2|⟨0⟩₀⟩₁⟩₁ : utree nat nat can be analyzed as:

in the type utree (nat+nat*utree nat nat) nat.

This can look simple here because unary trees are linear but imagine what is going on when the arity (number of sons) is allowed to be larger than 2. Then there is an exponential number of ways to analyses (displace information). However one can show that there are only finitely many ways to do it.

Then we say that an analysis in utree X' Y' is disappointing if either:

and an analysis is exceptional (denoted E t') if it contains a disappointing sub-tree. We say that an evaluation is exceptional (and write E t) if all its analyses are exceptional, ie. E t := ∀t', t' ⤇ t → E' t'.

We show the three following properties for ⤇ (or ev) and E'/E:

  1. fin(λ t', t' ⤇ t) (ev has finite inverse image);
  2. utree_embed R' T' s' t' → s' ⤇ s → t' ⤇ t → utree_embed R T s t ∨ E' s' (quasi morphism)
  3. E t → utree_embed R T ⟨α|τ⟩₁ t (exceptional evaluations embed ⟨α|τ⟩₁)

Actually, Item 3 holds in both directions but this is not needed. These 3 items are the conditions that constitute a quasi morphism (see af/af_quasi_morhism.v) and thus enable the transfer of the af property.

All this construction is performed:

They describe in the simple case of unary trees utree the very same steps that will also be performed for Kruskal's tree theorem. But for Kruskal's, the general setting is more complicated and the analysis/evaluation relation is much harder to implement.

Higman's lemma for lists

Lists are just unary trees where there is no information (eg of type unit) on the leaves. Hence there is an isomorphism between list X and utree unit X that we use as relational morphism to transfer af_utree_embed to lists.

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).

Theorem af_list_embed X (R : rel₂ X) : af R → af (list_embed R).

We derive Higman's lemma as eg stated on Wikipedia where the sub-list relation ≼ is abstracted by assuming its inductive rules. This allows the statement to be independent of an external inductive definition:

Variables (X : Type) (≼ : rel₂ (list X))
          (_ : ∃ l, ∀x : X, x ∈ l) 
          (_ :                    [] ≼ [])
          (_ : ∀ x l m, l ≼ m → x::l ≼ x::m)
          (_ : ∀ x l m, l ≼ m →    l ≼ x::m).

Theorem Higman_lemma : ∀ f : nat → list X, ∃ i j, i < j ∧ fᵢ ≼ fⱼ.

Lexicographic induction on AF predicates

To be completed, the reference files being: