Some tactics reflexivitysymmetrytransitivity work only on relations that respect the expected properties. Tnreflexive R x The ident gives a unique name to the morphism and it is used by the command to generate fresh names for automatically provided lemmas used internally.

Let R be a relation over A with n parameters. Besides, the extensibility allows the user to customize the proof search if necessary. The latter condition is proved by showing: The toolbox also extends the automatic rewriting capabilities of the system, allowing the specification of custom strategies for rewriting.

Example 2 Parametric relation cont. The expression A, which depends on x First-class morphisms and signatures. Typerelation list A satisfied by two lists with the same elements is a parametric relation over list A with one parameter A. Also notice that A is not required to be coq setoid re write anime term having the same parameters as Aeq, although that is often the case in practice this departs from the previous implementation.

However, the tactics that replace in a context C[] one term with another one related by R must verify that C[] is a morphism that respects the intended relation. Add Parametric Relation A: Toplevel input, characters Several tactics that deal with equality in Coq can also work with the registered relations.

Example 6 Parametric relation: As type class resolution is extensible using tactics, this allows users to define general ways to solve morphism constraints.

Each proof may refer to the introduced variables as well. The exact list of tactic will be given in this section. The proofs of reflexivity, symmetry and transitivity can be omitted if the relation is not an equivalence relation.

The remaining tactics replacerewrite and derived tactics such as autorewrite do not require any properties over the relation. Unfortunately, Leibniz equality is not always the intended equality for a given structure.

Similar definitions are given for parametric proofs of symmetry and transitivity. The proofs must be instances of the corresponding relation definitions: The parameters of the morphism as well as the signature may refer to the context of variables.

In the next section we will describe the commands to register terms as parametric relations and morphisms.

Currently the verification consists in checking whether C[] is a syntactic composition of morphism instances that respects some obvious compatibility constraints.

It is possible to rewrite under binders in the new implementation, if one provides the proper morphisms. Matthieu Sozeau This chapter presents the extension of several equality related tactics to work over user-defined structures called setoids that are equipped with ad-hoc equivalence relations meant to behave as equalities.

A term is a parametric proof of reflexivity for R if it has type forall x1: The signature relation instances and morphism will be typed in a context introducing variables for the parameters.

The division function Rdiv: Example 1 Parametric relation: The algorithm is separated in two parts: Signatures and morphisms are ordinary Coq terms, hence they can be manipulated inside Coq, put inside structures and lemmas about them can be proved inside the system.

An f that respects its input and output relations will be called a unary covariant morphism. It is possible to implement finite sets of elements of type A as unordered list of elements of type A.

Notice that division is covariant in its first argument and contravariant in its second argument. Example 4 Contravariant morphism:Library killarney10mile.com Require Export Relation_Definitions.

Set Implicit Arguments. It's unclear why setoid_rewrite works in the second case but not the first. For reference, ≡ is equiv and fmap is Proper. Other than that ≅, Functor and Cat are classes, I don't see any other relevant facts.

Also, setoid_replace works fine. setoid_rewrite uses rewrite_strat, with a specific selection strategy, you’re right. It can be made accessible. About delta, it’s configurable by hints on the “rewrite” database I think.

Indeed for failure I guess it should somehow follow the autorewrite semantics, I’ll have to see how this can work. move => A B H1 H2. (* ssreflect rewrite works, if the use ded_equiv *) have ->:= H1. (* but not if we use the unfolded definition *) Fail have. rewrite plus_comm.

reflexivity. Qed. I think using Require Import killarney10mile.com and then using rewrite plus_comm at 2. is supposed. Library killarney10mile.comonClasses Typeclass-based relations, tactics and standard instances This is the basic theory needed to formalize morphisms and setoids.

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