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rahmenlose Ansicht.rst DruckansichtHaskell {Haskell[174] Ada[391] Abap[530]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] .. _detailedexamplesoftactics: Detailed examples of tactics ============================ This chapter presents detailed examples of certain tactics, to illustrate their behavior. .. _dependent-induction: dependent induction ------------------- The tactics ``dependent induction`` and ``dependent destruction`` are another solution for inverting inductive predicate instances and potentially doing induction at the same time. It is based on the ``BasicElim`` tactic of Conor McBride which works by abstracting each argument of an inductive instance by a variable and constraining it by equalities afterwards. This way, the usual induction and destruct tactics can be applied to the abstracted instance and after simplification of the equalities we get the expected goals. The abstracting tactic is called generalize_eqs and it takes as argument a hypothesis to generalize. It uses the JMeq datatype defined in Coq.Logic.JMeq, hence we need to require it before. For example, revisiting the first example of the inversion documentation: .. coqtop:: in reset Require Import Coq.Logic.JMeq. Inductive Le : nat -> nat -> Set := | LeO : forall n:nat, Le 0 n | LeS : forall n m:nat, Le n m -> Le (S n) (S m). Parameter P : nat -> nat -> Prop. Goal forall n m:nat, Le (S n) m -> P n m. intros n m H. .. coqtop:: all generalize_eqs H. The index ``S n`` gets abstracted by a variable here, but a corresponding equality is added under the abstract instance so that no information is actually lost. The goal is now almost amenable to do induction or case analysis. One should indeed first move ``n`` into the goal to strengthen it before doing induction, or ``n`` will be fixed in the inductive hypotheses (this does not matter for case analysis). As a rule of thumb, all the variables that appear inside constructors in the indices of the hypothesis should be generalized. This is exactly what the ``generalize_eqs_vars`` variant does: .. coqtop:: all abort generalize_eqs_vars H. induction H. As the hypothesis itself did not appear in the goal, we did not need to use an heterogeneous equality to relate the new hypothesis to the old one (which just disappeared here). However, the tactic works just as well in this case, e.g.: .. coqtop:: none Require Import Coq.Program.Equality. .. coqtop:: in Parameter Q : forall (n m : nat), Le n m -> Prop. Goal forall n m (p : Le (S n) m), Q (S n) m p. .. coqtop:: all intros n m p. generalize_eqs_vars p. One drawback of this approach is that in the branches one will have to substitute the equalities back into the instance to get the right assumptions. Sometimes injection of constructors will also be needed to recover the needed equalities. Also, some subgoals should be directly solved because of inconsistent contexts arising from the constraints on indexes. The nice thing is that we can make a tactic based on discriminate, injection and variants of substitution to automatically do such simplifications (which may involve the axiom K). This is what the ``simplify_dep_elim`` tactic from ``Coq.Program.Equality`` does. For example, we might simplify the previous goals considerably: .. coqtop:: all abort induction p ; simplify_dep_elim. The higher-order tactic ``do_depind`` defined in ``Coq.Program.Equality`` takes a tactic and combines the building blocks we have seen with it: generalizing by equalities calling the given tactic with the generalized induction hypothesis as argument and cleaning the subgoals with respect to equalities. Its most important instantiations are ``dependent induction`` and ``dependent destruction`` that do induction or simply case analysis on the generalized hypothesis. For example we can redo what we’ve done manually with dependent destruction: .. coqtop:: in Lemma ex : forall n m:nat, Le (S n) m -> P n m. .. coqtop:: in intros n m H. .. coqtop:: all abort dependent destruction H. This gives essentially the same result as inversion. Now if the destructed hypothesis actually appeared in the goal, the tactic would still be able to invert it, contrary to dependent inversion. Consider the following example on vectors: .. coqtop:: in Set Implicit Arguments. .. coqtop:: in Parameter A : Set. .. coqtop:: in Inductive vector : nat -> Type := | vnil : vector 0 | vcons : A -> forall n, vector n -> vector (S n). .. coqtop:: in Goal forall n, forall v : vector (S n), exists v' : vector n, exists a : A, v = vcons a v'. .. coqtop:: in intros n v. .. coqtop:: all dependent destruction v. In this case, the ``v`` variable can be replaced in the goal by the generalized hypothesis only when it has a type of the form ``vector (S n)``, that is only in the second case of the destruct. The first one is dismissed because ``S n <> 0``. A larger example ~~~~~~~~~~~~~~~~ Let’s see how the technique works with induction on inductive predicates on a real example. We will develop an example application to the theory of simply-typed lambda-calculus formalized in a dependently-typed style: .. coqtop:: in reset Inductive type : Type := | base : type | arrow : type -> type -> type. .. coqtop:: in Notation " t --> t' " := (arrow t t') (at level 20, t' at next level). .. coqtop:: in Inductive ctx : Type := | empty : ctx | snoc : ctx -> type -> ctx. .. coqtop:: in Notation " G , tau " := (snoc G tau) (at level 20, tau at next level). .. coqtop:: in Fixpoint conc (G D : ctx) : ctx := match D with | empty => G | snoc D' x => snoc (conc G D') x end. .. coqtop:: in Notation " G ; D " := (conc G D) (at level 20). .. coqtop:: in Inductive term : ctx -> type -> Type := | ax : forall G tau, term (G, tau) tau | weak : forall G tau, term G tau -> forall tau', term (G, tau') tau | abs : forall G tau tau', term (G , tau) tau' -> term G (tau --> tau') | app : forall G tau tau', term G (tau --> tau') -> term G tau -> term G tau'. We have defined types and contexts which are snoc-lists of types. We also have a ``conc`` operation that concatenates two contexts. The ``term`` datatype represents in fact the possible typing derivations of the calculus, which are isomorphic to the well-typed terms, hence the name. A term is either an application of: + the axiom rule to type a reference to the first variable in a context + the weakening rule to type an object in a larger context + the abstraction or lambda rule to type a function + the application to type an application of a function to an argument Once we have this datatype we want to do proofs on it, like weakening: .. coqtop:: in abort Lemma weakening : forall G D tau, term (G ; D) tau -> forall tau', term (G , tau' ; D) tau. The problem here is that we can’t just use induction on the typing derivation because it will forget about the ``G ; D`` constraint appearing in the instance. A solution would be to rewrite the goal as: .. coqtop:: in abort Lemma weakening' : forall G' tau, term G' tau -> forall G D, (G ; D) = G' -> forall tau', term (G, tau' ; D) tau. With this proper separation of the index from the instance and the right induction loading (putting ``G`` and ``D`` after the inducted-on hypothesis), the proof will go through, but it is a very tedious process. One is also forced to make a wrapper lemma to get back the more natural statement. The ``dependent induction`` tactic alleviates this trouble by doing all of this plumbing of generalizing and substituting back automatically. Indeed we can simply write: .. coqtop:: in Require Import Coq.Program.Tactics. Require Import Coq.Program.Equality. .. coqtop:: in Lemma weakening : forall G D tau, term (G ; D) tau -> forall tau', term (G , tau' ; D) tau. .. coqtop:: in Proof with simpl in * ; simpl_depind ; auto. .. coqtop:: in intros G D tau H. dependent induction H generalizing G D ; intros. This call to dependent induction has an additional arguments which is a list of variables appearing in the instance that should be generalized in the goal, so that they can vary in the induction hypotheses. By default, all variables appearing inside constructors (except in a parameter position) of the instantiated hypothesis will be generalized automatically but one can always give the list explicitly. .. coqtop:: all Show. The ``simpl_depind`` tactic includes an automatic tactic that tries to simplify equalities appearing at the beginning of induction hypotheses, generally using trivial applications of ``reflexivity``. In cases where the equality is not between constructor forms though, one must help the automation by giving some arguments, using the ``specialize`` tactic for example. .. coqtop:: in destruct D... apply weak; apply ax. apply ax. .. coqtop:: in destruct D... .. coqtop:: all Show. .. coqtop:: all specialize (IHterm G0 empty eq_refl). Once the induction hypothesis has been narrowed to the right equality, it can be used directly. .. coqtop:: all apply weak, IHterm. Now concluding this subgoal is easy. .. coqtop:: in constructor; apply IHterm; reflexivity. .. seealso:: The :tacn:`induction`, :tacn:`case`, and :tacn:`inversion` tactics. autorewrite ----------- Here are two examples of ``autorewrite`` use. The first one ( *Ackermann function*) shows actually a quite basic use where there is no conditional rewriting. The second one ( *Mac Carthy function*) involves conditional rewritings and shows how to deal with them using the optional tactic of the ``Hint Rewrite`` command. .. example:: Ackermann function .. coqtop:: in reset Require Import Arith. .. coqtop:: in Parameter Ack : nat -> nat -> nat. .. coqtop:: in Axiom Ack0 : forall m:nat, Ack 0 m = S m. Axiom Ack1 : forall n:nat, Ack (S n) 0 = Ack n 1. Axiom Ack2 : forall n m:nat, Ack (S n) (S m) = Ack n (Ack (S n) m). .. coqtop:: in Hint Rewrite Ack0 Ack1 Ack2 : base0. .. coqtop:: all Lemma ResAck0 : Ack 3 2 = 29. .. coqtop:: all autorewrite with base0 using try reflexivity. .. example:: MacCarthy function .. coqtop:: in reset Require Import Omega. .. coqtop:: in Parameter g : nat -> nat -> nat. .. coqtop:: in Axiom g0 : forall m:nat, g 0 m = m. Axiom g1 : forall n m:nat, (n > 0) -> (m > 100) -> g n m = g (pred n) (m - 10). Axiom g2 : forall n m:nat, (n > 0) -> (m <= 100) -> g n m = g (S n) (m + 11). .. coqtop:: in Hint Rewrite g0 g1 g2 using omega : base1. .. coqtop:: in Lemma Resg0 : g 1 110 = 100. .. coqtop:: out Show. .. coqtop:: all autorewrite with base1 using reflexivity || simpl. .. coqtop:: none Qed. .. coqtop:: all Lemma Resg1 : g 1 95 = 91. .. coqtop:: all autorewrite with base1 using reflexivity || simpl. .. coqtop:: none Qed. [ Verzeichnis aufwärts0.191unsichere Verbindung ] |