AltAutoA Streamlined Treatment of Automation

So far, we've been doing all our proofs using just a small handful of Coq's tactics and completely ignoring its powerful facilities for constructing parts of proofs automatically. Getting used to them will take some work -- Coq's automation is a power tool -- but it will allow us to scale up our efforts to more complex definitions and more interesting properties without becoming overwhelmed by boring, repetitive, low-level details.
In this chapter, we'll learn about
  • tacticals, which allow tactics to be combined;
  • new tactics that make dealing with hypothesis names less fussy and more maintainable;
  • automatic solvers that can prove limited classes of theorems without any human assistance;
  • proof search with the auto tactic; and
  • the Ltac language for writing tactics.
These features enable startlingly short proofs. Used properly, they can also make proofs more maintainable and robust to changes in underlying definitions.
This chapter is an alternative to the combination of Imp and Auto, which cover roughly the same material about automation, but in the context of programming language metatheory. A deeper treatment of auto can be found in the UseAuto chapter in Programming Language Foundations.
Set Warnings "-notation-overridden,-parsing,-deprecated-hint-without-locality,-deprecated-syntactic-definition,-deprecated]".
From Coq Require Import Arith List.
From LF Require Import IndProp.
As a simple illustration of the benefits of automation, let's consider another problem on regular expressions, which we formalized in IndProp. A given set of strings can be denoted by many different regular expressions. For example, App EmptyString re matches exactly the same strings as re. We can write a function that "optimizes" any regular expression into a potentially simpler one by applying this fact throughout the r.e. (Note that, for simplicity, the function does not optimize expressions that arise as the result of other optimizations.)
Fixpoint re_opt_e {T:Type} (re: reg_exp T) : reg_exp T :=
  match re with
  | App EmptyStr re2 ⇒ re_opt_e re2
  | App re1 re2 ⇒ App (re_opt_e re1) (re_opt_e re2)
  | Union re1 re2 ⇒ Union (re_opt_e re1) (re_opt_e re2)
  | Star re ⇒ Star (re_opt_e re)
  | _ ⇒ re
  end.
We would like to show the equivalence of re's with their "optimized" form. One direction of this equivalence looks like this (the other is similar).
Lemma re_opt_e_match : ∀ T (re: reg_exp T) s,
  s =~ re → s =~ re_opt_e re.
Proof.
  intros T re s M.
  induction M
    as [| x'
        | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
        | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
        | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2].
  - (* MEmpty *) simpl. apply MEmpty.
  - (* MChar *) simpl. apply MChar.
  - (* MApp *) simpl.
    destruct re1.
    + apply MApp.
      × apply IH1.
      × apply IH2.
    + inversion Hmatch1. simpl. apply IH2.
    + apply MApp.
      × apply IH1.
      × apply IH2.
    + apply MApp.
      × apply IH1.
      × apply IH2.
    + apply MApp.
      × apply IH1.
      × apply IH2.
    + apply MApp.
      × apply IH1.
      × apply IH2.
  - (* MUnionL *) simpl. apply MUnionL. apply IH.
  - (* MUnionR *) simpl. apply MUnionR. apply IH.
  - (* MStar0 *) simpl. apply MStar0.
  - (* MStarApp *) simpl. apply MStarApp.
    × apply IH1.
    × apply IH2.
Qed.
The amount of repetition in that proof is annoying. And if we wanted to extend the optimization function to handle other, similar, rewriting opportunities, it would start to be a real problem. We can streamline the proof with tacticals, which we turn to, next.

Tacticals

Tacticals are tactics that take other tactics as arguments -- "higher-order tactics," if you will.

The try Tactical

If T is a tactic, then try T is a tactic that is just like T except that, if T fails, try T successfully does nothing at all instead of failing.
Theorem silly1 : ∀ n, 1 + n = S n.
Proof. try reflexivity. (* this just does reflexivity *) Qed.
Theorem silly2 : ∀ (P : Prop), P → P.
Proof.
  intros P HP.
  Fail reflexivity.
  try reflexivity. (* proof state is unchanged *)
  apply HP.
Qed.
There is no real reason to use try in completely manual proofs like these, but it is very useful for doing automated proofs in conjunction with the ; tactical, which we show next.

The Sequence Tactical ; (Simple Form)

In its most common form, the sequence tactical, written with semicolon ;, takes two tactics as arguments. The compound tactic T; T' first performs T and then performs T' on each subgoal generated by T.
For example, consider the following trivial lemma:
Lemma simple_semi : ∀ n, (n + 1 =? 0) = false.
Proof.
  intros n.
  destruct n eqn:E.
    (* Leaves two subgoals, which are discharged identically...  *)
    - (* n=0 *) simpl. reflexivity.
    - (* n=Sn' *) simpl. reflexivity.
Qed.
We can simplify this proof using the ; tactical:
Lemma simple_semi' : ∀ n, (n + 1 =? 0) = false.
Proof.
  intros n.
  (* destruct the current goal *)
  destruct n;
  (* then simpl each resulting subgoal *)
  simpl;
  (* and do reflexivity on each resulting subgoal *)
  reflexivity.
Qed.
Or even more tersely, destruct can do the intro, and simpl can be omitted:
Lemma simple_semi'' : ∀ n, (n + 1 =? 0) = false.
Proof.
  destruct n; reflexivity.
Qed.

Exercise: 3 stars, standard (try_sequence)

Prove the following theorems using try and ;. Like simple_semi'' above, each proof script should be a sequence t1; ...; tn. of tactics, and there should be only one period in between Proof. and Qed.. Let's call that a "one shot" proof.
Theorem andb_eq_orb :
  ∀ (b c : bool),
  (andb b c = orb b c) →
  b = c.
Proof. (* FILL IN HERE *) Admitted.
Theorem add_assoc : ∀ n m p : nat,
    n + (m + p) = (n + m) + p.
Proof. (* FILL IN HERE *) Admitted.
Fixpoint nonzeros (lst : list nat) :=
  match lst with
  | [] ⇒ []
  | 0 :: t ⇒ nonzeros t
  | h :: t ⇒ h :: nonzeros t
  end.
Lemma nonzeros_app : ∀ lst1 lst2 : list nat,
  nonzeros (lst1 ++ lst2) = (nonzeros lst1) ++ (nonzeros lst2).
Proof. (* FILL IN HERE *) Admitted.
☐
Using try and ; together, we can improve the proof about regular expression optimization.
Lemma re_opt_e_match' : ∀ T (re: reg_exp T) s,
  s =~ re → s =~ re_opt_e re.
Proof.
  intros T re s M.
  induction M
    as [| x'
        | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
        | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
        | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2];
    (* Do the simpl for every case here: *)
    simpl.
  - (* MEmpty *) apply MEmpty.
  - (* MChar *) apply MChar.
  - (* MApp *)
    destruct re1;
    (* Most cases follow by the same formula.  Notice that apply MApp gives two subgoals: try apply IH1 is run on _both_ of
       them and succeeds on the first but not the second; apply IH2
       is then run on this remaining goal. *)
    try (apply MApp; try apply IH1; apply IH2).
    (* The interesting case, on which try... does nothing, is when
       re1 = EmptyStr. In this case, we have to appeal to the fact
       that re1 matches only the empty string: *)
    inversion Hmatch1. simpl. apply IH2.
  - (* MUnionL *) apply MUnionL. apply IH.
  - (* MUnionR *) apply MUnionR. apply IH.
  - (* MStar0 *) apply MStar0.
  - (* MStarApp *) apply MStarApp. apply IH1. apply IH2.
Qed.

The Sequence Tactical ; (Local Form)

The sequence tactical ; also has a more general form than the simple T; T' we saw above. If T, T1, ..., Tn are tactics, then
[ T; [T1 | T2 | ... | Tn] ]
is a tactic that first performs T and then locally performs T1 on the first subgoal generated by T, locally performs T2 on the second subgoal, etc.
So T; T' is just special notation for the case when all of the Ti's are the same tactic; i.e., T; T' is shorthand for:

      T; [T' | T' | ... | T']
For example, the following proof makes it clear which tactics are used to solve the base case vs. the inductive case.
Theorem app_length : ∀ (X : Type) (lst1 lst2 : list X),
    length (lst1 ++ lst2) = (length lst1) + (length lst2).
Proof.
  intros; induction lst1;
    [reflexivity | simpl; rewrite IHlst1; reflexivity].
Qed.
The identity tactic idtac always succeeds without changing the proof state. We can use it to factor out reflexivity in the previous proof.
Theorem app_length' : ∀ (X : Type) (lst1 lst2 : list X),
    length (lst1 ++ lst2) = (length lst1) + (length lst2).
Proof.
  intros; induction lst1;
    [idtac | simpl; rewrite IHlst1];
    reflexivity.
Qed.

Exercise: 1 star, standard (notry_sequence)

Prove the following theorem with a one-shot proof, but this time, do not use try.
Theorem add_assoc' : ∀ n m p : nat,
    n + (m + p) = (n + m) + p.
Proof. (* FILL IN HERE *) Admitted.
☐
We can use the local form of the sequence tactical to give a slightly neater version of our optimization proof. Two lines change, as shown below with <===.
Lemma re_opt_e_match'' : ∀ T (re: reg_exp T) s,
  s =~ re → s =~ re_opt_e re.
Proof.
  intros T re s M.
  induction M
    as [| x'
        | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
        | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
        | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2];
    (* Do the simpl for every case here: *)
    simpl.
  - (* MEmpty *) apply MEmpty.
  - (* MChar *) apply MChar.
  - (* MApp *)
    destruct re1;
    try (apply MApp; [apply IH1 | apply IH2]). (* <=== *)
    inversion Hmatch1. simpl. apply IH2.
  - (* MUnionL *) apply MUnionL. apply IH.
  - (* MUnionR *) apply MUnionR. apply IH.
  - (* MStar0 *) apply MStar0.
  - (* MStarApp *) apply MStarApp; [apply IH1 | apply IH2]. (* <=== *)
Qed.

The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until it fails or stops making progress. Here is an example showing that 10 is in a long list:
Theorem In10 : In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
  repeat (try (left; reflexivity); right).
Qed.
The tactic repeat T never fails: if the tactic T doesn't apply to the original goal, then repeat still succeeds without changing the original goal (i.e., it repeats zero times).
Theorem In10' : In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
  repeat (left; reflexivity).
  repeat (right; try (left; reflexivity)).
Qed.
The tactic repeat T also does not have any upper bound on the number of times it applies T. If T is a tactic that always succeeds, then repeat T will loop forever (e.g., repeat simpl loops, since simpl always succeeds). Evaluation in Coq's term language, Gallina, is guaranteed to terminate, but tactic evaluation is not. This does not affect Coq's logical consistency, however, since the job of repeat and other tactics is to guide Coq in constructing proofs. If the construction process diverges, it simply means that we have failed to construct a proof, not that we have constructed an incorrect proof.

Exercise: 1 star, standard (ev100)

Prove that 100 is even. Your proof script should be quite short.
Theorem ev100: ev 100.
Proof. (* FILL IN HERE *) Admitted.
☐

An Optimization Exercise

Exercise: 4 stars, standard (re_opt)

Consider this more powerful version of the regular expression optimizer.
Fixpoint re_opt {T:Type} (re: reg_exp T) : reg_exp T :=
  match re with
  | App _ EmptySet ⇒ EmptySet
  | App EmptyStr re2 ⇒ re_opt re2
  | App re1 EmptyStr ⇒ re_opt re1
  | App re1 re2 ⇒ App (re_opt re1) (re_opt re2)
  | Union EmptySet re2 ⇒ re_opt re2
  | Union re1 EmptySet ⇒ re_opt re1
  | Union re1 re2 ⇒ Union (re_opt re1) (re_opt re2)
  | Star EmptySet ⇒ EmptyStr
  | Star EmptyStr ⇒ EmptyStr
  | Star re ⇒ Star (re_opt re)
  | EmptySet ⇒ EmptySet
  | EmptyStr ⇒ EmptyStr
  | Char x ⇒ Char x
  end.
(* Here is an incredibly tedious manual proof of (one direction of)
   its correctness: *)
Lemma re_opt_match : ∀ T (re: reg_exp T) s,
  s =~ re → s =~ re_opt re.
Proof.
  intros T re s M.
  induction M
    as [| x'
        | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
        | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
        | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2].
  - (* MEmpty *) simpl. apply MEmpty.
  - (* MChar *) simpl. apply MChar.
  - (* MApp *) simpl.
    destruct re1.
    + inversion IH1.
    + inversion IH1. simpl. destruct re2.
      × apply IH2.
      × apply IH2.
      × apply IH2.
      × apply IH2.
      × apply IH2.
      × apply IH2.
    + destruct re2.
      × inversion IH2.
      × inversion IH2. rewrite app_nil_r. apply IH1.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
    + destruct re2.
      × inversion IH2.
      × inversion IH2. rewrite app_nil_r. apply IH1.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
    + destruct re2.
      × inversion IH2.
      × inversion IH2. rewrite app_nil_r. apply IH1.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
    + destruct re2.
      × inversion IH2.
      × inversion IH2. rewrite app_nil_r. apply IH1.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
      × apply MApp.
        -- apply IH1.
        -- apply IH2.
  - (* MUnionL *) simpl.
    destruct re1.
    + inversion IH.
    + destruct re2.
      × apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
    + destruct re2.
      × apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
    + destruct re2.
      × apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
    + destruct re2.
      × apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
    + destruct re2.
      × apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
      × apply MUnionL. apply IH.
  - (* MUnionR *) simpl.
    destruct re1.
    + apply IH.
    + destruct re2.
      × inversion IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
    + destruct re2.
      × inversion IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
    + destruct re2.
      × inversion IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
    + destruct re2.
      × inversion IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
    + destruct re2.
      × inversion IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
      × apply MUnionR. apply IH.
 - (* MStar0 *) simpl.
    destruct re.
    + apply MEmpty.
    + apply MEmpty.
    + apply MStar0.
    + apply MStar0.
    + apply MStar0.
    + simpl.
      destruct re.
      × apply MStar0.
      × apply MStar0.
      × apply MStar0.
      × apply MStar0.
      × apply MStar0.
      × apply MStar0.
 - (* MStarApp *) simpl.
   destruct re.
   + inversion IH1.
   + inversion IH1. inversion IH2. apply MEmpty.
   + apply star_app.
     × apply MStar1. apply IH1.
     × apply IH2.
   + apply star_app.
     × apply MStar1. apply IH1.
     × apply IH2.
   + apply star_app.
     × apply MStar1. apply IH1.
     × apply IH2.
   + apply star_app.
     × apply MStar1. apply IH1.
     × apply IH2.
Qed.

(* Use the tacticals described so far to shorten the proof. The proof
   above is about 200 lines. Reduce it to 50 or fewer lines of similar
   density. Solve each of the seven top-level bullets with a one-shot
   proof.

   Hint: use a bottom-up approach. First copy-paste the entire proof
   below. Then automate the innermost bullets first, proceeding
   outwards. Frequently double-check that the entire proof still
   compiles. If it doesn't, undo the most recent changes you made
   until you get back to a compiling proof. *)
Lemma re_opt_match' : ∀ T (re: reg_exp T) s,
  s =~ re → s =~ re_opt re.
Proof.
(* FILL IN HERE *) Admitted.
(* Do not modify the following line: *)
Definition manual_grade_for_re_opt : option (nat×string) := None.
☐

Tactics that Make Mentioning Names Unnecessary

So far we have been dependent on knowing the names of hypotheses. For example, to prove the following simple theorem, we hardcode the name HP:
Theorem hyp_name : ∀ (P : Prop), P → P.
Proof.
  intros P HP. apply HP.
Qed.
We took the trouble to invent a name for HP, then we had to remember that name. If we later change the name in one place, we have to change it everywhere. Likewise, if we were to add new arguments to the theorem, we would have to adjust the intros list. That makes it challenging to maintain large proofs. So, Coq provides several tactics that make it possible to write proof scripts that do not hardcode names.

The assumption tactic

The assumption tactic is useful to streamline the proof above. It looks through the hypotheses and, if it finds the goal as one them, it uses that to finish the proof.
Theorem no_hyp_name : ∀ (P : Prop), P → P.
Proof.
  intros. assumption.
Qed.
Some might argue to the contrary that hypothesis names improve self-documention of proof scripts. Maybe they do, sometimes. But in the case of the two proofs above, the first mentions unnecessary detail, whereas the second could be paraphrased simply as "the conclusion follows from the assumptions."
Anyway, unlike informal (good) mathematical proofs, Coq proof scripts are generally not that illuminating to readers. Worries about rich, self-documenting names for hypotheses might be misplaced.

The contradiction tactic

The contradiction tactic handles some ad hoc situations where a hypothesis contains False, or two hypotheses derive False.
Theorem false_assumed : False → 0 = 1.
Proof.
  intros H. destruct H.
Qed.
Theorem false_assumed' : False → 0 = 1.
Proof.
  intros. contradiction.
Qed.
Theorem contras : ∀ (P : Prop), P → ¬P → 0 = 1.
Proof.
  intros P HP HNP. exfalso. apply HNP. apply HP.
Qed.
Theorem contras' : ∀ (P : Prop), P → ¬P → 0 = 1.
Proof.
  intros. contradiction.
Qed.

The subst tactic

The subst tactic substitutes away an identifier, replacing it everywhere and eliminating it from the context. That helps us to avoid naming hypotheses in rewrites.
Theorem many_eq : ∀ (n m o p : nat),
  n = m →
  o = p →
  [n; o] = [m; p].
Proof.
  intros n m o p Hnm Hop. rewrite Hnm. rewrite Hop. reflexivity.
Qed.
Theorem many_eq' : ∀ (n m o p : nat),
  n = m →
  o = p →
  [n; o] = [m; p].
Proof.
  intros. subst. reflexivity.
Qed.
Actually there are two forms of this tactic.
  • subst x finds an assumption x = e or e = x in the context, replaces x with e throughout the context and current goal, and removes the assumption from the context.
  • subst substitutes away all assumptions of the form x = e or e = x.

The constructor tactic

The constructor tactic tries to find a constructor c (from the appropriate Inductive definition in the current environment) that can be applied to solve the current goal.
Check ev_0 : ev 0.
Check ev_SS : ∀ n : nat, ev n → ev (S (S n)).
Example constructor_example: ∀ (n:nat),
    ev (n + n).
Proof.
  induction n; simpl.
  - constructor. (* applies ev_0 *)
  - rewrite add_comm. simpl. constructor. (* applies ev_SS *)
    assumption.
Qed.
Warning: if more than one constructor can apply, constructor picks the first one, in the order in which they were defined in the Inductive definition. That might not be the one you want.

Automatic Solvers

Coq has several special-purpose tactics that can solve certain kinds of goals in a completely automated way. These tactics are based on sophisticated algorithms developed for verification in specific mathematical or logical domains.
Some automatic solvers are decision procedures, which are algorithms that always terminate, and always give a correct answer. Here, that means that they always find a correct proof, or correctly determine that the goal is invalid. Other automatic solvers are incomplete: they might fail to find a proof of a valid goal.

Linear Integer Arithmetic: The lia Tactic

The lia tactic implements a decision procedure for integer linear arithmetic, a subset of propositional logic and arithmetic. As input it accepts goals constructed as follows:
  • variables and constants of type nat, Z, and other integer types;
  • arithmetic operators +, -, ×, and ^;
  • equality = and ordering <, >, ≤, ≥; and
  • the logical connectives ∧, ∨, ¬, →, and ↔; and constants True and False.
Linear goals involve (in)equalities over expressions of the form c1 × x1 + ... + cn × xn, where ci are constants and xi are variables.
  • For linear goals, lia will either solve the goal or fail, meaning that the goal is actually invalid.
  • For non-linear goals, lia will also either solve the goal or fail. But in this case, the failure does not necessarily mean that the goal is invalid -- it might just be beyond lia's reach to prove because of non-linearity.
Also, lia will do intros as necessary.
From Coq Require Import Lia.
Theorem lia_succeed1 : ∀ (n : nat),
  n > 0 → n × 2 > n.
Proof. lia. Qed.
Theorem lia_succeed2 : ∀ (n m : nat),
    n × m = m × n.
Proof.
  lia. (* solvable though non-linear *)
Qed.
Theorem lia_fail1 : 0 = 1.
Proof.
  Fail lia. (* goal is invalid *)
Abort.
Theorem lia_fail2 : ∀ (n : nat),
    n ≥ 1 → 2 ^ n = 2 × 2 ^ (n - 1).
Proof.
  Fail lia. (*goal is non-linear, valid, but unsolvable by lia *)
Abort.
There are other tactics that can solve arithmetic goals. The ring and field tactics, for example, can solve equations over the algebraic structures of rings and fields, from which the tactics get their names. These tactics do not do intros.
Require Import Ring.
Theorem mult_comm : ∀ (n m : nat),
    n × m = m × n.
Proof.
  intros n m. ring.
Qed.

Equalities: The congruence Tactic

The lia tactic makes use of facts about addition and multiplication to prove equalities. A more basic way of treating such formulas is to regard every function appearing in them as a black box: nothing is known about the function's behavior. Based on the properties of equality itself, it is still possible to prove some formulas. For example, y = f x → g y = g (f x), even if we know nothing about f or g:
Theorem eq_example1 :
  ∀ (A B C : Type) (f : A → B) (g : B → C) (x : A) (y : B),
    y = f x → g y = g (f x).
Proof.
  intros. rewrite H. reflexivity.
Qed.
The essential properties of equality are that it is:
  • reflexive
  • symmetric
  • transitive
  • a congruence: it respects function and predicate application.
It is that congruence property that we're using when we rewrite in the proof above: if a = b then f a = f b. (The ProofObjects chapter explores this idea further under the name "Leibniz equality".)
The congruence tactic is a decision procedure for equality with uninterpreted functions and other symbols.
Theorem eq_example1' :
  ∀ (A B C : Type) (f : A → B) (g : B → C) (x : A) (y : B),
    y = f x → g y = g (f x).
Proof.
  congruence.
Qed.
The congruence tactic is able to work with constructors, even taking advantage of their injectivity and distinctness.
Theorem eq_example2 : ∀ (n m o p : nat),
    n = m →
    o = p →
    (n, o) = (m, p).
Proof.
  congruence.
Qed.
Theorem eq_example3 : ∀ (X : Type) (h : X) (t : list X),
    [] ≠ h :: t.
Proof.
  congruence.
Qed.

Propositions: The intuition Tactic

A tautology is a logical formula that is always provable. A formula is propositional if it does not use quantifiers -- or at least, if quantifiers do not have to be instantiated to carry out the proof. The intuition tactic implements a decision procedure for propositional tautologies in Coq's constructive (that is, intuitionistic) logic. Even if a goal is not a propositional tautology, intuition will still attempt to reduce it to simpler subgoals.
Theorem intuition_succeed1 : ∀ (P : Prop),
    P → P.
Proof. intuition. Qed.
Theorem intuition_succeed2 : ∀ (P Q : Prop),
    ¬ (P ∨ Q) → ¬P ∧ ¬Q.
Proof. intuition. Qed.
Theorem intuition_simplify1 : ∀ (P : Prop),
    ~~P → P.
Proof.
  intuition. (* not a constructively valid formula *)
Abort.
Theorem intuition_simplify2 : ∀ (x y : nat) (P Q : nat → Prop),
  x = y ∧ (P x → Q x) ∧ P x → Q y.
Proof.
  Fail congruence. (* the propositions stump it *)
  intuition. (* the = stumps it, but it simplifies the propositions *)
  congruence.
Qed.
In the previous example, neither congruence nor intuition alone can solve the goal. But after intuition simplifies the propositions involved in the goal, congruence can succeed. For situations like this, intuition takes an optional argument, which is a tactic to apply to all the unsolved goals that intuition generated. Using that we can offer a shorter proof:
Theorem intuition_simplify2' : ∀ (x y : nat) (P Q : nat → Prop),
  x = y ∧ (P x → Q x) ∧ P x → Q y.
Proof.
  intuition congruence.
Qed.

Exercises with Automatic Solvers

Exercise: 2 stars, standard (automatic_solvers)

The exercises below are gleaned from previous chapters, where they were proved with (relatively) long proof scripts. Each should now be provable with just a single invocation of an automatic solver.
Theorem plus_id_exercise_from_basics : ∀ n m o : nat,
  n = m → m = o → n + m = m + o.
Proof. (* FILL IN HERE *) Admitted.
Theorem add_assoc_from_induction : ∀ n m p : nat,
    n + (m + p) = (n + m) + p.
Proof. (* FILL IN HERE *) Admitted.
Theorem S_injective_from_tactics : ∀ (n m : nat),
  S n = S m →
  n = m.
Proof. (* FILL IN HERE *) Admitted.
Theorem or_distributes_over_and_from_logic : ∀ P Q R : Prop,
    P ∨ (Q ∧ R) ↔ (P ∨ Q) ∧ (P ∨ R).
Proof. (* FILL IN HERE *) Admitted.
☐

Search Tactics

The automated solvers we just discussed are capable of finding proofs in specific domains. Some of them might pay attention to local hypotheses, but overall they don't make use of any custom lemmas we've proved, or that are provided by libraries that we load.
Another kind of automation that Coq provides does just that: the auto tactic and its variants search for proofs that can be assembled out of hypotheses and lemmas.

The auto Tactic

Until this chapter, our proof scripts mostly applied relevant hypotheses or lemmas by name, and one at a time.
Example auto_example_1 : ∀ (P Q R: Prop),
  (P → Q) → (Q → R) → P → R.
Proof.
  intros P Q R H1 H2 H3.
  apply H2. apply H1. apply H3.
Qed.
The auto tactic frees us from this drudgery by searching for a sequence of applications that will prove the goal:
Example auto_example_1' : ∀ (P Q R: Prop),
  (P → Q) → (Q → R) → P → R.
Proof.
  auto.
Qed.
The auto tactic solves goals that are solvable by any combination of
  • intros and
  • apply (of hypotheses from the local context, by default).
Using auto is always "safe" in the sense that it will never fail and will never change the proof state: either it completely solves the current goal, or it does nothing.
Here is a more interesting example showing auto's power:
Example auto_example_2 : ∀ P Q R S T U : Prop,
  (P → Q) →
  (P → R) →
  (T → R) →
  (S → T → U) →
  ((P → Q) → (P → S)) →
  T →
  P →
  U.
Proof. auto. Qed.
Proof search could, in principle, take an arbitrarily long time, so there are limits to how far auto will search by default.
Example auto_example_3 : ∀ (P Q R S T U: Prop),
  (P → Q) →
  (Q → R) →
  (R → S) →
  (S → T) →
  (T → U) →
  P →
  U.
Proof.
  (* When it cannot solve the goal, auto does nothing *)
  auto.
  (* Optional argument says how deep to search (default is 5) *)
  auto 6.
Qed.
The auto tactic considers the hypotheses in the current context together with a hint database of other lemmas and constructors. Some common facts about equality and logical operators are installed in the hint database by default.
Example auto_example_4 : ∀ P Q R : Prop,
  Q →
  (Q → R) →
  P ∨ (Q ∧ R).
Proof. auto. Qed.
If we want to see which facts auto is using, we can use info_auto instead.
Example auto_example_5 : 2 = 2.
Proof.
  (* auto subsumes reflexivity because eq_refl is in the hint
     database. *)
  info_auto.
Qed.
We can extend the hint database with theorem t just for the purposes of one application of auto by writing auto using t.
Lemma le_antisym : ∀ n m: nat, (n ≤ m ∧ m ≤ n) → n = m.
Proof. intros. lia. Qed.
Example auto_example_6 : ∀ n m p : nat,
  (n ≤ p → (n ≤ m ∧ m ≤ n)) →
  n ≤ p →
  n = m.
Proof.
  auto using le_antisym.
Qed.
Of course, in any given development there will probably be some specific constructors and lemmas that are used very often in proofs. We can add these to a hint database named db by writing
      Create HintDb db. to create the database, then
      Hint Resolve T : db. to add T to the database, where T is a top-level theorem or a constructor of an inductively defined proposition (i.e., anything whose type is an implication). We tell auto to use that database by writing auto with db. Technically creation of the database is optional; Coq will create it automatically the first time we use Hint.
Create HintDb le_db.
Hint Resolve le_antisym : le_db.
Example auto_example_6' : ∀ n m p : nat,
  (n ≤ p → (n ≤ m ∧ m ≤ n)) →
  n ≤ p →
  n = m.
Proof.
  auto with le_db.
Qed.
As a shorthand, we can write
      Hint Constructors c : db. to tell Coq to do a Hint Resolve for all of the constructors from the inductive definition of c.
It is also sometimes necessary to add
      Hint Unfold d : db. where d is a defined symbol, so that auto knows to expand uses of d, thus enabling further possibilities for applying lemmas that it knows about.
Definition is_fortytwo x := (x = 42).
Example auto_example_7: ∀ x,
  (x ≤ 42 ∧ 42 ≤ x) → is_fortytwo x.
Proof.
  auto. (* does nothing *)
Abort.
Hint Unfold is_fortytwo : le_db.
Example auto_example_7' : ∀ x,
  (x ≤ 42 ∧ 42 ≤ x) → is_fortytwo x.
Proof. info_auto with le_db. Qed.
The "global" database that auto always uses is named core. You can add your own hints to it, but the Coq manual discourages that, preferring instead to have specialized databases for specific developments. Many of the important libraries have their own hint databases that you can tag in: arith, bool, datatypes (including lists), etc.
Example auto_example_8 : ∀ (n m : nat),
    n + m = m + n.
Proof.
  auto. (* no progress *)
  info_auto with arith. (* uses Nat.add_comm *)
Qed.

Exercise: 3 stars, standard (re_opt_match_auto)

Use auto to shorten your proof of re_opt_match even more. Eliminate all uses of apply, thus removing the need to name specific constructors and lemmas about regular expressions. The number of lines of proof script won't decrease that much, because auto won't be able to find induction, destruct, or inversion opportunities by itself.
Hint: again, use a bottom-up approach. Always keep the proof compiling. You might find it easier to return to the original, very long proof, and shorten it, rather than starting with re_opt_match'; but, either way can work.
Lemma re_opt_match'' : ∀ T (re: reg_exp T) s,
  s =~ re → s =~ re_opt re.
Proof.
(* FILL IN HERE *) Admitted.
(* Do not modify the following line: *)
Definition manual_grade_for_re_opt_match'' : option (nat×string) := None.
☐

Exercise: 3 stars, advanced, optional (pumping_redux)

Use auto, lia, and any other useful tactics from this chapter to shorten your proof (or the "official" solution proof) of the weak Pumping Lemma exercise from IndProp.
Import Pumping.
Lemma weak_pumping : ∀ T (re : reg_exp T) s,
    s =~ re →
    pumping_constant re ≤ length s →
    ∃ s1 s2 s3,
      s = s1 ++ s2 ++ s3 ∧
        s2 ≠ [] ∧
        ∀ m, s1 ++ napp m s2 ++ s3 =~ re.
Proof.
(* FILL IN HERE *) Admitted.
(* Do not modify the following line: *)
Definition manual_grade_for_pumping_redux : option (nat×string) := None.
☐

Exercise: 3 stars, advanced, optional (pumping_redux_strong)

Use auto, lia, and any other useful tactics from this chapter to shorten your proof (or the "official" solution proof) of the stronger Pumping Lemma exercise from IndProp.
Lemma pumping : ∀ T (re : reg_exp T) s,
    s =~ re →
    pumping_constant re ≤ length s →
    ∃ s1 s2 s3,
      s = s1 ++ s2 ++ s3 ∧
        s2 ≠ [] ∧
        length s1 + length s2 ≤ pumping_constant re ∧
        ∀ m, s1 ++ napp m s2 ++ s3 =~ re.
Proof.
  intros T re s Hmatch.
  induction Hmatch
    as [ | x | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
       | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
       | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2 ];
    simpl; try lia;
    intros Hlen.
(* FILL IN HERE *) Admitted.
(* Do not modify the following line: *)
Definition manual_grade_for_pumping_redux_strong : option (nat×string) := None.
☐

The eauto variant

There is a variant of auto (and other tactics, such as apply) that makes it possible to delay instantiation of quantifiers. To motivate this feature, consider again this simple example:
Example trans_example1: ∀ a b c d,
    a ≤ b + b × c →
    (1 + c) × b ≤ d →
    a ≤ d.
Proof.
  intros a b c d H1 H2.
  apply Nat.le_trans with (b + b × c).
    (* ^ We must supply the intermediate value *)
  - apply H1.
  - simpl in H2. rewrite mul_comm. apply H2.
Qed.
In the first step of the proof, we had to explicitly provide a longish expression to help Coq instantiate a "hidden" argument to the le_trans constructor. This was needed because the definition of le_trans...
    le_trans : ∀ m n o : nat, m ≤ n → n ≤ o → m ≤ o is quantified over a variable, n, that does not appear in its conclusion, so unifying its conclusion with the goal state doesn't help Coq find a suitable value for this variable. If we leave out the with, this step fails ("Error: Unable to find an instance for the variable n").
We already know one way to avoid an explicit with clause, namely to provide H1 as the (first) explicit argument to le_trans. But here's another way, using the eapply tactic:
Example trans_example1': ∀ a b c d,
    a ≤ b + b × c →
    (1 + c) × b ≤ d →
    a ≤ d.
Proof.
  intros a b c d H1 H2.
  eapply Nat.le_trans. (* 1 *)
  - apply H1. (* 2 *)
  - simpl in H2. rewrite mul_comm. apply H2.
Qed.
The eapply H tactic behaves just like apply H except that, after it finishes unifying the goal state with the conclusion of H, it does not bother to check whether all the variables that were introduced in the process have been given concrete values during unification.
If you step through the proof above, you'll see that the goal state at position 1 mentions the existential variable ?n in both of the generated subgoals. The next step (which gets us to position 2) replaces ?n with a concrete value. When we start working on the second subgoal (position 3), we observe that the occurrence of ?n in this subgoal has been replaced by the value that it was given during the first subgoal.
Several of the tactics that we've seen so far, including ∃, constructor, and auto, have e... variants. For example, here's a proof using eauto:
Example trans_example2: ∀ a b c d,
    a ≤ b + b × c →
    b + b × c ≤ d →
    a ≤ d.
Proof.
  intros a b c d H1 H2.
  info_eauto using Nat.le_trans.
Qed.
The eauto tactic works just like auto, except that it uses eapply instead of apply.
Pro tip: One might think that, since eapply and eauto are more powerful than apply and auto, it would be a good idea to use them all the time. Unfortunately, they are also significantly slower -- especially eauto. Coq experts tend to use apply and auto most of the time, only switching to the e variants when the ordinary variants don't do the job.

Ltac: The Tactic Language

Most of the tactics we have been using are implemented in OCaml, where they are able to use an API to access Coq's internal structures at a low level. But this is seldom worth the trouble for ordinary Coq users.
Coq has a high-level language called Ltac for programming new tactics in Coq itself, without having to escape to OCaml. Actually we've been using Ltac all along -- anytime we are in proof mode, we've been writing Ltac programs. At their most basic, those programs are just invocations of built-in tactics. The tactical constructs we learned at the beginning of this chapter are also part of Ltac.
What we turn to, next, is ways to use Ltac to reduce the amount of proof script we have to write ourselves.

Ltac Functions

Here is a simple Ltac example:
Ltac simpl_and_try tac := simpl; try tac.
This defines a new tactic called simpl_and_try that takes one tactic tac as an argument and is defined to be equivalent to simpl; try tac. Now writing "simpl_and_try reflexivity." in a proof will be the same as writing "simpl; try reflexivity."
Example sat_ex1 : 1 + 1 = 2.
Proof. simpl_and_try reflexivity. Qed.
Example sat_ex2 : ∀ (n : nat), 1 - 1 + n + 1 = 1 + n.
Proof. simpl_and_try reflexivity. lia. Qed.
Of course, that little tactic is not so useful. But it demonstrates that we can parameterize Ltac-defined tactics, and that their bodies are themselves tactics that will be run in the context of a proof. So Ltac can be used to create functions on tactics.
For a more useful tactic, consider these three proofs from Basics, and how structurally similar they all are:
Theorem plus_1_neq_0 : ∀ n : nat,
  (n + 1) =? 0 = false.
Proof.
  intros n. destruct n.
  - reflexivity.
  - reflexivity.
Qed.
Theorem negb_involutive : ∀ b : bool,
  negb (negb b) = b.
Proof.
  intros b. destruct b.
  - reflexivity.
  - reflexivity.
Qed.
Theorem andb_commutative : ∀ b c, andb b c = andb c b.
Proof.
  intros b c. destruct b.
  - destruct c.
    + reflexivity.
    + reflexivity.
  - destruct c.
    + reflexivity.
    + reflexivity.
Qed.
We can factor out the common structure:
  • Do a destruct.
  • For each branch, finish with reflexivity -- if possible.
Ltac destructpf x :=
  destruct x; try reflexivity.
Theorem plus_1_neq_0' : ∀ n : nat,
    (n + 1) =? 0 = false.
Proof. intros n; destructpf n. Qed.
Theorem negb_involutive' : ∀ b : bool,
  negb (negb b) = b.
Proof. intros b; destructpf b. Qed.
Theorem andb_commutative' : ∀ b c, andb b c = andb c b.
Proof.
  intros b c; destructpf b; destructpf c.
Qed.

Exercise: 1 star, standard (andb3_exchange)

Re-prove the following theorem from Basics, using only intros and destructpf. You should have a one-shot proof.
Theorem andb3_exchange :
  ∀ b c d, andb (andb b c) d = andb (andb b d) c.
Proof. (* FILL IN HERE *) Admitted.
☐

Exercise: 2 stars, standard (andb_true_elim2)

The following theorem from Basics can't be proved with destructpf.
Theorem andb_true_elim2 : ∀ b c : bool,
  andb b c = true → c = true.
Proof.
  intros b c. destruct b eqn:Eb.
  - simpl. intros H. rewrite H. reflexivity.
  - simpl. intros H. destruct c eqn:Ec.
    + reflexivity.
    + rewrite H. reflexivity.
Qed.
Uncomment the definition of destructpf', below, and define your own, improved version of destructpf. Use it to prove the theorem.
(*
Ltac destructpf' x := ...
*)
Your one-shot proof should need only intros and destructpf'.
Theorem andb_true_elim2' : ∀ b c : bool,
    andb b c = true → c = true.
Proof. (* FILL IN HERE *) Admitted.
Double-check that intros and your new destructpf' still suffice to prove this earlier theorem -- i.e., that your improved tactic is general enough to still prove it in one shot:
Theorem andb3_exchange' :
  ∀ b c d, andb (andb b c) d = andb (andb b d) c.
Proof. (* FILL IN HERE *) Admitted.
☐

Ltac Pattern Matching

Here is another common proof pattern that we have seen in many simple proofs by induction:
Theorem app_nil_r : ∀ (X : Type) (lst : list X),
    lst ++ [] = lst.
Proof.
  intros X lst. induction lst as [ | h t IHt].
  - reflexivity.
  - simpl. rewrite IHt. reflexivity.
Qed.
At the point we rewrite, we can't substitute away t: it is present on both sides of the equality in the inductive hypothesis IHt : t ++ [] = t. How can we pick out which hypothesis to rewrite in an Ltac tactic?
To solve this and other problems, Ltac contains a pattern-matching tactic match goal. It allows us to match against the proof state rather than against a program.
Theorem match_ex1 : True.
Proof.
  match goal with
  | [ ⊢ True ] ⇒ apply I
  end.
Qed.
The syntax is similar to a match in Gallina (Coq's term language), but has some new features:
  • The word goal here is a keyword, rather than an expression being matched. It means to match against the proof state, rather than a program term.
  • The square brackets around the pattern can often be omitted, but they do make it easier to visually distinguish which part of the code is the pattern.
  • The turnstile ⊢ separates the hypothesis patterns (if any) from the conclusion pattern. It represents the big horizontal line shown by your IDE in the proof state: the hypotheses are to the left of it, the conclusion is to the right.
  • The hypotheses in the pattern need not completely describe all the hypotheses present in the proof state. It is fine for there to be additional hypotheses in the proof state that do not match any of the patterns. The point is for match goal to pick out particular hypotheses of interest, rather than fully specify the proof state.
  • The right-hand side of a branch is a tactic to run, rather than a program term.
The single branch above therefore specifies to match a goal whose conclusion is the term True and whose hypotheses may be anything. If such a match occurs, it will run apply I.
There may be multiple branches, which are tried in order.
Theorem match_ex2 : True ∧ True.
Proof.
  match goal with
  | [ ⊢ True ] ⇒ apply I
  | [ ⊢ True ∧ True ] ⇒ split; apply I
  end.
Qed.
To see what branches are being tried, it can help to insert calls to the identity tactic idtac. It optionally accepts an argument to print out as debugging information.
Theorem match_ex2' : True ∧ True.
Proof.
  match goal with
  | [ ⊢ True ] ⇒ idtac "branch 1"; apply I
  | [ ⊢ True ∧ True ] ⇒ idtac "branch 2"; split; apply I
  end.
Qed.
Only the second branch was tried. The first one did not match the goal.
The semantics of the tactic match goal have a big difference with the semantics of the term match. With the latter, the first matching pattern is chosen, and later branches are never considered. In fact, an error is produced if later branches are known to be redundant.
Fail Definition redundant_match (n : nat) : nat :=
  match n with
  | x ⇒ x
  | 0 ⇒ 1
  end.
But with match goal, if the tactic for the branch fails, pattern matching continues with the next branch, until a branch succeeds, or all branches have failed.
Theorem match_ex2'' : True ∧ True.
Proof.
  match goal with
  | [ ⊢ _ ] ⇒ idtac "branch 1"; apply I
  | [ ⊢ True ∧ True ] ⇒ idtac "branch 2"; split; apply I
  end.
Qed.
The first branch was tried but failed, then the second branch was tried and succeeded. If all the branches fail, the match goal fails.
Theorem match_ex2''' : True ∧ True.
Proof.
  Fail match goal with
  | [ ⊢ _ ] ⇒ idtac "branch 1"; apply I
  | [ ⊢ _ ] ⇒ idtac "branch 2"; apply I
  end.
Abort.
Next, let's try matching against hypotheses. We can bind a hypothesis name, as with H below, and use that name on the right-hand side of the branch.
Theorem match_ex3 : ∀ (P : Prop), P → P.
Proof.
  intros P HP.
  match goal with
  | [ H : _ ⊢ _ ] ⇒ apply H
  end.
Qed.
The actual name of the hypothesis is of course HP, but the pattern binds it as H. Using idtac, we can even observe the actual name: stepping through the following proof causes "HP" to be printed.
Theorem match_ex3' : ∀ (P : Prop), P → P.
Proof.
  intros P HP.
  match goal with
  | [ H : _ ⊢ _ ] ⇒ idtac H; apply H
  end.
Qed.
We'll keep using idtac for awhile to observe the behavior of match goal, but, note that it isn't necessary for the successful proof of any of the following examples.
If there are multiple hypotheses that match, which one does Ltac choose? Here is a big difference with regular match against terms: Ltac will try all possible matches until one succeeds (or all have failed).
Theorem match_ex4 : ∀ (P Q : Prop), P → Q → P.
Proof.
  intros P Q HP HQ.
  match goal with
  | [ H : _ ⊢ _ ] ⇒ idtac H; apply H
  end.
Qed.
That example prints "HQ" followed by "HP". Ltac first matched against the most recently introduced hypothesis HQ and tried applying it. That did not solve the goal. So Ltac backtracks and tries the next most-recent matching hypothesis, which is HP. Applying that does succeed.
But if there were no successful hypotheses, the entire match would fail:
Theorem match_ex5 : ∀ (P Q R : Prop), P → Q → R.
Proof.
  intros P Q R HP HQ.
  Fail match goal with
  | [ H : _ ⊢ _ ] ⇒ idtac H; apply H
  end.
Abort.
So far we haven't been very demanding in how to match hypotheses. The wildcard (aka joker) pattern we've used matches everything. We could be more specific by using metavariables:
Theorem match_ex5 : ∀ (P Q : Prop), P → Q → P.
Proof.
  intros P Q HP HQ.
  match goal with
  | [ H : ?X ⊢ ?X ] ⇒ idtac H; apply H
  end.
Qed.
Note that this time, the only hypothesis printed by idtac is HP. The HQ hypothesis is ruled out, because it does not have the form ?X ⊢ ?X.
The occurrences of ?X in that pattern are as metavariables that stand for the same term appearing both as the type of hypothesis H and as the conclusion of the goal.
(The syntax of match goal requires that ? to distinguish metavariables from other identifiers that might be in scope. However, the ? is used only in the pattern. On the right-hand side of the branch, it's actually required to drop the ?.)
Now we have seen yet another difference between match goal and regular match against terms: match goal allows a metavariable to be used multiple times in a pattern, each time standing for the same term. The regular match does not allow that:
Fail Definition dup_first_two_elts (lst : list nat) :=
  match lst with
  | x :: x :: _ ⇒ true
  | _ ⇒ false
  end.
The technical term for this is linearity: regular match requires pattern variables to be linear, meaning that they are used only once. Tactic match goal permits non-linear metavariables, meaning that they can be used multiple times in a pattern and must bind the same term each time.
Now that we've learned a bit about match goal, let's return to the proof pattern of some simple inductions:
Theorem app_nil_r' : ∀ (X : Type) (lst : list X),
    lst ++ [] = lst.
Proof.
  intros X lst. induction lst as [ | h t IHt].
  - reflexivity.
  - simpl. rewrite IHt. reflexivity.
Qed.
With match goal, we can automate that proof pattern:
Ltac simple_induction t :=
  induction t; simpl;
  try match goal with
      | [H : _ = _ ⊢ _] ⇒ rewrite H
      end;
  reflexivity.
Theorem app_nil_r'' : ∀ (X : Type) (lst : list X),
    lst ++ [] = lst.
Proof.
  intros X lst. simple_induction lst.
Qed.
That works great! Here are two other proofs that follow the same pattern.
Theorem add_assoc'' : ∀ n m p : nat,
    n + (m + p) = (n + m) + p.
Proof.
  intros n m p. induction n.
  - reflexivity.
  - simpl. rewrite IHn. reflexivity.
Qed.
Theorem add_assoc''' : ∀ n m p : nat,
    n + (m + p) = (n + m) + p.
Proof.
  intros n m p. simple_induction n.
Qed.
Theorem plus_n_Sm : ∀ n m : nat,
    S (n + m) = n + (S m).
Proof.
  intros n m. induction n.
  - reflexivity.
  - simpl. rewrite IHn. reflexivity.
Qed.
Theorem plus_n_Sm' : ∀ n m : nat,
    S (n + m) = n + (S m).
Proof.
  intros n m. simple_induction n.
Qed.

Using match goal to Prove Tautologies

The Ltac source code of intuition can be found in the GitHub repo for Coq in theories/Init/Tauto.v. At heart, it is a big loop that runs match goal to find propositions it can apply and destruct.
Let's build our own simplified "knock off" of intuition. Here's a start on implication:
Ltac imp_intuition :=
  repeat match goal with
         | [ H : ?P ⊢ ?P ] ⇒ apply H
         | [ ⊢ ∀ _, _ ] ⇒ intro
         | [ H1 : ?P → ?Q, H2 : ?P ⊢ _ ] ⇒ apply H1 in H2
         end.
That tactic repeatedly matches against the goal until the match fails to make progress. At each step, the match goal does one of three things:
  • Finds that the conclusion is already in the hypotheses, in which case the goal is finished.
  • Finds that the conclusion is a quantification, in which case it is introduced. Since implication P → Q is itself a quantification ∀ (_ : P), Q, this case handles introduction of implications, too.
  • Finds that two formulas of the form ?P → ?Q and ?P are in the hypotheses. This is the first time we've seen an example of matching against two hypotheses simultaneously. Note that the metavariable ?P is once more non-linear: the same formula must occur in two different hypotheses. In this case, the tactic uses forward reasoning to change hypothesis H2 into ?Q.
Already we can prove many theorems with this tactic:
Example imp1 : ∀ (P : Prop), P → P.
Proof. imp_intuition. Qed.
Example imp2 : ∀ (P Q : Prop), P → (P → Q) → Q.
Proof. imp_intuition. Qed.
Example imp3 : ∀ (P Q R : Prop), (P → Q → R) → (Q → P → R).
Proof. imp_intuition. Qed.
Suppose we were to add a new logical connective: nor, the "not or" connective.
Inductive nor (P Q : Prop) :=
| stroke : ¬P → ¬Q → nor P Q.
Classically, nor P Q would be equivalent to ~(P ∨ Q). But constructively, only one direction of that is provable.
Theorem nor_not_or : ∀ (P Q : Prop),
    nor P Q → ¬ (P ∨ Q).
Proof.
  intros. destruct H. unfold not. intros. destruct H. auto. auto.
Qed.
Some other usual theorems about nor are still provable, though.
Theorem nor_comm : ∀ (P Q : Prop),
    nor P Q ↔ nor Q P.
Proof.
  intros P Q. split.
  - intros H. destruct H. apply stroke; assumption.
  - intros H. destruct H. apply stroke; assumption.
Qed.
Theorem nor_not : ∀ (P : Prop),
    nor P P ↔ ¬P.
Proof.
  intros P. split.
  - intros H. destruct H. assumption.
  - intros H. apply stroke; assumption.
Qed.

Exercise: 4 stars, advanced (nor_intuition)

Create your own tactic nor_intuition. It should be able to prove the three theorems above -- nor_not_and, nor_comm, and nor_not -- fully automatically. You may not use intuition or any other automated solvers in your solution.
Begin by copying the code from imp_intuition. You will then need to expand it to handle conjunctions, negations, bi-implications, and nor.
(* Ltac nor_intuition := ... *)
Each of the three theorems below, and many others involving these logical connectives, should be provable with just Proof. nor_intuition. Qed.
Theorem nor_comm' : ∀ (P Q : Prop),
    nor P Q ↔ nor Q P.
Proof. (* FILL IN HERE *) Admitted.
Theorem nor_not' : ∀ (P : Prop),
    nor P P ↔ ¬P.
Proof. (* FILL IN HERE *) Admitted.
Theorem nor_not_and' : ∀ (P Q : Prop),
    nor P Q → ¬ (P ∧ Q).
Proof. (* FILL IN HERE *) Admitted.
(* Do not modify the following line: *)
Definition manual_grade_for_nor_intuition : option (nat×string) := None.
☐

Review

We've learned a lot of new features and tactics in this chapter:
  • try, ;, repeat
  • assumption, contradiction, subst, constructor
  • lia, congruence, intuition
  • auto, eauto, eapply
  • Ltac functions and match goal
(* 2024-12-26 02:02 *)