Library rt.util.tactics


(* *********************************************************************)
(*                                                                     *)
(*     Basic lemmas & tactics (based on Viktor Vafeiadis' Vbase.v)     *)
(*                                                                     *)
(* *********************************************************************)

This file collects a number of basic lemmas and tactics for better proof automation, structuring large proofs, or rewriting. Most of the rewriting support is ported from ssreflect.
Symbols starting with [vlib__] are internal.


Open Scope bool_scope.
Open Scope list_scope.

Set Implicit Arguments.

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

Very basic automation

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

Set up for basic simplification


Adaptation of the ss-reflect "[done]" tactic.

Ltac vlib__basic_done :=
  solve [trivial with vlib | apply sym_equal; trivial | discriminate | contradiction].

Ltac done := trivial with vlib; hnf; intros;
  solve [try vlib__basic_done; split;
         try vlib__basic_done; split;
         try vlib__basic_done; split;
         try vlib__basic_done; split;
         try vlib__basic_done; split; vlib__basic_done
    | match goal with H : ¬ _ |- _solve [case H; trivial] end].

A variant of the ssr "done" tactic that performs "eassumption".

Ltac edone := try eassumption; trivial; hnf; intros;
  solve [try eassumption; try vlib__basic_done; split;
         try eassumption; try vlib__basic_done; split;
         try eassumption; try vlib__basic_done; split;
         try eassumption; try vlib__basic_done; split;
         try eassumption; try vlib__basic_done; split;
         try eassumption; vlib__basic_done
    | match goal with H : ¬ _ |- _solve [case H; trivial] end].

Tactic Notation "by" tactic(tac) := (tac; done).
Tactic Notation "eby" tactic(tac) := (tac; edone).

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

Equality types

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

Lemma vlib__internal_eqP :
   (T: eqType) (x y : T), reflect (x = y) (x == y).

Lemma neqP : (T: eqType) (x y: T), reflect (x y) (x != y).

Lemma beq_refl : (T : eqType) (x : T), x == x.

Lemma beq_sym : (T : eqType) (x y : T), (x == y) = (y == x).

Hint Resolve beq_refl : vlib.
Hint Rewrite beq_refl : vlib_trivial.

Notation eqxx := beq_refl.

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

Basic simplification tactics

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

Lemma vlib__negb_rewrite : b, negb b b = false.

Lemma vlib__andb_split : b1 b2, b1 && b2 b1 b2.

Lemma vlib__nandb_split : b1 b2, b1 && b2 = false b1 = false b2 = false.

Lemma vlib__orb_split : b1 b2, b1 || b2 b1 b2.

Lemma vlib__norb_split : b1 b2, b1 || b2 = false b1 = false b2 = false.

Lemma vlib__eqb_split : b1 b2 : bool, (b1 b2) (b2 b1) b1 = b2.

Lemma vlib__beq_rewrite : (T : eqType) (x1 x2 : T), x1 == x2 x1 = x2.

Lemma vlib__leq_split : x1 x2 x3, x1 x2 x2 x3 x1 x2 x3.

Lemma vlib__ltn_split1 : x1 x2 x3, x1 x2 x2 < x3 x1 x2 < x3.

Lemma vlib__ltn_split2 : x1 x2 x3, x1 < x2 x2 x3 x1 < x2 x3.

Set up for basic simplification: database of reflection lemmas


Hint Resolve andP orP nandP norP negP vlib__internal_eqP neqP : vlib_refl.

(* Add x <= y <= z splitting to the core hint database. *)
Hint Immediate vlib__leq_split vlib__ltn_split1 vlib__ltn_split2.

Ltac vlib__complaining_inj f H :=
  let X := fresh in
  (match goal with | [|- ?P ] ⇒ set (X := P) end);
  injection H;
  (*  (lazymatch goal with | [ |- f _ = f _ -> _] => fail | _ => idtac end);  
      (* Previous statement no longer necessary in 8.4 *) *)

  clear H; intros; subst X;
  try subst.

Ltac vlib__clarify1 :=
  try subst;
  repeat match goal with
  | [H: is_true (andb _ _) |- _] ⇒
      let H' := fresh H in case (vlib__andb_split H); clear H; intros H' H
  | [H: is_true (negb ?x) |- _] ⇒ rewrite (vlib__negb_rewrite H) in ×
  | [H: is_true ?x |- _] ⇒ rewrite H in ×
  | [H: ?x = true |- _] ⇒ rewrite H in ×
  | [H: ?x = false |- _] ⇒ rewrite H in ×
  | [H: is_true (_ == _) |- _] ⇒ generalize (vlib__beq_rewrite H); clear H; intro H
  | [H: ?f _ = ?f _ |- _] ⇒ vlib__complaining_inj f H
  | [H: ?f _ _ = ?f _ _ |- _] ⇒ vlib__complaining_inj f H
  | [H: ?f _ _ _ = ?f _ _ _ |- _] ⇒ vlib__complaining_inj f H
  | [H: ?f _ _ _ _ = ?f _ _ _ _ |- _] ⇒ vlib__complaining_inj f H
  | [H: ?f _ _ _ _ _ = ?f _ _ _ _ _ |- _] ⇒ vlib__complaining_inj f H
  | [H: ?f _ _ _ _ _ _ = ?f _ _ _ _ _ _ |- _] ⇒ vlib__complaining_inj f H
  | [H: ?f _ _ _ _ _ _ _ = ?f _ _ _ _ _ _ _ |- _] ⇒ vlib__complaining_inj f H
  end; try done.

Perform injections & discriminations on all hypotheses

Ltac clarify :=
  vlib__clarify1;
  repeat match goal with
    | H1: ?x = Some _, H2: ?x = None |- _rewrite H2 in H1; discriminate
    | H1: ?x = Some _, H2: ?x = Some _ |- _rewrite H2 in H1; vlib__clarify1
  end; (* autorewrite with vlib_trivial; *) try done.

Ltac inv x := inversion x; clarify.
Ltac simpls := simpl in *; try done.
Ltac ins := simpl in *; try done; intros.

Tactic Notation "case_eq" constr(x) := case_eq (x).

Tactic Notation "case_eq" constr(x) "as" simple_intropattern(H) :=
  destruct x as [] eqn:H; try done.

Ltac vlib__clarsimp1 :=
  clarify; (autorewrite with vlib_trivial vlib in × );
  (autorewrite with vlib_trivial in × ); try done;
  clarify; auto 1 with vlib.

Ltac clarsimp := intros; simpl in *; vlib__clarsimp1.

Ltac autos := clarsimp; auto with vlib.

(* ************************************************************************** *)
Destruct but give useful names
(* ************************************************************************** *)

Destruct, but no case split
Ltac desc :=
  repeat match goal with
    | H: is_true (_ == _) |- _generalize (vlib__beq_rewrite H); clear H; intro H
    | H : x, ?p |- _
      let x' := fresh x in destruct H as [x' H]
    | H : ?p ?q |- _
      let x' := H in
      let y' := fresh H in
        destruct H as [x' y']
    | H : is_true (_ && _) |- _
          let H' := fresh H in case (vlib__andb_split H); clear H; intros H H'
    | H : (_ || _) = false |- _
          let H' := fresh H in case (vlib__norb_split H); clear H; intros H H'
    | H : ?x = ?x |- _clear H
  end.

Ltac des :=
  repeat match goal with
    | H: is_true (_ == _) |- _generalize (vlib__beq_rewrite H); clear H; intro H
    | H : x, ?p |- _
      let x' := fresh x in destruct H as [x' H]
    | H : exists2 x, ?p & ?q |- _
      let x' := fresh x in destruct H as [x' H1 H2]
    | H : ?p ?q |- _
      let x' := H in
      let y' := fresh H in
        destruct H as [x' y']
    | H : is_true (_ && _) |- _
        let H' := fresh H in case (vlib__andb_split H); clear H; intros H H'
    | H : (_ || _) = false |- _
        let H' := fresh H in case (vlib__norb_split H); clear H; intros H H'
    | H : ?x = ?x |- _clear H
    | H : ?p ?q |- _
      let x' := H in
      let y' := fresh H in
        destruct H as [x' y']
    | H : ?p ?q |- _
      let x' := H in
      let y' := H in
      destruct H as [x' | y']
    | H : is_true (_ || _) |- _case (vlib__orb_split H); clear H; intro H
    | H : (_ && _) = false |- _case (vlib__nandb_split H); clear H; intro H
  end.

Ltac des_if_asm :=
  clarify;
  repeat
    match goal with
      | H: context[ match ?x with __ end ] |- _
        match (type of x) with
          | { _ } + { _ }destruct x; clarify
          | bool
            let Heq := fresh "Heq" in
            let P := fresh in
            evar(P: Prop);
            assert (Heq: reflect P x) by (subst P; trivial with vlib_refl);
            subst P; destruct Heq as [Heq|Heq]
          | _let Heq := fresh "Heq" in destruct x as [] eqn:Heq; clarify
        end
    end.

Ltac des_if_goal :=
  clarify;
  repeat
    match goal with
      | |- context[match ?x with __ end] ⇒
        match (type of x) with
          | { _ } + { _ }destruct x; clarify
          | bool
            let Heq := fresh "Heq" in
            let P := fresh in
            evar(P: Prop);
            assert (Heq: reflect P x) by (subst P; trivial with vlib_refl);
            subst P; destruct Heq as [Heq|Heq]
          | _let Heq := fresh "Heq" in destruct x as [] eqn:Heq; clarify
        end
    end.

Ltac des_if :=
  clarify;
  repeat
    match goal with
      | |- context[match ?x with __ end] ⇒
        match (type of x) with
          | { _ } + { _ }destruct x; clarify
          | bool
            let Heq := fresh "Heq" in
            let P := fresh in
            evar(P: Prop);
            assert (Heq: reflect P x) by (subst P; trivial with vlib_refl);
            subst P; destruct Heq as [Heq|Heq]
          | _let Heq := fresh "Heq" in destruct x as [] eqn:Heq; clarify
        end
      | H: context[ match ?x with __ end ] |- _
        match (type of x) with
          | { _ } + { _ }destruct x; clarify
          | bool
            let Heq := fresh "Heq" in
            let P := fresh in
            evar(P: Prop);
            assert (Heq: reflect P x) by (subst P; trivial with vlib_refl);
            subst P; destruct Heq as [Heq|Heq]
          | _let Heq := fresh "Heq" in destruct x as [] eqn:Heq; clarify
        end
    end.

Ltac des_eqrefl :=
  match goal with
    | H: context[match ?X with |true_ | false_ end Logic.eq_refl] |- _
    let EQ := fresh "EQ" in
    let id' := fresh "x" in
    revert H;
    generalize (Logic.eq_refl X);
    try (generalize X at 1 3); try (generalize X at 2 3);
    intros id' EQ; destruct id'; intros H
    | |- context[match ?X with |true_ | false_ end Logic.eq_refl] ⇒
    let EQ := fresh "EQ" in
    let id' := fresh "x" in
    generalize (Logic.eq_refl X);
    try (generalize X at 1 3); try (generalize X at 2 3);
    intros id' EQ; destruct id'
  end.

Ltac desf_asm := clarify; des; des_if_asm.

Ltac desf := clarify; des; des_if.

Ltac clarassoc := clarsimp; autorewrite with vlib_trivial vlib vlibA in *; try done.

Ltac vlib__hacksimp1 :=
   clarsimp;
   match goal with
     | H: _ |- _solve [rewrite H; clear H; clarsimp
                         |rewrite <- H; clear H; clarsimp]
     | _solve [f_equal; clarsimp]
   end.

Ltac hacksimp :=
   clarsimp;
   try match goal with
   | H: _ |- _solve [rewrite H; clear H; clarsimp
                              |rewrite <- H; clear H; clarsimp]
   | |- context[match ?p with __ end] ⇒ solve [destruct p; vlib__hacksimp1]
   | _solve [f_equal; clarsimp]
   end.

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

Delineating cases in proofs

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

Named case tactics (taken from Libtactics)

Tactic Notation "assert_eq" ident(x) constr(v) :=
  let H := fresh in
  assert (x = v) as H by reflexivity;
  clear H.

Tactic Notation "Case_aux" ident(x) constr(name) :=
  first [
    set (x := name); move x at top
  | assert_eq x name
  | fail 1 "because we are working on a different case." ].

Ltac Case name := Case_aux case name.
Ltac SCase name := Case_aux subcase name.
Ltac SSCase name := Case_aux subsubcase name.
Ltac SSSCase name := Case_aux subsubsubcase name.
Ltac SSSSCase name := Case_aux subsubsubsubcase name.

Lightweight case tactics (without names)

Tactic Notation "--" tactic(c) :=
  first [
    assert (WithinCaseM := True); move WithinCaseM at top
  | fail 1 "because we are working on a different case." ]; c.

Tactic Notation "++" tactic(c) :=
  first [
    assert (WithinCaseP := True); move WithinCaseP at top
  | fail 1 "because we are working on a different case." ]; c.

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

Exploiting a hypothesis

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

Exploit an assumption (adapted from CompCert).

Ltac exploit x :=
    refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _ _) _)
 || refine ((fun x yy x) (x _ _ _) _)
 || refine ((fun x yy x) (x _ _) _)
 || refine ((fun x yy x) (x _) _).

(* This tactic feeds the precondition of an implication in order to derive the conclusion
   (taken from http://comments.gmane.org/gmane.science.mathematics.logic.coq.club/7013).

   Usage: feed H.

   H: P -> Q  ==becomes==>  H: P
                            ____
                            Q

   After completing this proof, Q becomes a hypothesis in the context. *)

Ltac feed H :=
  match type of H with
  | ?foo _
    let FOO := fresh in
    assert foo as FOO; [|specialize (H FOO); clear FOO]
  end.

(* Generalization of feed for multiple hypotheses.
   feed_n is useful for accessing conclusions of long implications.

   Usage: feed_n 3 H.
     H: P1 -> P2 -> P3 -> Q.

   We'll be asked to prove P1, P2 and P3, so that Q can be inferred. *)

Ltac feed_n n H := match constr:(n) with
  | Oidtac
  | (S ?m) ⇒ feed H ; [| feed_n m H]
  end.

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

New tactics for ssreflect

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

(* Tactic for simplifying a sum with constant term.
   Usage: simpl_sum_const in H.
   H: \sum_(2 <= x < 4) 5 > 0  ==becomes==>  H: 5 * (4 - 2) > 0 *)

Ltac simpl_sum_const :=
  rewrite ?big_const_nat ?big_const_ord ?big_const_seq iter_addn ?muln1 ?mul1n ?mul0n
          ?muln0 ?addn0 ?add0n.

(* Tactic for splitting all conjunctions in a hypothesis.
   Usage: split_conj H.
   H: A /\ (B /\ C)  ==becomes==>  H1: A
                                   H2: B
                                   H3: C *)

Ltac split_conj X :=
  hnf in X;
  repeat match goal with
    | H : ?p ?q |- _
      let x' := H in
      let y' := fresh H in
        destruct H as [x' y']
  end.