110 lines
3.6 KiB
Coq
110 lines
3.6 KiB
Coq
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(** A library of additional tactics. *)
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Require Export String.
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Open Scope string_scope.
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(* *********************************************************************** *)
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(** * Extensions of the standard library *)
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(** "[remember c as x in |-]" replaces the term [c] by the identifier
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[x] in the conclusion of the current goal and introduces the
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hypothesis [x=c] into the context. This tactic differs from a
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similar one in the standard library in that the replacmement is
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made only in the conclusion of the goal; the context is left
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unchanged. *)
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Tactic Notation "remember" constr(c) "as" ident(x) "in" "|-" :=
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let x := fresh x in
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let H := fresh "Heq" x in
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(set (x := c); assert (H : x = c) by reflexivity; clearbody x).
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(** "[unsimpl E]" replaces all occurence of [X] by [E], where [X] is
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the result that tactic [simpl] would give when used to evaluate
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[E]. *)
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Tactic Notation "unsimpl" constr(E) :=
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let F := (eval simpl in E) in change F with E.
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(** The following tactic calls the [apply] tactic with the first
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hypothesis that succeeds, "first" meaning the hypothesis that
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comes earlist in the context (i.e., higher up in the list). *)
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Ltac apply_first_hyp :=
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match reverse goal with
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| H : _ |- _ => apply H
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end.
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(* *********************************************************************** *)
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(** * Variations on [auto] *)
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(** The [auto*] and [eauto*] tactics are intended to be "stronger"
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versions of the [auto] and [eauto] tactics. Similar to [auto] and
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[eauto], they each take an optional "depth" argument. Note that
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if we declare these tactics using a single string, e.g., "auto*",
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then the resulting tactics are unusable since they fail to
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parse. *)
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Tactic Notation "auto" "*" :=
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try solve [ congruence | auto | intuition auto ].
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Tactic Notation "auto" "*" integer(n) :=
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try solve [ congruence | auto n | intuition (auto n) ].
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Tactic Notation "eauto" "*" :=
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try solve [ congruence | eauto | intuition eauto ].
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Tactic Notation "eauto" "*" integer(n) :=
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try solve [ congruence | eauto n | intuition (eauto n) ].
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(* *********************************************************************** *)
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(** * Delineating cases in proofs *)
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(** This section was taken from the POPLmark Wiki
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( http://alliance.seas.upenn.edu/~plclub/cgi-bin/poplmark/ ). *)
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(** ** Tactic definitions *)
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Ltac move_to_top x :=
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match reverse goal with
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| H : _ |- _ => try move x after H
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end.
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Tactic Notation "assert_eq" ident(x) constr(v) :=
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let H := fresh in
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assert (x = v) as H by reflexivity;
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clear H.
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Tactic Notation "Case_aux" ident(x) constr(name) :=
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first [
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set (x := name); move_to_top x
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| assert_eq x name
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| fail 1 "because we are working on a different case." ].
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Ltac Case name := Case_aux case name.
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Ltac SCase name := Case_aux subcase name.
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Ltac SSCase name := Case_aux subsubcase name.
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(** ** Example
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One mode of use for the above tactics is to wrap Coq's [induction]
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tactic such that automatically inserts "case" markers into each
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branch of the proof. For example:
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<<
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Tactic Notation "induction" "nat" ident(n) :=
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induction n; [ Case "O" | Case "S" ].
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Tactic Notation "sub" "induction" "nat" ident(n) :=
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induction n; [ SCase "O" | SCase "S" ].
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Tactic Notation "sub" "sub" "induction" "nat" ident(n) :=
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induction n; [ SSCase "O" | SSCase "S" ].
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>>
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If you use such customized versions of the induction tactics, then
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the [Case] tactic will verify that you are working on the case
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that you think you are. You may also use the [Case] tactic with
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the standard version of [induction], in which case no verification
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is done. *)
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