Library prosa.util.tactics
Lemmas & tactics adopted (with permission) from V. Vafeiadis' Vbase.v.
Lemma neqP: ∀ (T: eqType) (x y: T), reflect (x ≠ y) (x != y).
Ltac ins := simpl in *; try done; intros.
(* ************************************************************************** *)
(* ************************************************************************** *)
Exploit an assumption (adapted from CompCert).
Lemma modusponens : ∀ (P Q : Prop), P → (P → Q) → Q.
Ltac exploit x :=
refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _ _) _)
|| refine (modusponens _ _ (x _ _ _) _)
|| refine (modusponens _ _ (x _ _) _)
|| refine (modusponens _ _ (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
| O ⇒ idtac
| (S ?m) ⇒ feed H ; [| feed_n m H]
end.
We introduce tactics rt_auto and rt_eauto as a shorthand for
(e)auto with basic_rt_facts to facilitate automation. Here, we
use scope basic_rt_facts that contains a collection of basic
real-time theory lemmas. Note: constant 4 was chosen because most of the basic rt facts
have the structure A1 → A2 → ... B, where Ai is a hypothesis
usually present in the context, which gives the depth of the
search which is equal to two. Two additional levels of depth (4)
was added to support rare exceptions to this rule. In particular,
depth 4 is needed for automatic periodic->RBF arrival model
conversion. At the same time, the constant should not be too large
to avoid slowdowns in case of an unsuccessful application of
automation.
Ltac rt_auto := auto 4 with basic_rt_facts.
Ltac rt_eauto := eauto 4 with basic_rt_facts.
(* ************************************************************************** *)
Ltac rt_eauto := eauto 4 with basic_rt_facts.
(* ************************************************************************** *)
(* ************************************************************************** *)
Ltac move_neq_down H :=
exfalso;
(move: H; rewrite ltnNge; move ⇒ /negP H; apply: H; clear H)
|| (move: H; rewrite leqNgt; move ⇒ /negP H; apply: H; clear H).
Ltac move_neq_up H :=
(rewrite ltnNge; apply/negP; intros H)
|| (rewrite leqNgt; apply/negP; intros H).
Ltac move_neq_down H :=
exfalso;
(move: H; rewrite ltnNge; move ⇒ /negP H; apply: H; clear H)
|| (move: H; rewrite leqNgt; move ⇒ /negP H; apply: H; clear H).
Ltac move_neq_up H :=
(rewrite ltnNge; apply/negP; intros H)
|| (rewrite leqNgt; apply/negP; intros H).
The following tactic converts inequality t1 ≤ t2 into a constant
k such that t2 = t1 + k and substitutes all the occurrences of
t2.
Ltac interval_to_duration t1 t2 k :=
match goal with
| [ H: (t1 ≤ t2) = true |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto | ];
destruct EX as [k EQ]; subst t2; clear H
)
| [ H: (t1 < t2) = true |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto using ltnW | ];
destruct EX as [k EQ]; subst t2; clear H
)
| [ H: is_true (t1 ≤ t2) |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto | ];
destruct EX as [k EQ]; subst t2; clear H
)
| [ H: is_true (t1 < t2) |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto using ltnW | ];
destruct EX as [k EQ]; subst t2; clear H
)
end.
match goal with
| [ H: (t1 ≤ t2) = true |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto | ];
destruct EX as [k EQ]; subst t2; clear H
)
| [ H: (t1 < t2) = true |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto using ltnW | ];
destruct EX as [k EQ]; subst t2; clear H
)
| [ H: is_true (t1 ≤ t2) |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto | ];
destruct EX as [k EQ]; subst t2; clear H
)
| [ H: is_true (t1 < t2) |- _ ] ⇒
ltac:(
assert (EX : ∃ (k: nat), t2 = t1 + k);
[∃ (t2 - t1); rewrite subnKC; auto using ltnW | ];
destruct EX as [k EQ]; subst t2; clear H
)
end.