frap/IntroToProofScripting_template.v
2018-02-28 09:01:07 -05:00

553 lines
14 KiB
Coq

Require Import Frap.
Set Implicit Arguments.
(** * Ltac Programming Basics *)
Theorem hmm : forall (a b c : bool),
if a
then if b
then True
else True
else if c
then True
else True.
Proof.
Admitted.
Theorem hmm2 : forall (a b : bool),
(if a then 42 else 42) = (if b then 42 else 42).
Proof.
Admitted.
(** * Automating the two-thread locked-increment example from TransitionSystems *)
(* Let's experience the process of gradually automating the proof we finished
* the last lecture with. Here's the system-definition code, stripped of
* comments. *)
Inductive increment_program :=
| Lock
| Read
| Write (local : nat)
| Unlock
| Done.
Record inc_state := {
Locked : bool;
Global : nat
}.
Record threaded_state shared private := {
Shared : shared;
Private : private
}.
Definition increment_state := threaded_state inc_state increment_program.
Inductive increment_init : increment_state -> Prop :=
| IncInit :
increment_init {| Shared := {| Locked := false; Global := O |};
Private := Lock |}.
Inductive increment_step : increment_state -> increment_state -> Prop :=
| IncLock : forall g,
increment_step {| Shared := {| Locked := false; Global := g |};
Private := Lock |}
{| Shared := {| Locked := true; Global := g |};
Private := Read |}
| IncRead : forall l g,
increment_step {| Shared := {| Locked := l; Global := g |};
Private := Read |}
{| Shared := {| Locked := l; Global := g |};
Private := Write g |}
| IncWrite : forall l g v,
increment_step {| Shared := {| Locked := l; Global := g |};
Private := Write v |}
{| Shared := {| Locked := l; Global := S v |};
Private := Unlock |}
| IncUnlock : forall l g,
increment_step {| Shared := {| Locked := l; Global := g |};
Private := Unlock |}
{| Shared := {| Locked := false; Global := g |};
Private := Done |}.
Definition increment_sys := {|
Initial := increment_init;
Step := increment_step
|}.
Inductive parallel1 shared private1 private2
(init1 : threaded_state shared private1 -> Prop)
(init2 : threaded_state shared private2 -> Prop)
: threaded_state shared (private1 * private2) -> Prop :=
| Pinit : forall sh pr1 pr2,
init1 {| Shared := sh; Private := pr1 |}
-> init2 {| Shared := sh; Private := pr2 |}
-> parallel1 init1 init2 {| Shared := sh; Private := (pr1, pr2) |}.
Inductive parallel2 shared private1 private2
(step1 : threaded_state shared private1 -> threaded_state shared private1 -> Prop)
(step2 : threaded_state shared private2 -> threaded_state shared private2 -> Prop)
: threaded_state shared (private1 * private2)
-> threaded_state shared (private1 * private2) -> Prop :=
| Pstep1 : forall sh pr1 pr2 sh' pr1',
step1 {| Shared := sh; Private := pr1 |} {| Shared := sh'; Private := pr1' |}
-> parallel2 step1 step2 {| Shared := sh; Private := (pr1, pr2) |}
{| Shared := sh'; Private := (pr1', pr2) |}
| Pstep2 : forall sh pr1 pr2 sh' pr2',
step2 {| Shared := sh; Private := pr2 |} {| Shared := sh'; Private := pr2' |}
-> parallel2 step1 step2 {| Shared := sh; Private := (pr1, pr2) |}
{| Shared := sh'; Private := (pr1, pr2') |}.
Definition parallel shared private1 private2
(sys1 : trsys (threaded_state shared private1))
(sys2 : trsys (threaded_state shared private2)) := {|
Initial := parallel1 sys1.(Initial) sys2.(Initial);
Step := parallel2 sys1.(Step) sys2.(Step)
|}.
Definition increment2_sys := parallel increment_sys increment_sys.
Definition contribution_from (pr : increment_program) : nat :=
match pr with
| Unlock => 1
| Done => 1
| _ => 0
end.
Definition has_lock (pr : increment_program) : bool :=
match pr with
| Read => true
| Write _ => true
| Unlock => true
| _ => false
end.
Definition shared_from_private (pr1 pr2 : increment_program) :=
{| Locked := has_lock pr1 || has_lock pr2;
Global := contribution_from pr1 + contribution_from pr2 |}.
Definition instruction_ok (self other : increment_program) :=
match self with
| Lock => True
| Read => has_lock other = false
| Write n => has_lock other = false /\ n = contribution_from other
| Unlock => has_lock other = false
| Done => True
end.
Inductive increment2_invariant :
threaded_state inc_state (increment_program * increment_program) -> Prop :=
| Inc2Inv : forall pr1 pr2,
instruction_ok pr1 pr2
-> instruction_ok pr2 pr1
-> increment2_invariant {| Shared := shared_from_private pr1 pr2; Private := (pr1, pr2) |}.
Lemma Inc2Inv' : forall sh pr1 pr2,
sh = shared_from_private pr1 pr2
-> instruction_ok pr1 pr2
-> instruction_ok pr2 pr1
-> increment2_invariant {| Shared := sh; Private := (pr1, pr2) |}.
Proof.
simplify.
rewrite H.
apply Inc2Inv; assumption.
Qed.
(* OK, HERE is where prove the main theorem. *)
Theorem increment2_invariant_ok : invariantFor increment2_sys increment2_invariant.
Proof.
Admitted.
(** * Implementing some of [propositional] ourselves *)
Print True.
Print False.
Locate "/\".
Print and.
Locate "\/".
Print or.
(* Implication ([->]) is built into Coq, so nothing to look up there. *)
Section propositional.
Variables P Q R : Prop.
Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
Proof.
Admitted.
End propositional.
(* Backrtracking example #1 *)
Theorem m1 : True.
Proof.
match goal with
| [ |- _ ] => intro
| [ |- True ] => constructor
end.
Qed.
(* Backtracking example #2 *)
Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
Proof.
intros; match goal with
| [ H : _ |- _ ] => idtac H
end.
Admitted.
(* Let's try some more ambitious reasoning, with quantifiers. We'll be
* instantiating quantified facts heuristically. If we're not careful, we get
* in a loop repeating the same instantiation forever. *)
(* Spec: ensure that [P] doesn't follow trivially from hypotheses. *)
Ltac notHyp P := idtac.
(* Spec: add [pf] as hypothesis only if it doesn't already follow trivially. *)
Ltac extend pf := idtac.
(* Spec: add all simple consequences of known facts, including
* [forall]-quantified. *)
Ltac completer := idtac.
Section firstorder.
Variable A : Set.
Variables P Q R S : A -> Prop.
Hypothesis H1 : forall x, P x -> Q x /\ R x.
Hypothesis H2 : forall x, R x -> S x.
Theorem fo : forall (y x : A), P x -> S x.
Proof.
Admitted.
End firstorder.
(** * Functional Programming in Ltac *)
(* Spec: return length of list. *)
Ltac length ls := constr:(0).
Goal False.
let n := length (1 :: 2 :: 3 :: nil) in
pose n.
Abort.
(* Spec: map Ltac function over list. *)
Ltac map f ls := constr:(0).
Goal False.
(*let ls := map (nat * nat)%type ltac:(fun x => constr:((x, x))) (1 :: 2 :: 3 :: nil) in
pose ls.*)
Abort.
(* Now let's revisit [length] and see how we might implement "printf debugging"
* for it. *)
(** * Recursive Proof Search *)
(* Let's work on a tactic to try all possible instantiations of quantified
* hypotheses, attempting to find out where the goal becomes obvious. *)
Ltac inster n := idtac.
Section test_inster.
Variable A : Set.
Variables P Q : A -> Prop.
Variable f : A -> A.
Variable g : A -> A -> A.
Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
Theorem test_inster : forall x, P (g x x) -> Q (f x).
Proof.
inster 2.
Admitted.
Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
Proof.
inster 3.
Admitted.
End test_inster.
(** ** A fancier example of proof search (probably skipped on first
reading/run-through) *)
Definition imp (P1 P2 : Prop) := P1 -> P2.
Infix "-->" := imp (no associativity, at level 95).
Ltac imp := unfold imp; firstorder.
(** These lemmas about [imp] will be useful in the tactic that we will write. *)
Theorem and_True_prem : forall P Q,
(P /\ True --> Q)
-> (P --> Q).
Proof.
imp.
Qed.
Theorem and_True_conc : forall P Q,
(P --> Q /\ True)
-> (P --> Q).
Proof.
imp.
Qed.
Theorem pick_prem1 : forall P Q R S,
(P /\ (Q /\ R) --> S)
-> ((P /\ Q) /\ R --> S).
Proof.
imp.
Qed.
Theorem pick_prem2 : forall P Q R S,
(Q /\ (P /\ R) --> S)
-> ((P /\ Q) /\ R --> S).
Proof.
imp.
Qed.
Theorem comm_prem : forall P Q R,
(P /\ Q --> R)
-> (Q /\ P --> R).
Proof.
imp.
Qed.
Theorem pick_conc1 : forall P Q R S,
(S --> P /\ (Q /\ R))
-> (S --> (P /\ Q) /\ R).
Proof.
imp.
Qed.
Theorem pick_conc2 : forall P Q R S,
(S --> Q /\ (P /\ R))
-> (S --> (P /\ Q) /\ R).
Proof.
imp.
Qed.
Theorem comm_conc : forall P Q R,
(R --> P /\ Q)
-> (R --> Q /\ P).
Proof.
imp.
Qed.
Ltac search_prem tac :=
let rec search P :=
tac
|| (apply and_True_prem; tac)
|| match P with
| ?P1 /\ ?P2 =>
(apply pick_prem1; search P1)
|| (apply pick_prem2; search P2)
end
in match goal with
| [ |- ?P /\ _ --> _ ] => search P
| [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
| [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
end.
Ltac search_conc tac :=
let rec search P :=
tac
|| (apply and_True_conc; tac)
|| match P with
| ?P1 /\ ?P2 =>
(apply pick_conc1; search P1)
|| (apply pick_conc2; search P2)
end
in match goal with
| [ |- _ --> ?P /\ _ ] => search P
| [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
| [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
end.
Theorem False_prem : forall P Q,
False /\ P --> Q.
Proof.
imp.
Qed.
Theorem True_conc : forall P Q : Prop,
(P --> Q)
-> (P --> True /\ Q).
Proof.
imp.
Qed.
Theorem Match : forall P Q R : Prop,
(Q --> R)
-> (P /\ Q --> P /\ R).
Proof.
imp.
Qed.
Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
(forall x, P x /\ Q --> R)
-> (ex P /\ Q --> R).
Proof.
imp.
Qed.
Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
(Q --> P x /\ R)
-> (Q --> ex P /\ R).
Proof.
imp.
Qed.
Theorem imp_True : forall P,
P --> True.
Proof.
imp.
Qed.
Ltac matcher :=
intros;
repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
|| search_prem ltac:(simple apply Match));
try simple apply imp_True.
(* Our tactic succeeds at proving a simple example. *)
Theorem t2 : forall P Q : Prop,
Q /\ (P /\ False) /\ P --> P /\ Q.
Proof.
matcher.
Qed.
(* In the generated proof, we find a trace of the workings of the search tactics. *)
Print t2.
(* We can also see that [matcher] is well-suited for cases where some human
* intervention is needed after the automation finishes. *)
Theorem t3 : forall P Q R : Prop,
P /\ Q --> Q /\ R /\ P.
Proof.
matcher.
Abort.
(* The [matcher] tactic even succeeds at guessing quantifier instantiations. It
* is the unification that occurs in uses of the [Match] lemma that does the
* real work here. *)
Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
Proof.
matcher.
Qed.
Print t4.
(** * Creating Unification Variables *)
(* A final useful ingredient in tactic crafting is the ability to allocate new
* unification variables explicitly. Before we are ready to write a tactic, we
* can try out its ingredients one at a time. *)
Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
Proof.
intros.
evar (y : nat).
let y' := eval unfold y in y in
clear y; specialize (H y').
apply H.
Qed.
(* Spec: create new evar of type [T] and pass to [k]. *)
Ltac newEvar T k := idtac.
(* Spec: instantiate initial [forall]s of [H] with new evars. *)
Ltac insterU H := idtac.
Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
Proof.
Admitted.
(* This particular example is somewhat silly, since [apply] by itself would have
* solved the goal originally. Separate forward reasoning is more useful on
* hypotheses that end in existential quantifications. Before we go through an
* example, it is useful to define a variant of [insterU] that does not clear
* the base hypothesis we pass to it. *)
Ltac insterKeep H := idtac.
Section t6.
Variables A B : Type.
Variable P : A -> B -> Prop.
Variable f : A -> A -> A.
Variable g : B -> B -> B.
Hypothesis H1 : forall v, exists u, P v u.
Hypothesis H2 : forall v1 u1 v2 u2,
P v1 u1
-> P v2 u2
-> P (f v1 v2) (g u1 u2).
Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
Proof.
Admitted.
End t6.
(* Here's an example where something bad happens. *)
Section t7.
Variables A B : Type.
Variable Q : A -> Prop.
Variable P : A -> B -> Prop.
Variable f : A -> A -> A.
Variable g : B -> B -> B.
Hypothesis H1 : forall v, Q v -> exists u, P v u.
Hypothesis H2 : forall v1 u1 v2 u2,
P v1 u1
-> P v2 u2
-> P (f v1 v2) (g u1 u2).
Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
Proof.
(*intros; do 2 insterKeep H1;
repeat match goal with
| [ H : ex _ |- _ ] => destruct H
end; eauto.
(* Oh, two trivial goals remain. *)
Unshelve.
assumption.
assumption.*)
Admitted.
End t7.
Theorem t8 : exists p : nat * nat, fst p = 3.
Proof.
econstructor.
instantiate (1 := (3, 2)).
equality.
Qed.
(* A way that plays better with automation: *)
Theorem t9 : exists p : nat * nat, fst p = 3.
Proof.
econstructor; match goal with
| [ |- fst ?x = 3 ] => unify x (3, 2)
end; equality.
Qed.