mirror of
https://github.com/achlipala/frap.git
synced 2024-11-10 00:07:51 +00:00
IntroToProofScripting_template
This commit is contained in:
Adam Chlipala
parent
79a4b02b4c
commit
ddeb7b25fa
3 changed files with 558 additions and 0 deletions
|
|
@ -706,6 +706,7 @@ Print t4.
|
|||
* can try out its ingredients one at a time. *)
|
||||
|
||||
Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
|
||||
Proof.
|
||||
intros.
|
||||
|
||||
evar (y : nat).
|
||||
|
|
|
|||
556
IntroToProofScripting_template.v
Normal file
556
IntroToProofScripting_template.v
Normal file
|
|
@ -0,0 +1,556 @@
|
|||
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: *)
|
||||
|
||||
Ltac equate x y :=
|
||||
let dummy := constr:(eq_refl x : x = y) in idtac.
|
||||
|
||||
Theorem t9 : exists p : nat * nat, fst p = 3.
|
||||
Proof.
|
||||
econstructor; match goal with
|
||||
| [ |- fst ?x = 3 ] => equate x (3, 2)
|
||||
end; equality.
|
||||
Qed.
|
||||
|
|
@ -19,6 +19,7 @@ Interpreters.v
|
|||
TransitionSystems_template.v
|
||||
TransitionSystems.v
|
||||
IntroToProofScripting.v
|
||||
IntroToProofScripting_template.v
|
||||
ModelChecking_template.v
|
||||
ModelChecking.v
|
||||
OperationalSemantics_template.v
|
||||
|
|
|
|||
Loading…
Reference in a new issue