TypesType Systems
Set Warnings "-notation-overridden,-parsing,-deprecated-hint-without-locality".
From Coq Require Import Arith.Arith.
From PLF Require Import Maps.
From PLF Require Import Smallstep.
Set Default Goal Selector "!".
Hint Constructors multi : core.
From Coq Require Import Arith.Arith.
From PLF Require Import Maps.
From PLF Require Import Smallstep.
Set Default Goal Selector "!".
Hint Constructors multi : core.
Typed Arithmetic Expressions
Syntax
t ::= true
| false
| if t then t else t
| 0
| succ t
| pred t
| iszero t And here it is formally:
Module TM.
Inductive tm : Type :=
| tru : tm
| fls : tm
| ite : tm → tm → tm → tm
| zro : tm
| scc : tm → tm
| prd : tm → tm
| iszro : tm → tm.
Declare Custom Entry tm.
Declare Scope tm_scope.
Notation "'true'" := true (at level 1): tm_scope.
Notation "'true'" := (tru) (in custom tm at level 0): tm_scope.
Notation "'false'" := false (at level 1): tm_scope.
Notation "'false'" := (fls) (in custom tm at level 0): tm_scope.
Notation "<{ e }>" := e (e custom tm at level 99): tm_scope.
Notation "( x )" := x (in custom tm, x at level 99): tm_scope.
Notation "x" := x (in custom tm at level 0, x constr at level 0): tm_scope.
Notation "'0'" := (zro) (in custom tm at level 0): tm_scope.
Notation "'0'" := 0 (at level 1): tm_scope.
Notation "'succ' x" := (scc x) (in custom tm at level 90, x custom tm at level 80): tm_scope.
Notation "'pred' x" := (prd x) (in custom tm at level 90, x custom tm at level 80): tm_scope.
Notation "'iszero' x" := (iszro x) (in custom tm at level 80, x custom tm at level 70): tm_scope.
Notation "'if' c 'then' t 'else' e" := (ite c t e)
(in custom tm at level 90, c custom tm at level 80,
t custom tm at level 80, e custom tm at level 80): tm_scope.
Local Open Scope tm_scope.
Inductive tm : Type :=
| tru : tm
| fls : tm
| ite : tm → tm → tm → tm
| zro : tm
| scc : tm → tm
| prd : tm → tm
| iszro : tm → tm.
Declare Custom Entry tm.
Declare Scope tm_scope.
Notation "'true'" := true (at level 1): tm_scope.
Notation "'true'" := (tru) (in custom tm at level 0): tm_scope.
Notation "'false'" := false (at level 1): tm_scope.
Notation "'false'" := (fls) (in custom tm at level 0): tm_scope.
Notation "<{ e }>" := e (e custom tm at level 99): tm_scope.
Notation "( x )" := x (in custom tm, x at level 99): tm_scope.
Notation "x" := x (in custom tm at level 0, x constr at level 0): tm_scope.
Notation "'0'" := (zro) (in custom tm at level 0): tm_scope.
Notation "'0'" := 0 (at level 1): tm_scope.
Notation "'succ' x" := (scc x) (in custom tm at level 90, x custom tm at level 80): tm_scope.
Notation "'pred' x" := (prd x) (in custom tm at level 90, x custom tm at level 80): tm_scope.
Notation "'iszero' x" := (iszro x) (in custom tm at level 80, x custom tm at level 70): tm_scope.
Notation "'if' c 'then' t 'else' e" := (ite c t e)
(in custom tm at level 90, c custom tm at level 80,
t custom tm at level 80, e custom tm at level 80): tm_scope.
Local Open Scope tm_scope.
Values are <{true}>, <{false}>, and numeric values...
Inductive bvalue : tm → Prop :=
| bv_True : bvalue <{ true }>
| bv_false : bvalue <{ false }>.
Inductive nvalue : tm → Prop :=
| nv_0 : nvalue <{ 0 }>
| nv_succ : ∀ t, nvalue t → nvalue <{ succ t }>.
Definition value (t : tm) := bvalue t ∨ nvalue t.
Hint Constructors bvalue nvalue : core.
Hint Unfold value : core.
| bv_True : bvalue <{ true }>
| bv_false : bvalue <{ false }>.
Inductive nvalue : tm → Prop :=
| nv_0 : nvalue <{ 0 }>
| nv_succ : ∀ t, nvalue t → nvalue <{ succ t }>.
Definition value (t : tm) := bvalue t ∨ nvalue t.
Hint Constructors bvalue nvalue : core.
Hint Unfold value : core.
Operational Semantics
(ST_IfTrue) | |
if true then t1 else t2 --> t1 |
(ST_IfFalse) | |
if false then t1 else t2 --> t2 |
t1 --> t1' | (ST_If) |
if t1 then t2 else t3 --> if t1' then t2 else t3 |
t1 --> t1' | (ST_Succ) |
succ t1 --> succ t1' |
(ST_Pred0) | |
pred 0 --> 0 |
numeric value v | (ST_PredSucc) |
pred (succ v) --> v |
t1 --> t1' | (ST_Pred) |
pred t1 --> pred t1' |
(ST_IsZero0) | |
iszero 0 --> true |
numeric value v | (ST_IszeroSucc) |
iszero (succ v) --> false |
t1 --> t1' | (ST_Iszero) |
iszero t1 --> iszero t1' |
Reserved Notation "t '-->' t'" (at level 40).
Inductive step : tm → tm → Prop :=
| ST_IfTrue : ∀ t1 t2,
<{ if true then t1 else t2 }> --> t1
| ST_IfFalse : ∀ t1 t2,
<{ if false then t1 else t2 }> --> t2
| ST_If : ∀ c c' t2 t3,
c --> c' →
<{ if c then t2 else t3 }> --> <{ if c' then t2 else t3 }>
| ST_Succ : ∀ t1 t1',
t1 --> t1' →
<{ succ t1 }> --> <{ succ t1' }>
| ST_Pred0 :
<{ pred 0 }> --> <{ 0 }>
| ST_PredSucc : ∀ v,
nvalue v →
<{ pred (succ v) }> --> v
| ST_Pred : ∀ t1 t1',
t1 --> t1' →
<{ pred t1 }> --> <{ pred t1' }>
| ST_Iszero0 :
<{ iszero 0 }> --> <{ true }>
| ST_IszeroSucc : ∀ v,
nvalue v →
<{ iszero (succ v) }> --> <{ false }>
| ST_Iszero : ∀ t1 t1',
t1 --> t1' →
<{ iszero t1 }> --> <{ iszero t1' }>
where "t '-->' t'" := (step t t').
Hint Constructors step : core.
Inductive step : tm → tm → Prop :=
| ST_IfTrue : ∀ t1 t2,
<{ if true then t1 else t2 }> --> t1
| ST_IfFalse : ∀ t1 t2,
<{ if false then t1 else t2 }> --> t2
| ST_If : ∀ c c' t2 t3,
c --> c' →
<{ if c then t2 else t3 }> --> <{ if c' then t2 else t3 }>
| ST_Succ : ∀ t1 t1',
t1 --> t1' →
<{ succ t1 }> --> <{ succ t1' }>
| ST_Pred0 :
<{ pred 0 }> --> <{ 0 }>
| ST_PredSucc : ∀ v,
nvalue v →
<{ pred (succ v) }> --> v
| ST_Pred : ∀ t1 t1',
t1 --> t1' →
<{ pred t1 }> --> <{ pred t1' }>
| ST_Iszero0 :
<{ iszero 0 }> --> <{ true }>
| ST_IszeroSucc : ∀ v,
nvalue v →
<{ iszero (succ v) }> --> <{ false }>
| ST_Iszero : ∀ t1 t1',
t1 --> t1' →
<{ iszero t1 }> --> <{ iszero t1' }>
where "t '-->' t'" := (step t t').
Hint Constructors step : core.
Notice that the step relation doesn't care about whether the
expression being stepped makes global sense -- it just checks that
the operation in the next reduction step is being applied to the
right kinds of operands. For example, the term <{ succ true }> cannot
take a step, but the almost as obviously nonsensical term
<{ succ (if true then true else true) }> can take a step (once, before becoming stuck).
<{ succ (if true then true else true) }> can take a step (once, before becoming stuck).
Normal Forms and Values
Notation step_normal_form := (normal_form step).
Definition stuck (t : tm) : Prop :=
step_normal_form t ∧ ¬ value t.
Hint Unfold stuck : core.
Definition stuck (t : tm) : Prop :=
step_normal_form t ∧ ¬ value t.
Hint Unfold stuck : core.
Exercise: 3 stars, standard (value_is_nf)
(Hint: You will reach a point in this proof where you need to
use an induction to reason about a term that is known to be a
numeric value. This induction can be performed either over the
term itself or over the evidence that it is a numeric value. The
proof goes through in either case, but you will find that one way
is quite a bit shorter than the other. For the sake of the
exercise, try to complete the proof both ways.) ☐
Exercise: 3 stars, standard (step_deterministic)
Use value_is_nf to show that the step relation is also deterministic.Typing
In informal notation, the typing relation is often written
|-- t \in T and pronounced "t has type T." The |-- symbol
is called a "turnstile." Below, we're going to see richer typing
relations where one or more additional "context" arguments are
written to the left of the turnstile. For the moment, the context
is always empty.
(T_True) | |
|-- true ∈ Bool |
(T_False) | |
|-- false ∈ Bool |
|-- t1 ∈ Bool |-- t2 ∈ T |-- t3 ∈ T | (T_If) |
|-- if t1 then t2 else t3 ∈ T |
(T_0) | |
|-- 0 ∈ Nat |
|-- t1 ∈ Nat | (T_Succ) |
|-- succ t1 ∈ Nat |
|-- t1 ∈ Nat | (T_Pred) |
|-- pred t1 ∈ Nat |
|-- t1 ∈ Nat | (T_Iszero) |
|-- iszero t1 ∈ Bool |
Reserved Notation "'|--' t '∈' T" (at level 40).
Inductive has_type : tm → ty → Prop :=
| T_True :
|-- <{ true }> \in Bool
| T_False :
|-- <{ false }> \in Bool
| T_If : ∀ t1 t2 t3 T,
|-- t1 \in Bool →
|-- t2 \in T →
|-- t3 \in T →
|-- <{ if t1 then t2 else t3 }> \in T
| T_0 :
|-- <{ 0 }> \in Nat
| T_Succ : ∀ t1,
|-- t1 \in Nat →
|-- <{ succ t1 }> \in Nat
| T_Pred : ∀ t1,
|-- t1 \in Nat →
|-- <{ pred t1 }> \in Nat
| T_Iszero : ∀ t1,
|-- t1 \in Nat →
|-- <{ iszero t1 }> \in Bool
where "'|--' t '∈' T" := (has_type t T).
Hint Constructors has_type : core.
Example has_type_1 :
|-- <{ if false then 0 else (succ 0) }> \in Nat.
Proof.
apply T_If.
- apply T_False.
- apply T_0.
- apply T_Succ. apply T_0.
Qed.
Inductive has_type : tm → ty → Prop :=
| T_True :
|-- <{ true }> \in Bool
| T_False :
|-- <{ false }> \in Bool
| T_If : ∀ t1 t2 t3 T,
|-- t1 \in Bool →
|-- t2 \in T →
|-- t3 \in T →
|-- <{ if t1 then t2 else t3 }> \in T
| T_0 :
|-- <{ 0 }> \in Nat
| T_Succ : ∀ t1,
|-- t1 \in Nat →
|-- <{ succ t1 }> \in Nat
| T_Pred : ∀ t1,
|-- t1 \in Nat →
|-- <{ pred t1 }> \in Nat
| T_Iszero : ∀ t1,
|-- t1 \in Nat →
|-- <{ iszero t1 }> \in Bool
where "'|--' t '∈' T" := (has_type t T).
Hint Constructors has_type : core.
Example has_type_1 :
|-- <{ if false then 0 else (succ 0) }> \in Nat.
Proof.
apply T_If.
- apply T_False.
- apply T_0.
- apply T_Succ. apply T_0.
Qed.
(Since we've included all the constructors of the typing relation
in the hint database, the auto tactic can actually find this
proof automatically.)
It's important to realize that the typing relation is a
conservative (or static) approximation: it does not consider
what happens when the term is reduced -- in particular, it does
not calculate the type of its normal form.
Example has_type_not :
¬ ( |-- <{ if false then 0 else true}> \in Bool ).
¬ ( |-- <{ if false then 0 else true}> \in Bool ).
Proof.
intros Contra. solve_by_inverts 2. Qed.
intros Contra. solve_by_inverts 2. Qed.
Canonical forms
Lemma bool_canonical : ∀ t,
|-- t \in Bool → value t → bvalue t.
Lemma nat_canonical : ∀ t,
|-- t \in Nat → value t → nvalue t.
|-- t \in Bool → value t → bvalue t.
Proof.
intros t HT [Hb | Hn].
- assumption.
- destruct Hn as [ | Hs].
+ inversion HT.
+ inversion HT.
Qed.
intros t HT [Hb | Hn].
- assumption.
- destruct Hn as [ | Hs].
+ inversion HT.
+ inversion HT.
Qed.
Lemma nat_canonical : ∀ t,
|-- t \in Nat → value t → nvalue t.
Proof.
intros t HT [Hb | Hn].
- inversion Hb; subst; inversion HT.
- assumption.
Qed.
intros t HT [Hb | Hn].
- inversion Hb; subst; inversion HT.
- assumption.
Qed.
Progress
Exercise: 3 stars, standard (finish_progress)
Complete the formal proof of the progress property. (Make sure
you understand the parts we've given of the informal proof in the
following exercise before starting -- this will save you a lot of
time.)
Proof.
intros t T HT.
induction HT; auto.
(* The cases that were obviously values, like T_True and
T_False, are eliminated immediately by auto *)
- (* T_If *)
right. destruct IHHT1.
+ (* t1 is a value *)
apply (bool_canonical t1 HT1) in H.
destruct H.
× ∃ t2. auto.
× ∃ t3. auto.
+ (* t1 can take a step *)
destruct H as [t1' H1].
∃ (<{ if t1' then t2 else t3 }>). auto.
(* FILL IN HERE *) Admitted.
intros t T HT.
induction HT; auto.
(* The cases that were obviously values, like T_True and
T_False, are eliminated immediately by auto *)
- (* T_If *)
right. destruct IHHT1.
+ (* t1 is a value *)
apply (bool_canonical t1 HT1) in H.
destruct H.
× ∃ t2. auto.
× ∃ t3. auto.
+ (* t1 can take a step *)
destruct H as [t1' H1].
∃ (<{ if t1' then t2 else t3 }>). auto.
(* FILL IN HERE *) Admitted.
☐
Type Preservation
Exercise: 2 stars, standard (finish_preservation)
Complete the formal proof of the preservation property. (Again,
make sure you understand the informal proof fragment in the
following exercise first.)
Proof.
intros t t' T HT HE.
generalize dependent t'.
induction HT;
(* every case needs to introduce a couple of things *)
intros t' HE;
(* and we can deal with several impossible
cases all at once *)
try solve_by_invert.
- (* T_If *) inversion HE; subst; clear HE.
+ (* ST_IFTrue *) assumption.
+ (* ST_IfFalse *) assumption.
+ (* ST_If *) apply T_If; try assumption.
apply IHHT1; assumption.
(* FILL IN HERE *) Admitted.
intros t t' T HT HE.
generalize dependent t'.
induction HT;
(* every case needs to introduce a couple of things *)
intros t' HE;
(* and we can deal with several impossible
cases all at once *)
try solve_by_invert.
- (* T_If *) inversion HE; subst; clear HE.
+ (* ST_IFTrue *) assumption.
+ (* ST_IfFalse *) assumption.
+ (* ST_If *) apply T_If; try assumption.
apply IHHT1; assumption.
(* FILL IN HERE *) Admitted.
☐
Exercise: 3 stars, standard (preservation_alternate_proof)
Now prove the same property again by induction on the evaluation derivation instead of on the typing derivation. Begin by carefully reading and thinking about the first few lines of the above proofs to make sure you understand what each one is doing. The set-up for this proof is similar, but not exactly the same.
Theorem preservation' : ∀ t t' T,
|-- t \in T →
t --> t' →
|-- t' \in T.
Proof with eauto.
(* FILL IN HERE *) Admitted.
☐
|-- t \in T →
t --> t' →
|-- t' \in T.
Proof with eauto.
(* FILL IN HERE *) Admitted.
☐
Type Soundness
Definition multistep := (multi step).
Notation "t1 '-->*' t2" := (multistep t1 t2) (at level 40).
Corollary soundness : ∀ t t' T,
|-- t \in T →
t -->* t' →
~(stuck t').
Notation "t1 '-->*' t2" := (multistep t1 t2) (at level 40).
Corollary soundness : ∀ t t' T,
|-- t \in T →
t -->* t' →
~(stuck t').
Proof.
intros t t' T HT P. induction P; intros [R S].
- apply progress in HT. destruct HT; auto.
- apply IHP.
+ apply preservation with (t := x); auto.
+ unfold stuck. split; auto.
Qed.
intros t t' T HT P. induction P; intros [R S].
- apply progress in HT. destruct HT; auto.
- apply IHP.
+ apply preservation with (t := x); auto.
+ unfold stuck. split; auto.
Qed.