StlcPropProperties of STLC
Set Warnings "-notation-overridden,-parsing,-deprecated-hint-without-locality".
From PLF Require Import Maps.
From PLF Require Import Types.
From PLF Require Import Stlc.
From PLF Require Import Smallstep.
Module STLCProp.
Import STLC.
In this chapter, we develop the fundamental theory of the Simply
Typed Lambda Calculus -- in particular, the type safety
theorem.
Canonical Forms
Lemma canonical_forms_bool : ∀ t,
empty ⊢ t \in Bool →
value t →
(t = <{true}>) ∨ (t = <{false}>).
Proof.
intros t HT HVal.
destruct HVal; auto.
inversion HT.
Qed.
intros t HT HVal.
destruct HVal; auto.
inversion HT.
Qed.
Lemma canonical_forms_fun : ∀ t T1 T2,
empty ⊢ t \in (T1 → T2) →
value t →
∃ x u, t = <{\x:T1, u}>.
Proof.
intros t T1 T2 HT HVal.
destruct HVal; inversion HT; subst.
∃ x0, t1. reflexivity.
Qed.
intros t T1 T2 HT HVal.
destruct HVal; inversion HT; subst.
∃ x0, t1. reflexivity.
Qed.
Progress
Proof: By induction on the derivation of ⊢ t \in T.
- The last rule of the derivation cannot be T_Var, since a
variable is never well typed in an empty context.
- The T_True, T_False, and T_Abs cases are trivial, since in
each of these cases we can see by inspecting the rule that t
is a value.
- If the last rule of the derivation is T_App, then t has the
form t1 t2 for some t1 and t2, where ⊢ t1 \in T2 → T
and ⊢ t2 \in T2 for some type T2. The induction hypothesis
for the first subderivation says that either t1 is a value or
else it can take a reduction step.
- If t1 is a value, then consider t2, which by the
induction hypothesis for the second subderivation must also
either be a value or take a step.
- Suppose t2 is a value. Since t1 is a value with an
arrow type, it must be a lambda abstraction; hence t1
t2 can take a step by ST_AppAbs.
- Otherwise, t2 can take a step, and hence so can t1
t2 by ST_App2.
- Suppose t2 is a value. Since t1 is a value with an
arrow type, it must be a lambda abstraction; hence t1
t2 can take a step by ST_AppAbs.
- If t1 can take a step, then so can t1 t2 by ST_App1.
- If t1 is a value, then consider t2, which by the
induction hypothesis for the second subderivation must also
either be a value or take a step.
- If the last rule of the derivation is T_If, then t = if
t1 then t2 else t3, where t1 has type Bool. The first IH
says that t1 either is a value or takes a step.
- If t1 is a value, then since it has type Bool it must be
either true or false. If it is true, then t steps to
t2; otherwise it steps to t3.
- Otherwise, t1 takes a step, and therefore so does t (by ST_If).
- If t1 is a value, then since it has type Bool it must be
either true or false. If it is true, then t steps to
t2; otherwise it steps to t3.
Proof with eauto.
intros t T Ht.
remember empty as Gamma.
induction Ht; subst Gamma; auto.
(* auto solves all three cases in which t is a value *)
- (* T_Var *)
(* contradictory: variables cannot be typed in an
empty context *)
discriminate H.
- (* T_App *)
(* t = t1 t2. Proceed by cases on whether t1 is a
value or steps... *)
right. destruct IHHt1...
+ (* t1 is a value *)
destruct IHHt2...
× (* t2 is also a value *)
eapply canonical_forms_fun in Ht1; [|assumption].
destruct Ht1 as [x [t0 H1]]. subst.
∃ (<{ [x:=t2]t0 }>)...
× (* t2 steps *)
destruct H0 as [t2' Hstp]. ∃ (<{t1 t2'}>)...
+ (* t1 steps *)
destruct H as [t1' Hstp]. ∃ (<{t1' t2}>)...
- (* T_If *)
right. destruct IHHt1...
+ (* t1 is a value *)
destruct (canonical_forms_bool t1); subst; eauto.
+ (* t1 also steps *)
destruct H as [t1' Hstp]. ∃ <{if t1' then t2 else t3}>...
Qed.
intros t T Ht.
remember empty as Gamma.
induction Ht; subst Gamma; auto.
(* auto solves all three cases in which t is a value *)
- (* T_Var *)
(* contradictory: variables cannot be typed in an
empty context *)
discriminate H.
- (* T_App *)
(* t = t1 t2. Proceed by cases on whether t1 is a
value or steps... *)
right. destruct IHHt1...
+ (* t1 is a value *)
destruct IHHt2...
× (* t2 is also a value *)
eapply canonical_forms_fun in Ht1; [|assumption].
destruct Ht1 as [x [t0 H1]]. subst.
∃ (<{ [x:=t2]t0 }>)...
× (* t2 steps *)
destruct H0 as [t2' Hstp]. ∃ (<{t1 t2'}>)...
+ (* t1 steps *)
destruct H as [t1' Hstp]. ∃ (<{t1' t2}>)...
- (* T_If *)
right. destruct IHHt1...
+ (* t1 is a value *)
destruct (canonical_forms_bool t1); subst; eauto.
+ (* t1 also steps *)
destruct H as [t1' Hstp]. ∃ <{if t1' then t2 else t3}>...
Qed.
Exercise: 3 stars, advanced (progress_from_term_ind)
Show that progress can also be proved by induction on terms instead of induction on typing derivations.Theorem progress' : ∀ t T,
empty ⊢ t \in T →
value t ∨ ∃ t', t --> t'.
Proof.
intros t.
induction t; intros T Ht; auto.
(* FILL IN HERE *) Admitted.
☐
Preservation
- The preservation theorem is proved by induction on a typing
derivation, pretty much as we did in the Types chapter.
The one case that is significantly different is the one for
the ST_AppAbs rule, whose definition uses the substitution
operation. To see that this step preserves typing, we need to
know that the substitution itself does. So we prove a...
- substitution lemma, stating that substituting a (closed)
term s for a variable x in a term t preserves the type
of t. The proof goes by induction on the form of t and
requires looking at all the different cases in the definition
of substitition. This time, for the variables case, we
discover that we need to deduce from the fact that a term
s has type S in the empty context the fact that s has
type S in every context. For this we prove a...
- weakening lemma, showing that typing is preserved under "extensions" to the context Gamma.
The Weakening Lemma
Lemma weakening : ∀ Gamma Gamma' t T,
inclusion Gamma Gamma' →
Gamma ⊢ t \in T →
Gamma' ⊢ t \in T.
Proof.
intros Gamma Gamma' t T H Ht.
generalize dependent Gamma'.
induction Ht; eauto using inclusion_update.
Qed.
The following simple corollary is what we actually need below.
Lemma weakening_empty : ∀ Gamma t T,
empty ⊢ t \in T →
Gamma ⊢ t \in T.
Proof.
intros Gamma t T.
eapply weakening.
discriminate.
Qed.
A Substitution Lemma
Lemma substitution_preserves_typing : ∀ Gamma x U t v T,
x ⊢> U ; Gamma ⊢ t \in T →
empty ⊢ v \in U →
Gamma ⊢ [x:=v]t \in T.
The substitution lemma can be viewed as a kind of "commutation
property." Intuitively, it says that substitution and typing can
be done in either order: we can either assign types to the terms
t and v separately (under suitable contexts) and then combine
them using substitution, or we can substitute first and then
assign a type to [x:=v] t ; the result is the same either
way.
Proof: We show, by induction on t, that for all T and
Gamma, if x⊢>U; Gamma ⊢ t \in T and ⊢ v \in U, then
Gamma ⊢ [x:=v]t \in T.
- If t is a variable there are two cases to consider,
depending on whether t is x or some other variable.
- If t = x, then from the fact that x⊢>U; Gamma ⊢ x \in
T we conclude that U = T. We must show that [x:=v]x =
v has type T under Gamma, given the assumption that
v has type U = T under the empty context. This
follows from the weakening lemma.
- If t is some variable y that is not equal to x, then
we need only note that y has the same type under x⊢>U;
Gamma as under Gamma.
- If t = x, then from the fact that x⊢>U; Gamma ⊢ x \in
T we conclude that U = T. We must show that [x:=v]x =
v has type T under Gamma, given the assumption that
v has type U = T under the empty context. This
follows from the weakening lemma.
- If t is an abstraction \y:S, t0, then T = S→T1 and
the IH tells us, for all Gamma' and T0, that if x⊢>U;
Gamma' ⊢ t0 \in T0, then Gamma' ⊢ [x:=v]t0 \in T0.
Moreover, by inspecting the typing rules we see it must be
the case that y⊢>S; x⊢>U; Gamma ⊢ t0 \in T1.
- If t is an application t1 t2, the result follows
straightforwardly from the definition of substitution and the
induction hypotheses.
- The remaining cases are similar to the application case.
Proof.
intros Gamma x U t v T Ht Hv.
generalize dependent Gamma. generalize dependent T.
induction t; intros T Gamma H;
(* in each case, we'll want to get at the derivation of H *)
inversion H; clear H; subst; simpl; eauto.
- (* var *)
rename s into y. destruct (eqb_stringP x y); subst.
+ (* x=y *)
rewrite update_eq in H2.
injection H2 as H2; subst.
apply weakening_empty. assumption.
+ (* x<>y *)
apply T_Var. rewrite update_neq in H2; auto.
- (* abs *)
rename s into y, t into S.
destruct (eqb_stringP x y); subst; apply T_Abs.
+ (* x=y *)
rewrite update_shadow in H5. assumption.
+ (* x<>y *)
apply IHt.
rewrite update_permute; auto.
Qed.
intros Gamma x U t v T Ht Hv.
generalize dependent Gamma. generalize dependent T.
induction t; intros T Gamma H;
(* in each case, we'll want to get at the derivation of H *)
inversion H; clear H; subst; simpl; eauto.
- (* var *)
rename s into y. destruct (eqb_stringP x y); subst.
+ (* x=y *)
rewrite update_eq in H2.
injection H2 as H2; subst.
apply weakening_empty. assumption.
+ (* x<>y *)
apply T_Var. rewrite update_neq in H2; auto.
- (* abs *)
rename s into y, t into S.
destruct (eqb_stringP x y); subst; apply T_Abs.
+ (* x=y *)
rewrite update_shadow in H5. assumption.
+ (* x<>y *)
apply IHt.
rewrite update_permute; auto.
Qed.
One technical subtlety in the statement of the above lemma is that
we assume v has type U in the empty context -- in other
words, we assume v is closed. (Since we are using a simple
definition of substition that is not capture-avoiding, it doesn't
make sense to substitute non-closed terms into other terms.
Fortunately, closed terms are all we need!)
Exercise: 3 stars, advanced (substitution_preserves_typing_from_typing_ind)
Show that substitution_preserves_typing can also be proved by induction on typing derivations instead of induction on terms.
Lemma substitution_preserves_typing_from_typing_ind : ∀ Gamma x U t v T,
x ⊢> U ; Gamma ⊢ t \in T →
empty ⊢ v \in U →
Gamma ⊢ [x:=v]t \in T.
Proof.
intros Gamma x U t v T Ht Hv.
remember (x ⊢> U; Gamma) as Gamma'.
generalize dependent Gamma.
induction Ht; intros Gamma' G; simpl; eauto.
(* FILL IN HERE *) Admitted.
☐
x ⊢> U ; Gamma ⊢ t \in T →
empty ⊢ v \in U →
Gamma ⊢ [x:=v]t \in T.
Proof.
intros Gamma x U t v T Ht Hv.
remember (x ⊢> U; Gamma) as Gamma'.
generalize dependent Gamma.
induction Ht; intros Gamma' G; simpl; eauto.
(* FILL IN HERE *) Admitted.
☐
Main Theorem
Proof: By induction on the derivation of ⊢ t \in T.
- We can immediately rule out T_Var, T_Abs, T_True, and
T_False as final rules in the derivation, since in each of these
cases t cannot take a step.
- If the last rule in the derivation is T_App, then t = t1 t2,
and there are subderivations showing that ⊢ t1 \in T2→T and
⊢ t2 \in T2 plus two induction hypotheses: (1) t1 --> t1'
implies ⊢ t1' \in T2→T and (2) t2 --> t2' implies ⊢ t2'
\in T2. There are now three subcases to consider, one for
each rule that could be used to show that t1 t2 takes a step
to t'.
- If t1 t2 takes a step by ST_App1, with t1 stepping to
t1', then, by the first IH, t1' has the same type as
t1 (⊢ t1' \in T2→T), and hence by T_App t1' t2 has
type T.
- The ST_App2 case is similar, using the second IH.
- If t1 t2 takes a step by ST_AppAbs, then t1 =
\x:T0,t0 and t1 t2 steps to [x0:=t2]t0; the desired
result now follows from the substitution lemma.
- If t1 t2 takes a step by ST_App1, with t1 stepping to
t1', then, by the first IH, t1' has the same type as
t1 (⊢ t1' \in T2→T), and hence by T_App t1' t2 has
type T.
- If the last rule in the derivation is T_If, then t = if
t1 then t2 else t3, with ⊢ t1 \in Bool, ⊢ t2 \in T1, and
⊢ t3 \in T1, and with three induction hypotheses: (1) t1 -->
t1' implies ⊢ t1' \in Bool, (2) t2 --> t2' implies ⊢ t2'
\in T1, and (3) t3 --> t3' implies ⊢ t3' \in T1.
- If t steps to t2 or t3 by ST_IfTrue or
ST_IfFalse, the result is immediate, since t2 and t3
have the same type as t.
- Otherwise, t steps by ST_If, and the desired conclusion follows directly from the first induction hypothesis.
- If t steps to t2 or t3 by ST_IfTrue or
ST_IfFalse, the result is immediate, since t2 and t3
have the same type as t.
Proof with eauto.
intros t t' T HT. generalize dependent t'.
remember empty as Gamma.
induction HT;
intros t' HE; subst;
try solve [inversion HE; subst; auto].
- (* T_App *)
inversion HE; subst...
(* Most of the cases are immediate by induction,
and eauto takes care of them *)
+ (* ST_AppAbs *)
apply substitution_preserves_typing with T2...
inversion HT1...
Qed.
intros t t' T HT. generalize dependent t'.
remember empty as Gamma.
induction HT;
intros t' HE; subst;
try solve [inversion HE; subst; auto].
- (* T_App *)
inversion HE; subst...
(* Most of the cases are immediate by induction,
and eauto takes care of them *)
+ (* ST_AppAbs *)
apply substitution_preserves_typing with T2...
inversion HT1...
Qed.
Exercise: 2 stars, standard, especially useful (subject_expansion_stlc)
An exercise in the Types chapter asked about the subject expansion property for the simple language of arithmetic and boolean expressions. This property did not hold for that language, and it also fails for STLC. That is, it is not always the case that, if t --> t' and empty ⊢ t' \in T, then empty ⊢ t \in T. Show this by giving a counter-example that does not involve conditionals.(* FILL IN HERE *)
(* Do not modify the following line: *)
Definition manual_grade_for_subject_expansion_stlc : option (nat×string) := None.
☐
Type Soundness
Exercise: 2 stars, standard, optional (type_soundness)
Put progress and preservation together and show that a well-typed term can never reach a stuck state.Definition stuck (t:tm) : Prop :=
(normal_form step) t ∧ ¬ value t.
Corollary type_soundness : ∀ t t' T,
empty ⊢ t \in T →
t -->* t' →
~(stuck t').
Proof.
intros t t' T Hhas_type Hmulti. unfold stuck.
intros [Hnf Hnot_val]. unfold normal_form in Hnf.
induction Hmulti.
(* FILL IN HERE *) Admitted.
intros t t' T Hhas_type Hmulti. unfold stuck.
intros [Hnf Hnot_val]. unfold normal_form in Hnf.
induction Hmulti.
(* FILL IN HERE *) Admitted.
☐
Uniqueness of Types
Exercise: 3 stars, standard (unique_types)
Another nice property of the STLC is that types are unique: a given term (in a given context) has at most one type.Theorem unique_types : ∀ Gamma e T T',
Gamma ⊢ e \in T →
Gamma ⊢ e \in T' →
T = T'.
Proof.
(* FILL IN HERE *) Admitted.
☐
Additional Exercises
Exercise: 1 star, standard (progress_preservation_statement)
Without peeking at their statements above, write down the progress and preservation theorems for the simply typed lambda-calculus (as Coq theorems). You can write Admitted for the proofs.(* FILL IN HERE *)
(* Do not modify the following line: *)
Definition manual_grade_for_progress_preservation_statement : option (nat×string) := None.
☐
Exercise: 2 stars, standard (stlc_variation1)
Suppose we add a new term zap with the following reduction rule(ST_Zap) | |
t --> zap |
(T_Zap) | |
Gamma ⊢ zap ∈ T |
- Determinism of step
- Progress
- Preservation
(* Do not modify the following line: *)
Definition manual_grade_for_stlc_variation1 : option (nat×string) := None.
☐
Exercise: 2 stars, standard (stlc_variation2)
Suppose instead that we add a new term foo with the following reduction rules:(ST_Foo1) | |
(\x:A, x) --> foo |
(ST_Foo2) | |
foo --> true |
- Determinism of step
- Progress
- Preservation
(* Do not modify the following line: *)
Definition manual_grade_for_stlc_variation2 : option (nat×string) := None.
☐
Exercise: 2 stars, standard (stlc_variation3)
Suppose instead that we remove the rule ST_App1 from the step relation. Which of the following properties of the STLC remain true in the presence of this rule? For each one, write either "remains true" or else "becomes false." If a property becomes false, give a counterexample.- Determinism of step
- Progress
- Preservation
(* Do not modify the following line: *)
Definition manual_grade_for_stlc_variation3 : option (nat×string) := None.
☐
Exercise: 2 stars, standard, optional (stlc_variation4)
Suppose instead that we add the following new rule to the reduction relation:(ST_FunnyIfTrue) | |
(if true then t1 else t2) --> true |
- Determinism of step
- Progress
- Preservation
☐
Exercise: 2 stars, standard, optional (stlc_variation5)
Suppose instead that we add the following new rule to the typing relation:Gamma ⊢ t1 ∈ Bool->Bool->Bool | |
Gamma ⊢ t2 ∈ Bool | (T_FunnyApp) |
Gamma ⊢ t1 t2 ∈ Bool |
- Determinism of step
- Progress
- Preservation
☐
Exercise: 2 stars, standard, optional (stlc_variation6)
Suppose instead that we add the following new rule to the typing relation:Gamma ⊢ t1 ∈ Bool | |
Gamma ⊢ t2 ∈ Bool | (T_FunnyApp') |
Gamma ⊢ t1 t2 ∈ Bool |
- Determinism of step
- Progress
- Preservation
☐
Exercise: 2 stars, standard, optional (stlc_variation7)
Suppose we add the following new rule to the typing relation of the STLC:(T_FunnyAbs) | |
⊢ \x:Bool,t ∈ Bool |
- Determinism of step
- Progress
- Preservation
☐
Exercise: STLC with Arithmetic
To types, we add a base type of natural numbers (and remove
booleans, for brevity).
To terms, we add natural number constants, along with
successor, predecessor, multiplication, and zero-testing.
Inductive tm : Type :=
| tm_var : string → tm
| tm_app : tm → tm → tm
| tm_abs : string → ty → tm → tm
| tm_const : nat → tm
| tm_succ : tm → tm
| tm_pred : tm → tm
| tm_mult : tm → tm → tm
| tm_if0 : tm → tm → tm → tm.
Notation "{ x }" := x (in custom stlc at level 1, x constr).
Notation "<{ e }>" := e (e custom stlc at level 99).
Notation "( x )" := x (in custom stlc, x at level 99).
Notation "x" := x (in custom stlc at level 0, x constr at level 0).
Notation "S -> T" := (Ty_Arrow S T) (in custom stlc at level 50, right associativity).
Notation "x y" := (tm_app x y) (in custom stlc at level 1, left associativity).
Notation "\ x : t , y" :=
(tm_abs x t y) (in custom stlc at level 90, x at level 99,
t custom stlc at level 99,
y custom stlc at level 99,
left associativity).
Coercion tm_var : string >-> tm.
Notation "'Nat'" := Ty_Nat (in custom stlc at level 0).
Notation "'succ' x" := (tm_succ x) (in custom stlc at level 0,
x custom stlc at level 0).
Notation "'pred' x" := (tm_pred x) (in custom stlc at level 0,
x custom stlc at level 0).
Notation "x * y" := (tm_mult x y) (in custom stlc at level 1,
left associativity).
Notation "'if0' x 'then' y 'else' z" :=
(tm_if0 x y z) (in custom stlc at level 89,
x custom stlc at level 99,
y custom stlc at level 99,
z custom stlc at level 99,
left associativity).
Coercion tm_const : nat >-> tm.
Exercise: 5 stars, standard (stlc_arith)
Finish formalizing the definition and properties of the STLC extended with arithmetic. This is a longer exercise. Specifically:- Fixpoint subst
- Inductive value
- Inductive step
- Inductive has_type
- Inductive appears_free_in
- Lemma weakening
- Lemma weakening_empty
- Lemma substitution_preserves_typing
- Theorem preservation
- Theorem progress