EvaluationContexts: products and sums

EvaluationContexts.v

@@ -294,3 +294,590 @@ Module Stlc.
     eapply plug_func; eassumption.
   Qed.    
 End Stlc.
+
+(** * Some More Classic Features *)
+
+(* Here's how easy it is to extend those definitions and proofs to two other
+ * common features of functional-programming languages.  We'll use comments to
+ * mark the only places where code is added.  Very little old code needs to be
+ * changed!  The version in the book PDF shows even more clearly how evaluation
+ * contexts make for compact descriptions of features, since here we are
+ * manually writing [plug] relations, following clear conventions in
+ * evaluation-context grammars. *)
+
+Module StlcPairs.
+  Inductive exp : Set :=
+  | Var (x : var)
+  | Const (n : nat)
+  | Plus (e1 e2 : exp)
+  | Abs (x : var) (e1 : exp)
+  | App (e1 e2 : exp)
+
+  (* We can combine two values together into a pair, and then we can use
+   * projection functions to retrieve the first and second components,
+   * respectively. *)
+  | Pair (e1 e2 : exp)
+  | Fst (e1 : exp)
+  | Snd (e2 : exp).
+
+  Inductive value : exp -> Prop :=
+  | VConst : forall n, value (Const n)
+  | VAbs : forall x e1, value (Abs x e1)
+  (* A pair of values is a value.  (Now this relation finally becomes
+   * recursive.) *)
+  | VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2).
+
+  Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
+    match e2 with
+      | Var y => if y ==v x then e1 else Var y
+      | Const n => Const n
+      | Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
+      | Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
+      | App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
+      (* Some bureaucratic work here to add predictable cases *)
+      | Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
+      | Fst e2' => Fst (subst e1 x e2')
+      | Snd e2' => Snd (subst e1 x e2')
+    end.
+
+  Inductive context : Set :=
+  | Hole : context
+  | Plus1 : context -> exp -> context
+  | Plus2 : exp -> context -> context
+  | App1 : context -> exp -> context
+  | App2 : exp -> context -> context
+  (* Two new context kinds, indicating left-to-right evaluation order for
+   * pairs *)
+  | Pair1 : context -> exp -> context
+  | Pair2 : exp -> context -> context
+  (* And similar for projections *)
+  | Fst1 : context -> context
+  | Snd1 : context -> context.
+                                
+  Inductive plug : context -> exp -> exp -> Prop :=
+  | PlugHole : forall e, plug Hole e e
+  | PlugPlus1 : forall e e' C e2,
+    plug C e e'
+    -> plug (Plus1 C e2) e (Plus e' e2)
+  | PlugPlus2 : forall e e' v1 C,
+    value v1
+    -> plug C e e'
+    -> plug (Plus2 v1 C) e (Plus v1 e')
+  | PlugApp1 : forall e e' C e2,
+    plug C e e'
+    -> plug (App1 C e2) e (App e' e2)
+  | PlugApp2 : forall e e' v1 C,
+    value v1
+    -> plug C e e'
+    -> plug (App2 v1 C) e (App v1 e')
+
+  (* Our new plugging rules *)
+  | PlugPair1 : forall e e' C e2,
+    plug C e e'
+    -> plug (Pair1 C e2) e (Pair e' e2)
+  | PlugPair2 : forall e e' v1 C,
+    value v1
+    -> plug C e e'
+    -> plug (Pair2 v1 C) e (Pair v1 e')
+  | PlugFst1 : forall e e' C,
+    plug C e e'
+    -> plug (Fst1 C) e (Fst e')
+  | PlugSnd1 : forall e e' C,
+    plug C e e'
+    -> plug (Snd1 C) e (Snd e').
+
+  Inductive step0 : exp -> exp -> Prop :=
+  | Beta : forall x e v,
+    value v
+    -> step0 (App (Abs x e) v) (subst v x e)
+  | Add : forall n1 n2,
+      step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
+
+  (* Reducing projections *)
+  | FstPair : forall v1 v2,
+      value v1
+      -> value v2
+      -> step0 (Fst (Pair v1 v2)) v1
+  | SndPair : forall v1 v2,
+      value v1
+      -> value v2
+      -> step0 (Snd (Pair v1 v2)) v2.
+
+  Inductive step : exp -> exp -> Prop :=
+  | StepRule : forall C e1 e2 e1' e2',
+    plug C e1 e1'
+    -> plug C e2 e2'
+    -> step0 e1 e2
+    -> step e1' e2'.
+
+  Definition trsys_of (e : exp) := {|
+    Initial := {e};
+    Step := step
+  |}.
+
+
+  Inductive type :=
+  | Nat
+  | Fun (dom ran : type)
+  | Prod (t1 t2 : type) (* "Prod" for "product," as in Cartesian product *).
+
+  Inductive hasty : fmap var type -> exp -> type -> Prop :=
+  | HtVar : forall G x t,
+    G $? x = Some t
+    -> hasty G (Var x) t
+  | HtConst : forall G n,
+    hasty G (Const n) Nat
+  | HtPlus : forall G e1 e2,
+    hasty G e1 Nat
+    -> hasty G e2 Nat
+    -> hasty G (Plus e1 e2) Nat
+  | HtAbs : forall G x e1 t1 t2,
+    hasty (G $+ (x, t1)) e1 t2
+    -> hasty G (Abs x e1) (Fun t1 t2)
+  | HtApp : forall G e1 e2 t1 t2,
+    hasty G e1 (Fun t1 t2)
+    -> hasty G e2 t1
+    -> hasty G (App e1 e2) t2
+  | HtPair : forall G e1 e2 t1 t2,
+    hasty G e1 t1
+    -> hasty G e2 t2
+    -> hasty G (Pair e1 e2) (Prod t1 t2)
+  | HtFst : forall G e1 t1 t2,
+    hasty G e1 (Prod t1 t2)
+    -> hasty G (Fst e1) t1
+  | HtSnd : forall G e1 t1 t2,
+    hasty G e1 (Prod t1 t2)
+    -> hasty G (Snd e1) t2.
+
+  Local Hint Constructors value plug step0 step hasty : core.
+
+  Infix "-->" := Fun (at level 60, right associativity).
+  Coercion Const : nat >-> exp.
+  Infix "^+^" := Plus (at level 50).
+  Coercion Var : var >-> exp.
+  Notation "\ x , e" := (Abs x e) (at level 51).
+  Infix "@" := App (at level 49, left associativity).
+
+  Ltac t0 := match goal with
+             | [ H : ex _ |- _ ] => invert H
+             | [ H : _ /\ _ |- _ ] => invert H
+             | [ |- context[?x ==v ?y] ] => cases (x ==v y)
+             | [ H : Some _ = Some _ |- _ ] => invert H
+
+             | [ H : step _ _ |- _ ] => invert H
+             | [ H : step0 _ _ |- _ ] => invert1 H
+             | [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; []
+               (* Change here!  We need to enforce there is at most one
+                * remaining subgoal, or we'll keep doing useless [value]
+                * inversions ad infinitum. *)
+             | [ H : hasty _ _ _ |- _ ] => invert1 H
+             | [ H : plug _ _ _ |- _ ] => invert1 H
+             end; subst.
+
+  Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 6.
+
+  Lemma progress : forall e t,
+    hasty $0 e t
+    -> value e
+    \/ (exists e' : exp, step e e').
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Lemma weakening_override : forall (G G' : fmap var type) x t,
+    (forall x' t', G $? x' = Some t' -> G' $? x' = Some t')
+    -> (forall x' t', G $+ (x, t) $? x' = Some t'
+                      -> G' $+ (x, t) $? x' = Some t').
+  Proof.
+    simplify.
+    cases (x ==v x'); simplify; eauto.
+  Qed.
+
+  Local Hint Resolve weakening_override : core.
+
+  Lemma weakening : forall G e t,
+    hasty G e t
+    -> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
+      -> hasty G' e t.
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Local Hint Resolve weakening : core.
+
+  (* Replacing a typing context with an equal one has no effect (useful to guide
+   * proof search as a hint). *)
+  Lemma hasty_change : forall G e t,
+    hasty G e t
+    -> forall G', G' = G
+      -> hasty G' e t.
+  Proof.
+    t.
+  Qed.
+
+  Local Hint Resolve hasty_change : core.
+
+  Lemma substitution : forall G x t' e t e',
+    hasty (G $+ (x, t')) e t
+    -> hasty $0 e' t'
+    -> hasty G (subst e' x e) t.
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Local Hint Resolve substitution : core.
+
+  Lemma preservation0 : forall e1 e2,
+    step0 e1 e2
+    -> forall t, hasty $0 e1 t
+      -> hasty $0 e2 t.
+  Proof.
+    invert 1; t.
+  Qed.
+
+  Local Hint Resolve preservation0 : core.
+
+  Lemma preservation' : forall C e1 e1',
+      plug C e1 e1'
+      -> forall e2 e2' t, plug C e2 e2'
+      -> step0 e1 e2
+      -> hasty $0 e1' t
+      -> hasty $0 e2' t.
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Local Hint Resolve preservation' : core.
+  
+  Lemma preservation : forall e1 e2,
+    step e1 e2
+    -> forall t, hasty $0 e1 t
+      -> hasty $0 e2 t.
+  Proof.
+    invert 1; t.
+  Qed.
+
+  Local Hint Resolve progress preservation : core.
+
+  Theorem safety : forall e t, hasty $0 e t
+    -> invariantFor (trsys_of e)
+                    (fun e' => value e'
+                               \/ exists e'', step e' e'').
+  Proof.
+    simplify.
+    apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t); eauto.
+    apply invariant_induction; simplify; eauto; equality.
+  Qed.
+End StlcPairs.
+
+(* Next, the dual feature of *variants*, corresponding to the following type
+ * family from Coq's standard library. *)
+
+Print sum.
+
+Module StlcSums.
+  Inductive exp : Set :=
+  | Var (x : var)
+  | Const (n : nat)
+  | Plus (e1 e2 : exp)
+  | Abs (x : var) (e1 : exp)
+  | App (e1 e2 : exp)
+  | Pair (e1 e2 : exp)
+  | Fst (e1 : exp)
+  | Snd (e2 : exp)
+
+  (* New cases: *)
+  | Inl (e1 : exp)
+  | Inr (e2 : exp)
+  | Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp).
+  (* The last one roughly means "match e' with inl x1 => e1 | inr x2 => e2". *)
+
+  Inductive value : exp -> Prop :=
+  | VConst : forall n, value (Const n)
+  | VAbs : forall x e1, value (Abs x e1)
+  | VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
+  | VInl : forall v, value v -> value (Inl v)
+  | VInr : forall v, value v -> value (Inr v).
+
+  Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
+    match e2 with
+      | Var y => if y ==v x then e1 else Var y
+      | Const n => Const n
+      | Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
+      | Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
+      | App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
+      | Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
+      | Fst e2' => Fst (subst e1 x e2')
+      | Snd e2' => Snd (subst e1 x e2')
+      (* Some bureaucratic work here to add predictable cases *)
+      | Inl e2' => Inl (subst e1 x e2')
+      | Inr e2' => Inr (subst e1 x e2')
+      | Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
+                                         x1 (if x1 ==v x then e21 else subst e1 x e21)
+                                         x2 (if x2 ==v x then e22 else subst e1 x e22)
+    end.
+
+  Inductive context : Set :=
+  | Hole : context
+  | Plus1 : context -> exp -> context
+  | Plus2 : exp -> context -> context
+  | App1 : context -> exp -> context
+  | App2 : exp -> context -> context
+  | Pair1 : context -> exp -> context
+  | Pair2 : exp -> context -> context
+  | Fst1 : context -> context
+  | Snd1 : context -> context
+
+  (* New cases: *)
+  | Inl1 : context -> context
+  | Inr1 : context -> context
+  | Match1 : context -> var -> exp -> var -> exp -> context.
+                        
+  Inductive plug : context -> exp -> exp -> Prop :=
+  | PlugHole : forall e, plug Hole e e
+  | PlugPlus1 : forall e e' C e2,
+    plug C e e'
+    -> plug (Plus1 C e2) e (Plus e' e2)
+  | PlugPlus2 : forall e e' v1 C,
+    value v1
+    -> plug C e e'
+    -> plug (Plus2 v1 C) e (Plus v1 e')
+  | PlugApp1 : forall e e' C e2,
+    plug C e e'
+    -> plug (App1 C e2) e (App e' e2)
+  | PlugApp2 : forall e e' v1 C,
+    value v1
+    -> plug C e e'
+    -> plug (App2 v1 C) e (App v1 e')
+  | PlugPair1 : forall e e' C e2,
+    plug C e e'
+    -> plug (Pair1 C e2) e (Pair e' e2)
+  | PlugPair2 : forall e e' v1 C,
+    value v1
+    -> plug C e e'
+    -> plug (Pair2 v1 C) e (Pair v1 e')
+  | PlugFst1 : forall e e' C,
+    plug C e e'
+    -> plug (Fst1 C) e (Fst e')
+  | PlugSnd1 : forall e e' C,
+    plug C e e'
+    -> plug (Snd1 C) e (Snd e')
+
+  (* Our new plugging rules *)            
+  | PlugInl1 : forall e e' C,
+    plug C e e'
+    -> plug (Inl1 C) e (Inl e')
+  | PlugInr1 : forall e e' C,
+    plug C e e'
+    -> plug (Inr1 C) e (Inr e')
+  | PluMatch1 : forall e e' C x1 e1 x2 e2,
+    plug C e e'
+    -> plug (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2).
+
+  Inductive step0 : exp -> exp -> Prop :=
+  | Beta : forall x e v,
+    value v
+    -> step0 (App (Abs x e) v) (subst v x e)
+  | Add : forall n1 n2,
+      step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
+  | FstPair : forall v1 v2,
+      value v1
+      -> value v2
+      -> step0 (Fst (Pair v1 v2)) v1
+  | SndPair : forall v1 v2,
+      value v1
+      -> value v2
+      -> step0 (Snd (Pair v1 v2)) v2
+
+  (* Reducing a [Match] *)
+  | MatchInl : forall v x1 e1 x2 e2,
+      value v
+      -> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
+  | MatchInr : forall v x1 e1 x2 e2,
+      value v
+      -> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2).
+
+  Inductive step : exp -> exp -> Prop :=
+  | StepRule : forall C e1 e2 e1' e2',
+    plug C e1 e1'
+    -> plug C e2 e2'
+    -> step0 e1 e2
+    -> step e1' e2'.
+
+  Definition trsys_of (e : exp) := {|
+    Initial := {e};
+    Step := step
+  |}.
+
+
+  Inductive type :=
+  | Nat
+  | Fun (dom ran : type)
+  | Prod (t1 t2 : type)
+  (* New case: *)
+  | Sum (t1 t2 : type).
+
+  Inductive hasty : fmap var type -> exp -> type -> Prop :=
+  | HtVar : forall G x t,
+    G $? x = Some t
+    -> hasty G (Var x) t
+  | HtConst : forall G n,
+    hasty G (Const n) Nat
+  | HtPlus : forall G e1 e2,
+    hasty G e1 Nat
+    -> hasty G e2 Nat
+    -> hasty G (Plus e1 e2) Nat
+  | HtAbs : forall G x e1 t1 t2,
+    hasty (G $+ (x, t1)) e1 t2
+    -> hasty G (Abs x e1) (Fun t1 t2)
+  | HtApp : forall G e1 e2 t1 t2,
+    hasty G e1 (Fun t1 t2)
+    -> hasty G e2 t1
+    -> hasty G (App e1 e2) t2
+  | HtPair : forall G e1 e2 t1 t2,
+    hasty G e1 t1
+    -> hasty G e2 t2
+    -> hasty G (Pair e1 e2) (Prod t1 t2)
+  | HtFst : forall G e1 t1 t2,
+    hasty G e1 (Prod t1 t2)
+    -> hasty G (Fst e1) t1
+  | HtSnd : forall G e1 t1 t2,
+    hasty G e1 (Prod t1 t2)
+    -> hasty G (Snd e1) t2
+
+  (* New cases: *)
+  | HtInl : forall G e1 t1 t2,
+      hasty G e1 t1
+      -> hasty G (Inl e1) (Sum t1 t2)
+  | HtInr : forall G e1 t1 t2,
+      hasty G e1 t2
+      -> hasty G (Inr e1) (Sum t1 t2)
+  | HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
+      hasty G e (Sum t1 t2)
+      -> hasty (G $+ (x1, t1)) e1 t
+      -> hasty (G $+ (x2, t2)) e2 t
+      -> hasty G (Match e x1 e1 x2 e2) t.
+
+  Local Hint Constructors value plug step0 step hasty : core.
+
+  Infix "-->" := Fun (at level 60, right associativity).
+  Coercion Const : nat >-> exp.
+  Infix "^+^" := Plus (at level 50).
+  Coercion Var : var >-> exp.
+  Notation "\ x , e" := (Abs x e) (at level 51).
+  Infix "@" := App (at level 49, left associativity).
+
+  Ltac t0 := match goal with
+             | [ H : ex _ |- _ ] => invert H
+             | [ H : _ /\ _ |- _ ] => invert H
+             | [ |- context[?x ==v ?y] ] => cases (x ==v y)
+             | [ H : Some _ = Some _ |- _ ] => invert H
+
+             | [ H : step _ _ |- _ ] => invert H
+             | [ H : step0 _ _ |- _ ] => invert1 H
+             | [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; []
+
+             (* New case!  For sums, we sometimes need to consider two rules for
+              * one [value] inversion. *)
+             | [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; [|]
+
+             | [ H : hasty _ _ _ |- _ ] => invert1 H
+             | [ H : plug _ _ _ |- _ ] => invert1 H
+             end; subst.
+
+  Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
+                                                                 (* change! --^ *)
+
+  Lemma progress : forall e t,
+    hasty $0 e t
+    -> value e
+    \/ (exists e' : exp, step e e').
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Lemma weakening_override : forall (G G' : fmap var type) x t,
+    (forall x' t', G $? x' = Some t' -> G' $? x' = Some t')
+    -> (forall x' t', G $+ (x, t) $? x' = Some t'
+                      -> G' $+ (x, t) $? x' = Some t').
+  Proof.
+    simplify.
+    cases (x ==v x'); simplify; eauto.
+  Qed.
+
+  Local Hint Resolve weakening_override : core.
+
+  Lemma weakening : forall G e t,
+    hasty G e t
+    -> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
+      -> hasty G' e t.
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Local Hint Resolve weakening : core.
+
+  (* Replacing a typing context with an equal one has no effect (useful to guide
+   * proof search as a hint). *)
+  Lemma hasty_change : forall G e t,
+    hasty G e t
+    -> forall G', G' = G
+      -> hasty G' e t.
+  Proof.
+    t.
+  Qed.
+
+  Local Hint Resolve hasty_change : core.
+
+  Lemma substitution : forall G x t' e t e',
+    hasty (G $+ (x, t')) e t
+    -> hasty $0 e' t'
+    -> hasty G (subst e' x e) t.
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Local Hint Resolve substitution : core.
+
+  Lemma preservation0 : forall e1 e2,
+    step0 e1 e2
+    -> forall t, hasty $0 e1 t
+      -> hasty $0 e2 t.
+  Proof.
+    invert 1; t.
+  Qed.
+
+  Local Hint Resolve preservation0 : core.
+
+  Lemma preservation' : forall C e1 e1',
+      plug C e1 e1'
+      -> forall e2 e2' t, plug C e2 e2'
+      -> step0 e1 e2
+      -> hasty $0 e1' t
+      -> hasty $0 e2' t.
+  Proof.
+    induct 1; t.
+  Qed.
+
+  Local Hint Resolve preservation' : core.
+  
+  Lemma preservation : forall e1 e2,
+    step e1 e2
+    -> forall t, hasty $0 e1 t
+      -> hasty $0 e2 t.
+  Proof.
+    invert 1; t.
+  Qed.
+
+  Local Hint Resolve progress preservation : core.
+
+  Theorem safety : forall e t, hasty $0 e t
+    -> invariantFor (trsys_of e)
+                    (fun e' => value e'
+                               \/ exists e'', step e' e'').
+  Proof.
+    simplify.
+    apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t); eauto.
+    apply invariant_induction; simplify; eauto; equality.
+  Qed.
+End StlcSums.