DependentInductiveTypes_template

DependentInductiveTypes.v

@@ -492,6 +492,66 @@ Qed.
  * adapt automatically to changes in function definitions. *)
 
 
+(** * Interlude: The Convoy Pattern *)
+
+(* Here are some examples, contemplation of which may provoke enlightenment.
+ * See more discussion later of the idea behind the examples. *)
+
+Fail Definition firstElements n A B (ls1 : ilist A n) (ls2 : ilist B n) : option (A * B) :=
+  match ls1 with
+  | Cons _ v1 _ =>
+    Some (v1,
+          match ls2 in ilist _ N return match N with O => unit | S _ => B end with
+          | Cons _ v2 _ => v2
+          | Nil => tt
+          end)
+  | Nil => None
+  end.
+
+Definition firstElements n A B (ls1 : ilist A n) (ls2 : ilist B n) : option (A * B) :=
+  match ls1 in ilist _ N return ilist B N -> option (A * B) with
+  | Cons _ v1 _ => fun ls2 =>
+    Some (v1,
+          match ls2 in ilist _ N return match N with O => unit | S _ => B end with
+          | Cons _ v2 _ => v2
+          | Nil => tt
+          end)
+  | Nil => fun _ => None
+  end ls2.
+
+(* Note use of a [struct] annotation to tell Coq which argument should decrease
+ * across recursive calls.  It's an artificial choice here, since usually those
+ * annotations are inferred.  Here we are making an effort to demonstrate a
+ * decently common problem! *)
+Fail Fixpoint zip n A B (ls1 : ilist A n) (ls2 : ilist B n) {struct ls1} : ilist (A * B) n :=
+  match ls1 in ilist _ N return ilist B N -> ilist (A * B) N with
+  | Cons _ v1 ls1' => 
+    fun ls2 =>
+      match ls2 in ilist _ N return match N with
+                                    | O => unit
+                                    | S N' => ilist A N' -> ilist (A * B) N
+                                    end with
+      | Cons _ v2 ls2' => fun ls1' => Cons (v1, v2) (zip ls1' ls2')
+      | Nll => tt
+      end ls1'
+  | Nil => fun _ => Nil _
+  end ls2.
+
+Fixpoint zip n A B (ls1 : ilist A n) (ls2 : ilist B n) {struct ls1} : ilist (A * B) n :=
+  match ls1 in ilist _ N return ilist B N -> ilist (A * B) N with
+  | Cons _ v1 ls1' => 
+    fun ls2 =>
+      match ls2 in ilist _ N return match N with
+                                    | O => unit
+                                    | S N' => (ilist B N' -> ilist (A * B) N') -> ilist (A * B) N
+                                    end with
+      | Cons _ v2 ls2' => fun zip_ls1' => Cons (v1, v2) (zip_ls1' ls2')
+      | Nll => tt
+      end (zip ls1')
+  | Nil => fun _ => Nil _
+  end ls2.
+
+
 (** * Dependently Typed Red-Black Trees *)
 
 (* Red-black trees are a favorite purely functional data structure with an

DependentInductiveTypes_template.v

@@ -0,0 +1,895 @@
+Require Import FrapWithoutSets SubsetTypes.
+
+Set Implicit Arguments.
+Set Asymmetric Patterns.
+
+
+(** * Length-Indexed Lists *)
+
+Section ilist.
+  Variable A : Set.
+  Inductive ilist : nat -> Set :=
+  | Nil : ilist O
+  | Cons : forall n, A -> ilist n -> ilist (S n).
+
+  Fixpoint app n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) : ilist (n1 + n2) :=
+    match ls1 with
+      | Nil => ls2
+      | Cons _ x ls1' => Cons x (app ls1' ls2)
+    end.
+
+  Fixpoint inject (ls : list A) : ilist (length ls) :=
+    match ls with
+      | nil => Nil
+      | h :: t => Cons h (inject t)
+    end.
+
+  Fixpoint unject n (ls : ilist n) : list A :=
+    match ls with
+      | Nil => nil
+      | Cons _ h t => h :: unject t
+    end.
+
+  Theorem inject_inverse : forall ls, unject (inject ls) = ls.
+  Proof.
+    induct ls; simplify; equality.
+  Qed.
+
+  Fail Definition hd n (ls : ilist (S n)) : A :=
+    match ls with
+    | Nil => _
+    | Cons _ h _ => h
+    end.
+End ilist.
+
+
+(** * A Tagless Interpreter *)
+
+Inductive type : Set :=
+| Nat : type
+| Bool : type
+| Prod : type -> type -> type.
+
+Inductive exp : type -> Set :=
+| NConst : nat -> exp Nat
+| Plus : exp Nat -> exp Nat -> exp Nat
+| Eq : exp Nat -> exp Nat -> exp Bool
+
+| BConst : bool -> exp Bool
+| And : exp Bool -> exp Bool -> exp Bool
+| If : forall t, exp Bool -> exp t -> exp t -> exp t
+
+| Pair : forall t1 t2, exp t1 -> exp t2 -> exp (Prod t1 t2)
+| Fst : forall t1 t2, exp (Prod t1 t2) -> exp t1
+| Snd : forall t1 t2, exp (Prod t1 t2) -> exp t2.
+
+Fixpoint typeDenote (t : type) : Set :=
+  match t with
+    | Nat => nat
+    | Bool => bool
+    | Prod t1 t2 => typeDenote t1 * typeDenote t2
+  end%type.
+
+Fixpoint expDenote t (e : exp t) : typeDenote t :=
+  match e with
+    | NConst n => n
+    | Plus e1 e2 => expDenote e1 + expDenote e2
+    | Eq e1 e2 => if eq_nat_dec (expDenote e1) (expDenote e2) then true else false
+
+    | BConst b => b
+    | And e1 e2 => expDenote e1 && expDenote e2
+    | If _ e' e1 e2 => if expDenote e' then expDenote e1 else expDenote e2
+
+    | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
+    | Fst _ _ e' => fst (expDenote e')
+    | Snd _ _ e' => snd (expDenote e')
+  end.
+
+Fixpoint cfold t (e : exp t) : exp t :=
+  match e with
+    | NConst n => NConst n
+    | Plus e1 e2 =>
+      let e1' := cfold e1 in
+      let e2' := cfold e2 in
+      match e1', e2' return exp Nat with
+        | NConst n1, NConst n2 => NConst (n1 + n2)
+        | _, _ => Plus e1' e2'
+      end
+    | Eq e1 e2 =>
+      let e1' := cfold e1 in
+      let e2' := cfold e2 in
+      match e1', e2' return exp Bool with
+        | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
+        | _, _ => Eq e1' e2'
+      end
+
+    | BConst b => BConst b
+    | And e1 e2 =>
+      let e1' := cfold e1 in
+      let e2' := cfold e2 in
+      match e1', e2' return exp Bool with
+        | BConst b1, BConst b2 => BConst (b1 && b2)
+        | _, _ => And e1' e2'
+      end
+    | If _ e e1 e2 =>
+      let e' := cfold e in
+      match e' with
+        | BConst true => cfold e1
+        | BConst false => cfold e2
+        | _ => If e' (cfold e1) (cfold e2)
+      end
+
+    | Pair _ _ e1 e2 => Pair (cfold e1) (cfold e2)
+    | Fst _ _ e => Fst e
+    | Snd _ _ e => Snd e
+  end.
+
+Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
+Proof.
+
+
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+Admitted.
+  (*induct e; simplify;
+    repeat (match goal with
+              | [ |- context[match cfold ?E with NConst _ => _ | _ => _ end] ] =>
+                dep_cases (cfold E)
+              | [ |- context[match pairOut (cfold ?E) with Some _ => _
+                               | None => _ end] ] =>
+                dep_cases (cfold E)
+              | [ |- context[if ?E then _ else _] ] => cases E
+              | [ H : _ = _ |- _ ] => rewrite H
+            end; simplify); try equality.
+Qed.*)
+
+
+(** * Interlude: The Convoy Pattern *)
+
+Fail Definition firstElements n A B (ls1 : ilist A n) (ls2 : ilist B n) : option (A * B) :=
+  match ls1 with
+  | Cons _ v1 _ =>
+    Some (v1,
+          match ls2 in ilist _ N return match N with O => unit | S _ => B end with
+          | Cons _ v2 _ => v2
+          | Nil => tt
+          end)
+  | Nil => None
+  end.
+
+Fail Fixpoint zip n A B (ls1 : ilist A n) (ls2 : ilist B n) {struct ls1} : ilist (A * B) n :=
+  match ls1 in ilist _ N return ilist B N -> ilist (A * B) N with
+  | Cons _ v1 ls1' => 
+    fun ls2 =>
+      match ls2 in ilist _ N return match N with
+                                    | O => unit
+                                    | S N' => ilist A N' -> ilist (A * B) N
+                                    end with
+      | Cons _ v2 ls2' => fun ls1' => Cons (v1, v2) (zip ls1' ls2')
+      | Nll => tt
+      end ls1'
+  | Nil => fun _ => Nil _
+  end ls2.
+
+
+(** * Dependently Typed Red-Black Trees *)
+
+Inductive color : Set := Red | Black.
+
+Inductive rbtree : color -> nat -> Set :=
+| Leaf : rbtree Black 0
+| RedNode : forall n, rbtree Black n -> nat -> rbtree Black n -> rbtree Red n
+| BlackNode : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rbtree Black (S n).
+
+Section depth.
+  Variable f : nat -> nat -> nat.
+
+  Fixpoint depth c n (t : rbtree c n) : nat :=
+    match t with
+      | Leaf => 0
+      | RedNode _ t1 _ t2 => S (f (depth t1) (depth t2))
+      | BlackNode _ _ _ t1 _ t2 => S (f (depth t1) (depth t2))
+    end.
+End depth.
+
+Theorem depth_min : forall c n (t : rbtree c n), depth min t >= 0.
+Proof.
+Admitted.
+
+Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 0.
+Proof.
+Admitted.
+
+Theorem balanced : forall c n (t : rbtree c n), t = t.
+Proof.
+Admitted.
+
+Inductive rtree : nat -> Set :=
+| RedNode' : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rtree n.
+
+Section present.
+  Variable x : nat.
+
+  Fixpoint present c n (t : rbtree c n) : Prop :=
+    match t with
+      | Leaf => False
+      | RedNode _ a y b => present a \/ x = y \/ present b
+      | BlackNode _ _ _ a y b => present a \/ x = y \/ present b
+    end.
+
+  Definition rpresent n (t : rtree n) : Prop :=
+    match t with
+      | RedNode' _ _ _ a y b => present a \/ x = y \/ present b
+    end.
+End present.
+
+Locate "{ _ : _ & _ }".
+Print sigT.
+
+Notation "{< x >}" := (existT _ _ x).
+
+Definition balance1 n (a : rtree n) (data : nat) c2 :=
+  match a in rtree n return rbtree c2 n
+    -> { c : color & rbtree c (S n) } with
+    | RedNode' _ c0 _ t1 y t2 =>
+      match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
+        -> { c : color & rbtree c (S n) } with
+        | RedNode _ a x b => fun c d =>
+          {<RedNode (BlackNode a x b) y (BlackNode c data d)>}
+        | t1' => fun t2 =>
+          match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
+            -> { c : color & rbtree c (S n) } with
+            | RedNode _ b x c => fun a d =>
+              {<RedNode (BlackNode a y b) x (BlackNode c data d)>}
+            | b => fun a t => {<BlackNode (RedNode a y b) data t>}
+          end t1'
+      end t2
+  end.
+
+Definition balance2 n (a : rtree n) (data : nat) c2 :=
+  match a in rtree n return rbtree c2 n -> { c : color & rbtree c (S n) } with
+    | RedNode' _ c0 _ t1 z t2 =>
+      match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
+        -> { c : color & rbtree c (S n) } with
+        | RedNode _ b y c => fun d a =>
+          {<RedNode (BlackNode a data b) y (BlackNode c z d)>}
+        | t1' => fun t2 =>
+          match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
+            -> { c : color & rbtree c (S n) } with
+            | RedNode _ c z' d => fun b a =>
+              {<RedNode (BlackNode a data b) z (BlackNode c z' d)>}
+            | b => fun a t => {<BlackNode t data (RedNode a z b)>}
+          end t1'
+      end t2
+  end.
+
+Section insert.
+  Variable x : nat.
+
+  Definition insResult c n :=
+    match c with
+      | Red => rtree n
+      | Black => { c' : color & rbtree c' n }
+    end.
+
+  Fixpoint ins c n (t : rbtree c n) : insResult c n :=
+    match t with
+      | Leaf => {< RedNode Leaf x Leaf >}
+      | RedNode _ a y b =>
+        if le_lt_dec x y
+          then RedNode' (projT2 (ins a)) y b
+          else RedNode' a y (projT2 (ins b))
+      | BlackNode c1 c2 _ a y b =>
+        if le_lt_dec x y
+          then
+            match c1 return insResult c1 _ -> _ with
+              | Red => fun ins_a => balance1 ins_a y b
+              | _ => fun ins_a => {< BlackNode (projT2 ins_a) y b >}
+            end (ins a)
+          else
+            match c2 return insResult c2 _ -> _ with
+              | Red => fun ins_b => balance2 ins_b y a
+              | _ => fun ins_b => {< BlackNode a y (projT2 ins_b) >}
+            end (ins b)
+    end.
+
+  Definition insertResult c n :=
+    match c with
+      | Red => rbtree Black (S n)
+      | Black => { c' : color & rbtree c' n }
+    end.
+
+  Definition makeRbtree {c n} : insResult c n -> insertResult c n :=
+    match c with
+      | Red => fun r =>
+        match r with
+          | RedNode' _ _ _ a x b => BlackNode a x b
+        end
+      | Black => fun r => r
+    end.
+
+  Definition insert c n (t : rbtree c n) : insertResult c n :=
+    makeRbtree (ins t).
+
+  Section present.
+    Variable z : nat.
+
+    Ltac present_balance :=
+      simplify;
+      repeat (match goal with
+                | [ _ : context[match ?T with Leaf => _ | _ => _ end] |- _ ] =>
+                  dep_cases T
+                | [ |- context[match ?T with Leaf => _ | _ => _ end] ] => dep_cases T
+              end; simplify); propositional.
+
+    Lemma present_balance1 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
+      present z (projT2 (balance1 a y b))
+      <-> rpresent z a \/ z = y \/ present z b.
+    Proof.
+      simplify; cases a; present_balance.
+    Qed.
+
+    Lemma present_balance2 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
+      present z (projT2 (balance2 a y b))
+      <-> rpresent z a \/ z = y \/ present z b.
+    Proof.
+      simplify; cases a; present_balance.
+    Qed.
+
+    Definition present_insResult c n :=
+      match c return (rbtree c n -> insResult c n -> Prop) with
+        | Red => fun t r => rpresent z r <-> z = x \/ present z t
+        | Black => fun t r => present z (projT2 r) <-> z = x \/ present z t
+      end.
+
+    Theorem present_ins : forall c n (t : rbtree c n),
+      present_insResult t (ins t).
+    Proof.
+      induct t; simplify;
+        repeat (match goal with
+                  | [ _ : context[if ?E then _ else _] |- _ ] => cases E
+                  | [ |- context[if ?E then _ else _] ] => cases E
+                  | [ _ : context[match ?C with Red => _ | Black => _ end]
+                      |- _ ] => cases C
+                end; simplify);
+        try match goal with
+              | [ _ : context[balance1 ?A ?B ?C] |- _ ] =>
+                pose proof (present_balance1 A B C)
+            end;
+        try match goal with
+              | [ _ : context[balance2 ?A ?B ?C] |- _ ] =>
+                pose proof (present_balance2 A B C)
+            end;
+        try match goal with
+              | [ |- context[balance1 ?A ?B ?C] ] =>
+                pose proof (present_balance1 A B C)
+            end;
+        try match goal with
+              | [ |- context[balance2 ?A ?B ?C] ] =>
+                pose proof (present_balance2 A B C)
+            end;
+        simplify; propositional.
+    Qed.
+
+    Ltac present_insert :=
+      unfold insert; intros n t;
+        pose proof (present_ins t); simplify;
+          cases (ins t); propositional.
+
+    Theorem present_insert_Red : forall n (t : rbtree Red n),
+      present z (insert t)
+      <-> (z = x \/ present z t).
+    Proof.
+      present_insert.
+    Qed.
+
+    Theorem present_insert_Black : forall n (t : rbtree Black n),
+      present z (projT2 (insert t))
+      <-> (z = x \/ present z t).
+    Proof.
+      present_insert.
+    Qed.
+  End present.
+End insert.
+
+Recursive Extraction insert.
+
+
+(** * A Certified Regular Expression Matcher *)
+
+Require Import Ascii String.
+Open Scope string_scope.
+
+Section star.
+  Variable P : string -> Prop.
+
+  Inductive star : string -> Prop :=
+  | Empty : star ""
+  | Iter : forall s1 s2,
+    P s1
+    -> star s2
+    -> star (s1 ++ s2).
+End star.
+
+Fail Inductive regexp : (string -> Prop) -> Set :=
+| Char : forall ch : ascii,
+  regexp (fun s => s = String ch "")
+| Concat : forall (P1 P2 : string -> Prop) (r1 : regexp P1) (r2 : regexp P2),
+  regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2).
+
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+Inductive regexp : (string -> Prop) -> Type :=
+| Char : forall ch : ascii,
+  regexp (fun s => s = String ch "")
+| Concat : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
+  regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2)
+| Or : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
+  regexp (fun s => P1 s \/ P2 s)
+| Star : forall P (r : regexp P),
+  regexp (star P).
+
+(* Many theorems about strings are useful for implementing a certified regexp
+ * matcher, and few of them are in the [String] library.  Here they are.  Feel
+ * free to resume reading at "BOREDOM'S END". *)
+
+Lemma length_emp : length "" <= 0.
+Proof.
+  auto.
+Qed.
+
+Lemma append_emp : forall s, s = "" ++ s.
+Proof.
+  auto.
+Qed.
+
+Ltac substring :=
+  simplify;
+  repeat match goal with
+           | [ |- context[match ?N with O => _ | S _ => _ end] ] =>
+             destruct N; simplify
+         end; try linear_arithmetic; eauto; try equality.
+
+Hint Resolve le_n_S.
+
+Lemma substring_le : forall s n m,
+  length (substring n m s) <= m.
+Proof.
+  induct s; substring.
+Qed.
+
+Lemma substring_all : forall s,
+  substring 0 (length s) s = s.
+Proof.
+  induct s; substring.
+Qed.
+
+Lemma substring_none : forall s n,
+  substring n 0 s = "".
+Proof.
+  induct s; substring.
+Qed.
+
+Hint Rewrite substring_all substring_none.
+
+Lemma substring_split : forall s m,
+  substring 0 m s ++ substring m (length s - m) s = s.
+Proof.
+  induct s; substring.
+Qed.
+
+Lemma length_app1 : forall s1 s2,
+  length s1 <= length (s1 ++ s2).
+Proof.
+  induct s1; substring.
+Qed.
+
+Hint Resolve length_emp append_emp substring_le substring_split length_app1.
+
+Lemma substring_app_fst : forall s2 s1 n,
+  length s1 = n
+  -> substring 0 n (s1 ++ s2) = s1.
+Proof.
+  induct s1; simplify; subst; simplify; try equality.
+  rewrite IHs1; auto.
+Qed.
+
+Hint Rewrite <- minus_n_O.
+
+Lemma substring_app_snd : forall s2 s1 n,
+  length s1 = n
+  -> substring n (length (s1 ++ s2) - n) (s1 ++ s2) = s2.
+Proof.
+  induct s1; simplify; subst; simplify; auto.
+Qed.
+
+Hint Rewrite substring_app_fst substring_app_snd using solve [trivial].
+
+(* BOREDOM'S END! *)
+
+Section sumbool_and.
+  Variables P1 Q1 P2 Q2 : Prop.
+
+  Variable x1 : {P1} + {Q1}.
+  Variable x2 : {P2} + {Q2}.
+
+  Definition sumbool_and : {P1 /\ P2} + {Q1 \/ Q2} :=
+    match x1 with
+      | left HP1 =>
+        match x2 with
+          | left HP2 => left _ (conj HP1 HP2)
+          | right HQ2 => right _ (or_intror _ HQ2)
+        end
+      | right HQ1 => right _ (or_introl _ HQ1)
+    end.
+End sumbool_and.
+
+Infix "&&" := sumbool_and (at level 40, left associativity).
+
+Hint Extern 1 (_ <= _) => linear_arithmetic.
+
+Section split.
+  Variables P1 P2 : string -> Prop.
+  Variable P1_dec : forall s, {P1 s} + {~ P1 s}.
+  Variable P2_dec : forall s, {P2 s} + {~ P2 s}.
+
+  Variable s : string.
+
+  Definition split' : forall n : nat, n <= length s
+    -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
+    + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2}.
+    refine (fix F (n : nat) : n <= length s
+      -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
+      + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2} :=
+      match n with
+        | O => fun _ => Reduce (P1_dec "" && P2_dec s)
+        | S n' => fun _ => (P1_dec (substring 0 (S n') s)
+            && P2_dec (substring (S n') (length s - S n') s))
+          || F n' _
+      end); clear F; simplify;
+      repeat match goal with
+             | [ H : exists x, _ |- _ ] => invert H
+             end; propositional; eauto 7;
+      try match goal with
+          | [ _ : length ?S <= 0 |- _ ] => cases S; simplify
+          | [ _ : length ?S' <= S ?N |- _ ] => cases (length S' ==n S N)
+          end; subst; simplify; try equality; try linear_arithmetic; eauto.
+  Defined.
+
+  Definition split : {exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2}
+    + {forall s1 s2, s = s1 ++ s2 -> ~ P1 s1 \/ ~ P2 s2}.
+    refine (Reduce (split' (n := length s) _)); simplify; auto; first_order; subst; eauto.
+  Defined.
+End split.
+
+Implicit Arguments split [P1 P2].
+
+(* And now, a few more boring lemmas.  Rejoin at "BOREDOM VANQUISHED", if you
+ * like. *)
+
+Lemma app_empty_end : forall s, s ++ "" = s.
+Proof.
+  induct s; substring.
+Qed.
+
+Hint Rewrite app_empty_end.
+
+Lemma substring_self : forall s n,
+  n <= 0
+  -> substring n (length s - n) s = s.
+Proof.
+  induct s; substring.
+Qed.
+
+Lemma substring_empty : forall s n m,
+  m <= 0
+  -> substring n m s = "".
+Proof.
+  induct s; substring.
+Qed.
+
+Hint Rewrite substring_self substring_empty using linear_arithmetic.
+Hint Rewrite substring_split.
+
+Lemma substring_split' : forall s n m,
+  substring n m s ++ substring (n + m) (length s - (n + m)) s
+  = substring n (length s - n) s.
+Proof.
+  induct s; substring.
+Qed.
+
+Hint Extern 1 (String _ _ = String _ _) => f_equal.
+
+Lemma substring_stack : forall s n2 m1 m2,
+  m1 <= m2
+  -> substring 0 m1 (substring n2 m2 s)
+  = substring n2 m1 s.
+Proof.
+  induct s; substring.
+Qed.
+
+Ltac substring' :=
+  simplify;
+  repeat match goal with
+           | [ |- context[match ?N with O => _ | S _ => _ end] ] => cases N; simplify
+         end; try equality; try linear_arithmetic.
+
+Lemma substring_stack' : forall s n1 n2 m1 m2,
+  n1 + m1 <= m2
+  -> substring n1 m1 (substring n2 m2 s)
+  = substring (n1 + n2) m1 s.
+Proof.
+  induct s; substring';
+    match goal with
+    | [ H : _ |- _ ] => rewrite H by linear_arithmetic; f_equal; linear_arithmetic
+    end.
+Qed.
+
+Lemma substring_suffix : forall s n,
+  n <= length s
+  -> length (substring n (length s - n) s) = length s - n.
+Proof.
+  induct s; substring.
+Qed.
+
+Lemma substring_suffix_emp' : forall s n m,
+  substring n (S m) s = ""
+  -> n >= length s.
+Proof.
+  induct s; simplify; auto;
+    match goal with
+      | [ |- ?N >= _ ] => cases N; simplify; try equality
+    end;
+    match goal with
+      [ |- S ?N >= S ?E ] => assert (N >= E) by eauto; linear_arithmetic
+    end.
+Qed.
+
+Lemma substring_suffix_emp : forall s n m,
+  substring n m s = ""
+  -> m > 0
+  -> n >= length s.
+Proof.
+  simplify; cases m; simplify; eauto using substring_suffix_emp'.
+Qed.
+
+Hint Rewrite substring_stack substring_stack' substring_suffix using linear_arithmetic.
+
+Lemma minus_minus : forall n m1 m2,
+  m1 + m2 <= n
+  -> n - m1 - m2 = n - (m1 + m2).
+Proof.
+  linear_arithmetic.
+Qed.
+
+Lemma plus_n_Sm' : forall n m : nat, S (n + m) = m + S n.
+Proof.
+  linear_arithmetic.
+Qed.
+
+Hint Rewrite minus_minus plus_n_Sm' using linear_arithmetic.
+
+(* BOREDOM VANQUISHED! *)
+
+Section dec_star.
+  Variable P : string -> Prop.
+  Variable P_dec : forall s, {P s} + {~ P s}.
+
+  (* Some new lemmas and hints about the [star] type family are useful.  Rejoin
+   * at BOREDOM DEMOLISHED to skip the details. *)
+
+  Hint Constructors star.
+
+  Lemma star_empty : forall s,
+    length s = 0
+    -> star P s.
+  Proof.
+    simplify; cases s; simplify; try equality; eauto.
+  Qed.
+
+  Lemma star_singleton : forall s, P s -> star P s.
+  Proof.
+    simplify.
+    rewrite <- (app_empty_end s); auto.
+  Qed.
+
+  Lemma star_app : forall s n m,
+    P (substring n m s)
+    -> star P (substring (n + m) (length s - (n + m)) s)
+    -> star P (substring n (length s - n) s).
+  Proof.
+    induct n; substring;
+      match goal with
+        | [ H : P (substring ?N ?M ?S) |- _ ] =>
+          solve [ rewrite <- (substring_split S M); auto
+            | rewrite <- (substring_split' S N M); simplify; auto ]
+      end.
+  Qed.
+
+  Hint Resolve star_empty star_singleton star_app.
+
+  Variable s : string.
+
+  Hint Extern 1 (exists i : nat, _) =>
+    match goal with
+    | [ H : P (String _ ?S) |- _ ] => exists (length S); simplify
+    end.
+
+  Lemma star_inv : forall s,
+    star P s
+    -> s = ""
+    \/ exists i, i < length s
+      /\ P (substring 0 (S i) s)
+      /\ star P (substring (S i) (length s - S i) s).
+  Proof.
+    induct 1; simplify; first_order; subst.
+    cases s1; simplify; propositional; eauto 10.
+    cases s1; simplify; propositional; eauto 10.
+  Qed.
+
+  Lemma star_substring_inv : forall n,
+    n <= length s
+    -> star P (substring n (length s - n) s)
+    -> substring n (length s - n) s = ""
+    \/ exists l, l < length s - n
+      /\ P (substring n (S l) s)
+      /\ star P (substring (n + S l) (length s - (n + S l)) s).
+  Proof.
+    simplify;
+      match goal with
+        | [ H : star _ _ |- _ ] => pose proof (star_inv H); simplify;
+                                     first_order; simplify; eauto
+      end.
+  Qed.
+
+  (* BOREDOM DEMOLISHED! *)
+
+  Section dec_star''.
+    Variable n : nat.
+
+    Variable P' : string -> Prop.
+    Variable P'_dec : forall n' : nat, n' > n
+      -> {P' (substring n' (length s - n') s)}
+      + {~ P' (substring n' (length s - n') s)}.
+
+    Hint Extern 1 (_ \/ _) => linear_arithmetic.
+
+    Definition dec_star'' : forall l : nat,
+      {exists l', S l' <= l
+        /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
+      + {forall l', S l' <= l
+        -> ~ P (substring n (S l') s)
+        \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)}.
+      refine (fix F (l : nat) : {exists l', S l' <= l
+          /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
+        + {forall l', S l' <= l
+          -> ~ P (substring n (S l') s)
+          \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} :=
+        match l with
+          | O => _
+          | S l' =>
+            (P_dec (substring n (S l') s) && P'_dec (n' := n + S l') _)
+            || F l'
+        end); clear F; simplify; first_order; eauto 7;
+        match goal with
+        | [ H : ?X <= S ?Y |- _ ] => destruct (eq_nat_dec X (S Y)); simplify; eauto; equality
+        end.
+    Defined.
+  End dec_star''.
+
+  Lemma star_length_contra : forall n,
+    length s > n
+    -> n >= length s
+    -> False.
+  Proof.
+    linear_arithmetic.
+  Qed.
+
+  Lemma star_length_flip : forall n n',
+    length s - n <= S n'
+    -> length s > n
+    -> length s - n > 0.
+  Proof.
+    linear_arithmetic.
+  Qed.
+
+  Hint Resolve star_length_contra star_length_flip substring_suffix_emp.
+
+  Definition dec_star' : forall n n' : nat, length s - n' <= n
+    -> {star P (substring n' (length s - n') s)}
+    + {~ star P (substring n' (length s - n') s)}.
+    refine (fix F (n n' : nat) : length s - n' <= n
+      -> {star P (substring n' (length s - n') s)}
+      + {~ star P (substring n' (length s - n') s)} :=
+      match n with
+        | O => fun _ => Yes
+        | S n'' => fun _ =>
+          le_gt_dec (length s) n'
+          || dec_star'' (n := n') (star P)
+            (fun n0 _ => Reduce (F n'' n0 _)) (length s - n')
+      end); clear F; simplify; first_order; propositional; eauto;
+    match goal with
+      | [ H : star _ _ |- _ ] => apply star_substring_inv in H; simplify; eauto
+    end; first_order; eauto.
+  Defined.
+
+  Definition dec_star : {star P s} + {~ star P s}.
+    refine (Reduce (dec_star' (n := length s) 0 _)); simplify; auto.
+  Defined.
+End dec_star.
+
+Lemma app_cong : forall x1 y1 x2 y2,
+  x1 = x2
+  -> y1 = y2
+  -> x1 ++ y1 = x2 ++ y2.
+Proof.
+  equality.
+Qed.
+
+Hint Resolve app_cong.
+
+Definition matches : forall P (r : regexp P) s, {P s} + {~ P s}.
+  refine (fix F P (r : regexp P) s : {P s} + {~ P s} :=
+    match r with
+      | Char ch => string_dec s (String ch "")
+      | Concat _ _ r1 r2 => Reduce (split (F _ r1) (F _ r2) s)
+      | Or _ _ r1 r2 => F _ r1 s || F _ r2 s
+      | Star _ r => dec_star _ _ _
+    end); simplify; first_order.
+Defined.
+
+Definition toBool A B (x : {A} + {B}) :=
+  if x then true else false.
+
+Example hi := Concat (Char "h"%char) (Char "i"%char).
+Compute toBool (matches hi "hi").
+Compute toBool (matches hi "bye").
+
+Example a_b := Or (Char "a"%char) (Char "b"%char).
+Compute toBool (matches a_b "").
+Compute toBool (matches a_b "a").
+Compute toBool (matches a_b "aa").
+Compute toBool (matches a_b "b").
+
+Example a_star := Star (Char "a"%char).
+Compute toBool (matches a_star "").
+Compute toBool (matches a_star "a").
+Compute toBool (matches a_star "b").
+Compute toBool (matches a_star "aa").

_CoqProject

@@ -38,6 +38,7 @@ SubsetTypes.v
 SubsetTypes_template.v
 LambdaCalculusAndTypeSoundness_template.v
 LambdaCalculusAndTypeSoundness.v
+DependentInductiveTypes_template.v
 DependentInductiveTypes.v
 TypesAndMutation.v
 DeepAndShallowEmbeddings_template.v