Start of Featherweight Ur semantics

1 files changed, 156 insertions, 0 deletions

diff --git a/src/coq/Semantics.v b/src/coq/Semantics.v
new file mode 100644
index 00000000..c3e99b5d
--- /dev/null
+++ b/src/coq/Semantics.v
@@ -0,0 +1,156 @@
+(* Copyright (c) 2009, Adam Chlipala
+ * All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions are met:
+ *
+ * - Redistributions of source code must retain the above copyright notice,
+ *   this list of conditions and the following disclaimer.
+ * - Redistributions in binary form must reproduce the above copyright notice,
+ *   this list of conditions and the following disclaimer in the documentation
+ *   and/or other materials provided with the distribution.
+ * - The names of contributors may not be used to endorse or promote products
+ *   derived from this software without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+ * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+ * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 
+ * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+ * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+ * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+ * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+ * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+ * POSSIBILITY OF SUCH DAMAGE.
+ *)
+
+Require Import Arith List Omega TheoryList.
+
+Require Import Syntax.
+
+Set Implicit Arguments.
+
+
+Section row'.
+  Variable A : Type.
+
+  Inductive row' : list name -> Type :=
+  | Nil : row' nil
+  | Cons : forall n ls, A -> AllS (lt n) ls -> row' ls -> row' (n :: ls).
+End row'.
+
+Implicit Arguments Nil [A].
+
+Record row (A : Type) : Type := Row {
+  keys : list name;
+  data : row' A keys
+}.
+
+Inductive record' : forall ls, row' Set ls -> Set :=
+| RNil : record' Nil
+| RCons : forall n ls (T : Set) (pf : AllS (lt n) ls) r, T -> record' r -> record' (Cons T pf r).
+
+Definition record (r : row Set) := record' (data r).
+
+Fixpoint kDen (k : kind) : Type :=
+  match k with
+    | KType => Set
+    | KName => name
+    | KArrow k1 k2 => kDen k1 -> kDen k2
+    | KRecord k1 => row (kDen k1)
+  end.
+
+Axiom cheat : forall T, T.
+
+Fixpoint cinsert (n : name) (ls : list name) {struct ls} : list name :=
+  match ls with
+    | nil => n :: nil
+    | n' :: ls' =>
+      if eq_nat_dec n n'
+        then ls
+        else if le_lt_dec n n'
+          then n :: ls
+          else n' :: cinsert n ls'
+  end.
+
+Hint Constructors AllS.
+Hint Extern 1 (_ < _) => omega.
+
+Lemma insert_front' : forall n n',
+  n <> n'
+  -> n <= n'
+  -> forall ls, AllS (lt n') ls
+    -> AllS (lt n) ls.
+  induction 3; auto.
+Qed.
+
+Lemma insert_front : forall n n',
+  n <> n'
+  -> n <= n'
+  -> forall ls, AllS (lt n') ls
+    -> AllS (lt n) (n' :: ls).
+  Hint Resolve insert_front'.
+  eauto.
+Qed.
+
+Lemma insert_continue : forall n n',
+  n <> n'
+  -> n' < n
+  -> forall ls, AllS (lt n') ls
+    -> AllS (lt n') (cinsert n ls).
+  induction 3; simpl; auto;
+    repeat (match goal with
+              | [ |- context[if ?E then _ else _] ] => destruct E
+            end; auto).
+Qed.
+
+Fixpoint insert T (n : name) (v : T) ls (r : row' T ls) {struct r} : row' T (cinsert n ls) :=
+  match r in row' _ ls return row' T (cinsert n ls) with
+    | Nil => Cons (n := n) v (allS_nil _) Nil
+    | Cons n' ls' v' pf r' =>
+      match eq_nat_dec n n' as END
+        return row' _ (if END then _ else _) with
+        | left _ => Cons (n := n') v' pf r'
+        | right pfNe =>
+          match le_lt_dec n n' as LLD
+            return row' _ (if LLD then _ else _) with
+            | left pfLe => Cons (n := n) v (insert_front pfNe pfLe pf) (Cons (n := n') v' pf r')
+            | right pfLt => Cons (n := n') v' (insert_continue pfNe pfLt pf) (insert n v r')
+          end
+      end
+  end.
+
+Fixpoint cconcat (ls1 ls2 : list name) {struct ls1} : list name :=
+  match ls1 with
+    | nil => ls2
+    | n :: ls1' => cinsert n (cconcat ls1' ls2)
+  end.
+
+Fixpoint concat T ls1 ls2 (r1 : row' T ls1) (r2 : row' T ls2) {struct r1} : row' T (cconcat ls1 ls2) :=
+  match r1 in row' _ ls1 return row' _ (cconcat ls1 _) with
+    | Nil => r2
+    | Cons n _ v _ r1' => insert n v (concat r1' r2)
+  end.
+
+Fixpoint cfold T T' (f : name -> T -> T' -> T') (i : T') ls (r : row' T ls) {struct r} : T' :=
+  match r with
+    | Nil => i
+    | Cons n _ v _ r' => f n v (cfold f i r')
+  end.
+
+Fixpoint cDen k (c : con kDen k) {struct c} : kDen k :=
+  match c in con _ k return kDen k with
+    | CVar _ x => x
+    | Arrow c1 c2 => cDen c1 -> cDen c2
+    | Poly _ c1 => forall x, cDen (c1 x)
+    | CAbs _ _ c1 => fun x => cDen (c1 x)
+    | CApp _ _ c1 c2 => (cDen c1) (cDen c2)
+    | Name n => n
+    | TRecord c1 => record (cDen c1)
+    | CEmpty _ => Row Nil
+    | CSingle _ c1 c2 => Row (Cons (n := cDen c1) (cDen c2) (allS_nil _) Nil)
+    | CConcat _ c1 c2 => Row (concat (data (cDen c1)) (data (cDen c2)))
+    | CFold k1 k2 => fun f i r => cfold f i (data r)
+    | CGuarded _ _ _ _ c => cDen c
+  end.