Proper word-based vectorization

Basic wordization lemmas

Really complex derivation of wand correctness

1 files changed, 76 insertions, 0 deletions

diff --git a/src/Assembly/Vectorize.v b/src/Assembly/Vectorize.v
new file mode 100644
index 000000000..71d4d573d
--- /dev/null
+++ b/src/Assembly/Vectorize.v
@@ -0,0 +1,76 @@
+
+Require Export Bedrock.Word Bedrock.Nomega.
+Require Import NPeano NArith PArith Ndigits Compare_dec Arith.
+Require Import ProofIrrelevance Ring List.
+Require Import MultiBoundedWord.
+
+Section Vector.
+  Import ListNotations.
+
+  Definition vec T n := {x: list T | length x = n}.
+
+  Definition vget {n T} (x: vec T n) (i: {v: nat | (v < n)%nat}): T.
+    refine (
+      match (proj1_sig x) as x' return (proj1_sig x) = x' -> _ with
+      | [] => fun _ => _
+      | x0 :: xs => fun _ => nth (proj1_sig i) (proj1_sig x) x0
+      end eq_refl); abstract (
+      destruct x as [x xp], i as [i ip]; destruct x as [|x0 xs];
+      simpl in xp; subst; inversion _H; clear _H; omega).
+  Defined.
+
+  Lemma vget_spec: forall {T n} (x: vec T n) (i: {v: nat | (v < n)%nat}) (d: T),
+      vget x i = nth (proj1_sig i) (proj1_sig x) d.
+  Proof.
+    intros; destruct x as [x xp], i as [i ip];
+      destruct x as [|x0 xs]; induction i; unfold vget; simpl;
+      intuition; try (simpl in xp; subst; omega);
+      induction n; simpl in xp; try omega; clear IHi IHn.
+
+    apply nth_indep.
+    assert (length xs = n) by omega; subst.
+    omega.
+  Qed.
+
+  Definition vec0 {T} : vec T 0.
+    refine (exist _ [] _); abstract intuition.
+  Defined.
+
+  Lemma lift0: forall {T n} (x: T), vec (word n) 0 -> T.
+    intros; refine x.
+  Defined.
+
+  Lemma liftS: forall {T n m} (f: vec (word n) m -> word n -> T),
+    (vec (word n) (S m) -> T).
+  Proof.
+    intros T n m f v; destruct v as [v p]; induction m, v;
+      try (abstract (inversion p)).
+
+    - refine (f (exist _ [] _) w); abstract intuition.
+    - refine (f (exist _ v _) w); abstract intuition.
+  Defined.
+
+  Lemma vecToList: forall T n m (f: vec (word n) m -> T),
+    (list (word n) -> option T).
+  Proof.
+    intros T n m f x; destruct (Nat.eq_dec (length x) m).
+
+    - refine (Some (f (exist _ x _))); abstract intuition.
+    - refine None.
+  Defined.
+End Vector.
+
+Ltac vectorize :=
+  repeat match goal with
+   | [ |- context[?a - 1] ] =>
+     let c := eval simpl in (a - 1) in
+     replace (a - 1) with c by omega
+  | [ |- vec (word ?n) O -> ?T] => apply (@lift0 T n)
+  | [ |- vec (word ?n) ?m -> ?T] => apply (@liftS T n (m - 1))
+  end.
+
+Section Examples.
+  Lemma vectorize_example: (vec (word 32) 2 -> word 32).
+    vectorize; refine (@wplus 32).
+  Qed.
+End Examples.