IterAssocOp: now uses arbitrary representation of scalar that implements testbit

1 files changed, 29 insertions, 23 deletions

diff --git a/src/Util/IterAssocOp.v b/src/Util/IterAssocOp.v
index 238fdfe73..fc21b997b 100644
--- a/src/Util/IterAssocOp.v
+++ b/src/Util/IterAssocOp.v
@@ -9,7 +9,11 @@ Section IterAssocOp.
           (assoc: forall a b c, op a (op b c) === op (op a b) c)
           (neutral:T)
           (neutral_l : forall a, op neutral a === a)
-          (neutral_r : forall a, op a neutral === a).
+          (neutral_r : forall a, op a neutral === a)
+          {scalar : Type}
+          (testbit : scalar -> nat -> bool)
+          (scToN : scalar -> N)
+          (testbit_spec : forall x i, testbit x i = N.testbit_nat (scToN x) i).
   Existing Instance op_proper.
 
   Fixpoint nat_iter_op n a :=
@@ -51,19 +55,19 @@ Section IterAssocOp.
     | S exp' => f (funexp f a exp')
     end.
 
-  Definition test_and_op n a (state : nat * T) :=
+  Definition test_and_op sc a (state : nat * T) :=
     let '(i, acc) := state in
     let acc2 := op acc acc in
     match i with
     | O => (0, acc)
-    | S i' => (i', if N.testbit_nat n i' then op a acc2 else acc2)
+    | S i' => (i', if testbit sc i' then op a acc2 else acc2)
     end.
 
-  Definition iter_op n a bound : T := 
-    snd (funexp (test_and_op n a) (bound, neutral) bound).
+  Definition iter_op sc a bound : T := 
+    snd (funexp (test_and_op sc a) (bound, neutral) bound).
 
-  Definition test_and_op_inv n a (s : nat * T) :=
-    snd s === nat_iter_op (N.to_nat (N.shiftr_nat n (fst s))) a.
+  Definition test_and_op_inv sc a (s : nat * T) :=
+    snd s === nat_iter_op (N.to_nat (N.shiftr_nat (scToN sc) (fst s))) a.
 
   Hint Rewrite
     N.succ_double_spec
@@ -91,7 +95,7 @@ Section IterAssocOp.
     reflexivity.
   Qed.
 
-  Lemma shiftr_succ : forall n i,
+  Lemma Nshiftr_succ : forall n i,
     N.to_nat (N.shiftr_nat n i) =
     if N.testbit_nat n i
     then S (2 * N.to_nat (N.shiftr_nat n (S i)))
@@ -115,21 +119,23 @@ Section IterAssocOp.
       apply N2Nat.id.
   Qed.
 
-  Lemma test_and_op_inv_step : forall n a s,
-    test_and_op_inv n a s ->
-    test_and_op_inv n a (test_and_op n a s).
+  Lemma test_and_op_inv_step : forall sc a s,
+    test_and_op_inv sc a s ->
+    test_and_op_inv sc a (test_and_op sc a s).
   Proof.
     destruct s as [i acc].
     unfold test_and_op_inv, test_and_op; simpl; intro Hpre.
     destruct i; [ apply Hpre | ].
     simpl.
-    rewrite shiftr_succ.
-    case_eq (N.testbit_nat n i); intro; simpl; rewrite Hpre, <- plus_n_O, nat_iter_op_plus; reflexivity.
+    rewrite Nshiftr_succ.
+    case_eq (testbit sc i); intro testbit_eq; simpl;
+      rewrite testbit_spec in testbit_eq; rewrite testbit_eq;
+      rewrite Hpre, <- plus_n_O, nat_iter_op_plus; reflexivity.
   Qed.
 
-  Lemma test_and_op_inv_holds : forall n a i s,
-    test_and_op_inv n a s ->
-    test_and_op_inv n a (funexp (test_and_op n a) s i).
+  Lemma test_and_op_inv_holds : forall sc a i s,
+    test_and_op_inv sc a s ->
+    test_and_op_inv sc a (funexp (test_and_op sc a) s i).
   Proof.
     induction i; intros; auto; simpl; apply test_and_op_inv_step; auto.
   Qed.
@@ -144,14 +150,14 @@ Section IterAssocOp.
     destruct i; rewrite NPeano.Nat.sub_succ_r; subst; rewrite <- IHy; simpl; reflexivity.
   Qed.
 
-  Lemma iter_op_termination : forall n a bound, N.size_nat n <= bound ->
-    test_and_op_inv n a 
-      (funexp (test_and_op n a) (bound, neutral) bound) -> 
-    iter_op n a bound === nat_iter_op (N.to_nat n) a.
+  Lemma iter_op_termination : forall sc a bound,
+    N.size_nat (scToN sc) <= bound ->
+    test_and_op_inv sc a 
+      (funexp (test_and_op sc a) (bound, neutral) bound) -> 
+    iter_op sc a bound === nat_iter_op (N.to_nat (scToN sc)) a.
   Proof.
     unfold test_and_op_inv, iter_op; simpl; intros ? ? ? ? Hinv.
     rewrite Hinv, funexp_test_and_op_index, Minus.minus_diag.
-    replace (N.shiftr_nat n 0) with n by auto.
     reflexivity.
   Qed.
 
@@ -178,8 +184,8 @@ Section IterAssocOp.
     auto.
   Qed.
   
-  Lemma iter_op_spec : forall n a bound, N.size_nat n <= bound ->
-    iter_op n a bound === nat_iter_op (N.to_nat n) a.
+  Lemma iter_op_spec : forall sc a bound, N.size_nat (scToN sc) <= bound ->
+    iter_op sc a bound === nat_iter_op (N.to_nat (scToN sc)) a.
   Proof.
     intros.
     apply iter_op_termination; auto.