Add verified shrd, aka funnel shift right. (#444)

We prove two specs: One that uses bitwise operations, and one that uses arithmetic operations.

bedrock2/src/bedrock2Examples/shrd.v

@@ -0,0 +1,91 @@
+Require Import bedrock2.NotationsCustomEntry.
+
+Import Syntax BinInt String.
+Local Open Scope string_scope.
+Local Open Scope Z_scope.
+
+Definition shrd :=
+  func! (lo, hi, n) ~> (res) {
+      res = lo >> n;
+      if n {
+        res = hi << ($64 - n) | res
+      }
+    }.
+
+From bedrock2 Require Import WeakestPrecondition ProgramLogic BasicC64Semantics.
+Import ProgramLogic.Coercions.
+
+Local Instance spec_of_shrd : spec_of "shrd" :=
+  fnspec! "shrd" (lo hi n : word) ~> res,
+    { requires t m := n < 64;
+      ensures T M :=
+        M = m /\ T = t /\
+        word.unsigned res = Z.lor (Z.shiftl hi (64 - n)) (Z.shiftr lo n) mod 2^64
+    }.
+
+Require Import bedrock2.ZnWords Coq.ZArith.ZArith Lia.
+
+Lemma shrd_ok : program_logic_goal_for_function! shrd.
+Proof.
+  repeat (straightline || apply WeakestPreconditionProperties.dexpr_expr).
+  split; intro n_cond; repeat (straightline || unfold cmd_body).
+  {
+    cbv [v res].
+    rewrite word.unsigned_or, word.unsigned_slu, word.unsigned_sub, word.unsigned_of_Z, word.unsigned_sru by ZnWords.
+    cbv [word.wrap].
+    change (64 mod 2^64) with 64.
+    rewrite (Z.mod_small (64 - _)) by ZnWords.
+    rewrite <-!Z.land_ones, !Z.land_lor_distr_l, !Z.land_ones by lia.
+    rewrite !Zmod_mod.
+    trivial.
+  }
+  {
+    cbv [res].
+    rewrite word.unsigned_sru by lia.
+    cbv [word.wrap].
+    rewrite n_cond, Z.shiftr_0_r.
+    change (64 - 0) with 64.
+    rewrite <-!Z.land_ones, !Z.land_lor_distr_l, !Z.land_ones by lia.
+    rewrite Z.shiftl_mul_pow2, Z.mod_mul by lia.
+    rewrite Z.lor_0_l.
+    trivial.
+  }
+Qed.
+
+Section ArithmeticSpec.
+  Local Instance spec_of_shrd_arith : spec_of "shrd" :=
+    fnspec! "shrd" (lo hi n : word) ~> res,
+      { requires t m := word.unsigned n < 64;
+        ensures T M :=
+          M = m /\ T = t /\
+          res = (lo + 2^64 * hi) / 2^n mod 2^64 :> Z
+      }.
+
+  Lemma shdr_ok_arith : program_logic_goal_for_function! shrd.
+  Proof.
+    cbv [program_logic_goal_for spec_of_shrd_arith].
+    intros.
+    eapply WeakestPreconditionProperties.Proper_call; [|apply shrd_ok]; trivial.
+    intros ? ? ? spec.
+    destruct spec as (res & ? & ? & ? & spec).
+    exists res.
+    repeat split; trivial.
+    rewrite spec.
+    rewrite <-BitOps.or_to_plus.
+    {
+      rewrite Z.mul_comm.
+      rewrite <-Z.shiftl_mul_pow2 by lia.
+      rewrite <-Z.shiftr_div_pow2, Z.shiftr_lor, Z.shiftr_shiftl_l by ZnWords.
+      rewrite Z.lor_comm.
+      trivial.
+    }
+    {
+      rewrite <-Properties.word.wrap_unsigned.
+      rewrite <-Z.land_ones by lia.
+      rewrite <-Z.land_assoc, (Z.land_comm (Z.ones 64) _).
+      rewrite Z.land_ones, Z.mul_comm, Z.mod_mul by lia.
+      rewrite Z.land_0_r.
+      trivial.
+    }
+  Qed.
+End ArithmeticSpec.