new example: binary_multiplication with 63-bit integers

examples/binary_multiplication.mlw

 
-(* Ancient Egyptian Multiplication
+(** {1 Russian peasant multiplication}
 
    Multiply two integers a and b using only addition, multiplication by 2,
    and division by 2.
@@ -13,7 +13,7 @@ module BinaryMultiplication
   use mach.int.Int
   use ref.Ref
 
-  let binary_mult (a b: int)
+  let binary_mult (a b: int) : int
     requires { b >= 0 }
     ensures  { result = a * b }
   = let x = ref a in
@@ -30,3 +30,40 @@ module BinaryMultiplication
     !z
 
 end
+
+(** Now using machine integers.
+
+    Assuming that the product fits in machine integers, we can still
+    verify the code. The only exception is when `a*b = min_int`.
+
+    The code below makes no assumption on the sign of `b`.
+    Instead, it uses the fact that `!y % 2` has the sign of `!y`
+    so that `!x` is either added to or subtracted from the result.
+*)
+
+module BinaryMultiplication63
+
+  use int.Int
+  use int.Abs
+  use mach.int.Int63
+  use ref.Ref
+
+  let binary_mult (a b: int63) : int63
+    requires { min_int < a * b <= max_int }
+    ensures  { result = a * b }
+  = let x = ref a in
+    let y = ref b in
+    let z = ref 0 in
+    while !y <> 0 do
+      invariant { if a*b >= 0 then !x * !y >= 0 && !z >= 0
+                              else !x * !y <= 0 && !z <= 0 }
+      invariant { !z + !x * !y = a * b }
+      variant   { abs !y }
+      z := !z + !x * (!y % 2);
+      y := !y / 2;
+      (* be careful not to make the very last multiplication *)
+      if !y <> 0 then x := 2 * !x
+    done;
+    !z
+
+end

examples/binary_multiplication/why3session.xml

@@ -3,11 +3,66 @@
 "http://why3.lri.fr/why3session.dtd">
 <why3session shape_version="5">
 <prover id="0" name="Alt-Ergo" version="2.0.0" timelimit="10" steplimit="0" memlimit="1000"/>
+<prover id="3" name="CVC4" version="1.5" timelimit="1" steplimit="0" memlimit="1000"/>
 <file name="../binary_multiplication.mlw" proved="true">
 <theory name="BinaryMultiplication" proved="true">
  <goal name="VC binary_mult" expl="VC for binary_mult" proved="true">
  <proof prover="0"><result status="valid" time="0.05" steps="63"/></proof>
  </goal>
 </theory>
+<theory name="BinaryMultiplication63" proved="true">
+ <goal name="VC binary_mult" expl="VC for binary_mult" proved="true">
+ <transf name="split_vc" proved="true" >
+  <goal name="VC binary_mult.0" expl="loop invariant init" proved="true">
+  <proof prover="3"><result status="valid" time="0.01"/></proof>
+  </goal>
+  <goal name="VC binary_mult.1" expl="loop invariant init" proved="true">
+  <proof prover="3"><result status="valid" time="0.00"/></proof>
+  </goal>
+  <goal name="VC binary_mult.2" expl="division by zero" proved="true">
+  <proof prover="3"><result status="valid" time="0.00"/></proof>
+  </goal>
+  <goal name="VC binary_mult.3" expl="integer overflow" proved="true">
+  <proof prover="3"><result status="valid" time="0.02"/></proof>
+  </goal>
+  <goal name="VC binary_mult.4" expl="integer overflow" proved="true">
+  <proof prover="3"><result status="valid" time="0.08"/></proof>
+  </goal>
+  <goal name="VC binary_mult.5" expl="integer overflow" proved="true">
+  <proof prover="3"><result status="valid" time="0.07"/></proof>
+  </goal>
+  <goal name="VC binary_mult.6" expl="division by zero" proved="true">
+  <proof prover="3"><result status="valid" time="0.00"/></proof>
+  </goal>
+  <goal name="VC binary_mult.7" expl="integer overflow" proved="true">
+  <proof prover="3"><result status="valid" time="0.01"/></proof>
+  </goal>
+  <goal name="VC binary_mult.8" expl="integer overflow" proved="true">
+  <proof prover="3"><result status="valid" time="0.22"/></proof>
+  </goal>
+  <goal name="VC binary_mult.9" expl="loop variant decrease" proved="true">
+  <proof prover="3"><result status="valid" time="0.01"/></proof>
+  </goal>
+  <goal name="VC binary_mult.10" expl="loop invariant preservation" proved="true">
+  <proof prover="3"><result status="valid" time="0.17"/></proof>
+  </goal>
+  <goal name="VC binary_mult.11" expl="loop invariant preservation" proved="true">
+  <proof prover="3"><result status="valid" time="0.73"/></proof>
+  </goal>
+  <goal name="VC binary_mult.12" expl="loop variant decrease" proved="true">
+  <proof prover="3"><result status="valid" time="0.01"/></proof>
+  </goal>
+  <goal name="VC binary_mult.13" expl="loop invariant preservation" proved="true">
+  <proof prover="3"><result status="valid" time="0.08"/></proof>
+  </goal>
+  <goal name="VC binary_mult.14" expl="loop invariant preservation" proved="true">
+  <proof prover="3"><result status="valid" time="0.11"/></proof>
+  </goal>
+  <goal name="VC binary_mult.15" expl="postcondition" proved="true">
+  <proof prover="3"><result status="valid" time="0.00"/></proof>
+  </goal>
+ </transf>
+ </goal>
+</theory>
 </file>
 </why3session>