## VerifyThis @ FM 2012, problem 2

Computes prefix sums of an array, in-place

Authors: Claude Marché / François Bobot

Topics: Array Data Structure / Trees / Ghost code

Tools: Why3

References: VerifyThis @ FM2012

see also the index (by topic, by tool, by reference, by year)

# The VerifyThis competition at FM2012: Problem 2

```module PrefixSumRec

use int.Int
use int.ComputerDivision
use int.Power

use map.Map
use array.Array
use array.ArraySum

```

## Preliminary lemmas on division by 2 and power of 2

```  lemma Div_mod_2:
forall x:int. x >= 0 -> x >= 2 * div x 2 >= x-1
```

Needed for the proof of phase1_frame and phase1_frame2

```  predicate is_power_of_2 (x:int) = exists k:int. (k >= 0 /\ x = power 2 k)
```

The step must be a power of 2

```  (* needed *)
lemma is_power_of_2_1:
forall x:int. is_power_of_2 x -> x > 1 -> 2 * div x 2 = x

```

## Modeling the "upsweep" phase

```  function go_left (left right:int) : int =
let space = right - left in left - div space 2
```

Shortcuts

```  function go_right (left right:int) : int =
let space = right - left in right - div space 2

```

Description in a purely logic way the effect of the first phase "upsweep" of the algorithm. The second phase "downsweep" then traverses the array in the same way than the first phase. Hence, the inductive nature of this definition is not an issue.

```  inductive phase1 (left:int) (right:int) (a0: array int) (a: array int) =
| Leaf : forall left right:int, a0 a : array int.
right = left + 1 ->
a[left] = a0[left] ->
phase1 left right a0 a
| Node : forall left right:int, a0 a : array int.
right > left + 1 ->
phase1 (go_left left right) left a0 a ->
phase1 (go_right left right) right a0 a ->
a[left] = sum a0 (left-(right-left)+1) (left+1) ->
phase1 left right a0 a

```

Frame properties of the "phase1" inductive

```  let rec lemma phase1_frame (left right:int) (a0 a a' : array int) : unit
requires { forall i:int. left-(right-left) < i < right ->
a[i] = a'[i]}
requires { phase1 left right a0 a }
variant { right-left }
ensures { phase1 left right a0 a' } =
if right > left + 1 then begin
phase1_frame (go_left left right) left a0 a a';
phase1_frame (go_right left right) right a0 a a'
end
```

frame lemma for "phase1" on fourth argument. needed to prove both upsweep, downsweep and compute_sums

```  let rec lemma phase1_frame2 (left right:int) (a0 a0' a : array int) : unit
requires { forall i:int. left-(right-left) < i < right ->
a0[i] = a0'[i]}
requires { phase1 left right a0 a }
variant { right-left }
ensures { phase1 left right a0' a } =
if right > left + 1 then begin
phase1_frame2 (go_left left right) left a0 a0' a;
phase1_frame2 (go_right left right) right a0 a0' a
end
```

frame lemma for "phase1" on third argument. needed to prove upsweep and compute_sums

## The upsweep phase

First function: modify partially the table and compute some intermediate partial sums

```  let rec upsweep (left right: int) (a: array int)
requires { 0 <= left < right < a.length }
requires { -1 <= left - (right - left) }
requires { is_power_of_2 (right - left) }
variant { right - left }
ensures { phase1 left right (old a) a }
ensures { let space = right - left in
a[right] = sum (old a) (left-space+1) (right+1) /\
(forall i: int. i <= left-space -> a[i] = (old a)[i]) /\
(forall i: int. i > right -> a[i] = (old a)[i]) }
= let space = right - left in
if right > left+1 then begin
upsweep (left - div space 2) left a;
upsweep (right - div space 2) right a;
assert { phase1 (left - div space 2) left (old a) a };
assert { phase1 (right - div space 2) right (old a) a };
assert { a[left] = sum (old a) (left-(right-left)+1) (left+1) };
assert { a[right] = sum (old a) (left+1) (right+1) }
end;
a[right] <- a[left] + a[right];
assert {
right > left+1 -> phase1 (left - div space 2) left (old a) a };
assert {
right > left+1 -> phase1 (right - div space 2) right (old a) a }

```

## The downsweep phase

```  predicate partial_sum (left:int) (right:int) (a0 a : array int) =
forall i : int. (left-(right-left)) < i <= right -> a[i] = sum a0 0 i
```

The property we ultimately want to prove

```  let rec downsweep (left right: int) (ghost a0 : array int) (a : array int)
requires { 0 <= left < right < a.length }
requires { -1 <= left - (right - left) }
requires { is_power_of_2 (right - left) }
requires { a[right] = sum a0 0 (left-(right-left) + 1) }
requires { phase1 left right a0 a }
variant  { right - left }
ensures { partial_sum left right a0 a }
ensures { forall i: int. i <= left-(right-left) -> a[i] = (old a)[i] }
ensures { forall i: int. i > right -> a[i] = (old a)[i] }
= let tmp = a[right] in
assert { a[right] = sum a0 0 (left-(right-left) + 1) };
assert { a[left] = sum a0 (left-(right-left)+1) (left+1) };
a[right] <- a[right] + a[left];
a[left] <- tmp;
assert { a[right] = sum a0 0 (left + 1) };
if right > left+1 then
let space = right - left in
assert { phase1 (go_left left right) left a0 (old a) };
assert { phase1 (go_right left right) right a0 (old a) };
assert { phase1 (go_right left right) right a0 a };
downsweep (left - div space 2) left a0 a;
assert { phase1 (go_right left right) right a0 a };
downsweep (right - div space 2) right a0 a;
assert { partial_sum (left - div space 2) left a0 a };
assert { partial_sum (right - div space 2) right a0 a }
```

Second function: complete the partial using the remaining intial value and the partial sum already computed

## The main procedure

```  let compute_sums a
requires { a.length >= 2 }
requires { is_power_of_2 a.length }
ensures { forall i : int. 0 <= i < a.length -> a[i] = sum (old a) 0 i }
= let a0 = ghost (copy a) in
let l = a.length in
let left = div l 2 - 1 in
let right = l - 1 in
upsweep left right a;
(* needed for the precondition of downsweep *)
assert { phase1 left right a0 a };
a[right] <- 0;
downsweep left right a0 a;
(* needed to prove the post-condition *)
assert { forall i : int.
left-(right-left) < i <= right -> a[i] = sum a0 0 i }

```

## A simple test

```  let test_harness () =
let a = make 8 0 in
(* needed for the precondition of compute_sums *)
assert { power 2 3 = a.length };
a[0] <- 3; a[1] <- 1; a[2] <- 7; a[3] <- 0;
a[4] <- 4; a[5] <- 1; a[6] <- 6; a[7] <- 3;
compute_sums a;
assert { a[0] = 0 };
assert { a[1] = 3 };
assert { a[2] = 4 };
assert { a[3] = 11 };
assert { a[4] = 11 };
assert { a[5] = 15 };
assert { a[6] = 16 };
assert { a[7] = 22 }

exception BenchFailure

let bench () raises { BenchFailure -> true } =
let a = make 8 0 in
(* needed for the precondition of compute_sums *)
assert { power 2 3 = a.length };
a[0] <- 3; a[1] <- 1; a[2] <- 7; a[3] <- 0;
a[4] <- 4; a[5] <- 1; a[6] <- 6; a[7] <- 3;
compute_sums a;
if a[0] <> 0 then raise BenchFailure;
if a[1] <> 3 then raise BenchFailure;
if a[2] <> 4 then raise BenchFailure;
if a[3] <> 11 then raise BenchFailure;
if a[4] <> 11 then raise BenchFailure;
if a[5] <> 15 then raise BenchFailure;
if a[6] <> 16 then raise BenchFailure;
if a[7] <> 22 then raise BenchFailure;
a

end
```

# Why3 Proof Results for Project "verifythis_PrefixSumRec"

## Theory "verifythis_PrefixSumRec.PrefixSumRec": fully verified

 Obligations Alt-Ergo 2.3.0 Alt-Ergo 2.3.3 CVC3 2.4.1 CVC4 1.4 CVC4 1.5 CVC4 1.6 Z3 3.2 Z3 4.12.2 Z3 4.3.1 Div_mod_2 --- 0.01 0.02 0.03 --- --- 0.02 0.02 --- is_power_of_2_1 --- 0.01 0.03 0.02 --- --- --- --- --- VC for phase1_frame --- --- --- --- --- --- --- --- --- split_goal_right variant decrease --- 0.02 --- 0.02 --- --- --- --- --- precondition --- 0.02 --- 0.03 --- --- --- 0.01 --- precondition --- 0.01 --- 0.02 --- --- --- 0.02 --- variant decrease --- 0.01 --- 0.02 --- --- --- 0.01 --- precondition --- 0.01 --- 0.03 --- --- --- 0.01 --- precondition --- --- 0.04 --- --- --- --- --- --- postcondition 0.24 --- --- --- --- --- --- --- --- VC for phase1_frame2 --- --- --- --- --- --- --- --- --- split_goal_right variant decrease --- 0.02 --- 0.04 --- --- --- 0.01 --- precondition --- --- 0.03 --- --- --- 0.02 --- --- precondition --- 0.01 --- 0.02 --- --- --- 0.01 --- variant decrease --- 0.01 --- 0.02 --- --- --- --- --- precondition --- 0.02 --- 0.03 --- --- --- 0.01 --- precondition --- 0.02 --- 0.03 --- --- --- --- --- postcondition --- --- --- --- --- --- --- --- 0.04 VC for upsweep --- --- --- --- --- --- --- --- --- split_goal_right precondition --- 0.04 0.12 0.05 --- --- --- --- --- variant decrease --- 0.02 0.01 --- --- --- 0.09 0.11 --- precondition --- 0.10 0.01 0.05 --- --- --- --- --- precondition --- 0.01 0.00 0.02 --- --- 0.00 0.00 --- precondition --- 0.01 0.01 0.01 --- --- 16.09 --- --- precondition --- --- --- --- 0.03 --- --- --- --- variant decrease --- --- 0.02 --- --- --- 0.06 --- --- precondition --- --- 0.02 --- --- --- --- --- --- precondition --- 0.03 0.06 0.04 --- --- --- --- --- precondition --- 0.15 0.18 0.02 --- --- 10.72 --- --- assertion --- 0.02 --- --- --- --- --- --- --- assertion --- 0.02 --- --- --- --- --- --- --- assertion --- 0.00 0.01 --- --- --- 4.00 --- --- assertion --- --- 3.19 --- --- --- --- --- --- index in array bounds --- 0.02 0.02 0.02 --- --- 0.02 0.02 --- index in array bounds --- 0.02 0.02 0.02 --- --- 0.02 0.03 --- index in array bounds --- 0.01 --- 0.07 --- --- --- --- --- assertion --- 0.16 0.34 0.12 --- --- 0.15 --- --- assertion --- 0.14 0.14 0.12 --- --- 0.01 --- --- postcondition --- --- 0.24 --- --- --- 0.06 --- --- postcondition --- 0.67 --- 0.11 --- --- --- --- --- index in array bounds --- 0.04 --- 0.09 --- --- --- --- --- index in array bounds --- 0.01 0.02 0.02 --- --- --- --- --- index in array bounds --- 0.02 0.02 0.01 --- --- 0.02 0.05 --- assertion --- 0.00 0.04 0.07 --- --- --- --- --- assertion --- 0.00 0.00 0.01 --- --- 0.02 0.02 --- postcondition 0.01 --- --- --- --- --- --- --- --- postcondition --- 0.25 0.01 0.02 --- --- --- --- --- VC for downsweep --- --- --- --- --- --- --- --- --- split_goal_right index in array bounds --- 0.01 0.03 0.01 --- --- 0.02 0.04 --- assertion --- 0.03 0.02 0.02 --- --- --- --- --- assertion --- --- --- 0.06 --- --- 0.05 --- --- index in array bounds --- 0.01 0.02 0.02 --- --- 0.03 0.03 --- index in array bounds --- 0.00 0.02 0.07 --- --- --- --- --- index in array bounds --- 0.04 0.02 0.02 --- --- --- --- --- index in array bounds --- 0.08 --- 0.02 --- --- 0.02 --- --- assertion --- 0.04 --- 0.02 --- --- 4.79 --- --- assertion --- 0.01 0.02 0.01 --- --- 0.00 0.00 --- assertion --- 0.03 0.03 0.03 --- --- 0.04 0.04 --- assertion --- 0.01 1.45 0.01 --- --- 11.88 --- --- precondition --- 0.02 0.03 0.03 --- --- --- --- --- variant decrease --- 0.06 --- --- --- --- --- --- --- precondition --- 0.10 0.09 0.06 --- --- --- --- --- precondition --- 0.11 0.02 0.03 --- --- 0.02 --- --- precondition --- 0.44 --- --- --- 0.14 5.58 --- --- precondition --- --- --- 0.13 --- --- 0.04 --- --- precondition --- --- --- --- --- --- --- --- 0.97 assertion --- --- 0.08 --- --- --- 0.12 --- --- precondition --- 0.02 0.12 --- --- --- --- --- --- variant decrease --- 0.11 0.03 0.04 --- --- 0.08 0.11 --- precondition --- 0.02 0.06 0.02 --- --- 0.02 0.03 --- precondition --- --- --- --- 0.02 --- --- --- --- precondition --- 0.50 --- 0.27 --- --- 7.04 --- --- precondition --- 0.03 0.04 0.02 --- --- 0.02 --- --- precondition --- 0.02 0.02 --- --- --- 0.04 --- --- assertion --- --- --- --- 0.02 --- --- --- --- assertion --- --- 0.02 0.02 --- --- --- --- --- postcondition --- 0.04 0.68 0.04 --- --- --- --- --- postcondition --- --- --- 2.96 --- --- --- --- --- postcondition --- 0.02 --- --- --- --- --- --- --- postcondition --- 0.01 0.36 0.18 --- --- --- --- --- postcondition --- 0.00 0.06 0.01 --- --- 0.12 --- --- postcondition --- 0.12 --- 0.09 --- --- --- --- --- VC for compute_sums --- --- --- --- --- --- --- --- --- split_goal_right precondition --- --- 0.01 0.02 --- --- --- --- --- precondition --- 0.02 0.02 0.01 --- --- 1.35 0.03 --- precondition --- 0.02 0.03 0.02 --- --- 0.02 0.03 --- precondition --- --- --- 0.04 --- --- --- --- --- assertion --- 0.02 --- 0.04 --- --- --- --- --- index in array bounds --- 0.02 0.02 0.01 --- --- 0.03 0.03 --- precondition --- --- 0.03 0.04 --- --- 0.02 --- --- precondition --- --- --- --- 0.02 --- --- --- --- precondition --- --- 0.17 0.08 --- --- --- --- --- precondition --- 0.09 0.03 0.04 --- --- --- --- --- precondition --- 0.02 0.01 0.01 --- --- 0.63 --- --- assertion --- 0.02 0.03 0.02 --- --- 0.07 0.05 --- postcondition --- --- --- 8.32 --- --- --- --- --- VC for test_harness --- --- --- --- --- --- --- --- --- split_goal_right array creation size --- 0.02 0.01 0.01 --- --- 0.00 0.00 --- assertion --- 0.04 0.03 0.03 --- --- 0.05 --- --- index in array bounds --- 0.03 0.02 0.01 --- --- 0.02 0.04 --- index in array bounds --- 0.01 0.01 0.02 --- --- 0.00 0.00 --- index in array bounds --- 0.01 0.01 0.01 --- --- 0.00 0.00 --- index in array bounds --- 0.01 0.01 0.02 --- --- 0.00 0.00 --- index in array bounds --- 0.04 0.03 0.03 --- --- --- --- --- index in array bounds --- 0.02 0.03 0.02 --- --- 0.02 0.03 --- index in array bounds --- 0.01 0.02 0.01 --- --- 0.00 0.00 --- index in array bounds --- 0.01 0.01 0.01 --- --- 0.00 0.00 --- precondition --- 0.04 0.02 0.05 --- --- --- --- --- precondition --- 0.04 0.02 0.03 --- --- --- --- --- assertion --- 0.14 0.02 0.03 --- --- --- --- --- assertion --- 0.18 0.01 0.00 --- --- 0.18 1.06 --- assertion --- 0.20 0.02 0.04 --- --- 0.32 4.11 --- assertion --- 0.25 0.03 0.06 --- --- 0.33 4.24 --- assertion --- 0.23 0.03 0.04 --- --- 0.37 4.12 --- assertion --- 0.14 0.01 0.12 --- --- 0.34 3.69 --- assertion --- 0.16 0.02 0.13 --- --- 0.34 4.16 --- assertion --- 0.20 0.02 0.15 --- --- 0.18 0.78 --- VC for bench --- 0.12 --- --- --- --- --- --- ---