* [Ada] Further adapt proof of double arithmetic runtime unit
@ 2022-05-19 14:15 Pierre-Marie de Rodat
0 siblings, 0 replies; only message in thread
From: Pierre-Marie de Rodat @ 2022-05-19 14:15 UTC (permalink / raw)
To: gcc-patches; +Cc: Yannick Moy
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After changes in Why3 and generation of VCs, ghost code needs to be
adapted for proofs to remain automatic.
Tested on x86_64-pc-linux-gnu, committed on trunk
gcc/ada/
* libgnat/s-aridou.adb (Lemma_Abs_Range,
Lemma_Double_Shift_Left, Lemma_Shift_Left): New lemmas.
(Double_Divide): Add ghost code.
(Lemma_Concat_Definition, Lemma_Double_Shift_Left,
Lemma_Shift_Left, Lemma_Shift_Right): Define or complete lemmas.
(Scaled_Divide): Add ghost code.
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diff --git a/gcc/ada/libgnat/s-aridou.adb b/gcc/ada/libgnat/s-aridou.adb
--- a/gcc/ada/libgnat/s-aridou.adb
+++ b/gcc/ada/libgnat/s-aridou.adb
@@ -208,6 +208,13 @@ is
Ghost,
Post => abs (X * Y) = abs X * abs Y;
+ procedure Lemma_Abs_Range (X : Big_Integer)
+ with
+ Ghost,
+ Pre => In_Double_Int_Range (X),
+ Post => abs (X) <= Big_2xxDouble_Minus_1
+ and then In_Double_Int_Range (-abs (X));
+
procedure Lemma_Abs_Rem_Commutation (X, Y : Big_Integer)
with
Ghost,
@@ -306,6 +313,20 @@ is
Pre => S <= Double_Size - S1,
Post => Shift_Left (Shift_Left (X, S), S1) = Shift_Left (X, S + S1);
+ procedure Lemma_Double_Shift_Left (X : Double_Uns; S, S1 : Double_Uns)
+ with
+ Ghost,
+ Pre => S <= Double_Uns (Double_Size)
+ and then S1 <= Double_Uns (Double_Size),
+ Post => Shift_Left (Shift_Left (X, Natural (S)), Natural (S1)) =
+ Shift_Left (X, Natural (S + S1));
+
+ procedure Lemma_Double_Shift_Left (X : Double_Uns; S, S1 : Natural)
+ with
+ Ghost,
+ Pre => S <= Double_Size - S1,
+ Post => Shift_Left (Shift_Left (X, S), S1) = Shift_Left (X, S + S1);
+
procedure Lemma_Double_Shift_Right (X : Double_Uns; S, S1 : Double_Uns)
with
Ghost,
@@ -505,6 +526,13 @@ is
Pre => A = B * Q + R and then R < B,
Post => Q = A / B and then R = A rem B;
+ procedure Lemma_Shift_Left (X : Double_Uns; Shift : Natural)
+ with
+ Ghost,
+ Pre => Shift < Double_Size
+ and then Big (X) * Big_2xx (Shift) < Big_2xxDouble,
+ Post => Big (Shift_Left (X, Shift)) = Big (X) * Big_2xx (Shift);
+
procedure Lemma_Shift_Right (X : Double_Uns; Shift : Natural)
with
Ghost,
@@ -560,10 +588,10 @@ is
procedure Inline_Le3 (X1, X2, X3, Y1, Y2, Y3 : Single_Uns) is null;
procedure Lemma_Abs_Commutation (X : Double_Int) is null;
procedure Lemma_Abs_Mult_Commutation (X, Y : Big_Integer) is null;
+ procedure Lemma_Abs_Range (X : Big_Integer) is null;
procedure Lemma_Add_Commutation (X : Double_Uns; Y : Single_Uns) is null;
procedure Lemma_Add_One (X : Double_Uns) is null;
procedure Lemma_Bounded_Powers_Of_2_Increasing (M, N : Natural) is null;
- procedure Lemma_Concat_Definition (X, Y : Single_Uns) is null;
procedure Lemma_Deep_Mult_Commutation
(Factor : Big_Integer;
X, Y : Single_Uns)
@@ -581,6 +609,8 @@ is
procedure Lemma_Double_Big_2xxSingle is null;
procedure Lemma_Double_Shift (X : Double_Uns; S, S1 : Double_Uns) is null;
procedure Lemma_Double_Shift (X : Single_Uns; S, S1 : Natural) is null;
+ procedure Lemma_Double_Shift_Left (X : Double_Uns; S, S1 : Double_Uns)
+ is null;
procedure Lemma_Double_Shift_Right (X : Double_Uns; S, S1 : Double_Uns)
is null;
procedure Lemma_Ge_Commutation (A, B : Double_Uns) is null;
@@ -949,6 +979,7 @@ is
pragma Assert (if X = Double_Int'First and then Round then
Mult > Big_2xxDouble);
elsif Ylo > 0 then
+ pragma Assert (Double_Uns'(Ylo * Zhi) > 0);
pragma Assert (Big (Double_Uns'(Ylo * Zhi)) > 0);
pragma Assert (if X = Double_Int'First and then Round then
Mult > Big_2xxDouble);
@@ -1024,15 +1055,24 @@ is
pragma Assert (Big (Double_Uns (Hi (T2))) >= 1);
pragma Assert (Big (Double_Uns (Lo (T2))) >= 0);
pragma Assert (Big (Double_Uns (Lo (T1))) >= 0);
+ pragma Assert (Mult >= Big_2xxDouble * Big (Double_Uns (Hi (T2))));
pragma Assert (Mult >= Big_2xxDouble);
if Hi (T2) > 1 then
pragma Assert (Big (Double_Uns (Hi (T2))) > 1);
+ pragma Assert (if X = Double_Int'First and then Round then
+ Mult > Big_2xxDouble);
elsif Lo (T2) > 0 then
pragma Assert (Big (Double_Uns (Lo (T2))) > 0);
+ pragma Assert (if X = Double_Int'First and then Round then
+ Mult > Big_2xxDouble);
elsif Lo (T1) > 0 then
pragma Assert (Double_Uns (Lo (T1)) > 0);
Lemma_Gt_Commutation (Double_Uns (Lo (T1)), 0);
pragma Assert (Big (Double_Uns (Lo (T1))) > 0);
+ pragma Assert (if X = Double_Int'First and then Round then
+ Mult > Big_2xxDouble);
+ else
+ pragma Assert (not (X = Double_Int'First and then Round));
end if;
Prove_Quotient_Zero;
end if;
@@ -1172,6 +1212,18 @@ is
end if;
end Lemma_Abs_Rem_Commutation;
+ -----------------------------
+ -- Lemma_Concat_Definition --
+ -----------------------------
+
+ procedure Lemma_Concat_Definition (X, Y : Single_Uns) is
+ Hi : constant Double_Uns := Shift_Left (Double_Uns (X), Single_Size);
+ Lo : constant Double_Uns := Double_Uns (Y);
+ begin
+ pragma Assert (Hi = Double_Uns'(2 ** Single_Size) * Double_Uns (X));
+ pragma Assert ((Hi or Lo) = Hi + Lo);
+ end Lemma_Concat_Definition;
+
------------------------
-- Lemma_Double_Shift --
------------------------
@@ -1185,6 +1237,19 @@ is
= Shift_Left (X, Natural (Double_Uns (S + S1))));
end Lemma_Double_Shift;
+ -----------------------------
+ -- Lemma_Double_Shift_Left --
+ -----------------------------
+
+ procedure Lemma_Double_Shift_Left (X : Double_Uns; S, S1 : Natural) is
+ begin
+ Lemma_Double_Shift_Left (X, Double_Uns (S), Double_Uns (S1));
+ pragma Assert (Shift_Left (Shift_Left (X, S), S1)
+ = Shift_Left (Shift_Left (X, S), Natural (Double_Uns (S1))));
+ pragma Assert (Shift_Left (X, S + S1)
+ = Shift_Left (X, Natural (Double_Uns (S + S1))));
+ end Lemma_Double_Shift_Left;
+
------------------------------
-- Lemma_Double_Shift_Right --
------------------------------
@@ -1328,15 +1393,78 @@ is
Lemma_Neg_Rem (X, Y);
end Lemma_Rem_Abs;
+ ----------------------
+ -- Lemma_Shift_Left --
+ ----------------------
+
+ procedure Lemma_Shift_Left (X : Double_Uns; Shift : Natural) is
+
+ procedure Lemma_Mult_Pow2 (X : Double_Uns; I : Natural)
+ with
+ Ghost,
+ Pre => I < Double_Size - 1,
+ Post => X * Double_Uns'(2) ** I * Double_Uns'(2)
+ = X * Double_Uns'(2) ** (I + 1);
+
+ procedure Lemma_Mult_Pow2 (X : Double_Uns; I : Natural) is
+ Mul1 : constant Double_Uns := Double_Uns'(2) ** I;
+ Mul2 : constant Double_Uns := Double_Uns'(2);
+ Left : constant Double_Uns := X * Mul1 * Mul2;
+ begin
+ pragma Assert (Left = X * (Mul1 * Mul2));
+ pragma Assert (Mul1 * Mul2 = Double_Uns'(2) ** (I + 1));
+ end Lemma_Mult_Pow2;
+
+ XX : Double_Uns := X;
+
+ begin
+ for J in 1 .. Shift loop
+ declare
+ Cur_XX : constant Double_Uns := XX;
+ begin
+ XX := Shift_Left (XX, 1);
+ pragma Assert (XX = Cur_XX * Double_Uns'(2));
+ Lemma_Mult_Pow2 (X, J - 1);
+ end;
+ Lemma_Double_Shift_Left (X, J - 1, 1);
+ pragma Loop_Invariant (XX = Shift_Left (X, J));
+ pragma Loop_Invariant (XX = X * Double_Uns'(2) ** J);
+ end loop;
+ end Lemma_Shift_Left;
+
-----------------------
-- Lemma_Shift_Right --
-----------------------
procedure Lemma_Shift_Right (X : Double_Uns; Shift : Natural) is
+
+ procedure Lemma_Div_Pow2 (X : Double_Uns; I : Natural)
+ with
+ Ghost,
+ Pre => I < Double_Size - 1,
+ Post => X / Double_Uns'(2) ** I / Double_Uns'(2)
+ = X / Double_Uns'(2) ** (I + 1);
+
+ procedure Lemma_Div_Pow2 (X : Double_Uns; I : Natural) is
+ Div1 : constant Double_Uns := Double_Uns'(2) ** I;
+ Div2 : constant Double_Uns := Double_Uns'(2);
+ Left : constant Double_Uns := X / Div1 / Div2;
+ begin
+ pragma Assert (Left = X / (Div1 * Div2));
+ pragma Assert (Div1 * Div2 = Double_Uns'(2) ** (I + 1));
+ end Lemma_Div_Pow2;
+
XX : Double_Uns := X;
+
begin
for J in 1 .. Shift loop
- XX := Shift_Right (XX, 1);
+ declare
+ Cur_XX : constant Double_Uns := XX;
+ begin
+ XX := Shift_Right (XX, 1);
+ pragma Assert (XX = Cur_XX / Double_Uns'(2));
+ Lemma_Div_Pow2 (X, J - 1);
+ end;
Lemma_Double_Shift_Right (X, J - 1, 1);
pragma Loop_Invariant (XX = Shift_Right (X, J));
pragma Loop_Invariant (XX = X / Double_Uns'(2) ** J);
@@ -1607,6 +1735,7 @@ is
"Intentional Unsigned->Signed conversion");
else
Prove_Neg_Int;
+ Lemma_Abs_Range (Big (X) * Big (Y));
return To_Neg_Int (T2);
end if;
else -- X < 0
@@ -1617,6 +1746,7 @@ is
"Intentional Unsigned->Signed conversion");
else
Prove_Neg_Int;
+ Lemma_Abs_Range (Big (X) * Big (Y));
return To_Neg_Int (T2);
end if;
end if;
@@ -1901,6 +2031,9 @@ is
procedure Prove_Dividend_Scaling is
begin
+ Lemma_Shift_Left (D (1) & D (2), Scale);
+ Lemma_Shift_Left (Double_Uns (D (3)), Scale);
+ Lemma_Shift_Left (Double_Uns (D (4)), Scale);
Lemma_Hi_Lo (D (1) & D (2), D (1), D (2));
pragma Assert (Mult * Big_2xx (Scale) =
Big_2xxSingle
@@ -2116,6 +2249,7 @@ is
pragma Assert (Double_Uns (Lo (T1 rem Zlo)) = T1 rem Zlo);
Lemma_Hi_Lo (T2, Lo (T1 rem Zlo), D (4));
pragma Assert (T1 rem Zlo + Double_Uns'(1) <= Double_Uns (Zlo));
+ Lemma_Ge_Commutation (Double_Uns (Zlo), T1 rem Zlo + Double_Uns'(1));
Lemma_Add_Commutation (T1 rem Zlo, 1);
pragma Assert (Big (T1 rem Zlo) + 1 <= Big (Double_Uns (Zlo)));
Lemma_Div_Definition (T2, Zlo, T2 / Zlo, Ru);
@@ -2567,6 +2701,21 @@ is
elsif D (J) = Zhi then
Qd (J) := Single_Uns'Last;
+ Lemma_Concat_Definition (D (J), D (J + 1));
+ pragma Assert (Big_2xxSingle > Big (Double_Uns (D (J + 2))));
+ pragma Assert (Big3 (D (J), D (J + 1), 0) + Big_2xxSingle
+ > Big3 (D (J), D (J + 1), D (J + 2)));
+ pragma Assert (Big (Double_Uns'(0)) = 0);
+ pragma Assert (Big (D (J) & D (J + 1)) * Big_2xxSingle =
+ Big_2xxSingle * (Big_2xxSingle * Big (Double_Uns (D (J)))
+ + Big (Double_Uns (D (J + 1)))));
+ pragma Assert (Big (D (J) & D (J + 1)) * Big_2xxSingle =
+ Big_2xxSingle * Big_2xxSingle * Big (Double_Uns (D (J)))
+ + Big_2xxSingle * Big (Double_Uns (D (J + 1))));
+ pragma Assert (Big (D (J) & D (J + 1)) * Big_2xxSingle
+ = Big3 (D (J), D (J + 1), 0));
+ pragma Assert ((Big (D (J) & D (J + 1)) + 1) * Big_2xxSingle
+ = Big3 (D (J), D (J + 1), 0) + Big_2xxSingle);
Lemma_Gt_Mult (Big (Zu), Big (D (J) & D (J + 1)) + 1,
Big_2xxSingle,
Big3 (D (J), D (J + 1), D (J + 2)));
@@ -2617,6 +2766,8 @@ is
pragma Loop_Invariant (Qd (J)'Initialized);
pragma Loop_Invariant
(Big3 (S1, S2, S3) = Big (Double_Uns (Qd (J))) * Big (Zu));
+ pragma Loop_Invariant
+ (Big3 (S1, S2, S3) > Big3 (D (J), D (J + 1), D (J + 2)));
pragma Assert (Big3 (S1, S2, S3) > 0);
if Qd (J) = 0 then
pragma Assert (Big3 (S1, S2, S3) = 0);
@@ -2632,6 +2783,9 @@ is
(Big3 (S1, S2, S3) >
Big3 (D (J), D (J + 1), D (J + 2)) - Big (Zu));
Lemma_Subtract_Commutation (Double_Uns (Qd (J)), 1);
+ pragma Assert (Double_Uns (Qd (J)) - Double_Uns'(1)
+ = Double_Uns (Qd (J) - 1));
+ pragma Assert (Big (Double_Uns'(1)) = 1);
Lemma_Substitution (Big3 (S1, S2, S3), Big (Zu),
Big (Double_Uns (Qd (J))) - 1,
Big (Double_Uns (Qd (J) - 1)), 0);
@@ -2660,8 +2814,7 @@ is
pragma Assert (Big3 (D (J), D (J + 1), D (J + 2)) < Big (Zu));
if D (J) > 0 then
- pragma Assert
- (Big_2xxSingle * Big_2xxSingle = Big_2xxDouble);
+ Lemma_Double_Big_2xxSingle;
pragma Assert (Big3 (D (J), D (J + 1), D (J + 2)) =
Big_2xxSingle
* Big_2xxSingle * Big (Double_Uns (D (J)))
@@ -2671,9 +2824,22 @@ is
Big_2xxDouble * Big (Double_Uns (D (J)))
+ Big_2xxSingle * Big (Double_Uns (D (J + 1)))
+ Big (Double_Uns (D (J + 2))));
+ pragma Assert (Big_2xxSingle >= 0);
+ pragma Assert (Big (Double_Uns (D (J + 1))) >= 0);
+ pragma Assert
+ (Big_2xxSingle * Big (Double_Uns (D (J + 1))) >= 0);
+ pragma Assert
+ (Big_2xxSingle * Big (Double_Uns (D (J + 1)))
+ + Big (Double_Uns (D (J + 2))) >= 0);
pragma Assert (Big3 (D (J), D (J + 1), D (J + 2)) >=
Big_2xxDouble * Big (Double_Uns (D (J))));
Lemma_Ge_Commutation (Double_Uns (D (J)), Double_Uns'(1));
+ Lemma_Ge_Mult (Big (Double_Uns (D (J))),
+ Big (Double_Uns'(1)),
+ Big_2xxDouble,
+ Big (Double_Uns'(1)) * Big_2xxDouble);
+ pragma Assert
+ (Big_2xxDouble * Big (Double_Uns'(1)) = Big_2xxDouble);
pragma Assert
(Big3 (D (J), D (J + 1), D (J + 2)) >= Big_2xxDouble);
pragma Assert (False);
@@ -3039,6 +3205,7 @@ is
begin
pragma Assert (Ru = Double_Uns (X) - Double_Uns (Y));
if Ru < 2 ** (Double_Size - 1) then -- R >= 0
+ pragma Assert (To_Uns (Y) <= To_Uns (X));
Lemma_Subtract_Double_Uns (X => Y, Y => X);
pragma Assert (Ru = Double_Uns (X - Y));
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