Fix integer.mul_by_digit producing denormalized zeros. (#141)

mul_by_digit(0, Y) produced i(Len, [0, 0, ...]) instead of the
canonical integer.zero = i(0, []). Since is_zero/1 only matched
the canonical form, denormalized zeros were silently treated as
nonzero, causing incorrect results in rational.m (e.g., gcd_2
non-termination, division-by-zero crashes in big_quot_rem).

Three fixes applied:
- Guard mul_by_digit and printbase_mul_by_digit to return
  integer.zero when the digit is 0 (root cause fix).
- Make is_zero/1 recognize denormalized all-zero digit lists
  as defense in depth.
- Replace structural `= integer.zero` checks in rational.m
  with integer.is_zero/1 calls for robustness.

Co-authored-by: Claude Opus 4.6 <noreply@anthropic.com>

library/integer.m

@@ -642,6 +642,9 @@ sixteen = i(1, [16]).
 abs(N) = big_abs(N).
 
 is_zero(i(0, [])).
+is_zero(i(_, [D | Ds])) :-
+    D = 0,
+    list.all_true(pred(0::in) is semidet, Ds).
 
 %---------------------%
 
@@ -896,11 +899,15 @@ mul_base_2([H | T]) = [H | mul_base_2(T)].
 :- func mul_by_digit(digit, integer) = integer.
 
 mul_by_digit(Digit, i(Len, Digits0)) = Out :-
-    mul_by_digit_2(Digit, Mod, Digits0, Digits),
-    ( if Mod = 0 then
-        Out = i(Len, Digits)
+    ( if Digit = 0 then
+        Out = integer.zero
     else
-        Out = i(Len + 1, [Mod | Digits])
+        mul_by_digit_2(Digit, Mod, Digits0, Digits),
+        ( if Mod = 0 then
+            Out = i(Len, Digits)
+        else
+            Out = i(Len + 1, [Mod | Digits])
+        )
     ).
 
 :- pred mul_by_digit_2(digit::in, digit::out, list(digit)::in,
@@ -1758,11 +1765,15 @@ printbase_chop(printbase(Base), N, Div, Mod) :-
 :- func printbase_mul_by_digit(printbase, digit, integer) = integer.
 
 printbase_mul_by_digit(Base, D, i(Len, Ds)) = Out :-
-    printbase_mul_by_digit_2(Base, D, Div, Ds, DsOut),
-    ( if Div = 0 then
-        Out = i(Len, DsOut)
+    ( if D = 0 then
+        Out = integer.zero
     else
-        Out = i(Len + 1, [Div | DsOut])
+        printbase_mul_by_digit_2(Base, D, Div, Ds, DsOut),
+        ( if Div = 0 then
+            Out = i(Len, DsOut)
+        else
+            Out = i(Len + 1, [Div | DsOut])
+        )
     ).
 
 :- pred printbase_mul_by_digit_2(printbase::in, digit::in, digit::out,

library/rational.m

@@ -153,7 +153,7 @@ r(An, Ad) * r(Bn, Bd) = rational_norm(Numer, Denom) :-
 R1 / R2 = R1 * reciprocal(R2).
 
 reciprocal(r(Num, Den)) =
-    ( if Num = integer.zero then
+    ( if integer.is_zero(Num) then
         func_error($pred, "division by zero")
     else
         r(signum(Num) * Den, integer.abs(Num))
@@ -164,9 +164,9 @@ abs(r(Num, Den)) = r(integer.abs(Num), Den).
 :- func rational_norm(integer, integer) = rational.
 
 rational_norm(Num, Den) = Rat :-
-    ( if Den = integer.zero then
+    ( if integer.is_zero(Den) then
         error("rational.rational_norm: division by zero")
-    else if Num = integer.zero then
+    else if integer.is_zero(Num) then
         Rat = r(integer.zero, integer.one)
     else
         G    = gcd(Num, Den),
@@ -181,14 +181,14 @@ gcd(A, B) = gcd_2(integer.abs(A), integer.abs(B)).
 
 :- func gcd_2(integer, integer) = integer.
 
-gcd_2(A, B) = ( if B = integer.zero then A else gcd_2(B, A rem B) ).
+gcd_2(A, B) = ( if integer.is_zero(B) then A else gcd_2(B, A rem B) ).
 
 :- func lcm(integer, integer) = integer.
 
 lcm(A, B) =
-    ( if A = integer.zero then
+    ( if integer.is_zero(A) then
         integer.zero
-    else if B = integer.zero then
+    else if integer.is_zero(B) then
         integer.zero
     else
         integer.abs((A // gcd(A, B)) * B)
@@ -197,7 +197,7 @@ lcm(A, B) =
 :- func signum(integer) = integer.
 
 signum(N) =
-    ( if N = integer.zero then
+    ( if integer.is_zero(N) then
         integer.zero
     else if N < integer.zero then
         -integer.one
@@ -219,7 +219,7 @@ cmp(R1, R2) = Cmp :-
 
 :- pred is_zero(rational::in) is semidet.
 
-is_zero(r(integer.zero, _)).
+is_zero(r(Num, _)) :- integer.is_zero(Num).
 
 :- pred is_negative(rational::in) is semidet.