Commit 84d4ed32 by Martin Clochard

### euler003 example completed.

parent d49c6573
 ... ... @@ -12,21 +12,29 @@ module PrimeFactor use import int.ComputerDivision use import number.Divisibility use import number.Prime use import number.Coprime (** returns the smallest divisor of [n] greater or equal to [d] *) (** returns the smallest divisor of [n] greater then 1, assuming no divisor between 2 and [d]. *) let rec smallest_divisor (d n:int) : int requires { 2 <= n } requires { 1 <= d <= n } requires { 2 <= n } requires { 2 <= d <= n } requires { forall i:int. 2 <= i < d -> not divides i n } ensures { d <= result <= n } ensures { divides result n } ensures { forall i:int. d <= i < result -> not divides result i } ensures { forall i:int. 2 <= i < result -> not divides i n } variant { n - d } = if d * d > n then n else if d >= 2 && mod n d = 0 then d else = if d * d > n then begin assert { forall i:int. 2 <= i < n /\ divides i n -> i >= d && let u = div n i in u * i = n && divides u n && u * i = n && (u >= d -> n >= d * i >= d * d && false) && u >= 2 && u < n && false } ; n end else if d >= 2 && mod n d = 0 then d else smallest_divisor (d+1) n (* The tests are invalidated by the new precondition. *) let test2 () = smallest_divisor 2 13195 (* 5 *) (*let test2 () = smallest_divisor 2 13195 (* 5 *) let test5 () = smallest_divisor 5 13195 (* 5 *) ... ... @@ -36,7 +44,7 @@ let test8 () = smallest_divisor 8 13195 (* 13 *) let test14 () = smallest_divisor 14 13195 (* 29 *) let test30 () = smallest_divisor 30 13195 (* 35 *) let test30 () = smallest_divisor 30 13195 (* 35 *)*) use import ref.Ref ... ... @@ -49,21 +57,40 @@ let largest_prime_factor (n:int) : int ensures { prime result } ensures { divides result n } ensures { forall i:int. result < i <= n -> not (prime i /\ divides i n) } = let d = smallest_divisor 1 n in = let d = smallest_divisor 2 n in let factor = ref d in let target = ref (div n d) in factors := Cons d Nil; assert { !target * d = n && divides !target n } ; assert { forall i:int. prime i /\ divides i n /\ i > d -> coprime d i && divides i !target }; while !target >= 2 do invariant { !target = 1 \/ 2 <= !factor <= !target } (* invariant { 2 <= !factor <= n } *) (* invariant { divides !target n } *) invariant { 1 <= !target <= n } invariant { 2 <= !factor <= n } invariant { divides !factor n } invariant { prime !factor } variant { !target } invariant { forall i:int. divides i !target /\ i >= 2 -> i >= !factor /\ divides i n } invariant { forall i:int. prime i /\ divides i n /\ i > !factor -> divides i !target } (*invariant { !target = 1 \/ 2 <= !factor <= !target }*) (* invariant { 2 <= !factor <= n } *) (* invariant { divides !target n } *) (*invariant { divides !factor n }*) (*invariant { prime !factor }*) variant { !target } let ghost oldt = !target in let ghost oldf = !factor in assert { divides !target !target && !target >= 2 && !target >= !factor }; let d = smallest_divisor !factor !target in assert { prime d }; factor := d; factors := Cons d !factors; target := div !target d target := div !target d; assert { !target * d = oldt && divides !target oldt } ; assert { forall i:int. prime i /\ divides i n /\ i > d -> i > oldf && divides i oldt && 1 <= d < i && coprime d i && divides i !target } done; !factor ... ...
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