euler001.mlw 2.08 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(* Euler Project, problem 1

If we list all the natural numbers below 10 that are multiples of 3 or
5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

*)


theory SumMultiple

  use import int.Int
  use import int.EuclideanDivision

16
  (* [sum_multiple_3_5_lt n] is the sum of all the multiples
17
     of 3 or 5 below n] *)
Andrei Paskevich's avatar
Andrei Paskevich committed
18
  function sum_multiple_3_5_lt int : int
19 20 21 22

  axiom SumEmpty: sum_multiple_3_5_lt 0 = 0

  axiom SumNo : forall n:int. n >= 0 ->
Andrei Paskevich's avatar
Andrei Paskevich committed
23
    mod n 3 <> 0 /\ mod n 5 <> 0 ->
24 25 26
    sum_multiple_3_5_lt (n+1) = sum_multiple_3_5_lt n

  axiom SumYes: forall n:int. n >= 0 ->
Andrei Paskevich's avatar
Andrei Paskevich committed
27
    mod n 3 = 0 \/ mod n 5 = 0 ->
28 29 30
    sum_multiple_3_5_lt (n+1) = sum_multiple_3_5_lt n + n

  lemma div_minus1_1 :
Andrei Paskevich's avatar
Andrei Paskevich committed
31
    forall x y:int. x >= 0 /\ y > 0 ->
32 33 34
      mod (x+1) y = 0 -> div (x+1) y = (div x y) + 1

  lemma div_minus1_2 :
Andrei Paskevich's avatar
Andrei Paskevich committed
35
    forall x y:int. x >= 0 /\ y > 0 ->
36
      mod (x+1) y <> 0 -> div (x+1) y = (div x y)
37

Andrei Paskevich's avatar
Andrei Paskevich committed
38
  function closed_formula (n:int) : int =
39 40 41
    let n3 = div n 3 in
    let n5 = div n 5 in
    let n15 = div n 15 in
42 43 44
    div (3 * n3 * (n3+1) +
         5 * n5 * (n5+1) -
         15 * n15 * (n15+1)) 2
45 46 47

  lemma mod_15 :
    forall n:int. n >= 0 ->
Andrei Paskevich's avatar
Andrei Paskevich committed
48
      mod n 15 = 0 <-> (mod n 3 = 0 /\ mod n 5 = 0)
49

Andrei Paskevich's avatar
Andrei Paskevich committed
50
  predicate p (n:int) = sum_multiple_3_5_lt (n+1) = closed_formula n
51 52 53

  lemma Closed_formula_0: p 0

54 55
  lemma Closed_formula_n_1:
    forall n:int. n > 0 -> p (n-1) ->
Andrei Paskevich's avatar
Andrei Paskevich committed
56
      mod n 3 <> 0 /\ mod n 5 <> 0 -> p n
57

58
  lemma Closed_formula_n_2:
59
    forall n:int. n > 0 -> p (n-1) ->
Andrei Paskevich's avatar
Andrei Paskevich committed
60
      mod n 3 = 0 \/ mod n 5 = 0 -> p n
61

Andrei Paskevich's avatar
Andrei Paskevich committed
62
  clone int.Induction as I with predicate p = p
63

64
  lemma Closed_formula:
65 66 67 68 69 70 71 72 73 74
    forall n:int. 0 <= n -> p n

end

module Euler001

  use import SumMultiple
  use import int.Int
  use import int.EuclideanDivision

75
  let solve n =
76 77 78 79 80 81 82
    { n >= 1 }
    let n3 = div (n-1) 3 in
    let n5 = div (n-1) 5 in
    let n15 = div (n-1) 15 in
    div (3 * n3 * (n3+1) + 5 * n5 * (n5+1) - 15 * n15 * (n15+1)) 2
    { result = sum_multiple_3_5_lt n }

83 84 85 86 87 88 89 90
end

(*
Local Variables:
compile-command: "unset LANG; make -C ../.. examples/programs/euler001.gui"
End:
*)