(* 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 DivModHints use import int.Int use import int.ComputerDivision lemma mod_div_unique : forall x y q r:int. y > 0 /\ x = q*y + r /\ 0 <= r < y -> r = mod x y /\ q = div x y lemma mod_succ_1 : forall x y:int. x >= 0 /\ y > 0 -> mod (x+1) y <> 0 -> mod (x+1) y = (mod x y) + 1 lemma mod_succ_2 : forall x y:int. x >= 0 /\ y > 0 -> mod (x+1) y = 0 -> mod x y = y-1 lemma div_succ_1 : forall x y:int. x >= 0 /\ y > 0 -> mod (x+1) y = 0 -> div (x+1) y = (div x y) + 1 lemma div_succ_2 : forall x y:int. x >= 0 /\ y > 0 -> mod (x+1) y <> 0 -> div (x+1) y = (div x y) end theory SumMultiple use import int.Int use import int.ComputerDivision (* [sum_multiple_3_5_lt n] is the sum of all the multiples of 3 or 5 below n] *) function sum_multiple_3_5_lt int : int axiom SumEmpty: sum_multiple_3_5_lt 0 = 0 axiom SumNo : forall n:int. n >= 0 -> mod n 3 <> 0 /\ mod n 5 <> 0 -> sum_multiple_3_5_lt (n+1) = sum_multiple_3_5_lt n axiom SumYes: forall n:int. n >= 0 -> mod n 3 = 0 \/ mod n 5 = 0 -> sum_multiple_3_5_lt (n+1) = sum_multiple_3_5_lt n + n function closed_formula (n:int) : int = let n3 = div n 3 in let n5 = div n 5 in let n15 = div n 15 in div (3 * n3 * (n3+1) + 5 * n5 * (n5+1) - 15 * n15 * (n15+1)) 2 predicate p (n:int) = sum_multiple_3_5_lt (n+1) = closed_formula n lemma Closed_formula_0: p 0 use DivModHints lemma mod_15: forall n:int. mod n 15 = 0 <-> mod n 3 = 0 /\ mod n 5 = 0 lemma Closed_formula_n: forall n:int. n > 0 -> p (n-1) -> mod n 3 <> 0 /\ mod n 5 <> 0 -> p n lemma Closed_formula_n_3: forall n:int. n > 0 -> p (n-1) -> mod n 3 = 0 /\ mod n 5 <> 0 -> p n lemma Closed_formula_n_5: forall n:int. n > 0 -> p (n-1) -> mod n 3 <> 0 /\ mod n 5 = 0 -> p n lemma Closed_formula_n_15: forall n:int. n > 0 -> p (n-1) -> mod n 3 = 0 /\ mod n 5 = 0 -> p n clone int.Induction as I with predicate p = p lemma Closed_formula: forall n:int. 0 <= n -> p n end module Euler001 use import SumMultiple use import int.Int use import int.ComputerDivision let solve n = { 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 } end (* Local Variables: compile-command: "unset LANG; make -C ../.. examples/programs/euler001.gui" End: *)