Euler example fully proved

examples/programs/euler001.mlw

-(* Euler Project, problem 1
+(** {1 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.
@@ -14,7 +14,7 @@ theory DivModHints
 
   lemma mod_div_unique :
     forall x y q r:int. x >= 0 /\ y > 0 /\ x = q*y + r /\ 0 <= r < y ->
-      q = div x y /\ r = mod x y 
+      q = div x y /\ r = mod x y
 
   lemma mod_succ_1 :
     forall x y:int. x >= 0 /\ y > 0 ->
@@ -32,6 +32,49 @@ theory DivModHints
     forall x y:int. x >= 0 /\ y > 0 ->
       mod (x+1) y <> 0 -> div (x+1) y = (div x y)
 
+  lemma mod2_mul2:
+    forall x:int. mod (2 * x) 2 = 0
+
+  lemma mod2_mul2_aux:
+    forall x y:int. mod (y * (2 * x)) 2 = 0
+
+  lemma mod2_mul2_aux2:
+    forall x y z t:int. mod (y * (2 * x) + z * (2 * t)) 2 = 0
+
+  lemma div2_mul2:
+    forall x:int. div (2 * x) 2 = x
+
+  lemma div2_mul2_aux:
+    forall x y:int. div (y * (2 * x)) 2 = y * x
+
+  lemma div2_add:
+    forall x y:int. mod x 2 = 0 /\ mod y 2 = 0 ->
+      div (x+y) 2 = div x 2 + div y 2
+
+  lemma div2_sub:
+    forall x y:int. mod x 2 = 0 /\ mod y 2 = 0 ->
+      div (x-y) 2 = div x 2 - div y 2
+
+end
+
+theory TriangularNumbers
+
+  use import int.Int
+  use import int.ComputerDivision
+  use import int.Div2
+  use DivModHints
+
+  lemma tr_mod_2:
+    forall n:int. n >= 0 -> mod (n*(n+1)) 2 = 0
+
+  function tr (n:int) : int =  div (n*(n+1)) 2
+
+  lemma tr_repr:
+    forall n:int. n >= 0 -> n*(n+1) = 2 * tr n
+
+  lemma tr_succ:
+    forall n:int. n >= 0 -> tr (n+1) = tr n + n + 1
+
 end
 
 
@@ -54,49 +97,63 @@ theory SumMultiple
     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 =
+  use import TriangularNumbers
+
+  function closed_formula_aux (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
+    3 * tr n3 + 5 * tr n5 - 15 * tr n15
 
-  predicate p (n:int) = sum_multiple_3_5_lt (n+1) = closed_formula n
-
-  lemma Closed_formula_0: p 0
+  predicate p (n:int) = sum_multiple_3_5_lt (n+1) = closed_formula_aux n
 
   use DivModHints
+  use TriangularNumbers
 
   lemma mod_15:
     forall n:int.
       mod n 15 = 0 <-> mod n 3 = 0 /\ mod n 5 = 0
 
-  lemma triangle_numbers:
-    forall n:int.
-      2 * div (n*(n+1)) 2 = n*(n+1)
+  lemma Closed_formula_0: p 0
 
   lemma Closed_formula_n:
-    forall n:int. n >= 0 -> p n ->
-      mod (n+1) 3 <> 0 /\ mod (n+1) 5 <> 0 -> p (n+1)
+    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 ->
-      mod (n+1) 3 = 0 /\ mod (n+1) 5 <> 0 -> p (n+1)
+    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 ->
-      mod (n+1) 3 <> 0 /\ mod (n+1) 5 = 0 -> p (n+1)
+    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 ->
-      mod (n+1) 3 = 0 /\ mod (n+1) 5 = 0 -> p (n+1)
+    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
+  constant b : int = 0
 
-  lemma Closed_formula:
+  clone int.Induction as I with constant bound = b, predicate p = p
+
+  lemma Closed_formula_ind:
     forall n:int. 0 <= n -> p 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
+
+  lemma div_15: forall n:int. 0 <= n -> div n 15 >= 0
+  lemma div_5: forall n:int. 0 <= n -> div n 5 >= 0
+  lemma div_3: forall n:int. 0 <= n -> div n 3 >= 0
+
+  lemma Closed_Formula:
+    forall n:int. 0 <= n -> sum_multiple_3_5_lt (n+1) = closed_formula n
+
 end
 
 module Euler001
@@ -115,7 +172,7 @@ module Euler001
 
 end
 
-(*
+(***
 Local Variables:
 compile-command: "unset LANG; make -C ../.. examples/programs/euler001.gui"
 End:

examples/programs/euler001/euler001_DivModHints_add_div2_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import ZOdiv.
+Require int.Int.
+Require int.Abs.
+Require int.ComputerDivision.
+
+Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
+  ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
+  (r < y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+
+Axiom mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZOmod (x + 1%Z)%Z y) = ((ZOmod x y) + 1%Z)%Z)).
+
+Axiom mod_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  (((ZOmod (x + 1%Z)%Z y) = 0%Z) -> ((ZOmod x y) = (y - 1%Z)%Z)).
+
+Axiom div_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  (((ZOmod (x + 1%Z)%Z y) = 0%Z) ->
+  ((ZOdiv (x + 1%Z)%Z y) = ((ZOdiv x y) + 1%Z)%Z)).
+
+Axiom div_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZOdiv (x + 1%Z)%Z y) = (ZOdiv x y))).
+
+Axiom div2_simpl : forall (x:Z) (y:Z), ((2%Z * x)%Z = y) ->
+  (x = (ZOdiv y 2%Z)).
+
+(* Why3 goal *)
+Theorem add_div2 : forall (x:Z) (y:Z), ((ZOmod x 2%Z) = 0%Z) ->
+  ((ZOdiv (x + y)%Z 2%Z) = ((ZOdiv x 2%Z) + (ZOdiv y 2%Z))%Z).
+intros x y Hx.
+generalize (ZO_div_mod_eq x 2); intro h1.
+rewrite Hx in h1.
+rewrite h1.
+simpl.
+Qed.
+
+

examples/programs/euler001/euler001_SumMultiple_Closed_Formula_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import ZOdiv.
+Require int.Int.
+Require int.Abs.
+Require int.ComputerDivision.
+
+Parameter sum_multiple_3_5_lt: Z -> Z.
+
+Axiom SumEmpty : ((sum_multiple_3_5_lt 0%Z) = 0%Z).
+
+Axiom SumNo : forall (n:Z), (0%Z <= n)%Z -> (((~ ((ZOmod n 3%Z) = 0%Z)) /\
+  ~ ((ZOmod n 5%Z) = 0%Z)) ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = (sum_multiple_3_5_lt n))).
+
+Axiom SumYes : forall (n:Z), (0%Z <= n)%Z -> ((((ZOmod n 3%Z) = 0%Z) \/
+  ((ZOmod n 5%Z) = 0%Z)) ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = ((sum_multiple_3_5_lt n) + n)%Z)).
+
+Axiom div2 : forall (x:Z), exists y:Z, (x = (2%Z * y)%Z) \/
+  (x = ((2%Z * y)%Z + 1%Z)%Z).
+
+Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
+  ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
+  (r < y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+
+Axiom mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZOmod (x + 1%Z)%Z y) = ((ZOmod x y) + 1%Z)%Z)).
+
+Axiom mod_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  (((ZOmod (x + 1%Z)%Z y) = 0%Z) -> ((ZOmod x y) = (y - 1%Z)%Z)).
+
+Axiom div_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  (((ZOmod (x + 1%Z)%Z y) = 0%Z) ->
+  ((ZOdiv (x + 1%Z)%Z y) = ((ZOdiv x y) + 1%Z)%Z)).
+
+Axiom div_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZOdiv (x + 1%Z)%Z y) = (ZOdiv x y))).
+
+Axiom mod2_mul2 : forall (x:Z), ((ZOmod (2%Z * x)%Z 2%Z) = 0%Z).
+
+Axiom mod2_mul2_aux : forall (x:Z) (y:Z),
+  ((ZOmod (y * (2%Z * x)%Z)%Z 2%Z) = 0%Z).
+
+Axiom mod2_mul2_aux2 : forall (x:Z) (y:Z) (z:Z) (t:Z),
+  ((ZOmod ((y * (2%Z * x)%Z)%Z + (z * (2%Z * t)%Z)%Z)%Z 2%Z) = 0%Z).
+
+Axiom div2_mul2 : forall (x:Z), ((ZOdiv (2%Z * x)%Z 2%Z) = x).
+
+Axiom div2_mul2_aux : forall (x:Z) (y:Z),
+  ((ZOdiv (y * (2%Z * x)%Z)%Z 2%Z) = (y * x)%Z).
+
+Axiom div2_add : forall (x:Z) (y:Z), (((ZOmod x 2%Z) = 0%Z) /\
+  ((ZOmod y 2%Z) = 0%Z)) ->
+  ((ZOdiv (x + y)%Z 2%Z) = ((ZOdiv x 2%Z) + (ZOdiv y 2%Z))%Z).
+
+Axiom div2_sub : forall (x:Z) (y:Z), (((ZOmod x 2%Z) = 0%Z) /\
+  ((ZOmod y 2%Z) = 0%Z)) ->
+  ((ZOdiv (x - y)%Z 2%Z) = ((ZOdiv x 2%Z) - (ZOdiv y 2%Z))%Z).
+
+Axiom tr_mod_2 : forall (n:Z), (0%Z <= n)%Z ->
+  ((ZOmod (n * (n + 1%Z)%Z)%Z 2%Z) = 0%Z).
+
+(* Why3 assumption *)
+Definition tr(n:Z): Z := (ZOdiv (n * (n + 1%Z)%Z)%Z 2%Z).
+
+Axiom tr_repr : forall (n:Z), (0%Z <= n)%Z ->
+  ((n * (n + 1%Z)%Z)%Z = (2%Z * (tr n))%Z).
+
+Axiom tr_succ : forall (n:Z), (0%Z <= n)%Z ->
+  ((tr (n + 1%Z)%Z) = (((tr n) + n)%Z + 1%Z)%Z).
+
+(* Why3 assumption *)
+Definition closed_formula_aux(n:Z): Z := let n3 := (ZOdiv n 3%Z) in let n5 :=
+  (ZOdiv n 5%Z) in let n15 := (ZOdiv n 15%Z) in
+  (((3%Z * (tr n3))%Z + (5%Z * (tr n5))%Z)%Z - (15%Z * (tr n15))%Z)%Z.
+
+(* Why3 assumption *)
+Definition p(n:Z): Prop :=
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = (closed_formula_aux n)).
+
+Axiom mod_15 : forall (n:Z), ((ZOmod n 15%Z) = 0%Z) <->
+  (((ZOmod n 3%Z) = 0%Z) /\ ((ZOmod n 5%Z) = 0%Z)).
+
+Axiom Closed_formula_0 : (p 0%Z).
+
+Axiom Closed_formula_n : forall (n:Z), (0%Z < n)%Z -> ((p (n - 1%Z)%Z) ->
+  (((~ ((ZOmod n 3%Z) = 0%Z)) /\ ~ ((ZOmod n 5%Z) = 0%Z)) -> (p n))).
+
+Axiom Closed_formula_n_3 : forall (n:Z), (0%Z < n)%Z -> ((p (n - 1%Z)%Z) ->
+  ((((ZOmod n 3%Z) = 0%Z) /\ ~ ((ZOmod n 5%Z) = 0%Z)) -> (p n))).
+
+Axiom Closed_formula_n_5 : forall (n:Z), (0%Z < n)%Z -> ((p (n - 1%Z)%Z) ->
+  (((~ ((ZOmod n 3%Z) = 0%Z)) /\ ((ZOmod n 5%Z) = 0%Z)) -> (p n))).
+
+Axiom Closed_formula_n_15 : forall (n:Z), (0%Z < n)%Z -> ((p (n - 1%Z)%Z) ->
+  ((((ZOmod n 3%Z) = 0%Z) /\ ((ZOmod n 5%Z) = 0%Z)) -> (p n))).
+
+Axiom Induction : (forall (n:Z), (0%Z <= n)%Z -> ((forall (k:Z),
+  ((0%Z <= k)%Z /\ (k < n)%Z) -> (p k)) -> (p n))) -> forall (n:Z),
+  (0%Z <= n)%Z -> (p n).
+
+Axiom Induction_bound : (forall (n:Z), (0%Z <= n)%Z -> ((forall (k:Z),
+  ((0%Z <= k)%Z /\ (k < n)%Z) -> (p k)) -> (p n))) -> forall (n:Z),
+  (0%Z <= n)%Z -> (p n).
+
+Axiom Closed_formula_ind : forall (n:Z), (0%Z <= n)%Z -> (p n).
+
+(* Why3 assumption *)
+Definition closed_formula(n:Z): Z := let n3 := (ZOdiv n 3%Z) in let n5 :=
+  (ZOdiv n 5%Z) in let n15 := (ZOdiv n 15%Z) in
+  (ZOdiv (((3%Z * (n3 * (n3 + 1%Z)%Z)%Z)%Z + (5%Z * (n5 * (n5 + 1%Z)%Z)%Z)%Z)%Z - (15%Z * (n15 * (n15 + 1%Z)%Z)%Z)%Z)%Z 2%Z).
+
+Axiom div_15 : forall (n:Z), (0%Z <= n)%Z -> (0%Z <= (ZOdiv n 15%Z))%Z.
+
+Axiom div_5 : forall (n:Z), (0%Z <= n)%Z -> (0%Z <= (ZOdiv n 5%Z))%Z.
+
+Axiom div_3 : forall (n:Z), (0%Z <= n)%Z -> (0%Z <= (ZOdiv n 3%Z))%Z.
+
+(* Why3 goal *)
+Theorem Closed_Formula : forall (n:Z), (0%Z <= n)%Z ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = (closed_formula n)).
+intros n Hn.
+rewrite Closed_formula_ind; auto.
+unfold closed_formula_aux, closed_formula.
+repeat rewrite tr_repr.
+rewrite div2_sub.
+rewrite div2_mul2_aux.
+rewrite div2_add.
+do 2 rewrite div2_mul2_aux.
+auto.
+repeat rewrite mod2_mul2_aux; auto.
+rewrite mod2_mul2_aux; auto.
+rewrite mod2_mul2_aux2; auto.
+apply div_15; auto.
+apply div_5; auto.
+apply div_3; auto.
+Qed.
+
+

examples/programs/euler001/euler001_SumMultiple_Closed_formula_ind_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import ZOdiv.
+Require int.Int.
+Require int.Abs.
+Require int.ComputerDivision.
+
+Parameter sum_multiple_3_5_lt: Z -> Z.
+
+Axiom SumEmpty : ((sum_multiple_3_5_lt 0%Z) = 0%Z).
+
+Axiom SumNo : forall (n:Z), (0%Z <= n)%Z -> (((~ ((ZOmod n 3%Z) = 0%Z)) /\
+  ~ ((ZOmod n 5%Z) = 0%Z)) ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = (sum_multiple_3_5_lt n))).
+
+Axiom SumYes : forall (n:Z), (0%Z <= n)%Z -> ((((ZOmod n 3%Z) = 0%Z) \/
+  ((ZOmod n 5%Z) = 0%Z)) ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = ((sum_multiple_3_5_lt n) + n)%Z)).
+
+Axiom div2 : forall (x:Z), exists y:Z, (x = (2%Z * y)%Z) \/
+  (x = ((2%Z * y)%Z + 1%Z)%Z).
+
+Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
+  ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
+  (r < y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+
+Axiom mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZOmod (x + 1%Z)%Z y) = ((ZOmod x y) + 1%Z)%Z)).
+
+Axiom mod_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  (((ZOmod (x + 1%Z)%Z y) = 0%Z) -> ((ZOmod x y) = (y - 1%Z)%Z)).
+
+Axiom div_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  (((ZOmod (x + 1%Z)%Z y) = 0%Z) ->
+  ((ZOdiv (x + 1%Z)%Z y) = ((ZOdiv x y) + 1%Z)%Z)).
+
+Axiom div_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
+  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZOdiv (x + 1%Z)%Z y) = (ZOdiv x y))).
+
+Axiom mod2_mul2 : forall (x:Z), ((ZOmod (2%Z * x)%Z 2%Z) = 0%Z).
+
+Axiom mod2_mul2_aux : forall (x:Z) (y:Z),
+  ((ZOmod (y * (2%Z * x)%Z)%Z 2%Z) = 0%Z).
+
+Axiom mod2_mul2_aux2 : forall (x:Z) (y:Z) (z:Z),
+  ((ZOmod ((y * (2%Z * x)%Z)%Z + z)%Z 2%Z) = (ZOmod z 2%Z)).
+
+Axiom div2_mul2 : forall (x:Z), ((ZOdiv (2%Z * x)%Z 2%Z) = x).
+
+Axiom div2_mul2_aux : forall (x:Z) (y:Z),
+  ((ZOdiv (y * (2%Z * x)%Z)%Z 2%Z) = (y * x)%Z).
+
+Axiom div2_add : forall (x:Z) (y:Z), (((ZOmod x 2%Z) = 0%Z) /\
+  ((ZOmod y 2%Z) = 0%Z)) ->
+  ((ZOdiv (x + y)%Z 2%Z) = ((ZOdiv x 2%Z) + (ZOdiv y 2%Z))%Z).
+
+Axiom div2_sub : forall (x:Z) (y:Z), (((ZOmod x 2%Z) = 0%Z) /\
+  ((ZOmod y 2%Z) = 0%Z)) ->
+  ((ZOdiv (x - y)%Z 2%Z) = ((ZOdiv x 2%Z) - (ZOdiv y 2%Z))%Z).
+
+Axiom tr_mod_2 : forall (n:Z), (0%Z <= n)%Z ->
+  ((ZOmod (n * (n + 1%Z)%Z)%Z 2%Z) = 0%Z).
+
+(* Why3 assumption *)
+Definition tr(n:Z): Z := (ZOdiv (n * (n + 1%Z)%Z)%Z 2%Z).
+
+Axiom tr_repr : forall (n:Z), (0%Z <= n)%Z ->
+  ((n * (n + 1%Z)%Z)%Z = (2%Z * (tr n))%Z).
+
+Axiom tr_succ : forall (n:Z), (0%Z <= n)%Z ->
+  ((tr (n + 1%Z)%Z) = (((tr n) + n)%Z + 1%Z)%Z).
+
+(* Why3 assumption *)
+Definition closed_formula_aux(n:Z): Z := let n3 := (ZOdiv n 3%Z) in let n5 :=
+  (ZOdiv n 5%Z) in let n15 := (ZOdiv n 15%Z) in
+  (((3%Z * (tr n3))%Z + (5%Z * (tr n5))%Z)%Z - (15%Z * (tr n15))%Z)%Z.
+
+(* Why3 assumption *)
+Definition p(n:Z): Prop :=
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = (closed_formula_aux n)).
+
+Axiom mod_15 : forall (n:Z), ((ZOmod n 15%Z) = 0%Z) <->
+  (((ZOmod n 3%Z) = 0%Z) /\ ((ZOmod n 5%Z) = 0%Z)).
+
+Axiom Closed_formula_0 : (p 0%Z).
+
+Axiom Closed_formula_n : forall (n:Z), (0%Z <= n)%Z -> ((p n) ->
+  (((~ ((ZOmod (n + 1%Z)%Z 3%Z) = 0%Z)) /\
+  ~ ((ZOmod (n + 1%Z)%Z 5%Z) = 0%Z)) -> (p (n + 1%Z)%Z))).
+
+Axiom Closed_formula_n_3 : forall (n:Z), (0%Z <= n)%Z -> ((p n) ->
+  ((((ZOmod (n + 1%Z)%Z 3%Z) = 0%Z) /\ ~ ((ZOmod (n + 1%Z)%Z 5%Z) = 0%Z)) ->
+  (p (n + 1%Z)%Z))).
+
+Axiom Closed_formula_n_5 : forall (n:Z), (0%Z <= n)%Z -> ((p n) ->
+  (((~ ((ZOmod (n + 1%Z)%Z 3%Z) = 0%Z)) /\
+  ((ZOmod (n + 1%Z)%Z 5%Z) = 0%Z)) -> (p (n + 1%Z)%Z))).
+
+Axiom Closed_formula_n_15 : forall (n:Z), (0%Z <= n)%Z -> ((p n) ->
+  ((((ZOmod (n + 1%Z)%Z 3%Z) = 0%Z) /\ ((ZOmod (n + 1%Z)%Z 5%Z) = 0%Z)) ->
+  (p (n + 1%Z)%Z))).
+
+Axiom Induction : (forall (n:Z), (0%Z <= n)%Z -> ((forall (k:Z),
+  ((0%Z <= k)%Z /\ (k < n)%Z) -> (p k)) -> (p n))) -> forall (n:Z),
+  (0%Z <= n)%Z -> (p n).
+
+Axiom Induction_bound : (forall (n:Z), (0%Z <= n)%Z -> ((forall (k:Z),
+  ((0%Z <= k)%Z /\ (k < n)%Z) -> (p k)) -> (p n))) -> forall (n:Z),
+  (0%Z <= n)%Z -> (p n).
+
+(* Why3 goal *)
+Theorem Closed_formula_ind : forall (n:Z), (0%Z <= n)%Z -> (p n).
+apply Induction_bound.
+intros n Hn Hind.
+assert (h: (n=0 \/ n > 0)) by omega.
+destruct h.
+subst; apply Closed_formula_0.
+assert (h: (Z0mod n=0 \/ n > 0)) by omega.
+
+Qed.
+
+

examples/programs/euler001/euler001_SumMultiple_Closed_formula_n_3_1.v

@@ -3,12 +3,12 @@
 Require Import ZArith.
 Require Import Rbase.
 Require Import ZOdiv.
-Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
-  (x <= y)%Z).
+Require int.Int.
+Require int.Abs.
+Require int.ComputerDivision.
 
 Parameter sum_multiple_3_5_lt: Z -> Z.
 
-
 Axiom SumEmpty : ((sum_multiple_3_5_lt 0%Z) = 0%Z).
 
 Axiom SumNo : forall (n:Z), (0%Z <= n)%Z -> (((~ ((ZOmod n 3%Z) = 0%Z)) /\
@@ -19,31 +19,33 @@ Axiom SumYes : forall (n:Z), (0%Z <= n)%Z -> ((((ZOmod n 3%Z) = 0%Z) \/
   ((ZOmod n 5%Z) = 0%Z)) ->
   ((sum_multiple_3_5_lt (n + 1%Z)%Z) = ((sum_multiple_3_5_lt n) + n)%Z)).
 
+(* Why3 assumption *)
 Definition closed_formula(n:Z): Z := let n3 := (ZOdiv n 3%Z) in let n5 :=
   (ZOdiv n 5%Z) in let n15 := (ZOdiv n 15%Z) in
   (ZOdiv ((((3%Z * n3)%Z * (n3 + 1%Z)%Z)%Z + ((5%Z * n5)%Z * (n5 + 1%Z)%Z)%Z)%Z - ((15%Z * n15)%Z * (n15 + 1%Z)%Z)%Z)%Z 2%Z).
 
+(* Why3 assumption *)
 Definition p(n:Z): Prop :=
   ((sum_multiple_3_5_lt (n + 1%Z)%Z) = (closed_formula n)).
 
 Axiom Closed_formula_0 : (p 0%Z).
 
 Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
-  ((0%Z <  y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
-  (r <  y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+  ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
+  (r < y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
 
-Axiom mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+Axiom mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
   ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
   ((ZOmod (x + 1%Z)%Z y) = ((ZOmod x y) + 1%Z)%Z)).
 
-Axiom mod_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+Axiom mod_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
   (((ZOmod (x + 1%Z)%Z y) = 0%Z) -> ((ZOmod x y) = (y - 1%Z)%Z)).
 
-Axiom div_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+Axiom div_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
   (((ZOmod (x + 1%Z)%Z y) = 0%Z) ->
   ((ZOdiv (x + 1%Z)%Z y) = ((ZOdiv x y) + 1%Z)%Z)).
 
-Axiom div_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+Axiom div_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
   ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
   ((ZOdiv (x + 1%Z)%Z y) = (ZOdiv x y))).
 
@@ -51,16 +53,22 @@ Axiom mod_15 : forall (n:Z), ((ZOmod n 15%Z) = 0%Z) <->
   (((ZOmod n 3%Z) = 0%Z) /\ ((ZOmod n 5%Z) = 0%Z)).
 
 Axiom triangle_numbers : forall (n:Z),
-  ((2%Z * (ZOdiv (n * (n + 1%Z)%Z)%Z 2%Z))%Z = (n * (n + 1%Z)%Z)%Z).
+  ((ZOmod (n * (n + 1%Z)%Z)%Z 2%Z) = 0%Z).
+
+Axiom div2_simpl : forall (x:Z) (y:Z), ((2%Z * x)%Z = y) ->
+  (x = (ZOdiv y 2%Z)).
+
+Axiom add_div2 : forall (x:Z) (y:Z), ((ZOmod x 2%Z) = 0%Z) ->
+  ((ZOdiv (x + y)%Z 2%Z) = ((ZOdiv x 2%Z) + (ZOdiv y 2%Z))%Z).
 
 Axiom Closed_formula_n : forall (n:Z), (0%Z <= n)%Z -> ((p n) ->
   (((~ ((ZOmod (n + 1%Z)%Z 3%Z) = 0%Z)) /\