Coq proofs for euler001

examples/programs/euler001.mlw

@@ -13,8 +13,8 @@ theory DivModHints
   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
+    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 
 
   lemma mod_succ_1 :
     forall x y:int. x >= 0 /\ y > 0 ->
@@ -72,21 +72,25 @@ theory SumMultiple
     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_n:
-    forall n:int. n > 0 -> p (n-1) ->
-      mod n 3 <> 0 /\ mod n 5 <> 0 -> p n
+    forall n:int. n >= 0 -> p n ->
+      mod (n+1) 3 <> 0 /\ mod (n+1) 5 <> 0 -> p (n+1)
 
   lemma Closed_formula_n_3:
-    forall n:int. n > 0 -> p (n-1) ->
-      mod n 3 = 0 /\ mod n 5 <> 0 -> p n
+    forall n:int. n >= 0 -> p n ->
+      mod (n+1) 3 = 0 /\ mod (n+1) 5 <> 0 -> p (n+1)
 
   lemma Closed_formula_n_5:
-    forall n:int. n > 0 -> p (n-1) ->
-      mod n 3 <> 0 /\ mod n 5 = 0 -> p n
+    forall n:int. n >= 0 -> p n ->
+      mod (n+1) 3 <> 0 /\ mod (n+1) 5 = 0 -> p (n+1)
 
   lemma Closed_formula_n_15:
-    forall n:int. n > 0 -> p (n-1) ->
-      mod n 3 = 0 /\ mod n 5 = 0 -> p n
+    forall n:int. n >= 0 -> p n ->
+      mod (n+1) 3 = 0 /\ mod (n+1) 5 = 0 -> p (n+1)
 
   clone int.Induction as I with predicate p = p
 

examples/programs/euler001/euler001_DivModHints_mod_div_unique_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.
+Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%Z).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem 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))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y q r (H1,(H2,(H3,(H4,H5)))).
+apply ZOdiv_mod_unique_full.
+2: rewrite H3; ring.
+red.
+left.
+rewrite Zabs_eq; auto with zarith.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/euler001/euler001_DivModHints_mod_succ_1_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.
+Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%Z).
+
+Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <  y)%Z /\
+  ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\ (r <  y)%Z))) ->
+  ((r = (ZOmod x y)) /\ (q = (ZOdiv x y))).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+Open Scope Z_scope.
+(* DO NOT EDIT BELOW *)
+
+Theorem 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)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y (Hx,Hy) H.
+assert (h: y>0) by omega.
+generalize (ZO_div_mod_eq x y); intro h1.
+generalize (ZO_div_mod_eq (x+1) y); intro h2.
+assert (h3:x = y * ((x + 1) / y) + ((x + 1) mod y - 1)) by omega.
+generalize (mod_div_unique x y ((x + 1) / y) ((x + 1) mod y - 1)).
+intuition.
+destruct H1; auto with zarith.
+rewrite h3 at 1.
+ring.
+assert (0 <= (x + 1) mod y < y).
+  apply ZOmod_lt_pos_pos; omega.
+omega.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/euler001/euler001_DivModHints_mod_succ_2_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.
+Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%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)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem 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)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y (Hx,Hy) H.
+generalize (ZO_div_mod_eq x y); intro h1.
+generalize (ZO_div_mod_eq (x+1) y); intro h2.
+assert (h3:x = y * ((x + 1) / y - 1) + ((x + 1) mod y + y - 1)).
+  ring_simplify; omega.
+rewrite H in h3.
+generalize (mod_div_unique x y ((x + 1) / y - 1) (0 + y - 1)).
+intuition.
+destruct H1; auto with zarith.
+rewrite h3 at 1.
+ring.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/euler001/euler001_SumMultiple_Closed_formula_n_3_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.
+Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%Z).
+
+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)).
+
+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).
+
+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))).
+
+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 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).
+
+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))).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem 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))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros n Hn Hind (H3,H5).
+unfold p in *.
+rewrite SumYes; auto with zarith.
+rewrite Hind; clear Hind.
+unfold closed_formula.
+rewrite (div_succ_1 n 3); auto with zarith.
+rewrite (div_succ_2 n 5); auto with zarith.
+rewrite (div_succ_2 n 15); auto with zarith.
+2: generalize (mod_15 (n+1)); intuition.
+
+ring_simplify.
+
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/euler001/euler001_SumMultiple_div_minus1_2_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 Zdiv.
+Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%Z).
+
+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 -> (((~ ((Zmod n 3%Z) = 0%Z)) /\
+  ~ ((Zmod 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 -> ((((Zmod n 3%Z) = 0%Z) \/
+  ((Zmod n 5%Z) = 0%Z)) ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = ((sum_multiple_3_5_lt n) + n)%Z)).
+
+Axiom div_minus1_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  (((Zmod (x + 1%Z)%Z y) = 0%Z) ->
+  ((Zdiv (x + 1%Z)%Z y) = ((Zdiv x y) + 1%Z)%Z)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem div_minus1_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  ((~ ((Zmod (x + 1%Z)%Z y) = 0%Z)) -> ((Zdiv (x + 1%Z)%Z y) = (Zdiv x y))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y (Hx,Hy) H.
+pose (q := (x/ y)%Z).
+pose (r := (x mod y)%Z).
+assert (h2: (x = y * q + r)%Z).
+  apply (Z_div_mod_eq x y); auto with zarith.
+assert (h3: (0 <= r < y - 1)%Z).
+  admit.
+rewrite h2.
+rewrite Zmult_comm.
+rewrite <- Zplus_assoc.
+rewrite Z_div_plus_full_l; auto with zarith.
+rewrite Z_div_plus_full_l; auto with zarith.
+rewrite Zdiv_small; auto with zarith.
+rewrite Zdiv_small; auto with zarith.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/euler001/euler001_SumMultiple_div_minus1_2_2.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import Zdiv.
+Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%Z).
+
+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 -> (((~ ((Zmod n 3%Z) = 0%Z)) /\
+  ~ ((Zmod 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 -> ((((Zmod n 3%Z) = 0%Z) \/
+  ((Zmod n 5%Z) = 0%Z)) ->
+  ((sum_multiple_3_5_lt (n + 1%Z)%Z) = ((sum_multiple_3_5_lt n) + n)%Z)).
+
+Axiom div_minus1_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  (((Zmod (x + 1%Z)%Z y) = 0%Z) ->
+  ((Zdiv (x + 1%Z)%Z y) = ((Zdiv x y) + 1%Z)%Z)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem div_minus1_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  ((~ ((Zmod (x + 1%Z)%Z y) = 0%Z)) -> ((Zdiv (x + 1%Z)%Z y) = (Zdiv x y))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y (Hx,Hy) H.
+pose (q := (x/ y)%Z).
+pose (r := (x mod y)%Z).
+assert (h2: (x = y * q + r)%Z).
+  apply (Z_div_mod_eq x y); auto with zarith.
+assert (h3: (0 < r <= y - 1)%Z).
+  admit.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/euler001/why3session.xml

@@ -52,47 +52,26 @@
   expanded="true">
   <theory
    name="DivModHints"
-   verified="false"
+   verified="true"
    expanded="true">
    <goal
     name="mod_div_unique"
-    sum="dad7f26312903f3c0ce7ea31f08a8a94"
-    proved="false"
+    sum="f9938f131fc09755c1d3c9cb8bea2a18"
+    proved="true"
     expanded="true"
-    shape="ainfix =V2adivV0V1Aainfix =V3amodV0V1Iainfix <V3V1Aainfix <=c0V3Aainfix =V0ainfix +ainfix *V2V1V3Aainfix >V1c0F">
-    <proof
-     prover="simplify"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="unknown" time="0.05"/>
-    </proof>
-    <proof
-     prover="cvc3"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.13"/>
-    </proof>
-    <proof
-     prover="alt-ergo"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="unknown" time="0.37"/>
-    </proof>
+    shape="ainfix =V3amodV0V1Aainfix =V2adivV0V1Iainfix <V3V1Aainfix <=c0V3Aainfix =V0ainfix +ainfix *V2V1V3Aainfix >V1c0Aainfix >=V0c0F">
     <proof
-     prover="z3"
+     prover="coq"
      timelimit="5"
-     edited=""
+     edited="euler001_DivModHints_mod_div_unique_1.v"
      obsolete="false">
-     <result status="timeout" time="5.08"/>
+     <result status="valid" time="1.11"/>
     </proof>
    </goal>
    <goal
     name="mod_succ_1"
-    sum="0a9a4fed2057820683962116afbcf691"
-    proved="false"
+    sum="6f6edb13a7196ddd82da5b51501ddbf0"
+    proved="true"
     expanded="true"
     shape="ainfix =amodainfix +V0c1V1ainfix +amodV0V1c1Iainfix =amodainfix +V0c1V1c0NIainfix >V1c0Aainfix >=V0c0F">
     <proof
@@ -100,103 +79,40 @@
      timelimit="5"
      edited="euler001_DivModHints_mod_succ_1_1.v"
      obsolete="false">
-     <result status="unknown" time="1.54"/>
-    </proof>
-    <proof
-     prover="simplify"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.07"/>
-    </proof>
-    <proof
-     prover="cvc3"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.08"/>
-    </proof>
-    <proof
-     prover="alt-ergo"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.08"/>
-    </proof>
-    <proof
-     prover="z3"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.09"/>
-    </proof>
-    <proof
-     prover="gappa"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="unknown" time="0.02"/>
-    </proof>
-    <proof
-     prover="vampire"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.01"/>
-    </proof>
-    <proof
-     prover="spass"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.06"/>
+     <result status="valid" time="1.33"/>
     </proof>
    </goal>
    <goal
     name="mod_succ_2"
-    sum="d10d5aa2a6f8a1ece4e07b571cd6ab5a"
-    proved="false"
+    sum="7104913925174b07351cfd6bb290b525"
+    proved="true"
     expanded="true"
     shape="ainfix =amodV0V1ainfix -V1c1Iainfix =amodainfix +V0c1V1c0Iainfix >V1c0Aainfix >=V0c0F">
     <proof
-     prover="alt-ergo"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.08"/>
-    </proof>
-    <proof
-     prover="cvc3"
-     timelimit="5"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="5.09"/>
-    </proof>
-    <proof
-     prover="z3"
+     prover="coq"
      timelimit="5"
-     edited=""
+     edited="euler001_DivModHints_mod_succ_2_1.v"
      obsolete="false">
-     <result status="timeout" time="5.10"/>
+     <result status="valid" time="1.29"/>
     </proof>
    </goal>
    <goal
     name="div_succ_1"
-    sum="773313eff1da137611f6b6782b46b107"
+    sum="2fdd276de2d8937784e2bd42924d0a83"
     proved="true"
-    expanded="false"
+    expanded="true"
     shape="ainfix =adivainfix +V0c1V1ainfix +adivV0V1c1Iainfix =amodainfix +V0c1V1c0Iainfix >V1c0Aainfix >=V0c0F">
     <proof
      prover="cvc3"
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.10"/>
+     <result status="valid" time="0.08"/>
     </proof>
    </goal>
    <goal
     name="div_succ_2"
-    sum="500683558e28d63d764f874cc487b206"
+    sum="af570b10585b322352a5f00bc9f9fd12"
     proved="true"
     expanded="false"
     shape="ainfix =adivainfix +V0c1V1adivV0V1Iainfix =amodainfix +V0c1V1c0NIainfix >V1c0Aainfix >=V0c0F">
@@ -205,21 +121,21 @@
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="1.28"/>
+     <result status="valid" time="1.33"/>
     </proof>
     <proof
      prover="cvc3"
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.09"/>
+     <result status="valid" time="0.08"/>
     </proof>
     <proof
      prover="z3"
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.75"/>
+     <result status="valid" time="0.49"/>
     </proof>
    </goal>
   </theory>
@@ -238,7 +154,7 @@
      timelimit="2"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.07"/>
+     <result status="valid" time="0.12"/>
     </proof>
     <proof
      prover="alt-ergo"
@@ -252,12 +168,12 @@
      timelimit="2"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.07"/>
+     <result status="valid" time="0.09"/>
     </proof>
    </goal>
    <goal
     name="mod_15"
-    sum="071e467d1502abbcf3367fb8f2269e3c"
+    sum="7c10cad2c344640bfeff4b3a6a8fb07d"
     proved="true"
     expanded="false"
     shape="ainfix =amodV0c5c0Aainfix =amodV0c3c0qainfix =amodV0c15c0F">
@@ -266,160 +182,216 @@
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.37"/>
+     <result status="valid" time="0.53"/>
     </proof>
     <proof
      prover="alt-ergo"
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="3.05"/>
+     <result status="valid" time="2.05"/>
     </proof>
     <proof
      prover="z3"
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="timeout" time="5.10"/>
+     <result status="timeout" time="5.08"/>
+    </proof>
+   </goal>
+   <goal
+    name="triangle_numbers"
+    sum="eab47529f1d5363b4583acc38168c78f"
+    proved="true"
+    expanded="true"
+    shape="ainfix =ainfix *c2adivainfix *V0ainfix +V0c1c2ainfix *V0ainfix +V0c1F">
+    <proof
+     prover="cvc3"
+     timelimit="5"
+     edited=""
+     obsolete="false">
+     <result status="timeout" time="5.09"/>
+    </proof>
+    <proof
+     prover="alt-ergo"
+     timelimit="5"
+     edited=""
+     obsolete="false">
+     <result status="valid" time="0.04"/>
+    </proof>
+    <proof
+     prover="z3"
+     timelimit="5"
+     edited=""