preuve de power_sum

tests/bitvector1/bitvector1_Pow2int_Power_sum_1.v

@@ -20,9 +20,22 @@ Theorem Power_sum : forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
   ((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%Z).
 (* YOU MAY EDIT THE PROOF BELOW *)
 intros n m Hmn.
+cut (0 <= m); auto with zarith.
+apply Z_lt_induction with
+  (P:= fun m => 
+      0 <= m -> pow2 (n + m) = pow2 n * pow2 m);
+  auto with zarith.
+intros x Hind Hxpos.
+assert (h:(x = 0 \/ x > 0)) by omega.
+destruct h.
+subst x.
+rewrite Power_0.
+
+
+(*
+auto with zarith.
 elim n.
 replace (0+m) with m by omega.
-rewrite Power_0.
 replace (1* pow2 m) with (pow2 m) by omega.
 auto.
 intro.
@@ -37,7 +50,7 @@ replace (2 * Zpos p0) with (Zpos p0 * 2) by omega.
 rewrite<-Zplus_diag_eq_mult_2.
 rewrite Power_s.
 
-
+*)
 
 
 Qed.

tests/bitvector1/why3session.xml

@@ -1036,7 +1036,7 @@
   <theory
    name="Pow2real"
    verified="false"
-   expanded="true">
+   expanded="false">
    <goal
     name="Power_1"
     sum="05eff9d48281eb24cb06eb9caecf811e"
@@ -1118,7 +1118,7 @@
   <theory
    name="BitVector"
    verified="false"
-   expanded="true">
+   expanded="false">
    <goal
     name="Nth_bw_xor_v1true"
     sum="3ea193c399b136e4f0619cc666a98664"
@@ -1305,7 +1305,7 @@
     name="nth_from_int_low_even"
     sum="5de955fad899fb21842dc88dddf9eef7"
     proved="true"
-    expanded="true"
+    expanded="false"
     shape="ainfix =anthafrom_intV0c0aFalseIainfix =amodV0c2c0F">
     <proof
      prover="coq"
@@ -1417,28 +1417,28 @@
     name="nth_from_int2c_low_even_positive"
     sum="69f18f37edd8fc04bfdf09f8bb8a8f85"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =anthafrom_int2cV0c0aFalseIainfix =amodV0c2c0Aainfix >=V0c0F">
    </goal>
    <goal
     name="nth_from_int2c_low_odd_positive"
     sum="475988455b849e1fbc62bc87c2a4b147"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =anthafrom_int2cV0c0aTrueIainfix =amodV0c2c0NAainfix >=V0c0F">
    </goal>
    <goal
     name="nth_from_int2c_low_even_negative"
     sum="5ffbc4d5d31761cff206901e8b5f825a"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =anthafrom_int2cV0c0aTrueIainfix =amodV0c2c0Aainfix <V0c0F">
    </goal>
    <goal
     name="nth_from_int2c_low_odd_negative"
     sum="043edbe6d2a94a92420cb81b9b134329"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =anthafrom_int2cV0c0aFalseIainfix =amodV0c2c0NAainfix <V0c0F">
    </goal>
   </theory>
@@ -2969,14 +2969,14 @@
     name="x_negative"
     sum="75aca8e6393ca908f8163a9722aeb5e8"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =ato_nat_subavarV0c30c0aprefix -V0Iainfix =anthavarV0c31aFalseF">
    </goal>
    <goal
     name="mantissa_var_x_negative"
     sum="af80cf6d8e488e18405d3f91f4ce940f"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =amantissaavarV0ainfix -apow2c31V0Iainfix =anthavarV0c31aFalseF">
     <proof
      prover="coq"
@@ -3067,7 +3067,7 @@
     name="from_int_sum"
     sum="a1bb3937bfed62df0155e2517454c2d0"
     proved="false"
-    expanded="true"
+    expanded="false"
     shape="ainfix =afrom_intainfix +apow2c31V0ainfix +.afrom_intapow2c31afrom_intV0F">
    </goal>
    <goal