remove some (but not all) instances of coq tactic in example

  vstte12_tree_reconstruction

examples/vstte12_tree_reconstruction.mlw

@@ -222,7 +222,7 @@ module ZipperBased
   (* an easy lemma on [g] *)
 
   lemma g_append:
-    forall l1 l2: list (int, tree). g (l1 ++ l2) -> g l1
+    forall l1 [@induction] l2: list (int, tree). g (l1 ++ l2) -> g l1
 
   (* key lemma for completeness: whenever we fail because [right] is empty,
      we have to prove that there is no solution

examples/vstte12_tree_reconstruction/vstte12_tree_reconstruction_Harness_VC_harness2_1.v

@@ -8,9 +8,6 @@ Require list.Length.
 Require list.Mem.
 Require list.Append.
 
-(* Why3 assumption *)
-Definition unit := unit.
-
 (* Why3 assumption *)
 Inductive tree :=
   | Leaf : tree
@@ -22,47 +19,53 @@ Existing Instance tree_WhyType.
 Fixpoint depths (d:Z) (t:tree) {struct t}: (list Z) :=
   match t with
   | Leaf => (Init.Datatypes.cons d Init.Datatypes.nil)
-  | (Node l r) => (Init.Datatypes.app (depths (d + 1%Z)%Z
-      l) (depths (d + 1%Z)%Z r))
+  | Node l r =>
+      (Init.Datatypes.app (depths (d + 1%Z)%Z l) (depths (d + 1%Z)%Z r))
   end.
 
-Axiom depths_head : forall (t:tree) (d:Z), match (depths d
-  t) with
+Axiom depths_head :
+  forall (t:tree) (d:Z),
+  match depths d t with
   | (Init.Datatypes.cons x _) => (d <= x)%Z
   | Init.Datatypes.nil => False
   end.
 
-Axiom depths_unique : forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d
-  t1) s1) = (Init.Datatypes.app (depths d t2) s2)) -> ((t1 = t2) /\
-  (s1 = s2)).
+Axiom depths_unique :
+  forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d t1) s1) =
+   (Init.Datatypes.app (depths d t2) s2)) ->
+  (t1 = t2) /\ (s1 = s2).
 
-Axiom depths_prefix : forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d1
-  t) s1) = (Init.Datatypes.app (depths d2 t) s2)) -> (d1 = d2).
+Axiom depths_prefix :
+  forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d1 t) s1) =
+   (Init.Datatypes.app (depths d2 t) s2)) ->
+  (d1 = d2).
 
-Axiom depths_prefix_simple : forall (t:tree) (d1:Z) (d2:Z), ((depths d1
-  t) = (depths d2 t)) -> (d1 = d2).
+Axiom depths_prefix_simple :
+  forall (t:tree) (d1:Z) (d2:Z), ((depths d1 t) = (depths d2 t)) -> (d1 = d2).
 
-Axiom depths_subtree : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z)
-  (s1:(list Z)), ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) ->
-  (d2 <= d1)%Z.
+Axiom depths_subtree :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z) (s1:(list Z)),
+  ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) -> (d2 <= d1)%Z.
 
-Axiom depths_unique2 : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z), ((depths d1
-  t1) = (depths d2 t2)) -> ((d1 = d2) /\ (t1 = t2)).
+Axiom depths_unique2 :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z),
+  ((depths d1 t1) = (depths d2 t2)) -> (d1 = d2) /\ (t1 = t2).
 
 (* Why3 assumption *)
-Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type): Prop :=
+Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type) : Prop :=
   match x1 with
   | (s1, d1) =>
       match x2 with
-      | (s2, d2) => ((list.Length.length s1) < (list.Length.length s2))%Z \/
-          (((list.Length.length s1) = (list.Length.length s2)) /\ match (s1,
-          s2) with
-          | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
-              (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
-          | _ => False
-          end)
+      | (s2, d2) =>
+          ((list.Length.length s1) < (list.Length.length s2))%Z \/
+          (((list.Length.length s1) = (list.Length.length s2)) /\
+           match (s1, s2) with
+           | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
+               (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
+           | _ => False
+           end)
       end
   end.
 
@@ -78,8 +81,13 @@ Qed.
 
 
 (* Why3 goal *)
-Theorem VC_harness2 : forall (result:tree), ~ ((depths 0%Z
-  result) = (Init.Datatypes.cons 1%Z (Init.Datatypes.cons 3%Z (Init.Datatypes.cons 2%Z (Init.Datatypes.cons 2%Z Init.Datatypes.nil))))).
+Theorem VC_harness2 :
+  forall (result:tree),
+  ~ ((depths 0%Z result) =
+     (Init.Datatypes.cons 1%Z
+      (Init.Datatypes.cons 3%Z
+       (Init.Datatypes.cons 2%Z (Init.Datatypes.cons 2%Z Init.Datatypes.nil))))).
+(* Why3 intros result. *)
 
 intuition.
 destruct result; simpl in H.

examples/vstte12_tree_reconstruction/vstte12_tree_reconstruction_Harness_VC_harness_1.v

@@ -8,9 +8,6 @@ Require list.Length.
 Require list.Mem.
 Require list.Append.
 
-(* Why3 assumption *)
-Definition unit := unit.
-
 (* Why3 assumption *)
 Inductive tree :=
   | Leaf : tree
@@ -22,54 +19,65 @@ Existing Instance tree_WhyType.
 Fixpoint depths (d:Z) (t:tree) {struct t}: (list Z) :=
   match t with
   | Leaf => (Init.Datatypes.cons d Init.Datatypes.nil)
-  | (Node l r) => (Init.Datatypes.app (depths (d + 1%Z)%Z
-      l) (depths (d + 1%Z)%Z r))
+  | Node l r =>
+      (Init.Datatypes.app (depths (d + 1%Z)%Z l) (depths (d + 1%Z)%Z r))
   end.
 
-Axiom depths_head : forall (t:tree) (d:Z), match (depths d
-  t) with
+Axiom depths_head :
+  forall (t:tree) (d:Z),
+  match depths d t with
   | (Init.Datatypes.cons x _) => (d <= x)%Z
   | Init.Datatypes.nil => False
   end.
 
-Axiom depths_unique : forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d
-  t1) s1) = (Init.Datatypes.app (depths d t2) s2)) -> ((t1 = t2) /\
-  (s1 = s2)).
+Axiom depths_unique :
+  forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d t1) s1) =
+   (Init.Datatypes.app (depths d t2) s2)) ->
+  (t1 = t2) /\ (s1 = s2).
 
-Axiom depths_prefix : forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d1
-  t) s1) = (Init.Datatypes.app (depths d2 t) s2)) -> (d1 = d2).
+Axiom depths_prefix :
+  forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d1 t) s1) =
+   (Init.Datatypes.app (depths d2 t) s2)) ->
+  (d1 = d2).
 
-Axiom depths_prefix_simple : forall (t:tree) (d1:Z) (d2:Z), ((depths d1
-  t) = (depths d2 t)) -> (d1 = d2).
+Axiom depths_prefix_simple :
+  forall (t:tree) (d1:Z) (d2:Z), ((depths d1 t) = (depths d2 t)) -> (d1 = d2).
 
-Axiom depths_subtree : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z)
-  (s1:(list Z)), ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) ->
-  (d2 <= d1)%Z.
+Axiom depths_subtree :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z) (s1:(list Z)),
+  ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) -> (d2 <= d1)%Z.
 
-Axiom depths_unique2 : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z), ((depths d1
-  t1) = (depths d2 t2)) -> ((d1 = d2) /\ (t1 = t2)).
+Axiom depths_unique2 :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z),
+  ((depths d1 t1) = (depths d2 t2)) -> (d1 = d2) /\ (t1 = t2).
 
 (* Why3 assumption *)
-Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type): Prop :=
+Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type) : Prop :=
   match x1 with
   | (s1, d1) =>
       match x2 with
-      | (s2, d2) => ((list.Length.length s1) < (list.Length.length s2))%Z \/
-          (((list.Length.length s1) = (list.Length.length s2)) /\ match (s1,
-          s2) with
-          | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
-              (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
-          | _ => False
-          end)
+      | (s2, d2) =>
+          ((list.Length.length s1) < (list.Length.length s2))%Z \/
+          (((list.Length.length s1) = (list.Length.length s2)) /\
+           match (s1, s2) with
+           | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
+               (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
+           | _ => False
+           end)
       end
   end.
 
 (* Why3 goal *)
-Theorem VC_harness : forall (result:tree), ((depths 0%Z
-  result) = (Init.Datatypes.cons 1%Z (Init.Datatypes.cons 3%Z (Init.Datatypes.cons 3%Z (Init.Datatypes.cons 2%Z Init.Datatypes.nil))))) ->
+Theorem VC_harness :
+  forall (result:tree),
+  ((depths 0%Z result) =
+   (Init.Datatypes.cons 1%Z
+    (Init.Datatypes.cons 3%Z
+     (Init.Datatypes.cons 3%Z (Init.Datatypes.cons 2%Z Init.Datatypes.nil))))) ->
   (result = (Node Leaf (Node (Node Leaf Leaf) Leaf))).
+(* Why3 intros result h1. *)
 
 intuition.
 

examples/vstte12_tree_reconstruction/vstte12_tree_reconstruction_Harness_VC_harness_2.v

@@ -8,9 +8,6 @@ Require list.Length.
 Require list.Mem.
 Require list.Append.
 
-(* Why3 assumption *)
-Definition unit := unit.
-
 (* Why3 assumption *)
 Inductive tree :=
   | Leaf : tree
@@ -22,53 +19,64 @@ Existing Instance tree_WhyType.
 Fixpoint depths (d:Z) (t:tree) {struct t}: (list Z) :=
   match t with
   | Leaf => (Init.Datatypes.cons d Init.Datatypes.nil)
-  | (Node l r) => (Init.Datatypes.app (depths (d + 1%Z)%Z
-      l) (depths (d + 1%Z)%Z r))
+  | Node l r =>
+      (Init.Datatypes.app (depths (d + 1%Z)%Z l) (depths (d + 1%Z)%Z r))
   end.
 
-Axiom depths_head : forall (t:tree) (d:Z), match (depths d
-  t) with
+Axiom depths_head :
+  forall (t:tree) (d:Z),
+  match depths d t with
   | (Init.Datatypes.cons x _) => (d <= x)%Z
   | Init.Datatypes.nil => False
   end.
 
-Axiom depths_unique : forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d
-  t1) s1) = (Init.Datatypes.app (depths d t2) s2)) -> ((t1 = t2) /\
-  (s1 = s2)).
+Axiom depths_unique :
+  forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d t1) s1) =
+   (Init.Datatypes.app (depths d t2) s2)) ->
+  (t1 = t2) /\ (s1 = s2).
 
-Axiom depths_prefix : forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d1
-  t) s1) = (Init.Datatypes.app (depths d2 t) s2)) -> (d1 = d2).
+Axiom depths_prefix :
+  forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d1 t) s1) =
+   (Init.Datatypes.app (depths d2 t) s2)) ->
+  (d1 = d2).
 
-Axiom depths_prefix_simple : forall (t:tree) (d1:Z) (d2:Z), ((depths d1
-  t) = (depths d2 t)) -> (d1 = d2).
+Axiom depths_prefix_simple :
+  forall (t:tree) (d1:Z) (d2:Z), ((depths d1 t) = (depths d2 t)) -> (d1 = d2).
 
-Axiom depths_subtree : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z)
-  (s1:(list Z)), ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) ->
-  (d2 <= d1)%Z.
+Axiom depths_subtree :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z) (s1:(list Z)),
+  ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) -> (d2 <= d1)%Z.
 
-Axiom depths_unique2 : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z), ((depths d1
-  t1) = (depths d2 t2)) -> ((d1 = d2) /\ (t1 = t2)).
+Axiom depths_unique2 :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z),
+  ((depths d1 t1) = (depths d2 t2)) -> (d1 = d2) /\ (t1 = t2).
 
 (* Why3 assumption *)
-Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type): Prop :=
+Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type) : Prop :=
   match x1 with
   | (s1, d1) =>
       match x2 with
-      | (s2, d2) => ((list.Length.length s1) < (list.Length.length s2))%Z \/
-          (((list.Length.length s1) = (list.Length.length s2)) /\ match (s1,
-          s2) with
-          | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
-              (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
-          | _ => False
-          end)
+      | (s2, d2) =>
+          ((list.Length.length s1) < (list.Length.length s2))%Z \/
+          (((list.Length.length s1) = (list.Length.length s2)) /\
+           match (s1, s2) with
+           | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
+               (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
+           | _ => False
+           end)
       end
   end.
 
 (* Why3 goal *)
-Theorem VC_harness : ~ forall (t:tree), ~ ((depths 0%Z
-  t) = (Init.Datatypes.cons 1%Z (Init.Datatypes.cons 3%Z (Init.Datatypes.cons 3%Z (Init.Datatypes.cons 2%Z Init.Datatypes.nil))))).
+Theorem VC_harness :
+  ~ (forall (t:tree),
+     ~ ((depths 0%Z t) =
+        (Init.Datatypes.cons 1%Z
+         (Init.Datatypes.cons 3%Z
+          (Init.Datatypes.cons 3%Z
+           (Init.Datatypes.cons 2%Z Init.Datatypes.nil)))))).
 intuition.
 apply (H (Node Leaf (Node (Node Leaf Leaf) Leaf))).
 reflexivity.

examples/vstte12_tree_reconstruction/vstte12_tree_reconstruction_TreeReconstruction_VC_build_rec_1.v

@@ -8,9 +8,6 @@ Require list.Length.
 Require list.Mem.
 Require list.Append.
 
-(* Why3 assumption *)
-Definition unit := unit.
-
 (* Why3 assumption *)
 Inductive tree :=
   | Leaf : tree
@@ -22,57 +19,66 @@ Existing Instance tree_WhyType.
 Fixpoint depths (d:Z) (t:tree) {struct t}: (list Z) :=
   match t with
   | Leaf => (Init.Datatypes.cons d Init.Datatypes.nil)
-  | (Node l r) => (Init.Datatypes.app (depths (d + 1%Z)%Z
-      l) (depths (d + 1%Z)%Z r))
+  | Node l r =>
+      (Init.Datatypes.app (depths (d + 1%Z)%Z l) (depths (d + 1%Z)%Z r))
   end.
 
-Axiom depths_head : forall (t:tree) (d:Z), match (depths d
-  t) with
+Axiom depths_head :
+  forall (t:tree) (d:Z),
+  match depths d t with
   | (Init.Datatypes.cons x _) => (d <= x)%Z
   | Init.Datatypes.nil => False
   end.
 
-Axiom depths_unique : forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d
-  t1) s1) = (Init.Datatypes.app (depths d t2) s2)) -> ((t1 = t2) /\
-  (s1 = s2)).
+Axiom depths_unique :
+  forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d t1) s1) =
+   (Init.Datatypes.app (depths d t2) s2)) ->
+  (t1 = t2) /\ (s1 = s2).
 
-Axiom depths_prefix : forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d1
-  t) s1) = (Init.Datatypes.app (depths d2 t) s2)) -> (d1 = d2).
+Axiom depths_prefix :
+  forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d1 t) s1) =
+   (Init.Datatypes.app (depths d2 t) s2)) ->
+  (d1 = d2).
 
-Axiom depths_prefix_simple : forall (t:tree) (d1:Z) (d2:Z), ((depths d1
-  t) = (depths d2 t)) -> (d1 = d2).
+Axiom depths_prefix_simple :
+  forall (t:tree) (d1:Z) (d2:Z), ((depths d1 t) = (depths d2 t)) -> (d1 = d2).
 
-Axiom depths_subtree : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z)
-  (s1:(list Z)), ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) ->
-  (d2 <= d1)%Z.
+Axiom depths_subtree :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z) (s1:(list Z)),
+  ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) -> (d2 <= d1)%Z.
 
-Axiom depths_unique2 : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z), ((depths d1
-  t1) = (depths d2 t2)) -> ((d1 = d2) /\ (t1 = t2)).
+Axiom depths_unique2 :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z),
+  ((depths d1 t1) = (depths d2 t2)) -> (d1 = d2) /\ (t1 = t2).
 
 (* Why3 assumption *)
-Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type): Prop :=
+Definition lex (x1:((list Z)* Z)%type) (x2:((list Z)* Z)%type) : Prop :=
   match x1 with
   | (s1, d1) =>
       match x2 with
-      | (s2, d2) => ((list.Length.length s1) < (list.Length.length s2))%Z \/
-          (((list.Length.length s1) = (list.Length.length s2)) /\ match (s1,
-          s2) with
-          | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
-              (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
-          | _ => False
-          end)
+      | (s2, d2) =>
+          ((list.Length.length s1) < (list.Length.length s2))%Z \/
+          (((list.Length.length s1) = (list.Length.length s2)) /\
+           match (s1, s2) with
+           | ((Init.Datatypes.cons h1 _), (Init.Datatypes.cons h2 _)) =>
+               (d2 < d1)%Z /\ ((d1 <= h1)%Z /\ (h1 = h2))
+           | _ => False
+           end)
       end
   end.
 
 (* Why3 goal *)
-Theorem VC_build_rec : forall (d:Z) (s:(list Z)), forall (x:Z) (x1:(list Z)),
-  (s = (Init.Datatypes.cons x x1)) -> ((~ (x < d)%Z) -> ((~ (x = d)) ->
-  forall (o:tree) (o1:(list Z)), (s = (Init.Datatypes.app (depths (d + 1%Z)%Z
-  o) o1)) -> ((forall (t:tree) (s':(list Z)),
-  ~ ((Init.Datatypes.app (depths (d + 1%Z)%Z t) s') = o1)) -> forall (t:tree)
-  (s':(list Z)), ~ ((Init.Datatypes.app (depths d t) s') = s)))).
+Theorem VC_build_rec :
+  forall (d:Z) (s:(list Z)), forall (x:Z) (x1:(list Z)),
+  (s = (Init.Datatypes.cons x x1)) -> ~ (x < d)%Z -> ~ (x = d) ->
+  forall (o:tree) (o1:(list Z)),
+  (s = (Init.Datatypes.app (depths (d + 1%Z)%Z o) o1)) ->
+  (forall (t:tree) (s':(list Z)),
+   ~ ((Init.Datatypes.app (depths (d + 1%Z)%Z t) s') = o1)) ->
+  forall (t:tree) (s':(list Z)), ~ ((Init.Datatypes.app (depths d t) s') = s).
+(* Why3 intros d s x x1 h1 h2 h3 o o1 h4 h5 t s'. *)
 (*intros d s x x1 h1 h2 h3 o o1 h4 h5 t s'.*)
 intros d s x x1 h1 h2 h3 result result1 h4 h5 t s'.
 subst.

examples/vstte12_tree_reconstruction/vstte12_tree_reconstruction_TreeReconstruction_VC_build_rec_2.v

@@ -8,9 +8,6 @@ Require list.Length.
 Require list.Mem.
 Require list.Append.
 
-(* Why3 assumption *)
-Definition unit := unit.
-
 (* Why3 assumption *)
 Inductive tree :=
   | Leaf : tree
@@ -22,56 +19,64 @@ Existing Instance tree_WhyType.
 Fixpoint depths (d:Z) (t:tree) {struct t}: (list Z) :=
   match t with
   | Leaf => (Init.Datatypes.cons d Init.Datatypes.nil)
-  | (Node l r) => (Init.Datatypes.app (depths (d + 1%Z)%Z
-      l) (depths (d + 1%Z)%Z r))
+  | Node l r =>
+      (Init.Datatypes.app (depths (d + 1%Z)%Z l) (depths (d + 1%Z)%Z r))
   end.
 
-Axiom depths_head : forall (t:tree) (d:Z), match (depths d
-  t) with
+Axiom depths_head :
+  forall (t:tree) (d:Z),
+  match depths d t with
   | (Init.Datatypes.cons x _) => (d <= x)%Z
   | Init.Datatypes.nil => False
   end.
 
-Axiom depths_unique : forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d
-  t1) s1) = (Init.Datatypes.app (depths d t2) s2)) -> ((t1 = t2) /\
-  (s1 = s2)).
+Axiom depths_unique :
+  forall (t1:tree) (t2:tree) (d:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d t1) s1) =
+   (Init.Datatypes.app (depths d t2) s2)) ->
+  (t1 = t2) /\ (s1 = s2).
 
-Axiom depths_prefix : forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z))
-  (s2:(list Z)), ((Init.Datatypes.app (depths d1
-  t) s1) = (Init.Datatypes.app (depths d2 t) s2)) -> (d1 = d2).
+Axiom depths_prefix :
+  forall (t:tree) (d1:Z) (d2:Z) (s1:(list Z)) (s2:(list Z)),
+  ((Init.Datatypes.app (depths d1 t) s1) =
+   (Init.Datatypes.app (depths d2 t) s2)) ->
+  (d1 = d2).
 
-Axiom depths_prefix_simple : forall (t:tree) (d1:Z) (d2:Z), ((depths d1
-  t) = (depths d2 t)) -> (d1 = d2).
+Axiom depths_prefix_simple :
+  forall (t:tree) (d1:Z) (d2:Z), ((depths d1 t) = (depths d2 t)) -> (d1 = d2).
 
-Axiom depths_subtree : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z)
-  (s1:(list Z)), ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) ->
-  (d2 <= d1)%Z.
+Axiom depths_subtree :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z) (s1:(list Z)),
+  ((Init.Datatypes.app (depths d1 t1) s1) = (depths d2 t2)) -> (d2 <= d1)%Z.
 
-Axiom depths_unique2 : forall (t1:tree) (t2:tree) (d1:Z) (d2:Z), ((depths d1
-  t1) = (depths d2 t2)) -> ((d1 = d2) /\ (t1 = t2)).
+Axiom depths_unique2 :
+  forall (t1:tree) (t2:tree) (d1:Z) (d2:Z),
+  ((depths d1 t1) = (depths d2 t2)) -> (d1 = d2) /\ (t1 = t2).
 
 (* Why3 assumption *)