Update parts of Coq realizations whose printing looks sane.

9ef42e17 · Guillaume Melquiond · 5a92d21d · 9ef42e17 · 9ef42e17 · 9ef42e17
Commit 9ef42e17 authored Jan 19, 2018 by Guillaume Melquiond
17 changed files
--- a/lib/coq/bv/BV_Gen.v
+++ b/lib/coq/bv/BV_Gen.v
@@ -212,8 +212,8 @@ omega.
 Qed.

 (* Why3 goal *)
-Lemma nth_out_of_bound : forall (x:t) (n:Z), ((n < 0%Z)%Z \/
-  (size <= n)%Z) -> ((nth x n) = false).
+Lemma nth_out_of_bound :
+  forall (x:t) (n:Z), ((n < 0%Z)%Z \/ (size <= n)%Z) -> ((nth x n) = false).
 intros.
 unfold nth.
 rewrite nth_aux_out_of_bound; auto with zarith.
@@ -1202,8 +1202,8 @@ Lemma to_int_def : forall (x:t), ((is_signed_positive x) ->
 Qed.

 (* Why3 goal *)
-Lemma to_uint_extensionality : forall (v:t) (v':t),
-  ((to_uint v) = (to_uint v')) -> (v = v').
+Lemma to_uint_extensionality :
+  forall (v:t) (v':t), ((to_uint v) = (to_uint v')) -> (v = v').
  unfold to_uint.
  intros v v'.
  rewrite Nat2Z.inj_iff.
@@ -1211,8 +1211,8 @@ Lemma to_uint_extensionality : forall (v:t) (v':t),
 Qed.

 (* Why3 goal *)
-Lemma to_int_extensionality : forall (v:t) (v':t),
-  ((to_int v) = (to_int v')) -> (v = v').
+Lemma to_int_extensionality :
+  forall (v:t) (v':t), ((to_int v) = (to_int v')) -> (v = v').
  apply twos_complement_extensionality.
 Qed.

@@ -1220,8 +1220,8 @@ Qed.
 Definition uint_in_range (i:Z) : Prop := (0%Z <= i)%Z /\ (i <= max_int)%Z.

 (* Why3 goal *)
-Lemma to_uint_bounds : forall (v:t), (0%Z <= (to_uint v))%Z /\
-  ((to_uint v) < two_power_size)%Z.
+Lemma to_uint_bounds :
+  forall (v:t), (0%Z <= (to_uint v))%Z /\ ((to_uint v) < two_power_size)%Z.
  intros v.
  unfold to_uint, uint_in_range.
  split.
@@ -1430,8 +1430,8 @@ Lemma zeros_sign_false: Bsign last_bit zeros = false.
 Qed.

 (* Why3 goal *)
-Lemma positive_is_ge_zeros : forall (x:t), (is_signed_positive x) <-> (sge x
-  zeros).
+Lemma positive_is_ge_zeros :
+  forall (x:t), (is_signed_positive x) <-> (sge x zeros).
  intros.
  unfold is_signed_positive, sge, to_int, twos_complement, size_nat.
  rewrite zeros_sign_false. destruct Bsign.
@@ -1586,8 +1586,8 @@ Definition lsr_bv : t -> t -> t.
 Defined.

 (* Why3 goal *)
-Lemma lsr_bv_is_lsr : forall (x:t) (n:t), ((lsr_bv x n) = (lsr x
-  (to_uint n))).
+Lemma lsr_bv_is_lsr :
+  forall (x:t) (n:t), ((lsr_bv x n) = (lsr x (to_uint n))).
  easy.
 Qed.

@@ -1616,8 +1616,8 @@ Definition lsl_bv : t -> t -> t.
 Defined.

 (* Why3 goal *)
-Lemma lsl_bv_is_lsl : forall (x:t) (n:t), ((lsl_bv x n) = (lsl x
-  (to_uint n))).
+Lemma lsl_bv_is_lsl :
+  forall (x:t) (n:t), ((lsl_bv x n) = (lsl x (to_uint n))).
  easy.
 Qed.

@@ -1640,14 +1640,14 @@ Definition rotate_left_bv : t -> t -> t.
 Defined.

 (* Why3 goal *)
-Lemma rotate_left_bv_is_rotate_left : forall (v:t) (n:t), ((rotate_left_bv v
-  n) = (rotate_left v (to_uint n))).
+Lemma rotate_left_bv_is_rotate_left :
+  forall (v:t) (n:t), ((rotate_left_bv v n) = (rotate_left v (to_uint n))).
 trivial.
 Qed.

 (* Why3 goal *)
-Lemma rotate_right_bv_is_rotate_right : forall (v:t) (n:t),
-  ((rotate_right_bv v n) = (rotate_right v (to_uint n))).
+Lemma rotate_right_bv_is_rotate_right :
+  forall (v:t) (n:t), ((rotate_right_bv v n) = (rotate_right v (to_uint n))).
 trivial.
 Qed.


--- a/lib/coq/ieee_float/Float32.v
+++ b/lib/coq/ieee_float/Float32.v
@@ -255,8 +255,8 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma zero_to_real : forall (x:t), (is_zero x) <-> ((t'isFinite x) /\
-  ((t'real x) = 0%R)).
+Lemma zero_to_real :
+  forall (x:t), (is_zero x) <-> ((t'isFinite x) /\ ((t'real x) = 0%R)).
 Proof.
  apply zero_to_real.
 Qed.
@@ -274,8 +274,8 @@ Proof.
 Defined.

 (* Why3 goal *)
-Lemma zero_of_int : forall (m:ieee_float.RoundingMode.mode),
-  (zeroF = (of_int m 0%Z)).
+Lemma zero_of_int :
+  forall (m:ieee_float.RoundingMode.mode), (zeroF = (of_int m 0%Z)).
 Proof.
  apply zero_of_int.
 Qed.
@@ -302,7 +302,8 @@ Proof.
 Defined.

 (* Why3 goal *)
-Lemma max_real_int : ((33554430 * 10141204801825835211973625643008)%R = (BuiltIn.IZR max_int)).
+Lemma max_real_int :
+  ((33554430 * 10141204801825835211973625643008)%R = (BuiltIn.IZR max_int)).
 Proof.
  unfold max_int.
  now rewrite mult_IZR, <- !Z2R_IZR.
@@ -314,8 +315,8 @@ Definition in_range (x:R) : Prop :=
  (x <= (33554430 * 10141204801825835211973625643008)%R)%R.

 (* Why3 assumption *)
-Definition in_int_range (i:Z): Prop := ((-max_int)%Z <= i)%Z /\
-  (i <= max_int)%Z.
+Definition in_int_range (i:Z) : Prop :=
+  ((-max_int)%Z <= i)%Z /\ (i <= max_int)%Z.

 (* Why3 goal *)
 Lemma is_finite : forall (x:t), (t'isFinite x) -> (in_range (t'real x)).
@@ -362,15 +363,15 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma Round_down_le : forall (x:R), ((round ieee_float.RoundingMode.RTN
-  x) <= x)%R.
+Lemma Round_down_le :
+  forall (x:R), ((round ieee_float.RoundingMode.RTN x) <= x)%R.
 Proof.
  apply Round_down_le.
 Qed.

 (* Why3 goal *)
-Lemma Round_up_ge : forall (x:R), (x <= (round ieee_float.RoundingMode.RTP
-  x))%R.
+Lemma Round_up_ge :
+  forall (x:R), (x <= (round ieee_float.RoundingMode.RTP x))%R.
 Proof.
  apply Round_up_ge.
 Qed.
@@ -390,8 +391,8 @@ Proof.
 Qed.

 (* Why3 assumption *)
-Definition in_safe_int_range (i:Z): Prop := ((-16777216%Z)%Z <= i)%Z /\
-  (i <= 16777216%Z)%Z.
+Definition in_safe_int_range (i:Z) : Prop :=
+  ((-16777216%Z)%Z <= i)%Z /\ (i <= 16777216%Z)%Z.

 (* Why3 goal *)
 Lemma Exact_rounding_for_integers : forall (m:ieee_float.RoundingMode.mode)
@@ -403,12 +404,14 @@ Proof.
 Qed.

 (* Why3 assumption *)
-Definition same_sign (x:t) (y:t): Prop := ((is_positive x) /\ (is_positive
-  y)) \/ ((is_negative x) /\ (is_negative y)).
+Definition same_sign (x:t) (y:t) : Prop :=
+  ((is_positive x) /\ (is_positive y)) \/
+  ((is_negative x) /\ (is_negative y)).

 (* Why3 assumption *)
-Definition diff_sign (x:t) (y:t): Prop := ((is_positive x) /\ (is_negative
-  y)) \/ ((is_negative x) /\ (is_positive y)).
+Definition diff_sign (x:t) (y:t) : Prop :=
+  ((is_positive x) /\ (is_negative y)) \/
+  ((is_negative x) /\ (is_positive y)).

 (* Why3 goal *)
 Lemma feq_eq : forall (x:t) (y:t), (t'isFinite x) -> ((t'isFinite y) ->
@@ -530,50 +533,50 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma lt_lt_finite : forall (x:t) (y:t) (z:t), (lt x y) -> ((lt y z) ->
-  (t'isFinite y)).
+Lemma lt_lt_finite :
+  forall (x:t) (y:t) (z:t), (lt x y) -> ((lt y z) -> (t'isFinite y)).
 Proof.
  apply lt_lt_finite.
 Qed.

 (* Why3 goal *)
-Lemma positive_to_real : forall (x:t), (t'isFinite x) -> ((is_positive x) ->
-  (0%R <= (t'real x))%R).
+Lemma positive_to_real :
+  forall (x:t), (t'isFinite x) -> ((is_positive x) -> (0%R <= (t'real x))%R).
 Proof.
  apply positive_to_real.
 Qed.

 (* Why3 goal *)
-Lemma to_real_positive : forall (x:t), (t'isFinite x) ->
-  ((0%R < (t'real x))%R -> (is_positive x)).
+Lemma to_real_positive :
+  forall (x:t), (t'isFinite x) -> ((0%R < (t'real x))%R -> (is_positive x)).
 Proof.
  apply to_real_positive.
 Qed.

 (* Why3 goal *)
-Lemma negative_to_real : forall (x:t), (t'isFinite x) -> ((is_negative x) ->
-  ((t'real x) <= 0%R)%R).
+Lemma negative_to_real :
+  forall (x:t), (t'isFinite x) -> ((is_negative x) -> ((t'real x) <= 0%R)%R).
 Proof.
  apply negative_to_real.
 Qed.

 (* Why3 goal *)
-Lemma to_real_negative : forall (x:t), (t'isFinite x) ->
-  (((t'real x) < 0%R)%R -> (is_negative x)).
+Lemma to_real_negative :
+  forall (x:t), (t'isFinite x) -> (((t'real x) < 0%R)%R -> (is_negative x)).
 Proof.
  apply to_real_negative.
 Qed.

 (* Why3 goal *)
-Lemma negative_xor_positive : forall (x:t), ~ ((is_positive x) /\
-  (is_negative x)).
+Lemma negative_xor_positive :
+  forall (x:t), ~ ((is_positive x) /\ (is_negative x)).
 Proof.
  apply negative_xor_positive.
 Qed.

 (* Why3 goal *)
-Lemma negative_or_positive : forall (x:t), (is_not_nan x) -> ((is_positive
-  x) \/ (is_negative x)).
+Lemma negative_or_positive :
+  forall (x:t), (is_not_nan x) -> ((is_positive x) \/ (is_negative x)).
 Proof.
  apply negative_or_positive.
 Qed.
@@ -600,8 +603,9 @@ Proof.
 Qed.

 (* Why3 assumption *)
-Definition product_sign (z:t) (x:t) (y:t): Prop := ((same_sign x y) ->
-  (is_positive z)) /\ ((diff_sign x y) -> (is_negative z)).
+Definition product_sign (z:t) (x:t) (y:t) : Prop :=
+  ((same_sign x y) -> (is_positive z)) /\
+  ((diff_sign x y) -> (is_negative z)).

 (* Why3 assumption *)
 Definition overflow_value (m:ieee_float.RoundingMode.mode) (x:t): Prop :=
@@ -621,7 +625,7 @@ Definition overflow_value (m:ieee_float.RoundingMode.mode) (x:t): Prop :=
  end.

 (* Why3 assumption *)
-Definition sign_zero_result (m:ieee_float.RoundingMode.mode) (x:t): Prop :=
+Definition sign_zero_result (m:ieee_float.RoundingMode.mode) (x:t) : Prop :=
  (is_zero x) ->
  match m with
  | ieee_float.RoundingMode.RTN => (is_negative x)

--- a/lib/coq/ieee_float/Float64.v
+++ b/lib/coq/ieee_float/Float64.v
@@ -274,8 +274,8 @@ Proof.
 Defined.

 (* Why3 goal *)
-Lemma zero_of_int : forall (m:ieee_float.RoundingMode.mode),
-  (zeroF = (of_int m 0%Z)).
+Lemma zero_of_int :
+  forall (m:ieee_float.RoundingMode.mode), (zeroF = (of_int m 0%Z)).
 Proof.
  apply zero_of_int.
 Qed.
@@ -402,12 +402,14 @@ Proof.
 Qed.

 (* Why3 assumption *)
-Definition same_sign (x:t) (y:t): Prop := ((is_positive x) /\ (is_positive
-  y)) \/ ((is_negative x) /\ (is_negative y)).
+Definition same_sign (x:t) (y:t) : Prop :=
+  ((is_positive x) /\ (is_positive y)) \/
+  ((is_negative x) /\ (is_negative y)).

 (* Why3 assumption *)
-Definition diff_sign (x:t) (y:t): Prop := ((is_positive x) /\ (is_negative
-  y)) \/ ((is_negative x) /\ (is_positive y)).
+Definition diff_sign (x:t) (y:t) : Prop :=
+  ((is_positive x) /\ (is_negative y)) \/
+  ((is_negative x) /\ (is_positive y)).

 (* Why3 goal *)
 Lemma feq_eq : forall (x:t) (y:t), (t'isFinite x) -> ((t'isFinite y) ->
@@ -529,50 +531,50 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma lt_lt_finite : forall (x:t) (y:t) (z:t), (lt x y) -> ((lt y z) ->
-  (t'isFinite y)).
+Lemma lt_lt_finite :
+  forall (x:t) (y:t) (z:t), (lt x y) -> ((lt y z) -> (t'isFinite y)).
 Proof.
  apply lt_lt_finite.
 Qed.

 (* Why3 goal *)
-Lemma positive_to_real : forall (x:t), (t'isFinite x) -> ((is_positive x) ->
-  (0%R <= (t'real x))%R).
+Lemma positive_to_real :
+  forall (x:t), (t'isFinite x) -> ((is_positive x) -> (0%R <= (t'real x))%R).
 Proof.
  apply positive_to_real.
 Qed.

 (* Why3 goal *)
-Lemma to_real_positive : forall (x:t), (t'isFinite x) ->
-  ((0%R < (t'real x))%R -> (is_positive x)).
+Lemma to_real_positive :
+  forall (x:t), (t'isFinite x) -> ((0%R < (t'real x))%R -> (is_positive x)).
 Proof.
  apply to_real_positive.
 Qed.

 (* Why3 goal *)
-Lemma negative_to_real : forall (x:t), (t'isFinite x) -> ((is_negative x) ->
-  ((t'real x) <= 0%R)%R).
+Lemma negative_to_real :
+  forall (x:t), (t'isFinite x) -> ((is_negative x) -> ((t'real x) <= 0%R)%R).
 Proof.
  apply negative_to_real.
 Qed.

 (* Why3 goal *)
-Lemma to_real_negative : forall (x:t), (t'isFinite x) ->
-  (((t'real x) < 0%R)%R -> (is_negative x)).
+Lemma to_real_negative :
+  forall (x:t), (t'isFinite x) -> (((t'real x) < 0%R)%R -> (is_negative x)).
 Proof.
  apply to_real_negative.
 Qed.

 (* Why3 goal *)
-Lemma negative_xor_positive : forall (x:t), ~ ((is_positive x) /\
-  (is_negative x)).
+Lemma negative_xor_positive :
+  forall (x:t), ~ ((is_positive x) /\ (is_negative x)).
 Proof.
  apply negative_xor_positive.
 Qed.

 (* Why3 goal *)
-Lemma negative_or_positive : forall (x:t), (is_not_nan x) -> ((is_positive
-  x) \/ (is_negative x)).
+Lemma negative_or_positive :
+  forall (x:t), (is_not_nan x) -> ((is_positive x) \/ (is_negative x)).
 Proof.
  apply negative_or_positive.
 Qed.
@@ -599,8 +601,9 @@ Proof.
 Qed.

 (* Why3 assumption *)
-Definition product_sign (z:t) (x:t) (y:t): Prop := ((same_sign x y) ->
-  (is_positive z)) /\ ((diff_sign x y) -> (is_negative z)).
+Definition product_sign (z:t) (x:t) (y:t) : Prop :=
+  ((same_sign x y) -> (is_positive z)) /\
+  ((diff_sign x y) -> (is_negative z)).

 (* Why3 assumption *)
 Definition overflow_value (m:ieee_float.RoundingMode.mode) (x:t): Prop :=

--- a/lib/coq/ieee_float/GenericFloat.v
+++ b/lib/coq/ieee_float/GenericFloat.v
@@ -782,8 +782,8 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma zero_to_real : forall (x:t), (is_zero x) <-> ((is_finite x) /\
-  ((to_real x) = 0%R)).
+Lemma zero_to_real :
+  forall (x:t), (is_zero x) <-> ((is_finite x) /\ ((to_real x) = 0%R)).
 Proof.
  unfold is_zero.
  assert (is_finite zeroF) by easy.
@@ -856,8 +856,8 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma zero_of_int : forall (m:ieee_float.RoundingMode.mode),
-  (zeroF = (of_int m 0%Z)).
+Lemma zero_of_int :
+  forall (m:ieee_float.RoundingMode.mode), (zeroF = (of_int m 0%Z)).
 Proof.
 auto.
 Qed.
@@ -1699,8 +1699,8 @@ destruct x, y, z; destruct b, b0, b1; easy.
 Qed.

 (* Why3 goal *)
-Lemma positive_to_real : forall (x:t), (is_finite x) -> ((is_positive x) ->
-  (0%R <= (to_real x))%R).
+Lemma positive_to_real :
+  forall (x:t), (is_finite x) -> ((is_positive x) -> (0%R <= (to_real x))%R).
 Proof.
  intros x h1 h2.
  assert (is_finite zeroF) as zero_is_finite by easy.
@@ -1731,8 +1731,8 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma to_real_positive : forall (x:t), (is_finite x) ->
-  ((0%R < (to_real x))%R -> (is_positive x)).
+Lemma to_real_positive :
+  forall (x:t), (is_finite x) -> ((0%R < (to_real x))%R -> (is_positive x)).
 Proof.
  intros x h1 h2.
  assert (is_finite zeroF) as zero_is_finite by easy.
@@ -1741,8 +1741,8 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma negative_to_real : forall (x:t), (is_finite x) -> ((is_negative x) ->
-  ((to_real x) <= 0%R)%R).
+Lemma negative_to_real :
+  forall (x:t), (is_finite x) -> ((is_negative x) -> ((to_real x) <= 0%R)%R).
 Proof.
  intros x h1 h2.
  assert (is_finite zeroF) as zero_is_finite by easy.
@@ -1773,8 +1773,8 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma to_real_negative : forall (x:t), (is_finite x) ->
-  (((to_real x) < 0%R)%R -> (is_negative x)).
+Lemma to_real_negative :
+  forall (x:t), (is_finite x) -> (((to_real x) < 0%R)%R -> (is_negative x)).
 Proof.
  intros x h1 h2.
  assert (is_finite zeroF) as zero_is_finite by easy.
@@ -1783,16 +1783,16 @@ Proof.
 Qed.

 (* Why3 goal *)
-Lemma negative_xor_positive : forall (x:t), ~ ((is_positive x) /\
-  (is_negative x)).
+Lemma negative_xor_positive :
+  forall (x:t), ~ ((is_positive x) /\ (is_negative x)).
 Proof.
 intros x.
 destruct x ; destruct b; easy.
 Qed.

 (* Why3 goal *)
-Lemma negative_or_positive : forall (x:t), (is_not_nan x) -> ((is_positive
-  x) \/ (is_negative x)).
+Lemma negative_or_positive :
+  forall (x:t), (is_not_nan x) -> ((is_positive x) \/ (is_negative x)).
 Proof.
 intros x h1.
 destruct x ; destruct b; simpl; auto; now elim h1.
@@ -1868,7 +1868,7 @@ Definition overflow_value (m:ieee_float.RoundingMode.mode) (x:t) : Prop :=
  end.

 (* Why3 assumption *)
-Definition sign_zero_result (m:ieee_float.RoundingMode.mode) (x:t): Prop :=
+Definition sign_zero_result (m:ieee_float.RoundingMode.mode) (x:t) : Prop :=
  (is_zero x) ->
  match m with
  | ieee_float.RoundingMode.RTN => (is_negative x)
@@ -3507,7 +3507,7 @@ Proof.
 Qed.

 (* Why3 goal *)
-Definition is_int: t -> Prop.
+Definition is_int : t -> Prop.
 Proof.
 exact (fun x => is_finite x /\ is_intR (to_real x)).
 Defined.

--- a/lib/coq/int/Exponentiation.v
+++ b/lib/coq/int/Exponentiation.v
@@ -52,8 +52,9 @@ easy.
 Qed.

 (* Why3 goal *)
-Lemma Power_s : forall (x:t) (n:Z), (0%Z <= n)%Z -> ((power x
-  (n + 1%Z)%Z) = (infix_as x (power x n))).
+Lemma Power_s :
+  forall (x:t) (n:Z),
+  (0%Z <= n)%Z -> ((power x (n + 1%Z)%Z) = (infix_as x (power x n))).
 Proof.
 intros x n h1.
 unfold power.
@@ -62,8 +63,9 @@ now rewrite Zabs_nat_Zsucc.
 Qed.

 (* Why3 goal *)
-Lemma Power_s_alt : forall (x:t) (n:Z), (0%Z < n)%Z -> ((power x
-  n) = (infix_as x (power x (n - 1%Z)%Z))).
+Lemma Power_s_alt :
+  forall (x:t) (n:Z),
+  (0%Z < n)%Z -> ((power x n) = (infix_as x (power x (n - 1%Z)%Z))).
 Proof.
 intros x n h1.
 rewrite <- Power_s; auto with zarith.

--- a/lib/coq/int/Int.v
+++ b/lib/coq/int/Int.v
@@ -38,8 +38,8 @@ exact Zle_lt_or_eq_iff.
 Qed.

 (* Why3 goal *)
-Lemma Assoc : forall (x:Z) (y:Z) (z:Z),
-  (((x + y)%Z + z)%Z = (x + (y + z)%Z)%Z).
+Lemma Assoc :
+  forall (x:Z) (y:Z) (z:Z), (((x + y)%Z + z)%Z = (x + (y + z)%Z)%Z).
 Proof.
 intros x y z.
 apply sym_eq.
@@ -77,8 +77,8 @@ exact Zplus_comm.
 Qed.

 (* Why3 goal *)
-Lemma Assoc1 : forall (x:Z) (y:Z) (z:Z),
-  (((x * y)%Z * z)%Z = (x * (y * z)%Z)%Z).
+Lemma Assoc1 :
+  forall (x:Z) (y:Z) (z:Z), (((x * y)%Z * z)%Z = (x * (y * z)%Z)%Z).
 Proof.
 intros x y z.
 apply sym_eq.
@@ -86,16 +86,16 @@ apply Zmult_assoc.
 Qed.

 (* Why3 goal *)
-Lemma Mul_distr_l : forall (x:Z) (y:Z) (z:Z),
-  ((x * (y + z)%Z)%Z = ((x * y)%Z + (x * z)%Z)%Z).
+Lemma Mul_distr_l :
+  forall (x:Z) (y:Z) (z:Z), ((x * (y + z)%Z)%Z = ((x * y)%Z + (x * z)%Z)%Z).
 Proof.
 intros x y z.
 apply Zmult_plus_distr_r.
 Qed.

 (* Why3 goal *)
-Lemma Mul_distr_r : forall (x:Z) (y:Z) (z:Z),
-  (((y + z)%Z * x)%Z = ((y * x)%Z + (z * x)%Z)%Z).
+Lemma Mul_distr_r :
+  forall (x:Z) (y:Z) (z:Z), (((y + z)%Z * x)%Z = ((y * x)%Z + (z * x)%Z)%Z).
 Proof.
 intros x y z.
 apply Zmult_plus_distr_l.
@@ -127,8 +127,8 @@ apply Zle_refl.
 Qed.

 (* Why3 goal *)
-Lemma Trans : forall (x:Z) (y:Z) (z:Z), (x <= y)%Z -> ((y <= z)%Z ->
-  (x <= z)%Z).
+Lemma Trans :
+  forall (x:Z) (y:Z) (z:Z), (x <= y)%Z -> ((y <= z)%Z -> (x <= z)%Z).
 Proof.
 exact Zle_trans.
 Qed.
@@ -158,8 +158,8 @@ now left.
 Qed.

 (* Why3 goal *)
-Lemma CompatOrderAdd : forall (x:Z) (y:Z) (z:Z), (x <= y)%Z ->
-  ((x + z)%Z <= (y + z)%Z)%Z.
+Lemma CompatOrderAdd :
+  forall (x:Z) (y:Z) (z:Z), (x <= y)%Z -> ((x + z)%Z <= (y + z)%Z)%Z.
 Proof.
 exact Zplus_le_compat_r.
 Qed.

--- a/lib/coq/number/Divisibility.v
+++ b/lib/coq/number/Divisibility.v
@@ -22,7 +22,8 @@ Require number.Parity.
 (* Hack so that Why3 does not override the notation below.

 (* Why3 assumption *)
-Definition divides (d:Z) (n:Z): Prop := ((d = 0%Z) -> (n = 0%Z)) /\
+Definition divides (d:Z) (n:Z) : Prop :=
+  ((d = 0%Z) -> (n = 0%Z)) /\
  ((~ (d = 0%Z)) -> ((ZArith.BinInt.Z.rem n d) = 0%Z)).

 *)
@@ -31,8 +32,8 @@ Require Import Znumtheory.
 Notation divides := Zdivide (only parsing).