Update Coq realizations.

lib/coq/floating_point/Double.v

@@ -39,15 +39,15 @@ exact (model 53 1024).
 Defined.
 
 (* Why3 assumption *)
-Definition round_error(x:floating_point.DoubleFormat.double): R :=
+Definition round_error (x:floating_point.DoubleFormat.double): R :=
   (Rabs ((value x) - (exact x))%R).
 
 (* Why3 assumption *)
-Definition total_error(x:floating_point.DoubleFormat.double): R :=
+Definition total_error (x:floating_point.DoubleFormat.double): R :=
   (Rabs ((value x) - (model x))%R).
 
 (* Why3 assumption *)
-Definition no_overflow(m:floating_point.Rounding.mode) (x:R): Prop :=
+Definition no_overflow (m:floating_point.Rounding.mode) (x:R): Prop :=
   ((Rabs (round m
   x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
 
@@ -118,12 +118,12 @@ now apply Round_up_neg.
 Qed.
 
 (* Why3 assumption *)
-Definition of_real_post(m:floating_point.Rounding.mode) (x:R)
+Definition of_real_post (m:floating_point.Rounding.mode) (x:R)
   (res:floating_point.DoubleFormat.double): Prop := ((value res) = (round m
   x)) /\ (((exact res) = x) /\ ((model res) = x)).
 
 (* Why3 assumption *)
-Definition add_post(m:floating_point.Rounding.mode)
+Definition add_post (m:floating_point.Rounding.mode)
   (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double)
   (res:floating_point.DoubleFormat.double): Prop := ((value res) = (round m
@@ -132,7 +132,7 @@ Definition add_post(m:floating_point.Rounding.mode)
   ((model res) = ((model x) + (model y))%R)).
 
 (* Why3 assumption *)
-Definition sub_post(m:floating_point.Rounding.mode)
+Definition sub_post (m:floating_point.Rounding.mode)
   (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double)
   (res:floating_point.DoubleFormat.double): Prop := ((value res) = (round m
@@ -141,7 +141,7 @@ Definition sub_post(m:floating_point.Rounding.mode)
   ((model res) = ((model x) - (model y))%R)).
 
 (* Why3 assumption *)
-Definition mul_post(m:floating_point.Rounding.mode)
+Definition mul_post (m:floating_point.Rounding.mode)
   (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double)
   (res:floating_point.DoubleFormat.double): Prop := ((value res) = (round m
@@ -150,7 +150,7 @@ Definition mul_post(m:floating_point.Rounding.mode)
   ((model res) = ((model x) * (model y))%R)).
 
 (* Why3 assumption *)
-Definition div_post(m:floating_point.Rounding.mode)
+Definition div_post (m:floating_point.Rounding.mode)
   (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double)
   (res:floating_point.DoubleFormat.double): Prop := ((value res) = (round m
@@ -159,17 +159,17 @@ Definition div_post(m:floating_point.Rounding.mode)
   ((model res) = (Rdiv (model x) (model y))%R)).
 
 (* Why3 assumption *)
-Definition neg_post(x:floating_point.DoubleFormat.double)
+Definition neg_post (x:floating_point.DoubleFormat.double)
   (res:floating_point.DoubleFormat.double): Prop :=
   ((value res) = (-(value x))%R) /\ (((exact res) = (-(exact x))%R) /\
   ((model res) = (-(model x))%R)).
 
 (* Why3 assumption *)
-Definition lt(x:floating_point.DoubleFormat.double)
+Definition lt (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double): Prop := ((value x) < (value y))%R.
 
 (* Why3 assumption *)
-Definition gt(x:floating_point.DoubleFormat.double)
+Definition gt (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double): Prop := ((value y) < (value x))%R.
 
 

lib/coq/floating_point/Rounding.v

@@ -4,12 +4,12 @@ Require Import BuiltIn.
 Require BuiltIn.
 
 (* Why3 assumption *)
-Inductive mode  :=
-  | NearestTiesToEven : mode 
-  | ToZero : mode 
-  | Up : mode 
-  | Down : mode 
-  | NearestTiesToAway : mode .
+Inductive mode :=
+  | NearestTiesToEven : mode
+  | ToZero : mode
+  | Up : mode
+  | Down : mode
+  | NearestTiesToAway : mode.
 Axiom mode_WhyType : WhyType mode.
 Existing Instance mode_WhyType.
 

lib/coq/floating_point/Single.v

@@ -39,15 +39,15 @@ exact (model 24 128).
 Defined.
 
 (* Why3 assumption *)
-Definition round_error(x:floating_point.SingleFormat.single): R :=
+Definition round_error (x:floating_point.SingleFormat.single): R :=
   (Rabs ((value x) - (exact x))%R).
 
 (* Why3 assumption *)
-Definition total_error(x:floating_point.SingleFormat.single): R :=
+Definition total_error (x:floating_point.SingleFormat.single): R :=
   (Rabs ((value x) - (model x))%R).
 
 (* Why3 assumption *)
-Definition no_overflow(m:floating_point.Rounding.mode) (x:R): Prop :=
+Definition no_overflow (m:floating_point.Rounding.mode) (x:R): Prop :=
   ((Rabs (round m x)) <= (33554430 * 10141204801825835211973625643008)%R)%R.
 
 Lemma max_single_eq: (33554430 * 10141204801825835211973625643008 = max 24 128)%R.
@@ -126,12 +126,12 @@ now apply Round_up_neg.
 Qed.
 
 (* Why3 assumption *)
-Definition of_real_post(m:floating_point.Rounding.mode) (x:R)
+Definition of_real_post (m:floating_point.Rounding.mode) (x:R)
   (res:floating_point.SingleFormat.single): Prop := ((value res) = (round m
   x)) /\ (((exact res) = x) /\ ((model res) = x)).
 
 (* Why3 assumption *)
-Definition add_post(m:floating_point.Rounding.mode)
+Definition add_post (m:floating_point.Rounding.mode)
   (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single)
   (res:floating_point.SingleFormat.single): Prop := ((value res) = (round m
@@ -140,7 +140,7 @@ Definition add_post(m:floating_point.Rounding.mode)
   ((model res) = ((model x) + (model y))%R)).
 
 (* Why3 assumption *)
-Definition sub_post(m:floating_point.Rounding.mode)
+Definition sub_post (m:floating_point.Rounding.mode)
   (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single)
   (res:floating_point.SingleFormat.single): Prop := ((value res) = (round m
@@ -149,7 +149,7 @@ Definition sub_post(m:floating_point.Rounding.mode)
   ((model res) = ((model x) - (model y))%R)).
 
 (* Why3 assumption *)
-Definition mul_post(m:floating_point.Rounding.mode)
+Definition mul_post (m:floating_point.Rounding.mode)
   (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single)
   (res:floating_point.SingleFormat.single): Prop := ((value res) = (round m
@@ -158,7 +158,7 @@ Definition mul_post(m:floating_point.Rounding.mode)
   ((model res) = ((model x) * (model y))%R)).
 
 (* Why3 assumption *)
-Definition div_post(m:floating_point.Rounding.mode)
+Definition div_post (m:floating_point.Rounding.mode)
   (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single)
   (res:floating_point.SingleFormat.single): Prop := ((value res) = (round m
@@ -167,17 +167,17 @@ Definition div_post(m:floating_point.Rounding.mode)
   ((model res) = (Rdiv (model x) (model y))%R)).
 
 (* Why3 assumption *)
-Definition neg_post(x:floating_point.SingleFormat.single)
+Definition neg_post (x:floating_point.SingleFormat.single)
   (res:floating_point.SingleFormat.single): Prop :=
   ((value res) = (-(value x))%R) /\ (((exact res) = (-(exact x))%R) /\
   ((model res) = (-(model x))%R)).
 
 (* Why3 assumption *)
-Definition lt(x:floating_point.SingleFormat.single)
+Definition lt (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single): Prop := ((value x) < (value y))%R.
 
 (* Why3 assumption *)
-Definition gt(x:floating_point.SingleFormat.single)
+Definition gt (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single): Prop := ((value y) < (value x))%R.
 
 

lib/coq/map/MapPermut.v

@@ -6,7 +6,7 @@ Require int.Int.
 Require map.Map.
 
 (* Why3 assumption *)
-Definition exchange {a:Type} {a_WT:WhyType a}(a1:(map.Map.map Z a))
+Definition exchange {a:Type} {a_WT:WhyType a} (a1:(map.Map.map Z a))
   (a2:(map.Map.map Z a)) (i:Z) (j:Z): Prop := ((map.Map.get a1
   i) = (map.Map.get a2 j)) /\ (((map.Map.get a2 i) = (map.Map.get a1 j)) /\
   forall (k:Z), ((~ (k = i)) /\ ~ (k = j)) -> ((map.Map.get a1

lib/coq/number/Divisibility.v

@@ -12,7 +12,7 @@ Require number.Parity.
 (* Hack so that Why3 does not override the notation below.
 
 (* Why3 assumption *)
-Definition divides(d:Z) (n:Z): Prop := exists q:Z, (n = (q * d)%Z).
+Definition divides (d:Z) (n:Z): Prop := exists q:Z, (n = (q * d)%Z).
 
 *)
 
@@ -38,15 +38,15 @@ exact Zdivide_0.
 Qed.
 
 (* Why3 goal *)
-Lemma divides_left : forall (a:Z) (b:Z) (c:Z), (divides a b) ->
-  (divides (c * a)%Z (c * b)%Z).
+Lemma divides_left : forall (a:Z) (b:Z) (c:Z), (divides a b) -> (divides
+  (c * a)%Z (c * b)%Z).
 Proof.
 exact Zmult_divide_compat_l.
 Qed.
 
 (* Why3 goal *)
-Lemma divides_right : forall (a:Z) (b:Z) (c:Z), (divides a b) ->
-  (divides (a * c)%Z (b * c)%Z).
+Lemma divides_right : forall (a:Z) (b:Z) (c:Z), (divides a b) -> (divides
+  (a * c)%Z (b * c)%Z).
 Proof.
 exact Zmult_divide_compat_r.
 Qed.
@@ -160,8 +160,8 @@ ring.
 Qed.
 
 (* Why3 goal *)
-Lemma divides_mod_euclidean : forall (a:Z) (b:Z), (~ (b = 0%Z)) ->
-  ((divides b a) -> ((int.EuclideanDivision.mod1 a b) = 0%Z)).
+Lemma divides_mod_euclidean : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> ((divides
+  b a) -> ((int.EuclideanDivision.mod1 a b) = 0%Z)).
 Proof.
 intros a b Zb H.
 assert (Zmod a b = Z0).

lib/coq/number/Parity.v

@@ -5,10 +5,10 @@ Require BuiltIn.
 Require int.Int.
 
 (* Why3 assumption *)
-Definition even(n:Z): Prop := exists k:Z, (n = (2%Z * k)%Z).
+Definition even (n:Z): Prop := exists k:Z, (n = (2%Z * k)%Z).
 
 (* Why3 assumption *)
-Definition odd(n:Z): Prop := exists k:Z, (n = ((2%Z * k)%Z + 1%Z)%Z).
+Definition odd (n:Z): Prop := exists k:Z, (n = ((2%Z * k)%Z + 1%Z)%Z).
 
 Lemma even_is_Zeven :
   forall n, even n <-> Zeven n.

lib/coq/number/Prime.v

@@ -13,7 +13,7 @@ Require number.Divisibility.
 Import Znumtheory.
 
 (* Why3 assumption *)
-Definition prime(p:Z): Prop := (2%Z <= p)%Z /\ forall (n:Z), ((1%Z < n)%Z /\
+Definition prime (p:Z): Prop := (2%Z <= p)%Z /\ forall (n:Z), ((1%Z < n)%Z /\
   (n < p)%Z) -> ~ (number.Divisibility.divides n p).
 
 Lemma prime_is_Zprime :
@@ -58,8 +58,8 @@ Qed.
 (* Why3 goal *)
 Lemma small_divisors : forall (p:Z), (2%Z <= p)%Z -> ((forall (d:Z),
   (2%Z <= d)%Z -> ((prime d) -> (((1%Z < (d * d)%Z)%Z /\
-  ((d * d)%Z <= p)%Z) -> ~ (number.Divisibility.divides d p)))) ->
-  (prime p)).
+  ((d * d)%Z <= p)%Z) -> ~ (number.Divisibility.divides d p)))) -> (prime
+  p)).
 Proof.
 intros p Hp H.
 apply <- prime_is_Zprime.

lib/coq/real/ExpLog.v

@@ -48,9 +48,9 @@ exact exp_ln.
 Qed.
 
 (* Why3 assumption *)
-Definition log2(x:R): R := (Rdiv (ln x) (ln 2%R))%R.
+Definition log2 (x:R): R := (Rdiv (ln x) (ln 2%R))%R.
 
 (* Why3 assumption *)
-Definition log10(x:R): R := (Rdiv (ln x) (ln 10%R))%R.
+Definition log10 (x:R): R := (Rdiv (ln x) (ln 10%R))%R.
 
 

lib/coq/set/Set.v

@@ -35,7 +35,7 @@ Defined.
 Hint Unfold mem.
 
 (* Why3 assumption *)
-Definition infix_eqeq {a:Type} {a_WT:WhyType a}(s1:(set a)) (s2:(set
+Definition infix_eqeq {a:Type} {a_WT:WhyType a} (s1:(set a)) (s2:(set
   a)): Prop := forall (x:a), (mem x s1) <-> (mem x s2).
 
 Notation "x == y" := (infix_eqeq x y) (at level 70, no associativity).
@@ -50,8 +50,8 @@ apply h1.
 Qed.
 
 (* Why3 assumption *)
-Definition subset {a:Type} {a_WT:WhyType a}(s1:(set a)) (s2:(set a)): Prop :=
-  forall (x:a), (mem x s1) -> (mem x s2).
+Definition subset {a:Type} {a_WT:WhyType a} (s1:(set a)) (s2:(set
+  a)): Prop := forall (x:a), (mem x s1) -> (mem x s2).
 
 (* Why3 goal *)
 Lemma subset_refl : forall {a:Type} {a_WT:WhyType a}, forall (s:(set a)),
@@ -75,7 +75,7 @@ exact (fun _ => False).
 Defined.
 
 (* Why3 assumption *)
-Definition is_empty {a:Type} {a_WT:WhyType a}(s:(set a)): Prop :=
+Definition is_empty {a:Type} {a_WT:WhyType a} (s:(set a)): Prop :=
   forall (x:a), ~ (mem x s).
 
 (* Why3 goal *)