Revert Coq realizations to the versions before b6e25241.

lib/coq/bool/Bool.v

@@ -15,60 +15,55 @@ Require Import BuiltIn.
 Require BuiltIn.
 
 (* Why3 goal *)
-Lemma andb_def :
-  forall (x:bool) (y:bool),
-   ((Init.Datatypes.andb x y) = match x with
-                                | true => y
-                                | false => false
-                                end).
+Lemma andb_def : forall (x:bool) (y:bool),
+  ((Init.Datatypes.andb x y) = match x with
+  | true => y
+  | false => false
+  end).
 Proof.
 intros x y.
 apply refl_equal.
 Qed.
 
 (* Why3 goal *)
-Lemma orb_def :
-  forall (x:bool) (y:bool),
-   ((Init.Datatypes.orb x y) = match x with
-                               | false => y
-                               | true => true
-                               end).
+Lemma orb_def : forall (x:bool) (y:bool),
+  ((Init.Datatypes.orb x y) = match x with
+  | false => y
+  | true => true
+  end).
 Proof.
 intros x y.
 apply refl_equal.
 Qed.
 
 (* Why3 goal *)
-Lemma notb_def :
-  forall (x:bool),
-   ((Init.Datatypes.negb x) = match x with
-                              | false => true
-                              | true => false
-                              end).
+Lemma notb_def : forall (x:bool),
+  ((Init.Datatypes.negb x) = match x with
+  | false => true
+  | true => false
+  end).
 Proof.
 intros x.
 apply refl_equal.
 Qed.
 
 (* Why3 goal *)
-Lemma xorb_def :
-  forall (x:bool) (y:bool),
-   ((Init.Datatypes.xorb x y) = match x with
-                                | false => y
-                                | true => (Init.Datatypes.negb y)
-                                end).
+Lemma xorb_def : forall (x:bool) (y:bool),
+  ((Init.Datatypes.xorb x y) = match x with
+  | false => y
+  | true => (Init.Datatypes.negb y)
+  end).
 Proof.
 intros x y.
 destruct x; destruct y; auto.
 Qed.
 
 (* Why3 goal *)
-Lemma implb_def :
-  forall (x:bool) (y:bool),
-   ((Init.Datatypes.implb x y) = match x with
-                                 | false => true
-                                 | true => y
-                                 end).
+Lemma implb_def : forall (x:bool) (y:bool),
+  ((Init.Datatypes.implb x y) = match x with
+  | false => true
+  | true => y
+  end).
 Proof.
 now intros [|] [|].
 Qed.

lib/coq/bv/Pow2int.v

@@ -21,36 +21,31 @@ Definition pow2: Z -> Z.
 Defined.
 
 (* Why3 goal *)
-Lemma Power_0 :
-((pow2 0%Z) = 1%Z).
+Lemma Power_0 : ((pow2 0%Z) = 1%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma Power_s :
-forall (n:Z), (0%Z <= n)%Z -> ((pow2 (n + 1%Z)%Z) = (2%Z * (pow2 n))%Z).
+Lemma Power_s : forall (n:Z), (0%Z <= n)%Z ->
+  ((pow2 (n + 1%Z)%Z) = (2%Z * (pow2 n))%Z).
   apply two_p_S.
 Qed.
 
 (* Why3 goal *)
-Lemma Power_1 :
-((pow2 1%Z) = 2%Z).
+Lemma Power_1 : ((pow2 1%Z) = 2%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma Power_sum :
-forall (n:Z) (m:Z),
- ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
- ((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%Z).
+Lemma Power_sum : forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
+  ((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%Z).
   unfold pow2.
   intros n m [H1 H2].
   apply two_p_is_exp; easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2pos :
-forall (i:Z), (0%Z <= i)%Z -> (0%Z < (pow2 i))%Z.
+Lemma pow2pos : forall (i:Z), (0%Z <= i)%Z -> (0%Z < (pow2 i))%Z.
   intros i h1.
   Require Import Zorder.
   apply Zgt_lt.
@@ -58,393 +53,328 @@ forall (i:Z), (0%Z <= i)%Z -> (0%Z < (pow2 i))%Z.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_0 :
-((pow2 0%Z) = 1%Z).
+Lemma pow2_0 : ((pow2 0%Z) = 1%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_1 :
-((pow2 1%Z) = 2%Z).
+Lemma pow2_1 : ((pow2 1%Z) = 2%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_2 :
-((pow2 2%Z) = 4%Z).
+Lemma pow2_2 : ((pow2 2%Z) = 4%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_3 :
-((pow2 3%Z) = 8%Z).
+Lemma pow2_3 : ((pow2 3%Z) = 8%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_4 :
-((pow2 4%Z) = 16%Z).
+Lemma pow2_4 : ((pow2 4%Z) = 16%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_5 :
-((pow2 5%Z) = 32%Z).
+Lemma pow2_5 : ((pow2 5%Z) = 32%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_6 :
-((pow2 6%Z) = 64%Z).
+Lemma pow2_6 : ((pow2 6%Z) = 64%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_7 :
-((pow2 7%Z) = 128%Z).
+Lemma pow2_7 : ((pow2 7%Z) = 128%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_8 :
-((pow2 8%Z) = 256%Z).
+Lemma pow2_8 : ((pow2 8%Z) = 256%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_9 :
-((pow2 9%Z) = 512%Z).
+Lemma pow2_9 : ((pow2 9%Z) = 512%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_10 :
-((pow2 10%Z) = 1024%Z).
+Lemma pow2_10 : ((pow2 10%Z) = 1024%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_11 :
-((pow2 11%Z) = 2048%Z).
+Lemma pow2_11 : ((pow2 11%Z) = 2048%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_12 :
-((pow2 12%Z) = 4096%Z).
+Lemma pow2_12 : ((pow2 12%Z) = 4096%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_13 :
-((pow2 13%Z) = 8192%Z).
+Lemma pow2_13 : ((pow2 13%Z) = 8192%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_14 :
-((pow2 14%Z) = 16384%Z).
+Lemma pow2_14 : ((pow2 14%Z) = 16384%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_15 :
-((pow2 15%Z) = 32768%Z).
+Lemma pow2_15 : ((pow2 15%Z) = 32768%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_16 :
-((pow2 16%Z) = 65536%Z).
+Lemma pow2_16 : ((pow2 16%Z) = 65536%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_17 :
-((pow2 17%Z) = 131072%Z).
+Lemma pow2_17 : ((pow2 17%Z) = 131072%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_18 :
-((pow2 18%Z) = 262144%Z).
+Lemma pow2_18 : ((pow2 18%Z) = 262144%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_19 :
-((pow2 19%Z) = 524288%Z).
+Lemma pow2_19 : ((pow2 19%Z) = 524288%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_20 :
-((pow2 20%Z) = 1048576%Z).
+Lemma pow2_20 : ((pow2 20%Z) = 1048576%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_21 :
-((pow2 21%Z) = 2097152%Z).
+Lemma pow2_21 : ((pow2 21%Z) = 2097152%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_22 :
-((pow2 22%Z) = 4194304%Z).
+Lemma pow2_22 : ((pow2 22%Z) = 4194304%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_23 :
-((pow2 23%Z) = 8388608%Z).
+Lemma pow2_23 : ((pow2 23%Z) = 8388608%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_24 :
-((pow2 24%Z) = 16777216%Z).
+Lemma pow2_24 : ((pow2 24%Z) = 16777216%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_25 :
-((pow2 25%Z) = 33554432%Z).
+Lemma pow2_25 : ((pow2 25%Z) = 33554432%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_26 :
-((pow2 26%Z) = 67108864%Z).
+Lemma pow2_26 : ((pow2 26%Z) = 67108864%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_27 :
-((pow2 27%Z) = 134217728%Z).
+Lemma pow2_27 : ((pow2 27%Z) = 134217728%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_28 :
-((pow2 28%Z) = 268435456%Z).
+Lemma pow2_28 : ((pow2 28%Z) = 268435456%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_29 :
-((pow2 29%Z) = 536870912%Z).
+Lemma pow2_29 : ((pow2 29%Z) = 536870912%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_30 :
-((pow2 30%Z) = 1073741824%Z).
+Lemma pow2_30 : ((pow2 30%Z) = 1073741824%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_31 :
-((pow2 31%Z) = 2147483648%Z).
+Lemma pow2_31 : ((pow2 31%Z) = 2147483648%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_32 :
-((pow2 32%Z) = 4294967296%Z).
+Lemma pow2_32 : ((pow2 32%Z) = 4294967296%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_33 :
-((pow2 33%Z) = 8589934592%Z).
+Lemma pow2_33 : ((pow2 33%Z) = 8589934592%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_34 :
-((pow2 34%Z) = 17179869184%Z).
+Lemma pow2_34 : ((pow2 34%Z) = 17179869184%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_35 :
-((pow2 35%Z) = 34359738368%Z).
+Lemma pow2_35 : ((pow2 35%Z) = 34359738368%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_36 :
-((pow2 36%Z) = 68719476736%Z).
+Lemma pow2_36 : ((pow2 36%Z) = 68719476736%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_37 :
-((pow2 37%Z) = 137438953472%Z).
+Lemma pow2_37 : ((pow2 37%Z) = 137438953472%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_38 :
-((pow2 38%Z) = 274877906944%Z).
+Lemma pow2_38 : ((pow2 38%Z) = 274877906944%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_39 :
-((pow2 39%Z) = 549755813888%Z).
+Lemma pow2_39 : ((pow2 39%Z) = 549755813888%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_40 :
-((pow2 40%Z) = 1099511627776%Z).
+Lemma pow2_40 : ((pow2 40%Z) = 1099511627776%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_41 :
-((pow2 41%Z) = 2199023255552%Z).
+Lemma pow2_41 : ((pow2 41%Z) = 2199023255552%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_42 :
-((pow2 42%Z) = 4398046511104%Z).
+Lemma pow2_42 : ((pow2 42%Z) = 4398046511104%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_43 :
-((pow2 43%Z) = 8796093022208%Z).
+Lemma pow2_43 : ((pow2 43%Z) = 8796093022208%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_44 :
-((pow2 44%Z) = 17592186044416%Z).
+Lemma pow2_44 : ((pow2 44%Z) = 17592186044416%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_45 :
-((pow2 45%Z) = 35184372088832%Z).
+Lemma pow2_45 : ((pow2 45%Z) = 35184372088832%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_46 :
-((pow2 46%Z) = 70368744177664%Z).
+Lemma pow2_46 : ((pow2 46%Z) = 70368744177664%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_47 :
-((pow2 47%Z) = 140737488355328%Z).
+Lemma pow2_47 : ((pow2 47%Z) = 140737488355328%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_48 :
-((pow2 48%Z) = 281474976710656%Z).
+Lemma pow2_48 : ((pow2 48%Z) = 281474976710656%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_49 :
-((pow2 49%Z) = 562949953421312%Z).
+Lemma pow2_49 : ((pow2 49%Z) = 562949953421312%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_50 :
-((pow2 50%Z) = 1125899906842624%Z).
+Lemma pow2_50 : ((pow2 50%Z) = 1125899906842624%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_51 :
-((pow2 51%Z) = 2251799813685248%Z).
+Lemma pow2_51 : ((pow2 51%Z) = 2251799813685248%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_52 :
-((pow2 52%Z) = 4503599627370496%Z).
+Lemma pow2_52 : ((pow2 52%Z) = 4503599627370496%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_53 :
-((pow2 53%Z) = 9007199254740992%Z).
+Lemma pow2_53 : ((pow2 53%Z) = 9007199254740992%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_54 :
-((pow2 54%Z) = 18014398509481984%Z).
+Lemma pow2_54 : ((pow2 54%Z) = 18014398509481984%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_55 :
-((pow2 55%Z) = 36028797018963968%Z).
+Lemma pow2_55 : ((pow2 55%Z) = 36028797018963968%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_56 :
-((pow2 56%Z) = 72057594037927936%Z).
+Lemma pow2_56 : ((pow2 56%Z) = 72057594037927936%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_57 :
-((pow2 57%Z) = 144115188075855872%Z).
+Lemma pow2_57 : ((pow2 57%Z) = 144115188075855872%Z).
   easy.
 
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_58 :
-((pow2 58%Z) = 288230376151711744%Z).
+Lemma pow2_58 : ((pow2 58%Z) = 288230376151711744%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_59 :
-((pow2 59%Z) = 576460752303423488%Z).
+Lemma pow2_59 : ((pow2 59%Z) = 576460752303423488%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_60 :
-((pow2 60%Z) = 1152921504606846976%Z).
+Lemma pow2_60 : ((pow2 60%Z) = 1152921504606846976%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_61 :
-((pow2 61%Z) = 2305843009213693952%Z).
+Lemma pow2_61 : ((pow2 61%Z) = 2305843009213693952%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_62 :
-((pow2 62%Z) = 4611686018427387904%Z).
+Lemma pow2_62 : ((pow2 62%Z) = 4611686018427387904%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_63 :
-((pow2 63%Z) = 9223372036854775808%Z).
+Lemma pow2_63 : ((pow2 63%Z) = 9223372036854775808%Z).
   easy.
 Qed.
 
 (* Why3 goal *)
-Lemma pow2_64 :
-((pow2 64%Z) = 18446744073709551616%Z).
+Lemma pow2_64 : ((pow2 64%Z) = 18446744073709551616%Z).
   easy.
 Qed.
 

lib/coq/floating_point/Double.v

@@ -53,80 +53,72 @@ Definition total_error (x:floating_point.DoubleFormat.double): R :=
 
 (* Why3 assumption *)
 Definition no_overflow (m:floating_point.Rounding.mode) (x:R): Prop :=
-  ((Reals.Rbasic_fun.Rabs (round m x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
+  ((Reals.Rbasic_fun.Rabs (round m
+  x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
 
 (* Why3 goal *)
-Lemma Bounded_real_no_overflow :
-forall (m:floating_point.Rounding.mode) (x:R),
- ((Reals.Rbasic_fun.Rabs x) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R ->
- (no_overflow m x).
+Lemma Bounded_real_no_overflow : forall (m:floating_point.Rounding.mode)
+  (x:R),
+  ((Reals.Rbasic_fun.Rabs x) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R ->
+  (no_overflow m x).
 exact (Bounded_real_no_overflow 53 1024 (refl_equal true) (refl_equal true)).
 Qed.
 
 (* Why3 goal *)
-Lemma Round_monotonic :
-forall (m:floating_point.Rounding.mode) (x:R) (y:R),
- (x <= y)%R -> ((round m x) <= (round m y))%R.
+Lemma Round_monotonic : forall (m:floating_point.Rounding.mode) (x:R) (y:R),
+  (x <= y)%R -> ((round m x) <= (round m y))%R.
 now apply Round_monotonic.
 Qed.
 
 (* Why3 goal *)
-Lemma Round_idempotent :
-forall (m1:floating_point.Rounding.mode) (m2:floating_point.Rounding.mode)
-       (x:R), ((round m1 (round m2 x)) = (round m2 x)).
+Lemma Round_idempotent : forall (m1:floating_point.Rounding.mode)
+  (m2:floating_point.Rounding.mode) (x:R), ((round m1 (round m2
+  x)) = (round m2 x)).
 now apply Round_idempotent.
 Qed.
 
 (* Why3 goal *)
-Lemma Round_value :
-forall (m:floating_point.Rounding.mode)
-       (x:floating_point.DoubleFormat.double),
- ((round m (value x)) = (value x)).
+Lemma Round_value : forall (m:floating_point.Rounding.mode)
+  (x:floating_point.DoubleFormat.double), ((round m (value x)) = (value x)).
 now apply Round_value.
 Qed.
 
 (* Why3 goal *)
-Lemma Bounded_value :
-forall (x:floating_point.DoubleFormat.double),
- ((Reals.Rbasic_fun.Rabs (value x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
+Lemma Bounded_value : forall (x:floating_point.DoubleFormat.double),
+  ((Reals.Rbasic_fun.Rabs (value x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
 now apply Bounded_value.
 Qed.
 
 (* Why3 goal *)
-Lemma Exact_rounding_for_integers :
-forall (m:floating_point.Rounding.mode) (i:Z),
- (((-9007199254740992%Z)%Z <= i)%Z /\ (i <= 9007199254740992%Z)%Z) ->
- ((round m (BuiltIn.IZR i)) = (BuiltIn.IZR i)).
+Lemma Exact_rounding_for_integers : forall (m:floating_point.Rounding.mode)
+  (i:Z), (((-9007199254740992%Z)%Z <= i)%Z /\ (i <= 9007199254740992%Z)%Z) ->
+  ((round m (BuiltIn.IZR i)) = (BuiltIn.IZR i)).
 Proof.
 intros m i Hi.
 now apply Exact_rounding_for_integers.
 Qed.
 
 (* Why3 goal *)
-Lemma Round_down_le :
-forall (x:R), ((round floating_point.Rounding.Down x) <= x)%R.
+Lemma Round_down_le : forall (x:R), ((round floating_point.Rounding.Down
+  x) <= x)%R.
 now apply Round_down_le.
 Qed.
 
 (* Why3 goal *)
-Lemma Round_up_ge :
-forall (x:R), (x <= (round floating_point.Rounding.Up x))%R.
+Lemma Round_up_ge : forall (x:R), (x <= (round floating_point.Rounding.Up
+  x))%R.
 now apply Round_up_ge.
 Qed.
 
 (* Why3 goal *)
-Lemma Round_down_neg :
-forall (x:R),
- ((round floating_point.Rounding.Down (-x)%R) = (-(round floating_point.Rounding.Up
-                                                    x))%R).
+Lemma Round_down_neg : forall (x:R), ((round floating_point.Rounding.Down
+  (-x)%R) = (-(round floating_point.Rounding.Up x))%R).
 now apply Round_down_neg.
 Qed.
 
 (* Why3 goal *)
-Lemma Round_up_neg :
-forall (x:R),
- ((round floating_point.Rounding.Up (-x)%R) = (-(round floating_point.Rounding.Down
-                                                  x))%R).
+Lemma Round_up_neg : forall (x:R), ((round floating_point.Rounding.Up
+  (-x)%R) = (-(round floating_point.Rounding.Down x))%R).