Added implicit coercions for radix and ln_beta.

src/Appli/Fappli_Axpy.v

@@ -52,7 +52,7 @@ Qed.
 Theorem implies_MinOrMax_not_bpow:
   forall x f, format f ->
   (0 < f)%R ->
-  f <> bpow (projT1 (ln_beta beta f)) ->
+  f <> bpow (ln_beta beta f) ->
   (Rabs (f-x) < ulp f)%R -> 
   MinOrMax x f.
 intros x f Hf1 Hf2 Hf3 Hxf1.
@@ -84,7 +84,7 @@ Qed.
 
 Theorem implies_MinOrMax_bpow:
   forall x f, format f ->
-  f = bpow (projT1 (ln_beta beta f)) ->
+  f = bpow (ln_beta beta f) ->
   (Rabs (f-x) < /2* ulp f)%R -> 
   MinOrMax x f.
 intros x f Hf1 Hf2 Hxf.

src/Calc/Fcalc_bracket.v

@@ -507,17 +507,17 @@ Theorem inbetween_float_new_location :
   forall x m e l k,
   (0 < k)%Z ->
   inbetween_float m e x l ->
-  inbetween_float (Zdiv m (Zpower (radix_val beta) k)) (e + k) x (new_location (Zpower (radix_val beta) k) (Zmod m (Zpower (radix_val beta) k)) l).
+  inbetween_float (Zdiv m (Zpower beta k)) (e + k) x (new_location (Zpower beta k) (Zmod m (Zpower beta k)) l).
 Proof.
 intros x m e l k Hk Hx.
 unfold inbetween_float in *.
-assert (Hr: forall m, F2R (Float beta m (e + k)) = F2R (Float beta (m * Zpower (radix_val beta) k) e)).
+assert (Hr: forall m, F2R (Float beta m (e + k)) = F2R (Float beta (m * Zpower beta k) e)).
 clear -Hk. intros m.
 rewrite (F2R_change_exp beta e).
 apply (f_equal (fun r => F2R (Float beta (m * Zpower _ r) e))).
 ring.
 omega.
-assert (Hp: (Zpower (radix_val beta) k > 0)%Z).
+assert (Hp: (Zpower beta k > 0)%Z).
 apply Zlt_gt.
 now apply Zpower_gt_0.
 (* . *)
@@ -537,10 +537,10 @@ Qed.
 Theorem inbetween_float_new_location_single :
   forall x m e l,
   inbetween_float m e x l ->
-  inbetween_float (Zdiv m (radix_val beta)) (e + 1) x (new_location (radix_val beta) (Zmod m (radix_val beta)) l).
+  inbetween_float (Zdiv m beta) (e + 1) x (new_location beta (Zmod m beta) l).
 Proof.
 intros x m e l Hx.
-replace (radix_val beta) with (Zpower (radix_val beta) 1).
+replace (radix_val beta) with (Zpower beta 1).
 now apply inbetween_float_new_location.
 apply Zmult_1_r.
 Qed.

src/Calc/Fcalc_digits.v

@@ -35,15 +35,15 @@ Hypothesis Hp : (0 <= p)%Z.
 Fixpoint digits_aux (nb pow : Z) (n : nat) { struct n } : Z :=
   match n with
   | O => nb
-  | S n => if Zlt_bool p pow then nb else digits_aux (nb + 1) (Zmult (radix_val beta) pow) n
+  | S n => if Zlt_bool p pow then nb else digits_aux (nb + 1) (Zmult beta pow) n
   end.
 
 Lemma digits_aux_invariant :
   forall k n nb,
   (0 <= nb)%Z ->
-  (Zpower (radix_val beta) (nb + Z_of_nat k - 1) <= p)%Z ->
-  digits_aux (nb + Z_of_nat k) (Zpower (radix_val beta) (nb + Z_of_nat k)) n =
-  digits_aux nb (Zpower (radix_val beta) nb) (n + k).
+  (Zpower beta (nb + Z_of_nat k - 1) <= p)%Z ->
+  digits_aux (nb + Z_of_nat k) (Zpower beta (nb + Z_of_nat k)) n =
+  digits_aux nb (Zpower beta nb) (n + k).
 Proof.
 induction k ; intros n nb Hnb.
 now rewrite Zplus_0_r, plus_0_r.
@@ -54,7 +54,7 @@ rewrite (Zplus_comm _ 1), Zplus_assoc.
 rewrite IHk.
 rewrite <- plus_n_Sm.
 simpl.
-generalize (Zlt_cases p (Zpower (radix_val beta) nb)).
+generalize (Zlt_cases p (Zpower beta nb)).
 case Zlt_bool ; intros Hpp.
 elim (Zlt_irrefl p).
 apply Zlt_le_trans with (1 := Hpp).
@@ -81,8 +81,8 @@ End digits_aux.
 Definition digits n :=
   match n with
   | Z0 => Z0
-  | Zneg p => digits_aux (Zpos p) 1 (radix_val beta) (digits2_Pnat p)
-  | Zpos p => digits_aux n 1 (radix_val beta) (digits2_Pnat p)
+  | Zneg p => digits_aux (Zpos p) 1 beta (digits2_Pnat p)
+  | Zpos p => digits_aux n 1 beta (digits2_Pnat p)
   end.
 
 Theorem digits_abs :
@@ -94,7 +94,7 @@ Qed.
 Theorem digits_ln_beta :
   forall n,
   n <> Z0 ->
-  digits n = projT1 (ln_beta beta (Z2R n)).
+  digits n = ln_beta beta (Z2R n).
 Proof.
 intros n Hn.
 destruct (ln_beta beta (Z2R n)) as (e, He).
@@ -117,7 +117,7 @@ now apply (Zlt_le_succ 0).
 assert (He2: (Z_of_nat (Zabs_nat (e - 1)) = e - 1)%Z).
 rewrite inj_Zabs_nat.
 now apply Zabs_eq.
-replace (radix_val beta) with (Zpower (radix_val beta) 1).
+replace (radix_val beta) with (Zpower beta 1).
 replace (digits2_Pnat p) with (digits2_Pnat p - Zabs_nat (e - 1) + Zabs_nat (e - 1)).
 rewrite <- digits_aux_invariant.
 rewrite He2.
@@ -125,7 +125,7 @@ ring_simplify (1 + (e - 1))%Z.
 destruct (digits2_Pnat p - Zabs_nat (e - 1)) as [|n].
 easy.
 simpl.
-generalize (Zlt_cases (Zpos p) (Zpower (radix_val beta) e)).
+generalize (Zlt_cases (Zpos p) (Zpower beta e)).
 case Zlt_bool ; intros Hpp.
 easy.
 elim Rlt_not_le with (1 := proj2 He).
@@ -154,7 +154,7 @@ exact (proj2 (digits2_Pnat_correct p)).
 clear.
 induction (S (digits2_Pnat p)).
 easy.
-change (2 * Zpower_nat 2 n <= radix_val beta * Zpower_nat (radix_val beta) n)%Z.
+change (2 * Zpower_nat 2 n <= beta * Zpower_nat beta n)%Z.
 apply Zmult_le_compat ; try easy.
 apply Zle_bool_imp_le.
 apply beta.
@@ -188,7 +188,7 @@ Qed.
 Theorem digits_shift :
   forall m e,
   m <> Z0 -> (0 <= e)%Z ->
-  digits (m * Zpower (radix_val beta) e) = (digits m + e)%Z.
+  digits (m * Zpower beta e) = (digits m + e)%Z.
 Proof.
 intros m e Hm He.
 rewrite 2!digits_ln_beta.
@@ -316,4 +316,4 @@ rewrite Zmult_0_l, Zplus_0_r, 2!Zplus_0_l.
 apply Zle_refl.
 Qed.
 
-End Fcalc_digits.
\ No newline at end of file
+End Fcalc_digits.

src/Calc/Fcalc_div.v

@@ -21,7 +21,7 @@ Definition Fdiv_aux prec m1 e1 m2 e2 :=
   let e := (e1 - e2)%Z in
   let (m, e') :=
     match (d2 + prec - d1)%Z with
-    | Zpos p => (m1 * Zpower_pos (radix_val beta) p, e + Zneg p)%Z
+    | Zpos p => (m1 * Zpower_pos beta p, e + Zneg p)%Z
     | _ => (m1, e)
     end in
   let '(q, r) :=  Zdiv_eucl m m2 in
@@ -41,7 +41,7 @@ set (d1 := digits beta m1).
 set (d2 := digits beta m2).
 case_eq
  (match (d2 + prec - d1)%Z with
-  | Zpos p => ((m1 * Zpower_pos (radix_val beta) p)%Z, (e1 - e2 + Zneg p)%Z)
+  | Zpos p => ((m1 * Zpower_pos beta p)%Z, (e1 - e2 + Zneg p)%Z)
   | _ => (m1, (e1 - e2)%Z)
   end).
 intros m' e' Hme.
@@ -56,7 +56,7 @@ repeat split.
 now rewrite <- H0.
 apply Zle_refl.
 replace (e1 - e2 + Zneg p + e2)%Z with (e1 - Zpos p)%Z by (unfold Zminus ; simpl ; ring).
-fold (Zpower (radix_val beta) (Zpos p)).
+fold (Zpower beta (Zpos p)).
 split.
 pattern (Zpos p) at 1 ; replace (Zpos p) with (e1 - (e1 - Zpos p))%Z by ring.
 apply sym_eq.

src/Calc/Fcalc_ops.v

@@ -12,8 +12,8 @@ Definition Falign (f1 f2 : float beta) :=
   let '(Float m1 e1) := f1 in
   let '(Float m2 e2) := f2 in
   if Zle_bool e1 e2
-  then (m1, (m2 * Zpower (radix_val beta) (e2 - e1))%Z, e1)
-  else ((m1 * Zpower (radix_val beta) (e1 - e2))%Z, m2, e2).
+  then (m1, (m2 * Zpower beta (e2 - e1))%Z, e1)
+  else ((m1 * Zpower beta (e1 - e2))%Z, m2, e2).
 
 Theorem Falign_spec :
   forall f1 f2 : float beta,

src/Calc/Fcalc_round.v

@@ -15,7 +15,7 @@ Definition truncate t :=
   let '(m, e, l) := t in
   let k := (fexp (digits beta m + e) - e)%Z in
   if Zlt_bool 0 k then
-    let p := Zpower (radix_val beta) k in
+    let p := Zpower beta k in
     (Zdiv m p, (e + k)%Z, new_location p (Zmod m p) l)
   else t.
 
@@ -31,7 +31,7 @@ Proof.
 intros x m e l Hx H1 H2.
 unfold truncate.
 set (k := (fexp (digits beta m + e) - e)%Z).
-set (p := Zpower (radix_val beta) k).
+set (p := Zpower beta k).
 (* *)
 assert (Hx': (F2R (Float beta m e) <= x < F2R (Float beta (m + 1) e))%R).
 apply inbetween_bounds with (2 := H1).

src/Calc/Fcalc_round_FIX.v

@@ -13,7 +13,7 @@ Definition round_FIX t :=
   let '(m, e, l) := t in
   let k := (emin - e)%Z in
   if Zlt_bool 0 k then
-    let p := Zpower (radix_val beta) k in
+    let p := Zpower beta k in
     (Zdiv m p, (e + k)%Z, new_location p (Zmod m p) l)
   else t.
 
@@ -28,7 +28,7 @@ Proof.
 intros x m e l H1 H2.
 unfold round_FIX.
 set (k := (emin - e)%Z).
-set (p := Zpower (radix_val beta) k).
+set (p := Zpower beta k).
 unfold canonic_exponent, FIX_exp.
 generalize (Zlt_cases 0 k).
 case (Zlt_bool 0 k) ; intros Hk.

src/Calc/Fcalc_sqrt.v

@@ -124,7 +124,7 @@ Definition Fsqrt_aux prec m e :=
   let (s', e'') := if Zeven e' then (s, e') else (s + 1, e' - 1)%Z in
   let m' :=
     match s' with
-    | Zpos p => (m * Zpower_pos (radix_val beta) p)%Z
+    | Zpos p => (m * Zpower_pos beta p)%Z
     | _ => m
     end in
   let (q, r) := Zsqrt m' in
@@ -172,7 +172,7 @@ destruct He as (He1, (He2, (He3, He4))).
 (* . shift *)
 set (m' := match s' with
   | Z0 => m
-  | Zpos p => (m * Zpower_pos (radix_val beta) p)%Z
+  | Zpos p => (m * Zpower_pos beta p)%Z
   | Zneg _ => m
   end).
 assert (Hs: F2R (Float beta m' e') = F2R (Float beta m e) /\ (2 * prec <= digits beta m')%Z /\ (0 < m')%Z).
@@ -182,7 +182,7 @@ destruct s' as [|p|p].
 repeat split ; try easy.
 fold d.
 omega.
-fold (Zpower (radix_val beta) (Zpos p)).
+fold (Zpower beta (Zpos p)).
 split.
 replace (Zpos p) with (Zpos p + e' - e')%Z by ring.
 rewrite <- F2R_change_exp.

src/Core/Fcore_FLT.v

@@ -19,7 +19,7 @@ Variable Hp : Zlt 0 prec.
 (* floating-point format with gradual underflow *)
 Definition FLT_format (x : R) :=
   exists f : float beta,
-  x = F2R f /\ (Zabs (Fnum f) < Zpower (radix_val beta) prec)%Z /\ (emin <= Fexp f)%Z.
+  x = F2R f /\ (Zabs (Fnum f) < Zpower beta prec)%Z /\ (emin <= Fexp f)%Z.
 
 Definition FLT_RoundingModeP (rnd : R -> R):=
   Rounding_for_Format FLT_format rnd.
@@ -256,7 +256,7 @@ generalize (Zmax_spec (emin + 1 - prec) emin).
 omega.
 Qed.
 
-Hypothesis NE_prop : Zeven (radix_val beta) = false \/ (1 < prec)%Z.
+Hypothesis NE_prop : Zeven beta = false \/ (1 < prec)%Z.
 
 Theorem NE_ex_prop_FLT :
   NE_ex_prop beta FLT_exp.

src/Core/Fcore_FLX.v

@@ -18,7 +18,7 @@ Variable Hp : Zlt 0 prec.
 (* unbounded floating-point format *)
 Definition FLX_format (x : R) :=
   exists f : float beta,
-  x = F2R f /\ (Zabs (Fnum f) < Zpower (radix_val beta) prec)%Z.
+  x = F2R f /\ (Zabs (Fnum f) < Zpower beta prec)%Z.
 
 Definition FLX_RoundingModeP (rnd : R -> R):=
   Rounding_for_Format FLX_format rnd.
@@ -106,7 +106,7 @@ apply iff_trans with (FIX_format beta (ex - prec) x).
 apply FLX_format_FIX.
 exact (conj (proj1 Hx2) (Rlt_le _ _ (proj2 Hx2))).
 apply FIX_format_generic.
-assert (Hf: FLX_exp (projT1 (ln_beta beta x)) = FIX_exp (ex - prec) (projT1 (ln_beta beta x))).
+assert (Hf: FLX_exp (ln_beta beta x) = FIX_exp (ex - prec) (ln_beta beta x)).
 unfold FIX_exp, FLX_exp.
 now rewrite ln_beta_unique with (1 := Hx2).
 split ;
@@ -128,7 +128,7 @@ Qed.
 Definition FLXN_format (x : R) :=
   exists f : float beta,
   x = F2R f /\ (x <> R0 ->
-  Zpower (radix_val beta) (prec - 1) <= Zabs (Fnum f) < Zpower (radix_val beta) prec)%Z.
+  Zpower beta (prec - 1) <= Zabs (Fnum f) < Zpower beta prec)%Z.
 
 Definition FLXN_RoundingModeP (rnd : R -> R):=
   Rounding_for_Format FLXN_format rnd.
@@ -159,7 +159,7 @@ apply H4.
 rewrite <- Z2R_Zpower, <- abs_Z2R.
 now apply Z2R_lt.
 now apply Zlt_le_weak.
-exists (Float beta (xm * Zpower (radix_val beta) (prec - d)) (xe + d - prec)).
+exists (Float beta (xm * Zpower beta (prec - d)) (xe + d - prec)).
 split.
 unfold F2R. simpl.
 rewrite mult_Z2R, Z2R_Zpower.
@@ -228,7 +228,7 @@ unfold FLX_exp.
 omega.
 Qed.
 
-Hypothesis NE_prop : Zeven (radix_val beta) = false \/ (1 < prec)%Z.
+Hypothesis NE_prop : Zeven beta = false \/ (1 < prec)%Z.
 
 Theorem NE_ex_prop_FLX :
   NE_ex_prop beta FLX_exp.

src/Core/Fcore_FTZ.v

@@ -17,7 +17,7 @@ Variable Hp : Zlt 0 prec.
 (* floating-point format with abrupt underflow *)
 Definition FTZ_format (x : R) :=
   exists f : float beta,
-  x = F2R f /\ (x <> R0 -> Zpower (radix_val beta) (prec - 1) <= Zabs (Fnum f) < Zpower (radix_val beta) prec)%Z /\
+  x = F2R f /\ (x <> R0 -> Zpower beta (prec - 1) <= Zabs (Fnum f) < Zpower beta prec)%Z /\
   (emin <= Fexp f)%Z.
 
 Definition FTZ_RoundingModeP (rnd : R -> R):=

src/Core/Fcore_Raux.v

@@ -1269,7 +1269,7 @@ End Proof_Irrelevance.
 
 Section pow.
 
-Record radix := { radix_val : Z ; radix_prop : Zle_bool 2 radix_val = true }.
+Record radix := { radix_val :> Z ; radix_prop : Zle_bool 2 radix_val = true }.
 
 Theorem radix_val_inj :
   forall r1 r2, radix_val r1 = radix_val r2 -> r1 = r2.
@@ -1285,7 +1285,7 @@ Qed.
 
 Variable r : radix.
 
-Theorem radix_gt_1 : (1 < radix_val r)%Z.
+Theorem radix_gt_1 : (1 < r)%Z.
 Proof.
 destruct r as (v, Hr). simpl.
 apply Zlt_le_trans with 2%Z.
@@ -1293,7 +1293,7 @@ easy.
 now apply Zle_bool_imp_le.
 Qed.
 
-Theorem radix_pos : (0 < Z2R (radix_val r))%R.
+Theorem radix_pos : (0 < Z2R r)%R.
 Proof.
 destruct r as (v, Hr). simpl.
 apply (Z2R_lt 0).
@@ -1304,8 +1304,8 @@ Qed.
 
 Definition bpow e :=
   match e with
-  | Zpos p => Z2R (Zpower_pos (radix_val r) p)
-  | Zneg p => Rinv (Z2R (Zpower_pos (radix_val r) p))
+  | Zpos p => Z2R (Zpower_pos r p)
+  | Zneg p => Rinv (Z2R (Zpower_pos r p))
   | Z0 => R1
   end.
 
@@ -1327,7 +1327,7 @@ Qed.
 
 Theorem bpow_powerRZ :
   forall e,
-  bpow e = powerRZ (Z2R (radix_val r)) e.
+  bpow e = powerRZ (Z2R r) e.
 Proof.
 destruct e ; unfold bpow.
 reflexivity.
@@ -1364,14 +1364,14 @@ apply radix_pos.
 Qed.
 
 Theorem bpow_1 :
-  bpow 1 = Z2R (radix_val r).
+  bpow 1 = Z2R r.
 Proof.
 unfold bpow, Zpower_pos, iter_pos.
 now rewrite Zmult_1_r.
 Qed.
 
 Theorem bpow_add1 :
-  forall e : Z, (bpow (e+1) = Z2R (radix_val r) * bpow e)%R.
+  forall e : Z, (bpow (e + 1) = Z2R r * bpow e)%R.
 Proof.
 intros.
 rewrite <- bpow_1.
@@ -1393,7 +1393,7 @@ Qed.
 
 Theorem Z2R_Zpower_nat :
   forall e : nat,
-  Z2R (Zpower_nat (radix_val r) e) = bpow (Z_of_nat e).
+  Z2R (Zpower_nat r e) = bpow (Z_of_nat e).
 Proof.
 intros [|e].
 split.
@@ -1405,7 +1405,7 @@ Qed.
 Theorem Z2R_Zpower :
   forall e : Z,
   (0 <= e)%Z ->
-  Z2R (Zpower (radix_val r) e) = bpow e.
+  Z2R (Zpower r e) = bpow e.
 Proof.
 intros [|e|e] H.
 split.
@@ -1435,7 +1435,7 @@ elim (lt_irrefl 0).
 pattern O at 2 ; rewrite <- H.
 apply lt_O_nat_of_P.
 intros n _.
-assert (1 < Zpower_nat (radix_val r) 1)%Z.
+assert (1 < Zpower_nat r 1)%Z.
 unfold Zpower_nat, iter_nat.
 rewrite Zmult_1_r.
 apply radix_gt_1.
@@ -1502,10 +1502,10 @@ Qed.
 
 Theorem bpow_exp :
   forall e : Z,
-  bpow e = exp (Z2R e * ln (Z2R (radix_val r))).
+  bpow e = exp (Z2R e * ln (Z2R r)).
 Proof.
 (* positive case *)
-assert (forall e, bpow (Zpos e) = exp (Z2R (Zpos e) * ln (Z2R (radix_val r)))).
+assert (forall e, bpow (Zpos e) = exp (Z2R (Zpos e) * ln (Z2R r))).
 intros e.
 unfold bpow.
 rewrite Zpower_pos_nat.
@@ -1534,19 +1534,23 @@ rewrite Rmult_0_l.
 now rewrite exp_0.
 apply H.
 unfold bpow.
-change (Z2R (Zpower_pos (radix_val r) e)) with (bpow (Zpos e)).
+change (Z2R (Zpower_pos r e)) with (bpow (Zpos e)).
 rewrite H.
 rewrite <- exp_Ropp.
 rewrite <- Ropp_mult_distr_l_reverse.
 now rewrite <- opp_Z2R.
 Qed.
 
-Theorem ln_beta :
-  forall x : R,
-  {e | (x <> 0)%R -> (bpow (e - 1)%Z <= Rabs x < bpow e)%R}.
+Record ln_beta_prop x := {
+  ln_beta_val :> Z ;
+   _ : (x <> 0)%R -> (bpow (ln_beta_val - 1)%Z <= Rabs x < bpow ln_beta_val)%R
+}.
+
+Definition ln_beta :
+  forall x : R, ln_beta_prop x.
 Proof.
 intros x.
-set (fact := ln (Z2R (radix_val r))).
+set (fact := ln (Z2R r)).
 (* . *)
 assert (0 < fact)%R.
 apply exp_lt_inv.
@@ -1614,7 +1618,7 @@ Qed.
 Theorem ln_beta_unique :
   forall (x : R) (e : Z),
   (bpow (e - 1) <= Rabs x < bpow e)%R ->
-  projT1 (ln_beta x) = e.
+  ln_beta x = e :> Z.
 Proof.
 intros x e1 He.
 destruct (Req_dec x 0) as [Hx|Hx].
@@ -1629,7 +1633,7 @@ Qed.
 
 Theorem ln_beta_opp :
   forall x,
-  projT1 (ln_beta (-x)) = projT1 (ln_beta x).
+  ln_beta (-x) = ln_beta x :> Z.
 Proof.
 intros x.
 destruct (Req_dec x 0) as [Hx|Hx].
@@ -1643,20 +1647,19 @@ Qed.
 
 Theorem ln_beta_abs :
   forall x,
-  projT1 (ln_beta (Rabs x)) = projT1 (ln_beta x).
+  ln_beta (Rabs x) = ln_beta x :> Z.
 Proof.
 intros x.
-set (m := projT1 (ln_beta x)).
 unfold Rabs.
 case Rcase_abs ; intros _.
-now rewrite ln_beta_opp.
+apply ln_beta_opp.
 apply refl_equal.
 Qed.
 
 Theorem ln_beta_monotone_abs :
   forall x y,
   (x <> 0)%R -> (Rabs x <= Rabs y)%R ->
-  (projT1 (ln_beta x) <= projT1 (ln_beta y))%Z.
+  (ln_beta x <= ln_beta y)%Z.
 Proof.
 intros x y H0x Hxy.
 destruct (ln_beta x) as (ex, Hx).
@@ -1677,7 +1680,7 @@ Qed.
 Theorem ln_beta_monotone :
   forall x y,
   (0 < x)%R -> (x <= y)%R ->
-  (projT1 (ln_beta x) <= projT1 (ln_beta y))%Z.
+  (ln_beta x <= ln_beta y)%Z.
 Proof.
 intros x y H0x Hxy.
 apply ln_beta_monotone_abs.
@@ -1690,7 +1693,7 @@ now apply Rlt_le.
 Qed.
 
 Theorem ln_beta_bpow :
-  forall e, projT1 (ln_beta (bpow e)) = (e + 1)%Z.
+  forall e, (ln_beta (bpow e) = e + 1 :> Z)%Z.
 Proof.
 intros e.
 apply ln_beta_unique.
@@ -1718,7 +1721,7 @@ Qed.
 Theorem Zpower_gt_1 :
   forall p,
   (0 < p)%Z ->
-  (1 < Zpower (radix_val r) p)%Z.
+  (1 < Zpower r p)%Z.
 Proof.
 intros.
 apply lt_Z2R.
@@ -1730,7 +1733,7 @@ Qed.
 Theorem Zpower_gt_0 :
   forall p,
   (0 < p)%Z ->
-  (0 < Zpower (radix_val r) p)%Z.
+  (0 < Zpower r p)%Z.
 Proof.
 intros.
 apply Zlt_trans with 1%Z.

src/Core/Fcore_float_prop.v

@@ -261,7 +261,7 @@ Qed.
 Theorem ln_beta_F2R_bounds :
   forall x m e, (0 < m)%Z ->
   (F2R (Float beta m e) <= x < F2R (Float beta (m + 1) e))%R ->
-  projT1 (ln_beta beta x) = projT1 (ln_beta beta (F2R (Float beta m e))).
+  ln_beta beta x = ln_beta beta (F2R (Float beta m e)) :> Z.
 Proof.
 intros x m e Hp (Hx,Hx2).
 destruct (ln_beta beta (F2R (Float beta m e))) as (ex, He).
@@ -308,7 +308,7 @@ Qed.
 
 Theorem F2R_lt_bpow :
   forall f : float beta, forall e',
-  (Zabs (Fnum f) < Zpower (radix_val beta) (e' - Fexp f))%Z ->
+  (Zabs (Fnum f) < Zpower beta (e' - Fexp f))%Z ->
   (Rabs (F2R f) < bpow e')%R.
 Proof.
 intros (m, e) e' Hm.
@@ -332,7 +332,7 @@ Qed.
 Theorem F2R_change_exp :
   forall e' m e : Z,
   (e' <= e)%Z ->
-  F2R (Float beta m e) = F2R (Float beta (m * Zpower (radix_val beta) (e - e')) e').
+  F2R (Float beta m e) = F2R (Float beta (m * Zpower beta (e - e')) e').
 Proof.
 intros e' m e He.
 unfold F2R. simpl.
@@ -345,9 +345,9 @@ Qed.
 
 Theorem F2R_prec_normalize :
   forall m e e' p : Z,
-  (Zabs m < Zpower (radix_val beta) p)%Z ->
+  (Zabs m < Zpower beta p)%Z ->
   (bpow (e' - 1)%Z <= Rabs (F2R (Float beta m e)))%R ->
-  F2R (Float beta m e) = F2R (Float beta (m * Zpower (radix_val beta) (e - e' + p)) (e' - p)).
+  F2R (Float beta m e) = F2R (Float beta (m * Zpower beta (e - e' + p)) (e' - p)).
 Proof.
 intros m e e' p Hm Hf.
 assert (Hp: (0 <= p)%Z).
@@ -370,7 +370,7 @@ Qed.
 Theorem ln_beta_F2R :
   forall m e : Z,
   m <> Z0 ->
-  (projT1 (ln_beta beta (F2R (Float beta m e))) = projT1 (ln_beta beta (Z2R m)) + e)%Z.
+  (ln_beta beta (F2R (Float beta m e)) = ln_beta beta (Z2R m) + e :> Z)%Z.
 Proof.
 intros m e H.
 destruct (ln_beta beta (Z2R m)) as (d, Hd).
@@ -396,7 +396,7 @@ Theorem float_distribution_pos :
   forall m1 e1 m2 e2 : Z,
   (0 < m1)%Z ->
   (F2R (Float beta m1 e1) < F2R (Float beta m2 e2) < F2R (Float beta (m1 + 1) e1))%R ->
-  (e2 < e1)%Z /\ (e1 + projT1 (ln_beta beta (Z2R m1)) = e2 + projT1 (ln_beta beta (Z2R m2)))%Z.
+  (e2 < e1)%Z /\ (e1 + ln_beta beta (Z2R m1) = e2 + ln_beta beta (Z2R m2))%Z.
 Proof.
 intros m1 e1 m2 e2 Hp1 (H12, H21).
 assert (He: (e2 < e1)%Z).

src/Core/Fcore_generic_fmt.v

@@ -21,7 +21,7 @@ Definition valid_exp :=
 Variable prop_exp : valid_exp.
 
 Definition canonic_exponent x :=
-  fexp (projT1 (ln_beta beta x)).
+  fexp (ln_beta beta x).
 
 Definition canonic (f : float beta) :=
   Fexp f = canonic_exponent (F2R f).
@@ -85,7 +85,7 @@ unfold generic_format, scaled_mantissa.
 set (e' := canonic_exponent (F2R (Float beta m e))).
 intros He.
 unfold F2R at 3. simpl.
-assert (H: (Z2R m * bpow e * bpow (- e') = Z2R (m * Zpower (radix_val beta) (e + -e')))%R).
+assert (H: (Z2R m * bpow e * bpow (- e') = Z2R (m * Zpower beta (e + -e')))%R).
 rewrite Rmult_assoc, <- bpow_add, mult_Z2R.
 rewrite Z2R_Zpower.
 apply refl_equal.
@@ -312,10 +312,9 @@ rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r, Rmult_1_r.
 now rewrite Ztrunc_Z2R.
 Qed.
 
-
 Theorem canonic_exp_ge:
   forall prec,
-  (forall e, (e-fexp e <= prec)%Z) ->
+  (forall e, (e - fexp e <= prec)%Z) ->
   (* OK with FLX, FLT and FTZ *)
   forall x, generic_format x ->
   (Rabs x < bpow (prec + canonic_exponent x))%R.
@@ -324,15 +323,14 @@ case (Req_dec x 0); intros Hxz.
 rewrite Hxz, Rabs_R0.
 apply bpow_gt_0.
 unfold canonic_exponent.
-destruct (ln_beta beta x); simpl.
-specialize (a Hxz).
-apply Rlt_le_trans with (1:=proj2 a).
+destruct (ln_beta beta x) as (ex,Ex) ; simpl.
+specialize (Ex Hxz).
+apply Rlt_le_trans with (1 := proj2 Ex).
 apply -> bpow_le.
-specialize (Hp x0).
+specialize (Hp ex).
 omega.
 Qed.
 
-
 Theorem generic_format_bpow_inv :
   forall e,
     generic_format (bpow e) ->
@@ -368,7 +366,6 @@ unfold scaled_mantissa.
 apply Rmult_le_pos; apply bpow_ge_0.
 Qed.
 
-
 Section Fcore_generic_rounding_pos.
 
 Record Zrounding := mkZrounding {
@@ -526,7 +523,7 @@ replace (ex - 1)%Z with (ex - 1 + - fexp ex + fexp ex)%Z by ring.
 rewrite bpow_add.
 apply Rmult_le_compat_r.
 apply bpow_ge_0.
-assert (Hf: Z2R (Zpower (radix_val beta) (ex - 1 - fexp ex)) = bpow (ex - 1 + - fexp ex)).
+assert (Hf: Z2R (Zpower beta (ex - 1 - fexp ex)) = bpow (ex - 1 + - fexp ex)).
 apply Z2R_Zpower.
 omega.
 rewrite <- Hf.
@@ -553,7 +550,7 @@ pattern ex at 3 ; replace ex with (ex - fexp ex + fexp ex)%Z by ring.
 rewrite bpow_add.
 apply Rmult_le_compat_r.
 apply bpow_ge_0.
-assert (Hf: Z2R (Zpower (radix_val beta) (ex - fexp ex)) = bpow (ex - fexp ex)).
+assert (Hf: Z2R (Zpower beta (ex - fexp ex)) = bpow (ex - fexp ex)).
 apply Z2R_Zpower.
 omega.
 rewrite <- Hf.
@@ -724,7 +721,7 @@ apply F2R_le_0_compat. simpl.
 rewrite <- (Zrnd_Z2R rnd 0).
 apply Zrnd_monotone.
 simpl.
-rewrite <- (Rmult_0_l (bpow (- fexp (projT1 (ln_beta beta x))))).
+rewrite <- (Rmult_0_l (bpow (- fexp (ln_beta beta x)))).
 apply Rmult_le_compat_r.
 apply bpow_ge_0.
 now apply Rlt_le.
@@ -1031,10 +1028,10 @@ intros e x He Hx.
 pattern x at 2 ; rewrite Hx.
 unfold F2R at 2. simpl.
 rewrite Rmult_assoc, <- bpow_add.
-assert (H: Z2R (Zpower (radix_val beta) (canonic_exponent x + - fexp e)) = bpow (canonic_exponent x + - fexp e)).
+assert (H: Z2R (Zpower beta (canonic_exponent x + - fexp e)) = bpow (canonic_exponent x + - fexp e)).
 apply Z2R_Zpower.
 unfold canonic_exponent.
-set (ex := projT1 (ln_beta beta x)).
+set (ex := ln_beta beta x).
 generalize (not_FTZ ex).
 generalize (proj2 (proj2 (prop_exp _) He) (fexp ex + 1)%Z).