Make Zdigits <-> Zpower theorems axiom-free.

src/Calc/Fcalc_digits.v

@@ -60,80 +60,6 @@ apply sym_eq.
 now apply Zdigits_ln_beta.
 Qed.
 
-Theorem Zdigits_le :
-  forall x y,
-  (0 <= x)%Z -> (x <= y)%Z ->
-  (Zdigits beta x <= Zdigits beta y)%Z.
-Proof.
-intros x y Hx Hxy.
-case (Z_lt_le_dec 0 x).
-clear Hx. intros Hx.
-rewrite 2!Zdigits_ln_beta.
-apply ln_beta_le.
-now apply (Z2R_lt 0).
-now apply Z2R_le.
-apply Zgt_not_eq.
-now apply Zlt_le_trans with x.
-now apply Zgt_not_eq.
-intros Hx'.
-rewrite (Zle_antisym _ _ Hx' Hx).
-apply Zdigits_ge_0.
-Qed.
-
-Theorem lt_Zdigits :
-  forall x y,
-  (0 <= y)%Z ->
-  (Zdigits beta x < Zdigits beta y)%Z ->
-  (x < y)%Z.
-Proof.
-intros x y Hy.
-cut (y <= x -> Zdigits beta y <= Zdigits beta x)%Z. omega.
-now apply Zdigits_le.
-Qed.
-
-Theorem Zpower_le_Zdigits :
-  forall e x,
-  (e < Zdigits beta x)%Z ->
-  (Zpower beta e <= Zabs x)%Z.
-Proof.
-intros e x Hex.
-destruct (Zdigits_correct beta x) as (H1,H2).
-apply Zle_trans with (2 := H1).
-apply Zpower_le.
-clear -Hex ; omega.
-Qed.
-
-Theorem Zdigits_le_Zpower :
-  forall e x,
-  (Zabs x < Zpower beta e)%Z ->
-  (Zdigits beta x <= e)%Z.
-Proof.
-intros e x.
-generalize (Zpower_le_Zdigits e x).
-omega.
-Qed.
-
-Theorem Zpower_gt_Zdigits :
-  forall e x,
-  (Zdigits beta x <= e)%Z ->
-  (Zabs x < Zpower beta e)%Z.
-Proof.
-intros e x Hex.
-destruct (Zdigits_correct beta x) as (H1,H2).
-apply Zlt_le_trans with (1 := H2).
-now apply Zpower_le.
-Qed.
-
-Theorem Zdigits_gt_Zpower :
-  forall e x,
-  (Zpower beta e <= Zabs x)%Z ->
-  (e < Zdigits beta x)%Z.
-Proof.
-intros e x Hex.
-generalize (Zpower_gt_Zdigits e x).
-omega.
-Qed.
-
 (** Characterizes the number digits of a product.
 
 This strong version is needed for proofs of division and square root

src/Core/Fcore_digits.v

@@ -946,4 +946,70 @@ rewrite Zdigits_mult_Zpower ; try easy.
 apply Zplus_comm.
 Qed.
 
+Theorem Zdigits_le :
+  forall x y,
+  (0 <= x)%Z -> (x <= y)%Z ->
+  (Zdigits x <= Zdigits y)%Z.
+Proof.
+intros x y Zx Hxy.
+assert (Hx := Zdigits_correct x).
+assert (Hy := Zdigits_correct y).
+apply (Zpower_lt_Zpower beta).
+zify ; omega.
+Qed.
+
+Theorem lt_Zdigits :
+  forall x y,
+  (0 <= y)%Z ->
+  (Zdigits x < Zdigits y)%Z ->
+  (x < y)%Z.
+Proof.
+intros x y Hy.
+cut (y <= x -> Zdigits y <= Zdigits x)%Z. omega.
+now apply Zdigits_le.
+Qed.
+
+Theorem Zpower_le_Zdigits :
+  forall e x,
+  (e < Zdigits x)%Z ->
+  (Zpower beta e <= Zabs x)%Z.
+Proof.
+intros e x Hex.
+destruct (Zdigits_correct x) as [H1 H2].
+apply Zle_trans with (2 := H1).
+apply Zpower_le.
+clear -Hex ; omega.
+Qed.
+
+Theorem Zdigits_le_Zpower :
+  forall e x,
+  (Zabs x < Zpower beta e)%Z ->
+  (Zdigits x <= e)%Z.
+Proof.
+intros e x.
+generalize (Zpower_le_Zdigits e x).
+omega.
+Qed.
+
+Theorem Zpower_gt_Zdigits :
+  forall e x,
+  (Zdigits x <= e)%Z ->
+  (Zabs x < Zpower beta e)%Z.
+Proof.
+intros e x Hex.
+destruct (Zdigits_correct x) as [H1 H2].
+apply Zlt_le_trans with (1 := H2).
+now apply Zpower_le.
+Qed.
+
+Theorem Zdigits_gt_Zpower :
+  forall e x,
+  (Zpower beta e <= Zabs x)%Z ->
+  (e < Zdigits x)%Z.
+Proof.
+intros e x Hex.
+generalize (Zpower_gt_Zdigits e x).
+omega.
+Qed.
+
 End Fcore_digits.