Making those proofs which depend on names generated for the arguments

in Prop of constructors of inductive types independent of these names.

Incidentally upgraded/simplified a couple of proofs, mainly in Reals.

This prepares to the next commit about using names based on H for such
hypotheses in Prop.

1 files changed, 18 insertions, 32 deletions

diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v
index 3abdff982..99aff9a61 100644
--- a/theories/Arith/Euclid.v
+++ b/theories/Arith/Euclid.v
@@ -19,16 +19,12 @@ Inductive diveucl a b : Set :=
 
 Lemma eucl_dev : forall n, n > 0 -> forall m:nat, diveucl m n.
 Proof.
-  intros b H a; pattern a; apply gt_wf_rec; intros n H0.
-  elim (le_gt_dec b n).
-  intro lebn.
-  elim (H0 (n - b)); auto with arith.
-  intros q r g e.
-  apply divex with (S q) r; simpl; auto with arith.
-  elim plus_assoc.
-  elim e; auto with arith.
-  intros gtbn.
-  apply divex with 0 n; simpl; auto with arith.
+  induction m as (m,H0) using gt_wf_rec.
+  destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
+  destruct (H0 (m - n)) as (q,r,Hge0,Heq); auto with arith.
+  apply divex with (S q) r; trivial.
+  simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
+  apply divex with 0 m; simpl; trivial.
 Defined.
 
 Lemma quotient :
@@ -36,17 +32,12 @@ Lemma quotient :
     n > 0 ->
     forall m:nat, {q : nat |  exists r : nat, m = q * n + r /\ n > r}.
 Proof.
-  intros b H a; pattern a; apply gt_wf_rec; intros n H0.
-  elim (le_gt_dec b n).
-  intro lebn.
-  elim (H0 (n - b)); auto with arith.
-  intros q Hq; exists (S q).
-  elim Hq; intros r Hr.
-  exists r; simpl; elim Hr; intros.
-  elim plus_assoc.
-  elim H1; auto with arith.
-  intros gtbn.
-  exists 0; exists n; simpl; auto with arith.
+  induction m as (m,H0) using gt_wf_rec.
+  destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
+  destruct (H0 (m - n)) as (q & Hq); auto with arith; exists (S q).
+  destruct Hq as (r & Heq & Hgt); exists r; split; trivial.
+  simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
+  exists 0; exists m; simpl; auto with arith.
 Defined.
 
 Lemma modulo :
@@ -54,15 +45,10 @@ Lemma modulo :
     n > 0 ->
     forall m:nat, {r : nat |  exists q : nat, m = q * n + r /\ n > r}.
 Proof.
-  intros b H a; pattern a; apply gt_wf_rec; intros n H0.
-  elim (le_gt_dec b n).
-  intro lebn.
-  elim (H0 (n - b)); auto with arith.
-  intros r Hr; exists r.
-  elim Hr; intros q Hq.
-  elim Hq; intros; exists (S q); simpl.
-  elim plus_assoc.
-  elim H1; auto with arith.
-  intros gtbn.
-  exists n; exists 0; simpl; auto with arith.
+  induction m as (m,H0) using gt_wf_rec.
+  destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
+  destruct (H0 (m - n)) as (r & Hr); auto with arith; exists r.
+  destruct Hr as (q & Heq & Hgt); exists (S q); split; trivial.
+  simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
+  exists m; exists 0; simpl; auto with arith.
 Defined.