Modify Numbers/Integer/Abstract/ZAdd.v to compile with -mangle-names

All that really needed to be done was add an explicit intro before
nzinduct, but all the issues in this file could be fixed by moving n m p
before the colon, and I couldn't stop my self.

1 files changed, 85 insertions, 88 deletions

diff --git a/theories/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v
index 2361d59c26..0c097b6773 100644
--- a/theories/Numbers/Integer/Abstract/ZAdd.v
+++ b/theories/Numbers/Integer/Abstract/ZAdd.v
@@ -20,159 +20,157 @@ Include ZBaseProp Z.
 
 Hint Rewrite opp_0 : nz.
 
-Theorem add_pred_l : forall n m, P n + m == P (n + m).
+Theorem add_pred_l n m : P n + m == P (n + m).
 Proof.
-intros n m.
 rewrite <- (succ_pred n) at 2.
 now rewrite add_succ_l, pred_succ.
 Qed.
 
-Theorem add_pred_r : forall n m, n + P m == P (n + m).
+Theorem add_pred_r n m : n + P m == P (n + m).
 Proof.
-intros n m; rewrite 2 (add_comm n); apply add_pred_l.
+rewrite 2 (add_comm n); apply add_pred_l.
 Qed.
 
-Theorem add_opp_r : forall n m, n + (- m) == n - m.
+Theorem add_opp_r n m : n + (- m) == n - m.
 Proof.
 nzinduct m.
 now nzsimpl.
 intro m. rewrite opp_succ, sub_succ_r, add_pred_r. now rewrite pred_inj_wd.
 Qed.
 
-Theorem sub_0_l : forall n, 0 - n == - n.
+Theorem sub_0_l n : 0 - n == - n.
 Proof.
-intro n; rewrite <- add_opp_r; now rewrite add_0_l.
+rewrite <- add_opp_r; now rewrite add_0_l.
 Qed.
 
-Theorem sub_succ_l : forall n m, S n - m == S (n - m).
+Theorem sub_succ_l n m : S n - m == S (n - m).
 Proof.
-intros n m; rewrite <- 2 add_opp_r; now rewrite add_succ_l.
+rewrite <- 2 add_opp_r; now rewrite add_succ_l.
 Qed.
 
-Theorem sub_pred_l : forall n m, P n - m == P (n - m).
+Theorem sub_pred_l n m : P n - m == P (n - m).
 Proof.
-intros n m. rewrite <- (succ_pred n) at 2.
+rewrite <- (succ_pred n) at 2.
 rewrite sub_succ_l; now rewrite pred_succ.
 Qed.
 
-Theorem sub_pred_r : forall n m, n - (P m) == S (n - m).
+Theorem sub_pred_r n m : n - (P m) == S (n - m).
 Proof.
-intros n m. rewrite <- (succ_pred m) at 2.
+rewrite <- (succ_pred m) at 2.
 rewrite sub_succ_r; now rewrite succ_pred.
 Qed.
 
-Theorem opp_pred : forall n, - (P n) == S (- n).
+Theorem opp_pred n : - (P n) == S (- n).
 Proof.
-intro n. rewrite <- (succ_pred n) at 2.
+rewrite <- (succ_pred n) at 2.
 rewrite opp_succ. now rewrite succ_pred.
 Qed.
 
-Theorem sub_diag : forall n, n - n == 0.
+Theorem sub_diag n : n - n == 0.
 Proof.
 nzinduct n.
 now nzsimpl.
 intro n. rewrite sub_succ_r, sub_succ_l; now rewrite pred_succ.
 Qed.
 
-Theorem add_opp_diag_l : forall n, - n + n == 0.
+Theorem add_opp_diag_l n : - n + n == 0.
 Proof.
-intro n; now rewrite add_comm, add_opp_r, sub_diag.
+now rewrite add_comm, add_opp_r, sub_diag.
 Qed.
 
-Theorem add_opp_diag_r : forall n, n + (- n) == 0.
+Theorem add_opp_diag_r n : n + (- n) == 0.
 Proof.
-intro n; rewrite add_comm; apply add_opp_diag_l.
+rewrite add_comm; apply add_opp_diag_l.
 Qed.
 
-Theorem add_opp_l : forall n m, - m + n == n - m.
+Theorem add_opp_l n m : - m + n == n - m.
 Proof.
-intros n m; rewrite <- add_opp_r; now rewrite add_comm.
+rewrite <- add_opp_r; now rewrite add_comm.
 Qed.
 
-Theorem add_sub_assoc : forall n m p, n + (m - p) == (n + m) - p.
+Theorem add_sub_assoc n m p : n + (m - p) == (n + m) - p.
 Proof.
-intros n m p; rewrite <- 2 add_opp_r; now rewrite add_assoc.
+rewrite <- 2 add_opp_r; now rewrite add_assoc.
 Qed.
 
-Theorem opp_involutive : forall n, - (- n) == n.
+Theorem opp_involutive n : - (- n) == n.
 Proof.
 nzinduct n.
 now nzsimpl.
 intro n. rewrite opp_succ, opp_pred. now rewrite succ_inj_wd.
 Qed.
 
-Theorem opp_add_distr : forall n m, - (n + m) == - n + (- m).
+Theorem opp_add_distr n m : - (n + m) == - n + (- m).
 Proof.
-intros n m; nzinduct n.
+nzinduct n.
 now nzsimpl.
 intro n. rewrite add_succ_l; do 2 rewrite opp_succ; rewrite add_pred_l.
 now rewrite pred_inj_wd.
 Qed.
 
-Theorem opp_sub_distr : forall n m, - (n - m) == - n + m.
+Theorem opp_sub_distr n m : - (n - m) == - n + m.
 Proof.
-intros n m; rewrite <- add_opp_r, opp_add_distr.
+rewrite <- add_opp_r, opp_add_distr.
 now rewrite opp_involutive.
 Qed.
 
-Theorem opp_inj : forall n m, - n == - m -> n == m.
+Theorem opp_inj n m : - n == - m -> n == m.
 Proof.
-intros n m H. apply opp_wd in H. now rewrite 2 opp_involutive in H.
+intros H. apply opp_wd in H. now rewrite 2 opp_involutive in H.
 Qed.
 
-Theorem opp_inj_wd : forall n m, - n == - m <-> n == m.
+Theorem opp_inj_wd n m : - n == - m <-> n == m.
 Proof.
-intros n m; split; [apply opp_inj | intros; now f_equiv].
+split; [apply opp_inj | intros; now f_equiv].
 Qed.
 
-Theorem eq_opp_l : forall n m, - n == m <-> n == - m.
+Theorem eq_opp_l n m : - n == m <-> n == - m.
 Proof.
-intros n m. now rewrite <- (opp_inj_wd (- n) m), opp_involutive.
+now rewrite <- (opp_inj_wd (- n) m), opp_involutive.
 Qed.
 
-Theorem eq_opp_r : forall n m, n == - m <-> - n == m.
+Theorem eq_opp_r n m : n == - m <-> - n == m.
 Proof.
 symmetry; apply eq_opp_l.
 Qed.
 
-Theorem sub_add_distr : forall n m p, n - (m + p) == (n - m) - p.
+Theorem sub_add_distr n m p : n - (m + p) == (n - m) - p.
 Proof.
-intros n m p; rewrite <- add_opp_r, opp_add_distr, add_assoc.
+rewrite <- add_opp_r, opp_add_distr, add_assoc.
 now rewrite 2 add_opp_r.
 Qed.
 
-Theorem sub_sub_distr : forall n m p, n - (m - p) == (n - m) + p.
+Theorem sub_sub_distr n m p : n - (m - p) == (n - m) + p.
 Proof.
-intros n m p; rewrite <- add_opp_r, opp_sub_distr, add_assoc.
+rewrite <- add_opp_r, opp_sub_distr, add_assoc.
 now rewrite add_opp_r.
 Qed.
 
-Theorem sub_opp_l : forall n m, - n - m == - m - n.
+Theorem sub_opp_l n m : - n - m == - m - n.
 Proof.
-intros n m. rewrite <- 2 add_opp_r. now rewrite add_comm.
+rewrite <- 2 add_opp_r. now rewrite add_comm.
 Qed.
 
-Theorem sub_opp_r : forall n m, n - (- m) == n + m.
+Theorem sub_opp_r n m : n - (- m) == n + m.
 Proof.
-intros n m; rewrite <- add_opp_r; now rewrite opp_involutive.
+rewrite <- add_opp_r; now rewrite opp_involutive.
 Qed.
 
-Theorem add_sub_swap : forall n m p, n + m - p == n - p + m.
+Theorem add_sub_swap n m p : n + m - p == n - p + m.
 Proof.
-intros n m p. rewrite <- add_sub_assoc, <- (add_opp_r n p), <- add_assoc.
+rewrite <- add_sub_assoc, <- (add_opp_r n p), <- add_assoc.
 now rewrite add_opp_l.
 Qed.
 
-Theorem sub_cancel_l : forall n m p, n - m == n - p <-> m == p.
+Theorem sub_cancel_l n m p : n - m == n - p <-> m == p.
 Proof.
-intros n m p. rewrite <- (add_cancel_l (n - m) (n - p) (- n)).
+rewrite <- (add_cancel_l (n - m) (n - p) (- n)).
 rewrite 2 add_sub_assoc. rewrite add_opp_diag_l; rewrite 2 sub_0_l.
 apply opp_inj_wd.
 Qed.
 
-Theorem sub_cancel_r : forall n m p, n - p == m - p <-> n == m.
+Theorem sub_cancel_r n m p : n - p == m - p <-> n == m.
 Proof.
-intros n m p.
 stepl (n - p + p == m - p + p) by apply add_cancel_r.
 now do 2 rewrite <- sub_sub_distr, sub_diag, sub_0_r.
 Qed.
@@ -182,16 +180,15 @@ Qed.
     in the original equation ([add] or [sub]) and the indication
     whether the left or right term is moved. *)
 
-Theorem add_move_l : forall n m p, n + m == p <-> m == p - n.
+Theorem add_move_l n m p : n + m == p <-> m == p - n.
 Proof.
-intros n m p.
 stepl (n + m - n == p - n) by apply sub_cancel_r.
 now rewrite add_comm, <- add_sub_assoc, sub_diag, add_0_r.
 Qed.
 
-Theorem add_move_r : forall n m p, n + m == p <-> n == p - m.
+Theorem add_move_r n m p : n + m == p <-> n == p - m.
 Proof.
-intros n m p; rewrite add_comm; now apply add_move_l.
+rewrite add_comm; now apply add_move_l.
 Qed.
 
 (** The two theorems above do not allow rewriting subformulas of the
@@ -199,98 +196,98 @@ Qed.
     right-hand side of the equation. Hence the following two
     theorems. *)
 
-Theorem sub_move_l : forall n m p, n - m == p <-> - m == p - n.
+Theorem sub_move_l n m p : n - m == p <-> - m == p - n.
 Proof.
-intros n m p; rewrite <- (add_opp_r n m); apply add_move_l.
+rewrite <- (add_opp_r n m); apply add_move_l.
 Qed.
 
-Theorem sub_move_r : forall n m p, n - m == p <-> n == p + m.
+Theorem sub_move_r n m p : n - m == p <-> n == p + m.
 Proof.
-intros n m p; rewrite <- (add_opp_r n m). now rewrite add_move_r, sub_opp_r.
+rewrite <- (add_opp_r n m). now rewrite add_move_r, sub_opp_r.
 Qed.
 
-Theorem add_move_0_l : forall n m, n + m == 0 <-> m == - n.
+Theorem add_move_0_l n m : n + m == 0 <-> m == - n.
 Proof.
-intros n m; now rewrite add_move_l, sub_0_l.
+now rewrite add_move_l, sub_0_l.
 Qed.
 
-Theorem add_move_0_r : forall n m, n + m == 0 <-> n == - m.
+Theorem add_move_0_r n m : n + m == 0 <-> n == - m.
 Proof.
-intros n m; now rewrite add_move_r, sub_0_l.
+now rewrite add_move_r, sub_0_l.
 Qed.
 
-Theorem sub_move_0_l : forall n m, n - m == 0 <-> - m == - n.
+Theorem sub_move_0_l n m : n - m == 0 <-> - m == - n.
 Proof.
-intros n m. now rewrite sub_move_l, sub_0_l.
+now rewrite sub_move_l, sub_0_l.
 Qed.
 
-Theorem sub_move_0_r : forall n m, n - m == 0 <-> n == m.
+Theorem sub_move_0_r n m : n - m == 0 <-> n == m.
 Proof.
-intros n m. now rewrite sub_move_r, add_0_l.
+now rewrite sub_move_r, add_0_l.
 Qed.
 
 (** The following section is devoted to cancellation of like
     terms. The name includes the first operator and the position of
     the term being canceled. *)
 
-Theorem add_simpl_l : forall n m, n + m - n == m.
+Theorem add_simpl_l n m : n + m - n == m.
 Proof.
-intros; now rewrite add_sub_swap, sub_diag, add_0_l.
+now rewrite add_sub_swap, sub_diag, add_0_l.
 Qed.
 
-Theorem add_simpl_r : forall n m, n + m - m == n.
+Theorem add_simpl_r n m : n + m - m == n.
 Proof.
-intros; now rewrite <- add_sub_assoc, sub_diag, add_0_r.
+now rewrite <- add_sub_assoc, sub_diag, add_0_r.
 Qed.
 
-Theorem sub_simpl_l : forall n m, - n - m + n == - m.
+Theorem sub_simpl_l n m : - n - m + n == - m.
 Proof.
-intros; now rewrite <- add_sub_swap, add_opp_diag_l, sub_0_l.
+now rewrite <- add_sub_swap, add_opp_diag_l, sub_0_l.
 Qed.
 
-Theorem sub_simpl_r : forall n m, n - m + m == n.
+Theorem sub_simpl_r n m : n - m + m == n.
 Proof.
-intros; now rewrite <- sub_sub_distr, sub_diag, sub_0_r.
+now rewrite <- sub_sub_distr, sub_diag, sub_0_r.
 Qed.
 
-Theorem sub_add : forall n m, m - n + n == m.
+Theorem sub_add n m : m - n + n == m.
 Proof.
- intros. now rewrite <- add_sub_swap, add_simpl_r.
+now rewrite <- add_sub_swap, add_simpl_r.
 Qed.
 
 (** Now we have two sums or differences; the name includes the two
     operators and the position of the terms being canceled *)
 
-Theorem add_add_simpl_l_l : forall n m p, (n + m) - (n + p) == m - p.
+Theorem add_add_simpl_l_l n m p : (n + m) - (n + p) == m - p.
 Proof.
-intros n m p. now rewrite (add_comm n m), <- add_sub_assoc,
+now rewrite (add_comm n m), <- add_sub_assoc,
 sub_add_distr, sub_diag, sub_0_l, add_opp_r.
 Qed.
 
-Theorem add_add_simpl_l_r : forall n m p, (n + m) - (p + n) == m - p.
+Theorem add_add_simpl_l_r n m p : (n + m) - (p + n) == m - p.
 Proof.
-intros n m p. rewrite (add_comm p n); apply add_add_simpl_l_l.
+rewrite (add_comm p n); apply add_add_simpl_l_l.
 Qed.
 
-Theorem add_add_simpl_r_l : forall n m p, (n + m) - (m + p) == n - p.
+Theorem add_add_simpl_r_l n m p : (n + m) - (m + p) == n - p.
 Proof.
-intros n m p. rewrite (add_comm n m); apply add_add_simpl_l_l.
+rewrite (add_comm n m); apply add_add_simpl_l_l.
 Qed.
 
-Theorem add_add_simpl_r_r : forall n m p, (n + m) - (p + m) == n - p.
+Theorem add_add_simpl_r_r n m p : (n + m) - (p + m) == n - p.
 Proof.
-intros n m p. rewrite (add_comm p m); apply add_add_simpl_r_l.
+rewrite (add_comm p m); apply add_add_simpl_r_l.
 Qed.
 
-Theorem sub_add_simpl_r_l : forall n m p, (n - m) + (m + p) == n + p.
+Theorem sub_add_simpl_r_l n m p : (n - m) + (m + p) == n + p.
 Proof.
-intros n m p. now rewrite <- sub_sub_distr, sub_add_distr, sub_diag,
+now rewrite <- sub_sub_distr, sub_add_distr, sub_diag,
 sub_0_l, sub_opp_r.
 Qed.
 
-Theorem sub_add_simpl_r_r : forall n m p, (n - m) + (p + m) == n + p.
+Theorem sub_add_simpl_r_r n m p : (n - m) + (p + m) == n + p.
 Proof.
-intros n m p. rewrite (add_comm p m); apply sub_add_simpl_r_l.
+rewrite (add_comm p m); apply sub_add_simpl_r_l.
 Qed.
 
 (** Of course, there are many other variants *)