arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_plus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
index e7702b80d6ca656b5987b6290d1e549b8f3bc1c6..d9c82ae33f9f8fdb05978cb80e1aa5b95ff68126 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
@@ -14,14 +14,14 @@
 
 include "ground/arith/nat_succ_iter.ma".
 
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* ADDITION FOR NON-NEGATIVE INTEGERS ***************************************)
 
 (*** plus *)
 definition nplus: nat → nat → nat ≝
            λm,n. nsucc^n m.
 
 interpretation
-  "plus (positive integers"
+  "plus (positive integers)"
   'plus m n = (nplus m n).
 
 (* Basic rewrites ***********************************************************)
@@ -33,13 +33,14 @@ lemma nplus_zero_dx (m): m = m + 𝟎.
 lemma nplus_one_dx (n): ↑n = n + 𝟏.
 // qed.
 
-(* Semigroup properties *****************************************************)
+(* Advanved rewrites (semigroup properties) *********************************)
 
 (*** plus_n_Sm *)
 lemma nplus_succ_dx (m) (n): ↑(m+n) = m + ↑n.
 #m #n @(niter_succ … nsucc)
 qed.
 
+(*** plus_S1 *)
 lemma nplus_succ_sn (m) (n): ↑(m+n) = ↑m + n.
 #m #n @(niter_appl … nsucc)
 qed.
@@ -54,15 +55,20 @@ lemma nplus_comm: commutative … nplus.
 #m @(nat_ind … m) -m //
 qed-.
 
-lemma nplus_one_sn (n): ↑n = 𝟏 + n.
-#n <nplus_comm // qed.
-
 (*** associative_plus *)
 lemma nplus_assoc: associative … nplus.
 #m #n #o @(nat_ind … o) -o //
 #o #IH <nplus_succ_dx <nplus_succ_dx <nplus_succ_dx <IH -IH //
 qed.
 
+(* Advanced constructions ***************************************************)
+
+lemma nplus_one_sn (n): ↑n = 𝟏 + n.
+#n <nplus_comm // qed.
+
+lemma nplus_succ_shift (m) (n): ↑m + n = m + ↑n.
+// qed-.
+
 (*** assoc_plus1 *)
 lemma nplus_plus_comm_12 (o) (m) (n): m + n + o = n + (m + o).
 #o #m #n <nplus_comm in ⊢ (??(?%?)?); // qed.
@@ -88,7 +94,8 @@ qed-.
 (*** injective_plus_r *)
 lemma eq_inv_nplus_bi_sn (o) (m) (n): o + m = o + n → m = n.
 #o #m #n <nplus_comm <nplus_comm in ⊢ (???%→?);
-/2 width=2 by eq_inv_nplus_bi_dx/ qed-.
+/2 width=2 by eq_inv_nplus_bi_dx/
+qed-.
 
 (* Advanced eliminations ****************************************************)