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 ffb181cc4908dea0a5048498cb6c2a11a453f56a..f7d79ccdf3914095f44acde0d5689a4c48d6082c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
@@ -12,57 +12,74 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/arith/nat_iter_succ.ma".
+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 ***********************************************************)
+(* Basic constructions ******************************************************)
 
 (*** plus_n_O *)
 lemma nplus_zero_dx (m): m = m + 𝟎.
 // qed.
 
+(*** plus_SO_dx *)
 lemma nplus_one_dx (n): ↑n = n + 𝟏.
 // qed.
 
-(* Semigroup properties *****************************************************)
-
 (*** plus_n_Sm *)
 lemma nplus_succ_dx (m) (n): ↑(m+n) = m + ↑n.
 #m #n @(niter_succ … nsucc)
 qed.
 
+(* Constructions with niter *************************************************)
+
+(*** iter_plus *)
+lemma niter_plus (A) (f) (a) (n1) (n2):
+      f^n1 (f^n2 a) = f^{A}(n1+n2) a.
+#A #f #a #n1 #n2 @(nat_ind_succ … n2) -n2 //
+#n2 #IH <nplus_succ_dx <niter_succ <niter_succ <niter_appl //
+qed.
+
+(* Advanved constructions (semigroup properties) ****************************)
+
+(*** plus_S1 *)
 lemma nplus_succ_sn (m) (n): ↑(m+n) = ↑m + n.
 #m #n @(niter_appl … nsucc)
 qed.
 
 (*** plus_O_n.con *)
 lemma nplus_zero_sn (m): m = 𝟎 + m.
-#m @(nat_ind … m) -m //
+#m @(nat_ind_succ … m) -m //
 qed.
 
 (*** commutative_plus *)
 lemma nplus_comm: commutative … nplus.
-#m @(nat_ind … m) -m //
+#m @(nat_ind_succ … 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 //
+#m #n #o @(nat_ind_succ … o) -o //
 #o #IH <nplus_succ_dx <nplus_succ_dx <nplus_succ_dx <IH -IH //
 qed.
 
+(* Helper constructions *****************************************************)
+
+(*** plus_SO_sn *)
+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.
@@ -75,25 +92,28 @@ qed-.
 (* Basic inversions *********************************************************)
 
 lemma eq_inv_nzero_plus (m) (n): 𝟎 = m + n → ∧∧ 𝟎 = m & 𝟎 = n.
-#m #n @(nat_ind … n) -n /2 width=1 by conj/
-#n #_ <nplus_succ_dx #H
-elim (eq_inv_nzero_succ … H)
+#m #n @(nat_ind_succ … n) -n
+[ /2 width=1 by conj/
+| #n #_ <nplus_succ_dx #H
+  elim (eq_inv_nzero_succ … H)
+]
 qed-.
 
 (*** injective_plus_l *)
 lemma eq_inv_nplus_bi_dx (o) (m) (n): m + o = n + o → m = n.
-#o @(nat_ind … o) -o /3 width=1 by eq_inv_nsucc_bi/
+#o @(nat_ind_succ … o) -o /3 width=1 by eq_inv_nsucc_bi/
 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 ****************************************************)
 
 (*** nat_ind_plus *)
 lemma nat_ind_plus (Q:predicate …):
       Q (𝟎) → (∀n. Q n → Q (𝟏+n)) → ∀n. Q n.
-#Q #IH1 #IH2 #n @(nat_ind … n) -n /2 width=1 by/
+#Q #IH1 #IH2 #n @(nat_ind_succ … n) -n /2 width=1 by/
 qed-.