propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / rtmap_nexts.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_nexts.ma b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_nexts.ma
index 08851ebe6670fcec05738458e018546574846858..128c26d6438b6b754ec79c8f109bd2ff891727b5 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_nexts.ma
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_nexts.ma
@@ -13,20 +13,42 @@
 (**************************************************************************)
 
 include "ground/notation/functions/uparrowstar_2.ma".
+include "ground/arith/nat_succ_iter.ma".
 include "ground/relocation/rtmap_eq.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
-rec definition nexts (f:rtmap) (n:nat) on n: rtmap ≝ match n with
-[ O ⇒ f | S m ⇒ ↑(nexts f m) ].
+definition nexts (f:rtmap) (n:nat) ≝ next^n f.
 
 interpretation "nexts (rtmap)" 'UpArrowStar n f = (nexts f n).
 
+(* Basic properties *********************************************************)
+
+lemma nexts_O: ∀f. f = ↑*[𝟎] f.
+// qed.
+
+lemma nexts_S: ∀f,n. ↑↑*[n] f = ↑*[↑n] f.
+#f #n @(niter_succ … next)
+qed.
+
+lemma nexts_eq_repl: ∀n. eq_repl (λf1,f2. ↑*[n] f1 ≡ ↑*[n] f2).
+#n @(nat_ind_succ … n) -n /3 width=5 by eq_next/
+qed.
+
+(* Advanced properties ******************************************************)
+
+lemma nexts_swap: ∀f,n. ↑↑*[n] f = ↑*[n] ↑f.
+#f #n @(niter_appl … next)
+qed.
+
+lemma nexts_xn: ∀n,f. ↑*[n] ↑f = ↑*[↑n] f.
+// qed.
+
 (* Basic_inversion lemmas *****************************************************)
 
 lemma eq_inv_nexts_sn: ∀n,f1,g2. ↑*[n] f1 ≡ g2 →
                        ∃∃f2. f1 ≡ f2 & ↑*[n] f2 = g2.
-#n elim n -n /2 width=3 by ex2_intro/
+#n @(nat_ind_succ … n) -n /2 width=3 by ex2_intro/
 #n #IH #f1 #g2 #H elim (eq_inv_nx … H) -H [|*: // ]
 #f0 #Hf10 #H1 elim (IH … Hf10) -IH -Hf10 #f2 #Hf12 #H2 destruct
 /2 width=3 by ex2_intro/
@@ -34,26 +56,8 @@ qed-.
 
 lemma eq_inv_nexts_dx: ∀n,f2,g1. g1 ≡ ↑*[n] f2 →
                        ∃∃f1. f1 ≡ f2 & ↑*[n] f1 = g1.
-#n elim n -n /2 width=3 by ex2_intro/
+#n @(nat_ind_succ … n) -n /2 width=3 by ex2_intro/
 #n #IH #f2 #g1 #H elim (eq_inv_xn … H) -H [|*: // ]
 #f0 #Hf02 #H1 elim (IH … Hf02) -IH -Hf02 #f1 #Hf12 #H2 destruct
 /2 width=3 by ex2_intro/
 qed-.
-
-(* Basic properties *********************************************************)
-
-lemma nexts_O: ∀f. f = ↑*[0] f.
-// qed.
-
-lemma nexts_S: ∀f,n. ↑↑*[n] f = ↑*[↑n] f.
-// qed.
-
-lemma nexts_eq_repl: ∀n. eq_repl (λf1,f2. ↑*[n] f1 ≡ ↑*[n] f2).
-#n elim n -n /3 width=5 by eq_next/
-qed.
-
-(* Advanced properties ******************************************************)
-
-lemma nexts_xn: ∀n,f. ↑*[n] ↑f = ↑*[↑n] f.
-#n elim n -n //
-qed.