freescale porting, work in progress

[helm.git] / helm / software / matita / contribs / ng_assembly / freescale / nat_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/freescale/nat_lemmas.ma b/helm/software/matita/contribs/ng_assembly/freescale/nat_lemmas.ma
index 9e6b6e8a5da8a436cb16a239d3184e6d51651cf6..72ceb6188e680d5c8f701a19432128645340bdd9 100644 (file)
--- a/helm/software/matita/contribs/ng_assembly/freescale/nat_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/freescale/nat_lemmas.ma
@@ -31,7 +31,7 @@ include "freescale/nat.ma".
 (* NATURALI *)
 (* ******** *)
 
-nlemma nat_destruct : ∀n1,n2:nat.S n1 = S n2 → n1 = n2.
+nlemma nat_destruct_S_S : ∀n1,n2:nat.S n1 = S n2 → n1 = n2.
  #n1; #n2; #H;
  nchange with (match S n2 with [ O ⇒ False | S a ⇒ n1 = a ]);
  nrewrite < H;
@@ -89,7 +89,7 @@ nlemma eq_to_eqnat : ∀n1,n2:nat.n1 = n2 → eq_nat n1 n2 = true.
           nnormalize;
           ##[ ##1: #H1; nelim (nat_destruct_S_0 ? H1)
           ##| ##2: #n4; #H1;
-                   napply (H n4 (nat_destruct ?? H1))
+                   napply (H n4 (nat_destruct_S_S ?? H1))
           ##]
  ##]
 nqed. 
@@ -113,3 +113,53 @@ nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
           ##]
  ##]
 nqed.
+
+nlemma Sn_p_n_to_S_npn : ∀n1,n2.(S n1) + n2 = S (n1 + n2).
+ #n1;
+ nelim n1;
+ ##[ ##1: nnormalize; #n2; napply (refl_eq ??)
+ ##| ##2: #n; #H; #n2; nrewrite > (H n2);
+          ncases n in H:(%) ⊢ %;
+          ##[ ##1: nnormalize; #H; napply (refl_eq ??)
+          ##| ##2: #n3; nnormalize; #H; napply (refl_eq ??)
+          ##]
+ ##]
+nqed.
+
+nlemma n_p_Sn_to_S_npn : ∀n1,n2.n1 + (S n2) = S (n1 + n2).
+ #n1;
+ nelim n1;
+ ##[ ##1: nnormalize; #n2; napply (refl_eq ??)
+ ##| ##2: #n; #H; #n2;
+          nrewrite > (Sn_p_n_to_S_npn n (S n2));
+          nrewrite > (H n2);
+          napply (refl_eq ??)
+ ##]
+nqed.
+
+nlemma Opn_to_n : ∀n.O + n = n.
+ #n; nnormalize; napply (refl_eq ??).
+nqed.
+
+nlemma npO_to_n : ∀n.n + O = n.
+ #n;
+ nelim n;
+ ##[ ##1: nnormalize; napply (refl_eq ??)
+ ##| ##2: #n1; #H;
+          nrewrite > (Sn_p_n_to_S_npn n1 O); 
+          nrewrite > H;
+          napply (refl_eq ??)
+ ##]
+nqed.
+
+nlemma symmetric_plusnat : symmetricT nat nat plus.
+ #n1;
+ nelim n1;
+ ##[ ##1: #n2; nrewrite > (npO_to_n n2); nnormalize; napply (refl_eq ??)
+ ##| ##2: #n2; #H; #n3;
+          nrewrite > (Sn_p_n_to_S_npn n2 n3);
+          nrewrite > (n_p_Sn_to_S_npn n3 n2);
+          nrewrite > (H n3);
+          napply (refl_eq ??)
+ ##]
+nqed.