X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fassembly%2Fassembly.ma;h=fd01ff8cb72a5d5105057ee8811e762a99369872;hb=4e813c00fb151bae5b20e7abeb14d9f02a720891;hp=35455f9dc1cfa7774b84ccb0900c5a35de5ace4f;hpb=3bd823a4d795ecd8929b55d4e4a698b411a97ab9;p=helm.git

diff --git a/helm/software/matita/library/assembly/assembly.ma b/helm/software/matita/library/assembly/assembly.ma
index 35455f9dc..fd01ff8cb 100644
--- a/helm/software/matita/library/assembly/assembly.ma
+++ b/helm/software/matita/library/assembly/assembly.ma
@@ -15,7 +15,6 @@
 set "baseuri" "cic:/matita/assembly/".
 
 include "nat/div_and_mod.ma".
-(*include "nat/compare.ma".*)
 include "list/list.ma".
 
 inductive exadecimal : Type ≝
@@ -145,6 +144,604 @@ definition eqex ≝
 definition eqbyte ≝
  λb,b'. eqex (bh b) (bh b') ∧ eqex (bl b) (bl b').
 
+inductive cartesian_product (A,B: Type) : Type ≝
+ couple: ∀a:A.∀b:B. cartesian_product A B.
+
+definition plusex ≝
+ λb1,b2,c.
+  match c with
+   [ true ⇒
+      match b1 with
+       [ x0 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x1 false
+            | x1 ⇒ couple exadecimal bool x2 false
+            | x2 ⇒ couple exadecimal bool x3 false
+            | x3 ⇒ couple exadecimal bool x4 false
+            | x4 ⇒ couple exadecimal bool x5 false
+            | x5 ⇒ couple exadecimal bool x6 false
+            | x6 ⇒ couple exadecimal bool x7 false
+            | x7 ⇒ couple exadecimal bool x8 false
+            | x8 ⇒ couple exadecimal bool x9 false
+            | x9 ⇒ couple exadecimal bool xA false
+            | xA ⇒ couple exadecimal bool xB false
+            | xB ⇒ couple exadecimal bool xC false
+            | xC ⇒ couple exadecimal bool xD false
+            | xD ⇒ couple exadecimal bool xE false
+            | xE ⇒ couple exadecimal bool xF false
+            | xF ⇒ couple exadecimal bool x0 true ] 
+       | x1 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x2 false
+            | x1 ⇒ couple exadecimal bool x3 false
+            | x2 ⇒ couple exadecimal bool x4 false
+            | x3 ⇒ couple exadecimal bool x5 false
+            | x4 ⇒ couple exadecimal bool x6 false
+            | x5 ⇒ couple exadecimal bool x7 false
+            | x6 ⇒ couple exadecimal bool x8 false
+            | x7 ⇒ couple exadecimal bool x9 false
+            | x8 ⇒ couple exadecimal bool xA false
+            | x9 ⇒ couple exadecimal bool xB false
+            | xA ⇒ couple exadecimal bool xC false
+            | xB ⇒ couple exadecimal bool xD false
+            | xC ⇒ couple exadecimal bool xE false
+            | xD ⇒ couple exadecimal bool xF false
+            | xE ⇒ couple exadecimal bool x0 true
+            | xF ⇒ couple exadecimal bool x1 true ] 
+       | x2 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x3 false
+            | x1 ⇒ couple exadecimal bool x4 false
+            | x2 ⇒ couple exadecimal bool x5 false
+            | x3 ⇒ couple exadecimal bool x6 false
+            | x4 ⇒ couple exadecimal bool x7 false
+            | x5 ⇒ couple exadecimal bool x8 false
+            | x6 ⇒ couple exadecimal bool x9 false
+            | x7 ⇒ couple exadecimal bool xA false
+            | x8 ⇒ couple exadecimal bool xB false
+            | x9 ⇒ couple exadecimal bool xC false
+            | xA ⇒ couple exadecimal bool xD false
+            | xB ⇒ couple exadecimal bool xE false
+            | xC ⇒ couple exadecimal bool xF false
+            | xD ⇒ couple exadecimal bool x0 true
+            | xE ⇒ couple exadecimal bool x1 true
+            | xF ⇒ couple exadecimal bool x2 true ] 
+       | x3 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x4 false
+            | x1 ⇒ couple exadecimal bool x5 false
+            | x2 ⇒ couple exadecimal bool x6 false
+            | x3 ⇒ couple exadecimal bool x7 false
+            | x4 ⇒ couple exadecimal bool x8 false
+            | x5 ⇒ couple exadecimal bool x9 false
+            | x6 ⇒ couple exadecimal bool xA false
+            | x7 ⇒ couple exadecimal bool xB false
+            | x8 ⇒ couple exadecimal bool xC false
+            | x9 ⇒ couple exadecimal bool xD false
+            | xA ⇒ couple exadecimal bool xE false
+            | xB ⇒ couple exadecimal bool xF false
+            | xC ⇒ couple exadecimal bool x0 true
+            | xD ⇒ couple exadecimal bool x1 true
+            | xE ⇒ couple exadecimal bool x2 true
+            | xF ⇒ couple exadecimal bool x3 true ] 
+       | x4 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x5 false
+            | x1 ⇒ couple exadecimal bool x6 false
+            | x2 ⇒ couple exadecimal bool x7 false
+            | x3 ⇒ couple exadecimal bool x8 false
+            | x4 ⇒ couple exadecimal bool x9 false
+            | x5 ⇒ couple exadecimal bool xA false
+            | x6 ⇒ couple exadecimal bool xB false
+            | x7 ⇒ couple exadecimal bool xC false
+            | x8 ⇒ couple exadecimal bool xD false
+            | x9 ⇒ couple exadecimal bool xE false
+            | xA ⇒ couple exadecimal bool xF false
+            | xB ⇒ couple exadecimal bool x0 true
+            | xC ⇒ couple exadecimal bool x1 true
+            | xD ⇒ couple exadecimal bool x2 true
+            | xE ⇒ couple exadecimal bool x3 true
+            | xF ⇒ couple exadecimal bool x4 true ] 
+       | x5 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x6 false
+            | x1 ⇒ couple exadecimal bool x7 false
+            | x2 ⇒ couple exadecimal bool x8 false
+            | x3 ⇒ couple exadecimal bool x9 false
+            | x4 ⇒ couple exadecimal bool xA false
+            | x5 ⇒ couple exadecimal bool xB false
+            | x6 ⇒ couple exadecimal bool xC false
+            | x7 ⇒ couple exadecimal bool xD false
+            | x8 ⇒ couple exadecimal bool xE false
+            | x9 ⇒ couple exadecimal bool xF false
+            | xA ⇒ couple exadecimal bool x0 true
+            | xB ⇒ couple exadecimal bool x1 true
+            | xC ⇒ couple exadecimal bool x2 true
+            | xD ⇒ couple exadecimal bool x3 true
+            | xE ⇒ couple exadecimal bool x4 true
+            | xF ⇒ couple exadecimal bool x5 true ] 
+       | x6 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x7 false
+            | x1 ⇒ couple exadecimal bool x8 false
+            | x2 ⇒ couple exadecimal bool x9 false
+            | x3 ⇒ couple exadecimal bool xA false
+            | x4 ⇒ couple exadecimal bool xB false
+            | x5 ⇒ couple exadecimal bool xC false
+            | x6 ⇒ couple exadecimal bool xD false
+            | x7 ⇒ couple exadecimal bool xE false
+            | x8 ⇒ couple exadecimal bool xF false
+            | x9 ⇒ couple exadecimal bool x0 true
+            | xA ⇒ couple exadecimal bool x1 true
+            | xB ⇒ couple exadecimal bool x2 true
+            | xC ⇒ couple exadecimal bool x3 true
+            | xD ⇒ couple exadecimal bool x4 true
+            | xE ⇒ couple exadecimal bool x5 true
+            | xF ⇒ couple exadecimal bool x6 true ] 
+       | x7 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x8 false
+            | x1 ⇒ couple exadecimal bool x9 false
+            | x2 ⇒ couple exadecimal bool xA false
+            | x3 ⇒ couple exadecimal bool xB false
+            | x4 ⇒ couple exadecimal bool xC false
+            | x5 ⇒ couple exadecimal bool xD false
+            | x6 ⇒ couple exadecimal bool xE false
+            | x7 ⇒ couple exadecimal bool xF false
+            | x8 ⇒ couple exadecimal bool x0 true
+            | x9 ⇒ couple exadecimal bool x1 true
+            | xA ⇒ couple exadecimal bool x2 true
+            | xB ⇒ couple exadecimal bool x3 true
+            | xC ⇒ couple exadecimal bool x4 true
+            | xD ⇒ couple exadecimal bool x5 true
+            | xE ⇒ couple exadecimal bool x6 true
+            | xF ⇒ couple exadecimal bool x7 true ] 
+       | x8 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x9 false
+            | x1 ⇒ couple exadecimal bool xA false
+            | x2 ⇒ couple exadecimal bool xB false
+            | x3 ⇒ couple exadecimal bool xC false
+            | x4 ⇒ couple exadecimal bool xD false
+            | x5 ⇒ couple exadecimal bool xE false
+            | x6 ⇒ couple exadecimal bool xF false
+            | x7 ⇒ couple exadecimal bool x0 true
+            | x8 ⇒ couple exadecimal bool x1 true
+            | x9 ⇒ couple exadecimal bool x2 true
+            | xA ⇒ couple exadecimal bool x3 true
+            | xB ⇒ couple exadecimal bool x4 true
+            | xC ⇒ couple exadecimal bool x5 true
+            | xD ⇒ couple exadecimal bool x6 true
+            | xE ⇒ couple exadecimal bool x7 true
+            | xF ⇒ couple exadecimal bool x8 true ] 
+       | x9 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xA false
+            | x1 ⇒ couple exadecimal bool xB false
+            | x2 ⇒ couple exadecimal bool xC false
+            | x3 ⇒ couple exadecimal bool xD false
+            | x4 ⇒ couple exadecimal bool xE false
+            | x5 ⇒ couple exadecimal bool xF false
+            | x6 ⇒ couple exadecimal bool x0 true
+            | x7 ⇒ couple exadecimal bool x1 true
+            | x8 ⇒ couple exadecimal bool x2 true
+            | x9 ⇒ couple exadecimal bool x3 true
+            | xA ⇒ couple exadecimal bool x4 true
+            | xB ⇒ couple exadecimal bool x5 true
+            | xC ⇒ couple exadecimal bool x6 true
+            | xD ⇒ couple exadecimal bool x7 true
+            | xE ⇒ couple exadecimal bool x8 true
+            | xF ⇒ couple exadecimal bool x9 true ] 
+       | xA ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xB false
+            | x1 ⇒ couple exadecimal bool xC false
+            | x2 ⇒ couple exadecimal bool xD false
+            | x3 ⇒ couple exadecimal bool xE false
+            | x4 ⇒ couple exadecimal bool xF false
+            | x5 ⇒ couple exadecimal bool x0 true
+            | x6 ⇒ couple exadecimal bool x1 true
+            | x7 ⇒ couple exadecimal bool x2 true
+            | x8 ⇒ couple exadecimal bool x3 true
+            | x9 ⇒ couple exadecimal bool x4 true
+            | xA ⇒ couple exadecimal bool x5 true
+            | xB ⇒ couple exadecimal bool x6 true
+            | xC ⇒ couple exadecimal bool x7 true
+            | xD ⇒ couple exadecimal bool x8 true
+            | xE ⇒ couple exadecimal bool x9 true
+            | xF ⇒ couple exadecimal bool xA true ] 
+       | xB ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xC false
+            | x1 ⇒ couple exadecimal bool xD false
+            | x2 ⇒ couple exadecimal bool xE false
+            | x3 ⇒ couple exadecimal bool xF false
+            | x4 ⇒ couple exadecimal bool x0 true
+            | x5 ⇒ couple exadecimal bool x1 true
+            | x6 ⇒ couple exadecimal bool x2 true
+            | x7 ⇒ couple exadecimal bool x3 true
+            | x8 ⇒ couple exadecimal bool x4 true
+            | x9 ⇒ couple exadecimal bool x5 true
+            | xA ⇒ couple exadecimal bool x6 true
+            | xB ⇒ couple exadecimal bool x7 true
+            | xC ⇒ couple exadecimal bool x8 true
+            | xD ⇒ couple exadecimal bool x9 true
+            | xE ⇒ couple exadecimal bool xA true
+            | xF ⇒ couple exadecimal bool xB true ] 
+       | xC ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xD false
+            | x1 ⇒ couple exadecimal bool xE false
+            | x2 ⇒ couple exadecimal bool xF false
+            | x3 ⇒ couple exadecimal bool x0 true
+            | x4 ⇒ couple exadecimal bool x1 true
+            | x5 ⇒ couple exadecimal bool x2 true
+            | x6 ⇒ couple exadecimal bool x3 true
+            | x7 ⇒ couple exadecimal bool x4 true
+            | x8 ⇒ couple exadecimal bool x5 true
+            | x9 ⇒ couple exadecimal bool x6 true
+            | xA ⇒ couple exadecimal bool x7 true
+            | xB ⇒ couple exadecimal bool x8 true
+            | xC ⇒ couple exadecimal bool x9 true
+            | xD ⇒ couple exadecimal bool xA true
+            | xE ⇒ couple exadecimal bool xB true
+            | xF ⇒ couple exadecimal bool xC true ] 
+       | xD ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xE false
+            | x1 ⇒ couple exadecimal bool xF false
+            | x2 ⇒ couple exadecimal bool x0 true
+            | x3 ⇒ couple exadecimal bool x1 true
+            | x4 ⇒ couple exadecimal bool x2 true
+            | x5 ⇒ couple exadecimal bool x3 true
+            | x6 ⇒ couple exadecimal bool x4 true
+            | x7 ⇒ couple exadecimal bool x5 true
+            | x8 ⇒ couple exadecimal bool x6 true
+            | x9 ⇒ couple exadecimal bool x7 true
+            | xA ⇒ couple exadecimal bool x8 true
+            | xB ⇒ couple exadecimal bool x9 true
+            | xC ⇒ couple exadecimal bool xA true
+            | xD ⇒ couple exadecimal bool xB true
+            | xE ⇒ couple exadecimal bool xC true
+            | xF ⇒ couple exadecimal bool xD true ] 
+       | xE ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xF false
+            | x1 ⇒ couple exadecimal bool x0 true
+            | x2 ⇒ couple exadecimal bool x1 true
+            | x3 ⇒ couple exadecimal bool x2 true
+            | x4 ⇒ couple exadecimal bool x3 true
+            | x5 ⇒ couple exadecimal bool x4 true
+            | x6 ⇒ couple exadecimal bool x5 true
+            | x7 ⇒ couple exadecimal bool x6 true
+            | x8 ⇒ couple exadecimal bool x7 true
+            | x9 ⇒ couple exadecimal bool x8 true
+            | xA ⇒ couple exadecimal bool x9 true
+            | xB ⇒ couple exadecimal bool xA true
+            | xC ⇒ couple exadecimal bool xB true
+            | xD ⇒ couple exadecimal bool xC true
+            | xE ⇒ couple exadecimal bool xD true
+            | xF ⇒ couple exadecimal bool xE true ]
+       | xF ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x0 true
+            | x1 ⇒ couple exadecimal bool x1 true
+            | x2 ⇒ couple exadecimal bool x2 true
+            | x3 ⇒ couple exadecimal bool x3 true
+            | x4 ⇒ couple exadecimal bool x4 true
+            | x5 ⇒ couple exadecimal bool x5 true
+            | x6 ⇒ couple exadecimal bool x6 true
+            | x7 ⇒ couple exadecimal bool x7 true
+            | x8 ⇒ couple exadecimal bool x8 true
+            | x9 ⇒ couple exadecimal bool x9 true
+            | xA ⇒ couple exadecimal bool xA true
+            | xB ⇒ couple exadecimal bool xB true
+            | xC ⇒ couple exadecimal bool xC true
+            | xD ⇒ couple exadecimal bool xD true
+            | xE ⇒ couple exadecimal bool xE true
+            | xF ⇒ couple exadecimal bool xF true ] 
+       ]
+   | false ⇒
+      match b1 with
+       [ x0 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x0 false
+            | x1 ⇒ couple exadecimal bool x1 false
+            | x2 ⇒ couple exadecimal bool x2 false
+            | x3 ⇒ couple exadecimal bool x3 false
+            | x4 ⇒ couple exadecimal bool x4 false
+            | x5 ⇒ couple exadecimal bool x5 false
+            | x6 ⇒ couple exadecimal bool x6 false
+            | x7 ⇒ couple exadecimal bool x7 false
+            | x8 ⇒ couple exadecimal bool x8 false
+            | x9 ⇒ couple exadecimal bool x9 false
+            | xA ⇒ couple exadecimal bool xA false
+            | xB ⇒ couple exadecimal bool xB false
+            | xC ⇒ couple exadecimal bool xC false
+            | xD ⇒ couple exadecimal bool xD false
+            | xE ⇒ couple exadecimal bool xE false
+            | xF ⇒ couple exadecimal bool xF false ] 
+       | x1 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x1 false
+            | x1 ⇒ couple exadecimal bool x2 false
+            | x2 ⇒ couple exadecimal bool x3 false
+            | x3 ⇒ couple exadecimal bool x4 false
+            | x4 ⇒ couple exadecimal bool x5 false
+            | x5 ⇒ couple exadecimal bool x6 false
+            | x6 ⇒ couple exadecimal bool x7 false
+            | x7 ⇒ couple exadecimal bool x8 false
+            | x8 ⇒ couple exadecimal bool x9 false
+            | x9 ⇒ couple exadecimal bool xA false
+            | xA ⇒ couple exadecimal bool xB false
+            | xB ⇒ couple exadecimal bool xC false
+            | xC ⇒ couple exadecimal bool xD false
+            | xD ⇒ couple exadecimal bool xE false
+            | xE ⇒ couple exadecimal bool xF false
+            | xF ⇒ couple exadecimal bool x0 true ] 
+       | x2 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x2 false
+            | x1 ⇒ couple exadecimal bool x3 false
+            | x2 ⇒ couple exadecimal bool x4 false
+            | x3 ⇒ couple exadecimal bool x5 false
+            | x4 ⇒ couple exadecimal bool x6 false
+            | x5 ⇒ couple exadecimal bool x7 false
+            | x6 ⇒ couple exadecimal bool x8 false
+            | x7 ⇒ couple exadecimal bool x9 false
+            | x8 ⇒ couple exadecimal bool xA false
+            | x9 ⇒ couple exadecimal bool xB false
+            | xA ⇒ couple exadecimal bool xC false
+            | xB ⇒ couple exadecimal bool xD false
+            | xC ⇒ couple exadecimal bool xE false
+            | xD ⇒ couple exadecimal bool xF false
+            | xE ⇒ couple exadecimal bool x0 true
+            | xF ⇒ couple exadecimal bool x1 true ] 
+       | x3 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x3 false
+            | x1 ⇒ couple exadecimal bool x4 false
+            | x2 ⇒ couple exadecimal bool x5 false
+            | x3 ⇒ couple exadecimal bool x6 false
+            | x4 ⇒ couple exadecimal bool x7 false
+            | x5 ⇒ couple exadecimal bool x8 false
+            | x6 ⇒ couple exadecimal bool x9 false
+            | x7 ⇒ couple exadecimal bool xA false
+            | x8 ⇒ couple exadecimal bool xB false
+            | x9 ⇒ couple exadecimal bool xC false
+            | xA ⇒ couple exadecimal bool xD false
+            | xB ⇒ couple exadecimal bool xE false
+            | xC ⇒ couple exadecimal bool xF false
+            | xD ⇒ couple exadecimal bool x0 true
+            | xE ⇒ couple exadecimal bool x1 true
+            | xF ⇒ couple exadecimal bool x2 true ] 
+       | x4 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x4 false
+            | x1 ⇒ couple exadecimal bool x5 false
+            | x2 ⇒ couple exadecimal bool x6 false
+            | x3 ⇒ couple exadecimal bool x7 false
+            | x4 ⇒ couple exadecimal bool x8 false
+            | x5 ⇒ couple exadecimal bool x9 false
+            | x6 ⇒ couple exadecimal bool xA false
+            | x7 ⇒ couple exadecimal bool xB false
+            | x8 ⇒ couple exadecimal bool xC false
+            | x9 ⇒ couple exadecimal bool xD false
+            | xA ⇒ couple exadecimal bool xE false
+            | xB ⇒ couple exadecimal bool xF false
+            | xC ⇒ couple exadecimal bool x0 true
+            | xD ⇒ couple exadecimal bool x1 true
+            | xE ⇒ couple exadecimal bool x2 true
+            | xF ⇒ couple exadecimal bool x3 true ] 
+       | x5 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x5 false
+            | x1 ⇒ couple exadecimal bool x6 false
+            | x2 ⇒ couple exadecimal bool x7 false
+            | x3 ⇒ couple exadecimal bool x8 false
+            | x4 ⇒ couple exadecimal bool x9 false
+            | x5 ⇒ couple exadecimal bool xA false
+            | x6 ⇒ couple exadecimal bool xB false
+            | x7 ⇒ couple exadecimal bool xC false
+            | x8 ⇒ couple exadecimal bool xD false
+            | x9 ⇒ couple exadecimal bool xE false
+            | xA ⇒ couple exadecimal bool xF false
+            | xB ⇒ couple exadecimal bool x0 true
+            | xC ⇒ couple exadecimal bool x1 true
+            | xD ⇒ couple exadecimal bool x2 true
+            | xE ⇒ couple exadecimal bool x3 true
+            | xF ⇒ couple exadecimal bool x4 true ] 
+       | x6 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x6 false
+            | x1 ⇒ couple exadecimal bool x7 false
+            | x2 ⇒ couple exadecimal bool x8 false
+            | x3 ⇒ couple exadecimal bool x9 false
+            | x4 ⇒ couple exadecimal bool xA false
+            | x5 ⇒ couple exadecimal bool xB false
+            | x6 ⇒ couple exadecimal bool xC false
+            | x7 ⇒ couple exadecimal bool xD false
+            | x8 ⇒ couple exadecimal bool xE false
+            | x9 ⇒ couple exadecimal bool xF false
+            | xA ⇒ couple exadecimal bool x0 true
+            | xB ⇒ couple exadecimal bool x1 true
+            | xC ⇒ couple exadecimal bool x2 true
+            | xD ⇒ couple exadecimal bool x3 true
+            | xE ⇒ couple exadecimal bool x4 true
+            | xF ⇒ couple exadecimal bool x5 true ] 
+       | x7 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x7 false
+            | x1 ⇒ couple exadecimal bool x8 false
+            | x2 ⇒ couple exadecimal bool x9 false
+            | x3 ⇒ couple exadecimal bool xA false
+            | x4 ⇒ couple exadecimal bool xB false
+            | x5 ⇒ couple exadecimal bool xC false
+            | x6 ⇒ couple exadecimal bool xD false
+            | x7 ⇒ couple exadecimal bool xE false
+            | x8 ⇒ couple exadecimal bool xF false
+            | x9 ⇒ couple exadecimal bool x0 true
+            | xA ⇒ couple exadecimal bool x1 true
+            | xB ⇒ couple exadecimal bool x2 true
+            | xC ⇒ couple exadecimal bool x3 true
+            | xD ⇒ couple exadecimal bool x4 true
+            | xE ⇒ couple exadecimal bool x5 true
+            | xF ⇒ couple exadecimal bool x6 true ] 
+       | x8 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x8 false
+            | x1 ⇒ couple exadecimal bool x9 false
+            | x2 ⇒ couple exadecimal bool xA false
+            | x3 ⇒ couple exadecimal bool xB false
+            | x4 ⇒ couple exadecimal bool xC false
+            | x5 ⇒ couple exadecimal bool xD false
+            | x6 ⇒ couple exadecimal bool xE false
+            | x7 ⇒ couple exadecimal bool xF false
+            | x8 ⇒ couple exadecimal bool x0 true
+            | x9 ⇒ couple exadecimal bool x1 true
+            | xA ⇒ couple exadecimal bool x2 true
+            | xB ⇒ couple exadecimal bool x3 true
+            | xC ⇒ couple exadecimal bool x4 true
+            | xD ⇒ couple exadecimal bool x5 true
+            | xE ⇒ couple exadecimal bool x6 true
+            | xF ⇒ couple exadecimal bool x7 true ] 
+       | x9 ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool x9 false
+            | x1 ⇒ couple exadecimal bool xA false
+            | x2 ⇒ couple exadecimal bool xB false
+            | x3 ⇒ couple exadecimal bool xC false
+            | x4 ⇒ couple exadecimal bool xD false
+            | x5 ⇒ couple exadecimal bool xE false
+            | x6 ⇒ couple exadecimal bool xF false
+            | x7 ⇒ couple exadecimal bool x0 true
+            | x8 ⇒ couple exadecimal bool x1 true
+            | x9 ⇒ couple exadecimal bool x2 true
+            | xA ⇒ couple exadecimal bool x3 true
+            | xB ⇒ couple exadecimal bool x4 true
+            | xC ⇒ couple exadecimal bool x5 true
+            | xD ⇒ couple exadecimal bool x6 true
+            | xE ⇒ couple exadecimal bool x7 true
+            | xF ⇒ couple exadecimal bool x8 true ] 
+       | xA ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xA false
+            | x1 ⇒ couple exadecimal bool xB false
+            | x2 ⇒ couple exadecimal bool xC false
+            | x3 ⇒ couple exadecimal bool xD false
+            | x4 ⇒ couple exadecimal bool xE false
+            | x5 ⇒ couple exadecimal bool xF false
+            | x6 ⇒ couple exadecimal bool x0 true
+            | x7 ⇒ couple exadecimal bool x1 true
+            | x8 ⇒ couple exadecimal bool x2 true
+            | x9 ⇒ couple exadecimal bool x3 true
+            | xA ⇒ couple exadecimal bool x4 true
+            | xB ⇒ couple exadecimal bool x5 true
+            | xC ⇒ couple exadecimal bool x6 true
+            | xD ⇒ couple exadecimal bool x7 true
+            | xE ⇒ couple exadecimal bool x8 true
+            | xF ⇒ couple exadecimal bool x9 true ] 
+       | xB ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xB false
+            | x1 ⇒ couple exadecimal bool xC false
+            | x2 ⇒ couple exadecimal bool xD false
+            | x3 ⇒ couple exadecimal bool xE false
+            | x4 ⇒ couple exadecimal bool xF false
+            | x5 ⇒ couple exadecimal bool x0 true
+            | x6 ⇒ couple exadecimal bool x1 true
+            | x7 ⇒ couple exadecimal bool x2 true
+            | x8 ⇒ couple exadecimal bool x3 true
+            | x9 ⇒ couple exadecimal bool x4 true
+            | xA ⇒ couple exadecimal bool x5 true
+            | xB ⇒ couple exadecimal bool x6 true
+            | xC ⇒ couple exadecimal bool x7 true
+            | xD ⇒ couple exadecimal bool x8 true
+            | xE ⇒ couple exadecimal bool x9 true
+            | xF ⇒ couple exadecimal bool xA true ] 
+       | xC ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xC false
+            | x1 ⇒ couple exadecimal bool xD false
+            | x2 ⇒ couple exadecimal bool xE false
+            | x3 ⇒ couple exadecimal bool xF false
+            | x4 ⇒ couple exadecimal bool x0 true
+            | x5 ⇒ couple exadecimal bool x1 true
+            | x6 ⇒ couple exadecimal bool x2 true
+            | x7 ⇒ couple exadecimal bool x3 true
+            | x8 ⇒ couple exadecimal bool x4 true
+            | x9 ⇒ couple exadecimal bool x5 true
+            | xA ⇒ couple exadecimal bool x6 true
+            | xB ⇒ couple exadecimal bool x7 true
+            | xC ⇒ couple exadecimal bool x8 true
+            | xD ⇒ couple exadecimal bool x9 true
+            | xE ⇒ couple exadecimal bool xA true
+            | xF ⇒ couple exadecimal bool xB true ] 
+       | xD ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xD false
+            | x1 ⇒ couple exadecimal bool xE false
+            | x2 ⇒ couple exadecimal bool xF false
+            | x3 ⇒ couple exadecimal bool x0 true
+            | x4 ⇒ couple exadecimal bool x1 true
+            | x5 ⇒ couple exadecimal bool x2 true
+            | x6 ⇒ couple exadecimal bool x3 true
+            | x7 ⇒ couple exadecimal bool x4 true
+            | x8 ⇒ couple exadecimal bool x5 true
+            | x9 ⇒ couple exadecimal bool x6 true
+            | xA ⇒ couple exadecimal bool x7 true
+            | xB ⇒ couple exadecimal bool x8 true
+            | xC ⇒ couple exadecimal bool x9 true
+            | xD ⇒ couple exadecimal bool xA true
+            | xE ⇒ couple exadecimal bool xB true
+            | xF ⇒ couple exadecimal bool xC true ] 
+       | xE ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xE false
+            | x1 ⇒ couple exadecimal bool xF false
+            | x2 ⇒ couple exadecimal bool x0 true
+            | x3 ⇒ couple exadecimal bool x1 true
+            | x4 ⇒ couple exadecimal bool x2 true
+            | x5 ⇒ couple exadecimal bool x3 true
+            | x6 ⇒ couple exadecimal bool x4 true
+            | x7 ⇒ couple exadecimal bool x5 true
+            | x8 ⇒ couple exadecimal bool x6 true
+            | x9 ⇒ couple exadecimal bool x7 true
+            | xA ⇒ couple exadecimal bool x8 true
+            | xB ⇒ couple exadecimal bool x9 true
+            | xC ⇒ couple exadecimal bool xA true
+            | xD ⇒ couple exadecimal bool xB true
+            | xE ⇒ couple exadecimal bool xC true
+            | xF ⇒ couple exadecimal bool xD true ] 
+       | xF ⇒
+           match b2 with
+            [ x0 ⇒ couple exadecimal bool xF false
+            | x1 ⇒ couple exadecimal bool x0 true
+            | x2 ⇒ couple exadecimal bool x1 true
+            | x3 ⇒ couple exadecimal bool x2 true
+            | x4 ⇒ couple exadecimal bool x3 true
+            | x5 ⇒ couple exadecimal bool x4 true
+            | x6 ⇒ couple exadecimal bool x5 true
+            | x7 ⇒ couple exadecimal bool x6 true
+            | x8 ⇒ couple exadecimal bool x7 true
+            | x9 ⇒ couple exadecimal bool x8 true
+            | xA ⇒ couple exadecimal bool x9 true
+            | xB ⇒ couple exadecimal bool xA true
+            | xC ⇒ couple exadecimal bool xB true
+            | xD ⇒ couple exadecimal bool xC true
+            | xE ⇒ couple exadecimal bool xD true
+            | xF ⇒ couple exadecimal bool xE true ]
+       ]
+   ]
+.
+
+definition plusbyte ≝
+ λb1,b2,c.
+  match plusex (bl b1) (bl b2) c with
+   [ couple l c' ⇒
+      match plusex (bh b1) (bh b2) c' with
+       [ couple h c'' ⇒ couple ? ? (mk_byte h l) c'' ]].
+
 alias num (instance 0) = "natural number".
 definition nat_of_exadecimal ≝
  λb.
@@ -159,12 +756,12 @@ definition nat_of_exadecimal ≝
    | x7 ⇒ 7
    | x8 ⇒ 8
    | x9 ⇒ 9
-   | x10 ⇒ 10
-   | x11 ⇒ 11
-   | x12 ⇒ 12
-   | x13 ⇒ 13
-   | x14 ⇒ 14
-   | x15 ⇒ 15
+   | xA ⇒ 10
+   | xB ⇒ 11
+   | xC ⇒ 12
+   | xD ⇒ 13
+   | xE ⇒ 14
+   | xF ⇒ 15
    ].
 
 coercion cic:/matita/assembly/nat_of_exadecimal.con.
@@ -173,8 +770,7 @@ definition nat_of_byte ≝ λb:byte. 16*(bh b) + (bl b).
 
 coercion cic:/matita/assembly/nat_of_byte.con.
 
-definition exadecimal_of_nat ≝
- λb.
+let rec exadecimal_of_nat b ≝
   match b with [ O ⇒ x0 | S b ⇒
   match b with [ O ⇒ x1 | S b ⇒
   match b with [ O ⇒ x2 | S b ⇒ 
@@ -190,11 +786,11 @@ definition exadecimal_of_nat ≝
   match b with [ O ⇒ xC | S b ⇒ 
   match b with [ O ⇒ xD | S b ⇒ 
   match b with [ O ⇒ xE | S b ⇒ 
-  match b with [ O ⇒ xF | S b ⇒ x0]]]]]]]]]]]]]]]]. 
+  match b with [ O ⇒ xF | S b ⇒ exadecimal_of_nat b ]]]]]]]]]]]]]]]]. 
 
 definition byte_of_nat ≝
- λn. mk_byte (exadecimal_of_nat ((n / 16) \mod 16)) (exadecimal_of_nat (n \mod 16)).
- 
+ λn. mk_byte (exadecimal_of_nat (n / 16)) (exadecimal_of_nat n).
+
 lemma byte_of_nat_nat_of_byte: ∀b. byte_of_nat (nat_of_byte b) = b.
  intros;
  elim b;
@@ -203,10 +799,211 @@ lemma byte_of_nat_nat_of_byte: ∀b. byte_of_nat (nat_of_byte b) = b.
  reflexivity.
 qed.
 
-lemma sign_ok: byte_of_nat 257 = mk_byte x0 x1.
+lemma lt_nat_of_exadecimal_16: ∀b. nat_of_exadecimal b < 16.
+ intro;
+ elim b;
+ simplify;
+ autobatch.
+qed.
+
+lemma lt_nat_of_byte_256: ∀b. nat_of_byte b < 256.
+ intro;
+ unfold nat_of_byte;
+ letin H ≝ (lt_nat_of_exadecimal_16 (bh b)); clearbody H;
+ letin K ≝ (lt_nat_of_exadecimal_16 (bl b)); clearbody K;
+ unfold lt in H K ⊢ %;
+ letin H' ≝ (le_S_S_to_le ? ? H); clearbody H'; clear H;
+ letin K' ≝ (le_S_S_to_le ? ? K); clearbody K'; clear K;
+ apply le_S_S;
+ cut (16*bh b ≤ 16*15);
+  [ letin Hf ≝ (le_plus ? ? ? ? Hcut K'); clearbody Hf;
+    simplify in Hf:(? ? %);
+    assumption
+  | autobatch
+  ]
+qed.
+
+lemma le_to_lt: ∀n,m. n ≤ m → n < S m.
+ intros;
+ autobatch.
+qed.
+
+axiom daemon: False.
+
+lemma exadecimal_of_nat_mod:
+ ∀n.exadecimal_of_nat n = exadecimal_of_nat (n \mod 16).
+ elim daemon.
+(*
+ intros;
+ cases n; [ reflexivity | ];
+ cases n1; [ reflexivity | ];
+ cases n2; [ reflexivity | ];
+ cases n3; [ reflexivity | ];
+ cases n4; [ reflexivity | ];
+ cases n5; [ reflexivity | ];
+ cases n6; [ reflexivity | ];  
+ cases n7; [ reflexivity | ];
+ cases n8; [ reflexivity | ];
+ cases n9; [ reflexivity | ];
+ cases n10; [ reflexivity | ];
+ cases n11; [ reflexivity | ];
+ cases n12; [ reflexivity | ];
+ cases n13; [ reflexivity | ];
+ cases n14; [ reflexivity | ];
+ cases n15; [ reflexivity | ];
+ change in ⊢ (? ? ? (? (? % ?))) with (16 + n16);
+ cut ((16 + n16) \mod 16 = n16 \mod 16);
+  [ rewrite > Hcut;
+    simplify in ⊢ (? ? % ?);
+    
+  | unfold mod;
+    change with (mod_aux (16+n16) (16+n16) 15 = n16);
+    unfold mod_aux;
+    change with
+     (match leb (16+n16) 15 with
+       [true ⇒ 16+n16
+       | false ⇒ mod_aux (15+n16) ((16+n16) - 16) 15
+       ] = n16);
+    cut (leb (16+n16) 15 = false);
+     [ rewrite > Hcut;
+       change with (mod_aux (15+n16) (16+n16-16) 15 = n16);
+       cut (16+n16-16 = n16);
+        [ rewrite > Hcut1; clear Hcut1;
+          
+        |
+        ]
+     |
+     ]
+  ]*)
+qed.
+
+(*lemma exadecimal_of_nat_elim:
+ ∀P:exadecimal → Prop.
+  (∀m. m < 16 → P (exadecimal_of_nat m)) →
+    ∀n. P (exadecimal_of_nat n).
+ intros;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1; 
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ cases n; [ apply H; autobatch | ]; clear n;
+ cases n1; [ apply H; autobatch | ]; clear n1;
+ simplify;
+ elim daemon.
+qed.
+*)
+      
+axiom nat_of_exadecimal_exadecimal_of_nat:
+ ∀n. nat_of_exadecimal (exadecimal_of_nat n) = n \mod 16.
+(*
+ intro;
+ apply (exadecimal_of_nat_elim (λn.;
+ 
+ 
+ 
+ elim n 0; [ reflexivity | intro ];
+ elim n1 0; [ intros; reflexivity | intros 2 ];
+ elim n2 0; [ intros; reflexivity | intros 2 ];
+ elim n3 0; [ intros; reflexivity | intros 2 ];
+ elim n4 0; [ intros; reflexivity | intros 2 ];
+ elim n5 0; [ intros; reflexivity | intros 2 ];
+ elim n6 0; [ intros; reflexivity | intros 2 ];
+ elim n7 0; [ intros; reflexivity | intros 2 ];
+ elim n8 0; [ intros; reflexivity | intros 2 ];
+ elim n9 0; [ intros; reflexivity | intros 2 ];
+ elim n10 0; [ intros; reflexivity | intros 2 ];
+ elim n11 0; [ intros; reflexivity | intros 2 ];
+ elim n12 0; [ intros; reflexivity | intros 2 ];
+ elim n13 0; [ intros; reflexivity | intros 2 ];
+ elim n14 0; [ intros; reflexivity | intros 2 ];
+ elim n15 0; [ intros; reflexivity | intros 2 ];
+ intro;
+ simplify;
+ rewrite < H15;
+ change in ⊢ (? ? % ?) with (nat_of_exadecimal (exadecimal_of_nat n16));
+qed.
+*)
+
+lemma nat_of_byte_byte_of_nat: ∀n. nat_of_byte (byte_of_nat n) = n \mod 256.
+ intro;
+ unfold byte_of_nat;
+ unfold nat_of_byte;
+ change with (16*(exadecimal_of_nat (n/16)) + exadecimal_of_nat n = n \mod 256);
+ rewrite > nat_of_exadecimal_exadecimal_of_nat in ⊢ (? ? (? (? ? %) ?) ?);
+ rewrite > nat_of_exadecimal_exadecimal_of_nat;
+ elim daemon.
+qed.
+
+definition nat_of_bool ≝
+ λb. match b with [ true ⇒ 1 | false ⇒ 0 ].
+
+lemma plusex_ok:
+ ∀b1,b2,c.
+  match plusex b1 b2 c with
+   [ couple r c' ⇒ b1 + b2 + nat_of_bool c = nat_of_exadecimal r + nat_of_bool c' * 16 ].
+ intros;
+ elim c;
+ elim b1;
+ elim b2;
+ normalize;
  reflexivity.
 qed.
-  
+
+lemma plusbyte_ok:
+ ∀b1,b2,c.
+  match plusbyte b1 b2 c with
+   [ couple r c' ⇒ b1 + b2 + nat_of_bool c = nat_of_byte r + nat_of_bool c' * 256
+   ].
+ intros;
+ unfold plusbyte;
+ generalize in match (plusex_ok (bl b1) (bl b2) c);
+ elim (plusex (bl b1) (bl b2) c);
+ simplify in H ⊢ %;
+ generalize in match (plusex_ok (bh b1) (bh b2) t1);
+ elim (plusex (bh b1) (bh b2) t1);
+ simplify in H1 ⊢ %;
+ change in ⊢ (? ? ? (? (? % ?) ?)) with (16 * t2);
+ unfold nat_of_byte;
+ letin K ≝ (eq_f ? ? (λy.16*y) ? ? H1); clearbody K; clear H1;
+ rewrite > distr_times_plus in K:(? ? ? %);
+ rewrite > symmetric_times in K:(? ? ? (? ? (? ? %)));
+ rewrite < associative_times in K:(? ? ? (? ? %));
+ normalize in K:(? ? ? (? ? (? % ?)));
+ rewrite > symmetric_times in K:(? ? ? (? ? %));
+ rewrite > sym_plus in ⊢ (? ? ? (? % ?));
+ rewrite > associative_plus in ⊢ (? ? ? %);
+ letin K' ≝ (eq_f ? ? (plus t) ? ? K); clearbody K'; clear K;
+  apply transitive_eq; [3: apply K' | skip | ];
+ clear K';
+ rewrite > sym_plus in ⊢ (? ? (? (? ? %) ?) ?);
+ rewrite > associative_plus in ⊢ (? ? (? % ?) ?);
+ rewrite > associative_plus in ⊢ (? ? % ?);
+ rewrite > associative_plus in ⊢ (? ? (? ? %) ?);
+ rewrite > associative_plus in ⊢ (? ? (? ? (? ? %)) ?);
+ rewrite > sym_plus in ⊢ (? ? (? ? (? ? (? ? %))) ?);
+ rewrite < associative_plus in ⊢ (? ? (? ? (? ? %)) ?);
+ rewrite < associative_plus in ⊢ (? ? (? ? %) ?);
+ rewrite < associative_plus in ⊢ (? ? (? ? (? % ?)) ?);
+ rewrite > H; clear H;
+ autobatch paramodulation.
+qed.
+
+(*
+lemma sign_ok: ∀ n:nat. nat_of_byte (byte_of_nat n) = n \mod 256.
+ intros; elim n; [ reflexivity | unfold byte_of_nat. 
+qed.
+*)
+
 definition addr ≝ nat.
 
 definition xpred ≝
@@ -236,7 +1033,7 @@ definition bpred ≝
    | false ⇒ mk_byte (bh b) (xpred (bl b))
    ]. 
 
-(* way too slow!
+(* Way too slow and subsumed by previous theorem
 lemma bpred_pred:
  ∀b.
   match eqbyte b (mk_byte x0 x0) with
@@ -246,7 +1043,6 @@ lemma bpred_pred:
  elim b;
  elim e;
  elim e1;
- whd;
  reflexivity.
 qed.
 *)
@@ -275,9 +1071,6 @@ let rec cycles_of_opcode op : nat ≝
   | STAd ⇒ 3
   ].
 
-inductive cartesian_product (A,B: Type) : Type ≝
- couple: ∀a:A.∀b:B. cartesian_product A B.
-
 definition opcodemap ≝
  [ couple ? ? ADDd (mk_byte xA xB);
    couple ? ? BEQ (mk_byte x3 x7);
@@ -331,25 +1124,6 @@ definition byte_of_opcode : opcode → byte ≝
   in
    aux opcodemap.
 
-notation "hvbox(# break a)"
-  non associative with precedence 80
-for @{ 'byte_of_opcode $a }.
-interpretation "byte_of_opcode" 'byte_of_opcode a =
- (cic:/matita/assembly/byte_of_opcode.con a).
-
-definition mult_source : list byte ≝
-  [#LDAi; mk_byte x0 x0;
-   #STAd; mk_byte x2 x0; (* 18 = locazione $12 *)
-   #LDAd; mk_byte x1 xF; (* 17 = locazione $11 *)
-   #BEQ;  mk_byte x0 xC;
-   #LDAd; mk_byte x1 x2;
-   #DECd; mk_byte x1 xF;
-   #ADDd; mk_byte x1 xE; (* 16 = locazione $10 *)
-   #STAd; mk_byte x2 x0;
-   #LDAd; mk_byte x1 xF;
-   #BRA;  mk_byte xF x2; (* 242 = -14 *)
-   #LDAd; mk_byte x2 x0].
-
 record status : Type ≝ {
   acc : byte;
   pc  : addr;
@@ -360,68 +1134,94 @@ record status : Type ≝ {
   clk : nat
 }.
 
-definition mult_status : status ≝
- mk_status (mk_byte x0 x0) 0 0 false false
-  (λa:addr. nth ? mult_source (mk_byte x0 x0) a) 0.
- 
 definition update ≝
  λf: addr → byte.λa.λv.λx.
   match eqb x a with
    [ true ⇒ v
    | false ⇒ f x ].
 
+lemma update_update_a_a:
+ ∀s,a,v1,v2,b.
+  update (update s a v1) a v2 b = update s a v2 b.
+ intros;
+ unfold update;
+ unfold update;
+ elim (eqb b a);
+ reflexivity.
+qed. 
+
+lemma update_update_a_b:
+ ∀s,a1,v1,a2,v2,b.
+  a1 ≠ a2 →
+   update (update s a1 v1) a2 v2 b = update (update s a2 v2) a1 v1 b.
+ intros;
+ unfold update;
+ unfold update;
+ apply (bool_elim ? (eqb b a1)); intros;
+ apply (bool_elim ? (eqb b a2)); intros;
+ simplify;
+ [ elim H;
+   rewrite < (eqb_true_to_eq ? ? H1);
+   apply eqb_true_to_eq;
+   assumption
+ |*: reflexivity
+ ].
+qed.
+
+definition mmod16 ≝ λn. nat_of_byte (byte_of_nat n).
+
 definition tick ≝
- λs:status.
-  (* fetch *)
-  let opc ≝ opcode_of_byte (mem s (pc s)) in
-  let op1 ≝ mem s (S (pc s)) in
+ λs:status. match s with [ mk_status acc pc spc zf cf mem clk ⇒
+  let opc ≝ opcode_of_byte (mem pc) in
+  let op1 ≝ mem (S pc) in
   let clk' ≝ cycles_of_opcode opc in
-  match eqb (S (clk s)) clk' with
+  match eqb (S clk) clk' with
    [ true ⇒
       match opc with
        [ ADDd ⇒
-          let x ≝ nat_of_byte (mem s op1) in
-          let acc' ≝ x + acc s in (* signed!!! *)
-           mk_status (byte_of_nat acc') (2 + pc s) (spc s)
-            (eqb O acc') (cf s) (mem s) 0
+          let res ≝ plusbyte acc (mem op1) false in (* verify carrier! *)
+          let acc' ≝ match res with [ couple acc' _ ⇒ acc' ] in
+          let c' ≝ match res with [ couple _ c' ⇒ c'] in
+           mk_status acc' (2 + pc) spc
+            (eqbyte (mk_byte x0 x0) acc') c' mem 0 (* verify carrier! *)
        | BEQ ⇒
           mk_status
-           (acc s)
-           (match zf s with
-             [ true ⇒ 2 + op1 + pc s   (* signed!!! *)
-             | false ⇒ 2 + pc s
+           acc
+           (match zf with
+             [ true ⇒ mmod16 (2 + op1 + pc) (*\mod 256*)   (* signed!!! *)
+             | false ⇒ 2 + pc
              ])
-           (spc s)
-           (zf s)
-           (cf s)
-           (mem s)
+           spc
+           zf
+           cf
+           mem
            0
        | BRA ⇒
           mk_status
-           (acc s) (2 + op1 + pc s) (* signed!!! *)
-           (spc s)
-           (zf s)
-           (cf s)
-           (mem s)
+           acc (mmod16 (2 + op1 + pc) (*\mod 256*)) (* signed!!! *)
+           spc
+           zf
+           cf
+           mem
            0
        | DECd ⇒
-          let x ≝ bpred (mem s op1) in (* signed!!! *)
-          let mem' ≝ update (mem s) op1 x in
-           mk_status (acc s) (2 + pc s) (spc s)
-            (eqb O x) (cf s) mem' 0 (* check zb!!! *)
+          let x ≝ bpred (mem op1) in (* signed!!! *)
+          let mem' ≝ update mem op1 x in
+           mk_status acc (2 + pc) spc
+            (eqbyte (mk_byte x0 x0) x) cf mem' 0 (* check zb!!! *)
        | LDAi ⇒
-          mk_status op1 (2 + pc s) (spc s) (eqb O op1) (cf s) (mem s) 0
+          mk_status op1 (2 + pc) spc (eqbyte (mk_byte x0 x0) op1) cf mem 0
        | LDAd ⇒
-          let x ≝ bpred (mem s op1) in
-           mk_status x (2 + pc s) (spc s) (eqb O x) (cf s) (mem s) 0
+          let x ≝ mem op1 in
+           mk_status x (2 + pc) spc (eqbyte (mk_byte x0 x0) x) cf mem 0
        | STAd ⇒
-          mk_status (acc s) (2 + pc s) (spc s) (zf s) (cf s)
-           (update (mem s) op1 (acc s)) 0
+          mk_status acc (2 + pc) spc zf cf
+           (update mem op1 acc) 0
        ]
    | false ⇒
        mk_status
-        (acc s) (pc s) (spc s) (zf s) (cf s) (mem s) (S (clk s))
-   ].
+        acc pc spc zf cf mem (S clk)
+   ]].
 
 let rec execute s n on n ≝
  match n with
@@ -429,41 +1229,500 @@ let rec execute s n on n ≝
   | S n' ⇒ execute (tick s) n'
   ].
   
-lemma foo: ∀s,n. execute s (S n) = execute (tick s) n.
- intros; reflexivity.
+lemma breakpoint:
+ ∀s,n1,n2. execute s (n1 + n2) = execute (execute s n1) n2.
+ intros;
+ generalize in match s; clear s;
+ elim n1;
+  [ reflexivity
+  | simplify;
+    apply H;
+  ]
 qed.
 
-lemma goo: True.
- letin s0 ≝ mult_status;
- letin pc0 ≝ (pc s0);
- 
- reduce in pc0;
- letin i0 ≝ (opcode_of_byte (mem s0 pc0));
- reduce in i0;
+notation "hvbox(# break a)"
+  non associative with precedence 80
+for @{ 'byte_of_opcode $a }.
+interpretation "byte_of_opcode" 'byte_of_opcode a =
+ (cic:/matita/assembly/byte_of_opcode.con a).
+
+definition mult_source : list byte ≝
+  [#LDAi; mk_byte x0 x0; (* A := 0 *)
+   #STAd; mk_byte x2 x0; (* Z := A *)
+   #LDAd; mk_byte x1 xF; (* (l1) A := Y *)
+   #BEQ;  mk_byte x0 xA; (* if A == 0 then goto l2 *)
+   #LDAd; mk_byte x2 x0; (* A := Z *)
+   #DECd; mk_byte x1 xF; (* Y := Y - 1 *)
+   #ADDd; mk_byte x1 xE; (* A += X *)
+   #STAd; mk_byte x2 x0; (* Z := A *)
+   #BRA;  mk_byte xF x2; (* goto l1 *)
+   #LDAd; mk_byte x2 x0].(* (l2) *)
+
+definition mult_memory ≝
+ λx,y.λa:addr.
+     match leb a 29 with
+      [ true ⇒ nth ? mult_source (mk_byte x0 x0) a
+      | false ⇒
+         match eqb a 30 with
+          [ true ⇒ x
+          | false ⇒ y
+          ]
+      ].
+
+definition mult_status ≝
+ λx,y.
+  mk_status (mk_byte x0 x0) 0 0 false false (mult_memory x y) 0.
+
+lemma plusbyte_O_x:
+ ∀b. plusbyte (mk_byte x0 x0) b false = couple ? ? b false.
+ intros;
+ elim b;
+ elim e;
+ elim e1;
+ reflexivity.
+qed.
+
+definition plusbytenc ≝
+ λx,y.
+  match plusbyte x y false with
+   [couple res _ ⇒ res].
+
+definition plusbytec ≝
+ λx,y.
+  match plusbyte x y false with
+   [couple _ c ⇒ c].
+
+lemma plusbytenc_O_x:
+ ∀x. plusbytenc (mk_byte x0 x0) x = x.
+ intros;
+ unfold plusbytenc;
+ rewrite > plusbyte_O_x;
+ reflexivity.
+qed.
+
+axiom mod_plus: ∀a,b,m. (a + b) \mod m = a \mod m + b \mod m.
+axiom eq_mod_times_n_m_m_O: ∀n,m. O < m → n * m \mod m = O.
+
+axiom eq_nat_of_byte_mod: ∀b. nat_of_byte b = nat_of_byte b \mod 256.
+
+theorem plusbytenc_ok:
+ ∀b1,b2:byte. nat_of_byte (plusbytenc b1 b2) = (b1 + b2) \mod 256.
+ intros;
+ unfold plusbytenc;
+ generalize in match (plusbyte_ok b1 b2 false);
+ elim (plusbyte b1 b2 false);
+ simplify in H ⊢ %;
+ change with (nat_of_byte t = (b1 + b2) \mod 256);
+ rewrite < plus_n_O in H;
+ rewrite > H; clear H;
+ rewrite > mod_plus;
+ letin K ≝ (eq_nat_of_byte_mod t); clearbody K;
+ letin K' ≝ (eq_mod_times_n_m_m_O (nat_of_bool t1) 256 ?); clearbody K';
+  [ autobatch | ];
+ autobatch paramodulation.
+qed.
+
+lemma test_O_O:
+  let i ≝ 14 in
+  let s ≝ execute (mult_status (mk_byte x0 x0) (mk_byte x0 x0)) i in
+   pc s = 20 ∧ mem s 32 = byte_of_nat 0.
+ normalize;
+ split;
+ reflexivity.
+qed.
+
+
+lemma test_0_2:
+  let x ≝ mk_byte x0 x0 in
+  let y ≝ mk_byte x0 x2 in
+  let i ≝ 14 + 23 * nat_of_byte y in
+  let s ≝ execute (mult_status x y) i in
+   pc s = 20 ∧ mem s 32 = plusbytenc x x.
+ intros;
+ split;
+ reflexivity.
+qed.
+
+lemma test_x_1:
+ ∀x.
+  let y ≝ mk_byte x0 x1 in
+  let i ≝ 14 + 23 * nat_of_byte y in
+  let s ≝ execute (mult_status x y) i in
+   pc s = 20 ∧ mem s 32 = x.
+ intros;
+ split;
+  [ reflexivity
+  | change in ⊢ (? ? % ?) with (plusbytenc (mk_byte x0 x0) x);
+    rewrite > plusbytenc_O_x;
+    reflexivity
+  ].
+qed.
+
+lemma test_x_2:
+ ∀x.
+  let y ≝ mk_byte x0 x2 in
+  let i ≝ 14 + 23 * nat_of_byte y in
+  let s ≝ execute (mult_status x y) i in
+   pc s = 20 ∧ mem s 32 = plusbytenc x x.
+ intros;
+ split;
+  [ reflexivity
+  | change in ⊢ (? ? % ?) with
+     (plusbytenc (plusbytenc (mk_byte x0 x0) x) x);
+    rewrite > plusbytenc_O_x;
+    reflexivity
+  ].
+qed.
+
+theorem lt_trans: ∀x,y,z. x < y → y < z → x < z.
+ unfold lt;
+ intros;
+ autobatch.
+qed.
+
+axiom status_eq:
+ ∀s,s'.
+  acc s = acc s' →
+  pc s = pc s' →
+  spc s = spc s' →
+  zf s = zf s' →
+  cf s = cf s' →
+  (∀a. mem s a = mem s' a) →
+  clk s = clk s' →
+   s=s'.
+
+lemma eq_eqex_S_x0_false:
+ ∀n. n < 15 → eqex x0 (exadecimal_of_nat (S n)) = false.
+ intro;
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ cases n1 0; [ intro; simplify; reflexivity | clear n1];
+ cases n 0; [ intro; simplify; reflexivity | clear n];
+ intro;
+ unfold lt in H;
+ cut (S n1 ≤ 0);
+  [ elim (not_le_Sn_O ? Hcut)
+  | do 15 (apply le_S_S_to_le);
+    assumption
+  ]
+qed.
+
+lemma leq_m_n_to_eq_div_n_m_S: ∀n,m:nat. 0 < m → m ≤ n → ∃z. n/m = S z.
+ intros;
+ unfold div;
+ apply (ex_intro ? ? (div_aux (pred n) (n-m) (pred m)));
+ cut (∃w.m = S w);
+  [ elim Hcut;
+    rewrite > H2;
+    rewrite > H2 in H1;
+    clear Hcut; clear H2; clear H; (*clear m;*)
+    simplify;
+    unfold in ⊢ (? ? % ?);
+    cut (∃z.n = S z);
+     [ elim Hcut; clear Hcut;
+       rewrite > H in H1;
+       rewrite > H; clear m;
+       change in ⊢ (? ? % ?)  with
+        (match leb (S a1) a with
+         [ true ⇒ O
+         | false ⇒ S (div_aux a1 ((S a1) - S a) a)]);
+       cut (S a1 ≰ a);
+        [ apply (leb_elim (S a1) a);
+           [ intro;
+             elim (Hcut H2)
+           | intro;
+             simplify;
+             reflexivity
+           ]
+        | intro;
+          autobatch
+        ]
+     | elim H1; autobatch
+     ]
+  | autobatch
+  ].
+qed.
+
+lemma eq_eqbyte_x0_x0_byte_of_nat_S_false:
+ ∀b. b < 255 → eqbyte (mk_byte x0 x0) (byte_of_nat (S b)) = false.
+ intros;
+ unfold byte_of_nat;
+ cut (b < 15 ∨ b ≥ 15);
+  [ elim Hcut;
+    [ unfold eqbyte;
+      change in ⊢ (? ? (? ? %) ?) with (eqex x0 (exadecimal_of_nat (S b))); 
+      rewrite > eq_eqex_S_x0_false;
+       [ alias id "andb_sym" = "cic:/matita/nat/propr_div_mod_lt_le_totient1_aux/andb_sym.con".
+         rewrite > andb_sym;
+         reflexivity
+       | assumption
+       ]
+    | unfold eqbyte;
+      change in ⊢ (? ? (? % ?) ?) with (eqex x0 (exadecimal_of_nat (S b/16)));
+      letin K ≝ (leq_m_n_to_eq_div_n_m_S (S b) 16 ? ?);
+       [ autobatch
+       | unfold in H1;
+         apply le_S_S;
+         assumption
+       | clearbody K;
+         elim K; clear K;
+         rewrite > H2;
+         rewrite > eq_eqex_S_x0_false;
+          [ reflexivity
+          | unfold lt;
+            unfold lt in H;
+            rewrite < H2;
+            clear H2; clear a; clear H1; clear Hcut;
+            elim daemon (* trivial arithmetic property over <= and div *)
+          ] 
+       ]
+    ]
+  | elim daemon
+  ].
+qed.
+
+lemma eq_bpred_S_a_a:
+ ∀a. a < 255 → bpred (byte_of_nat (S a)) = byte_of_nat a.
+elim daemon. (*
+ intros;
+ unfold byte_of_nat;
+ cut (a \mod 16 = 15 ∨ a \mod 16 < 15);
+  [ elim Hcut;
+     [ 
+     |
+     ]
+  | autobatch
+  ].*)
+qed.
  
- letin s1 ≝ (execute s0 (cycles_of_opcode i0));
- letin pc1 ≝ (pc s1);
- reduce in pc1;
- letin i1 ≝ (opcode_of_byte (mem s1 pc1));
- reduce in i1;
-
- letin s2 ≝ (execute s1 (cycles_of_opcode i1));
- letin pc2 ≝ (pc s2);
- reduce in pc2;
- letin i2 ≝ (opcode_of_byte (mem s2 pc2));
- reduce in i2;
-
- letin s3 ≝ (execute s2 (cycles_of_opcode i2));
- letin pc3 ≝ (pc s3);
- reduce in pc3;
- letin i3 ≝ (opcode_of_byte (mem s3 pc3));
- reduce in i3;
-
- letin s4 ≝ (execute s3 (cycles_of_opcode i3));
- letin pc4 ≝ (pc s4);
- reduce in pc4;
- letin i4 ≝ (opcode_of_byte (mem s4 pc4));
- reduce in i4;
+lemma plusbyteenc_S:
+ ∀x:byte.∀n.plusbytenc (byte_of_nat (x*n)) x = byte_of_nat (x * S n).
+ intros;
+ rewrite < byte_of_nat_nat_of_byte;
+ rewrite > (plusbytenc_ok (byte_of_nat (x*n)) x);
+ rewrite > na
+  
+(*CSC*)
+ intros;
+ unfold byte_of_nat;
+ unfold plusbytenc;
+ unfold plusbyte;
  
- exact I.
+ elim daemon.
+qed. 
+
+lemma eq_plusbytec_x0_x0_x_false:
+ ∀x.plusbytec (mk_byte x0 x0) x = false.
+ intro;
+ elim x;
+ elim e;
+ elim e1;
+ reflexivity.
+qed.
+
+lemma loop_invariant':
+ ∀x,y:byte.∀j:nat. j ≤ y →
+  execute (mult_status x y) (5 + 23*j)
+   =
+    mk_status (byte_of_nat (x * j)) 4 0 (eqbyte (mk_byte x0 x0) (byte_of_nat (x*j)))
+     (plusbytec (byte_of_nat (x*pred j)) x)
+     (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (y - j))) 32
+      (byte_of_nat (x * j)))
+     0.
+ intros 3;
+ elim j;
+  [ do 2 (rewrite < times_n_O);
+    apply status_eq;
+    [1,2,3,4,7: normalize; reflexivity
+    | rewrite > eq_plusbytec_x0_x0_x_false;
+      normalize;
+      reflexivity 
+    | intro;
+      elim daemon
+    ]
+  | cut (5 + 23 * S n = 5 + 23 * n + 23);
+    [ letin K ≝ (breakpoint (mult_status x y) (5 + 23 * n) 23); clearbody K;
+      letin H' ≝ (H ?); clearbody H'; clear H;
+      [ autobatch
+      | letin xxx ≝ (eq_f ? ? (λz. execute (mult_status x y) z) ? ? Hcut); clearbody xxx;
+        clear Hcut;
+        rewrite > xxx;
+        clear xxx;
+        apply (transitive_eq ? ? ? ? K);
+        clear K; 
+        rewrite > H';
+        clear H';
+        cut (∃z.y-n=S z ∧ z < 255);
+         [ elim Hcut; clear Hcut;
+           elim H; clear H;
+           rewrite > H2;
+           (* instruction LDAd *)
+           letin K ≝
+            (breakpoint
+              (mk_status (byte_of_nat (x*n)) 4 O
+               (eqbyte (mk_byte x0 x0) (byte_of_nat (x*n)))
+               (plusbytec (byte_of_nat (x*pred n)) x)
+               (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (S a))) 32
+               (byte_of_nat (x*n))) O)
+              3 20); clearbody K;
+           normalize in K:(? ? (? ? %) ?);
+           apply transitive_eq; [2: apply K | skip | ]; clear K;
+           whd in ⊢ (? ? (? % ?) ?);
+           normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
+           change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?)
+            with (byte_of_nat (S a));
+           change in ⊢ (? ? (? (? ? ? ? (? ? %) ? ? ?) ?) ?) with
+            (byte_of_nat (S a));
+           (* instruction BEQ *)
+           letin K ≝
+            (breakpoint
+              (mk_status (byte_of_nat (S a)) 6 O
+               (eqbyte (mk_byte x0 x0) (byte_of_nat (S a)))
+               (plusbytec (byte_of_nat (x*pred n)) x)
+                (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (S a))) 32
+                 (byte_of_nat (x*n))) O)
+              3 17); clearbody K;
+           normalize in K:(? ? (? ? %) ?);
+           apply transitive_eq; [2: apply K | skip | ]; clear K;
+           whd in ⊢ (? ? (? % ?) ?);
+           letin K ≝ (eq_eqbyte_x0_x0_byte_of_nat_S_false ? H3); clearbody K;
+           rewrite > K; clear K;
+           simplify in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
+           (* instruction LDAd *)
+           letin K ≝
+            (breakpoint
+              (mk_status (byte_of_nat (S a)) 8 O
+               (eqbyte (mk_byte x0 x0) (byte_of_nat (S a)))
+               (plusbytec (byte_of_nat (x*pred n)) x)
+                (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (S a))) 32
+                 (byte_of_nat (x*n))) O)
+              3 14); clearbody K;
+           normalize in K:(? ? (? ? %) ?);
+           apply transitive_eq; [2: apply K | skip | ]; clear K;
+           whd in ⊢ (? ? (? % ?) ?);
+           change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?) with (byte_of_nat (x*n));
+           normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
+           change in ⊢ (? ? (? (? ? ? ? % ? ? ?) ?) ?) with (eqbyte (mk_byte x0 x0) (byte_of_nat (x*n)));
+           (* instruction DECd *)
+           letin K ≝
+            (breakpoint
+             (mk_status (byte_of_nat (x*n)) 10 O
+              (eqbyte (mk_byte x0 x0) (byte_of_nat (x*n)))
+              (plusbytec (byte_of_nat (x*pred n)) x)
+               (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (S a))) 32
+                (byte_of_nat (x*n))) O)
+             5 9); clearbody K;
+           normalize in K:(? ? (? ? %) ?);
+           apply transitive_eq; [2: apply K | skip | ]; clear K;
+           whd in ⊢ (? ? (? % ?) ?);
+           change in ⊢ (? ? (? (? ? ? ? (? ? %) ? ? ?) ?) ?) with (bpred (byte_of_nat (S a)));
+           rewrite > (eq_bpred_S_a_a ? H3);
+           normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
+           normalize in ⊢ (? ? (? (? ? ? ? ? ? (? ? % ?) ?) ?) ?);
+           cut (y - S n = a);
+            [2: elim daemon | ];
+           rewrite < Hcut; clear Hcut; clear H3; clear H2; clear a;          
+           (* instruction ADDd *)
+           letin K ≝
+            (breakpoint
+             (mk_status (byte_of_nat (x*n)) 12
+              O (eqbyte (mk_byte x0 x0) (byte_of_nat (y-S n)))
+              (plusbytec (byte_of_nat (x*pred n)) x)
+              (update
+               (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (S (y-S n))))
+               32 (byte_of_nat (x*n))) 31
+               (byte_of_nat (y-S n))) O)
+             3 6); clearbody K;
+           normalize in K:(? ? (? ? %) ?);            
+           apply transitive_eq; [2: apply K | skip | ]; clear K;
+           whd in ⊢ (? ? (? % ?) ?);
+           change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?) with
+            (plusbytenc (byte_of_nat (x*n)) x);
+           change in ⊢ (? ? (? (? ? ? ? (? ? %) ? ? ?) ?) ?) with
+            (plusbytenc (byte_of_nat (x*n)) x);
+           normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
+           change in ⊢ (? ? (? (? ? ? ? ? % ? ?) ?) ?)
+            with (plusbytec (byte_of_nat (x*n)) x);
+           rewrite > plusbyteenc_S;
+           (* instruction STAd *)
+           letin K ≝
+            (breakpoint
+             (mk_status (byte_of_nat (x*S n)) 14 O
+              (eqbyte (mk_byte x0 x0) (byte_of_nat (x*S n)))
+              (plusbytec (byte_of_nat (x*n)) x)
+               (update
+                (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat (S (y-S n))))
+                32 (byte_of_nat (x*n))) 31
+                (byte_of_nat (y-S n))) O)
+            3 3); clearbody K;
+           normalize in K:(? ? (? ? %) ?);            
+           apply transitive_eq; [2: apply K | skip | ]; clear K;
+           whd in ⊢ (? ? (? % ?) ?);
+           normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
+           (* instruction BRA *)
+           whd in ⊢ (? ? % ?);
+           normalize in ⊢ (? ? (? ? % ? ? ? ? ?) ?);
+           rewrite < pred_Sn;        
+           apply status_eq;
+            [1,2,3,4,7: normalize; reflexivity
+            | change with (plusbytec (byte_of_nat (x*n)) x =
+                             plusbytec (byte_of_nat (x*n)) x);
+              reflexivity
+            |6: intro;
+              elim daemon
+            ]
+         | exists;
+            [ apply (y - S n)
+            | split;
+               [ rewrite < (minus_S_S y n);
+                 autobatch
+               | letin K ≝ (lt_nat_of_byte_256 y); clearbody K;
+                 letin K' ≝ (lt_minus_m y (S n) ? ?); clearbody K';
+                 autobatch
+               ]
+            ]
+         ]
+      ]
+    | rewrite > associative_plus;
+      autobatch paramodulation
+    ] 
+  ]   
+qed.
+
+theorem test_x_y:
+ ∀x,y:byte.
+  let i ≝ 14 + 23 * y in
+   execute (mult_status x y) i =
+    mk_status (byte_of_nat (x*y)) 20 0
+     (eqbyte (mk_byte x0 x0) (byte_of_nat (x*y)))
+     (plusbytec (byte_of_nat (x*pred y)) x)
+     (update
+       (update (mult_memory x y) 31 (mk_byte x0 x0))
+       32 (byte_of_nat (x*y)))
+     0.
+ intros;
+ cut (14 + 23 * y = 5 + 23*y + 9);
+  [2: autobatch paramodulation;
+  | rewrite > Hcut; (* clear Hcut; *)
+    rewrite > (breakpoint (mult_status x y) (5 + 23*y) 9);
+    rewrite > loop_invariant';
+     [2: apply le_n
+     | rewrite < minus_n_n;
+       apply status_eq;
+        [1,2,3,4,5,7: normalize; reflexivity
+        | elim daemon
+        ]
+     ]
+  ].
 qed.