set "baseuri" "cic:/matita/assembly/".
-include "nat/div_and_mod_new.ma".
-(*include "nat/compare.ma".*)
+include "nat/div_and_mod.ma".
include "list/list.ma".
inductive exadecimal : Type ≝
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.
| 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.
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 ⇒
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;
reflexivity.
qed.
-lemma sign_ok: byte_of_nat 257 = mk_byte x0 x1.
+definition nat_of_bool ≝
+ λb. match b with [ true ⇒ 1 | false ⇒ 0 ].
+
+(* Way too slow. Handles 2^32 goals!
+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'
+ ].
+ intros;
+ elim c;
+ elim b1;
+ elim e;
+ elim e1;
+ elim b2;
+ elim e2;
+ elim e3;
reflexivity.
qed.
-
+*)
+
+notation "14" non associative with precedence 80 for @{ 'x14 }.
+interpretation "natural number" 'x14 =
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)))))))))))))))).
+
+notation "22" non associative with precedence 80 for @{ 'x22 }.
+interpretation "natural number" 'x22 =
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)))))))))))))))))))))))).
+
+notation "256" non associative with precedence 80 for @{ 'x256 }.
+interpretation "natural number" 'x256 =
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
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+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))).
+
+(*
+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 ≝
| 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
elim b;
elim e;
elim e1;
- whd;
reflexivity.
qed.
*)
| 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);
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;
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 ].
+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
+ (eqb O (nat_of_byte 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
+ (eqb O 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 (eqb O 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 (eqb O 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
intros; reflexivity.
qed.
+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_status ≝
+ λx,y.
+ mk_status (mk_byte x0 x0) 0 0 false false
+ (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
+ ]
+ ]) 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].
+
+lemma plusbytenc_O_x:
+ ∀x. plusbytenc (mk_byte x0 x0) x = x.
+ intros;
+ unfold plusbytenc;
+ rewrite > plusbyte_O_x;
+ reflexivity.
+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.
+
+(*
+lemma test_x_y:
+ ∀x,y.
+ let i ≝ 14 + 23 * nat_of_byte y in
+ let s ≝ execute (mult_status x y) i in
+ pc s = 20 ∧ mem s 32 = byte_of_nat (nat_of_byte x * nat_of_byte y).
+ intros;
+
+qed.
+*)
+
+(*
+ letin w ≝ 22;
+ letin opc ≝ (let s ≝ execute (mult_status x y) w in opcode_of_byte (mem s (pc s))); whd in opc;
+ letin acc' ≝ (acc (execute (mult_status x y) w));
+ normalize in acc';
+ change in acc' with x;
+ letin z ≝ (let s ≝ (execute (mult_status x y) w) in mem s 32); whd in z;
+ letin x ≝ (let s ≝ (execute (mult_status x y) w) in mem s 30); whd in x;
+ (*letin xxx ≝ (byte_of_nat (x+y)); normalize in xxx;*)
+ split;
+ [ normalize; reflexivity
+ | change with (byte_of_nat x = x);
+ normalize;
+ split;
+ [ reflexivity
+ | change with (byte_of_nat (x + 0));
+ letin www ≝ (nat_of_byte (byte_of_nat 260)); whd in www;
+ letin xxx ≝ (260 \mod 256); reduce in xxx;
+ letin xxx ≝ ((18 + 242) \mod 256);
+ whd in xxx;
+ letin pc' ≝ (pc s);
+ normalize in pc';
+ letin opcode ≝ (let s ≝ s in opcode_of_byte (mem s (pc s)));
+ normalize in opcode;
+ csc.
+ split;
+ reduce in s;
+ reflexivity.
+qed.
+
+lemma goo1:
+ ∀x,y.
+ let i ≝ 14 + 23 * nat_of_byte y in
+ let s ≝ execute (mult_status x y) i in
+ pc s = 22 ∧ mem s 32 = byte_of_nat (nat_of_byte x * nat_of_byte y).
+ intros;
+qed.
+
lemma goo: True.
letin s0 ≝ mult_status;
- letin pc0 ≝ (pc s0);
-
+ letin pc0 ≝ (pc s0);
reduce in pc0;
letin i0 ≝ (opcode_of_byte (mem s0 pc0));
reduce in i0;
reduce in pc3;
letin i3 ≝ (opcode_of_byte (mem s3 pc3));
reduce in i3;
+ letin zf3 ≝ (zf s3);
+ reduce in zf3;
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;
+
+ letin s5 ≝ (execute s4 (cycles_of_opcode i4));
+ letin pc5 ≝ (pc s5);
+ reduce in pc5;
+ letin i5 ≝ (opcode_of_byte (mem s5 pc5));
+ reduce in i5;
+
+ letin s6 ≝ (execute s5 (cycles_of_opcode i5));
+ letin pc6 ≝ (pc s6);
+ reduce in pc6;
+ letin i6 ≝ (opcode_of_byte (mem s6 pc6));
+ reduce in i6;
+
+ letin s7 ≝ (execute s6 (cycles_of_opcode i6));
+ letin pc7 ≝ (pc s7);
+ reduce in pc7;
+ letin i7 ≝ (opcode_of_byte (mem s7 pc7));
+ reduce in i7;
+
+ letin s8 ≝ (execute s7 (cycles_of_opcode i7));
+ letin pc8 ≝ (pc s8);
+ reduce in pc8;
+ letin i8 ≝ (opcode_of_byte (mem s8 pc8));
+ reduce in i8;
+
+ letin s9 ≝ (execute s8 (cycles_of_opcode i8));
+ letin pc9 ≝ (pc s9);
+ reduce in pc9;
+ letin i9 ≝ (opcode_of_byte (mem s9 pc9));
+ reduce in i9;
exact I.
-qed.
\ No newline at end of file
+qed.
+*)
projects / helm.git / blobdiff