+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/assembly/".
-
-include "nat/div_and_mod.ma".
-include "list/list.ma".
-
-inductive exadecimal : Type ≝
- x0: exadecimal
- | x1: exadecimal
- | x2: exadecimal
- | x3: exadecimal
- | x4: exadecimal
- | x5: exadecimal
- | x6: exadecimal
- | x7: exadecimal
- | x8: exadecimal
- | x9: exadecimal
- | xA: exadecimal
- | xB: exadecimal
- | xC: exadecimal
- | xD: exadecimal
- | xE: exadecimal
- | xF: exadecimal.
-
-record byte : Type ≝ {
- bh: exadecimal;
- bl: exadecimal
-}.
-
-definition eqex ≝
- λb1,b2.
- match b1 with
- [ x0 ⇒
- match b2 with
- [ x0 ⇒ true | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x1 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ true | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x2 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ true | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x3 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ true
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x4 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ true | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x5 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ true | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x6 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ true | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x7 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ true
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x8 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ true | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | x9 ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ true | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | xA ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ true | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | xB ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ true
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | xC ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ true | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
- | xD ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ true | xE ⇒ false | xF ⇒ false ]
- | xE ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ true | xF ⇒ false ]
- | xF ⇒
- match b2 with
- [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
- | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
- | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
- | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ true ]].
-
-
-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.
- match b with
- [ x0 ⇒ 0
- | x1 ⇒ 1
- | x2 ⇒ 2
- | x3 ⇒ 3
- | x4 ⇒ 4
- | x5 ⇒ 5
- | x6 ⇒ 6
- | x7 ⇒ 7
- | x8 ⇒ 8
- | x9 ⇒ 9
- | xA ⇒ 10
- | xB ⇒ 11
- | xC ⇒ 12
- | xD ⇒ 13
- | xE ⇒ 14
- | xF ⇒ 15
- ].
-
-coercion cic:/matita/assembly/nat_of_exadecimal.con.
-
-definition nat_of_byte ≝ λb:byte. 16*(bh b) + (bl b).
-
-coercion cic:/matita/assembly/nat_of_byte.con.
-
-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 ⇒ x3 | S b ⇒
- match b with [ O ⇒ x4 | S b ⇒
- match b with [ O ⇒ x5 | S b ⇒
- match b with [ O ⇒ x6 | S b ⇒
- match b with [ O ⇒ x7 | S b ⇒
- match b with [ O ⇒ x8 | S b ⇒
- match b with [ O ⇒ x9 | S b ⇒
- match b with [ O ⇒ xA | S b ⇒
- match b with [ O ⇒ xB | 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 ⇒ exadecimal_of_nat b ]]]]]]]]]]]]]]]].
-
-definition byte_of_nat ≝
- 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;
- elim e;
- elim e1;
- reflexivity.
-qed.
-
-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 ≝
- λb.
- match b with
- [ x0 ⇒ xF
- | x1 ⇒ x0
- | x2 ⇒ x1
- | x3 ⇒ x2
- | x4 ⇒ x3
- | x5 ⇒ x4
- | x6 ⇒ x5
- | x7 ⇒ x6
- | x8 ⇒ x7
- | x9 ⇒ x8
- | xA ⇒ x9
- | xB ⇒ xA
- | xC ⇒ xB
- | xD ⇒ xC
- | xE ⇒ xD
- | xF ⇒ xE ].
-
-definition bpred ≝
- λb.
- match eqex (bl b) x0 with
- [ true ⇒ mk_byte (xpred (bh b)) (xpred (bl b))
- | false ⇒ mk_byte (bh b) (xpred (bl b))
- ].
-
-(* Way too slow and subsumed by previous theorem
-lemma bpred_pred:
- ∀b.
- match eqbyte b (mk_byte x0 x0) with
- [ true ⇒ nat_of_byte (bpred b) = mk_byte xF xF
- | false ⇒ nat_of_byte (bpred b) = pred (nat_of_byte b)].
- intros;
- elim b;
- elim e;
- elim e1;
- reflexivity.
-qed.
-*)
-
-definition addr_of_byte : byte → addr ≝ λb. nat_of_byte b.
-
-coercion cic:/matita/assembly/addr_of_byte.con.
-
-inductive opcode: Type ≝
- ADDd: opcode (* 3 clock, 171 *)
- | BEQ: opcode (* 3, 55 *)
- | BRA: opcode (* 3, 48 *)
- | DECd: opcode (* 5, 58 *)
- | LDAi: opcode (* 2, 166 *)
- | LDAd: opcode (* 3, 182 *)
- | STAd: opcode. (* 3, 183 *)
-
-let rec cycles_of_opcode op : nat ≝
- match op with
- [ ADDd ⇒ 3
- | BEQ ⇒ 3
- | BRA ⇒ 3
- | DECd ⇒ 5
- | LDAi ⇒ 2
- | LDAd ⇒ 3
- | STAd ⇒ 3
- ].
-
-definition opcodemap ≝
- [ couple ? ? ADDd (mk_byte xA xB);
- couple ? ? BEQ (mk_byte x3 x7);
- couple ? ? BRA (mk_byte x3 x0);
- couple ? ? DECd (mk_byte x3 xA);
- couple ? ? LDAi (mk_byte xA x6);
- couple ? ? LDAd (mk_byte xB x6);
- couple ? ? STAd (mk_byte xB x7) ].
-
-definition opcode_of_byte ≝
- λb.
- let rec aux l ≝
- match l with
- [ nil ⇒ ADDd
- | cons c tl ⇒
- match c with
- [ couple op n ⇒
- match eqbyte n b with
- [ true ⇒ op
- | false ⇒ aux tl
- ]]]
- in
- aux opcodemap.
-
-definition magic_of_opcode ≝
- λop1.
- match op1 with
- [ ADDd ⇒ 0
- | BEQ ⇒ 1
- | BRA ⇒ 2
- | DECd ⇒ 3
- | LDAi ⇒ 4
- | LDAd ⇒ 5
- | STAd ⇒ 6 ].
-
-definition opcodeeqb ≝
- λop1,op2. eqb (magic_of_opcode op1) (magic_of_opcode op2).
-
-definition byte_of_opcode : opcode → byte ≝
- λop.
- let rec aux l ≝
- match l with
- [ nil ⇒ mk_byte x0 x0
- | cons c tl ⇒
- match c with
- [ couple op' n ⇒
- match opcodeeqb op op' with
- [ true ⇒ n
- | false ⇒ aux tl
- ]]]
- in
- aux opcodemap.
-
-record status : Type ≝ {
- acc : byte;
- pc : addr;
- spc : addr;
- zf : bool;
- cf : bool;
- mem : addr → byte;
- clk : nat
-}.
-
-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. 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) clk' with
- [ true ⇒
- match opc with
- [ ADDd ⇒
- 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
- (match zf with
- [ true ⇒ mmod16 (2 + op1 + pc) (*\mod 256*) (* signed!!! *)
- (* FIXME: can't work - address truncated to 8 bits *)
- | false ⇒ 2 + pc
- ])
- spc
- zf
- cf
- mem
- 0
- | BRA ⇒
- mk_status
- acc (mmod16 (2 + op1 + pc) (*\mod 256*)) (* signed!!! *)
- (* FIXME: same as above *)
- spc
- zf
- cf
- mem
- 0
- | DECd ⇒
- 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) spc (eqbyte (mk_byte x0 x0) op1) cf mem 0
- | LDAd ⇒
- let x ≝ mem op1 in
- mk_status x (2 + pc) spc (eqbyte (mk_byte x0 x0) x) cf mem 0
- | STAd ⇒
- mk_status acc (2 + pc) spc zf cf
- (update mem op1 acc) 0
- ]
- | false ⇒
- mk_status
- acc pc spc zf cf mem (S clk)
- ]].
-
-let rec execute s n on n ≝
- match n with
- [ O ⇒ s
- | S n' ⇒ execute (tick s) n'
- ].
-
-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.
-
-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 mod_plus: \forall a1,a2,b1,b2,m.
- a1 \mod m = b1 \mod m \to
- a2 \mod m = b2 \mod m \to
- (a1 + a2) \mod m = (b1 + b2) \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;
- letin K ≝ (eq_nat_of_byte_mod t); clearbody K;
- rewrite > K in ⊢ (? ? % ?);
- letin K' ≝ (eq_mod_times_n_m_m_O (nat_of_bool t1) 256 ?); clearbody K';
- [ autobatch
- | cut (O = O \mod 256);
- [ rewrite > Hcut in K':(? ? ? %);
- rewrite > K in K:(? ? % ?);
- rewrite > (mod_plus ? ? ? ? ? K K') in ⊢ (? ? ? %);
- rewrite < plus_n_O;reflexivity
- |simplify;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.
-
-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;
- [ elim (eqex (bh (mk_byte x0 x0))
-(bh (mk_byte (exadecimal_of_nat (S b/16)) (exadecimal_of_nat (S b)))));simplify;
-(*
- 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.
-
-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;
-
- 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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/byte".
+
+include "exadecimal.ma".
+
+record byte : Type ≝ {
+ bh: exadecimal;
+ bl: exadecimal
+}.
+
+definition eqbyte ≝
+ λb,b'. eqex (bh b) (bh b') ∧ eqex (bl b) (bl b').
+
+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'' ]].
+
+definition nat_of_byte ≝ λb:byte. 16*(bh b) + (bl b).
+
+coercion cic:/matita/assembly/byte/nat_of_byte.con.
+
+definition byte_of_nat ≝
+ 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;
+ elim e;
+ elim e1;
+ reflexivity.
+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 nat_of_byte_byte_of_nat: ∀n. nat_of_byte (byte_of_nat n) = n \mod 256.
+ intro;
+ letin H ≝ (lt_nat_of_byte_256 (byte_of_nat n)); clearbody H;
+ rewrite < (lt_to_eq_mod ? ? H); clear H;
+ unfold byte_of_nat;
+ unfold nat_of_byte;
+ change with ((16*(exadecimal_of_nat (n/16)) + exadecimal_of_nat n) \mod 256 = n \mod 256);
+ letin H ≝ (div_mod n 16 ?); clearbody H; [ autobatch | ];
+ rewrite > symmetric_times in H;
+ rewrite > nat_of_exadecimal_exadecimal_of_nat in ⊢ (? ? (? (? % ?) ?) ?);
+ rewrite > nat_of_exadecimal_exadecimal_of_nat in ⊢ (? ? (? (? ? %) ?) ?);
+ rewrite > H in ⊢ (? ? ? (? % ?)); clear H;
+ rewrite > mod_plus in ⊢ (? ? % ?);
+ rewrite > mod_plus in ⊢ (? ? ? %);
+ apply eq_mod_to_eq_plus_mod;
+ rewrite < mod_mod in ⊢ (? ? ? %); [ | autobatch];
+ rewrite < mod_mod in ⊢ (? ? % ?); [ | autobatch];
+ rewrite < (eq_mod_times_times_mod ? ? 16 256) in ⊢ (? ? (? % ?) ?); [2: reflexivity | ];
+ rewrite < mod_mod in ⊢ (? ? % ?);
+ [ reflexivity
+ | autobatch
+ ].
+qed.
+
+axiom eq_nat_of_byte_n_nat_of_byte_mod_n_256:
+ ∀n. byte_of_nat n = byte_of_nat (n \mod 256).
+
+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.
+
+definition bpred ≝
+ λb.
+ match eqex (bl b) x0 with
+ [ true ⇒ mk_byte (xpred (bh b)) (xpred (bl b))
+ | false ⇒ mk_byte (bh b) (xpred (bl b))
+ ].
+
+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 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 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;
+ [ elim (eqex (bh (mk_byte x0 x0))
+(bh (mk_byte (exadecimal_of_nat (S b/16)) (exadecimal_of_nat (S b)))));simplify;
+(*
+ 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.
+
+lemma plusbytenc_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 < times_n_Sm;
+ rewrite > mod_plus;
+ rewrite < eq_nat_of_byte_mod in ⊢ (? ? (? (? (? ? %) ?)) ?);
+ rewrite > nat_of_byte_byte_of_nat;
+ rewrite < mod_mod in ⊢ (? ? (? (? (? % ?) ?)) ?);
+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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/exadecimal/".
+
+include "extra.ma".
+
+inductive exadecimal : Type ≝
+ x0: exadecimal
+ | x1: exadecimal
+ | x2: exadecimal
+ | x3: exadecimal
+ | x4: exadecimal
+ | x5: exadecimal
+ | x6: exadecimal
+ | x7: exadecimal
+ | x8: exadecimal
+ | x9: exadecimal
+ | xA: exadecimal
+ | xB: exadecimal
+ | xC: exadecimal
+ | xD: exadecimal
+ | xE: exadecimal
+ | xF: exadecimal.
+
+definition eqex ≝
+ λb1,b2.
+ match b1 with
+ [ x0 ⇒
+ match b2 with
+ [ x0 ⇒ true | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x1 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ true | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x2 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ true | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x3 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ true
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x4 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ true | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x5 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ true | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x6 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ true | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x7 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ true
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x8 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ true | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | x9 ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ true | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | xA ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ true | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | xB ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ true
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | xC ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ true | xD ⇒ false | xE ⇒ false | xF ⇒ false ]
+ | xD ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ true | xE ⇒ false | xF ⇒ false ]
+ | xE ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ true | xF ⇒ false ]
+ | xF ⇒
+ match b2 with
+ [ x0 ⇒ false | x1 ⇒ false | x2 ⇒ false | x3 ⇒ false
+ | x4 ⇒ false | x5 ⇒ false | x6 ⇒ false | x7 ⇒ false
+ | x8 ⇒ false | x9 ⇒ false | xA ⇒ false | xB ⇒ false
+ | xC ⇒ false | xD ⇒ false | xE ⇒ false | xF ⇒ true ]].
+
+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 nat_of_exadecimal ≝
+ λb.
+ match b with
+ [ x0 ⇒ 0
+ | x1 ⇒ 1
+ | x2 ⇒ 2
+ | x3 ⇒ 3
+ | x4 ⇒ 4
+ | x5 ⇒ 5
+ | x6 ⇒ 6
+ | x7 ⇒ 7
+ | x8 ⇒ 8
+ | x9 ⇒ 9
+ | xA ⇒ 10
+ | xB ⇒ 11
+ | xC ⇒ 12
+ | xD ⇒ 13
+ | xE ⇒ 14
+ | xF ⇒ 15
+ ].
+
+coercion cic:/matita/assembly/exadecimal/nat_of_exadecimal.con.
+
+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 ⇒ x3 | S b ⇒
+ match b with [ O ⇒ x4 | S b ⇒
+ match b with [ O ⇒ x5 | S b ⇒
+ match b with [ O ⇒ x6 | S b ⇒
+ match b with [ O ⇒ x7 | S b ⇒
+ match b with [ O ⇒ x8 | S b ⇒
+ match b with [ O ⇒ x9 | S b ⇒
+ match b with [ O ⇒ xA | S b ⇒
+ match b with [ O ⇒ xB | 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 ⇒ exadecimal_of_nat b ]]]]]]]]]]]]]]]].
+
+lemma lt_nat_of_exadecimal_16: ∀b. nat_of_exadecimal b < 16.
+ intro;
+ elim b;
+ simplify;
+ [
+ |*: alias id "lt_S_S" = "cic:/matita/algebra/finite_groups/lt_S_S.con".
+ repeat (apply lt_S_S)
+ ];
+ autobatch.
+qed.
+
+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.
+
+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 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.
+
+definition xpred ≝
+ λb.
+ match b with
+ [ x0 ⇒ xF
+ | x1 ⇒ x0
+ | x2 ⇒ x1
+ | x3 ⇒ x2
+ | x4 ⇒ x3
+ | x5 ⇒ x4
+ | x6 ⇒ x5
+ | x7 ⇒ x6
+ | x8 ⇒ x7
+ | x9 ⇒ x8
+ | xA ⇒ x9
+ | xB ⇒ xA
+ | xC ⇒ xB
+ | xD ⇒ xC
+ | xE ⇒ xD
+ | xF ⇒ xE ].
+
+(* Way too slow and subsumed by previous theorem
+lemma bpred_pred:
+ ∀b.
+ match eqbyte b (mk_byte x0 x0) with
+ [ true ⇒ nat_of_byte (bpred b) = mk_byte xF xF
+ | false ⇒ nat_of_byte (bpred b) = pred (nat_of_byte b)].
+ intros;
+ elim b;
+ elim e;
+ elim e1;
+ reflexivity.
+qed.
+*)
+
+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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/extra".
+
+include "nat/div_and_mod.ma".
+include "nat/primes.ma".
+include "list/list.ma".
+
+axiom mod_plus: ∀a,b,m. (a + b) \mod m = (a \mod m + b \mod m) \mod m.
+axiom mod_mod: ∀a,n,m. n∣m → a \mod n = a \mod n \mod m.
+axiom eq_mod_times_n_m_m_O: ∀n,m. O < m → n * m \mod m = O.
+axiom eq_mod_to_eq_plus_mod: ∀a,b,c,m. a \mod m = b \mod m → (a+c) \mod m = (b+c) \mod m.
+axiom eq_mod_times_times_mod: ∀a,b,n,m. m = a*n → (a*b) \mod m = a * (b \mod n).
+
+inductive cartesian_product (A,B: Type) : Type ≝
+ couple: ∀a:A.∀b:B. cartesian_product A B.
+
+lemma le_to_lt: ∀n,m. n ≤ m → n < S m.
+ intros;
+ autobatch.
+qed.
+
+alias num (instance 0) = "natural number".
+definition nat_of_bool ≝
+ λb. match b with [ true ⇒ 1 | false ⇒ 0 ].
+
+theorem lt_trans: ∀x,y,z. x < y → y < z → x < z.
+ unfold lt;
+ intros;
+ autobatch.
+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.
+
+axiom daemon: False.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/test/".
+
+include "vm.ma".
+
+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/vm/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 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 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 > plusbytenc_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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/vm/".
+
+include "byte.ma".
+
+definition addr ≝ nat.
+
+definition addr_of_byte : byte → addr ≝ λb. nat_of_byte b.
+
+coercion cic:/matita/assembly/vm/addr_of_byte.con.
+
+inductive opcode: Type ≝
+ ADDd: opcode (* 3 clock, 171 *)
+ | BEQ: opcode (* 3, 55 *)
+ | BRA: opcode (* 3, 48 *)
+ | DECd: opcode (* 5, 58 *)
+ | LDAi: opcode (* 2, 166 *)
+ | LDAd: opcode (* 3, 182 *)
+ | STAd: opcode. (* 3, 183 *)
+
+let rec cycles_of_opcode op : nat ≝
+ match op with
+ [ ADDd ⇒ 3
+ | BEQ ⇒ 3
+ | BRA ⇒ 3
+ | DECd ⇒ 5
+ | LDAi ⇒ 2
+ | LDAd ⇒ 3
+ | STAd ⇒ 3
+ ].
+
+definition opcodemap ≝
+ [ couple ? ? ADDd (mk_byte xA xB);
+ couple ? ? BEQ (mk_byte x3 x7);
+ couple ? ? BRA (mk_byte x3 x0);
+ couple ? ? DECd (mk_byte x3 xA);
+ couple ? ? LDAi (mk_byte xA x6);
+ couple ? ? LDAd (mk_byte xB x6);
+ couple ? ? STAd (mk_byte xB x7) ].
+
+definition opcode_of_byte ≝
+ λb.
+ let rec aux l ≝
+ match l with
+ [ nil ⇒ ADDd
+ | cons c tl ⇒
+ match c with
+ [ couple op n ⇒
+ match eqbyte n b with
+ [ true ⇒ op
+ | false ⇒ aux tl
+ ]]]
+ in
+ aux opcodemap.
+
+definition magic_of_opcode ≝
+ λop1.
+ match op1 with
+ [ ADDd ⇒ 0
+ | BEQ ⇒ 1
+ | BRA ⇒ 2
+ | DECd ⇒ 3
+ | LDAi ⇒ 4
+ | LDAd ⇒ 5
+ | STAd ⇒ 6 ].
+
+definition opcodeeqb ≝
+ λop1,op2. eqb (magic_of_opcode op1) (magic_of_opcode op2).
+
+definition byte_of_opcode : opcode → byte ≝
+ λop.
+ let rec aux l ≝
+ match l with
+ [ nil ⇒ mk_byte x0 x0
+ | cons c tl ⇒
+ match c with
+ [ couple op' n ⇒
+ match opcodeeqb op op' with
+ [ true ⇒ n
+ | false ⇒ aux tl
+ ]]]
+ in
+ aux opcodemap.
+
+record status : Type ≝ {
+ acc : byte;
+ pc : addr;
+ spc : addr;
+ zf : bool;
+ cf : bool;
+ mem : addr → byte;
+ clk : nat
+}.
+
+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. 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) clk' with
+ [ true ⇒
+ match opc with
+ [ ADDd ⇒
+ 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
+ (match zf with
+ [ true ⇒ mmod16 (2 + op1 + pc) (*\mod 256*) (* signed!!! *)
+ (* FIXME: can't work - address truncated to 8 bits *)
+ | false ⇒ 2 + pc
+ ])
+ spc
+ zf
+ cf
+ mem
+ 0
+ | BRA ⇒
+ mk_status
+ acc (mmod16 (2 + op1 + pc) (*\mod 256*)) (* signed!!! *)
+ (* FIXME: same as above *)
+ spc
+ zf
+ cf
+ mem
+ 0
+ | DECd ⇒
+ 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) spc (eqbyte (mk_byte x0 x0) op1) cf mem 0
+ | LDAd ⇒
+ let x ≝ mem op1 in
+ mk_status x (2 + pc) spc (eqbyte (mk_byte x0 x0) x) cf mem 0
+ | STAd ⇒
+ mk_status acc (2 + pc) spc zf cf
+ (update mem op1 acc) 0
+ ]
+ | false ⇒
+ mk_status
+ acc pc spc zf cf mem (S clk)
+ ]].
+
+let rec execute s n on n ≝
+ match n with
+ [ O ⇒ s
+ | S n' ⇒ execute (tick s) n'
+ ].
+
+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.
+
+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'.
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