0. less nice solution by Enrico reverted

[helm.git] / helm / software / matita / library / assembly / assembly.ma

diff --git a/helm/software/matita/library/assembly/assembly.ma b/helm/software/matita/library/assembly/assembly.ma
index 3497de4e32df2e29fd6f4f6635a88172375cdef1..578112e7d23145b1c9b9c4725cde852ad646482a 100644 (file)
--- a/helm/software/matita/library/assembly/assembly.ma
+++ b/helm/software/matita/library/assembly/assembly.ma
@@ -14,8 +14,7 @@
 
 set "baseuri" "cic:/matita/assembly/".
 
-include "nat/div_and_mod_new.ma".
-(*include "nat/compare.ma".*)
+include "nat/div_and_mod.ma".
 include "list/list.ma".
 
 inductive exadecimal : Type ≝
@@ -159,12 +158,12 @@ definition nat_of_exadecimal ≝
    | x7 ⇒ 7
    | x8 ⇒ 8
    | x9 ⇒ 9
-   | x10 ⇒ 10
-   | x11 ⇒ 11
-   | x12 ⇒ 12
-   | x13 ⇒ 13
-   | x14 ⇒ 14
-   | x15 ⇒ 15
+   | xA ⇒ 10
+   | xB ⇒ 11
+   | xC ⇒ 12
+   | xD ⇒ 13
+   | xE ⇒ 14
+   | xF ⇒ 15
    ].
 
 coercion cic:/matita/assembly/nat_of_exadecimal.con.
@@ -173,8 +172,7 @@ definition nat_of_byte ≝ λb:byte. 16*(bh b) + (bl b).
 
 coercion cic:/matita/assembly/nat_of_byte.con.
 
-definition exadecimal_of_nat ≝
- λb.
+let rec exadecimal_of_nat b ≝
   match b with [ O ⇒ x0 | S b ⇒
   match b with [ O ⇒ x1 | S b ⇒
   match b with [ O ⇒ x2 | S b ⇒ 
@@ -190,11 +188,11 @@ definition exadecimal_of_nat ≝
   match b with [ O ⇒ xC | S b ⇒ 
   match b with [ O ⇒ xD | S b ⇒ 
   match b with [ O ⇒ xE | S b ⇒ 
-  match b with [ O ⇒ xF | S b ⇒ x0]]]]]]]]]]]]]]]]. 
+  match b with [ O ⇒ xF | S b ⇒ exadecimal_of_nat b ]]]]]]]]]]]]]]]]. 
 
 definition byte_of_nat ≝
- λn. mk_byte (exadecimal_of_nat ((n / 16) \mod 16)) (exadecimal_of_nat (n \mod 16)).
- 
+ λn. mk_byte (exadecimal_of_nat (n / 16)) (exadecimal_of_nat n).
+
 lemma byte_of_nat_nat_of_byte: ∀b. byte_of_nat (nat_of_byte b) = b.
  intros;
  elim b;
@@ -203,10 +201,317 @@ lemma byte_of_nat_nat_of_byte: ∀b. byte_of_nat (nat_of_byte b) = b.
  reflexivity.
 qed.
 
-lemma sign_ok: byte_of_nat 257 = mk_byte x0 x1.
- reflexivity.
+notation "14" non associative with precedence 80 for @{ 'x14 }.
+interpretation "natural number" 'x14 = 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)))))))))))))))).
+
+notation "22" non associative with precedence 80 for @{ 'x22 }.
+interpretation "natural number" 'x22 = 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
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+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
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+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)))))))))))))))))))))))).
+ 
+notation "256" non associative with precedence 80 for @{ 'x256 }.
+interpretation "natural number" 'x256 = 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
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+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/2) 
+(cic:/matita/nat/nat/nat.ind#xpointer(1/1/1) 
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))).
+
+(*
+lemma sign_ok: ∀ n:nat. nat_of_byte (byte_of_nat n) = n \mod 256.
+ intros; elim n; [ reflexivity | unfold byte_of_nat. 
 qed.
-  
+*)
+
 definition addr ≝ nat.
 
 definition xpred ≝
@@ -236,7 +541,7 @@ definition bpred ≝
    | false ⇒ mk_byte (bh b) (xpred (bl b))
    ]. 
 
-(* way too slow!
+(* Way too slow and subsumed by previous theorem
 lemma bpred_pred:
  ∀b.
   match eqbyte b (mk_byte x0 x0) with
@@ -246,7 +551,6 @@ lemma bpred_pred:
  elim b;
  elim e;
  elim e1;
- whd;
  reflexivity.
 qed.
 *)
@@ -331,25 +635,6 @@ definition byte_of_opcode : opcode → byte ≝
   in
    aux opcodemap.
 
-notation "hvbox(# break a)"
-  non associative with precedence 80
-for @{ 'byte_of_opcode $a }.
-interpretation "byte_of_opcode" 'byte_of_opcode a =
- (cic:/matita/assembly/byte_of_opcode.con a).
-
-definition mult_source : list byte ≝
-  [#LDAi; mk_byte x0 x0;
-   #STAd; mk_byte x2 x0; (* 18 = locazione $12 *)
-   #LDAd; mk_byte x1 xF; (* 17 = locazione $11 *)
-   #BEQ;  mk_byte x0 xC;
-   #LDAd; mk_byte x1 x2;
-   #DECd; mk_byte x1 xF;
-   #ADDd; mk_byte x1 xE; (* 16 = locazione $10 *)
-   #STAd; mk_byte x2 x0;
-   #LDAd; mk_byte x1 xF;
-   #BRA;  mk_byte xF x2; (* 242 = -14 *)
-   #LDAd; mk_byte x2 x0].
-
 record status : Type ≝ {
   acc : byte;
   pc  : addr;
@@ -360,16 +645,14 @@ record status : Type ≝ {
   clk : nat
 }.
 
-definition mult_status : status ≝
- mk_status (mk_byte x0 x0) 0 0 false false
-  (λa:addr. nth ? mult_source (mk_byte x0 x0) a) 0.
- 
 definition update ≝
  λf: addr → byte.λa.λv.λx.
   match eqb x a with
    [ true ⇒ v
    | false ⇒ f x ].
 
+definition mmod16 ≝ λn. nat_of_byte (byte_of_nat n).
+
 definition tick ≝
  λs:status.
   (* fetch *)
@@ -381,14 +664,14 @@ definition tick ≝
       match opc with
        [ ADDd ⇒
           let x ≝ nat_of_byte (mem s op1) in
-          let acc' ≝ x + acc s in (* signed!!! *)
+          let acc' ≝ acc s + x in (* signed!!! *)
            mk_status (byte_of_nat acc') (2 + pc s) (spc s)
             (eqb O acc') (cf s) (mem s) 0
        | BEQ ⇒
           mk_status
            (acc s)
            (match zf s with
-             [ true ⇒ 2 + op1 + pc s   (* signed!!! *)
+             [ true ⇒ mmod16 (2 + op1 + pc s) (*\mod 256*)   (* signed!!! *)
              | false ⇒ 2 + pc s
              ])
            (spc s)
@@ -398,7 +681,7 @@ definition tick ≝
            0
        | BRA ⇒
           mk_status
-           (acc s) (2 + op1 + pc s) (* signed!!! *)
+           (acc s) (mmod16 (2 + op1 + pc s) (*\mod 256*)) (* signed!!! *)
            (spc s)
            (zf s)
            (cf s)
@@ -412,7 +695,7 @@ definition tick ≝
        | LDAi ⇒
           mk_status op1 (2 + pc s) (spc s) (eqb O op1) (cf s) (mem s) 0
        | LDAd ⇒
-          let x ≝ bpred (mem s op1) in
+          let x ≝ mem s op1 in
            mk_status x (2 + pc s) (spc s) (eqb O x) (cf s) (mem s) 0
        | STAd ⇒
           mk_status (acc s) (2 + pc s) (spc s) (zf s) (cf s)
@@ -433,10 +716,83 @@ lemma foo: ∀s,n. execute s (S n) = execute (tick s) n.
  intros; reflexivity.
 qed.
 
+notation "hvbox(# break a)"
+  non associative with precedence 80
+for @{ 'byte_of_opcode $a }.
+interpretation "byte_of_opcode" 'byte_of_opcode a =
+ (cic:/matita/assembly/byte_of_opcode.con a).
+
+definition mult_source : list byte ≝
+  [#LDAi; mk_byte x0 x0; (* A := 0 *)
+   #STAd; mk_byte x2 x0; (* Z := A *)
+   #LDAd; mk_byte x1 xF; (* (l1) A := Y *)
+   #BEQ;  mk_byte x0 xA; (* if A == 0 then goto l2 *)
+   #LDAd; mk_byte x2 x0; (* A := Z *)
+   #DECd; mk_byte x1 xF; (* Y := Y - 1 *)
+   #ADDd; mk_byte x1 xE; (* A += X *)
+   #STAd; mk_byte x2 x0; (* Z := A *)
+   #BRA;  mk_byte xF x2; (* goto l1 *)
+   #LDAd; mk_byte x2 x0].(* (l2) *)
+
+definition mult_status ≝
+ λx,y.
+  mk_status (mk_byte x0 x0) 0 0 false false
+   (λa:addr.
+     match leb a 29 with
+      [ true ⇒ nth ? mult_source (mk_byte x0 x0) a
+      | false ⇒
+         match eqb a 30 with
+          [ true ⇒ x
+          | false ⇒ y
+          ]
+      ]) 0.
+
+(*
+lemma foobar:
+ ∀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;
+ letin w ≝ 22;
+ letin opc ≝ (let s ≝ execute (mult_status x y) w in opcode_of_byte (mem s (pc s))); whd in opc;
+ letin acc' ≝ (acc (execute (mult_status x y) w)); change in acc' with (byte_of_nat x);
+ letin z ≝ (let s ≝ (execute (mult_status x y) w) in mem s 32); whd in z;
+ letin x ≝ (let s ≝ (execute (mult_status x y) w) in mem s 30); whd in x;
+ (*letin xxx ≝ (byte_of_nat (x+y)); normalize in xxx;*)
+ split;
+  [ normalize; reflexivity
+  | change with (byte_of_nat x = x);
+ normalize;
+ split;
+  [ reflexivity
+  | change with (byte_of_nat (x + 0));
+ letin www ≝ (nat_of_byte (byte_of_nat 260)); whd in www;
+ letin xxx ≝ (260 \mod 256); reduce in xxx;
+ letin xxx ≝ ((18 + 242) \mod 256);
+ whd in xxx;
+ letin pc' ≝ (pc s);
+ normalize in pc';
+ letin opcode ≝ (let s ≝ s in opcode_of_byte (mem s (pc s)));
+ normalize in opcode;
+ csc.
+ split;
+ reduce in s;
+ reflexivity.
+qed.
+
+lemma goo1:
+ ∀x,y.
+  let i ≝ 14 + 23 * nat_of_byte y in
+  let s ≝ execute (mult_status x y) i in
+   pc s = 22 ∧ mem s 32 = byte_of_nat (nat_of_byte x * nat_of_byte y).
+ intros;
+qed.
+
 lemma goo: True.
  letin s0 ≝ mult_status;
- letin pc0 ≝ (pc s0);
- 
+ letin pc0 ≝ (pc s0); 
  reduce in pc0;
  letin i0 ≝ (opcode_of_byte (mem s0 pc0));
  reduce in i0;
@@ -458,12 +814,45 @@ lemma goo: True.
  reduce in pc3;
  letin i3 ≝ (opcode_of_byte (mem s3 pc3));
  reduce in i3;
+ letin zf3 ≝ (zf s3);
+ reduce in zf3;
 
  letin s4 ≝ (execute s3 (cycles_of_opcode i3));
  letin pc4 ≝ (pc s4);
  reduce in pc4;
  letin i4 ≝ (opcode_of_byte (mem s4 pc4));
  reduce in i4;
+
+ letin s5 ≝ (execute s4 (cycles_of_opcode i4));
+ letin pc5 ≝ (pc s5);
+ reduce in pc5;
+ letin i5 ≝ (opcode_of_byte (mem s5 pc5));
+ reduce in i5;
+ 
+ letin s6 ≝ (execute s5 (cycles_of_opcode i5));
+ letin pc6 ≝ (pc s6);
+ reduce in pc6;
+ letin i6 ≝ (opcode_of_byte (mem s6 pc6));
+ reduce in i6;
+ 
+ letin s7 ≝ (execute s6 (cycles_of_opcode i6));
+ letin pc7 ≝ (pc s7);
+ reduce in pc7;
+ letin i7 ≝ (opcode_of_byte (mem s7 pc7));
+ reduce in i7;
+ 
+ letin s8 ≝ (execute s7 (cycles_of_opcode i7));
+ letin pc8 ≝ (pc s8);
+ reduce in pc8;
+ letin i8 ≝ (opcode_of_byte (mem s8 pc8));
+ reduce in i8;
+
+ letin s9 ≝ (execute s8 (cycles_of_opcode i8));
+ letin pc9 ≝ (pc s9);
+ reduce in pc9;
+ letin i9 ≝ (opcode_of_byte (mem s9 pc9));
+ reduce in i9;
  
  exact I.
-qed.
\ No newline at end of file
+qed.
+*)
\ No newline at end of file