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).
+include "assembly/vm.ma".
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) *)
+ [#LDAi; 〈x0, x0〉; (* A := 0 *)
+ #STAd; 〈x2, x0〉; (* Z := A *)
+ #LDAd; 〈x1, xF〉; (* (l1) A := Y *)
+ #BEQ; 〈x0, xA〉; (* if A == 0 then goto l2 *)
+ #LDAd; 〈x2, x0〉; (* A := Z *)
+ #DECd; 〈x1, xF〉; (* Y := Y - 1 *)
+ #ADDd; 〈x1, xE〉; (* A += X *)
+ #STAd; 〈x2, x0〉; (* Z := A *)
+ #BRA; 〈xF, x2〉; (* goto l1 *)
+ #LDAd; 〈x2, x0〉].(* (l2) *)
definition mult_memory ≝
λx,y.λa:addr.
match leb a 29 with
- [ true ⇒ nth ? mult_source (mk_byte x0 x0) a
+ [ true ⇒ nth ? mult_source 〈x0, x0〉 a
| false ⇒
match eqb a 30 with
[ true ⇒ x
definition mult_status ≝
λx,y.
- mk_status (mk_byte x0 x0) 0 0 false false (mult_memory x y) 0.
+ mk_status 〈x0, x0〉 0 0 false false (mult_memory x y) 0.
+
+notation " 'M' \sub (x y)" non associative with precedence 80 for
+ @{ 'memory $x $y }.
+
+interpretation "mult_memory" 'memory x y =
+ (cic:/matita/assembly/test/mult_memory.con x y).
+
+notation " 'M' \sub (x y) \nbsp a" non associative with precedence 80 for
+ @{ 'memory4 $x $y $a }.
+
+interpretation "mult_memory4" 'memory4 x y a =
+ (cic:/matita/assembly/test/mult_memory.con x y a).
+
+notation " \Sigma \sub (x y)" non associative with precedence 80 for
+ @{ 'status $x $y }.
+
+interpretation "mult_status" 'status x y =
+ (cic:/matita/assembly/test/mult_status.con x y).
lemma test_O_O:
let i ≝ 14 in
- let s ≝ execute (mult_status (mk_byte x0 x0) (mk_byte x0 x0)) i in
+ let s ≝ execute (mult_status 〈x0, x0〉 〈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 x ≝ 〈x0, x0〉 in
+ let y ≝ 〈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.
lemma test_x_1:
∀x.
- let y ≝ mk_byte x0 x1 in
+ let y ≝ 〈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);
+ | change in ⊢ (? ? % ?) with (plusbytenc 〈x0, x0〉 x);
rewrite > plusbytenc_O_x;
reflexivity
].
lemma test_x_2:
∀x.
- let y ≝ mk_byte x0 x2 in
+ let y ≝ 〈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.
split;
[ reflexivity
| change in ⊢ (? ? % ?) with
- (plusbytenc (plusbytenc (mk_byte x0 x0) x) x);
+ (plusbytenc (plusbytenc 〈x0, x0〉 x) x);
rewrite > plusbytenc_O_x;
reflexivity
].
∀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)))
+ mk_status (byte_of_nat (x * j)) 4 0 (eqbyte 〈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)))
| intro;
rewrite < minus_n_O;
normalize in ⊢ (? ? (? (? ? %) ?) ?);
- whd in ⊢ (? ? % ?);
- change in ⊢ (? ? % ?) with (update (mult_memory x y) 32 (mk_byte x0 x0) a);
- change in ⊢ (? ? ? %) with (update (update (update (mult_memory x y) 30 x) 31
- (byte_of_nat y)) 32 (byte_of_nat 0) a);
+ change in ⊢ (? ? % ?) with (update (mult_memory x y) 32 〈x0, x0〉 a);
+ simplify in ⊢ (? ? ? %);
change in ⊢ (? ? ? (? (? (? ? ? %) ? ?) ? ? ?)) with (mult_memory x y 30);
rewrite > byte_of_nat_nat_of_byte;
change in ⊢ (? ? ? (? (? ? ? %) ? ? ?)) with (mult_memory x y 31);
apply inj_update;
intro;
rewrite > (eq_update_s_a_sa (update (mult_memory x y) 30 (mult_memory x y 30))
- 31 a) in ⊢ (? ? ? %);
+ 31 a);
rewrite > eq_update_s_a_sa;
reflexivity
]
| 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);
+ [ rewrite > Hcut; clear Hcut;
+ rewrite > breakpoint;
+ rewrite > H; clear H;
+ [2: apply le_S_S_to_le;
+ apply le_S;
+ apply H1
+ | 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;
+ change in ⊢ (? ? (? ? %) ?) with (3+20);
+ rewrite > breakpoint in ⊢ (? ? % ?);
whd in ⊢ (? ? (? % ?) ?);
normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?)
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;
+ change in ⊢ (? ? (? ? %) ?) with (3+17);
+ rewrite > breakpoint in ⊢ (? ? % ?);
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;
+ change in ⊢ (? ? (? ? %) ?) with (3+14);
+ rewrite > breakpoint in ⊢ (? ? % ?);
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)));
+ change in ⊢ (? ? (? (? ? ? ? % ? ? ?) ?) ?) with (eqbyte 〈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;
+ change in ⊢ (? ? (? ? %) ?) with (5+9);
+ rewrite > breakpoint in ⊢ (? ? % ?);
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 | ];
+ [2: rewrite > eq_minus_S_pred;
+ rewrite > H2;
+ reflexivity | ];
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;
+ change in ⊢ (? ? (? ? %) ?) with (3+6);
+ rewrite > breakpoint in ⊢ (? ? % ?);
whd in ⊢ (? ? (? % ?) ?);
change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?) with
(plusbytenc (byte_of_nat (x*n)) x);
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;
+ rewrite > (breakpoint ? 3 3);
whd in ⊢ (? ? (? % ?) ?);
normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
(* instruction BRA *)
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);
+ | change with (plusbytec #(x*n) x = plusbytec #(x*n) x);
reflexivity
|6: intro;
simplify in ⊢ (? ? ? %);
+ normalize in ⊢ (? ? (? (? ? ? ? ? ? (? ? (? %) ?) ?) ?) ?);
change in ⊢ (? ? % ?) with
- (update
- (update
- (update
- (update (update (mult_memory x y) 30 x) 31
- (byte_of_nat (S (nat_of_byte y-S n)))) 32 (byte_of_nat (nat_of_byte x*n))) 31
- (byte_of_nat (nat_of_byte y-S n)))
- (nat_of_byte
- (update
- (update
- (update (update (mult_memory x y) 30 x) 31
- (byte_of_nat (S (nat_of_byte y-S n)))) 32 (byte_of_nat (nat_of_byte x*n))) 31
- (byte_of_nat (nat_of_byte y-S n)) 15))
- (byte_of_nat (nat_of_byte x*S n)) a);
- normalize in ⊢ (? ? (? ? % ? ?) ?);
+ ((mult_memory x y){30↦x}{31↦#(S (y-S n))}{32↦#(x*n)}{31↦#(y-S n)}
+ {〈x2,x0〉↦ #(x*S n)} a);
apply inj_update;
intro;
apply inj_update;
intro;
- rewrite > not_eq_a_b_to_eq_update_a_b in ⊢ (? ? % ?); [2: apply H | ];
- rewrite > not_eq_a_b_to_eq_update_a_b in ⊢ (? ? % ?);
+ rewrite > not_eq_a_b_to_eq_update_a_b; [2: apply H | ];
+ rewrite > not_eq_a_b_to_eq_update_a_b;
[ reflexivity
| assumption
]
[ apply (y - S n)
| split;
[ rewrite < (minus_S_S y n);
- autobatch
+ apply (minus_Sn_m (nat_of_byte y) (S n) H1)
| letin K ≝ (lt_nat_of_byte_256 y); clearbody K;
letin K' ≝ (lt_minus_m y (S n) ? ?); clearbody K';
- autobatch
+ [ apply (lt_to_le_to_lt O (S n) (nat_of_byte y) ? ?);
+ autobatch
+ | autobatch
+ | autobatch
+ ]
]
]
]
]
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)))
+ mk_status (#(x*y)) 20 0
+ (eqbyte 〈x0, x0〉 (#(x*y)))
(plusbytec (byte_of_nat (x*pred y)) x)
(update
- (update (mult_memory x y) 31 (mk_byte x0 x0))
+ (update (mult_memory x y) 31 〈x0, x0〉)
32 (byte_of_nat (x*y)))
0.
intros;
| rewrite < minus_n_n;
apply status_eq;
[1,2,3,4,5,7: normalize; reflexivity
- | elim daemon
+ | intro;
+ simplify in ⊢ (? ? ? %);
+ change in ⊢ (? ? % ?) with
+ (update (update (update (mult_memory x y) 30 x) 31 (byte_of_nat O)) 32
+(byte_of_nat (nat_of_byte x*nat_of_byte y)) a);
+ repeat (apply inj_update; intro);
+ apply (eq_update_s_a_sa ? 30)
]
]
].
-qed.
+qed.
\ No newline at end of file
projects / helm.git / blobdiff