rewrite < minus_n_O;
normalize in ⊢ (? ? (? (? ? %) ?) ?);
change in ⊢ (? ? % ?) with (update (mult_memory x y) 32 〈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);
+ 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);
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;
- [ apply le_S_S_to_le;
+ [ rewrite > Hcut; clear Hcut;
+ rewrite > breakpoint;
+ rewrite > H; clear H;
+ [2: apply le_S_S_to_le;
apply le_S;
apply H1
- | 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);
+ | 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 〈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:(? ? (? ? %) ?);
- rewrite > K; 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 〈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:(? ? (? ? %) ?);
- rewrite > K; 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 *)
- rewrite > (breakpoint ? 3 14);
+ change in ⊢ (? ? (? ? %) ?) with (3+14);
+ rewrite > breakpoint in ⊢ (? ? % ?);
whd in ⊢ (? ? (? % ?) ?);
change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?) with (byte_of_nat (x*n));
normalize in ⊢ (? ? (? (? ? % ? ? ? ? ?) ?) ?);
change in ⊢ (? ? (? (? ? ? ? % ? ? ?) ?) ?) with (eqbyte 〈x0, x0〉 (byte_of_nat (x*n)));
(* instruction DECd *)
- rewrite > (breakpoint ? 5 9);
+ 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);
reflexivity | ];
rewrite < Hcut; clear Hcut; clear H3; clear H2; clear a;
(* instruction ADDd *)
- rewrite > (breakpoint ? 3 6);
+ change in ⊢ (? ? (? ? %) ?) with (3+6);
+ rewrite > breakpoint in ⊢ (? ? % ?);
whd in ⊢ (? ? (? % ?) ?);
change in ⊢ (? ? (? (? % ? ? ? ? ? ?) ?) ?) with
(plusbytenc (byte_of_nat (x*n)) x);
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;
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
+ ]
]
]
]
apply status_eq;
[1,2,3,4,5,7: normalize; reflexivity
| intro;
- letin xxx \def ((mult_memory x y) { a ↦ x }).
- change 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 =
- update (update (mult_memory x y) 31 〈x0, x0〉) 32
+ 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);
- apply inj_update; intro;
- apply inj_update; intro;
- change in ⊢ (? ? (? ? ? % ?) ?) with (mult_memory x y 30);
- apply eq_update_s_a_sa
+ repeat (apply inj_update; intro);
+ apply (eq_update_s_a_sa ? 30)
]
]
].
-qed.
+qed.
\ No newline at end of file