(* WORD *)
(* **** *)
-nlemma word16_destruct_1 :
-∀x1,x2,y1,y2.
- mk_word16 x1 y1 = mk_word16 x2 y2 → x1 = x2.
- #x1; #x2; #y1; #y2; #H;
- nchange with (match mk_word16 x2 y2 with [ mk_word16 a _ ⇒ x1 = a ]);
- nrewrite < H;
- nnormalize;
- napply refl_eq.
-nqed.
+ndefinition word16_destruct_1 ≝ cn_destruct_1 word16.
+ndefinition word16_destruct_2 ≝ cn_destruct_2 word16.
-nlemma word16_destruct_2 :
-∀x1,x2,y1,y2.
- mk_word16 x1 y1 = mk_word16 x2 y2 → y1 = y2.
- #x1; #x2; #y1; #y2; #H;
- nchange with (match mk_word16 x2 y2 with [ mk_word16 _ b ⇒ y1 = b ]);
- nrewrite < H;
- nnormalize;
- napply refl_eq.
-nqed.
+ndefinition symmetric_eqw16 ≝ symmetric_eqcn ? eq_b8 symmetric_eqb8.
-nlemma symmetric_eqw16 : symmetricT word16 bool eq_w16.
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- nchange with (((eq_b8 b3 b1)⊗(eq_b8 b4 b2)) = ((eq_b8 b1 b3)⊗(eq_b8 b2 b4)));
- nrewrite > (symmetric_eqb8 b1 b3);
- nrewrite > (symmetric_eqb8 b2 b4);
- napply refl_eq.
-nqed.
+ndefinition eqw16_to_eq ≝ eqcn_to_eq ? eq_b8 eqb8_to_eq.
+ndefinition eq_to_eqw16 ≝ eq_to_eqcn ? eq_b8 eq_to_eqb8.
-nlemma symmetric_andw16 : symmetricT word16 word16 and_w16.
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- nchange with ((mk_word16 (and_b8 b3 b1) (and_b8 b4 b2)) = (mk_word16 (and_b8 b1 b3) (and_b8 b2 b4)));
- nrewrite > (symmetric_andb8 b1 b3);
- nrewrite > (symmetric_andb8 b2 b4);
- napply refl_eq.
-nqed.
+ndefinition decidable_w16 ≝ decidable_cn ? decidable_b8.
-nlemma associative_andw16 : ∀w1,w2,w3.(and_w16 (and_w16 w1 w2) w3) = (and_w16 w1 (and_w16 w2 w3)).
- #w1; #w2; #w3;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nelim w3;
- #b5; #b6;
- nchange with (mk_word16 (and_b8 (and_b8 b1 b3) b5) (and_b8 (and_b8 b2 b4) b6) =
- mk_word16 (and_b8 b1 (and_b8 b3 b5)) (and_b8 b2 (and_b8 b4 b6)));
- nrewrite < (associative_andb8 b1 b3 b5);
- nrewrite < (associative_andb8 b2 b4 b6);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_orw16 : symmetricT word16 word16 or_w16.
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- nchange with ((mk_word16 (or_b8 b3 b1) (or_b8 b4 b2)) = (mk_word16 (or_b8 b1 b3) (or_b8 b2 b4)));
- nrewrite > (symmetric_orb8 b1 b3);
- nrewrite > (symmetric_orb8 b2 b4);
- napply refl_eq.
-nqed.
-
-nlemma associative_orw16 : ∀w1,w2,w3.(or_w16 (or_w16 w1 w2) w3) = (or_w16 w1 (or_w16 w2 w3)).
- #w1; #w2; #w3;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nelim w3;
- #b5; #b6;
- nchange with (mk_word16 (or_b8 (or_b8 b1 b3) b5) (or_b8 (or_b8 b2 b4) b6) =
- mk_word16 (or_b8 b1 (or_b8 b3 b5)) (or_b8 b2 (or_b8 b4 b6)));
- nrewrite < (associative_orb8 b1 b3 b5);
- nrewrite < (associative_orb8 b2 b4 b6);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_xorw16 : symmetricT word16 word16 xor_w16.
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- nchange with ((mk_word16 (xor_b8 b3 b1) (xor_b8 b4 b2)) = (mk_word16 (xor_b8 b1 b3) (xor_b8 b2 b4)));
- nrewrite > (symmetric_xorb8 b1 b3);
- nrewrite > (symmetric_xorb8 b2 b4);
- napply refl_eq.
-nqed.
-
-nlemma associative_xorw16 : ∀w1,w2,w3.(xor_w16 (xor_w16 w1 w2) w3) = (xor_w16 w1 (xor_w16 w2 w3)).
- #w1; #w2; #w3;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nelim w3;
- #b5; #b6;
- nchange with (mk_word16 (xor_b8 (xor_b8 b1 b3) b5) (xor_b8 (xor_b8 b2 b4) b6) =
- mk_word16 (xor_b8 b1 (xor_b8 b3 b5)) (xor_b8 b2 (xor_b8 b4 b6)));
- nrewrite < (associative_xorb8 b1 b3 b5);
- nrewrite < (associative_xorb8 b2 b4 b6);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_plusw16_dc_dc : ∀w1,w2,c.plus_w16_dc_dc w1 w2 c = plus_w16_dc_dc w2 w1 c.
- #w1; #w2; #c;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nchange with (
- match plus_b8_dc_dc b2 b4 c with [ pair l c ⇒ match plus_b8_dc_dc b1 b3 c with [ pair h c' ⇒ pair … 〈h:l〉 c' ]] =
- match plus_b8_dc_dc b4 b2 c with [ pair l c ⇒ match plus_b8_dc_dc b3 b1 c with [ pair h c' ⇒ pair … 〈h:l〉 c' ]]);
- nrewrite > (symmetric_plusb8_dc_dc b4 b2 c);
- ncases (plus_b8_dc_dc b2 b4 c);
- #b5; #c1;
- nchange with (
- match plus_b8_dc_dc b1 b3 c1 with [ pair h c' ⇒ pair … 〈h:b5〉 c' ] =
- match plus_b8_dc_dc b3 b1 c1 with [ pair h c' ⇒ pair … 〈h:b5〉 c' ]);
- nrewrite > (symmetric_plusb8_dc_dc b1 b3 c1);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_plusw16_dc_d : ∀w1,w2,c.plus_w16_dc_d w1 w2 c = plus_w16_dc_d w2 w1 c.
- #w1; #w2; #c;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nchange with (
- match plus_b8_dc_dc b2 b4 c with [ pair l c ⇒ 〈plus_b8_dc_d b1 b3 c:l〉 ] =
- match plus_b8_dc_dc b4 b2 c with [ pair l c ⇒ 〈plus_b8_dc_d b3 b1 c:l〉 ]);
- nrewrite > (symmetric_plusb8_dc_dc b4 b2 c);
- ncases (plus_b8_dc_dc b2 b4 c);
- #b5; #c1;
- nchange with (〈plus_b8_dc_d b1 b3 c1:b5〉 = 〈plus_b8_dc_d b3 b1 c1:b5〉);
- nrewrite > (symmetric_plusb8_dc_d b1 b3 c1);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_plusw16_dc_c : ∀w1,w2,c.plus_w16_dc_c w1 w2 c = plus_w16_dc_c w2 w1 c.
- #w1; #w2; #c;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nchange with (
- plus_b8_dc_c b1 b3 (plus_b8_dc_c b2 b4 c) =
- plus_b8_dc_c b3 b1 (plus_b8_dc_c b4 b2 c));
- nrewrite > (symmetric_plusb8_dc_c b4 b2 c);
- nrewrite > (symmetric_plusb8_dc_c b3 b1 (plus_b8_dc_c b2 b4 c));
- napply refl_eq.
-nqed.
-
-nlemma symmetric_plusw16_d_dc : ∀w1,w2.plus_w16_d_dc w1 w2 = plus_w16_d_dc w2 w1.
- #w1; #w2;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nchange with (
- match plus_b8_d_dc b2 b4 with [ pair l c ⇒ match plus_b8_dc_dc b1 b3 c with [ pair h c' ⇒ pair … 〈h:l〉 c' ]] =
- match plus_b8_d_dc b4 b2 with [ pair l c ⇒ match plus_b8_dc_dc b3 b1 c with [ pair h c' ⇒ pair … 〈h:l〉 c' ]]);
- nrewrite > (symmetric_plusb8_d_dc b4 b2);
- ncases (plus_b8_d_dc b2 b4);
- #b5; #c;
- nchange with (
- match plus_b8_dc_dc b1 b3 c with [ pair h c' ⇒ pair … 〈h:b5〉 c' ] =
- match plus_b8_dc_dc b3 b1 c with [ pair h c' ⇒ pair … 〈h:b5〉 c' ]);
- nrewrite > (symmetric_plusb8_dc_dc b1 b3 c);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_plusw16_d_d : ∀w1,w2.plus_w16_d_d w1 w2 = plus_w16_d_d w2 w1.
- #w1; #w2;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nchange with (
- match plus_b8_d_dc b2 b4 with [ pair l c ⇒ 〈plus_b8_dc_d b1 b3 c:l〉 ] =
- match plus_b8_d_dc b4 b2 with [ pair l c ⇒ 〈plus_b8_dc_d b3 b1 c:l〉 ]);
- nrewrite > (symmetric_plusb8_d_dc b4 b2);
- ncases (plus_b8_d_dc b2 b4);
- #b5; #c;
- nchange with (〈plus_b8_dc_d b1 b3 c:b5〉 = 〈plus_b8_dc_d b3 b1 c:b5〉);
- nrewrite > (symmetric_plusb8_dc_d b1 b3 c);
- napply refl_eq.
-nqed.
-
-nlemma symmetric_plusw16_d_c : ∀w1,w2.plus_w16_d_c w1 w2 = plus_w16_d_c w2 w1.
- #w1; #w2;
- nelim w1;
- #b1; #b2;
- nelim w2;
- #b3; #b4;
- nchange with (
- plus_b8_dc_c b1 b3 (plus_b8_d_c b2 b4) =
- plus_b8_dc_c b3 b1 (plus_b8_d_c b4 b2));
- nrewrite > (symmetric_plusb8_d_c b4 b2);
- nrewrite > (symmetric_plusb8_dc_c b3 b1 (plus_b8_d_c b2 b4));
- napply refl_eq.
-nqed.
-
-nlemma symmetric_mulb8 : symmetricT byte8 word16 mul_b8.
- #b1; #b2;
- nelim b1;
- #e1; #e2;
- nelim b2;
- #e3; #e4;
- nchange with (match mul_ex e2 e4 with
- [ mk_byte8 t1_h t1_l ⇒ match mul_ex e1 e4 with
- [ mk_byte8 t2_h t2_l ⇒ match mul_ex e3 e2 with
- [ mk_byte8 t3_h t3_l ⇒ match mul_ex e1 e3 with
- [ mk_byte8 t4_h t4_l ⇒ plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,t3_h〉:〈t3_l,x0〉〉 〈〈x0,t2_h〉:〈t2_l,x0〉〉) 〈〈t4_h,t4_l〉:〈x0,x0〉〉)〈〈x0,x0〉:〈t1_h,t1_l〉〉
- ]]]] = match mul_ex e4 e2 with
- [ mk_byte8 t1_h t1_l ⇒ match mul_ex e3 e2 with
- [ mk_byte8 t2_h t2_l ⇒ match mul_ex e1 e4 with
- [ mk_byte8 t3_h t3_l ⇒ match mul_ex e3 e1 with
- [ mk_byte8 t4_h t4_l ⇒ plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,t3_h〉:〈t3_l,x0〉〉 〈〈x0,t2_h〉:〈t2_l,x0〉〉) 〈〈t4_h,t4_l〉:〈x0,x0〉〉)〈〈x0,x0〉:〈t1_h,t1_l〉〉
- ]]]]);
- nrewrite < (symmetric_mulex e2 e4);
- ncases (mul_ex e2 e4);
- #e5; #e6;
- nchange with (match mul_ex e1 e4 with
- [ mk_byte8 t2_h t2_l ⇒ match mul_ex e3 e2 with
- [ mk_byte8 t3_h t3_l ⇒ match mul_ex e1 e3 with
- [ mk_byte8 t4_h t4_l ⇒ plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,t3_h〉:〈t3_l,x0〉〉 〈〈x0,t2_h〉:〈t2_l,x0〉〉) 〈〈t4_h,t4_l〉:〈x0,x0〉〉)〈〈x0,x0〉:〈e5,e6〉〉
- ]]] = match mul_ex e3 e2 with
- [ mk_byte8 t2_h t2_l ⇒ match mul_ex e1 e4 with
- [ mk_byte8 t3_h t3_l ⇒ match mul_ex e3 e1 with
- [ mk_byte8 t4_h t4_l ⇒ plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,t3_h〉:〈t3_l,x0〉〉 〈〈x0,t2_h〉:〈t2_l,x0〉〉) 〈〈t4_h,t4_l〉:〈x0,x0〉〉)〈〈x0,x0〉:〈e5,e6〉〉
- ]]]);
- ncases (mul_ex e3 e2);
- #e7; #e8;
- ncases (mul_ex e1 e4);
- #e9; #e10;
- nchange with (match mul_ex e1 e3 with
- [ mk_byte8 t4_h t4_l ⇒ plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,e7〉:〈e8,x0〉〉 〈〈x0,e9〉:〈e10,x0〉〉) 〈〈t4_h,t4_l〉:〈x0,x0〉〉)〈〈x0,x0〉:〈e5,e6〉〉
- ] = match mul_ex e3 e1 with
- [ mk_byte8 t4_h t4_l ⇒ plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,e9〉:〈e10,x0〉〉 〈〈x0,e7〉:〈e8,x0〉〉) 〈〈t4_h,t4_l〉:〈x0,x0〉〉)〈〈x0,x0〉:〈e5,e6〉〉
- ]);
- nrewrite < (symmetric_mulex e1 e3);
- ncases (mul_ex e1 e3);
- #e11; #e12;
- nchange with ((plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,e7〉:〈e8,x0〉〉 〈〈x0,e9〉:〈e10,x0〉〉) 〈〈e11,e12〉:〈x0,x0〉〉)〈〈x0,x0〉:〈e5,e6〉〉) =
- (plus_w16_d_d (plus_w16_d_d (plus_w16_d_d 〈〈x0,e9〉:〈e10,x0〉〉 〈〈x0,e7〉:〈e8,x0〉〉) 〈〈e11,e12〉:〈x0,x0〉〉)〈〈x0,x0〉:〈e5,e6〉〉));
- nrewrite > (symmetric_plusw16_d_d 〈〈x0,e7〉:〈e8,x0〉〉 〈〈x0,e9〉:〈e10,x0〉〉);
- napply refl_eq.
-nqed.
-
-nlemma eqw16_to_eq : ∀w1,w2:word16.(eq_w16 w1 w2 = true) → (w1 = w2).
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- nchange in ⊢ (% → ?) with (((eq_b8 b3 b1)⊗(eq_b8 b4 b2)) = true);
- #H;
- nrewrite < (eqb8_to_eq … (andb_true_true_l … H));
- nrewrite < (eqb8_to_eq … (andb_true_true_r … H));
- napply refl_eq.
-nqed.
-
-nlemma eq_to_eqw16 : ∀w1,w2.w1 = w2 → eq_w16 w1 w2 = true.
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- #H;
- nrewrite < (word16_destruct_1 … H);
- nrewrite < (word16_destruct_2 … H);
- nchange with (((eq_b8 b3 b3)⊗(eq_b8 b4 b4)) = true);
- nrewrite > (eq_to_eqb8 b3 b3 (refl_eq …));
- nrewrite > (eq_to_eqb8 b4 b4 (refl_eq …));
- nnormalize;
- napply refl_eq.
-nqed.
-
-nlemma decidable_w16_aux1 : ∀b1,b2,b3,b4.b1 ≠ b3 → 〈b1:b2〉 ≠ 〈b3:b4〉.
- #b1; #b2; #b3; #b4;
- nnormalize; #H; #H1;
- napply (H (word16_destruct_1 … H1)).
-nqed.
-
-nlemma decidable_w16_aux2 : ∀b1,b2,b3,b4.b2 ≠ b4 → 〈b1:b2〉 ≠ 〈b3:b4〉.
- #b1; #b2; #b3; #b4;
- nnormalize; #H; #H1;
- napply (H (word16_destruct_2 … H1)).
-nqed.
-
-nlemma decidable_w16 : ∀x,y:word16.decidable (x = y).
- #x; nelim x; #b1; #b2;
- #y; nelim y; #b3; #b4;
- nnormalize;
- napply (or2_elim (b1 = b3) (b1 ≠ b3) ? (decidable_b8 b1 b3) …);
- ##[ ##2: #H; napply (or2_intro2 … (decidable_w16_aux1 b1 b2 b3 b4 H))
- ##| ##1: #H; napply (or2_elim (b2 = b4) (b2 ≠ b4) ? (decidable_b8 b2 b4) …);
- ##[ ##2: #H1; napply (or2_intro2 … (decidable_w16_aux2 b1 b2 b3 b4 H1))
- ##| ##1: #H1; nrewrite > H; nrewrite > H1;
- napply (or2_intro1 … (refl_eq ? 〈b3:b4〉))
- ##]
- ##]
-nqed.
-
-nlemma neqw16_to_neq : ∀w1,w2:word16.(eq_w16 w1 w2 = false) → (w1 ≠ w2).
- #w1; #w2;
- nelim w1;
- nelim w2;
- #b1; #b2; #b3; #b4;
- nchange with ((((eq_b8 b3 b1) ⊗ (eq_b8 b4 b2)) = false) → ?);
- #H;
- napply (or2_elim ((eq_b8 b3 b1) = false) ((eq_b8 b4 b2) = false) ? (andb_false2 … H) …);
- ##[ ##1: #H1; napply (decidable_w16_aux1 … (neqb8_to_neq … H1))
- ##| ##2: #H1; napply (decidable_w16_aux2 … (neqb8_to_neq … H1))
- ##]
-nqed.
-
-nlemma word16_destruct : ∀b1,b2,b3,b4.〈b1:b2〉 ≠ 〈b3:b4〉 → b1 ≠ b3 ∨ b2 ≠ b4.
- #b1; #b2; #b3; #b4;
- nnormalize; #H;
- napply (or2_elim (b1 = b3) (b1 ≠ b3) ? (decidable_b8 b1 b3) …);
- ##[ ##2: #H1; napply (or2_intro1 … H1)
- ##| ##1: #H1; napply (or2_elim (b2 = b4) (b2 ≠ b4) ? (decidable_b8 b2 b4) …);
- ##[ ##2: #H2; napply (or2_intro2 … H2)
- ##| ##1: #H2; nrewrite > H1 in H:(%);
- nrewrite > H2;
- #H; nelim (H (refl_eq …))
- ##]
- ##]
-nqed.
-
-nlemma neq_to_neqw16 : ∀w1,w2.w1 ≠ w2 → eq_w16 w1 w2 = false.
- #w1; #w2;
- nelim w1; #b1; #b2;
- nelim w2; #b3; #b4;
- #H; nchange with (((eq_b8 b1 b3) ⊗ (eq_b8 b2 b4)) = false);
- napply (or2_elim (b1 ≠ b3) (b2 ≠ b4) ? (word16_destruct … H) …);
- ##[ ##1: #H1; nrewrite > (neq_to_neqb8 … H1); nnormalize; napply refl_eq
- ##| ##2: #H1; nrewrite > (neq_to_neqb8 … H1);
- nrewrite > (symmetric_andbool (eq_b8 b1 b3) false);
- nnormalize; napply refl_eq
- ##]
-nqed.
+ndefinition neqw16_to_neq ≝ neqcn_to_neq ? eq_b8 neqb8_to_neq.
+ndefinition neq_to_neqw16 ≝ neq_to_neqcn ? eq_b8 neq_to_neqb8 decidable_b8.
projects / helm.git / blobdiff