--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/word16.ma".
+include "num/byte8_lemmas.ma".
+
+(* **** *)
+(* 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.
+
+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.
+
+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.
+
+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.
+
+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.
projects / helm.git / blobdiff