--- /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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "emulator/status/HC08_status_base.ma".
+
+(* *********************************** *)
+(* STATUS INTERNO DEL PROCESSORE (ALU) *)
+(* *********************************** *)
+
+nlemma aluHC08_destruct_1 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x1 = y1.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 a _ _ _ _ _ _ _ _ _ _ _ ⇒ x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_2 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x2 = y2.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ a _ _ _ _ _ _ _ _ _ _ ⇒ x2 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_3 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x3 = y3.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ a _ _ _ _ _ _ _ _ _ ⇒ x3 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_4 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x4 = y4.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ a _ _ _ _ _ _ _ _ ⇒ x4 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_5 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x5 = y5.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ a _ _ _ _ _ _ _ ⇒ x5 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_6 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x6 = y6.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ a _ _ _ _ _ _ ⇒ x6 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_7 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x7 = y7.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ _ a _ _ _ _ _ ⇒ x7 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_8 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x8 = y8.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ _ _ a _ _ _ _ ⇒ x8 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_9 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x9 = y9.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ _ _ _ a _ _ _ ⇒ x9 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_10 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x10 = y10.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ _ _ _ _ a _ _ ⇒ x10 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_11 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x11 = y11.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ _ _ _ _ _ a _ ⇒ x11 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_destruct_12 :
+∀x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12.
+ mk_alu_HC08 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 = mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 →
+ x12 = y12.
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange with (match mk_alu_HC08 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
+ with [ mk_alu_HC08 _ _ _ _ _ _ _ _ _ _ _ a ⇒ x12 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma eq_to_eqaluHC08 : ∀alu1,alu2.alu1 = alu2 → eq_HC08_alu alu1 alu2 = true.
+ #alu1; #alu2;
+ ncases alu1;
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ ncases alu2;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nrewrite > (aluHC08_destruct_1 … H);
+ nrewrite > (aluHC08_destruct_2 … H);
+ nrewrite > (aluHC08_destruct_3 … H);
+ nrewrite > (aluHC08_destruct_4 … H);
+ nrewrite > (aluHC08_destruct_5 … H);
+ nrewrite > (aluHC08_destruct_6 … H);
+ nrewrite > (aluHC08_destruct_7 … H);
+ nrewrite > (aluHC08_destruct_8 … H);
+ nrewrite > (aluHC08_destruct_9 … H);
+ nrewrite > (aluHC08_destruct_10 … H);
+ nrewrite > (aluHC08_destruct_11 … H);
+ nrewrite > (aluHC08_destruct_12 … H);
+ nchange with (
+ ((eqc ? y1 y1) ⊗ (eqc ? y2 y2) ⊗ (eqc ? y3 y3) ⊗ (eqc ? y4 y4) ⊗
+ (eqc ? y5 y5) ⊗ (eqc ? y6 y6) ⊗ (eqc ? y7 y7) ⊗ (eqc ? y8 y8) ⊗
+ (eqc ? y9 y9) ⊗ (eqc ? y10 y10) ⊗ (eqc ? y11 y11) ⊗ (eqc ? y12 y12)) = true);
+ nrewrite > (eq_to_eqc ? y1 y1 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y2 y2 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y3 y3 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y4 y4 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y5 y5 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y6 y6 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y7 y7 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y8 y8 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y9 y9 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y10 y10 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y11 y11 (refl_eq …));
+ nrewrite > (eq_to_eqc ? y12 y12 (refl_eq …));
+ napply refl_eq.
+nqed.
+
+nlemma neqaluHC08_to_neq : ∀alu1,alu2.eq_HC08_alu alu1 alu2 = false → alu1 ≠ alu2.
+ #s1; #s2; #H;
+ napply (not_to_not (s1 = s2) (eq_HC08_alu s1 s2 = true) …);
+ ##[ ##1: napply (eq_to_eqaluHC08 s1 s2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqaluHC08_to_eq : ∀alu1,alu2.eq_HC08_alu alu1 alu2 = true → alu1 = alu2.
+ #alu1; #alu2;
+ ncases alu1;
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ ncases alu2;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12; #H;
+ nchange in H:(%) with (
+ ((eqc ? x1 y1) ⊗ (eqc ? x2 y2) ⊗ (eqc ? x3 y3) ⊗ (eqc ? x4 y4) ⊗
+ (eqc ? x5 y5) ⊗ (eqc ? x6 y6) ⊗ (eqc ? x7 y7) ⊗ (eqc ? x8 y8) ⊗
+ (eqc ? x9 y9) ⊗ (eqc ? x10 y10) ⊗ (eqc ? x11 y11) ⊗ (eqc ? x12 y12)) = true);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H));
+ nletin H1 ≝ (andb_true_true_l … H);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H1));
+ nletin H2 ≝ (andb_true_true_l … H1);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H2));
+ nletin H3 ≝ (andb_true_true_l … H2);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H3));
+ nletin H4 ≝ (andb_true_true_l … H3);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H4));
+ nletin H5 ≝ (andb_true_true_l … H4);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H5));
+ nletin H6 ≝ (andb_true_true_l … H5);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H6));
+ nletin H7 ≝ (andb_true_true_l … H6);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H7));
+ nletin H8 ≝ (andb_true_true_l … H7);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H8));
+ nletin H9 ≝ (andb_true_true_l … H8);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H9));
+ nletin H10 ≝ (andb_true_true_l … H9);
+ nrewrite > (eqc_to_eq … (andb_true_true_r … H10));
+ nrewrite > (eqc_to_eq … (andb_true_true_l … H10));
+ napply refl_eq.
+nqed.
+
+nlemma neq_to_neqaluHC08 : ∀alu1,alu2.alu1 ≠ alu2 → eq_HC08_alu alu1 alu2 = false.
+ #s1; #s2; #H;
+ napply (neqtrue_to_eqfalse (eq_HC08_alu s1 s2));
+ napply (not_to_not (eq_HC08_alu s1 s2 = true) (s1 = s2) ? H);
+ napply (eqaluHC08_to_eq s1 s2).
+nqed.
+
+nlemma decidable_aluHC08 : ∀x,y:alu_HC08.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_HC08_alu x y = true) (eq_HC08_alu x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqaluHC08_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqaluHC08_to_neq … H))
+ ##]
+nqed.
+
+nlemma symmetric_eqaluHC08 : symmetricT alu_HC08 bool eq_HC08_alu.
+ #alu1; #alu2;
+ ncases alu1;
+ #x1; #x2; #x3; #x4; #x5; #x6; #x7; #x8; #x9; #x10; #x11; #x12;
+ ncases alu2;
+ #y1; #y2; #y3; #y4; #y5; #y6; #y7; #y8; #y9; #y10; #y11; #y12;
+ nchange with (
+ ((eqc ? x1 y1) ⊗ (eqc ? x2 y2) ⊗ (eqc ? x3 y3) ⊗ (eqc ? x4 y4) ⊗
+ (eqc ? x5 y5) ⊗ (eqc ? x6 y6) ⊗ (eqc ? x7 y7) ⊗ (eqc ? x8 y8) ⊗
+ (eqc ? x9 y9) ⊗ (eqc ? x10 y10) ⊗ (eqc ? x11 y11) ⊗ (eqc ? x12 y12)) =
+ ((eqc ? y1 x1) ⊗ (eqc ? y2 x2) ⊗ (eqc ? y3 x3) ⊗ (eqc ? y4 x4) ⊗
+ (eqc ? y5 x5) ⊗ (eqc ? y6 x6) ⊗ (eqc ? y7 x7) ⊗ (eqc ? y8 x8) ⊗
+ (eqc ? y9 x9) ⊗ (eqc ? y10 x10) ⊗ (eqc ? y11 x11) ⊗ (eqc ? y12 x12)));
+ nrewrite > (symmetric_eqc ? x1 y1);
+ nrewrite > (symmetric_eqc ? x2 y2);
+ nrewrite > (symmetric_eqc ? x3 y3);
+ nrewrite > (symmetric_eqc ? x4 y4);
+ nrewrite > (symmetric_eqc ? x5 y5);
+ nrewrite > (symmetric_eqc ? x6 y6);
+ nrewrite > (symmetric_eqc ? x7 y7);
+ nrewrite > (symmetric_eqc ? x8 y8);
+ nrewrite > (symmetric_eqc ? x9 y9);
+ nrewrite > (symmetric_eqc ? x10 y10);
+ nrewrite > (symmetric_eqc ? x11 y11);
+ nrewrite > (symmetric_eqc ? x12 y12);
+ napply refl_eq.
+nqed.
+
+nlemma aluHC08_is_comparable : comparable.
+ @ alu_HC08
+ ##[ napply (mk_alu_HC08 (zeroc ?) (zeroc ?) (zeroc ?) (zeroc ?)
+ (zeroc ?) (zeroc ?) (zeroc ?) (zeroc ?)
+ (zeroc ?) (zeroc ?) (zeroc ?) (zeroc ?))
+ ##| napply forall_HC08_alu
+ ##| napply eq_HC08_alu
+ ##| napply eqaluHC08_to_eq
+ ##| napply eq_to_eqaluHC08
+ ##| napply neqaluHC08_to_neq
+ ##| napply neq_to_neqaluHC08
+ ##| napply decidable_aluHC08
+ ##| napply symmetric_eqaluHC08
+ ##]
+nqed.
+
+unification hint 0 ≔ ⊢ carr aluHC08_is_comparable ≡ alu_HC08.
projects / helm.git / blobdiff