--- /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 "compiler/ast_type.ma".
+include "common/list_utility_lemmas.ma".
+
+(* ************************* *)
+(* dimensioni degli elementi *)
+(* ************************* *)
+
+(*
+ndefinition astbasetype_destruct_aux ≝
+Πb1,b2:ast_base_type.ΠP:Prop.b1 = b2 →
+ match eq_astbasetype b1 b2 with [ true ⇒ P → P | false ⇒ P ].
+
+ndefinition astbasetype_destruct : astbasetype_destruct_aux.
+ #b1; #b2; #P; #H;
+ nrewrite < H;
+ nelim b1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+*)
+
+nlemma eq_to_eqastbasetype : ∀n1,n2.n1 = n2 → eq_astbasetype n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqastbasetype_to_neq : ∀n1,n2.eq_astbasetype n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_astbasetype n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqastbasetype n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqastbasetype_to_eq : ∀b1,b2.eq_astbasetype b1 b2 = true → b1 = b2.
+ #b1; #b2; ncases b1; ncases b2; nnormalize;
+ ##[ ##1,5,9: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma neq_to_neqastbasetype : ∀n1,n2.n1 ≠ n2 → eq_astbasetype n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_astbasetype n1 n2));
+ napply (not_to_not (eq_astbasetype n1 n2 = true) (n1 = n2) ? H);
+ napply (eqastbasetype_to_eq n1 n2).
+nqed.
+
+nlemma decidable_astbasetype : ∀x,y:ast_base_type.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_astbasetype x y = true) (eq_astbasetype x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqastbasetype_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqastbasetype_to_neq … H))
+ ##]
+nqed.
+
+nlemma symmetric_eqastbasetype : symmetricT ast_base_type bool eq_astbasetype.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_astbasetype n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqastbasetype n1 n2 H);
+ napply (symmetric_eq ? (eq_astbasetype n2 n1) false);
+ napply (neq_to_neqastbasetype n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma asttype_destruct_base_base : ∀b1,b2.AST_TYPE_BASE b1 = AST_TYPE_BASE b2 → b1 = b2.
+ #b1; #b2; #H;
+ nchange with (match AST_TYPE_BASE b2 with [ AST_TYPE_BASE a ⇒ b1 = a | _ ⇒ False ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma asttype_destruct_array_array_1 : ∀x1,x2,y1,y2.AST_TYPE_ARRAY x1 y1 = AST_TYPE_ARRAY x2 y2 → x1 = x2.
+ #x1; #x2; #y1; #y2; #H;
+ nchange with (match AST_TYPE_ARRAY x2 y2 with [ AST_TYPE_ARRAY a _ ⇒ x1 = a | _ ⇒ False ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma asttype_destruct_array_array_2 : ∀x1,x2,y1,y2.AST_TYPE_ARRAY x1 y1 = AST_TYPE_ARRAY x2 y2 → y1 = y2.
+ #x1; #x2; #y1; #y2; #H;
+ nchange with (match AST_TYPE_ARRAY x2 y2 with [ AST_TYPE_ARRAY _ b ⇒ y1 = b | _ ⇒ False ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma asttype_destruct_struct_struct : ∀b1,b2.AST_TYPE_STRUCT b1 = AST_TYPE_STRUCT b2 → b1 = b2.
+ #b1; #b2; #H;
+ nchange with (match AST_TYPE_STRUCT b2 with [ AST_TYPE_STRUCT a ⇒ b1 = a | _ ⇒ False ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+(*
+ndefinition asttype_destruct_aux ≝
+Πb1,b2:ast_type.ΠP:Prop.b1 = b2 →
+ match eq_asttype b1 b2 with [ true ⇒ P → P | false ⇒ P ].
+
+ndefinition asttype_destruct : asttype_destruct_aux.
+ #b1; #b2; #P; #H;
+ nrewrite > H;
+ napply (ast_type_index … b2);
+ ##[ ##1: #e; nchange with (match eq_astbasetype e e with [ true ⇒ P → P | false ⇒ P ]);
+ nrewrite > (eq_to_eqastbasetype e e (refl_eq …));
+ nnormalize; napply (λx.x);
+ ##| ##2: #e; #n; #H; nchange with (match (eq_asttype e e)⊗(eq_nat n n) with [ true ⇒ P → P | false ⇒ P]);
+ nrewrite > (eq_to_eqnat n n (refl_eq …));
+ nrewrite > (symmetric_andbool (eq_asttype e e) true);
+ nchange with (match eq_asttype e e with [ true ⇒ P → P | false ⇒ P]);
+ napply H;
+ ##| ##3: #e; #H; nchange with (match eq_asttype e e with [ true ⇒ P → P | false ⇒ P]);
+ napply H;
+ ##| ##4: #hh; #tt; #H;
+ nchange with (match bfold_right_neList2 ?? tt tt with [ true ⇒ P → P | false ⇒ P ] →
+ match (eq_asttype hh hh)⊗(bfold_right_neList2 ?? tt tt) with [ true ⇒ P → P | false ⇒ P ]);
+ #H1;
+ ncases (eq_asttype hh hh) in H:(%) ⊢ %; #H;
+ ncases (bfold_right_neList2 ? (λx1,x2.eq_asttype x1 x2) tt tt) in H1:(%) ⊢ %; #H1;
+ ##[ ##1: nnormalize; napply (λx.x);
+ ##| ##3: nnormalize in H:(%) ⊢ %; napply H
+ ##| ##*: nnormalize in H1:(%) ⊢ %; napply H1
+ ##]
+ ##]
+nqed.
+*)
+
+nlemma eq_to_eqasttype_aux1
+ : ∀nl1,nl2.
+ ((eq_asttype (AST_TYPE_STRUCT nl1) (AST_TYPE_STRUCT nl2)) = true) →
+ ((bfold_right_neList2 ? (λx,y.eq_asttype x y) nl1 nl2) = true).
+ #nl1; #nl2; #H;
+ napply H.
+nqed.
+
+nlemma eq_to_eqasttype : ∀t1,t2.t1 = t2 → eq_asttype t1 t2 = true.
+ #t1;
+ napply (ast_type_index … t1);
+ ##[ ##1: #b1; #t2; ncases t2;
+ ##[ ##1: #b2; #H; nrewrite > (asttype_destruct_base_base … H);
+ nchange with ((eq_astbasetype b2 b2) = true);
+ nrewrite > (eq_to_eqastbasetype b2 b2 (refl_eq …));
+ napply refl_eq
+ ##| ##2: #st2; #n2; #H; ndestruct (*napply (asttype_destruct … H)*)
+ ##| ##3: #nl2; #H; ndestruct (*napply (asttype_destruct … H)*)
+ ##]
+ ##| ##2: #st1; #n1; #H; #t2; ncases t2;
+ ##[ ##2: #st2; #n2; #H1; nchange with (((eq_asttype st1 st2)⊗(eq_nat n1 n2)) = true);
+ nrewrite > (H st2 (asttype_destruct_array_array_1 … H1));
+ nrewrite > (eq_to_eqnat n1 n2 (asttype_destruct_array_array_2 … H1));
+ nnormalize;
+ napply refl_eq
+ ##| ##1: #b2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##| ##3: #nl2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##]
+ ##| ##3: #hh1; #H; #t2; ncases t2;
+ ##[ ##3: #nl2; ncases nl2;
+ ##[ ##1: #hh2; #H1; nchange with ((eq_asttype hh1 hh2) = true);
+ nrewrite > (H hh2 (nelist_destruct_nil_nil ? hh1 hh2 (asttype_destruct_struct_struct … H1)));
+ napply refl_eq
+ ##| ##2: #hh2; #ll2; #H1;
+ (* !!! ndestruct non va *)
+ nelim (nelist_destruct_nil_cons ? hh1 hh2 ll2 (asttype_destruct_struct_struct … H1))
+ ##]
+ ##| ##1: #b2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##| ##2: #st2; #n2; #H1; ndestruct (*napply (asttype_destruct … H1)*)
+ ##]
+ ##| ##4: #hh1; #ll1; #H; #H1; #t2; ncases t2;
+ ##[ ##3: #nl2; ncases nl2;
+ ##[ ##1: #hh2; #H2;
+ (* !!! ndestruct non va *)
+ nelim (nelist_destruct_cons_nil ? hh1 hh2 ll1 (asttype_destruct_struct_struct … H2))
+ ##| ##2: #hh2; #ll2; #H2; nchange with (((eq_asttype hh1 hh2)⊗(bfold_right_neList2 ? (λx,y.eq_asttype x y) ll1 ll2)) = true);
+ nrewrite > (H hh2 (nelist_destruct_cons_cons_1 … (asttype_destruct_struct_struct … H2)));
+ nrewrite > (eq_to_eqasttype_aux1 ll1 ll2 (H1 (AST_TYPE_STRUCT ll2) ?));
+ ##[ ##1: nnormalize; napply refl_eq
+ ##| ##2: nrewrite > (nelist_destruct_cons_cons_2 … (asttype_destruct_struct_struct … H2));
+ napply refl_eq
+ ##]
+ ##]
+ ##| ##1: #b2; #H2; ndestruct (*napply (asttype_destruct … H2)*)
+ ##| ##2: #st2; #n2; #H2; ndestruct (*napply (asttype_destruct … H2)*)
+ ##]
+ ##]
+nqed.
+
+nlemma neqasttype_to_neq : ∀n1,n2.eq_asttype n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_asttype n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqasttype n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqasttype_to_eq : ∀t1,t2.eq_asttype t1 t2 = true → t1 = t2.
+ #t1;
+ napply (ast_type_index … t1);
+ ##[ ##1: #b1; #t2; ncases t2;
+ ##[ ##1: #b2; #H; nchange in H:(%) with ((eq_astbasetype b1 b2) = true);
+ nrewrite > (eqastbasetype_to_eq b1 b2 H);
+ napply refl_eq
+ ##| ##2: #st2; #n2; nnormalize; #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##3: #nl2; nnormalize; #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+ ##| ##2: #st1; #n1; #H; #t2; ncases t2;
+ ##[ ##2: #st2; #n2; #H1; nchange in H1:(%) with (((eq_asttype st1 st2)⊗(eq_nat n1 n2)) = true);
+ nrewrite > (H st2 (andb_true_true_l … H1));
+ nrewrite > (eqnat_to_eq n1 n2 (andb_true_true_r … H1));
+ napply refl_eq
+ ##| ##1: #b2; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##| ##3: #nl2; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##]
+ ##| ##3: #hh1; #H; #t2; ncases t2;
+ ##[ ##3: #nl2; ncases nl2;
+ ##[ ##1: #hh2; #H1; nchange in H1:(%) with ((eq_asttype hh1 hh2) = true);
+ nrewrite > (H hh2 H1);
+ napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##]
+ ##| ##1: #b2; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##| ##2: #st2; #n2; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##]
+ ##| ##4: #hh1; #ll1; #H; #H1; #t2; ncases t2;
+ ##[ ##3: #nl2; ncases nl2;
+ ##[ ##1: #hh2; nnormalize; #H2; ndestruct (*napply (bool_destruct … H2)*)
+ ##| ##2: #hh2; #ll2; #H2; nchange in H2:(%) with (((eq_asttype hh1 hh2)⊗(bfold_right_neList2 ? (λx,y.eq_asttype x y) ll1 ll2)) = true);
+ nrewrite > (H hh2 (andb_true_true_l … H2));
+ nrewrite > (asttype_destruct_struct_struct ll1 ll2 (H1 (AST_TYPE_STRUCT ll2) (andb_true_true_r … H2)));
+ napply refl_eq
+ ##]
+ ##| ##1: #b2; nnormalize; #H2; ndestruct (*napply (bool_destruct … H2)*)
+ ##| ##2: #st2; #n2; nnormalize; #H2; ndestruct (*napply (bool_destruct … H2)*)
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqasttype : ∀n1,n2.n1 ≠ n2 → eq_asttype n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_asttype n1 n2));
+ napply (not_to_not (eq_asttype n1 n2 = true) (n1 = n2) ? H);
+ napply (eqasttype_to_eq n1 n2).
+nqed.
+
+nlemma decidable_asttype : ∀x,y:ast_type.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_asttype x y = true) (eq_asttype x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqasttype_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqasttype_to_neq … H))
+ ##]
+nqed.
+
+nlemma symmetric_eqasttype : symmetricT ast_type bool eq_asttype.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_asttype n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqasttype n1 n2 H);
+ napply (symmetric_eq ? (eq_asttype n2 n1) false);
+ napply (neq_to_neqasttype n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma isbastbasetype_to_isastbasetype : ∀ast.isb_ast_base_type ast = true → is_ast_base_type ast.
+ #ast;
+ ncases ast;
+ nnormalize;
+ ##[ ##1: #t; #H; napply I
+ ##| ##2: #t; #n; #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##3: #t; #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma isntbastbasetype_to_isntastbasetype : ∀ast.isntb_ast_base_type ast = true → isnt_ast_base_type ast.
+ #ast;
+ ncases ast;
+ nnormalize;
+ ##[ ##1: #t; #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##2: #t; #n; #H; napply I
+ ##| ##3: #l; #H; napply I
+ ##]
+nqed.
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