+
+(*include "compiler/ast_type_lemmas.ma".
+
+nlemma symmetric_eqasttype_aux1
+ : ∀x1,x2,y2.eq_nat (len_neList ast_type («£x1»)) (len_neList ast_type (x2§§y2)) = false.
+ #x1; #x2; #y2; ncases y2; nnormalize;
+ ##[ ##1: #t; napply (refl_eq ??)
+ ##| ##2: #t; #l; napply (refl_eq ??)
+ ##]
+nqed.
+
+nlemma symmetric_eqasttype_aux2
+ : ∀x1,y1,x2.eq_nat (len_neList ast_type (x1§§y1)) (len_neList ast_type («£x2»)) = false.
+ #x1; #y1; #x2; ncases y1; nnormalize;
+ ##[ ##1: #t; napply (refl_eq ??)
+ ##| ##2: #t; #l; napply (refl_eq ??)
+ ##]
+nqed.
+
+ndefinition symmetric_eqasttype_aux3 ≝
+λnelSubType1,nelSubType2:ne_list ast_type.
+ match eq_nat (len_neList ast_type nelSubType1) (len_neList ast_type nelSubType2)
+ return λx.eq_nat (len_neList ast_type nelSubType1) (len_neList ast_type nelSubType2) = x → bool
+ with
+ [ true ⇒ λp:(eq_nat (len_neList ast_type nelSubType1) (len_neList ast_type nelSubType2) = true).
+ fold_right_neList2 ?? (λx1,x2,acc.(eq_ast_type x1 x2)⊗acc)
+ true nelSubType1 nelSubType2
+ (eqnat_to_eq (len_neList ? nelSubType1) (len_neList ? nelSubType2) p)
+ | false ⇒ λp:(eq_nat (len_neList ast_type nelSubType1) (len_neList ast_type nelSubType2) = false).false
+ ] (refl_eq ? (eq_nat (len_neList ast_type nelSubType1) (len_neList ast_type nelSubType2))).
+
+nlemma symmetric_eqasttype : symmetricT ast_type bool eq_ast_type.
+ #t1;
+ napply (ast_type_ind ????? t1);
+ ##[ ##1: #b1; #t2; ncases t2;
+ ##[ ##1: #b2; nchange with ((eq_ast_base_type b1 b2) = (eq_ast_base_type b2 b1));
+ nrewrite > (symmetric_eqastbasetype b1 b2);
+ napply (refl_eq ??)
+ ##| ##2: #tt; #n; nnormalize; napply (refl_eq ??)
+ ##| ##3: #l; nnormalize; napply (refl_eq ??)
+ ##]
+ ##| ##2: #tt1; #n1; #H; #t2; ncases t2;
+ ##[ ##2: #tt2; #n2; nchange with (((eq_ast_type tt1 tt2)⊗(eq_nat n1 n2)) = ((eq_ast_type tt2 tt1)⊗(eq_nat n2 n1)));
+ nrewrite > (H tt2);
+ nrewrite > (symmetric_eqnat n1 n2);
+ napply (refl_eq ??)
+ ##| ##1: #b2; nnormalize; napply (refl_eq ??)
+ ##| ##3: #l; nnormalize; napply (refl_eq ??)
+ ##]
+ ##| ##3: #tt1; #H; #t2; ncases t2;
+ ##[ ##3: #l; ncases l;
+ ##[ ##1: #tt2; nnormalize; nrewrite > (H tt2); napply (refl_eq ??)
+ ##| ##2: #tt2; #l1;
+ nchange with (
+ (match eq_nat (len_neList ast_type «£tt1») (len_neList ast_type (tt2§§l1))
+ return λx.eq_nat (len_neList ast_type «£tt1») (len_neList ast_type (tt2§§l1)) = x → bool
+ with
+ [ true ⇒ λp:(eq_nat (len_neList ast_type «£tt1») (len_neList ast_type (tt2§§l1)) = true).
+ fold_right_neList2 ?? (λx1,x2,acc.(eq_ast_type x1 x2)⊗acc)
+ true «£tt1» (tt2§§l1)
+ (eqnat_to_eq (len_neList ? «£tt1») (len_neList ? (tt2§§l1)) p)
+ | false ⇒ λp:(eq_nat (len_neList ast_type «£tt1») (len_neList ast_type (tt2§§l1)) = false).false
+ ] (refl_eq ? (eq_nat (len_neList ast_type «£tt1») (len_neList ast_type (tt2§§l1))))) = ?);
+
+ (* eseguendo questa sequenza e' ok *)
+ nrewrite > (symmetric_eqasttype_aux1 tt1 tt2 l1);
+ nchange with (
+ false = (match eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1»)
+ return λx.eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1») = x → bool
+ with
+ [ true ⇒ λp:(eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1») = true).
+ fold_right_neList2 ?? (λx1,x2,acc.(eq_ast_type x1 x2)⊗acc)
+ true (tt2§§l1) «£tt1»
+ (eqnat_to_eq (len_neList ? (tt2§§l1)) (len_neList ? «£tt1») p)
+ | false ⇒ λp:(eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1») = false).false
+ ] (refl_eq ? (eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1»)))));
+ nrewrite > (symmetric_eqasttype_aux2 tt2 l1 tt1);
+ nnormalize;
+
+ (* se commentiamo invece la prima sequenza ed eseguiamo questa *)
+ (* come e' possibile che rigetti la seconda rewrite? *)
+ nchange with (
+ ? = (match eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1»)
+ return λx.eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1») = x → bool
+ with
+ [ true ⇒ λp:(eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1») = true).
+ fold_right_neList2 ?? (λx1,x2,acc.(eq_ast_type x1 x2)⊗acc)
+ true (tt2§§l1) «£tt1»
+ (eqnat_to_eq (len_neList ? (tt2§§l1)) (len_neList ? «£tt1») p)
+ | false ⇒ λp:(eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1») = false).false
+ ] (refl_eq ? (eq_nat (len_neList ast_type (tt2§§l1)) (len_neList ast_type «£tt1»)))));
+ nrewrite > (symmetric_eqasttype_aux1 tt1 tt2 l1);
+ nrewrite > (symmetric_eqasttype_aux2 tt2 l1 tt1);
+*)