ndefinition astbasetype_destruct_aux ≝
Πb1,b2:ast_base_type.ΠP:Prop.b1 = b2 →
- match b1 with
- [ AST_BASE_TYPE_BYTE8 ⇒ match b2 with [ AST_BASE_TYPE_BYTE8 ⇒ P → P | _ ⇒ P ]
- | AST_BASE_TYPE_WORD16 ⇒ match b2 with [ AST_BASE_TYPE_WORD16 ⇒ P → P | _ ⇒ P ]
- | AST_BASE_TYPE_WORD32 ⇒ match b2 with [ AST_BASE_TYPE_WORD32 ⇒ P → P | _ ⇒ P ]
- ].
+ match eq_astbasetype b1 b2 with [ true ⇒ P → P | false ⇒ P ].
ndefinition astbasetype_destruct : astbasetype_destruct_aux.
- #b1; #b2; #P;
+ #b1; #b2; #P; #H;
+ nrewrite < H;
nelim b1;
- nelim b2;
nnormalize;
- #H;
- ##[ ##1,5,9: napply (λx:P.x)
- ##| ##2,3: napply False_ind;
- nchange with (match AST_BASE_TYPE_BYTE8 with [ AST_BASE_TYPE_BYTE8 ⇒ False | _ ⇒ True]);
- nrewrite > H;
- nnormalize;
- napply I
- ##| ##4,6: napply False_ind;
- nchange with (match AST_BASE_TYPE_WORD16 with [ AST_BASE_TYPE_WORD16 ⇒ False | _ ⇒ True]);
- nrewrite > H;
- nnormalize;
- napply I
- ##| ##7,8: napply False_ind;
- nchange with (match AST_BASE_TYPE_WORD32 with [ AST_BASE_TYPE_WORD32 ⇒ False | _ ⇒ True]);
- nrewrite > H;
- nnormalize;
- napply I
- ##]
+ napply (λx.x).
nqed.
-nlemma symmetric_eqastbasetype : symmetricT ast_base_type bool eq_ast_base_type.
+nlemma symmetric_eqastbasetype : symmetricT ast_base_type bool eq_astbasetype.
#b1; #b2; ncases b1; ncases b2; nnormalize; napply refl_eq. nqed.
-nlemma eqastbasetype_to_eq : ∀b1,b2.eq_ast_base_type b1 b2 = true → b1 = b2.
+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; napply (bool_destruct … H)
##]
nqed.
-nlemma eq_to_eqastbasetype : ∀b1,b2.b1 = b2 → eq_ast_base_type b1 b2 = true.
+nlemma eq_to_eqastbasetype : ∀b1,b2.b1 = b2 → eq_astbasetype b1 b2 = true.
#b1; #b2; ncases b1; ncases b2; nnormalize;
##[ ##1,5,9: #H; napply refl_eq
##| ##*: #H; napply (astbasetype_destruct … H)
##]
nqed.
+nlemma decidable_astbasetype : ∀x,y:ast_base_type.decidable (x = y).
+ #x; #y;
+ nnormalize;
+ nelim x;
+ nelim y;
+ ##[ ##1,5,9: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
+ ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H;
+ napply False_ind;
+ napply (astbasetype_destruct … H)
+ ##]
+nqed.
+
+nlemma neqastbasetype_to_neq : ∀b1,b2.(eq_astbasetype b1 b2 = false) → (b1 ≠ b2).
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,5,9: #H; napply (bool_destruct … H)
+ ##| ##*: #H; #H1; napply (astbasetype_destruct … H1)
+ ##]
+nqed.
+
+nlemma neq_to_neqastbasetype : ∀b1,b2.b1 ≠ b2 → eq_astbasetype b1 b2 = false.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,5,9: #H; nelim (H (refl_eq …))
+ ##| ##*: #H; napply refl_eq
+ ##]
+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 ]);
ndefinition asttype_destruct_aux ≝
Πb1,b2:ast_type.ΠP:Prop.b1 = b2 →
- match b1 with
- [ AST_TYPE_BASE s1 ⇒ match b2 with
- [ AST_TYPE_BASE s2 ⇒ match s1 with
- [ AST_BASE_TYPE_BYTE8 ⇒ match s2 with [ AST_BASE_TYPE_BYTE8 ⇒ P → P | _ ⇒ P ]
- | AST_BASE_TYPE_WORD16 ⇒ match s2 with [ AST_BASE_TYPE_WORD16 ⇒ P → P | _ ⇒ P ]
- | AST_BASE_TYPE_WORD32 ⇒ match s2 with [ AST_BASE_TYPE_WORD32 ⇒ P → P | _ ⇒ P ]
- ] | _ ⇒ P ]
- | AST_TYPE_ARRAY _ _ ⇒ match b2 with [ AST_TYPE_ARRAY _ _ ⇒ P → P | _ ⇒ P ]
- | AST_TYPE_STRUCT _ ⇒ match b2 with [ AST_TYPE_STRUCT _ ⇒ P → P | _ ⇒ P ]
- ].
+ match eq_asttype b1 b2 with [ true ⇒ P → P | false ⇒ P ].
ndefinition asttype_destruct : asttype_destruct_aux.
- #b1; #b2; #P;
- ncases b1;
- ##[ ##1: ncases b2;
- ##[ ##1: nnormalize; #s1; #s2; ncases s1; ncases s2; nnormalize;
- ##[ ##1,5,9: #H; napply (λx:P.x)
- ##| ##*: #H; napply (astbasetype_destruct … (asttype_destruct_base_base … H))
- ##]
- ##| ##2: #t; #n; #b; nnormalize; #H
- ##| ##3: #l; #b; nnormalize; #H
- ##]
- napply False_ind;
- nchange with (match AST_TYPE_BASE b with [ AST_TYPE_BASE _ ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##| ##2: ncases b2;
- ##[ ##2: #t1; #n1; #t2; #n2; nnormalize; #H; napply (λx:P.x)
- ##| ##1: #b; #t; #n; nnormalize; #H
- ##| ##3: #l; #t; #n; nnormalize; #H
- ##]
- napply False_ind;
- nchange with (match AST_TYPE_ARRAY t n with [ AST_TYPE_ARRAY _ _ ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##| ##3: ncases b2;
- ##[ ##3: #l1; #l2; nnormalize; #H; napply (λx:P.x)
- ##| ##1: #b; #l; nnormalize; #H
- ##| ##2: #t; #n; #l; nnormalize; #H
+ #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
##]
- napply False_ind;
- nchange with (match AST_TYPE_STRUCT l with [ AST_TYPE_STRUCT _ ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
##]
nqed.
nlemma symmetric_eqasttype_aux1
: ∀nl1,nl2.
- (eq_ast_type (AST_TYPE_STRUCT nl1) (AST_TYPE_STRUCT nl2)) = (eq_ast_type (AST_TYPE_STRUCT nl2) (AST_TYPE_STRUCT nl1)) →
- (bfold_right_neList2 ? (λx,y.eq_ast_type x y) nl1 nl2) = (bfold_right_neList2 ? (λx,y.eq_ast_type x y) nl2 nl1).
+ (eq_asttype (AST_TYPE_STRUCT nl1) (AST_TYPE_STRUCT nl2)) = (eq_asttype (AST_TYPE_STRUCT nl2) (AST_TYPE_STRUCT nl1)) →
+ (bfold_right_neList2 ? (λx,y.eq_asttype x y) nl1 nl2) = (bfold_right_neList2 ? (λx,y.eq_asttype x y) nl2 nl1).
#nl1; #nl2; #H;
napply H.
nqed.
-nlemma symmetric_eqasttype : symmetricT ast_type bool eq_ast_type.
+nlemma symmetric_eqasttype : symmetricT ast_type bool eq_asttype.
#t1; napply (ast_type_index … t1);
##[ ##1: #b1; #t2; ncases t2;
- ##[ ##1: #b2; nchange with ((eq_ast_base_type b1 b2) = (eq_ast_base_type b2 b1));
+ ##[ ##1: #b2; nchange with ((eq_astbasetype b1 b2) = (eq_astbasetype b2 b1));
nrewrite > (symmetric_eqastbasetype b1 b2);
napply refl_eq
##| ##2: #st2; #n2; nnormalize; napply refl_eq
##| ##3: #nl2; nnormalize; napply refl_eq
##]
##| ##2: #st1; #n1; #H; #t2; ncases t2;
- ##[ ##2: #st2; #n2; nchange with (((eq_ast_type st1 st2)⊗(eq_nat n1 n2)) = ((eq_ast_type st2 st1)⊗(eq_nat n2 n1)));
+ ##[ ##2: #st2; #n2; nchange with (((eq_asttype st1 st2)⊗(eq_nat n1 n2)) = ((eq_asttype st2 st1)⊗(eq_nat n2 n1)));
nrewrite > (symmetric_eqnat n1 n2);
nrewrite > (H st2);
napply refl_eq
##]
##| ##3: #hh1; #H; #t2; ncases t2;
##[ ##3: #nl2; ncases nl2;
- ##[ ##1: #hh2; nchange with ((eq_ast_type hh1 hh2) = (eq_ast_type hh2 hh1));
+ ##[ ##1: #hh2; nchange with ((eq_asttype hh1 hh2) = (eq_asttype hh2 hh1));
nrewrite > (H hh2);
napply refl_eq
##| ##2: #hh2; #ll2; nnormalize; napply refl_eq
##]
nqed.
-nlemma eqasttype_to_eq : ∀t1,t2.eq_ast_type t1 t2 = true → t1 = t2.
+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_ast_base_type b1 b2) = true);
+ ##[ ##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; napply (bool_destruct … H)
##| ##3: #nl2; nnormalize; #H; napply (bool_destruct … H)
##]
##| ##2: #st1; #n1; #H; #t2; ncases t2;
- ##[ ##2: #st2; #n2; #H1; nchange in H1:(%) with (((eq_ast_type st1 st2)⊗(eq_nat n1 n2)) = true);
+ ##[ ##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
##]
##| ##3: #hh1; #H; #t2; ncases t2;
##[ ##3: #nl2; ncases nl2;
- ##[ ##1: #hh2; #H1; nchange in H1:(%) with ((eq_ast_type hh1 hh2) = true);
+ ##[ ##1: #hh2; #H1; nchange in H1:(%) with ((eq_asttype hh1 hh2) = true);
nrewrite > (H hh2 H1);
napply refl_eq
##| ##2: #hh2; #ll2; nnormalize; #H1; napply (bool_destruct … H1)
##| ##4: #hh1; #ll1; #H; #H1; #t2; ncases t2;
##[ ##3: #nl2; ncases nl2;
##[ ##1: #hh2; nnormalize; #H2; napply (bool_destruct … H2)
- ##| ##2: #hh2; #ll2; #H2; nchange in H2:(%) with (((eq_ast_type hh1 hh2)⊗(bfold_right_neList2 ? (λx,y.eq_ast_type x y) ll1 ll2)) = true);
+ ##| ##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
nlemma eq_to_eqasttype_aux1
: ∀nl1,nl2.
- ((eq_ast_type (AST_TYPE_STRUCT nl1) (AST_TYPE_STRUCT nl2)) = true) →
- ((bfold_right_neList2 ? (λx,y.eq_ast_type x y) nl1 nl2) = true).
+ ((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_ast_type t1 t2 = true.
+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_ast_base_type b2 b2) = true);
+ nchange with ((eq_astbasetype b2 b2) = true);
nrewrite > (eq_to_eqastbasetype b2 b2 (refl_eq …));
napply refl_eq
##| ##2: #st2; #n2; #H; napply (asttype_destruct … H)
##| ##3: #nl2; #H; napply (asttype_destruct … H)
##]
##| ##2: #st1; #n1; #H; #t2; ncases t2;
- ##[ ##2: #st2; #n2; #H1; nchange with (((eq_ast_type st1 st2)⊗(eq_nat n1 n2)) = true);
+ ##[ ##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;
##]
##| ##3: #hh1; #H; #t2; ncases t2;
##[ ##3: #nl2; ncases nl2;
- ##[ ##1: #hh2; #H1; nchange with ((eq_ast_type hh1 hh2) = true);
+ ##[ ##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; nelim (nelist_destruct_nil_cons ? hh1 hh2 ll2 (asttype_destruct_struct_struct … H1))
##| ##4: #hh1; #ll1; #H; #H1; #t2; ncases t2;
##[ ##3: #nl2; ncases nl2;
##[ ##1: #hh2; #H2; nelim (nelist_destruct_cons_nil ? hh1 hh2 ll1 (asttype_destruct_struct_struct … H2))
- ##| ##2: #hh2; #ll2; #H2; nchange with (((eq_ast_type hh1 hh2)⊗(bfold_right_neList2 ? (λx,y.eq_ast_type x y) ll1 ll2)) = true);
+ ##| ##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
##]
nqed.
+nlemma decidable_asttype : ∀x,y:ast_type.decidable (x = y).
+ #x;
+ napply (ast_type_index … x);
+ ##[ ##1: #b1; #y; ncases y;
+ ##[ ##1: #b2; nnormalize; napply (or2_elim (? = ?) (? ≠ ?) ? (decidable_astbasetype b1 b2));
+ ##[ ##1: #H; nrewrite > H; napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+ ##| ##2: #H; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H1; napply (H (asttype_destruct_base_base … H1))
+ ##]
+ ##| ##2: #b2; #n2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H; napply (asttype_destruct … H)
+ ##| ##3: #n2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H; napply (asttype_destruct … H)
+ ##]
+ ##| ##2: #b1; #n1; #H; #y; ncases y;
+ ##[ ##2: #b2; #n2; nnormalize; napply (or2_elim (? = ?) (? ≠ ?) ? (H b2));
+ ##[ ##2: #H1; napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize;
+ #H2; napply (H1 (asttype_destruct_array_array_1 … H2))
+ ##| ##1: #H1; napply (or2_elim (? = ?) (? ≠ ?) ? (decidable_nat n1 n2));
+ ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize;
+ #H3; napply (H2 (asttype_destruct_array_array_2 … H3))
+ ##| ##1: #H2; nrewrite > H1; nrewrite > H2;
+ napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+ ##]
+ ##]
+ ##| ##1: #b2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H1; napply (asttype_destruct … H1)
+ ##| ##3: #n2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H1; napply (asttype_destruct … H1)
+ ##]
+ ##| ##3: #hh1; #H; #y; ncases y;
+ ##[ ##3: #n2; ncases n2;
+ ##[ ##1: #hh2; nnormalize; napply (or2_elim (? = ?) (? ≠ ?) ? (H hh2));
+ ##[ ##2: #H1; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H2;
+ napply (H1 (nelist_destruct_nil_nil … (asttype_destruct_struct_struct … H2)))
+ ##| ##1: #H1; nrewrite > H1; napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+ ##]
+ ##| ##2: #hh2; #tt2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H1; napply (nelist_destruct_nil_cons ast_type … (asttype_destruct_struct_struct … H1))
+ ##]
+ ##| ##1: #b2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H1; napply (asttype_destruct … H1)
+ ##| ##2: #b2; #n2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H1; napply (asttype_destruct … H1)
+ ##]
+ ##| ##4: #hh1; #tt1; #H; #H1; #y; ncases y;
+ ##[ ##3: #n2; ncases n2;
+ ##[ ##2: #hh2; #tt2; nnormalize; napply (or2_elim (? = ?) (? ≠ ?) ? (H hh2));
+ ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H3; napply (H2 (nelist_destruct_cons_cons_1 ast_type … (asttype_destruct_struct_struct … H3)))
+ ##| ##1: #H2; napply (or2_elim (? = ?) (? ≠ ?) ? (H1 (AST_TYPE_STRUCT tt2)));
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H4; napply H3;
+ nrewrite > (nelist_destruct_cons_cons_2 ast_type … (asttype_destruct_struct_struct … H4));
+ napply refl_eq
+ ##| ##1: #H3; nrewrite > H2; nrewrite > (asttype_destruct_struct_struct … H3);
+ napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+ ##]
+ ##]
+ ##| ##1: #hh2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H2; napply (nelist_destruct_cons_nil ast_type … (asttype_destruct_struct_struct … H2))
+ ##]
+ ##| ##1: #b2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H2; napply (asttype_destruct … H2)
+ ##| ##2: #b2; #n2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H2; napply (asttype_destruct … H2)
+ ##]
+ ##]
+nqed.
+
+nlemma neqasttype_to_neq : ∀t1,t2.eq_asttype t1 t2 = false → t1 ≠ t2.
+ #t1;
+ napply (ast_type_index … t1);
+ ##[ ##1: #b1; #t2; ncases t2;
+ ##[ ##1: #b2; nchange with ((eq_astbasetype b1 b2 = false) → ?); #H;
+ nnormalize; #H1; napply (neqastbasetype_to_neq b1 b2 H);
+ napply (asttype_destruct_base_base … H1)
+ ##| ##2: #b2; #n2; nnormalize; #H; #H1; napply (asttype_destruct … H1)
+ ##| ##3: #n2; nnormalize; #H; #H1; napply (asttype_destruct … H1)
+ ##]
+ ##| ##2: #b1; #n1; #H; #t2; ncases t2;
+ ##[ ##2: #b2; #n2; nchange with ((((eq_asttype b1 b2)⊗(eq_nat n1 n2)) = false) → ?); #H1;
+ napply (or2_elim ??? (andb_false2 … H1));
+ ##[ ##1: #H2; nnormalize; #H3; napply (H b2 H2); napply (asttype_destruct_array_array_1 … H3)
+ ##| ##2: #H2; nnormalize; #H3; napply (neqnat_to_neq n1 n2 H2); napply (asttype_destruct_array_array_2 … H3)
+ ##]
+ ##| ##1: #b2; nnormalize; #H1; #H2; napply (asttype_destruct … H2)
+ ##| ##3: #n2; nnormalize; #H1; #H2; napply (asttype_destruct … H2)
+ ##]
+ ##| ##3: #hh1; #H; #t2; ncases t2;
+ ##[ ##3: #n2; ncases n2;
+ ##[ ##1: #hh2; nchange with ((eq_asttype hh1 hh2 = false) → ?); #H1;
+ nnormalize; #H2; napply (H hh2 H1);
+ napply (nelist_destruct_nil_nil ast_type … (asttype_destruct_struct_struct … H2))
+ ##| ##2: #hh2; #tt2; nnormalize; #H1; #H2;
+ napply (nelist_destruct_nil_cons ast_type … (asttype_destruct_struct_struct … H2))
+ ##]
+ ##| ##1: #b2; nnormalize; #H1; #H2; napply (asttype_destruct … H2)
+ ##| ##2: #b2; #n2; nnormalize; #H1; #H2; napply (asttype_destruct … H2)
+ ##]
+ ##| ##4: #hh1; #tt1; #H; #H1; #t2; ncases t2;
+ ##[ ##3: #n2; ncases n2;
+ ##[ ##2: #hh2; #tt2; nchange with ((((eq_asttype hh1 hh2)⊗(bfold_right_neList2 ?? tt1 tt2)) = false) → ?);
+ #H2; napply (or2_elim ??? (andb_false2 … H2));
+ ##[ ##1: #H3; nnormalize; #H4; napply (H hh2 H3);
+ napply (nelist_destruct_cons_cons_1 ast_type … (asttype_destruct_struct_struct … H4))
+ ##| ##2: #H3; nnormalize; #H4; napply (H1 (AST_TYPE_STRUCT tt2) H3);
+ nrewrite > (nelist_destruct_cons_cons_2 ast_type … (asttype_destruct_struct_struct … H4));
+ napply refl_eq
+ ##]
+ ##| ##1: #hh2; nnormalize; #H2; #H3;
+ napply (nelist_destruct_cons_nil ast_type … (asttype_destruct_struct_struct … H3))
+ ##]
+ ##| ##1: #b2; nnormalize; #H2; #H3; napply (asttype_destruct … H3)
+ ##| ##2: #b2; #n2; nnormalize; #H2; #H3; napply (asttype_destruct … H3)
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqasttype : ∀t1,t2.t1 ≠ t2 → eq_asttype t1 t2 = false.
+ #t1;
+ napply (ast_type_index … t1);
+ ##[ ##1: #b1; #t2; ncases t2;
+ ##[ ##1: #b2; nchange with (? → (eq_astbasetype b1 b2 = false)); #H;
+ napply (neq_to_neqastbasetype b1 b2 ?); nnormalize; #H1;
+ napply H; nrewrite > H1; napply refl_eq
+ ##| ##2: #b2; #n2; nnormalize; #H; napply refl_eq
+ ##| ##3: #n2; nnormalize; #H; napply refl_eq
+ ##]
+ ##| ##2: #b1; #n1; #H; #t2; ncases t2;
+ ##[ ##2: #b2; #n2; nchange with (? → ((eq_asttype b1 b2)⊗(eq_nat n1 n2)) = false); #H1;
+ napply (or2_elim ??? (decidable_asttype b1 b2));
+ ##[ ##2: #H2; nrewrite > (H b2 H2); nnormalize; napply refl_eq
+ ##| ##1: #H2; napply (or2_elim ??? (decidable_nat n1 n2))
+ ##[ ##2: #H3; nrewrite > (neq_to_neqnat n1 n2 H3);
+ nrewrite > (andb_false2_2 (eq_asttype b1 b2));
+ napply refl_eq
+ ##| ##1: #H3; nrewrite > H2 in H1:(%); nrewrite > H3; #H1;
+ nelim (H1 (refl_eq …))
+ ##]
+ ##]
+ ##| ##1: #b2; nnormalize; #H1; napply refl_eq
+ ##| ##3: #n2; nnormalize; #H1; napply refl_eq
+ ##]
+ ##| ##3: #hh1; #H; #t2; ncases t2;
+ ##[ ##3: #n2; ncases n2;
+ ##[ ##1: #hh2; nchange with (? → (eq_asttype hh1 hh2 = false)); #H1;
+ napply (H hh2); nnormalize; #H2; nrewrite > H2 in H1:(%);
+ nnormalize; #H1; napply (H1 (refl_eq …))
+ ##| ##2: #hh2; #tt2; nnormalize; #H1; napply refl_eq
+ ##]
+ ##| ##1: #b2; nnormalize; #H1; napply refl_eq
+ ##| ##2: #b2; #n2; nnormalize; #H1; napply refl_eq
+ ##]
+ ##| ##4: #hh1; #tt1; #H; #H1; #t2; ncases t2;
+ ##[ ##3: #n2; ncases n2;
+ ##[ ##2: #hh2; #tt2; nchange with (? → ((eq_asttype hh1 hh2)⊗(bfold_right_neList2 ?? tt1 tt2)) = false);
+ #H2; napply (or2_elim (hh1 ≠ hh2) (tt1 ≠ tt2) …);
+ ##[ ##2: #H3; nrewrite > (H hh2 H3); nnormalize; napply refl_eq
+ ##| ##3: #H3; nchange with (((eq_asttype hh1 hh2)⊗(eq_asttype (AST_TYPE_STRUCT tt1) (AST_TYPE_STRUCT tt2))) = false);
+ nrewrite > (H1 (AST_TYPE_STRUCT tt2) ?);
+ ##[ ##1: nrewrite > (andb_false2_2 (eq_asttype hh1 hh2)); napply refl_eq
+ ##| ##2: nnormalize; #H4; napply (H3 (asttype_destruct_struct_struct … H4))
+ ##]
+ ##| ##1: napply (or2_elim ??? (decidable_asttype hh1 hh2));
+ ##[ ##2: #H3; napply (or2_intro1 … H3)
+ ##| ##1: #H3; napply (or2_elim ??? (decidable_nelist ast_type decidable_asttype tt1 tt2));
+ ##[ ##2: #H4; napply (or2_intro2 … H4)
+ ##| ##1: #H4; nrewrite > H3 in H2:(%); nrewrite > H4; #H2;
+ nelim (H2 (refl_eq …))
+ ##]
+ ##]
+ ##]
+ ##| ##1: #hh2; nnormalize; #H2; napply refl_eq
+ ##]
+ ##| ##1: #b2; nnormalize; #H2; napply refl_eq
+ ##| ##2: #b2; #n2; nnormalize; #H2; napply refl_eq
+ ##]
+ ##]
+nqed.
+
nlemma isbastbasetype_to_isastbasetype : ∀ast.isb_ast_base_type ast = true → is_ast_base_type ast.
#ast;
ncases ast;
projects / helm.git / blobdiff