(* *)
(**************************************************************************)
+include "ground/xoa/ex_1_3.ma".
include "static_2/notation/functions/item0_1.ma".
include "static_2/notation/functions/snitem2_3.ma".
include "static_2/notation/functions/snbind2_4.ma".
(* terms *)
inductive term: Type[0] ≝
- | TAtom: item0 → term (* atomic item construction *)
- | TPair: item2 → term → term → term (* binary item construction *)
+| TAtom: item0 → term (* atomic item construction *)
+| TPair: item2 → term → term → term (* binary item construction *)
.
-interpretation "term construction (atomic)"
- 'Item0 I = (TAtom I).
+interpretation
+ "term construction (atomic)"
+ 'Item0 I = (TAtom I).
-interpretation "term construction (binary)"
- 'SnItem2 I T1 T2 = (TPair I T1 T2).
+interpretation
+ "term construction (binary)"
+ 'SnItem2 I T1 T2 = (TPair I T1 T2).
-interpretation "term binding construction (binary)"
- 'SnBind2 p I T1 T2 = (TPair (Bind2 p I) T1 T2).
+interpretation
+ "term binding construction (binary)"
+ 'SnBind2 p I T1 T2 = (TPair (Bind2 p I) T1 T2).
-interpretation "term positive binding construction (binary)"
- 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
+interpretation
+ "term positive binding construction (binary)"
+ 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
-interpretation "term negative binding construction (binary)"
- 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
+interpretation
+ "term negative binding construction (binary)"
+ 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
-interpretation "term flat construction (binary)"
- 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
+interpretation
+ "term flat construction (binary)"
+ 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
-interpretation "sort (term)"
- 'Star s = (TAtom (Sort s)).
+interpretation
+ "sort (term)"
+ 'Star s = (TAtom (Sort s)).
-interpretation "local reference (term)"
- 'LRef i = (TAtom (LRef i)).
+interpretation
+ "local reference (term)"
+ 'LRef i = (TAtom (LRef i)).
-interpretation "global reference (term)"
- 'GRef l = (TAtom (GRef l)).
+interpretation
+ "global reference (term)"
+ 'GRef l = (TAtom (GRef l)).
-interpretation "abbreviation (term)"
- 'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
+interpretation
+ "abbreviation (term)"
+ 'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
-interpretation "positive abbreviation (term)"
- 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
+interpretation
+ "positive abbreviation (term)"
+ 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
-interpretation "negative abbreviation (term)"
- 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
+interpretation
+ "negative abbreviation (term)"
+ 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
-interpretation "abstraction (term)"
- 'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
+interpretation
+ "abstraction (term)"
+ 'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
-interpretation "positive abstraction (term)"
- 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
+interpretation
+ "positive abstraction (term)"
+ 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
-interpretation "negative abstraction (term)"
- 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
+interpretation
+ "negative abstraction (term)"
+ 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
-interpretation "application (term)"
- 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
+interpretation
+ "application (term)"
+ 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
-interpretation "native type annotation (term)"
- 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
+interpretation
+ "native type annotation (term)"
+ 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
(* Basic properties *********************************************************)
+lemma abst_dec (X):
+ ∨∨ ∃∃p,W,T. X = ⓛ[p]W.T
+ | (∀p,W,T. X = ⓛ[p]W.T → ⊥).
+* [ #I | * [ #p * | #I ] #V #T ]
+[3: /3 width=4 by ex1_3_intro, or_introl/ ]
+@or_intror #q #W #U #H destruct
+qed-.
+
(* Basic_1: was: term_dec *)
-lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
+lemma eq_term_dec:
+ ∀T1,T2:term. Decidable (T1 = T2).
#T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
[1,4: @or_intror #H destruct
| elim (eq_item2_dec I1 I2) #HI
(* Basic inversion lemmas ***************************************************)
-fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪{I1} = ⓪{I2} → I1 = I2.
+fact destruct_tatom_tatom_aux:
+ ∀I1,I2. ⓪[I1] = ⓪[I2] → I1 = I2.
#I1 #I2 #H destruct //
qed-.
-fact destruct_tpair_tpair_aux: ∀I1,I2,T1,T2,V1,V2. ②{I1}T1.V1 = ②{I2}T2.V2 →
- ∧∧T1 = T2 & I1 = I2 & V1 = V2.
+fact destruct_tpair_tpair_aux:
+ ∀I1,I2,T1,T2,V1,V2. ②[I1]T1.V1 = ②[I2]T2.V2 →
+ ∧∧ T1 = T2 & I1 = I2 & V1 = V2.
#I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
qed-.
-lemma discr_tpair_xy_x: ∀I,T,V. ②{I}V.T = V → ⊥.
+lemma discr_tpair_xy_x:
+ ∀I,T,V. ②[I]V.T = V → ⊥.
#I #T #V elim V -V
[ #J #H destruct
| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
qed-.
(* Basic_1: was: thead_x_y_y *)
-lemma discr_tpair_xy_y: ∀I,V,T. ②{I}V.T = T → ⊥.
+lemma discr_tpair_xy_y:
+ ∀I,V,T. ②[I]V.T = T → ⊥.
#I #V #T elim T -T
[ #J #H destruct
| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
-lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
- (②{I}V1.T1 = ②{I}V2.T2 → ⊥) →
- (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
+lemma eq_false_inv_tpair_sn:
+ ∀I,V1,T1,V2,T2.
+ (②[I]V1.T1 = ②[I]V2.T2 → ⊥) →
+ (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
#I #V1 #T1 #V2 #T2 #H
elim (eq_term_dec V1 V2) /3 width=1 by or_introl/ #HV12 destruct
@or_intror @conj // #HT12 destruct /2 width=1 by/
qed-.
-lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I}V2.T2 → ⊥) →
- (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
+lemma eq_false_inv_tpair_dx:
+ ∀I,V1,T1,V2,T2.
+ (②[I] V1. T1 = ②[I]V2.T2 → ⊥) →
+ (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
#I #V1 #T1 #V2 #T2 #H
elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct
@or_intror @conj // #HT12 destruct /2 width=1 by/