- lambda_delta: programmed renaming to lambdadelta

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / grammar / term.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/grammar/item.ma".
+
+(* TERMS ********************************************************************)
+
+(* terms *)
+inductive term: Type[0] ≝
+  | 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 (binary)"
+   'SnItem2 I T1 T2 = (TPair I T1 T2).
+
+interpretation "term binding construction (binary)"
+   'SnBind2 a I T1 T2 = (TPair (Bind2 a 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 flat construction (binary)"
+   'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
+
+interpretation "sort (term)"
+   'Star k = (TAtom (Sort k)).
+
+interpretation "local reference (term)"
+   'LRef i = (TAtom (LRef i)).
+
+interpretation "global reference (term)"
+   'GRef p = (TAtom (GRef p)).
+
+interpretation "abbreviation (term)"
+   'SnAbbr a T1 T2 = (TPair (Bind2 a 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 "abstraction (term)"
+   'SnAbst a T1 T2 = (TPair (Bind2 a 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 "application (term)"
+   'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
+
+interpretation "native type annotation (term)"
+   'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: term_dec *)
+axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+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 destruct
+  -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
+  /2 width=1/
+]
+qed-.
+
+(* Basic_1: was: thead_x_y_y *)
+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 destruct
+  -H (**) (* destruct: the destucted equality is not erased *)
+  /2 width=1/
+]
+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 → ⊥)).
+#I #V1 #T1 #V2 #T2 #H
+elim (term_eq_dec V1 V2) /3 width=1/ #HV12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1/ 
+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 → ⊥)).
+#I #V1 #T1 #V2 #T2 #H
+elim (term_eq_dec T1 T2) /3 width=1/ #HT12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1/
+qed-.
+
+lemma eq_false_inv_beta: ∀a,V1,V2,W1,W2,T1,T2.
+                         (ⓐV1. ⓛ{a}W1. T1 = ⓐV2. ⓛ{a}W2 .T2 → ⊥) →
+                         (W1 = W2 → ⊥) ∨
+                         (W1 = W2 ∧ (ⓓ{a}V1. T1 = ⓓ{a}V2. T2 → ⊥)).
+#a #V1 #V2 #W1 #W2 #T1 #T2 #H
+elim (eq_false_inv_tpair_sn … H) -H
+[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/
+  #H destruct @or_intror @conj // #H destruct /2 width=1/
+| * #H1 #H2 destruct
+  elim (eq_false_inv_tpair_sn … H2) -H2 /3 width=1/
+  * #H #HT12 destruct
+  @or_intror @conj // #H destruct /2 width=1/
+]
+qed.
+
+(* Basic_1: removed theorems 3:
+            not_void_abst not_abbr_void not_abst_void
+*)