update in basic_2 ...

[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
index 9e7d860b4348b631df54670b7b8b1c301882cc9c..ff00f4f84c45721632cac4321209c24f8de38347 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma
@@ -12,6 +12,23 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/constructors/item0_1.ma".
+include "basic_2/notation/constructors/snitem2_3.ma".
+include "basic_2/notation/constructors/snbind2_4.ma".
+include "basic_2/notation/constructors/snbind2pos_3.ma".
+include "basic_2/notation/constructors/snbind2neg_3.ma".
+include "basic_2/notation/constructors/snflat2_3.ma".
+include "basic_2/notation/constructors/star_1.ma".
+include "basic_2/notation/constructors/lref_1.ma".
+include "basic_2/notation/constructors/gref_1.ma".
+include "basic_2/notation/constructors/snabbr_3.ma".
+include "basic_2/notation/constructors/snabbrpos_2.ma".
+include "basic_2/notation/constructors/snabbrneg_2.ma".
+include "basic_2/notation/constructors/snabst_3.ma".
+include "basic_2/notation/constructors/snabstpos_2.ma".
+include "basic_2/notation/constructors/snabstneg_2.ma".
+include "basic_2/notation/constructors/snappl_2.ma".
+include "basic_2/notation/constructors/sncast_2.ma".
 include "basic_2/grammar/item.ma".
 
 (* TERMS ********************************************************************)
@@ -29,7 +46,7 @@ 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).
+   '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).
@@ -41,16 +58,16 @@ interpretation "term flat construction (binary)"
    'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
 
 interpretation "sort (term)"
-   'Star k = (TAtom (Sort k)).
+   'Star s = (TAtom (Sort s)).
 
 interpretation "local reference (term)"
    'LRef i = (TAtom (LRef i)).
 
 interpretation "global reference (term)"
-   'GRef p = (TAtom (GRef p)).
+   'GRef l = (TAtom (GRef l)).
 
 interpretation "abbreviation (term)"
-   'SnAbbr a T1 T2 = (TPair (Bind2 a Abbr) T1 T2).
+   'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
 
 interpretation "positive abbreviation (term)"
    'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
@@ -59,7 +76,7 @@ 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).
+   'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
 
 interpretation "positive abstraction (term)"
    'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
@@ -76,43 +93,59 @@ interpretation "native type annotation (term)"
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: term_dec *)
-axiom term_eq_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
+  [ elim (IHV1 V2) -IHV1 #HV
+    [ elim (IHT1 T2) -IHT1 /2 width=1 by or_introl/ #HT ]
+  ]
+  @or_intror #H destruct /2 width=1 by/
+| elim (eq_item0_dec I1 I2) /2 width=1 by or_introl/ #HI
+  @or_intror #H destruct /2 width=1 by/
+]
+qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → ⊥.
+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.
+#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 → ⊥.
 #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/
+| #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 destruct
-  -H (**) (* destruct: the destucted equality is not erased *)
-  /2 width=1/
+| #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 → ⊥) →
+                             (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/
+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 → ⊥) →
+                             (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/
+elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: removed theorems 3: