X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsyntax%2Ftdeq.ma;h=d431b7677a55bd28dc55f1e47578c4c7d06cbcd9;hp=21d140e51abf96759559b202c163d89dd76685f7;hb=222044da28742b24584549ba86b1805a87def070;hpb=b5ded5b0c305b30349339b24760820154f7de390

diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma
index 21d140e51..d431b7677 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma
@@ -12,9 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/lazyeq_4.ma".
+include "basic_2/notation/relations/stareq_4.ma".
 include "basic_2/syntax/item_sd.ma".
-include "basic_2/syntax/lenv.ma".
+include "basic_2/syntax/term.ma".
 
 (* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
 
@@ -26,28 +26,12 @@ inductive tdeq (h) (o): relation term ≝
 .
 
 interpretation
-   "degree-based equivalence (terms)"
-   'LazyEq h o T1 T2 = (tdeq h o T1 T2).
-
-definition cdeq: ∀h. sd h → relation3 lenv term term ≝
-                 λh,o,L. tdeq h o.
-
-(* Basic properties *********************************************************)
-
-lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
-#h #o #T elim T -T /2 width=1 by tdeq_pair/
-* /2 width=1 by tdeq_lref, tdeq_gref/
-#s elim (deg_total h o s) /2 width=3 by tdeq_sort/
-qed.
-
-lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
-#h #o #T1 #T2 #H elim H -T1 -T2
-/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
-qed-.
+   "context-free degree-based equivalence (term)"
+   'StarEq h o T1 T2 = (tdeq h o T1 T2).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀s1. X = ⋆s1 →
+fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀s1. X = ⋆s1 →
                          ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
 #h #o #X #Y * -X -Y
 [ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
@@ -57,32 +41,32 @@ fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀s1. X = ⋆s1 →
 ]
 qed-.
 
-lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y →
+lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y →
                       ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
 /2 width=3 by tdeq_inv_sort1_aux/ qed-.
 
-fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀i. X = #i → Y = #i.
+fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀i. X = #i → Y = #i.
 #h #o #X #Y * -X -Y //
 [ #s1 #s2 #d #_ #_ #j #H destruct
 | #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
 ]
 qed-.
 
-lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≡[h, o] Y → Y = #i.
+lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≛[h, o] Y → Y = #i.
 /2 width=5 by tdeq_inv_lref1_aux/ qed-.
 
-fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀l. X = §l → Y = §l.
+fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀l. X = §l → Y = §l.
 #h #o #X #Y * -X -Y //
 [ #s1 #s2 #d #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
 ]
 qed-.
 
-lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≡[h, o] Y → Y = §l.
+lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≛[h, o] Y → Y = §l.
 /2 width=5 by tdeq_inv_gref1_aux/ qed-.
 
-fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 →
-                         ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2.
+fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 →
+                         ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
 #h #o #X #Y * -X -Y
 [ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
 | #i #J #W1 #U1 #H destruct
@@ -91,27 +75,113 @@ fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀I,V1,T1. X = ②{I}V1.
 ]
 qed-.
 
-lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≡[h, o] Y →
-                      ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2.
+lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≛[h, o] Y →
+                      ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
 /2 width=3 by tdeq_inv_pair1_aux/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y → ∀d. deg h o s1 d →
+lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y → ∀d. deg h o s1 d →
                           ∃∃s2. deg h o s2 d & Y = ⋆s2.
 #h #o #Y #s1 #H #d #Hs1 elim (tdeq_inv_sort1 … H) -H
-#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/  
+#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
+qed-.
+
+lemma tdeq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ≛[h, o] ⋆s2 →
+                         ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
+                         d1 = d2.
+#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
+elim (tdeq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
+<(deg_mono h o … Hy … Hs2) -s2 -d1 //
 qed-.
 
-lemma tdeq_inv_pair: ∀h,o,I,V1,V2,T1,T2. ②{I}V1.T1 ≡[h, o] ②{I}V2.T2 →
-                     V1 ≡[h, o] V2 ∧ T1 ≡[h, o] T2.
-#h #o #I #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
-#V0 #T0 #HV #HT #H destruct /2 width=1 by conj/
-qed-. 
+lemma tdeq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 →
+                     ∧∧ I1 = I2 & V1 ≛[h, o] V2 & T1 ≛[h, o] T2.
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
+#V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma tdeq_inv_pair_xy_x: ∀h,o,I,V,T. ②{I}V.T ≛[h, o] V → ⊥.
+#h #o #I #V elim V -V
+[ #J #T #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
+| #J #X #Y #IHX #_ #T #H elim (tdeq_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
+]
+qed-.
+
+lemma tdeq_inv_pair_xy_y: ∀h,o,I,T,V. ②{I}V.T ≛[h, o] T → ⊥.
+#h #o #I #T elim T -T
+[ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
+| #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
+]
+qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≡[h, o] Y → ∃J. Y = ⓪{J}.
+lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≛[h, o] Y → ∃J. Y = ⓪{J}.
 #h #o * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
 /3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/
 qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
+#h #o #T elim T -T /2 width=1 by tdeq_pair/
+* /2 width=1 by tdeq_lref, tdeq_gref/
+#s elim (deg_total h o s) /2 width=3 by tdeq_sort/
+qed.
+
+lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
+#h #o #T1 #T2 #H elim H -T1 -T2
+/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
+qed-.
+
+lemma tdeq_dec: ∀h,o,T1,T2. Decidable (T1 ≛[h, o] T2).
+#h #o #T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ elim (deg_total h o s1) #d1 #H1
+  elim (deg_total h o s2) #d2 #H2
+  elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by tdeq_sort, or_introl/
+  @or_intror #H
+  lapply (tdeq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
+|2,3,13:
+  @or_intror #H
+  elim (tdeq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
+|4,6,14:
+  @or_intror #H
+  lapply (tdeq_inv_lref1 … H) -H #H destruct
+|5:
+  elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+  @or_intror #H
+  lapply (tdeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+  @or_intror #H
+  lapply (tdeq_inv_gref1 … H) -H #H destruct
+|9:
+  elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+  @or_intror #H
+  lapply (tdeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+  @or_intror #H
+  elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
+|16:
+  elim (eq_item2_dec I1 I2) #HI12 destruct
+  [ elim (IHV V2) -IHV #HV12
+    elim (IHT T2) -IHT #HT12
+    [ /3 width=1 by tdeq_pair, or_introl/ ]
+  ]
+  @or_intror #H
+  elim (tdeq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+lemma tdneq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
+                      (②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 → ⊥) → 
+                      ∨∨ I1 = I2 → ⊥
+                      |  (V1 ≛[h, o] V2 → ⊥)
+                      |  (T1 ≛[h, o] T2 → ⊥).
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H12
+elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
+elim (tdeq_dec h o V1 V2) /3 width=1 by or3_intro1/
+elim (tdeq_dec h o T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
+qed-.