notational update in lambdadelta completed

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lifts_vector.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma
index 26d395f19fc833ba9dadb90596e8b798816106de..dc7c7d309c13a3f75e373e40c8eae9e0d1bb19ce 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma
@@ -21,31 +21,31 @@ include "basic_2/relocation/lifts.ma".
 inductive liftsv (f:rtmap): relation (list term) ≝
 | liftsv_nil : liftsv f (◊) (◊)
 | liftsv_cons: ∀T1s,T2s,T1,T2.
-               â¬\86*[f] T1 â\89¡ T2 → liftsv f T1s T2s →
-               liftsv f (T1 @ T1s) (T2 @ T2s)
+               â¬\86*[f] T1 â\89\98 T2 → liftsv f T1s T2s →
+               liftsv f (T1 ⨮ T1s) (T2 ⨮ T2s)
 .
 
-interpretation "uniform relocation (vector)"
+interpretation "uniform relocation (term vector)"
    'RLiftStar i T1s T2s = (liftsv (uni i) T1s T2s).
 
-interpretation "generic relocation (vector)"
+interpretation "generic relocation (term vector)"
    'RLiftStar f T1s T2s = (liftsv f T1s T2s).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact liftsv_inv_nil1_aux: â\88\80f,X,Y. â¬\86*[f] X â\89¡ Y → X = ◊ → Y = ◊.
+fact liftsv_inv_nil1_aux: â\88\80f,X,Y. â¬\86*[f] X â\89\98 Y → X = ◊ → Y = ◊.
 #f #X #Y * -X -Y //
 #T1s #T2s #T1 #T2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes: liftv_inv_nil1 *)
-lemma liftsv_inv_nil1: â\88\80f,Y. â¬\86*[f] â\97\8a â\89¡ Y → Y = ◊.
+lemma liftsv_inv_nil1: â\88\80f,Y. â¬\86*[f] â\97\8a â\89\98 Y → Y = ◊.
 /2 width=5 by liftsv_inv_nil1_aux/ qed-.
 
-fact liftsv_inv_cons1_aux: â\88\80f:rtmap. â\88\80X,Y. â¬\86*[f] X â\89¡ Y →
-                           ∀T1,T1s. X = T1 @ T1s →
-                           â\88\83â\88\83T2,T2s. â¬\86*[f] T1 â\89¡ T2 & â¬\86*[f] T1s â\89¡ T2s &
-                                     Y = T2 @ T2s.
+fact liftsv_inv_cons1_aux: â\88\80f:rtmap. â\88\80X,Y. â¬\86*[f] X â\89\98 Y →
+                           ∀T1,T1s. X = T1 ⨮ T1s →
+                           â\88\83â\88\83T2,T2s. â¬\86*[f] T1 â\89\98 T2 & â¬\86*[f] T1s â\89\98 T2s &
+                                     Y = T2 ⨮ T2s.
 #f #X #Y * -X -Y
 [ #U1 #U1s #H destruct
 | #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
@@ -53,37 +53,37 @@ fact liftsv_inv_cons1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≡ Y →
 qed-.
 
 (* Basic_2A1: includes: liftv_inv_cons1 *)
-lemma liftsv_inv_cons1: ∀f:rtmap. ∀T1,T1s,Y. ⬆*[f] T1 @ T1s ≡ Y →
-                        â\88\83â\88\83T2,T2s. â¬\86*[f] T1 â\89¡ T2 & â¬\86*[f] T1s â\89¡ T2s &
-                                  Y = T2 @ T2s.
+lemma liftsv_inv_cons1: ∀f:rtmap. ∀T1,T1s,Y. ⬆*[f] T1 ⨮ T1s ≘ Y →
+                        â\88\83â\88\83T2,T2s. â¬\86*[f] T1 â\89\98 T2 & â¬\86*[f] T1s â\89\98 T2s &
+                                  Y = T2 ⨮ T2s.
 /2 width=3 by liftsv_inv_cons1_aux/ qed-.
 
-fact liftsv_inv_nil2_aux: â\88\80f,X,Y. â¬\86*[f] X â\89¡ Y → Y = ◊ → X = ◊.
+fact liftsv_inv_nil2_aux: â\88\80f,X,Y. â¬\86*[f] X â\89\98 Y → Y = ◊ → X = ◊.
 #f #X #Y * -X -Y //
 #T1s #T2s #T1 #T2 #_ #_ #H destruct
 qed-.
 
-lemma liftsv_inv_nil2: â\88\80f,X. â¬\86*[f] X â\89¡ ◊ → X = ◊.
+lemma liftsv_inv_nil2: â\88\80f,X. â¬\86*[f] X â\89\98 ◊ → X = ◊.
 /2 width=5 by liftsv_inv_nil2_aux/ qed-.
 
-fact liftsv_inv_cons2_aux: â\88\80f:rtmap. â\88\80X,Y. â¬\86*[f] X â\89¡ Y →
-                           ∀T2,T2s. Y = T2 @ T2s →
-                           â\88\83â\88\83T1,T1s. â¬\86*[f] T1 â\89¡ T2 & â¬\86*[f] T1s â\89¡ T2s &
-                                     X = T1 @ T1s.
+fact liftsv_inv_cons2_aux: â\88\80f:rtmap. â\88\80X,Y. â¬\86*[f] X â\89\98 Y →
+                           ∀T2,T2s. Y = T2 ⨮ T2s →
+                           â\88\83â\88\83T1,T1s. â¬\86*[f] T1 â\89\98 T2 & â¬\86*[f] T1s â\89\98 T2s &
+                                     X = T1 ⨮ T1s.
 #f #X #Y * -X -Y
 [ #U2 #U2s #H destruct
 | #T1s #T2s #T1 #T2 #HT12 #HT12s #U2 #U2s #H destruct /2 width=5 by ex3_2_intro/
 ]
 qed-.
 
-lemma liftsv_inv_cons2: â\88\80f:rtmap. â\88\80X,T2,T2s. â¬\86*[f] X â\89¡ T2 @ T2s →
-                        â\88\83â\88\83T1,T1s. â¬\86*[f] T1 â\89¡ T2 & â¬\86*[f] T1s â\89¡ T2s &
-                                  X = T1 @ T1s.
+lemma liftsv_inv_cons2: â\88\80f:rtmap. â\88\80X,T2,T2s. â¬\86*[f] X â\89\98 T2 ⨮ T2s →
+                        â\88\83â\88\83T1,T1s. â¬\86*[f] T1 â\89\98 T2 & â¬\86*[f] T1s â\89\98 T2s &
+                                  X = T1 ⨮ T1s.
 /2 width=3 by liftsv_inv_cons2_aux/ qed-.
 
 (* Basic_1: was: lifts1_flat (left to right) *)
-lemma lifts_inv_applv1: â\88\80f:rtmap. â\88\80V1s,U1,T2. â¬\86*[f] â\92¶ V1s.U1 â\89¡ T2 →
-                        â\88\83â\88\83V2s,U2. â¬\86*[f] V1s â\89¡ V2s & â¬\86*[f] U1 â\89¡ U2 &
+lemma lifts_inv_applv1: â\88\80f:rtmap. â\88\80V1s,U1,T2. â¬\86*[f] â\92¶ V1s.U1 â\89\98 T2 →
+                        â\88\83â\88\83V2s,U2. â¬\86*[f] V1s â\89\98 V2s & â¬\86*[f] U1 â\89\98 U2 &
                                   T2 = Ⓐ V2s.U2.
 #f #V1s elim V1s -V1s
 [ /3 width=5 by ex3_2_intro, liftsv_nil/
@@ -93,8 +93,8 @@ lemma lifts_inv_applv1: ∀f:rtmap. ∀V1s,U1,T2. ⬆*[f] Ⓐ V1s.U1 ≡ T2 →
 ]
 qed-.
 
-lemma lifts_inv_applv2: â\88\80f:rtmap. â\88\80V2s,U2,T1. â¬\86*[f] T1 â\89¡ Ⓐ V2s.U2 →
-                        â\88\83â\88\83V1s,U1. â¬\86*[f] V1s â\89¡ V2s & â¬\86*[f] U1 â\89¡ U2 &
+lemma lifts_inv_applv2: â\88\80f:rtmap. â\88\80V2s,U2,T1. â¬\86*[f] T1 â\89\98 Ⓐ V2s.U2 →
+                        â\88\83â\88\83V1s,U1. â¬\86*[f] V1s â\89\98 V2s & â¬\86*[f] U1 â\89\98 U2 &
                                   T1 = Ⓐ V1s.U1.
 #f #V2s elim V2s -V2s
 [ /3 width=5 by ex3_2_intro, liftsv_nil/
@@ -107,7 +107,7 @@ qed-.
 (* Basic properties *********************************************************)
 
 (* Basic_2A1: includes: liftv_total *)
-lemma liftsv_total: â\88\80f. â\88\80T1s:list term. â\88\83T2s. â¬\86*[f] T1s â\89¡ T2s.
+lemma liftsv_total: â\88\80f. â\88\80T1s:list term. â\88\83T2s. â¬\86*[f] T1s â\89\98 T2s.
 #f #T1s elim T1s -T1s
 [ /2 width=2 by liftsv_nil, ex_intro/
 | #T1 #T1s * #T2s #HT12s
@@ -116,10 +116,22 @@ lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⬆*[f] T1s ≡ T2s.
 qed-.
 
 (* Basic_1: was: lifts1_flat (right to left) *)
-lemma lifts_applv: â\88\80f:rtmap. â\88\80V1s,V2s. â¬\86*[f] V1s â\89¡ V2s →
-                   â\88\80T1,T2. â¬\86*[f] T1 â\89¡ T2 →
-                   â¬\86*[f] â\92¶ V1s.T1 â\89¡ Ⓐ V2s.T2.
+lemma lifts_applv: â\88\80f:rtmap. â\88\80V1s,V2s. â¬\86*[f] V1s â\89\98 V2s →
+                   â\88\80T1,T2. â¬\86*[f] T1 â\89\98 T2 →
+                   â¬\86*[f] â\92¶ V1s.T1 â\89\98 Ⓐ V2s.T2.
 #f #V1s #V2s #H elim H -V1s -V2s /3 width=1 by lifts_flat/
 qed.
 
+lemma liftsv_split_trans: ∀f,T1s,T2s. ⬆*[f] T1s ≘ T2s →
+                          ∀f1,f2. f2 ⊚ f1 ≘ f →
+                          ∃∃Ts. ⬆*[f1] T1s ≘ Ts & ⬆*[f2] Ts ≘ T2s.
+#f #T1s #T2s #H elim H -T1s -T2s
+[ /2 width=3 by liftsv_nil, ex2_intro/
+| #T1s #T2s #T1 #T2 #HT12 #_ #IH #f1 #f2 #Hf
+  elim (IH … Hf) -IH
+  elim (lifts_split_trans … HT12 … Hf) -HT12 -Hf
+  /3 width=5 by liftsv_cons, ex2_intro/
+]
+qed-.
+
 (* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *)