syntactic components detached from basic_2 become static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / relocation / lifts_vector.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_vector.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts_vector.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 "static_2/syntax/term_vector.ma".
+include "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
+
+(* Basic_2A1: includes: liftv_nil liftv_cons *)
+inductive liftsv (f:rtmap): relation (list term) ≝
+| liftsv_nil : liftsv f (Ⓔ) (Ⓔ)
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+               ⬆*[f] T1 ≘ T2 → liftsv f T1s T2s →
+               liftsv f (T1 ⨮ T1s) (T2 ⨮ T2s)
+.
+
+interpretation "uniform relocation (term vector)"
+   'RLiftStar i T1s T2s = (liftsv (uni i) T1s T2s).
+
+interpretation "generic relocation (term vector)"
+   'RLiftStar f T1s T2s = (liftsv f T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact liftsv_inv_nil1_aux: ∀f,X,Y. ⬆*[f] X ≘ 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: ∀f,Y. ⬆*[f] Ⓔ ≘ Y → Y = Ⓔ.
+/2 width=5 by liftsv_inv_nil1_aux/ qed-.
+
+fact liftsv_inv_cons1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
+                           ∀T1,T1s. X = T1 ⨮ T1s →
+                           ∃∃T2,T2s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ 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/
+]
+qed-.
+
+(* Basic_2A1: includes: liftv_inv_cons1 *)
+lemma liftsv_inv_cons1: ∀f:rtmap. ∀T1,T1s,Y. ⬆*[f] T1 ⨮ T1s ≘ Y →
+                        ∃∃T2,T2s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
+                                  Y = T2 ⨮ T2s.
+/2 width=3 by liftsv_inv_cons1_aux/ qed-.
+
+fact liftsv_inv_nil2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → Y = Ⓔ → X = Ⓔ.
+#f #X #Y * -X -Y //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed-.
+
+lemma liftsv_inv_nil2: ∀f,X. ⬆*[f] X ≘ Ⓔ → X = Ⓔ.
+/2 width=5 by liftsv_inv_nil2_aux/ qed-.
+
+fact liftsv_inv_cons2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
+                           ∀T2,T2s. Y = T2 ⨮ T2s →
+                           ∃∃T1,T1s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ 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: ∀f:rtmap. ∀X,T2,T2s. ⬆*[f] X ≘ T2 ⨮ T2s →
+                        ∃∃T1,T1s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ 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: ∀f:rtmap. ∀V1s,U1,T2. ⬆*[f] Ⓐ V1s.U1 ≘ T2 →
+                        ∃∃V2s,U2. ⬆*[f] V1s ≘ V2s & ⬆*[f] U1 ≘ U2 &
+                                  T2 = Ⓐ V2s.U2.
+#f #V1s elim V1s -V1s
+[ /3 width=5 by ex3_2_intro, liftsv_nil/
+| #V1 #V1s #IHV1s #T1 #X #H elim (lifts_inv_flat1 … H) -H
+  #V2 #Y #HV12 #HY #H destruct elim (IHV1s … HY) -IHV1s -HY
+  #V2s #T2 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+]
+qed-.
+
+lemma lifts_inv_applv2: ∀f:rtmap. ∀V2s,U2,T1. ⬆*[f] T1 ≘ Ⓐ V2s.U2 →
+                        ∃∃V1s,U1. ⬆*[f] V1s ≘ V2s & ⬆*[f] U1 ≘ U2 &
+                                  T1 = Ⓐ V1s.U1.
+#f #V2s elim V2s -V2s
+[ /3 width=5 by ex3_2_intro, liftsv_nil/
+| #V2 #V2s #IHV2s #T2 #X #H elim (lifts_inv_flat2 … H) -H
+  #V1 #Y #HV12 #HY #H destruct elim (IHV2s … HY) -IHV2s -HY
+  #V1s #T1 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: includes: liftv_total *)
+lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⬆*[f] T1s ≘ T2s.
+#f #T1s elim T1s -T1s
+[ /2 width=2 by liftsv_nil, ex_intro/
+| #T1 #T1s * #T2s #HT12s
+  elim (lifts_total T1 f) /3 width=2 by liftsv_cons, ex_intro/
+]
+qed-.
+
+(* Basic_1: was: lifts1_flat (right to left) *)
+lemma lifts_applv: ∀f:rtmap. ∀V1s,V2s. ⬆*[f] V1s ≘ V2s →
+                   ∀T1,T2. ⬆*[f] T1 ≘ T2 →
+                   ⬆*[f] Ⓐ V1s.T1 ≘ Ⓐ 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 *)