the support for candidates of reducibility continues ...

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / unfold / lifts.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/lifts.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/lifts.ma
index 9994f44a2cab71f94d7df92c1ff72a078f43d029..27ddc5174854a14bb167c60a943fb8d8291e7a18 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/unfold/lifts.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/lifts.ma
@@ -12,14 +12,139 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic_2/substitution/lift_vector.ma".
+include "Basic_2/substitution/lift.ma".
+include "Basic_2/unfold/gr2.ma".
 
-(* GENERIC RELOCATION *******************************************************)
+(* GENERIC TERM RELOCATION **************************************************)
 
 inductive lifts: list2 nat nat → relation term ≝
 | lifts_nil : ∀T. lifts ⟠ T T
 | lifts_cons: ∀T1,T,T2,des,d,e.
-              â\87\91[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} :: des) T1 T2
+              â\87§[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} :: des) T1 T2
 .
 
-interpretation "generic relocation" 'RLift des T1 T2 = (lifts des T1 T2).
+interpretation "generic relocation (term)"
+   'RLiftStar des T1 T2 = (lifts des T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2.
+#T1 #T2 #des * -T1 -T2 -des //
+#T1 #T #T2 #d #e #des #_ #_ #H destruct
+qed.
+
+lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2.
+/2 width=3/ qed-.
+
+fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 →
+                         ∀d,e,tl. des = {d, e} :: tl →
+                         ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2.
+#T1 #T2 #des * -T1 -T2 -des
+[ #T #d #e #tl #H destruct
+| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+  /2 width=3/
+qed.
+
+lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} :: des] T1 ≡ T2 →
+                      ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2.
+/2 width=3/ qed-.
+
+(* Basic_1: was: lift1_sort *)
+lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k.
+#T2 #k #des elim des -des
+[ #H <(lifts_inv_nil … H) -H //
+| #d #e #des #IH #H
+  elim (lifts_inv_cons … H) -H #X #H
+  >(lift_inv_sort1 … H) -H /2 width=1/
+]
+qed-.
+
+(* Basic_1: was: lift1_lref *)
+lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 →
+                       ∃∃i2. @[i1] des ≡ i2 & T2 = #i2.
+#T2 #des elim des -des
+[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3/
+| #d #e #des #IH #i1 #H
+  elim (lifts_inv_cons … H) -H #X #H1 #H2
+  elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct
+  elim (IH … H2) -IH -H2 /3 width=3/
+]
+qed-.
+
+lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p.
+#T2 #p #des elim des -des
+[ #H <(lifts_inv_nil … H) -H //
+| #d #e #des #IH #H
+  elim (lifts_inv_cons … H) -H #X #H
+  >(lift_inv_gref1 … H) -H /2 width=1/
+]
+qed-.
+
+(* Basic_1: was: lift1_bind *)
+lemma lifts_inv_bind1: ∀I,T2,des,V1,U1. ⇧*[des] 𝕓{I} V1. U1 ≡ T2 →
+                       ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 &
+                                T2 = 𝕓{I} V2. U2.
+#I #T2 #des elim des -des
+[ #V1 #U1 #H
+  <(lifts_inv_nil … H) -H /2 width=5/
+| #d #e #des #IHdes #V1 #U1 #H
+  elim (lifts_inv_cons … H) -H #X #H #HT2
+  elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct
+  elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+  /3 width=5/
+]
+qed-.
+
+(* Basic_1: was: lift1_flat *)
+lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] 𝕗{I} V1. U1 ≡ T2 →
+                       ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 &
+                                T2 = 𝕗{I} V2. U2.
+#I #T2 #des elim des -des
+[ #V1 #U1 #H
+  <(lifts_inv_nil … H) -H /2 width=5/
+| #d #e #des #IHdes #V1 #U1 #H
+  elim (lifts_inv_cons … H) -H #X #H #HT2
+  elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct
+  elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+  /3 width=5/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝕊[T1] → 𝕊[T2].
+#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_dx/
+qed-.
+
+lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝕊[T2] → 𝕊[T1].
+#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_sn/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_bind: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
+                  ∀T1. ⇧*[des + 1] T1 ≡ T2 →
+                  ⇧*[des] 𝕓{I} V1. T1 ≡ 𝕓{I} V2. T2.
+#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+  elim (lifts_inv_cons … H) -H /3 width=3/
+]
+qed.
+
+lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
+                  ∀T1. ⇧*[des] T1 ≡ T2 →
+                  ⇧*[des] 𝕗{I} V1. T1 ≡ 𝕗{I} V2. T2.
+#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+  elim (lifts_inv_cons … H) -H /3 width=3/
+]
+qed.
+
+lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2.
+#des elim des -des /2 width=2/
+#d #e #des #IH #T1
+elim (lift_total T1 d e) #T #HT1
+elim (IH T) -IH /3 width=4/
+qed.