--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "basic_2A/notation/relations/rliftstar_3.ma".
+include "basic_2A/substitution/lift.ma".
+include "basic_2A/multiple/mr2_plus.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+inductive lifts: list2 nat nat → relation term ≝
+| lifts_nil : ∀T. lifts (◊) T T
+| lifts_cons: ∀T1,T,T2,cs,l,m.
+ ⬆[l,m] T1 ≡ T → lifts cs T T2 → lifts ({l, m} @ cs) T1 T2
+.
+
+interpretation "generic relocation (term)"
+ 'RLiftStar cs T1 T2 = (lifts cs T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lifts_inv_nil_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → cs = ◊ → T1 = T2.
+#T1 #T2 #cs * -T1 -T2 -cs //
+#T1 #T #T2 #l #m #cs #_ #_ #H destruct
+qed-.
+
+lemma lifts_inv_nil: ∀T1,T2. ⬆*[◊] T1 ≡ T2 → T1 = T2.
+/2 width=3 by lifts_inv_nil_aux/ qed-.
+
+fact lifts_inv_cons_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 →
+ ∀l,m,tl. cs = {l, m} @ tl →
+ ∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[tl] T ≡ T2.
+#T1 #T2 #cs * -T1 -T2 -cs
+[ #T #l #m #tl #H destruct
+| #T1 #T #T2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
+ /2 width=3 by ex2_intro/
+qed-.
+
+lemma lifts_inv_cons: ∀T1,T2,l,m,cs. ⬆*[{l, m} @ cs] T1 ≡ T2 →
+ ∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[cs] T ≡ T2.
+/2 width=3 by lifts_inv_cons_aux/ qed-.
+
+(* Basic_1: was: lift1_sort *)
+lemma lifts_inv_sort1: ∀T2,k,cs. ⬆*[cs] ⋆k ≡ T2 → T2 = ⋆k.
+#T2 #k #cs elim cs -cs
+[ #H <(lifts_inv_nil … H) -H //
+| #l #m #cs #IH #H
+ elim (lifts_inv_cons … H) -H #X #H
+ >(lift_inv_sort1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: was: lift1_lref *)
+lemma lifts_inv_lref1: ∀T2,cs,i1. ⬆*[cs] #i1 ≡ T2 →
+ ∃∃i2. @⦃i1, cs⦄ ≡ i2 & T2 = #i2.
+#T2 #cs elim cs -cs
+[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/
+| #l #m #cs #IH #i1 #H
+ elim (lifts_inv_cons … H) -H #X #H1 #H2
+ elim (lift_inv_lref1 … H1) -H1 * #Hli1 #H destruct
+ elim (IH … H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/
+]
+qed-.
+
+lemma lifts_inv_gref1: ∀T2,p,cs. ⬆*[cs] §p ≡ T2 → T2 = §p.
+#T2 #p #cs elim cs -cs
+[ #H <(lifts_inv_nil … H) -H //
+| #l #m #cs #IH #H
+ elim (lifts_inv_cons … H) -H #X #H
+ >(lift_inv_gref1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: was: lift1_bind *)
+lemma lifts_inv_bind1: ∀a,I,T2,cs,V1,U1. ⬆*[cs] ⓑ{a,I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⬆*[cs] V1 ≡ V2 & ⬆*[cs + 1] U1 ≡ U2 &
+ T2 = ⓑ{a,I} V2. U2.
+#a #I #T2 #cs elim cs -cs
+[ #V1 #U1 #H
+ <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
+| #l #m #cs #IHcs #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 (IHcs … HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ /3 width=5 by ex3_2_intro, lifts_cons/
+]
+qed-.
+
+(* Basic_1: was: lift1_flat *)
+lemma lifts_inv_flat1: ∀I,T2,cs,V1,U1. ⬆*[cs] ⓕ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⬆*[cs] V1 ≡ V2 & ⬆*[cs] U1 ≡ U2 &
+ T2 = ⓕ{I} V2. U2.
+#I #T2 #cs elim cs -cs
+[ #V1 #U1 #H
+ <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
+| #l #m #cs #IHcs #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 (IHcs … HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ /3 width=5 by ex3_2_intro, lifts_cons/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lifts_simple_dx: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 #cs #H elim H -T1 -T2 -cs /3 width=5 by lift_simple_dx/
+qed-.
+
+lemma lifts_simple_sn: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#T1 #T2 #cs #H elim H -T1 -T2 -cs /3 width=5 by lift_simple_sn/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_bind: ∀a,I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 →
+ ∀T1. ⬆*[cs + 1] T1 ≡ T2 →
+ ⬆*[cs] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
+#a #I #T2 #V1 #V2 #cs #H elim H -V1 -V2 -cs
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #cs #l #m #HV1 #_ #IHV #T1 #H
+ elim (lifts_inv_cons … H) -H /3 width=3 by lift_bind, lifts_cons/
+]
+qed.
+
+lemma lifts_flat: ∀I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 →
+ ∀T1. ⬆*[cs] T1 ≡ T2 →
+ ⬆*[cs] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #T2 #V1 #V2 #cs #H elim H -V1 -V2 -cs
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #cs #l #m #HV1 #_ #IHV #T1 #H
+ elim (lifts_inv_cons … H) -H /3 width=3 by lift_flat, lifts_cons/
+]
+qed.
+
+lemma lifts_total: ∀cs,T1. ∃T2. ⬆*[cs] T1 ≡ T2.
+#cs elim cs -cs /2 width=2 by lifts_nil, ex_intro/
+#l #m #cs #IH #T1 elim (lift_total T1 l m)
+#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_intro/
+qed.
projects / helm.git / blobdiff