(* GENERIC TERM RELOCATION **************************************************)
-inductive lifts: list2 nat nat → relation term ≝
-| lifts_nil : â\88\80T. lifts (â\9f ) T T
-| lifts_cons: ∀T1,T,T2,des,d,e.
- â\87§[d,e] T1 â\89¡ T â\86\92 lifts des T T2 â\86\92 lifts ({d, e} @ des) T1 T2
+inductive lifts: list2 ynat nat → relation term ≝
+| lifts_nil : â\88\80T. lifts (â\97\8a) T T
+| lifts_cons: ∀T1,T,T2,cs,l,m.
+ â¬\86[l,m] T1 â\89¡ T â\86\92 lifts cs T T2 â\86\92 lifts ({l, m} @ cs) T1 T2
.
interpretation "generic relocation (term)"
- 'RLiftStar des T1 T2 = (lifts des T1 T2).
+ 'RLiftStar cs T1 T2 = (lifts cs 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
+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: â\88\80T1,T2. â\87§*[â\9f ] T1 ≡ T2 → T1 = T2.
+lemma lifts_inv_nil: â\88\80T1,T2. â¬\86*[â\97\8a] T1 ≡ T2 → T1 = T2.
/2 width=3 by lifts_inv_nil_aux/ qed-.
-fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 →
- ∀d,e,tl. des = {d, e} @ tl →
- â\88\83â\88\83T. â\87§[d, e] T1 â\89¡ T & â\87§*[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
+fact lifts_inv_cons_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 →
+ ∀l,m,tl. cs = {l, m} @ tl →
+ â\88\83â\88\83T. â¬\86[l, m] T1 â\89¡ T & â¬\86*[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,d,e,des. ⇧*[{d, e} @ des] T1 ≡ T2 →
- â\88\83â\88\83T. â\87§[d, e] T1 â\89¡ T & â\87§*[des] T ≡ T2.
+lemma lifts_inv_cons: ∀T1,T2,l,m,cs. ⬆*[{l, m} @ cs] T1 ≡ T2 →
+ â\88\83â\88\83T. â¬\86[l, m] T1 â\89¡ T & â¬\86*[cs] T ≡ T2.
/2 width=3 by lifts_inv_cons_aux/ qed-.
(* Basic_1: was: lift1_sort *)
-lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k.
-#T2 #k #des elim des -des
+lemma lifts_inv_sort1: ∀T2,k,cs. ⬆*[cs] ⋆k ≡ T2 → T2 = ⋆k.
+#T2 #k #cs elim cs -cs
[ #H <(lifts_inv_nil … H) -H //
-| #d #e #des #IH #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,des,i1. ⇧*[des] #i1 ≡ T2 →
- ∃∃i2. @⦃i1, des⦄ ≡ i2 & T2 = #i2.
-#T2 #des elim des -des
+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/
-| #d #e #des #IH #i1 #H
+| #l #m #cs #IH #i1 #H
elim (lifts_inv_cons … H) -H #X #H1 #H2
- elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct
+ 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,des. ⇧*[des] §p ≡ T2 → T2 = §p.
-#T2 #p #des elim des -des
+lemma lifts_inv_gref1: ∀T2,p,cs. ⬆*[cs] §p ≡ T2 → T2 = §p.
+#T2 #p #cs elim cs -cs
[ #H <(lifts_inv_nil … H) -H //
-| #d #e #des #IH #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,des,V1,U1. ⇧*[des] ⓑ{a,I} V1. U1 ≡ T2 →
- â\88\83â\88\83V2,U2. â\87§*[des] V1 â\89¡ V2 & â\87§*[des + 1] U1 ≡ U2 &
+lemma lifts_inv_bind1: ∀a,I,T2,cs,V1,U1. ⬆*[cs] ⓑ{a,I} V1. U1 ≡ T2 →
+ â\88\83â\88\83V2,U2. â¬\86*[cs] V1 â\89¡ V2 & â¬\86*[cs + 1] U1 ≡ U2 &
T2 = ⓑ{a,I} V2. U2.
-#a #I #T2 #des elim des -des
+#a #I #T2 #cs elim cs -cs
[ #V1 #U1 #H
<(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
-| #d #e #des #IHdes #V1 #U1 #H
+| #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 (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #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,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 →
- â\88\83â\88\83V2,U2. â\87§*[des] V1 â\89¡ V2 & â\87§*[des] U1 ≡ U2 &
+lemma lifts_inv_flat1: ∀I,T2,cs,V1,U1. ⬆*[cs] ⓕ{I} V1. U1 ≡ T2 →
+ â\88\83â\88\83V2,U2. â¬\86*[cs] V1 â\89¡ V2 & â¬\86*[cs] U1 ≡ U2 &
T2 = ⓕ{I} V2. U2.
-#I #T2 #des elim des -des
+#I #T2 #cs elim cs -cs
[ #V1 #U1 #H
<(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
-| #d #e #des #IHdes #V1 #U1 #H
+| #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 (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #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,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_dx/
+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,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_sn/
+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,des. ⇧*[des] V1 ≡ V2 →
- â\88\80T1. â\87§*[des + 1] T1 ≡ T2 →
- â\87§*[des] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
-#a #I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+lemma lifts_bind: ∀a,I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 →
+ â\88\80T1. â¬\86*[cs + 1] T1 ≡ T2 →
+ â¬\86*[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 #des #d #e #HV1 #_ #IHV #T1 #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,des. ⇧*[des] V1 ≡ V2 →
- â\88\80T1. â\87§*[des] T1 ≡ T2 →
- â\87§*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
-#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+lemma lifts_flat: ∀I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 →
+ â\88\80T1. â¬\86*[cs] T1 ≡ T2 →
+ â¬\86*[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 #des #d #e #HV1 #_ #IHV #T1 #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: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2.
-#des elim des -des /2 width=2 by lifts_nil, ex_intro/
-#d #e #des #IH #T1 elim (lift_total T1 d e)
+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.
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