(* GENERIC TERM RELOCATION **************************************************)
inductive lifts: list2 nat nat → relation term ≝
-| lifts_nil : ∀T. lifts ⟠ T T
+| lifts_nil : ∀T. lifts (⟠) T T
| lifts_cons: ∀T1,T,T2,des,d,e.
⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} @ des) T1 T2
.
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.
+qed-.
lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2.
-/2 width=3/ qed-.
+/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 →
#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.
+ /2 width=3 by ex2_intro/
+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-.
+/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.
[ #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/
+ >(lift_inv_sort1 … H) -H /2 width=1 by/
]
qed-.
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/
+[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/
| #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/
+ elim (IH … H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/
]
qed-.
[ #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/
+ >(lift_inv_gref1 … H) -H /2 width=1 by/
]
qed-.
T2 = ⓑ{a,I} V2. U2.
#a #I #T2 #des elim des -des
[ #V1 #U1 #H
- <(lifts_inv_nil … H) -H /2 width=5/
+ <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
| #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/
+ /3 width=5 by ex3_2_intro, lifts_cons/
]
qed-.
T2 = ⓕ{I} V2. U2.
#I #T2 #des elim des -des
[ #V1 #U1 #H
- <(lifts_inv_nil … H) -H /2 width=5/
+ <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
| #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/
+ /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/
+#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/
+#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_sn/
qed-.
(* Basic properties *********************************************************)
#a #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/
+ elim (lifts_inv_cons … H) -H /3 width=3 by lift_bind, lifts_cons/
]
qed.
#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/
+ 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/
-#d #e #des #IH #T1
-elim (lift_total T1 d e) #T #HT1
-elim (IH T) -IH /3 width=4/
+#des elim des -des /2 width=2 by lifts_nil, ex_intro/
+#d #e #des #IH #T1 elim (lift_total T1 d e)
+#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_intro/
qed.