(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
(* Basic_2A1: includes: liftv_nil liftv_cons *)
-inductive liftsv (f:rtmap): relation (list term) ≝
-| liftsv_nil : liftsv f (â\92º) (â\92º)
+inductive liftsv (f): relation … ≝
+| liftsv_nil : liftsv f (â\93\94) (â\93\94)
| 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).
+interpretation "uniform relocation (term vector)"
+ 'RLift i T1s T2s = (liftsv (pr_uni i) T1s T2s).
+
(* Basic inversion lemmas ***************************************************)
-fact liftsv_inv_nil1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → X = Ⓔ → Y = Ⓔ.
+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 = Ⓔ.
+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.
+fact liftsv_inv_cons1_aux (f):
+ ∀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.
+lemma liftsv_inv_cons1 (f):
+ ∀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 = Ⓔ.
+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 = Ⓔ.
+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.
+fact liftsv_inv_cons2_aux (f):
+ ∀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.
+lemma liftsv_inv_cons2 (f):
+ ∀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.
+lemma lifts_inv_applv1 (f):
+ ∀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
]
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.
+lemma lifts_inv_applv2 (f):
+ ∀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
(* Basic properties *********************************************************)
(* Basic_2A1: includes: liftv_total *)
-lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⇧*[f] T1s ≘ T2s.
+lemma liftsv_total (f):
+ ∀T1s. ∃T2s. ⇧*[f] T1s ≘ T2s.
#f #T1s elim T1s -T1s
[ /2 width=2 by liftsv_nil, ex_intro/
| #T1 #T1s * #T2s #HT12s
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.
+lemma lifts_applv (f):
+ ∀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.
+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