(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
(* Basic_2A1: includes: liftv_nil liftv_cons *)
-inductive liftsv (t:trace) : relation (list term) ≝
-| liftsv_nil : liftsv t (◊) (◊)
-| liftsv_cons: ∀T1s,T2s,T1,T2.
- ⬆*[t] T1 ≡ T2 → liftsv t T1s T2s →
- liftsv t (T1 @ T1s) (T2 @ T2s)
+inductive liftsv (f): relation (list term) ≝
+| liftsv_nil : liftsv f (◊) (◊)
+| liftsv_cons: ∀T1c,T2c,T1,T2.
+ ⬆*[f] T1 ≡ T2 → liftsv f T1c T2c →
+ liftsv f (T1 @ T1c) (T2 @ T2c)
.
interpretation "generic relocation (vector)"
- 'RLiftStar t T1s T2s = (liftsv t T1s T2s).
+ 'RLiftStar f T1c T2c = (liftsv f T1c T2c).
(* Basic inversion lemmas ***************************************************)
-fact liftsv_inv_nil1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → X = ◊ → Y = ◊.
-#X #Y #t * -X -Y //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
+fact liftsv_inv_nil1_aux: ∀X,Y,f. ⬆*[f] X ≡ Y → X = ◊ → Y = ◊.
+#X #Y #f * -X -Y //
+#T1c #T2c #T1 #T2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes: liftv_inv_nil1 *)
-lemma liftsv_inv_nil1: ∀Y,t. ⬆*[t] ◊ ≡ Y → Y = ◊.
+lemma liftsv_inv_nil1: ∀Y,f. ⬆*[f] ◊ ≡ Y → Y = ◊.
/2 width=5 by liftsv_inv_nil1_aux/ qed-.
-fact liftsv_inv_cons1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y →
- ∀T1,T1s. X = T1 @ T1s →
- ∃∃T2,T2s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s &
- Y = T2 @ T2s.
-#X #Y #t * -X -Y
-[ #U1 #U1s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
+fact liftsv_inv_cons1_aux: ∀X,Y,f. ⬆*[f] X ≡ Y →
+ ∀T1,T1c. X = T1 @ T1c →
+ ∃∃T2,T2c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
+ Y = T2 @ T2c.
+#X #Y #f * -X -Y
+[ #U1 #U1c #H destruct
+| #T1c #T2c #T1 #T2 #HT12 #HT12c #U1 #U1c #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_2A1: includes: liftv_inv_cons1 *)
-lemma liftsv_inv_cons1: ∀T1,T1s,Y,t. ⬆*[t] T1 @ T1s ≡ Y →
- ∃∃T2,T2s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s &
- Y = T2 @ T2s.
+lemma liftsv_inv_cons1: ∀T1,T1c,Y,f. ⬆*[f] T1 @ T1c ≡ Y →
+ ∃∃T2,T2c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
+ Y = T2 @ T2c.
/2 width=3 by liftsv_inv_cons1_aux/ qed-.
-fact liftsv_inv_nil2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → Y = ◊ → X = ◊.
-#X #Y #t * -X -Y //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
+fact liftsv_inv_nil2_aux: ∀X,Y,f. ⬆*[f] X ≡ Y → Y = ◊ → X = ◊.
+#X #Y #f * -X -Y //
+#T1c #T2c #T1 #T2 #_ #_ #H destruct
qed-.
-lemma liftsv_inv_nil2: ∀X,t. ⬆*[t] X ≡ ◊ → X = ◊.
+lemma liftsv_inv_nil2: ∀X,f. ⬆*[f] X ≡ ◊ → X = ◊.
/2 width=5 by liftsv_inv_nil2_aux/ qed-.
-fact liftsv_inv_cons2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y →
- ∀T2,T2s. Y = T2 @ T2s →
- ∃∃T1,T1s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s &
- X = T1 @ T1s.
-#X #Y #t * -X -Y
-[ #U2 #U2s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U2 #U2s #H destruct /2 width=5 by ex3_2_intro/
+fact liftsv_inv_cons2_aux: ∀X,Y,f. ⬆*[f] X ≡ Y →
+ ∀T2,T2c. Y = T2 @ T2c →
+ ∃∃T1,T1c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
+ X = T1 @ T1c.
+#X #Y #f * -X -Y
+[ #U2 #U2c #H destruct
+| #T1c #T2c #T1 #T2 #HT12 #HT12c #U2 #U2c #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma liftsv_inv_cons2: ∀X,T2,T2s,t. ⬆*[t] X ≡ T2 @ T2s →
- ∃∃T1,T1s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s &
- X = T1 @ T1s.
+lemma liftsv_inv_cons2: ∀X,T2,T2c,f. ⬆*[f] X ≡ T2 @ T2c →
+ ∃∃T1,T1c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
+ X = T1 @ T1c.
/2 width=3 by liftsv_inv_cons2_aux/ qed-.
(* Basic_1: was: lifts1_flat (left to right) *)
-lemma lifts_inv_applv1: ∀V1s,U1,T2,t. ⬆*[t] Ⓐ V1s.U1 ≡ T2 →
- ∃∃V2s,U2. ⬆*[t] V1s ≡ V2s & ⬆*[t] U1 ≡ U2 &
- T2 = Ⓐ V2s.U2.
-#V1s elim V1s -V1s
+lemma lifts_inv_applv1: ∀V1c,U1,T2,f. ⬆*[f] Ⓐ V1c.U1 ≡ T2 →
+ ∃∃V2c,U2. ⬆*[f] V1c ≡ V2c & ⬆*[f] U1 ≡ U2 &
+ T2 = Ⓐ V2c.U2.
+#V1c elim V1c -V1c
[ /3 width=5 by ex3_2_intro, liftsv_nil/
-| #V1 #V1s #IHV1s #T1 #X #t #H elim (lifts_inv_flat1 … H) -H
- #V2 #Y #HV12 #HY #H destruct elim (IHV1s … HY) -IHV1s -HY
- #V2s #T2 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+| #V1 #V1c #IHV1c #T1 #X #f #H elim (lifts_inv_flat1 … H) -H
+ #V2 #Y #HV12 #HY #H destruct elim (IHV1c … HY) -IHV1c -HY
+ #V2c #T2 #HV12c #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
]
qed-.
-lemma lifts_inv_applv2: ∀V2s,U2,T1,t. ⬆*[t] T1 ≡ Ⓐ V2s.U2 →
- ∃∃V1s,U1. ⬆*[t] V1s ≡ V2s & ⬆*[t] U1 ≡ U2 &
- T1 = Ⓐ V1s.U1.
-#V2s elim V2s -V2s
+lemma lifts_inv_applv2: ∀V2c,U2,T1,f. ⬆*[f] T1 ≡ Ⓐ V2c.U2 →
+ ∃∃V1c,U1. ⬆*[f] V1c ≡ V2c & ⬆*[f] U1 ≡ U2 &
+ T1 = Ⓐ V1c.U1.
+#V2c elim V2c -V2c
[ /3 width=5 by ex3_2_intro, liftsv_nil/
-| #V2 #V2s #IHV2s #T2 #X #t #H elim (lifts_inv_flat2 … H) -H
- #V1 #Y #HV12 #HY #H destruct elim (IHV2s … HY) -IHV2s -HY
- #V1s #T1 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+| #V2 #V2c #IHV2c #T2 #X #f #H elim (lifts_inv_flat2 … H) -H
+ #V1 #Y #HV12 #HY #H destruct elim (IHV2c … HY) -IHV2c -HY
+ #V1c #T1 #HV12c #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
]
qed-.
(* Basic properties *********************************************************)
+(* Basic_2A1: includes: liftv_total *)
+lemma liftsv_total: ∀f. ∀T1c:list term. ∃T2c. ⬆*[f] T1c ≡ T2c.
+#f #T1c elim T1c -T1c
+[ /2 width=2 by liftsv_nil, ex_intro/
+| #T1 #T1c * #T2c #HT12c
+ elim (lifts_total T1 f) /3 width=2 by liftsv_cons, ex_intro/
+]
+qed-.
+
(* Basic_1: was: lifts1_flat (right to left) *)
-lemma lifts_applv: ∀V1s,V2s,t. ⬆*[t] V1s ≡ V2s →
- ∀T1,T2. ⬆*[t] T1 ≡ T2 →
- ⬆*[t] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
-#V1s #V2s #t #H elim H -V1s -V2s /3 width=1 by lifts_flat/
+lemma lifts_applv: ∀V1c,V2c,f. ⬆*[f] V1c ≡ V2c →
+ ∀T1,T2. ⬆*[f] T1 ≡ T2 →
+ ⬆*[f] Ⓐ V1c. T1 ≡ Ⓐ V2c. T2.
+#V1c #V2c #f #H elim H -V1c -V2c /3 width=1 by lifts_flat/
qed.
-(* Basic_2A1: removed theorems 1: liftv_total *)
(* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *)
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