(* *)
(**************************************************************************)
-include "basic_2/grammar/term_vector.ma".
+include "basic_2/syntax/term_vector.ma".
include "basic_2/relocation/lifts.ma".
(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
(* Basic_2A1: includes: liftv_nil liftv_cons *)
-inductive liftsv (f): relation (list term) ≝
-| liftsv_nil : liftsv f (â\97\8a) (â\97\8a)
-| liftsv_cons: ∀T1c,T2c,T1,T2.
- â¬\86*[f] T1 â\89¡ T2 â\86\92 liftsv f T1c T2c →
- liftsv f (T1 @ T1c) (T2 @ T2c)
+inductive liftsv (f:rtmap): relation (list term) ≝
+| liftsv_nil : liftsv f (â\92º) (â\92º)
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+ â¬\86*[f] T1 â\89\98 T2 â\86\92 liftsv f T1s T2s →
+ liftsv f (T1 ⨮ T1s) (T2 ⨮ T2s)
.
-interpretation "generic relocation (vector)"
- 'RLiftStar f T1c T2c = (liftsv f T1c T2c).
+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).
(* Basic inversion lemmas ***************************************************)
-fact liftsv_inv_nil1_aux: ∀X,Y,f. ⬆*[f] X ≡ Y → X = ◊ → Y = ◊.
-#X #Y #f * -X -Y //
-#T1c #T2c #T1 #T2 #_ #_ #H destruct
+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: ∀Y,f. ⬆*[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: ∀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/
+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.
+#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: ∀T1,T1c,Y,f. ⬆*[f] T1 @ T1c ≡ Y →
- ∃∃T2,T2c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
- Y = T2 @ T2c.
+lemma liftsv_inv_cons1: ∀f:rtmap. ∀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: ∀X,Y,f. ⬆*[f] X ≡ Y → Y = ◊ → X = ◊.
-#X #Y #f * -X -Y //
-#T1c #T2c #T1 #T2 #_ #_ #H destruct
+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: ∀X,f. ⬆*[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: ∀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/
+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.
+#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: ∀X,T2,T2c,f. ⬆*[f] X ≡ T2 @ T2c →
- ∃∃T1,T1c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
- X = T1 @ T1c.
+lemma liftsv_inv_cons2: ∀f:rtmap. ∀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: ∀V1c,U1,T2,f. ⬆*[f] Ⓐ V1c.U1 ≡ T2 →
- ∃∃V2c,U2. ⬆*[f] V1c ≡ V2c & ⬆*[f] U1 ≡ U2 &
- T2 = Ⓐ V2c.U2.
-#V1c elim V1c -V1c
+lemma lifts_inv_applv1: ∀f:rtmap. ∀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 #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/
+| #V1 #V1s #IHV1s #T1 #X #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/
]
qed-.
-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
+lemma lifts_inv_applv2: ∀f:rtmap. ∀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 #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/
+| #V2 #V2s #IHV2s #T2 #X #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/
]
qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: includes: liftv_total *)
-lemma liftsv_total: ∀f. ∀T1c:list term. ∃T2c. ⬆*[f] T1c ≡ T2c.
-#f #T1c elim T1c -T1c
+lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⬆*[f] T1s ≘ T2s.
+#f #T1s elim T1s -T1s
[ /2 width=2 by liftsv_nil, ex_intro/
-| #T1 #T1c * #T2c #HT12c
+| #T1 #T1s * #T2s #HT12s
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: ∀V1c,V2c,f. ⬆*[f] V1c ≡ V2c →
- â\88\80T1,T2. â¬\86*[f] T1 â\89¡ T2 →
- ⬆*[f] Ⓐ V1c. T1 ≡ Ⓐ V2c. T2.
-#V1c #V2c #f #H elim H -V1c -V2c /3 width=1 by lifts_flat/
+lemma lifts_applv: ∀f:rtmap. ∀V1s,V2s. ⬆*[f] V1s ≘ V2s →
+ â\88\80T1,T2. â¬\86*[f] T1 â\89\98 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.
+#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
+ elim (IH … Hf) -IH
+ elim (lifts_split_trans … HT12 … Hf) -HT12 -Hf
+ /3 width=5 by liftsv_cons, ex2_intro/
+]
+qed-.
+
(* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *)
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