(* DELIFTING SUBSTITUTION ***************************************************)
(* Policy: depth (level) metavariables: d, e (as for lift) *)
-let rec dsubst C d M on M ≝ match M with
-[ VRef i ⇒ tri … i d (#i) (↑[i] C) (#(i-1))
-| Abst A ⇒ 𝛌. (dsubst C (d+1) A)
-| Appl B A ⇒ @ (dsubst C d B). (dsubst C d A)
+let rec dsubst D d M on M ≝ match M with
+[ VRef i ⇒ tri … i d (#i) (↑[i] D) (#(i-1))
+| Abst A ⇒ 𝛌. (dsubst D (d+1) A)
+| Appl B A ⇒ @ (dsubst D d B). (dsubst D d A)
].
interpretation "delifting substitution"
- 'DSubst C d M = (dsubst C d M).
+ 'DSubst D d M = (dsubst D d M).
(* Note: the notation with "/" does not work *)
-notation "hvbox( [ term 46 d ⬐ break term 46 C ] break term 46 M )"
+notation "hvbox( [ term 46 d ⬐ break term 46 D ] break term 46 M )"
non associative with precedence 46
- for @{ 'DSubst $C $d $M }.
+ for @{ 'DSubst $D $d $M }.
-notation > "hvbox( [ ⬐ term 46 C ] break term 46 M )"
+notation > "hvbox( [ ⬐ term 46 D ] break term 46 M )"
non associative with precedence 46
- for @{ 'DSubst $C 0 $M }.
+ for @{ 'DSubst $D 0 $M }.
-lemma dsubst_vref_lt: ∀i,d,C. i < d → [d ⬐ C] #i = #i.
+lemma dsubst_vref_lt: ∀i,d,D. i < d → [d ⬐ D] #i = #i.
normalize /2 width=1/
qed.
-lemma dsubst_vref_eq: ∀d,C. [d ⬐ C] #d = ↑[d]C.
+lemma dsubst_vref_eq: ∀d,D. [d ⬐ D] #d = ↑[d]D.
normalize //
qed.
-lemma dsubst_vref_gt: ∀i,d,C. d < i → [d ⬐ C] #i = #(i-1).
+lemma dsubst_vref_gt: ∀i,d,D. d < i → [d ⬐ D] #i = #(i-1).
normalize /2 width=1/
qed.
-theorem dsubst_lift_le: ∀h,C,M,d1,d2. d2 ≤ d1 →
- [d2 ⬐ ↑[d1 - d2, h] C] ↑[d1 + 1, h] M = ↑[d1, h] [d2 ⬐ C] M.
-#h #C #M elim M -M
+theorem dsubst_lift_le: ∀h,D,M,d1,d2. d2 ≤ d1 →
+ [d2 ⬐ ↑[d1 - d2, h] D] ↑[d1 + 1, h] M = ↑[d1, h] [d2 ⬐ D] M.
+#h #D #M elim M -M
[ #i #d1 #d2 #Hd21 elim (lt_or_eq_or_gt i d2) #Hid2
[ lapply (lt_to_le_to_lt … Hid2 Hd21) -Hd21 #Hid1
>(dsubst_vref_lt … Hid2) >(lift_vref_lt … Hid1) >lift_vref_lt /2 width=1/
]
qed.
-theorem dsubst_lift_be: ∀h,C,M,d1,d2. d1 ≤ d2 → d2 ≤ d1 + h →
- [d2 ⬐ C] ↑[d1, h + 1] M = ↑[d1, h] M.
-#h #C #M elim M -M
+theorem dsubst_lift_be: ∀h,D,M,d1,d2. d1 ≤ d2 → d2 ≤ d1 + h →
+ [d2 ⬐ D] ↑[d1, h + 1] M = ↑[d1, h] M.
+#h #D #M elim M -M
[ #i #d1 #d2 #Hd12 #Hd21 elim (lt_or_ge i d1) #Hid1
[ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 -Hd21 #Hid2
>(lift_vref_lt … Hid1) >(lift_vref_lt … Hid1) /2 width=1/
]
qed.
-theorem dsubst_lift_ge: ∀h,C,M,d1,d2. d1 + h ≤ d2 →
- [d2 ⬐ C] ↑[d1, h] M = ↑[d1, h] [d2 - h ⬐ C] M.
-#h #C #M elim M -M
+theorem dsubst_lift_ge: ∀h,D,M,d1,d2. d1 + h ≤ d2 →
+ [d2 ⬐ D] ↑[d1, h] M = ↑[d1, h] [d2 - h ⬐ D] M.
+#h #D #M elim M -M
[ #i #d1 #d2 #Hd12 elim (lt_or_eq_or_gt i (d2-h)) #Hid2h
[ >(dsubst_vref_lt … Hid2h) elim (lt_or_ge i d1) #Hid1
[ lapply (lt_to_le_to_lt … (d1+h) Hid1 ?) -Hid2h // #Hid1h
]
qed.
-theorem subst_subst_ge: ∀C1,C2,M,d1,d2. d1 ≤ d2 →
- [d2 ⬐ C2] [d1 ⬐ C1] M = [d1 ⬐ [d2 - d1 ⬐ C2] C1] [d2 + 1 ⬐ C2] M.
-#C1 #C2 #M elim M -M
+theorem subst_subst_ge: ∀D1,D2,M,d1,d2. d1 ≤ d2 →
+ [d2 ⬐ D2] [d1 ⬐ D1] M = [d1 ⬐ [d2 - d1 ⬐ D2] D1] [d2 + 1 ⬐ D2] M.
+#D1 #D2 #M elim M -M
[ #i #d1 #d2 #Hd12 elim (lt_or_eq_or_gt i d1) #Hid1
[ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
>(dsubst_vref_lt … Hid1) >(dsubst_vref_lt … Hid2) >dsubst_vref_lt /2 width=1/
]
qed.
-theorem subst_subst_lt: ∀C1,C2,M,d1,d2. d2 < d1 →
- [d2 ⬐ [d1 - d2 -1 ⬐ C1] C2] [d1 ⬐ C1] M = [d1 - 1 ⬐ C1] [d2 ⬐ C2] M.
-#C1 #C2 #M #d1 #d2 #Hd21
+theorem subst_subst_lt: ∀D1,D2,M,d1,d2. d2 < d1 →
+ [d2 ⬐ [d1 - d2 -1 ⬐ D1] D2] [d1 ⬐ D1] M = [d1 - 1 ⬐ D1] [d2 ⬐ D2] M.
+#D1 #D2 #M #d1 #d2 #Hd21
lapply (ltn_to_ltO … Hd21) #Hd1
>subst_subst_ge in ⊢ (???%); /2 width=1/ <plus_minus_m_m //
qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "term.ma".
+
+(* REDEX POINTER ************************************************************)
+
+(* Policy: boolean metavariables: a, b
+ pointer metavariables: p, q
+*)
+(* Note: this is a path in the tree representation of a term
+ in which abstraction nodes are omitted;
+ on application nodes, "false" means "proceed left"
+ and "true" means "proceed right"
+*)
+definition rpointer: Type[0] ≝ list bool.
+
+(* Note: precedence relation on redex pointers: p ≺ q
+ to serve as base for the order relations: p < q and p ≤ q *)
+inductive rpprec: relation rpointer ≝
+| rpprec_nil : ∀b,q. rpprec ([]) (b::q)
+| rppprc_cons: ∀p,q. rpprec (false::p) (true::q)
+| rpprec_comp: ∀b,p,q. rpprec p q → rpprec (b::p) (b::q)
+.
+
+interpretation "'precedes' on redex pointers"
+ 'prec p q = (rpprec p q).
+
+(* Note: this should go to core_notation *)
+notation "hvbox(a break ≺ b)"
+ non associative with precedence 45
+ for @{ 'prec $a $b }.
+
+(* Note: this is p < q *)
+interpretation "'less than' on redex pointers"
+ 'lt p q = (TC rpointer rpprec p q).
+
+(* Note: this is p ≤ q, that *really* is p < q ∨ p = q *)
+interpretation "'less or equal to' on redex pointers"
+ 'leq x y = (RC rpointer (TC rpointer rpprec) x y).
projects / helm.git / commitdiff