(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( ⊗ term 60 p )"
- non associative with precedence 60
+notation "hvbox( ⊗ term 70 p )"
+ non associative with precedence 70
for @{ 'CircledTimes $p }.
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( # break term 90 n )"
+notation "hvbox( ⧣ break term 90 n )"
non associative with precedence 70
for @{ 'Hash $n }.
@(subset_eq_trans … (lift_fsubst …))
[ <lift_rmap_append <lift_rmap_A_sn <lift_rmap_append <lift_rmap_L_sn
<structure_append <structure_A_sn <structure_append <structure_L_sn
- <depth_plus <depth_L_sn <depth_structure <depth_structure
+ <depth_append <depth_L_sn <depth_structure <depth_structure
@fsubst_eq_repl [ // ]
@(subset_eq_trans … (lift_iref …))
@(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
(* Advanced constructions with proj_rmap and path_rcons *********************)
+lemma lift_rmap_d_dx (f) (p) (n):
+ (↑[p]f)∘𝐮❨ninj n❩ = ↑[p◖𝗱n]f.
+// qed.
+
lemma lift_rmap_m_dx (f) (p):
↑[p]f = ↑[p◖𝗺]f.
// qed.
#p elim p -p
[ #f <lift_rmap_empty <lift_rmap_empty <tr_pushs_zero
| * [ #n ] #p #IH #f //
- <lift_rmap_d_sn <lift_rmap_d_sn <depth_d
+ <lift_rmap_d_sn <lift_rmap_d_sn <depth_d_sn
@(trans_eq … (IH …)) -IH
(**************************************************************************)
include "delayed_updating/substitution/lift.ma".
+include "ground/relocation/tr_uni_compose.ma".
include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_pn.ma".
include "ground/relocation/tr_compose_eq.ma".
include "ground/relocation/tr_pn_eq.ma".
>tr_compose_push_bi //
]
qed.
+
+(* Advanced constructions with proj_rmap and stream_tls *********************)
+
+lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
+ ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
+#f #p #m #n
+<lift_rmap_d_dx >nrplus_inj_dx
+/2 width=1 by tr_tls_compose_uni_dx/
+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 "delayed_updating/substitution/lift_eq.ma".
+include "delayed_updating/syntax/path_update.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+(* Constructions with update ***********************************************)
+
+axiom arith1 (p,q:pnat) (n:nat):
+ nrplus (pplus p q) n = pplus p (nrplus q n).
+
+lemma pippo (f) (p) (m:pnat):
+ ⇂*[ninj (m+⧣p)]f ≗ ⇂*[ninj (m+❘p❘)]↑[p]f.
+#f #p @(list_ind_rcons … p) -p [ // ]
+#p * [ #n ] #IH #m
+[ <update_d_dx <depth_d_dx
+ <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <arith1
+ @(stream_eq_trans … (lift_rmap_tls_d_dx …))
+ @(stream_eq_canc_dx … (IH …)) -IH //
+| //
+| <update_L_dx <depth_L_dx
+ <nrplus_succ_dx >nsucc_inj //
+| //
+| //
+]
+qed.
(* BY-DEPTH DELAYED (BDD) TERM **********************************************)
inductive bdd: 𝒫❨prototerm❩ ≝
-| bdd_oref: ∀n. bdd (#n)
+| bdd_oref: ∀n. bdd (⧣n)
| bdd_iref: ∀t,n. bdd t → bdd (𝛗n.t)
| bdd_abst: ∀t. bdd t → bdd (𝛌.t)
| bdd_appl: ∀u,t. bdd u → bdd t → bdd (@u.t)
lemma depth_S_sn (q): ❘q❘ = ❘𝗦◗q❘.
// qed.
-(* Advanced constructions with nplus ****************************************)
+(* Main constructions *******************************************************)
-lemma depth_plus (p1) (p2):
- ❘p2❘+❘p1❘ = ❘p1●p2❘.
+theorem depth_append (p1) (p2):
+ ❘p2❘+❘p1❘ = ❘p1●p2❘.
#p1 elim p1 -p1 //
* [ #n ] #p1 #IH #p2 <list_append_lcons_sn
[ <depth_d_sn <depth_d_sn //
| <depth_S_sn <depth_S_sn //
]
qed.
+
+(* Constructions with list_rcons ********************************************)
+
+lemma depth_d_dx (p) (n):
+ ❘p❘ = ❘p◖𝗱n❘.
+// qed.
+
+lemma depth_m_dx (p):
+ ❘p❘ = ❘p◖𝗺❘.
+// qed.
+
+lemma depth_L_dx (p):
+ ↑❘p❘ = ❘p◖𝗟❘.
+// qed.
+
+lemma depth_A_dx (p):
+ ❘p❘ = ❘p◖𝗔❘.
+// qed.
+
+lemma depth_S_dx (p):
+ ❘p❘ = ❘p◖𝗦❘.
+// 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 "ground/arith/nat_plus.ma".
+include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/hash_1.ma".
+
+(* UPDATE COUNT FOR PATH ****************************************************)
+
+rec definition update (p) on p: nat ≝
+match p with
+[ list_empty ⇒ 𝟎
+| list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒ n + update q
+ | label_m ⇒ update q
+ | label_L ⇒ update q
+ | label_A ⇒ update q
+ | label_S ⇒ update q
+ ]
+].
+
+interpretation
+ "update count (path)"
+ 'Hash p = (update p).
+
+(* Basic constructions ******************************************************)
+
+lemma update_empty: 𝟎 = ⧣𝐞.
+// qed.
+
+lemma update_d_sn (q) (n): ninj n+⧣q = ⧣(𝗱n◗q).
+// qed.
+
+lemma update_m_sn (q): ⧣q = ⧣(𝗺◗q).
+// qed.
+
+lemma update_L_sn (q): ⧣q = ⧣(𝗟◗q).
+// qed.
+
+lemma update_A_sn (q): ⧣q = ⧣(𝗔◗q).
+// qed.
+
+lemma update_S_sn (q): ⧣q = ⧣(𝗦◗q).
+// qed.
+
+(* Main constructions *******************************************************)
+
+theorem update_append (p1) (p2):
+ (⧣p2+⧣p1) = ⧣(p1●p2).
+#p1 elim p1 -p1 //
+* [ #n ] #p1 #IH #p2 <list_append_lcons_sn
+[ <update_d_sn <update_d_sn //
+| <update_m_sn <update_m_sn //
+| <update_L_sn <update_L_sn //
+| <update_A_sn <update_A_sn //
+| <update_S_sn <update_S_sn //
+]
+qed.
+
+(* Constructions with list_rcons ********************************************)
+
+lemma update_d_dx (p) (n):
+ (⧣p)+(ninj n) = ⧣(p◖𝗱n).
+// qed.
+
+lemma update_m_dx (p):
+ (⧣p) = ⧣(p◖𝗺).
+// qed.
+
+lemma update_L_dx (p):
+ (⧣p) = ⧣(p◖𝗟).
+// qed.
+
+lemma update_A_dx (p):
+ (⧣p) = ⧣(p◖𝗔).
+// qed.
+
+lemma update_S_dx (p):
+ (⧣p) = ⧣(p◖𝗦).
+// qed.
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