--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation < "hvbox( ▼❨ term 46 k, break term 46 p, break term 46 f ❩ )"
+ non associative with precedence 70
+ for @{ 'BlackDownTriangle $S $k $p $f }.
+
+notation > "hvbox( ▼❨ term 46 k, break term 46 p, break term 46 f ❩ )"
+ non associative with precedence 70
+ for @{ 'BlackDownTriangle ? $k $p $f }.
+
+notation > "hvbox( ▼{ term 46 S }❨ break term 46 k, break term 46 p, break term 46 f ❩ )"
+ non associative with precedence 70
+ for @{ 'BlackDownTriangle $S $k $p $f }.
--- /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/notation/functions/uparrow_4.ma".
+include "delayed_updating/notation/functions/uparrow_2.ma".
+include "delayed_updating/syntax/path.ma".
+include "ground/relocation/tr_id_pap.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+definition lift_continuation (A:Type[0]) ≝
+ tr_map → path → A.
+
+rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
+match p with
+[ list_empty ⇒ k f (𝐞)
+| list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (𝐢) q
+ | label_m ⇒ lift_gen (A) (λg,p. k g (𝗺◗p)) f q
+ | label_L ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
+ | label_A ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
+ | label_S ⇒ lift_gen (A) (λg,p. k g (𝗦◗p)) f q
+ ]
+].
+
+interpretation
+ "lift (gneric)"
+ 'UpArrow A k f p = (lift_gen A k f p).
+
+definition proj_path: lift_continuation … ≝
+ λf,p.p.
+
+definition proj_rmap: lift_continuation … ≝
+ λf,p.f.
+
+interpretation
+ "lift (path)"
+ 'UpArrow f p = (lift_gen ? proj_path f p).
+
+interpretation
+ "lift (relocation map)"
+ 'UpArrow p f = (lift_gen ? proj_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma lift_empty (A) (k) (f):
+ k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
+// qed.
+
+lemma lift_d_sn (A) (k) (p) (n) (f):
+ ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), 𝐢, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
+// qed.
+
+lemma lift_m_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗺◗p)), f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
+// qed.
+
+lemma lift_L_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ↑{A}❨k, f, 𝗟◗p❩.
+// qed.
+
+lemma lift_A_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗔◗p)), f, p❩ = ↑{A}❨k, f, 𝗔◗p❩.
+// qed.
+
+lemma lift_S_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗦◗p)), f, p❩ = ↑{A}❨k, f, 𝗦◗p❩.
+// qed.
+
+(* Basic constructions with proj_path ***************************************)
+
+lemma lift_path_empty (f):
+ (𝐞) = ↑[f]𝐞.
+// qed.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma lift_rmap_empty (f):
+ f = ↑[𝐞]f.
+// qed.
+
+lemma lift_rmap_d_sn (f) (p) (n):
+ ↑[p]𝐢 = ↑[𝗱n◗p]f.
+// qed.
+
+lemma lift_rmap_m_sn (f) (p):
+ ↑[p]f = ↑[𝗺◗p]f.
+// qed.
+
+lemma lift_rmap_L_sn (f) (p):
+ ↑[p](⫯f) = ↑[𝗟◗p]f.
+// qed.
+
+lemma lift_rmap_A_sn (f) (p):
+ ↑[p]f = ↑[𝗔◗p]f.
+// qed.
+
+lemma lift_rmap_S_sn (f) (p):
+ ↑[p]f = ↑[𝗦◗p]f.
+// qed.
+
+(* Advanced cinstructionswith proj_rmap and tr_id ***************************)
+
+lemma lift_rmap_id (p):
+ (𝐢) = ↑[p]𝐢.
+#p elim p -p //
+* [ #n ] #p #IH //
+qed.
+
+(* Advanced constructions with proj_rmap and path_append ********************)
+
+lemma lift_rmap_append (p2) (p1) (f):
+ ↑[p2]↑[p1]f = ↑[p1●p2]f.
+#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
+[ <lift_rmap_d_sn <lift_rmap_d_sn //
+| <lift_rmap_m_sn <lift_rmap_m_sn //
+| <lift_rmap_A_sn <lift_rmap_A_sn //
+| <lift_rmap_S_sn <lift_rmap_S_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma lift_rmap_d_dx (f) (p) (n):
+ (𝐢) = ↑[p◖𝗱n]f.
+// qed.
+
+lemma lift_rmap_m_dx (f) (p):
+ ↑[p]f = ↑[p◖𝗺]f.
+// qed.
+
+lemma lift_rmap_L_dx (f) (p):
+ (⫯↑[p]f) = ↑[p◖𝗟]f.
+// qed.
+
+lemma lift_rmap_A_dx (f) (p):
+ ↑[p]f = ↑[p◖𝗔]f.
+// qed.
+
+lemma lift_rmap_S_dx (f) (p):
+ ↑[p]f = ↑[p◖𝗦]f.
+// qed.
+
+lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
+ m = ↑[p◖𝗱n]f@❨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 "delayed_updating/substitution/lift_eq.ma".
+include "ground/relocation/tr_compose_pap.ma".
+include "ground/relocation/tr_compose_pn.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+(* Constructions with tr_after *********************************************)
+
+lemma lift_path_after (p) (f1) (f2):
+ ↑[f2]↑[f1]p = ↑[f2∘f1]p.
+#p elim p -p [| * ] // #p #IH #f1 #f2
+<lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
+>tr_compose_push_bi //
+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_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_uni.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+lemma lift_iref_bi (t1) (t2) (n):
+ t1 ⇔ t2 → 𝛗n.t1 ⇔ 𝛗n.t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
+ (𝛗f@❨n❩.t) ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q #Hq #H0 destruct
+@(ex2_intro … (𝗱n◗𝗺◗q))
+/2 width=1 by in_comp_iref/
+qed-.
+
+lemma lift_iref_dx (f) (t) (n:pnat):
+ ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
+/2 width=1 by in_comp_iref/
+qed-.
+
+lemma lift_iref (f) (t) (n:pnat):
+ (𝛗f@❨n❩.t) ⇔ ↑[f](𝛗n.t).
+/3 width=1 by conj, lift_iref_sn, lift_iref_dx/
+qed.
+
+lemma lift_iref_uni (t) (m) (n):
+ (𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
+// 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.ma".
+include "ground/relocation/tr_pap_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
+ λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
+
+interpretation
+ "extensional equivalence (lift continuation)"
+ 'RingEq A k1 k2 = (lift_exteq A k1 k2).
+
+(* Constructions with lift_exteq ********************************************)
+
+lemma lift_eq_repl (A) (p) (k1) (k2):
+ k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
+#A #p elim p -p [| * [ #n ] #q #IH ]
+#k1 #k2 #Hk #f1 #f2 #Hf
+[ <lift_empty <lift_empty /2 width=1 by/
+| <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
+ /3 width=1 by stream_eq_refl/
+| /3 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /3 width=1 by/
+| /3 width=1 by/
+]
+qed-.
+
+(* Advanced constructions ***************************************************)
+
+lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
+ ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
+#A #k #f #p #l #Hk
+@lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
+qed.
+
+lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
+ ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
+#A #k #f #p1 #p #l #Hk
+@lift_eq_repl // #g1 #g2 #p2 #Hg
+<list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
+qed.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma proj_path_proper:
+ proj_path ≗ proj_path.
+// qed.
+
+lemma lift_path_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
+/2 width=1 by lift_eq_repl/ qed.
+
+lemma lift_path_append_sn (p) (f) (q):
+ q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
+#p elim p -p // * [ #n ] #p #IH #f #q
+[ <lift_d_sn <lift_d_sn
+| <lift_m_sn <lift_m_sn
+| <lift_L_sn <lift_L_sn
+| <lift_A_sn <lift_A_sn
+| <lift_S_sn <lift_S_sn
+]
+>lift_lcons_alt // >lift_append_rcons_sn //
+<IH <IH -IH <list_append_rcons_sn //
+qed.
+
+lemma lift_path_lcons (f) (p) (l):
+ l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
+#f #p #l
+>lift_lcons_alt <lift_path_append_sn //
+qed.
+
+lemma lift_path_d_sn (f) (p) (n):
+ (𝗱(f@❨n❩)◗↑[𝐢]p) = ↑[f](𝗱n◗p).
+// qed.
+
+lemma lift_path_m_sn (f) (p):
+ (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
+// qed.
+
+lemma lift_path_L_sn (f) (p):
+ (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
+// qed.
+
+lemma lift_path_A_sn (f) (p):
+ (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
+// qed.
+
+lemma lift_path_S_sn (f) (p):
+ (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
+// qed.
+
+lemma lift_path_id (p):
+ p = ↑[𝐢]p.
+#p elim p -p //
+* [ #n ] #p #IH //
+[ <lift_path_d_sn //
+| <lift_path_L_sn //
+]
+qed.
+
+lemma lift_path_append (p2) (p1) (f):
+ (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
+#p2 #p1 elim p1 -p1 //
+* [ #n1 ] #p1 #IH #f
+[ <lift_path_d_sn <lift_path_d_sn <IH //
+| <lift_path_m_sn <lift_path_m_sn <IH //
+| <lift_path_L_sn <lift_path_L_sn <IH //
+| <lift_path_A_sn <lift_path_A_sn <IH //
+| <lift_path_S_sn <lift_path_S_sn <IH //
+]
+qed.
+
+lemma lift_path_d_dx (n) (p) (f):
+ (↑[f]p)◖𝗱((↑[p]f)@❨n❩) = ↑[f](p◖𝗱n).
+#n #p #f <lift_path_append //
+qed.
+
+lemma lift_path_m_dx (p) (f):
+ (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
+#p #f <lift_path_append //
+qed.
+
+lemma lift_path_L_dx (p) (f):
+ (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
+#p #f <lift_path_append //
+qed.
+
+lemma lift_path_A_dx (p) (f):
+ (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
+#p #f <lift_path_append //
+qed.
+
+lemma lift_path_S_dx (p) (f):
+ (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
+#p #f <lift_path_append //
+qed.
+
+(* COMMENT
+
+(* 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 "ground/lib/list_length.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+(* Constructions with list_length ******************************************)
+
+lemma lift_path_length (p) (f):
+ ❘p❘ = ❘↑[f]p❘.
+#p elim p -p // * [ #n ] #p #IH #f //
+[ <lift_path_m_sn
+| <lift_path_L_sn
+| <lift_path_A_sn
+| <lift_path_S_sn
+]
+>list_length_lcons >list_length_lcons //
+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.ma".
+include "delayed_updating/syntax/prototerm.ma".
+include "ground/lib/subset_ext.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+interpretation
+ "lift (prototerm)"
+ 'UpArrow f t = (subset_ext_f1 ? ? (lift_gen ? proj_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_lift_bi (f) (p) (t):
+ p ϵ t → ↑[f]p ϵ ↑[f]t.
+/2 width=1 by subset_in_ext_f1_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 "ground/lib/subset_ext_equivalence.ma".
+include "delayed_updating/substitution/lift_after.ma".
+include "delayed_updating/substitution/lift_prototerm.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma lift_term_eq_repl_sn (f1) (f2) (t):
+ f1 ≗ f2 → ↑[f1]t ⇔ ↑[f2]t.
+/3 width=1 by subset_equivalence_ext_f1_exteq, lift_path_eq_repl/
+qed.
+
+lemma lift_term_eq_repl_dx (f) (t1) (t2):
+ t1 ⇔ t2 → ↑[f]t1 ⇔ ↑[f]t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma lift_term_after (f1) (f2) (t):
+ ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
+#f1 #f2 #t @subset_eq_trans
+[
+| @subset_inclusion_ext_f1_compose
+| @subset_equivalence_ext_f1_exteq /2 width=5/
+]
+qed.
+
+lemma lift_term_id_sn (t):
+ t ⊆ ↑[𝐢]t.
+#t #p #Hp
+>(lift_path_id p)
+/2 width=1 by in_comp_lift_bi/
+qed-.
+
+lemma lift_term_id_dx (t):
+ ↑[𝐢]t ⊆ t.
+#t #p * #q #Hq #H destruct //
+qed-.
+
+lemma lift_term_id (t):
+ t ⇔ ↑[𝐢]t.
+/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
+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_structure.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+(* Constructions with structure ********************************************)
+
+lemma structure_lift (p) (f):
+ ⊗p = ⊗↑[f]p.
+#p elim p -p //
+* [ #n ] #p #IH #f //
+[ <lift_path_m_sn <structure_m_sn <structure_m_sn //
+| <lift_path_L_sn <structure_L_sn <structure_L_sn //
+]
+qed.
+
+lemma lift_structure (p) (f):
+ ⊗p = ↑[f]⊗p.
+#p elim p -p //
+* [ #n ] #p #IH #f //
+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 "ground/relocation/tr_uni_pap.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+(* Constructions with tr_uni ***********************************************)
+
+lemma lift_path_d_sn_uni (p) (m) (n):
+ (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
+// 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_length.ma".
+include "delayed_updating/notation/functions/black_downtriangle_1.ma".
+include "ground/relocation/tr_uni.ma".
+
+(* BASIC UNWIND FOR PATH ****************************************************)
+
+rec definition unwind1_path_pnat (n) (p) on n ≝
+match n with
+[ punit ⇒ p
+| psucc m ⇒
+ match p with
+ [ list_empty ⇒ 𝐞
+ | list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒
+ match q with
+ [ list_empty ⇒ l◗(unwind1_path_pnat m q)
+ | list_lcons _ _ ⇒ unwind1_path_pnat m (↑[𝐮❨n❩]q)
+ ]
+ | label_m ⇒ unwind1_path_pnat m q
+ | label_L ⇒ 𝗟◗(unwind1_path_pnat m q)
+ | label_A ⇒ 𝗔◗(unwind1_path_pnat m q)
+ | label_S ⇒ 𝗦◗(unwind1_path_pnat m q)
+ ]
+ ]
+].
+
+definition unwind1_path (p) ≝
+ unwind1_path_pnat (↑❘p❘) p.
+
+interpretation
+ "basic unwind (path)"
+ 'BlackDownTriangle p = (unwind1_path p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind1_path_unfold (p):
+ unwind1_path_pnat (↑❘p❘) p = ▼p.
+// qed-.
+
+lemma unwind1_path_empty:
+ (𝐞) = ▼𝐞.
+// qed.
+
+lemma unwind1_path_d_empty (n):
+ (𝗱n◗𝐞) = ▼(𝗱n◗𝐞).
+// qed.
+
+lemma unwind1_path_d_lcons (p) (l) (n:pnat):
+ ▼(↑[𝐮❨n❩](l◗p)) = ▼(𝗱n◗l◗p).
+#p #l #n
+<unwind1_path_unfold <lift_path_length //
+qed.
+
+lemma unwind1_path_m_sn (p):
+ ▼p = ▼(𝗺◗p).
+// qed.
+
+lemma unwind1_path_L_sn (p):
+ (𝗟◗▼p) = ▼(𝗟◗p).
+// qed.
+
+lemma unwind1_path_A_sn (p):
+ (𝗔◗▼p) = ▼(𝗔◗p).
+// qed.
+
+lemma unwind1_path_S_sn (p):
+ (𝗦◗▼p) = ▼(𝗦◗p).
+// qed.
+
+(* Main constructions *******************************************************)
+
+fact unwind1_path_fix_aux (k) (p):
+ k = ❘p❘ → ▼p = ▼▼p.
+#k @(nat_ind_succ … k) -k
+[ #p #H0 >(list_length_inv_zero_sn … H0) -p //
+| #k #IH *
+ [ #H0 elim (eq_inv_nsucc_zero … H0)
+ | * [ #n ] #p #H0
+ lapply (eq_inv_nsucc_bi … H0) -H0
+ [ cases p -p [ -IH | #l #p ] #H0 destruct //
+ <unwind1_path_d_lcons <IH -IH //
+ | #H0 destruct <unwind1_path_m_sn <IH -IH //
+ | #H0 destruct <unwind1_path_L_sn <unwind1_path_L_sn <IH -IH //
+ | #H0 destruct <unwind1_path_A_sn <unwind1_path_A_sn <IH -IH //
+ | #H0 destruct <unwind1_path_S_sn <unwind1_path_S_sn <IH -IH //
+ ]
+ ]
+]
+qed-.
+
+theorem unwind1_path_fix (p):
+ ▼p = ▼▼p.
+/2 width=2 by unwind1_path_fix_aux/ 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/unwind0/unwind1_rmap_eq.ma".
+include "delayed_updating/unwind0/unwind1_path.ma".
+include "delayed_updating/substitution/lift_structure.ma".
+
+(* BASIC UNWIND FOR PATH ****************************************************)
+
+(* Constructions with structure *********************************************)
+
+fact unwind1_path_d_dx_aux (k) (p) (n):
+ k = ❘p❘ → (⊗p)◖𝗱((▶p)@❨n❩) = ▼(p◖𝗱n).
+#k @(nat_ind_succ … k) -k
+[ #p #n #H0 >(list_length_inv_zero_sn … H0) -p //
+| #k #IH *
+ [ #n #H0 elim (eq_inv_nsucc_zero … H0)
+ | * [ #m ] #p #n #H0
+ lapply (eq_inv_nsucc_bi … H0) -H0
+ [ cases p -p [ -IH | #l #p ] #H0 destruct <unwind1_path_d_lcons
+ [ <lift_path_d_sn <lift_path_empty <unwind1_path_d_empty
+ <list_cons_comm <(tr_pap_eq_repl … (unwind1_rmap_d_empty …)) //
+ | >list_cons_shift <lift_path_d_dx <IH -IH //
+ <structure_lift <structure_d_sn /3 width=1 by eq_f2/
+ ]
+ | #H0 destruct <unwind1_path_m_sn <IH -IH //
+ | #H0 destruct <unwind1_path_L_sn <IH -IH //
+ | #H0 destruct <unwind1_path_A_sn <IH -IH //
+ | #H0 destruct <unwind1_path_S_sn <IH -IH //
+ ]
+ ]
+]
+qed-.
+
+lemma unwind1_path_d_dx (p) (n):
+ (⊗p)◖𝗱((▶p)@❨n❩) = ▼(p◖𝗱n).
+/2 width=2 by unwind1_path_d_dx_aux/ 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_length.ma".
+include "delayed_updating/notation/functions/black_righttriangle_1.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/relocation/tr_compose.ma".
+
+(* BASIC UNWIND FOR RELOCATION MAP ******************************************)
+
+rec definition unwind1_rmap_pnat (n) (p) on n ≝
+match n with
+[ punit ⇒ 𝐢
+| psucc m ⇒
+ match p with
+ [ list_empty ⇒ 𝐢
+ | list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒ (unwind1_rmap_pnat m (↑[𝐮❨n❩]q))∘(↑[q]𝐮❨n❩)
+ | label_m ⇒ unwind1_rmap_pnat m q
+ | label_L ⇒ unwind1_rmap_pnat m q
+ | label_A ⇒ unwind1_rmap_pnat m q
+ | label_S ⇒ unwind1_rmap_pnat m q
+ ]
+ ]
+].
+
+definition unwind1_rmap (p) ≝
+ unwind1_rmap_pnat (↑❘p❘) p.
+
+interpretation
+ "basic unwind (relocation map)"
+ 'BlackRightTriangle p = (unwind1_rmap p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind1_rmap_unfold (p):
+ unwind1_rmap_pnat (↑❘p❘) p = ▶p.
+// qed-.
+
+lemma unwind1_rmap_empty:
+ (𝐢) = ▶(𝐞).
+// qed.
+
+lemma unwind1_rmap_d_sn (p) (n:pnat):
+ (▶(↑[𝐮❨n❩]p))∘(↑[p]𝐮❨n❩) = ▶(𝗱n◗p).
+#p #n
+<unwind1_rmap_unfold <lift_path_length //
+qed.
+
+lemma unwind1_rmap_m_sn (p):
+ ▶p = ▶(𝗺◗p).
+// qed.
+
+lemma unwind1_rmap_L_sn (p):
+ ▶p = ▶(𝗟◗p).
+// qed.
+
+lemma unwind1_rmap_A_sn (p):
+ ▶p = ▶(𝗔◗p).
+// qed.
+
+lemma unwind1_rmap_S_sn (p):
+ ▶p = ▶(𝗦◗p).
+// 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/unwind0/unwind1_rmap.ma".
+include "ground/relocation/tr_id_compose.ma".
+
+(* BASIC UNWIND FOR RELOCATION MAP ******************************************)
+
+(* Constructions with stream_eq *********************************************)
+
+lemma unwind1_rmap_d_empty (n:pnat):
+ (𝐮❨n❩) ≗ ▶(𝗱n◗𝐞).
+// 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/unwind0/unwind1_path.ma".
+include "delayed_updating/notation/functions/black_downtriangle_2.ma".
+
+(* EXTENDED UNWIND FOR PATH *************************************************)
+
+definition unwind2_path (f) (p) ≝
+ ▼↑[f]p.
+
+interpretation
+ "extended unwind (path)"
+ 'BlackDownTriangle f p = (unwind2_path f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind2_path_unfold (f) (p):
+ ▼↑[f]p = ▼[f]p.
+// qed.
+
+lemma unwind2_path_id (p):
+ ▼p = ▼[𝐢]p.
+// qed.
+
+lemma unwind2_path_empty (f):
+ (𝐞) = ▼[f]𝐞.
+// qed.
+
+lemma unwind2_path_d_empty (f) (n):
+ (𝗱(f@❨n❩)◗𝐞) = ▼[f](𝗱n◗𝐞).
+// qed.
+
+lemma unwind2_path_d_lcons (f) (p) (l) (n:pnat):
+ ▼[𝐮❨f@❨n❩❩](l◗p) = ▼[f](𝗱n◗l◗p).
+#f #p #l #n
+<unwind2_path_unfold in ⊢ (???%);
+<lift_path_d_sn <lift_path_id <unwind1_path_d_lcons //
+qed.
+
+lemma unwind2_path_m_sn (f) (p):
+ ▼[f]p = ▼[f](𝗺◗p).
+#f #p
+<unwind2_path_unfold in ⊢ (???%);
+<lift_path_m_sn //
+qed.
+
+lemma unwind2_path_L_sn (f) (p):
+ (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
+#f #p
+<unwind2_path_unfold in ⊢ (???%);
+<lift_path_L_sn //
+qed.
+
+lemma unwind2_path_A_sn (f) (p):
+ (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
+#f #p
+<unwind2_path_unfold in ⊢ (???%);
+<lift_path_A_sn //
+qed.
+
+lemma unwind2_path_S_sn (f) (p):
+ (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
+#f #p
+<unwind2_path_unfold in ⊢ (???%);
+<lift_path_S_sn //
+qed.
+
+(* Main constructions *******************************************************)
+
+theorem unwind2_path_after_id_sn (p) (f):
+ ▼[f]p = ▼▼[f]p.
+// qed.
+
+(* Constructions with stream_eq *********************************************)
+
+lemma unwind2_path_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
+#p #f1 #f2 #Hf
+<unwind2_path_unfold <unwind2_path_unfold
+<(lift_path_eq_repl … Hf) -Hf //
+qed.
+
+(* Advanced eliminations with path ******************************************)
+
+lemma path_ind_unwind2 (Q:predicate …):
+ Q (𝐞) →
+ (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
+ (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
+ (∀p. Q p → Q (𝗺◗p)) →
+ (∀p. Q p → Q (𝗟◗p)) →
+ (∀p. Q p → Q (𝗔◗p)) →
+ (∀p. Q p → Q (𝗦◗p)) →
+ ∀p. Q p.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
+elim p -p [| * [ #n * ] ]
+/2 width=1 by/
+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/unwind0/unwind2_rmap.ma".
+include "delayed_updating/unwind0/unwind2_path.ma".
+include "delayed_updating/unwind0/unwind1_path_structure.ma".
+
+(* EXTENDED UNWIND FOR PATH *************************************************)
+
+(* Constructions with structure *********************************************)
+
+lemma unwind2_path_d_dx (f) (p) (n):
+ (⊗p)◖𝗱((▶[f]p)@❨n❩) = ▼[f](p◖𝗱n).
+#f #p #n
+<unwind2_path_unfold <lift_path_d_dx <unwind1_path_d_dx
+<structure_lift >tr_compose_pap //
+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/unwind0/unwind1_rmap.ma".
+include "delayed_updating/notation/functions/black_righttriangle_2.ma".
+
+(* EXTENDED UNWIND FOR RELOCATION MAP ***************************************)
+
+definition unwind2_rmap (f) (p): tr_map ≝
+ (▶↑[f]p)∘(↑[p]f).
+
+interpretation
+ "extended unwind (relocation map)"
+ 'BlackRightTriangle f p = (unwind2_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind2_rmap_unfold (f) (p):
+ (▶↑[f]p)∘(↑[p]f) = ▶[f]p.
+// qed.
+
+lemma unwind2_rmap_m_sn (f) (p):
+ ▶[f]p = ▶[f](𝗺◗p).
+#f #p
+<unwind2_rmap_unfold in ⊢ (???%);
+<lift_rmap_m_sn <lift_path_m_sn //
+qed.
+
+lemma unwind2_rmap_L_sn (f) (p):
+ ▶[⫯f]p = ▶[f](𝗟◗p).
+#f #p
+<unwind2_rmap_unfold in ⊢ (???%);
+<lift_rmap_L_sn <lift_path_L_sn //
+qed.
+
+lemma unwind2_rmap_A_sn (f) (p):
+ ▶[f]p = ▶[f](𝗔◗p).
+#f #p
+<unwind2_rmap_unfold in ⊢ (???%);
+<lift_rmap_A_sn <lift_path_A_sn //
+qed.
+
+lemma unwind2_rmap_S_sn (f) (p):
+ ▶[f]p = ▶[f](𝗦◗p).
+#f #p
+<unwind2_rmap_unfold in ⊢ (???%);
+<lift_rmap_S_sn <lift_path_S_sn //
+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/unwind0/unwind2_rmap.ma".
+include "ground/relocation/tr_id_compose.ma".
+
+(* EXTENDED UNWIND FOR RELOCATION MAP ***************************************)
+
+(* Advanced constructions ***************************************************)
+
+lemma unwind2_rmap_id (p):
+ ▶p ≗ ▶[𝐢]p.
+// qed.
+
+lemma unwind2_rmap_empty (f):
+ f ≗ ▶[f]𝐞.
+// qed.
+
+lemma unwind2_rmap_d_sn (f) (p) (n):
+ ▶[𝐮❨f@❨n❩❩]p ≗ ▶[f](𝗱n◗p).
+#f #p #n
+<unwind2_rmap_unfold <unwind2_rmap_unfold
+<lift_rmap_d_sn <lift_rmap_id
+<lift_path_d_sn <lift_path_id <unwind1_rmap_d_sn //
+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/notation/functions/uparrow_4.ma".
-include "delayed_updating/notation/functions/uparrow_2.ma".
-include "delayed_updating/syntax/path.ma".
-include "ground/relocation/tr_uni.ma".
-include "ground/relocation/tr_pap_tls.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-definition lift_continuation (A:Type[0]) ≝
- tr_map → path → A.
-
-rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
-match p with
-[ list_empty ⇒ k f (𝐞)
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (⇂*[n]f) q
- | label_m ⇒ lift_gen (A) (λg,p. k g (𝗺◗p)) f q
- | label_L ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
- | label_A ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
- | label_S ⇒ lift_gen (A) (λg,p. k g (𝗦◗p)) f q
- ]
-].
-
-interpretation
- "lift (gneric)"
- 'UpArrow A k f p = (lift_gen A k f p).
-
-definition proj_path: lift_continuation … ≝
- λf,p.p.
-
-definition proj_rmap: lift_continuation … ≝
- λf,p.f.
-
-interpretation
- "lift (path)"
- 'UpArrow f p = (lift_gen ? proj_path f p).
-
-interpretation
- "lift (relocation map)"
- 'UpArrow p f = (lift_gen ? proj_rmap f p).
-
-(* Basic constructions ******************************************************)
-
-lemma lift_empty (A) (k) (f):
- k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
-// qed.
-
-lemma lift_d_sn (A) (k) (p) (n) (f):
- ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), ⇂*[n]f, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
-// qed.
-
-lemma lift_m_sn (A) (k) (p) (f):
- ↑❨(λg,p. k g (𝗺◗p)), f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
-// qed.
-
-lemma lift_L_sn (A) (k) (p) (f):
- ↑❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ↑{A}❨k, f, 𝗟◗p❩.
-// qed.
-
-lemma lift_A_sn (A) (k) (p) (f):
- ↑❨(λg,p. k g (𝗔◗p)), f, p❩ = ↑{A}❨k, f, 𝗔◗p❩.
-// qed.
-
-lemma lift_S_sn (A) (k) (p) (f):
- ↑❨(λg,p. k g (𝗦◗p)), f, p❩ = ↑{A}❨k, f, 𝗦◗p❩.
-// qed.
-
-(* Basic constructions with proj_path ***************************************)
-
-lemma lift_path_empty (f):
- (𝐞) = ↑[f]𝐞.
-// qed.
-
-(* Basic constructions with proj_rmap ***************************************)
-
-lemma lift_rmap_empty (f):
- f = ↑[𝐞]f.
-// qed.
-
-lemma lift_rmap_d_sn (f) (p) (n):
- ↑[p](⇂*[ninj n]f) = ↑[𝗱n◗p]f.
-// qed.
-
-lemma lift_rmap_m_sn (f) (p):
- ↑[p]f = ↑[𝗺◗p]f.
-// qed.
-
-lemma lift_rmap_L_sn (f) (p):
- ↑[p](⫯f) = ↑[𝗟◗p]f.
-// qed.
-
-lemma lift_rmap_A_sn (f) (p):
- ↑[p]f = ↑[𝗔◗p]f.
-// qed.
-
-lemma lift_rmap_S_sn (f) (p):
- ↑[p]f = ↑[𝗦◗p]f.
-// qed.
-
-(* Advanced constructions with proj_rmap and path_append ********************)
-
-lemma lift_rmap_append (p2) (p1) (f):
- ↑[p2]↑[p1]f = ↑[p1●p2]f.
-#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <lift_rmap_m_sn <lift_rmap_m_sn //
-| <lift_rmap_A_sn <lift_rmap_A_sn //
-| <lift_rmap_S_sn <lift_rmap_S_sn //
-]
-qed.
-
-(* Advanced constructions with proj_rmap and path_rcons *********************)
-
-lemma lift_rmap_d_dx (f) (p) (n):
- ⇂*[ninj n](↑[p]f) = ↑[p◖𝗱n]f.
-// qed.
-
-lemma lift_rmap_m_dx (f) (p):
- ↑[p]f = ↑[p◖𝗺]f.
-// qed.
-
-lemma lift_rmap_L_dx (f) (p):
- (⫯↑[p]f) = ↑[p◖𝗟]f.
-// qed.
-
-lemma lift_rmap_A_dx (f) (p):
- ↑[p]f = ↑[p◖𝗔]f.
-// qed.
-
-lemma lift_rmap_S_dx (f) (p):
- ↑[p]f = ↑[p◖𝗦]f.
-// qed.
-
-lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
- ↑[p]f@❨m+n❩ = ↑[p◖𝗱n]f@❨m❩+↑[p]f@❨n❩.
-// 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 "ground/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_compose_pn.ma".
-include "ground/relocation/tr_compose_tls.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with tr_after *********************************************)
-
-lemma lift_path_after (p) (f1) (f2):
- ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p elim p -p [| * ] // [ #n ] #p #IH #f1 #f2
-[ <lift_path_d_sn <lift_path_d_sn <lift_path_d_sn //
-| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
- >tr_compose_push_bi //
-]
-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_prototerm_id.ma".
-include "delayed_updating/substitution/lift_uni.ma".
-include "delayed_updating/syntax/prototerm_constructors.ma".
-
-(* LIFT FOR PROTOTERM *******************************************************)
-
-lemma lift_iref_bi (t1) (t2) (n):
- t1 ⇔ t2 → 𝛗n.t1 ⇔ 𝛗n.t2.
-/2 width=1 by subset_equivalence_ext_f1_bi/
-qed.
-
-lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
- (𝛗f@❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro … (𝗱n◗𝗺◗r))
-/2 width=1 by in_comp_iref/
-qed-.
-
-lemma lift_iref_dx (f) (t) (n:pnat):
- ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.↑[⇂*[n]f]t.
-#f #t #n #p * #q #Hq #H0 destruct
-elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
-/3 width=1 by in_comp_iref, in_comp_lift_bi/
-qed-.
-
-lemma lift_iref (f) (t) (n:pnat):
- (𝛗f@❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
-/3 width=1 by conj, lift_iref_sn, lift_iref_dx/
-qed.
-
-lemma lift_iref_uni (t) (m) (n):
- (𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
-#t #m #n
-@(subset_eq_trans … (lift_iref …))
-<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
-/3 width=1 by lift_iref_bi, lift_term_id/
-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.ma".
-(*
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_eq.ma".
-*)
-include "ground/relocation/tr_pap_eq.ma".
-include "ground/relocation/tr_pn_eq.ma".
-include "ground/lib/stream_tls_eq.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
- λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
-
-interpretation
- "extensional equivalence (lift continuation)"
- 'RingEq A k1 k2 = (lift_exteq A k1 k2).
-
-(* Constructions with lift_exteq ********************************************)
-
-lemma lift_eq_repl (A) (p) (k1) (k2):
- k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
-#A #p elim p -p [| * [ #n ] #q #IH ]
-#k1 #k2 #Hk #f1 #f2 #Hf
-[ <lift_empty <lift_empty /2 width=1 by/
-| <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
- /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
-| /3 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /3 width=1 by/
-| /3 width=1 by/
-]
-qed-.
-
-(* Advanced constructions ***************************************************)
-
-lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
- ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
-#A #k #f #p #l #Hk
-@lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
-qed.
-
-lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
- ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
-#A #k #f #p1 #p #l #Hk
-@lift_eq_repl // #g1 #g2 #p2 #Hg
-<list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
-qed.
-
-(* Advanced constructions with proj_path ************************************)
-
-lemma proj_path_proper:
- proj_path ≗ proj_path.
-// qed.
-
-lemma lift_path_eq_repl (p):
- stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
-/2 width=1 by lift_eq_repl/ qed.
-
-lemma lift_path_append_sn (p) (f) (q):
- q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
-#p elim p -p // * [ #n ] #p #IH #f #q
-[ <lift_d_sn <lift_d_sn
-| <lift_m_sn <lift_m_sn
-| <lift_L_sn <lift_L_sn
-| <lift_A_sn <lift_A_sn
-| <lift_S_sn <lift_S_sn
-]
->lift_lcons_alt // >lift_append_rcons_sn //
-<IH <IH -IH <list_append_rcons_sn //
-qed.
-
-lemma lift_path_lcons (f) (p) (l):
- l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
-#f #p #l
->lift_lcons_alt <lift_path_append_sn //
-qed.
-
-lemma lift_path_d_sn (f) (p) (n):
- (𝗱(f@❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
-// qed.
-
-lemma lift_path_m_sn (f) (p):
- (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
-// qed.
-
-lemma lift_path_L_sn (f) (p):
- (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
-// qed.
-
-lemma lift_path_A_sn (f) (p):
- (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
-// qed.
-
-lemma lift_path_S_sn (f) (p):
- (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
-// qed.
-(* COMMENT
-
-(* 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 "ground/relocation/tr_id_pap.ma".
-include "ground/relocation/tr_id_tls.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with tr_id ************************************************)
-
-lemma lift_path_id (p):
- p = ↑[𝐢]p.
-#p elim p -p //
-* [ #n ] #p #IH //
-[ <lift_path_d_sn //
-| <lift_path_L_sn //
-]
-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.ma".
-include "delayed_updating/syntax/prototerm.ma".
-include "ground/lib/subset_ext.ma".
-
-(* LIFT FOR PROTOTERM *******************************************************)
-
-interpretation
- "lift (prototerm)"
- 'UpArrow f t = (subset_ext_f1 ? ? (lift_gen ? proj_path f) t).
-
-(* Basic constructions ******************************************************)
-
-lemma in_comp_lift_bi (f) (p) (t):
- p ϵ t → ↑[f]p ϵ ↑[f]t.
-/2 width=1 by subset_in_ext_f1_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 "ground/lib/subset_ext_equivalence.ma".
-include "delayed_updating/substitution/lift_after.ma".
-include "delayed_updating/substitution/lift_prototerm.ma".
-
-(* LIFT FOR PROTOTERM *******************************************************)
-
-(* Constructions with subset_equivalence ************************************)
-
-lemma lift_term_eq_repl_sn (f1) (f2) (t):
- f1 ≗ f2 → ↑[f1]t ⇔ ↑[f2]t.
-/3 width=1 by subset_equivalence_ext_f1_exteq, lift_path_eq_repl/
-qed.
-
-lemma lift_term_eq_repl_dx (f) (t1) (t2):
- t1 ⇔ t2 → ↑[f]t1 ⇔ ↑[f]t2.
-/2 width=1 by subset_equivalence_ext_f1_bi/
-qed.
-
-lemma lift_term_after (f1) (f2) (t):
- ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
-#f1 #f2 #t @subset_eq_trans
-[
-| @subset_inclusion_ext_f1_compose
-| @subset_equivalence_ext_f1_exteq /2 width=5/
-]
-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_prototerm_eq.ma".
-include "delayed_updating/substitution/lift_id.ma".
-
-lemma lift_term_id_sn (t):
- t ⊆ ↑[𝐢]t.
-#t #p #Hp
->(lift_path_id p)
-/2 width=1 by in_comp_lift_bi/
-qed-.
-
-lemma lift_term_id_dx (t):
- ↑[𝐢]t ⊆ t.
-#t #p * #q #Hq #H destruct //
-qed-.
-
-lemma lift_term_id (t):
- t ⇔ ↑[𝐢]t.
-/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
-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_id.ma".
-include "ground/relocation/tr_uni_pap.ma".
-include "ground/relocation/tr_uni_tls.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with tr_uni ***********************************************)
-
-lemma lift_path_d_sn_uni (p) (m) (n):
- (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
-#p #m #n
-<lift_path_d_sn <tr_uni_pap >nsucc_pnpred
-<tr_tls_succ_uni //
-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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( ◆[ term 46 f ] break term 70 p )"
+ non associative with precedence 70
+ for @{ 'BlackDiamond $f $p }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR DELAYED UPDATING ********************************************)
-
-notation < "hvbox( ▼❨ term 46 k, break term 46 p, break term 46 f ❩ )"
- non associative with precedence 70
- for @{ 'BlackDownTriangle $S $k $p $f }.
-
-notation > "hvbox( ▼❨ term 46 k, break term 46 p, break term 46 f ❩ )"
- non associative with precedence 70
- for @{ 'BlackDownTriangle ? $k $p $f }.
-
-notation > "hvbox( ▼{ term 46 S }❨ break term 46 k, break term 46 p, break term 46 f ❩ )"
- non associative with precedence 70
- for @{ 'BlackDownTriangle $S $k $p $f }.
include "delayed_updating/notation/functions/uparrow_4.ma".
include "delayed_updating/notation/functions/uparrow_2.ma".
include "delayed_updating/syntax/path.ma".
-include "ground/relocation/tr_id_pap.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/relocation/tr_pap_tls.ma".
(* LIFT FOR PATH ***********************************************************)
[ list_empty ⇒ k f (𝐞)
| list_lcons l q ⇒
match l with
- [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (𝐢) q
+ [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (⇂*[n]f) q
| label_m ⇒ lift_gen (A) (λg,p. k g (𝗺◗p)) f q
| label_L ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
| label_A ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
// qed.
lemma lift_d_sn (A) (k) (p) (n) (f):
- ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), 𝐢, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
+ ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), ⇂*[n]f, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
// qed.
lemma lift_m_sn (A) (k) (p) (f):
// qed.
lemma lift_rmap_d_sn (f) (p) (n):
- ↑[p]𝐢 = ↑[𝗱n◗p]f.
+ ↑[p](⇂*[ninj n]f) = ↑[𝗱n◗p]f.
// qed.
lemma lift_rmap_m_sn (f) (p):
↑[p]f = ↑[𝗦◗p]f.
// qed.
-(* Advanced cinstructionswith proj_rmap and tr_id ***************************)
-
-lemma lift_rmap_id (p):
- (𝐢) = ↑[p]𝐢.
-#p elim p -p //
-* [ #n ] #p #IH //
-qed.
-
(* Advanced constructions with proj_rmap and path_append ********************)
lemma lift_rmap_append (p2) (p1) (f):
↑[p2]↑[p1]f = ↑[p1●p2]f.
#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <lift_rmap_d_sn <lift_rmap_d_sn //
-| <lift_rmap_m_sn <lift_rmap_m_sn //
+[ <lift_rmap_m_sn <lift_rmap_m_sn //
| <lift_rmap_A_sn <lift_rmap_A_sn //
| <lift_rmap_S_sn <lift_rmap_S_sn //
]
(* Advanced constructions with proj_rmap and path_rcons *********************)
lemma lift_rmap_d_dx (f) (p) (n):
- (𝐢) = ↑[p◖𝗱n]f.
+ ⇂*[ninj n](↑[p]f) = ↑[p◖𝗱n]f.
// qed.
lemma lift_rmap_m_dx (f) (p):
// qed.
lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
- m = ↑[p◖𝗱n]f@❨m❩.
+ ↑[p]f@❨m+n❩ = ↑[p◖𝗱n]f@❨m❩+↑[p]f@❨n❩.
// qed.
include "delayed_updating/substitution/lift_eq.ma".
include "ground/relocation/tr_compose_pap.ma".
include "ground/relocation/tr_compose_pn.ma".
+include "ground/relocation/tr_compose_tls.ma".
(* LIFT FOR PATH ***********************************************************)
lemma lift_path_after (p) (f1) (f2):
↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p elim p -p [| * ] // #p #IH #f1 #f2
-<lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
->tr_compose_push_bi //
+#p elim p -p [| * ] // [ #n ] #p #IH #f1 #f2
+[ <lift_path_d_sn <lift_path_d_sn <lift_path_d_sn //
+| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
+ >tr_compose_push_bi //
+]
qed.
(* *)
(**************************************************************************)
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_id.ma".
include "delayed_updating/substitution/lift_uni.ma".
include "delayed_updating/syntax/prototerm_constructors.ma".
qed.
lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
- (𝛗f@❨n❩.t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q #Hq #H0 destruct
-@(ex2_intro … (𝗱n◗𝗺◗q))
+ (𝛗f@❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro … (𝗱n◗𝗺◗r))
/2 width=1 by in_comp_iref/
qed-.
lemma lift_iref_dx (f) (t) (n:pnat):
- ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.t.
+ ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.↑[⇂*[n]f]t.
#f #t #n #p * #q #Hq #H0 destruct
elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
-/2 width=1 by in_comp_iref/
+/3 width=1 by in_comp_iref, in_comp_lift_bi/
qed-.
lemma lift_iref (f) (t) (n:pnat):
- (𝛗f@❨n❩.t) ⇔ ↑[f](𝛗n.t).
+ (𝛗f@❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
/3 width=1 by conj, lift_iref_sn, lift_iref_dx/
qed.
lemma lift_iref_uni (t) (m) (n):
(𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
-// qed.
+#t #m #n
+@(subset_eq_trans … (lift_iref …))
+<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
+/3 width=1 by lift_iref_bi, lift_term_id/
+qed.
(**************************************************************************)
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_eq.ma".
+*)
include "ground/relocation/tr_pap_eq.ma".
include "ground/relocation/tr_pn_eq.ma".
+include "ground/lib/stream_tls_eq.ma".
(* LIFT FOR PATH ***********************************************************)
#k1 #k2 #Hk #f1 #f2 #Hf
[ <lift_empty <lift_empty /2 width=1 by/
| <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
- /3 width=1 by stream_eq_refl/
+ /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
| /3 width=1 by/
| /3 width=1 by tr_push_eq_repl/
| /3 width=1 by/
qed.
lemma lift_path_d_sn (f) (p) (n):
- (𝗱(f@❨n❩)◗↑[𝐢]p) = ↑[f](𝗱n◗p).
+ (𝗱(f@❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
// qed.
lemma lift_path_m_sn (f) (p):
lemma lift_path_S_sn (f) (p):
(𝗦◗↑[f]p) = ↑[f](𝗦◗p).
// qed.
-
-lemma lift_path_id (p):
- p = ↑[𝐢]p.
-#p elim p -p //
-* [ #n ] #p #IH //
-[ <lift_path_d_sn //
-| <lift_path_L_sn //
-]
-qed.
-
-lemma lift_path_append (p2) (p1) (f):
- (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
-#p2 #p1 elim p1 -p1 //
-* [ #n1 ] #p1 #IH #f
-[ <lift_path_d_sn <lift_path_d_sn <IH //
-| <lift_path_m_sn <lift_path_m_sn <IH //
-| <lift_path_L_sn <lift_path_L_sn <IH //
-| <lift_path_A_sn <lift_path_A_sn <IH //
-| <lift_path_S_sn <lift_path_S_sn <IH //
-]
-qed.
-
-lemma lift_path_d_dx (n) (p) (f):
- (↑[f]p)◖𝗱((↑[p]f)@❨n❩) = ↑[f](p◖𝗱n).
-#n #p #f <lift_path_append //
-qed.
-
-lemma lift_path_m_dx (p) (f):
- (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
-#p #f <lift_path_append //
-qed.
-
-lemma lift_path_L_dx (p) (f):
- (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
-#p #f <lift_path_append //
-qed.
-
-lemma lift_path_A_dx (p) (f):
- (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
-#p #f <lift_path_append //
-qed.
-
-lemma lift_path_S_dx (p) (f):
- (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
-#p #f <lift_path_append //
-qed.
-
(* COMMENT
(* Advanced constructions with proj_rmap and stream_tls *********************)
--- /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 "ground/relocation/tr_id_pap.ma".
+include "ground/relocation/tr_id_tls.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+(* Constructions with tr_id ************************************************)
+
+lemma lift_path_id (p):
+ p = ↑[𝐢]p.
+#p elim p -p //
+* [ #n ] #p #IH //
+[ <lift_path_d_sn //
+| <lift_path_L_sn //
+]
+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 "ground/lib/list_length.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with list_length ******************************************)
-
-lemma lift_path_length (p) (f):
- ❘p❘ = ❘↑[f]p❘.
-#p elim p -p // * [ #n ] #p #IH #f //
-[ <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-]
->list_length_lcons >list_length_lcons //
-qed.
| @subset_equivalence_ext_f1_exteq /2 width=5/
]
qed.
-
-lemma lift_term_id_sn (t):
- t ⊆ ↑[𝐢]t.
-#t #p #Hp
->(lift_path_id p)
-/2 width=1 by in_comp_lift_bi/
-qed-.
-
-lemma lift_term_id_dx (t):
- ↑[𝐢]t ⊆ t.
-#t #p * #q #Hq #H destruct //
-qed-.
-
-lemma lift_term_id (t):
- t ⇔ ↑[𝐢]t.
-/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
-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_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_id.ma".
+
+lemma lift_term_id_sn (t):
+ t ⊆ ↑[𝐢]t.
+#t #p #Hp
+>(lift_path_id p)
+/2 width=1 by in_comp_lift_bi/
+qed-.
+
+lemma lift_term_id_dx (t):
+ ↑[𝐢]t ⊆ t.
+#t #p * #q #Hq #H destruct //
+qed-.
+
+lemma lift_term_id (t):
+ t ⇔ ↑[𝐢]t.
+/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
+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_structure.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with structure ********************************************)
-
-lemma structure_lift (p) (f):
- ⊗p = ⊗↑[f]p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-[ <lift_path_m_sn <structure_m_sn <structure_m_sn //
-| <lift_path_L_sn <structure_L_sn <structure_L_sn //
-]
-qed.
-
-lemma lift_structure (p) (f):
- ⊗p = ↑[f]⊗p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-qed.
(* *)
(**************************************************************************)
-include "delayed_updating/substitution/lift_eq.ma".
+include "delayed_updating/substitution/lift_id.ma".
include "ground/relocation/tr_uni_pap.ma".
+include "ground/relocation/tr_uni_tls.ma".
(* LIFT FOR PATH ***********************************************************)
lemma lift_path_d_sn_uni (p) (m) (n):
(𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
-// qed.
+#p #m #n
+<lift_path_d_sn <tr_uni_pap >nsucc_pnpred
+<tr_tls_succ_uni //
+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/syntax/path_depth.ma".
+include "delayed_updating/syntax/path_labels.ma".
+
+(* DEPTH FOR PATH ***********************************************************)
+
+(* Constructions with labels ************************************************)
+
+lemma depth_labels_L (n):
+ n = ♭(𝗟∗∗n).
+#n @(nat_ind_succ … n) -n //
+#n #IH <labels_succ <depth_L_sn //
+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/syntax/path_labels.ma".
+include "delayed_updating/notation/functions/downarrowright_2.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_pred_succ.ma".
+
+(* HEAD FOR PATH ************************************************************)
+
+rec definition path_head (m) (p) on p: path ≝
+match m with
+[ nzero ⇒ 𝐞
+| ninj o ⇒
+ match p with
+ [ list_empty ⇒ 𝗟∗∗m
+ | list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒ l◗(path_head (m+n) q)
+ | label_m ⇒ l◗(path_head m q)
+ | label_L ⇒ l◗(path_head (↓o) q)
+ | label_A ⇒ l◗(path_head m q)
+ | label_S ⇒ l◗(path_head m q)
+ ]
+ ]
+].
+
+interpretation
+ "head (reversed path)"
+ 'DownArrowRight n p = (path_head n p).
+
+(* basic constructions ****************************************************)
+
+lemma path_head_zero (p):
+ (𝐞) = ↳[𝟎]p.
+* // qed.
+
+lemma path_head_empty (n):
+ (𝗟∗∗n) = ↳[n]𝐞.
+* // qed.
+
+lemma path_head_d_sn (p) (n) (m:pnat):
+ (𝗱m◗↳[↑n+m]p) = ↳[↑n](𝗱m◗p).
+// qed.
+
+lemma path_head_m_sn (p) (n):
+ (𝗺◗↳[↑n]p) = ↳[↑n](𝗺◗p).
+// qed.
+
+lemma path_head_L_sn (p) (n):
+ (𝗟◗↳[n]p) = ↳[↑n](𝗟◗p).
+#p #n
+whd in ⊢ (???%); //
+qed.
+
+lemma path_head_A_sn (p) (n):
+ (𝗔◗↳[↑n]p) = ↳[↑n](𝗔◗p).
+// qed.
+
+lemma path_head_S_sn (p) (n):
+ (𝗦◗↳[↑n]p) = ↳[↑n](𝗦◗p).
+// 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/syntax/path_head.ma".
+include "delayed_updating/syntax/path_depth_labels.ma".
+include "delayed_updating/syntax/path_height_labels.ma".
+
+(* HEAD FOR PATH ************************************************************)
+
+(* Constructions with depth and height **************************************)
+
+lemma path_head_depth (p) (n):
+ n + ♯(↳[n]p) = ♭↳[n]p.
+#p elim p -p // #l #p #IH
+#n @(nat_ind_succ … n) -n // #n #_
+cases l [ #m ]
+[ <path_head_d_sn <height_d_sn <depth_d_sn //
+| <path_head_m_sn <height_m_sn <depth_m_sn //
+| <path_head_L_sn <height_L_sn <depth_L_sn
+ <nplus_succ_sn //
+| <path_head_A_sn <height_A_sn <depth_A_sn //
+| <path_head_S_sn <height_S_sn <depth_S_sn //
+]
+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/syntax/path_height.ma".
+include "delayed_updating/syntax/path_labels.ma".
+
+(* HEIGHT FOR PATH **********************************************************)
+
+(* Constructions with labels ************************************************)
+
+lemma height_labels_L (n):
+ (𝟎) = ♯(𝗟∗∗n).
+#n @(nat_ind_succ … n) -n //
+#n #IH <labels_succ <height_L_sn //
+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/syntax/path_labels.ma".
-include "delayed_updating/notation/functions/downarrowright_2.ma".
-include "ground/arith/nat_plus.ma".
-include "ground/arith/nat_pred_succ.ma".
-
-(* TAIL FOR PATH ************************************************************)
-
-rec definition tail (m) (p) on p: path ≝
-match m with
-[ nzero ⇒ 𝐞
-| ninj o ⇒
- match p with
- [ list_empty ⇒ 𝗟∗∗m
- | list_lcons l q ⇒
- match l with
- [ label_d n ⇒ l◗(tail (m+n) q)
- | label_m ⇒ l◗(tail m q)
- | label_L ⇒ l◗(tail (↓o) q)
- | label_A ⇒ l◗(tail m q)
- | label_S ⇒ l◗(tail m q)
- ]
- ]
-].
-
-interpretation
- "tail (reversed path)"
- 'DownArrowRight n p = (tail n p).
-
-(* basic constructions ****************************************************)
-
-lemma tail_zero (p):
- (𝐞) = ↳[𝟎]p.
-* // qed.
-
-lemma tail_empty (n):
- (𝗟∗∗n) = ↳[n]𝐞.
-* // qed.
-
-lemma tail_d_sn (p) (n) (m:pnat):
- (𝗱m◗↳[↑n+m]p) = ↳[↑n](𝗱m◗p).
-// qed.
-
-lemma tail_m_sn (p) (n):
- (𝗺◗↳[↑n]p) = ↳[↑n](𝗺◗p).
-// qed.
-
-lemma tail_L_sn (p) (n):
- (𝗟◗↳[n]p) = ↳[↑n](𝗟◗p).
-#p #n
-whd in ⊢ (???%); //
-qed.
-
-lemma tail_A_sn (p) (n):
- (𝗔◗↳[↑n]p) = ↳[↑n](𝗔◗p).
-// qed.
-
-lemma tail_S_sn (p) (n):
- (𝗦◗↳[↑n]p) = ↳[↑n](𝗦◗p).
-// 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/syntax/path_tail.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
-(*
-include "ground/arith/nat_plus.ma".
-include "ground/arith/nat_pred_succ.ma".
-*)
-(* TAIL FOR PATH ************************************************************)
-
-(* Constructions with depth and height **************************************)
-
-axiom depth_lablels_L (n):
- n = ♭(𝗟∗∗n).
-
-axiom height_lablels_L (n):
- (𝟎) = ♯(𝗟∗∗n).
-
-lemma tail_depth (p) (n):
- n + ♯(↳[n]p) = ♭↳[n]p.
-#p elim p -p // #l #p #IH
-#n @(nat_ind_succ … n) -n // #n #_
-cases l [ #m ]
-[ <tail_d_sn <height_d_sn <depth_d_sn //
-| <tail_m_sn <height_m_sn <depth_m_sn //
-| <tail_L_sn <height_L_sn <depth_L_sn
- <nplus_succ_sn //
-| <tail_A_sn <height_A_sn <depth_A_sn //
-| <tail_S_sn <height_S_sn <depth_S_sn //
-]
-qed.
--- /dev/null
+include "ground/relocation/tr_uni_pap.ma".
+include "ground/relocation/tr_compose_pap.ma".
+include "ground/relocation/tr_pap_pn.ma".
+include "ground/notation/functions/applysucc_2.ma".
+include "ground/arith/nat_lt.ma".
+include "ground/arith/nat_plus_rplus.ma".
+include "ground/arith/nat_pred_succ.ma".
+
+lemma nlt_npsucc_bi (n1) (n2):
+ n1 < n2 → npsucc n1 < npsucc n2.
+#n1 #n2 #Hn elim Hn -n2 //
+#n2 #_ #IH /2 width=1 by plt_succ_dx_trans/
+qed.
+
+definition tr_nap (f) (l:nat): nat ≝
+ ↓(f@❨↑l❩).
+
+interpretation
+ "functional non-negative application (total relocation maps)"
+ 'ApplySucc f l = (tr_nap f l).
+
+lemma tr_nap_unfold (f) (l):
+ ↓(f@❨↑l❩) = f@↑❨l❩.
+// qed.
+
+lemma tr_compose_nap (f2) (f1) (l):
+ f2@↑❨f1@↑❨l❩❩ = (f2∘f1)@↑❨l❩.
+#f2 #f1 #l
+<tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
+<tr_compose_pap <npsucc_pred //
+qed.
+
+lemma tr_uni_nap (n) (m):
+ m + n = 𝐮❨n❩@↑❨m❩.
+#n #m
+<tr_nap_unfold
+<tr_uni_pap <nrplus_npsucc_sn //
+qed.
+
+lemma tr_nap_push (f):
+ ∀l. ↑(f@↑❨l❩) = (⫯f)@↑❨↑l❩.
+#f #l
+<tr_nap_unfold <tr_nap_unfold
+<tr_pap_push <pnpred_psucc //
+qed.
+
+lemma tr_nap_pushs_lt (f) (n) (m):
+ m < n → m = (⫯*[n]f)@↑❨m❩.
+#f #n #m #Hmn
+<tr_nap_unfold <tr_pap_pushs_le
+/2 width=1 by nlt_npsucc_bi/
+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/syntax/path.ma".
-include "delayed_updating/notation/functions/black_righttriangle_1.ma".
-include "ground/relocation/tr_uni.ma".
-include "ground/relocation/tr_compose.ma".
-
-(* BASIC UNWIND MAP FOR PATH ************************************************)
-
-rec definition unwind1_rmap p on p: tr_map ≝
-match p with
-[ list_empty ⇒ 𝐢
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒ (unwind1_rmap q)∘𝐮❨n❩
- | label_m ⇒ unwind1_rmap q
- | label_L ⇒ ⫯(unwind1_rmap q)
- | label_A ⇒ unwind1_rmap q
- | label_S ⇒ unwind1_rmap q
- ]
-].
-
-interpretation
- "basic unwind map (reversed path)"
- 'BlackRightTriangle p = (unwind1_rmap p).
-
-(* Basic constructions *******************************************************)
-
-lemma unwind1_rmap_empty:
- (𝐢) = ▶(𝐞).
-// qed.
-
-lemma unwind1_rmap_d_sn (p) (n:pnat):
- (▶p∘𝐮❨n❩) = ▶(𝗱n◗p).
-// qed.
-
-lemma unwind1_rmap_m_sn (p):
- ▶p = ▶(𝗺◗p).
-// qed.
-
-lemma unwind1_rmap_L_sn (p):
- (⫯▶p) = ▶(𝗟◗p).
-// qed.
-
-lemma unwind1_rmap_A_sn (p):
- ▶p = ▶(𝗔◗p).
-// qed.
-
-lemma unwind1_rmap_S_sn (p):
- ▶p = ▶(𝗦◗p).
-// 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/unwind/unwind1_rmap.ma".
-include "delayed_updating/syntax/path_tail_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
-
-(* BASIC UNWIND MAP FOR PATH ************************************************)
-
-include "ground/relocation/tr_uni_pap.ma".
-include "ground/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_pap_pn.ma".
-include "ground/notation/functions/applysucc_2.ma".
-include "ground/arith/nat_plus_rplus.ma".
-include "ground/arith/nat_pred_succ.ma".
-
-definition tr_nap (f) (l:nat): nat ≝
- ↓(f@❨↑l❩).
-
-interpretation
- "functional non-negative application (total relocation maps)"
- 'ApplySucc f l = (tr_nap f l).
-
-lemma tr_nap_unfold (f) (l):
- ↓(f@❨↑l❩) = f@↑❨l❩.
-// qed.
-
-lemma tr_compose_nap (f2) (f1) (l):
- f2@↑❨f1@↑❨l❩❩ = (f2∘f1)@↑❨l❩.
-#f2 #f1 #l
-<tr_nap_unfold <tr_nap_unfold <tr_nap_unfold
-<tr_compose_pap <npsucc_pred //
-qed.
-
-lemma tr_uni_nap (n) (m):
- m + n = 𝐮❨n❩@↑❨m❩.
-#n #m
-<tr_nap_unfold
-<tr_uni_pap <nrplus_npsucc_sn //
-qed.
-
-lemma tr_nap_push (f):
- ∀l. ↑(f@↑❨l❩) = (⫯f)@↑❨↑l❩.
-#f #l
-<tr_nap_unfold <tr_nap_unfold
-<tr_pap_push <pnpred_psucc //
-qed.
-
-(****)
-
-lemma unwind1_rmap_labels_L (n):
- (𝐢) = ▶(𝗟∗∗n).
-#n @(nat_ind_succ … n) -n //
-#n #IH
-<labels_succ <unwind1_rmap_L_sn //
-qed.
-
-lemma unwind1_rmap_tail (p) (n):
- n + ♯(↳[n]p) = (▶↳[n]p)@↑❨n❩.
-#p elim p -p //
-#l #p #IH #n @(nat_ind_succ … n) -n //
-#n #_ cases l [ #m ]
-[ <unwind1_rmap_d_sn <tail_d_sn <height_d_sn
- <nplus_assoc >IH -IH <tr_compose_nap //
-| <unwind1_rmap_m_sn <tail_m_sn <height_m_sn //
-| <unwind1_rmap_L_sn <tail_L_sn <height_L_sn
- <tr_nap_push <npred_succ //
-| <unwind1_rmap_A_sn <tail_A_sn <height_A_sn //
-| <unwind1_rmap_S_sn <tail_S_sn <height_S_sn //
-]
-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/syntax/path.ma".
+include "delayed_updating/notation/functions/black_righttriangle_2.ma".
+include "delayed_updating/notation/functions/black_righttriangle_1.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/relocation/tr_compose.ma".
+
+(* UNWIND MAP FOR PATH ******************************************************)
+
+rec definition unwind2_rmap (f) (p) on p: tr_map ≝
+match p with
+[ list_empty ⇒ f
+| list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒ (unwind2_rmap f q)∘𝐮❨n❩
+ | label_m ⇒ unwind2_rmap f q
+ | label_L ⇒ ⫯(unwind2_rmap f q)
+ | label_A ⇒ unwind2_rmap f q
+ | label_S ⇒ unwind2_rmap f q
+ ]
+].
+
+interpretation
+ "tailed unwind map (reversed path)"
+ 'BlackRightTriangle f p = (unwind2_rmap f p).
+
+interpretation
+ "unwind map (reversed path)"
+ 'BlackRightTriangle p = (unwind2_rmap tr_id p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind2_rmap_empty (f):
+ f = ▶[f]𝐞.
+// qed.
+
+lemma unwind2_rmap_d_sn (f) (p) (n:pnat):
+ (▶[f]p∘𝐮❨n❩) = ▶[f](𝗱n◗p).
+// qed.
+
+lemma unwind2_rmap_m_sn (f) (p):
+ ▶[f]p = ▶[f](𝗺◗p).
+// qed.
+
+lemma unwind2_rmap_L_sn (f) (p):
+ (⫯▶[f]p) = ▶[f](𝗟◗p).
+// qed.
+
+lemma unwind2_rmap_A_sn (f) (p):
+ ▶[f]p = ▶[f](𝗔◗p).
+// qed.
+
+lemma unwind2_rmap_S_sn (f) (p):
+ ▶[f]p = ▶[f](𝗦◗p).
+// qed.
+
+(* Constructions with list_append *******************************************)
+
+lemma unwind2_rmap_append (f) (p1) (p2):
+ ▶[▶[f]p2]p1 = ▶[f](p1●p2).
+#f #p1 elim p1 -p1 //
+* [ #n ] #p1 #IH #p2 //
+[ <unwind2_rmap_m_sn <unwind2_rmap_m_sn //
+| <unwind2_rmap_L_sn <unwind2_rmap_L_sn //
+| <unwind2_rmap_A_sn <unwind2_rmap_A_sn //
+| <unwind2_rmap_S_sn <unwind2_rmap_S_sn //
+]
+qed.
+
+(* Constructions with list_rcons ********************************************)
+
+lemma unwind2_rmap_d_dx (f) (p) (n:pnat):
+ ▶[f∘𝐮❨n❩]p = ▶[f](p◖𝗱n).
+// qed.
+
+lemma unwind2_rmap_m_dx (f) (p):
+ ▶[f]p = ▶[f](p◖𝗺).
+// qed.
+
+lemma unwind2_rmap_L_dx (f) (p):
+ ▶[⫯f]p = ▶[f](p◖𝗟).
+// qed.
+
+lemma unwind2_rmap_A_dx (f) (p):
+ ▶[f]p = ▶[f](p◖𝗔).
+// qed.
+
+lemma unwind2_rmap_S_dx (f) (p):
+ ▶[f]p = ▶[f](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 "delayed_updating/unwind/unwind2_rmap.ma".
+include "ground/relocation/tr_compose_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+
+(* UNWIND MAP FOR PATH ******************************************************)
+
+(* Constructions with stream_eq *********************************************)
+
+lemma unwind2_rmap_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ▶[f1]p ≗ ▶[f2]p).
+#p elim p -p //
+* [ #n ] #p #IH #f1 #f2 #Hf
+[ /3 width=1 by tr_compose_eq_repl/
+| /2 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /2 width=1 by/
+| /2 width=1 by/
+]
+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/unwind/unwind2_rmap_labels.ma".
+include "delayed_updating/unwind/nap.ma".
+include "delayed_updating/syntax/path_head_depth.ma".
+include "delayed_updating/syntax/path_height.ma".
+
+(* UNWIND MAP FOR PATH ******************************************************)
+
+(* Constructions with path_head *********************************************)
+
+lemma unwind2_rmap_head_push (f) (p) (n):
+ n + ♯(↳[n]p) = (▶[⫯f]↳[n]p)@↑❨n❩.
+#f #p elim p -p
+[ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
+ >tr_pushs_swap <tr_nap_pushs_lt //
+| #l #p #IH #n @(nat_ind_succ … n) -n //
+ #n #_ cases l [ #m ]
+ [ <unwind2_rmap_d_sn <path_head_d_sn <height_d_sn
+ <nplus_assoc >IH -IH <tr_compose_nap //
+ | <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
+ | <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
+ <tr_nap_push <npred_succ //
+ | <unwind2_rmap_A_sn <path_head_A_sn <height_A_sn //
+ | <unwind2_rmap_S_sn <path_head_S_sn <height_S_sn //
+ ]
+]
+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/unwind/unwind2_rmap.ma".
+include "delayed_updating/syntax/path_labels.ma".
+include "ground/relocation/tr_pushs.ma".
+
+(* UNWIND MAP FOR PATH ******************************************************)
+
+(* Constructions with labels ************************************************)
+
+lemma unwind2_rmap_labels_L (f) (n):
+ (⫯*[n]f) = ▶[f](𝗟∗∗n).
+#f #n @(nat_ind_succ … n) -n //
+#n #IH
+<labels_succ <unwind2_rmap_L_sn //
+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/unwind/unwind2_rmap.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/relocation/tr_pushs.ma".
+
+(* UNWIND MAP FOR PATH ******************************************************)
+
+(* Constructions with structure and depth ***********************************)
+
+lemma unwind2_rmap_structure (p) (f):
+ (⫯*[♭p]f) = ▶[f]⊗p.
+#p elim p -p //
+* [ #n ] #p #IH #f //
+[ <unwind2_rmap_A_sn //
+| <unwind2_rmap_S_sn //
+]
+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/syntax/path.ma".
+include "delayed_updating/notation/functions/black_diamond_2.ma".
+include "ground/relocation/tr_pap.ma".
+
+(* GENERIC UNWIND FOR PATH **************************************************)
+
+rec definition unwind_gen (f) (p) on p ≝
+match p with
+[ list_empty ⇒ 𝐞
+| list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒
+ match q with
+ [ list_empty ⇒ 𝗱(f@❨n❩)◗(unwind_gen f q)
+ | list_lcons _ _ ⇒ unwind_gen f q
+ ]
+ | label_m ⇒ unwind_gen f q
+ | label_L ⇒ l◗(unwind_gen f q)
+ | label_A ⇒ l◗(unwind_gen f q)
+ | label_S ⇒ l◗(unwind_gen f q)
+ ]
+].
+
+interpretation
+ "generic unwind (path)"
+ 'BlackDiamond f p = (unwind_gen f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind_gen_empty (f):
+ (𝐞) = ◆[f]𝐞.
+// qed.
+
+lemma unwind_gen_d_empty (f) (n):
+ 𝗱(f@❨n❩)◗𝐞 = ◆[f](𝗱n◗𝐞).
+// qed.
+
+lemma unwind_gen_d_lcons (f) (p) (l) (n):
+ ◆[f](l◗p) = ◆[f](𝗱n◗l◗p).
+// qed.
+
+lemma unwind_gen_m_sn (f) (p):
+ ◆[f]p = ◆[f](𝗺◗p).
+// qed.
+
+lemma unwind_gen_L_sn (f) (p):
+ (𝗟◗◆[f]p) = ◆[f](𝗟◗p).
+// qed.
+
+lemma unwind_gen_A_sn (f) (p):
+ (𝗔◗◆[f]p) = ◆[f](𝗔◗p).
+// qed.
+
+lemma unwind_gen_S_sn (f) (p):
+ (𝗦◗◆[f]p) = ◆[f](𝗦◗p).
+// qed.
+
+(* Advanced eliminations with path ******************************************)
+
+lemma path_ind_unwind (Q:predicate …):
+ Q (𝐞) →
+ (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
+ (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
+ (∀p. Q p → Q (𝗺◗p)) →
+ (∀p. Q p → Q (𝗟◗p)) →
+ (∀p. Q p → Q (𝗔◗p)) →
+ (∀p. Q p → Q (𝗦◗p)) →
+ ∀p. Q p.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
+elim p -p [| * [ #n * ] ]
+/2 width=1 by/
+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/unwind/unwind_gen.ma".
+include "ground/relocation/tr_compose_pap.ma".
+
+(* GENERIC UNWIND FOR PATH **************************************************)
+
+(* Properties with tr_compose ***********************************************)
+
+lemma unwind_gen_after (f2) (f1) (p):
+ ◆[f2]◆[f1]p = ◆[f2∘f1]p.
+#f2 #f1 #p @(path_ind_unwind … p) -p //
+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/unwind/unwind_gen.ma".
+include "ground/relocation/tr_pap_eq.ma".
+
+(* GENERIC UNWIND FOR PATH **************************************************)
+
+(* Constructions with stream_eq *********************************************)
+
+lemma unwind_gen_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ◆[f1]p = ◆[f2]p).
+#p @(path_ind_unwind … p) -p // [ #n |*: #p ] #IH #f1 #f2 #Hf
+[ <unwind_gen_d_empty <unwind_gen_d_empty
+ <(tr_pap_eq_repl … Hf) -f2 -IH //
+| <unwind_gen_L_sn <unwind_gen_L_sn
+ <(IH … Hf) -f2 -IH //
+| <unwind_gen_A_sn <unwind_gen_A_sn
+ <(IH … Hf) -f2 -IH //
+| <unwind_gen_S_sn <unwind_gen_S_sn
+ <(IH … Hf) -f2 -IH //
+]
+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/unwind/unwind_gen.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_inner.ma".
+include "delayed_updating/syntax/path_proper.ma".
+include "ground/xoa/ex_4_2.ma".
+include "ground/xoa/ex_3_2.ma".
+
+(* GENERIC UNWIND FOR PATH *************************************************)
+
+(* Basic constructions with structure **************************************)
+
+lemma structure_unwind_gen (p) (f):
+ ⊗p = ⊗◆[f]p.
+#p @(path_ind_unwind … p) -p //
+qed.
+
+lemma unwind_gen_structure (p) (f):
+ ⊗p = ◆[f]⊗p.
+#p @(path_ind_unwind … p) -p //
+qed.
+
+(* Destructions with structure **********************************************)
+
+lemma unwind_gen_des_structure (q) (p) (f):
+ ⊗q = ◆[f]p → ⊗q = ⊗p.
+// qed-.
+
+(* Constructions with proper condition for path *****************************)
+
+lemma unwind_gen_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
+ (⊗p1)●(◆[f]p2) = ◆[f](p1●p2).
+#p2 #p1 @(path_ind_unwind … p1) -p1 //
+[ #n | #n #l #p1 |*: #p1 ] #IH #f #Hp2
+[ elim (ppc_inv_lcons … Hp2) -Hp2 #l #q #H0 destruct //
+| <unwind_gen_d_lcons <IH //
+| <unwind_gen_m_sn <IH //
+| <unwind_gen_L_sn <IH //
+| <unwind_gen_A_sn <IH //
+| <unwind_gen_S_sn <IH //
+]
+qed-.
+
+(* Constructions with inner condition for path ******************************)
+
+lemma unwind_gen_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
+ (⊗p1)●(◆[f]p2) = ◆[f](p1●p2).
+#p1 @(list_ind_rcons … p1) -p1 //
+#p1 * [ #n ] #_ #p2 #f #Hp1
+[ elim (pic_inv_d_dx … Hp1)
+| <list_append_rcons_sn <unwind_gen_append_proper_dx //
+| <list_append_rcons_sn <unwind_gen_append_proper_dx //
+ <structure_L_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_gen_append_proper_dx //
+ <structure_A_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_gen_append_proper_dx //
+ <structure_S_dx <list_append_rcons_sn //
+]
+qed-.
+
+(* Advanced constructions with list_rcons ***********************************)
+
+lemma unwind_gen_d_empty_dx (n) (p) (f):
+ (⊗p)◖𝗱(f@❨n❩) = ◆[f](p◖𝗱n).
+#n #p #f <unwind_gen_append_proper_dx //
+qed.
+
+lemma unwind_gen_m_dx (p) (f):
+ ⊗p = ◆[f](p◖𝗺).
+#p #f <unwind_gen_append_proper_dx //
+qed.
+
+lemma unwind_gen_L_dx (p) (f):
+ (⊗p)◖𝗟 = ◆[f](p◖𝗟).
+#p #f <unwind_gen_append_proper_dx //
+qed.
+
+lemma unwind_gen_A_dx (p) (f):
+ (⊗p)◖𝗔 = ◆[f](p◖𝗔).
+#p #f <unwind_gen_append_proper_dx //
+qed.
+
+lemma unwind_gen_S_dx (p) (f):
+ (⊗p)◖𝗦 = ◆[f](p◖𝗦).
+#p #f <unwind_gen_append_proper_dx //
+qed.
+
+lemma unwind_gen_root (f) (p):
+ ∃∃r. 𝐞 = ⊗r & ⊗p●r = ◆[f]p.
+#f #p @(list_ind_rcons … p) -p
+[ /2 width=3 by ex2_intro/
+| #p * [ #n ] /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma unwind_gen_inv_d_sn (k) (q) (p) (f):
+ (𝗱k◗q) = ◆[f]p →
+ ∃∃r,h. 𝐞 = ⊗r & f@❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
+#k #q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_gen_empty #H destruct
+| <unwind_gen_d_empty #H destruct -IH
+ /2 width=5 by ex4_2_intro/
+| <unwind_gen_d_lcons #H
+ elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+ /2 width=5 by ex4_2_intro/
+| <unwind_gen_m_sn #H
+ elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+ /2 width=5 by ex4_2_intro/
+| <unwind_gen_L_sn #H destruct
+| <unwind_gen_A_sn #H destruct
+| <unwind_gen_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_gen_inv_m_sn (q) (p) (f):
+ (𝗺◗q) = ◆[f]p → ⊥.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_gen_empty #H destruct
+| <unwind_gen_d_empty #H destruct
+| <unwind_gen_d_lcons #H /2 width=2 by/
+| <unwind_gen_m_sn #H /2 width=2 by/
+| <unwind_gen_L_sn #H destruct
+| <unwind_gen_A_sn #H destruct
+| <unwind_gen_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_gen_inv_L_sn (q) (p) (f):
+ (𝗟◗q) = ◆[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ◆[f]r2 & r1●𝗟◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_gen_empty #H destruct
+| <unwind_gen_d_empty #H destruct
+| <unwind_gen_d_lcons #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_L_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_A_sn #H destruct
+| <unwind_gen_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_gen_inv_A_sn (q) (p) (f):
+ (𝗔◗q) = ◆[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ◆[f]r2 & r1●𝗔◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_gen_empty #H destruct
+| <unwind_gen_d_empty #H destruct
+| <unwind_gen_d_lcons #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_L_sn #H destruct
+| <unwind_gen_A_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_gen_inv_S_sn (q) (p) (f):
+ (𝗦◗q) = ◆[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ◆[f]r2 & r1●𝗦◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_gen_empty #H destruct
+| <unwind_gen_d_empty #H destruct
+| <unwind_gen_d_lcons #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_gen_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/| <unwind_gen_L_sn #H destruct
+| <unwind_gen_A_sn #H destruct
+| <unwind_gen_S_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma unwind_gen_inv_append_proper_dx (q2) (q1) (p) (f):
+ q2 ϵ 𝐏 → q1●q2 = ◆[f]p →
+ ∃∃p1,p2. ⊗p1 = q1 & ◆[f]p2 = q2 & p1●p2 = p.
+#q2 #q1 elim q1 -q1
+[ #p #f #Hq2 <list_append_empty_sn #H destruct
+ /2 width=5 by ex3_2_intro/
+| * [ #n1 ] #q1 #IH #p #f #Hq2 <list_append_lcons_sn #H
+ [ elim (unwind_gen_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
+ elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
+ elim Hq2 -Hq2 //
+ | elim (unwind_gen_inv_m_sn … H)
+ | elim (unwind_gen_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+ elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … (r1●𝗟◗p1)) //
+ <structure_append <Hr1 -Hr1 //
+ | elim (unwind_gen_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+ elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … (r1●𝗔◗p1)) //
+ <structure_append <Hr1 -Hr1 //
+ | elim (unwind_gen_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+ elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … (r1●𝗦◗p1)) //
+ <structure_append <Hr1 -Hr1 //
+ ]
+]
+qed-.
+
+(* Inversions with inner condition for path *********************************)
+
+lemma unwind_gen_inv_append_inner_sn (q1) (q2) (p) (f):
+ q1 ϵ 𝐈 → q1●q2 = ◆[f]p →
+ ∃∃p1,p2. ⊗p1 = q1 & ◆[f]p2 = q2 & p1●p2 = p.
+#q1 @(list_ind_rcons … q1) -q1
+[ #q2 #p #f #Hq1 <list_append_empty_sn #H destruct
+ /2 width=5 by ex3_2_intro/
+| #q1 * [ #n1 ] #_ #q2 #p #f #Hq2
+ [ elim (pic_inv_d_dx … Hq2)
+ | <list_append_rcons_sn #H0
+ elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_gen_inv_m_sn … (sym_eq … H2))
+ | <list_append_rcons_sn #H0
+ elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_gen_inv_L_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+ @(ex3_2_intro … (p1●r2◖𝗟)) [1,3: // ]
+ [ <structure_append <structure_L_dx <Hr2 -Hr2 //
+ | <list_append_assoc <list_append_rcons_sn //
+ ]
+ | <list_append_rcons_sn #H0
+ elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_gen_inv_A_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+ @(ex3_2_intro … (p1●r2◖𝗔)) [1,3: // ]
+ [ <structure_append <structure_A_dx <Hr2 -Hr2 //
+ | <list_append_assoc <list_append_rcons_sn //
+ ]
+ | <list_append_rcons_sn #H0
+ elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_gen_inv_S_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+ @(ex3_2_intro … (p1●r2◖𝗦)) [1,3: // ]
+ [ <structure_append <structure_S_dx <Hr2 -Hr2 //
+ | <list_append_assoc <list_append_rcons_sn //
+ ]
+ ]
+]
+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_length.ma".
-include "delayed_updating/notation/functions/black_downtriangle_1.ma".
-include "ground/relocation/tr_uni.ma".
-
-(* BASIC UNWIND FOR PATH ****************************************************)
-
-rec definition unwind1_path_pnat (n) (p) on n ≝
-match n with
-[ punit ⇒ p
-| psucc m ⇒
- match p with
- [ list_empty ⇒ 𝐞
- | list_lcons l q ⇒
- match l with
- [ label_d n ⇒
- match q with
- [ list_empty ⇒ l◗(unwind1_path_pnat m q)
- | list_lcons _ _ ⇒ unwind1_path_pnat m (↑[𝐮❨n❩]q)
- ]
- | label_m ⇒ unwind1_path_pnat m q
- | label_L ⇒ 𝗟◗(unwind1_path_pnat m q)
- | label_A ⇒ 𝗔◗(unwind1_path_pnat m q)
- | label_S ⇒ 𝗦◗(unwind1_path_pnat m q)
- ]
- ]
-].
-
-definition unwind1_path (p) ≝
- unwind1_path_pnat (↑❘p❘) p.
-
-interpretation
- "basic unwind (path)"
- 'BlackDownTriangle p = (unwind1_path p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind1_path_unfold (p):
- unwind1_path_pnat (↑❘p❘) p = ▼p.
-// qed-.
-
-lemma unwind1_path_empty:
- (𝐞) = ▼𝐞.
-// qed.
-
-lemma unwind1_path_d_empty (n):
- (𝗱n◗𝐞) = ▼(𝗱n◗𝐞).
-// qed.
-
-lemma unwind1_path_d_lcons (p) (l) (n:pnat):
- ▼(↑[𝐮❨n❩](l◗p)) = ▼(𝗱n◗l◗p).
-#p #l #n
-<unwind1_path_unfold <lift_path_length //
-qed.
-
-lemma unwind1_path_m_sn (p):
- ▼p = ▼(𝗺◗p).
-// qed.
-
-lemma unwind1_path_L_sn (p):
- (𝗟◗▼p) = ▼(𝗟◗p).
-// qed.
-
-lemma unwind1_path_A_sn (p):
- (𝗔◗▼p) = ▼(𝗔◗p).
-// qed.
-
-lemma unwind1_path_S_sn (p):
- (𝗦◗▼p) = ▼(𝗦◗p).
-// qed.
-
-(* Main constructions *******************************************************)
-
-fact unwind1_path_fix_aux (k) (p):
- k = ❘p❘ → ▼p = ▼▼p.
-#k @(nat_ind_succ … k) -k
-[ #p #H0 >(list_length_inv_zero_sn … H0) -p //
-| #k #IH *
- [ #H0 elim (eq_inv_nsucc_zero … H0)
- | * [ #n ] #p #H0
- lapply (eq_inv_nsucc_bi … H0) -H0
- [ cases p -p [ -IH | #l #p ] #H0 destruct //
- <unwind1_path_d_lcons <IH -IH //
- | #H0 destruct <unwind1_path_m_sn <IH -IH //
- | #H0 destruct <unwind1_path_L_sn <unwind1_path_L_sn <IH -IH //
- | #H0 destruct <unwind1_path_A_sn <unwind1_path_A_sn <IH -IH //
- | #H0 destruct <unwind1_path_S_sn <unwind1_path_S_sn <IH -IH //
- ]
- ]
-]
-qed-.
-
-theorem unwind1_path_fix (p):
- ▼p = ▼▼p.
-/2 width=2 by unwind1_path_fix_aux/ 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/unwind0/unwind1_rmap_eq.ma".
-include "delayed_updating/unwind0/unwind1_path.ma".
-include "delayed_updating/substitution/lift_structure.ma".
-
-(* BASIC UNWIND FOR PATH ****************************************************)
-
-(* Constructions with structure *********************************************)
-
-fact unwind1_path_d_dx_aux (k) (p) (n):
- k = ❘p❘ → (⊗p)◖𝗱((▶p)@❨n❩) = ▼(p◖𝗱n).
-#k @(nat_ind_succ … k) -k
-[ #p #n #H0 >(list_length_inv_zero_sn … H0) -p //
-| #k #IH *
- [ #n #H0 elim (eq_inv_nsucc_zero … H0)
- | * [ #m ] #p #n #H0
- lapply (eq_inv_nsucc_bi … H0) -H0
- [ cases p -p [ -IH | #l #p ] #H0 destruct <unwind1_path_d_lcons
- [ <lift_path_d_sn <lift_path_empty <unwind1_path_d_empty
- <list_cons_comm <(tr_pap_eq_repl … (unwind1_rmap_d_empty …)) //
- | >list_cons_shift <lift_path_d_dx <IH -IH //
- <structure_lift <structure_d_sn /3 width=1 by eq_f2/
- ]
- | #H0 destruct <unwind1_path_m_sn <IH -IH //
- | #H0 destruct <unwind1_path_L_sn <IH -IH //
- | #H0 destruct <unwind1_path_A_sn <IH -IH //
- | #H0 destruct <unwind1_path_S_sn <IH -IH //
- ]
- ]
-]
-qed-.
-
-lemma unwind1_path_d_dx (p) (n):
- (⊗p)◖𝗱((▶p)@❨n❩) = ▼(p◖𝗱n).
-/2 width=2 by unwind1_path_d_dx_aux/ 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_length.ma".
-include "delayed_updating/notation/functions/black_righttriangle_1.ma".
-include "ground/relocation/tr_uni.ma".
-include "ground/relocation/tr_compose.ma".
-
-(* BASIC UNWIND FOR RELOCATION MAP ******************************************)
-
-rec definition unwind1_rmap_pnat (n) (p) on n ≝
-match n with
-[ punit ⇒ 𝐢
-| psucc m ⇒
- match p with
- [ list_empty ⇒ 𝐢
- | list_lcons l q ⇒
- match l with
- [ label_d n ⇒ (unwind1_rmap_pnat m (↑[𝐮❨n❩]q))∘(↑[q]𝐮❨n❩)
- | label_m ⇒ unwind1_rmap_pnat m q
- | label_L ⇒ unwind1_rmap_pnat m q
- | label_A ⇒ unwind1_rmap_pnat m q
- | label_S ⇒ unwind1_rmap_pnat m q
- ]
- ]
-].
-
-definition unwind1_rmap (p) ≝
- unwind1_rmap_pnat (↑❘p❘) p.
-
-interpretation
- "basic unwind (relocation map)"
- 'BlackRightTriangle p = (unwind1_rmap p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind1_rmap_unfold (p):
- unwind1_rmap_pnat (↑❘p❘) p = ▶p.
-// qed-.
-
-lemma unwind1_rmap_empty:
- (𝐢) = ▶(𝐞).
-// qed.
-
-lemma unwind1_rmap_d_sn (p) (n:pnat):
- (▶(↑[𝐮❨n❩]p))∘(↑[p]𝐮❨n❩) = ▶(𝗱n◗p).
-#p #n
-<unwind1_rmap_unfold <lift_path_length //
-qed.
-
-lemma unwind1_rmap_m_sn (p):
- ▶p = ▶(𝗺◗p).
-// qed.
-
-lemma unwind1_rmap_L_sn (p):
- ▶p = ▶(𝗟◗p).
-// qed.
-
-lemma unwind1_rmap_A_sn (p):
- ▶p = ▶(𝗔◗p).
-// qed.
-
-lemma unwind1_rmap_S_sn (p):
- ▶p = ▶(𝗦◗p).
-// 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/unwind0/unwind1_rmap.ma".
-include "ground/relocation/tr_id_compose.ma".
-
-(* BASIC UNWIND FOR RELOCATION MAP ******************************************)
-
-(* Constructions with stream_eq *********************************************)
-
-lemma unwind1_rmap_d_empty (n:pnat):
- (𝐮❨n❩) ≗ ▶(𝗱n◗𝐞).
-// 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/unwind0/unwind1_path.ma".
-include "delayed_updating/notation/functions/black_downtriangle_2.ma".
-
-(* EXTENDED UNWIND FOR PATH *************************************************)
-
-definition unwind2_path (f) (p) ≝
- ▼↑[f]p.
-
-interpretation
- "extended unwind (path)"
- 'BlackDownTriangle f p = (unwind2_path f p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind2_path_unfold (f) (p):
- ▼↑[f]p = ▼[f]p.
-// qed.
-
-lemma unwind2_path_id (p):
- ▼p = ▼[𝐢]p.
-// qed.
-
-lemma unwind2_path_empty (f):
- (𝐞) = ▼[f]𝐞.
-// qed.
-
-lemma unwind2_path_d_empty (f) (n):
- (𝗱(f@❨n❩)◗𝐞) = ▼[f](𝗱n◗𝐞).
-// qed.
-
-lemma unwind2_path_d_lcons (f) (p) (l) (n:pnat):
- ▼[𝐮❨f@❨n❩❩](l◗p) = ▼[f](𝗱n◗l◗p).
-#f #p #l #n
-<unwind2_path_unfold in ⊢ (???%);
-<lift_path_d_sn <lift_path_id <unwind1_path_d_lcons //
-qed.
-
-lemma unwind2_path_m_sn (f) (p):
- ▼[f]p = ▼[f](𝗺◗p).
-#f #p
-<unwind2_path_unfold in ⊢ (???%);
-<lift_path_m_sn //
-qed.
-
-lemma unwind2_path_L_sn (f) (p):
- (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
-#f #p
-<unwind2_path_unfold in ⊢ (???%);
-<lift_path_L_sn //
-qed.
-
-lemma unwind2_path_A_sn (f) (p):
- (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
-#f #p
-<unwind2_path_unfold in ⊢ (???%);
-<lift_path_A_sn //
-qed.
-
-lemma unwind2_path_S_sn (f) (p):
- (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
-#f #p
-<unwind2_path_unfold in ⊢ (???%);
-<lift_path_S_sn //
-qed.
-
-(* Main constructions *******************************************************)
-
-theorem unwind2_path_after_id_sn (p) (f):
- ▼[f]p = ▼▼[f]p.
-// qed.
-
-(* Constructions with stream_eq *********************************************)
-
-lemma unwind2_path_eq_repl (p):
- stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
-#p #f1 #f2 #Hf
-<unwind2_path_unfold <unwind2_path_unfold
-<(lift_path_eq_repl … Hf) -Hf //
-qed.
-
-(* Advanced eliminations with path ******************************************)
-
-lemma path_ind_unwind2 (Q:predicate …):
- Q (𝐞) →
- (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
- (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
- (∀p. Q p → Q (𝗺◗p)) →
- (∀p. Q p → Q (𝗟◗p)) →
- (∀p. Q p → Q (𝗔◗p)) →
- (∀p. Q p → Q (𝗦◗p)) →
- ∀p. Q p.
-#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
-elim p -p [| * [ #n * ] ]
-/2 width=1 by/
-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/unwind0/unwind2_rmap.ma".
-include "delayed_updating/unwind0/unwind2_path.ma".
-include "delayed_updating/unwind0/unwind1_path_structure.ma".
-
-(* EXTENDED UNWIND FOR PATH *************************************************)
-
-(* Constructions with structure *********************************************)
-
-lemma unwind2_path_d_dx (f) (p) (n):
- (⊗p)◖𝗱((▶[f]p)@❨n❩) = ▼[f](p◖𝗱n).
-#f #p #n
-<unwind2_path_unfold <lift_path_d_dx <unwind1_path_d_dx
-<structure_lift >tr_compose_pap //
-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/unwind0/unwind1_rmap.ma".
-include "delayed_updating/notation/functions/black_righttriangle_2.ma".
-
-(* EXTENDED UNWIND FOR RELOCATION MAP ***************************************)
-
-definition unwind2_rmap (f) (p): tr_map ≝
- (▶↑[f]p)∘(↑[p]f).
-
-interpretation
- "extended unwind (relocation map)"
- 'BlackRightTriangle f p = (unwind2_rmap f p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind2_rmap_unfold (f) (p):
- (▶↑[f]p)∘(↑[p]f) = ▶[f]p.
-// qed.
-
-lemma unwind2_rmap_m_sn (f) (p):
- ▶[f]p = ▶[f](𝗺◗p).
-#f #p
-<unwind2_rmap_unfold in ⊢ (???%);
-<lift_rmap_m_sn <lift_path_m_sn //
-qed.
-
-lemma unwind2_rmap_L_sn (f) (p):
- ▶[⫯f]p = ▶[f](𝗟◗p).
-#f #p
-<unwind2_rmap_unfold in ⊢ (???%);
-<lift_rmap_L_sn <lift_path_L_sn //
-qed.
-
-lemma unwind2_rmap_A_sn (f) (p):
- ▶[f]p = ▶[f](𝗔◗p).
-#f #p
-<unwind2_rmap_unfold in ⊢ (???%);
-<lift_rmap_A_sn <lift_path_A_sn //
-qed.
-
-lemma unwind2_rmap_S_sn (f) (p):
- ▶[f]p = ▶[f](𝗦◗p).
-#f #p
-<unwind2_rmap_unfold in ⊢ (???%);
-<lift_rmap_S_sn <lift_path_S_sn //
-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/unwind0/unwind2_rmap.ma".
-include "ground/relocation/tr_id_compose.ma".
-
-(* EXTENDED UNWIND FOR RELOCATION MAP ***************************************)
-
-(* Advanced constructions ***************************************************)
-
-lemma unwind2_rmap_id (p):
- ▶p ≗ ▶[𝐢]p.
-// qed.
-
-lemma unwind2_rmap_empty (f):
- f ≗ ▶[f]𝐞.
-// qed.
-
-lemma unwind2_rmap_d_sn (f) (p) (n):
- ▶[𝐮❨f@❨n❩❩]p ≗ ▶[f](𝗱n◗p).
-#f #p #n
-<unwind2_rmap_unfold <unwind2_rmap_unfold
-<lift_rmap_d_sn <lift_rmap_id
-<lift_path_d_sn <lift_path_id <unwind1_rmap_d_sn //
-qed.
["{"; "❴"; "⦃" ] ;
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["□"; "◽"; "▪"; "◾"; ];
- ["◊"; "⬦"; "♢"; "⧫"; "♦"; "⟐"; "⟠"; ] ;
+ ["â\97\8a"; "⬦"; "â\99¢"; "â\97\86"; "⧫"; "â\99¦"; "â\9f\90"; "â\9f "; ] ;
[">"; "⭃"; "⧁"; "〉"; "»"; "❭"; "❯"; "❱"; "▸"; "►"; "▶"; "⊃"; "⊐"; ] ;
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["∨"; "⩖"; "∪"; "∩"; "⋓"; "⋒" ] ;
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