+++ /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/prototerm_constructors.ma".
-include "delayed_updating/syntax/prototerm_equivalence.ma".
-include "delayed_updating/substitution/fsubst.ma".
-include "delayed_updating/substitution/unwind.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_balanced.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
-include "ground/xoa/ex_1_2.ma".
-include "ground/xoa/and_4.ma".
-
-(* DELAYED FOCUSED REDUCTION ************************************************)
-
-definition dfr (p) (q): relation2 prototerm prototerm ≝
- λt1,t2. ∃∃b,n.
- let r ≝ p●𝗔◗b●𝗟◗q in
- ∧∧ ⊗b ϵ 𝐁 & ∀f. ↑❘q❘ = (↑[q]⫯f)@❨n❩ & r◖𝗱n ϵ t1 &
- t1[⋔r←𝛗(n+❘b❘).(t1⋔(p◖𝗦))] ⇔ t2
-.
-
-interpretation
- "focused balanced reduction with delayed updating (prototerm)"
- 'BlackRightArrowDF t1 p q t2 = (dfr p q t1 t2).
+++ /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/reduction/dfr.ma".
-include "delayed_updating/reduction/ifr.ma".
-include "delayed_updating/substitution/fsubst_unwind.ma".
-include "delayed_updating/substitution/fsubst_eq.ma".
-include "delayed_updating/substitution/unwind_constructors.ma".
-include "delayed_updating/substitution/unwind_preterm_eq.ma".
-include "delayed_updating/substitution/unwind_structure_depth.ma".
-include "delayed_updating/substitution/unwind_depth.ma".
-include "delayed_updating/syntax/prototerm_proper_constructors.ma".
-include "delayed_updating/syntax/path_structure_depth.ma".
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_pap_pushs.ma".
-
-(* DELAYED FOCUSED REDUCTION ************************************************)
-
-lemma tr_uni_eq_repl (n1) (n2):
- n1 = n2 → 𝐮❨n1❩ ≗ 𝐮❨n2❩.
-// qed.
-
-axiom pippo (b) (q) (n):
- ↑❘q❘ = (↑[q]𝐢)@❨n❩ →
- ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)@❨n+❘b❘❩.
-
-lemma unwind_rmap_tls_eq_id (p) (n):
- ❘p❘ = ↑[p]𝐢@❨n❩ →
- (𝐢) ≗ ⇂*[n]↑[p]𝐢.
-#p @(list_ind_rcons … p) -p
-[ #n <depth_empty #H destruct
-| #p * [ #m ] #IH #n
- [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
- @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
- @(stream_eq_trans … (IH …)) -IH //
- | /2 width=1 by/
- | <depth_L_dx <unwind_rmap_L_dx
- cases n -n [| #n ] #H0
- [
- |
- ]
- | /2 width=1 by/
- | /2 width=1 by/
- ]
-]
-
-
-(* (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢@❨n+❘b❘❩ *)
-(* [↑[p]𝐢@❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
-lemma unwind_rmap_tls_eq (f) (p) (n):
- ❘p❘ = ↑[p]𝐢@❨n❩ →
- f ≗ ⇂*[n]↑[p]f.
-#f #p #n #Hp
-@(stream_eq_canc_dx … (stream_tls_eq_repl …))
-[| @unwind_rmap_decompose | skip ]
-<tr_compose_tls <Hp
-
-@(stream_eq_canc_dx) … (unwind_rmap_decompose …))
-
-lemma dfr_unwind_bi (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
- t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[⊗p,⊗q] ↑[f]t2.
-#f #p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro … (⊗b) (↑❘⊗q❘)) @and4_intro
-[ //
-| #g <unwind_rmap_structure <depth_structure
- >tr_pushs_swap <tr_pap_pushs_le //
-| lapply (in_comp_unwind_bi f … Ht1) -Ht1 -H0t1 -Hb -Ht2
- <unwind_path_d_empty_dx //
-| lapply (unwind_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
- @(subset_eq_trans … Ht2) -t2
- @(subset_eq_trans … (unwind_fsubst …))
- [ <unwind_rmap_append <unwind_rmap_A_sn (* <unwind_rmap_append <unwind_rmap_L_sn *)
- <structure_append <structure_A_sn <structure_append <structure_L_sn
- <depth_append <depth_L_sn <depth_structure <depth_structure
- @fsubst_eq_repl [ // ]
- @(subset_eq_trans … (unwind_iref …))
- @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
- [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
- @(subset_eq_trans … (unwind_term_after …))
- @(subset_eq_canc_dx … (unwind_term_after …))
- @unwind_term_eq_repl_sn -t1
- @(stream_eq_trans … (tr_compose_uni_dx …))
- lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
- lapply (pippo … b … Hn) -Hn #Hn
- @tr_compose_eq_repl
- [ <unwind_rmap_pap_le //
- <Hn <nrplus_inj_sn //
- |
- ]
- | //
- | /2 width=2 by ex_intro/
- | //
- ]
-]
+++ /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/prototerm.ma".
-include "delayed_updating/notation/functions/pitchforkleftarrow_3.ma".
-
-(* FOCALIZED SUBSTITUTION ***************************************************)
-
-definition fsubst (p) (u): prototerm → prototerm ≝
- λt,q.
- ∨∨ ∃∃r. r ϵ u & p●r = q
- | ∧∧ q ϵ t & (∀r. p●r = q → ⊥)
-.
-
-interpretation
- "focalized substitution (prototerm)"
- 'PitchforkLeftArrow t p u = (fsubst p u t).
+++ /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/fsubst.ma".
-include "delayed_updating/syntax/prototerm_equivalence.ma".
-
-(* Constructions with subset_equivalence ************************************)
-
-lemma subset_inclusion_fsubst_bi (t1) (t2) (u1) (u2) (p):
- t1 ⊆ t2 → u1 ⊆ u2 → t1[⋔p←u1] ⊆ t2[⋔p←u2].
-#t1 #t2 #u1 #u2 #p #Ht #Hu #q * *
-[ #r #Hr #H0 destruct
- /4 width=3 by ex2_intro, or_introl/
-| /4 width=2 by or_intror, conj/
-]
-qed.
-
-lemma fsubst_eq_repl (t1) (t2) (u1) (u2) (p):
- t1 ⇔ t2 → u1 ⇔ u2 → t1[⋔p←u1] ⇔ t2[⋔p←u2].
-#t1 #t2 #u1 #u2 #p * #H1t #H2t * #H1u #H2u
-/3 width=5 by conj, subset_inclusion_fsubst_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/fsubst.ma".
-include "delayed_updating/substitution/unwind_prototerm.ma".
-include "delayed_updating/syntax/prototerm_equivalence.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_balanced.ma".
-include "delayed_updating/notation/relations/black_rightarrow_f_4.ma".
-include "ground/xoa/ex_1_2.ma".
-include "ground/xoa/and_4.ma".
-
-(* IMMEDIATE FOCUSED REDUCTION ************************************************)
-
-definition ifr (p) (q): relation2 prototerm prototerm ≝
- λt1,t2. ∃∃b,n.
- let r ≝ p●𝗔◗b●𝗟◗q in
- ∧∧ ⊗b ϵ 𝐁 & ∀f. ↑❘q❘ = (↑[q]⫯f)@❨n❩ & r◖𝗱n ϵ t1 &
- t1[⋔r←↑[𝐮❨❘b●𝗟◗q❘❩](t1⋔(p◖𝗦))] ⇔ t2
-.
-
-interpretation
- "focused balanced reduction with immediate updating (prototerm)"
- 'BlackRightArrowF t1 p q t2 = (ifr p q t1 t2).
+++ /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 *)
+(* *)
+(**************************************************************************)
+
+include "ground/arith/pnat_plus.ma".
+include "ground/relocation/tr_map.ma".
+include "delayed_updating/notation/functions/verticalbarat_2.ma".
+
+(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)
+
+rec definition sbr_pap (i: pnat) on i: tr_map → pnat.
+* #p #f cases i -i
+[ @(p)
+| #i cases p -p
+ [ lapply (sbr_pap i f) -sbr_pap -i -f
+ #i @(↑i)
+ | #p @(i+↑p)
+ ]
+]
+defined.
+
+interpretation
+ "functional positive application (stack of relocation maps)"
+ 'VerticalBarAt f i = (sbr_pap i f).
+
+(* Basic constructions ******************************************************)
+
+lemma sbr_pap_unfold_unit (f):
+ ∀p. p = (p⨮f)❘@❨𝟏❩.
+// qed.
+
+lemma sbr_pap_unfold_unit_succ (f):
+ ∀i. ↑(f❘@❨i❩) = (𝟏⨮f)❘@❨↑i❩.
+// qed.
+
+lemma sbr_pap_unfold_succ_bi (f):
+ ∀p,i. i+↑p = (↑p⨮f)❘@❨↑i❩.
+// 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/relocation/sbr_pap.ma".
+include "ground/relocation/tr_eq.ma".
+
+(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)
+
+(* Main constructions with stream_eq ****************************************)
+
+theorem sbr_pap_eq_repl_sn (i):
+ stream_eq_repl … (λf1,f2. f1❘@❨i❩ = f2❘@❨i❩).
+#i elim i -i [| #i #IH ]
+* [ #p1 | * [| #p1 ]] #f1
+* #p2 #f2 #H
+elim (stream_eq_inv_cons_bi … H) -H [1,8,15: |*: // ] #Hp #Hf destruct //
+<sbr_pap_unfold_unit_succ <sbr_pap_unfold_unit_succ
+/3 width=1 by eq_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/relocation/sbr_pap.ma".
+include "ground/relocation/tr_id.ma".
+
+(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)
+
+(* Constructions with tr_id *************************************************)
+
+lemma sbr_pap_id (p):
+ p = 𝐢❘@❨p❩.
+#p elim p -p [| #p #IH ] <tr_id_unfold //
+<sbr_pap_unfold_unit_succ //
+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/relocation/sbr_pap.ma".
+include "delayed_updating/relocation/sbr_push.ma".
+
+(* POSITIVE APPLICATION FOR BASIC STACKS OF RELOCATION MAPS *****************)
+
+(* constructions with sbr_push **********************************************)
+
+lemma sbr_push_pap_unit (f) (p):
+ ↑p = (p⫯❘f)❘@❨𝟏❩.
+// qed.
+
+lemma sbr_push_pap (f) (q) (p):
+ q + p = (p⫯❘f)❘@❨q❩.
+#f * //
+#q * //
+#p <sbr_pap_unfold_succ_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 "ground/relocation/tr_map.ma".
+include "delayed_updating/notation/functions/upspoonverticalbar_2.ma".
+
+(* PUSH FOR STACKS OF BASIC RELOCATION MAPS *********************************)
+
+definition sbr_push (p:pnat) (f): tr_map ≝ (↑p)⨮f.
+
+interpretation
+ "push (stack of basic relocation maps)"
+ 'UpSpoonVerticalBar n f = (sbr_push n f).
+
+(* Basic properties *********************************************************)
+
+lemma sbr_push_unfold (p) (f):
+ (↑p)⨮f = p⫯❘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/relocation/sbr_push.ma".
+include "ground/relocation/tr_eq.ma".
+
+(* PUSH FOR STACKS OF BASIC RELOCATION MAPS *********************************)
+
+(* Main constructions with stream_eq ****************************************)
+
+theorem sbr_push_eq_repl_dx (p):
+ stream_eq_repl … (λf1,f2. p⫯❘f1 ≗ p⫯❘f2).
+/2 width=1 by stream_eq_cons/ 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/relocation/sbr_push.ma".
+include "ground/relocation/tr_uni.ma".
+
+(* PUSH FOR STACKS OF BASIC RELOCATION MAPS *********************************)
+
+(* Constructions with tr_uni ************************************************)
+
+theorem sbr_push_id (p:pnat):
+ (𝐮❨p❩) = p⫯❘𝐢.
+// 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 *)
+(* *)
+(**************************************************************************)
+
+(* GROUND NOTATION **********************************************************)
+
+notation "hvbox( hd ⫯❘ break tl )"
+ right associative with precedence 47
+ for @{ 'UpSpoonVerticalBar $hd $tl }.
--- /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( f ❘@❨ break term 46 a ❩ )"
+ non associative with precedence 60
+ for @{ 'VerticalBarAt $f $a }.
--- /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/relocation/tr_compose_pap.ma".
+include "ground/relocation/tr_uni_pap.ma".
+include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/black_downtriangle_4.ma".
+include "delayed_updating/notation/functions/black_downtriangle_2.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind_continuation (A:Type[0]) ≝
+tr_map → path → A.
+
+rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f) (p) on p ≝
+match p with
+[ list_empty ⇒ k f (𝐞)
+| list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒
+ match q with
+ [ list_empty ⇒ unwind_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (f∘𝐮❨n❩) q
+ | list_lcons _ _ ⇒ unwind_gen (A) k (f∘𝐮❨n❩) q
+ ]
+ | label_m ⇒ unwind_gen (A) k f q
+ | label_L ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
+ | label_A ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
+ | label_S ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
+ ]
+].
+
+interpretation
+ "unwind (gneric)"
+ 'BlackDownTriangle A k f p = (unwind_gen A k f p).
+
+definition proj_path: unwind_continuation … ≝
+ λf,p.p.
+
+definition proj_rmap: unwind_continuation … ≝
+ λf,p.f.
+
+interpretation
+ "unwind (path)"
+ 'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
+
+interpretation
+ "unwind (relocation map)"
+ 'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind_empty (A) (k) (f):
+ k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
+// qed.
+
+lemma unwind_d_empty_sn (A) (k) (n) (f):
+ ▼❨(λg,p. k g (𝗱(f@❨n❩)◗p)), f∘𝐮❨ninj n❩, 𝐞❩ = ▼{A}❨k, f,
+𝗱n◗𝐞❩.
+// qed.
+
+lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
+ ▼❨k, f∘𝐮❨ninj n❩, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
+// qed.
+
+lemma unwind_m_sn (A) (k) (p) (f):
+ ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
+// qed.
+
+lemma unwind_L_sn (A) (k) (p) (f):
+ ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
+// qed.
+
+lemma unwind_A_sn (A) (k) (p) (f):
+ ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
+// qed.
+
+lemma unwind_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 unwind_path_empty (f):
+ (𝐞) = ▼[f]𝐞.
+// qed.
+
+lemma unwind_path_d_empty_sn (f) (n):
+ 𝗱(f@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
+// qed.
+
+lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
+ ▼[f∘𝐮❨ninj n❩](l◗p) = ▼[f](𝗱n◗l◗p).
+// qed.
+
+lemma unwind_path_m_sn (f) (p):
+ ▼[f]p = ▼[f](𝗺◗p).
+// qed.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma unwind_rmap_empty (f):
+ f = ▼[𝐞]f.
+// qed.
+
+lemma unwind_rmap_d_sn (f) (p) (n):
+ ▼[p](f∘𝐮❨ninj n❩) = ▼[𝗱n◗p]f.
+#f * // qed.
+
+lemma unwind_rmap_m_sn (f) (p):
+ ▼[p]f = ▼[𝗺◗p]f.
+// qed.
+
+lemma unwind_rmap_L_sn (f) (p):
+ ▼[p](⫯f) = ▼[𝗟◗p]f.
+// qed.
+
+lemma unwind_rmap_A_sn (f) (p):
+ ▼[p]f = ▼[𝗔◗p]f.
+// qed.
+
+lemma unwind_rmap_S_sn (f) (p):
+ ▼[p]f = ▼[𝗦◗p]f.
+// qed.
+
+(* Advanced constructions with proj_rmap and path_append ********************)
+
+lemma unwind_rmap_append (p2) (p1) (f):
+ ▼[p2]▼[p1]f = ▼[p1●p2]f.
+#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
+[ <unwind_rmap_m_sn <unwind_rmap_m_sn //
+| <unwind_rmap_A_sn <unwind_rmap_A_sn //
+| <unwind_rmap_S_sn <unwind_rmap_S_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma unwind_rmap_d_dx (f) (p) (n):
+ (▼[p]f)∘𝐮❨ninj n❩ = ▼[p◖𝗱n]f.
+// qed.
+
+lemma unwind_rmap_m_dx (f) (p):
+ ▼[p]f = ▼[p◖𝗺]f.
+// qed.
+
+lemma unwind_rmap_L_dx (f) (p):
+ (⫯▼[p]f) = ▼[p◖𝗟]f.
+// qed.
+
+lemma unwind_rmap_A_dx (f) (p):
+ ▼[p]f = ▼[p◖𝗔]f.
+// qed.
+
+lemma unwind_rmap_S_dx (f) (p):
+▼[p]f = ▼[p◖𝗦]f.
+// qed.
+
+lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
+ ▼[p]f@❨m+n❩ = ▼[p◖𝗱n]f@❨m❩.
+#f #p #n #m
+<unwind_rmap_d_dx <tr_compose_pap <tr_uni_pap //
+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/unwind2/unwind_prototerm_eq.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+lemma unwind_iref_after_sn (f) (t:prototerm) (n:pnat):
+ ▼[f∘𝐮❨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 unwind_iref_after_dx (f) (t) (n:pnat):
+ ▼[f](𝛗n.t) ⊆ ▼[f∘𝐮❨n❩]t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_iref_after (f) (t) (n:pnat):
+ ▼[f∘𝐮❨n❩]t ⇔ ▼[f](𝛗n.t).
+/3 width=1 by conj, unwind_iref_after_sn, unwind_iref_after_dx/
+qed.
+
+lemma unwind_iref (f) (t) (n:pnat):
+ ▼[f]▼[𝐮❨n❩]t ⇔ ▼[f](𝛗n.t).
+/3 width=3 by unwind_term_after, subset_eq_trans/
+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/unwind2/unwind.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/relocation/tr_id_compose.ma".
+include "ground/relocation/tr_compose_compose.ma".
+include "ground/relocation/tr_compose_pn.ma".
+include "ground/relocation/tr_compose_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with depth *************************************************)
+
+lemma unwind_rmap_decompose (p) (f):
+ ▼[p]f ≗ (⫯*[❘p❘]f)∘(▼[p]𝐢).
+#p @(list_ind_rcons … p) -p
+[ #f <unwind_rmap_empty <unwind_rmap_empty <tr_pushs_zero //
+| #p * [ #n ] #IH #f //
+ [ <unwind_rmap_d_dx <unwind_rmap_d_dx <depth_d_dx
+ @(stream_eq_trans … (tr_compose_assoc …))
+ /2 width=1 by tr_compose_eq_repl/
+ | <unwind_rmap_L_dx <unwind_rmap_L_dx <depth_L_dx
+ <tr_pushs_succ <tr_compose_push_bi
+ /2 width=1 by tr_push_eq_repl/
+ ]
+]
+qed.
+
+lemma unwind_rmap_pap_le (f) (p) (n):
+ (▼[p]𝐢)@❨n❩ < ↑❘p❘ → (▼[p]𝐢)@❨n❩ = (▼[p]f)@❨n❩.
+#f #p #n #Hn
+>(tr_pap_eq_repl … (▼[p]f) … (unwind_rmap_decompose …))
+<tr_compose_pap @tr_pap_pushs_le //
+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/unwind2/unwind.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_pn_eq.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind_exteq (A): relation2 (unwind_continuation A) (unwind_continuation A) ≝
+ λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
+
+interpretation
+ "extensional equivalence (unwind continuation)"
+ 'RingEq A k1 k2 = (unwind_exteq A k1 k2).
+
+(* Constructions with unwind_exteq ******************************************)
+
+lemma unwind_eq_repl (A) (p) (k1) (k2):
+ k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ▼❨k1, f1, p❩ = ▼❨k2, f2, p❩).
+#A #p @(path_ind_unwind … p) -p [| #n #IH | #n #l0 #q #IH |*: #q #IH ]
+#k1 #k2 #Hk #f1 #f2 #Hf
+[ <unwind_empty <unwind_empty /2 width=1 by/
+| <unwind_d_empty_sn <unwind_d_empty_sn <(tr_pap_eq_repl … Hf)
+ /3 width=1 by tr_compose_eq_repl, stream_eq_refl/
+| <unwind_d_lcons_sn <unwind_d_lcons_sn
+ /3 width=1 by tr_compose_eq_repl, stream_eq_refl/
+| /2 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /3 width=1 by/
+| /3 width=1 by/
+]
+qed-.
+
+(* Advanced constructions ***************************************************)
+
+lemma unwind_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
+@unwind_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
+qed.
+
+lemma unwind_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
+@unwind_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 unwind_path_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
+/2 width=1 by unwind_eq_repl/ qed.
+
+lemma unwind_path_append_sn (p) (f) (q):
+ q●▼[f]p = ▼❨(λg,p. proj_path g (q●p)), f, p❩.
+#p @(path_ind_unwind … p) -p // [ #n #l #p |*: #p ] #IH #f #q
+[ <unwind_d_lcons_sn <unwind_d_lcons_sn <IH -IH //
+| <unwind_m_sn <unwind_m_sn //
+| <unwind_L_sn <unwind_L_sn >unwind_lcons_alt // >unwind_append_rcons_sn //
+ <IH <IH -IH <list_append_rcons_sn //
+| <unwind_A_sn <unwind_A_sn >unwind_lcons_alt >unwind_append_rcons_sn //
+ <IH <IH -IH <list_append_rcons_sn //
+| <unwind_S_sn <unwind_S_sn >unwind_lcons_alt >unwind_append_rcons_sn //
+ <IH <IH -IH <list_append_rcons_sn //
+]
+qed.
+
+lemma unwind_path_lcons (f) (p) (l):
+ l◗▼[f]p = ▼❨(λg,p. proj_path g (l◗p)), f, p❩.
+#f #p #l
+>unwind_lcons_alt <unwind_path_append_sn //
+qed.
+
+lemma unwind_path_L_sn (f) (p):
+ (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
+// qed.
+
+lemma unwind_path_A_sn (f) (p):
+ (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
+// qed.
+
+lemma unwind_path_S_sn (f) (p):
+ (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
+// qed.
+
+lemma unwind_path_after (p) (f1) (f2):
+ ▼[f2]▼[f1]p = ▼[f2∘f1]p.
+#p @(path_ind_unwind … p) -p // [ #n #l #p | #p ] #IH #f1 #f2
+[ <unwind_path_d_lcons_sn <unwind_path_d_lcons_sn
+ >(unwind_path_eq_repl … (tr_compose_assoc …)) //
+| <unwind_path_L_sn <unwind_path_L_sn <unwind_path_L_sn
+ >tr_compose_push_bi //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and stream_tls *********************)
+
+lemma unwind_rmap_tls_d_dx (f) (p) (m) (n):
+ ⇂*[m+n]▼[p]f ≗ ⇂*[m]▼[p◖𝗱n]f.
+#f #p #m #n
+<unwind_rmap_d_dx >nrplus_inj_dx
+/2 width=1 by tr_tls_compose_uni_dx/
+qed.
--- /dev/null
+include "delayed_updating/unwind2/unwind_depth.ma".
+include "ground/relocation/tr_uni_compose.ma".
+
+lemma pippo (q) (p):
+ ▼[p]𝐢 ≗ ⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢.
+#q @(list_ind_rcons … q) -q //
+#q * [ #n ] #IH #p //
+<depth_d_dx >list_cons_shift <list_append_assoc
+<unwind_rmap_d_dx
+check tr_tls_compose_uni_sn
\ No newline at end of file
--- /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/unwind2/unwind_prototerm_eq.ma".
+include "delayed_updating/unwind2/unwind_structure.ma".
+include "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with fsubst ************************************************)
+
+lemma unwind_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
+ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⊆ ▼[f](t[⋔p←u]).
+#f #t #u #p #Hu #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind_append_proper_dx
+ /4 width=5 by in_comp_unwind_bi, in_ppc_comp_trans, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro … H1) @or_intror @conj // *
+ [ <list_append_empty_dx #H2 destruct
+ elim (unwind_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ @H0 -H0 [| <unwind_append_proper_dx /2 width=3 by ppc_lcons/ ]
+ ]
+]
+qed-.
+
+lemma unwind_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+ ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u].
+#f #t #u #p #Hu #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ @or_introl @ex2_intro
+ [|| <unwind_append_proper_dx /2 width=5 by in_ppc_comp_trans/ ]
+ /2 width=3 by ex2_intro/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind_bi/ ] *
+ [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
+ <list_append_empty_dx @H2p -H2p /2 width=1 by prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⇔ ▼[f](t[⋔p←u]).
+/4 width=3 by unwind_fsubst_sn, conj, unwind_fsubst_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/unwind2/unwind_eq.ma".
+include "delayed_updating/syntax/path_height.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with height ************************************************)
+
+lemma unwind_rmap_pap_le (f1) (f2) (p) (m:pnat) (l:nat):
+ ninj (m+⧣p+l) = ❘p❘ →
+ (▼[p]f1)@❨m❩ = (▼[p]f2)@❨m❩.
+#f1 #f2 #p @(list_ind_rcons … p) -p
+[ #m #l <depth_empty #H0 destruct
+| #p * [ #n ] #IH #m #l
+ [ <height_d_dx <depth_d_dx <unwind_rmap_pap_d_dx <unwind_rmap_pap_d_dx
+ <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
+ #H0 <(IH … l) -IH //
+ | /2 width=2 by/
+ | <height_L_dx <depth_L_dx <unwind_rmap_L_dx <unwind_rmap_L_dx
+ cases m -m // #m
+ <nrplus_succ_sn <nrplus_succ_sn >nsucc_inj #H0
+ <tr_pap_push <tr_pap_push
+ <(IH … l) -IH //
+ | /2 width=2 by/
+ | /2 width=2 by/
+ ]
+]
+qed.
+
+lemma unwind_rmap_pap_gt (f) (p) (m):
+ f@❨m+⧣p❩+❘p❘ = (▼[p]f)@❨m+❘p❘❩.
+#f #p @(list_ind_rcons … p) -p [ // ]
+#p * [ #n ] #IH #m
+[ <height_d_dx <depth_d_dx
+ <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
+ <unwind_rmap_pap_d_dx >IH -IH //
+| //
+| <height_L_dx <depth_L_dx
+ <nrplus_succ_dx <nrplus_succ_dx <unwind_rmap_L_dx <tr_pap_push //
+| //
+| //
+]
+qed.
+
+lemma unwind_rmap_tls_gt (f) (p) (m:pnat):
+ ⇂*[ninj (m+⧣p)]f ≗ ⇂*[ninj (m+❘p❘)]▼[p]f.
+#f #p @(list_ind_rcons … p) -p [ // ]
+#p * [ #n ] #IH #m
+[ <height_d_dx <depth_d_dx
+ <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
+ @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
+ @(stream_eq_canc_dx … (IH …)) -IH //
+| //
+| <height_L_dx <depth_L_dx
+ <nrplus_succ_dx >nsucc_inj //
+| //
+| //
+]
+qed.
+
+lemma unwind_rmap_tls_eq (f) (p):
+ ⇂*[⧣p]f ≗ ⇂*[↑❘p❘]▼[p]⫯f.
+(*
+#f #p @(list_ind_rcons … p) -p //
+#p * [ #n ] #IH //
+<height_d_dx <depth_d_dx <unwind_rmap_d_dx
+@(stream_eq_trans … (tr_tls_compose_uni_dx …))
+*)
--- /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_equivalence.ma".
+include "delayed_updating/syntax/path_structure_inner.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/unwind2/unwind_structure.ma".
+include "delayed_updating/unwind2/unwind_prototerm.ma".
+
+(* UNWIND FOR PRETERM *******************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind_grafted_sn (f) (t) (p): p ϵ 𝐈 →
+ ▼[▼[p]f](t⋔p) ⊆ (▼[f]t)⋔(⊗p).
+#f #t #p #Hp #q * #r #Hr #H0 destruct
+@(ex2_intro … Hr) -Hr
+<unwind_append_inner_sn //
+qed-.
+
+lemma unwind_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)⋔(⊗p) ⊆ ▼[▼[p]f](t⋔p).
+#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
+elim (unwind_inv_append_inner_sn … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ ▼[▼[p]f](t⋔p) ⇔ (▼[f]t)⋔(⊗p).
+/3 width=1 by unwind_grafted_sn, conj, unwind_grafted_dx/ qed.
+
+
+lemma unwind_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)⋔((⊗p)◖𝗦) ⊆ ▼[▼[p]f](t⋔(p◖𝗦)).
+#f #t #p #Hp #Ht #q * #r #Hr
+<list_append_rcons_sn #H0
+elim (unwind_inv_append_proper_dx … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (unwind_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
+#r1 #r2 #Hr1 #Hr2 #H0 destruct
+lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
+[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ ▼[▼[p]f](t⋔(p◖𝗦)) ⇔ (▼[f]t)⋔((⊗p)◖𝗦).
+#f #t #p #Hp #Ht
+@conj
+[ >unwind_rmap_S_dx >structure_S_dx
+ @unwind_grafted_sn //
+| /2 width=1 by unwind_grafted_S_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/unwind2/unwind.ma".
+include "delayed_updating/syntax/prototerm.ma".
+include "ground/lib/subset_ext.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+interpretation
+ "unwind (prototerm)"
+ 'BlackDownTriangle f t = (subset_ext_f1 ? ? (unwind_gen ? proj_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_unwind_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/unwind2/unwind_eq.ma".
+include "delayed_updating/unwind2/unwind_prototerm.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind_term_eq_repl_sn (f1) (f2) (t):
+ f1 ≗ f2 → ▼[f1]t ⇔ ▼[f2]t.
+/3 width=1 by subset_equivalence_ext_f1_exteq, unwind_path_eq_repl/
+qed.
+
+lemma unwind_term_eq_repl_dx (f) (t1) (t2):
+ t1 ⇔ t2 → ▼[f]t1 ⇔ ▼[f]t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma unwind_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/unwind2/unwind_eq.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".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Basic constructions with structure ***************************************)
+
+lemma structure_unwind (p) (f):
+ ⊗p = ⊗▼[f]p.
+#p @(path_ind_unwind … p) -p // #p #IH #f
+<unwind_path_L_sn //
+qed.
+
+lemma unwind_structure (p) (f):
+ ⊗p = ▼[f]⊗p.
+#p @(path_ind_unwind … p) -p //
+qed.
+
+(* Destructions with structure **********************************************)
+
+lemma unwind_des_structure (q) (p) (f):
+ ⊗q = ▼[f]p → ⊗q = ⊗p.
+// qed-.
+
+(* Constructions with proper condition for path *****************************)
+
+lemma unwind_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
+ (⊗p1)●(▼[▼[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 #H destruct //
+| <unwind_path_d_lcons_sn <IH //
+| <unwind_path_m_sn <IH //
+| <unwind_path_L_sn <IH //
+| <unwind_path_A_sn <IH //
+| <unwind_path_S_sn <IH //
+]
+qed-.
+
+(* Constructions with inner condition for path ******************************)
+
+lemma unwind_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
+ (⊗p1)●(▼[▼[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_append_proper_dx //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+ <structure_L_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+ <structure_A_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+ <structure_S_dx <list_append_rcons_sn //
+]
+qed-.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma unwind_path_d_empty_dx (n) (p) (f):
+ (⊗p)◖𝗱((▼[p]f)@❨n❩) = ▼[f](p◖𝗱n).
+#n #p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_m_dx (p) (f):
+ ⊗p = ▼[f](p◖𝗺).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_L_dx (p) (f):
+ (⊗p)◖𝗟 = ▼[f](p◖𝗟).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_A_dx (p) (f):
+ (⊗p)◖𝗔 = ▼[f](p◖𝗔).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_S_dx (p) (f):
+ (⊗p)◖𝗦 = ▼[f](p◖𝗦).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_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 with proj_path ***************************************)
+
+lemma unwind_path_inv_d_sn (k) (q) (p) (f):
+ (𝗱k◗q) = ▼[f]p →
+ ∃∃r,h. 𝐞 = ⊗r & (▼[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_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct -IH
+ /2 width=5 by ex4_2_intro/
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+ /2 width=5 by ex4_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+ /2 width=5 by ex4_2_intro/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_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_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H /2 width=2 by/
+| <unwind_path_m_sn #H /2 width=2 by/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_L_sn (q) (p) (f):
+ (𝗟◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▼[r1]f]r2 & r1●𝗟◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_L_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_A_sn (q) (p) (f):
+ (𝗔◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗔◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_S_sn (q) (p) (f):
+ (𝗦◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗦◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma unwind_inv_append_proper_dx (q2) (q1) (p) (f):
+ q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]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_path_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
+ elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
+ elim Hq2 -Hq2 //
+ | elim (unwind_path_inv_m_sn … H)
+ | elim (unwind_path_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_path_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_path_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_inv_append_inner_sn (q1) (q2) (p) (f):
+ q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_inv_m_sn … (sym_eq … H2))
+ | <list_append_rcons_sn #H0
+ elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_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/unwind2/unwind.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/arith/nat_pred_succ.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Basic constructions with structure and depth *****************************)
+
+lemma unwind_rmap_structure (p) (f):
+ (⫯*[❘p❘]f) = ▼[⊗p]f.
+#p elim p -p //
+* [ #n ] #p #IH #f //
+[ <unwind_rmap_A_sn //
+| <unwind_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/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 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/relocation/sbr_pap_push.ma".
+include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/black_downtriangle_4.ma".
+include "delayed_updating/notation/functions/black_downtriangle_2.ma".
+include "ground/relocation/tr_pn.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind_continuation (A:Type[0]) ≝
+ tr_map → path → A.
+
+(* Note: f is a stack of update functions *)
+rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f:tr_map) (p) on p ≝
+match p with
+[ list_empty ⇒ k f (𝐞)
+| list_lcons l q ⇒
+ match l with
+ [ label_d n ⇒
+ match q with
+ [ list_empty ⇒ unwind_gen (A) (λg,p. k g (𝗱(f❘@❨n❩)◗p)) (f❘@❨n❩⫯❘f) q
+ | list_lcons _ _ ⇒ unwind_gen (A) k (f❘@❨n❩⫯❘f) q
+ ]
+ | label_m ⇒ unwind_gen (A) k f q
+ | label_L ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
+ | label_A ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
+ | label_S ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
+ ]
+].
+
+interpretation
+ "unwind (gneric)"
+ 'BlackDownTriangle A k f p = (unwind_gen A k f p).
+
+definition proj_path: unwind_continuation … ≝
+ λf,p.p.
+
+definition proj_rmap: unwind_continuation … ≝
+ λf,p.f.
+
+interpretation
+ "unwind (path)"
+ 'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
+
+interpretation
+ "unwind (relocation map)"
+ 'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind_empty (A) (k) (f):
+ k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
+// qed.
+
+lemma unwind_d_empty_sn (A) (k) (n) (f):
+ ▼❨(λg,p. k g (𝗱(f❘@❨n❩)◗p)), f❘@❨n❩⫯❘f, 𝐞❩ = ▼{A}❨k, f, 𝗱n◗𝐞❩.
+// qed.
+
+lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
+ ▼❨k, f❘@❨n❩⫯❘f, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
+// qed.
+
+lemma unwind_m_sn (A) (k) (p) (f):
+ ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
+// qed.
+
+lemma unwind_L_sn (A) (k) (p) (f):
+ ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
+// qed.
+
+lemma unwind_A_sn (A) (k) (p) (f):
+ ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
+// qed.
+
+lemma unwind_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 unwind_path_empty (f):
+ (𝐞) = ▼[f]𝐞.
+// qed.
+
+lemma unwind_path_d_empty_sn (f) (n):
+ 𝗱(f❘@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
+// qed.
+
+lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
+ ▼[f❘@❨n❩⫯❘f](l◗p) = ▼[f](𝗱n◗l◗p).
+// qed.
+
+lemma unwind_path_m_sn (f) (p):
+ ▼[f]p = ▼[f](𝗺◗p).
+// qed.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma unwind_rmap_empty (f):
+ f = ▼[𝐞]f.
+// qed.
+
+lemma unwind_rmap_d_sn (f) (p) (n):
+ ▼[p](f❘@❨n❩⫯❘f) = ▼[𝗱n◗p]f.
+#f * // qed.
+
+lemma unwind_rmap_m_sn (f) (p):
+ ▼[p]f = ▼[𝗺◗p]f.
+// qed.
+
+lemma unwind_rmap_L_sn (f) (p):
+ ▼[p](⫯f) = ▼[𝗟◗p]f.
+// qed.
+
+lemma unwind_rmap_A_sn (f) (p):
+ ▼[p]f = ▼[𝗔◗p]f.
+// qed.
+
+lemma unwind_rmap_S_sn (f) (p):
+ ▼[p]f = ▼[𝗦◗p]f.
+// qed.
+
+(* Advanced constructions with proj_rmap and path_append ********************)
+
+lemma unwind_rmap_append (p2) (p1) (f):
+ ▼[p2]▼[p1]f = ▼[p1●p2]f.
+#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
+[ <unwind_rmap_m_sn <unwind_rmap_m_sn //
+| <unwind_rmap_A_sn <unwind_rmap_A_sn //
+| <unwind_rmap_S_sn <unwind_rmap_S_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma unwind_rmap_d_dx (f) (p) (n):
+ (▼[p]f)❘@❨n❩⫯❘▼[p]f = ▼[p◖𝗱n]f.
+// qed.
+
+lemma unwind_rmap_m_dx (f) (p):
+ ▼[p]f = ▼[p◖𝗺]f.
+// qed.
+
+lemma unwind_rmap_L_dx (f) (p):
+ (⫯▼[p]f) = ▼[p◖𝗟]f.
+// qed.
+
+lemma unwind_rmap_A_dx (f) (p):
+ ▼[p]f = ▼[p◖𝗔]f.
+// qed.
+
+lemma unwind_rmap_S_dx (f) (p):
+▼[p]f = ▼[p◖𝗦]f.
+// qed.
+
+lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
+ m+▼[p]f❘@❨n❩ = ▼[p◖𝗱n]f❘@❨m❩.
+#f #p #n #m
+<unwind_rmap_d_dx <sbr_push_pap //
+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/unwind3/unwind_prototerm_eq.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/relocation/sbr_push_uni.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with constructors for prototerm ****************************)
+
+lemma unwind_iref_sn (f) (t:prototerm) (n:pnat):
+ ▼[f❘@❨n❩⫯❘f]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 unwind_iref_dx (f) (t) (n:pnat):
+ ▼[f](𝛗n.t) ⊆ ▼[f❘@❨n❩⫯❘f]t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_iref (f) (t) (n:pnat):
+ ▼[f❘@❨n❩⫯❘f]t ⇔ ▼[f](𝛗n.t).
+/3 width=1 by conj, unwind_iref_sn, unwind_iref_dx/
+qed.
+
+lemma unwind_iref_id (t) (n:pnat):
+ ▼[𝐮❨n❩]t ⇔ ▼[𝐢](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/unwind3/unwind.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/relocation/tr_uni_compose.ma".
+include "ground/lib/stream_tls_plus.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with depth *************************************************)
+
+lemma tls_succ_unwind (q) (p) (f):
+ ▼[p]f ≗ ⇂*[↑❘q❘]▼[p●𝗟◗q]f.
+#q @(list_ind_rcons … q) -q //
+#q * [ #n ] #IH #p #f //
+<depth_d_dx >list_cons_shift <list_append_assoc <unwind_rmap_d_dx
+/3 width=3 by tr_tls_compose_uni_sn, stream_eq_trans/
+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/unwind3/unwind.ma".
+include "delayed_updating/relocation/sbr_pap_id.ma".
+include "delayed_updating/relocation/sbr_pap_eq.ma".
+include "delayed_updating/relocation/sbr_push_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+(*
+include "ground/lib/stream_tls.ma".
+*)
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind_exteq (A): relation2 (unwind_continuation A) (unwind_continuation A) ≝
+ λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
+
+interpretation
+ "extensional equivalence (unwind continuation)"
+ 'RingEq A k1 k2 = (unwind_exteq A k1 k2).
+
+(* Constructions with unwind_exteq ******************************************)
+
+lemma unwind_eq_repl (A) (p) (k1) (k2):
+ k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ▼❨k1, f1, p❩ = ▼❨k2, f2, p❩).
+#A #p @(path_ind_unwind … p) -p [| #n #IH | #n #l0 #q #IH |*: #q #IH ]
+#k1 #k2 #Hk #f1 #f2 #Hf
+[ <unwind_empty <unwind_empty /2 width=1 by/
+| <unwind_d_empty_sn <unwind_d_empty_sn <(sbr_pap_eq_repl_sn … Hf)
+ /3 width=1 by sbr_push_eq_repl_dx/
+| <unwind_d_lcons_sn <unwind_d_lcons_sn <(sbr_pap_eq_repl_sn … Hf)
+ /3 width=1 by sbr_push_eq_repl_dx/
+| /2 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /3 width=1 by/
+| /3 width=1 by/
+]
+qed-.
+
+(* Advanced constructions ***************************************************)
+
+lemma unwind_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
+@unwind_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
+qed.
+
+lemma unwind_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
+@unwind_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 unwind_path_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
+/2 width=1 by unwind_eq_repl/ qed.
+
+lemma unwind_path_append_sn (p) (f) (q):
+ q●▼[f]p = ▼❨(λg,p. proj_path g (q●p)), f, p❩.
+#p @(path_ind_unwind … p) -p // [ #n #l #p |*: #p ] #IH #f #q
+[ <unwind_d_lcons_sn <unwind_d_lcons_sn <IH -IH //
+| <unwind_m_sn <unwind_m_sn //
+| <unwind_L_sn <unwind_L_sn >unwind_lcons_alt // >unwind_append_rcons_sn //
+ <IH <IH -IH <list_append_rcons_sn //
+| <unwind_A_sn <unwind_A_sn >unwind_lcons_alt >unwind_append_rcons_sn //
+ <IH <IH -IH <list_append_rcons_sn //
+| <unwind_S_sn <unwind_S_sn >unwind_lcons_alt >unwind_append_rcons_sn //
+ <IH <IH -IH <list_append_rcons_sn //
+]
+qed.
+
+lemma unwind_path_lcons (f) (p) (l):
+ l◗▼[f]p = ▼❨(λg,p. proj_path g (l◗p)), f, p❩.
+#f #p #l
+>unwind_lcons_alt <unwind_path_append_sn //
+qed.
+
+lemma unwind_path_L_sn (f) (p):
+ (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
+// qed.
+
+lemma unwind_path_A_sn (f) (p):
+ (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
+// qed.
+
+lemma unwind_path_S_sn (f) (p):
+ (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
+// qed.
+
+lemma unwind_path_after_id_sn (p) (f):
+ ▼[𝐢]▼[f]p = ▼[f]p.
+#p @(path_ind_unwind … p) -p // #p #IH #f
+[ <unwind_path_d_empty_sn <unwind_path_d_empty_sn //
+| <unwind_path_L_sn <unwind_path_L_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and stream_tls *********************)
+(* COMMENT
+lemma unwind_rmap_tls_d_dx (f) (p) (m:pnat) (n):
+ ⇂*[m]▼[p]f ≗ ⇂*[m]▼[p◖𝗱n]f.
+#f #p #m #n
+<unwind_rmap_d_dx
+/2 width=1 by tr_tls_compose_uni_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/unwind3/unwind_prototerm_eq.ma".
+include "delayed_updating/unwind3/unwind_structure.ma".
+include "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with fsubst ************************************************)
+
+lemma unwind_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
+ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⊆ ▼[f](t[⋔p←u]).
+#f #t #u #p #Hu #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind_append_proper_dx
+ /4 width=5 by in_comp_unwind_bi, in_ppc_comp_trans, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro … H1) @or_intror @conj // *
+ [ <list_append_empty_dx #H2 destruct
+ elim (unwind_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ @H0 -H0 [| <unwind_append_proper_dx /2 width=3 by ppc_lcons/ ]
+ ]
+]
+qed-.
+
+lemma unwind_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+ ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u].
+#f #t #u #p #Hu #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ @or_introl @ex2_intro
+ [|| <unwind_append_proper_dx /2 width=5 by in_ppc_comp_trans/ ]
+ /2 width=3 by ex2_intro/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind_bi/ ] *
+ [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
+ <list_append_empty_dx @H2p -H2p /2 width=1 by prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⇔ ▼[f](t[⋔p←u]).
+/4 width=3 by unwind_fsubst_sn, conj, unwind_fsubst_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/unwind3/unwind.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/lib/stream_tls_plus.ma".
+include "ground/lib/stream_eq.ma".
+(*
+include "ground/relocation/tr_uni_compose.ma".
+include "ground/lib/stream_eq_eq.ma".
+*)
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with list_length ********************************************)
+
+axiom tls_unwind (p) (f):
+ f ≗ ⇂*[❘p❘]▼[p]f.
+(* COMMENT
+#q @(list_ind_rcons … q) -q //
+#q * [ #n ] #IH #f //
+<depth_d_dx >list_cons_shift <list_append_assoc <unwind_rmap_d_dx
+/3 width=3 by tr_tls_compose_uni_sn, stream_eq_trans/
+qed.
+*)
\ No newline at end of file
--- /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/unwind3/unwind_structure.ma".
+include "delayed_updating/substitution/lift_structure.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with lift **************************************************)
+
+lemma pippo (p) (n:pnat) (f):
+ ↑[𝐮❨n❩]▼[⇂*[n]f]p = ▼[𝐮❨n❩∘f]p.
+#p @(list_ind_rcons … p) -p //
+#p * [ #m ] #IH #n #f //
+[
+| <unwind_path_m_dx
+
+ //
+<depth_d_dx >list_cons_shift <list_append_assoc <unwind_rmap_d_dx
+/3 width=3 by tr_tls_compose_uni_sn, stream_eq_trans/
+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_equivalence.ma".
+include "delayed_updating/syntax/path_structure_inner.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/unwind3/unwind_structure.ma".
+include "delayed_updating/unwind3/unwind_prototerm.ma".
+
+(* UNWIND FOR PRETERM *******************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind_grafted_sn (f) (t) (p): p ϵ 𝐈 →
+ ▼[▼[p]f](t⋔p) ⊆ (▼[f]t)⋔(⊗p).
+#f #t #p #Hp #q * #r #Hr #H0 destruct
+@(ex2_intro … Hr) -Hr
+<unwind_append_inner_sn //
+qed-.
+
+lemma unwind_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)⋔(⊗p) ⊆ ▼[▼[p]f](t⋔p).
+#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
+elim (unwind_inv_append_inner_sn … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ ▼[▼[p]f](t⋔p) ⇔ (▼[f]t)⋔(⊗p).
+/3 width=1 by unwind_grafted_sn, conj, unwind_grafted_dx/ qed.
+
+
+lemma unwind_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)⋔((⊗p)◖𝗦) ⊆ ▼[▼[p]f](t⋔(p◖𝗦)).
+#f #t #p #Hp #Ht #q * #r #Hr
+<list_append_rcons_sn #H0
+elim (unwind_inv_append_proper_dx … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (unwind_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
+#r1 #r2 #Hr1 #Hr2 #H0 destruct
+lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
+[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ ▼[▼[p]f](t⋔(p◖𝗦)) ⇔ (▼[f]t)⋔((⊗p)◖𝗦).
+#f #t #p #Hp #Ht
+@conj
+[ >unwind_rmap_S_dx >structure_S_dx
+ @unwind_grafted_sn //
+| /2 width=1 by unwind_grafted_S_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/unwind3/unwind.ma".
+include "delayed_updating/syntax/prototerm.ma".
+include "ground/lib/subset_ext.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+interpretation
+ "unwind (prototerm)"
+ 'BlackDownTriangle f t = (subset_ext_f1 ? ? (unwind_gen ? proj_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_unwind_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/unwind3/unwind_eq.ma".
+include "delayed_updating/unwind3/unwind_prototerm.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind_term_eq_repl_sn (f1) (f2) (t):
+ f1 ≗ f2 → ▼[f1]t ⇔ ▼[f2]t.
+/3 width=1 by subset_equivalence_ext_f1_exteq, unwind_path_eq_repl/
+qed.
+
+lemma unwind_term_eq_repl_dx (f) (t1) (t2):
+ t1 ⇔ t2 → ▼[f]t1 ⇔ ▼[f]t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma unwind_term_after_id_sn (f) (t):
+ ▼[𝐢]▼[f]t ⇔ ▼[f]t.
+#f #t @subset_eq_trans
+[
+| @subset_inclusion_ext_f1_compose
+| @subset_equivalence_ext_f1_exteq
+ #p @unwind_path_after_id_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/unwind3/unwind_eq.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".
+
+(* UNWIND FOR PATH *********************************************************)
+
+(* Basic constructions with structure **************************************)
+
+lemma structure_unwind (p) (f):
+ ⊗p = ⊗▼[f]p.
+#p @(path_ind_unwind … p) -p // #p #IH #f
+<unwind_path_L_sn //
+qed.
+
+lemma unwind_structure (p) (f):
+ ⊗p = ▼[f]⊗p.
+#p @(path_ind_unwind … p) -p //
+qed.
+
+(* Destructions with structure **********************************************)
+
+lemma unwind_des_structure (q) (p) (f):
+ ⊗q = ▼[f]p → ⊗q = ⊗p.
+// qed-.
+
+(* Constructions with proper condition for path *****************************)
+
+lemma unwind_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
+ (⊗p1)●(▼[▼[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 #H destruct //
+| <unwind_path_d_lcons_sn <IH //
+| <unwind_path_m_sn <IH //
+| <unwind_path_L_sn <IH //
+| <unwind_path_A_sn <IH //
+| <unwind_path_S_sn <IH //
+]
+qed-.
+
+(* Constructions with inner condition for path ******************************)
+
+lemma unwind_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
+ (⊗p1)●(▼[▼[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_append_proper_dx //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+ <structure_L_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+ <structure_A_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+ <structure_S_dx <list_append_rcons_sn //
+]
+qed-.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma unwind_path_d_empty_dx (n) (p) (f):
+ (⊗p)◖𝗱((▼[p]f)❘@❨n❩) = ▼[f](p◖𝗱n).
+#n #p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_m_dx (p) (f):
+ ⊗p = ▼[f](p◖𝗺).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_L_dx (p) (f):
+ (⊗p)◖𝗟 = ▼[f](p◖𝗟).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_A_dx (p) (f):
+ (⊗p)◖𝗔 = ▼[f](p◖𝗔).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_S_dx (p) (f):
+ (⊗p)◖𝗦 = ▼[f](p◖𝗦).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_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 with proj_path ***************************************)
+
+lemma unwind_path_inv_d_sn (k) (q) (p) (f):
+ (𝗱k◗q) = ▼[f]p →
+ ∃∃r,h. 𝐞 = ⊗r & (▼[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_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct -IH
+ /2 width=5 by ex4_2_intro/
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+ /2 width=5 by ex4_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+ /2 width=5 by ex4_2_intro/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_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_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H /2 width=2 by/
+| <unwind_path_m_sn #H /2 width=2 by/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_L_sn (q) (p) (f):
+ (𝗟◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▼[r1]f]r2 & r1●𝗟◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_L_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_A_sn (q) (p) (f):
+ (𝗔◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗔◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_S_sn (q) (p) (f):
+ (𝗦◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗦◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+ elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+ /2 width=5 by ex3_2_intro/| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct -IH
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma unwind_inv_append_proper_dx (q2) (q1) (p) (f):
+ q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]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_path_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
+ elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
+ elim Hq2 -Hq2 //
+ | elim (unwind_path_inv_m_sn … H)
+ | elim (unwind_path_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_path_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_path_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_inv_append_inner_sn (q1) (q2) (p) (f):
+ q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_inv_m_sn … (sym_eq … H2))
+ | <list_append_rcons_sn #H0
+ elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (unwind_path_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/unwind3/unwind.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/arith/nat_pred_succ.ma".
+
+(* UNWIND FOR PATH *********************************************************)
+
+(* Basic constructions with structure and depth ****************************)
+
+lemma unwind_rmap_structure (p) (f):
+ (⫯*[❘p❘]f) = ▼[⊗p]f.
+#p elim p -p //
+* [ #n ] #p #IH #f //
+[ <unwind_rmap_A_sn //
+| <unwind_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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( ▼ term 70 p )"
+ non associative with precedence 70
+ for @{ 'BlackDownTriangle $p }.
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( ▼[ term 46 t1 ] break term 70 t2 )"
+notation "hvbox( ▼[ term 46 f ] break term 70 p )"
non associative with precedence 70
- for @{ 'BlackDownTriangle $t1 $t2 }.
+ for @{ 'BlackDownTriangle $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 70 p )"
+ non associative with precedence 70
+ for @{ 'BlackRightTriangle $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 f ] break term 70 p )"
+ non associative with precedence 70
+ for @{ 'BlackRightTriangle $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 70 p )"
+ non associative with precedence 70
+ for @{ 'Flat $p }.
. replace.sh . "/substitution/" "/unwind/"
. replace.sh . ↑ ▼
+
+266D;MUSIC FLAT SIGN;So;0;ON;;;;;N;FLAT;;;;
+266E;MUSIC NATURAL SIGN;So;0;ON;;;;;N;NATURAL;;;;
+266F;MUSIC SHARP SIGN;Sm;0;ON;;;;;N;SHARP;;;;
definition dfr (p) (q): relation2 prototerm prototerm ≝
λt1,t2. ∃∃b,n.
let r ≝ p●𝗔◗b●𝗟◗q in
- â\88§â\88§ (â\8a\97b ϵ ð\9d\90\81 â\88§ ð\9d\9f\8e = â\9d\98bâ\9d\98) & â\86\91â\9d\98qâ\9d\98 = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
- t1[â\8b\94râ\86\90ð\9d\9b\97(n+â\9d\98bâ\9d\98).(t1⋔(p◖𝗦))] ⇔ t2
+ â\88§â\88§ (â\8a\97b ϵ ð\9d\90\81 â\88§ ð\9d\9f\8e = â\99b) & â\86\91â\99q = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
+ t1[â\8b\94râ\86\90ð\9d\9b\97(n+â\99b).(t1⋔(p◖𝗦))] ⇔ t2
.
interpretation
include "delayed_updating/reduction/dfr.ma".
include "delayed_updating/reduction/ifr.ma".
+
include "delayed_updating/unwind1/unwind_fsubst.ma".
include "delayed_updating/unwind1/unwind_constructors.ma".
include "delayed_updating/unwind1/unwind_preterm_eq.ma".
include "delayed_updating/unwind1/unwind_structure_depth.ma".
include "delayed_updating/unwind1/unwind_depth.ma".
+
include "delayed_updating/substitution/fsubst_eq.ma".
include "delayed_updating/substitution/lift_prototerm_eq.ma".
include "delayed_updating/syntax/prototerm_proper_constructors.ma".
t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
#p #q #t1 #t2 #H0t1
* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro â\80¦ (â\8a\97b) (â\86\91â\9d\98â\8a\97qâ\9d\98)) @and4_intro
+@(ex1_2_intro â\80¦ (â\8a\97b) (â\86\91â\99â\8a\97q)) @and4_intro
[ //
-| //
+| (*//*)
| lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
<unwind_path_d_empty_dx <depth_structure //
| lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
@fsubst_eq_repl [ // ]
@(subset_eq_trans … (unwind_iref …))
- elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <Hn
+ elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <nsucc_unfold
+ >Hn
@(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
[ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
-
+ <Hn <Hn
+(*
+ @(subset_eq_trans … (lift_term_eq_repl_dx …))
+ [ @(unwind_term_eq_repl_sn … (tls_succ_unwind q …)) | skip ]
+*)
(*
+
+ @subset_eq_trans
+ [2: @unwind_term_eq_repl_dx
+ @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
+
@(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
[ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
---------------------------
↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢@❨n+❘b❘❩❩] t
*)
+(*
+(↑[𝐮❨↑❘q❘❩]▼[⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢](t1⋔(p◖𝗦))⇔▼[𝐮❨↑❘q❘❩∘▼[p●𝗔◗b●𝗟◗q]𝐢](t1⋔(p◖𝗦))
+*)
definition ifr (p) (q): relation2 prototerm prototerm ≝
λt1,t2. ∃∃b,n.
let r ≝ p●𝗔◗b●𝗟◗q in
- â\88§â\88§ (â\8a\97b ϵ ð\9d\90\81 â\88§ ð\9d\9f\8e = â\9d\98bâ\9d\98) & â\86\91â\9d\98qâ\9d\98 = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
- t1[â\8b\94râ\86\90â\86\91[ð\9d\90®â\9d¨â\9d\98bâ\97\8fð\9d\97\9fâ\97\97qâ\9d\98❩](t1⋔(p◖𝗦))] ⇔ t2
+ â\88§â\88§ (â\8a\97b ϵ ð\9d\90\81 â\88§ ð\9d\9f\8e = â\99b) & â\86\91â\99q = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
+ t1[â\8b\94râ\86\90â\86\91[ð\9d\90®â\9d¨â\99(bâ\97\8fð\9d\97\9fâ\97\97q)❩](t1⋔(p◖𝗦))] ⇔ t2
.
interpretation
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".
+include "ground/relocation/tr_id_pap.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)) (⇂*[n]f) q
+ [ 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
// 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❩.
+ ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), 𝐢, 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](⇂*[ninj n]f) = ↑[𝗱n◗p]f.
+ ↑[p]𝐢 = ↑[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_m_sn <lift_rmap_m_sn //
+[ <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 //
]
(* Advanced constructions with proj_rmap and path_rcons *********************)
lemma lift_rmap_d_dx (f) (p) (n):
- ⇂*[ninj n](↑[p]f) = ↑[p◖𝗱n]f.
+ (𝐢) = ↑[p◖𝗱n]f.
// qed.
lemma lift_rmap_m_dx (f) (p):
// qed.
lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
- ↑[p]f@❨m+n❩ = ↑[p◖𝗱n]f@❨m❩+↑[p]f@❨n❩.
+ m = ↑[p◖𝗱n]f@❨m❩.
// 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 [| * ] // [ #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 //
-]
+#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.
(* *)
(**************************************************************************)
-include "delayed_updating/substitution/lift_prototerm_id.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.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❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro … (𝗱n◗𝗺◗r))
+ (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❩.↑[⇂*[n]f]t.
+ ↑[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
-/3 width=1 by in_comp_iref, in_comp_lift_bi/
+/2 width=1 by in_comp_iref/
qed-.
lemma lift_iref (f) (t) (n:pnat):
- (𝛗f@❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
+ (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).
-#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.
+// 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=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
+ /3 width=1 by stream_eq_refl/
| /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❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
+ (𝗱(f@❨n❩)◗↑[𝐢]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_id.ma".
+include "delayed_updating/substitution/lift_eq.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).
-#p #m #n
-<lift_path_d_sn <tr_uni_pap >nsucc_pnpred
-<tr_tls_succ_uni //
-qed.
+// qed.
(**************************************************************************)
include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/flat_1.ma".
include "ground/arith/nat_plus.ma".
-include "ground/notation/functions/verticalbars_1.ma".
(* DEPTH FOR PATH ***********************************************************)
interpretation
"depth (path)"
- 'VerticalBars p = (depth p).
+ 'Flat p = (depth p).
(* Basic constructions ******************************************************)
-lemma depth_empty: ð\9d\9f\8e = â\9d\98ð\9d\90\9eâ\9d\98.
+lemma depth_empty: ð\9d\9f\8e = â\99ð\9d\90\9e.
// qed.
-lemma depth_d_sn (q) (n): â\9d\98qâ\9d\98 = â\9d\98ð\9d\97±nâ\97\97qâ\9d\98.
+lemma depth_d_sn (q) (n): â\99q = â\99(ð\9d\97±nâ\97\97q).
// qed.
-lemma depth_m_sn (q): â\9d\98qâ\9d\98 = â\9d\98ð\9d\97ºâ\97\97qâ\9d\98.
+lemma depth_m_sn (q): â\99q = â\99(ð\9d\97ºâ\97\97q).
// qed.
-lemma depth_L_sn (q): â\86\91â\9d\98qâ\9d\98 = â\9d\98ð\9d\97\9fâ\97\97qâ\9d\98.
+lemma depth_L_sn (q): â\86\91â\99q = â\99(ð\9d\97\9fâ\97\97q).
// qed.
-lemma depth_A_sn (q): â\9d\98qâ\9d\98 = â\9d\98ð\9d\97\94â\97\97qâ\9d\98.
+lemma depth_A_sn (q): â\99q = â\99(ð\9d\97\94â\97\97q).
// qed.
-lemma depth_S_sn (q): â\9d\98qâ\9d\98 = â\9d\98ð\9d\97¦â\97\97qâ\9d\98.
+lemma depth_S_sn (q): â\99q = â\99(ð\9d\97¦â\97\97q).
// qed.
(* Main constructions *******************************************************)
theorem depth_append (p1) (p2):
- ❘p2❘+❘p1❘ = ❘p1●p2❘.
+ (♭p2)+(♭p1) = ♭(p1●p2).
#p1 elim p1 -p1 //
* [ #n ] #p1 #IH #p2 <list_append_lcons_sn
[ <depth_d_sn <depth_d_sn //
(* Constructions with list_rcons ********************************************)
lemma depth_d_dx (p) (n):
- â\9d\98pâ\9d\98 = â\9d\98pâ\97\96ð\9d\97±nâ\9d\98.
+ â\99p = â\99(pâ\97\96ð\9d\97±n).
// qed.
lemma depth_m_dx (p):
- â\9d\98pâ\9d\98 = â\9d\98pâ\97\96ð\9d\97ºâ\9d\98.
+ â\99p = â\99(pâ\97\96ð\9d\97º).
// qed.
lemma depth_L_dx (p):
- â\86\91â\9d\98pâ\9d\98 = â\9d\98pâ\97\96ð\9d\97\9fâ\9d\98.
+ â\86\91â\99p = â\99(pâ\97\96ð\9d\97\9f).
// qed.
lemma depth_A_dx (p):
- â\9d\98pâ\9d\98 = â\9d\98pâ\97\96ð\9d\97\94â\9d\98.
+ â\99p = â\99(pâ\97\96ð\9d\97\94).
// qed.
lemma depth_S_dx (p):
- â\9d\98pâ\9d\98 = â\9d\98pâ\97\96ð\9d\97¦â\9d\98.
+ â\99p = â\99(pâ\97\96ð\9d\97¦).
// qed.
(* Constructions with depth *************************************************)
lemma depth_structure (p):
- â\9d\98pâ\9d\98 = â\9d\98â\8a\97pâ\9d\98.
+ â\99p = â\99â\8a\97p.
#p elim p -p //
* [ #n ] #p #IH //
[ <structure_L_sn <depth_L_sn <depth_L_sn //
--- /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.
k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
// qed.
-lemma unwind_d_empty_sn (A) (k) (n) (f):
+lemma unwind_d_empty (A) (k) (n) (f):
▼❨(λg,p. k g (𝗱(f@❨n❩)◗p)), 𝐮❨f@❨n❩❩, 𝐞❩ = ▼{A}❨k, f, 𝗱n◗𝐞❩.
// qed.
-lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
+lemma unwind_d_lcons (A) (k) (p) (l) (n) (f):
▼❨k, 𝐮❨f@❨n❩❩, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
// qed.
(𝐞) = ▼[f]𝐞.
// qed.
-lemma unwind_path_d_empty_sn (f) (n):
+lemma unwind_path_d_empty (f) (n):
𝗱(f@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
// qed.
-lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
+lemma unwind_path_d_lcons (f) (p) (l) (n):
▼[𝐮❨f@❨n❩❩](l◗p) = ▼[f](𝗱n◗l◗p).
// qed.
#A #p @(path_ind_unwind … p) -p [| #n #IH | #n #l0 #q #IH |*: #q #IH ]
#k1 #k2 #Hk #f1 #f2 #Hf
[ <unwind_empty <unwind_empty /2 width=1 by/
-| <unwind_d_empty_sn <unwind_d_empty_sn <(tr_pap_eq_repl … Hf)
+| <unwind_d_empty <unwind_d_empty <(tr_pap_eq_repl … Hf)
/2 width=1 by stream_eq_refl/
-| <unwind_d_lcons_sn <unwind_d_lcons_sn
+| <unwind_d_lcons <unwind_d_lcons
/5 width=1 by tr_uni_eq_repl, tr_pap_eq_repl, eq_f/
| /2 width=1 by/
| /3 width=1 by tr_push_eq_repl/
lemma unwind_path_append_sn (p) (f) (q):
q●▼[f]p = ▼❨(λg,p. proj_path g (q●p)), f, p❩.
#p @(path_ind_unwind … p) -p // [ #n #l #p |*: #p ] #IH #f #q
-[ <unwind_d_lcons_sn <unwind_d_lcons_sn <IH -IH //
+[ <unwind_d_lcons <unwind_d_lcons <IH -IH //
| <unwind_m_sn <unwind_m_sn //
| <unwind_L_sn <unwind_L_sn >unwind_lcons_alt // >unwind_append_rcons_sn //
<IH <IH -IH <list_append_rcons_sn //
lemma unwind_path_after_id_sn (p) (f):
▼[𝐢]▼[f]p = ▼[f]p.
#p @(path_ind_unwind … p) -p // [ #n | #n #l #p | #p ] #IH #f
-[ <unwind_path_d_empty_sn //
-| <unwind_path_d_lcons_sn //
+[ <unwind_path_d_empty //
+| <unwind_path_d_lcons //
| <unwind_path_L_sn <unwind_path_L_sn //
]
qed.
#p2 #p1 @(path_ind_unwind … p1) -p1 //
[ #n | #n #l #p1 |*: #p1 ] #IH #f #Hp2
[ elim (ppc_inv_lcons … Hp2) -Hp2 #l #q #H destruct //
-| <unwind_path_d_lcons_sn <IH //
+| <unwind_path_d_lcons <IH //
| <unwind_path_m_sn <IH //
| <unwind_path_L_sn <IH //
| <unwind_path_A_sn <IH //
#k #q #p @(path_ind_unwind … p) -p
[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct -IH
+| <unwind_path_d_empty #H destruct -IH
/2 width=5 by ex4_2_intro/
-| <unwind_path_d_lcons_sn #H
+| <unwind_path_d_lcons #H
elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
/2 width=5 by ex4_2_intro/
| <unwind_path_m_sn #H
#q #p @(path_ind_unwind … p) -p
[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H /2 width=2 by/
+| <unwind_path_d_empty #H destruct
+| <unwind_path_d_lcons #H /2 width=2 by/
| <unwind_path_m_sn #H /2 width=2 by/
| <unwind_path_L_sn #H destruct
| <unwind_path_A_sn #H destruct
#q #p @(path_ind_unwind … p) -p
[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H
+| <unwind_path_d_empty #H destruct
+| <unwind_path_d_lcons #H
elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
/2 width=5 by ex3_2_intro/
| <unwind_path_m_sn #H
#q #p @(path_ind_unwind … p) -p
[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H
+| <unwind_path_d_empty #H destruct
+| <unwind_path_d_lcons #H
elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
/2 width=5 by ex3_2_intro/
| <unwind_path_m_sn #H
#q #p @(path_ind_unwind … p) -p
[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H
+| <unwind_path_d_empty #H destruct
+| <unwind_path_d_lcons #H
elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
/2 width=5 by ex3_2_intro/
| <unwind_path_m_sn #H
(* Basic constructions with structure and depth ****************************)
lemma unwind_rmap_structure (p) (f):
- (⫯*[â\9d\98pâ\9d\98]f) = ▼[⊗p]f.
+ (⫯*[â\99p]f) = ▼[⊗p]f.
#p elim p -p //
* [ #n ] #p #IH #f //
[ <unwind_rmap_A_sn //
+++ /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/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_uni_pap.ma".
-include "delayed_updating/syntax/path.ma".
-include "delayed_updating/notation/functions/black_downtriangle_4.ma".
-include "delayed_updating/notation/functions/black_downtriangle_2.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-definition unwind_continuation (A:Type[0]) ≝
-tr_map → path → A.
-
-rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f) (p) on p ≝
-match p with
-[ list_empty ⇒ k f (𝐞)
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒
- match q with
- [ list_empty ⇒ unwind_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (f∘𝐮❨n❩) q
- | list_lcons _ _ ⇒ unwind_gen (A) k (f∘𝐮❨n❩) q
- ]
- | label_m ⇒ unwind_gen (A) k f q
- | label_L ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
- | label_A ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
- | label_S ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
- ]
-].
-
-interpretation
- "unwind (gneric)"
- 'BlackDownTriangle A k f p = (unwind_gen A k f p).
-
-definition proj_path: unwind_continuation … ≝
- λf,p.p.
-
-definition proj_rmap: unwind_continuation … ≝
- λf,p.f.
-
-interpretation
- "unwind (path)"
- 'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
-
-interpretation
- "unwind (relocation map)"
- 'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind_empty (A) (k) (f):
- k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
-// qed.
-
-lemma unwind_d_empty_sn (A) (k) (n) (f):
- ▼❨(λg,p. k g (𝗱(f@❨n❩)◗p)), f∘𝐮❨ninj n❩, 𝐞❩ = ▼{A}❨k, f,
-𝗱n◗𝐞❩.
-// qed.
-
-lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
- ▼❨k, f∘𝐮❨ninj n❩, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
-// qed.
-
-lemma unwind_m_sn (A) (k) (p) (f):
- ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
-// qed.
-
-lemma unwind_L_sn (A) (k) (p) (f):
- ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
-// qed.
-
-lemma unwind_A_sn (A) (k) (p) (f):
- ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
-// qed.
-
-lemma unwind_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 unwind_path_empty (f):
- (𝐞) = ▼[f]𝐞.
-// qed.
-
-lemma unwind_path_d_empty_sn (f) (n):
- 𝗱(f@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
-// qed.
-
-lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
- ▼[f∘𝐮❨ninj n❩](l◗p) = ▼[f](𝗱n◗l◗p).
-// qed.
-
-lemma unwind_path_m_sn (f) (p):
- ▼[f]p = ▼[f](𝗺◗p).
-// qed.
-
-(* Basic constructions with proj_rmap ***************************************)
-
-lemma unwind_rmap_empty (f):
- f = ▼[𝐞]f.
-// qed.
-
-lemma unwind_rmap_d_sn (f) (p) (n):
- ▼[p](f∘𝐮❨ninj n❩) = ▼[𝗱n◗p]f.
-#f * // qed.
-
-lemma unwind_rmap_m_sn (f) (p):
- ▼[p]f = ▼[𝗺◗p]f.
-// qed.
-
-lemma unwind_rmap_L_sn (f) (p):
- ▼[p](⫯f) = ▼[𝗟◗p]f.
-// qed.
-
-lemma unwind_rmap_A_sn (f) (p):
- ▼[p]f = ▼[𝗔◗p]f.
-// qed.
-
-lemma unwind_rmap_S_sn (f) (p):
- ▼[p]f = ▼[𝗦◗p]f.
-// qed.
-
-(* Advanced constructions with proj_rmap and path_append ********************)
-
-lemma unwind_rmap_append (p2) (p1) (f):
- ▼[p2]▼[p1]f = ▼[p1●p2]f.
-#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <unwind_rmap_m_sn <unwind_rmap_m_sn //
-| <unwind_rmap_A_sn <unwind_rmap_A_sn //
-| <unwind_rmap_S_sn <unwind_rmap_S_sn //
-]
-qed.
-
-(* Advanced constructions with proj_rmap and path_rcons *********************)
-
-lemma unwind_rmap_d_dx (f) (p) (n):
- (▼[p]f)∘𝐮❨ninj n❩ = ▼[p◖𝗱n]f.
-// qed.
-
-lemma unwind_rmap_m_dx (f) (p):
- ▼[p]f = ▼[p◖𝗺]f.
-// qed.
-
-lemma unwind_rmap_L_dx (f) (p):
- (⫯▼[p]f) = ▼[p◖𝗟]f.
-// qed.
-
-lemma unwind_rmap_A_dx (f) (p):
- ▼[p]f = ▼[p◖𝗔]f.
-// qed.
-
-lemma unwind_rmap_S_dx (f) (p):
-▼[p]f = ▼[p◖𝗦]f.
-// qed.
-
-lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
- ▼[p]f@❨m+n❩ = ▼[p◖𝗱n]f@❨m❩.
-#f #p #n #m
-<unwind_rmap_d_dx <tr_compose_pap <tr_uni_pap //
-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/unwind2/unwind_prototerm_eq.ma".
-include "delayed_updating/syntax/prototerm_constructors.ma".
-
-(* UNWIND FOR PROTOTERM *****************************************************)
-
-lemma unwind_iref_after_sn (f) (t:prototerm) (n:pnat):
- ▼[f∘𝐮❨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 unwind_iref_after_dx (f) (t) (n:pnat):
- ▼[f](𝛗n.t) ⊆ ▼[f∘𝐮❨n❩]t.
-#f #t #n #p * #q #Hq #H0 destruct
-elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
-/2 width=1 by in_comp_unwind_bi/
-qed-.
-
-lemma unwind_iref_after (f) (t) (n:pnat):
- ▼[f∘𝐮❨n❩]t ⇔ ▼[f](𝛗n.t).
-/3 width=1 by conj, unwind_iref_after_sn, unwind_iref_after_dx/
-qed.
-
-lemma unwind_iref (f) (t) (n:pnat):
- ▼[f]▼[𝐮❨n❩]t ⇔ ▼[f](𝛗n.t).
-/3 width=3 by unwind_term_after, subset_eq_trans/
-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/unwind2/unwind.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "ground/relocation/tr_id_compose.ma".
-include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_pn.ma".
-include "ground/relocation/tr_compose_eq.ma".
-include "ground/relocation/tr_pn_eq.ma".
-include "ground/lib/stream_eq_eq.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-(* Constructions with depth *************************************************)
-
-lemma unwind_rmap_decompose (p) (f):
- ▼[p]f ≗ (⫯*[❘p❘]f)∘(▼[p]𝐢).
-#p @(list_ind_rcons … p) -p
-[ #f <unwind_rmap_empty <unwind_rmap_empty <tr_pushs_zero //
-| #p * [ #n ] #IH #f //
- [ <unwind_rmap_d_dx <unwind_rmap_d_dx <depth_d_dx
- @(stream_eq_trans … (tr_compose_assoc …))
- /2 width=1 by tr_compose_eq_repl/
- | <unwind_rmap_L_dx <unwind_rmap_L_dx <depth_L_dx
- <tr_pushs_succ <tr_compose_push_bi
- /2 width=1 by tr_push_eq_repl/
- ]
-]
-qed.
-
-lemma unwind_rmap_pap_le (f) (p) (n):
- (▼[p]𝐢)@❨n❩ < ↑❘p❘ → (▼[p]𝐢)@❨n❩ = (▼[p]f)@❨n❩.
-#f #p #n #Hn
->(tr_pap_eq_repl … (▼[p]f) … (unwind_rmap_decompose …))
-<tr_compose_pap @tr_pap_pushs_le //
-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/unwind2/unwind.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_pn_eq.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-definition unwind_exteq (A): relation2 (unwind_continuation A) (unwind_continuation A) ≝
- λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
-
-interpretation
- "extensional equivalence (unwind continuation)"
- 'RingEq A k1 k2 = (unwind_exteq A k1 k2).
-
-(* Constructions with unwind_exteq ******************************************)
-
-lemma unwind_eq_repl (A) (p) (k1) (k2):
- k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ▼❨k1, f1, p❩ = ▼❨k2, f2, p❩).
-#A #p @(path_ind_unwind … p) -p [| #n #IH | #n #l0 #q #IH |*: #q #IH ]
-#k1 #k2 #Hk #f1 #f2 #Hf
-[ <unwind_empty <unwind_empty /2 width=1 by/
-| <unwind_d_empty_sn <unwind_d_empty_sn <(tr_pap_eq_repl … Hf)
- /3 width=1 by tr_compose_eq_repl, stream_eq_refl/
-| <unwind_d_lcons_sn <unwind_d_lcons_sn
- /3 width=1 by tr_compose_eq_repl, stream_eq_refl/
-| /2 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /3 width=1 by/
-| /3 width=1 by/
-]
-qed-.
-
-(* Advanced constructions ***************************************************)
-
-lemma unwind_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
-@unwind_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
-qed.
-
-lemma unwind_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
-@unwind_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 unwind_path_eq_repl (p):
- stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
-/2 width=1 by unwind_eq_repl/ qed.
-
-lemma unwind_path_append_sn (p) (f) (q):
- q●▼[f]p = ▼❨(λg,p. proj_path g (q●p)), f, p❩.
-#p @(path_ind_unwind … p) -p // [ #n #l #p |*: #p ] #IH #f #q
-[ <unwind_d_lcons_sn <unwind_d_lcons_sn <IH -IH //
-| <unwind_m_sn <unwind_m_sn //
-| <unwind_L_sn <unwind_L_sn >unwind_lcons_alt // >unwind_append_rcons_sn //
- <IH <IH -IH <list_append_rcons_sn //
-| <unwind_A_sn <unwind_A_sn >unwind_lcons_alt >unwind_append_rcons_sn //
- <IH <IH -IH <list_append_rcons_sn //
-| <unwind_S_sn <unwind_S_sn >unwind_lcons_alt >unwind_append_rcons_sn //
- <IH <IH -IH <list_append_rcons_sn //
-]
-qed.
-
-lemma unwind_path_lcons (f) (p) (l):
- l◗▼[f]p = ▼❨(λg,p. proj_path g (l◗p)), f, p❩.
-#f #p #l
->unwind_lcons_alt <unwind_path_append_sn //
-qed.
-
-lemma unwind_path_L_sn (f) (p):
- (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
-// qed.
-
-lemma unwind_path_A_sn (f) (p):
- (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
-// qed.
-
-lemma unwind_path_S_sn (f) (p):
- (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
-// qed.
-
-lemma unwind_path_after (p) (f1) (f2):
- ▼[f2]▼[f1]p = ▼[f2∘f1]p.
-#p @(path_ind_unwind … p) -p // [ #n #l #p | #p ] #IH #f1 #f2
-[ <unwind_path_d_lcons_sn <unwind_path_d_lcons_sn
- >(unwind_path_eq_repl … (tr_compose_assoc …)) //
-| <unwind_path_L_sn <unwind_path_L_sn <unwind_path_L_sn
- >tr_compose_push_bi //
-]
-qed.
-
-(* Advanced constructions with proj_rmap and stream_tls *********************)
-
-lemma unwind_rmap_tls_d_dx (f) (p) (m) (n):
- ⇂*[m+n]▼[p]f ≗ ⇂*[m]▼[p◖𝗱n]f.
-#f #p #m #n
-<unwind_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/unwind2/unwind_prototerm_eq.ma".
-include "delayed_updating/unwind2/unwind_structure.ma".
-include "delayed_updating/substitution/fsubst.ma".
-include "delayed_updating/syntax/preterm.ma".
-include "delayed_updating/syntax/prototerm_proper.ma".
-
-(* UNWIND FOR PROTOTERM *****************************************************)
-
-(* Constructions with fsubst ************************************************)
-
-lemma unwind_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
- (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⊆ ▼[f](t[⋔p←u]).
-#f #t #u #p #Hu #ql * *
-[ #rl * #r #Hr #H1 #H2 destruct
- >unwind_append_proper_dx
- /4 width=5 by in_comp_unwind_bi, in_ppc_comp_trans, or_introl, ex2_intro/
-| * #q #Hq #H1 #H0
- @(ex2_intro … H1) @or_intror @conj // *
- [ <list_append_empty_dx #H2 destruct
- elim (unwind_path_root f q) #r #_ #Hr /2 width=2 by/
- | #l #r #H2 destruct
- @H0 -H0 [| <unwind_append_proper_dx /2 width=3 by ppc_lcons/ ]
- ]
-]
-qed-.
-
-lemma unwind_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
- ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u].
-#f #t #u #p #Hu #H1p #H2p #ql * #q * *
-[ #r #Hu #H1 #H2 destruct
- @or_introl @ex2_intro
- [|| <unwind_append_proper_dx /2 width=5 by in_ppc_comp_trans/ ]
- /2 width=3 by ex2_intro/
-| #Hq #H0 #H1 destruct
- @or_intror @conj [ /2 width=1 by in_comp_unwind_bi/ ] *
- [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
- <list_append_empty_dx @H2p -H2p /2 width=1 by prototerm_in_comp_root/
- | #l #r #Hr
- elim (unwind_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
- lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
- ]
-]
-qed-.
-
-lemma unwind_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⇔ ▼[f](t[⋔p←u]).
-/4 width=3 by unwind_fsubst_sn, conj, unwind_fsubst_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/unwind2/unwind_eq.ma".
-include "delayed_updating/syntax/path_height.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "ground/lib/stream_eq_eq.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-(* Constructions with height ************************************************)
-
-lemma unwind_rmap_pap_le (f1) (f2) (p) (m:pnat) (l:nat):
- ninj (m+⧣p+l) = ❘p❘ →
- (▼[p]f1)@❨m❩ = (▼[p]f2)@❨m❩.
-#f1 #f2 #p @(list_ind_rcons … p) -p
-[ #m #l <depth_empty #H0 destruct
-| #p * [ #n ] #IH #m #l
- [ <height_d_dx <depth_d_dx <unwind_rmap_pap_d_dx <unwind_rmap_pap_d_dx
- <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
- #H0 <(IH … l) -IH //
- | /2 width=2 by/
- | <height_L_dx <depth_L_dx <unwind_rmap_L_dx <unwind_rmap_L_dx
- cases m -m // #m
- <nrplus_succ_sn <nrplus_succ_sn >nsucc_inj #H0
- <tr_pap_push <tr_pap_push
- <(IH … l) -IH //
- | /2 width=2 by/
- | /2 width=2 by/
- ]
-]
-qed.
-
-lemma unwind_rmap_pap_gt (f) (p) (m):
- f@❨m+⧣p❩+❘p❘ = (▼[p]f)@❨m+❘p❘❩.
-#f #p @(list_ind_rcons … p) -p [ // ]
-#p * [ #n ] #IH #m
-[ <height_d_dx <depth_d_dx
- <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
- <unwind_rmap_pap_d_dx >IH -IH //
-| //
-| <height_L_dx <depth_L_dx
- <nrplus_succ_dx <nrplus_succ_dx <unwind_rmap_L_dx <tr_pap_push //
-| //
-| //
-]
-qed.
-
-lemma unwind_rmap_tls_gt (f) (p) (m:pnat):
- ⇂*[ninj (m+⧣p)]f ≗ ⇂*[ninj (m+❘p❘)]▼[p]f.
-#f #p @(list_ind_rcons … p) -p [ // ]
-#p * [ #n ] #IH #m
-[ <height_d_dx <depth_d_dx
- <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
- @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
- @(stream_eq_canc_dx … (IH …)) -IH //
-| //
-| <height_L_dx <depth_L_dx
- <nrplus_succ_dx >nsucc_inj //
-| //
-| //
-]
-qed.
-
-lemma unwind_rmap_tls_eq (f) (p):
- ⇂*[⧣p]f ≗ ⇂*[↑❘p❘]▼[p]⫯f.
-(*
-#f #p @(list_ind_rcons … p) -p //
-#p * [ #n ] #IH //
-<height_d_dx <depth_d_dx <unwind_rmap_d_dx
-@(stream_eq_trans … (tr_tls_compose_uni_dx …))
-*)
+++ /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_equivalence.ma".
-include "delayed_updating/syntax/path_structure_inner.ma".
-include "delayed_updating/syntax/preterm.ma".
-include "delayed_updating/unwind2/unwind_structure.ma".
-include "delayed_updating/unwind2/unwind_prototerm.ma".
-
-(* UNWIND FOR PRETERM *******************************************************)
-
-(* Constructions with subset_equivalence ************************************)
-
-lemma unwind_grafted_sn (f) (t) (p): p ϵ 𝐈 →
- ▼[▼[p]f](t⋔p) ⊆ (▼[f]t)⋔(⊗p).
-#f #t #p #Hp #q * #r #Hr #H0 destruct
-@(ex2_intro … Hr) -Hr
-<unwind_append_inner_sn //
-qed-.
-
-lemma unwind_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)⋔(⊗p) ⊆ ▼[▼[p]f](t⋔p).
-#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
-elim (unwind_inv_append_inner_sn … (sym_eq … H0)) -H0 //
-#p0 #q0 #Hp0 #Hq0 #H0 destruct
-<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
-/2 width=1 by in_comp_unwind_bi/
-qed-.
-
-lemma unwind_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
- ▼[▼[p]f](t⋔p) ⇔ (▼[f]t)⋔(⊗p).
-/3 width=1 by unwind_grafted_sn, conj, unwind_grafted_dx/ qed.
-
-
-lemma unwind_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)⋔((⊗p)◖𝗦) ⊆ ▼[▼[p]f](t⋔(p◖𝗦)).
-#f #t #p #Hp #Ht #q * #r #Hr
-<list_append_rcons_sn #H0
-elim (unwind_inv_append_proper_dx … (sym_eq … H0)) -H0 //
-#p0 #q0 #Hp0 #Hq0 #H0 destruct
-<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
-elim (unwind_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
-#r1 #r2 #Hr1 #Hr2 #H0 destruct
-lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
-[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
-/2 width=1 by in_comp_unwind_bi/
-qed-.
-
-lemma unwind_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
- ▼[▼[p]f](t⋔(p◖𝗦)) ⇔ (▼[f]t)⋔((⊗p)◖𝗦).
-#f #t #p #Hp #Ht
-@conj
-[ >unwind_rmap_S_dx >structure_S_dx
- @unwind_grafted_sn //
-| /2 width=1 by unwind_grafted_S_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/unwind2/unwind.ma".
-include "delayed_updating/syntax/prototerm.ma".
-include "ground/lib/subset_ext.ma".
-
-(* UNWIND FOR PROTOTERM *****************************************************)
-
-interpretation
- "unwind (prototerm)"
- 'BlackDownTriangle f t = (subset_ext_f1 ? ? (unwind_gen ? proj_path f) t).
-
-(* Basic constructions ******************************************************)
-
-lemma in_comp_unwind_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/unwind2/unwind_eq.ma".
-include "delayed_updating/unwind2/unwind_prototerm.ma".
-
-(* UNWIND FOR PROTOTERM *****************************************************)
-
-(* Constructions with subset_equivalence ************************************)
-
-lemma unwind_term_eq_repl_sn (f1) (f2) (t):
- f1 ≗ f2 → ▼[f1]t ⇔ ▼[f2]t.
-/3 width=1 by subset_equivalence_ext_f1_exteq, unwind_path_eq_repl/
-qed.
-
-lemma unwind_term_eq_repl_dx (f) (t1) (t2):
- t1 ⇔ t2 → ▼[f]t1 ⇔ ▼[f]t2.
-/2 width=1 by subset_equivalence_ext_f1_bi/
-qed.
-
-lemma unwind_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/unwind2/unwind_eq.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".
-
-(* UNWIND FOR PATH **********************************************************)
-
-(* Basic constructions with structure ***************************************)
-
-lemma structure_unwind (p) (f):
- ⊗p = ⊗▼[f]p.
-#p @(path_ind_unwind … p) -p // #p #IH #f
-<unwind_path_L_sn //
-qed.
-
-lemma unwind_structure (p) (f):
- ⊗p = ▼[f]⊗p.
-#p @(path_ind_unwind … p) -p //
-qed.
-
-(* Destructions with structure **********************************************)
-
-lemma unwind_des_structure (q) (p) (f):
- ⊗q = ▼[f]p → ⊗q = ⊗p.
-// qed-.
-
-(* Constructions with proper condition for path *****************************)
-
-lemma unwind_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
- (⊗p1)●(▼[▼[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 #H destruct //
-| <unwind_path_d_lcons_sn <IH //
-| <unwind_path_m_sn <IH //
-| <unwind_path_L_sn <IH //
-| <unwind_path_A_sn <IH //
-| <unwind_path_S_sn <IH //
-]
-qed-.
-
-(* Constructions with inner condition for path ******************************)
-
-lemma unwind_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
- (⊗p1)●(▼[▼[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_append_proper_dx //
-| <list_append_rcons_sn <unwind_append_proper_dx //
- <structure_L_dx <list_append_rcons_sn //
-| <list_append_rcons_sn <unwind_append_proper_dx //
- <structure_A_dx <list_append_rcons_sn //
-| <list_append_rcons_sn <unwind_append_proper_dx //
- <structure_S_dx <list_append_rcons_sn //
-]
-qed-.
-
-(* Advanced constructions with proj_path ************************************)
-
-lemma unwind_path_d_empty_dx (n) (p) (f):
- (⊗p)◖𝗱((▼[p]f)@❨n❩) = ▼[f](p◖𝗱n).
-#n #p #f <unwind_append_proper_dx //
-qed.
-
-lemma unwind_path_m_dx (p) (f):
- ⊗p = ▼[f](p◖𝗺).
-#p #f <unwind_append_proper_dx //
-qed.
-
-lemma unwind_path_L_dx (p) (f):
- (⊗p)◖𝗟 = ▼[f](p◖𝗟).
-#p #f <unwind_append_proper_dx //
-qed.
-
-lemma unwind_path_A_dx (p) (f):
- (⊗p)◖𝗔 = ▼[f](p◖𝗔).
-#p #f <unwind_append_proper_dx //
-qed.
-
-lemma unwind_path_S_dx (p) (f):
- (⊗p)◖𝗦 = ▼[f](p◖𝗦).
-#p #f <unwind_append_proper_dx //
-qed.
-
-lemma unwind_path_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 with proj_path ***************************************)
-
-lemma unwind_path_inv_d_sn (k) (q) (p) (f):
- (𝗱k◗q) = ▼[f]p →
- ∃∃r,h. 𝐞 = ⊗r & (▼[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_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct -IH
- /2 width=5 by ex4_2_intro/
-| <unwind_path_d_lcons_sn #H
- elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
- /2 width=5 by ex4_2_intro/
-| <unwind_path_m_sn #H
- elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
- /2 width=5 by ex4_2_intro/
-| <unwind_path_L_sn #H destruct
-| <unwind_path_A_sn #H destruct
-| <unwind_path_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_path_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_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H /2 width=2 by/
-| <unwind_path_m_sn #H /2 width=2 by/
-| <unwind_path_L_sn #H destruct
-| <unwind_path_A_sn #H destruct
-| <unwind_path_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_path_inv_L_sn (q) (p) (f):
- (𝗟◗q) = ▼[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▼[r1]f]r2 & r1●𝗟◗r2 = p.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_path_m_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_path_L_sn #H destruct -IH
- /2 width=5 by ex3_2_intro/
-| <unwind_path_A_sn #H destruct
-| <unwind_path_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_path_inv_A_sn (q) (p) (f):
- (𝗔◗q) = ▼[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗔◗r2 = p.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_path_m_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_path_L_sn #H destruct
-| <unwind_path_A_sn #H destruct -IH
- /2 width=5 by ex3_2_intro/
-| <unwind_path_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_path_inv_S_sn (q) (p) (f):
- (𝗦◗q) = ▼[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗦◗r2 = p.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_path_empty #H destruct
-| <unwind_path_d_empty_sn #H destruct
-| <unwind_path_d_lcons_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_path_m_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/| <unwind_path_L_sn #H destruct
-| <unwind_path_A_sn #H destruct
-| <unwind_path_S_sn #H destruct -IH
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Inversions with proper condition for path ********************************)
-
-lemma unwind_inv_append_proper_dx (q2) (q1) (p) (f):
- q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
- ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]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_path_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
- elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
- elim Hq2 -Hq2 //
- | elim (unwind_path_inv_m_sn … H)
- | elim (unwind_path_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_path_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_path_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_inv_append_inner_sn (q1) (q2) (p) (f):
- q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
- ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_path_inv_m_sn … (sym_eq … H2))
- | <list_append_rcons_sn #H0
- elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_path_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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_path_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_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_path_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/unwind2/unwind.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "ground/arith/nat_pred_succ.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-(* Basic constructions with structure and depth *****************************)
-
-lemma unwind_rmap_structure (p) (f):
- (⫯*[❘p❘]f) = ▼[⊗p]f.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-[ <unwind_rmap_A_sn //
-| <unwind_rmap_S_sn //
-]
-qed.
lemma npsucc_inj (p): (↑p) = ↑(ninj p).
// qed.
+lemma nsucc_unfold (n): ninj (↑n) = ↑n.
+// qed-.
+
lemma nsucc_zero: ninj (𝟏) = ↑𝟎.
// qed.
// qed.
lemma list_append_rcons_dx (A):
- ∀l1,l2,a. l1 ⨁ l2 ⨭ a = l1 ⨁{A} (l2 ⨭ a).
+ ∀l1,l2,a. (l1 ⨁ l2) ⨭ a = l1 ⨁{A} (l2 ⨭ a).
// qed.
(* Basic inversions *********************************************************)
(* Constructions with tr_compose and tr_tls *********************************)
+lemma tr_tls_compose_uni_sn (f) (n) (p:pnat):
+ ⇂*[p]f ≗ ⇂*[p](𝐮❨n❩∘f).
+#f #n #p elim p -p //
+#p #IH /2 width=1 by stream_tl_eq_repl/
+qed.
+
lemma tr_tl_compose_uni_dx (f) (n):
⇂*[↑n]f ≗ ⇂(f∘𝐮❨n❩).
// qed.
["∨"; "⩖"; "∪"; "∩"; "⋓"; "⋒" ] ;
["a"; "α"; "𝕒"; "𝐚"; "𝛂"; "ⓐ"; ] ;
["A"; "ℵ"; "𝔸"; "𝐀"; "Ⓐ"; "𝗔"; ] ;
- ["b"; "β"; "ß"; "𝕓"; "𝐛"; "𝛃"; "ⓑ"; ] ;
+ ["b"; "β"; "ß"; "𝕓"; "𝐛"; "𝛃"; "ⓑ"; "♭"; ] ;
["B"; "ℶ"; "ℬ"; "𝔹"; "𝐁"; "Ⓑ"; ] ;
["c"; "𝕔"; "𝐜"; "ⓒ"; ] ;
["C"; "ℭ"; "∁"; "𝐂"; "ℂ"; "Ⓒ"; ] ;