--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( ◆[ term 46 f ] break term 70 p )"
+ non associative with precedence 70
+ for @{ 'BlackDiamond $f $p }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( ▼ term 70 p )"
+ non associative with precedence 70
+ for @{ 'BlackDownTriangle $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
+include "delayed_updating/notation/functions/black_righttriangle_1.ma".
+
+interpretation
+ "unwind map (reversed path)"
+ 'BlackRightTriangle p = (unwind2_rmap tr_id 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 @{ 'BlackDiamond $f $p }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR DELAYED UPDATING ********************************************)
-
-notation "hvbox( ▼ term 70 p )"
- non associative with precedence 70
- for @{ 'BlackDownTriangle $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 k, break term 46 p, break term 46 f ❩ )"
- non associative with precedence 70
- for @{ 'UpArrow $S $k $p $f }.
-
-notation > "hvbox( ↑❨ term 46 k, break term 46 p, break term 46 f ❩ )"
- non associative with precedence 70
- for @{ 'UpArrow ? $k $p $f }.
-
-notation > "hvbox( ↑{ term 46 S }❨ break term 46 k, break term 46 p, break term 46 f ❩ )"
- non associative with precedence 70
- for @{ 'UpArrow $S $k $p $f }.
include "delayed_updating/substitution/prelift_label.ma".
include "delayed_updating/substitution/lift_rmap.ma".
-include "ground/xoa/ex_3_2.ma".
(* LIFT FOR PATH ************************************************************)
(**************************************************************************)
include "delayed_updating/substitution/lift_rmap.ma".
-include "ground/relocation/tr_pn_eq.ma".
+include "delayed_updating/substitution/prelift_rmap_eq.ma".
include "ground/lib/stream_tls_plus.ma".
-include "ground/lib/stream_tls_eq.ma".
include "ground/arith/nat_plus_rplus.ma".
include "ground/arith/nat_rplus_pplus.ma".
lemma lift_rmap_eq_repl (p):
stream_eq_repl … (λf1,f2. ↑[p]f1 ≗ ↑[p]f2).
#p elim p -p //
-* [ #k ] #p #IH #f1 #f2 #Hf
-[ /3 width=1 by stream_tls_eq_repl/
-| /2 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /2 width=1 by/
-| /2 width=1 by/
-]
+#l #p #IH #f1 #f2 #Hf
+/3 width=1 by prelift_rmap_eq_repl/
qed.
lemma tls_lift_rmap_d_dx (f) (p) (n) (k):
p◖𝗱k ϵ 𝐈 → ⊥.
#p #k #H0 @H0 -H0 //
qed-.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma pic_m_sn (p):
+ p ϵ 𝐈 → 𝗺◗p ϵ 𝐈.
+* [| * [ #k ] #p #Hp <list_cons_shift ] //
+[ #_ <list_cons_comm //
+| elim (pic_inv_d_dx … Hp)
+]
+qed.
+
+lemma pic_L_sn (p):
+ p ϵ 𝐈 → 𝗟◗p ϵ 𝐈.
+* [| * [ #k ] #p #Hp <list_cons_shift ] //
+[ #_ <list_cons_comm //
+| elim (pic_inv_d_dx … Hp)
+]
+qed.
+
+lemma pic_A_sn (p):
+ p ϵ 𝐈 → 𝗔◗p ϵ 𝐈.
+* [| * [ #k ] #p #Hp <list_cons_shift ] //
+[ #_ <list_cons_comm //
+| elim (pic_inv_d_dx … Hp)
+]
+qed.
+
+lemma pic_S_sn (p):
+ p ϵ 𝐈 → 𝗦◗p ϵ 𝐈.
+* [| * [ #k ] #p #Hp <list_cons_shift ] //
+[ #_ <list_cons_comm //
+| elim (pic_inv_d_dx … Hp)
+]
+qed.
include "delayed_updating/syntax/path.ma".
include "delayed_updating/notation/functions/circled_times_1.ma".
+include "ground/xoa/ex_3_2.ma".
(* STRUCTURE FOR PATH *******************************************************)
(𝗦◗⊗p) = ⊗(𝗦◗p).
#p <structure_append //
qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_d_dx_structure (h) (q) (p):
+ q◖𝗱h = ⊗p → ⊥.
+#h #q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0 /2 width=1 by/
+| <structure_m_dx #H0 /2 width=1 by/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_m_dx_structure (q) (p):
+ q◖𝗺 = ⊗p → ⊥.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0 /2 width=1 by/
+| <structure_m_dx #H0 /2 width=1 by/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_L_dx_structure (q) (p):
+ q◖𝗟 = ⊗p →
+ ∃∃r1,r2. q = ⊗r1 & 𝐞 = ⊗r2 & r1●𝗟◗r2 = p.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_m_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_L_dx #H0 destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_A_dx_structure (q) (p):
+ q◖𝗔 = ⊗p →
+ ∃∃r1,r2. q = ⊗r1 & 𝐞 = ⊗r2 & r1●𝗔◗r2 = p.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_m_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_S_dx_structure (q) (p):
+ q◖𝗦 = ⊗p →
+ ∃∃r1,r2. q = ⊗r1 & 𝐞 = ⊗r2 & r1●𝗦◗r2 = p.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_m_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct -IH
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Main inversions **********************************************************)
+
+theorem eq_inv_append_structure (p) (q) (r):
+ p●q = ⊗r →
+ ∃∃r1,r2.p = ⊗r1 & q = ⊗r2 & r1●r2 = r.
+#p #q elim q -q [| * [ #k ] #q #IH ] #r
+[ <list_append_empty_sn #H0 destruct
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_d_dx_structure … H0)
+| #H0 elim (eq_inv_m_dx_structure … H0)
+| #H0 elim (eq_inv_L_dx_structure … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH … Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗟◗r2)) //
+ <structure_append <structure_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_structure … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH … Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗔◗r2)) //
+ <structure_append <structure_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_structure … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH … Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗦◗r2)) //
+ <structure_append <structure_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with path_lcons ***********************************************)
+
+lemma eq_inv_d_sn_structure (h) (q) (p):
+ (𝗱h◗q) = ⊗p → ⊥.
+#h #q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_d_dx_structure … H0)
+qed-.
+
+lemma eq_inv_m_sn_structure (q) (p):
+ (𝗺 ◗q) = ⊗p → ⊥.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_m_dx_structure … H0)
+qed-.
+
+lemma eq_inv_L_sn_structure (q) (p):
+ (𝗟◗q) = ⊗p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ⊗r2 & r1●𝗟◗r2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_L_dx_structure … H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro … s1 (s2●r2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
+
+lemma eq_inv_A_sn_structure (q) (p):
+ (𝗔◗q) = ⊗p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ⊗r2 & r1●𝗔◗r2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_A_dx_structure … H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro … s1 (s2●r2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
+
+lemma eq_inv_S_sn_structure (q) (p):
+ (𝗦◗q) = ⊗p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ⊗r2 & r1●𝗦◗r2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_S_dx_structure … H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro … s1 (s2●r2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/syntax/label.ma".
+include "delayed_updating/notation/functions/black_righttriangle_2.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/relocation/tr_compose.ma".
+
+(* TAILED PREUNWIND FOR RELOCATION MAP **************************************)
+
+definition preunwind2_rmap (f) (l): tr_map ≝
+match l with
+[ label_d k ⇒ f∘𝐮❨k❩
+| label_m ⇒ f
+| label_L ⇒ ⫯f
+| label_A ⇒ f
+| label_S ⇒ f
+].
+
+interpretation
+ "tailed preunwind (relocation map)"
+ 'BlackRightTriangle f l = (preunwind2_rmap f l).
+
+(* Basic constructions ******************************************************)
+
+lemma preunwind2_rmap_d (f) (k:pnat):
+ f∘𝐮❨k❩ = ▶[f]𝗱k.
+// qed.
+
+lemma preunwind2_rmap_m (f):
+ f = ▶[f]𝗺.
+// qed.
+
+lemma preunwind2_rmap_L (f):
+ (⫯f) = ▶[f]𝗟.
+// qed.
+
+lemma preunwind2_rmap_A (f):
+ f = ▶[f]𝗔.
+// qed.
+
+lemma preunwind2_rmap_S (f):
+ f = ▶[f]𝗦.
+// qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind/preunwind2_rmap.ma".
+include "ground/relocation/tr_compose_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+
+(* TAILED PREUNWIND FOR RELOCATION MAP **************************************)
+
+(* Constructions with tr_map_eq *********************************************)
+
+lemma preunwind2_rmap_eq_repl (l):
+ stream_eq_repl … (λf1,f2. ▶[f1]l ≗ ▶[f2]l).
+* // #k #f1 #f2 #Hf
+/2 width=1 by tr_compose_eq_repl/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind/preunwind2_rmap.ma".
+include "delayed_updating/substitution/prelift_label.ma".
+include "delayed_updating/substitution/prelift_rmap.ma".
+include "ground/relocation/tr_uni_compose.ma".
+include "ground/relocation/tr_compose_compose.ma".
+include "ground/relocation/tr_compose_eq.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* TAILED PREUNWIND FOR RELOCATION MAP **************************************)
+
+(* Constructions with lift_path *********************************************)
+
+lemma preunwind2_lift_rmap_after (g) (f) (l):
+ ▶[g]↑[f]l∘↑[l]f ≗ ▶[g∘f]l.
+#g #f * // #k
+<prelift_label_d <prelift_rmap_d <preunwind2_rmap_d <preunwind2_rmap_d
+@(stream_eq_trans … (tr_compose_assoc …))
+@(stream_eq_canc_dx … (tr_compose_assoc …))
+/2 width=1 by tr_compose_eq_repl/
+qed.
(**************************************************************************)
include "delayed_updating/unwind/unwind2_prototerm_eq.ma".
+include "delayed_updating/unwind/unwind2_path_append.ma".
include "delayed_updating/syntax/prototerm_constructors.ma".
-(* UNWIND FOR PROTOTERM *****************************************************)
+(* TAILED UNWIND FOR PROTOTERM **********************************************)
(* Constructions with constructors ******************************************)
-lemma unwind2_term_iref_sn (f) (t) (n:pnat):
- ▼[f∘𝐮❨n❩]t ⊆ ▼[f](𝛕n.t).
-#f #t #n #p * #q #Hq #H0 destruct
-@(ex2_intro … (𝗱n◗𝗺◗q))
+lemma unwind2_term_iref_sn (f) (t) (k:pnat):
+ ▼[f∘𝐮❨k❩]t ⊆ ▼[f](𝛕k.t).
+#f #t #k #p * #q #Hq #H0 destruct
+@(ex2_intro … (𝗱k◗𝗺◗q))
/2 width=1 by in_comp_iref/
qed-.
-lemma unwind2_term_iref_dx (f) (t) (n:pnat):
- ▼[f](𝛕n.t) ⊆ ▼[f∘𝐮❨n❩]t.
-#f #t #n #p * #q #Hq #H0 destruct
+lemma unwind2_term_iref_dx (f) (t) (k:pnat):
+ ▼[f](𝛕k.t) ⊆ ▼[f∘𝐮❨k❩]t.
+#f #t #k #p * #q #Hq #H0 destruct
elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
/2 width=1 by in_comp_unwind2_path_term/
qed-.
-lemma unwind2_term_iref (f) (t) (n:pnat):
- ▼[f∘𝐮❨n❩]t ⇔ ▼[f](𝛕n.t).
+lemma unwind2_term_iref (f) (t) (k:pnat):
+ ▼[f∘𝐮❨k❩]t ⇔ ▼[f](𝛕k.t).
/3 width=2 by conj, unwind2_term_iref_sn, unwind2_term_iref_dx/
qed.
(* *)
(**************************************************************************)
-include "delayed_updating/unwind/unwind_gen.ma".
include "delayed_updating/unwind/unwind2_rmap.ma".
-include "delayed_updating/syntax/path_reverse.ma".
+include "delayed_updating/syntax/path_structure.ma".
include "delayed_updating/notation/functions/black_downtriangle_2.ma".
-include "ground/lib/list_tl.ma".
-(* UNWIND FOR PATH **********************************************************)
+(* TAILED UNWIND FOR PATH ***************************************************)
definition unwind2_path (f) (p): path ≝
- ◆[▶[f]⇂(pᴿ)]p
-.
+match p with
+[ list_empty ⇒ (𝐞)
+| list_lcons l q ⇒
+ match l with
+ [ label_d k ⇒ (⊗q)◖𝗱(▶[f]q＠⧣❨k❩)
+ | label_m ⇒ ⊗p
+ | label_L ⇒ ⊗p
+ | label_A ⇒ ⊗p
+ | label_S ⇒ ⊗p
+ ]
+].
interpretation
- "unwind (path)"
+ "tailed unwind (path)"
'BlackDownTriangle f p = (unwind2_path f p).
(* Basic constructions ******************************************************)
-lemma unwind2_path_unfold (f) (p):
- ◆[▶[f]⇂(pᴿ)]p = ▼[f]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).
+lemma unwind2_path_d_dx (f) (p) (k) :
+ (⊗p)◖𝗱((▶[f]p)＠⧣❨k❩) = ▼[f](p◖𝗱k).
// qed.
-lemma unwind2_path_d_lcons (f) (p) (l) (n:pnat):
- ▼[f∘𝐮❨n❩](l◗p) = ▼[f](𝗱n◗l◗p).
-#f #p #l #n <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_d_lcons <reverse_lcons
-@(list_ind_rcons … p) -p // #p #l0 #_
-<reverse_rcons <reverse_lcons <reverse_lcons <reverse_rcons
-<list_tl_lcons <list_tl_lcons //
-qed.
+lemma unwind2_path_m_dx (f) (p):
+ ⊗p = ▼[f](p◖𝗺).
+// qed.
-lemma unwind2_path_m_sn (f) (p):
- ▼[f]p = ▼[f](𝗺◗p).
-#f #p <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_m_sn <reverse_lcons
-@(list_ind_rcons … p) -p // #p #l #_
-<reverse_rcons <list_tl_lcons <list_tl_lcons //
-qed.
+lemma unwind2_path_L_dx (f) (p):
+ (⊗p)◖𝗟 = ▼[f](p◖𝗟).
+// qed.
-lemma unwind2_path_L_sn (f) (p):
- (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
-#f #p <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_L_sn <reverse_lcons
-@(list_ind_rcons … p) -p // #p #l #_
-<reverse_rcons <list_tl_lcons <list_tl_lcons //
-qed.
+lemma unwind2_path_A_dx (f) (p):
+ (⊗p)◖𝗔 = ▼[f](p◖𝗔).
+// qed.
+
+lemma unwind2_path_S_dx (f) (p):
+ (⊗p)◖𝗦 = ▼[f](p◖𝗦).
+// qed.
-lemma unwind2_path_A_sn (f) (p):
- (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
-#f #p <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_A_sn <reverse_lcons
-@(list_ind_rcons … p) -p // #p #l #_
-<reverse_rcons <list_tl_lcons <list_tl_lcons //
+(* Constructions with structure *********************************************)
+
+lemma structure_unwind2_path (f) (p):
+ ⊗p = ⊗▼[f]p.
+#f * // * [ #k ] #p //
qed.
-lemma unwind2_path_S_sn (f) (p):
- (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
-#f #p <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_S_sn <reverse_lcons
-@(list_ind_rcons … p) -p // #p #l #_
-<reverse_rcons <list_tl_lcons <list_tl_lcons //
+lemma unwind2_path_structure (f) (p):
+ ⊗p = ▼[f]⊗p.
+#f #p elim p -p // * [ #k ] #p #IH //
+[ <structure_L_dx <unwind2_path_L_dx //
+| <structure_A_dx <unwind2_path_A_dx //
+| <structure_S_dx <unwind2_path_S_dx //
+]
qed.
+
+lemma unwind2_path_root (f) (p):
+ ∃∃r. 𝐞 = ⊗r & ⊗p●r = ▼[f]p.
+#f * [| * [ #k ] #p ]
+/2 width=3 by ex2_intro/
+<unwind2_path_d_dx <structure_d_dx
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Destructions with structure **********************************************)
+
+lemma unwind2_path_des_structure (f) (q) (p):
+ ⊗q = ▼[f]p → ⊗q = ⊗p.
+// qed-.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_d_dx_unwind2_path (f) (q) (p) (h):
+ q◖𝗱h = ▼[f]p →
+ ∃∃r,k. q = ⊗r & h = ▶[f]r＠⧣❨k❩ & r◖𝗱k = p.
+#f #q * [| * [ #k ] #p ] #h
+[ <unwind2_path_empty #H0 destruct
+| <unwind2_path_d_dx #H0 destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind2_path_m_dx #H0
+ elim (eq_inv_d_dx_structure … H0)
+| <unwind2_path_L_dx #H0 destruct
+| <unwind2_path_A_dx #H0 destruct
+| <unwind2_path_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_m_dx_unwind2_path (f) (q) (p):
+ q◖𝗺 = ▼[f]p → ⊥.
+#f #q * [| * [ #k ] #p ]
+[ <unwind2_path_empty #H0 destruct
+| <unwind2_path_d_dx #H0 destruct
+| <unwind2_path_m_dx #H0
+ elim (eq_inv_m_dx_structure … H0)
+| <unwind2_path_L_dx #H0 destruct
+| <unwind2_path_A_dx #H0 destruct
+| <unwind2_path_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_L_dx_unwind2_path (f) (q) (p):
+ q◖𝗟 = ▼[f]p →
+ ∃∃r1,r2. q = ⊗r1 & ∀g. 𝐞 = ▼[g]r2 & r1●𝗟◗r2 = p.
+#f #q * [| * [ #k ] #p ]
+[ <unwind2_path_empty #H0 destruct
+| <unwind2_path_d_dx #H0 destruct
+| <unwind2_path_m_dx #H0
+ elim (eq_inv_L_dx_structure … H0) -H0 #r1 #r2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind2_path_L_dx #H0 destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind2_path_A_dx #H0 destruct
+| <unwind2_path_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_A_dx_unwind2_path (f) (q) (p):
+ q◖𝗔 = ▼[f]p →
+ ∃∃r1,r2. q = ⊗r1 & ∀g. 𝐞 = ▼[g]r2 & r1●𝗔◗r2 = p.
+#f #q * [| * [ #k ] #p ]
+[ <unwind2_path_empty #H0 destruct
+| <unwind2_path_d_dx #H0 destruct
+| <unwind2_path_m_dx #H0
+ elim (eq_inv_A_dx_structure … H0) -H0 #r1 #r2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind2_path_L_dx #H0 destruct
+| <unwind2_path_A_dx #H0 destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind2_path_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_S_dx_unwind2_path (f) (q) (p):
+ q◖𝗦 = ▼[f]p →
+ ∃∃r1,r2. q = ⊗r1 & ∀g. 𝐞 = ▼[g]r2 & r1●𝗦◗r2 = p.
+#f #q * [| * [ #k ] #p ]
+[ <unwind2_path_empty #H0 destruct
+| <unwind2_path_d_dx #H0 destruct
+| <unwind2_path_m_dx #H0
+ elim (eq_inv_S_dx_structure … H0) -H0 #r1 #r2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| <unwind2_path_L_dx #H0 destruct
+| <unwind2_path_A_dx #H0 destruct
+| <unwind2_path_S_dx #H0 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind/unwind2_path_eq.ma".
+include "delayed_updating/unwind/unwind2_rmap_after.ma".
+
+(* TAILED UNWIND FOR PATH ***************************************************)
+
+(* Properties with tr_after *************************************************)
+
+lemma unwind2_path_after (g) (f) (p):
+ ▼[g]▼[f]p = ▼[g∘f]p.
+#g #f * // * [ #k ] #p //
+<unwind2_path_d_dx <unwind2_path_d_dx
+@eq_f2 // @eq_f >tr_compose_pap
+/2 width=3 by tr_pap_eq_repl/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind/unwind2_path.ma".
+include "delayed_updating/syntax/path_inner.ma".
+include "delayed_updating/syntax/path_proper.ma".
+include "ground/xoa/ex_4_2.ma".
+
+(* TAILED UNWIND FOR PATH ***************************************************)
+
+(* Constructions with inner condition for path ******************************)
+
+lemma unwind2_path_inner (f) (p):
+ p ϵ 𝐈 → ⊗p = ▼[f]p.
+#f * // * // #k #q #Hq
+elim (pic_inv_d_dx … Hq)
+qed-.
+
+(* Constructions with append and inner condition for path *******************)
+
+lemma unwind2_path_append_inner_sn (f) (p) (q): p ϵ 𝐈 →
+ (⊗p)●(▼[▶[f]p]q) = ▼[f](p●q).
+#f #p * [ #Hp | * [ #k ] #q #_ ] //
+[ <(unwind2_path_inner … Hp) -Hp //
+| <unwind2_path_d_dx <unwind2_path_d_dx
+ /2 width=3 by trans_eq/
+| <unwind2_path_L_dx <unwind2_path_L_dx //
+| <unwind2_path_A_dx <unwind2_path_A_dx //
+| <unwind2_path_S_dx <unwind2_path_S_dx //
+]
+qed.
+
+(* Constructions with append and proper condition for path ******************)
+
+lemma unwind2_path_append_proper_dx (f) (p) (q): q ϵ 𝐏 →
+ (⊗p)●(▼[▶[f]p]q) = ▼[f](p●q).
+#f #p * [ #Hq | * [ #k ] #q #_ ] //
+[ elim (ppc_inv_empty … Hq)
+| <unwind2_path_d_dx <unwind2_path_d_dx
+ /2 width=3 by trans_eq/
+| <unwind2_path_L_dx <unwind2_path_L_dx //
+| <unwind2_path_A_dx <unwind2_path_A_dx //
+| <unwind2_path_S_dx <unwind2_path_S_dx //
+]
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma unwind2_path_d_empty (f) (k):
+ (𝗱(f＠⧣❨k❩)◗𝐞) = ▼[f](𝗱k◗𝐞).
+// qed.
+
+lemma unwind2_path_d_lcons (f) (p) (l) (k:pnat):
+ ▼[f∘𝐮❨k❩](l◗p) = ▼[f](𝗱k◗l◗p).
+#f #p #l #k <unwind2_path_append_proper_dx in ⊢ (???%); //
+qed.
+
+lemma unwind2_path_m_sn (f) (p):
+ ▼[f]p = ▼[f](𝗺◗p).
+#f #p <unwind2_path_append_inner_sn //
+qed.
+
+lemma unwind2_path_L_sn (f) (p):
+ (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
+#f #p <unwind2_path_append_inner_sn //
+qed.
+
+lemma unwind2_path_A_sn (f) (p):
+ (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
+#f #p <unwind2_path_append_inner_sn //
+qed.
+
+lemma unwind2_path_S_sn (f) (p):
+ (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
+#f #p <unwind2_path_append_inner_sn //
+qed.
+
+(* Destructions with inner condition for path *******************************)
+
+lemma unwind2_path_des_inner (f) (p):
+ ▼[f]p ϵ 𝐈 → p ϵ 𝐈.
+#f * // * [ #k ] #p //
+<unwind2_path_d_dx #H0
+elim (pic_inv_d_dx … H0)
+qed-.
+
+(* Destructions with append and inner condition for path ********************)
+
+lemma unwind2_path_des_append_inner_sn (f) (p) (q1) (q2):
+ q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. q1 = ⊗p1 & q2 = ▼[▶[f]p1]p2 & p1●p2 = p.
+#f #p #q1 * [| * [ #k ] #q2 ] #Hq1
+[ <list_append_empty_sn #H0 destruct
+ lapply (unwind2_path_des_inner … Hq1) -Hq1 #Hp
+ <(unwind2_path_inner … Hp) -Hp
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_d_dx_unwind2_path … H0) -H0 #r #h #Hr #H1 #H2 destruct
+ elim (eq_inv_append_structure … Hr) -Hr #s1 #s2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_m_dx_unwind2_path … H0)
+| #H0 elim (eq_inv_L_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗟◗r2)) //
+ <unwind2_path_append_proper_dx //
+ <unwind2_path_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗔◗r2)) //
+ <unwind2_path_append_proper_dx //
+ <unwind2_path_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗦◗r2)) //
+ <unwind2_path_append_proper_dx //
+ <unwind2_path_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with append and proper condition for path *********************)
+
+lemma unwind2_path_inv_append_proper_dx (f) (p) (q1) (q2):
+ q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. q1 = ⊗p1 & q2 = ▼[▶[f]p1]p2 & p1●p2 = p.
+#f #p #q1 * [| * [ #k ] #q2 ] #Hq1
+[ <list_append_empty_sn #H0 destruct
+ elim (ppc_inv_empty … Hq1)
+| #H0 elim (eq_inv_d_dx_unwind2_path … H0) -H0 #r #h #Hr #H1 #H2 destruct
+ elim (eq_inv_append_structure … Hr) -Hr #s1 #s2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_m_dx_unwind2_path … H0)
+| #H0 elim (eq_inv_L_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗟◗r2)) //
+ <unwind2_path_append_proper_dx //
+ <unwind2_path_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗔◗r2)) //
+ <unwind2_path_append_proper_dx //
+ <unwind2_path_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗦◗r2)) //
+ <unwind2_path_append_proper_dx //
+ <unwind2_path_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with path_lcons ***********************************************)
+
+lemma eq_inv_d_sn_unwind2_path (f) (q) (p) (k):
+ (𝗱k◗q) = ▼[f]p →
+ ∃∃r,h. 𝐞 = ⊗r & ▶[f]r＠⧣❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
+#f * [| #l #q ] #p #k
+[ <list_cons_comm #H0
+ elim (eq_inv_d_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+ /2 width=5 by ex4_2_intro/
+| >list_cons_comm #H0
+ elim (unwind2_path_inv_append_proper_dx … H0) -H0 // #r1 #r2 #Hr1 #_ #_ -r2
+ elim (eq_inv_d_dx_structure … Hr1)
+]
+qed-.
+
+lemma eq_inv_m_sn_unwind2_path (f) (q) (p):
+ (𝗺◗q) = ▼[f]p → ⊥.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_m_dx_structure … Hr1)
+qed-.
+
+lemma eq_inv_L_sn_unwind2_path (f) (q) (p):
+ (𝗟◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▶[f]r1]r2 & r1●𝗟◗r2 = p.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_L_dx_structure … Hr1) -Hr1 #s1 #s2 #H1 #_ #H3 destruct
+<list_append_assoc in H0; <list_append_assoc
+<unwind2_path_append_proper_dx //
+<unwind2_path_L_sn <H1 <list_append_empty_dx #H0
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H0 #_
+/2 width=5 by ex3_2_intro/
+qed-.
+
+lemma eq_inv_A_sn_unwind2_path (f) (q) (p):
+ (𝗔◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1]r2 & r1●𝗔◗r2 = p.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_A_dx_structure … Hr1) -Hr1 #s1 #s2 #H1 #_ #H3 destruct
+<list_append_assoc in H0; <list_append_assoc
+<unwind2_path_append_proper_dx //
+<unwind2_path_A_sn <H1 <list_append_empty_dx #H0
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H0 #_
+/2 width=5 by ex3_2_intro/
+qed-.
+
+lemma eq_inv_S_sn_unwind2_path (f) (q) (p):
+ (𝗦◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1]r2 & r1●𝗦◗r2 = p.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_S_dx_structure … Hr1) -Hr1 #s1 #s2 #H1 #_ #H3 destruct
+<list_append_assoc in H0; <list_append_assoc
+<unwind2_path_append_proper_dx //
+<unwind2_path_S_sn <H1 <list_append_empty_dx #H0
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H0 #_
+/2 width=5 by ex3_2_intro/
+qed-.
+
+(* Advanced eliminations with path ******************************************)
+
+lemma path_ind_unwind (Q:predicate …):
+ Q (𝐞) →
+ (∀k. Q (𝐞) → Q (𝗱k◗𝐞)) →
+ (∀k,l,p. Q (l◗p) → Q (𝗱k◗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
+@(list_ind_rcons … p) -p // #p * [ #k ]
+[ @(list_ind_rcons … p) -p ]
+/2 width=1 by/
+qed-.
include "delayed_updating/unwind/unwind2_path.ma".
include "delayed_updating/unwind/unwind2_rmap_eq.ma".
-include "delayed_updating/unwind/unwind_gen_eq.ma".
-include "ground/relocation/tr_compose_compose.ma".
-(* UNWIND FOR PATH **********************************************************)
+(* TAILED UNWIND FOR PATH ***************************************************)
-(* Constructions with stream_eq *********************************************)
+(* Constructions with tr_map_eq *********************************************)
lemma unwind2_path_eq_repl (p):
stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
-/3 width=1 by unwind2_rmap_eq_repl, unwind_gen_eq_repl/
+* // * [ #k ] #p #f1 #f2 #Hf //
+<unwind2_path_d_dx <unwind2_path_d_dx
+lapply (unwind2_rmap_eq_repl … Hf) -Hf
+[| #Hf <(tr_pap_eq_repl … Hf) -f2 ] //
qed-.
-(* Properties with tr_compose ***********************************************)
-
-lemma unwind2_path_after (p) (f1) (f2):
- ▼[f2]▼[f1]p = ▼[f2∘f1]p.
-#p @(path_ind_unwind … p) -p // [ #n #l #p | #p ] #IH #f1 #f2
-[ <unwind2_path_d_lcons <unwind2_path_d_lcons
- >(unwind2_path_eq_repl … (tr_compose_assoc …)) //
-| <(unwind2_path_L_sn … f1) <unwind2_path_L_sn <unwind2_path_L_sn
- >tr_compose_push_bi //
-]
-qed.
(* *)
(**************************************************************************)
-include "delayed_updating/unwind/unwind2_path_eq.ma".
-include "delayed_updating/substitution/lift_path_prelift.ma".
+include "delayed_updating/unwind/unwind2_path.ma".
+include "delayed_updating/unwind/unwind2_rmap_lift.ma".
+include "delayed_updating/substitution/lift_path_structure.ma".
-(* UNWIND FOR PATH **********************************************************)
+(* TAILED UNWIND FOR PATH ***************************************************)
(* Constructions with lift_path *********************************************)
-lemma lift_unwind2_path_after (p) (f1) (f2):
- ↑[f2]▼[f1]p = ▼[f2∘f1]p.
-#p @(path_ind_unwind … p) -p // [ #n | #p ] #IH #f1 #f2
-[ <unwind2_path_d_empty <unwind2_path_d_empty
- <lift_path_d_sn <lift_path_empty //
-| <unwind2_path_L_sn <unwind2_path_L_sn <lift_path_L_sn
- >tr_compose_push_bi //
-]
+lemma lift_unwind2_path_after (g) (f) (p):
+ ↑[g]▼[f]p = ▼[g∘f]p.
+#g #f * // * [ #k ] #p //
+<unwind2_path_d_dx <unwind2_path_d_dx <lift_path_d_dx
+<lift_path_structure >tr_compose_pap
+/4 width=1 by tr_pap_eq_repl, eq_f2, eq_f/
qed.
-lemma unwind2_path_after_lift (p) (f1) (f2):
- ▼[f2]↑[f1]p = ▼[f2∘f1]p.
-#p @(path_ind_unwind … p) -p // [ #n #l ] #p #IH #f1 #f2
-[ <lift_path_d_sn <unwind2_path_d_lcons
- <lift_path_lcons_prelift <unwind2_path_d_lcons >lift_path_lcons_prelift
- >IH -IH
- >(unwind2_path_eq_repl … (tr_compose_assoc …))
- >(unwind2_path_eq_repl … (tr_compose_assoc …))
- <unwind2_path_after <unwind2_path_after in ⊢ (???%);
- /3 width=1 by unwind2_path_eq_repl, eq_f/
-| <lift_path_m_sn <unwind2_path_m_sn <unwind2_path_m_sn //
-| <lift_path_L_sn <unwind2_path_L_sn <unwind2_path_L_sn
- >tr_compose_push_bi //
+lemma unwind2_lift_path_after (g) (f) (p):
+ ▼[g]↑[f]p = ▼[g∘f]p.
+#g #f * // * [ #k ] #p
+[ <unwind2_path_d_dx <unwind2_path_d_dx
+ <structure_lift_path >tr_compose_pap
+ /4 width=1 by tr_pap_eq_repl, eq_f2, eq_f/
+| <unwind2_path_m_dx <unwind2_path_m_dx //
+| <unwind2_path_L_dx <unwind2_path_L_dx //
+| <unwind2_path_A_dx <unwind2_path_A_dx //
+| <unwind2_path_S_dx <unwind2_path_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/unwind/unwind2_path.ma".
-include "delayed_updating/unwind/unwind_gen_structure.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-(* Constructions with list_rcons ********************************************)
-
-lemma unwind2_path_d_dx (f) (p) (n) :
- (⊗p)◖𝗱((▶[f](pᴿ))＠⧣❨n❩) = ▼[f](p◖𝗱n).
-#f #p #n <unwind2_path_unfold
-<unwind_gen_d_dx //
-qed.
-
-lemma unwind2_path_m_dx (f) (p):
- ⊗p = ▼[f](p◖𝗺).
-#f #p <unwind2_path_unfold //
-qed.
-
-lemma unwind2_path_L_dx (f) (p):
- (⊗p)◖𝗟 = ▼[f](p◖𝗟).
-#f #p <unwind2_path_unfold //
-qed.
-
-lemma unwind2_path_A_dx (f) (p):
- (⊗p)◖𝗔 = ▼[f](p◖𝗔).
-#f #p <unwind2_path_unfold //
-qed.
-
-lemma unwind2_path_S_dx (f) (p):
- (⊗p)◖𝗦 = ▼[f](p◖𝗦).
-#f #p <unwind2_path_unfold //
-qed.
-
-lemma unwind2_path_root (f) (p):
- ∃∃r. 𝐞 = ⊗r & ⊗p●r = ▼[f]p.
-#f #p
-elim (unwind_gen_root)
-/2 width=3 by ex2_intro/
-qed-.
-
-(* Constructions with proper condition for path *****************************)
-
-lemma unwind2_path_append_proper_dx (f) (p1) (p2): p2 ϵ 𝐏 →
- (⊗p1)●(▼[▶[f]p1ᴿ]p2) = ▼[f](p1●p2).
-#f #p1 #p2 #Hp2 <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_append_proper_dx // -Hp2 <reverse_append
-@(list_ind_rcons … p2) -p2 // #q2 #l2 #_
-<reverse_rcons <list_tl_lcons <list_tl_lcons //
-qed-.
-
-(* Constructions with inner condition for path ******************************)
-
-lemma unwind2_path_append_inner_sn (f) (p1) (p2): p1 ϵ 𝐈 →
- (⊗p1)●(▼[▶[f]p1ᴿ]p2) = ▼[f](p1●p2).
-#f #p1 #p2 #Hp1 <unwind2_path_unfold <unwind2_path_unfold
-<unwind_gen_append_inner_sn // -Hp1 <reverse_append
-@(list_ind_rcons … p2) -p2 // #q2 #l2 #_
-<reverse_rcons <list_tl_lcons <list_tl_lcons //
-qed-.
-
-(* Inversions with list_lcons ***********************************************)
-
-lemma unwind2_path_inv_S_sn (f) (p) (q):
- (𝗦◗q) = ▼[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1ᴿ]r2 & r1●𝗦◗r2 = p.
-#f #p #q #H0
-elim (unwind_gen_inv_S_sn … H0) -H0 #r1 #r2 #Hr1 #H1 #H2 destruct
-<reverse_append <reverse_lcons
-@(list_ind_rcons … r2) -r2 [ /2 width=5 by ex3_2_intro/ ] #r2 #l2 #_
-<reverse_rcons <list_append_lcons_sn <list_append_rcons_sn
-<list_tl_lcons <unwind2_rmap_append <unwind2_rmap_S_sn
-/2 width=5 by ex3_2_intro/
-qed-.
-
-(* Inversions with proper condition for path ********************************)
-
-lemma unwind2_path_inv_append_proper_dx (f) (p) (q1) (q2):
- q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
- ∃∃p1,p2. ⊗p1 = q1 & ▼[▶[f]p1ᴿ]p2 = q2 & p1●p2 = p.
-#f #p #q1 #q2 #Hq2 #H0
-elim (unwind_gen_inv_append_proper_dx … Hq2 H0) -Hq2 -H0
-#p1 #p2 #H1 #H2 #H3 destruct <reverse_append
-@(list_ind_rcons … p2) -p2 [ /2 width=5 by ex3_2_intro/ ] #q2 #l2 #_
-<reverse_rcons <list_tl_lcons <unwind2_rmap_append
-@ex3_2_intro [4: |*: // ] <unwind2_path_unfold // (**) (* auto fails *)
-qed-.
-
-(* Inversions with inner condition for path *********************************)
-
-lemma unwind2_path_inv_append_inner_sn (f) (p) (q1) (q2):
- q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
- ∃∃p1,p2. ⊗p1 = q1 & ▼[▶[f]p1ᴿ]p2 = q2 & p1●p2 = p.
-#f #p #q1 #q2 #Hq1 #H0
-elim (unwind_gen_inv_append_inner_sn … Hq1 H0) -Hq1 -H0
-#p1 #p2 #H1 #H2 #H3 destruct <reverse_append
-@(list_ind_rcons … p2) -p2 [ /2 width=5 by ex3_2_intro/ ] #q2 #l2 #_
-<reverse_rcons <list_tl_lcons <unwind2_rmap_append
-@ex3_2_intro [4: |*: // ] <unwind2_path_unfold // (**) (* auto fails *)
-qed-.
-
-(* Destructions with inner condition for path *******************************)
-
-lemma unwind2_path_des_inner (f) (p):
- ▼[f]p ϵ 𝐈 → p ϵ 𝐈.
-#f #p @(list_ind_rcons … p) -p //
-#p * [ #n ] #_ //
-<unwind2_path_d_dx #H0
-elim (pic_inv_d_dx … H0)
-qed-.
(**************************************************************************)
include "delayed_updating/unwind/unwind2_prototerm.ma".
-include "delayed_updating/unwind/unwind2_path_structure.ma".
+include "delayed_updating/unwind/unwind2_path_append.ma".
include "delayed_updating/syntax/preterm.ma".
include "delayed_updating/syntax/path_structure_inner.ma".
include "ground/lib/subset_equivalence.ma".
-(* UNWIND FOR PRETERM *******************************************************)
+(* TAILED UNWIND FOR PRETERM ************************************************)
(* Constructions with subset_equivalence ************************************)
lemma unwind2_term_grafted_sn (f) (t) (p): p ϵ 𝐈 →
- ▼[▶[f]pᴿ](t⋔p) ⊆ (▼[f]t)⋔(⊗p).
+ ▼[▶[f]p](t⋔p) ⊆ (▼[f]t)⋔(⊗p).
#f #t #p #Hp #q * #r #Hr #H0 destruct
@(ex2_intro … Hr) -Hr
<unwind2_path_append_inner_sn //
qed-.
lemma unwind2_term_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)⋔(⊗p) ⊆ ▼[▶[f]pᴿ](t⋔p).
+ (▼[f]t)⋔(⊗p) ⊆ ▼[▶[f]p](t⋔p).
#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
-elim (unwind2_path_inv_append_inner_sn … (sym_eq … H0)) -H0 //
+elim (unwind2_path_des_append_inner_sn … (sym_eq … H0)) -H0 //
#p0 #q0 #Hp0 #Hq0 #H0 destruct
-<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+>(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
/2 width=1 by in_comp_unwind2_path_term/
qed-.
lemma unwind2_term_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
- ▼[▶[f]pᴿ](t⋔p) ⇔ (▼[f]t)⋔(⊗p).
+ ▼[▶[f]p](t⋔p) ⇔ (▼[f]t)⋔(⊗p).
/3 width=1 by unwind2_term_grafted_sn, unwind2_term_grafted_dx, conj/ qed.
lemma unwind2_term_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)⋔((⊗p)◖𝗦) ⊆ ▼[▶[f]pᴿ](t⋔(p◖𝗦)).
+ (▼[f]t)⋔((⊗p)◖𝗦) ⊆ ▼[▶[f]p](t⋔(p◖𝗦)).
#f #t #p #Hp #Ht #q * #r #Hr
-<list_append_rcons_sn #H0
+>list_append_rcons_sn #H0
elim (unwind2_path_inv_append_proper_dx … (sym_eq … H0)) -H0 //
#p0 #q0 #Hp0 #Hq0 #H0 destruct
-<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
-elim (unwind2_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
+>(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (eq_inv_S_sn_unwind2_path … 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
qed-.
lemma unwind2_term_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
- ▼[▶[f]pᴿ](t⋔(p◖𝗦)) ⇔ (▼[f]t)⋔((⊗p)◖𝗦).
+ ▼[▶[f]p](t⋔(p◖𝗦)) ⇔ (▼[f]t)⋔((⊗p)◖𝗦).
#f #t #p #Hp #Ht
@conj
-[ >unwind2_rmap_S_sn >reverse_rcons >structure_S_dx
+[ >unwind2_rmap_S_dx >structure_S_dx
@unwind2_term_grafted_sn // (**) (* auto fails *)
| /2 width=1 by unwind2_term_grafted_S_dx/
]
(**************************************************************************)
include "delayed_updating/unwind/unwind2_prototerm_eq.ma".
-include "delayed_updating/unwind/unwind2_path_structure.ma".
+include "delayed_updating/unwind/unwind2_path_append.ma".
include "delayed_updating/substitution/fsubst.ma".
include "delayed_updating/syntax/preterm.ma".
include "delayed_updating/syntax/prototerm_proper.ma".
-(* UNWIND FOR PRETERM *******************************************************)
+(* TAILED UNWIND FOR PRETERM ************************************************)
(* Constructions with fsubst ************************************************)
lemma unwind2_term_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
- (▼[f]t)[⋔(⊗p)←▼[▶[f]pᴿ]u] ⊆ ▼[f](t[⋔p←u]).
+ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u] ⊆ ▼[f](t[⋔p←u]).
#f #t #u #p #Hu #ql * *
[ #rl * #r #Hr #H1 #H2 destruct
>unwind2_path_append_proper_dx
/4 width=5 by in_comp_unwind2_path_term, in_comp_tpc_trans, or_introl, ex2_intro/
| * #q #Hq #H1 #H0
@(ex2_intro … H1) @or_intror @conj // *
- [ <list_append_empty_dx #H2 destruct
+ [ <list_append_empty_sn #H2 destruct
elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
| #l #r #H2 destruct
- @H0 -H0 [| <unwind2_path_append_proper_dx /2 width=3 by ppc_lcons/ ]
+ @H0 -H0 [| <unwind2_path_append_proper_dx /2 width=3 by ppc_rcons/ ]
]
]
qed-.
lemma unwind2_term_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
- ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▶[f]pᴿ]u].
+ ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u].
#f #t #u #p #Hu #H1p #H2p #ql * #q * *
[ #r #Hu #H1 #H2 destruct
@or_introl @ex2_intro
/2 width=3 by ex2_intro/
| #Hq #H0 #H1 destruct
@or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
- [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
- <list_append_empty_dx @H2p -H2p
- /2 width=2 by unwind_gen_des_structure, prototerm_in_comp_root/
+ [ <list_append_empty_sn #Hr @(H0 … (𝐞)) -H0
+ <list_append_empty_sn @H2p -H2p
+ /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
| #l #r #Hr
elim (unwind2_path_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
qed-.
lemma unwind2_term_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)[⋔(⊗p)←▼[▶[f]pᴿ]u] ⇔ ▼[f](t[⋔p←u]).
+ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u] ⇔ ▼[f](t[⋔p←u]).
/4 width=3 by unwind2_term_fsubst_sn, conj, unwind2_term_fsubst_dx/ qed.
include "delayed_updating/syntax/prototerm.ma".
include "ground/lib/subset_ext.ma".
-(* UNWIND FOR PROTOTERM *****************************************************)
+(* TAILED UNWIND FOR PROTOTERM **********************************************)
interpretation
"unwind (prototerm)"
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind/unwind2_prototerm_eq.ma".
+include "delayed_updating/unwind/unwind2_path_after.ma".
+
+(* TAILED UNWIND FOR PROTOTERM **********************************************)
+
+(* Constructions with tr_after **********************************************)
+
+lemma unwind2_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.
include "delayed_updating/unwind/unwind2_path_eq.ma".
include "delayed_updating/unwind/unwind2_prototerm.ma".
-(* UNWIND FOR PROTOTERM *****************************************************)
+(* TAILED UNWIND FOR PROTOTERM **********************************************)
(* Constructions with subset_equivalence ************************************)
t1 ⇔ t2 → ▼[f]t1 ⇔ ▼[f]t2.
/2 width=1 by subset_equivalence_ext_f1_bi/
qed.
-
-lemma unwind2_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.
(**************************************************************************)
include "delayed_updating/unwind/unwind2_prototerm.ma".
-include "delayed_updating/unwind/unwind2_path_structure.ma".
+include "delayed_updating/unwind/unwind2_path_append.ma".
include "ground/lib/subset_overlap.ma".
-(* UNWIND FOR PROTOTERM *****************************************************)
+(* TAILED UNWIND FOR PROTOTERM **********************************************)
(* Destructions with inner condition for path *******************************)
include "delayed_updating/unwind/unwind2_path_lift.ma".
include "delayed_updating/unwind/unwind2_prototerm.ma".
-(* UNWIND FOR PROTOTERM *****************************************************)
+(* TAILED UNWIND FOR PROTOTERM **********************************************)
(* Constructions with lift_prototerm ****************************************)
@lift_unwind2_path_after
qed.
-lemma unwind2_term_after_lift (f1) (f2) (t):
+lemma unwind2_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 #p
-@unwind2_path_after_lift
+@unwind2_lift_path_after
qed.
(* *)
(**************************************************************************)
+include "delayed_updating/unwind/preunwind2_rmap.ma".
include "delayed_updating/syntax/path.ma".
-include "delayed_updating/notation/functions/black_righttriangle_2.ma".
-include "delayed_updating/notation/functions/black_righttriangle_1.ma".
-include "ground/relocation/tr_uni.ma".
-include "ground/relocation/tr_compose.ma".
-(* UNWIND MAP FOR PATH ******************************************************)
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
rec definition unwind2_rmap (f) (p) on p: tr_map ≝
match p with
[ list_empty ⇒ f
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒ (unwind2_rmap f q)∘𝐮❨n❩
- | label_m ⇒ unwind2_rmap f q
- | label_L ⇒ ⫯(unwind2_rmap f q)
- | label_A ⇒ unwind2_rmap f q
- | label_S ⇒ unwind2_rmap f q
- ]
+| list_lcons l q ⇒ ▶[unwind2_rmap f q]l
].
interpretation
- "tailed unwind map (reversed path)"
+ "tailed unwind (relocation map)"
'BlackRightTriangle f p = (unwind2_rmap f p).
-interpretation
- "unwind map (reversed path)"
- 'BlackRightTriangle p = (unwind2_rmap tr_id p).
-
(* Basic constructions ******************************************************)
lemma unwind2_rmap_empty (f):
f = ▶[f]𝐞.
// qed.
-lemma unwind2_rmap_d_sn (f) (p) (n:pnat):
- (▶[f]p∘𝐮❨n❩) = ▶[f](𝗱n◗p).
+lemma unwind2_rmap_rcons (f) (p) (l):
+ ▶[▶[f]p]l = ▶[f](p◖l).
// qed.
-lemma unwind2_rmap_m_sn (f) (p):
- ▶[f]p = ▶[f](𝗺◗p).
+lemma unwind2_rmap_d_dx (f) (p) (k:pnat):
+ ▶[f]p∘𝐮❨k❩ = ▶[f](p◖𝗱k).
// qed.
-lemma unwind2_rmap_L_sn (f) (p):
- (⫯▶[f]p) = ▶[f](𝗟◗p).
+lemma unwind2_rmap_m_dx (f) (p):
+ ▶[f]p = ▶[f](p◖𝗺).
// qed.
-lemma unwind2_rmap_A_sn (f) (p):
- ▶[f]p = ▶[f](𝗔◗p).
+lemma unwind2_rmap_L_dx (f) (p):
+ (⫯▶[f]p) = ▶[f](p◖𝗟).
// qed.
-lemma unwind2_rmap_S_sn (f) (p):
- ▶[f]p = ▶[f](𝗦◗p).
+lemma unwind2_rmap_A_dx (f) (p):
+ ▶[f]p = ▶[f](p◖𝗔).
+// qed.
+
+lemma unwind2_rmap_S_dx (f) (p):
+ ▶[f]p = ▶[f](p◖𝗦).
// qed.
-(* Constructions with list_append *******************************************)
-
-lemma unwind2_rmap_append (f) (p1) (p2):
- ▶[▶[f]p2]p1 = ▶[f](p1●p2).
-#f #p1 elim p1 -p1 //
-* [ #n ] #p1 #IH #p2 //
-[ <unwind2_rmap_m_sn <unwind2_rmap_m_sn //
-| <unwind2_rmap_L_sn <unwind2_rmap_L_sn //
-| <unwind2_rmap_A_sn <unwind2_rmap_A_sn //
-| <unwind2_rmap_S_sn <unwind2_rmap_S_sn //
-]
+(* Constructions with path_append *******************************************)
+
+lemma unwind2_rmap_append (f) (p) (q):
+ ▶[▶[f]p]q = ▶[f](p●q).
+#f #p #q elim q -q // #l #q #IH
+<unwind2_rmap_rcons <unwind2_rmap_rcons //
qed.
-(* Constructions with list_rcons ********************************************)
+(* Constructions with path_lcons ********************************************)
-lemma unwind2_rmap_d_dx (f) (p) (n:pnat):
- ▶[f∘𝐮❨n❩]p = ▶[f](p◖𝗱n).
+lemma unwind2_rmap_lcons (f) (p) (l):
+ ▶[▶[f]l]p = ▶[f](l◗p).
// qed.
-lemma unwind2_rmap_m_dx (f) (p):
- ▶[f]p = ▶[f](p◖𝗺).
+lemma unwind2_rmap_d_sn (f) (p) (k:pnat):
+ ▶[f∘𝐮❨k❩]p = ▶[f](𝗱k◗p).
// qed.
-lemma unwind2_rmap_L_dx (f) (p):
- ▶[⫯f]p = ▶[f](p◖𝗟).
+lemma unwind2_rmap_m_sn (f) (p):
+ ▶[f]p = ▶[f](𝗺◗p).
// qed.
-lemma unwind2_rmap_A_dx (f) (p):
- ▶[f]p = ▶[f](p◖𝗔).
+lemma unwind2_rmap_L_sn (f) (p):
+ ▶[⫯f]p = ▶[f](𝗟◗p).
// qed.
-lemma unwind2_rmap_S_dx (f) (p):
- ▶[f]p = ▶[f](p◖𝗦).
+lemma unwind2_rmap_A_sn (f) (p):
+ ▶[f]p = ▶[f](𝗔◗p).
+// qed.
+
+lemma unwind2_rmap_S_sn (f) (p):
+ ▶[f]p = ▶[f](𝗦◗p).
// qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind/unwind2_rmap.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "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".
+
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
+
+(* Constructions with tr_after **********************************************)
+
+lemma unwind2_rmap_after (g) (f) (p:path):
+ ▶[g]⊗p∘▶[f]p ≗ ▶[g∘f]p.
+#g #f #p elim p -p // * [ #k ] #p #IH //
+[ <structure_d_dx <unwind2_rmap_d_dx <unwind2_rmap_d_dx
+ @(stream_eq_canc_sn … (tr_compose_assoc …))
+ /2 width=1 by tr_compose_eq_repl/
+| <structure_L_dx <unwind2_rmap_L_dx <unwind2_rmap_L_dx <unwind2_rmap_L_dx
+ /2 width=1 by tr_push_eq_repl/
+]
+qed.
(**************************************************************************)
include "delayed_updating/unwind/unwind2_rmap.ma".
+include "delayed_updating/unwind/preunwind2_rmap_eq.ma".
include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_compose_eq.ma".
-include "ground/relocation/tr_pn_eq.ma".
include "ground/arith/nat_rplus_pplus.ma".
-(* UNWIND MAP FOR PATH ******************************************************)
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
-(* Constructions with stream_eq *********************************************)
+(* Constructions with tr_map_eq *********************************************)
lemma unwind2_rmap_eq_repl (p):
stream_eq_repl … (λf1,f2. ▶[f1]p ≗ ▶[f2]p).
#p elim p -p //
-* [ #n ] #p #IH #f1 #f2 #Hf
-[ /3 width=1 by tr_compose_eq_repl/
-| /2 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /2 width=1 by/
-| /2 width=1 by/
-]
+#l #p #IH #f1 #f2 #Hf
+/3 width=1 by preunwind2_rmap_eq_repl/
qed-.
-lemma tls_unwind2_rmap_d_sn (f) (p) (m) (n):
- ⇂*[m+n]▶[f]p ≗ ⇂*[m]▶[f](𝗱n◗p).
-#f #p #m #n
-<unwind2_rmap_d_sn >nrplus_inj_dx
+lemma tls_unwind2_rmap_d_dx (f) (p) (n) (k):
+ ⇂*[n+k]▶[f]p ≗ ⇂*[n]▶[f](p◖𝗱k).
+#f #p #n #k
+<unwind2_rmap_d_dx >nrplus_inj_dx
/2 width=1 by tr_tls_compose_uni_dx/
qed.
include "ground/lib/stream_eq_eq.ma".
include "ground/arith/nat_le_plus.ma".
-(* UNWIND MAP FOR PATH ******************************************************)
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
(* Constructions with path_head *********************************************)
-lemma unwind2_rmap_head_xap_le_closed (f) (p) (q) (n) (k):
- p = ↳[n]p → k ≤ n →
- ▶[f](p●q)＠❨k❩ = ▶[f]↳[n](p●q)＠❨k❩.
-#f #p elim p -p
-[ #q #n #k #Hq
- <(eq_inv_path_empty_head … Hq) -n #Hk
- <(nle_inv_zero_dx … Hk) -k //
-| #l #p #IH #q #n @(nat_ind_succ … n) -n
- [ #k #_ #Hk <(nle_inv_zero_dx … Hk) -k -IH //
- | #n #_ #k cases l [ #m ]
- [ <path_head_d_sn #Hq #Hk
+lemma unwind2_rmap_head_xap_le_closed (f) (p) (q) (n) (m):
+ q = ↳[n]q → m ≤ n →
+ ▶[f](p●q)＠❨m❩ = ▶[f]↳[n](p●q)＠❨m❩.
+#f #p #q elim q -q
+[ #n #m #Hq
+ <(eq_inv_path_empty_head … Hq) -n #Hm
+ <(nle_inv_zero_dx … Hm) -m //
+| #l #q #IH #n @(nat_ind_succ … n) -n
+ [ #m #_ #Hm <(nle_inv_zero_dx … Hm) -m -IH //
+ | #n #_ #m cases l [ #k ]
+ [ <path_head_d_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_d_sn <unwind2_rmap_d_sn
+ <unwind2_rmap_d_dx <unwind2_rmap_d_dx
<tr_compose_xap <tr_compose_xap
@(IH … Hq) -IH -Hq (**) (* auto too slow *)
@nle_trans [| @tr_uni_xap ]
/2 width=1 by nle_plus_bi_dx/
- | <path_head_m_sn #Hq #Hk
+ | <path_head_m_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_m_sn <unwind2_rmap_m_sn
+ <unwind2_rmap_m_dx <unwind2_rmap_m_dx
/2 width=2 by/
- | <path_head_L_sn #Hq
- @(nat_ind_succ … k) -k // #k #_ #Hk
- lapply (nle_inv_succ_bi … Hk) -Hk #Hk
+ | <path_head_L_dx #Hq
+ @(nat_ind_succ … m) -m // #m #_ #Hm
+ lapply (nle_inv_succ_bi … Hm) -Hm #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_L_sn <unwind2_rmap_L_sn
+ <unwind2_rmap_L_dx <unwind2_rmap_L_dx
<tr_xap_push <tr_xap_push
/3 width=2 by eq_f/
- | <path_head_A_sn #Hq #Hk
+ | <path_head_A_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_A_sn <unwind2_rmap_A_sn
+ <unwind2_rmap_A_dx <unwind2_rmap_A_dx
/2 width=2 by/
- | <path_head_S_sn #Hq #Hk
+ | <path_head_S_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_S_sn <unwind2_rmap_S_sn
+ <unwind2_rmap_S_dx <unwind2_rmap_S_dx
/2 width=2 by/
]
]
qed-.
lemma unwind2_rmap_head_xap_closed (f) (p) (q) (n):
- p = ↳[n]p →
+ q = ↳[n]q →
▶[f](p●q)＠❨n❩ = ▶[f]↳[n](p●q)＠❨n❩.
/2 width=2 by unwind2_rmap_head_xap_le_closed/
qed-.
[ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
<tr_xap_pushs_le //
| #l #p #IH #n @(nat_ind_succ … n) -n //
- #n #_ cases l [ #m ]
- [ <unwind2_rmap_d_sn <path_head_d_sn <height_d_sn
- <nplus_assoc >IH -IH <tr_compose_xap <tr_uni_xap_succ //
- | <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
- | <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
- <tr_xap_push <npred_succ //
- | <unwind2_rmap_A_sn <path_head_A_sn <height_A_sn //
- | <unwind2_rmap_S_sn <path_head_S_sn <height_S_sn //
+ #n #_ cases l [ #k ]
+ [ <unwind2_rmap_d_dx <path_head_d_dx <height_d_dx
+ <nplus_comm in ⊢ (??(??%)?); <nplus_assoc
+ >IH -IH <tr_compose_xap <tr_uni_xap_succ //
+ | <unwind2_rmap_m_dx <path_head_m_dx <height_m_dx //
+ | <unwind2_rmap_L_dx <path_head_L_dx <height_L_dx
+ <tr_xap_push <npred_succ <nplus_succ_sn //
+ | <unwind2_rmap_A_dx <path_head_A_dx <height_A_dx //
+ | <unwind2_rmap_S_dx <path_head_S_dx <height_S_dx //
]
]
qed.
lemma unwind2_rmap_append_pap_closed (f) (p) (q) (n:pnat):
- p = ↳[n]p →
- ♭p = ninj (▶[f](p●q)＠⧣❨n❩).
+ q = ↳[n]q →
+ ♭q = ninj (▶[f](p●q)＠⧣❨n❩).
#f #p #q #n #Hn
->tr_xap_ninj >(path_head_refl_append q … Hn) in ⊢ (??%?);
+>tr_xap_ninj >(path_head_refl_append p … Hn) in ⊢ (??%?);
>(unwind2_rmap_head_xap_closed … Hn) -Hn
<path_head_depth //
qed.
-lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (k):
- p = ↳[n]p →
- ⇂*[k]▶[f]q ≗ ⇂*[n+k]▶[f](p●q).
-#f #p elim p -p
-[ #q #n #k #Hq
+lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (m):
+ q = ↳[n]q →
+ ⇂*[m]▶[f]p ≗ ⇂*[n+m]▶[f](p●q).
+#f #p #q elim q -q
+[ #n #m #Hq
<(eq_inv_path_empty_head … Hq) -n //
-| #l #p #IH #q #n @(nat_ind_succ … n) -n //
- #n #_ #k cases l [ #m ]
- [ <path_head_d_sn #Hq
+| #l #q #IH #n @(nat_ind_succ … n) -n //
+ #n #_ #m cases l [ #k ]
+ [ <path_head_d_dx #Hq
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
- @(stream_eq_trans … (tls_unwind2_rmap_d_sn …))
+ @(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
>nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
/2 width=1 by/
- | <path_head_m_sn #Hq
+ | <path_head_m_dx #Hq
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
<unwind2_rmap_m_sn /2 width=1 by/
- | <path_head_L_sn #Hq
+ | <path_head_L_dx #Hq
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_L_sn <nplus_succ_sn /2 width=1 by/
- | <path_head_A_sn #Hq
+ <unwind2_rmap_L_dx <nplus_succ_sn /2 width=1 by/
+ | <path_head_A_dx #Hq
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_A_sn /2 width=2 by/
- | <path_head_S_sn #Hq
+ <unwind2_rmap_A_dx /2 width=2 by/
+ | <path_head_S_dx #Hq
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_S_sn /2 width=2 by/
+ <unwind2_rmap_S_dx /2 width=2 by/
]
]
qed-.
lemma tls_unwind2_rmap_closed (f) (p) (q) (n):
- p = ↳[n]p →
- ▶[f]q ≗ ⇂*[n]▶[f](p●q).
+ q = ↳[n]q →
+ ▶[f]p ≗ ⇂*[n]▶[f](p●q).
/2 width=1 by tls_unwind2_rmap_plus_closed/
qed.
include "delayed_updating/syntax/path_labels.ma".
include "ground/relocation/tr_pushs.ma".
-(* UNWIND MAP FOR PATH ******************************************************)
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
(* Constructions with labels ************************************************)
(⫯*[n]f) = ▶[f](𝗟∗∗n).
#f #n @(nat_ind_succ … n) -n //
#n #IH
-<labels_succ <unwind2_rmap_L_sn //
+<labels_succ <unwind2_rmap_L_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/unwind/unwind2_rmap.ma".
+include "delayed_updating/unwind/preunwind2_rmap_lift.ma".
+include "delayed_updating/unwind/preunwind2_rmap_eq.ma".
+include "delayed_updating/substitution/lift_path.ma".
+include "delayed_updating/syntax/path_structure.ma".
+
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
+
+(* Constructions with lift_path *********************************************)
+
+lemma lift_unwind2_rmap_after (g) (f) (p):
+ ↑[⊗p]g∘▶[f]p ≗ ▶[g∘f]p.
+#g #f #p elim p -p //
+* [ #k ] #p #IH //
+[ <unwind2_rmap_d_dx <unwind2_rmap_d_dx
+ @(stream_eq_canc_sn … (tr_compose_assoc …))
+ /2 width=1 by tr_compose_eq_repl/
+| <unwind2_rmap_L_dx <unwind2_rmap_L_dx <lift_rmap_L_dx
+ /2 width=1 by tr_push_eq_repl/
+]
+qed.
+
+lemma unwind2_lift_rmap_after (g) (f) (p:path):
+ ▶[g]↑[f]p∘↑[p]f ≗ ▶[g∘f]p.
+#g #f #p elim p -p // #l #p #IH
+<lift_path_rcons <lift_rmap_rcons <unwind2_rmap_rcons <unwind2_rmap_rcons
+@(stream_eq_trans … (preunwind2_lift_rmap_after …))
+/2 width=1 by preunwind2_rmap_eq_repl/
+qed.
include "delayed_updating/syntax/path_depth.ma".
include "ground/relocation/tr_pushs.ma".
-(* UNWIND MAP FOR PATH ******************************************************)
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
(* Constructions with structure and depth ***********************************)
-lemma unwind2_rmap_structure (p) (f):
+lemma unwind2_rmap_structure (f) (p):
(⫯*[♭p]f) = ▶[f]⊗p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-[ <unwind2_rmap_A_sn //
-| <unwind2_rmap_S_sn //
+#f #p elim p -p //
+* [ #k ] #p #IH //
+[ <unwind2_rmap_L_dx //
+| <unwind2_rmap_A_dx //
+| <unwind2_rmap_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/syntax/path.ma".
-include "delayed_updating/notation/functions/black_diamond_2.ma".
-include "ground/relocation/tr_pap.ma".
-
-(* GENERIC UNWIND FOR PATH **************************************************)
-
-rec definition unwind_gen (f) (p) on p ≝
-match p with
-[ list_empty ⇒ 𝐞
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒
- match q with
- [ list_empty ⇒ 𝗱(f＠⧣❨n❩)◗(unwind_gen f q)
- | list_lcons _ _ ⇒ unwind_gen f q
- ]
- | label_m ⇒ unwind_gen f q
- | label_L ⇒ l◗(unwind_gen f q)
- | label_A ⇒ l◗(unwind_gen f q)
- | label_S ⇒ l◗(unwind_gen f q)
- ]
-].
-
-interpretation
- "generic unwind (path)"
- 'BlackDiamond f p = (unwind_gen f p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind_gen_empty (f):
- (𝐞) = ◆[f]𝐞.
-// qed.
-
-lemma unwind_gen_d_empty (f) (n):
- 𝗱(f＠⧣❨n❩)◗𝐞 = ◆[f](𝗱n◗𝐞).
-// qed.
-
-lemma unwind_gen_d_lcons (f) (p) (l) (n):
- ◆[f](l◗p) = ◆[f](𝗱n◗l◗p).
-// qed.
-
-lemma unwind_gen_m_sn (f) (p):
- ◆[f]p = ◆[f](𝗺◗p).
-// qed.
-
-lemma unwind_gen_L_sn (f) (p):
- (𝗟◗◆[f]p) = ◆[f](𝗟◗p).
-// qed.
-
-lemma unwind_gen_A_sn (f) (p):
- (𝗔◗◆[f]p) = ◆[f](𝗔◗p).
-// qed.
-
-lemma unwind_gen_S_sn (f) (p):
- (𝗦◗◆[f]p) = ◆[f](𝗦◗p).
-// qed.
-
-(* Advanced eliminations with path ******************************************)
-
-lemma path_ind_unwind (Q:predicate …):
- Q (𝐞) →
- (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
- (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
- (∀p. Q p → Q (𝗺◗p)) →
- (∀p. Q p → Q (𝗟◗p)) →
- (∀p. Q p → Q (𝗔◗p)) →
- (∀p. Q p → Q (𝗦◗p)) →
- ∀p. Q p.
-#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
-elim p -p [| * [ #n * ] ]
-/2 width=1 by/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "delayed_updating/unwind/unwind_gen.ma".
-include "ground/relocation/tr_compose_pap.ma".
-
-(* GENERIC UNWIND FOR PATH **************************************************)
-
-(* Properties with tr_compose ***********************************************)
-
-lemma unwind_gen_after (f2) (f1) (p):
- ◆[f2]◆[f1]p = ◆[f2∘f1]p.
-#f2 #f1 #p @(path_ind_unwind … p) -p //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "delayed_updating/unwind/unwind_gen.ma".
-include "ground/relocation/tr_pap_eq.ma".
-
-(* GENERIC UNWIND FOR PATH **************************************************)
-
-(* Constructions with stream_eq *********************************************)
-
-lemma unwind_gen_eq_repl (p):
- stream_eq_repl … (λf1,f2. ◆[f1]p = ◆[f2]p).
-#p @(path_ind_unwind … p) -p // [ #n |*: #p ] #IH #f1 #f2 #Hf
-[ <unwind_gen_d_empty <unwind_gen_d_empty
- <(tr_pap_eq_repl … Hf) -f2 -IH //
-| <unwind_gen_L_sn <unwind_gen_L_sn
- <(IH … Hf) -f2 -IH //
-| <unwind_gen_A_sn <unwind_gen_A_sn
- <(IH … Hf) -f2 -IH //
-| <unwind_gen_S_sn <unwind_gen_S_sn
- <(IH … Hf) -f2 -IH //
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "delayed_updating/unwind/unwind_gen.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_inner.ma".
-include "delayed_updating/syntax/path_proper.ma".
-include "ground/xoa/ex_4_2.ma".
-include "ground/xoa/ex_3_2.ma".
-
-(* GENERIC UNWIND FOR PATH **************************************************)
-
-(* Basic constructions with structure ***************************************)
-
-lemma structure_unwind_gen (p) (f):
- ⊗p = ⊗◆[f]p.
-#p @(path_ind_unwind … p) -p //
-qed.
-
-lemma unwind_gen_structure (p) (f):
- ⊗p = ◆[f]⊗p.
-#p @(path_ind_unwind … p) -p //
-qed.
-
-(* Destructions with structure **********************************************)
-
-lemma unwind_gen_des_structure (q) (p) (f):
- ⊗q = ◆[f]p → ⊗q = ⊗p.
-// qed-.
-
-(* Constructions with proper condition for path *****************************)
-
-lemma unwind_gen_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
- (⊗p1)●(◆[f]p2) = ◆[f](p1●p2).
-#p2 #p1 @(path_ind_unwind … p1) -p1 //
-[ #n | #n #l #p1 |*: #p1 ] #IH #f #Hp2
-[ elim (ppc_inv_lcons … Hp2) -Hp2 #l #q #H0 destruct //
-| <unwind_gen_d_lcons <IH //
-| <unwind_gen_m_sn <IH //
-| <unwind_gen_L_sn <IH //
-| <unwind_gen_A_sn <IH //
-| <unwind_gen_S_sn <IH //
-]
-qed-.
-
-(* Constructions with inner condition for path ******************************)
-
-lemma unwind_gen_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
- (⊗p1)●(◆[f]p2) = ◆[f](p1●p2).
-#p1 @(list_ind_rcons … p1) -p1 //
-#p1 * [ #n ] #_ #p2 #f #Hp1
-[ elim (pic_inv_d_dx … Hp1)
-| <list_append_rcons_sn <unwind_gen_append_proper_dx //
-| <list_append_rcons_sn <unwind_gen_append_proper_dx //
- <structure_L_dx <list_append_rcons_sn //
-| <list_append_rcons_sn <unwind_gen_append_proper_dx //
- <structure_A_dx <list_append_rcons_sn //
-| <list_append_rcons_sn <unwind_gen_append_proper_dx //
- <structure_S_dx <list_append_rcons_sn //
-]
-qed-.
-
-(* Advanced constructions with list_rcons ***********************************)
-
-lemma unwind_gen_d_dx (f) (p) (n):
- (⊗p)◖𝗱(f＠⧣❨n❩) = ◆[f](p◖𝗱n).
-#f #p #n <unwind_gen_append_proper_dx //
-qed.
-
-lemma unwind_gen_m_dx (f) (p):
- ⊗p = ◆[f](p◖𝗺).
-#f #p <unwind_gen_append_proper_dx //
-qed.
-
-lemma unwind_gen_L_dx (f) (p):
- (⊗p)◖𝗟 = ◆[f](p◖𝗟).
-#f #p <unwind_gen_append_proper_dx //
-qed.
-
-lemma unwind_gen_A_dx (f) (p):
- (⊗p)◖𝗔 = ◆[f](p◖𝗔).
-#f #p <unwind_gen_append_proper_dx //
-qed.
-
-lemma unwind_gen_S_dx (f) (p):
- (⊗p)◖𝗦 = ◆[f](p◖𝗦).
-#f #p <unwind_gen_append_proper_dx //
-qed.
-
-lemma unwind_gen_root (f) (p):
- ∃∃r. 𝐞 = ⊗r & ⊗p●r = ◆[f]p.
-#f #p @(list_ind_rcons … p) -p
-[ /2 width=3 by ex2_intro/
-| #p * [ #n ] /2 width=3 by ex2_intro/
-]
-qed-.
-
-(* Advanced inversions ******************************************************)
-
-lemma unwind_gen_inv_d_sn (k) (q) (p) (f):
- (𝗱k◗q) = ◆[f]p →
- ∃∃r,h. 𝐞 = ⊗r & f＠⧣❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
-#k #q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_gen_empty #H destruct
-| <unwind_gen_d_empty #H destruct -IH
- /2 width=5 by ex4_2_intro/
-| <unwind_gen_d_lcons #H
- elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
- /2 width=5 by ex4_2_intro/
-| <unwind_gen_m_sn #H
- elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
- /2 width=5 by ex4_2_intro/
-| <unwind_gen_L_sn #H destruct
-| <unwind_gen_A_sn #H destruct
-| <unwind_gen_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_gen_inv_m_sn (q) (p) (f):
- (𝗺◗q) = ◆[f]p → ⊥.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_gen_empty #H destruct
-| <unwind_gen_d_empty #H destruct
-| <unwind_gen_d_lcons #H /2 width=2 by/
-| <unwind_gen_m_sn #H /2 width=2 by/
-| <unwind_gen_L_sn #H destruct
-| <unwind_gen_A_sn #H destruct
-| <unwind_gen_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_gen_inv_L_sn (q) (p) (f):
- (𝗟◗q) = ◆[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ◆[f]r2 & r1●𝗟◗r2 = p.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_gen_empty #H destruct
-| <unwind_gen_d_empty #H destruct
-| <unwind_gen_d_lcons #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_m_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_L_sn #H destruct -IH
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_A_sn #H destruct
-| <unwind_gen_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_gen_inv_A_sn (q) (p) (f):
- (𝗔◗q) = ◆[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ◆[f]r2 & r1●𝗔◗r2 = p.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_gen_empty #H destruct
-| <unwind_gen_d_empty #H destruct
-| <unwind_gen_d_lcons #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_m_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_L_sn #H destruct
-| <unwind_gen_A_sn #H destruct -IH
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_S_sn #H destruct
-]
-qed-.
-
-lemma unwind_gen_inv_S_sn (q) (p) (f):
- (𝗦◗q) = ◆[f]p →
- ∃∃r1,r2. 𝐞 = ⊗r1 & q = ◆[f]r2 & r1●𝗦◗r2 = p.
-#q #p @(path_ind_unwind … p) -p
-[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
-[ <unwind_gen_empty #H destruct
-| <unwind_gen_d_empty #H destruct
-| <unwind_gen_d_lcons #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/
-| <unwind_gen_m_sn #H
- elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
- /2 width=5 by ex3_2_intro/| <unwind_gen_L_sn #H destruct
-| <unwind_gen_A_sn #H destruct
-| <unwind_gen_S_sn #H destruct -IH
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Inversions with proper condition for path ********************************)
-
-lemma unwind_gen_inv_append_proper_dx (q2) (q1) (p) (f):
- q2 ϵ 𝐏 → q1●q2 = ◆[f]p →
- ∃∃p1,p2. ⊗p1 = q1 & ◆[f]p2 = q2 & p1●p2 = p.
-#q2 #q1 elim q1 -q1
-[ #p #f #Hq2 <list_append_empty_sn #H destruct
- /2 width=5 by ex3_2_intro/
-| * [ #n1 ] #q1 #IH #p #f #Hq2 <list_append_lcons_sn #H
- [ elim (unwind_gen_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
- elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
- elim Hq2 -Hq2 //
- | elim (unwind_gen_inv_m_sn … H)
- | elim (unwind_gen_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
- elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
- @(ex3_2_intro … (r1●𝗟◗p1)) //
- <structure_append <Hr1 -Hr1 //
- | elim (unwind_gen_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
- elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
- @(ex3_2_intro … (r1●𝗔◗p1)) //
- <structure_append <Hr1 -Hr1 //
- | elim (unwind_gen_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
- elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
- @(ex3_2_intro … (r1●𝗦◗p1)) //
- <structure_append <Hr1 -Hr1 //
- ]
-]
-qed-.
-
-(* Inversions with inner condition for path *********************************)
-
-lemma unwind_gen_inv_append_inner_sn (q1) (q2) (p) (f):
- q1 ϵ 𝐈 → q1●q2 = ◆[f]p →
- ∃∃p1,p2. ⊗p1 = q1 & ◆[f]p2 = q2 & p1●p2 = p.
-#q1 @(list_ind_rcons … q1) -q1
-[ #q2 #p #f #Hq1 <list_append_empty_sn #H destruct
- /2 width=5 by ex3_2_intro/
-| #q1 * [ #n1 ] #_ #q2 #p #f #Hq2
- [ elim (pic_inv_d_dx … Hq2)
- | <list_append_rcons_sn #H0
- elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_gen_inv_m_sn … (sym_eq … H2))
- | <list_append_rcons_sn #H0
- elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_gen_inv_L_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
- @(ex3_2_intro … (p1●r2◖𝗟)) [1,3: // ]
- [ <structure_append <structure_L_dx <Hr2 -Hr2 //
- | <list_append_assoc <list_append_rcons_sn //
- ]
- | <list_append_rcons_sn #H0
- elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_gen_inv_A_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
- @(ex3_2_intro … (p1●r2◖𝗔)) [1,3: // ]
- [ <structure_append <structure_A_dx <Hr2 -Hr2 //
- | <list_append_assoc <list_append_rcons_sn //
- ]
- | <list_append_rcons_sn #H0
- elim (unwind_gen_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
- elim (unwind_gen_inv_S_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
- @(ex3_2_intro … (p1●r2◖𝗦)) [1,3: // ]
- [ <structure_append <structure_S_dx <Hr2 -Hr2 //
- | <list_append_assoc <list_append_rcons_sn //
- ]
- ]
-]
-qed-.
projects / helm.git / commitdiff