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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground/relocation/tr_compose_pap.ma".
16 include "ground/relocation/tr_uni_pap.ma".
17 include "delayed_updating/syntax/path.ma".
18 include "delayed_updating/notation/functions/black_downtriangle_4.ma".
19 include "delayed_updating/notation/functions/black_downtriangle_2.ma".
21 (* UNWIND FOR PATH **********************************************************)
23 definition unwind_continuation (A:Type[0]) ≝
26 rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f) (p) on p ≝
28 [ list_empty ⇒ k f (𝐞)
33 [ list_empty ⇒ unwind_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (f∘𝐮❨n❩) q
34 | list_lcons _ _ ⇒ unwind_gen (A) k (f∘𝐮❨n❩) q
36 | label_m ⇒ unwind_gen (A) k f q
37 | label_L ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
38 | label_A ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
39 | label_S ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
45 'BlackDownTriangle A k f p = (unwind_gen A k f p).
47 definition proj_path: unwind_continuation … ≝
50 definition proj_rmap: unwind_continuation … ≝
55 'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
58 "unwind (relocation map)"
59 'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
61 (* Basic constructions ******************************************************)
63 lemma unwind_empty (A) (k) (f):
64 k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
67 lemma unwind_d_empty_sn (A) (k) (n) (f):
68 ▼❨(λg,p. k g (𝗱(f@❨n❩)◗p)), f∘𝐮❨ninj n❩, 𝐞❩ = ▼{A}❨k, f,
72 lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
73 ▼❨k, f∘𝐮❨ninj n❩, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
76 lemma unwind_m_sn (A) (k) (p) (f):
77 ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
80 lemma unwind_L_sn (A) (k) (p) (f):
81 ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
84 lemma unwind_A_sn (A) (k) (p) (f):
85 ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
88 lemma unwind_S_sn (A) (k) (p) (f):
89 ▼❨(λg,p. k g (𝗦◗p)), f, p❩ = ▼{A}❨k, f, 𝗦◗p❩.
92 (* Basic constructions with proj_path ***************************************)
94 lemma unwind_path_empty (f):
98 lemma unwind_path_d_empty_sn (f) (n):
99 𝗱(f@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
102 lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
103 ▼[f∘𝐮❨ninj n❩](l◗p) = ▼[f](𝗱n◗l◗p).
106 lemma unwind_path_m_sn (f) (p):
110 (* Basic constructions with proj_rmap ***************************************)
112 lemma unwind_rmap_empty (f):
116 lemma unwind_rmap_d_sn (f) (p) (n):
117 ▼[p](f∘𝐮❨ninj n❩) = ▼[𝗱n◗p]f.
120 lemma unwind_rmap_m_sn (f) (p):
124 lemma unwind_rmap_L_sn (f) (p):
128 lemma unwind_rmap_A_sn (f) (p):
132 lemma unwind_rmap_S_sn (f) (p):
136 (* Advanced constructions with proj_rmap and path_append ********************)
138 lemma unwind_rmap_append (p2) (p1) (f):
139 ▼[p2]▼[p1]f = ▼[p1●p2]f.
140 #p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
141 [ <unwind_rmap_m_sn <unwind_rmap_m_sn //
142 | <unwind_rmap_A_sn <unwind_rmap_A_sn //
143 | <unwind_rmap_S_sn <unwind_rmap_S_sn //
147 (* Advanced constructions with proj_rmap and path_rcons *********************)
149 lemma unwind_rmap_d_dx (f) (p) (n):
150 (▼[p]f)∘𝐮❨ninj n❩ = ▼[p◖𝗱n]f.
153 lemma unwind_rmap_m_dx (f) (p):
157 lemma unwind_rmap_L_dx (f) (p):
161 lemma unwind_rmap_A_dx (f) (p):
165 lemma unwind_rmap_S_dx (f) (p):
169 lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
170 ▼[p]f@❨m+n❩ = ▼[p◖𝗱n]f@❨m❩.
172 <unwind_rmap_d_dx <tr_compose_pap <tr_uni_pap //
175 (* Advanced eliminations with path ******************************************)
177 lemma path_ind_unwind (Q:predicate …):
179 (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
180 (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
181 (∀p. Q p → Q (𝗺◗p)) →
182 (∀p. Q p → Q (𝗟◗p)) →
183 (∀p. Q p → Q (𝗔◗p)) →
184 (∀p. Q p → Q (𝗦◗p)) →
186 #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
187 elim p -p [| * [ #n * ] ]