1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "delayed_updating/substitution/lift.ma".
16 include "ground/relocation/tr_pap_pap.ma".
17 include "ground/relocation/tr_pap_eq.ma".
18 include "ground/relocation/tr_pn_eq.ma".
19 include "ground/lib/stream_tls_eq.ma".
21 (* LIFT FOR PATH ************************************************************)
23 definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
24 λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
27 "extensional equivalence (lift continuation)"
28 'RingEq A k1 k2 = (lift_exteq A k1 k2).
30 (* Constructions with lift_exteq ********************************************)
32 lemma lift_eq_repl (A) (p) (k1) (k2):
33 k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
34 #A #p elim p -p [| * [ #n ] #q #IH ]
35 #k1 #k2 #Hk #f1 #f2 #Hf
36 [ <lift_empty <lift_empty /2 width=1 by/
37 | <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
38 /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
40 | /3 width=1 by tr_push_eq_repl/
46 (* Advanced constructions ***************************************************)
48 lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
49 ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
51 @lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
54 lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
55 ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
56 #A #k #f #p1 #p #l #Hk
57 @lift_eq_repl // #g1 #g2 #p2 #Hg
58 <list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
61 (* Advanced constructions with proj_path ************************************)
63 lemma proj_path_proper:
64 proj_path ≗ proj_path.
67 lemma lift_path_eq_repl (p):
68 stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
69 /2 width=1 by lift_eq_repl/ qed.
71 lemma lift_path_append_sn (p) (f) (q):
72 q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
73 #p elim p -p // * [ #n ] #p #IH #f #q
74 [ <lift_d_sn <lift_d_sn
75 | <lift_m_sn <lift_m_sn
76 | <lift_L_sn <lift_L_sn
77 | <lift_A_sn <lift_A_sn
78 | <lift_S_sn <lift_S_sn
80 >lift_lcons_alt // >lift_append_rcons_sn //
81 <IH <IH -IH <list_append_rcons_sn //
84 lemma lift_path_lcons (f) (p) (l):
85 l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
87 >lift_lcons_alt <lift_path_append_sn //
90 lemma lift_path_d_sn (f) (p) (n):
91 (𝗱(f@⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
94 lemma lift_path_m_sn (f) (p):
95 (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
98 lemma lift_path_L_sn (f) (p):
99 (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
102 lemma lift_path_A_sn (f) (p):
103 (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
106 lemma lift_path_S_sn (f) (p):
107 (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
110 lemma lift_path_append (p2) (p1) (f):
111 (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
112 #p2 #p1 elim p1 -p1 //
114 [ <lift_path_d_sn <lift_path_d_sn <IH //
115 | <lift_path_m_sn <lift_path_m_sn <IH //
116 | <lift_path_L_sn <lift_path_L_sn <IH //
117 | <lift_path_A_sn <lift_path_A_sn <IH //
118 | <lift_path_S_sn <lift_path_S_sn <IH //
122 lemma lift_path_d_dx (f) (p) (n):
123 (↑[f]p)◖𝗱((↑[p]f)@⧣❨n❩) = ↑[f](p◖𝗱n).
124 #f #p #n <lift_path_append //
127 lemma lift_path_m_dx (f) (p):
128 (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
129 #f #p <lift_path_append //
132 lemma lift_path_L_dx (f) (p):
133 (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
134 #f #p <lift_path_append //
137 lemma lift_path_A_dx (f) (p):
138 (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
139 #f #p <lift_path_append //
142 lemma lift_path_S_dx (f) (p):
143 (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
144 #f #p <lift_path_append //
147 (* Advanced inversions ******************************************************)
149 lemma lift_path_inv_empty (f) (p):
160 lemma lift_path_inv_d_sn (f) (p) (q) (k):
162 ∃∃r,h. k = f@⧣❨h❩ & q = ↑[⇂*[h]f]r & 𝗱h◗r = p.
163 #f * [| * [ #n ] #p ] #q #k
171 /2 width=5 by ex3_2_intro/
174 lemma lift_path_inv_m_sn (f) (p) (q):
176 ∃∃r. q = ↑[f]r & 𝗺◗r = p.
177 #f * [| * [ #n ] #p ] #q
185 /2 width=3 by ex2_intro/
188 lemma lift_path_inv_L_sn (f) (p) (q):
190 ∃∃r. q = ↑[⫯f]r & 𝗟◗r = p.
191 #f * [| * [ #n ] #p ] #q
199 /2 width=3 by ex2_intro/
202 lemma lift_path_inv_A_sn (f) (p) (q):
204 ∃∃r. q = ↑[f]r & 𝗔◗r = p.
205 #f * [| * [ #n ] #p ] #q
213 /2 width=3 by ex2_intro/
216 lemma lift_path_inv_S_sn (f) (p) (q):
218 ∃∃r. q = ↑[f]r & 𝗦◗r = p.
219 #f * [| * [ #n ] #p ] #q
227 /2 width=3 by ex2_intro/
230 lemma lift_path_inv_append_dx (q2) (q1) (p) (f):
232 ∃∃p1,p2. q1 = ↑[f]p1 & q2 = ↑[↑[p1]f]p2 & p1●p2 = p.
234 [| * [ #n1 ] #q1 #IH ] #p #f
235 [ <list_append_empty_sn #H0 destruct
236 /2 width=5 by ex3_2_intro/
237 | <list_append_lcons_sn #H0
238 elim (lift_path_inv_d_sn … H0) -H0 #r1 #m1 #_ #_ #H0 #_ -IH
239 elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
241 | elim (lift_path_inv_m_sn … H)
242 | elim (lift_path_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
243 elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
244 @(ex3_2_intro … (r1●𝗟◗p1)) //
245 <structure_append <Hr1 -Hr1 //
246 | elim (lift_path_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
247 elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
248 @(ex3_2_intro … (r1●𝗔◗p1)) //
249 <structure_append <Hr1 -Hr1 //
250 | elim (lift_path_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
251 elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
252 @(ex3_2_intro … (r1●𝗦◗p1)) //
253 <structure_append <Hr1 -Hr1 //
258 (* Main inversions **********************************************************)
260 theorem lift_path_inj (q:path) (p) (f):
261 ↑[f]q = ↑[f]p → q = p.
262 #q elim q -q [| * [ #k ] #q #IH ] #p #f
263 [ <lift_path_empty #H0
264 <(lift_path_inv_empty … H0) -H0 //
265 | <lift_path_d_sn #H0
266 elim (lift_path_inv_d_sn … H0) -H0 #r #h #H0
267 <(tr_pap_inj ????? H0) -h [1,3: // ] #Hr #H0 destruct
268 | <lift_path_m_sn #H0
269 elim (lift_path_inv_m_sn … H0) -H0 #r #Hr #H0 destruct
270 | <lift_path_L_sn #H0
271 elim (lift_path_inv_L_sn … H0) -H0 #r #Hr #H0 destruct
272 | <lift_path_A_sn #H0
273 elim (lift_path_inv_A_sn … H0) -H0 #r #Hr #H0 destruct
274 | <lift_path_S_sn #H0
275 elim (lift_path_inv_S_sn … H0) -H0 #r #Hr #H0 destruct
282 (* Advanced constructions with proj_rmap and stream_tls *********************)
284 lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
285 ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
287 <lift_rmap_d_dx >nrplus_inj_dx
288 /2 width=1 by tr_tls_compose_uni_dx/