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_gen.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_plus.ma".
20 include "ground/lib/stream_tls_eq.ma".
21 include "ground/arith/nat_plus_rplus.ma".
22 include "ground/arith/nat_rplus_pplus.ma".
24 (* LIFT FOR PATH ************************************************************)
26 definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
27 λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
30 "extensional equivalence (lift continuation)"
31 'RingEq A k1 k2 = (lift_exteq A k1 k2).
33 (* Constructions with lift_exteq ********************************************)
35 lemma lift_eq_repl (A) (p) (k1) (k2):
36 k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
37 #A #p elim p -p [| * [ #n ] #q #IH ]
38 #k1 #k2 #Hk #f1 #f2 #Hf
39 [ <lift_empty <lift_empty /2 width=1 by/
40 | <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
41 /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
43 | /3 width=1 by tr_push_eq_repl/
49 (* Advanced constructions ***************************************************)
51 lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
52 ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
54 @lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
57 lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
58 ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
59 #A #k #f #p1 #p #l #Hk
60 @lift_eq_repl // #g1 #g2 #p2 #Hg
61 <list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
64 (* Advanced constructions with proj_path ************************************)
66 lemma proj_path_proper:
67 proj_path ≗ proj_path.
70 lemma lift_path_eq_repl (p):
71 stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
72 /2 width=1 by lift_eq_repl/ qed.
74 lemma lift_path_append_sn (p) (f) (q):
75 q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
76 #p elim p -p // * [ #n ] #p #IH #f #q
77 [ <lift_d_sn <lift_d_sn
78 | <lift_m_sn <lift_m_sn
79 | <lift_L_sn <lift_L_sn
80 | <lift_A_sn <lift_A_sn
81 | <lift_S_sn <lift_S_sn
83 >lift_lcons_alt // >lift_append_rcons_sn //
84 <IH <IH -IH <list_append_rcons_sn //
87 lemma lift_path_lcons (f) (p) (l):
88 l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
90 >lift_lcons_alt <lift_path_append_sn //
93 lemma lift_path_d_sn (f) (p) (n):
94 (𝗱(f@⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
97 lemma lift_path_m_sn (f) (p):
98 (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
101 lemma lift_path_L_sn (f) (p):
102 (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
105 lemma lift_path_A_sn (f) (p):
106 (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
109 lemma lift_path_S_sn (f) (p):
110 (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
113 lemma lift_path_append (p2) (p1) (f):
114 (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
115 #p2 #p1 elim p1 -p1 //
117 [ <lift_path_d_sn <lift_path_d_sn <IH //
118 | <lift_path_m_sn <lift_path_m_sn <IH //
119 | <lift_path_L_sn <lift_path_L_sn <IH //
120 | <lift_path_A_sn <lift_path_A_sn <IH //
121 | <lift_path_S_sn <lift_path_S_sn <IH //
125 lemma lift_path_d_dx (f) (p) (n):
126 (↑[f]p)◖𝗱((↑[p]f)@⧣❨n❩) = ↑[f](p◖𝗱n).
127 #f #p #n <lift_path_append //
130 lemma lift_path_m_dx (f) (p):
131 (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
132 #f #p <lift_path_append //
135 lemma lift_path_L_dx (f) (p):
136 (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
137 #f #p <lift_path_append //
140 lemma lift_path_A_dx (f) (p):
141 (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
142 #f #p <lift_path_append //
145 lemma lift_path_S_dx (f) (p):
146 (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
147 #f #p <lift_path_append //
150 (* Advanced constructions with proj_rmap ************************************)
152 lemma lift_rmap_eq_repl (p):
153 stream_eq_repl … (λf1,f2. ↑[p]f1 ≗ ↑[p]f2).
155 * [ #n ] #p #IH #f1 #f2 #Hf
156 [ /3 width=1 by stream_tls_eq_repl/
158 | /3 width=1 by tr_push_eq_repl/
164 lemma tls_lift_rmap_d_dx (f) (p) (m) (n):
165 ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
167 <lift_rmap_d_dx >nrplus_inj_dx >nrplus_inj_sn //
170 (* Advanced inversions with proj_path ***************************************)
172 lemma lift_path_inv_empty (f) (p):
183 lemma lift_path_inv_d_sn (f) (p) (q) (k):
185 ∃∃r,h. k = f@⧣❨h❩ & q = ↑[⇂*[h]f]r & 𝗱h◗r = p.
186 #f * [| * [ #n ] #p ] #q #k
194 /2 width=5 by ex3_2_intro/
197 lemma lift_path_inv_m_sn (f) (p) (q):
199 ∃∃r. q = ↑[f]r & 𝗺◗r = p.
200 #f * [| * [ #n ] #p ] #q
208 /2 width=3 by ex2_intro/
211 lemma lift_path_inv_L_sn (f) (p) (q):
213 ∃∃r. q = ↑[⫯f]r & 𝗟◗r = p.
214 #f * [| * [ #n ] #p ] #q
222 /2 width=3 by ex2_intro/
225 lemma lift_path_inv_A_sn (f) (p) (q):
227 ∃∃r. q = ↑[f]r & 𝗔◗r = p.
228 #f * [| * [ #n ] #p ] #q
236 /2 width=3 by ex2_intro/
239 lemma lift_path_inv_S_sn (f) (p) (q):
241 ∃∃r. q = ↑[f]r & 𝗦◗r = p.
242 #f * [| * [ #n ] #p ] #q
250 /2 width=3 by ex2_intro/
253 lemma lift_path_inv_append_sn (q2) (q1) (p) (f):
255 ∃∃p1,p2. q1 = ↑[f]p1 & q2 = ↑[↑[p1]f]p2 & p1●p2 = p.
257 [| * [ #n1 ] #q1 #IH ] #p #f
258 [ <list_append_empty_sn #H0 destruct
259 /2 width=5 by ex3_2_intro/
260 | <list_append_lcons_sn #H0
261 elim (lift_path_inv_d_sn … H0) -H0 #r1 #m1 #H1 #H0 #H2 destruct
262 elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
263 /2 width=5 by ex3_2_intro/
264 | <list_append_lcons_sn #H0
265 elim (lift_path_inv_m_sn … H0) -H0 #r1 #H0 #H1 destruct
266 elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
267 /2 width=5 by ex3_2_intro/
268 | <list_append_lcons_sn #H0
269 elim (lift_path_inv_L_sn … H0) -H0 #r1 #H0 #H1 destruct
270 elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
271 /2 width=5 by ex3_2_intro/
272 | <list_append_lcons_sn #H0
273 elim (lift_path_inv_A_sn … H0) -H0 #r1 #H0 #H1 destruct
274 elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
275 /2 width=5 by ex3_2_intro/
276 | <list_append_lcons_sn #H0
277 elim (lift_path_inv_S_sn … H0) -H0 #r1 #H0 #H1 destruct
278 elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
279 /2 width=5 by ex3_2_intro/
283 (* Main inversions with proj_path *******************************************)
285 theorem lift_path_inj (q:path) (p) (f):
286 ↑[f]q = ↑[f]p → q = p.
287 #q elim q -q [| * [ #k ] #q #IH ] #p #f
288 [ <lift_path_empty #H0
289 <(lift_path_inv_empty … H0) -H0 //
290 | <lift_path_d_sn #H0
291 elim (lift_path_inv_d_sn … H0) -H0 #r #h #H0
292 >(tr_pap_inj ???? H0) -k [1,3: // ] #Hr #H0 destruct
293 | <lift_path_m_sn #H0
294 elim (lift_path_inv_m_sn … H0) -H0 #r #Hr #H0 destruct
295 | <lift_path_L_sn #H0
296 elim (lift_path_inv_L_sn … H0) -H0 #r #Hr #H0 destruct
297 | <lift_path_A_sn #H0
298 elim (lift_path_inv_A_sn … H0) -H0 #r #Hr #H0 destruct
299 | <lift_path_S_sn #H0
300 elim (lift_path_inv_S_sn … H0) -H0 #r #Hr #H0 destruct