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/syntax/path.ma".
16 include "delayed_updating/notation/functions/class_c_2.ma".
17 include "ground/arith/nat_plus_pred.ma".
18 include "ground/lib/subset.ma".
19 include "ground/lib/bool_and.ma".
20 include "ground/generated/insert_eq_1.ma".
21 include "ground/xoa/ex_3_2.ma".
23 (* CLOSED CONDITION FOR PATH ************************************************)
25 inductive pcc (o): relation2 nat path ≝
28 | pcc_d_dx (p) (n) (k):
30 pcc o (n+ninj k) p → pcc o n (p◖𝗱k)
32 pcc o n p → pcc o n (p◖𝗺)
34 pcc o n p → pcc o (↑n) (p◖𝗟)
36 pcc o n p → pcc o n (p◖𝗔)
38 pcc o n p → pcc o n (p◖𝗦)
42 "closed condition (path)"
43 'ClassC o n = (pcc o n).
45 (* Advanced constructions ***************************************************)
47 lemma pcc_false_d_dx (p) (n) (k:pnat):
48 p ϵ 𝐂❨Ⓕ,n+k❩ → p◖𝗱k ϵ 𝐂❨Ⓕ,n❩.
54 lemma pcc_true_d_dx (p) (n:pnat) (k:pnat):
55 p ϵ 𝐂❨Ⓣ,n+k❩ → p◖𝗱k ϵ 𝐂❨Ⓣ,n❩.
56 /2 width=1 by pcc_d_dx/
59 (* Basic inversions ********************************************************)
61 lemma pcc_inv_empty (o) (n):
63 #o #n @(insert_eq_1 … (𝐞))
65 #p #n [ #k #_ ] #_ #H0 destruct
69 alias symbol "DownArrow" (instance 4) = "predecessor (non-negative integers)".
70 alias symbol "UpArrow" (instance 3) = "successor (non-negative integers)".
71 alias symbol "and" (instance 1) = "logical and".
73 lemma pcc_inv_d_dx (o) (p) (n) (k):
77 #o #p #n #h @(insert_eq_1 … (p◖𝗱h))
79 [|*: #x #n [ #k #Ho ] #Hx ] #H0 destruct
83 lemma pcc_inv_m_dx (o) (p) (n):
84 p◖𝗺 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
85 #o #p #n @(insert_eq_1 … (p◖𝗺))
87 [|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
90 lemma pcc_inv_L_dx (o) (p) (n):
92 ∧∧ p ϵ 𝐂❨o,↓n❩ & n = ↑↓n.
93 #o #p #n @(insert_eq_1 … (p◖𝗟))
95 [|*: #x #n [ #k #_ ] #Hx ] #H0 destruct
96 <npred_succ /2 width=1 by conj/
99 lemma pcc_inv_A_dx (o) (p) (n):
100 p◖𝗔 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
101 #o #p #n @(insert_eq_1 … (p◖𝗔))
103 [|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
106 lemma pcc_inv_S_dx (o) (p) (n):
107 p◖𝗦 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
108 #o #p #n @(insert_eq_1 … (p◖𝗦))
110 [|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
113 (* Advanced destructions ****************************************************)
115 lemma pcc_des_d_dx (o) (p) (n) (k):
116 p◖𝗱k ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n+k❩.
118 elim (pcc_inv_d_dx … H0) -H0 #H1 #H2 //
121 lemma pcc_des_gen (o) (p) (n):
122 p ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨Ⓕ,n❩.
123 #o #p #n #H0 elim H0 -p -n //
124 #p #n [ #k #Ho ] #_ #IH
125 /2 width=1 by pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx, pcc_false_d_dx/
128 (* Advanced inversions ******************************************************)
130 lemma pcc_inv_empty_succ (o) (n):
133 lapply (pcc_inv_empty … H0) -H0 #H0
134 /2 width=7 by eq_inv_zero_nsucc/
137 lemma pcc_true_inv_d_dx_zero (p) (k):
140 elim (pcc_inv_d_dx … H0) -H0 #H0 #_
141 elim (eq_inv_zero_nsucc … (H0 ?)) -H0 //
144 lemma pcc_inv_L_dx_zero (o) (p):
147 elim (pcc_inv_L_dx … H0) -H0 #_ #H0
148 /2 width=7 by eq_inv_zero_nsucc/
151 lemma pcc_inv_L_dx_succ (o) (p) (n):
152 p◖𝗟 ϵ 𝐂❨o,↑n❩ → p ϵ 𝐂❨o,n❩.
154 elim (pcc_inv_L_dx … H0) -H0 //
157 (* Constructions with land **************************************************)
159 lemma pcc_land_dx (o1) (o2) (p) (n):
160 p ϵ 𝐂❨o1,n❩ → p ϵ 𝐂❨o1∧o2,n❩.
161 #o1 * /2 width=2 by pcc_des_gen/
164 lemma pcc_land_sn (o1) (o2) (p) (n):
165 p ϵ 𝐂❨o2,n❩ → p ϵ 𝐂❨o1∧o2,n❩.
166 * /2 width=2 by pcc_des_gen/
169 (* Main constructions with path_append **************************************)
171 theorem pcc_append_bi (o1) (o2) (p) (q) (m) (n):
172 p ϵ 𝐂❨o1,m❩ → q ϵ 𝐂❨o2,n❩ → p●q ϵ 𝐂❨o1∧o2,m+n❩.
173 #o1 #o2 #p #q #m #n #Hm #Hn elim Hn -q -n
174 /2 width=1 by pcc_m_dx, pcc_A_dx, pcc_S_dx, pcc_land_dx/
175 #q #n [ #k #Ho2 ] #_ #IH
177 elim (andb_inv_true_sn … H0) -H0 #_ #H0 >Ho2 //
178 <nplus_succ_dx <npred_succ //
179 | <nplus_succ_dx /2 width=1 by pcc_L_dx/
183 (* Inversions with path_append **********************************************)
185 lemma pcc_false_inv_append_bi (x) (m) (n):
187 ∃∃p,q. p ϵ 𝐂❨Ⓕ,m❩ & q ϵ 𝐂❨Ⓕ,n❩ & p●q = x.
189 @(insert_eq_1 … (m+n) … Hx) -Hx #y #Hy
190 generalize in match n; -n
191 generalize in match m; -m
192 elim Hy -x -y [|*: #x #y [ #k #_ ] #Hx #IH ] #m #n #Hy destruct
193 [ elim (eq_inv_nplus_zero … Hy) -Hy #H1 #H2 destruct
194 /2 width=5 by pcc_empty, ex3_2_intro/
195 | elim (IH m (n+k)) -IH // #p #q #Hp #Hq #H0 destruct -Hx
196 /3 width=5 by pcc_false_d_dx, ex3_2_intro/
197 | elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
198 /3 width=5 by pcc_m_dx, ex3_2_intro/
199 | elim (eq_inv_succ_nplus_dx … (sym_eq … Hy)) -Hy * #H1 #H2 (**) (* sym_eq *)
201 /3 width=5 by pcc_empty, pcc_L_dx, ex3_2_intro/
202 | elim (IH m (↓n)) -IH // #p #q #Hp #Hq #H0 destruct -Hx
203 /3 width=5 by pcc_L_dx, ex3_2_intro/
205 | elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
206 /3 width=5 by pcc_A_dx, ex3_2_intro/
207 | elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
208 /3 width=5 by pcc_S_dx, ex3_2_intro/
213 (* Constructions with path_lcons ********************************************)
215 lemma pcc_m_sn (o) (q) (n):
216 q ϵ 𝐂❨o,n❩ → (𝗺◗q) ϵ 𝐂❨o,n❩.
218 lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗺) … Hq) -Hq
219 /2 width=3 by pcc_m_dx/
222 lemma pcc_L_sn (o) (q) (n):
223 q ϵ 𝐂❨o,n❩ → (𝗟◗q) ϵ 𝐂❨o,↑n❩.
225 lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗟) … Hq) -Hq
226 /2 width=3 by pcc_L_dx/
229 lemma pcc_A_sn (o) (q) (n):
230 q ϵ 𝐂❨o,n❩ → (𝗔◗q) ϵ 𝐂❨o,n❩.
232 lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗔) … Hq) -Hq
233 /2 width=3 by pcc_A_dx/
236 lemma pcc_S_sn (o) (q) (n):
237 q ϵ 𝐂❨o,n❩ → (𝗦◗q) ϵ 𝐂❨o,n❩.
239 lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗦) … Hq) -Hq
240 /2 width=3 by pcc_S_dx/
243 (* Main inversions **********************************************************)
245 theorem pcc_mono (o1) (o2) (q) (n1):
246 q ϵ 𝐂❨o1,n1❩ → ∀n2. q ϵ 𝐂❨o2,n2❩ → n1 = n2.
247 #o1 #o2 #q1 #n1 #Hn1 elim Hn1 -q1 -n1
248 [|*: #q1 #n1 [ #k1 #_ ] #_ #IH ] #n2 #Hn2
249 [ <(pcc_inv_empty … Hn2) -n2 //
250 | lapply (pcc_des_d_dx … Hn2) -Hn2 #Hn2
251 lapply (IH … Hn2) -q1 #H0
252 /2 width=2 by eq_inv_nplus_bi_dx/
253 | lapply (pcc_inv_m_dx … Hn2) -Hn2 #Hn2
254 <(IH … Hn2) -q1 -n2 //
255 | elim (pcc_inv_L_dx … Hn2) -Hn2 #Hn2 #H0
257 | lapply (pcc_inv_A_dx … Hn2) -Hn2 #Hn2
258 <(IH … Hn2) -q1 -n2 //
259 | lapply (pcc_inv_S_dx … Hn2) -Hn2 #Hn2
260 <(IH … Hn2) -q1 -n2 //
264 theorem pcc_inj_L_sn (o1) (o2) (p1) (p2) (q1) (n):
265 q1 ϵ 𝐂❨o1,n❩ → ∀q2. q2 ϵ 𝐂❨o2,n❩ →
266 p1●𝗟◗q1 = p2●𝗟◗q2 → q1 = q2.
267 #o1 #o2 #p1 #p2 #q1 #n #Hq1 elim Hq1 -q1 -n
268 [|*: #q1 #n1 [ #k1 #_ ] #_ #IH ] * //
269 [1,3,5,7,9,11: #l2 #q2 ] #Hq2
270 <list_append_lcons_sn <list_append_lcons_sn #H0
271 elim (eq_inv_list_lcons_bi ????? H0) -H0 #H0 #H1 destruct
272 [ elim (pcc_inv_L_dx_zero … Hq2)
273 | lapply (pcc_des_d_dx … Hq2) -Hq2 #Hq2
275 | lapply (pcc_inv_m_dx … Hq2) -Hq2 #Hq2
277 | lapply (pcc_inv_L_dx_succ … Hq2) -Hq2 #Hq2
279 | lapply (pcc_inv_A_dx … Hq2) -Hq2 #Hq2
281 | lapply (pcc_inv_S_dx … Hq2) -Hq2 #Hq2
283 | elim (pcc_inv_empty_succ … Hq2)
287 theorem pcc_inv_L_sn (o) (q) (n) (m):
288 (𝗟◗q) ϵ 𝐂❨o,n❩ → q ϵ 𝐂❨o,m❩ →
290 #o #q #n #m #H1q #H2q
291 lapply (pcc_L_sn … H2q) -H2q #H2q
292 <(pcc_mono … H2q … H1q) -q -n