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15 include "ground_2/notation/relations/rcoafter_3.ma".
16 include "ground_2/relocation/rtmap_sor.ma".
17 include "ground_2/relocation/rtmap_after.ma".
19 (* RELOCATION MAP ***********************************************************)
21 coinductive coafter: relation3 rtmap rtmap rtmap ≝
22 | coafter_refl: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
23 ↑f1 = g1 → ↑f2 = g2 → ↑f = g → coafter g1 g2 g
24 | coafter_push: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
25 ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → coafter g1 g2 g
26 | coafter_next: ∀f1,f2,f,g1,g. coafter f1 f2 f →
27 ⫯f1 = g1 → ↑f = g → coafter g1 f2 g
30 interpretation "relational co-composition (rtmap)"
31 'RCoAfter f1 f2 f = (coafter f1 f2 f).
33 definition H_coafter_inj: predicate rtmap ≝
35 ∀f,f21,f22. f1 ~⊚ f21 ≡ f → f1 ~⊚ f22 ≡ f → f21 ≗ f22.
37 definition H_coafter_fwd_isid2: predicate rtmap ≝
38 λf1. ∀f2,f. f1 ~⊚ f2 ≡ f → 𝐓⦃f1⦄ → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
40 definition H_coafter_isfin2_fwd: predicate rtmap ≝
41 λf1. ∀f2. 𝐅⦃f2⦄ → 𝐓⦃f1⦄ → ∀f. f1 ~⊚ f2 ≡ f → 𝐅⦃f⦄.
43 (* Basic inversion lemmas ***************************************************)
45 lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
46 ∃∃f. f1 ~⊚ f2 ≡ f & ↑f = g.
47 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
48 [ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
49 >(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
50 /2 width=3 by ex2_intro/
51 | #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
52 elim (discr_push_next … Hx2)
53 | #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
54 elim (discr_push_next … Hx1)
58 lemma coafter_inv_pnx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
59 ∃∃f. f1 ~⊚ f2 ≡ f & ⫯f = g.
60 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
61 [ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
62 elim (discr_next_push … Hx2)
63 | #g2 #g #Hf #H1 #H2 #H3 #x1 #x2 #Hx1 #Hx2 destruct
64 >(injective_push … Hx1) >(injective_next … Hx2) -x2 -x1
65 /2 width=3 by ex2_intro/
66 | #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
67 elim (discr_push_next … Hx1)
71 lemma coafter_inv_nxx: ∀g1,f2,g. g1 ~⊚ f2 ≡ g → ∀f1. ⫯f1 = g1 →
72 ∃∃f. f1 ~⊚ f2 ≡ f & ↑f = g.
73 #g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
74 [ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
75 elim (discr_next_push … Hx1)
76 | #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
77 elim (discr_next_push … Hx1)
78 | #g #Hf #H1 #H #x1 #Hx1 destruct
79 >(injective_next … Hx1) -x1
80 /2 width=3 by ex2_intro/
84 (* Advanced inversion lemmas ************************************************)
86 lemma coafter_inv_ppp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
87 ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ~⊚ f2 ≡ f.
88 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
89 elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
90 <(injective_push … Hx) -f //
93 lemma coafter_inv_ppn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
94 ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
95 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
96 elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
97 elim (discr_push_next … Hx)
100 lemma coafter_inv_pnn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
101 ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ~⊚ f2 ≡ f.
102 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
103 elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
104 <(injective_next … Hx) -f //
107 lemma coafter_inv_pnp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
108 ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
109 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
110 elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
111 elim (discr_next_push … Hx)
114 lemma coafter_inv_nxp: ∀g1,f2,g. g1 ~⊚ f2 ≡ g →
115 ∀f1,f. ⫯f1 = g1 → ↑f = g → f1 ~⊚ f2 ≡ f.
116 #g1 #f2 #g #Hg #f1 #f #H1 #H
117 elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
118 <(injective_push … Hx) -f //
121 lemma coafter_inv_nxn: ∀g1,f2,g. g1 ~⊚ f2 ≡ g →
122 ∀f1,f. ⫯f1 = g1 → ⫯f = g → ⊥.
123 #g1 #f2 #g #Hg #f1 #f #H1 #H
124 elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
125 elim (discr_push_next … Hx)
128 lemma coafter_inv_pxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
129 ∀f1,f. ↑f1 = g1 → ↑f = g →
130 ∃∃f2. f1 ~⊚ f2 ≡ f & ↑f2 = g2.
131 #g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
132 [ lapply (coafter_inv_ppp … Hg … H1 H2 H) -g1 -g /2 width=3 by ex2_intro/
133 | elim (coafter_inv_pnp … Hg … H1 H2 H)
137 lemma coafter_inv_pxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
138 ∀f1,f. ↑f1 = g1 → ⫯f = g →
139 ∃∃f2. f1 ~⊚ f2 ≡ f & ⫯f2 = g2.
140 #g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
141 [ elim (coafter_inv_ppn … Hg … H1 H2 H)
142 | lapply (coafter_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
146 lemma coafter_inv_xxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ⫯f = g →
147 ∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ⫯f2 = g2.
148 #g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
149 [ elim (coafter_inv_pxn … Hg … H1 H) -g /2 width=5 by ex3_2_intro/
150 | elim (coafter_inv_nxn … Hg … H1 H)
154 lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ↑f = g →
155 (∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2) ∨
156 ∃∃f1. f1 ~⊚ g2 ≡ f & ⫯f1 = g1.
157 #g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
158 [ elim (coafter_inv_pxp … Hg … H1 H) -g
159 /3 width=5 by or_introl, ex3_2_intro/
160 | /4 width=5 by coafter_inv_nxp, or_intror, ex2_intro/
164 lemma coafter_inv_pxx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1. ↑f1 = g1 →
165 (∃∃f2,f. f1 ~⊚ f2 ≡ f & ↑f2 = g2 & ↑f = g) ∨
166 (∃∃f2,f. f1 ~⊚ f2 ≡ f & ⫯f2 = g2 & ⫯f = g).
167 #g1 #g2 #g #Hg #f1 #H1 elim (pn_split g2) * #f2 #H2
168 [ elim (coafter_inv_ppx … Hg … H1 H2) -g1
169 /3 width=5 by or_introl, ex3_2_intro/
170 | elim (coafter_inv_pnx … Hg … H1 H2) -g1
171 /3 width=5 by or_intror, ex3_2_intro/
175 (* Basic properties *********************************************************)
177 corec lemma coafter_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ~⊚ f1 ≡ f).
178 #f1 #f #f2 * -f2 -f1 -f
179 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
180 [ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_refl/
181 | cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_push/
182 | cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by coafter_next/
186 lemma coafter_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ~⊚ f1 ≡ f).
187 #f1 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back2/
190 corec lemma coafter_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ~⊚ f1 ≡ f).
191 #f2 #f #f1 * -f2 -f1 -f
192 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
193 [ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by coafter_refl/
194 | cases (eq_inv_nx … H0 … H11) -g11 /3 width=7 by coafter_push/
195 | @(coafter_next … H2 H) /2 width=5 by/
199 lemma coafter_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ~⊚ f1 ≡ f).
200 #f2 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back1/
203 corec lemma coafter_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ~⊚ f1 ≡ f).
204 #f2 #f1 #f * -f2 -f1 -f
205 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
206 [ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by coafter_refl/
207 | cases (eq_inv_nx … H0 … H01) -g01 /3 width=7 by coafter_push/
208 | cases (eq_inv_px … H0 … H01) -g01 /3 width=5 by coafter_next/
212 lemma coafter_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ~⊚ f1 ≡ f).
213 #f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
216 (* Main inversion lemmas ****************************************************)
218 corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y → x ≗ y.
219 #f1 #f2 #x #y * -f1 -f2 -x
220 #f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
221 [ cases (coafter_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
222 | cases (coafter_inv_pnx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_next/
223 | cases (coafter_inv_nxx … Hy … H1) -g1 /3 width=8 by eq_push/
227 lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
228 f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
229 /4 width=4 by coafter_mono, coafter_eq_repl_back1, coafter_eq_repl_back2/ qed-.
231 (* Inversion lemmas with pushs **********************************************)
233 lemma coafter_fwd_pushs: ∀n,g2,g1,g. g2 ~⊚ g1 ≡ g → @⦃0, g2⦄ ≡ n →
235 #n elim n -n /2 width=2 by ex_intro/
236 #n #IH #g2 #g1 #g #Hg #Hg2
237 cases (at_inv_pxn … Hg2) -Hg2 [ |*: // ] #f2 #Hf2 #H2
238 cases (coafter_inv_nxx … Hg … H2) -Hg -H2 #f #Hf #H0 destruct
239 elim (IH … Hf Hf2) -g1 -g2 -f2 /2 width=2 by ex_intro/
242 (* Inversion lemmas with tail ***********************************************)
244 lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≡ g →
245 ∃∃f. ↑g2 ~⊚ g1 ≡ f & ⫱f = g.
246 #g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
247 [ /3 width=7 by coafter_refl, ex2_intro/
248 | @(ex2_intro … (⫯g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
252 lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≡ ⫱g →
253 ∃∃f1. ↑g2 ~⊚ f1 ≡ g & ⫱f1 = g1.
254 #g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
255 [ /3 width=7 by coafter_refl, ex2_intro/
256 | @(ex2_intro … (⫯g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
260 (* Properties on tls ********************************************************)
262 lemma coafter_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n →
263 f1 ~⊚ f2 ≡ f → ⫱*[n]f1 ~⊚ f2 ≡ ⫱*[n]f.
265 #n #IH #f1 #f2 #f #Hf1 #Hf
266 cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
267 cases (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
268 <tls_xn <tls_xn /2 width=1 by/
271 lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≡ g →
272 ∀n. @⦃0, g2⦄ ≡ n → ⫱*[⫯n]g2 ~⊚ ⫱g1 ≡ ⫱*[⫯n]g.
273 #g2 #g1 #g #Hg #n #Hg2
274 lapply (coafter_tls … Hg2 … Hg) -Hg #Hg
275 lapply (at_pxx_tls … Hg2) -Hg2 #H
276 elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
277 elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
278 <tls_S <tls_S <H2 <H0 -g2 -g -n //
281 lemma coafter_fwd_xpx_pushs: ∀g2,f1,g,n. g2 ~⊚ ↑f1 ≡ g → @⦃0, g2⦄ ≡ n →
283 #g2 #g1 #g #n #Hg #Hg2
284 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
285 lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs #Hf
286 lapply (at_pxx_tls … Hg2) -Hg2 #H
287 elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
288 elim (coafter_inv_pxx … Hf … H2) -Hf -H2 * #f1 #g #_ #H1 #H0 destruct
289 [ /2 width=2 by ex_intro/
290 | elim (discr_next_push … H1)
294 lemma coafter_fwd_xnx_pushs: ∀g2,f1,g,n. g2 ~⊚ ⫯f1 ≡ g → @⦃0, g2⦄ ≡ n →
296 #g2 #g1 #g #n #Hg #Hg2
297 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
298 lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs #Hf
299 lapply (at_pxx_tls … Hg2) -Hg2 #H
300 elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
301 elim (coafter_inv_pxx … Hf … H2) -Hf -H2 * #f1 #g #_ #H1 #H0 destruct
302 [ elim (discr_push_next … H1)
303 | /2 width=2 by ex_intro/
307 (* Properties on isid *******************************************************)
309 corec lemma coafter_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ~⊚ f2 ≡ f2.
310 #f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
311 /3 width=7 by coafter_push, coafter_refl/
314 corec lemma coafter_isid_dx: ∀f2,f. 𝐈⦃f2⦄ → 𝐈⦃f⦄ → ∀f1. f1 ~⊚ f2 ≡ f.
315 #f2 #f * -f2 #f2 #g2 #Hf2 #H2 * -f #f #g #Hf #H #f1 cases (pn_split f1) * #g1 #H1
316 [ /3 width=7 by coafter_refl/
317 | @(coafter_next … H1 … H) /3 width=3 by isid_push/
321 (* Inversion lemmas on isid *************************************************)
323 lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
324 /3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
326 lemma coafter_isid_inv_dx: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f2⦄ → 𝐈⦃f⦄.
327 /4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
329 (* Properties on isuni ******************************************************)
331 lemma coafter_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ~⊚ ⫯f2 ≡ ⫯f1.
332 #f1 #f2 #Hf2 #H elim H -H
333 /5 width=7 by coafter_isid_dx, coafter_eq_repl_back2, coafter_next, coafter_push, eq_push_inv_isid/
336 lemma coafter_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ~⊚ f1 ≡ f → f2 ~⊚ ⫯f1 ≡ f.
338 [ #f2 #Hf2 #f1 #f #Hf
339 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
340 /4 width=7 by coafter_isid_inv_sn, coafter_isid_sn, coafter_eq_repl_back0, eq_next/
341 | #f2 #_ #g2 #H2 #IH #f1 #f #Hf
342 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
343 /3 width=5 by coafter_next/
347 (* Properties on uni ********************************************************)
349 lemma coafter_uni: ∀n1,n2. 𝐔❴n1❵ ~⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
351 /4 width=5 by coafter_uni_next2, coafter_isid_sn, coafter_isid_dx, coafter_next/
354 (* Forward lemmas on at *****************************************************)
356 lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ~⊚ f1 ≡ f →
357 ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
358 #i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
359 [ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
360 [1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
361 | elim (at_inv_xxp … Hf) -Hf //
364 [2: elim (coafter_inv_xxn … Hf21 … H) -f *
365 [ #g2 #g1 #Hg21 #H2 #H1 | #g2 #Hg21 #H2 ]
366 |*: elim (coafter_inv_xxp … Hf21 … H) -f
367 #g2 #g1 #Hg21 #H2 #H1
369 [4: -Hg21 |*: elim (IH … Hg … Hg21) -g -IH ]
370 /3 width=9 by at_refl, at_push, at_next, ex2_intro/
373 lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
374 ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
375 #i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
376 [ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
377 #g2 [ #j2 ] #Hg2 [ #H22 ] #H20
378 [ elim (at_inv_xxn … Hf1 … H22) -i2 *
379 #g1 [ #j1 ] #Hg1 [ #H11 ] #H10
380 [ elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /3 width=7 by at_push/
381 | elim (coafter_inv_pnx … Hf … H20 H10) -f1 -f2 /3 width=6 by at_next/
383 | elim (coafter_inv_nxx … Hf … H20) -f2 /3 width=7 by at_next/
385 | elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H22 #H20
386 elim (at_inv_xxp … Hf1 … H22) -i2 #g1 #H11 #H10
387 elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /2 width=2 by at_refl/
391 lemma coafter_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
392 ∀f2. f2 ~⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
393 #f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (coafter_at_fwd … Hf … H) -f
394 #j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
397 lemma coafter_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≡ i → @⦃i2, f2⦄ ≡ i →
398 ∀f1. f2 ~⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
399 #i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
400 [ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
401 #g [ #j1 ] #Hg [ #H01 ] #H00
402 elim (at_inv_xxn … Hf2) -Hf2 [1,3,5,7: * |*: // ]
403 #g2 [1,3: #j2 ] #Hg2 [1,2: #H22 ] #H20
404 [ elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=7 by at_push/
405 | elim (coafter_inv_pxn … Hf1 … H20 H00) -f2 -f /3 width=5 by at_next/
406 | elim (coafter_inv_nxp … Hf1 … H20 H00)
407 | /4 width=9 by coafter_inv_nxn, at_next/
409 | elim (at_inv_xxp … Hf) -Hf // #g #H01 #H00
410 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H21 #H20
411 elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=2 by at_refl/
415 (* Properties with at *******************************************************)
417 lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
418 ∀f. f2 ~⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f.
420 [ #i1 #f2 #Hf2 #f #Hf
421 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
422 lapply (coafter_isid_inv_dx … Hf ?) -Hf
423 /3 width=3 by coafter_isid_sn, coafter_eq_repl_back0/
424 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
425 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
426 [ #g2 #j1 #Hg2 #H1 #H2 destruct
427 elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
428 /3 width=5 by coafter_next/
429 | #g2 #Hg2 #H2 destruct
430 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
431 /3 width=5 by coafter_next/
436 lemma coafter_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
437 ∀f. 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f → f2 ~⊚ 𝐔❴i1❵ ≡ f.
439 [ #i1 #f2 #Hf2 #f #Hf
440 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
441 lapply (coafter_isid_inv_sn … Hf ?) -Hf
442 /3 width=3 by coafter_isid_dx, coafter_eq_repl_back0/
443 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
444 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
445 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
446 [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by coafter_push/
447 | #g2 #Hg2 #H2 destruct /3 width=5 by coafter_next/
452 lemma coafter_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
453 ∀f. f2 ~⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f.
455 [ #i1 #f2 #Hf2 #f #Hf
456 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
457 elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
458 lapply (coafter_isid_inv_dx … Hg ?) -Hg
459 /4 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next/
460 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
461 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
462 [ #g2 #j1 #Hg2 #H1 #H2 destruct
463 elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
464 /3 width=5 by coafter_next/
465 | #g2 #Hg2 #H2 destruct
466 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
467 /3 width=5 by coafter_next/
472 lemma coafter_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
473 ∀f. 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f → f2 ~⊚ 𝐔❴⫯i1❵ ≡ f.
475 [ #i1 #f2 #Hf2 #f #Hf
476 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
477 elim (coafter_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
478 lapply (coafter_isid_inv_sn … Hg ?) -Hg
479 /4 width=7 by coafter_isid_dx, coafter_eq_repl_back0, coafter_push/
480 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
481 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
482 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
483 [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by coafter_push/
484 | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by coafter_next/
489 lemma coafter_uni_one_dx: ∀f2,f. ↑f2 ~⊚ 𝐔❴⫯O❵ ≡ f → 𝐔❴⫯O❵ ~⊚ f2 ≡ f.
490 #f2 #f #H @(coafter_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
493 lemma coafter_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ~⊚ f1 ≡ f → ↑f1 ~⊚ 𝐔❴⫯O❵ ≡ f.
494 /3 width=3 by coafter_uni_succ_sn, at_refl/ qed-.
496 (* Forward lemmas with istot ************************************************)
498 lemma coafter_istot_fwd: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
499 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
500 #i2 #Hf1 elim (Hf2 i2) -Hf2
501 /3 width=7 by coafter_fwd_at, ex_intro/
504 lemma coafter_fwd_istot_dx: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
505 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
506 #i2 #Hf elim (coafter_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
509 lemma coafter_fwd_istot_sn: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
510 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
511 #i #Hf elim (coafter_at_fwd … Hf … H) -f
512 #i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
513 #Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
516 lemma coafter_inv_istot: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
517 /3 width=4 by coafter_fwd_istot_sn, coafter_fwd_istot_dx, conj/ qed-.
519 lemma coafter_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ~⊚ f1 ≡ f →
520 ∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
521 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
522 /3 width=8 by coafter_fwd_at, ex2_intro/
525 lemma coafter_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f1 ≗ f → 𝐈⦃f2⦄.
526 #f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
527 #Hf2 #Hf1 #H @isid_at_total // -Hf2
528 #i2 #i #Hf2 elim (Hf1 i2) -Hf1
529 #i0 #Hf1 lapply (at_increasing … Hf1)
530 #Hi20 lapply (coafter_fwd_at2 … i0 … Hf1 … Hf) -Hf
531 /3 width=7 by at_eq_repl_back, at_mono, at_id_le/
534 lemma coafter_fwd_isid_dx: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f2 ≗ f → 𝐈⦃f1⦄.
535 #f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
536 #Hf2 #Hf1 #H2 @isid_at_total // -Hf1
537 #i1 #i2 #Hi12 elim (coafter_at1_fwd … Hi12 … Hf) -f1
538 /3 width=8 by at_inj, at_eq_repl_back/
541 corec fact coafter_inj_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_inj f1.
542 #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
543 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
544 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
545 cases (H2g1 0) #n #Hn
546 cases (coafter_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
547 [ cases (coafter_inv_pxp … H2f … H1 H) -f1 -f #g22 #H2g #H22
548 @(eq_push … H21 H22) -f21 -f22
549 | cases (coafter_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
550 @(eq_next … H21 H22) -f21 -f22
552 @(coafter_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -coafter_inj_O_aux
553 /2 width=1 by coafter_tls, istot_tls, at_pxx_tls/
556 fact coafter_inj_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_inj f1) →
557 ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_inj f1.
558 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
559 #i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
560 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
561 elim (coafter_inv_nxx … H1f … H1) -H1f #g #H1g #H
562 lapply (coafter_inv_nxp … H2f … H1 H) -f #H2g
563 /3 width=6 by istot_inv_next/
566 theorem coafter_inj: ∀f1. H_coafter_inj f1.
567 #f1 #H cases (H 0) /3 width=7 by coafter_inj_aux, coafter_inj_O_aux/
570 corec fact coafter_fwd_isid2_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
571 H_coafter_fwd_isid2 f1.
572 #f1 #H1f1 #f2 #f #H #H2f1 #Hf
573 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
574 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
575 cases (H2g1 0) #n #Hn
576 cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #H0
577 [ lapply (isid_inv_push … Hf … H0) -Hf #Hg
579 /3 width=7 by coafter_tls, istot_tls, at_pxx_tls, isid_tls/
580 | cases (isid_inv_next … Hf … H0)
584 fact coafter_fwd_isid2_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_fwd_isid2 f1) →
585 ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_fwd_isid2 f1.
586 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
587 #i2 #IH #f1 #H1f1 #f2 #f #H #H2f1 #Hf
588 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
589 elim (coafter_inv_nxx … H … H1) -H #g #Hg #H0
590 @(IH … Hg1 … Hg) /2 width=3 by istot_inv_next, isid_inv_push/ (**) (* full auto fails *)
593 lemma coafter_fwd_isid2: ∀f1. H_coafter_fwd_isid2 f1.
594 #f1 #f2 #f #Hf #H cases (H 0)
595 /3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
598 fact coafter_isfin2_fwd_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
599 H_coafter_isfin2_fwd f1.
601 generalize in match Hf1; generalize in match f1; -f1
603 [ /3 width=4 by coafter_isid_inv_dx, isfin_isid/ ]
604 #f2 #_ #IH #f1 #H #Hf1 #f #Hf
605 elim (at_inv_pxp … H) -H [ |*: // ] #g1 #H1
606 lapply (istot_inv_push … Hf1 … H1) -Hf1 #Hg1
608 [ elim (coafter_inv_ppx … Hf) | elim (coafter_inv_pnx … Hf)
609 ] -Hf [1,6: |*: // ] #g #Hg #H0 destruct
610 /5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls/
613 fact coafter_isfin2_fwd_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_isfin2_fwd f1) →
614 ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
615 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
616 #i2 #IH #f1 #H1f1 #f2 #Hf2 #H2f1 #f #Hf
617 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
618 elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0
619 lapply (IH … Hg1 … Hg) -i2 -Hg
620 /2 width=4 by istot_inv_next, isfin_push/ (**) (* full auto fails *)
623 lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
624 #f1 #f2 #Hf2 #Hf1 cases (Hf1 0)
625 /3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
628 lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀fa,fb. fa ⋓ fb ≡ f →
629 ∃∃f1a,f1b. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & f1a ⋓ f1b ≡ f1.
631 [ #f #Hf #f2 #Hf2 #f1 #H1f #fa #fb #H2f
632 elim (sor_inv_isid3 … H2f) -H2f //
633 lapply (coafter_fwd_isid2 … H1f ??) -H1f //
634 /3 width=5 by ex3_2_intro, coafter_isid_dx, sor_isid/
635 | #f #_ #IH #f2 #Hf2 #f1 #H1 #fa #fb #H2
636 elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #H2f
637 elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ] #g2 [ #g1 ] #H1f #Hgf2
638 [ lapply (istot_inv_push … Hf2 … Hgf2) | lapply (istot_inv_next … Hf2 … Hgf2) ] -Hf2 #Hg2
639 elim (IH … Hg2 … H1f … H2f) -f -Hg2
640 /3 width=11 by sor_pp, ex3_2_intro, coafter_refl, coafter_next/
641 | #f #_ #IH #f2 #Hf2 #f1 #H1 #fa #fb #H2
642 elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #H1f #Hgf2
643 lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
644 elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #H2f
645 elim (IH … Hg2 … H1f … H2f) -f -Hg2
646 /3 width=11 by sor_np, sor_pn, sor_nn, ex3_2_intro, coafter_refl, coafter_push/
650 (* Properties with istot ****************************************************)
652 lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀f1a,f1b. f1a ⋓ f1b ≡ f1 →
653 ∃∃fa,fb. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & fa ⋓ fb ≡ f.
655 [ #f #Hf #f2 #Hf2 #f1 #Hf #f1a #f1b #Hf1
656 lapply (coafter_fwd_isid2 … Hf ??) -Hf // #H2f1
657 elim (sor_inv_isid3 … Hf1) -Hf1 //
658 /3 width=5 by coafter_isid_dx, sor_refl, ex3_2_intro/
659 | #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
660 elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ]
661 [ #g2 #g1 #Hf #Hgf2 #Hgf1
662 elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #Hg1
663 lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
664 elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
665 /3 width=11 by coafter_refl, sor_pp, ex3_2_intro/
667 lapply (istot_inv_next … Hf2 … Hgf2) -Hf2 #Hg2
668 elim (IH … Hf … H2) // -f1 -IH -Hg2
669 /3 width=11 by coafter_next, sor_pp, ex3_2_intro/
671 | #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
672 elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #Hf #Hgf2 #Hgf1
673 lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
674 elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #Hg1
675 elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
676 /3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
680 (* Properties with after ****************************************************)
682 corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
683 ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
684 ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
685 #f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
686 [ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
687 cases (coafter_inv_xxp … Hg0 … H0) -g0
689 cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
690 #f #Hf #H /3 width=7 by coafter_refl/
691 | #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
692 cases (coafter_inv_xxp … Hg0 … H0) -g0
694 cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
695 #f #Hf #H /3 width=7 by coafter_push/
696 | #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
697 cases (coafter_inv_xxn … Hg0 … H0) -g0 *
698 [ #f1 #f2 #Hf0 #H1 #H2
699 cases (coafter_inv_nxx … Hg … H2) -g2
700 #f #Hf #H /3 width=7 by coafter_push/
701 | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
706 corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
707 ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
708 ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
709 #f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
710 [ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
711 cases (coafter_inv_xxp … Hg0 … H0) -g0
713 cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
714 #f #Hf #H /3 width=7 by coafter_refl/
715 | #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
716 cases (coafter_inv_xxn … Hg0 … H0) -g0 *
717 [ #f2 #f3 #Hf0 #H2 #H3
718 cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
719 #f #Hf #H /3 width=7 by coafter_push/
721 cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
722 #f #Hf #H /3 width=6 by coafter_next/
724 | #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
725 cases (coafter_inv_nxx … Hg … H1) -g1
726 #f #Hg #H /3 width=6 by coafter_next/