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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_xnn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
155 ∀f2,f. ↑f2 = g2 → ↑f = g →
156 ∃∃f1. f1 ~⊚ f2 ≘ f & ⫯f1 = g1.
157 #g1 #g2 #g #Hg #f2 #f #H2 destruct #H
158 elim (coafter_inv_xxn … Hg … H) -g
159 #z1 #z2 #Hf #H1 #H2 destruct /2 width=3 by ex2_intro/
162 lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f. ⫯f = g →
163 (∃∃f1,f2. f1 ~⊚ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2) ∨
164 ∃∃f1. f1 ~⊚ g2 ≘ f & ↑f1 = g1.
165 #g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
166 [ elim (coafter_inv_pxp … Hg … H1 H) -g
167 /3 width=5 by or_introl, ex3_2_intro/
168 | /4 width=5 by coafter_inv_nxp, or_intror, ex2_intro/
172 lemma coafter_inv_pxx: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f1. ⫯f1 = g1 →
173 (∃∃f2,f. f1 ~⊚ f2 ≘ f & ⫯f2 = g2 & ⫯f = g) ∨
174 (∃∃f2,f. f1 ~⊚ f2 ≘ f & ↑f2 = g2 & ↑f = g).
175 #g1 #g2 #g #Hg #f1 #H1 elim (pn_split g2) * #f2 #H2
176 [ elim (coafter_inv_ppx … Hg … H1 H2) -g1
177 /3 width=5 by or_introl, ex3_2_intro/
178 | elim (coafter_inv_pnx … Hg … H1 H2) -g1
179 /3 width=5 by or_intror, ex3_2_intro/
183 (* Basic properties *********************************************************)
185 corec lemma coafter_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ~⊚ f1 ≘ f).
186 #f1 #f #f2 * -f2 -f1 -f
187 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
188 [ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_refl/
189 | cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_push/
190 | cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by coafter_next/
194 lemma coafter_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ~⊚ f1 ≘ f).
195 #f1 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back2/
198 corec lemma coafter_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ~⊚ f1 ≘ f).
199 #f2 #f #f1 * -f2 -f1 -f
200 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
201 [ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by coafter_refl/
202 | cases (eq_inv_nx … H0 … H11) -g11 /3 width=7 by coafter_push/
203 | @(coafter_next … H2 H) /2 width=5 by/
207 lemma coafter_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ~⊚ f1 ≘ f).
208 #f2 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back1/
211 corec lemma coafter_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ~⊚ f1 ≘ f).
212 #f2 #f1 #f * -f2 -f1 -f
213 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
214 [ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by coafter_refl/
215 | cases (eq_inv_nx … H0 … H01) -g01 /3 width=7 by coafter_push/
216 | cases (eq_inv_px … H0 … H01) -g01 /3 width=5 by coafter_next/
220 lemma coafter_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ~⊚ f1 ≘ f).
221 #f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
224 (* Main inversion lemmas ****************************************************)
226 corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≘ x → f1 ~⊚ f2 ≘ y → x ≡ y.
227 #f1 #f2 #x #y * -f1 -f2 -x
228 #f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
229 [ cases (coafter_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
230 | cases (coafter_inv_pnx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_next/
231 | cases (coafter_inv_nxx … Hy … H1) -g1 /3 width=8 by eq_push/
235 lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
236 f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
237 /4 width=4 by coafter_mono, coafter_eq_repl_back1, coafter_eq_repl_back2/ qed-.
239 (* Forward lemmas with pushs ************************************************)
241 lemma coafter_fwd_pushs: ∀j,i,g2,f1,g. g2 ~⊚ ⫯*[i]f1 ≘ g → @⦃i, g2⦄ ≘ j →
244 [ #i #g2 #f1 #g #Hg #H
245 elim (at_inv_xxp … H) -H [|*: // ] #f2 #H1 #H2 destruct
246 /2 width=2 by ex_intro/
247 | #j #IH * [| #i ] #g2 #f1 #g #Hg #H
248 [ elim (at_inv_pxn … H) -H [|*: // ] #f2 #Hij #H destruct
249 elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
250 elim (IH … Hf Hij) -f1 -f2 -IH /2 width=2 by ex_intro/
251 | elim (at_inv_nxn … H) -H [1,4: * |*: // ] #f2 #Hij #H destruct
252 [ elim (coafter_inv_ppx … Hg) -Hg [|*: // ] #f #Hf #H destruct
253 elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
254 | elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
255 elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
261 (* Inversion lemmas with tail ***********************************************)
263 lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≘ g →
264 ∃∃f. ⫯g2 ~⊚ g1 ≘ f & ⫱f = g.
265 #g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
266 [ /3 width=7 by coafter_refl, ex2_intro/
267 | @(ex2_intro … (↑g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
271 lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≘ ⫱g →
272 ∃∃f1. ⫯g2 ~⊚ f1 ≘ g & ⫱f1 = g1.
273 #g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
274 [ /3 width=7 by coafter_refl, ex2_intro/
275 | @(ex2_intro … (↑g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
279 (* Properties with iterated tail ********************************************)
281 lemma coafter_tls: ∀j,i,f1,f2,f. @⦃i, f1⦄ ≘ j →
282 f1 ~⊚ f2 ≘ f → ⫱*[j]f1 ~⊚ ⫱*[i]f2 ≘ ⫱*[j]f.
283 #j elim j -j [ #i | #j #IH * [| #i ] ] #f1 #f2 #f #Hf1 #Hf
284 [ elim (at_inv_xxp … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1 destruct //
285 | elim (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
286 elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
287 lapply (IH … Hg1 Hg) -IH -Hg1 -Hg //
288 | elim (at_inv_nxn … Hf1) -Hf1 [1,4: * |*: // ] #g1 #Hg1 #H1
289 [ elim (coafter_inv_pxx … Hf … H1) -Hf * #g2 #g #Hg #H2 #H0 destruct
290 lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
291 | elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
292 lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
297 lemma coafter_tls_O: ∀n,f1,f2,f. @⦃0, f1⦄ ≘ n →
298 f1 ~⊚ f2 ≘ f → ⫱*[n]f1 ~⊚ f2 ≘ ⫱*[n]f.
299 /2 width=1 by coafter_tls/ qed.
301 lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≘ g →
302 ∀n. @⦃0, g2⦄ ≘ n → ⫱*[↑n]g2 ~⊚ ⫱g1 ≘ ⫱*[↑n]g.
303 #g2 #g1 #g #Hg #n #Hg2
304 lapply (coafter_tls … Hg2 … Hg) -Hg #Hg
305 lapply (at_pxx_tls … Hg2) -Hg2 #H
306 elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
307 elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
308 <tls_S <tls_S <H2 <H0 -g2 -g -n //
311 lemma coafter_fwd_xpx_pushs: ∀g2,f1,g,i,j. @⦃i, g2⦄ ≘ j → g2 ~⊚ ⫯*[↑i]f1 ≘ g →
312 ∃∃f. ⫱*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[↑j]f = g.
313 #g2 #g1 #g #i #j #Hg2 <pushs_xn #Hg
314 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
315 lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
316 lapply (at_inv_tls … Hg2) -Hg2 #H
317 lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
318 elim (coafter_inv_ppx … Hf) [|*: // ] -Hf #g #Hg #H destruct
319 /2 width=3 by ex2_intro/
322 lemma coafter_fwd_xnx_pushs: ∀g2,f1,g,i,j. @⦃i, g2⦄ ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
323 ∃∃f. ⫱*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j] ↑f = g.
324 #g2 #g1 #g #i #j #Hg2 #Hg
325 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
326 lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
327 lapply (at_inv_tls … Hg2) -Hg2 #H
328 lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
329 elim (coafter_inv_pnx … Hf) [|*: // ] -Hf #g #Hg #H destruct
330 /2 width=3 by ex2_intro/
333 (* Properties with test for identity ****************************************)
335 corec lemma coafter_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ~⊚ f2 ≘ f2.
336 #f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
337 /3 width=7 by coafter_push, coafter_refl/
340 corec lemma coafter_isid_dx: ∀f2,f. 𝐈⦃f2⦄ → 𝐈⦃f⦄ → ∀f1. f1 ~⊚ f2 ≘ f.
341 #f2 #f * -f2 #f2 #g2 #Hf2 #H2 * -f #f #g #Hf #H #f1 cases (pn_split f1) * #g1 #H1
342 [ /3 width=7 by coafter_refl/
343 | @(coafter_next … H1 … H) /3 width=3 by isid_push/
347 (* Inversion lemmas with test for identity **********************************)
349 lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → 𝐈⦃f1⦄ → f2 ≡ f.
350 /3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
352 lemma coafter_isid_inv_dx: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → 𝐈⦃f2⦄ → 𝐈⦃f⦄.
353 /4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
355 (* Properties with test for uniform relocations *****************************)
357 lemma coafter_isuni_isid: ∀f2. 𝐈⦃f2⦄ → ∀f1. 𝐔⦃f1⦄ → f1 ~⊚ f2 ≘ f2.
358 #f #Hf #g #H elim H -g
359 /3 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next, eq_push_inv_isid/
364 lemma coafter_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ~⊚ ↑f2 ≘ ↑f1.
365 #f1 #f2 #Hf2 #H elim H -H
366 /5 width=7 by coafter_isid_dx, coafter_eq_repl_back2, coafter_next, coafter_push, eq_push_inv_isid/
369 lemma coafter_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ↑f2 ~⊚ f1 ≘ f → f2 ~⊚ ↑f1 ≘ f.
371 [ #f2 #Hf2 #f1 #f #Hf
372 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
373 /4 width=7 by coafter_isid_inv_sn, coafter_isid_sn, coafter_eq_repl_back0, eq_next/
374 | #f2 #_ #g2 #H2 #IH #f1 #f #Hf
375 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
376 /3 width=5 by coafter_next/
381 (* Properties with uniform relocations **************************************)
383 lemma coafter_uni_sn: ∀i,f. 𝐔❴i❵ ~⊚ f ≘ ⫯*[i] f.
384 #i elim i -i /2 width=5 by coafter_isid_sn, coafter_next/
388 lemma coafter_uni: ∀n1,n2. 𝐔❴n1❵ ~⊚ 𝐔❴n2❵ ≘ 𝐔❴n1+n2❵.
390 /4 width=5 by coafter_uni_next2, coafter_isid_sn, coafter_isid_dx, coafter_next/
393 (* Forward lemmas on at *****************************************************)
395 lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≘ i → ∀f2,f1. f2 ~⊚ f1 ≘ f →
396 ∃∃i2. @⦃i1, f1⦄ ≘ i2 & @⦃i2, f2⦄ ≘ i.
397 #i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
398 [ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
399 [1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
400 | elim (at_inv_xxp … Hf) -Hf //
403 [2: elim (coafter_inv_xxn … Hf21 … H) -f *
404 [ #g2 #g1 #Hg21 #H2 #H1 | #g2 #Hg21 #H2 ]
405 |*: elim (coafter_inv_xxp … Hf21 … H) -f
406 #g2 #g1 #Hg21 #H2 #H1
408 [4: -Hg21 |*: elim (IH … Hg … Hg21) -g -IH ]
409 /3 width=9 by at_refl, at_push, at_next, ex2_intro/
412 lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≘ i2 → @⦃i2, f2⦄ ≘ i →
413 ∀f. f2 ~⊚ f1 ≘ f → @⦃i1, f⦄ ≘ i.
414 #i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
415 [ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
416 #g2 [ #j2 ] #Hg2 [ #H22 ] #H20
417 [ elim (at_inv_xxn … Hf1 … H22) -i2 *
418 #g1 [ #j1 ] #Hg1 [ #H11 ] #H10
419 [ elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /3 width=7 by at_push/
420 | elim (coafter_inv_pnx … Hf … H20 H10) -f1 -f2 /3 width=6 by at_next/
422 | elim (coafter_inv_nxx … Hf … H20) -f2 /3 width=7 by at_next/
424 | elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H22 #H20
425 elim (at_inv_xxp … Hf1 … H22) -i2 #g1 #H11 #H10
426 elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /2 width=2 by at_refl/
430 lemma coafter_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≘ i → ∀f1,i2. @⦃i1, f1⦄ ≘ i2 →
431 ∀f2. f2 ~⊚ f1 ≘ f → @⦃i2, f2⦄ ≘ i.
432 #f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (coafter_at_fwd … Hf … H) -f
433 #j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
436 lemma coafter_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≘ i → @⦃i2, f2⦄ ≘ i →
437 ∀f1. f2 ~⊚ f1 ≘ f → @⦃i1, f1⦄ ≘ i2.
438 #i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
439 [ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
440 #g [ #j1 ] #Hg [ #H01 ] #H00
441 elim (at_inv_xxn … Hf2) -Hf2 [1,3,5,7: * |*: // ]
442 #g2 [1,3: #j2 ] #Hg2 [1,2: #H22 ] #H20
443 [ elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=7 by at_push/
444 | elim (coafter_inv_pxn … Hf1 … H20 H00) -f2 -f /3 width=5 by at_next/
445 | elim (coafter_inv_nxp … Hf1 … H20 H00)
446 | /4 width=9 by coafter_inv_nxn, at_next/
448 | elim (at_inv_xxp … Hf) -Hf // #g #H01 #H00
449 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H21 #H20
450 elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=2 by at_refl/
454 (* Properties with at *******************************************************)
456 lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
457 ∀f. f2 ~⊚ 𝐔❴i1❵ ≘ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≘ f.
459 [ #i1 #f2 #Hf2 #f #Hf
460 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
461 lapply (coafter_isid_inv_dx … Hf ?) -Hf
462 /3 width=3 by coafter_isid_sn, coafter_eq_repl_back0/
463 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
464 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
465 [ #g2 #j1 #Hg2 #H1 #H2 destruct
466 elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
467 /3 width=5 by coafter_next/
468 | #g2 #Hg2 #H2 destruct
469 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
470 /3 width=5 by coafter_next/
475 lemma coafter_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
476 ∀f. 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≘ f → f2 ~⊚ 𝐔❴i1❵ ≘ f.
478 [ #i1 #f2 #Hf2 #f #Hf
479 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
480 lapply (coafter_isid_inv_sn … Hf ?) -Hf
481 /3 width=3 by coafter_isid_dx, coafter_eq_repl_back0/
482 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
483 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
484 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
485 [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by coafter_push/
486 | #g2 #Hg2 #H2 destruct /3 width=5 by coafter_next/
491 lemma coafter_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
492 ∀f. f2 ~⊚ 𝐔❴↑i1❵ ≘ f → 𝐔❴↑i2❵ ~⊚ ⫱*[↑i2] f2 ≘ f.
494 [ #i1 #f2 #Hf2 #f #Hf
495 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
496 elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
497 lapply (coafter_isid_inv_dx … Hg ?) -Hg
498 /4 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next/
499 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
500 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
501 [ #g2 #j1 #Hg2 #H1 #H2 destruct
502 elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
503 /3 width=5 by coafter_next/
504 | #g2 #Hg2 #H2 destruct
505 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
506 /3 width=5 by coafter_next/
511 lemma coafter_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
512 ∀f. 𝐔❴↑i2❵ ~⊚ ⫱*[↑i2] f2 ≘ f → f2 ~⊚ 𝐔❴↑i1❵ ≘ f.
514 [ #i1 #f2 #Hf2 #f #Hf
515 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
516 elim (coafter_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
517 lapply (coafter_isid_inv_sn … Hg ?) -Hg
518 /4 width=7 by coafter_isid_dx, coafter_eq_repl_back0, coafter_push/
519 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
520 elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
521 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
522 [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by coafter_push/
523 | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by coafter_next/
528 lemma coafter_uni_one_dx: ∀f2,f. ⫯f2 ~⊚ 𝐔❴↑O❵ ≘ f → 𝐔❴↑O❵ ~⊚ f2 ≘ f.
529 #f2 #f #H @(coafter_uni_succ_dx … (⫯f2)) /2 width=3 by at_refl/
532 lemma coafter_uni_one_sn: ∀f1,f. 𝐔❴↑O❵ ~⊚ f1 ≘ f → ⫯f1 ~⊚ 𝐔❴↑O❵ ≘ f.
533 /3 width=3 by coafter_uni_succ_sn, at_refl/ qed-.
535 (* Forward lemmas with istot ************************************************)
537 lemma coafter_istot_fwd: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
538 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
539 #i2 #Hf1 elim (Hf2 i2) -Hf2
540 /3 width=7 by coafter_fwd_at, ex_intro/
543 lemma coafter_fwd_istot_dx: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
544 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
545 #i2 #Hf elim (coafter_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
548 lemma coafter_fwd_istot_sn: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
549 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
550 #i #Hf elim (coafter_at_fwd … Hf … H) -f
551 #i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
552 #Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
555 lemma coafter_inv_istot: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
556 /3 width=4 by coafter_fwd_istot_sn, coafter_fwd_istot_dx, conj/ qed-.
558 lemma coafter_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≘ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ~⊚ f1 ≘ f →
559 ∃∃i. @⦃i2, f2⦄ ≘ i & @⦃i1, f⦄ ≘ i.
560 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
561 /3 width=8 by coafter_fwd_at, ex2_intro/
564 lemma coafter_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≘ f → f1 ≡ f → 𝐈⦃f2⦄.
565 #f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
566 #Hf2 #Hf1 #H @isid_at_total // -Hf2
567 #i2 #i #Hf2 elim (Hf1 i2) -Hf1
568 #i0 #Hf1 lapply (at_increasing … Hf1)
569 #Hi20 lapply (coafter_fwd_at2 … i0 … Hf1 … Hf) -Hf
570 /3 width=7 by at_eq_repl_back, at_mono, at_id_le/
573 lemma coafter_fwd_isid_dx: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≘ f → f2 ≡ f → 𝐈⦃f1⦄.
574 #f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
575 #Hf2 #Hf1 #H2 @isid_at_total // -Hf1
576 #i1 #i2 #Hi12 elim (coafter_at1_fwd … Hi12 … Hf) -f1
577 /3 width=8 by at_inj, at_eq_repl_back/
580 corec fact coafter_inj_O_aux: ∀f1. @⦃0, f1⦄ ≘ 0 → H_coafter_inj f1.
581 #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
582 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
583 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
584 cases (H2g1 0) #n #Hn
585 cases (coafter_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
586 [ cases (coafter_inv_pxp … H2f … H1 H) -f1 -f #g22 #H2g #H22
587 @(eq_push … H21 H22) -f21 -f22
588 | cases (coafter_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
589 @(eq_next … H21 H22) -f21 -f22
591 @(coafter_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -coafter_inj_O_aux
592 /2 width=1 by coafter_tls, istot_tls, at_pxx_tls/
595 fact coafter_inj_aux: (∀f1. @⦃0, f1⦄ ≘ 0 → H_coafter_inj f1) →
596 ∀i2,f1. @⦃0, f1⦄ ≘ i2 → H_coafter_inj f1.
597 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
598 #i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
599 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
600 elim (coafter_inv_nxx … H1f … H1) -H1f #g #H1g #H
601 lapply (coafter_inv_nxp … H2f … H1 H) -f #H2g
602 /3 width=6 by istot_inv_next/
605 theorem coafter_inj: ∀f1. H_coafter_inj f1.
606 #f1 #H cases (H 0) /3 width=7 by coafter_inj_aux, coafter_inj_O_aux/
609 corec fact coafter_fwd_isid2_O_aux: ∀f1. @⦃0, f1⦄ ≘ 0 →
610 H_coafter_fwd_isid2 f1.
611 #f1 #H1f1 #f2 #f #H #H2f1 #Hf
612 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
613 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
614 cases (H2g1 0) #n #Hn
615 cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #H0
616 [ lapply (isid_inv_push … Hf … H0) -Hf #Hg
617 @(isid_push … H2) -H2
618 /3 width=7 by coafter_tls_O, at_pxx_tls, istot_tls, isid_tls/
619 | cases (isid_inv_next … Hf … H0)
623 fact coafter_fwd_isid2_aux: (∀f1. @⦃0, f1⦄ ≘ 0 → H_coafter_fwd_isid2 f1) →
624 ∀i2,f1. @⦃0, f1⦄ ≘ i2 → H_coafter_fwd_isid2 f1.
625 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
626 #i2 #IH #f1 #H1f1 #f2 #f #H #H2f1 #Hf
627 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
628 elim (coafter_inv_nxx … H … H1) -H #g #Hg #H0
629 @(IH … Hg1 … Hg) /2 width=3 by istot_inv_next, isid_inv_push/ (**) (* full auto fails *)
632 lemma coafter_fwd_isid2: ∀f1. H_coafter_fwd_isid2 f1.
633 #f1 #f2 #f #Hf #H cases (H 0)
634 /3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
637 fact coafter_isfin2_fwd_O_aux: ∀f1. @⦃0, f1⦄ ≘ 0 →
638 H_coafter_isfin2_fwd f1.
640 generalize in match Hf1; generalize in match f1; -f1
642 [ /3 width=4 by coafter_isid_inv_dx, isfin_isid/ ]
643 #f2 #_ #IH #f1 #H #Hf1 #f #Hf
644 elim (at_inv_pxp … H) -H [ |*: // ] #g1 #H1
645 lapply (istot_inv_push … Hf1 … H1) -Hf1 #Hg1
647 [ elim (coafter_inv_ppx … Hf) | elim (coafter_inv_pnx … Hf)
648 ] -Hf [1,6: |*: // ] #g #Hg #H0 destruct
649 /5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls_O/
652 fact coafter_isfin2_fwd_aux: (∀f1. @⦃0, f1⦄ ≘ 0 → H_coafter_isfin2_fwd f1) →
653 ∀i2,f1. @⦃0, f1⦄ ≘ i2 → H_coafter_isfin2_fwd f1.
654 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
655 #i2 #IH #f1 #H1f1 #f2 #Hf2 #H2f1 #f #Hf
656 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
657 elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0
658 lapply (IH … Hg1 … Hg) -i2 -Hg
659 /2 width=4 by istot_inv_next, isfin_push/ (**) (* full auto fails *)
662 lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
663 #f1 #f2 #Hf2 #Hf1 cases (Hf1 0)
664 /3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
667 lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≘ f → ∀fa,fb. fa ⋓ fb ≘ f →
668 ∃∃f1a,f1b. f2 ~⊚ f1a ≘ fa & f2 ~⊚ f1b ≘ fb & f1a ⋓ f1b ≘ f1.
670 [ #f #Hf #f2 #Hf2 #f1 #H1f #fa #fb #H2f
671 elim (sor_inv_isid3 … H2f) -H2f //
672 lapply (coafter_fwd_isid2 … H1f ??) -H1f //
673 /3 width=5 by ex3_2_intro, coafter_isid_dx, sor_isid/
674 | #f #_ #IH #f2 #Hf2 #f1 #H1 #fa #fb #H2
675 elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #H2f
676 elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ] #g2 [ #g1 ] #H1f #Hgf2
677 [ lapply (istot_inv_push … Hf2 … Hgf2) | lapply (istot_inv_next … Hf2 … Hgf2) ] -Hf2 #Hg2
678 elim (IH … Hg2 … H1f … H2f) -f -Hg2
679 /3 width=11 by sor_pp, ex3_2_intro, coafter_refl, coafter_next/
680 | #f #_ #IH #f2 #Hf2 #f1 #H1 #fa #fb #H2
681 elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #H1f #Hgf2
682 lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
683 elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #H2f
684 elim (IH … Hg2 … H1f … H2f) -f -Hg2
685 /3 width=11 by sor_np, sor_pn, sor_nn, ex3_2_intro, coafter_refl, coafter_push/
689 (* Properties with istot ****************************************************)
691 lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≘ f → ∀f1a,f1b. f1a ⋓ f1b ≘ f1 →
692 ∃∃fa,fb. f2 ~⊚ f1a ≘ fa & f2 ~⊚ f1b ≘ fb & fa ⋓ fb ≘ f.
694 [ #f #Hf #f2 #Hf2 #f1 #Hf #f1a #f1b #Hf1
695 lapply (coafter_fwd_isid2 … Hf ??) -Hf // #H2f1
696 elim (sor_inv_isid3 … Hf1) -Hf1 //
697 /3 width=5 by coafter_isid_dx, sor_idem, ex3_2_intro/
698 | #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
699 elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ]
700 [ #g2 #g1 #Hf #Hgf2 #Hgf1
701 elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #Hg1
702 lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
703 elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
704 /3 width=11 by coafter_refl, sor_pp, ex3_2_intro/
706 lapply (istot_inv_next … Hf2 … Hgf2) -Hf2 #Hg2
707 elim (IH … Hf … H2) // -f1 -IH -Hg2
708 /3 width=11 by coafter_next, sor_pp, ex3_2_intro/
710 | #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
711 elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #Hf #Hgf2 #Hgf1
712 lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
713 elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #Hg1
714 elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
715 /3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
719 (* Properties with after ****************************************************)
721 corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≘ f4 →
722 ∀f1,f2. f1 ~⊚ f2 ≘ f0 →
723 ∀f. f2 ~⊚ f3 ≘ f → f1 ~⊚ f ≘ f4.
724 #f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
725 [ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
726 cases (coafter_inv_xxp … Hg0 … H0) -g0
728 cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
729 #f #Hf #H /3 width=7 by coafter_refl/
730 | #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
731 cases (coafter_inv_xxp … Hg0 … H0) -g0
733 cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
734 #f #Hf #H /3 width=7 by coafter_push/
735 | #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
736 cases (coafter_inv_xxn … Hg0 … H0) -g0 *
737 [ #f1 #f2 #Hf0 #H1 #H2
738 cases (coafter_inv_nxx … Hg … H2) -g2
739 #f #Hf #H /3 width=7 by coafter_push/
740 | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
745 corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≘ f4 →
746 ∀f2, f3. f2 ~⊚ f3 ≘ f0 →
747 ∀f. f1 ~⊚ f2 ≘ f → f ~⊚ f3 ≘ f4.
748 #f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
749 [ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
750 cases (coafter_inv_xxp … Hg0 … H0) -g0
752 cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
753 #f #Hf #H /3 width=7 by coafter_refl/
754 | #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
755 cases (coafter_inv_xxn … Hg0 … H0) -g0 *
756 [ #f2 #f3 #Hf0 #H2 #H3
757 cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
758 #f #Hf #H /3 width=7 by coafter_push/
760 cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
761 #f #Hf #H /3 width=6 by coafter_next/
763 | #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
764 cases (coafter_inv_nxx … Hg … H1) -g1
765 #f #Hg #H /3 width=6 by coafter_next/