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15 include "ground_2/notation/relations/rafter_3.ma".
16 include "ground_2/relocation/rtmap_istot.ma".
18 (* RELOCATION MAP ***********************************************************)
20 coinductive after: relation3 rtmap rtmap rtmap ≝
21 | after_refl: ∀f1,f2,f,g1,g2,g.
22 after f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → after g1 g2 g
23 | after_push: ∀f1,f2,f,g1,g2,g.
24 after f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → after g1 g2 g
25 | after_next: ∀f1,f2,f,g1,g.
26 after f1 f2 f → ↑f1 = g1 → ↑f = g → after g1 f2 g
29 interpretation "relational composition (rtmap)"
30 'RAfter f1 f2 f = (after f1 f2 f).
32 definition H_after_inj: predicate rtmap ≝
34 ∀f,f21,f22. f1 ⊚ f21 ≘ f → f1 ⊚ f22 ≘ f → f21 ≡ f22.
36 (* Basic inversion lemmas ***************************************************)
38 lemma after_inv_ppx: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
39 ∃∃f. f1 ⊚ f2 ≘ f & ⫯f = g.
40 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
41 [ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
42 >(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
43 /2 width=3 by ex2_intro/
44 | #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
45 elim (discr_push_next … Hx2)
46 | #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
47 elim (discr_push_next … Hx1)
51 lemma after_inv_pnx: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
52 ∃∃f. f1 ⊚ f2 ≘ f & ↑f = g.
53 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
54 [ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
55 elim (discr_next_push … Hx2)
56 | #g2 #g #Hf #H1 #H2 #H3 #x1 #x2 #Hx1 #Hx2 destruct
57 >(injective_push … Hx1) >(injective_next … Hx2) -x2 -x1
58 /2 width=3 by ex2_intro/
59 | #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
60 elim (discr_push_next … Hx1)
64 lemma after_inv_nxx: ∀g1,f2,g. g1 ⊚ f2 ≘ g → ∀f1. ↑f1 = g1 →
65 ∃∃f. f1 ⊚ f2 ≘ f & ↑f = g.
66 #g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
67 [ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
68 elim (discr_next_push … Hx1)
69 | #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
70 elim (discr_next_push … Hx1)
71 | #g #Hf #H1 #H #x1 #Hx1 destruct
72 >(injective_next … Hx1) -x1
73 /2 width=3 by ex2_intro/
77 (* Advanced inversion lemmas ************************************************)
79 lemma after_inv_ppp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
80 ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⊚ f2 ≘ f.
81 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_ppx … Hg … H1 H2) -g1 -g2
82 #x #Hf #Hx destruct <(injective_push … Hx) -f //
85 lemma after_inv_ppn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
86 ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
87 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_ppx … Hg … H1 H2) -g1 -g2
88 #x #Hf #Hx destruct elim (discr_push_next … Hx)
91 lemma after_inv_pnn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
92 ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⊚ f2 ≘ f.
93 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_pnx … Hg … H1 H2) -g1 -g2
94 #x #Hf #Hx destruct <(injective_next … Hx) -f //
97 lemma after_inv_pnp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
98 ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
99 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_pnx … Hg … H1 H2) -g1 -g2
100 #x #Hf #Hx destruct elim (discr_next_push … Hx)
103 lemma after_inv_nxn: ∀g1,f2,g. g1 ⊚ f2 ≘ g →
104 ∀f1,f. ↑f1 = g1 → ↑f = g → f1 ⊚ f2 ≘ f.
105 #g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
106 #x #Hf #Hx destruct <(injective_next … Hx) -f //
109 lemma after_inv_nxp: ∀g1,f2,g. g1 ⊚ f2 ≘ g →
110 ∀f1,f. ↑f1 = g1 → ⫯f = g → ⊥.
111 #g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
112 #x #Hf #Hx destruct elim (discr_next_push … Hx)
115 lemma after_inv_pxp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
116 ∀f1,f. ⫯f1 = g1 → ⫯f = g →
117 ∃∃f2. f1 ⊚ f2 ≘ f & ⫯f2 = g2.
118 #g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
119 [ elim (after_inv_pnp … Hg … H1 … H) -g1 -g -f1 -f //
120 | lapply (after_inv_ppp … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
124 lemma after_inv_pxn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
125 ∀f1,f. ⫯f1 = g1 → ↑f = g →
126 ∃∃f2. f1 ⊚ f2 ≘ f & ↑f2 = g2.
127 #g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
128 [ lapply (after_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
129 | elim (after_inv_ppn … Hg … H1 … H) -g1 -g -f1 -f //
133 lemma after_inv_xxp: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f. ⫯f = g →
134 ∃∃f1,f2. f1 ⊚ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2.
135 * * [2: #m1 ] #g1 #g2 #g #Hg #f #H
136 [ elim (after_inv_nxp … Hg … H) -g2 -g -f //
137 | elim (after_inv_pxp … Hg … H) -g /2 width=5 by ex3_2_intro/
141 lemma after_inv_xxn: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f. ↑f = g →
142 (∃∃f1,f2. f1 ⊚ f2 ≘ f & ⫯f1 = g1 & ↑f2 = g2) ∨
143 ∃∃f1. f1 ⊚ g2 ≘ f & ↑f1 = g1.
144 * * [2: #m1 ] #g1 #g2 #g #Hg #f #H
145 [ /4 width=5 by after_inv_nxn, or_intror, ex2_intro/
146 | elim (after_inv_pxn … Hg … H) -g
147 /3 width=5 by or_introl, ex3_2_intro/
151 lemma after_inv_pxx: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f1. ⫯f1 = g1 →
152 (∃∃f2,f. f1 ⊚ f2 ≘ f & ⫯f2 = g2 & ⫯f = g) ∨
153 (∃∃f2,f. f1 ⊚ f2 ≘ f & ↑f2 = g2 & ↑f = g).
154 #g1 * * [2: #m2 ] #g2 #g #Hg #f1 #H
155 [ elim (after_inv_pnx … Hg … H) -g1
156 /3 width=5 by or_intror, ex3_2_intro/
157 | elim (after_inv_ppx … Hg … H) -g1
158 /3 width=5 by or_introl, ex3_2_intro/
162 (* Basic properties *********************************************************)
164 corec lemma after_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ⊚ f1 ≘ f).
165 #f1 #f #f2 * -f2 -f1 -f
166 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
167 [ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by after_refl/
168 | cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by after_push/
169 | cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by after_next/
173 lemma after_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ⊚ f1 ≘ f).
174 #f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back2/
177 corec lemma after_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ⊚ f1 ≘ f).
178 #f2 #f #f1 * -f2 -f1 -f
179 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
180 [ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by after_refl/
181 | cases (eq_inv_nx … H0 … H11) -g11 /3 width=7 by after_push/
182 | @(after_next … H2 H) /2 width=5 by/
186 lemma after_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ⊚ f1 ≘ f).
187 #f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back1/
190 corec lemma after_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ⊚ f1 ≘ f).
191 #f2 #f1 #f * -f2 -f1 -f
192 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
193 [ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by after_refl/
194 | cases (eq_inv_nx … H0 … H01) -g01 /3 width=7 by after_push/
195 | cases (eq_inv_nx … H0 … H01) -g01 /3 width=5 by after_next/
199 lemma after_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ⊚ f1 ≘ f).
200 #f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back0/
203 (* Main properties **********************************************************)
205 corec theorem after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≘ f4 →
206 ∀f1,f2. f1 ⊚ f2 ≘ f0 →
207 ∀f. f2 ⊚ f3 ≘ f → f1 ⊚ f ≘ f4.
208 #f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
209 [ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
210 cases (after_inv_xxp … Hg0 … H0) -g0
212 cases (after_inv_ppx … Hg … H2 H3) -g2 -g3
213 #f #Hf #H /3 width=7 by after_refl/
214 | #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
215 cases (after_inv_xxp … Hg0 … H0) -g0
217 cases (after_inv_pnx … Hg … H2 H3) -g2 -g3
218 #f #Hf #H /3 width=7 by after_push/
219 | #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
220 cases (after_inv_xxn … Hg0 … H0) -g0 *
221 [ #f1 #f2 #Hf0 #H1 #H2
222 cases (after_inv_nxx … Hg … H2) -g2
223 #f #Hf #H /3 width=7 by after_push/
224 | #f1 #Hf0 #H1 /3 width=6 by after_next/
229 corec theorem after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≘ f4 →
230 ∀f2, f3. f2 ⊚ f3 ≘ f0 →
231 ∀f. f1 ⊚ f2 ≘ f → f ⊚ f3 ≘ f4.
232 #f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
233 [ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
234 cases (after_inv_xxp … Hg0 … H0) -g0
236 cases (after_inv_ppx … Hg … H1 H2) -g1 -g2
237 #f #Hf #H /3 width=7 by after_refl/
238 | #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
239 cases (after_inv_xxn … Hg0 … H0) -g0 *
240 [ #f2 #f3 #Hf0 #H2 #H3
241 cases (after_inv_ppx … Hg … H1 H2) -g1 -g2
242 #f #Hf #H /3 width=7 by after_push/
244 cases (after_inv_pnx … Hg … H1 H2) -g1 -g2
245 #f #Hf #H /3 width=6 by after_next/
247 | #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
248 cases (after_inv_nxx … Hg … H1) -g1
249 #f #Hg #H /3 width=6 by after_next/
253 (* Main inversion lemmas ****************************************************)
255 corec theorem after_mono: ∀f1,f2,x,y. f1 ⊚ f2 ≘ x → f1 ⊚ f2 ≘ y → x ≡ y.
256 #f1 #f2 #x #y * -f1 -f2 -x
257 #f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
258 [ cases (after_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
259 | cases (after_inv_pnx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_next/
260 | cases (after_inv_nxx … Hy … H1) -g1 /3 width=8 by eq_next/
264 lemma after_mono_eq: ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g →
265 f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
266 /4 width=4 by after_mono, after_eq_repl_back1, after_eq_repl_back2/ qed-.
268 (* Properties on tls ********************************************************)
270 lemma after_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≘ n →
271 f1 ⊚ f2 ≘ f → ⫱*[n]f1 ⊚ f2 ≘ ⫱*[n]f.
273 #n #IH #f1 #f2 #f #Hf1 #Hf
274 cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
275 cases (after_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
276 <tls_xn <tls_xn /2 width=1 by/
279 (* Properties on isid *******************************************************)
281 corec lemma after_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊚ f2 ≘ f2.
282 #f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
283 /3 width=7 by after_push, after_refl/
286 corec lemma after_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⊚ f2 ≘ f1.
287 #f2 * -f2 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) * #g1 #H1
288 [ /3 width=7 by after_refl/
289 | @(after_next … H1 H1) /3 width=3 by isid_push/
293 (* Inversion lemmas on isid *************************************************)
295 lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f1⦄ → f2 ≡ f.
296 /3 width=6 by after_isid_sn, after_mono/ qed-.
298 lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f2⦄ → f1 ≡ f.
299 /3 width=6 by after_isid_dx, after_mono/ qed-.
301 corec lemma after_fwd_isid1: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
302 #f1 #f2 #f * -f1 -f2 -f
303 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
304 [ /4 width=6 by isid_inv_push, isid_push/ ]
305 cases (isid_inv_next … H … H0)
308 corec lemma after_fwd_isid2: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
309 #f1 #f2 #f * -f1 -f2 -f
310 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
311 [ /4 width=6 by isid_inv_push, isid_push/ ]
312 cases (isid_inv_next … H … H0)
315 lemma after_inv_isid3: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
316 /3 width=4 by after_fwd_isid2, after_fwd_isid1, conj/ qed-.
318 (* Properties on isuni ******************************************************)
320 lemma after_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ⊚ ↑f2 ≘ ↑f1.
321 #f1 #f2 #Hf2 #H elim H -H
322 /5 width=7 by after_isid_dx, after_eq_repl_back2, after_next, after_push, eq_push_inv_isid/
325 lemma after_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ↑f2 ⊚ f1 ≘ f → f2 ⊚ ↑f1 ≘ f.
327 [ #f2 #Hf2 #f1 #f #Hf
328 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
329 /4 width=7 by after_isid_inv_sn, after_isid_sn, after_eq_repl_back0, eq_next/
330 | #f2 #_ #g2 #H2 #IH #f1 #f #Hf
331 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
332 /3 width=5 by after_next/
336 (* Properties on uni ********************************************************)
338 lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≘ 𝐔❴n1+n2❵.
339 @nat_elim2 [3: #n #m <plus_n_Sm ] (**) (* full auto fails *)
340 /4 width=5 by after_uni_next2, after_isid_dx, after_isid_sn, after_next/
343 (* Forward lemmas on at *****************************************************)
345 lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≘ i → ∀f2,f1. f2 ⊚ f1 ≘ f →
346 ∃∃i2. @⦃i1, f1⦄ ≘ i2 & @⦃i2, f2⦄ ≘ i.
347 #i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
348 [ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
349 [1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
350 | elim (at_inv_xxp … Hf) -Hf //
353 [2: elim (after_inv_xxn … Hf21 … H) -f *
354 [ #g2 #g1 #Hg21 #H2 #H1 | #g2 #Hg21 #H2 ]
355 |*: elim (after_inv_xxp … Hf21 … H) -f
356 #g2 #g1 #Hg21 #H2 #H1
358 [4: -Hg21 |*: elim (IH … Hg … Hg21) -g -IH ]
359 /3 width=9 by at_refl, at_push, at_next, ex2_intro/
362 lemma after_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≘ i2 → @⦃i2, f2⦄ ≘ i →
363 ∀f. f2 ⊚ f1 ≘ f → @⦃i1, f⦄ ≘ i.
364 #i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
365 [ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
366 #g2 [ #j2 ] #Hg2 [ #H22 ] #H20
367 [ elim (at_inv_xxn … Hf1 … H22) -i2 *
368 #g1 [ #j1 ] #Hg1 [ #H11 ] #H10
369 [ elim (after_inv_ppx … Hf … H20 H10) -f1 -f2 /3 width=7 by at_push/
370 | elim (after_inv_pnx … Hf … H20 H10) -f1 -f2 /3 width=6 by at_next/
372 | elim (after_inv_nxx … Hf … H20) -f2 /3 width=7 by at_next/
374 | elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H22 #H20
375 elim (at_inv_xxp … Hf1 … H22) -i2 #g1 #H11 #H10
376 elim (after_inv_ppx … Hf … H20 H10) -f1 -f2 /2 width=2 by at_refl/
380 lemma after_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≘ i → ∀f1,i2. @⦃i1, f1⦄ ≘ i2 →
381 ∀f2. f2 ⊚ f1 ≘ f → @⦃i2, f2⦄ ≘ i.
382 #f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (after_at_fwd … Hf … H) -f
383 #j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
386 lemma after_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≘ i → @⦃i2, f2⦄ ≘ i →
387 ∀f1. f2 ⊚ f1 ≘ f → @⦃i1, f1⦄ ≘ i2.
388 #i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
389 [ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
390 #g [ #j1 ] #Hg [ #H01 ] #H00
391 elim (at_inv_xxn … Hf2) -Hf2 [1,3,5,7: * |*: // ]
392 #g2 [1,3: #j2 ] #Hg2 [1,2: #H22 ] #H20
393 [ elim (after_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=7 by at_push/
394 | elim (after_inv_pxn … Hf1 … H20 H00) -f2 -f /3 width=5 by at_next/
395 | elim (after_inv_nxp … Hf1 … H20 H00)
396 | /4 width=9 by after_inv_nxn, at_next/
398 | elim (at_inv_xxp … Hf) -Hf // #g #H01 #H00
399 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H21 #H20
400 elim (after_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=2 by at_refl/
404 (* Properties with at *******************************************************)
406 lemma after_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
407 ∀f. f2 ⊚ 𝐔❴i1❵ ≘ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≘ f.
409 [ #i1 #f2 #Hf2 #f #Hf
410 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
411 lapply (after_isid_inv_dx … Hf ?) -Hf
412 /3 width=3 by after_isid_sn, after_eq_repl_back0/
413 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
414 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
415 [ #g2 #j1 #Hg2 #H1 #H2 destruct
416 elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
417 <tls_xn /3 width=5 by after_next/
418 | #g2 #Hg2 #H2 destruct
419 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
420 <tls_xn /3 width=5 by after_next/
425 lemma after_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
426 ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≘ f → f2 ⊚ 𝐔❴i1❵ ≘ f.
428 [ #i1 #f2 #Hf2 #f #Hf
429 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
430 lapply (after_isid_inv_sn … Hf ?) -Hf
431 /3 width=3 by after_isid_dx, after_eq_repl_back0/
432 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
433 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
434 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
435 [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by after_push/
436 | #g2 #Hg2 #H2 destruct /3 width=5 by after_next/
441 lemma after_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
442 ∀f. f2 ⊚ 𝐔❴↑i1❵ ≘ f → 𝐔❴↑i2❵ ⊚ ⫱*[↑i2] f2 ≘ f.
444 [ #i1 #f2 #Hf2 #f #Hf
445 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
446 elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
447 lapply (after_isid_inv_dx … Hg ?) -Hg
448 /4 width=5 by after_isid_sn, after_eq_repl_back0, after_next/
449 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
450 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
451 [ #g2 #j1 #Hg2 #H1 #H2 destruct
452 elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
453 <tls_xn /3 width=5 by after_next/
454 | #g2 #Hg2 #H2 destruct
455 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
456 <tls_xn /3 width=5 by after_next/
461 lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
462 ∀f. 𝐔❴↑i2❵ ⊚ ⫱*[↑i2] f2 ≘ f → f2 ⊚ 𝐔❴↑i1❵ ≘ f.
464 [ #i1 #f2 #Hf2 #f #Hf
465 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
466 elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
467 lapply (after_isid_inv_sn … Hg ?) -Hg
468 /4 width=7 by after_isid_dx, after_eq_repl_back0, after_push/
469 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
470 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
471 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
472 [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
473 | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
478 lemma after_uni_one_dx: ∀f2,f. ⫯f2 ⊚ 𝐔❴↑O❵ ≘ f → 𝐔❴↑O❵ ⊚ f2 ≘ f.
479 #f2 #f #H @(after_uni_succ_dx … (⫯f2)) /2 width=3 by at_refl/
482 lemma after_uni_one_sn: ∀f1,f. 𝐔❴↑O❵ ⊚ f1 ≘ f → ⫯f1 ⊚ 𝐔❴↑O❵ ≘ f.
483 /3 width=3 by after_uni_succ_sn, at_refl/ qed-.
485 (* Forward lemmas on istot **************************************************)
487 lemma after_istot_fwd: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
488 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
489 #i2 #Hf1 elim (Hf2 i2) -Hf2
490 /3 width=7 by after_fwd_at, ex_intro/
493 lemma after_fwd_istot_dx: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
494 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
495 #i2 #Hf elim (after_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
498 lemma after_fwd_istot_sn: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
499 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
500 #i #Hf elim (after_at_fwd … Hf … H) -f
501 #i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
502 #Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
505 lemma after_inv_istot: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
506 /3 width=4 by after_fwd_istot_sn, after_fwd_istot_dx, conj/ qed-.
508 lemma after_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≘ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ⊚ f1 ≘ f →
509 ∃∃i. @⦃i2, f2⦄ ≘ i & @⦃i1, f⦄ ≘ i.
510 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
511 /3 width=8 by after_fwd_at, ex2_intro/
514 lemma after_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ⊚ f1 ≘ f → f1 ≡ f → 𝐈⦃f2⦄.
515 #f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
516 #Hf2 #Hf1 #H @isid_at_total // -Hf2
517 #i2 #i #Hf2 elim (Hf1 i2) -Hf1
518 #i0 #Hf1 lapply (at_increasing … Hf1)
519 #Hi20 lapply (after_fwd_at2 … i0 … Hf1 … Hf) -Hf
520 /3 width=7 by at_eq_repl_back, at_mono, at_id_le/
523 lemma after_fwd_isid_dx: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ⊚ f1 ≘ f → f2 ≡ f → 𝐈⦃f1⦄.
524 #f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
525 #Hf2 #Hf1 #H2 @isid_at_total // -Hf1
526 #i1 #i2 #Hi12 elim (after_at1_fwd … Hi12 … Hf) -f1
527 /3 width=8 by at_inj, at_eq_repl_back/
530 corec fact after_inj_O_aux: ∀f1. @⦃0, f1⦄ ≘ 0 → H_after_inj f1.
531 #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
532 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
533 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
534 cases (H2g1 0) #n #Hn
535 cases (after_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
536 [ cases (after_inv_pxp … H2f … H1 H) -f1 -f #g22 #H2g #H22
537 @(eq_push … H21 H22) -f21 -f22
538 | cases (after_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
539 @(eq_next … H21 H22) -f21 -f22
541 @(after_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -after_inj_O_aux
542 /2 width=1 by after_tls, istot_tls, at_pxx_tls/
545 fact after_inj_aux: (∀f1. @⦃0, f1⦄ ≘ 0 → H_after_inj f1) →
546 ∀i2,f1. @⦃0, f1⦄ ≘ i2 → H_after_inj f1.
547 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
548 #i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
549 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
550 elim (after_inv_nxx … H1f … H1) -H1f #g #H1g #H
551 lapply (after_inv_nxn … H2f … H1 H) -f #H2g
552 /3 width=6 by istot_inv_next/
555 theorem after_inj: ∀f1. H_after_inj f1.
556 #f1 #H cases (H 0) /3 width=7 by after_inj_aux, after_inj_O_aux/