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15 include "ground_2/notation/relations/runion_3.ma".
16 include "ground_2/relocation/rtmap_isfin.ma".
17 include "ground_2/relocation/rtmap_sle.ma".
19 coinductive sor: relation3 rtmap rtmap rtmap ≝
20 | sor_pp: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → sor g1 g2 g
21 | sor_np: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → sor g1 g2 g
22 | sor_pn: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → sor g1 g2 g
23 | sor_nn: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → sor g1 g2 g
26 interpretation "union (rtmap)"
27 'RUnion f1 f2 f = (sor f1 f2 f).
29 (* Basic inversion lemmas ***************************************************)
31 lemma sor_inv_ppx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
32 ∃∃f. f1 ⋓ f2 ≡ f & ↑f = g.
33 #g1 #g2 #g * -g1 -g2 -g
34 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
35 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
36 try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
37 try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
38 try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
39 /2 width=3 by ex2_intro/
42 lemma sor_inv_npx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
43 ∃∃f. f1 ⋓ f2 ≡ f & ⫯f = g.
44 #g1 #g2 #g * -g1 -g2 -g
45 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
46 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
47 try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
48 try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
49 try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
50 /2 width=3 by ex2_intro/
53 lemma sor_inv_pnx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
54 ∃∃f. f1 ⋓ f2 ≡ f & ⫯f = g.
55 #g1 #g2 #g * -g1 -g2 -g
56 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
57 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
58 try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
59 try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
60 try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
61 /2 width=3 by ex2_intro/
64 lemma sor_inv_nnx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
65 ∃∃f. f1 ⋓ f2 ≡ f & ⫯f = g.
66 #g1 #g2 #g * -g1 -g2 -g
67 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
68 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
69 try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
70 try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
71 try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
72 /2 width=3 by ex2_intro/
75 (* Advanced inversion lemmas ************************************************)
77 lemma sor_inv_ppn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
78 ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
79 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
80 elim (sor_inv_ppx … H … H1 H2) -g1 -g2 #x #_ #H destruct
81 /2 width=3 by discr_push_next/
84 lemma sor_inv_nxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
85 ∀f1,f. ⫯f1 = g1 → ↑f = g → ⊥.
86 #g1 #g2 #g #H #f1 #f #H1 #H0
87 elim (pn_split g2) * #f2 #H2
88 [ elim (sor_inv_npx … H … H1 H2)
89 | elim (sor_inv_nnx … H … H1 H2)
90 ] -g1 -g2 #x #_ #H destruct
91 /2 width=3 by discr_next_push/
94 lemma sor_inv_xnp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
95 ∀f2,f. ⫯f2 = g2 → ↑f = g → ⊥.
96 #g1 #g2 #g #H #f2 #f #H2 #H0
97 elim (pn_split g1) * #f1 #H1
98 [ elim (sor_inv_pnx … H … H1 H2)
99 | elim (sor_inv_nnx … H … H1 H2)
100 ] -g1 -g2 #x #_ #H destruct
101 /2 width=3 by discr_next_push/
104 lemma sor_inv_ppp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
105 ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≡ f.
106 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
107 elim (sor_inv_ppx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
108 <(injective_push … H) -f //
111 lemma sor_inv_npn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
112 ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
113 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
114 elim (sor_inv_npx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
115 <(injective_next … H) -f //
118 lemma sor_inv_pnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
119 ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
120 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
121 elim (sor_inv_pnx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
122 <(injective_next … H) -f //
125 lemma sor_inv_nnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
126 ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
127 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
128 elim (sor_inv_nnx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
129 <(injective_next … H) -f //
132 lemma sor_inv_pxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
133 ∀f1,f. ↑f1 = g1 → ↑f = g →
134 ∃∃f2. f1 ⋓ f2 ≡ f & ↑f2 = g2.
135 #g1 #g2 #g #H #f1 #f #H1 #H0
136 elim (pn_split g2) * #f2 #H2
137 [ /3 width=7 by sor_inv_ppp, ex2_intro/
138 | elim (sor_inv_xnp … H … H2 H0)
142 lemma sor_inv_xpp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
143 ∀f2,f. ↑f2 = g2 → ↑f = g →
144 ∃∃f1. f1 ⋓ f2 ≡ f & ↑f1 = g1.
145 #g1 #g2 #g #H #f2 #f #H2 #H0
146 elim (pn_split g1) * #f1 #H1
147 [ /3 width=7 by sor_inv_ppp, ex2_intro/
148 | elim (sor_inv_nxp … H … H1 H0)
152 lemma sor_inv_pxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
153 ∀f1,f. ↑f1 = g1 → ⫯f = g →
154 ∃∃f2. f1 ⋓ f2 ≡ f & ⫯f2 = g2.
155 #g1 #g2 #g #H #f1 #f #H1 #H0
156 elim (pn_split g2) * #f2 #H2
157 [ elim (sor_inv_ppn … H … H1 H2 H0)
158 | /3 width=7 by sor_inv_pnn, ex2_intro/
162 lemma sor_inv_xpn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
163 ∀f2,f. ↑f2 = g2 → ⫯f = g →
164 ∃∃f1. f1 ⋓ f2 ≡ f & ⫯f1 = g1.
165 #g1 #g2 #g #H #f2 #f #H2 #H0
166 elim (pn_split g1) * #f1 #H1
167 [ elim (sor_inv_ppn … H … H1 H2 H0)
168 | /3 width=7 by sor_inv_npn, ex2_intro/
172 lemma sor_inv_xxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ↑f = g →
173 ∃∃f1,f2. f1 ⋓ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2.
175 elim (pn_split g1) * #f1 #H1
176 [ elim (sor_inv_pxp … H … H1 H0) -g /2 width=5 by ex3_2_intro/
177 | elim (sor_inv_nxp … H … H1 H0)
181 lemma sor_inv_nxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
182 ∀f1,f. ⫯f1 = g1 → ⫯f = g →
183 (∃∃f2. f1 ⋓ f2 ≡ f & ↑f2 = g2) ∨
184 ∃∃f2. f1 ⋓ f2 ≡ f & ⫯f2 = g2.
185 #g1 #g2 elim (pn_split g2) *
186 /4 width=7 by sor_inv_npn, sor_inv_nnn, ex2_intro, or_intror, or_introl/
189 lemma sor_inv_xnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
190 ∀f2,f. ⫯f2 = g2 → ⫯f = g →
191 (∃∃f1. f1 ⋓ f2 ≡ f & ↑f1 = g1) ∨
192 ∃∃f1. f1 ⋓ f2 ≡ f & ⫯f1 = g1.
193 #g1 elim (pn_split g1) *
194 /4 width=7 by sor_inv_pnn, sor_inv_nnn, ex2_intro, or_intror, or_introl/
197 lemma sor_inv_xxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ⫯f = g →
198 ∨∨ ∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ↑f2 = g2
199 | ∃∃f1,f2. f1 ⋓ f2 ≡ f & ↑f1 = g1 & ⫯f2 = g2
200 | ∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ⫯f2 = g2.
202 elim (pn_split g1) * #f1 #H1
203 [ elim (sor_inv_pxn … H … H1 H0) -g
204 /3 width=5 by or3_intro1, ex3_2_intro/
205 | elim (sor_inv_nxn … H … H1 H0) -g *
206 /3 width=5 by or3_intro0, or3_intro2, ex3_2_intro/
210 (* Main inversion lemmas ****************************************************)
212 corec theorem sor_mono: ∀f1,f2,x,y. f1 ⋓ f2 ≡ x → f1 ⋓ f2 ≡ y → x ≗ y.
213 #f1 #f2 #x #y * -f1 -f2 -x
214 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #H
215 [ cases (sor_inv_ppx … H … H1 H2)
216 | cases (sor_inv_npx … H … H1 H2)
217 | cases (sor_inv_pnx … H … H1 H2)
218 | cases (sor_inv_nnx … H … H1 H2)
220 /3 width=5 by eq_push, eq_next/
223 (* Basic properties *********************************************************)
225 corec lemma sor_eq_repl_back1: ∀f2,f. eq_repl_back … (λf1. f1 ⋓ f2 ≡ f).
226 #f2 #f #f1 * -f1 -f2 -f
227 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x #Hx
228 try cases (eq_inv_px … Hx … H1) try cases (eq_inv_nx … Hx … H1) -g1
229 /3 width=7 by sor_pp, sor_np, sor_pn, sor_nn/
232 lemma sor_eq_repl_fwd1: ∀f2,f. eq_repl_fwd … (λf1. f1 ⋓ f2 ≡ f).
233 #f2 #f @eq_repl_sym /2 width=3 by sor_eq_repl_back1/
236 corec lemma sor_eq_repl_back2: ∀f1,f. eq_repl_back … (λf2. f1 ⋓ f2 ≡ f).
237 #f1 #f #f2 * -f1 -f2 -f
238 #f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
239 try cases (eq_inv_px … Hx … H2) try cases (eq_inv_nx … Hx … H2) -g2
240 /3 width=7 by sor_pp, sor_np, sor_pn, sor_nn/
243 lemma sor_eq_repl_fwd2: ∀f1,f. eq_repl_fwd … (λf2. f1 ⋓ f2 ≡ f).
244 #f1 #f @eq_repl_sym /2 width=3 by sor_eq_repl_back2/
247 corec lemma sor_eq_repl_back3: ∀f1,f2. eq_repl_back … (λf. f1 ⋓ f2 ≡ f).
248 #f1 #f2 #f * -f1 -f2 -f
249 #f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
250 try cases (eq_inv_px … Hx … H0) try cases (eq_inv_nx … Hx … H0) -g
251 /3 width=7 by sor_pp, sor_np, sor_pn, sor_nn/
254 lemma sor_eq_repl_fwd3: ∀f1,f2. eq_repl_fwd … (λf. f1 ⋓ f2 ≡ f).
255 #f1 #f2 @eq_repl_sym /2 width=3 by sor_eq_repl_back3/
258 corec lemma sor_idem: ∀f. f ⋓ f ≡ f.
259 #f cases (pn_split f) * #g #H
260 [ @(sor_pp … H H H) | @(sor_nn … H H H) ] -H //
263 corec lemma sor_comm: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f2 ⋓ f1 ≡ f.
264 #f1 #f2 #f * -f1 -f2 -f
265 #f1 #f2 #f #g1 #g2 #g #Hf * * * -g1 -g2 -g
266 [ @sor_pp | @sor_pn | @sor_np | @sor_nn ] /2 width=7 by/
269 (* Properties with tail *****************************************************)
271 lemma sor_tl: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ⫱f1 ⋓ ⫱f2 ≡ ⫱f.
272 #f1 cases (pn_split f1) * #g1 #H1
273 #f2 cases (pn_split f2) * #g2 #H2
275 [ cases (sor_inv_ppx … Hf … H1 H2)
276 | cases (sor_inv_pnx … Hf … H1 H2)
277 | cases (sor_inv_npx … Hf … H1 H2)
278 | cases (sor_inv_nnx … Hf … H1 H2)
279 ] -Hf #g #Hg #H destruct //
282 lemma sor_xxn_tl: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ⫯f = g →
283 (∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ⫱g2 = f2) ∨
284 (∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫱g1 = f1 & ⫯f2 = g2).
285 #g1 #g2 #g #H #f #H0 elim (sor_inv_xxn … H … H0) -H -H0 *
286 /3 width=5 by ex3_2_intro, or_introl, or_intror/
289 lemma sor_xnx_tl: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f2. ⫯f2 = g2 →
290 ∃∃f1,f. f1 ⋓ f2 ≡ f & ⫱g1 = f1 & ⫯f = g.
291 #g1 elim (pn_split g1) * #f1 #H1 #g2 #g #H #f2 #H2
292 [ elim (sor_inv_pnx … H … H1 H2) | elim (sor_inv_nnx … H … H1 H2) ] -g2 #f #Hf #H0
293 /3 width=5 by ex3_2_intro/
296 (* Properties with iterated tail ********************************************)
298 lemma sor_tls: ∀f1,f2,f. f1 ⋓ f2 ≡ f →
299 ∀n. ⫱*[n]f1 ⋓ ⫱*[n]f2 ≡ ⫱*[n]f.
300 #f1 #f2 #f #Hf #n elim n -n /2 width=1 by sor_tl/
303 (* Properies with test for identity *****************************************)
305 corec lemma sor_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⋓ f2 ≡ f2.
307 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
308 /3 width=7 by sor_pp, sor_pn/
311 corec lemma sor_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⋓ f2 ≡ f1.
313 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
314 /3 width=7 by sor_pp, sor_np/
317 lemma sor_isid: ∀f1,f2,f. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → 𝐈⦃f⦄ → f1 ⋓ f2 ≡ f.
318 /4 width=3 by sor_eq_repl_back2, sor_eq_repl_back1, isid_inv_eq_repl/ qed.
320 (* Inversion lemmas with tail ***********************************************)
322 lemma sor_inv_tl_sn: ∀f1,f2,f. ⫱f1 ⋓ f2 ≡ f → f1 ⋓ ⫯f2 ≡ ⫯f.
323 #f1 #f2 #f elim (pn_split f1) *
324 #g1 #H destruct /2 width=7 by sor_pn, sor_nn/
327 lemma sor_inv_tl_dx: ∀f1,f2,f. f1 ⋓ ⫱f2 ≡ f → ⫯f1 ⋓ f2 ≡ ⫯f.
328 #f1 #f2 #f elim (pn_split f2) *
329 #g2 #H destruct /2 width=7 by sor_np, sor_nn/
332 (* Inversion lemmas with test for identity **********************************)
334 lemma sor_isid_inv_sn: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
335 /3 width=4 by sor_isid_sn, sor_mono/
338 lemma sor_isid_inv_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f2⦄ → f1 ≗ f.
339 /3 width=4 by sor_isid_dx, sor_mono/
342 corec lemma sor_fwd_isid1: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
343 #f1 #f2 #f * -f1 -f2 -f
344 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H #Hg
345 [ /4 width=6 by isid_inv_push, isid_push/ ]
346 cases (isid_inv_next … Hg … H)
349 corec lemma sor_fwd_isid2: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
350 #f1 #f2 #f * -f1 -f2 -f
351 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H #Hg
352 [ /4 width=6 by isid_inv_push, isid_push/ ]
353 cases (isid_inv_next … Hg … H)
356 lemma sor_inv_isid3: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
357 /3 width=4 by sor_fwd_isid2, sor_fwd_isid1, conj/ qed-.
359 (* Properties with finite colength assignment *******************************)
361 lemma sor_fcla_ex: ∀f1,n1. 𝐂⦃f1⦄ ≡ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≡ n2 →
362 ∃∃f,n. f1 ⋓ f2 ≡ f & 𝐂⦃f⦄ ≡ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
363 #f1 #n1 #Hf1 elim Hf1 -f1 -n1 /3 width=6 by sor_isid_sn, ex4_2_intro/
364 #f1 #n1 #Hf1 #IH #f2 #n2 * -f2 -n2 /3 width=6 by fcla_push, fcla_next, ex4_2_intro, sor_isid_dx/
365 #f2 #n2 #Hf2 elim (IH … Hf2) -IH -Hf2 -Hf1 [2,4: #f #n <plus_n_Sm ] (**) (* full auto fails *)
366 [ /3 width=7 by fcla_next, sor_pn, max_S2_le_S, le_S_S, ex4_2_intro/
367 | /4 width=7 by fcla_next, sor_nn, le_S, le_S_S, ex4_2_intro/
368 | /3 width=7 by fcla_push, sor_pp, ex4_2_intro/
369 | /3 width=7 by fcla_next, sor_np, max_S1_le_S, le_S_S, ex4_2_intro/
373 lemma sor_fcla: ∀f1,n1. 𝐂⦃f1⦄ ≡ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≡ n2 → ∀f. f1 ⋓ f2 ≡ f →
374 ∃∃n. 𝐂⦃f⦄ ≡ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
375 #f1 #n1 #Hf1 #f2 #n2 #Hf2 #f #Hf elim (sor_fcla_ex … Hf1 … Hf2) -Hf1 -Hf2
376 /4 width=6 by sor_mono, fcla_eq_repl_back, ex3_intro/
379 (* Forward lemmas with finite colength **************************************)
381 lemma sor_fwd_fcla_sn_ex: ∀f,n. 𝐂⦃f⦄ ≡ n → ∀f1,f2. f1 ⋓ f2 ≡ f →
382 ∃∃n1. 𝐂⦃f1⦄ ≡ n1 & n1 ≤ n.
383 #f #n #H elim H -f -n
384 [ /4 width=4 by sor_fwd_isid1, fcla_isid, ex2_intro/
385 | #f #n #_ #IH #f1 #f2 #H
386 elim (sor_inv_xxp … H) -H [ |*: // ] #g1 #g2 #Hf #H1 #H2 destruct
387 elim (IH … Hf) -f /3 width=3 by fcla_push, ex2_intro/
388 | #f #n #_ #IH #f1 #f2 #H
389 elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g1 #g2 #Hf #H1 #H2 destruct
390 elim (IH … Hf) -f /3 width=3 by fcla_push, fcla_next, le_S_S, le_S, ex2_intro/
394 lemma sor_fwd_fcla_dx_ex: ∀f,n. 𝐂⦃f⦄ ≡ n → ∀f1,f2. f1 ⋓ f2 ≡ f →
395 ∃∃n2. 𝐂⦃f2⦄ ≡ n2 & n2 ≤ n.
396 /3 width=4 by sor_fwd_fcla_sn_ex, sor_comm/ qed-.
398 (* Properties with test for finite colength *********************************)
400 lemma sor_isfin_ex: ∀f1,f2. 𝐅⦃f1⦄ → 𝐅⦃f2⦄ → ∃∃f. f1 ⋓ f2 ≡ f & 𝐅⦃f⦄.
401 #f1 #f2 * #n1 #H1 * #n2 #H2 elim (sor_fcla_ex … H1 … H2) -H1 -H2
402 /3 width=4 by ex2_intro, ex_intro/
405 lemma sor_isfin: ∀f1,f2. 𝐅⦃f1⦄ → 𝐅⦃f2⦄ → ∀f. f1 ⋓ f2 ≡ f → 𝐅⦃f⦄.
406 #f1 #f2 #Hf1 #Hf2 #f #Hf elim (sor_isfin_ex … Hf1 … Hf2) -Hf1 -Hf2
407 /3 width=6 by sor_mono, isfin_eq_repl_back/
410 (* Forward lemmas with test for finite colength *****************************)
412 lemma sor_fwd_isfin_sn: ∀f. 𝐅⦃f⦄ → ∀f1,f2. f1 ⋓ f2 ≡ f → 𝐅⦃f1⦄.
413 #f * #n #Hf #f1 #f2 #H
414 elim (sor_fwd_fcla_sn_ex … Hf … H) -f -f2 /2 width=2 by ex_intro/
417 lemma sor_fwd_isfin_dx: ∀f. 𝐅⦃f⦄ → ∀f1,f2. f1 ⋓ f2 ≡ f → 𝐅⦃f2⦄.
418 #f * #n #Hf #f1 #f2 #H
419 elim (sor_fwd_fcla_dx_ex … Hf … H) -f -f1 /2 width=2 by ex_intro/
422 (* Inversion lemmas with test for finite colength ***************************)
424 lemma sor_inv_isfin3: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐅⦃f⦄ → 𝐅⦃f1⦄ ∧ 𝐅⦃f2⦄.
425 /3 width=4 by sor_fwd_isfin_dx, sor_fwd_isfin_sn, conj/ qed-.
427 (* Inversion lemmas with inclusion ******************************************)
429 corec lemma sor_inv_sle_sn: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f1 ⊆ f.
430 #f1 #f2 #f * -f1 -f2 -f
431 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0
432 /3 width=5 by sle_push, sle_next, sle_weak/
435 corec lemma sor_inv_sle_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f2 ⊆ f.
436 #f1 #f2 #f * -f1 -f2 -f
437 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0
438 /3 width=5 by sle_push, sle_next, sle_weak/
441 lemma sor_inv_sle_sn_trans: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ∀g. g ⊆ f1 → g ⊆ f.
442 /3 width=4 by sor_inv_sle_sn, sle_trans/ qed-.
444 lemma sor_inv_sle_dx_trans: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ∀g. g ⊆ f2 → g ⊆ f.
445 /3 width=4 by sor_inv_sle_dx, sle_trans/ qed-.
447 axiom sor_inv_sle: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ∀g. f1 ⊆ g → f2 ⊆ g → f ⊆ g.
449 (* Properties with inclusion ************************************************)
451 corec lemma sor_sle_dx: ∀f1,f2. f1 ⊆ f2 → f1 ⋓ f2 ≡ f2.
452 #f1 #f2 * -f1 -f2 /3 width=7 by sor_pp, sor_nn, sor_pn/
455 corec lemma sor_sle_sn: ∀f1,f2. f1 ⊆ f2 → f2 ⋓ f1 ≡ f2.
456 #f1 #f2 * -f1 -f2 /3 width=7 by sor_pp, sor_nn, sor_np/
459 (* Main properties **********************************************************)
461 axiom monotonic_sle_sor: ∀f1,g1. f1 ⊆ g1 → ∀f2,g2. f2 ⊆ g2 →
462 ∀f. f1 ⋓ f2 ≡ f → ∀g. g1 ⋓ g2 ≡ g → f ⊆ g.
464 axiom sor_assoc_dx: ∀f0,f3,f4. f0 ⋓ f3 ≡ f4 →
465 ∀f1,f2. f1 ⋓ f2 ≡ f0 →
466 ∀f. f2 ⋓ f3 ≡ f → f1 ⋓ f ≡ f4.
468 axiom sor_assoc_sn: ∀f1,f0,f4. f1 ⋓ f0 ≡ f4 →
469 ∀f2, f3. f2 ⋓ f3 ≡ f0 →
470 ∀f. f1 ⋓ f2 ≡ f → f ⋓ f3 ≡ f4.
472 lemma sor_comm_23: ∀f0,f1,f2,f3,f4,f.
473 f0⋓f4 ≡ f1 → f1⋓f2 ≡ f → f0⋓f2 ≡ f3 → f3⋓f4 ≡ f.
474 /4 width=6 by sor_comm, sor_assoc_dx/ qed-.
476 corec theorem sor_comm_23_idem: ∀f0,f1,f2. f0 ⋓ f1 ≡ f2 →
477 ∀f. f1 ⋓ f2 ≡ f → f1 ⋓ f0 ≡ f.
478 #f0 #f1 #f2 * -f0 -f1 -f2
479 #f0 #f1 #f2 #g0 #g1 #g2 #Hf2 #H0 #H1 #H2 #g #Hg
480 [ cases (sor_inv_ppx … Hg … H1 H2)
481 | cases (sor_inv_pnx … Hg … H1 H2)
482 | cases (sor_inv_nnx … Hg … H1 H2)
483 | cases (sor_inv_nnx … Hg … H1 H2)
485 /3 width=7 by sor_nn, sor_np, sor_pn, sor_pp/
488 corec theorem sor_coll_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ∀g1,g2,g. g1 ⋓ g2 ≡ g →
489 ∀g0. g1 ⋓ g0 ≡ f1 → g2 ⋓ g0 ≡ f2 → g ⋓ g0 ≡ f.
490 #f1 #f2 #f cases (pn_split f) * #x #Hx #Hf #g1 #g2 #g #Hg #g0 #Hf1 #Hf2
491 [ cases (sor_inv_xxp … Hf … Hx) -Hf #x1 #x2 #Hf #Hx1 #Hx2
492 cases (sor_inv_xxp … Hf1 … Hx1) -f1 #y1 #y0 #Hf1 #Hy1 #Hy0
493 cases (sor_inv_xpp … Hf2 … Hy0 … Hx2) -f2 #y2 #Hf2 #Hy2
494 cases (sor_inv_ppx … Hg … Hy1 Hy2) -g1 -g2 #y #Hg #Hy
495 @(sor_pp … Hy Hy0 Hx) -g -g0 -f /2 width=8 by/
496 | cases (pn_split g) * #y #Hy
497 [ cases (sor_inv_xxp … Hg … Hy) -Hg #y1 #y2 #Hg #Hy1 #Hy2
498 cases (sor_xxn_tl … Hf … Hx) * #x1 #x2 #_ #Hx1 #Hx2
499 [ cases (sor_inv_pxn … Hf1 … Hy1 Hx1) -g1 #y0 #Hf1 #Hy0
500 cases (sor_inv_pnx … Hf2 … Hy2 Hy0) -g2 -x2 #x2 #Hf2 #Hx2
501 | cases (sor_inv_pxn … Hf2 … Hy2 Hx2) -g2 #y0 #Hf2 #Hy0
502 cases (sor_inv_pnx … Hf1 … Hy1 Hy0) -g1 -x1 #x1 #Hf1 #Hx1
504 lapply (sor_inv_nnn … Hf … Hx1 Hx2 Hx) -f1 -f2 #Hf
505 @(sor_pn … Hy Hy0 Hx) -g -g0 -f /2 width=8 by/
506 | lapply (sor_tl … Hf) -Hf #Hf
507 lapply (sor_tl … Hg) -Hg #Hg
508 lapply (sor_tl … Hf1) -Hf1 #Hf1
509 lapply (sor_tl … Hf2) -Hf2 #Hf2
510 cases (pn_split g0) * #y0 #Hy0
511 [ @(sor_np … Hy Hy0 Hx) /2 width=8 by/
512 | @(sor_nn … Hy Hy0 Hx) /2 width=8 by/
518 corec theorem sor_distr_dx: ∀g0,g1,g2,g. g1 ⋓ g2 ≡ g →
519 ∀f1,f2,f. g1 ⋓ g0 ≡ f1 → g2 ⋓ g0 ≡ f2 → g ⋓ g0 ≡ f →
521 #g0 cases (pn_split g0) * #y0 #H0 #g1 #g2 #g
522 [ * -g1 -g2 -g #y1 #y2 #y #g1 #g2 #g #Hy #Hy1 #Hy2 #Hy #f1 #f2 #f #Hf1 #Hf2 #Hf
523 [ cases (sor_inv_ppx … Hf1 … Hy1 H0) -g1
524 cases (sor_inv_ppx … Hf2 … Hy2 H0) -g2
525 cases (sor_inv_ppx … Hf … Hy H0) -g
526 | cases (sor_inv_npx … Hf1 … Hy1 H0) -g1
527 cases (sor_inv_ppx … Hf2 … Hy2 H0) -g2
528 cases (sor_inv_npx … Hf … Hy H0) -g
529 | cases (sor_inv_ppx … Hf1 … Hy1 H0) -g1
530 cases (sor_inv_npx … Hf2 … Hy2 H0) -g2
531 cases (sor_inv_npx … Hf … Hy H0) -g
532 | cases (sor_inv_npx … Hf1 … Hy1 H0) -g1
533 cases (sor_inv_npx … Hf2 … Hy2 H0) -g2
534 cases (sor_inv_npx … Hf … Hy H0) -g
535 ] -g0 #y #Hy #H #y2 #Hy2 #H2 #y1 #Hy1 #H1
536 /3 width=8 by sor_nn, sor_np, sor_pn, sor_pp/
537 | #H #f1 #f2 #f #Hf1 #Hf2 #Hf
538 cases (sor_xnx_tl … Hf1 … H0) -Hf1
539 cases (sor_xnx_tl … Hf2 … H0) -Hf2
540 cases (sor_xnx_tl … Hf … H0) -Hf
541 -g0 #y #x #Hx #Hy #H #y2 #x2 #Hx2 #Hy2 #H2 #y1 #x1 #Hx1 #Hy1 #H1
542 /4 width=8 by sor_tl, sor_nn/