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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground/relocation/pr_pat_lt.ma".
16 include "ground/relocation/pr_nat.ma".
17 include "ground/relocation/pr_ist.ma".
18 include "ground/relocation/pr_after_pat.ma".
20 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
22 (* Destructions with pr_ist *************************************************)
24 (*** after_istot_fwd *)
25 lemma pr_after_ist_des:
26 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❨f2❩ → 𝐓❨f1❩ → 𝐓❨f❩.
27 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
28 #i2 #Hf1 elim (Hf2 i2) -Hf2
29 /3 width=7 by pr_after_des_pat, ex_intro/
32 (*** after_fwd_istot_dx *)
33 lemma pr_after_des_ist_dx:
34 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❨f❩ → 𝐓❨f1❩.
35 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
36 #i2 #Hf elim (pr_after_pat_des … Hf … H) -f /2 width=2 by ex_intro/
39 (*** after_fwd_istot_sn *)
40 lemma pr_after_des_ist_sn:
41 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❨f❩ → 𝐓❨f2❩.
42 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
43 #i #Hf elim (pr_after_pat_des … Hf … H) -f
44 #i2 #Hf1 #Hf2 lapply (pr_pat_increasing … Hf1) -f1
45 #Hi12 elim (pr_pat_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
49 lemma pr_after_des_ist_pat:
50 ∀f1,i1,i2. @⧣❨i1, f1❩ ≘ i2 → ∀f2. 𝐓❨f2❩ → ∀f. f2 ⊚ f1 ≘ f →
51 ∃∃i. @⧣❨i2, f2❩ ≘ i & @⧣❨i1, f❩ ≘ i.
52 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
53 /3 width=8 by pr_after_des_pat, ex2_intro/
56 lemma pr_after_des_ist_nat:
57 ∀f1,l1,l2. @§❨l1, f1❩ ≘ l2 → ∀f2. 𝐓❨f2❩ → ∀f. f2 ⊚ f1 ≘ f →
58 ∃∃l. @§❨l2, f2❩ ≘ l & @§❨l1, f❩ ≘ l.
59 #f1 #l1 #l2 #H1 #f2 #H2 #f #Hf
60 elim (pr_after_des_ist_pat … H1 … H2 … Hf) -f1 -H2
61 /2 width=3 by ex2_intro/
64 (* Inversions with pr_ist ***************************************************)
66 (*** after_inv_istot *)
67 lemma pr_after_inv_ist:
68 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❨f❩ → ∧∧ 𝐓❨f2❩ & 𝐓❨f1❩.
69 /3 width=4 by pr_after_des_ist_sn, pr_after_des_ist_dx, conj/ qed-.