1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "ground/relocation/pr_pushs.ma".
16 include "ground/relocation/pr_tls.ma".
17 include "ground/relocation/pr_nat.ma".
18 include "ground/relocation/pr_coafter.ma".
20 (* RELATIONAL CO-COMPOSITION FOR PARTIAL RELOCATION MAPS ********************)
22 (* Destructions with pr_nat and pr_tls and pr_pushs *************************)
24 (*** coafter_fwd_pushs *)
25 lemma pr_coafter_des_pushs_dx (n) (m):
26 ∀g2,f1,g. g2 ~⊚ ⫯*[m]f1 ≘ g → @§❨m, g2❩ ≘ n →
27 ∃∃f. ⫰*[n]g2 ~⊚ f1 ≘ f & ⫯*[n] f = g.
28 #n @(nat_ind_succ … n) -n
29 [ #m #g2 #f1 #g #Hg #H
30 elim (pr_nat_inv_zero_dx … H) -H [|*: // ] #f2 #H1 #H2 destruct
31 /2 width=3 by ex2_intro/
32 | #n #IH * [| #m ] #g2 #f1 #g #Hg #H
33 [ elim (pr_nat_inv_zero_succ … H) -H [|*: // ] #f2 #Hmn #H destruct
34 elim (pr_coafter_inv_next_sn … Hg) -Hg [|*: // ] #f #Hf #H destruct
35 elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
36 | elim (pr_nat_inv_succ_bi … H) -H [1,4: * |*: // ] #f2 #Hmn #H destruct
37 [ elim (pr_coafter_inv_push_bi … Hg) -Hg [|*: // ] #f #Hf #H destruct
38 elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
39 | elim (pr_coafter_inv_next_sn … Hg) -Hg [|*: // ] #f #Hf #H destruct
40 elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/