matita/matita/contribs/lambdadelta/ground/relocation/pr_compose.ma

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   3 (*      ||M||                                                             *)
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  13 (**************************************************************************)
  14
  15 include "ground/relocation/pr_after_after.ma".
  16
  17 (* FUNCTIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
  18
  19 corec definition pr_compose: pr_map → pr_map → pr_map.
  20 * * #g2 [ #f1 | * * #g1 ]
  21 [ @pr_next @(pr_compose g2 f1)
  22 | @pr_next @(pr_compose g2 g1)
  23 | @pr_push @(pr_compose g2 g1)
  24 ]
  25 defined.
  26
  27 interpretation
  28   "functional composition (partial relocation maps)"
  29   'compose f2 f1 = (pr_compose f2 f1).
  30
  31 (* Basic constructions ******************************************************)
  32
  33 lemma pr_compose_unfold_refl (f2) (f1): ⫯(f2∘f1) = (⫯f2)∘(⫯f1).
  34 #f2 #f1
  35 <(stream_unfold … ((⫯f2)∘(⫯f1))) in ⊢ (???%); //
  36 qed.
  37
  38 lemma pr_compose_unfold_push (f2) (f1): ↑(f2∘f1) = (⫯f2)∘(↑f1).
  39 #f2 #f1
  40 <(stream_unfold … ((⫯f2)∘(↑f1))) in ⊢ (???%); //
  41 qed.
  42
  43 (*** compose_next *)
  44 lemma pr_compose_unfold_next (f2) (f1):
  45       ↑(f2∘f1) = (↑f2)∘f1.
  46 #f2 #f1
  47 <(stream_unfold … ((↑f2)∘f1)) in ⊢ (???%); //
  48 qed.
  49
  50 (* Main constructions *******************************************************)
  51
  52 (*** after_total *)
  53 corec theorem pr_after_total (f2) (f1):
  54               f2 ⊚ f1 ≘ f2∘f1.
  55 cases (pr_map_split_tl f2)*
  56 [ cases (pr_map_split_tl f1) * ]
  57 [ @pr_after_refl /2 width=7 by/
  58 | @pr_after_push /2 width=7 by/
  59 | @pr_after_next /2 width=5 by/
  60 ]
  61 qed.
  62
  63 (* Main inversions **********************************************************)
  64
  65 (*** after_inv_total *)
  66 lemma pr_after_inv_total: ∀f2,f1,f. f2 ⊚ f1 ≘ f → f2 ∘ f1 ≐ f.
  67 /2 width=4 by pr_after_mono/ qed-.