matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_fsubst.etc

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "delayed_updating/unwind3/unwind_prototerm_eq.ma".
  16 include "delayed_updating/unwind3/unwind_structure.ma".
  17 include "delayed_updating/substitution/fsubst.ma".
  18 include "delayed_updating/syntax/preterm.ma".
  19 include "delayed_updating/syntax/prototerm_proper.ma".
  20
  21 (* UNWIND FOR PROTOTERM *****************************************************)
  22
  23 (* Constructions with fsubst ************************************************)
  24
  25 lemma unwind_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
  26       (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⊆ ▼[f](t[⋔p←u]).
  27 #f #t #u #p #Hu #ql * *
  28 [ #rl * #r #Hr #H1 #H2 destruct
  29   >unwind_append_proper_dx
  30   /4 width=5 by in_comp_unwind_bi, in_ppc_comp_trans, or_introl, ex2_intro/
  31 | * #q #Hq #H1 #H0
  32   @(ex2_intro … H1) @or_intror @conj // *
  33   [ <list_append_empty_dx #H2 destruct
  34     elim (unwind_path_root f q) #r #_ #Hr /2 width=2 by/
  35   | #l #r #H2 destruct
  36     @H0 -H0 [| <unwind_append_proper_dx /2 width=3 by ppc_lcons/ ]
  37   ]
  38 ]
  39 qed-.
  40
  41 lemma unwind_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
  42       ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u].
  43 #f #t #u #p #Hu #H1p #H2p #ql * #q * *
  44 [ #r #Hu #H1 #H2 destruct
  45   @or_introl @ex2_intro
  46   [|| <unwind_append_proper_dx /2 width=5 by in_ppc_comp_trans/ ]
  47   /2 width=3 by ex2_intro/
  48 | #Hq #H0 #H1 destruct
  49   @or_intror @conj [ /2 width=1 by in_comp_unwind_bi/ ] *
  50   [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
  51     <list_append_empty_dx @H2p -H2p /2 width=1 by prototerm_in_comp_root/
  52   | #l #r #Hr
  53     elim (unwind_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
  54     lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
  55   ]
  56 ]
  57 qed-.
  58
  59 lemma unwind_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
  60       (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⇔ ▼[f](t[⋔p←u]).
  61 /4 width=3 by unwind_fsubst_sn, conj, unwind_fsubst_dx/ qed.