matita/matita/contribs/lambdadelta/basic_2/grammar/tshf.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   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 "basic_2/grammar/term_simple.ma".
  16
  17 (* SAME HEAD TERM FORMS *****************************************************)
  18
  19 inductive tshf: relation term ≝
  20    | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I})
  21    | tshf_abbr: ∀V1,V2,T1,T2. tshf (-ⓓV1. T1) (-ⓓV2. T2)
  22    | tshf_abst: ∀a,V1,V2,T1,T2. tshf (ⓛ{a}V1. T1) (ⓛ{a}V2. T2)
  23    | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄ →
  24                 tshf (ⓐV1. T1) (ⓐV2. T2)
  25 .
  26
  27 interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1.
  32 #T1 #T2 #H elim H -T1 -T2 /2 width=1/
  33 qed.
  34
  35 lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2.
  36 #T1 #T2 #H elim H -T1 -T2 // /2 width=1/
  37 qed.
  38
  39 lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1.
  40 /3 width=2/ qed.
  41
  42 lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
  43 #T1 #T2 #H elim H -T1 -T2 //
  44 [ #V1 #V2 #T1 #T2 #H
  45   elim (simple_inv_bind … H)
  46 | #a #V1 #V2 #T1 #T2 #H
  47   elim (simple_inv_bind … H)
  48 ]
  49 qed. (**) (* remove from index *)
  50
  51 lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
  52 /3 width=3/ qed-.
  53
  54 (* Basic inversion lemmas ***************************************************)
  55
  56 fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀a,I,W1,U1. T1 = ⓑ{a,I}W1.U1 →
  57                          ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
  58                                   (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
  59 #T1 #T2 * -T1 -T2
  60 [ #J #a #I #W1 #U1 #H destruct
  61 | #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
  62 | #b #V1 #V2 #T1 #T2 #a #I #W1 #U1 #H destruct /3 width=3/
  63 | #V1 #V2 #T1 #T2 #_ #_ #_ #a #I #W1 #U1 #H destruct
  64 ]
  65 qed.
  66
  67 lemma tshf_inv_bind1: ∀a,I,W1,U1,T2. ⓑ{a,I}W1.U1 ≈ T2 →
  68                       ∃∃W2,U2. T2 = ⓑ{a,I}W2. U2 &
  69                                (Bind2 a I = Bind2 false Abbr ∨ I = Abst).
  70 /2 width=5/ qed-.
  71
  72 fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 →
  73                          ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
  74                                   I = Appl & T2 = ⓐW2. U2.
  75 #T1 #T2 * -T1 -T2
  76 [ #J #I #W1 #U1 #H destruct
  77 | #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
  78 | #a #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
  79 | #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/
  80 ]
  81 qed.
  82
  83 lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 →
  84                       ∃∃W2,U2. U1 ≈ U2 & 𝐒⦃U1⦄ & 𝐒⦃U2⦄ &
  85                                I = Appl & T2 = ⓐW2. U2.
  86 /2 width=4/ qed-.