matita/matita/contribs/lambdadelta/static_2/static/reqx_reqx.ma

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   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
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  13 (**************************************************************************)
  14
  15 include "static_2/static/reqg_reqg.ma".
  16 include "static_2/static/reqx.ma".
  17
  18 (* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
  19
  20 (* Advanced properties ******************************************************)
  21 (*
  22 lemma reqx_sym: ∀T. symmetric … (reqx T).
  23 /3 width=3 by reqx_fsge_comp, rex_sym, teqx_sym/ qed-.
  24 *)
  25 (* Basic_2A1: uses: lleq_dec *)
  26 lemma reqx_dec: ∀L1,L2. ∀T:term. Decidable (L1 ≅[T] L2).
  27 /2 width=1 by reqg_dec/ qed-.
  28 (*
  29 (* Main properties **********************************************************)
  30
  31 (* Basic_2A1: uses: lleq_bind lleq_bind_O *)
  32 theorem reqx_bind: ∀p,I,L1,L2,V1,V2,T.
  33                    L1 ≛[V1] L2 → L1.ⓑ[I]V1 ≛[T] L2.ⓑ[I]V2 →
  34                    L1 ≛[ⓑ[p,I]V1.T] L2.
  35 /2 width=2 by rex_bind/ qed.
  36
  37 (* Basic_2A1: uses: lleq_flat *)
  38 theorem reqx_flat: ∀I,L1,L2,V,T.
  39                    L1 ≛[V] L2 → L1 ≛[T] L2 → L1 ≛[ⓕ[I]V.T] L2.
  40 /2 width=1 by rex_flat/ qed.
  41
  42 theorem reqx_bind_void: ∀p,I,L1,L2,V,T.
  43                         L1 ≛[V] L2 → L1.ⓧ ≛[T] L2.ⓧ → L1 ≛[ⓑ[p,I]V.T] L2.
  44 /2 width=1 by rex_bind_void/ qed.
  45
  46 (* Basic_2A1: uses: lleq_trans *)
  47 theorem reqx_trans: ∀T. Transitive … (reqx T).
  48 #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
  49 lapply (frees_teqx_conf_reqx … Hf1 T … HL1) // #H0
  50 lapply (frees_mono … Hf2 … H0) -Hf2 -H0
  51 /5 width=7 by sex_trans, sex_eq_repl_back, teqx_trans, ext2_trans, ex2_intro/
  52 qed-.
  53
  54 (* Basic_2A1: uses: lleq_canc_sn *)
  55 theorem reqx_canc_sn: ∀T. left_cancellable … (reqx T).
  56 /3 width=3 by reqx_trans, reqx_sym/ qed-.
  57
  58 (* Basic_2A1: uses: lleq_canc_dx *)
  59 theorem reqx_canc_dx: ∀T. right_cancellable … (reqx T).
  60 /3 width=3 by reqx_trans, reqx_sym/ qed-.
  61
  62 theorem reqx_repl: ∀L1,L2. ∀T:term. L1 ≅[T] L2 →
  63                    ∀K1. L1 ≅[T] K1 → ∀K2. L2 ≅[T] K2 → K1 ≅[T] K2.
  64 /2 width=5 by reqg_repl/ qed-.
  65
  66 (* Negated properties *******************************************************)
  67
  68 (* Note: auto works with /4 width=8/ so reqx_canc_sn is preferred **********)
  69 (* Basic_2A1: uses: lleq_nlleq_trans *)
  70 lemma reqx_rneqx_trans: ∀T:term.∀L1,L. L1 ≛[T] L →
  71                         ∀L2. (L ≛[T] L2 → ⊥) → (L1 ≛[T] L2 → ⊥).
  72 /3 width=3 by reqx_canc_sn/ qed-.
  73
  74 (* Basic_2A1: uses: nlleq_lleq_div *)
  75 lemma rneqx_reqx_div: ∀T:term.∀L2,L. L2 ≛[T] L →
  76                       ∀L1. (L1 ≛[T] L → ⊥) → (L1 ≛[T] L2 → ⊥).
  77 /3 width=3 by reqx_trans/ qed-.
  78
  79 theorem rneqx_reqx_canc_dx: ∀L1,L. ∀T:term. (L1 ≛[T] L → ⊥) →
  80                             ∀L2. L2 ≛[T] L → L1 ≛[T] L2 → ⊥.
  81 /3 width=3 by reqx_trans/ qed-.
  82
  83 (* Negated inversion lemmas *************************************************)
  84
  85 (* Basic_2A1: uses: nlleq_inv_bind nlleq_inv_bind_O *)
  86 lemma rneqx_inv_bind: ∀p,I,L1,L2,V,T. (L1 ≛[ⓑ[p,I]V.T] L2 → ⊥) →
  87                       (L1 ≛[V] L2 → ⊥) ∨ (L1.ⓑ[I]V ≛[T] L2.ⓑ[I]V → ⊥).
  88 /3 width=2 by rnex_inv_bind, teqx_dec/ qed-.
  89
  90 (* Basic_2A1: uses: nlleq_inv_flat *)
  91 lemma rneqx_inv_flat: ∀I,L1,L2,V,T. (L1 ≛[ⓕ[I]V.T] L2 → ⊥) →
  92                       (L1 ≛[V] L2 → ⊥) ∨ (L1 ≛[T] L2 → ⊥).
  93 /3 width=2 by rnex_inv_flat, teqx_dec/ qed-.
  94
  95 lemma rneqx_inv_bind_void: ∀p,I,L1,L2,V,T. (L1 ≛[ⓑ[p,I]V.T] L2 → ⊥) →
  96                            (L1 ≛[V] L2 → ⊥) ∨ (L1.ⓧ ≛[T] L2.ⓧ → ⊥).
  97 /3 width=3 by rnex_inv_bind_void, teqx_dec/ qed-.
  98 *)