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

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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "static_2/notation/relations/ideqsn_3.ma".
  16 include "static_2/syntax/teq_ext.ma".
  17 include "static_2/static/reqg.ma".
  18
  19 (* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
  20
  21 (* Basic_2A1: was: lleq *)
  22 definition req: relation3 term lenv lenv ≝
  23            reqg (eq …).
  24
  25 interpretation
  26   "syntactic equivalence on referred entries (local environment)"
  27   'IdEqSn T L1 L2 = (req T L1 L2).
  28
  29 (* Note: "R_transitive_req R" is equivalent to "R_transitive_rex ceq R R" *)
  30 (* Basic_2A1: uses: lleq_transitive *)
  31 definition R_transitive_req: predicate (relation3 lenv term term) ≝
  32            λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
  33
  34 (* Basic inversion lemmas ***************************************************)
  35
  36 lemma req_inv_bind:
  37       ∀p,I,L1,L2,V,T. L1 ≡[ⓑ[p,I]V.T] L2 →
  38       ∧∧ L1 ≡[V] L2 & L1.ⓑ[I]V ≡[T] L2.ⓑ[I]V.
  39 /2 width=2 by reqg_inv_bind_refl/ qed-.
  40
  41 lemma req_inv_flat:
  42       ∀I,L1,L2,V,T. L1 ≡[ⓕ[I]V.T] L2 →
  43       ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2.
  44 /2 width=2 by reqg_inv_flat/ qed-.
  45
  46 (* Advanced inversion lemmas ************************************************)
  47
  48 lemma req_inv_zero_pair_sn:
  49       ∀I,L2,K1,V. K1.ⓑ[I]V ≡[#0] L2 →
  50       ∃∃K2. K1 ≡[V] K2 & L2 = K2.ⓑ[I]V.
  51 #I #L2 #K1 #V #H
  52 elim (reqg_inv_zero_pair_sn … H) -H #K2 #X #HK12 #HX #H destruct
  53 @(teq_repl_1 … HX) -X
  54 @(ex2_intro … HK12) // (**) (* auto fails because a δ-expamsion gets in the way *)
  55 qed-.
  56
  57 lemma req_inv_zero_pair_dx:
  58       ∀I,L1,K2,V. L1 ≡[#0] K2.ⓑ[I]V →
  59       ∃∃K1. K1 ≡[V] K2 & L1 = K1.ⓑ[I]V.
  60 #I #L1 #K2 #V #H
  61 elim (reqg_inv_zero_pair_dx … H) -H #K1 #X #HK12 #HX #H destruct
  62 @(teq_repl_1 … HX) -V
  63 @(ex2_intro … HK12) // (**) (* auto fails because a δ-expamsion gets in the way *)
  64 qed-.
  65
  66 lemma req_inv_lref_bind_sn:
  67       ∀I1,K1,L2,i. K1.ⓘ[I1] ≡[#↑i] L2 →
  68       ∃∃I2,K2. K1 ≡[#i] K2 & L2 = K2.ⓘ[I2].
  69 /2 width=2 by reqg_inv_lref_bind_sn/ qed-.
  70
  71 lemma req_inv_lref_bind_dx:
  72       ∀I2,K2,L1,i. L1 ≡[#↑i] K2.ⓘ[I2] →
  73       ∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ[I1].
  74 /2 width=2 by reqg_inv_lref_bind_dx/ qed-.
  75
  76 (* Basic forward lemmas *****************************************************)
  77
  78 (* Basic_2A1: was: llpx_sn_lrefl *)
  79 (* Basic_2A1: this should have been lleq_fwd_llpx_sn *)
  80 lemma req_fwd_rex (R):
  81       c_reflexive … R →
  82       ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R,T] L2.
  83 #R #HR #L1 #L2 #T * #f #Hf #HL12
  84 /5 width=7 by sex_co, cext2_co, teq_repl_1, ex2_intro/
  85 qed-.
  86
  87 lemma req_fwd_reqg (S) (T:term):
  88       reflexive … S →
  89       ∀L1,L2. L1 ≡[T] L2 → L1 ≛[S,T] L2.
  90 /3 width=1 by req_fwd_rex, teqg_refl/ qed-.
  91
  92 lemma transitive_req_fwd_rex (R):
  93       R_transitive_req R → R_transitive_rex ceq R R.
  94 #R #HR #L1 #L #T1 #HL1 #T #HT #T2 #HT2
  95 @(HR … HL1) -HR -HL1 >(teq_inv_eq … HT) -T1 // (**)
  96 qed-.
  97
  98 (* Basic_2A1: removed theorems 10:
  99               lleq_ind lleq_fwd_lref
 100               lleq_fwd_drop_sn lleq_fwd_drop_dx
 101               lleq_skip lleq_lref lleq_free
 102               lleq_Y lleq_ge_up lleq_ge
 103
 104 *)