lambdadelta/basic_2/relocation/llor.ma

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  14
  15 include "basic_2/notation/relations/lazyor_4.ma".
  16 include "basic_2/relocation/lpx_sn_alt.ma".
  17
  18 (* LAZY UNION FOR LOCAL ENVIRONMENTS ****************************************)
  19
  20 inductive clor (T) (L2) (K1) (V1): predicate term ≝
  21 | clor_sn: ∀U. |K1| < |L2| → ⇧[|L2|-|K1|-1, 1] U ≡ T → clor T L2 K1 V1 V1
  22 | clor_dx: ∀I,K2,V2. |K1| < |L2| → (∀U. ⇧[|L2|-|K1|-1, 1] U ≡ T → ⊥) →
  23            ⇩[|L2|-|K1|-1] L2 ≡ K2.ⓑ{I}V2 → clor T L2 K1 V1 V2
  24 .
  25
  26 definition llor: relation4 term lenv lenv lenv ≝
  27                  λT,L2. lpx_sn (clor T L2).
  28
  29 interpretation
  30    "lazy union (local environment)"
  31    'LazyOr L1 T L2 L = (llor T L2 L1 L).
  32
  33 (* Basic properties *********************************************************)
  34
  35 lemma llor_pair_sn: ∀I,L1,L2,L,V,T,U. L1 ⩖[T] L2 ≡ L →
  36                     |L1| < |L2| → ⇧[|L2|-|L1|-1, 1] U ≡ T →
  37                     L1.ⓑ{I}V ⩖[T] L2 ≡ L.ⓑ{I}V.
  38 /3 width=2 by clor_sn, lpx_sn_pair/ qed.
  39
  40 lemma llor_pair_dx: ∀I,J,L1,L2,L,K2,V1,V2,T. L1 ⩖[T] L2 ≡ L →
  41                     |L1| < |L2| → (∀U. ⇧[|L2|-|L1|-1, 1] U ≡ T → ⊥) →
  42                     ⇩[|L2|-|L1|-1] L2 ≡ K2.ⓑ{J}V2 →
  43                     L1.ⓑ{I}V1 ⩖[T] L2 ≡ L.ⓑ{I}V2.
  44 /4 width=3 by clor_dx, lpx_sn_pair/ qed.
  45
  46 lemma llor_total: ∀T,L2,L1. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.
  47 #T #L2 #L1 elim L1 -L1 /2 width=2 by ex_intro/
  48 #L1 #I1 #V1 #IHL1 normalize
  49 #H elim IHL1 -IHL1 /2 width=3 by transitive_le/
  50 #L #HT elim (is_lift_dec T (|L2|-|L1|-1) 1)
  51 [ * /3 width=2 by llor_pair_sn, ex_intro/
  52 | elim (ldrop_O1_lt L2 (|L2|-|L1|-1))
  53   /5 width=4 by llor_pair_dx, monotonic_lt_minus_l, ex_intro/
  54 ]
  55 qed-.