matita/matita/contribs/lambdadelta/basic_2/etc/leq/leq_nat.etc

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  14
  15 include "Basic_2/grammar/lenv_length.ma".
  16
  17 (* LOCAL ENVIRONMENT EQUALITY ***********************************************)
  18
  19 notation "hvbox( T1 break [ d , break e ] ≈ break T2 )"
  20    non associative with precedence 45
  21    for @{ 'Eq $T1 $d $e $T2 }.
  22
  23 inductive leq: nat → nat → relation lenv ≝
  24 | leq_sort: ∀d,e. leq d e (⋆) (⋆)
  25 | leq_OO:   ∀L1,L2. leq 0 0 L1 L2
  26 | leq_eq:   ∀L1,L2,I,V,e. leq 0 e L1 L2 →
  27             leq 0 (e + 1) (L1. 𝕓{I} V) (L2.𝕓{I} V)
  28 | leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
  29             leq d e L1 L2 → leq (d + 1) e (L1. 𝕓{I1} V1) (L2. 𝕓{I2} V2)
  30 .
  31
  32 interpretation "local environment equality" 'Eq L1 d e L2 = (leq d e L1 L2).
  33
  34 definition leq_repl_dx: ∀S. (lenv → relation S) → Prop ≝ λS,R.
  35                         ∀L1,s1,s2. R L1 s1 s2 →
  36                         ∀L2,d,e. L1 [d, e]≈ L2 → R L2 s1 s2.
  37
  38 (* Basic properties *********************************************************)
  39
  40 lemma TC_leq_repl_dx: ∀S,R. leq_repl_dx S R → leq_repl_dx S (λL. (TC … (R L))).
  41 #S #R #HR #L1 #s1 #s2 #H elim H -H s2
  42 [ /3 width=5/
  43 | #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
  44   lapply (HR … Hs2 … HL12) -HR Hs2 HL12 /3/
  45 ]
  46 qed.
  47
  48 lemma leq_refl: ∀d,e,L. L [d, e] ≈ L.
  49 #d elim d -d
  50 [ #e elim e -e // #e #IHe #L elim L -L /2/
  51 | #d #IHd #e #L elim L -L /2/
  52 ]
  53 qed.
  54
  55 lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
  56 #L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
  57 qed.
  58
  59 lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
  60                    ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
  61
  62 #L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/
  63 qed.
  64
  65 (* Basic inversion lemmas ***************************************************)