matita/matita/contribs/lambdadelta/basic_2/examples/ex_cpr_omega.ma

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  14
  15 include "basic_2/reduction/cpr.ma".
  16
  17 (* EXAMPLES *****************************************************************)
  18
  19 (* A reduction cycle in two steps: the term Omega ***************************)
  20
  21 definition Delta: term → term ≝ λW. +ⓛW.ⓐ#0.#0.
  22
  23 definition Omega1: term → term ≝ λW. ⓐ(Delta W).(Delta W).
  24
  25 definition Omega2: term → term ≝ λW. +ⓓⓝW.(Delta W).ⓐ#0.#0.
  26
  27 (* Basic properties *********************************************************)
  28
  29 lemma Delta_lift: ∀W1,W2,d,e. ⇧[d, e] W1 ≡ W2 →
  30                   ⇧[d, e] (Delta W1) ≡ (Delta W2).
  31 /4 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
  32
  33 (* Main properties **********************************************************)
  34
  35 theorem cpr_Omega_12: ∀W. ⦃⋆, ⋆⦄ ⊢ Omega1 W ➡ Omega2 W.
  36 /2 width=1 by cpr_beta/ qed.
  37
  38 theorem cpr_Omega_21: ∀W. ⦃⋆, ⋆⦄ ⊢ Omega2 W ➡ Omega1 W.
  39 #W1 elim (lift_total W1 0 1) #W2 #HW12
  40 @(cpr_zeta … (Omega1 W2)) /3 width=1 by Delta_lift, lift_flat/
  41 @cpr_flat @(cpr_delta … (Delta W1) ? 0)
  42 [3,5,8,10: /2 width=2 by Delta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
  43 qed.