matita/matita/contribs/lambdadelta/ground/lib/list_eq.ma

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   3 (*      ||M||                                                             *)
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  14
  15 include "ground/notation/relations/ringeq_3.ma".
  16 include "ground/lib/list.ma".
  17
  18 (* EXTENSIONAL EQUIVALENCE OF LISTS *****************************************)
  19
  20 rec definition list_eq A (l1,l2:list A) on l1 ≝
  21 match l1 with
  22 [ list_nil        ⇒
  23   match l2 with
  24   [ list_nil      ⇒ ⊤
  25   | list_cons _ _ ⇒ ⊥
  26   ]
  27 | list_cons a1 l1 ⇒
  28   match l2 with
  29   [ list_nil        ⇒ ⊥
  30   | list_cons a2 l2 ⇒ a1 = a2 ∧ list_eq A l1 l2
  31   ]
  32 ].
  33
  34 interpretation
  35   "extensional equivalence (lists)"
  36   'RingEq A l1 l2 = (list_eq A l1 l2).
  37
  38 (* Basic constructions ******************************************************)
  39
  40 lemma list_eq_refl (A): reflexive … (list_eq A).
  41 #A #l elim l -l /2 width=1 by conj/
  42 qed.
  43
  44 (* Main constructions *******************************************************)
  45
  46 theorem eq_list_eq (A,l1,l2): l1 = l2 → l1 ≗{A} l2.
  47 // qed.
  48
  49 (* Main inversions **********************************************************)
  50
  51 theorem list_eq_inv_eq (A,l1,l2): l1 ≗{A} l2 → l1 = l2.
  52 #A #l1 elim l1 -l1 [| #a1 #l1 #IH ] *
  53 [ //
  54 | #a2 #l2 #H elim H
  55 | #H elim H
  56 | #a2 #l2 * #Ha #Hl /3 width=1 by eq_f2/
  57 ]
  58 qed-.