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

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  14
  15 include "ground/notation/functions/oplus_3.ma".
  16 include "ground/lib/list.ma".
  17
  18 (* APPEND FOR LISTS *********************************************************)
  19
  20 rec definition list_append A (l1:list A) (l2:list A) on l1 ≝ match l1 with
  21 [ list_empty       ⇒ l2
  22 | list_lcons hd tl ⇒ hd ⨮ (list_append A tl l2)
  23 ].
  24
  25 interpretation
  26   "append (lists)"
  27   'OPlus A l1 l2 = (list_append A l1 l2).
  28
  29 (* Basic constructions ******************************************************)
  30
  31 lemma list_append_empty_sn (A):
  32       ∀l2. l2 = ⓔ ⨁{A} l2.
  33 // qed.
  34
  35 lemma list_append_lcons_sn (A):
  36       ∀a,l1,l2. a ⨮ l1 ⨁ l2 = (a⨮l1) ⨁{A} l2.
  37 // qed.
  38
  39 (* Advanced constructions ***************************************************)
  40
  41 lemma list_append_empty_dx (A):
  42       ∀l1. l1 = l1 ⨁{A} ⓔ.
  43 #A #l1 elim l1 -l1
  44 [ <list_append_empty_sn //
  45 | #hd #tl #IH <list_append_lcons_sn <IH //
  46 ]
  47 qed.
  48
  49 lemma list_append_assoc (A):
  50       associative … (list_append A).
  51 #A #l1 elim l1 -l1
  52 [ <list_append_empty_sn //
  53 | #a1 #l1 #IH *
  54   [ #l3 <list_append_empty_dx <list_append_empty_sn //
  55   | #a2 #l2 #l3 <list_append_lcons_sn <list_append_lcons_sn <IH //
  56   ]
  57 ]
  58 qed.