helm/software/matita/contribs/dama/dama/nat_ordered_set.ma

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  14
  15 include "ordered_set.ma".
  16
  17 include "nat/compare.ma".
  18 include "cprop_connectives.ma".
  19
  20 definition nat_excess : nat → nat → CProp ≝ λn,m. m<n.
  21
  22 lemma nat_elim2:
  23   ∀R:nat → nat → CProp.
  24   (∀ n:nat. R O n) → (∀n:nat. R (S n) O) → (∀n,m:nat. R n m → R (S n) (S m)) →
  25     ∀n,m:nat. R n m.
  26 intros 5;elim n; [apply H]
  27 cases m;[ apply H1| apply H2; apply H3 ]
  28 qed.
  29
  30 lemma nat_discriminable: ∀x,y:nat.x < y ∨ x = y ∨ y < x.
  31 intros (x y); apply (nat_elim2 ???? x y);
  32 [1: intro;left;cases n; [right;reflexivity] left; apply lt_O_S;
  33 |2: intro;right;apply lt_O_S;
  34 |3: intros; cases H;
  35     [1: cases H1; [left; left; apply le_S_S; assumption]
  36         left;right;rewrite > H2; reflexivity;
  37     |2: right;apply le_S_S; assumption]]
  38 qed.
  39
  40 lemma nat_excess_cotransitive: cotransitive ? nat_excess.
  41 intros 3 (x y z); unfold nat_excess; simplify; intros;
  42 cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1;
  43 [1: right; apply (trans_lt ??? H H2);
  44 |2: right; rewrite < H2; assumption;]
  45 qed.
  46
  47 lemma nat_ordered_set : ordered_set.
  48 apply (mk_ordered_set ? nat_excess);
  49 [1: intro x; intro; apply (not_le_Sn_n ? H);
  50 |2: apply nat_excess_cotransitive]
  51 qed.