helm/software/matita/library/dama/models/nat_order_continuous.ma

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  14
  15 include "dama/models/increasing_supremum_stabilizes.ma".
  16 include "dama/models/nat_ordered_uniform.ma".
  17
  18 lemma nat_us_is_oc: ∀s:‡ℕ.order_continuity (segment_ordered_uniform_space ℕ s).
  19 intros 3; split; intros 3; cases H1; cases H2; clear H2 H1; cases H; clear H;
  20 [1: cases (increasing_supremum_stabilizes s a H1 ? H2);
  21     exists [apply w1]; intros;
  22     apply (restrict nat_ordered_uniform_space s ??? H4);
  23     generalize in match (H ? H5);
  24     cases x; intro;
  25     generalize in match (refl_eq ? (a i):a i = a i);
  26     generalize in match (a i) in ⊢ (?? % ? → %); intro X; cases X; clear X;
  27     intro; rewrite < H9 in H7; simplify; rewrite < H7; simplify;
  28     apply (us_phi1 nat_uniform_space ? H3); simplify; apply (eq_reflexive (us_carr nat_uniform_space));
  29 |2: cases (increasing_supremum_stabilizes_r s a H1 ? H2);
  30     exists [apply w1]; intros;
  31     apply (restrict nat_ordered_uniform_space s ??? H4);
  32     generalize in match (H ? H5);
  33     cases x; intro;
  34     generalize in match (refl_eq ? (a i):a i = a i);
  35     generalize in match (a i) in ⊢ (?? % ? → %); intro X; cases X; clear X;
  36     intro; rewrite < H9 in H7; simplify; rewrite < H7; simplify;
  37     apply (us_phi1 nat_uniform_space ? H3); simplify; apply (eq_reflexive (us_carr nat_uniform_space));]
  38 qed.
  39