helm/software/matita/contribs/limits/Class/eq.ma

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  14
  15 include "Class/defs.ma".
  16
  17 (* ACZEL CATEGORIES:
  18    - Here we prove the standard properties of the equality.
  19 *)
  20
  21 theorem ceq_cin_sx: ∀C,c1,c2. ceq C c1 c2 → cin ? c1.
  22 intros; elim H; clear H c1 c2; autobatch.
  23 qed.
  24
  25 theorem ceq_cin_dx: ∀C,c1,c2. ceq C c1 c2 → cin ? c2.
  26 intros; elim H; clear H c1 c2; autobatch.
  27 qed.
  28
  29 theorem ceq_trans: ∀C,c1,c2. ceq C c1 c2 → ∀c3. ceq ? c2 c3 → ceq ? c1 c3.
  30 intros 4; elim H; clear H c1 c2; autobatch.
  31 qed.
  32
  33 theorem ceq_conf: ∀C,c1,c2. ceq C c1 c2 → ∀c3. ceq ? c1 c3 → ceq ? c2 c3.
  34 intros 4; elim H; clear H c1 c2; autobatch depth = 4.
  35 qed.
  36
  37 theorem ceq_sym: ∀ C,c1,c2. ceq C c1 c2 → ceq ? c2 c1.
  38 intros; autobatch depth = 4.
  39 qed.
  40
  41 theorem ceq_repl: ∀C,c1,c2. ceq C c1 c2 →
  42                       ∀d1. ceq ? c1 d1 → ∀d2. ceq ? c2 d2 → ceq ? d1 d2.
  43 intros; autobatch.
  44 qed.