Thm* eq:{T }, f:{T= }, x,y:T. eq(x,y)  f(x,y) | [not_equivalence_implies_not_discrete_eq] |
Thm* eq:{T= }, f:{T }, x,y:T. eq(x,y)  f(x,y) | [equivalence_weakening_lemma] |
Thm* eq:{T }, u:T. eq(u,u) | [equivalence_reflexive] |
Thm* eq:{T }, x,y:T. eq(x,y)  eq(y,x) | [equivalence_symmetric] |
Thm* f:{T }. ( x,y,z:T. f(x,y)  f(y,z)  f(x,z)) & ( x,y:T. f(x,y)  f(y,x)) & ( x:T. f(x,x)) & f T T   | [equivalence_properties] |
Thm* P:(T Z Prop). ( x:T, y:Z. Dec(P(x,y)))  ( f_p:(T Z  ). x:T, y:Z. P(x,y)  f_p(x,y)) | [decidable_iff_exists_bool_function_2] |
Thm* P:(T Prop). ( x:T. Dec(P(x)))  ( f_p:(T  ). x:T. P(x)  f_p(x)) | [decidable_iff_exists_bool_function] |
Def {T } == {f:(T T  )| ( x:T. (f(x,x))) & ( x,y:T. (f(x,y))  (f(y,x))) & ( x,y,z:T. (f(x,y))  (f(y,z))  (f(x,z))) } | [equivalence] |