Thm* P:(T Prop). ( x:T. Dec(P(x)))  ( x:T.  P(x)  P(x)) | [not_not_rw] |
Thm* P:(T U Prop). ( x:T, y:U. Dec(P(x,y)))  ( p:(T U). Dec(p/l,r. P(l,r))) | [decidable__spread] |
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* 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* t1,t2:T, u1,u2:U. t1 = t2  u1 = u2  < t1,u1 > = < t2,u2 > | [pair_functionality_wrt_equal] |
Thm* Dec(P)  Dec(Q)  ((P  Q)  ( Q  P)) | [contrapositive] |
Thm* Dec(P)  Dec(Q)  ( (P & Q)  P Q) | [demorgan2] |
Thm* Dec(P)  Dec(Q)  ( (P Q)  P & Q) | [demorgan1] |
Thm* Dec(P)  Dec(Q)  ( (P Q)  P & Q) | [demorgan] |
Thm* (P  Q)  (Dec(P)  Dec(Q)) | [decidable_functionality_wrt_iff] |
Thm* Dec(P)  Dec(Q)  SqStable(P Q) | [sq_stable__or] |
Thm* SqStable(P)  SqStable(Q)  SqStable(P Q)  SqStable(P) SqStable(Q) | [sq_stable_functionality_wrt_or] |
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] |