Theory: operator

Parents


Type constants


Term constants


Axioms


Definitions

ASSOC_DEF
|- !f. ASSOC f = (!x y z. f x (f y z) = f (f x y) z)
COMM_DEF
|- !f. COMM f = (!x y. f x y = f y x)
FCOMM_DEF
|- !f g. FCOMM f g = (!x y z. g x (f y z) = f (g x y) z)
RIGHT_ID_DEF
|- !f e. RIGHT_ID f e = (!x. f x e = x)
LEFT_ID_DEF
|- !f e. LEFT_ID f e = (!x. f e x = x)
MONOID_DEF
|- !f e. MONOID f e = ASSOC f /\ RIGHT_ID f e /\ LEFT_ID f e

Theorems

ASSOC_CONJ
|- ASSOC $/\
ASSOC_DISJ
|- ASSOC $\/
FCOMM_ASSOC
|- !f. FCOMM f f = ASSOC f
MONOID_CONJ_T
|- MONOID $/\ T
MONOID_DISJ_F
|- MONOID $\/ F