# Combinator calculus

THEOREM

Let $$Ix = x$$, $$Kxy = x$$ and $$Sxyz = xz(yz)$$. Then $$I = SKK$$.

Proof available

THEOREM

Let $$Kxy = x$$, $$Sxyz = xz(yz)$$, and $$Bxyz = x(yz)$$. Then $$B = S(KS)K$$.

Proof available

THEOREM

Let $$Kxy = x$$, $$Sxyz = xz(yz)$$, and $$Cxyz = xzy$$. Then $$C = S(S(K(S(KS)K))S)(KK)$$.

Proof available

THEOREM

Let $$Kxy = x$$, $$Sxyz = xz(yz)$$, and $$Wxy = xyy$$. Then $$W = SS(SK)$$.

Proof available

THEOREM

Let $$Kxy = x$$, $$Wxy = xyy$$, and $$Ix = x$$. Then $$I = WK$$.

Proof available

THEOREM

Let $$Bxyz = x(yz)$$, $$Cxyz = xzy$$, and $$Wxy = xyy$$. Then $$S = B(B(BW)C)(BB)$$.

Proof available