Exercise 1 Which of the following implications are valid in intuitionistic logic? Give either a tableau proof or a counterexample.
(a) cp a f -> f a (p
(b) y/^ ((</>ai//)vi//)
(c) (--f -»• (<p -»• f)
(d) 9    —<p
(e) ((<pAi//)vi//)^i//
(f) —9 9
(g)
(h) ->(->f a -.1//) -»• (9 v f) (This one is tricky.)
(i) (p -*■ 3xcp (j) Vx<f> -> cp
(k) VxR(x,x) ^ Vx3yR(f(x),y)
(1) 3x(<p v 1//) -»• 3xcp v 3xi//
(m) 3xcp v 3xi// -»■ 3x(<p v 1//)
(n) Vx<p a Vxf -*■ Vx(<p a 1//)
(o) Vx(<p a 1//) -»• Vx<p a Vxi//
(p) VxV;/[<p(x) «-»• 9(7)] a 3x<p(x) -»• \/xq>(x)
1