logic p,q : int -> prop

goal arith3a : 
  (forall k:int. 0 <= k <= 10 -> p(k)) ->
  p(11) -> 
  forall k:int. 0 <= k <= 11 -> p(k)

goal arith3b : 
  (forall k:int. 0 <= k <= 10 -> (p(k) and q(k))) ->
  p(11) -> q(11) ->
  forall k:int. 0 <= k <= 11 -> (p(k) and q(k))

goal arith3c : 
  (forall k:int. 0 <= k <= 10 -> p(k) -> q(k)) ->
  (p(11) -> q(11)) ->
  forall k:int. 0 <= k <= 11 -> p(k) -> q(k)

logic a,b : int

goal arith4a : 
  (forall k:int. a <= k <= b -> p(k)) ->
  p(b+1) -> 
  forall k:int. a <= k <= b+1 -> p(k)

goal arith4b : 
  (forall k:int. a <= k <= b -> (p(k) and q(k))) ->
  p(b+1) -> q(b+1) ->
  forall k:int. a <= k <= b+1 -> (p(k) and q(k))