macro "conj_elim" hp:ident hq:ident "from" h:ident : tactic =>
  `(tactic|(apply And.elim _ $h; intro $hp $hq))

macro "conj_intro" h:ident "from" hp:ident hq:ident : tactic =>
  `(tactic|(have $h : _∧_ := And.intro $hp $hq))

macro "disj_elim" hp:ident hq:ident "from" h:term:arg : tactic =>
  `(tactic|(refine Or.elim $h (fun $hp => ?_) (fun $hq => ?_)))

macro "disj_intro_left" h:ident "from" hp:ident q:term:arg : tactic =>
  `(tactic|(have $h : _∨($q) := Or.inl $hp))

macro "disj_intro_right" h:ident "from" p:term:arg hq:ident : tactic =>
  `(tactic|(have $h : ($p)∨_ := Or.inr $hq))

macro "impl_elim" hq:ident "from" hp:ident hpq:ident : tactic =>
  `(tactic|(have $hq := $hpq $hp))

namespace And
theorem elim' {P Q : Prop} {R : Sort} (h : P ∧ Q) (f : P → Q → R) : R :=
  And.elim f h
end And


variable (P Q R S : Prop)


example : ¬ ( P ∧ ¬ P ) := by
  sorry


example ( h : P → Q ) ( hnq : ¬ Q ) : ¬ P := by
  sorry

/- goal-oriented reasoning -/
example (h : P ) : P ∨ Q := by
  sorry

/- hypothesis-oriented reasoning -/
example (h : P ) : P ∨ Q := by
  sorry

/- hypothesis-oriented reasoning -/
example ( h : P ∨ Q ) : Q ∨ P := by
  sorry

/- a little bit more goal-oriented, using the cases tactic -/
example ( h : P ∨ Q ) : Q ∨ P := by
  sorry


/- ex falso -/
example : False → P := by
  sorry


example ( h : ¬ P ∨ Q ) : P → Q := by
  sorry


open Classical
example : ( P ∨ ¬ P ) := by
  sorry

/- a proof using em P -/
example ( h : P → Q ) : ¬ P ∨ Q := by
  sorry

/- another proof, using em Q -/
example ( h : P → Q ) : ¬ P ∨ Q := by
  sorry