natural-number-game

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example (P Q : Prop) (p : P) (q : Q) : P ∧ Q :=
begin
  split,
  exact p,
  exact q,
end

lemma and_symm (P Q : Prop) : P ∧ Q → Q ∧ P :=
begin
  intro h,
  cases h with p q,
  split,
  exact q,
  exact p,
end

lemma and_trans (P Q R : Prop) : P ∧ Q → Q ∧ R → P ∧ R :=
begin
  intro h1,
  cases h1 with p _,
  intro h2,
  cases h2 with _ r,
  split,
  exact p,
  exact r,
end

lemma iff_trans (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) :=
begin
  intro p_iff_q,
  cases p_iff_q with hpq hqp,
  intro q_iff_r,
  cases q_iff_r with hqr hrq,
  split,
  intro p,
  apply hqr,
  apply hpq,
  exact p,
  intro r,
  apply hqp,
  apply hrq,
  exact r,
end

lemma iff_trans (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) :=
begin
  intro p_iff_q,
  cases p_iff_q with hpq hqp,
  intro q_iff_r,
  cases q_iff_r with hqr hrq,
  split,
  intro p,
  apply hqr,
  apply hpq,
  exact p,
  intro r,
  apply hqp,
  apply hrq,
  exact r,
end

example (P Q : Prop) : Q → (P ∨ Q) :=
  begin
  intro q,
  right,
  exact q,
end

lemma or_symm (P Q : Prop) : P ∨ Q → Q ∨ P :=
begin
  intro h,
  cases h with p q,
  right,
  exact p,
  left,
  exact q,
end

lemma and_or_distrib_left (P Q R : Prop) : P ∧ (Q ∨ R) ↔ (P ∧ Q) ∨ (P ∧ R) :=
begin
  split,
  intro h,
  have p := h.1,
  cases h.2 with q r,
  left,
  split,
  exact p,
  exact q,
  right,
  split,
  exact p,
  exact r,
  intro h,
  cases h with p_and_q p_and_r,
  cases p_and_q with p q,
  split,
  exact p,
  left,
  exact q,
  cases p_and_r with p r,
  split,
  exact p,
  right,
  exact r,
end

lemma contra (P Q : Prop) : (P ∧ ¬ P) → Q :=
begin
  intro h,
  rw not_iff_imp_false at h,
  exfalso,
  apply h.2,
  exact h.1,
end

lemma contrapositive2 (P Q : Prop) : (¬ Q → ¬ P) → (P → Q) :=
begin
  by_cases p : P; by_cases q : Q,
  intros _ _,
  exact q,
  intro h,
  have not_p := h q,
  exfalso,
  rw not_iff_imp_false at not_p,
  apply not_p,
  exact p,
  intros _ _,
  exact q,
  intros h p,
  have not_p := h q,
  rw not_iff_imp_false at not_p,
  exfalso,
  apply not_p,
  exact p,
end