Discrete-Math/Exercise 4 at master · A454545/Discrete-Math · GitHub

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Theorem (4.2) “Left-monotonicity of ∨” “Monotonicity of ∨”:
    (p ⇒ q) ⇒ (p ∨ r) ⇒ (q ∨ r)
Proof:
    (p ⇒ q) ⇒ (p ∨ r) ⇒ (q ∨ r)
  ≡⟨ “Definition of Implication” ⟩
    (p ⇒ q) ⇒ ((p ∨ r) ∨ (q ∨ r) ≡ q ∨ r)
  ≡⟨ “Idempotency of ∨” ⟩
    (p ⇒ q) ⇒ (p ∨ q ∨ r ≡ q ∨ r)
  ≡⟨ “Distributivity of ∨ over ≡” ⟩
    (p ⇒ q) ⇒ ((p ∨ q ≡ q) ∨ r)
  ≡⟨ “Definition of Implication” ⟩
    (p ⇒ q) ⇒ ((p ⇒ q) ∨ r)
  ≡⟨ “Strengthening” ⟩
    true

Theorem “Distributivity of ∨ over ⇒”:
    p ∨ (q ⇒ r) ≡ p ∨ q ⇒ p ∨ r
Proof:
    p ∨ (q ⇒ r) ≡ p ∨ q ⇒ p ∨ r
  ≡⟨ “Definition of Implication” ⟩
    p ∨ (q ⇒ r) ≡ p ∨ q ∨ p ∨ r ≡ p ∨ r
  ≡⟨ “Idempotency of ∨” ⟩
    p ∨ (q ⇒ r) ≡ p ∨ q ∨ r ≡ p ∨ r
  ≡⟨ “Distributivity of ∨ over ≡” ⟩
    p ∨ (q ⇒ r) ≡ p ∨ (q ∨ r ≡ r)
  ≡⟨ “Definition of Implication” ⟩
    p ∨ (q ⇒ r) ≡ p ∨ (q ⇒ r)
  ≡⟨ “Reflexivity of ≡” ⟩
    true

Theorem “Left-monotonicity of ∨” “Monotonicity of ∨”:
    (p ⇒ q) ⇒ (p ∨ r) ⇒ (q ∨ r)
Proof:
    (p ∨ r) ⇒ (q ∨ r)
  ≡⟨ “Distributivity of ∨ over ⇒” ⟩
    (p ⇒ q) ∨ r
  ⇐⟨ “Weakening” ⟩
    p ⇒ q

Theorem “Antitonicity of ¬”: (p ⇒ q) ⇒ (¬ q ⇒ ¬ p)
Proof:
    (p ⇒ q) ⇒ (¬ q ⇒ ¬ p)
  ≡⟨ “Definition of Implication” ⟩
    ¬ (¬ p ∨ q) ∨  (¬ ¬ q ∨ ¬ p)
  ≡⟨ “Double negation” ⟩
    ¬ (¬ p ∨ q) ∨ (q ∨ ¬ p)
  ≡⟨ “Distributivity of ∨ over ∨” ⟩
    (¬ (¬ p ∨ q) ∨ q) ∨ (¬ (¬ p ∨ q) ∨ ¬ p)
  ≡⟨ “Idempotency of ∨” ⟩
    (¬ (¬ p ∨ q) ∨ q ∨ ¬ p)
  ≡⟨ “Reflexivity of ≡”  ⟩
    ¬ (¬ p ∨ q) ∨ (¬ p ∨ q)
  ≡⟨ “Excluded middle”  ⟩
    true

Theorem “Left-monotonicity of ∨” “Monotonicity of ∨”:
    (p ⇒ q) ⇒ (p ∨ r) ⇒ (q ∨ r)
Proof:
  Assuming `p ⇒ q`:
      p ∨ r
    ≡⟨ Assumption `p ⇒ q` with “Definition of Implication” ⟩
      (p ∧ q) ∨ r
    ≡⟨ “Distributivity of ∨ over ∧” ⟩
      (p ∨ r) ∧ (q ∨ r)
    ⇒⟨ “Weakening” ⟩
      (q ∨ r)

Theorem (4.2) “Left-monotonicity of ∨” “Monotonicity of ∨”:
    (p ⇒ q) ⇒ (p ∨ r) ⇒ (q ∨ r)
Proof:
  Assuming `p ⇒ q`, `p ∨ r`:
    By cases: `p`, `r`
      Completeness: By Assumption `p ∨ r`
      Case `p`:
          true
        ≡⟨ Assumption `p ∨ r` ⟩
          p ∨ r
        ≡⟨ Assumption `p ⇒ q` with “Definition of Implication” ⟩
          (p ∧ q) ∨ r
        ≡⟨ “Distributivity of ∨ over ∧” ⟩
          (p ∨ r) ∧ (q ∨ r)
        ⇒⟨ “Weakening” ⟩
          q ∨ r
      Case `r`:
          true
        ≡⟨ Assumption `r` ⟩
          r
        ≡⟨ “Left-identity of ⇒” ⟩
          true ⇒ r
        ≡⟨ Assumption `p ⇒ q` ⟩
          (p ⇒ q) ⇒ r
        ≡⟨ “Definition of Implication” ⟩
          (p ⇒ q) ∨ r ≡ r
        ≡⟨ Assumption `r` ⟩
          (p ⇒ q) ∨ r ≡ true
        ≡⟨ “Reflexivity of ≡” ⟩
          (p ⇒ q) ∨ r
        ≡⟨ “Distributivity of ∨ over ⇒” ⟩
          p ∨ r ⇒ q ∨ r
        ≡⟨ Assumption `p ∨ r` ⟩
          true ⇒ q ∨ r
        ≡⟨ “Left-identity of ⇒” ⟩
          q ∨ r


Theorem (4.2) “Left-monotonicity of ∨” “Monotonicity of ∨”:
    (p ⇒ q) ⇒ (p ∨ r) ⇒ (q ∨ r)
Proof:
  Assuming `p ⇒ q`, `p ∨ r`:
    By cases: `p`, `r`
      Completeness: By Assumption `p ∨ r`
      Case `p`:
          true
        ≡⟨ Assumption `p` ⟩
          p
        ⇒⟨ “Weakening” ⟩
          p ∨ q ∨ r
        ≡⟨ Assumption `p ⇒ q` with “Definition of Implication” ⟩
          q ∨ r
      Case `r`:
          true
        ≡⟨ Assumption `r`⟩
          r
        ⇒⟨ “Strengthening” ⟩
           q ∨ (p ∨ r)
        ≡⟨ “Identity of ≡” ⟩
          q ∨ (p ∨ r ≡ true)
        ≡⟨ “Right-zero of ⇒” ⟩
          q ∨ (p ∨ r ≡ p ⇒ true)
        ≡⟨ Assumption `r` ⟩
          q ∨ (p ∨ r ≡ p ⇒ r)
        ≡⟨ Assumption `p ∨ r` with “Definition of Implication” ⟩
          q ∨ r

    (p ⇒ q) ⇒ (r ⇒ s) ⇒ (p ∨ r) ⇒ (q ∨ s)
Proof:
  Assuming `p ⇒ q`, `r ⇒ s`:
      p ∨ r
    ≡⟨ Assumption `p ⇒ q` with “Definition of Implication” ⟩
      (p ∧ q) ∨ r
    ⇒⟨ “Strengthening” ⟩
      q ∨ r
    ≡⟨ Assumption `r ⇒ s` with “Definition of Implication” ⟩
      q ∨ (r ∧ s)
    ⇒⟨ “Strengthening” ⟩

      q ∨ s

Theorem (4.3) “Left-monotonicity of ∧” “Monotonicity of ∧”:
    (p ⇒ q) ⇒ ((p ∧ r) ⇒ (q ∧ r))
Proof:
  Assuming `p ⇒ q`:
      p ∧ r
    ≡⟨ Assumption `p ⇒ q` with “Definition of Implication” ⟩
      p ∧ q ∧ r
    ⇒⟨ “Strengthening” ⟩
      q ∧ r

Theorem (4.3) “Left-monotonicity of ∧” “Monotonicity of ∧”:
    (p ⇒ q) ⇒ ((p ∧ r) ⇒ (q ∧ r))
Proof:
  Assuming `p ⇒ q` and using with “Definition of ⇒” (3.60):
      p ∧ r
    ≡⟨ Assumption `p ⇒ q` ⟩
      p ∧ q ∧ r
    ⇒⟨ “Strengthening” ⟩

      q ∧ r

Theorem “Monotonicity of ∧”:
    (p ⇒ p') ⇒ (q ⇒ q') ⇒ ((p ∧ q) ⇒ (p' ∧ q'))
Proof:
  Assuming `p ⇒ p'`, `q ⇒ q'`:
      p ∧ q
    ≡⟨ Assumption `p ⇒ p'` with “Definition of Implication” ⟩
      p ∧ p' ∧ q
    ≡⟨ Assumption `q ⇒ q'` with “Definition of Implication” ⟩
      p ∧ p' ∧ q ∧ q'
    ⇒⟨ “Strengthening” ⟩

      p' ∧ q'

Theorem “Proof by contradiction”:  ¬ p ⇒ false  ≡  p
Proof:
    ¬ p ⇒ false  ≡  p
  ≡⟨ “Definition of Implication” ⟩
    ¬ ¬ p ∨ false ≡ p
  ≡⟨ “Double negation” ⟩
    p ∨ false ≡ p
  ≡⟨ “Identity of ∨” ⟩
    p ≡ p
  ≡⟨ “Reflexivity of ≡” ⟩
    true