lean2/library/logic/examples/double_negation_translation.lean

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/-
Simulating classical reasoning without assuming excluded middle.
The idea is to use the double-negation translation.
We define several "helper" theorems for double negated formulas.
-/
variables {p q r : Prop}
theorem not_and_of_or_not : ¬p ¬q → ¬(p ∧ q) :=
λ h hpq, or.elim h
(λ hnp : ¬p, absurd (and.elim_left hpq) hnp)
(λ hnq : ¬q, absurd (and.elim_right hpq) hnq)
theorem not_or_elim_left : ¬(p q) → ¬p :=
λ hpq hp, absurd (or.inl hp) hpq
theorem not_or_elim_right : ¬(p q) → ¬q :=
λ hpq hq, absurd (or.inr hq) hpq
theorem not_imp_elim_right : ¬(p → q) → ¬q :=
λ h₁ hq, absurd (λ h, hq) h₁
theorem not_imp_elim_left : ¬(p → q) → ¬¬p :=
λ h₁ hnp, absurd (λ hp, by contradiction) h₁
theorem not_imp_intro : ¬¬p → ¬q → ¬(p → q) :=
λ hnnp hnq hpq,
have hnp : ¬ p, from λ hp, absurd (hpq hp) hnq,
absurd hnp hnnp
/- Double negation introduction -/
theorem nn_intro : p → ¬¬p :=
λ hp hnp, absurd hp hnp
/- Double negated implication -/
-- Introduction
theorem nn_imp_intro : (¬¬p → ¬¬q) → ¬¬(p → q) :=
λ h hnpq,
have hnnp : ¬¬p, from not_imp_elim_left hnpq,
have hnq : ¬q, from not_imp_elim_right hnpq,
have hnnq : ¬¬q, from h hnnp,
absurd hnq hnnq
-- Elimination (modus ponens)
theorem nn_mp : ¬¬(p → q) → p → ¬¬q :=
λ hpq hp hnq,
have aux : ¬(p → q), from not_imp_intro (nn_intro hp) hnq,
absurd aux hpq
-- Double negated modus tollens
theorem nn_mt : ¬¬(p → q) → ¬q → ¬p :=
λ hpq hnq hp, absurd hnq (nn_mp hpq hp)
/- Double negated disjuction -/
lemma not_or_of_not_of_not : ¬p → ¬q → ¬(p q) :=
λ hnp hnq hpq, or.elim hpq (λ hp, absurd hp hnp) (λ hq, absurd hq hnq)
-- Elimination
theorem nn_or_elim : ¬¬(p q) → (p → ¬¬r) → (q → ¬¬r) → ¬¬r :=
λ hpq hpr hqr hnr,
have hnp : ¬p, from λhp, absurd hnr (hpr hp),
have hnq : ¬q, from λhq, absurd hnr (hqr hq),
have aux : ¬(p q), from not_or_of_not_of_not hnp hnq,
absurd aux hpq
-- Introduction
theorem nn_or_inl : ¬¬p → ¬¬(p q) :=
λ h hnpq, absurd (not_or_elim_left hnpq) h
theorem nn_or_inr : ¬¬q → ¬¬(p q) :=
λ h hnpq, absurd (not_or_elim_right hnpq) h
/- Double negated conjunction -/
-- Elimination
theorem nn_and_elim_left : ¬¬(p ∧ q) → ¬¬p :=
λ h hnp, absurd (not_and_of_or_not (or.inl hnp)) h
theorem nn_and_elim_right : ¬¬(p ∧ q) → ¬¬q :=
λ h hnq, absurd (not_and_of_or_not (or.inr hnq)) h
-- Introduction
theorem nn_and_intro : ¬¬p → ¬¬q → ¬¬(p ∧ q) :=
λ hnnp hnnq hnpq,
have h₁ : ¬(p → ¬q), from not_imp_intro hnnp hnnq,
have h₂ : p → ¬q, from λ hp hq, absurd (and.intro hp hq) hnpq,
absurd h₂ h₁
/- Double negated excluded middle -/
theorem nn_em : ¬¬(p ¬p) :=
λ hn,
have hnp : ¬p, from not_or_elim_left hn,
have hnnp : ¬¬p, from not_or_elim_right hn,
absurd hnp hnnp
/- Examples: the following two examples are classically valid.
We can "simulate" the classical proofs using double negation.
-/
example : ¬¬((p → q) → (¬p q)) :=
nn_imp_intro (λ h, nn_or_elim (@nn_em p)
(λ hp : p,
have hnnq : ¬¬q, from nn_mp h hp,
nn_or_inr hnnq)
(λ hnp : ¬p, nn_intro (or.inl hnp)))
/- "Prove" Peirce's law -/
example : ¬¬(((p → q) → p) → p) :=
nn_imp_intro (λ h, nn_or_elim (@nn_em p)
(λ hp : p, nn_intro hp)
(λ hnp : ¬p,
have h₁ : ¬(p → q), from nn_mt h hnp,
have hnnp : ¬¬p, from not_imp_elim_left h₁,
absurd hnp hnnp))