From 1238f43575c6c5912dc43cecf3bd97dd7618db1e Mon Sep 17 00:00:00 2001
From: Leonardo de Moura <leonardo@microsoft.com>
Date: Sun, 31 May 2015 18:18:10 -0700
Subject: [PATCH] feat(library/logic/examples): add "double-negation
 translation" example

---
 .../examples/double_negation_translation.lean | 112 ++++++++++++++++++
 1 file changed, 112 insertions(+)
 create mode 100644 library/logic/examples/double_negation_translation.lean

diff --git a/library/logic/examples/double_negation_translation.lean b/library/logic/examples/double_negation_translation.lean
new file mode 100644
index 000000000..487df2073
--- /dev/null
+++ b/library/logic/examples/double_negation_translation.lean
@@ -0,0 +1,112 @@
+/-
+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))