Theorem Proving in Lean. Finish exercises 5.

src/theorem-proving-in-lean/exercises-4.lean

@@ -149,41 +149,38 @@ example : (∃ x : α, r) → r :=
 
 example (a : α) : r → (∃ x : α, r) :=
   assume hr,
-  exists.intro a hr
+  ⟨a, hr⟩
 
 example : (∃ x, p x ∧ r) ↔ (∃ x, p x) ∧ r :=
   iff.intro
-    (assume ⟨hx, ⟨hp, hr⟩⟩, ⟨exists.intro hx hp, hr⟩)
+    (assume ⟨hx, ⟨hp, hr⟩⟩, ⟨⟨hx, hp⟩, hr⟩)
-    (assume ⟨⟨hx, hp⟩, hr⟩, exists.intro hx ⟨hp, hr⟩)
+    (assume ⟨⟨hx, hp⟩, hr⟩, ⟨hx, ⟨hp, hr⟩⟩)
 
 example : (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ (∃ x, q x) :=
   iff.intro
     (assume ⟨hx, hpq⟩, hpq.elim
-      (assume hp, or.inl (exists.intro hx hp))
+      (assume hp, or.inl (⟨hx, hp⟩))
-      (assume hq, or.inr (exists.intro hx hq)))
+      (assume hq, or.inr (⟨hx, hq⟩)))
     (assume h, h.elim
-      (assume ⟨hx, hp⟩, exists.intro hx (or.inl hp))
+      (assume ⟨hx, hp⟩, ⟨hx, or.inl hp⟩)
-      (assume ⟨hx, hq⟩, exists.intro hx (or.inr hq)))
+      (assume ⟨hx, hq⟩, ⟨hx, or.inr hq⟩))
 
 example : (∀ x, p x) ↔ ¬ (∃ x, ¬ p x) :=
   iff.intro
     (assume h ⟨hx, np⟩, np (h hx))
-    (assume h hx, classical.by_contradiction (
+    (assume h hx, classical.by_contradiction (assume np, h ⟨hx, np⟩))
-      assume np,
-      h (exists.intro hx np)
-    ))
 
 example : (∃ x, p x) ↔ ¬ (∀ x, ¬ p x) :=
   iff.intro
     (assume ⟨hx, hp⟩ h, absurd hp (h hx))
     (assume h, classical.by_contradiction (
       assume h',
-      h (λ (x : α), assume hp, h' (exists.intro x hp))
+      h (λ (x : α), assume hp, h' ⟨x, hp⟩)
     ))
 
 example : (¬ ∃ x, p x) ↔ (∀ x, ¬ p x) :=
   iff.intro
-    (assume h hx hp, h (exists.intro hx hp))
+    (assume h hx hp, h ⟨hx, hp⟩)
     (assume h ⟨hx, hp⟩, absurd hp (h hx))
 
 lemma forall_negation : (¬ ∀ x, p x) ↔ (∃ x, ¬ p x) :=
@@ -192,7 +189,7 @@ lemma forall_negation : (¬ ∀ x, p x) ↔ (∃ x, ¬ p x) :=
       assume h',
       h (λ (x : α), classical.by_contradiction (
         assume np,
-        h' (exists.intro x np)
+        h' ⟨x, np⟩
       ))
     ))
     (assume ⟨hx, np⟩ h, absurd (h hx) np)
@@ -209,21 +206,21 @@ example (a : α) : (∃ x, p x → r) ↔ (∀ x, p x) → r :=
   iff.intro
     (assume ⟨hx, hp⟩ h, hp (h hx))
     (assume h₁, or.elim (classical.em (∀ x, p x))
-      (assume h₂, exists.intro a (assume hp, h₁ h₂))
+      (assume h₂, ⟨a, assume hp, h₁ h₂⟩)
       (assume h₂,
         have h₃ : (∃ x, ¬p x), from iff.elim_left (forall_negation α p) h₂,
         match h₃ with
-          ⟨hx, hp⟩ := exists.intro hx (assume hp', absurd hp' hp)
+          ⟨hx, hp⟩ := ⟨hx, (assume hp', absurd hp' hp)⟩
         end))
 
 example (a : α) : (∃ x, r → p x) ↔ (r → ∃ x, p x) :=
   iff.intro
-    (assume ⟨hx, hrp⟩ hr, exists.intro hx (hrp hr))
+    (assume ⟨hx, hrp⟩ hr, ⟨hx, hrp hr⟩)
     (assume h, or.elim (classical.em r)
       (assume hr, match h hr with
-        ⟨hx, hp⟩ := exists.intro hx (assume hr, hp)
+        ⟨hx, hp⟩ := ⟨hx, (assume hr, hp)⟩
       end)
-      (assume nr, exists.intro a (assume hr, absurd hr nr)))
+      (assume nr, ⟨a, (assume hr, absurd hr nr)⟩))
 
 end ex_5
 

src/theorem-proving-in-lean/exercises-5.lean

@@ -0,0 +1,499 @@
+/- Exercises 5.8
+ -
+ - Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+-/
+
+import tactic.rcases
+
+-- Exercise 1
+--
+-- Go back to the exercises in Chapter 3 and Chapter 4 and redo as many as you
+-- can now with tactic proofs, using also `rw` and `simp` as appropriate.
+section ex_1_3_1
+
+variables p q r : Prop
+
+-- Commutativity of ∧ and ∨
+example : p ∧ q ↔ q ∧ p :=
+begin
+  split,
+  { rintro ⟨p, q⟩, exact ⟨q, p⟩ },
+  { rintro ⟨q, p⟩, exact ⟨p, q⟩ },
+end
+
+example : p ∨ q ↔ q ∨ p :=
+begin
+  split,
+  { rintro (p | q),
+    { exact or.inr p },
+    { exact or.inl q },
+  },
+  { rintro (q | p),
+    { exact or.inr q },
+    { exact or.inl p },
+  },
+end
+
+-- Associativity of ∧ and ∨
+example : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) :=
+begin
+  split,
+  { rintro ⟨⟨p, q⟩, r⟩, exact ⟨p, q, r⟩ },
+  { rintro ⟨p, q, r⟩, exact ⟨⟨p, q⟩, r⟩ },
+end
+
+example : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) :=
+begin
+  split,
+  { rintro ((p | q) | r),
+    { exact or.inl p },
+    { exact or.inr (or.inl q) },
+    { exact or.inr (or.inr r) },
+  },
+  { rintro (p | q | r),
+    { exact or.inl (or.inl p) },
+    { exact or.inl (or.inr q) },
+    { exact or.inr r },
+  },
+end
+
+-- Distributivity
+example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) :=
+begin
+  split,
+  { rintro ⟨p, (q | r)⟩,
+    { exact or.inl ⟨p, q⟩ },
+    { exact or.inr ⟨p, r⟩ },
+  },
+  { rintro (⟨p, q⟩ | ⟨p, r⟩),
+    { exact ⟨p, or.inl q⟩ },
+    { exact ⟨p, or.inr r⟩ },
+  },
+end
+
+example : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) :=
+begin
+  split,
+  { rintro (p | ⟨q, r⟩),
+    { exact ⟨or.inl p, or.inl p⟩ },
+    { exact ⟨or.inr q, or.inr r⟩ },
+  },
+  { rintro ⟨(p | q), (p | r)⟩;
+    exact or.inl p <|> exact or.inr ⟨q, r⟩,
+  },
+end
+
+-- Other properties
+example : (p → (q → r)) ↔ (p ∧ q → r) :=
+begin
+  split,
+  { rintros h ⟨hp, hq⟩, exact h hp hq },
+  { intros h hp hq, exact h ⟨hp, hq⟩ },
+end
+
+example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) :=
+begin
+  split,
+  { intros h₁,
+    split,
+    { intro p, exact h₁ (or.inl p) },
+    { intro q, exact h₁ (or.inr q) },
+  },
+  { rintros ⟨hp, hq⟩ h,
+    apply or.elim h,
+    { intro p, exact hp p },
+    { intro q, exact hq q },
+  },
+end
+
+example : ¬(p ∨ q) ↔ ¬p ∧ ¬q :=
+begin
+  split,
+  { intro h,
+    split,
+    { intro hp, apply h, exact or.inl hp },
+    { intro hq, apply h, exact or.inr hq },
+  },
+  { rintros ⟨np, nq⟩ (p | q),
+    { apply np, assumption },
+    { apply nq, assumption },
+  },
+end
+
+example : ¬p ∨ ¬q → ¬(p ∧ q) :=
+begin
+  rintro (np | nq) ⟨hp, hq⟩,
+  { apply np, assumption },
+  { apply nq, assumption },
+end
+
+example : ¬(p ∧ ¬p) :=
+begin
+  rintro ⟨hp, np⟩,
+  exact absurd hp np
+end
+
+example : p ∧ ¬q → ¬(p → q) :=
+begin
+  rintro ⟨hp, nq⟩ hpq,
+  apply nq,
+  exact hpq hp
+end
+
+example : ¬p → (p → q) :=
+begin
+  intros np hp,
+  exact absurd hp np,
+end
+
+example : (¬p ∨ q) → (p → q) :=
+begin
+  rintros (np | hq) hp,
+  { exact absurd hp np },
+  { exact hq },
+end
+
+example : p ∨ false ↔ p :=
+begin
+  split,
+  { rintro (hp | hf),
+    { exact hp },
+    { exact false.elim hf },
+  },
+  { intro hp,
+    exact or.inl hp,
+  },
+end
+
+example : p ∧ false ↔ false :=
+begin
+  split,
+  { rintro ⟨hp, hf⟩, assumption },
+  { intro hf, exact false.elim hf },
+end
+
+example : (p → q) → (¬q → ¬p) :=
+begin
+  rintro hpq nq hp,
+  apply nq,
+  exact absurd (hpq hp) nq,
+end
+
+end ex_1_3_1
+
+section ex_1_3_2
+
+open classical
+
+variables p q r s : Prop
+
+example (hp : p) : (p → r ∨ s) → ((p → r) ∨ (p → s)) :=
+begin
+  intro h,
+  apply or.elim (h hp),
+  { intro hr, exact or.inl (assume hp, hr) },
+  { intro hs, exact or.inr (assume hp, hs) }
+end
+
+example : ¬(p ∧ q) → ¬p ∨ ¬q :=
+begin
+  intro h₁,
+  apply or.elim (classical.em p),
+  { intro hp,
+    apply or.elim (classical.em q),
+    { assume hq, exact false.elim (h₁ ⟨hp, hq⟩) },
+    { assume nq, exact or.inr nq },
+  },
+  { intro np,
+    exact or.inl np,
+  },
+end
+
+example : ¬(p → q) → p ∧ ¬q :=
+begin
+  intro h₁,
+  split,
+  { by_contradiction np, apply h₁, intro hp, exact absurd hp np },
+  { intro hq, apply h₁, intro hp, exact hq },
+end
+
+example : (p → q) → (¬p ∨ q) :=
+begin
+  intro hpq,
+  apply or.elim (classical.em p),
+  { assume hp, exact or.inr (hpq hp) },
+  { assume np, exact or.inl np },
+end
+
+example : (¬q → ¬p) → (p → q) :=
+begin
+  intros hqp hp,
+  by_contradiction nq,
+  exact absurd hp (hqp nq),
+end
+
+example : p ∨ ¬p :=
+by apply classical.em
+
+example : (((p → q) → p) → p) :=
+begin
+  intro h,
+  apply or.elim (classical.em p),
+  { intro hp, assumption },
+  { intro np, apply h, intro hp, exact absurd hp np },
+end
+
+end ex_1_3_2
+
+section ex_1_3_3
+
+variable p : Prop
+
+example (hp : p) : ¬(p ↔ ¬p) :=
+begin
+  intro h₁,
+  cases h₁ with h₂,
+  exact absurd hp (h₂ hp),
+end
+
+end ex_1_3_3
+
+section ex_1_4_1
+
+variables (α : Type*) (p q : α → Prop)
+
+example : (∀ x, p x ∧ q x) ↔ (∀ x, p x) ∧ (∀ x, q x) :=
+begin
+  split,
+  { intro h,
+    split,
+    { intro hx, exact and.left (h hx) },
+    { intro hx, exact and.right (h hx) },
+  },
+  { intros h hx,
+    have hl : ∀ (x : α), p x, from and.left h,
+    have hr : ∀ (x : α), q x, from and.right h,
+    exact ⟨hl hx, hr hx⟩
+  },
+end
+
+example : (∀ x, p x → q x) → (∀ x, p x) → (∀ x, q x) :=
+begin
+  intros h₁ h₂ hx,
+  exact h₁ hx (h₂ hx)
+end
+
+example : (∀ x, p x) ∨ (∀ x, q x) → ∀ x, p x ∨ q x :=
+begin
+  rintro (h₁ | h₂),
+  { intro hx, exact or.inl (h₁ hx) },
+  { intro hx, exact or.inr (h₂ hx) },
+end
+
+end ex_1_4_1
+
+section ex_1_4_2
+
+variables (α : Type*) (p q : α → Prop)
+variable r : Prop
+
+example : α → ((∀ x : α, r) ↔ r) :=
+begin
+  intro hα,
+  split,
+  { intro hαr, apply hαr, exact hα },
+  { intros hr hα, exact hr },
+end
+
+section
+
+open classical
+
+example : (∀ x, p x ∨ r) ↔ (∀ x, p x) ∨ r :=
+begin
+  split,
+  { intro h₁,
+    apply or.elim (classical.em r),
+    { intro hr, exact or.inr hr },
+    { intro nr,
+      exact or.inl begin
+        intro hx,
+        have h₂ : p hx ∨ r, from h₁ hx,
+        apply or.elim h₂,
+        { intro hp, exact hp },
+        { intro hr, exact absurd hr nr },
+      end
+    },
+  },
+  { assume h₁ hx,
+    apply or.elim h₁,
+    { assume h₂, exact or.inl (h₂ hx) },
+    { assume hr, exact or.inr hr },
+  },
+end
+
+end
+
+example : (∀ x, r → p x) ↔ (r → ∀ x, p x) :=
+begin
+  split,
+  { intros h hr hx, exact h hx hr },
+  { intros h hx hr, exact h hr hx },
+end
+
+end ex_1_4_2
+
+section ex_1_4_3
+
+open classical
+
+variables (men : Type*) (barber : men)
+variable (shaves : men → men → Prop)
+
+example (h : ∀ x : men, shaves barber x ↔ ¬ shaves x x) :
+  false :=
+begin
+  apply or.elim (classical.em (shaves barber barber)),
+  { intro hb,
+    apply iff.elim_left (h barber) hb,
+    exact hb,
+  },
+  { intro nb,
+    have s, from iff.elim_right (h barber) nb,
+    exact absurd s nb
+  },
+end
+
+end ex_1_4_3
+
+section ex_1_4_5
+
+open classical
+
+variables (α : Type*) (p q : α → Prop)
+variables r s : Prop
+
+example : (∃ x : α, r) → r :=
+begin
+  rintro ⟨hx, hr⟩,
+  exact hr,
+end
+
+example (a : α) : r → (∃ x : α, r) :=
+begin
+  intro hr,
+  exact ⟨a, hr⟩
+end
+
+example : (∃ x, p x ∧ r) ↔ (∃ x, p x) ∧ r :=
+begin
+  split,
+  { rintro ⟨hx, ⟨hp, hr⟩⟩, exact ⟨⟨hx, hp⟩, hr⟩ },
+  { rintro ⟨⟨hx, hp⟩, hr⟩, exact ⟨hx, ⟨hp, hr⟩⟩ },
+end
+
+example : (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ (∃ x, q x) :=
+begin
+  split,
+  { rintro ⟨hx, (hp | hq)⟩,
+    { exact or.inl ⟨hx, hp⟩ },
+    { exact or.inr ⟨hx, hq⟩ },
+  },
+  { rintro (⟨hx, hp⟩ | ⟨hx, hq⟩),
+    { exact ⟨hx, or.inl hp⟩ },
+    { exact ⟨hx, or.inr hq⟩ },
+  },
+end
+
+example : (∀ x, p x) ↔ ¬ (∃ x, ¬ p x) :=
+begin
+  split,
+  { rintros ha ⟨hx, np⟩, exact absurd (ha hx) np },
+  { intros he hx, by_contradiction np, exact he ⟨hx, np⟩ },
+end
+
+example : (∃ x, p x) ↔ ¬ (∀ x, ¬ p x) :=
+begin
+  split,
+  { rintro ⟨hx, hp⟩ h, exact absurd hp (h hx) },
+  { intro h₁,
+    by_contradiction h₂,
+    apply h₁,
+    intros hx hp,
+    exact h₂ ⟨hx, hp⟩,
+  },
+end
+
+example : (¬ ∃ x, p x) ↔ (∀ x, ¬ p x) :=
+begin
+  split,
+  { intros h hx hp,
+    exact h ⟨hx, hp⟩
+  },
+  { rintros h ⟨hx, hp⟩,
+    exact absurd hp (h hx)
+  },
+end
+
+lemma forall_negation : (¬ ∀ x, p x) ↔ (∃ x, ¬ p x) :=
+begin
+  split,
+  { intro h₁,
+    by_contradiction h₂,
+    exact h₁ (λ (x : α), begin
+      by_contradiction np,
+      exact h₂ ⟨x, np⟩
+    end)
+  },
+  { rintros ⟨hx, np⟩ h, exact absurd (h hx) np },
+end
+
+example : (¬ ∀ x, p x) ↔ (∃ x, ¬ p x) := forall_negation α p
+
+example : (∀ x, p x → r) ↔ (∃ x, p x) → r :=
+begin
+  split,
+  { rintros h ⟨hx, hp⟩, exact h hx hp },
+  { intros h hx hp,
+    exact h ⟨hx, hp⟩
+  },
+end
+
+example (a : α) : (∃ x, p x → r) ↔ (∀ x, p x) → r :=
+begin
+  split,
+  { rintros ⟨hx, hp⟩ h, apply hp, exact h hx },
+  { intro h₁,
+    apply or.elim (classical.em (∀ x, p x)),
+    { intro h₂, exact ⟨a, (assume hp, h₁ h₂)⟩ },
+    { intro h₂,
+      have h₃ : (∃ x, ¬p x), from iff.elim_left (forall_negation α p) h₂,
+      exact let ⟨hx, hp⟩ := h₃ in ⟨hx, (assume hp', absurd hp' hp)⟩,
+    },
+  },
+end
+
+example (a : α) : (∃ x, r → p x) ↔ (r → ∃ x, p x) :=
+begin
+  split,
+  { rintros ⟨hx, h⟩ hr, exact ⟨hx, h hr⟩ },
+  { intro h,
+    apply or.elim (classical.em r),
+    { intro hr,
+      exact let ⟨hx, hp⟩ := h hr in ⟨hx, (assume hr, hp)⟩
+    },
+    { intro nr, exact ⟨a, (assume hr, absurd hr nr)⟩ },
+  },
+end
+
+end ex_1_4_5
+
+-- Exercise 2
+--
+-- Use tactic combinators to obtain a one line proof of the following:
+section ex_2
+
+example (p q r : Prop) (hp : p) : (p ∨ q ∨ r) ∧ (q ∨ p ∨ r) ∧ (q ∨ r ∨ p) :=
+by simp *
+
+end ex_2