(*  Title:      HOL/Examples/Cantor.thy
    Author:     Makarius
*)

section ‹Cantor's Theorem›

theory Cantor
  imports Main
begin

subsection ‹Mathematical statement and proof›

text ‹
 Cantor's Theorem states that there is no surjection from
 a set to its powerset. The proof works by diagonalization. E.g.\ see
 ▪ 🌐‹http://mathworld.wolfram.com/CantorDiagonalMethod.html›
 ▪ 🌐‹https://en.wikipedia.org/wiki/Cantor's_diagonal_argument›
 ›

theorem Cantor: "∄f :: 'a → 'a set. ∀A. ∃x. A = f x"
proof
  assume "∃f :: 'a → 'a set. ∀A. ∃x. A = f x"
  then obtain f :: "'a → 'a set" where *: "∀A. ∃x. A = f x" ..
  let ?D = "{x. x ∉ f x}"
  from * obtain a where "?D = f a" by blast
  moreover have "a ∈ ?D ⟷ a ∉ f a" by blast
  ultimately show False by blast
qed


subsection ‹Automated proofs›

text ‹
 These automated proofs are much shorter, but lack information why and how it
 works.
 ›

theorem "∄f :: 'a → 'a set. ∀A. ∃x. f x = A"
  by best

theorem "∄f :: 'a → 'a set. ∀A. ∃x. f x = A"
  by force


subsection ‹Elementary version in higher-order predicate logic›

text ‹
 The subsequent formulation bypasses set notation of HOL; it uses elementary
 ‹λ›-calculus and predicate logic, with standard introduction and elimination
 rules. This also shows that the proof does not require classical reasoning.
 ›

lemma iff_contradiction:
  assumes *: "¬ A ⟷ A"
  shows False
proof (rule notE)
  show "¬ A"
  proof
    assume A
    with * have "¬ A" ..
    from this and ‹A› show False ..
  qed
  with * show A ..
qed

theorem Cantor': "∄f :: 'a → 'a → bool. ∀A. ∃x. A = f x"
proof
  assume "∃f :: 'a → 'a → bool. ∀A. ∃x. A = f x"
  then obtain f :: "'a → 'a → bool" where *: "∀A. ∃x. A = f x" ..
  let ?D = "λx. ¬ f x x"
  from * have "∃x. ?D = f x" ..
  then obtain a where "?D = f a" ..
  then have "?D a ⟷ f a a" by (rule arg_cong)
  then have "¬ f a a ⟷ f a a" .
  then show False by (rule iff_contradiction)
qed


subsection ‹Classic Isabelle/HOL example›

text ‹
 The following treatment of Cantor's Theorem follows the classic example from
 the early 1990s, e.g.\ see the file ▩‹92/HOL/ex/set.ML› in
 Isabelle92 or cite‹‹\S18.7› in "paulson-isa-book"›. The old tactic scripts
 synthesize key information of the proof by refinement of schematic goal
 states. In contrast, the Isar proof needs to say explicitly what is proven.

 ━
 Cantor's Theorem states that every set has more subsets than it has
 elements. It has become a favourite basic example in pure higher-order logic
 since it is so easily expressed:

 @{text [display]
 ‹∀f::α → α → bool. ∃S::α → bool. ∀x::α. f x ≠ S›}

 Viewing types as sets, ‹α → bool› represents the powerset of ‹α›. This
 version of the theorem states that for every function from ‹α› to its
 powerset, some subset is outside its range. The Isabelle/Isar proofs below
 uses HOL's set theory, with the type ‹α set› and the operator ‹range :: (α →
 β) → β set›.
 ›

theorem "∃S. S ∉ range (f :: 'a → 'a set)"
proof
  let ?S = "{x. x ∉ f x}"
  show "?S ∉ range f"
  proof
    assume "?S ∈ range f"
    then obtain y where "?S = f y" ..
    then show False
    proof (rule equalityCE)
      assume "y ∈ f y"
      assume "y ∈ ?S"
      then have "y ∉ f y" ..
      with ‹y ∈ f y› show ?thesis by contradiction
    next
      assume "y ∉ ?S"
      assume "y ∉ f y"
      then have "y ∈ ?S" ..
      with ‹y ∉ ?S› show ?thesis by contradiction
    qed
  qed
qed

text ‹
 How much creativity is required? As it happens, Isabelle can prove this
 theorem automatically using best-first search. Depth-first search would
 diverge, but best-first search successfully navigates through the large
 search space. The context of Isabelle's classical prover contains rules for
 the relevant constructs of HOL's set theory.
 ›

theorem "∃S. S ∉ range (f :: 'a → 'a set)"
  by best

end