Isabelle is an actively used programming language created in 1986.

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Example code from Linguist:

theory HelloWorld
imports Main
begin

section{*Playing around with Isabelle*}

text{* creating a lemma with the name hello_world*}
lemma hello_world: "True" by simp

(*inspecting it*)
thm hello_world

text{* defining a string constant HelloWorld *}

definition HelloWorld :: "string" where
  "HelloWorld \<equiv> ''Hello World!''"

(*reversing HelloWorld twice yilds HelloWorld again*)
theorem "rev (rev HelloWorld) = HelloWorld"
  by (fact List.rev_rev_ident)

text{*now we delete the already proven List.rev_rev_ident lema and show it by hand*}
declare List.rev_rev_ident[simp del]
hide_fact List.rev_rev_ident

(*It's trivial since we can just 'execute' it*)
corollary "rev (rev HelloWorld) = HelloWorld"
  apply(simp add: HelloWorld_def)
  done

text{*does it hold in general?*}
theorem rev_rev_ident:"rev (rev l) = l"
  proof(induction l)
  case Nil thus ?case by simp
  next
  case (Cons l ls)
    assume IH: "rev (rev ls) = ls"
    have "rev (l#ls) = (rev ls) @ [l]" by simp
    hence "rev (rev (l#ls)) = rev ((rev ls) @ [l])" by simp
    also have "\<dots> = [l] @ rev (rev ls)" by simp
    finally show "rev (rev (l#ls)) = l#ls" using IH by simp
  qed

corollary "\<forall>(l::string). rev (rev l) = l" by(fastforce intro: rev_rev_ident)

end

Example code from Wikipedia:

theorem sqrt2_not_rational:
  "sqrt (real 2) ∉ ℚ"
proof
  let ?x = "sqrt (real 2)"
  assume "?x ∈ ℚ"
  then obtain m n :: nat where
    sqrt_rat: "¦?x¦ = real m / real n" and lowest_terms: "coprime m n"
    by (rule Rats_abs_nat_div_natE)
  hence "real (m^2) = ?x^2 * real (n^2)" by (auto simp add: power2_eq_square)
  hence eq: "m^2 = 2 * n^2" using of_nat_eq_iff power2_eq_square by fastforce
  hence "2 dvd m^2" by simp
  hence "2 dvd m" by simp
  have "2 dvd n" proof-
    from ‹2 dvd m› obtain k where "m = 2 * k" ..
    with eq have "2 * n^2 = 2^2 * k^2" by simp
    hence "2 dvd n^2" by simp
    thus "2 dvd n" by simp
  qed
  with ‹2 dvd m› have "2 dvd gcd m n" by (rule gcd_greatest)
  with lowest_terms have "2 dvd 1" by simp
  thus False using odd_one by blast
qed
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Last updated February 11th, 2019