header {* Various examples for transfer procedure *}
theory Transfer_Ex
imports Main
begin
lemma ex1: "(x::nat) + y = y + x"
by auto
lemma "(0::int) ≤ (y::int) ==> (0::int) ≤ (x::int) ==> x + y = y + x"
by (fact ex1 [transferred])
lemma ex2: "(a::nat) div b * b + a mod b = a"
by (rule mod_div_equality)
lemma "(0::int) ≤ (b::int) ==> (0::int) ≤ (a::int) ==> a div b * b + a mod b = a"
by (fact ex2 [transferred])
lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y"
by auto
lemma "∀x≥0::int. ∀y≥0::int. ∃xa≥0::int. x + y ≤ xa"
by (fact ex3 [transferred nat_int])
lemma ex4: "(x::nat) >= y ==> (x - y) + y = x"
by auto
lemma "(0::int) ≤ (x::int) ==> (0::int) ≤ (y::int) ==> y ≤ x ==> tsub x y + y = x"
by (fact ex4 [transferred])
lemma ex5: "(2::nat) * ∑{..n} = n * (n + 1)"
by (induct n rule: nat_induct, auto)
lemma "(0::int) ≤ (n::int) ==> (2::int) * ∑{0::int..n} = n * (n + (1::int))"
by (fact ex5 [transferred])
lemma "(0::nat) ≤ (n::nat) ==> (2::nat) * ∑{0::nat..n} = n * (n + (1::nat))"
by (fact ex5 [transferred, transferred])
theorem ex6: "0 <= (n::int) ==> 2 * ∑{0..n} = n * (n + 1)"
by (rule ex5 [transferred])
lemma "(0::nat) ≤ (n::nat) ==> (2::nat) * ∑{0::nat..n} = n * (n + (1::nat))"
by (fact ex6 [transferred])
end