replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
header {* Various examples for transfer procedure *}
theory Transfer_Ex
imports Main
begin
lemma ex1: "(x::nat) + y = y + x"
by auto
lemma "(0\<Colon>int) \<le> (y\<Colon>int) \<Longrightarrow> (0\<Colon>int) \<le> (x\<Colon>int) \<Longrightarrow> 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\<Colon>int) \<le> (b\<Colon>int) \<Longrightarrow> (0\<Colon>int) \<le> (a\<Colon>int) \<Longrightarrow> 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 "\<forall>x\<ge>0\<Colon>int. \<forall>y\<ge>0\<Colon>int. \<exists>xa\<ge>0\<Colon>int. x + y \<le> xa"
by (fact ex3 [transferred nat_int])
lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x"
by auto
lemma "(0\<Colon>int) \<le> (x\<Colon>int) \<Longrightarrow> (0\<Colon>int) \<le> (y\<Colon>int) \<Longrightarrow> y \<le> x \<Longrightarrow> tsub x y + y = x"
by (fact ex4 [transferred])
lemma ex5: "(2::nat) * \<Sum>{..n} = n * (n + 1)"
by (induct n rule: nat_induct, auto)
lemma "(0\<Colon>int) \<le> (n\<Colon>int) \<Longrightarrow> (2\<Colon>int) * \<Sum>{0\<Colon>int..n} = n * (n + (1\<Colon>int))"
by (fact ex5 [transferred])
lemma "(0\<Colon>nat) \<le> (n\<Colon>nat) \<Longrightarrow> (2\<Colon>nat) * \<Sum>{0\<Colon>nat..n} = n * (n + (1\<Colon>nat))"
by (fact ex5 [transferred, transferred])
theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)"
by (rule ex5 [transferred])
lemma "(0\<Colon>nat) \<le> (n\<Colon>nat) \<Longrightarrow> (2\<Colon>nat) * \<Sum>{0\<Colon>nat..n} = n * (n + (1\<Colon>nat))"
by (fact ex6 [transferred])
end