mirror of
https://github.com/supleed2/ELEC70056-HSV-CW1.git
synced 2024-09-19 20:46:16 +00:00
59 lines
1.8 KiB
Plaintext
59 lines
1.8 KiB
Plaintext
theory HSV_chapter3 imports Complex_Main begin
|
|
|
|
(* We use the following command to search Isabelle's library
|
|
for theorems that contain a particular pattern. *)
|
|
find_theorems "_ \<in> \<rat>"
|
|
thm Rats_abs_nat_div_natE
|
|
|
|
find_theorems "(_ / _)^_"
|
|
thm power_divide
|
|
|
|
(* A proof that the square root of 2 is irrational. *)
|
|
theorem sqrt2_irrational: "sqrt 2 \<notin> \<rat>"
|
|
proof auto
|
|
assume "sqrt 2 \<in> \<rat>"
|
|
then obtain m n where
|
|
"n \<noteq> 0" and "\<bar>sqrt 2\<bar> = real m / real n" and "coprime m n"
|
|
by (rule Rats_abs_nat_div_natE)
|
|
hence "\<bar>sqrt 2\<bar>^2 = (real m / real n)^2" by auto
|
|
hence "2 = (real m / real n)^2" by simp
|
|
hence "2 = (real m)^2 / (real n)^2" unfolding power_divide by auto
|
|
hence "2 * (real n)^2 = (real m)^2"
|
|
by (simp add: nonzero_eq_divide_eq `n \<noteq> 0`)
|
|
hence "real (2 * n^2) = (real m)^2" by auto
|
|
hence *: "2 * n^2 = m^2"
|
|
using of_nat_power_eq_of_nat_cancel_iff by blast
|
|
hence "even (m^2)" by presburger
|
|
hence "even m" by simp
|
|
then obtain m' where "m = 2 * m'" by auto
|
|
with * have "2 * n^2 = (2 * m')^2" by auto
|
|
hence "2 * n^2 = 4 * m'^2" by simp
|
|
hence "n^2 = 2 * m'^2" by simp
|
|
hence "even (n^2)" by presburger
|
|
hence "even n" by simp
|
|
with `even m` and `coprime m n` show False by auto
|
|
qed
|
|
|
|
(* A proof that there is no greatest even number. *)
|
|
theorem "\<forall>n::int. even n \<longrightarrow> (\<exists>m. even m \<and> m > n)"
|
|
proof clarify
|
|
fix n::int
|
|
assume "even n"
|
|
hence "even (n+2)" by simp
|
|
moreover
|
|
have "n < (n+2)" by simp
|
|
ultimately
|
|
show "\<exists>m. even m \<and> n < m" by blast
|
|
qed
|
|
|
|
(* The same proof in the procedural style. *)
|
|
theorem "\<forall>n::int. even n \<longrightarrow> (\<exists>m. even m \<and> m > n)"
|
|
apply clarify
|
|
apply (rule_tac x="n+2" in exI)
|
|
apply (rule conjI)
|
|
apply simp
|
|
apply (thin_tac "even n")
|
|
apply simp
|
|
done
|
|
|
|
end |