ELEC70056-HSV-CW1/isabelle/2020/HSV_tasks_2020.thy

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2021-11-01 06:34:44 +00:00
theory HSV_tasks_2020 imports Complex_Main begin
section \<open>Task 1: proving that "3 / sqrt 2" is irrational.\<close>
(* In case it is helpful, the following theorem is copied from Chapter 3 of the worksheet. *)
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
theorem "3 / sqrt 2 \<notin> \<rat>"
sorry (* TODO: Complete this proof. *)
section \<open>Task 2: Centred pentagonal numbers.\<close>
fun pent :: "nat \<Rightarrow> nat" where
"pent n = (if n = 0 then 1 else 5 * n + pent (n - 1))"
value "pent 0" (* should be 1 *)
value "pent 1" (* should be 6 *)
value "pent 2" (* should be 16 *)
value "pent 3" (* should be 31 *)
theorem "pent n = (5 * n^2 + 5 * n + 2) div 2"
sorry (* TODO: Complete this proof. *)
section \<open>Task 3: Lucas numbers.\<close>
fun fib :: "nat \<Rightarrow> nat" where
"fib n = (if n = 0 then 0 else if n = 1 then 1 else fib (n - 1) + fib (n - 2))"
value "fib 0" (* should be 0 *)
value "fib 1" (* should be 1 *)
value "fib 2" (* should be 1 *)
value "fib 3" (* should be 2 *)
thm fib.induct (* rule induction theorem for fib *)
(* TODO: Complete this task. *)
section \<open>Task 4: Balancing circuits.\<close>
(* Here is a datatype for representing circuits, copied from the worksheet *)
datatype "circuit" =
NOT "circuit"
| AND "circuit" "circuit"
| OR "circuit" "circuit"
| TRUE
| FALSE
| INPUT "int"
text \<open>Delay (assuming all gates have a delay of 1)\<close>
(* The following "delay" function also appeared in the 2019 coursework exercises. *)
fun delay :: "circuit \<Rightarrow> nat" where
"delay (NOT c) = 1 + delay c"
| "delay (AND c1 c2) = 1 + max (delay c1) (delay c2)"
| "delay (OR c1 c2) = 1 + max (delay c1) (delay c2)"
| "delay _ = 0"
(* TODO: Complete this task. *)
section \<open>Task 5: Extending with NAND gates.\<close>
(* TODO: Complete this task. *)
end