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92 lines
2.7 KiB
Plaintext
92 lines
2.7 KiB
Plaintext
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theory HSV_tasks_2020 imports Complex_Main begin
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section \<open>Task 1: proving that "3 / sqrt 2" is irrational.\<close>
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(* In case it is helpful, the following theorem is copied from Chapter 3 of the worksheet. *)
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theorem sqrt2_irrational: "sqrt 2 \<notin> \<rat>"
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proof auto
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assume "sqrt 2 \<in> \<rat>"
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then obtain m n where
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"n \<noteq> 0" and "\<bar>sqrt 2\<bar> = real m / real n" and "coprime m n"
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by (rule Rats_abs_nat_div_natE)
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hence "\<bar>sqrt 2\<bar>^2 = (real m / real n)^2" by auto
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hence "2 = (real m / real n)^2" by simp
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hence "2 = (real m)^2 / (real n)^2" unfolding power_divide by auto
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hence "2 * (real n)^2 = (real m)^2"
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by (simp add: nonzero_eq_divide_eq `n \<noteq> 0`)
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hence "real (2 * n^2) = (real m)^2" by auto
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hence *: "2 * n^2 = m^2"
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using of_nat_power_eq_of_nat_cancel_iff by blast
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hence "even (m^2)" by presburger
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hence "even m" by simp
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then obtain m' where "m = 2 * m'" by auto
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with * have "2 * n^2 = (2 * m')^2" by auto
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hence "2 * n^2 = 4 * m'^2" by simp
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hence "n^2 = 2 * m'^2" by simp
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hence "even (n^2)" by presburger
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hence "even n" by simp
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with `even m` and `coprime m n` show False by auto
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qed
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theorem "3 / sqrt 2 \<notin> \<rat>"
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sorry (* TODO: Complete this proof. *)
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section \<open>Task 2: Centred pentagonal numbers.\<close>
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fun pent :: "nat \<Rightarrow> nat" where
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"pent n = (if n = 0 then 1 else 5 * n + pent (n - 1))"
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value "pent 0" (* should be 1 *)
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value "pent 1" (* should be 6 *)
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value "pent 2" (* should be 16 *)
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value "pent 3" (* should be 31 *)
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theorem "pent n = (5 * n^2 + 5 * n + 2) div 2"
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sorry (* TODO: Complete this proof. *)
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section \<open>Task 3: Lucas numbers.\<close>
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fun fib :: "nat \<Rightarrow> nat" where
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"fib n = (if n = 0 then 0 else if n = 1 then 1 else fib (n - 1) + fib (n - 2))"
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value "fib 0" (* should be 0 *)
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value "fib 1" (* should be 1 *)
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value "fib 2" (* should be 1 *)
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value "fib 3" (* should be 2 *)
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thm fib.induct (* rule induction theorem for fib *)
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(* TODO: Complete this task. *)
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section \<open>Task 4: Balancing circuits.\<close>
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(* Here is a datatype for representing circuits, copied from the worksheet *)
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datatype "circuit" =
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NOT "circuit"
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| AND "circuit" "circuit"
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| OR "circuit" "circuit"
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| TRUE
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| FALSE
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| INPUT "int"
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text \<open>Delay (assuming all gates have a delay of 1)\<close>
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(* The following "delay" function also appeared in the 2019 coursework exercises. *)
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fun delay :: "circuit \<Rightarrow> nat" where
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"delay (NOT c) = 1 + delay c"
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| "delay (AND c1 c2) = 1 + max (delay c1) (delay c2)"
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| "delay (OR c1 c2) = 1 + max (delay c1) (delay c2)"
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| "delay _ = 0"
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(* TODO: Complete this task. *)
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section \<open>Task 5: Extending with NAND gates.\<close>
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(* TODO: Complete this task. *)
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end
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