ELEC70056-HSV-CW1/isabelle/2022/HSV_tasks_2022.thy

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2022-11-17 11:15:35 +00:00
theory HSV_tasks_2022 imports Complex_Main begin
section {* Task 1: Full adders *}
fun fulladder :: "bool * bool * bool \<Rightarrow> bool * bool"
where
"fulladder (a,b,cin) = (
let (tmp1, tmp2) = halfadder(a,b) in
let (tmp3, s) = halfadder(cin,tmp2) in
let cout = tmp1 | tmp3 in
(cout, s))"
section {* Task 2: Fifth powers *}
theorem "(n::nat) ^ 5 mod 10 = n mod 10"
oops
section {* Task 3: Logic optimisation *}
(* Datatype for representing simple circuits. *)
datatype "circuit" =
NOT "circuit"
| AND "circuit" "circuit"
| OR "circuit" "circuit"
| TRUE
| FALSE
| INPUT "int"
(* Simulates a circuit given a valuation for each input wire. *)
fun simulate where
"simulate (AND c1 c2) \<rho> = ((simulate c1 \<rho>) \<and> (simulate c2 \<rho>))"
| "simulate (OR c1 c2) \<rho> = ((simulate c1 \<rho>) \<or> (simulate c2 \<rho>))"
| "simulate (NOT c) \<rho> = (\<not> (simulate c \<rho>))"
| "simulate TRUE \<rho> = True"
| "simulate FALSE \<rho> = False"
| "simulate (INPUT i) \<rho> = \<rho> i"
(* Equivalence between circuits. *)
fun circuits_equiv (infix "\<sim>" 50) (* the "50" indicates the operator precedence *) where
"c1 \<sim> c2 = (\<forall>\<rho>. simulate c1 \<rho> = simulate c2 \<rho>)"
(* An optimisation that exploits the following Boolean identities:
`a | a = a`
`a & a = a`
*)
fun opt_ident where
"opt_ident (NOT c) = NOT (opt_ident c)"
| "opt_ident (AND c1 c2) = (
let c1' = opt_ident c1 in
let c2' = opt_ident c2 in
if c1' = c2' then c1' else AND c1' c2')"
| "opt_ident (OR c1 c2) = (
let c1' = opt_ident c1 in
let c2' = opt_ident c2 in
if c1' = c2' then c1' else OR c1' c2')"
| "opt_ident TRUE = TRUE"
| "opt_ident FALSE = FALSE"
| "opt_ident (INPUT i) = INPUT i"
lemma (* test case *)
"opt_ident (AND (INPUT 1) (OR (INPUT 1) (INPUT 1))) = INPUT 1"
by eval
theorem opt_ident_is_sound: "opt_ident c \<sim> c"
oops
fun area :: "circuit \<Rightarrow> nat" where
"area (NOT c) = 1 + area c"
| "area (AND c1 c2) = 1 + area c1 + area c2"
| "area (OR c1 c2) = 1 + area c1 + area c2"
| "area _ = 0"
section {* Task 4: More logic optimisation *}
lemma (* test case *)
"opt_redundancy (AND (INPUT 1) (OR (INPUT 1) (INPUT 2)))
= INPUT 1"
(* by eval *) oops
lemma (* test case *)
"opt_redundancy (AND (AND (INPUT 1) (OR (INPUT 1) (INPUT 2)))
(OR (AND (INPUT 1) (OR (INPUT 1) (INPUT 2))) (INPUT 2)))
= INPUT 1"
(* by eval *) oops
lemma (* test case *)
"opt_redundancy (AND (AND (INPUT 1) (OR (INPUT 1) (INPUT 2)))
(OR (INPUT 2) (AND (INPUT 1) (OR (INPUT 1) (INPUT 2)))))
= INPUT 1"
(* by eval *) oops
lemma (* test case *)
"opt_redundancy (AND (AND (AND (INPUT 1) (OR (INPUT 1) (INPUT 2)))
(OR (INPUT 2) (AND (INPUT 1) (OR (INPUT 1) (INPUT 2)))))
(OR (INPUT 1) (INPUT 2)))
= INPUT 1"
(* by eval *) oops
end