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Isabelle Task 3 complete

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Aadi Desai 2022-12-08 14:38:12 +00:00
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1 changed files with 44 additions and 7 deletions
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51
isabelle/2022/HSV_tasks_2022.thy
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@ -51,7 +51,7 @@ theorem "\<exists>n::nat.(n^6 mod 10 \<noteq> n mod 10)" by (rule exI[where x =
section "Task 3: Logic optimisation"
(* Datatype for representing simple circuits. *)
datatype "circuit" =
datatype "circuit" =
NOT "circuit"
| AND "circuit" "circuit"
| OR "circuit" "circuit"
@ -60,7 +60,7 @@ datatype "circuit" =
| INPUT "int"
(* Simulates a circuit given a valuation for each input wire. *)
fun simulate where
fun simulate :: "circuit \<Rightarrow> (int \<Rightarrow> bool) \<Rightarrow> bool" 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>))"
@ -76,7 +76,7 @@ fun circuits_equiv (infix "\<sim>" 50) (* the "50" indicates the operator preced
`a | a = a`
`a & a = a`
*)
fun opt_ident where
fun opt_ident :: "circuit \<Rightarrow> circuit" where
"opt_ident (NOT c) = NOT (opt_ident c)"
| "opt_ident (AND c1 c2) = (
let c1' = opt_ident c1 in
@ -90,12 +90,16 @@ fun opt_ident where
| "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
lemma "opt_ident (AND (INPUT 1) (OR (INPUT 1) (INPUT 1))) = INPUT 1" by eval (* test case *)
theorem opt_ident_is_sound: "opt_ident c \<sim> c"
oops
proof (induct c)
case (AND c1 c2)
thus ?case by (smt circuits_equiv.simps opt_ident.simps(2) simulate.simps(1))
next
case (OR c1 c2)
thus ?case by (smt circuits_equiv.simps opt_ident.simps(3) simulate.simps(2))
qed(simp+)
fun area :: "circuit \<Rightarrow> nat" where
"area (NOT c) = 1 + area c"
@ -103,6 +107,39 @@ fun area :: "circuit \<Rightarrow> nat" where
| "area (OR c1 c2) = 1 + area c1 + area c2"
| "area _ = 0"
theorem opt_ident_never_inc_area: "area (opt_ident c) \<le> area c"
proof (induct c)
case (AND c1 c2)
{
assume "opt_ident c1 = opt_ident c2"
hence "area (opt_ident (AND c1 c2)) = area (opt_ident c1)" by simp
hence "area (opt_ident (AND c1 c2)) \<le> area (AND (opt_ident c1) (opt_ident c2))" by simp
hence ?case using AND.hyps(2) \<open>opt_ident c1 = opt_ident c2\<close> by auto
}
moreover
{
assume "opt_ident c1 \<noteq> opt_ident c2"
hence "area (opt_ident (AND c1 c2)) = area (AND (opt_ident c1) (opt_ident c2))" by simp
hence ?case by (simp add: AND.hyps(1) AND.hyps(2) add_mono_thms_linordered_semiring(1))
}
ultimately show ?case by fastforce
next
case (OR c1 c2)
{
assume "opt_ident c1 = opt_ident c2"
hence "area (opt_ident (OR c1 c2)) = area (opt_ident c1)" by simp
hence "area (opt_ident (OR c1 c2)) \<le> area (OR (opt_ident c1) (opt_ident c2))" by simp
hence ?case using OR.hyps(2) \<open>opt_ident c1 = opt_ident c2\<close> by auto
}
moreover
{
assume "opt_ident c1 \<noteq> opt_ident c2"
hence "area (opt_ident (OR c1 c2)) = area (OR (opt_ident c1) (opt_ident c2))" by simp
hence ?case by (simp add: OR.hyps(1) OR.hyps(2) add_mono_thms_linordered_semiring(1))
}
ultimately show ?case by fastforce
qed(simp+)
section "Task 4: More logic optimisation"
lemma (* test case *)