Cookies disclaimer

I agree Our site saves small pieces of text information (cookies) on your device in order to deliver better content and for statistical purposes. You can disable the usage of cookies by changing the settings of your browser. By browsing our website without changing the browser settings you grant us permission to store that information on your device.

Homework 03

This is the task corresponding to homework 3.

Resources

Download Files

Definitions File

theory Defs
  imports Main
begin

abbreviation a where "a ≡ CHR ''a''"
abbreviation b where "b ≡ CHR ''b''"

definition
  "L = {w. ∃n. w = replicate n a @ replicate n b}"



datatype instr = LDI int | LD nat | ST nat | ADD nat

type_synonym rstate = "nat ⇒ int"

datatype expr = C int | V nat | A expr expr

fun val :: "expr ⇒ (nat ⇒ int) ⇒ int" where
"val(C i) s = i" |
"val(V n) s = s n" |
"val(A e1 e2) s = val e1 s + val e2 s"

end

Template File

theory Submission
  imports Defs
begin

inductive_set G :: "string set"

theorem G_is_replicate:
  "w ∈ G ⟹ ∃n. w = replicate n a @ replicate n b"
  sorry

theorem replicate_G:
  "w = replicate n a @ replicate n b ⟹ w ∈ G"
  sorry

corollary L_eq_G:
  "L = G"
  unfolding L_def using G_is_replicate replicate_G by auto



fun exec :: "instr ⇒ rstate ⇒ rstate" where
  "exec _ _ = undefined"

fun execs :: "instr list ⇒ rstate ⇒ rstate" where
  "execs _ _= undefined"

theorem execs_append[simp]: "⋀s. execs (xs @ ys) s = execs ys (execs xs s)"
  sorry

fun cmp :: "expr ⇒ nat ⇒ instr list" where
  "cmp _ _ = undefined"

fun maxvar :: "expr ⇒ nat" where
  "maxvar _ = undefined"

theorem val_maxvar_same[simp]:
  "ALL n <= maxvar e. s n = s' n ⟹ val e s = val e s'"
  sorry

theorem compiler_correct: "execs (cmp e (maxvar e + 1)) s 0 = val e (s o Suc)"
  sorry

end

Check File

theory Check
  imports Submission
begin

theorem G_is_replicate:
  assumes "w ∈ G"
  shows "∃n. w = replicate n a @ replicate n b"
  using assms by (rule Submission.G_is_replicate)

theorem replicate_G:
  assumes "w = replicate n a @ replicate n b"
  shows "w ∈ G"
  using assms by (rule Submission.replicate_G)

theorem execs_append: "⋀s. execs (xs @ ys) s = execs ys (execs xs s)"
  by (rule Submission.execs_append)

theorem val_maxvar_same:
  "ALL n <= maxvar e. s n = s' n ⟹ val e s = val e s'"
  by (rule Submission.val_maxvar_same)

theorem compiler_correct: "execs (cmp e (maxvar e + 1)) s 0 = val e (s o Suc)"
  by (rule Submission.compiler_correct)

end

Terms and Conditions