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### Definitions File

### Template File

### Check File

theory Defs imports "HOL-IMP.AExp" "HOL-IMP.BExp" begin datatype aexp = N int | V vname | Plus aexp aexp | Mult int aexp fun aval :: "aexp ⇒ state ⇒ val" where "aval (N n) s = n" | "aval (V x) s = s x" | "aval (Plus a⇩1 a⇩2) s = aval a⇩1 s + aval a⇩2 s" | "aval (Mult i a) s = i * aval a s" end

theory Submission imports Defs begin fun rlenc :: "'a ⇒ nat ⇒ 'a list ⇒ ('a × nat) list" where "rlenc _ = undefined" value "replicate (3::nat) (1::nat) = [1,1,1]" theorem test1: ‹rlenc 0 0 ([1,3,3,8] :: int list) = [(0,0),(1,1),(3,2),(8,1)]› by eval theorem test2: ‹rlenc 1 0 ([3,4,5] :: int list) = [(1,0),(3,1),(4,1),(5,1)]› by eval fun rldec :: "('a × nat) list ⇒ 'a list" where "rldec _ = undefined" theorem enc_dec: "rldec (rlenc a 0 l) = l" sorry lemmas [simp] = algebra_simps fun normal :: "aexp ⇒ bool" where "normal _ = undefined" fun normalize :: "aexp ⇒ aexp" where "normalize _ = undefined" theorem semantics_unchanged: "aval (normalize a) s = aval a s" sorry theorem normalize_normalizes: "normal (normalize a)" sorry end

theory Check imports Submission begin theorem test1: ‹rlenc 0 0 ([1,3,3,8] :: int list) = [(0,0),(1,1),(3,2),(8,1)]› by (rule Submission.test1) theorem test2: ‹rlenc 1 0 ([3,4,5] :: int list) = [(1,0),(3,1),(4,1),(5,1)]› by (rule Submission.test2) theorem enc_dec: "rldec (rlenc a 0 l) = l" by (rule Submission.enc_dec) theorem semantics_unchanged: "aval (normalize a) s = aval a s" by (rule Submission.semantics_unchanged) theorem normalize_normalizes: "normal (normalize a)" by (rule Submission.normalize_normalizes) end

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