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### Definitions File

### Template File

### Check 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

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

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

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