(* Author: Tobias Nipkow *) (* FIXME adjust List.sorted to work like below; [code] is different! *) theory Sorting imports Complex_Main "HOL-Library.Multiset" begin hide_const List.sorted List.insort declare Let_def [simp] fun sorted :: "'a::linorder list ⇒ bool" where "sorted [] = True" | "sorted (x # xs) = ((∀y∈set xs. x ≤ y) & sorted xs)" lemma sorted_append: "sorted (xs@ys) = (sorted xs & sorted ys & (∀x ∈ set xs. ∀y ∈ set ys. x≤y))" by (induct xs) (auto) subsection "Insertion Sort" fun insort :: "'a::linorder ⇒ 'a list ⇒ 'a list" where "insort x [] = [x]" | "insort x (y#ys) = (if x ≤ y then x#y#ys else y#(insort x ys))" fun isort :: "'a::linorder list ⇒ 'a list" where "isort [] = []" | "isort (x#xs) = insort x (isort xs)" subsubsection "Functional Correctness" lemma mset_insort: "mset (insort x xs) = add_mset x (mset xs)" apply(induction xs) apply auto done lemma mset_isort: "mset (isort xs) = mset xs" apply(induction xs) apply simp apply (simp add: mset_insort) done lemma "sorted (insort a xs) = sorted xs" apply(induction xs) apply (auto) oops lemma set_insort: "set (insort x xs) = insert x (set xs)" by (metis mset_insort set_mset_add_mset_insert set_mset_mset) lemma set_isort: "set (isort xs) = set xs" by (metis mset_isort set_mset_mset) lemma sorted_insort: "sorted (insort a xs) = sorted xs" apply(induction xs) apply(auto simp add: set_insort) done lemma "sorted (isort xs)" apply(induction xs) apply(auto simp: sorted_insort) done subsection "Time Complexity" text ‹We count the number of function calls.› text‹ ‹insort x [] = [x]› ‹insort x (y#ys) = (if x ≤ y then x#y#ys else y#(insort x ys))› › fun t_insort :: "'a::linorder ⇒ 'a list ⇒ nat" where "t_insort x [] = 1" | "t_insort x (y#ys) = (if x ≤ y then 0 else t_insort x ys) + 1" text‹ ‹isort [] = []› ‹isort (x#xs) = insort x (isort xs)› › fun t_isort :: "'a::linorder list ⇒ nat" where "t_isort [] = 1" | "t_isort (x#xs) = t_isort xs + t_insort x (isort xs) + 1" lemma t_insort_length: "t_insort x xs ≤ length xs + 1" apply(induction xs) apply auto done lemma length_insort: "length (insort x xs) = length xs + 1" apply(induction xs) apply auto done lemma length_isort: "length (isort xs) = length xs" apply(induction xs) apply (auto simp: length_insort) done lemma t_isort_length: "t_isort xs ≤ (length xs + 1) ^ 2" proof(induction xs) case Nil show ?case by simp next case (Cons x xs) have "t_isort (x#xs) = t_isort xs + t_insort x (isort xs) + 1" by simp also have "… ≤ (length xs + 1) ^ 2 + t_insort x (isort xs) + 1" using Cons.IH by simp also have "… ≤ (length xs + 1) ^ 2 + length xs + 1 + 1" using t_insort_length[of x "isort xs"] by (simp add: length_isort) also have "… ≤ (length(x#xs) + 1) ^ 2" by (simp add: power2_eq_square) finally show ?case . qed subsection "Merge Sort" fun merge :: "'a::linorder list ⇒ 'a list ⇒ 'a list" where "merge [] ys = ys" | "merge xs [] = xs" | "merge (x#xs) (y#ys) = (if x ≤ y then x # merge xs (y#ys) else y # merge (x#xs) ys)" fun msort :: "'a::linorder list ⇒ 'a list" where "msort xs = (let n = length xs in if n ≤ 1 then xs else merge (msort (take (n div 2) xs)) (msort (drop (n div 2) xs)))" declare msort.simps [simp del] (* We count the number of comparisons between list elements only *) fun c_merge :: "'a::linorder list ⇒ 'a list ⇒ nat" where "c_merge [] ys = 0" | "c_merge xs [] = 0" | "c_merge (x#xs) (y#ys) = 1 + (if x ≤ y then c_merge xs (y#ys) else c_merge (x#xs) ys)" lemma c_merge_ub: "c_merge xs ys ≤ length xs + length ys" by (induction xs ys rule: c_merge.induct) auto fun c_msort :: "'a::linorder list ⇒ nat" where "c_msort xs = (let n = length xs; ys = take (n div 2) xs; zs = drop (n div 2) xs in if n ≤ 1 then 0 else c_msort ys + c_msort zs + c_merge (msort ys) (msort zs))" declare c_msort.simps [simp del] lemma length_merge: "length(merge xs ys) = length xs + length ys" apply (induction xs ys rule: merge.induct) apply auto done lemma length_msort: "length(msort xs) = length xs" proof (induction xs rule: msort.induct) case (1 xs) thus ?case by (auto simp: msort.simps[of xs] length_merge) qed text ‹Why structured proof? To have the name "xs" to specialize msort.simps with xs to ensure that msort.simps cannot be used recursively. Also works without this precaution, but that is just luck.› lemma c_msort_le: "length xs = 2^k ⟹ c_msort xs ≤ k * 2^k" proof(induction k arbitrary: xs) case 0 thus ?case by (simp add: c_msort.simps) next case (Suc k) let ?n = "length xs" let ?ys = "take (?n div 2) xs" let ?zs = "drop (?n div 2) xs" show ?case proof (cases "?n ≤ 1") case True thus ?thesis by(simp add: c_msort.simps) next case False have "c_msort(xs) = c_msort ?ys + c_msort ?zs + c_merge (msort ?ys) (msort ?zs)" by (simp add: c_msort.simps msort.simps) also have "… ≤ c_msort ?ys + c_msort ?zs + length ?ys + length ?zs" using c_merge_ub[of "msort ?ys" "msort ?zs"] length_msort[of ?ys] length_msort[of ?zs] by arith also have "… ≤ k * 2^k + c_msort ?zs + length ?ys + length ?zs" using Suc.IH[of ?ys] Suc.prems by simp also have "… ≤ k * 2^k + k * 2^k + length ?ys + length ?zs" using Suc.IH[of ?zs] Suc.prems by simp also have "… = 2 * k * 2^k + 2 * 2 ^ k" using Suc.prems by simp finally show ?thesis by simp qed qed (* Beware of conversions: *) lemma "length xs = 2^k ⟹ c_msort xs ≤ length xs * log 2 (length xs)" using c_msort_le[of xs k] apply (simp add: log_nat_power algebra_simps) by (metis (mono_tags) numeral_power_eq_real_of_nat_cancel_iff of_nat_le_iff of_nat_mult) end