{"full_name": "prop_01", "prop_defn": "theorem prop_01 (n: Nat) (xs: List \u03b1) :\n List.take n xs ++ List.drop n xs = xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:19", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nxs : List \u03b1\n\u22a2 List.take n xs ++ List.drop n xs = xs", "file_locs": "LeanSrc/LeanSrc/Properties.lean:20"} {"full_name": "prop_02", "prop_defn": "theorem prop_02 (n: Nat) (xs: List Nat) (ys: List Nat) :\n List.count n xs + List.count n ys = List.count n (xs ++ ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:23", "score": 5, "deps": "import Mathlib", "proof_state": "n : \u2115\nxs ys : List \u2115\n\u22a2 List.count n xs + List.count n ys = List.count n (xs ++ ys)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:24"} {"full_name": "prop_03", "prop_defn": "theorem prop_03 (n: Nat) (xs: List Nat) (ys: List Nat) :\n List.count n xs <= List.count n (xs ++ ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:27", "score": 4, "deps": "import Mathlib", "proof_state": "n : \u2115\nxs ys : List \u2115\n\u22a2 List.count n xs \u2264 List.count n (xs ++ ys)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:28"} {"full_name": "prop_04", "prop_defn": "theorem prop_04 (n: Nat) (xs: List Nat) :\n (List.count n xs).succ = List.count n (n :: xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:31", "score": 3, "deps": "import Mathlib", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 (List.count n xs).succ = List.count n (n :: xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:32"} {"full_name": "prop_05", "prop_defn": "theorem prop_05 (n: Nat) (x: Nat) (xs: List Nat) :\n (n = x) \u2192 (List.count n xs).succ = List.count n (x :: xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:35", "score": 4, "deps": "import Mathlib", "proof_state": "n x : \u2115\nxs : List \u2115\n\u22a2 n = x \u2192 (List.count n xs).succ = List.count n (x :: xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:36"} {"full_name": "prop_06", "prop_defn": "theorem prop_06 (n: Nat) (m: Nat) :\n (n - (n + m) = 0):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:39", "score": 2, "deps": "import Mathlib", "proof_state": "n m : \u2115\n\u22a2 n - (n + m) = 0", "file_locs": "LeanSrc/LeanSrc/Properties.lean:40"} {"full_name": "prop_07", "prop_defn": "theorem prop_07 (n: Nat) (m: Nat) :\n ((n + m) - n = m):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:43", "score": 1, "deps": "import Mathlib", "proof_state": "n m : \u2115\n\u22a2 n + m - n = m", "file_locs": "LeanSrc/LeanSrc/Properties.lean:44"} {"full_name": "prop_08", "prop_defn": "theorem prop_08 (k:Nat) (m: Nat) (n: Nat) :\n ((k + m) - (k + n) = m - n):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:47", "score": 3, "deps": "import Mathlib", "proof_state": "k m n : \u2115\n\u22a2 k + m - (k + n) = m - n", "file_locs": "LeanSrc/LeanSrc/Properties.lean:48"} {"full_name": "prop_09", "prop_defn": "theorem prop_09 (i: Nat) (j: Nat) (k: Nat) :\n ((i - j) - k = i - (j + k)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:51", "score": 2, "deps": "import Mathlib", "proof_state": "i j k : \u2115\n\u22a2 i - j - k = i - (j + k)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:52"} {"full_name": "prop_10", "prop_defn": "theorem prop_10 (m: Nat) :\n (m - m = 0):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:55", "score": 1, "deps": "import Mathlib", "proof_state": "m : \u2115\n\u22a2 m - m = 0", "file_locs": "LeanSrc/LeanSrc/Properties.lean:56"} {"full_name": "prop_11", "prop_defn": "theorem prop_11 (xs: List \u03b1) :\n (List.drop 0 xs = xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:59", "score": 3, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 List.drop 0 xs = xs", "file_locs": "LeanSrc/LeanSrc/Properties.lean:60"} {"full_name": "prop_12", "prop_defn": "theorem prop_12 (n: Nat) (f: \u03b1 \u2192 \u03b1) (xs: List \u03b1) :\n (List.drop n (List.map f xs) = List.map f (List.drop n xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:63", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nf : \u03b1 \u2192 \u03b1\nxs : List \u03b1\n\u22a2 List.drop n (List.map f xs) = List.map f (List.drop n xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:64"} {"full_name": "prop_13", "prop_defn": "theorem prop_13 (n: Nat) (x: \u03b1) (xs: List \u03b1) :\n (List.drop n.succ (x :: xs) = List.drop n xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:67", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nxs : List \u03b1\n\u22a2 List.drop n.succ (x :: xs) = List.drop n xs", "file_locs": "LeanSrc/LeanSrc/Properties.lean:68"} {"full_name": "prop_14", "prop_defn": "theorem prop_14 (p: \u03b1 \u2192 Bool) (xs: List \u03b1) (ys: List \u03b1) :\n (List.filter p (xs ++ ys) = (List.filter p xs) ++ (List.filter p ys)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:71", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\n\u22a2 List.filter p (xs ++ ys) = List.filter p xs ++ List.filter p ys", "file_locs": "LeanSrc/LeanSrc/Properties.lean:72"} {"full_name": "prop_15", "prop_defn": "theorem prop_15 (x: Nat) (xs: List Nat) :\n (List.length (ins x xs)) = (List.length xs).succ:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:75", "score": 5, "deps": "import Mathlib\n\ndef ins: Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if (n < x) then n :: x :: xs else x :: (ins n xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (ins x xs).length = xs.length.succ", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:202&&LeanSrc/LeanSrc/Properties.lean:76"} {"full_name": "prop_16", "prop_defn": "theorem prop_16 (x: Nat) (xs: List Nat) :\n xs = [] \u2192 last (x::xs) = x:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:79", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 xs = [] \u2192 last (x :: xs) = x", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:207&&LeanSrc/LeanSrc/Properties.lean:80"} {"full_name": "prop_17", "prop_defn": "theorem prop_17 (n: Nat) :\n n <= 0 \u2194 n = 0:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:83", "score": 1, "deps": "import Mathlib", "proof_state": "n : \u2115\n\u22a2 n \u2264 0 \u2194 n = 0", "file_locs": "LeanSrc/LeanSrc/Properties.lean:84"} {"full_name": "prop_18", "prop_defn": "theorem prop_18 i (m: Nat) :\n i < (i + m).succ:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:87", "score": 2, "deps": "import Mathlib", "proof_state": "i m : \u2115\n\u22a2 i < (i + m).succ", "file_locs": "LeanSrc/LeanSrc/Properties.lean:88"} {"full_name": "prop_19", "prop_defn": "theorem prop_19 (n: Nat) (xs: List Nat) :\n (List.length (List.drop n xs) = List.length xs - n):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:91", "score": 5, "deps": "import Mathlib", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 (List.drop n xs).length = xs.length - n", "file_locs": "LeanSrc/LeanSrc/Properties.lean:92"} {"full_name": "prop_20", "prop_defn": "theorem prop_20 (xs: List Nat) :\n (List.length (sort xs) = List.length xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:96", "score": 5, "deps": "import Mathlib\n\ndef insort : Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if n <= x then n::x::xs else x::(insort n xs)\n\n\ndef sort : List Nat \u2192 List Nat\n | [] => []\n | x::xs => insort x (sort xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (sort xs).length = xs.length", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:215&&LeanSrc/LeanSrc/Properties.lean:97"} {"full_name": "prop_21", "prop_defn": "theorem prop_21 (n: Nat) (m: Nat) :\n n <= (n + m):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:100", "score": 1, "deps": "import Mathlib", "proof_state": "n m : \u2115\n\u22a2 n \u2264 n + m", "file_locs": "LeanSrc/LeanSrc/Properties.lean:101"} {"full_name": "prop_22", "prop_defn": "theorem prop_22 (a: Nat) (b: Nat) (c: Nat) :\n (max (max a b) c = max a (max b c)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:104", "score": 2, "deps": "import Mathlib", "proof_state": "a b c : \u2115\n\u22a2 max (max a b) c = max a (max b c)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:105"} {"full_name": "prop_23", "prop_defn": "theorem prop_23 (a: Nat) (b: Nat) :\n (max a b = max b a):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:108", "score": 1, "deps": "import Mathlib", "proof_state": "a b : \u2115\n\u22a2 max a b = max b a", "file_locs": "LeanSrc/LeanSrc/Properties.lean:109"} {"full_name": "prop_24", "prop_defn": "theorem prop_24 (a: Nat) (b: Nat) :\n (((max a b) = a) \u2194 b <= a):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:112", "score": 2, "deps": "import Mathlib", "proof_state": "a b : \u2115\n\u22a2 max a b = a \u2194 b \u2264 a", "file_locs": "LeanSrc/LeanSrc/Properties.lean:113"} {"full_name": "prop_25", "prop_defn": "theorem prop_25 (a: Nat) (b: Nat) :\n (((max a b) = b) \u2194 a <= b):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:116", "score": 2, "deps": "import Mathlib", "proof_state": "a b : \u2115\n\u22a2 max a b = b \u2194 a \u2264 b", "file_locs": "LeanSrc/LeanSrc/Properties.lean:117"} {"full_name": "prop_26", "prop_defn": "theorem prop_26 (x: \u03b1) (xs: List \u03b1) (ys: List \u03b1) :\n x \u2208 xs \u2192 x \u2208 (xs ++ ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:120", "score": 4, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs ys : List \u03b1\n\u22a2 x \u2208 xs \u2192 x \u2208 xs ++ ys", "file_locs": "LeanSrc/LeanSrc/Properties.lean:121"} {"full_name": "prop_27", "prop_defn": "theorem prop_27 (x: \u03b1) (xs: List \u03b1) (ys: List \u03b1) :\n x \u2208 ys \u2192 x \u2208 (xs ++ ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:124", "score": 4, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs ys : List \u03b1\n\u22a2 x \u2208 ys \u2192 x \u2208 xs ++ ys", "file_locs": "LeanSrc/LeanSrc/Properties.lean:125"} {"full_name": "prop_28", "prop_defn": "theorem prop_28 (x: \u03b1) (xs: List \u03b1) :\n x \u2208 (xs ++ [x]):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:128", "score": 4, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\n\u22a2 x \u2208 xs ++ [x]", "file_locs": "LeanSrc/LeanSrc/Properties.lean:129"} {"full_name": "prop_29", "prop_defn": "theorem prop_29 (x: Nat) (xs: List Nat) :\n x \u2208 ins1 x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:132", "score": 5, "deps": "import Mathlib\n\ndef ins1 : Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if n == x then x::xs else x::(ins1 n xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 x \u2208 ins1 x xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:224&&LeanSrc/LeanSrc/Properties.lean:133"} {"full_name": "prop_30", "prop_defn": "theorem prop_30 (x: Nat) (xs: List Nat) :\n x \u2208 ins x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:136", "score": 5, "deps": "import Mathlib\n\ndef ins: Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if (n < x) then n :: x :: xs else x :: (ins n xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 x \u2208 ins x xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:202&&LeanSrc/LeanSrc/Properties.lean:137"} {"full_name": "prop_31", "prop_defn": "theorem prop_31 (a: Nat) (b: Nat) (c: Nat) :\n min (min a b) c = min a (min b c):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:140", "score": 2, "deps": "import Mathlib", "proof_state": "a b c : \u2115\n\u22a2 min (min a b) c = min a (min b c)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:141"} {"full_name": "prop_32", "prop_defn": "theorem prop_32 (a: Nat) (b: Nat) :\n min a b = min b a:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:144", "score": 1, "deps": "import Mathlib", "proof_state": "a b : \u2115\n\u22a2 min a b = min b a", "file_locs": "LeanSrc/LeanSrc/Properties.lean:145"} {"full_name": "prop_33", "prop_defn": "theorem prop_33 (a: Nat) (b: Nat) :\n min a b = a \u2194 a <= b:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:148", "score": 2, "deps": "import Mathlib", "proof_state": "a b : \u2115\n\u22a2 min a b = a \u2194 a \u2264 b", "file_locs": "LeanSrc/LeanSrc/Properties.lean:149"} {"full_name": "prop_34", "prop_defn": "theorem prop_34 (a: Nat) (b: Nat) :\n min a b = b \u2194 b <= a:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:152", "score": 2, "deps": "import Mathlib", "proof_state": "a b : \u2115\n\u22a2 min a b = b \u2194 b \u2264 a", "file_locs": "LeanSrc/LeanSrc/Properties.lean:153"} {"full_name": "prop_35", "prop_defn": "theorem prop_35 (xs: List \u03b1) :\n dropWhile (fun _ => False) xs = xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:156", "score": 5, "deps": "import Mathlib\n\ndef dropWhile : (\u03b1 \u2192 Bool) \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | p, x::xs => if p x then dropWhile p xs else x::xs\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 dropWhile (fun x => decide False) xs = xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:228&&LeanSrc/LeanSrc/Properties.lean:157"} {"full_name": "prop_36", "prop_defn": "theorem prop_36 (xs: List \u03b1) :\n takeWhile (fun _ => True) xs = xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:160", "score": 5, "deps": "import Mathlib\n\ndef takeWhile : (\u03b1 \u2192 Bool) \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | p, x::xs => if p x then x ::(takeWhile p xs) else []\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 takeWhile (fun x => decide True) xs = xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:232&&LeanSrc/LeanSrc/Properties.lean:161"} {"full_name": "prop_37", "prop_defn": "theorem prop_37 (x: Nat) (xs: List Nat) :\n not (x \u2208 delete x xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:164", "score": 5, "deps": "import Mathlib\n\ndef delete : Nat \u2192 List Nat \u2192 List Nat\n | _, [] => []\n | n, x::xs => if n == x then (delete n xs) else x::(delete n xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (!decide (x \u2208 delete x xs)) = true", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:236&&LeanSrc/LeanSrc/Properties.lean:165"} {"full_name": "prop_38", "prop_defn": "theorem prop_38 (n: Nat) (xs: List Nat) :\n List.count n (xs ++ [n]) = (List.count n xs).succ:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:168", "score": 4, "deps": "import Mathlib", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 List.count n (xs ++ [n]) = (List.count n xs).succ", "file_locs": "LeanSrc/LeanSrc/Properties.lean:169"} {"full_name": "prop_39", "prop_defn": "theorem prop_39 (n: Nat) (x: Nat) (xs: List Nat) :\n List.count n [x] + List.count n xs = List.count n (x::xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:172", "score": 4, "deps": "import Mathlib", "proof_state": "n x : \u2115\nxs : List \u2115\n\u22a2 List.count n [x] + List.count n xs = List.count n (x :: xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:173"} {"full_name": "prop_40", "prop_defn": "theorem prop_40 (xs: List \u03b1) :\n (List.take 0 xs = []):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:176", "score": 3, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 List.take 0 xs = []", "file_locs": "LeanSrc/LeanSrc/Properties.lean:177"} {"full_name": "prop_41", "prop_defn": "theorem prop_41 (n: Nat) (f: \u03b1 \u2192 \u03b1) (xs: List \u03b1) :\n (List.take n (List.map f xs) = List.map f (List.take n xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:180", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nf : \u03b1 \u2192 \u03b1\nxs : List \u03b1\n\u22a2 List.take n (List.map f xs) = List.map f (List.take n xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:181"} {"full_name": "prop_42", "prop_defn": "theorem prop_42 (n: Nat) (x: \u03b1) (xs: List \u03b1) :\n (List.take n.succ (x::xs) = x :: (List.take n xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:184", "score": 4, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nxs : List \u03b1\n\u22a2 List.take n.succ (x :: xs) = x :: List.take n xs", "file_locs": "LeanSrc/LeanSrc/Properties.lean:185"} {"full_name": "prop_43", "prop_defn": "theorem prop_43 (p: Nat \u2192 Bool) (xs: List Nat) :\n (takeWhile p xs ++ dropWhile p xs = xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:188", "score": 5, "deps": "import Mathlib\n\ndef dropWhile : (\u03b1 \u2192 Bool) \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | p, x::xs => if p x then dropWhile p xs else x::xs\n\n\ndef takeWhile : (\u03b1 \u2192 Bool) \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | p, x::xs => if p x then x ::(takeWhile p xs) else []\n", "proof_state": "p : \u2115 \u2192 Bool\nxs : List \u2115\n\u22a2 takeWhile p xs ++ dropWhile p xs = xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:232&&LeanSrc/LeanSrc/Properties.lean:189"} {"full_name": "prop_44", "prop_defn": "theorem prop_44 (x: \u03b1) (xs: List \u03b1) (ys: List \u03b2) :\n zip' (x::xs) ys = zipConcat x xs ys:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:192", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n\n\ndef zipConcat : \u03b1 \u2192 List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | _, _, [] => []\n | x, xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : \u03b1\nxs : List \u03b1\nys : List \u03b2\n\u22a2 zip' (x :: xs) ys = zipConcat x xs ys", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:245&&LeanSrc/LeanSrc/Properties.lean:193"} {"full_name": "prop_45", "prop_defn": "theorem prop_45 (x: \u03b1) (y: \u03b2) (xs: List \u03b1) (ys: List \u03b2) :\n zip' (x::xs) (y::ys) = (x, y) :: zip' xs ys:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:196", "score": 4, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : \u03b1\ny : \u03b2\nxs : List \u03b1\nys : List \u03b2\n\u22a2 zip' (x :: xs) (y :: ys) = (x, y) :: zip' xs ys", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:197"} {"full_name": "prop_46", "prop_defn": "theorem prop_46 {\u03b1 \u03b2: Type} (xs: List \u03b2) :\n zip' ([]: List \u03b1) xs = []:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:200", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 \u03b2 : Type\nxs : List \u03b2\n\u22a2 zip' [] xs = []", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:201"} {"full_name": "prop_47", "prop_defn": "theorem prop_47 (a: MyTree \u03b1) :\n (height' (mirror a) = height' a):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:204", "score": 5, "deps": "import Mathlib\n\ninductive MyTree (\u03b1: Type) where\n| leaf : MyTree \u03b1\n| node : MyTree \u03b1 \u2192 \u03b1 \u2192 MyTree \u03b1 \u2192 MyTree \u03b1\n\n\ndef height' : MyTree \u03b1 \u2192 \u2115\n | .leaf => 0\n | .node l _x r => (max (height' l) (height' r)).succ\n\n\ndef mirror : MyTree \u03b1 \u2192 MyTree \u03b1\n | MyTree.leaf => MyTree.leaf\n | MyTree.node l x r => MyTree.node r x l\n", "proof_state": "\u03b1 : Type\na : MyTree \u03b1\n\u22a2 height' (mirror a) = height' a", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:270&&LeanSrc/LeanSrc/Properties.lean:205"} {"full_name": "prop_48", "prop_defn": "theorem prop_48 (xs: List Nat) :\n not (null xs) \u2192 butlast xs ++ [last xs] = xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:208", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n\n\ndef null : List \u03b1 \u2192 Bool\n | [] => True\n | _ => False\n\n\ndef butlast : List \u03b1 \u2192 List \u03b1\n | [] => []\n | [_x] => []\n | x::xs => x::(butlast xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (!null xs) = true \u2192 butlast xs ++ [last xs] = xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:254&&LeanSrc/LeanSrc/Properties.lean:209"} {"full_name": "prop_49", "prop_defn": "theorem prop_49 (xs: List Nat) (ys: List Nat) :\n (butlast (xs ++ ys) = butlastConcat xs ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:212", "score": 4, "deps": "import Mathlib\n\ndef butlast : List \u03b1 \u2192 List \u03b1\n | [] => []\n | [_x] => []\n | x::xs => x::(butlast xs)\n\n\ndef butlastConcat : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | xs, [] => butlast xs\n | xs, ys => xs ++ butlast ys\n", "proof_state": "xs ys : List \u2115\n\u22a2 butlast (xs ++ ys) = butlastConcat xs ys", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:258&&LeanSrc/LeanSrc/Properties.lean:213"} {"full_name": "prop_50", "prop_defn": "theorem prop_50 (xs: List \u03b1) :\n (butlast xs = List.take (List.length xs - 1) xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:216", "score": 5, "deps": "import Mathlib\n\ndef butlast : List \u03b1 \u2192 List \u03b1\n | [] => []\n | [_x] => []\n | x::xs => x::(butlast xs)\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 butlast xs = List.take (xs.length - 1) xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:254&&LeanSrc/LeanSrc/Properties.lean:217"} {"full_name": "prop_51", "prop_defn": "theorem prop_51 (xs: List \u03b1) (x: \u03b1) :\n (butlast (xs ++ [x]) = xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:220", "score": 5, "deps": "import Mathlib\n\ndef butlast : List \u03b1 \u2192 List \u03b1\n | [] => []\n | [_x] => []\n | x::xs => x::(butlast xs)\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\nx : \u03b1\n\u22a2 butlast (xs ++ [x]) = xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:254&&LeanSrc/LeanSrc/Properties.lean:221"} {"full_name": "prop_52", "prop_defn": "theorem prop_52 (n: Nat) xs :\n (List.count n xs = List.count n (List.reverse xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:224", "score": 5, "deps": "import Mathlib", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 List.count n xs = List.count n xs.reverse", "file_locs": "LeanSrc/LeanSrc/Properties.lean:225"} {"full_name": "prop_53", "prop_defn": "theorem prop_53 (n: Nat) xs :\n (List.count n xs = List.count n (sort xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:229", "score": 5, "deps": "import Mathlib\n\ndef insort : Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if n <= x then n::x::xs else x::(insort n xs)\n\n\ndef sort : List Nat \u2192 List Nat\n | [] => []\n | x::xs => insort x (sort xs)\n", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 List.count n xs = List.count n (sort xs)", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:215&&LeanSrc/LeanSrc/Properties.lean:230"} {"full_name": "prop_54", "prop_defn": "theorem prop_54 (n: Nat) (m: Nat) :\n ((m + n) - n = m):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:233", "score": 2, "deps": "import Mathlib", "proof_state": "n m : \u2115\n\u22a2 m + n - n = m", "file_locs": "LeanSrc/LeanSrc/Properties.lean:234"} {"full_name": "prop_55", "prop_defn": "theorem prop_55 (n: Nat) (xs: List \u03b1) (ys: List \u03b1) :\n (List.drop n (xs ++ ys) = List.drop n xs ++ List.drop (n - List.length xs) ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:237", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nxs ys : List \u03b1\n\u22a2 List.drop n (xs ++ ys) = List.drop n xs ++ List.drop (n - xs.length) ys", "file_locs": "LeanSrc/LeanSrc/Properties.lean:238"} {"full_name": "prop_56", "prop_defn": "theorem prop_56 (n: Nat) (m: Nat) (xs: List \u03b1) :\n (List.drop n (List.drop m xs) = List.drop (n + m) xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:241", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn m : \u2115\nxs : List \u03b1\n\u22a2 List.drop n (List.drop m xs) = List.drop (n + m) xs", "file_locs": "LeanSrc/LeanSrc/Properties.lean:242"} {"full_name": "prop_57", "prop_defn": "theorem prop_57 (n: Nat) (m: Nat) (xs: List \u03b1) :\n (List.drop n (List.take m xs) = List.take (m - n) (List.drop n xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:245", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn m : \u2115\nxs : List \u03b1\n\u22a2 List.drop n (List.take m xs) = List.take (m - n) (List.drop n xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:246"} {"full_name": "prop_58", "prop_defn": "theorem prop_58 (n: Nat) (xs: List \u03b1) (ys: List \u03b2) :\n (List.drop n (zip' xs ys) = zip' (List.drop n xs) (List.drop n ys)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:249", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nxs : List \u03b1\nys : List \u03b2\n\u22a2 List.drop n (zip' xs ys) = zip' (List.drop n xs) (List.drop n ys)", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:250"} {"full_name": "prop_59", "prop_defn": "theorem prop_59 (xs: List Nat) (ys: List Nat) :\n ys = [] \u2192 last (xs ++ ys) = last xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:253", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n", "proof_state": "xs ys : List \u2115\n\u22a2 ys = [] \u2192 last (xs ++ ys) = last xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:207&&LeanSrc/LeanSrc/Properties.lean:254"} {"full_name": "prop_60", "prop_defn": "theorem prop_60 (xs: List Nat) (ys: List Nat) :\n not (null ys) \u2192 last (xs ++ ys) = last ys:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:257", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n\n\ndef null : List \u03b1 \u2192 Bool\n | [] => True\n | _ => False\n", "proof_state": "xs ys : List \u2115\n\u22a2 (!null ys) = true \u2192 last (xs ++ ys) = last ys", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:249&&LeanSrc/LeanSrc/Properties.lean:258"} {"full_name": "prop_61", "prop_defn": "theorem prop_61 (xs: List Nat) (ys: List Nat) :\n (last (xs ++ ys) = lastOfTwo xs ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:261", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n\n\ndef lastOfTwo : List \u2115 \u2192 List \u2115 \u2192 \u2115\n | xs, [] => last xs\n | _, ys => last ys\n", "proof_state": "xs ys : List \u2115\n\u22a2 last (xs ++ ys) = lastOfTwo xs ys", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:262&&LeanSrc/LeanSrc/Properties.lean:262"} {"full_name": "prop_62", "prop_defn": "theorem prop_62 (xs: List Nat) (x: Nat) :\n not (null xs) \u2192 last (x::xs) = last xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:265", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n\n\ndef null : List \u03b1 \u2192 Bool\n | [] => True\n | _ => False\n", "proof_state": "xs : List \u2115\nx : \u2115\n\u22a2 (!null xs) = true \u2192 last (x :: xs) = last xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:249&&LeanSrc/LeanSrc/Properties.lean:266"} {"full_name": "prop_63", "prop_defn": "theorem prop_63 (n: Nat) (xs: List Nat) :\n n < List.length xs \u2192 last (List.drop n xs) = last xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:269", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 n < xs.length \u2192 last (List.drop n xs) = last xs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:207&&LeanSrc/LeanSrc/Properties.lean:270"} {"full_name": "prop_64", "prop_defn": "theorem prop_64 x xs :\n (last (xs ++ [x]) = x):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:273", "score": 5, "deps": "import Mathlib\n\ndef last: List Nat \u2192 Nat\n | [] => 0\n | [x] => x\n | _x::xs => (last xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 last (xs ++ [x]) = x", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:207&&LeanSrc/LeanSrc/Properties.lean:274"} {"full_name": "prop_65", "prop_defn": "theorem prop_65 (i: Nat) (m: Nat) :\n i < (m + i).succ:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:277", "score": 2, "deps": "import Mathlib", "proof_state": "i m : \u2115\n\u22a2 i < (m + i).succ", "file_locs": "LeanSrc/LeanSrc/Properties.lean:278"} {"full_name": "prop_66", "prop_defn": "theorem prop_66 (p: \u03b1 \u2192 Bool) (xs: List \u03b1) :\n List.length (List.filter p xs) <= List.length xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:281", "score": 4, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nxs : List \u03b1\n\u22a2 (List.filter p xs).length \u2264 xs.length", "file_locs": "LeanSrc/LeanSrc/Properties.lean:282"} {"full_name": "prop_67", "prop_defn": "theorem prop_67 (xs: List Nat) :\n List.length (butlast xs) = List.length xs - 1:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:285", "score": 5, "deps": "import Mathlib\n\ndef butlast : List \u03b1 \u2192 List \u03b1\n | [] => []\n | [_x] => []\n | x::xs => x::(butlast xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (butlast xs).length = xs.length - 1", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:254&&LeanSrc/LeanSrc/Properties.lean:286"} {"full_name": "prop_68", "prop_defn": "theorem prop_68 (n: Nat) (xs: List Nat) :\n List.length (delete n xs) <= List.length xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:289", "score": 5, "deps": "import Mathlib\n\ndef delete : Nat \u2192 List Nat \u2192 List Nat\n | _, [] => []\n | n, x::xs => if n == x then (delete n xs) else x::(delete n xs)\n", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 (delete n xs).length \u2264 xs.length", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:236&&LeanSrc/LeanSrc/Properties.lean:290"} {"full_name": "prop_69", "prop_defn": "theorem prop_69 (n: Nat) (m: Nat) :\n n <= (m + n):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:293", "score": 1, "deps": "import Mathlib", "proof_state": "n m : \u2115\n\u22a2 n \u2264 m + n", "file_locs": "LeanSrc/LeanSrc/Properties.lean:294"} {"full_name": "prop_70", "prop_defn": "theorem prop_70 m (n: Nat) :\n m <= n \u2192 m <= n.succ:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:297", "score": 1, "deps": "import Mathlib", "proof_state": "m n : \u2115\n\u22a2 m \u2264 n \u2192 m \u2264 n.succ", "file_locs": "LeanSrc/LeanSrc/Properties.lean:298"} {"full_name": "prop_71", "prop_defn": "theorem prop_71 (x:Nat) (y :Nat) (xs: List Nat) :\n (x == y) = False \u2192 ((x \u2208 (ins y xs)) == (x \u2208 xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:301", "score": 5, "deps": "import Mathlib\n\ndef ins: Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if (n < x) then n :: x :: xs else x :: (ins n xs)\n", "proof_state": "x y : \u2115\nxs : List \u2115\n\u22a2 ((x == y) = true) = False \u2192 (decide (x \u2208 ins y xs) == decide (x \u2208 xs)) = true", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:202&&LeanSrc/LeanSrc/Properties.lean:302"} {"full_name": "prop_72", "prop_defn": "theorem prop_72 (i: Nat) (xs: List \u03b1) :\n (List.reverse (List.drop i xs) = List.take (List.length xs - i) (List.reverse xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:305", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\ni : \u2115\nxs : List \u03b1\n\u22a2 (List.drop i xs).reverse = List.take (xs.length - i) xs.reverse", "file_locs": "LeanSrc/LeanSrc/Properties.lean:306"} {"full_name": "prop_73", "prop_defn": "theorem prop_73 (p: \u03b1 \u2192 Bool) (xs: List \u03b1) :\n (List.reverse (List.filter p xs) = List.filter p (List.reverse xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:309", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nxs : List \u03b1\n\u22a2 (List.filter p xs).reverse = List.filter p xs.reverse", "file_locs": "LeanSrc/LeanSrc/Properties.lean:310"} {"full_name": "prop_74", "prop_defn": "theorem prop_74 (i: Nat) (xs: List \u03b1) :\n (List.reverse (List.take i xs) = List.drop (List.length xs - i) (List.reverse xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:313", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\ni : \u2115\nxs : List \u03b1\n\u22a2 (List.take i xs).reverse = List.drop (xs.length - i) xs.reverse", "file_locs": "LeanSrc/LeanSrc/Properties.lean:314"} {"full_name": "prop_75", "prop_defn": "theorem prop_75 (n: Nat) (m: Nat ) (xs: List Nat) :\n (List.count n xs + List.count n [m] = List.count n (m :: xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:317", "score": 5, "deps": "import Mathlib", "proof_state": "n m : \u2115\nxs : List \u2115\n\u22a2 List.count n xs + List.count n [m] = List.count n (m :: xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:318"} {"full_name": "prop_76", "prop_defn": "theorem prop_76 (n: Nat) (m: Nat) (xs: List Nat) :\n (n == m) = False \u2192 List.count n (xs ++ [m]) = List.count n xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:321", "score": 5, "deps": "import Mathlib", "proof_state": "n m : \u2115\nxs : List \u2115\n\u22a2 ((n == m) = true) = False \u2192 List.count n (xs ++ [m]) = List.count n xs", "file_locs": "LeanSrc/LeanSrc/Properties.lean:322"} {"full_name": "prop_77", "prop_defn": "theorem prop_77 (x: Nat) (xs: List Nat) :\n sorted xs \u2192 sorted (insort x xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:325", "score": 5, "deps": "import Mathlib\n\ndef insort : Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if n <= x then n::x::xs else x::(insort n xs)\n\n\ndef sorted : List Nat \u2192 Bool\n | [] => True\n | [_x] => True\n | x::y::xs => and (x <= y) (sorted (y::xs))\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 sorted xs = true \u2192 sorted (insort x xs) = true", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:220&&LeanSrc/LeanSrc/Properties.lean:326"} {"full_name": "prop_78", "prop_defn": "theorem prop_78 (xs: List Nat) :\n sorted (sort xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:330", "score": 5, "deps": "import Mathlib\n\ndef insort : Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if n <= x then n::x::xs else x::(insort n xs)\n\n\ndef sort : List Nat \u2192 List Nat\n | [] => []\n | x::xs => insort x (sort xs)\n\n\ndef sorted : List Nat \u2192 Bool\n | [] => True\n | [_x] => True\n | x::y::xs => and (x <= y) (sorted (y::xs))\n", "proof_state": "xs : List \u2115\n\u22a2 sorted (sort xs) = true", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:220&&LeanSrc/LeanSrc/Properties.lean:331"} {"full_name": "prop_79", "prop_defn": "theorem prop_79 (m: Nat) (n: Nat) (k: Nat) :\n ((m.succ - n) - k.succ = (m - n) - k):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:334", "score": 2, "deps": "import Mathlib", "proof_state": "m n k : \u2115\n\u22a2 m.succ - n - k.succ = m - n - k", "file_locs": "LeanSrc/LeanSrc/Properties.lean:335"} {"full_name": "prop_80", "prop_defn": "theorem prop_80 (n: Nat) (xs: List \u03b1) (ys: List \u03b1) :\n (List.take n (xs ++ ys) = List.take n xs ++ List.take (n - List.length xs) ys):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:338", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn : \u2115\nxs ys : List \u03b1\n\u22a2 List.take n (xs ++ ys) = List.take n xs ++ List.take (n - xs.length) ys", "file_locs": "LeanSrc/LeanSrc/Properties.lean:339"} {"full_name": "prop_81", "prop_defn": "theorem prop_81 (n: Nat) (m: Nat) (xs: List \u03b1) :\n (List.take n (List.drop m xs) = List.drop m (List.take (n + m) xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:343", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nn m : \u2115\nxs : List \u03b1\n\u22a2 List.take n (List.drop m xs) = List.drop m (List.take (n + m) xs)", "file_locs": "LeanSrc/LeanSrc/Properties.lean:344"} {"full_name": "prop_82", "prop_defn": "theorem prop_82 (n: Nat) (xs: List \u03b1) (ys: List \u03b2) :\n (List.take n (zip' xs ys) = zip' (List.take n xs) (List.take n ys)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:347", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nxs : List \u03b1\nys : List \u03b2\n\u22a2 List.take n (zip' xs ys) = zip' (List.take n xs) (List.take n ys)", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:348"} {"full_name": "prop_83", "prop_defn": "theorem prop_83 (xs: List \u03b1) (ys: List \u03b1) (zs: List \u03b2) :\n (zip' (xs ++ ys) zs =\n zip' xs (List.take (List.length xs) zs) ++ zip' ys (List.drop (List.length xs) zs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:351", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs ys : List \u03b1\nzs : List \u03b2\n\u22a2 zip' (xs ++ ys) zs = zip' xs (List.take xs.length zs) ++ zip' ys (List.drop xs.length zs)", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:353"} {"full_name": "prop_84", "prop_defn": "theorem prop_84 (xs: List \u03b1) (ys: List \u03b2) (zs: List \u03b2) :\n (zip' xs (ys ++ zs) =\n zip' (List.take (List.length ys) xs) ys ++ zip' (List.drop (List.length ys) xs) zs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:356", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs : List \u03b1\nys zs : List \u03b2\n\u22a2 zip' xs (ys ++ zs) = zip' (List.take ys.length xs) ys ++ zip' (List.drop ys.length xs) zs", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:358"} {"full_name": "prop_85", "prop_defn": "theorem prop_85 (xs: List \u03b1) (ys: List \u03b2) :\n (List.length xs = List.length ys) \u2192\n (zip' (List.reverse xs) (List.reverse ys) = List.reverse (zip' xs ys)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:363", "score": 5, "deps": "import Mathlib\n\ndef zip' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 \u03b2)\n | [], _ => []\n | _, [] => []\n | x::xs, y::ys => \u27e8x, y\u27e9 :: zip' xs ys\n", "proof_state": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs : List \u03b1\nys : List \u03b2\n\u22a2 xs.length = ys.length \u2192 zip' xs.reverse ys.reverse = (zip' xs ys).reverse", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:241&&LeanSrc/LeanSrc/Properties.lean:365"} {"full_name": "prop_86", "prop_defn": "theorem prop_86 (x: Nat) (y: Nat) (xs: List Nat) :\n x < y \u2192 ((x \u2208 (ins y xs)) == (x \u2208 xs)):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:368", "score": 5, "deps": "import Mathlib\n\ndef ins: Nat \u2192 List Nat \u2192 List Nat\n | n, [] => [n]\n | n, x::xs => if (n < x) then n :: x :: xs else x :: (ins n xs)\n", "proof_state": "x y : \u2115\nxs : List \u2115\n\u22a2 x < y \u2192 (decide (x \u2208 ins y xs) == decide (x \u2208 xs)) = true", "file_locs": "LeanSrc/LeanSrc/Definitions.lean:202&&LeanSrc/LeanSrc/Properties.lean:369"} {"full_name": "prop_ISortSorts", "prop_defn": "theorem prop_ISortSorts (xs: List Nat) : ordered (isort xs) == True:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:54", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (isort xs) == decide True) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:54"} {"full_name": "prop_ISortCount", "prop_defn": "theorem prop_ISortCount (x: Nat) (xs: List Nat) : count x (isort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:55", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (isort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:55"} {"full_name": "prop_ISortPermutes", "prop_defn": "theorem prop_ISortPermutes (xs: List Nat) : isPermutation (isort xs) xs == True:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:56", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (isort xs) xs == decide True) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:56"} {"full_name": "prop_BubSortSorts", "prop_defn": "theorem prop_BubSortSorts (xs: List Nat) : ordered (bubsort xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:89", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef bubsort (xs : List Nat) : {l' : List Nat // xs.length = l'.length} :=\n match xs with\n | [] => \u27e8[], rfl\u27e9\n | x :: xs =>\n match bubsort xs with\n | \u27e8[], h\u27e9 => \u27e8[x], by simp[h]\u27e9\n | \u27e8y :: ys, h\u27e9 =>\n if y < x then\n have : Nat.succ (List.length ys) < Nat.succ (List.length xs) := by rw [h, List.length_cons]; apply Nat.lt_succ_self\n let \u27e8zs, he\u27e9 := bubsort (x :: ys)\n \u27e8y :: zs, by simp[h, \u2190 he]\u27e9\n else\n \u27e8x :: y :: ys, by simp[h]\u27e9\ntermination_by xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered \u2191(bubsort xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:89"} {"full_name": "prop_BubSortCount", "prop_defn": "theorem prop_BubSortCount (x: Nat) (xs: List Nat) : count x (bubsort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:90", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef bubsort (xs : List Nat) : {l' : List Nat // xs.length = l'.length} :=\n match xs with\n | [] => \u27e8[], rfl\u27e9\n | x :: xs =>\n match bubsort xs with\n | \u27e8[], h\u27e9 => \u27e8[x], by simp[h]\u27e9\n | \u27e8y :: ys, h\u27e9 =>\n if y < x then\n have : Nat.succ (List.length ys) < Nat.succ (List.length xs) := by rw [h, List.length_cons]; apply Nat.lt_succ_self\n let \u27e8zs, he\u27e9 := bubsort (x :: ys)\n \u27e8y :: zs, by simp[h, \u2190 he]\u27e9\n else\n \u27e8x :: y :: ys, by simp[h]\u27e9\ntermination_by xs.length\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x \u2191(bubsort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:90"} {"full_name": "prop_BubSortPermutes", "prop_defn": "theorem prop_BubSortPermutes (xs: List Nat) : isPermutation (bubsort xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:91", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef bubsort (xs : List Nat) : {l' : List Nat // xs.length = l'.length} :=\n match xs with\n | [] => \u27e8[], rfl\u27e9\n | x :: xs =>\n match bubsort xs with\n | \u27e8[], h\u27e9 => \u27e8[x], by simp[h]\u27e9\n | \u27e8y :: ys, h\u27e9 =>\n if y < x then\n have : Nat.succ (List.length ys) < Nat.succ (List.length xs) := by rw [h, List.length_cons]; apply Nat.lt_succ_self\n let \u27e8zs, he\u27e9 := bubsort (x :: ys)\n \u27e8y :: zs, by simp[h, \u2190 he]\u27e9\n else\n \u27e8x :: y :: ys, by simp[h]\u27e9\ntermination_by xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (\u2191(bubsort xs)) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:91"} {"full_name": "prop_BubSortIsSort", "prop_defn": "theorem prop_BubSortIsSort (xs: List Nat) : bubblesort xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:92", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef bubsort (xs : List Nat) : {l' : List Nat // xs.length = l'.length} :=\n match xs with\n | [] => \u27e8[], rfl\u27e9\n | x :: xs =>\n match bubsort xs with\n | \u27e8[], h\u27e9 => \u27e8[x], by simp[h]\u27e9\n | \u27e8y :: ys, h\u27e9 =>\n if y < x then\n have : Nat.succ (List.length ys) < Nat.succ (List.length xs) := by rw [h, List.length_cons]; apply Nat.lt_succ_self\n let \u27e8zs, he\u27e9 := bubsort (x :: ys)\n \u27e8y :: zs, by simp[h, \u2190 he]\u27e9\n else\n \u27e8x :: y :: ys, by simp[h]\u27e9\ntermination_by xs.length\n\n\ndef bubblesort (xs: List Nat) : List Nat :=\n bubsort xs\n", "proof_state": "xs : List \u2115\n\u22a2 (bubblesort xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:92"} {"full_name": "prop_HSortSorts", "prop_defn": "theorem prop_HSortSorts (xs: List Nat) : ordered (hsort xs) == True:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:195", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef hpairwise : List MyHeap \u2192 List MyHeap\n| p::q::qs => (hmerge p q)::hpairwise qs\n| ps => ps\n\n\nlemma hpairwise_desc (ps: List MyHeap): List.length (hpairwise ps) \u2264 List.length ps := by\n generalize hl: ps.length = len\n induction len using Nat.strong_induction_on generalizing ps with\n | h len2 ih =>\n match ps with\n | [] => unfold hpairwise; simp\n | q1::qs1 => match qs1 with\n | [] => unfold hpairwise; rw [\u2190hl]\n | q2::qs2 => unfold hpairwise; rw [\u2190 hl, List.length_cons, List.length_cons, List.length_cons,Nat.succ_le_succ_iff]\n rw [List.length_cons, List.length_cons] at hl\n have hl3 := Nat.lt_of_succ_lt (Nat.lt_of_succ_le (Nat.le_of_eq hl))\n exact Nat.le.step (ih (qs2.length) hl3 qs2 rfl)\n\n\ndef hmerging : List MyHeap \u2192 MyHeap\n| [] => MyHeap.nil\n| [p] => p\n| p::q::ps =>\n have : List.length (hpairwise (p :: q :: ps)) < Nat.succ (Nat.succ (List.length ps)) := by\n unfold hpairwise\n rw [List.length_cons, Nat.succ_lt_succ_iff, Nat.lt_succ]\n exact hpairwise_desc _\n hmerging (hpairwise (p::q::ps))\ntermination_by ps => ps.length\n\n\ndef toHeap : List Nat \u2192 MyHeap\n| xs => hmerging (xs.map (fun x => MyHeap.node MyHeap.nil x MyHeap.nil))\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hsort : List Nat \u2192 List Nat\n | xs => toList (toHeap xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (hsort xs) == decide True) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:195"} {"full_name": "prop_HSortCount", "prop_defn": "theorem prop_HSortCount (x: Nat) (xs: List Nat) : count x (hsort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:196", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef hpairwise : List MyHeap \u2192 List MyHeap\n| p::q::qs => (hmerge p q)::hpairwise qs\n| ps => ps\n\n\nlemma hpairwise_desc (ps: List MyHeap): List.length (hpairwise ps) \u2264 List.length ps := by\n generalize hl: ps.length = len\n induction len using Nat.strong_induction_on generalizing ps with\n | h len2 ih =>\n match ps with\n | [] => unfold hpairwise; simp\n | q1::qs1 => match qs1 with\n | [] => unfold hpairwise; rw [\u2190hl]\n | q2::qs2 => unfold hpairwise; rw [\u2190 hl, List.length_cons, List.length_cons, List.length_cons,Nat.succ_le_succ_iff]\n rw [List.length_cons, List.length_cons] at hl\n have hl3 := Nat.lt_of_succ_lt (Nat.lt_of_succ_le (Nat.le_of_eq hl))\n exact Nat.le.step (ih (qs2.length) hl3 qs2 rfl)\n\n\ndef hmerging : List MyHeap \u2192 MyHeap\n| [] => MyHeap.nil\n| [p] => p\n| p::q::ps =>\n have : List.length (hpairwise (p :: q :: ps)) < Nat.succ (Nat.succ (List.length ps)) := by\n unfold hpairwise\n rw [List.length_cons, Nat.succ_lt_succ_iff, Nat.lt_succ]\n exact hpairwise_desc _\n hmerging (hpairwise (p::q::ps))\ntermination_by ps => ps.length\n\n\ndef toHeap : List Nat \u2192 MyHeap\n| xs => hmerging (xs.map (fun x => MyHeap.node MyHeap.nil x MyHeap.nil))\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hsort : List Nat \u2192 List Nat\n | xs => toList (toHeap xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (hsort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:196"} {"full_name": "prop_HSortPermutes", "prop_defn": "theorem prop_HSortPermutes (xs: List Nat) : isPermutation (hsort xs) xs == True:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:197", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef hpairwise : List MyHeap \u2192 List MyHeap\n| p::q::qs => (hmerge p q)::hpairwise qs\n| ps => ps\n\n\nlemma hpairwise_desc (ps: List MyHeap): List.length (hpairwise ps) \u2264 List.length ps := by\n generalize hl: ps.length = len\n induction len using Nat.strong_induction_on generalizing ps with\n | h len2 ih =>\n match ps with\n | [] => unfold hpairwise; simp\n | q1::qs1 => match qs1 with\n | [] => unfold hpairwise; rw [\u2190hl]\n | q2::qs2 => unfold hpairwise; rw [\u2190 hl, List.length_cons, List.length_cons, List.length_cons,Nat.succ_le_succ_iff]\n rw [List.length_cons, List.length_cons] at hl\n have hl3 := Nat.lt_of_succ_lt (Nat.lt_of_succ_le (Nat.le_of_eq hl))\n exact Nat.le.step (ih (qs2.length) hl3 qs2 rfl)\n\n\ndef hmerging : List MyHeap \u2192 MyHeap\n| [] => MyHeap.nil\n| [p] => p\n| p::q::ps =>\n have : List.length (hpairwise (p :: q :: ps)) < Nat.succ (Nat.succ (List.length ps)) := by\n unfold hpairwise\n rw [List.length_cons, Nat.succ_lt_succ_iff, Nat.lt_succ]\n exact hpairwise_desc _\n hmerging (hpairwise (p::q::ps))\ntermination_by ps => ps.length\n\n\ndef toHeap : List Nat \u2192 MyHeap\n| xs => hmerging (xs.map (fun x => MyHeap.node MyHeap.nil x MyHeap.nil))\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hsort : List Nat \u2192 List Nat\n | xs => toList (toHeap xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (hsort xs) xs == decide True) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:197"} {"full_name": "prop_HSortIsSort", "prop_defn": "theorem prop_HSortIsSort (xs: List Nat) : hsort xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:198", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef hpairwise : List MyHeap \u2192 List MyHeap\n| p::q::qs => (hmerge p q)::hpairwise qs\n| ps => ps\n\n\nlemma hpairwise_desc (ps: List MyHeap): List.length (hpairwise ps) \u2264 List.length ps := by\n generalize hl: ps.length = len\n induction len using Nat.strong_induction_on generalizing ps with\n | h len2 ih =>\n match ps with\n | [] => unfold hpairwise; simp\n | q1::qs1 => match qs1 with\n | [] => unfold hpairwise; rw [\u2190hl]\n | q2::qs2 => unfold hpairwise; rw [\u2190 hl, List.length_cons, List.length_cons, List.length_cons,Nat.succ_le_succ_iff]\n rw [List.length_cons, List.length_cons] at hl\n have hl3 := Nat.lt_of_succ_lt (Nat.lt_of_succ_le (Nat.le_of_eq hl))\n exact Nat.le.step (ih (qs2.length) hl3 qs2 rfl)\n\n\ndef hmerging : List MyHeap \u2192 MyHeap\n| [] => MyHeap.nil\n| [p] => p\n| p::q::ps =>\n have : List.length (hpairwise (p :: q :: ps)) < Nat.succ (Nat.succ (List.length ps)) := by\n unfold hpairwise\n rw [List.length_cons, Nat.succ_lt_succ_iff, Nat.lt_succ]\n exact hpairwise_desc _\n hmerging (hpairwise (p::q::ps))\ntermination_by ps => ps.length\n\n\ndef toHeap : List Nat \u2192 MyHeap\n| xs => hmerging (xs.map (fun x => MyHeap.node MyHeap.nil x MyHeap.nil))\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hsort : List Nat \u2192 List Nat\n | xs => toList (toHeap xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (hsort xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:198"} {"full_name": "prop_HSort2Sorts", "prop_defn": "theorem prop_HSort2Sorts (xs: List Nat) : ordered (hsort2 xs) == True:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:211", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hinsert : Nat \u2192 MyHeap \u2192 MyHeap\n| x, h => hmerge (MyHeap.node MyHeap.nil x MyHeap.nil) h\n\n\ndef toHeap2 : List Nat \u2192 MyHeap\n| [] => MyHeap.nil\n| x::xs => hinsert x (toHeap2 xs)\n\n\ndef hsort2 : List Nat \u2192 List Nat\n| xs => toList (toHeap2 xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (hsort2 xs) == decide True) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:211"} {"full_name": "prop_HSort2Count", "prop_defn": "theorem prop_HSort2Count (x: Nat) (xs: List Nat) : count x (hsort2 xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:212", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hinsert : Nat \u2192 MyHeap \u2192 MyHeap\n| x, h => hmerge (MyHeap.node MyHeap.nil x MyHeap.nil) h\n\n\ndef toHeap2 : List Nat \u2192 MyHeap\n| [] => MyHeap.nil\n| x::xs => hinsert x (toHeap2 xs)\n\n\ndef hsort2 : List Nat \u2192 List Nat\n| xs => toList (toHeap2 xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (hsort2 xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:212"} {"full_name": "prop_HSort2Permutes", "prop_defn": "theorem prop_HSort2Permutes (xs: List Nat) : isPermutation (hsort2 xs) xs == True:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:213", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hinsert : Nat \u2192 MyHeap \u2192 MyHeap\n| x, h => hmerge (MyHeap.node MyHeap.nil x MyHeap.nil) h\n\n\ndef toHeap2 : List Nat \u2192 MyHeap\n| [] => MyHeap.nil\n| x::xs => hinsert x (toHeap2 xs)\n\n\ndef hsort2 : List Nat \u2192 List Nat\n| xs => toList (toHeap2 xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (hsort2 xs) xs == decide True) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:213"} {"full_name": "prop_HSort2IsSort", "prop_defn": "theorem prop_HSort2IsSort (xs: List Nat) : hsort2 xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:214", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n\n\nlemma numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by\n have h': numElem (MyHeap.node q x r) = 1 + numElem q + numElem r; rfl\n rw [h'];\n exact \u27e8by linarith, by linarith\u27e9;\n\n\nlemma merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by\n generalize hsp: numElem p = sp\n generalize hsq: numElem q = sq\n generalize hspq: numElem (hmerge p q) = spq\n induction sp using Nat.strong_induction_on generalizing p q sq spq with\n | h sp2 ih => induction sq using Nat.strong_induction_on generalizing p q sp2 spq with\n | h sq2 ih2 =>\n rw [\u2190hsp, \u2190 hsq, \u2190 hspq];\n unfold hmerge;\n split;\n case h_1 _ _;\n unfold numElem; rw [Nat.add_comm, Nat.add_zero];\n case h_2 _ _;\n unfold numElem; rw [Nat.add_zero];\n case h_3 _ _ pl x pr ql y qr;\n split;\n unfold numElem;\n suffices h': numElem (hmerge pr (MyHeap.node ql y qr)) = numElem pr + (1 + numElem ql + numElem qr);\n rw[h']; linarith;\n rw [\u2190hsp] at ih;\n exact Eq.symm (ih (numElem pr) (numElem_lt_subHeaps _ _).2 pr (MyHeap.node ql y qr) rfl\n (numElem (MyHeap.node ql y qr)) rfl (numElem (hmerge pr (MyHeap.node ql y qr))) rfl);\n unfold numElem;\n suffices h': numElem (hmerge (MyHeap.node pl x pr) qr) = numElem qr + (1 + numElem pl + numElem pr);\n rw[h']; linarith;\n rw [\u2190hsq] at ih2;\n have h':= ih2 (numElem qr) (numElem_lt_subHeaps _ _).2 sp2 ih (MyHeap.node pl x pr) qr hsp rfl\n (numElem (hmerge (MyHeap.node pl x pr) qr)) rfl;\n rw [\u2190hsp] at h';\n suffices h'': 1 + numElem pl + numElem pr = numElem (MyHeap.node pl x pr);\n rw [h'']; linarith;\n rfl;\n\n\nlemma numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by\n rw [\u2190merge_elems _ _];\n have h': numElem (MyHeap.node p x q) = 1 + numElem p + numElem q; rfl\n rw [h']\n linarith;\n\n\ndef toList : MyHeap \u2192 List Nat\n| MyHeap.nil => []\n| MyHeap.node p x q =>\n have _h := numElem_merge_branches_lt p q x\n x :: toList (hmerge p q)\ntermination_by hp => numElem hp\n\n\ndef hinsert : Nat \u2192 MyHeap \u2192 MyHeap\n| x, h => hmerge (MyHeap.node MyHeap.nil x MyHeap.nil) h\n\n\ndef toHeap2 : List Nat \u2192 MyHeap\n| [] => MyHeap.nil\n| x::xs => hinsert x (toHeap2 xs)\n\n\ndef hsort2 : List Nat \u2192 List Nat\n| xs => toList (toHeap2 xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (hsort2 xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:214"} {"full_name": "prop_MSortBU2Sorts", "prop_defn": "theorem prop_MSortBU2Sorts (xs: List Nat) : ordered (msortbu2 xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:324", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef risers : List Nat \u2192 List (List Nat)\n| [] => []\n| [x] => [[x]]\n| x::y::xs => if x <= y then\n match (risers (y::xs)) with\n | ys::yss => (x::ys)::yss\n | _ => []\n else\n [x]::(risers (y::xs))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu2 : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu2 (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu2 : List Nat \u2192 List Nat\n| xs => mergingbu2 (risers xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (msortbu2 xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:324"} {"full_name": "prop_MSortBU2Count", "prop_defn": "theorem prop_MSortBU2Count (x: Nat) (xs: List Nat) : count x (msortbu2 xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:325", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef risers : List Nat \u2192 List (List Nat)\n| [] => []\n| [x] => [[x]]\n| x::y::xs => if x <= y then\n match (risers (y::xs)) with\n | ys::yss => (x::ys)::yss\n | _ => []\n else\n [x]::(risers (y::xs))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu2 : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu2 (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu2 : List Nat \u2192 List Nat\n| xs => mergingbu2 (risers xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (msortbu2 xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:325"} {"full_name": "prop_MSortBU2Permutes", "prop_defn": "theorem prop_MSortBU2Permutes (xs: List Nat) : isPermutation (msortbu2 xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:326", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef risers : List Nat \u2192 List (List Nat)\n| [] => []\n| [x] => [[x]]\n| x::y::xs => if x <= y then\n match (risers (y::xs)) with\n | ys::yss => (x::ys)::yss\n | _ => []\n else\n [x]::(risers (y::xs))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu2 : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu2 (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu2 : List Nat \u2192 List Nat\n| xs => mergingbu2 (risers xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (msortbu2 xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:326"} {"full_name": "prop_MSortBU2IsSort", "prop_defn": "theorem prop_MSortBU2IsSort (xs: List Nat) : msortbu2 xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:327", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef risers : List Nat \u2192 List (List Nat)\n| [] => []\n| [x] => [[x]]\n| x::y::xs => if x <= y then\n match (risers (y::xs)) with\n | ys::yss => (x::ys)::yss\n | _ => []\n else\n [x]::(risers (y::xs))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu2 : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu2 (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu2 : List Nat \u2192 List Nat\n| xs => mergingbu2 (risers xs)\n", "proof_state": "xs : List \u2115\n\u22a2 (msortbu2 xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:327"} {"full_name": "prop_MSortBUSorts", "prop_defn": "theorem prop_MSortBUSorts (xs: List Nat) : ordered (msortbu xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:352", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu : List Nat \u2192 List Nat\n| xs => mergingbu (xs.map (fun x => [x]))\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (msortbu xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:352"} {"full_name": "prop_MSortBUCount", "prop_defn": "theorem prop_MSortBUCount (x: Nat) (xs: List Nat) : count x (msortbu xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:353", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu : List Nat \u2192 List Nat\n| xs => mergingbu (xs.map (fun x => [x]))\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (msortbu xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:353"} {"full_name": "prop_MSortBUPermutes", "prop_defn": "theorem prop_MSortBUPermutes (xs: List Nat) : isPermutation (msortbu xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:354", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu : List Nat \u2192 List Nat\n| xs => mergingbu (xs.map (fun x => [x]))\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (msortbu xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:354"} {"full_name": "prop_MSortBUIsSort", "prop_defn": "theorem prop_MSortBUIsSort (xs: List Nat) : msortbu xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:355", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n\n\nlemma len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by\n generalize hxl : xs.length = xl;\n split_ifs with h1;\n case pos;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => simp at h1;\n | cons head2 tail2 =>\n cases tail2 with\n | nil => unfold pairwise; simp at hxl; rw [\u2190hxl]; simp;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _]\n ring_nf;\n simp;\n have hodd: Odd tail3.length := by\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.add_assoc, Nat.odd_add] at h1;\n apply h1.2; simp;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl hodd\n ring_nf at tmp;\n rw [tmp];\n ring_nf;\n case neg;\n induction xl using Nat.strong_induction_on generalizing xs with\n | h xls ih =>\n rw [\u2190 hxl] at h1;\n cases xs with\n | nil => unfold pairwise; simp [\u2190 hxl]\n | cons head2 tail2 =>\n cases tail2 with\n | nil => simp at h1;\n | cons head3 tail3 =>\n unfold pairwise;\n rw [\u2190hxl, List.length_cons, List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n simp;\n simp at h1;\n have heven: Even tail3.length := by\n rw [Nat.add_assoc, Nat.even_add] at h1;\n apply h1.2; simp;\n simp at ih;\n have tmp := ih tail3.length (by rw [\u2190hxl]; simp; linarith;) tail3 rfl heven\n ring_nf at tmp;\n exact tmp;\n\n\ndef mergingbu : List (List Nat) \u2192 List Nat\n| [] => []\n| [xs] => xs\n| xs::ys::xss =>\n have _h: (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by\n suffices h': 2* List.length (pairwise (xs :: ys :: xss)) < 2 * List.length (xs :: ys :: xss)\n case h';\n rw [ len_pairwise _];\n split_ifs with hparity;\n case pos;\n simp;\n ring_nf;\n linarith;\n case neg;\n simp;\n linarith [h'];\n mergingbu (pairwise (xs::ys::xss))\ntermination_by xss => xss.length\n\n\ndef msortbu : List Nat \u2192 List Nat\n| xs => mergingbu (xs.map (fun x => [x]))\n", "proof_state": "xs : List \u2115\n\u22a2 (msortbu xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:355"} {"full_name": "prop_MSortTDSorts", "prop_defn": "theorem prop_MSortTDSorts (xs: List Nat) : ordered (msorttd xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:372", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef msorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= (x::y::xs).length/2\n have _h: Nat.succ (Nat.succ (List.length xs)) / 2 < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n simp [Nat.succ_eq_add_one _];\n ring_nf;\n calc 1 + xs.length/2 \u2264 1 + xs.length := by simp; exact Nat.div_le_self (List.length xs) 2;\n _ < 2 + xs.length := by simp;\n lmerge (msorttd ((x::y::xs).take k)) (msorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (msorttd xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:372"} {"full_name": "prop_MSortTDCount", "prop_defn": "theorem prop_MSortTDCount (x: Nat) (xs: List Nat) : count x (msorttd xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:373", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef msorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= (x::y::xs).length/2\n have _h: Nat.succ (Nat.succ (List.length xs)) / 2 < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n simp [Nat.succ_eq_add_one _];\n ring_nf;\n calc 1 + xs.length/2 \u2264 1 + xs.length := by simp; exact Nat.div_le_self (List.length xs) 2;\n _ < 2 + xs.length := by simp;\n lmerge (msorttd ((x::y::xs).take k)) (msorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (msorttd xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:373"} {"full_name": "prop_MSortTDPermutes", "prop_defn": "theorem prop_MSortTDPermutes (xs: List Nat) : isPermutation (msorttd xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:374", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef msorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= (x::y::xs).length/2\n have _h: Nat.succ (Nat.succ (List.length xs)) / 2 < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n simp [Nat.succ_eq_add_one _];\n ring_nf;\n calc 1 + xs.length/2 \u2264 1 + xs.length := by simp; exact Nat.div_le_self (List.length xs) 2;\n _ < 2 + xs.length := by simp;\n lmerge (msorttd ((x::y::xs).take k)) (msorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (msorttd xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:374"} {"full_name": "prop_MSortTDIsSort", "prop_defn": "theorem prop_MSortTDIsSort (xs: List Nat) : msorttd xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:375", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef msorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= (x::y::xs).length/2\n have _h: Nat.succ (Nat.succ (List.length xs)) / 2 < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n simp [Nat.succ_eq_add_one _];\n ring_nf;\n calc 1 + xs.length/2 \u2264 1 + xs.length := by simp; exact Nat.div_le_self (List.length xs) 2;\n _ < 2 + xs.length := by simp;\n lmerge (msorttd ((x::y::xs).take k)) (msorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (msorttd xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:375"} {"full_name": "prop_NMSortTDSorts", "prop_defn": "theorem prop_NMSortTDSorts (xs: List Nat) : ordered (nmsorttd xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:420", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef half : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| x + 2 => 1 + (half x)\n\n\nlemma half_lt: half x \u2264 x := by\n induction x using Nat.strong_induction_on with\n | h n ih =>\n cases n with\n | zero => unfold half; simp;\n | succ nm1 =>\n cases nm1 with\n | zero => unfold half; simp;\n | succ nm2 =>\n unfold half;\n have tmp := ih nm2 (by exact Nat.le.step Nat.le.refl);\n ring_nf;\n calc 1 + half nm2 \u2264 1 + nm2 := Nat.add_le_add (@Nat.le.refl 1) tmp\n _ \u2264 2 + nm2 := by simp\n\n\ndef nmsorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= half ((x::y::xs).length)\n have _h: half (Nat.succ (Nat.succ (List.length xs))) < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n rw [Nat.add_comm];\n unfold half;\n ring_nf;\n calc 1 + half (xs.length) \u2264 1 + xs.length := by simp; exact half_lt;\n _ < 2 + xs.length := by simp;\n have _h': Nat.succ (Nat.succ (List.length xs)) - half (Nat.succ (Nat.succ (List.length xs))) <\n Nat.succ (Nat.succ (List.length xs)) := by\n suffices h': 0 < half (Nat.succ (Nat.succ (List.length xs)))\n case h';\n unfold half;simp;\n refine Nat.sub_lt ?h h'\n simp;\n lmerge (nmsorttd ((x::y::xs).take k)) (nmsorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (nmsorttd xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:420"} {"full_name": "prop_NMSortTDCount", "prop_defn": "theorem prop_NMSortTDCount (x: Nat) (xs: List Nat) : count x (nmsorttd xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:421", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef half : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| x + 2 => 1 + (half x)\n\n\nlemma half_lt: half x \u2264 x := by\n induction x using Nat.strong_induction_on with\n | h n ih =>\n cases n with\n | zero => unfold half; simp;\n | succ nm1 =>\n cases nm1 with\n | zero => unfold half; simp;\n | succ nm2 =>\n unfold half;\n have tmp := ih nm2 (by exact Nat.le.step Nat.le.refl);\n ring_nf;\n calc 1 + half nm2 \u2264 1 + nm2 := Nat.add_le_add (@Nat.le.refl 1) tmp\n _ \u2264 2 + nm2 := by simp\n\n\ndef nmsorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= half ((x::y::xs).length)\n have _h: half (Nat.succ (Nat.succ (List.length xs))) < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n rw [Nat.add_comm];\n unfold half;\n ring_nf;\n calc 1 + half (xs.length) \u2264 1 + xs.length := by simp; exact half_lt;\n _ < 2 + xs.length := by simp;\n have _h': Nat.succ (Nat.succ (List.length xs)) - half (Nat.succ (Nat.succ (List.length xs))) <\n Nat.succ (Nat.succ (List.length xs)) := by\n suffices h': 0 < half (Nat.succ (Nat.succ (List.length xs)))\n case h';\n unfold half;simp;\n refine Nat.sub_lt ?h h'\n simp;\n lmerge (nmsorttd ((x::y::xs).take k)) (nmsorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (nmsorttd xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:421"} {"full_name": "prop_NMSortTDPermutes", "prop_defn": "theorem prop_NMSortTDPermutes (xs: List Nat) : isPermutation (nmsorttd xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:422", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef half : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| x + 2 => 1 + (half x)\n\n\nlemma half_lt: half x \u2264 x := by\n induction x using Nat.strong_induction_on with\n | h n ih =>\n cases n with\n | zero => unfold half; simp;\n | succ nm1 =>\n cases nm1 with\n | zero => unfold half; simp;\n | succ nm2 =>\n unfold half;\n have tmp := ih nm2 (by exact Nat.le.step Nat.le.refl);\n ring_nf;\n calc 1 + half nm2 \u2264 1 + nm2 := Nat.add_le_add (@Nat.le.refl 1) tmp\n _ \u2264 2 + nm2 := by simp\n\n\ndef nmsorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= half ((x::y::xs).length)\n have _h: half (Nat.succ (Nat.succ (List.length xs))) < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n rw [Nat.add_comm];\n unfold half;\n ring_nf;\n calc 1 + half (xs.length) \u2264 1 + xs.length := by simp; exact half_lt;\n _ < 2 + xs.length := by simp;\n have _h': Nat.succ (Nat.succ (List.length xs)) - half (Nat.succ (Nat.succ (List.length xs))) <\n Nat.succ (Nat.succ (List.length xs)) := by\n suffices h': 0 < half (Nat.succ (Nat.succ (List.length xs)))\n case h';\n unfold half;simp;\n refine Nat.sub_lt ?h h'\n simp;\n lmerge (nmsorttd ((x::y::xs).take k)) (nmsorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (nmsorttd xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:422"} {"full_name": "prop_NMSortTDIsSort", "prop_defn": "theorem prop_NMSortTDIsSort (xs: List Nat) : nmsorttd xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:423", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef half : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| x + 2 => 1 + (half x)\n\n\nlemma half_lt: half x \u2264 x := by\n induction x using Nat.strong_induction_on with\n | h n ih =>\n cases n with\n | zero => unfold half; simp;\n | succ nm1 =>\n cases nm1 with\n | zero => unfold half; simp;\n | succ nm2 =>\n unfold half;\n have tmp := ih nm2 (by exact Nat.le.step Nat.le.refl);\n ring_nf;\n calc 1 + half nm2 \u2264 1 + nm2 := Nat.add_le_add (@Nat.le.refl 1) tmp\n _ \u2264 2 + nm2 := by simp\n\n\ndef nmsorttd : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n let k:= half ((x::y::xs).length)\n have _h: half (Nat.succ (Nat.succ (List.length xs))) < Nat.succ (Nat.succ (List.length xs)) := by\n rw [Nat.succ_eq_add_one _];\n ring_nf;\n rw [Nat.add_comm];\n unfold half;\n ring_nf;\n calc 1 + half (xs.length) \u2264 1 + xs.length := by simp; exact half_lt;\n _ < 2 + xs.length := by simp;\n have _h': Nat.succ (Nat.succ (List.length xs)) - half (Nat.succ (Nat.succ (List.length xs))) <\n Nat.succ (Nat.succ (List.length xs)) := by\n suffices h': 0 < half (Nat.succ (Nat.succ (List.length xs)))\n case h';\n unfold half;simp;\n refine Nat.sub_lt ?h h'\n simp;\n lmerge (nmsorttd ((x::y::xs).take k)) (nmsorttd ((x::y::xs).drop k))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (nmsorttd xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:423"} {"full_name": "prop_BSortSorts", "prop_defn": "theorem prop_BSortSorts (xs: List Nat) : ordered (bsort xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:530", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n\n\nlemma len_evens_le : (evens xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold evens; simp;\n | cons head1 tail1 =>\n unfold evens;\n cases tail1 with\n | nil => unfold odds; rw[\u2190 hxsl];\n | cons head2 tail2 =>\n unfold odds;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\nlemma len_odds_le : (odds xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold odds; simp;\n | cons head1 tail1 =>\n unfold odds;\n cases tail1 with\n | nil => unfold evens; simp;\n | cons head2 tail2 =>\n unfold evens;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\ndef sort2 (a b: Nat): List Nat := if a \u2264 b then [a,b] else [b, a]\n\n\ndef pairs : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => (sort2 x y) ++ (pairs xs ys)\n\n\ndef stitch : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| x::xs, ys => x::(pairs xs ys)\n\n\nlemma bmerge_term (a b: Nat) (as bs: List Nat) (hlen: \u00ac(List.length as == 0 && List.length bs == 0) = true): List.length (evens (a :: as)) + List.length (evens (b :: bs)) < Nat.succ (List.length as) + Nat.succ (List.length bs) := by\n unfold evens;\n simp at hlen;\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n cases as with\n | nil => cases bs with\n | nil => simp at hlen;\n | cons bhead btail => unfold odds; simp; linarith [@len_evens_le btail];\n | cons ahead atail =>\n rw [add_comm, add_comm (2 + List.length (ahead :: atail)) (List.length bs)]\n refine add_lt_add_of_le_of_lt (@len_odds_le bs) ?h; unfold odds; simp; linarith [@len_evens_le atail];\n\n\nlemma bmerge_term2 (x y: Nat) (xs ys: List Nat) : List.length (odds (x :: xs)) + List.length (odds (y :: ys)) < Nat.succ (List.length xs) + Nat.succ (List.length ys) := by\n unfold odds;\n exact add_lt_add_of_lt_of_lt (Nat.lt_succ_of_le (@len_evens_le xs)) (Nat.lt_succ_of_le (@len_evens_le ys))\n\n\ndef bmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], bs => bs -- I changed this from TIP. I don't believe this case is ever hit, though.\n| as, [] => as\n| x::xs, y::ys =>\n if hlen: xs.length == 0 && ys.length == 0 then sort2 x y else\n have _h := bmerge_term x y xs ys hlen;\n have _h2 := bmerge_term2 x y xs ys;\n stitch (bmerge (evens (x::xs)) (evens (y::ys))) (bmerge (odds (x::xs)) (odds (y::ys)))\ntermination_by xs ys => xs.length + ys.length\n\n\nlemma bsort_term1 (x y: Nat) (xs: List Nat): List.length (evens (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold evens; unfold odds;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_evens_le xs);\n\n\nlemma bsort_term2 (x y: Nat) (xs: List Nat): List.length (odds (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold odds; unfold evens;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_odds_le xs);\n\n\ndef bsort : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n have _h := bsort_term1 x y xs\n have _h2 := bsort_term2 x y xs\n bmerge (bsort (evens (x::y::xs))) (bsort (odds (x::y::xs)))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (bsort xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:530"} {"full_name": "prop_BSortCount", "prop_defn": "theorem prop_BSortCount (x: Nat) (xs: List Nat) : count x (bsort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:531", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n\n\nlemma len_evens_le : (evens xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold evens; simp;\n | cons head1 tail1 =>\n unfold evens;\n cases tail1 with\n | nil => unfold odds; rw[\u2190 hxsl];\n | cons head2 tail2 =>\n unfold odds;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\nlemma len_odds_le : (odds xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold odds; simp;\n | cons head1 tail1 =>\n unfold odds;\n cases tail1 with\n | nil => unfold evens; simp;\n | cons head2 tail2 =>\n unfold evens;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\ndef sort2 (a b: Nat): List Nat := if a \u2264 b then [a,b] else [b, a]\n\n\ndef pairs : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => (sort2 x y) ++ (pairs xs ys)\n\n\ndef stitch : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| x::xs, ys => x::(pairs xs ys)\n\n\nlemma bmerge_term (a b: Nat) (as bs: List Nat) (hlen: \u00ac(List.length as == 0 && List.length bs == 0) = true): List.length (evens (a :: as)) + List.length (evens (b :: bs)) < Nat.succ (List.length as) + Nat.succ (List.length bs) := by\n unfold evens;\n simp at hlen;\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n cases as with\n | nil => cases bs with\n | nil => simp at hlen;\n | cons bhead btail => unfold odds; simp; linarith [@len_evens_le btail];\n | cons ahead atail =>\n rw [add_comm, add_comm (2 + List.length (ahead :: atail)) (List.length bs)]\n refine add_lt_add_of_le_of_lt (@len_odds_le bs) ?h; unfold odds; simp; linarith [@len_evens_le atail];\n\n\nlemma bmerge_term2 (x y: Nat) (xs ys: List Nat) : List.length (odds (x :: xs)) + List.length (odds (y :: ys)) < Nat.succ (List.length xs) + Nat.succ (List.length ys) := by\n unfold odds;\n exact add_lt_add_of_lt_of_lt (Nat.lt_succ_of_le (@len_evens_le xs)) (Nat.lt_succ_of_le (@len_evens_le ys))\n\n\ndef bmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], bs => bs -- I changed this from TIP. I don't believe this case is ever hit, though.\n| as, [] => as\n| x::xs, y::ys =>\n if hlen: xs.length == 0 && ys.length == 0 then sort2 x y else\n have _h := bmerge_term x y xs ys hlen;\n have _h2 := bmerge_term2 x y xs ys;\n stitch (bmerge (evens (x::xs)) (evens (y::ys))) (bmerge (odds (x::xs)) (odds (y::ys)))\ntermination_by xs ys => xs.length + ys.length\n\n\nlemma bsort_term1 (x y: Nat) (xs: List Nat): List.length (evens (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold evens; unfold odds;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_evens_le xs);\n\n\nlemma bsort_term2 (x y: Nat) (xs: List Nat): List.length (odds (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold odds; unfold evens;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_odds_le xs);\n\n\ndef bsort : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n have _h := bsort_term1 x y xs\n have _h2 := bsort_term2 x y xs\n bmerge (bsort (evens (x::y::xs))) (bsort (odds (x::y::xs)))\ntermination_by xs => xs.length\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (bsort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:531"} {"full_name": "prop_BSortPermutes", "prop_defn": "theorem prop_BSortPermutes (xs: List Nat) : isPermutation (bsort xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:532", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n\n\nlemma len_evens_le : (evens xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold evens; simp;\n | cons head1 tail1 =>\n unfold evens;\n cases tail1 with\n | nil => unfold odds; rw[\u2190 hxsl];\n | cons head2 tail2 =>\n unfold odds;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\nlemma len_odds_le : (odds xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold odds; simp;\n | cons head1 tail1 =>\n unfold odds;\n cases tail1 with\n | nil => unfold evens; simp;\n | cons head2 tail2 =>\n unfold evens;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\ndef sort2 (a b: Nat): List Nat := if a \u2264 b then [a,b] else [b, a]\n\n\ndef pairs : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => (sort2 x y) ++ (pairs xs ys)\n\n\ndef stitch : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| x::xs, ys => x::(pairs xs ys)\n\n\nlemma bmerge_term (a b: Nat) (as bs: List Nat) (hlen: \u00ac(List.length as == 0 && List.length bs == 0) = true): List.length (evens (a :: as)) + List.length (evens (b :: bs)) < Nat.succ (List.length as) + Nat.succ (List.length bs) := by\n unfold evens;\n simp at hlen;\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n cases as with\n | nil => cases bs with\n | nil => simp at hlen;\n | cons bhead btail => unfold odds; simp; linarith [@len_evens_le btail];\n | cons ahead atail =>\n rw [add_comm, add_comm (2 + List.length (ahead :: atail)) (List.length bs)]\n refine add_lt_add_of_le_of_lt (@len_odds_le bs) ?h; unfold odds; simp; linarith [@len_evens_le atail];\n\n\nlemma bmerge_term2 (x y: Nat) (xs ys: List Nat) : List.length (odds (x :: xs)) + List.length (odds (y :: ys)) < Nat.succ (List.length xs) + Nat.succ (List.length ys) := by\n unfold odds;\n exact add_lt_add_of_lt_of_lt (Nat.lt_succ_of_le (@len_evens_le xs)) (Nat.lt_succ_of_le (@len_evens_le ys))\n\n\ndef bmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], bs => bs -- I changed this from TIP. I don't believe this case is ever hit, though.\n| as, [] => as\n| x::xs, y::ys =>\n if hlen: xs.length == 0 && ys.length == 0 then sort2 x y else\n have _h := bmerge_term x y xs ys hlen;\n have _h2 := bmerge_term2 x y xs ys;\n stitch (bmerge (evens (x::xs)) (evens (y::ys))) (bmerge (odds (x::xs)) (odds (y::ys)))\ntermination_by xs ys => xs.length + ys.length\n\n\nlemma bsort_term1 (x y: Nat) (xs: List Nat): List.length (evens (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold evens; unfold odds;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_evens_le xs);\n\n\nlemma bsort_term2 (x y: Nat) (xs: List Nat): List.length (odds (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold odds; unfold evens;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_odds_le xs);\n\n\ndef bsort : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n have _h := bsort_term1 x y xs\n have _h2 := bsort_term2 x y xs\n bmerge (bsort (evens (x::y::xs))) (bsort (odds (x::y::xs)))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (bsort xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:532"} {"full_name": "prop_BSortIsSort", "prop_defn": "theorem prop_BSortIsSort (xs: List Nat) : bsort xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:533", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n\n\nlemma len_evens_le : (evens xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold evens; simp;\n | cons head1 tail1 =>\n unfold evens;\n cases tail1 with\n | nil => unfold odds; rw[\u2190 hxsl];\n | cons head2 tail2 =>\n unfold odds;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\nlemma len_odds_le : (odds xs).length \u2264 xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl using Nat.strong_induction_on generalizing xs with\n | h n ih =>\n cases xs with\n | nil => unfold odds; simp;\n | cons head1 tail1 =>\n unfold odds;\n cases tail1 with\n | nil => unfold evens; simp;\n | cons head2 tail2 =>\n unfold evens;\n rw [List.length_cons, List.length_cons] at hxsl;\n rw [\u2190 hxsl, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n have h': List.length tail2 < n := by rw [\u2190 hxsl, Nat.succ_eq_add_one _]; linarith;\n exact Nat.le.step (ih (List.length tail2) h' rfl);\n\n\ndef sort2 (a b: Nat): List Nat := if a \u2264 b then [a,b] else [b, a]\n\n\ndef pairs : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => (sort2 x y) ++ (pairs xs ys)\n\n\ndef stitch : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| x::xs, ys => x::(pairs xs ys)\n\n\nlemma bmerge_term (a b: Nat) (as bs: List Nat) (hlen: \u00ac(List.length as == 0 && List.length bs == 0) = true): List.length (evens (a :: as)) + List.length (evens (b :: bs)) < Nat.succ (List.length as) + Nat.succ (List.length bs) := by\n unfold evens;\n simp at hlen;\n rw [List.length_cons, List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n ring_nf;\n cases as with\n | nil => cases bs with\n | nil => simp at hlen;\n | cons bhead btail => unfold odds; simp; linarith [@len_evens_le btail];\n | cons ahead atail =>\n rw [add_comm, add_comm (2 + List.length (ahead :: atail)) (List.length bs)]\n refine add_lt_add_of_le_of_lt (@len_odds_le bs) ?h; unfold odds; simp; linarith [@len_evens_le atail];\n\n\nlemma bmerge_term2 (x y: Nat) (xs ys: List Nat) : List.length (odds (x :: xs)) + List.length (odds (y :: ys)) < Nat.succ (List.length xs) + Nat.succ (List.length ys) := by\n unfold odds;\n exact add_lt_add_of_lt_of_lt (Nat.lt_succ_of_le (@len_evens_le xs)) (Nat.lt_succ_of_le (@len_evens_le ys))\n\n\ndef bmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], bs => bs -- I changed this from TIP. I don't believe this case is ever hit, though.\n| as, [] => as\n| x::xs, y::ys =>\n if hlen: xs.length == 0 && ys.length == 0 then sort2 x y else\n have _h := bmerge_term x y xs ys hlen;\n have _h2 := bmerge_term2 x y xs ys;\n stitch (bmerge (evens (x::xs)) (evens (y::ys))) (bmerge (odds (x::xs)) (odds (y::ys)))\ntermination_by xs ys => xs.length + ys.length\n\n\nlemma bsort_term1 (x y: Nat) (xs: List Nat): List.length (evens (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold evens; unfold odds;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_evens_le xs);\n\n\nlemma bsort_term2 (x y: Nat) (xs: List Nat): List.length (odds (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by\n unfold odds; unfold evens;\n rw [List.length_cons, Nat.succ_eq_add_one _, Nat.succ_eq_add_one _];\n simp;\n exact Nat.lt_succ_of_le (@len_odds_le xs);\n\n\ndef bsort : List Nat \u2192 List Nat\n| [] => []\n| [x] => [x]\n| x::y::xs =>\n have _h := bsort_term1 x y xs\n have _h2 := bsort_term2 x y xs\n bmerge (bsort (evens (x::y::xs))) (bsort (odds (x::y::xs)))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (bsort xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:533"} {"full_name": "prop_QSortSorts", "prop_defn": "theorem prop_QSortSorts (xs: List Nat) : ordered (qsort xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:570", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n\n\nlemma filter_len_le {f: Nat \u2192 Bool} {xs: List Nat}: (filter xs f).length <= xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; rw [hxsl]; unfold filter; simp;\n | succ n ih =>\n cases xs with\n | nil => unfold filter; simp;\n | cons head tail =>\n rw [List.length_cons] at hxsl; simp at hxsl;\n unfold filter; split_ifs with h1;\n case pos;\n rw [List.length_cons];\n exact Nat.pred_le_iff.mp (ih hxsl)\n case neg;\n exact Nat.le.step (ih hxsl)\n\n\nlemma qsort_term (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y \u2264 x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\nlemma qsort_term2 (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y > x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\ndef qsort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n have _h:= qsort_term x xs\n have _h2:= qsort_term2 x xs\n (qsort (filter xs (fun y => y <= x))) ++ [x] ++ (qsort (filter xs (fun y => y > x)))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (qsort xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:570"} {"full_name": "prop_QSortCount", "prop_defn": "theorem prop_QSortCount (x: Nat) (xs: List Nat) : count x (qsort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:571", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n\n\nlemma filter_len_le {f: Nat \u2192 Bool} {xs: List Nat}: (filter xs f).length <= xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; rw [hxsl]; unfold filter; simp;\n | succ n ih =>\n cases xs with\n | nil => unfold filter; simp;\n | cons head tail =>\n rw [List.length_cons] at hxsl; simp at hxsl;\n unfold filter; split_ifs with h1;\n case pos;\n rw [List.length_cons];\n exact Nat.pred_le_iff.mp (ih hxsl)\n case neg;\n exact Nat.le.step (ih hxsl)\n\n\nlemma qsort_term (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y \u2264 x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\nlemma qsort_term2 (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y > x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\ndef qsort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n have _h:= qsort_term x xs\n have _h2:= qsort_term2 x xs\n (qsort (filter xs (fun y => y <= x))) ++ [x] ++ (qsort (filter xs (fun y => y > x)))\ntermination_by xs => xs.length\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (qsort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:571"} {"full_name": "prop_QSortPermutes", "prop_defn": "theorem prop_QSortPermutes (xs: List Nat) : isPermutation (qsort xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:572", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n\n\nlemma filter_len_le {f: Nat \u2192 Bool} {xs: List Nat}: (filter xs f).length <= xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; rw [hxsl]; unfold filter; simp;\n | succ n ih =>\n cases xs with\n | nil => unfold filter; simp;\n | cons head tail =>\n rw [List.length_cons] at hxsl; simp at hxsl;\n unfold filter; split_ifs with h1;\n case pos;\n rw [List.length_cons];\n exact Nat.pred_le_iff.mp (ih hxsl)\n case neg;\n exact Nat.le.step (ih hxsl)\n\n\nlemma qsort_term (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y \u2264 x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\nlemma qsort_term2 (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y > x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\ndef qsort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n have _h:= qsort_term x xs\n have _h2:= qsort_term2 x xs\n (qsort (filter xs (fun y => y <= x))) ++ [x] ++ (qsort (filter xs (fun y => y > x)))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (qsort xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:572"} {"full_name": "prop_QSortIsSort", "prop_defn": "theorem prop_QSortIsSort (xs: List Nat) : qsort xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:573", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n\n\nlemma filter_len_le {f: Nat \u2192 Bool} {xs: List Nat}: (filter xs f).length <= xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; rw [hxsl]; unfold filter; simp;\n | succ n ih =>\n cases xs with\n | nil => unfold filter; simp;\n | cons head tail =>\n rw [List.length_cons] at hxsl; simp at hxsl;\n unfold filter; split_ifs with h1;\n case pos;\n rw [List.length_cons];\n exact Nat.pred_le_iff.mp (ih hxsl)\n case neg;\n exact Nat.le.step (ih hxsl)\n\n\nlemma qsort_term (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y \u2264 x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\nlemma qsort_term2 (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y > x)) < Nat.succ (List.length xs) := by\n exact Nat.lt_succ_of_le (filter_len_le);\n\n\ndef qsort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n have _h:= qsort_term x xs\n have _h2:= qsort_term2 x xs\n (qsort (filter xs (fun y => y <= x))) ++ [x] ++ (qsort (filter xs (fun y => y > x)))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (qsort xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:573"} {"full_name": "prop_SSortSorts", "prop_defn": "theorem prop_SSortSorts (xs: List Nat) : ordered (ssort xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:624", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef minimum : Nat \u2192 List Nat \u2192 Nat\n| x, [] => x\n| x, y::ys => if y <= x then minimum y ys else minimum x ys\n\n\nlemma min_in_list : minimum x xs \u2208 (x::xs) := by\n induction xs generalizing x with\n | nil => unfold minimum; simp;\n | cons head tail ih =>\n unfold minimum; split_ifs with h1;\n case pos;\n rw [List.mem_cons]; right; exact ih;\n case neg;\n rw [List.mem_cons];\n cases (List.mem_cons.1 (@ih x)) with\n | inl h2 => left; exact h2;\n | inr h2 => right; rw [List.mem_cons]; right; exact h2;\n\n\nlemma delete_len_eq {x: Nat} {xs: List Nat} (h: x \u2208 xs): (deleteFirst x xs).length + 1 = xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; simp [hxsl] at h;\n | succ n ih =>\n cases xs with\n | nil => simp at hxsl;\n | cons head tail =>\n unfold deleteFirst;\n split_ifs with h1;\n case pos;\n simp at hxsl;\n simp [hxsl];\n case neg;\n rw [List.mem_cons] at h;\n simp at h1;\n simp at hxsl;\n simp;\n exact ih ((or_iff_right h1).1 h) hxsl;\n\n\ndef ssort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n let m := minimum x xs\n have _h: List.length (deleteFirst (minimum x xs) (x :: xs)) < Nat.succ (List.length xs) := by\n have tmp := delete_len_eq (@min_in_list x xs)\n simp at tmp;\n simp [tmp];\n m :: ssort (deleteFirst m (x::xs))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (ssort xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:624"} {"full_name": "prop_SSortCount", "prop_defn": "theorem prop_SSortCount (x: Nat) (xs: List Nat) : count x (ssort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:625", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef minimum : Nat \u2192 List Nat \u2192 Nat\n| x, [] => x\n| x, y::ys => if y <= x then minimum y ys else minimum x ys\n\n\nlemma min_in_list : minimum x xs \u2208 (x::xs) := by\n induction xs generalizing x with\n | nil => unfold minimum; simp;\n | cons head tail ih =>\n unfold minimum; split_ifs with h1;\n case pos;\n rw [List.mem_cons]; right; exact ih;\n case neg;\n rw [List.mem_cons];\n cases (List.mem_cons.1 (@ih x)) with\n | inl h2 => left; exact h2;\n | inr h2 => right; rw [List.mem_cons]; right; exact h2;\n\n\nlemma delete_len_eq {x: Nat} {xs: List Nat} (h: x \u2208 xs): (deleteFirst x xs).length + 1 = xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; simp [hxsl] at h;\n | succ n ih =>\n cases xs with\n | nil => simp at hxsl;\n | cons head tail =>\n unfold deleteFirst;\n split_ifs with h1;\n case pos;\n simp at hxsl;\n simp [hxsl];\n case neg;\n rw [List.mem_cons] at h;\n simp at h1;\n simp at hxsl;\n simp;\n exact ih ((or_iff_right h1).1 h) hxsl;\n\n\ndef ssort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n let m := minimum x xs\n have _h: List.length (deleteFirst (minimum x xs) (x :: xs)) < Nat.succ (List.length xs) := by\n have tmp := delete_len_eq (@min_in_list x xs)\n simp at tmp;\n simp [tmp];\n m :: ssort (deleteFirst m (x::xs))\ntermination_by xs => xs.length\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (ssort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:625"} {"full_name": "prop_SSortPermutes", "prop_defn": "theorem prop_SSortPermutes (xs: List Nat) : isPermutation (ssort xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:626", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef minimum : Nat \u2192 List Nat \u2192 Nat\n| x, [] => x\n| x, y::ys => if y <= x then minimum y ys else minimum x ys\n\n\nlemma min_in_list : minimum x xs \u2208 (x::xs) := by\n induction xs generalizing x with\n | nil => unfold minimum; simp;\n | cons head tail ih =>\n unfold minimum; split_ifs with h1;\n case pos;\n rw [List.mem_cons]; right; exact ih;\n case neg;\n rw [List.mem_cons];\n cases (List.mem_cons.1 (@ih x)) with\n | inl h2 => left; exact h2;\n | inr h2 => right; rw [List.mem_cons]; right; exact h2;\n\n\nlemma delete_len_eq {x: Nat} {xs: List Nat} (h: x \u2208 xs): (deleteFirst x xs).length + 1 = xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; simp [hxsl] at h;\n | succ n ih =>\n cases xs with\n | nil => simp at hxsl;\n | cons head tail =>\n unfold deleteFirst;\n split_ifs with h1;\n case pos;\n simp at hxsl;\n simp [hxsl];\n case neg;\n rw [List.mem_cons] at h;\n simp at h1;\n simp at hxsl;\n simp;\n exact ih ((or_iff_right h1).1 h) hxsl;\n\n\ndef ssort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n let m := minimum x xs\n have _h: List.length (deleteFirst (minimum x xs) (x :: xs)) < Nat.succ (List.length xs) := by\n have tmp := delete_len_eq (@min_in_list x xs)\n simp at tmp;\n simp [tmp];\n m :: ssort (deleteFirst m (x::xs))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (ssort xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:626"} {"full_name": "prop_SSortIsSort", "prop_defn": "theorem prop_SSortIsSort (xs: List Nat) : ssort xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:627", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef minimum : Nat \u2192 List Nat \u2192 Nat\n| x, [] => x\n| x, y::ys => if y <= x then minimum y ys else minimum x ys\n\n\nlemma min_in_list : minimum x xs \u2208 (x::xs) := by\n induction xs generalizing x with\n | nil => unfold minimum; simp;\n | cons head tail ih =>\n unfold minimum; split_ifs with h1;\n case pos;\n rw [List.mem_cons]; right; exact ih;\n case neg;\n rw [List.mem_cons];\n cases (List.mem_cons.1 (@ih x)) with\n | inl h2 => left; exact h2;\n | inr h2 => right; rw [List.mem_cons]; right; exact h2;\n\n\nlemma delete_len_eq {x: Nat} {xs: List Nat} (h: x \u2208 xs): (deleteFirst x xs).length + 1 = xs.length := by\n generalize hxsl: xs.length = xsl\n induction xsl generalizing xs with\n | zero => rw [List.length_eq_zero] at hxsl; simp [hxsl] at h;\n | succ n ih =>\n cases xs with\n | nil => simp at hxsl;\n | cons head tail =>\n unfold deleteFirst;\n split_ifs with h1;\n case pos;\n simp at hxsl;\n simp [hxsl];\n case neg;\n rw [List.mem_cons] at h;\n simp at h1;\n simp at hxsl;\n simp;\n exact ih ((or_iff_right h1).1 h) hxsl;\n\n\ndef ssort : List Nat \u2192 List Nat\n| [] => []\n| x::xs =>\n let m := minimum x xs\n have _h: List.length (deleteFirst (minimum x xs) (x :: xs)) < Nat.succ (List.length xs) := by\n have tmp := delete_len_eq (@min_in_list x xs)\n simp at tmp;\n simp [tmp];\n m :: ssort (deleteFirst m (x::xs))\ntermination_by xs => xs.length\n", "proof_state": "xs : List \u2115\n\u22a2 (ssort xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:627"} {"full_name": "prop_TSortSorts", "prop_defn": "theorem prop_TSortSorts (xs: List Nat) : ordered (tsort xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:649", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ninductive MyTree where\n| nil : MyTree\n| node : MyTree \u2192 Nat \u2192 MyTree \u2192 MyTree\n\n\ndef add : Nat \u2192 MyTree \u2192 MyTree\n| x, .nil => .node .nil x .nil\n| x, .node p y q => if x <= y then .node (add x p) y q else .node p y (add x q)\n\n\ndef toTree : List Nat \u2192 MyTree\n| [] => .nil\n| x::xs => add x (toTree xs)\n\n\ndef flatten : MyTree \u2192 List Nat \u2192 List Nat\n| .nil, ys => ys\n| .node p x q, ys => flatten p (x :: flatten q ys)\n\n\ndef tsort : List Nat \u2192 List Nat\n| xs => flatten (toTree xs) []\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered (tsort xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:649"} {"full_name": "prop_TSortCount", "prop_defn": "theorem prop_TSortCount (x: Nat) (xs: List Nat) : count x (tsort xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:650", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ninductive MyTree where\n| nil : MyTree\n| node : MyTree \u2192 Nat \u2192 MyTree \u2192 MyTree\n\n\ndef add : Nat \u2192 MyTree \u2192 MyTree\n| x, .nil => .node .nil x .nil\n| x, .node p y q => if x <= y then .node (add x p) y q else .node p y (add x q)\n\n\ndef toTree : List Nat \u2192 MyTree\n| [] => .nil\n| x::xs => add x (toTree xs)\n\n\ndef flatten : MyTree \u2192 List Nat \u2192 List Nat\n| .nil, ys => ys\n| .node p x q, ys => flatten p (x :: flatten q ys)\n\n\ndef tsort : List Nat \u2192 List Nat\n| xs => flatten (toTree xs) []\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x (tsort xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:650"} {"full_name": "prop_TSortPermutes", "prop_defn": "theorem prop_TSortPermutes (xs: List Nat) : isPermutation (tsort xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:651", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ninductive MyTree where\n| nil : MyTree\n| node : MyTree \u2192 Nat \u2192 MyTree \u2192 MyTree\n\n\ndef add : Nat \u2192 MyTree \u2192 MyTree\n| x, .nil => .node .nil x .nil\n| x, .node p y q => if x <= y then .node (add x p) y q else .node p y (add x q)\n\n\ndef toTree : List Nat \u2192 MyTree\n| [] => .nil\n| x::xs => add x (toTree xs)\n\n\ndef flatten : MyTree \u2192 List Nat \u2192 List Nat\n| .nil, ys => ys\n| .node p x q, ys => flatten p (x :: flatten q ys)\n\n\ndef tsort : List Nat \u2192 List Nat\n| xs => flatten (toTree xs) []\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (tsort xs) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:651"} {"full_name": "prop_TSortIsSort", "prop_defn": "theorem prop_TSortIsSort (xs: List Nat) : tsort xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:652", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ninductive MyTree where\n| nil : MyTree\n| node : MyTree \u2192 Nat \u2192 MyTree \u2192 MyTree\n\n\ndef add : Nat \u2192 MyTree \u2192 MyTree\n| x, .nil => .node .nil x .nil\n| x, .node p y q => if x <= y then .node (add x p) y q else .node p y (add x q)\n\n\ndef toTree : List Nat \u2192 MyTree\n| [] => .nil\n| x::xs => add x (toTree xs)\n\n\ndef flatten : MyTree \u2192 List Nat \u2192 List Nat\n| .nil, ys => ys\n| .node p x q, ys => flatten p (x :: flatten q ys)\n\n\ndef tsort : List Nat \u2192 List Nat\n| xs => flatten (toTree xs) []\n", "proof_state": "xs : List \u2115\n\u22a2 (tsort xs == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:652"} {"full_name": "prop_StoogeSortSorts", "prop_defn": "theorem prop_StoogeSortSorts (xs: List Nat) : ordered (stoogesort' xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:766", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\ndef stoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((x1::x2::x3::xs).length/3) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@stoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (s1s2a.length/3) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@stoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (s1s1.length/3) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@stoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort' (xs: List Nat) := stoogesort \u27e8xs, Eq.refl xs.length\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered \u2191(stoogesort' xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:766"} {"full_name": "prop_StoogeSortCount", "prop_defn": "theorem prop_StoogeSortCount (x: Nat) (xs: List Nat) : count x (stoogesort' xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:767", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\ndef stoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((x1::x2::x3::xs).length/3) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@stoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (s1s2a.length/3) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@stoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (s1s1.length/3) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@stoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort' (xs: List Nat) := stoogesort \u27e8xs, Eq.refl xs.length\u27e9\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x \u2191(stoogesort' xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:767"} {"full_name": "prop_StoogeSortPermutes", "prop_defn": "theorem prop_StoogeSortPermutes (xs: List Nat) : isPermutation (stoogesort' xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:768", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\ndef stoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((x1::x2::x3::xs).length/3) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@stoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (s1s2a.length/3) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@stoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (s1s1.length/3) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@stoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort' (xs: List Nat) := stoogesort \u27e8xs, Eq.refl xs.length\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (\u2191(stoogesort' xs)) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:768"} {"full_name": "prop_StoogeSortIsSort", "prop_defn": "theorem prop_StoogeSortIsSort (xs: List Nat) : stoogesort' xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:769", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\ndef stoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((x1::x2::x3::xs).length/3) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@stoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (s1s2a.length/3) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@stoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (s1s1.length/3) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@stoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort' (xs: List Nat) := stoogesort \u27e8xs, Eq.refl xs.length\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (\u2191(stoogesort' xs) == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:769"} {"full_name": "prop_StoogeSort2Sorts", "prop_defn": "theorem prop_StoogeSort2Sorts (xs: List Nat) : ordered (stoogesort2' xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:850", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma splitAt_first_len_lt (hn: n < xl) (hlen': xl = xs.length): (splitAt n xs).1.length < xl := by\n suffices heq : (splitAt n xs).1.length = n\n simp [heq, hn]\n induction xs generalizing n xl with\n | nil => simp [hlen'] at hn;\n | cons head tail ih =>\n cases n with\n | zero => simp [splitAt];\n | succ nm1 =>\n simp [splitAt];\n exact @ih nm1 tail.length (by simp [hlen'] at hn; exact hn) rfl\n\n\nlemma twon_lt (n: Nat): (2*n.succ.succ.succ + 1)/ 3 < n.succ.succ.succ := by\n simp [Nat.succ_eq_add_one];\n ring_nf;\n rw [Nat.div_lt_iff_lt_mul (by simp)];\n ring_nf;\n linarith;\n\n\ndef stoogesort2 (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((2*(x1::x2::x3::xs).length + 1) / 3) (x1::x2::x3::xs)\n let \u27e8tmp2, hlen2\u27e9 := @stoogesort2 yzs1.1.length \u27e8yzs1.1, by rfl\u27e9\n let s2s2a := tmp2 ++ yzs1.2\n let yzs2 := splitAt (s2s2a.length / 3) s2s2a\n let \u27e8tmp3, hlen3\u27e9 := @stoogesort2 yzs2.2.length \u27e8yzs2.2, by rfl\u27e9\n let s2s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt ((2*s2s1.length + 1) / 3) s2s1\n let \u27e8tmp4, hlen4\u27e9 := @stoogesort2 yzs3.1.length \u27e8yzs3.1, by rfl\u27e9\n \u27e8tmp4 ++ yzs3.2, by\n simp [hlen4];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf;\n simp at hlen;\n rw [\u2190 hlen]\n apply splitAt_first_len_lt (twon_lt _) (by simp)\n simp_wf;\n suffices hyzs2 : yzs2.2.length < n\n unfold_let yzs2 at hyzs2\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let yzs2\n suffices hs2s2a : s2s2a.length = n\n rw [hs2s2a, \u2190 hlen]\n apply splitAt_second_len_lt'' (by simp [Nat.succ_eq_add_one, Nat.add_assoc]) (by simp [hs2s2a, \u2190hlen]) (by simp [hs2s2a, \u2190hlen])\n unfold_let s2s2a\n simp [hlen2, splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n simp_wf;\n suffices hdone: yzs3.1.length < n\n unfold_let yzs3 s2s1 yzs2 s2s2a yzs1 at hdone\n simp [List.length_append] at hdone;\n exact hdone;\n unfold_let yzs3\n suffices hs2s1: s2s1.length = xs.length.succ.succ.succ\n refine splitAt_first_len_lt (by simp [hs2s1, \u2190 hlen]; exact twon_lt _) (by simp [hs2s1, \u2190 hlen])\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort2' (xs: List Nat) := stoogesort2 \u27e8xs, by rfl\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered \u2191(stoogesort2' xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:850"} {"full_name": "prop_StoogeSort2Count", "prop_defn": "theorem prop_StoogeSort2Count (x: Nat) (xs: List Nat) : count x (stoogesort2' xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:851", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma splitAt_first_len_lt (hn: n < xl) (hlen': xl = xs.length): (splitAt n xs).1.length < xl := by\n suffices heq : (splitAt n xs).1.length = n\n simp [heq, hn]\n induction xs generalizing n xl with\n | nil => simp [hlen'] at hn;\n | cons head tail ih =>\n cases n with\n | zero => simp [splitAt];\n | succ nm1 =>\n simp [splitAt];\n exact @ih nm1 tail.length (by simp [hlen'] at hn; exact hn) rfl\n\n\nlemma twon_lt (n: Nat): (2*n.succ.succ.succ + 1)/ 3 < n.succ.succ.succ := by\n simp [Nat.succ_eq_add_one];\n ring_nf;\n rw [Nat.div_lt_iff_lt_mul (by simp)];\n ring_nf;\n linarith;\n\n\ndef stoogesort2 (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((2*(x1::x2::x3::xs).length + 1) / 3) (x1::x2::x3::xs)\n let \u27e8tmp2, hlen2\u27e9 := @stoogesort2 yzs1.1.length \u27e8yzs1.1, by rfl\u27e9\n let s2s2a := tmp2 ++ yzs1.2\n let yzs2 := splitAt (s2s2a.length / 3) s2s2a\n let \u27e8tmp3, hlen3\u27e9 := @stoogesort2 yzs2.2.length \u27e8yzs2.2, by rfl\u27e9\n let s2s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt ((2*s2s1.length + 1) / 3) s2s1\n let \u27e8tmp4, hlen4\u27e9 := @stoogesort2 yzs3.1.length \u27e8yzs3.1, by rfl\u27e9\n \u27e8tmp4 ++ yzs3.2, by\n simp [hlen4];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf;\n simp at hlen;\n rw [\u2190 hlen]\n apply splitAt_first_len_lt (twon_lt _) (by simp)\n simp_wf;\n suffices hyzs2 : yzs2.2.length < n\n unfold_let yzs2 at hyzs2\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let yzs2\n suffices hs2s2a : s2s2a.length = n\n rw [hs2s2a, \u2190 hlen]\n apply splitAt_second_len_lt'' (by simp [Nat.succ_eq_add_one, Nat.add_assoc]) (by simp [hs2s2a, \u2190hlen]) (by simp [hs2s2a, \u2190hlen])\n unfold_let s2s2a\n simp [hlen2, splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n simp_wf;\n suffices hdone: yzs3.1.length < n\n unfold_let yzs3 s2s1 yzs2 s2s2a yzs1 at hdone\n simp [List.length_append] at hdone;\n exact hdone;\n unfold_let yzs3\n suffices hs2s1: s2s1.length = xs.length.succ.succ.succ\n refine splitAt_first_len_lt (by simp [hs2s1, \u2190 hlen]; exact twon_lt _) (by simp [hs2s1, \u2190 hlen])\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort2' (xs: List Nat) := stoogesort2 \u27e8xs, by rfl\u27e9\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x \u2191(stoogesort2' xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:851"} {"full_name": "prop_StoogeSort2Permutes", "prop_defn": "theorem prop_StoogeSort2Permutes (xs: List Nat) : isPermutation (stoogesort2' xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:852", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma splitAt_first_len_lt (hn: n < xl) (hlen': xl = xs.length): (splitAt n xs).1.length < xl := by\n suffices heq : (splitAt n xs).1.length = n\n simp [heq, hn]\n induction xs generalizing n xl with\n | nil => simp [hlen'] at hn;\n | cons head tail ih =>\n cases n with\n | zero => simp [splitAt];\n | succ nm1 =>\n simp [splitAt];\n exact @ih nm1 tail.length (by simp [hlen'] at hn; exact hn) rfl\n\n\nlemma twon_lt (n: Nat): (2*n.succ.succ.succ + 1)/ 3 < n.succ.succ.succ := by\n simp [Nat.succ_eq_add_one];\n ring_nf;\n rw [Nat.div_lt_iff_lt_mul (by simp)];\n ring_nf;\n linarith;\n\n\ndef stoogesort2 (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((2*(x1::x2::x3::xs).length + 1) / 3) (x1::x2::x3::xs)\n let \u27e8tmp2, hlen2\u27e9 := @stoogesort2 yzs1.1.length \u27e8yzs1.1, by rfl\u27e9\n let s2s2a := tmp2 ++ yzs1.2\n let yzs2 := splitAt (s2s2a.length / 3) s2s2a\n let \u27e8tmp3, hlen3\u27e9 := @stoogesort2 yzs2.2.length \u27e8yzs2.2, by rfl\u27e9\n let s2s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt ((2*s2s1.length + 1) / 3) s2s1\n let \u27e8tmp4, hlen4\u27e9 := @stoogesort2 yzs3.1.length \u27e8yzs3.1, by rfl\u27e9\n \u27e8tmp4 ++ yzs3.2, by\n simp [hlen4];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf;\n simp at hlen;\n rw [\u2190 hlen]\n apply splitAt_first_len_lt (twon_lt _) (by simp)\n simp_wf;\n suffices hyzs2 : yzs2.2.length < n\n unfold_let yzs2 at hyzs2\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let yzs2\n suffices hs2s2a : s2s2a.length = n\n rw [hs2s2a, \u2190 hlen]\n apply splitAt_second_len_lt'' (by simp [Nat.succ_eq_add_one, Nat.add_assoc]) (by simp [hs2s2a, \u2190hlen]) (by simp [hs2s2a, \u2190hlen])\n unfold_let s2s2a\n simp [hlen2, splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n simp_wf;\n suffices hdone: yzs3.1.length < n\n unfold_let yzs3 s2s1 yzs2 s2s2a yzs1 at hdone\n simp [List.length_append] at hdone;\n exact hdone;\n unfold_let yzs3\n suffices hs2s1: s2s1.length = xs.length.succ.succ.succ\n refine splitAt_first_len_lt (by simp [hs2s1, \u2190 hlen]; exact twon_lt _) (by simp [hs2s1, \u2190 hlen])\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort2' (xs: List Nat) := stoogesort2 \u27e8xs, by rfl\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (\u2191(stoogesort2' xs)) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:852"} {"full_name": "prop_StoogeSort2IsSort", "prop_defn": "theorem prop_StoogeSort2IsSort (xs: List Nat) : stoogesort2' xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:853", "score": 5, "deps": "import Mathlib\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma splitAt_first_len_lt (hn: n < xl) (hlen': xl = xs.length): (splitAt n xs).1.length < xl := by\n suffices heq : (splitAt n xs).1.length = n\n simp [heq, hn]\n induction xs generalizing n xl with\n | nil => simp [hlen'] at hn;\n | cons head tail ih =>\n cases n with\n | zero => simp [splitAt];\n | succ nm1 =>\n simp [splitAt];\n exact @ih nm1 tail.length (by simp [hlen'] at hn; exact hn) rfl\n\n\nlemma twon_lt (n: Nat): (2*n.succ.succ.succ + 1)/ 3 < n.succ.succ.succ := by\n simp [Nat.succ_eq_add_one];\n ring_nf;\n rw [Nat.div_lt_iff_lt_mul (by simp)];\n ring_nf;\n linarith;\n\n\ndef stoogesort2 (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt ((2*(x1::x2::x3::xs).length + 1) / 3) (x1::x2::x3::xs)\n let \u27e8tmp2, hlen2\u27e9 := @stoogesort2 yzs1.1.length \u27e8yzs1.1, by rfl\u27e9\n let s2s2a := tmp2 ++ yzs1.2\n let yzs2 := splitAt (s2s2a.length / 3) s2s2a\n let \u27e8tmp3, hlen3\u27e9 := @stoogesort2 yzs2.2.length \u27e8yzs2.2, by rfl\u27e9\n let s2s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt ((2*s2s1.length + 1) / 3) s2s1\n let \u27e8tmp4, hlen4\u27e9 := @stoogesort2 yzs3.1.length \u27e8yzs3.1, by rfl\u27e9\n \u27e8tmp4 ++ yzs3.2, by\n simp [hlen4];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf;\n simp at hlen;\n rw [\u2190 hlen]\n apply splitAt_first_len_lt (twon_lt _) (by simp)\n simp_wf;\n suffices hyzs2 : yzs2.2.length < n\n unfold_let yzs2 at hyzs2\n rw [List.length_append] at hyzs2\n exact hyzs2;\n unfold_let yzs2\n suffices hs2s2a : s2s2a.length = n\n rw [hs2s2a, \u2190 hlen]\n apply splitAt_second_len_lt'' (by simp [Nat.succ_eq_add_one, Nat.add_assoc]) (by simp [hs2s2a, \u2190hlen]) (by simp [hs2s2a, \u2190hlen])\n unfold_let s2s2a\n simp [hlen2, splitAt_sum_preserves_len _ _ (Eq.refl yzs1), hlen]\n simp_wf;\n suffices hdone: yzs3.1.length < n\n unfold_let yzs3 s2s1 yzs2 s2s2a yzs1 at hdone\n simp [List.length_append] at hdone;\n exact hdone;\n unfold_let yzs3\n suffices hs2s1: s2s1.length = xs.length.succ.succ.succ\n refine splitAt_first_len_lt (by simp [hs2s1, \u2190 hlen]; exact twon_lt _) (by simp [hs2s1, \u2190 hlen])\n unfold_let s2s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s2s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef stoogesort2' (xs: List Nat) := stoogesort2 \u27e8xs, by rfl\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (\u2191(stoogesort2' xs) == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:853"} {"full_name": "prop_NStoogeSortSorts", "prop_defn": "theorem prop_NStoogeSortSorts (xs: List Nat) : ordered (nstoogesort' xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:923", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered \u2191(nstoogesort' xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:923"} {"full_name": "prop_NStoogeSortCount", "prop_defn": "theorem prop_NStoogeSortCount (x: Nat) (xs: List Nat) : count x (nstoogesort' xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:924", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x \u2191(nstoogesort' xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:924"} {"full_name": "prop_NStoogeSortPermutes", "prop_defn": "theorem prop_NStoogeSortPermutes (xs: List Nat) : isPermutation (nstoogesort' xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:925", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (\u2191(nstoogesort' xs)) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:925"} {"full_name": "prop_NStoogeSortIsSort", "prop_defn": "theorem prop_NStoogeSortIsSort (xs: List Nat) : nstoogesort' xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:926", "score": 5, "deps": "import Mathlib\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (\u2191(nstoogesort' xs) == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:926"} {"full_name": "prop_NStoogeSort2Sorts", "prop_defn": "theorem prop_NStoogeSort2Sorts (xs: List Nat) : ordered (nstoogesort2' xs) == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:996", "score": 5, "deps": "import Mathlib\n\ndef ordered : List Nat -> Bool\n| [] => True\n| [_x] => True\n| x::y::xs => x <= y && ordered (y::xs)\n\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort2' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (ordered \u2191(nstoogesort2' xs) == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:996"} {"full_name": "prop_NStoogeSort2Count", "prop_defn": "theorem prop_NStoogeSort2Count (x: Nat) (xs: List Nat) : count x (nstoogesort2' xs) == count x xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:997", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1] : \u03b1 \u2192 List \u03b1 \u2192 Nat\n| _x, [] => 0\n| x, y::ys => if x == y then 1 + (count x ys) else count x ys\n\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort2' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (count x \u2191(nstoogesort2' xs) == count x xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:997"} {"full_name": "prop_NStoogeSort2Permutes", "prop_defn": "theorem prop_NStoogeSort2Permutes (xs: List Nat) : isPermutation (nstoogesort2' xs) xs == true:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:998", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort2' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (isPermutation (\u2191(nstoogesort2' xs)) xs == true) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:998"} {"full_name": "prop_NStoogeSort2IsSort", "prop_defn": "theorem prop_NStoogeSort2IsSort (xs: List Nat) : nstoogesort2' xs == isort xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:999", "score": 5, "deps": "import Mathlib\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n\n\ndef insert' : Nat \u2192 List Nat \u2192 List Nat\n| x, [] => [x]\n| x, y::xs => if x <= y then x::y::xs else y::(insert x xs)\n\n\ndef isort: List Nat \u2192 List Nat\n| [] => []\n| x::xs => insert' x (isort xs)\n\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n\n\nlemma len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by\n induction l with\n | nil => simp [reverse]\n | cons head tail ih => unfold reverse; simp [ih];\n\n\nlemma splitAt_len_le : (splitAt n xs).2.length \u2264 xs.length := by\n induction xs generalizing n with\n | nil => unfold splitAt; simp;\n | cons head tail ih =>\n cases n with\n | zero => unfold splitAt; simp;\n | succ nm1 => unfold splitAt; simp; apply Nat.le.step; exact ih;\n\n\nlemma splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by\n unfold splitAt;\n simp;\n calc List.length (splitAt n xs).2 \u2264 (List.length xs) := splitAt_len_le\n _ < Nat.succ (List.length xs) := Nat.lt_succ_self _\n\n\nlemma splitAt_second_len_lt' (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by\n cases xs with\n | nil => simp at hlen;\n | cons x xs => exact splitAt_second_len_lt _;\n\n\nlemma splitAt_second_len_lt'' (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by\n cases n with\n | zero => simp at hn;\n | succ nm1 => rw [hlen']; exact splitAt_second_len_lt' _ hlen;\n\n\nlemma splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs):\n (spl.1.length + spl.2.length = xs.length) := by\n induction xs generalizing n spl with\n | nil => simp [splitAt] at hspl; simp [hspl];\n | cons head tail ih => cases n with\n | zero => simp [splitAt] at hspl; simp [hspl];\n | succ nm1 =>\n simp [splitAt] at hspl;\n simp [hspl]\n rw [Nat.succ_add];\n apply Order.succ_eq_succ_iff.2\n exact ih nm1 (by rfl);\n\n\nlemma third_eq_div_3 : (x/3) = third x := by\n induction x using Nat.strongInductionOn with\n | ind x ih =>\n unfold third\n match x with\n | 0 => simp;\n | 1 => simp;\n | 2 => simp;\n | n + 3 => simp [Nat.succ_eq_add_one]; ring_nf; linarith [ih n (by linarith)]\n\n\ndef nstoogesort (xs : {xs : List Nat // xs.length = n}) : {ys: List Nat // ys.length = n} := match xs with\n| \u27e8[], h\u27e9 => \u27e8[], h\u27e9\n| \u27e8[x], h\u27e9 => \u27e8[x], h\u27e9\n| \u27e8[x, y], h\u27e9 => if x <= y then \u27e8[x, y], h\u27e9 else \u27e8[y, x], by simp [\u2190h]\u27e9\n| \u27e8x1::x2::x3::xs, hlen\u27e9 =>\n let yzs1 := splitAt (third (x1::x2::x3::xs).length) (reverse (x1::x2::x3::xs))\n let \u27e8tmp2, hlen2\u27e9 := (@nstoogesort yzs1.2.length \u27e8yzs1.2, by rfl\u27e9)\n let s1s2a := tmp2 ++ (reverse yzs1.1)\n let yzs2 := splitAt (third s1s2a.length) s1s2a\n let \u27e8tmp3, hlen3\u27e9 := (@nstoogesort yzs2.2.length \u27e8yzs2.2, by rfl\u27e9)\n let s1s1 := yzs2.1 ++ tmp3\n let yzs3 := splitAt (third s1s1.length) (reverse s1s1)\n let \u27e8tmp4, hlen4\u27e9 := (@nstoogesort yzs3.2.length \u27e8yzs3.2, by rfl\u27e9)\n \u27e8tmp4 ++ (reverse yzs3.1), by\n simp [hlen4, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs3), len_rev_eq_len];\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len];\n rw [add_comm];\n simp [splitAt_sum_preserves_len _ _ (Eq.refl yzs1), len_rev_eq_len, hlen]\n \u27e9\ntermination_by match xs with | \u27e8lst, _h\u27e9 => lst.length\ndecreasing_by\n simp_wf\n apply splitAt_second_len_lt''\n (by simp [\u2190third_eq_div_3, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, Nat.succ_eq_add_one, Nat.add_assoc])\n (by simp [len_rev_eq_len, hlen])\n simp_wf\n suffices hyzs2 : (splitAt (List.length s1s2a / 3) s1s2a).2.length < n\n rw [List.length_append, third_eq_div_3] at hyzs2\n exact hyzs2;\n unfold_let s1s2a\n suffices hyzs1 : List.length (tmp2 ++ reverse yzs1.1) = List.length xs + 3\n exact splitAt_second_len_lt'' (by simp [hyzs1]) (by simp [hyzs1]) (by simp [hyzs1, \u2190hlen])\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n simp_wf\n suffices hyzs3 : (splitAt (List.length s1s1 / 3) (reverse s1s1)).2.length < n\n unfold_let s1s1 yzs2 s1s2a yzs1 at hyzs3\n simp [List.length_append, third_eq_div_3] at hyzs3;\n exact hyzs3;\n suffices hs1s1: s1s1.length = xs.length + 3\n exact splitAt_second_len_lt'' (by simp [hs1s1]) (by simp [len_rev_eq_len, hs1s1]) (by simp [len_rev_eq_len, hs1s1, \u2190hlen])\n unfold_let s1s1\n simp [hlen3, splitAt_sum_preserves_len _ _ (Eq.refl yzs2)]\n unfold_let s1s2a\n simp [hlen2, len_rev_eq_len, Nat.add_comm, splitAt_sum_preserves_len _ _ (Eq.refl yzs1)]\n ring_nf\n\n\ndef nstoogesort2' (xs: List Nat) := nstoogesort \u27e8xs, (by rfl)\u27e9\n", "proof_state": "xs : List \u2115\n\u22a2 (\u2191(nstoogesort2' xs) == isort xs) = true", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:999"} {"full_name": "hpairwise_desc", "prop_defn": "theorem hpairwise_desc (ps: List MyHeap): List.length (hpairwise ps) \u2264 List.length ps := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:109", "score": 5, "deps": "import Mathlib\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef hpairwise : List MyHeap \u2192 List MyHeap\n| p::q::qs => (hmerge p q)::hpairwise qs\n| ps => ps\n", "proof_state": "ps : List MyHeap\n\u22a2 (hpairwise ps).length \u2264 ps.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:120"} {"full_name": "numElem_lt_subHeaps", "prop_defn": "theorem numElem_lt_subHeaps (q r: MyHeap) {x: Nat}: numElem q < numElem (MyHeap.node q x r) \u2227 numElem r < numElem (MyHeap.node q x r) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:140", "score": 5, "deps": "import Mathlib\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n", "proof_state": "q r : MyHeap\nx : \u2115\n\u22a2 numElem q < numElem (q.node x r) \u2227 numElem r < numElem (q.node x r)", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:143"} {"full_name": "merge_elems", "prop_defn": "theorem merge_elems (p q: MyHeap): numElem p + numElem q = numElem (hmerge p q) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:145", "score": 5, "deps": "import Mathlib\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n", "proof_state": "p q : MyHeap\n\u22a2 numElem p + numElem q = numElem (hmerge p q)", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:177"} {"full_name": "numElem_merge_branches_lt", "prop_defn": "theorem numElem_merge_branches_lt (p q: MyHeap) (x: Nat): numElem (hmerge p q) < numElem (MyHeap.node p x q) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:179", "score": 5, "deps": "import Mathlib\n\ninductive MyHeap where\n| nil : MyHeap\n| node : MyHeap \u2192 Nat \u2192 MyHeap \u2192 MyHeap\n\n\ndef hmerge : MyHeap \u2192 MyHeap \u2192 MyHeap\n| MyHeap.nil, q => q\n| p, MyHeap.nil => p\n| MyHeap.node p x q, MyHeap.node r y s =>\n if x <= y then MyHeap.node (hmerge q (MyHeap.node r y s)) x p\n else MyHeap.node (hmerge (MyHeap.node p x q) s) y r\n\n\ndef numElem : MyHeap \u2192 Nat\n| MyHeap.nil => 0\n| MyHeap.node p _x q => 1 + numElem p + numElem q\n", "proof_state": "p q : MyHeap\nx : \u2115\n\u22a2 numElem (hmerge p q) < numElem (p.node x q)", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:183"} {"full_name": "len_pairwise", "prop_defn": "theorem len_pairwise (xs: List (List Nat)): 2 * (pairwise xs).length = (if (Odd xs.length) then xs.length + 1 else xs.length) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:241", "score": 5, "deps": "import Mathlib\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n", "proof_state": "xs : List (List \u2115)\n\u22a2 2 * (pairwise xs).length = if Odd xs.length then xs.length + 1 else xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:286"} {"full_name": "merge_term", "prop_defn": "theorem merge_term : (pairwise (xs::ys::xss)).length < (xs::ys::xss).length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:288", "score": 5, "deps": "import Mathlib\n\ndef lmerge : List Nat \u2192 List Nat \u2192 List Nat\n| [], ys => ys\n| xs, [] => xs\n| x::xs, y::ys => if x <= y then\n x::(lmerge xs (y::ys))\n else\n y::(lmerge (x::xs) ys)\n\n\ndef pairwise : List (List Nat) \u2192 List (List Nat)\n| xs::ys::xss => lmerge xs ys :: pairwise xss\n| xss => xss\n", "proof_state": "xs ys : List \u2115\nxss : List (List \u2115)\n\u22a2 (pairwise (xs :: ys :: xss)).length < (xs :: ys :: xss).length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:299"} {"full_name": "half_lt", "prop_defn": "theorem half_lt: half x \u2264 x := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:382", "score": 5, "deps": "import Mathlib\n\ndef half : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| x + 2 => 1 + (half x)\n", "proof_state": "x : \u2115\n\u22a2 half x \u2264 x", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:395"} {"full_name": "len_evens_le", "prop_defn": "theorem len_evens_le {xs: List Nat}: (evens xs).length \u2264 xs.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:435", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "xs : List \u2115\n\u22a2 (evens xs).length \u2264 xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:451"} {"full_name": "len_odds_le", "prop_defn": "theorem len_odds_le {xs: List Nat}: (odds xs).length \u2264 xs.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:453", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "xs : List \u2115\n\u22a2 (odds xs).length \u2264 xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:469"} {"full_name": "bmerge_term", "prop_defn": "theorem bmerge_term (a b: Nat) (as bs: List Nat) (hlen: \u00ac(List.length as == 0 && List.length bs == 0) = true): List.length (evens (a :: as)) + List.length (evens (b :: bs)) < Nat.succ (List.length as) + Nat.succ (List.length bs) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:482", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "a b : \u2115\nas bs : List \u2115\nhlen : \u00ac(as.length == 0 && bs.length == 0) = true\n\u22a2 (evens (a :: as)).length + (evens (b :: bs)).length < as.length.succ + bs.length.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:493"} {"full_name": "bmerge_term2", "prop_defn": "theorem bmerge_term2 (x y: Nat) (xs ys: List Nat) : List.length (odds (x :: xs)) + List.length (odds (y :: ys)) < Nat.succ (List.length xs) + Nat.succ (List.length ys) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:495", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "x y : \u2115\nxs ys : List \u2115\n\u22a2 (odds (x :: xs)).length + (odds (y :: ys)).length < xs.length.succ + ys.length.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:497"} {"full_name": "bsort_term1", "prop_defn": "theorem bsort_term1 (x y: Nat) (xs: List Nat): List.length (evens (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:509", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "x y : \u2115\nxs : List \u2115\n\u22a2 (evens (x :: y :: xs)).length < xs.length.succ.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:513"} {"full_name": "bsort_term2", "prop_defn": "theorem bsort_term2 (x y: Nat) (xs: List Nat): List.length (odds (x :: y :: xs)) < Nat.succ (Nat.succ (List.length xs)) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:515", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List Nat \u2192 List Nat\n | [] => []\n | x::xs => x::(odds xs)\n--\n def odds : List Nat \u2192 List Nat\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "x y : \u2115\nxs : List \u2115\n\u22a2 (odds (x :: y :: xs)).length < xs.length.succ.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:519"} {"full_name": "filter_len_le", "prop_defn": "theorem filter_len_le {f: Nat \u2192 Bool} {xs: List Nat}: (filter xs f).length <= xs.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:539", "score": 5, "deps": "import Mathlib\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n", "proof_state": "f : \u2115 \u2192 Bool\nxs : List \u2115\n\u22a2 (filter xs f).length \u2264 xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:553"} {"full_name": "qsort_term", "prop_defn": "theorem qsort_term (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y \u2264 x)) < Nat.succ (List.length xs) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:556", "score": 5, "deps": "import Mathlib\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (filter xs fun y => decide (y \u2264 x)).length < xs.length.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:557"} {"full_name": "qsort_term2", "prop_defn": "theorem qsort_term2 (x:Nat) (xs: List Nat) : List.length (filter xs fun y => decide (y > x)) < Nat.succ (List.length xs) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:559", "score": 5, "deps": "import Mathlib\n\ndef filter : List Nat \u2192 (Nat \u2192 Bool) \u2192 List Nat\n| [], _f => []\n| x::xs, f => if f x then x::(filter xs f) else (filter xs f)\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 (filter xs fun y => decide (y > x)).length < xs.length.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:560"} {"full_name": "min_in_list", "prop_defn": "theorem min_in_list : minimum x xs \u2208 (x::xs) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:579", "score": 5, "deps": "import Mathlib\n\ndef minimum : Nat \u2192 List Nat \u2192 Nat\n| x, [] => x\n| x, y::ys => if y <= x then minimum y ys else minimum x ys\n", "proof_state": "x : \u2115\nxs : List \u2115\n\u22a2 minimum x xs \u2208 x :: xs", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:590"} {"full_name": "delete_len_eq", "prop_defn": "theorem delete_len_eq {x: Nat} {xs: List Nat} (h: x \u2208 xs): (deleteFirst x xs).length + 1 = xs.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:592", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n", "proof_state": "x : \u2115\nxs : List \u2115\nh : x \u2208 xs\n\u22a2 (deleteFirst x xs).length + 1 = xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:610"} {"full_name": "len_rev_eq_len", "prop_defn": "theorem len_rev_eq_len {l: List Nat} : (reverse l).length = l.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:664", "score": 5, "deps": "import Mathlib\n\ndef reverse : List Nat \u2192 List Nat\n| [] => []\n| x::xs => (reverse xs) ++ [x]\n", "proof_state": "l : List \u2115\n\u22a2 (reverse l).length = l.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:667"} {"full_name": "splitAt_len_le", "prop_defn": "theorem splitAt_len_le {xs: List Nat}: (splitAt n xs).2.length \u2264 xs.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:669", "score": 5, "deps": "import Mathlib\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n", "proof_state": "n : \u2115\nxs : List \u2115\n\u22a2 (splitAt n xs).2.length \u2264 xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:675"} {"full_name": "splitAt_second_len_lt", "prop_defn": "theorem splitAt_second_len_lt (n: Nat): (splitAt n.succ (x::xs)).2.length < (x::xs).length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:677", "score": 5, "deps": "import Mathlib\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n", "proof_state": "x : \u2115\nxs : List \u2115\nn : \u2115\n\u22a2 (splitAt n.succ (x :: xs)).2.length < (x :: xs).length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:681"} {"full_name": "splitAt_second_len_lt'", "prop_defn": "theorem splitAt_second_len_lt' {xs: List Nat} (n: Nat) (hlen: xs.length > 0): (splitAt n.succ xs).2.length < xs.length := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:683", "score": 5, "deps": "import Mathlib\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n", "proof_state": "xs : List \u2115\nn : \u2115\nhlen : xs.length > 0\n\u22a2 (splitAt n.succ xs).2.length < xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:686"} {"full_name": "splitAt_second_len_lt''", "prop_defn": "theorem splitAt_second_len_lt'' {xs: List Nat} (hn: n > 0) (hlen: xs.length > 0) (hlen': xl = xs.length): (splitAt n xs).2.length < xl := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:688", "score": 5, "deps": "import Mathlib\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n", "proof_state": "n xl : \u2115\nxs : List \u2115\nhn : n > 0\nhlen : xs.length > 0\nhlen' : xl = xs.length\n\u22a2 (splitAt n xs).2.length < xl", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:691"} {"full_name": "splitAt_sum_preserves_len", "prop_defn": "theorem splitAt_sum_preserves_len (n: Nat) (xs: List Nat) (hspl: spl = splitAt n xs): (spl.1.length + spl.2.length = xs.length) := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:693", "score": 5, "deps": "import Mathlib\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n", "proof_state": "spl : List \u2115 \u00d7 List \u2115\nn : \u2115\nxs : List \u2115\nhspl : spl = splitAt n xs\n\u22a2 spl.1.length + spl.2.length = xs.length", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:703"} {"full_name": "splitAt_first_len_lt", "prop_defn": "theorem splitAt_first_len_lt {xs: List Nat} (hn: n < xl) (hlen': xl = xs.length): (splitAt n xs).1.length < xl := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:771", "score": 5, "deps": "import Mathlib\n\ndef splitAt : Nat \u2192 List Nat \u2192 (List Nat \u00d7 List Nat)\n| _n, [] => ([], [])\n| 0, xs => ([], xs)\n| n + 1, x::xs => match splitAt n xs with\n | (l1, l2) => (x::l1, l2)\n", "proof_state": "n xl : \u2115\nxs : List \u2115\nhn : n < xl\nhlen' : xl = xs.length\n\u22a2 (splitAt n xs).1.length < xl", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:781"} {"full_name": "twon_lt", "prop_defn": "theorem twon_lt (n: Nat): (2*n.succ.succ.succ + 1)/ 3 < n.succ.succ.succ := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:783", "score": 5, "deps": "import Mathlib", "proof_state": "n : \u2115\n\u22a2 (2 * n.succ.succ.succ + 1) / 3 < n.succ.succ.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:789"} {"full_name": "third_eq_div_3", "prop_defn": "theorem third_eq_div_3 : (x/3) = third x := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:854", "score": 5, "deps": "import Mathlib\n\ndef third : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 1 + (third n)\n", "proof_state": "x : \u2115\n\u22a2 x / 3 = third x", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:862"} {"full_name": "twon_lt'", "prop_defn": "theorem twon_lt' (n: Nat): twoThirds (n.succ.succ.succ) < n.succ.succ.succ := by sorry", "prop_loc": "LeanSrc/LeanSrc/Sorts.lean:927", "score": 5, "deps": "import Mathlib\n\ndef twoThirds : Nat \u2192 Nat\n| 0 => 0\n| 1 => 0\n| 2 => 0\n| n + 3 => 2 + (twoThirds n)\n", "proof_state": "n : \u2115\n\u22a2 twoThirds n.succ.succ.succ < n.succ.succ.succ", "file_locs": "LeanSrc/LeanSrc/Sorts.lean:935"} {"full_name": "prop_Select", "prop_defn": "theorem prop_Select (xs: List \u03b1) [DecidableEq \u03b1] :\n List.map Prod.fst (select xs) == xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:372", "score": 5, "deps": "import Mathlib\n\ndef select : List \u03b1 \u2192 List (\u03b1 \u00d7 (List \u03b1))\n | [] => []\n | x :: xs =>\n \u27e8x, xs\u27e9:: List.map (fun (p: \u03b1 \u00d7 (List \u03b1)) => (p.1, x::p.2)) (select xs)\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (List.map Prod.fst (select xs) == xs) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 279], ["LeanSrc/LeanSrc/Properties.lean", 373]]} {"full_name": "prop_SelectPermutations", "prop_defn": "theorem prop_SelectPermutations (xs: List \u03b1) [DecidableEq \u03b1] :\n (List.all\n (List.map\n (fun (p: \u03b1 \u00d7 List \u03b1) => isPermutation xs (p.1::p.2))\n (select xs)\n )\n (fun x => x)\n ):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:376", "score": 5, "deps": "import Mathlib\n\ndef select : List \u03b1 \u2192 List (\u03b1 \u00d7 (List \u03b1))\n | [] => []\n | x :: xs =>\n \u27e8x, xs\u27e9:: List.map (fun (p: \u03b1 \u00d7 (List \u03b1)) => (p.1, x::p.2)) (select xs)\n\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 ((List.map (fun p => isPermutation xs (p.1 :: p.2)) (select xs)).all fun x => x) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 331], ["LeanSrc/LeanSrc/Properties.lean", 383]]} {"full_name": "prop_SelectPermutations'", "prop_defn": "theorem prop_SelectPermutations' (xs: List \u03b1) (z: \u03b1) [DecidableEq \u03b1] :\n let n := count z xs\n (List.all\n (List.map\n (fun (p: \u03b1 \u00d7 List \u03b1) => n == (count z (p.1::p.2)))\n (select xs)\n )\n (fun x => x)\n ):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:386", "score": 5, "deps": "import Mathlib\n\ndef select : List \u03b1 \u2192 List (\u03b1 \u00d7 (List \u03b1))\n | [] => []\n | x :: xs =>\n \u27e8x, xs\u27e9:: List.map (fun (p: \u03b1 \u00d7 (List \u03b1)) => (p.1, x::p.2)) (select xs)\n\n\ndef count [DecidableEq \u03b1]: \u03b1 -> List \u03b1 -> Nat\n | _z, [] => 0\n | z, x::xs => if x==z then (count z xs).succ else count z xs\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\nz : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 let n := count z xs;\n ((List.map (fun p => n == count z (p.1 :: p.2)) (select xs)).all fun x => x) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 311], ["LeanSrc/LeanSrc/Properties.lean", 394]]} {"full_name": "prop_PairUnpair", "prop_defn": "theorem prop_PairUnpair (xs: List \u03b1) [DecidableEq \u03b1] :\n Even (xs.length) \u2192 ((unpair (pairs xs)) == xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:397", "score": 5, "deps": "import Mathlib\n\ndef pairs : List \u03b1 \u2192 List (\u03b1 \u00d7 \u03b1)\n | x::y::xs => (x, y):: (pairs xs)\n | _ => []\n\n\ndef unpair : List (\u03b1 \u00d7 \u03b1) \u2192 List \u03b1\n | [] => []\n | (x, y)::xs => x :: y :: (unpair xs)\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Even xs.length \u2192 (unpair (pairs xs) == xs) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 287], ["LeanSrc/LeanSrc/Properties.lean", 398]]} {"full_name": "prop_PairEvens", "prop_defn": "theorem prop_PairEvens (xs: List \u03b1) [DecidableEq \u03b1] :\n Even (xs.length) \u2192 List.map Prod.fst (pairs xs) == evens xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:401", "score": 5, "deps": "import Mathlib\n\ndef pairs : List \u03b1 \u2192 List (\u03b1 \u00d7 \u03b1)\n | x::y::xs => (x, y):: (pairs xs)\n | _ => []\n\n\nmutual\n def evens : List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(odds xs)\n def odds : List \u03b1 \u2192 List \u03b1\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Even xs.length \u2192 (List.map Prod.fst (pairs xs) == evens xs) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 297], ["LeanSrc/LeanSrc/Properties.lean", 402]]} {"full_name": "prop_PairOdds", "prop_defn": "theorem prop_PairOdds (xs: List \u03b1) [DecidableEq \u03b1] :\n List.map Prod.snd (pairs xs) == odds xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:405", "score": 5, "deps": "import Mathlib\n\ndef pairs : List \u03b1 \u2192 List (\u03b1 \u00d7 \u03b1)\n | x::y::xs => (x, y):: (pairs xs)\n | _ => []\n\n\nmutual\n def evens : List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(odds xs)\n def odds : List \u03b1 \u2192 List \u03b1\n | [] => []\n | _x::xs => evens xs\nend\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (List.map Prod.snd (pairs xs) == odds xs) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 297], ["LeanSrc/LeanSrc/Properties.lean", 406]]} {"full_name": "prop_interleave", "prop_defn": "theorem prop_interleave (xs: List \u03b1) [DecidableEq \u03b1] :\n interleave (evens xs) (odds xs) == xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:409", "score": 5, "deps": "import Mathlib\n\nmutual\n def evens : List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(odds xs)\n def odds : List \u03b1 \u2192 List \u03b1\n | [] => []\n | _x::xs => evens xs\nend\n\n\ndef interleave : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | (x::xs), ys => x :: interleave ys xs\n | [], ys => ys\ntermination_by xs ys => xs.length + ys.length\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (interleave (evens xs) (odds xs) == xs) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 302], ["LeanSrc/LeanSrc/Properties.lean", 410]]} {"full_name": "prop_append_inj_1", "prop_defn": "theorem prop_append_inj_1 (xs ys zs: List \u03b1) [DecidableEq \u03b1] :\n (xs ++ zs == ys ++ zs) \u2192 xs == ys:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:414", "score": 3, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nxs ys zs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (xs ++ zs == ys ++ zs) = true \u2192 (xs == ys) = true", "file_locs": [["LeanSrc/LeanSrc/Properties.lean", 415]]} {"full_name": "prop_append_inj_2", "prop_defn": "theorem prop_append_inj_2 (xs ys zs: List \u03b1) [DecidableEq \u03b1] :\n (xs ++ ys == xs ++ zs) \u2192 ys == zs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:418", "score": 3, "deps": "import Mathlib", "proof_state": "\u03b1 : Type u_1\nxs ys zs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (xs ++ ys == xs ++ zs) = true \u2192 (ys == zs) = true", "file_locs": [["LeanSrc/LeanSrc/Properties.lean", 419]]} {"full_name": "prop_nub_nub", "prop_defn": "theorem prop_nub_nub (xs: List \u03b1) [DecidableEq \u03b1] :\n nub (nub xs) == nub xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:422", "score": 5, "deps": "import Mathlib\n\ndef nub [DecidableEq \u03b1]: List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(nub (xs.filter (fun y => x != y)))\ntermination_by xs => xs.length\ndecreasing_by\n simp_wf\n rw [Nat.lt_succ]\n exact List.length_filter_le _ xs\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (nub (nub xs) == nub xs) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 322], ["LeanSrc/LeanSrc/Properties.lean", 423]]} {"full_name": "prop_elem_nub_l", "prop_defn": "theorem prop_elem_nub_l (x: \u03b1) (xs: List \u03b1) [DecidableEq \u03b1] :\n x \u2208 xs \u2192 x \u2208 nub xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:426", "score": 5, "deps": "import Mathlib\n\ndef nub [DecidableEq \u03b1]: List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(nub (xs.filter (fun y => x != y)))\ntermination_by xs => xs.length\ndecreasing_by\n simp_wf\n rw [Nat.lt_succ]\n exact List.length_filter_le _ xs\n", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 x \u2208 xs \u2192 x \u2208 nub xs", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 322], ["LeanSrc/LeanSrc/Properties.lean", 427]]} {"full_name": "prop_elem_nub_r", "prop_defn": "theorem prop_elem_nub_r (x: \u03b1) (xs: List \u03b1) [DecidableEq \u03b1] :\n x \u2208 nub xs \u2192 x \u2208 xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:430", "score": 5, "deps": "import Mathlib\n\ndef nub [DecidableEq \u03b1]: List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(nub (xs.filter (fun y => x != y)))\ntermination_by xs => xs.length\ndecreasing_by\n simp_wf\n rw [Nat.lt_succ]\n exact List.length_filter_le _ xs\n", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 x \u2208 nub xs \u2192 x \u2208 xs", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 322], ["LeanSrc/LeanSrc/Properties.lean", 431]]} {"full_name": "prop_count_nub", "prop_defn": "theorem prop_count_nub (x: \u03b1) (xs: List \u03b1) [DecidableEq \u03b1] :\n x \u2208 xs \u2192 (count x (nub xs) == 1):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:434", "score": 5, "deps": "import Mathlib\n\ndef count [DecidableEq \u03b1]: \u03b1 -> List \u03b1 -> Nat\n | _z, [] => 0\n | z, x::xs => if x==z then (count z xs).succ else count z xs\n\n\ndef nub [DecidableEq \u03b1]: List \u03b1 \u2192 List \u03b1\n | [] => []\n | x::xs => x::(nub (xs.filter (fun y => x != y)))\ntermination_by xs => xs.length\ndecreasing_by\n simp_wf\n rw [Nat.lt_succ]\n exact List.length_filter_le _ xs\n", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 x \u2208 xs \u2192 (count x (nub xs) == 1) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 322], ["LeanSrc/LeanSrc/Properties.lean", 435]]} {"full_name": "prop_perm_trans", "prop_defn": "theorem prop_perm_trans (xs ys zs: List \u03b1) [DecidableEq \u03b1] :\n isPermutation xs ys \u2192 isPermutation ys zs \u2192 isPermutation xs zs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:438", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n", "proof_state": "\u03b1 : Type u_1\nxs ys zs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 isPermutation xs ys = true \u2192 isPermutation ys zs = true \u2192 isPermutation xs zs = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 331], ["LeanSrc/LeanSrc/Properties.lean", 439]]} {"full_name": "prop_perm_refl", "prop_defn": "theorem prop_perm_refl (xs: List \u03b1) [DecidableEq \u03b1] :\n isPermutation xs xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:442", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n", "proof_state": "\u03b1 : Type u_1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 isPermutation xs xs = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 331], ["LeanSrc/LeanSrc/Properties.lean", 443]]} {"full_name": "prop_perm_symm", "prop_defn": "theorem prop_perm_symm (xs ys: List \u03b1) [DecidableEq \u03b1] :\n isPermutation xs ys \u2192 isPermutation ys xs:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:446", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n", "proof_state": "\u03b1 : Type u_1\nxs ys : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 isPermutation xs ys = true \u2192 isPermutation ys xs = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 331], ["LeanSrc/LeanSrc/Properties.lean", 447]]} {"full_name": "prop_perm_elem", "prop_defn": "theorem prop_perm_elem (x: \u03b1) (xs ys: List \u03b1) [DecidableEq \u03b1] :\n x \u2208 xs \u2192 isPermutation xs ys \u2192 x \u2208 ys:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:450", "score": 5, "deps": "import Mathlib\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n\n\ndef isPermutation [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Bool\n| [], ys => (ys == [])\n| x::xs, ys => x \u2208 ys && (isPermutation xs (deleteFirst x ys))\n", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs ys : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 x \u2208 xs \u2192 isPermutation xs ys = true \u2192 x \u2208 ys", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 331], ["LeanSrc/LeanSrc/Properties.lean", 451]]} {"full_name": "prop_deleteAll_count", "prop_defn": "theorem prop_deleteAll_count (x: \u03b1) (xs: List \u03b1) [DecidableEq \u03b1]:\n (delete x xs == deleteFirst x xs) \u2192 count x xs <= 1:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:454", "score": 5, "deps": "import Mathlib\n\ndef delete [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then (delete n xs) else x::(delete n xs)\n\n\ndef count [DecidableEq \u03b1]: \u03b1 -> List \u03b1 -> Nat\n | _z, [] => 0\n | z, x::xs => if x==z then (count z xs).succ else count z xs\n\n\ndef deleteFirst [DecidableEq \u03b1]: \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n | _, [] => []\n | n, x::xs => if n == x then xs else x::(deleteFirst n xs)\n", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (delete x xs == deleteFirst x xs) = true \u2192 count x xs \u2264 1", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 326], ["LeanSrc/LeanSrc/Properties.lean", 455]]} {"full_name": "prop_elem", "prop_defn": "theorem prop_elem (x: \u03b1) (xs: List \u03b1) [DecidableEq \u03b1] :\n x \u2208 xs \u2192 \u2203i, x == at' xs i:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:458", "score": 5, "deps": "import Mathlib\n\ndef at' : List \u03b1 \u2192 Nat \u2192 Option \u03b1\n | x::_, 0 => x\n | _::xs, n => at' xs (n - 1)\n | [], _ => none\n", "proof_state": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 x \u2208 xs \u2192 \u2203 i, (some x == at' xs i) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 307], ["LeanSrc/LeanSrc/Properties.lean", 459]]} {"full_name": "prop_elem_map", "prop_defn": "theorem prop_elem_map (y: \u03b2) (f: \u03b1 \u2192 \u03b2) (xs: List \u03b1) [DecidableEq \u03b2] :\n y \u2208 xs.map f \u2192 (\u2203x, (f x) == y \u2227 x \u2208 xs):= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:467", "score": 5, "deps": "import Mathlib", "proof_state": "\u03b2 : Type u_1\n\u03b1 : Type u_2\ny : \u03b2\nf : \u03b1 \u2192 \u03b2\nxs : List \u03b1\ninst\u271d : DecidableEq \u03b2\n\u22a2 y \u2208 List.map f xs \u2192 \u2203 x, (f x == y) = true \u2227 x \u2208 xs", "file_locs": [["LeanSrc/LeanSrc/Properties.lean", 468]]} {"full_name": "prop_Flatten1", "prop_defn": "theorem prop_Flatten1 (p: MyTree \u03b1) [DecidableEq \u03b1] :\n flatten1 [p] == flatten0 p:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:474", "score": 5, "deps": "import Mathlib\n\ninductive MyTree (\u03b1: Type) where\n| leaf : MyTree \u03b1\n| node : MyTree \u03b1 \u2192 \u03b1 \u2192 MyTree \u03b1 \u2192 MyTree \u03b1\n\n\ndef f1Size : MyTree \u03b1 \u2192 Nat\n| MyTree.leaf => 1\n| MyTree.node p _x q => f1Size p + f1Size q +\n (match p with\n | MyTree.leaf => 0\n | MyTree.node _a _b _c=> 2)\n\n\nlemma f1Size_gt_zero (t: MyTree \u03b1): f1Size t > 0 := by\n induction t with\n | leaf => simp [f1Size]\n | node p _x q ih1 => simp [f1Size, ih1]\n\n\nlemma f1Size_lt_subTrees (q r: MyTree \u03b1) {x: \u03b1}: f1Size q < f1Size (MyTree.node q x r) \u2227 f1Size r < f1Size (MyTree.node q x r) := by\n simp [f1Size]\n exact \u27e8by linarith [f1Size_gt_zero r], by linarith [f1Size_gt_zero q]\u27e9;\n\n\ndef flatten0 : MyTree \u03b1 \u2192 List \u03b1\n | MyTree.leaf => []\n | MyTree.node p x q => flatten0 p ++ [x] ++ flatten0 q\n\n\ndef flatten1 : List (MyTree \u03b1) \u2192 List \u03b1\n | [] => []\n | MyTree.leaf::ps => flatten1 ps\n | (MyTree.node MyTree.leaf x q)::ps => x::(flatten1 (q::ps))\n | (MyTree.node (MyTree.node a b c) x q)::ps => flatten1 ((MyTree.node a b c)::(MyTree.node MyTree.leaf x q)::ps)\ntermination_by ps => List.sum (ps.map (fun (t: MyTree \u03b1 ) => f1Size t))\ndecreasing_by\n simp_wf\n simp [f1Size]\n simp_wf\n simp [f1Size_lt_subTrees]\n simp_wf\n simp [f1Size]\n linarith\n", "proof_state": "\u03b1 : Type\np : MyTree \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (flatten1 [p] == flatten0 p) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 373], ["LeanSrc/LeanSrc/Properties.lean", 475]]} {"full_name": "prop_Flatten1List", "prop_defn": "theorem prop_Flatten1List (ps: List (MyTree \u03b1)) [DecidableEq \u03b1] :\n flatten1 ps == List.foldl (fun (ps2: List \u03b1) (t: MyTree \u03b1) => ps2 ++ (flatten0 t) ) [] ps:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:478", "score": 5, "deps": "import Mathlib\n\ninductive MyTree (\u03b1: Type) where\n| leaf : MyTree \u03b1\n| node : MyTree \u03b1 \u2192 \u03b1 \u2192 MyTree \u03b1 \u2192 MyTree \u03b1\n\n\ndef f1Size : MyTree \u03b1 \u2192 Nat\n| MyTree.leaf => 1\n| MyTree.node p _x q => f1Size p + f1Size q +\n (match p with\n | MyTree.leaf => 0\n | MyTree.node _a _b _c=> 2)\n\n\nlemma f1Size_gt_zero (t: MyTree \u03b1): f1Size t > 0 := by\n induction t with\n | leaf => simp [f1Size]\n | node p _x q ih1 => simp [f1Size, ih1]\n\n\nlemma f1Size_lt_subTrees (q r: MyTree \u03b1) {x: \u03b1}: f1Size q < f1Size (MyTree.node q x r) \u2227 f1Size r < f1Size (MyTree.node q x r) := by\n simp [f1Size]\n exact \u27e8by linarith [f1Size_gt_zero r], by linarith [f1Size_gt_zero q]\u27e9;\n\n\ndef flatten0 : MyTree \u03b1 \u2192 List \u03b1\n | MyTree.leaf => []\n | MyTree.node p x q => flatten0 p ++ [x] ++ flatten0 q\n\n\ndef flatten1 : List (MyTree \u03b1) \u2192 List \u03b1\n | [] => []\n | MyTree.leaf::ps => flatten1 ps\n | (MyTree.node MyTree.leaf x q)::ps => x::(flatten1 (q::ps))\n | (MyTree.node (MyTree.node a b c) x q)::ps => flatten1 ((MyTree.node a b c)::(MyTree.node MyTree.leaf x q)::ps)\ntermination_by ps => List.sum (ps.map (fun (t: MyTree \u03b1 ) => f1Size t))\ndecreasing_by\n simp_wf\n simp [f1Size]\n simp_wf\n simp [f1Size_lt_subTrees]\n simp_wf\n simp [f1Size]\n linarith\n", "proof_state": "\u03b1 : Type\nps : List (MyTree \u03b1)\ninst\u271d : DecidableEq \u03b1\n\u22a2 (flatten1 ps == List.foldl (fun ps2 t => ps2 ++ flatten0 t) [] ps) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 373], ["LeanSrc/LeanSrc/Properties.lean", 479]]} {"full_name": "prop_Flatten2", "prop_defn": "theorem prop_Flatten2 (p: MyTree \u03b1) [DecidableEq \u03b1] :\n flatten2 p [] == flatten0 p:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:482", "score": 5, "deps": "import Mathlib\n\ninductive MyTree (\u03b1: Type) where\n| leaf : MyTree \u03b1\n| node : MyTree \u03b1 \u2192 \u03b1 \u2192 MyTree \u03b1 \u2192 MyTree \u03b1\n\n\ndef flatten0 : MyTree \u03b1 \u2192 List \u03b1\n | MyTree.leaf => []\n | MyTree.node p x q => flatten0 p ++ [x] ++ flatten0 q\n\n\ndef flatten2 : MyTree \u03b1 -> List \u03b1 -> List \u03b1\n| MyTree.leaf, ys => ys\n| MyTree.node p x q, ys => flatten2 p (x:: flatten2 q ys)\n", "proof_state": "\u03b1 : Type\np : MyTree \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (flatten2 p [] == flatten0 p) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 377], ["LeanSrc/LeanSrc/Properties.lean", 483]]} {"full_name": "prop_Flatten3", "prop_defn": "theorem prop_Flatten3 (p: MyTree \u03b1) [DecidableEq \u03b1] :\n flatten3 p == flatten0 p:= by sorry", "prop_loc": "LeanSrc/LeanSrc/Properties.lean:486", "score": 5, "deps": "import Mathlib\n\ninductive MyTree (\u03b1: Type) where\n| leaf : MyTree \u03b1\n| node : MyTree \u03b1 \u2192 \u03b1 \u2192 MyTree \u03b1 \u2192 MyTree \u03b1\n\n\ndef flatten0 : MyTree \u03b1 \u2192 List \u03b1\n | MyTree.leaf => []\n | MyTree.node p x q => flatten0 p ++ [x] ++ flatten0 q\n\n\ndef f3Size : MyTree \u03b1 \u2192 Nat\n| MyTree.leaf => 1\n| MyTree.node p _x q => (f3Size p) * 2 + f3Size q\n\n\nlemma f3Size_gt_zero (t: MyTree \u03b1): f3Size t > 0 := by\n induction t with\n | leaf => simp [f3Size]\n | node p _ q ih1 => simp [f3Size, ih1]\n\n\ndef flatten3 : MyTree \u03b1 \u2192 List \u03b1\n| MyTree.leaf => []\n| MyTree.node (MyTree.node p x q) y r => flatten3 (MyTree.node p x (MyTree.node q y r))\n| MyTree.node MyTree.leaf x q => x :: flatten3 q\ntermination_by t => f3Size t\ndecreasing_by\n simp_wf\n simp [f3Size]\n linarith [f3Size_gt_zero p]\n simp_wf\n simp [f3Size]\n", "proof_state": "\u03b1 : Type\np : MyTree \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (flatten3 p == flatten0 p) = true", "file_locs": [["LeanSrc/LeanSrc/Definitions.lean", 399], ["LeanSrc/LeanSrc/Properties.lean", 487]]}