69 (#275)

This PR solves problem 69.

Author: datokrat

HumanEvalLean/HumanEval69.lean

@@ -1,7 +1,141 @@
 module
 
-def search : Unit :=
-  ()
+public import Std
+public import Init.Data.Iterators.Lemmas.Basic
+open Std
+
+public section
+
+/-! ## Implementation -/
+
+def frequencies (xs : List Nat) : TreeMap Nat Nat (fun a b => compare b a) :=
+  xs.foldl (init := ∅) (fun freq (x : Nat) => freq.alter x (fun v? => some (v?.getD 0 + 1)))
+
+def search (xs : List Nat) : Int :=
+  let frequencies := frequencies xs
+  let kv := frequencies.iter
+    |>.filter (fun (k, v) => 0 < k ∧ k ≤ v)
+    |>.map (fun (k, _) => k)
+    |>.first?
+  kv.getD (-1)
+
+/-! ## Tests -/
+
+example : search [5, 5, 5, 5, 1] = 1 := by native_decide
+example : search [4, 1, 4, 1, 4, 4] = 4 := by native_decide
+example : search [3, 3] = -1 := by native_decide
+example : search [8, 8, 8, 8, 8, 8, 8, 8] = 8 := by native_decide
+example : search [2, 3, 3, 2, 2] = 2 := by native_decide
+example : search [2, 7, 8, 8, 4, 8, 7, 3, 9, 6, 5, 10, 4, 3, 6, 7, 1, 7, 4, 10, 8, 1] = 1 := by native_decide
+example : search [3, 2, 8, 2] = 2 := by native_decide
+example : search [6, 7, 1, 8, 8, 10, 5, 8, 5, 3, 10] = 1 := by native_decide
+example : search [8, 8, 3, 6, 5, 6, 4] = -1 := by native_decide
+example : search [6, 9, 6, 7, 1, 4, 7, 1, 8, 8, 9, 8, 10, 10, 8, 4, 10, 4, 10, 1, 2, 9, 5, 7, 9] = 1 := by native_decide
+example : search [1, 9, 10, 1, 3] = 1 := by native_decide
+example : search [6, 9, 7, 5, 8, 7, 5, 3, 7, 5, 10, 10, 3, 6, 10, 2, 8, 6, 5, 4, 9, 5, 3, 10] = 5 := by native_decide
+example : search [1] = 1 := by native_decide
+example : search [8, 8, 10, 6, 4, 3, 5, 8, 2, 4, 2, 8, 4, 6, 10, 4, 2, 1, 10, 2, 1, 1, 5] = 4 := by native_decide
+example : search [2, 10, 4, 8, 2, 10, 5, 1, 2, 9, 5, 5, 6, 3, 8, 6, 4, 10] = 2 := by native_decide
+example : search [1, 6, 10, 1, 6, 9, 10, 8, 6, 8, 7, 3] = 1 := by native_decide
+example : search [9, 2, 4, 1, 5, 1, 5, 2, 5, 7, 7, 7, 3, 10, 1, 5, 4, 2, 8, 4, 1, 9, 10, 7, 10, 2, 8, 10, 9, 4] = 4 := by native_decide
+example : search [2, 6, 4, 2, 8, 7, 5, 6, 4, 10, 4, 6, 3, 7, 8, 8, 3, 1, 4, 2, 2, 10, 7] = 4 := by native_decide
+example : search [9, 8, 6, 10, 2, 6, 10, 2, 7, 8, 10, 3, 8, 2, 6, 2, 3, 1] = 2 := by native_decide
+example : search [5, 5, 3, 9, 5, 6, 3, 2, 8, 5, 6, 10, 10, 6, 8, 4, 10, 7, 7, 10, 8] = -1 := by native_decide
+example : search [10] = -1 := by native_decide
+example : search [9, 7, 7, 2, 4, 7, 2, 10, 9, 7, 5, 7, 2] = 2 := by native_decide
+example : search [5, 4, 10, 2, 1, 1, 10, 3, 6, 1, 8] = 1 := by native_decide
+example : search [7, 9, 9, 9, 3, 4, 1, 5, 9, 1, 2, 1, 1, 10, 7, 5, 6, 7, 6, 7, 7, 6] = 1 := by native_decide
+example : search [3, 10, 10, 9, 2] = -1 := by native_decide
+
+/-! ## Missing API -/
+
+@[simp, grind =]
+theorem Std.Iter.first?_toList {α β : Type w} [Iterator α Id β] [IteratorLoop α Id Id]
+    [Iterators.Finite α Id] [LawfulIteratorLoop α Id Id] {it : Iter (α := α) β} :
+    it.toList.head? = it.first? := by
+  induction it using Iter.inductSteps with | step it ihy ihs
+  rw [first?_eq_match_step, toList_eq_match_step]
+  cases it.step using PlausibleIterStep.casesOn <;> simp [*]
+
+theorem List.find?_eq_some_iff_of_pairwise_eq_lt {xs : List α} {cmp : α → α → Ordering}
+    [OrientedCmp cmp] (h : xs.Pairwise (cmp · · = .lt)) :
+    xs.find? P = some x ↔ x ∈ xs ∧ P x ∧ ∀ y ∈ xs, P y → (cmp x y).isLE := by
+  rw [List.find?_eq_some_iff_append]
+  constructor
+  · intro h'
+    refine ⟨by grind, by grind, ?_⟩
+    intro y hm hy
+    obtain ⟨as, bs, rfl, h''⟩ := h'.2
+    have : y ∈ x :: bs := by grind
+    simp only [pairwise_append, pairwise_cons] at h
+    grind [Ordering.isLE_of_eq_eq, Ordering.isLE_of_eq_lt]
+  · intro h'
+    refine ⟨by grind, ?_⟩
+    obtain ⟨as, bs, rfl⟩ := List.mem_iff_append.mp h'.1
+    refine ⟨as, bs, rfl, ?_⟩
+    intro a ha
+    simp only [pairwise_append, pairwise_cons] at h
+    replace h' := h'.2.2
+    specialize h' a (by grind)
+    have : cmp x a = .gt := by
+      rw [OrientedCmp.eq_swap (cmp := cmp), Ordering.swap_eq_gt]
+      grind
+    grind [Ordering.ne_gt_iff_isLE]
+
+theorem List.find?_eq_find? {xs : List α} {p q : α → Bool} (h : ∀ x ∈ xs, p x = q x) :
+    xs.find? p = xs.find? q := by
+  ext
+  grind [List.find?_eq_some_iff_append]
+
+theorem Option.getD_map_eq_elim {o : Option α} {f : α → β} {fallback : β} :
+    (o.map f).getD fallback = o.elim fallback f := by
+  cases o <;> simp
+
+/-! ## Verification -/
+
+theorem getElem?_frequencies {k : Nat} :
+    (frequencies xs)[k]? = (some (xs.count k)).filter (0 < ·) := by
+  -- We express the statement in terms of `List.foldr`, which makes the induction over `xs`
+  -- easier because the initial tree map of the fold never changes. If the statement is expressed in
+  -- terms of `List.foldl`, we would need to generalize the statement for general initial tree maps.
+  rw [← List.reverse_reverse (as := xs), frequencies, List.foldl_reverse]
+  generalize xs.reverse = xs
+  induction xs <;> grind
+
+theorem mem_frequencies :
+    x ∈ frequencies xs ↔ x ∈ xs := by
+  rw [← TreeMap.isSome_getElem?_iff_mem, getElem?_frequencies]
+  simp
+
+theorem search_eq_getD_find? :
+    search xs =
+      (frequencies xs
+        |>.toList.find? (fun x => 0 < x.1 ∧ x.1 ≤ xs.count x.1)
+        |>.map (↑·.fst)
+        |>.getD (-1 : Int)) := by
+  simp only [search, ← Iter.first?_toList, Iter.toList_map, Iter.toList_filter,
+    TreeMap.toList_iter, List.head?_map]
+  grind [List.find?_eq_find?, getElem?_frequencies]
+
+theorem search_eq_neg_one_iff :
+    search xs = -1 ↔ ∀ x, x = 0 ∨ xs.count x < x := by
+  simp [search_eq_getD_find?, Option.getD_eq_iff, getElem?_frequencies]
+  grind [List.length_filter_pos_iff]
+
+theorem search_eq_iff_of_ne_neg_one (h : y ≠ -1) :
+    search xs = y ↔ 0 < y ∧ y ≤ xs.count y.toNat ∧ ∀ z, 0 < z → z ≤ xs.count z → z ≤ y := by
+  -- Prepare `OrientedCmp` instance to be used by `List.find?_eq_some_iff_of_pairwise_eq_lt`
+  have : OrientedCmp (fun x y : Nat × Nat => compare y.1 x.1) := by
+    constructor
+    grind [Nat.compare_swap]
+  simp only [search_eq_getD_find?, Option.getD_map_eq_elim, Option.elim]
+  split <;> rename_i heq
+  · simp only [List.find?_eq_some_iff_of_pairwise_eq_lt (frequencies xs).ordered_keys_toList,
+      Prod.forall, TreeMap.mem_toList_iff_getElem?_eq_some, getElem?_frequencies] at heq
+    grind [List.count_pos_iff, isLE_compare]
+  · simp only [List.find?_eq_none, Prod.forall, TreeMap.mem_toList_iff_getElem?_eq_some,
+      getElem?_frequencies, Option.filter_eq_some_iff, and_imp] at heq
+    grind [mem_frequencies]
 
 /-!
 ## Prompt
@@ -10,8 +144,8 @@ def search : Unit :=
 
 def search(lst):
     '''
-    You are given a non-empty list of positive integers. Return the greatest integer that is greater than 
-    zero, and has a frequency greater than or equal to the value of the integer itself. 
+    You are given a non-empty list of positive integers. Return the greatest integer that is greater than
+    zero, and has a frequency greater than or equal to the value of the integer itself.
     The frequency of an integer is the number of times it appears in the list.
     If no such a value exist, return -1.
     Examples:
@@ -32,7 +166,7 @@ def search(lst):
     for i in range(1, len(frq)):
         if frq[i] >= i:
             ans = i
-    
+
     return ans
 ```
 
@@ -71,4 +205,4 @@ def check(candidate):
     assert candidate([3, 10, 10, 9, 2]) == -1
 
 ```
--/
+-/