HumanEval12

HumanEvalLean/HumanEval12.lean

@@ -1,5 +1,236 @@
-def longest : Unit :=
-  ()
+module
+
+open Std
+
+/-!
+## Implementation
+-/
+
+def argmax [LE β] [DecidableLE β] (f : α → β) (x y : α) : α :=
+  if f y ≤ f x then x else y
+
+/-
+`List.argmax` exists in mathlib, but:
+* it returns an `Option`, so it should actually be named `argmax?`
+* it relies on mathlib's `Preorder` type class and `DecidableLT`. In the standard library,
+  it would be more consistent to use `LE` and `DecidableLE`.
+
+Moreover, lemmas such as `List.index_of_argmax` aren't easily applicable because one would need
+`BEq α` and `LawfulBEq α` in order to use `idxOf`. Moreover, some API about `idxOf` and `findIdx`
+is still missing. In this file, we avoid these difficulties by not relying on `idxOf` and `findIdx`
+at all.
+-/
+def List.argmax [LE β] [DecidableLE β] (xs : List α) (f : α → β) (h : xs ≠ []) : α :=
+  match xs with
+  | x :: xs => xs.foldl (init := x) (_root_.argmax f)
+
+def List.argmax? [LE β] [DecidableLE β] (xs : List α) (f : α → β) : Option α :=
+  if h : xs ≠ [] then
+    some (xs.argmax f h)
+  else
+    none
+
+def longest? (xs : List String) : Option String :=
+  xs.argmax? String.length
+
+/-!
+## Tests
+-/
+
+example : longest? [] = none := by native_decide
+example : longest? ["x", "y", "z"] = some "x" := by native_decide
+example : longest? ["x", "yyy", "zzzz", "www", "kkkk", "abc"] = some "zzzz" := by native_decide
+
+/-!
+## Verification
+-/
+
+@[grind =]
+theorem List.argmax_singleton [LE β] [DecidableLE β] {x : α} {f : α → β} :
+    [x].argmax f (by grind) = x := by
+  grind [argmax]
+
+@[grind =]
+theorem argmax_assoc [LE β] [DecidableLE β] [IsLinearPreorder β] {f : α → β} {x y z : α} :
+    argmax f (argmax f x y) z = argmax f x (argmax f y z) := by
+  grind [argmax]
+
+instance [LE β] [DecidableLE β] [IsLinearPreorder β] {f : α → β} :
+    Associative (argmax f) where
+  assoc := by apply argmax_assoc
+
+theorem List.argmax_cons
+    [LE β] [DecidableLE β] [IsLinearPreorder β] {x : α} {xs : List α} {f : α → β} :
+    (x :: xs).argmax f (by grind) =
+      if h : xs = [] then x else _root_.argmax f x (xs.argmax f h) := by
+  simp only [argmax]
+  match xs with
+  | [] => simp
+  | y :: xs => simp [foldl_assoc]
+
+theorem argmax_eq_or [LE β] [DecidableLE β] {f : α → β} {x y : α} :
+    argmax f x y = x ∨ argmax f x y = y := by
+  grind [argmax]
+
+@[grind =]
+theorem argmax_self [LE β] [DecidableLE β] [IsLinearPreorder β] {f : α → β} {x : α} :
+    argmax f x x = x := by
+  grind [argmax]
+
+@[grind =]
+theorem argmax_eq_left [LE β] [DecidableLE β] {f : α → β} {x y : α} (h : f y ≤ f x) :
+    argmax f x y = x := by
+  grind [argmax]
+
+@[grind =]
+theorem argmax_eq_right [LE β] [DecidableLE β] {f : α → β} {x y : α} (h : ¬ f y ≤ f x) :
+    argmax f x y = y := by
+  grind [argmax]
+
+@[grind =>]
+theorem apply_left_le_apply_argmax [LE β] [DecidableLE β] [IsLinearPreorder β] {f : α → β}
+    {x y : α} : f x ≤ f (argmax f x y) := by
+  grind [argmax]
+
+@[grind =>]
+theorem apply_right_le_apply_argmax [LE β] [DecidableLE β] [IsLinearPreorder β]
+    {f : α → β} {x y : α} : f y ≤ f (argmax f x y) := by
+  grind [argmax]
+
+@[grind .]
+theorem List.argmax_mem [LE β] [DecidableLE β] [IsLinearPreorder β] {xs : List α}
+    {f : α → β} {h : xs ≠ []} : xs.argmax f h ∈ xs := by
+  simp only [List.argmax]
+  match xs with
+  | x :: xs =>
+    fun_induction xs.foldl (init := x) (_root_.argmax f) <;> grind [argmax_eq_or]
+
+@[grind =>]
+theorem List.le_apply_argmax_of_mem [LE β] [DecidableLE β] [IsLinearPreorder β]
+    {xs : List α} {f : α → β} {y : α} (hx : y ∈ xs) :
+    f y ≤ f (xs.argmax f (List.ne_nil_of_mem hx)) := by
+  have h : xs ≠ [] := List.ne_nil_of_mem hx
+  simp only [List.argmax]
+  match xs with
+  | x :: xs =>
+    fun_induction xs.foldl (init := x) (_root_.argmax f) generalizing y <;> grind
+
+@[grind =]
+theorem List.argmax_append [LE β] [DecidableLE β] [IsLinearPreorder β] {xs ys : List α}
+    {f : α → β} (hxs : xs ≠ []) (hys : ys ≠ []) :
+    (xs ++ ys).argmax f (by simp [hxs]) = _root_.argmax f (xs.argmax f hxs) (ys.argmax f hys) := by
+  match xs, ys with
+  | x :: xs, y :: ys => simp [argmax, foldl_assoc]
+
+/--
+`List.argmax xs f h` comes before any other element in `xs` where `f` attains its maximum.
+-/
+theorem List.argmax_left_leaning
+    [LE β] [DecidableLE β] [IsLinearPreorder β] {xs : List α} {f : α → β} (h : xs ≠ []) :
+    ∃ j : Fin xs.length, xs[j] = xs.argmax f h ∧
+      ∀ i : Fin j, ¬ f (xs.argmax f h) ≤ f xs[i] := by
+  simp only [List.argmax]
+  match xs with
+  | x :: xs =>
+    simp only
+    clear h
+    fun_induction xs.foldl (init := x) (_root_.argmax f)
+    · exact ⟨⟨0, by grind⟩, by grind⟩
+    · rename_i x y xs ih
+      obtain ⟨j, ih⟩ := ih
+      by_cases hj : j.val = 0
+      · by_cases hm : f y ≤ f x
+        · exact ⟨⟨0, by grind⟩, by grind⟩
+        · exact ⟨⟨1, by grind⟩, by grind⟩
+      · refine ⟨⟨j + 1, by grind⟩, ?_⟩
+        obtain ⟨j, _⟩ := Nat.exists_eq_succ_of_ne_zero hj
+        apply And.intro
+        · grind
+        · intro i hi
+          have : i.val ≥ 2 := by have := ih.2 ⟨0, by grind⟩; grind
+          obtain ⟨i, _⟩ := Nat.exists_eq_add_of_le this
+          have := ih.2 ⟨i + 1, by grind⟩
+          grind
+
+/-- `List.argmax?` returns `none` when applied to an empty list. -/
+@[grind =]
+theorem List.argmax?_nil [LE β] [DecidableLE β] {f : α → β} :
+    ([] : List α).argmax? f = none := by
+  simp [argmax?]
+
+@[grind =]
+theorem List.argmax?_cons
+    [LE β] [DecidableLE β] [IsLinearPreorder β] {f : α → β} {x : α} {xs : List α} :
+    (x :: xs).argmax? f = (xs.argmax? f).elim x (_root_.argmax f x) := by
+  grind [argmax?, argmax_cons]
+
+@[grind =>]
+theorem List.isSome_argmax?_of_mem
+    [LE β] [DecidableLE β] {f : α → β} {xs : List α} {x : α} (h : x ∈ xs) :
+    (xs.argmax? f).isSome := by
+  grind [argmax?]
+
+theorem List.le_apply_argmax?_get_of_mem
+    [LE β] [DecidableLE β] [IsLinearPreorder β] {f : α → β} {xs : List α} {x : α} (h : x ∈ xs) :
+    f x ≤ f ((xs.argmax? f).get (isSome_argmax?_of_mem h)) := by
+  grind [argmax?]
+
+-- The suggested patterns are not useful because all involve `IsLinearPreorder`.
+grind_pattern List.le_apply_argmax?_get_of_mem => x ∈ xs, (xs.argmax? f).get _
+
+theorem List.argmax?_left_leaning [LE β] [DecidableLE β] [IsLinearPreorder β] {xs : List α} {f : α → β} {x : α}
+    (hx : xs.argmax? f = some x) :
+    ∃ j : Fin xs.length, xs[j] = x ∧ ∀ i : Fin j, ¬ f x ≤ f xs[i] := by
+  simp only [argmax?] at hx
+  split at hx
+  · simp only [Option.some.injEq] at hx
+    rw [← hx]
+    apply argmax_left_leaning
+  · grind
+
+@[grind =]
+theorem List.argmax?_append [LE β] [DecidableLE β] [IsLinearPreorder β] (xs ys : List α) (f : α → β) :
+    (xs ++ ys).argmax? f =
+      (xs.argmax? f).merge (_root_.argmax f) (ys.argmax? f) := by
+  grind [argmax?, append_eq_nil_iff]
+
+/-!
+### Main theorems
+
+The following theorems verify important properties of `longest?`.
+The requirements from the prompt are verified by `le_length_longest?_get_of_mem` and
+`longest?_left_leaning`.
+
+Some other useful properties are proved by the remaining lemmas.
+-/
+
+theorem longest?_nil : longest? [] = none := by
+  grind [longest?]
+
+theorem longest?_singleton : longest? [x] = some x := by
+  grind [longest?]
+
+theorem longest?_append {xs ys : List String} :
+    longest? (xs ++ ys) = (longest? xs).merge (_root_.argmax String.length) (longest? ys) := by
+  grind [longest?]
+
+theorem isSome_longest?_of_mem {xs : List String} {x : String} (h : x ∈ xs) :
+    (longest? xs).isSome := by
+  grind [longest?]
+
+theorem le_length_longest?_get_of_mem {xs : List String} {x : String} (h : x ∈ xs) :
+    x.length ≤ ((longest? xs).get (isSome_longest?_of_mem h)).length := by
+  grind [longest?]
+
+/--
+`longest?` returns the first string with maximum length: any other string with maximum length
+appears at an index greater than the returned string's index.
+-/
+theorem longest?_left_leaning {xs : List String} {x : String} (h : longest? xs = some x) :
+    ∃ j : Fin xs.length, xs[j] = x ∧ ∀ i : Fin j, xs[i].length < x.length := by
+  rw [longest?] at h
+  have := List.argmax?_left_leaning h
+  grind
 
 /-!
 ## Prompt
@@ -48,4 +279,4 @@ def check(candidate):
     assert candidate(['x', 'y', 'z']) == 'x'
     assert candidate(['x', 'yyy', 'zzzz', 'www', 'kkkk', 'abc']) == 'zzzz'
 ```
--/
+-/