add custom-set

config.json

@@ -586,6 +586,14 @@
         "prerequisites": [],
         "difficulty": 7
       },
+      {
+        "slug": "custom-set",
+        "name": "Custom Set",
+        "uuid": "d643def0-b87f-44b9-ab31-787d94d4f6a0",
+        "practices": [],
+        "prerequisites": [],
+        "difficulty": 7
+      },
       {
         "slug": "pythagorean-triplet",
         "name": "Pythagorean Triplet",

exercises/practice/custom-set/.docs/instructions.md

@@ -0,0 +1,7 @@
+# Instructions
+
+Create a custom set type.
+
+Sometimes it is necessary to define a custom data structure of some type, like a set.
+In this exercise you will define your own set.
+How it works internally doesn't matter, as long as it behaves like a set of unique elements.

exercises/practice/custom-set/.meta/Example.lean

@@ -0,0 +1,78 @@
+namespace CustomSet
+
+abbrev Size := Nat
+
+inductive Set where
+  | nil  : Set
+  | node : Size → Nat → Set → Set → Set
+  deriving Repr, Inhabited
+
+instance : EmptyCollection Set :=
+  ⟨ .nil ⟩ -- takes care of ∅
+
+private def Set.size : Set → Size
+  | .nil          => 0
+  | .node s _ _ _ => s
+
+private def Set.contains : Set → Nat → Bool
+  | .nil, _ => false
+  | .node _ x l r, e => e == x || if e < x then l.contains e else r.contains e
+
+private def Set.foldl {α} (fn : α → Nat → α) (init : α) (set : Set) : α :=
+  match set with
+  | .nil => init
+  | .node _ x l r =>
+    let leftAcc := l.foldl fn init
+    let acc := fn leftAcc x
+    r.foldl fn acc
+
+instance : BEq Set where -- takes care of == and !=
+  beq s1 s2 := s1.size == s2.size && s1.foldl (fun acc x => acc && s2.contains x) true
+
+instance : Membership Nat Set where -- takes care of ∈ and ∉
+  mem s n := s.contains n
+
+instance (a : Nat) (s : Set) : Decidable (a ∈ s) := by
+  simp [Membership.mem]
+  exact inferInstance
+
+private def Set.subset (set1 set2 : Set) : Bool :=
+  set1.foldl (fun acc x => acc && set2.contains x) true
+
+instance : HasSubset Set where
+  Subset s1 s2 := s1.subset s2
+
+instance (s1 s2 : Set) : Decidable (s1 ⊆ s2) := by
+  simp [HasSubset.Subset]
+  exact inferInstance
+
+notation a " ⊈ " b => ¬(a ⊆ b)
+
+private def Set.addHelper (e : Nat) (s : Set) : Set :=
+  match s with
+  | .nil => .node 1 e .nil .nil
+  | .node s x l r =>
+    if e < x
+    then Set.node (s + 1) x (l.addHelper e) r
+    else Set.node (s + 1) x l (r.addHelper e)
+
+def Set.add (elem : Nat) (set : Set) : Set :=
+  if elem ∈ set then set
+  else set.addHelper elem
+
+def Set.ofList (xs : List Nat) : Set :=
+  xs.foldl (fun acc x => acc.add x) ∅
+
+instance : Inter Set where -- takes care of ∩
+  inter s1 s2 := s1.foldl (fun acc x => if s2.contains x then acc.add x else acc) ∅
+
+instance : Union Set where -- takes care of ∪
+  union s1 s2 := s1.foldl (fun acc x => acc.add x) s2
+
+instance : SDiff Set where -- takes care of \
+  sdiff s1 s2 := s1.foldl (fun acc x => if s2.contains x then acc else acc.add x) ∅
+
+def Set.disjoint (set1 set2 : Set) : Bool :=
+  set1.foldl (fun acc x => acc && !set2.contains x) true
+
+end CustomSet

exercises/practice/custom-set/.meta/config.json

@@ -0,0 +1,17 @@
+{
+  "authors": [
+    "oxe-i"
+  ],
+  "files": {
+    "solution": [
+      "CustomSet.lean"
+    ],
+    "test": [
+      "CustomSetTest.lean"
+    ],
+    "example": [
+      ".meta/Example.lean"
+    ]
+  },
+  "blurb": "Create a custom set type."
+}

exercises/practice/custom-set/.meta/tests.toml

@@ -0,0 +1,130 @@
+# This is an auto-generated file.
+#
+# Regenerating this file via `configlet sync` will:
+# - Recreate every `description` key/value pair
+# - Recreate every `reimplements` key/value pair, where they exist in problem-specifications
+# - Remove any `include = true` key/value pair (an omitted `include` key implies inclusion)
+# - Preserve any other key/value pair
+#
+# As user-added comments (using the # character) will be removed when this file
+# is regenerated, comments can be added via a `comment` key.
+
+[20c5f855-f83a-44a7-abdd-fe75c6cf022b]
+description = "Returns true if the set contains no elements -> sets with no elements are empty"
+
+[d506485d-5706-40db-b7d8-5ceb5acf88d2]
+description = "Returns true if the set contains no elements -> sets with elements are not empty"
+
+[759b9740-3417-44c3-8ca3-262b3c281043]
+description = "Sets can report if they contain an element -> nothing is contained in an empty set"
+
+[f83cd2d1-2a85-41bc-b6be-80adbff4be49]
+description = "Sets can report if they contain an element -> when the element is in the set"
+
+[93423fc0-44d0-4bc0-a2ac-376de8d7af34]
+description = "Sets can report if they contain an element -> when the element is not in the set"
+
+[c392923a-637b-4495-b28e-34742cd6157a]
+description = "A set is a subset if all of its elements are contained in the other set -> empty set is a subset of another empty set"
+
+[5635b113-be8c-4c6f-b9a9-23c485193917]
+description = "A set is a subset if all of its elements are contained in the other set -> empty set is a subset of non-empty set"
+
+[832eda58-6d6e-44e2-92c2-be8cf0173cee]
+description = "A set is a subset if all of its elements are contained in the other set -> non-empty set is not a subset of empty set"
+
+[c830c578-8f97-4036-b082-89feda876131]
+description = "A set is a subset if all of its elements are contained in the other set -> set is a subset of set with exact same elements"
+
+[476a4a1c-0fd1-430f-aa65-5b70cbc810c5]
+description = "A set is a subset if all of its elements are contained in the other set -> set is a subset of larger set with same elements"
+
+[d2498999-3e46-48e4-9660-1e20c3329d3d]
+description = "A set is a subset if all of its elements are contained in the other set -> set is not a subset of set that does not contain its elements"
+
+[7d38155e-f472-4a7e-9ad8-5c1f8f95e4cc]
+description = "Sets are disjoint if they share no elements -> the empty set is disjoint with itself"
+
+[7a2b3938-64b6-4b32-901a-fe16891998a6]
+description = "Sets are disjoint if they share no elements -> empty set is disjoint with non-empty set"
+
+[589574a0-8b48-48ea-88b0-b652c5fe476f]
+description = "Sets are disjoint if they share no elements -> non-empty set is disjoint with empty set"
+
+[febeaf4f-f180-4499-91fa-59165955a523]
+description = "Sets are disjoint if they share no elements -> sets are not disjoint if they share an element"
+
+[0de20d2f-c952-468a-88c8-5e056740f020]
+description = "Sets are disjoint if they share no elements -> sets are disjoint if they share no elements"
+
+[4bd24adb-45da-4320-9ff6-38c044e9dff8]
+description = "Sets with the same elements are equal -> empty sets are equal"
+
+[f65c0a0e-6632-4b2d-b82c-b7c6da2ec224]
+description = "Sets with the same elements are equal -> empty set is not equal to non-empty set"
+
+[81e53307-7683-4b1e-a30c-7e49155fe3ca]
+description = "Sets with the same elements are equal -> non-empty set is not equal to empty set"
+
+[d57c5d7c-a7f3-48cc-a162-6b488c0fbbd0]
+description = "Sets with the same elements are equal -> sets with the same elements are equal"
+
+[dd61bafc-6653-42cc-961a-ab071ee0ee85]
+description = "Sets with the same elements are equal -> sets with different elements are not equal"
+
+[06059caf-9bf4-425e-aaff-88966cb3ea14]
+description = "Sets with the same elements are equal -> set is not equal to larger set with same elements"
+
+[d4a1142f-09aa-4df9-8b83-4437dcf7ec24]
+description = "Sets with the same elements are equal -> set is equal to a set constructed from an array with duplicates"
+
+[8a677c3c-a658-4d39-bb88-5b5b1a9659f4]
+description = "Unique elements can be added to a set -> add to empty set"
+
+[0903dd45-904d-4cf2-bddd-0905e1a8d125]
+description = "Unique elements can be added to a set -> add to non-empty set"
+
+[b0eb7bb7-5e5d-4733-b582-af771476cb99]
+description = "Unique elements can be added to a set -> adding an existing element does not change the set"
+
+[893d5333-33b8-4151-a3d4-8f273358208a]
+description = "Intersection returns a set of all shared elements -> intersection of two empty sets is an empty set"
+
+[d739940e-def2-41ab-a7bb-aaf60f7d782c]
+description = "Intersection returns a set of all shared elements -> intersection of an empty set and non-empty set is an empty set"
+
+[3607d9d8-c895-4d6f-ac16-a14956e0a4b7]
+description = "Intersection returns a set of all shared elements -> intersection of a non-empty set and an empty set is an empty set"
+
+[b5120abf-5b5e-41ab-aede-4de2ad85c34e]
+description = "Intersection returns a set of all shared elements -> intersection of two sets with no shared elements is an empty set"
+
+[af21ca1b-fac9-499c-81c0-92a591653d49]
+description = "Intersection returns a set of all shared elements -> intersection of two sets with shared elements is a set of the shared elements"
+
+[c5e6e2e4-50e9-4bc2-b89f-c518f015b57e]
+description = "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of two empty sets is an empty set"
+
+[2024cc92-5c26-44ed-aafd-e6ca27d6fcd2]
+description = "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of empty set and non-empty set is an empty set"
+
+[e79edee7-08aa-4c19-9382-f6820974b43e]
+description = "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of a non-empty set and an empty set is the non-empty set"
+
+[c5ac673e-d707-4db5-8d69-7082c3a5437e]
+description = "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of two non-empty sets is a set of elements that are only in the first set"
+
+[20d0a38f-7bb7-4c4a-ac15-90c7392ecf2b]
+description = "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference removes all duplicates in the first set"
+
+[c45aed16-5494-455a-9033-5d4c93589dc6]
+description = "Union returns a set of all elements in either set -> union of empty sets is an empty set"
+
+[9d258545-33c2-4fcb-a340-9f8aa69e7a41]
+description = "Union returns a set of all elements in either set -> union of an empty set and non-empty set is the non-empty set"
+
+[3aade50c-80c7-4db8-853d-75bac5818b83]
+description = "Union returns a set of all elements in either set -> union of a non-empty set and empty set is the non-empty set"
+
+[a00bb91f-c4b4-4844-8f77-c73e2e9df77c]
+description = "Union returns a set of all elements in either set -> union of non-empty sets contains all unique elements"

exercises/practice/custom-set/CustomSet.lean

@@ -0,0 +1,21 @@
+namespace CustomSet
+
+/-
+  Define here a Set type
+
+  It must have operators defined with the following semantics:
+  * ∈, ∉ : an element is, or is not, in a Set
+  * ⊆, ⊈ : a Set is, or is not, a subset of another
+  * ==, != : two Sets have, or don't have, the same elements
+  * ∩ : gets the intersection of two Sets
+  * ∪ : gets the union of two Sets
+  * \ : gets the difference of two Sets
+  * ∅ : references an empty Set
+
+  In addition, at least three other functions must be defined:
+  * Set.ofList, that takes a List Nat and returns a Set
+  * Set.add, that adds a Nat to a Set, returning the new Set
+  * Set.disjoint, that checks if two Sets are, or are not, disjoint
+-/
+
+end CustomSet

exercises/practice/custom-set/CustomSetTest.lean

@@ -0,0 +1,90 @@
+import LeanTest
+import CustomSet
+
+open LeanTest
+
+def customSetTests : TestSuite :=
+  (TestSuite.empty "CustomSet")
+    |>.addTest "Returns true if the set contains no elements -> sets with no elements are empty" (do
+        return assertEqual ∅ <| CustomSet.Set.ofList [])
+    |>.addTest "Returns true if the set contains no elements -> sets with elements are not empty" (do
+        return assertNotEqual ∅ <| CustomSet.Set.ofList [1])
+    |>.addTest "Sets can report if they contain an element -> nothing is contained in an empty set" (do
+        return assert <| decide <| 1 ∉ CustomSet.Set.ofList [])
+    |>.addTest "Sets can report if they contain an element -> when the element is in the set" (do
+        return assert <| decide <| 1 ∈ CustomSet.Set.ofList [1, 2, 3])
+    |>.addTest "Sets can report if they contain an element -> when the element is not in the set" (do
+        return assert <| decide <| 4 ∉ CustomSet.Set.ofList [1, 2, 3])
+    |>.addTest "A set is a subset if all of its elements are contained in the other set -> empty set is a subset of another empty set" (do
+        return assert <| decide <| CustomSet.Set.ofList [] ⊆ CustomSet.Set.ofList [])
+    |>.addTest "A set is a subset if all of its elements are contained in the other set -> empty set is a subset of non-empty set" (do
+        return assert <| decide <| CustomSet.Set.ofList [] ⊆ CustomSet.Set.ofList [1])
+    |>.addTest "A set is a subset if all of its elements are contained in the other set -> non-empty set is not a subset of empty set" (do
+        return assert <| decide <| CustomSet.Set.ofList [1] ⊈ CustomSet.Set.ofList [])
+    |>.addTest "A set is a subset if all of its elements are contained in the other set -> set is a subset of set with exact same elements" (do
+        return assert <| decide <| CustomSet.Set.ofList [1, 2, 3] ⊆ CustomSet.Set.ofList [1, 2, 3])
+    |>.addTest "A set is a subset if all of its elements are contained in the other set -> set is a subset of larger set with same elements" (do
+        return assert <| decide <| CustomSet.Set.ofList [1, 2, 3] ⊆ CustomSet.Set.ofList [4, 1, 2, 3])
+    |>.addTest "A set is a subset if all of its elements are contained in the other set -> set is not a subset of set that does not contain its elements" (do
+        return assert <| decide <| CustomSet.Set.ofList [1, 2, 3] ⊈ CustomSet.Set.ofList [4, 1, 3])
+    |>.addTest "Sets are disjoint if they share no elements -> the empty set is disjoint with itself" (do
+        return assertTrue <| (CustomSet.Set.ofList []).disjoint <| CustomSet.Set.ofList [])
+    |>.addTest "Sets are disjoint if they share no elements -> empty set is disjoint with non-empty set" (do
+        return assertTrue <| (CustomSet.Set.ofList []).disjoint <| CustomSet.Set.ofList [1])
+    |>.addTest "Sets are disjoint if they share no elements -> non-empty set is disjoint with empty set" (do
+        return assertTrue <| (CustomSet.Set.ofList [1]).disjoint <| CustomSet.Set.ofList [])
+    |>.addTest "Sets are disjoint if they share no elements -> sets are not disjoint if they share an element" (do
+        return assertFalse <| (CustomSet.Set.ofList [1, 2]).disjoint <| CustomSet.Set.ofList [2, 3])
+    |>.addTest "Sets are disjoint if they share no elements -> sets are disjoint if they share no elements" (do
+        return assertTrue <| (CustomSet.Set.ofList [1, 2]).disjoint <| CustomSet.Set.ofList [3, 4])
+    |>.addTest "Sets with the same elements are equal -> empty sets are equal" (do
+        return assert <| CustomSet.Set.ofList [] == CustomSet.Set.ofList [])
+    |>.addTest "Sets with the same elements are equal -> empty set is not equal to non-empty set" (do
+        return assert <| CustomSet.Set.ofList [] != CustomSet.Set.ofList [1, 2, 3])
+    |>.addTest "Sets with the same elements are equal -> non-empty set is not equal to empty set" (do
+        return assert <| CustomSet.Set.ofList [1, 2, 3] != CustomSet.Set.ofList [])
+    |>.addTest "Sets with the same elements are equal -> sets with the same elements are equal" (do
+        return assert <| CustomSet.Set.ofList [1, 2] == CustomSet.Set.ofList [2, 1])
+    |>.addTest "Sets with the same elements are equal -> sets with different elements are not equal" (do
+        return assert <| CustomSet.Set.ofList [1, 2, 3] != CustomSet.Set.ofList [1, 2, 4])
+    |>.addTest "Sets with the same elements are equal -> set is not equal to larger set with same elements" (do
+        return assert <| CustomSet.Set.ofList [1, 2, 3] != CustomSet.Set.ofList [1, 2, 3, 4])
+    |>.addTest "Sets with the same elements are equal -> set is equal to a set constructed from an array with duplicates" (do
+        return assert <| CustomSet.Set.ofList [1] == CustomSet.Set.ofList [1, 1])
+    |>.addTest "Unique elements can be added to a set -> add to empty set" (do
+        return assertEqual (CustomSet.Set.ofList [3]) <| (CustomSet.Set.ofList []).add 3)
+    |>.addTest "Unique elements can be added to a set -> add to non-empty set" (do
+        return assertEqual (CustomSet.Set.ofList [1, 2, 3, 4]) <| (CustomSet.Set.ofList [1, 2, 4]).add 3)
+    |>.addTest "Unique elements can be added to a set -> adding an existing element does not change the set" (do
+        return assertEqual (CustomSet.Set.ofList [1, 2, 3]) <| (CustomSet.Set.ofList [1, 2, 3]).add 3)
+    |>.addTest "Intersection returns a set of all shared elements -> intersection of two empty sets is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [] ∩ CustomSet.Set.ofList [])
+    |>.addTest "Intersection returns a set of all shared elements -> intersection of an empty set and non-empty set is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [] ∩ CustomSet.Set.ofList [3, 2, 5])
+    |>.addTest "Intersection returns a set of all shared elements -> intersection of a non-empty set and an empty set is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [1, 2, 3, 4] ∩ CustomSet.Set.ofList [])
+    |>.addTest "Intersection returns a set of all shared elements -> intersection of two sets with no shared elements is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [1, 2, 3] ∩ CustomSet.Set.ofList [4, 5, 6])
+    |>.addTest "Intersection returns a set of all shared elements -> intersection of two sets with shared elements is a set of the shared elements" (do
+        return assertEqual (CustomSet.Set.ofList [2, 3]) <| CustomSet.Set.ofList [1, 2, 3, 4] ∩ CustomSet.Set.ofList [3, 2, 5])
+    |>.addTest "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of two empty sets is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [] \ CustomSet.Set.ofList [])
+    |>.addTest "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of empty set and non-empty set is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [] \ CustomSet.Set.ofList [3, 2, 5])
+    |>.addTest "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of a non-empty set and an empty set is the non-empty set" (do
+        return assertEqual (CustomSet.Set.ofList [1, 2, 3, 4]) <| CustomSet.Set.ofList [1, 2, 3, 4] \ CustomSet.Set.ofList [])
+    |>.addTest "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference of two non-empty sets is a set of elements that are only in the first set" (do
+        return assertEqual (CustomSet.Set.ofList [1, 3]) <| CustomSet.Set.ofList [3, 2, 1] \ CustomSet.Set.ofList [2, 4])
+    |>.addTest "Difference (or Complement) of a set is a set of all elements that are only in the first set -> difference removes all duplicates in the first set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [1, 1] \ CustomSet.Set.ofList [1])
+    |>.addTest "Union returns a set of all elements in either set -> union of empty sets is an empty set" (do
+        return assertEqual (CustomSet.Set.ofList []) <| CustomSet.Set.ofList [] ∪ CustomSet.Set.ofList [])
+    |>.addTest "Union returns a set of all elements in either set -> union of an empty set and non-empty set is the non-empty set" (do
+        return assertEqual (CustomSet.Set.ofList [2]) <| CustomSet.Set.ofList [] ∪ CustomSet.Set.ofList [2])
+    |>.addTest "Union returns a set of all elements in either set -> union of a non-empty set and empty set is the non-empty set" (do
+        return assertEqual (CustomSet.Set.ofList [1, 3]) <| CustomSet.Set.ofList [1, 3] ∪ CustomSet.Set.ofList [])
+    |>.addTest "Union returns a set of all elements in either set -> union of non-empty sets contains all unique elements" (do
+        return assertEqual (CustomSet.Set.ofList [3, 2, 1]) <| CustomSet.Set.ofList [1, 3] ∪ CustomSet.Set.ofList [2, 3])
+
+def main : IO UInt32 := do
+  runTestSuitesWithExitCode [customSetTests]