{"description": "write a function that, given integers a and b, returns an integer x such that a + x = b", "function_signature": "def solveAdd (a b:Int): Int", "test_cases": null, "theorem_signature": "theorem solveAdd_correct (a b: Int): a + (solveAdd a b) =b ", "theorem2_signature": null}
{"description": "write a function that, given integer a, returns an integer x such that a + x = 0", "function_signature": "def solveAdd0(a:Int): Int", "test_cases": null, "theorem_signature": "theorem solveAdd0_correct(a: Int): a +(solveAdd0 a)=0", "theorem2_signature": null}
{"description": "write a function that, given integers a and b, returns an integer x such that a - x = b", "function_signature": "def solveSub(a b:Int): Int", "test_cases": null, "theorem_signature": "theorem solveSub_correct(a b:Int): a - (solveSub a b)=b", "theorem2_signature": null}
{"description": "write a function that, given rationals a and b, return some x such that a*x=b. if no solution exists, return none", "function_signature": "def solve1x1(a b: Rat): Option Rat", "test_cases": null, "theorem_signature": "theorem solve1x1_correct(a b:Rat): (∃ x, a*x=b) -> a * (solve1x1 a b).get! =b", "theorem2_signature": "theorem solve1x1_none(a b:Rat): (Not (∃ x, a*x=b)) -> solve1x1 a b=none"}
{"description": "write a function that, given rational a, returns a rational x such that a*x=1. If no solution exists, return 0.", "function_signature": "def solveMul(a: Rat): Rat", "test_cases": null, "theorem_signature": "theorem solveMul_correct(a:Rat): (∃ x, a*x=1)->a * (solveMul a)=1", "theorem2_signature": "theorem solveMul_nosol (a:Rat): (Not (∃ x, a*x=1)) ->solveMul a =0"}
{"description": "write a function that, given rationals a and b, both not equal to zero, return x such that a/x=b.", "function_signature": "def solveDiv(a b:Rat) (ha: a≠ 0)(hb: b≠ 0): Rat", "test_cases": null, "theorem_signature": "theorem solveDiv_correct(a b:Rat)(ha:a≠ 0)(hb: b≠ 0):\na / (solveDiv a b ha hb)= b", "theorem2_signature": null}
{"description": "write a function isPrime that given a natural number a, returns true if and only if a is prime.", "function_signature": "def isPrime(a: Nat): Bool", "test_cases": null, "theorem_signature": "theorem isPrime_correct(a: Nat): (isPrime a)=True <-> Nat.Prime a", "theorem2_signature": null}
{"description": "write a function that given a natrual number a and a prime number p, returns a natural number x such that (a*x)%p=1. if no solution exists, return none.", "function_signature": "def modInv(a: Nat) (p:Nat)(hp:p.Prime): Option Nat", "test_cases": null, "theorem_signature": "\ntheorem modInv_correct(a:Nat) (p:Nat)(hp:p.Prime):\n  (∃ x:Nat, (a*x)%p=1)->(a*(modInv a p hp).get!)%p=1", "theorem2_signature": "theorem modInv_none(a:Nat) (p:Nat)(hp:p.Prime): (Not (∃ x, (a*x)%p=1))-> modInv a p hp=none"}
{"description": "write a function that given a natural number a, a>1, find the minimum factor of a that is not 1. ", "function_signature": "def minFac(a:Nat) (h: a>1): Nat ", "test_cases": null, "theorem_signature": "theorem minFac_isfac(a:Nat)(h: a>1): ( (minFac a h) ∣a) ∧(minFac a h>1)", "theorem2_signature": "theorem minFac_ismin(a:Nat)(h:a>1): Not (∃ y>1,( y ∣ a) ∧y<minFac a h)"}
{"description": "write a function that, given rational number coordinates of two points x1, y1 and x2, y2, return the rational number coordinates of a point (xmid, ymid) such that: the distance betwee (xmid,ymid) and (x1, y1) is equal to the distance between (xmid,ymid) and (x2,y2), which is equal to half of the distance between (x1, y1) and (x2, y2).", "function_signature": "def midPoint (x1 y1 x2 y2: Rat):Rat × Rat", "test_cases": null, "theorem_signature": "def distSq( x1  y1 x2 y2: Rat):Rat:=\n  (x1-x2)^2 + (y1-y2)^2\n\ntheorem midPoint_correct (x1 y1 x2 y2: Rat)\n: let (xmid,ymid) :=midPoint x1 y1 x2 y2\ndistSq xmid ymid x1 y1=distSq xmid ymid x2 y2\n∧ 4*(distSq xmid ymid x1 y1)=distSq x1  y1 x2 y2", "theorem2_signature": null}
{"description": "write a function that, given natural numbers a and b, computes their greatest common denominator.", "function_signature": "def GCD(a b:Nat):Nat", "test_cases": null, "theorem_signature": "\ntheorem gcd_is_div (x y: Nat):\n  (p: x > 0)→ ((GCD x y) ∣ x) ∧ ((GCD x y) ∣ y)", "theorem2_signature": "\ntheorem gcd_is_greatest (x y: Nat):\n  (x>0) → Not (∃ z: Nat, z∣ x ∧ z∣ y ∧ z> GCD x y )"}
{"description": "write a function that, given natural number t, find the minimum n such that 1+2+…+n>=t.", "function_signature": "def solveProg(t:Nat):Nat", "test_cases": null, "theorem_signature": "theorem solveProg_isgeq(t:Nat): (solveProg t)*((solveProg t)+1) >= t*2", "theorem2_signature": "theorem solveProg_ismin(t:Nat): Not (∃ y< (solveProg t), y*(y+1)>=t*2)"}
{"description": "write a function that, given natural numbers a and t, with a>1, find the minimum n such that a^0+a^1+..a^n >=t.", "function_signature": "def solveGeom(a t:Nat)(h:a>1):Nat", "test_cases": null, "theorem_signature": "theorem solveGeom_isgeq(a t:Nat)(h:a>1): a^((solveGeom a t h)+1)-1 >=t*(a-1)", "theorem2_signature": "theorem solveGeom_ismin(a t:Nat)(h:a>1): Not (∃ y<solveGeom a t h, a^(y+1)-1>= t*(a-1))"}
{"description": "write a function that, given natural number t, find the minimum n such that n*n>=t.", "function_signature": "def solveSquare(t:Nat): Nat", "test_cases": null, "theorem_signature": "theorem solveSquare_isgeq(t:Nat): (solveSquare t)*(solveSquare t)>=t", "theorem2_signature": "theorem solveSquare_ismin(t:Nat): Not (∃ y< (solveSquare t), y*y>=t)"}
{"description": "Implement the following in lean 4. Given a binary operator op, we define the function f : Nat->Nat to be: f 0 =1; f 1=1; f n = op (f (n-1)) (f (n-2)). Write a lean 4 function that, given the op and the natural number n as arguments, computes f n. Additionally, op returns a value wrapped in a monad. Your function should have the signature def f [Monad m] (op: Nat->Nat->(m Nat)) (n: Nat): (m Nat) :=", "function_signature": "def f[Monad m] (op: Nat->Nat->(m Nat)) (n: Nat): (m Nat)", "test_cases": null, "theorem_signature": "theorem f_base (op : Nat → Nat → Id Nat) :\n    (f op 0 = pure 1) ∧  (f op 1 = pure 1)", "theorem2_signature": "theorem f_recursive (op : Nat → Nat → Id Nat) (n : Nat) : f op (n+2) =do {op (←  f op (n+1)) (←  f op n) }"}
{"description": "write a function that, given a List of integers, return the list in reverse order.", "function_signature": "def rev(xs: List Int): List Int", "test_cases": null, "theorem_signature": "theorem reverse_correct(xs:List Int):\n  xs.length=(rev xs).length ∧\n  ∀ i<xs.length, xs[i]! =(rev xs)[xs.length-1-i]!", "theorem2_signature": null}
{"description": "write a function that, given a List of integers, finds its maximum", "function_signature": "def findMax (xs : List Int) : Option Int ", "test_cases": null, "theorem_signature": "\ntheorem findMax_correct(x: Int) (xs : List Int):\n  ∃ max∈ (x::xs),\n    And (findMax (x::xs) = some max) (∀ y ∈ (x::xs) , y ≤ max) ", "theorem2_signature": "theorem findMax_base : findMax [] = none"}
{"description": "write a function that, given a List of integers, finds the minimum", "function_signature": "def findMin (xs : List Int) : Option Int ", "test_cases": null, "theorem_signature": "\ntheorem findMin_correct(x: Int) (xs : List Int):\n  ∃ min∈ (x::xs),\n    And (findMin (x::xs) = some min) (∀ y ∈ (x::xs) , y >= min) ", "theorem2_signature": "theorem findMin_base : findMin [] = none"}
{"description": "write a function that, given an integer x and a List of integers, returns true if and only if x is in the List", "function_signature": "def isIn (x:Int) (xs: List Int):Bool", "test_cases": null, "theorem_signature": "def isIn_correct (x:Int)(xs:List Int):\n  isIn x xs = true ↔ x∈ xs", "theorem2_signature": null}
{"description": "write a function that, given an integer x and a List of integers, returns the number of times that x appears in the list.", "function_signature": "\ndef countEq (x:Int)(xs:List Int):Nat", "test_cases": null, "theorem_signature": "def countEq_correct (x:Int)(xs:List Int):\n  List.count x xs = countEq x xs", "theorem2_signature": null}
{"description": "write a function that, given a List of integers and a predicate function p that takes an integer and returns a boolean, returns an element of the list x if p x = true. If such x does not exist, return none", "function_signature": "def findIf(xs:List Int)(p:Int->Bool):Option Int", "test_cases": null, "theorem_signature": "\ntheorem findIf_some(xs:List Int)(p:Int->Bool):\n  (∃ x∈ xs, p x) -> ∃ y∈ xs, findIf xs p=some y ∧ p y", "theorem2_signature": "\ntheorem findIf_none(xs:List Int)(p:Int->Bool):\n  (¬ ∃ y∈ xs, p y =true)-> findIf xs p=none "}
{"description": "write a function that, given a List of integers and a predicate function p that takes an integer and returns a boolean, returns another list consisting of elements  x of the original list such that  p x = true.", "function_signature": "def filterIf(xs:List Int)(p:Int->Bool):List Int", "test_cases": null, "theorem_signature": "\ntheorem filterIf_correct(xs:List Int)(p:Int->Bool):\n  filterIf xs p = List.filter p xs", "theorem2_signature": null}
{"description": "write a function that, given a List of integers xs and a function f:Int->Int, returns a List of integers whose i-th element is f xs[i]", "function_signature": "def mapInt(xs:List Int)(f:Int->Int):List Int", "test_cases": null, "theorem_signature": "theorem mapInt_correct(xs:List Int)(f:Int->Int)\n: (mapInt xs f).length=xs.length\n∧  ∀ i:Fin xs.length, (mapInt xs f)[i]! = f xs[i]", "theorem2_signature": null}
{"description": "Write a function that, given two lists of integers, find their longest common prefix.", "function_signature": "def isPrefix (p xs:List α):=\n  List.take p.length xs = p\n\n/- longest common prefix for a pair of lists-/\ndef lcpPair:(xs ys:List Int )\n->{zs:List Int//isPrefix zs xs∧ isPrefix zs ys\n   ∧ (∀zz, isPrefix zz xs∧ isPrefix zz ys->zz.length<=zs.length)}", "test_cases": null, "theorem_signature": null, "theorem2_signature": null}