Add builtin arithmetic operators

* also allow choosing evaluator
* ensures equivalence of evaluators up-to α-renaming
* checks server accepts answers from challenge

1 files changed, 158 insertions, 44 deletions

diff --git a/lambda-calcul/haskell/challenges.json b/lambda-calcul/haskell/challenges.json
index 119115b..75561c5 100644
--- a/lambda-calcul/haskell/challenges.json
+++ b/lambda-calcul/haskell/challenges.json
@@ -73,7 +73,7 @@
   },
   {
     "level": 12,
-    "description": "Church-encoded booleans: False = lam (f t) f, True = lam (f t) t. A boolean selects one of two arguments — False picks the first, True picks the second. Not flips a boolean by applying it to True and False in swapped order. What is (Not True)?",
+    "description": "Church-encoded booleans: False = lam (f t) f, True = lam (f t) t. A boolean selects one of two arguments \u2014 False picks the first, True picks the second. Not flips a boolean by applying it to True and False in swapped order. What is (Not True)?",
     "expr": "(let ((False (lam (f t) f))\n  (True (lam (f t) t))\n      (Not (lam (b) (b True False))))\n (Not True))",
     "answer": "(lam (f t) f)"
   },
@@ -91,7 +91,7 @@
   },
   {
     "level": 15,
-    "description": "Three def declarations with no final expression to evaluate. def statements produce no output on their own — they only extend the environment.",
+    "description": "Three def declarations with no final expression to evaluate. def statements produce no output on their own \u2014 they only extend the environment.",
     "expr": "(def False (lam (f t) f))\n  (def True (lam (f t) t))\n  (def Not (lam (b) (b True False)))",
     "answer": ""
   },
@@ -109,7 +109,7 @@
   },
   {
     "level": 18,
-    "description": "Integer literals are native values in this interpreter — they are already fully reduced and need no evaluation. What does the integer 42 reduce to?",
+    "description": "Integer literals are native values in this interpreter \u2014 they are already fully reduced and need no evaluation. What does the integer 42 reduce to?",
     "expr": "42",
     "answer": "42"
   },
@@ -145,13 +145,13 @@
   },
   {
     "level": 24,
-    "description": "An integer passed as argument to a free variable f produces a stuck neutral term — f has no definition, so (f 42) cannot reduce further. What does (f 42) reduce to?",
+    "description": "An integer passed as argument to a free variable f produces a stuck neutral term \u2014 f has no definition, so (f 42) cannot reduce further. What does (f 42) reduce to?",
     "expr": "(f 42)",
     "answer": "(f 42)"
   },
   {
     "level": 25,
-    "description": "An integer in function position applied to an argument cannot reduce — 42 is not a lambda. The result is a stuck neutral application. What does (42 x) reduce to?",
+    "description": "An integer in function position applied to an argument cannot reduce \u2014 42 is not a lambda. The result is a stuck neutral application. What does (42 x) reduce to?",
     "expr": "(42 x)",
     "answer": "(42 x)"
   },
@@ -169,200 +169,314 @@
   },
   {
     "level": 28,
+    "description": "The interpreter now has built-in arithmetic operators. + takes two integers and returns their sum. What does (+ 3 4) reduce to?",
+    "expr": "(+ 3 4)",
+    "answer": "7"
+  },
+  {
+    "level": 29,
+    "description": "The built-in - subtracts its second argument from its first. What does (- 10 3) reduce to?",
+    "expr": "(- 10 3)",
+    "answer": "7"
+  },
+  {
+    "level": 30,
+    "description": "The built-in * multiplies two integers. What does (* 6 7) reduce to?",
+    "expr": "(* 6 7)",
+    "answer": "42"
+  },
+  {
+    "level": 31,
+    "description": "div performs integer division (truncated toward negative infinity). What does (div 17 5) reduce to?",
+    "expr": "(div 17 5)",
+    "answer": "3"
+  },
+  {
+    "level": 32,
+    "description": "mod returns the remainder of integer division. What does (mod 17 5) reduce to?",
+    "expr": "(mod 17 5)",
+    "answer": "2"
+  },
+  {
+    "level": 33,
+    "description": "neg is a unary built-in that negates its argument. What does (neg 5) reduce to?",
+    "expr": "(neg 5)",
+    "answer": "-5"
+  },
+  {
+    "level": 34,
+    "description": "Built-ins compose freely. * and + can be nested just like lambda applications. What does (* 2 (+ 3 4)) reduce to?",
+    "expr": "(* 2 (+ 3 4))",
+    "answer": "14"
+  },
+  {
+    "level": 35,
+    "description": "Pythagorean-style expression: (+ (* 3 3) (* 4 4)) computes 9 + 16. What is the result?",
+    "expr": "(+ (* 3 3) (* 4 4))",
+    "answer": "25"
+  },
+  {
+    "level": 36,
+    "description": "= is a built-in equality test. It returns the Church boolean True (lam (f t) t) when both arguments are equal and False (lam (f t) f) otherwise. What does (= 42 42) reduce to?",
+    "expr": "(= 42 42)",
+    "answer": "(lam (f t) t)"
+  },
+  {
+    "level": 37,
+    "description": "What does (= 42 0) reduce to? The two integers are not equal, so = returns the Church boolean False.",
+    "expr": "(= 42 0)",
+    "answer": "(lam (f t) f)"
+  },
+  {
+    "level": 38,
+    "description": "< is a built-in less-than test. It returns the Church boolean True when the first argument is strictly less than the second, False otherwise. What does (< 3 5) reduce to?",
+    "expr": "(< 3 5)",
+    "answer": "(lam (f t) t)"
+  },
+  {
+    "level": 39,
+    "description": "What does (< 5 3) reduce to? 5 is not less than 3, so < returns the Church boolean False.",
+    "expr": "(< 5 3)",
+    "answer": "(lam (f t) f)"
+  },
+  {
+    "level": 40,
+    "description": "Built-ins support partial application just like lambdas. Applying + to only one argument produces a function waiting for its second argument. What does (+ 1) reduce to?",
+    "expr": "(+ 1)",
+    "answer": "(+ 1)"
+  },
+  {
+    "level": 41,
+    "description": "A partially applied built-in is a first-class value that can be passed to and invoked by a lambda. What does ((lam (f) (f 10)) (+ 5)) reduce to?",
+    "expr": "((lam (f) (f 10)) (+ 5))",
+    "answer": "15"
+  },
+  {
+    "level": 42,
+    "description": "When a built-in operator receives a free variable as one of its arguments it cannot reduce and stays as a stuck neutral term. What does (+ x 1) reduce to?",
+    "expr": "(+ x 1)",
+    "answer": "(+ x 1)"
+  },
+  {
+    "level": 43,
+    "description": "Division by zero cannot be reduced \u2014 it stays as a stuck neutral application. What does (div 7 0) reduce to?",
+    "expr": "(div 7 0)",
+    "answer": "(div 7 0)"
+  },
+  {
+    "level": 44,
+    "description": "Use def to bind a lambda that uses * to square its argument, then apply it. What does (square 9) produce in this program?",
+    "expr": "(def square (lam (x) (* x x)))\n(square 9)",
+    "answer": "81"
+  },
+  {
+    "level": 45,
+    "description": "Use let to bind two integers and compute the sum of their squares. What does (let ((x 3) (y 4)) (+ (* x x) (* y y))) reduce to?",
+    "expr": "(let ((x 3) (y 4)) (+ (* x x) (* y y)))",
+    "answer": "25"
+  },
+  {
+    "level": 46,
+    "description": "Built-in operators are first-class values: they can be passed as arguments to lambdas. Here a lambda takes an operator f and two arguments x y and applies (f x y). What does ((lam (f x y) (f x y)) + 10 20) reduce to?",
+    "expr": "((lam (f x y) (f x y)) + 10 20)",
+    "answer": "30"
+  },
+  {
+    "level": 47,
     "description": "Introduce And. And a b = a False b: if a is True it selects its second argument (b); if a is False it selects its first argument (False). Evaluate (And True True).",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Or (lam (a b) (a b True)))\n(def And (lam (a b) (a False b)))\n(And True True)",
     "answer": "(lam (f t) t)"
   },
   {
-    "level": 29,
+    "level": 48,
     "description": "Combine Or, And and Not. Evaluate (And (Or False True) (Not False)). Both sub-expressions must reduce to True for And to return True.",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def Or (lam (a b) (a b True)))\n(def And (lam (a b) (a False b)))\n(And (Or False True) (Not False))",
     "answer": "(lam (f t) t)"
   },
   {
-    "level": 30,
+    "level": 49,
     "description": "Xor (exclusive or) is True when exactly one argument is True. It can be defined as And (Or a b) (Not (And a b)). Evaluate (Xor True False).",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def Xor (lam (a b) (And (Or a b) (Not (And a b)))))\n(Xor True False)",
     "answer": "(lam (f t) t)"
   },
   {
-    "level": 31,
+    "level": 50,
     "description": "Evaluate (Xor False False). Two identical inputs to Xor should always yield False.",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def Xor (lam (a b) (And (Or a b) (Not (And a b)))))\n(Xor False False)",
     "answer": "(lam (f t) f)"
   },
   {
-    "level": 32,
-    "description": "Church-encoded if-then-else. Because our True selects its second argument and False selects its first, ite must swap the branches: ite c t e = c e t. Evaluate (ite True a b) — which branch is returned?",
+    "level": 51,
+    "description": "Church-encoded if-then-else. Because our True selects its second argument and False selects its first, ite must swap the branches: ite c t e = c e t. Evaluate (ite True a b) \u2014 which branch is returned?",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def ite (lam (c t e) (c e t)))\n(ite True a b)",
     "answer": "a"
   },
   {
-    "level": 33,
+    "level": 52,
     "description": "Evaluate (ite False a b). When the condition is False, the else-branch should be returned.",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def ite (lam (c t e) (c e t)))\n(ite False a b)",
     "answer": "b"
   },
   {
-    "level": 34,
+    "level": 53,
     "description": "Church-encoded pairs. pair x y = lam f -> f x y stores two values by passing them to a selector function. fst p = p (lam x y -> x) retrieves the first. Evaluate (fst (pair a b)).",
     "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(fst (pair a b))",
     "answer": "a"
   },
   {
-    "level": 35,
+    "level": 54,
     "description": "snd p = p (lam x y -> y) retrieves the second element of a Church pair. Evaluate (snd (pair a b)).",
     "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(snd (pair a b))",
     "answer": "b"
   },
   {
-    "level": 36,
+    "level": 55,
     "description": "Pairs can be nested. (pair a (pair b c)) is a pair whose second element is itself a pair. Evaluate (snd (snd (pair a (pair b c)))).",
     "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(snd (snd (pair a (pair b c))))",
     "answer": "c"
   },
   {
-    "level": 37,
+    "level": 56,
     "description": "swap exchanges the two elements of a pair: swap p = pair (snd p) (fst p). Evaluate (fst (swap (pair a b))). After swapping, fst should return what was previously snd.",
     "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(def swap (lam (p) (pair (snd p) (fst p))))\n(fst (swap (pair a b)))",
     "answer": "b"
   },
   {
-    "level": 38,
+    "level": 57,
     "description": "Church numerals encode natural numbers as higher-order functions: n applies its first argument (the successor step) to its second argument (the base) exactly n times. zero = lam s z -> z applies s zero times. Evaluate (zero f x).",
     "expr": "(def zero (lam (s z) z))\n(zero f x)",
     "answer": "x"
   },
   {
-    "level": 39,
-    "description": "two = lam s z -> s (s z) applies s twice. Evaluate (two f x) — you should see f applied to x twice.",
+    "level": 58,
+    "description": "two = lam s z -> s (s z) applies s twice. Evaluate (two f x) \u2014 you should see f applied to x twice.",
     "expr": "(def two (lam (s z) (s (s z))))\n(two f x)",
     "answer": "(f (f x))"
   },
   {
-    "level": 40,
+    "level": 59,
     "description": "three = lam s z -> s (s (s z)) applies s three times. Evaluate (three f x).",
     "expr": "(def three (lam (s z) (s (s (s z)))))\n(three f x)",
     "answer": "(f (f (f x)))"
   },
   {
-    "level": 41,
+    "level": 60,
     "description": "isZero tests whether a Church numeral is zero. isZero n = n (lam x -> False) True: if n > 0 the step function (lam x -> False) is applied at least once, discarding its argument and returning False; if n = 0 the base True is returned. Evaluate (isZero zero).",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def zero (lam (s z) z))\n(def isZero (lam (n) (n (lam (x) False) True)))\n(isZero zero)",
     "answer": "(lam (f t) t)"
   },
   {
-    "level": 42,
+    "level": 61,
     "description": "Evaluate (isZero three). Three is not zero, so the step function fires and the result is False.",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def three (lam (s z) (s (s (s z)))))\n(def isZero (lam (n) (n (lam (x) False) True)))\n(isZero three)",
     "answer": "(lam (f t) f)"
   },
   {
-    "level": 43,
+    "level": 62,
     "description": "The I combinator (identity): I x = x. It is the simplest combinator and the building block of the SKI calculus. Evaluate (I a).",
     "expr": "(def I (lam (x) x))\n(I a)",
     "answer": "a"
   },
   {
-    "level": 44,
+    "level": 63,
     "description": "The K combinator (constant): K x y = x. It ignores its second argument and always returns its first. Evaluate (K a b).",
     "expr": "(def K (lam (x y) x))\n(K a b)",
     "answer": "a"
   },
   {
-    "level": 45,
-    "description": "The S combinator (substitution): S f g x = f x (g x). It distributes its third argument over two functions and applies the results. Evaluate (S f g x) — f and g are free variables here.",
+    "level": 64,
+    "description": "The S combinator (substitution): S f g x = f x (g x). It distributes its third argument over two functions and applies the results. Evaluate (S f g x) \u2014 f and g are free variables here.",
     "expr": "(def S (lam (f g x) ((f x) (g x))))\n(S f g x)",
     "answer": "(f x (g x))"
   },
   {
-    "level": 46,
+    "level": 65,
     "description": "SKI completeness: S K K encodes the identity function. S K K x = K x (K x) = x. Evaluate (S K K a) and confirm it equals a.",
     "expr": "(def S (lam (f g x) ((f x) (g x))))\n(def K (lam (x y) x))\n(S K K a)",
     "answer": "a"
   },
   {
-    "level": 47,
-    "description": "The B combinator (composition): B f g x = f (g x). It is function composition. Evaluate (B f g x) — you should see f applied to the result of g applied to x.",
+    "level": 66,
+    "description": "The B combinator (composition): B f g x = f (g x). It is function composition. Evaluate (B f g x) \u2014 you should see f applied to the result of g applied to x.",
     "expr": "(def B (lam (f g x) (f (g x))))\n(B f g x)",
     "answer": "(f (g x))"
   },
   {
-    "level": 48,
+    "level": 67,
     "description": "The C combinator (flip): C f x y = f y x. It flips the order of the last two arguments of a function. Evaluate (C f a b).",
     "expr": "(def C (lam (f x y) (f y x)))\n(C f a b)",
     "answer": "(f b a)"
   },
   {
-    "level": 49,
+    "level": 68,
     "description": "The W combinator (duplication): W f x = f x x. It applies f to x twice, duplicating the argument. Evaluate (W f a).",
     "expr": "(def W (lam (f x) (f x x)))\n(W f a)",
     "answer": "(f a a)"
   },
   {
-    "level": 50,
+    "level": 69,
     "description": "Scott-encoded lists. nil = lam c n -> n (the empty list). cons h t = lam c n -> c h t stores head and tail directly. head l = l (lam h t -> h) nil extracts the first element. Evaluate (head (cons a nil)).",
     "expr": "(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def head (lam (l) (l (lam (h t) h) nil)))\n(head (cons a nil))",
     "answer": "a"
   },
   {
-    "level": 51,
-    "description": "tail l = l (lam h t -> t) nil retrieves the rest of a Scott-encoded list. Evaluate (head (tail (cons a (cons b nil)))) — the head of the tail of a two-element list is its second element.",
+    "level": 70,
+    "description": "tail l = l (lam h t -> t) nil retrieves the rest of a Scott-encoded list. Evaluate (head (tail (cons a (cons b nil)))) \u2014 the head of the tail of a two-element list is its second element.",
     "expr": "(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def head (lam (l) (l (lam (h t) h) nil)))\n(def tail (lam (l) (l (lam (h t) t) nil)))\n(head (tail (cons a (cons b nil))))",
     "answer": "b"
   },
   {
-    "level": 52,
+    "level": 71,
     "description": "isNil tests whether a Scott-encoded list is empty. isNil l = l (lam h t -> False) True: a non-empty list immediately calls its cons branch returning False; nil returns its base True. Evaluate (isNil nil).",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def isNil (lam (l) (l (lam (h t) False) True)))\n(isNil nil)",
     "answer": "(lam (f t) t)"
   },
   {
-    "level": 53,
+    "level": 72,
     "description": "Evaluate (isNil (cons a nil)). A non-empty list is not nil, so the result should be False.",
     "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def isNil (lam (l) (l (lam (h t) False) True)))\n(isNil (cons a nil))",
     "answer": "(lam (f t) f)"
   },
   {
-    "level": 54,
+    "level": 73,
     "description": "Church-encoded sum types (Either). Left x = lam l r -> l x tags a value as Left; Right x = lam l r -> r x tags it as Right. either f g e = e f g eliminates the sum by applying the matching handler. Evaluate (either f g (Left a)).",
     "expr": "(def Left (lam (x l r) (l x)))\n(def Right (lam (x l r) (r x)))\n(def either (lam (f g e) (e f g)))\n(either f g (Left a))",
     "answer": "(f a)"
   },
   {
-    "level": 55,
+    "level": 74,
     "description": "Evaluate (either f g (Right a)). The Right handler g should be applied to a.",
     "expr": "(def Left (lam (x l r) (l x)))\n(def Right (lam (x l r) (r x)))\n(def either (lam (f g e) (e f g)))\n(either f g (Right a))",
     "answer": "(g a)"
   },
   {
-    "level": 56,
+    "level": 75,
     "description": "Church-encoded option (Maybe). None = lam n s -> n is the absent case. Some x = lam n s -> s x wraps a value. fromOption default opt = opt default (lam x -> x) returns the wrapped value or the default. Evaluate (fromOption a None).",
     "expr": "(def None (lam (n s) n))\n(def Some (lam (x n s) (s x)))\n(def fromOption (lam (default opt) (opt default (lam (x) x))))\n(fromOption a None)",
     "answer": "a"
   },
   {
-    "level": 57,
+    "level": 76,
     "description": "Evaluate (fromOption a (Some b)). The option contains b, so the default a is ignored and b is returned.",
     "expr": "(def None (lam (n s) n))\n(def Some (lam (x n s) (s x)))\n(def fromOption (lam (default opt) (opt default (lam (x) x))))\n(fromOption a (Some b))",
     "answer": "b"
   },
   {
-    "level": 58,
+    "level": 77,
     "description": "mapOption applies a function inside an Option without unwrapping it. mapOption f opt = opt None (lam x -> Some (f x)): None maps to None; Some x maps to Some (f x). Evaluate (mapOption f (Some a)) and give its fully reduced form.",
     "expr": "(def None (lam (n s) n))\n(def Some (lam (x n s) (s x)))\n(def mapOption (lam (f opt) (opt None (lam (x) (Some (f x))))))\n(mapOption f (Some a))",
     "answer": "(lam (n s) (s (f a)))"
   },
   {
-    "level": 59,
-    "description": "The Kleene predecessor uses a pair-stepping trick to compute pred n = n - 1 (with pred zero = zero). Starting from (pair zero zero), each step shifts: (pair (snd p)) (succ (snd p)). After n steps fst of the pair is n-1. Evaluate (pred zero) — it should return zero.",
+    "level": 78,
+    "description": "The Kleene predecessor uses a pair-stepping trick to compute pred n = n - 1 (with pred zero = zero). Starting from (pair zero zero), each step shifts: (pair (snd p)) (succ (snd p)). After n steps fst of the pair is n-1. Evaluate (pred zero) \u2014 it should return zero.",
     "expr": "(def zero (lam (s z) z))\n(def one (lam (s z) (s z)))\n(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(def succ (lam (n s z) (s ((n s) z))))\n(def pred (lam (n) (fst (n (lam (p) (pair (snd p) (succ (snd p)))) (pair zero zero)))))\n(pred zero)",
     "answer": "(lam (s z) z)"
   },
   {
-    "level": 60,
+    "level": 79,
     "description": "Evaluate (pred one) using the same Kleene predecessor. After one pair-step starting from (pair zero zero) the fst is still zero. Confirm that the predecessor of one is zero.",
     "expr": "(def zero (lam (s z) z))\n(def one (lam (s z) (s z)))\n(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(def succ (lam (n s z) (s ((n s) z))))\n(def pred (lam (n) (fst (n (lam (p) (pair (snd p) (succ (snd p)))) (pair zero zero)))))\n(pred one)",
     "answer": "(lam (s z) z)"
   }
-]
+]
\ No newline at end of file