X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=code%2Flambda.js;h=50880fc6585648d5b16c4173e53aba0f97105120;hp=c28a0c17223e34e874dde5472a63c19450168125;hb=e6c00ad32d58107e9861e2d121db7e03e626e184;hpb=ef0c9e389a02522583df53477f5a4622debc4a1b;ds=sidebyside

diff --git a/code/lambda.js b/code/lambda.js
index c28a0c17..50880fc6 100644
--- a/code/lambda.js
+++ b/code/lambda.js
@@ -1,172 +1,661 @@
-// Basic data structure, essentially a LISP/Scheme-like cons
-// pre-terminal nodes are expected to be of the form new cons(null, "string")
-function cons(car, cdr) {
-  this.car = car;
-  this.cdr = cdr;
-}
-
-// takes a stack of symbols, returns a pair: a tree and the remaining symbols
-function fold(split) {
-  if (split == null) return (new cons (null, null));
-  if (split.length == 0) return (new cons (null, null));
-  var token = split.shift();
-  if (token == ")") return (new cons (null, split));
-  var next = fold(split);
-  if (token == "(") {
-    var nextnext = fold(next.cdr);
-    return (new cons ((new cons (next.car, nextnext.car)),
-                      nextnext.cdr));
-  }
-  return (new cons ((new cons ((new cons (null, token)),
-                               next.car)),
-                    next.cdr))
-}
-
-// substitute arg in for v in tree
-function sub(tree, v, arg) {
-  if (tree == null) return (null);
-  if (tree.car == null) if (tree.cdr == v) return (arg);
-  if (tree.car == null) return (tree);
-
-  // perform alpha reduction to prevent variable collision
-  if (isBindingForm(tree)) 
-    return (new cons (tree.car, 
-                      sub(sub(tree.cdr,         // inner sub = alpha reduc.
-                              tree.cdr.car.cdr, 
-                              fresh(tree.cdr.car.cdr)),
-                          v,
-                          arg)));
-
-  return (new cons ((sub (tree.car, v, arg)),
-                    (sub (tree.cdr, v, arg))))
-}
-
-// Guaranteed unique for each call as long as string is not empty.
-var i = 0;
-function fresh(string) {
-  i = i+1;
-  if (typeof(string) != "string") return (string);
-  return (new cons (null,  
-                    string.substring(0,1) + (i).toString()));
-}
-
-// Keep reducing until there is no more change
-function fixedPoint (tree) {
-  var t2 = reduce(tree);
-  if (treeToString(tree) == treeToString(t2)) return (tree);
-  return (fixedPoint (t2));
-}
-
-// Reduce all the arguments, then try to do beta conversion on the whole
-function reduce(tree) {
-  if (tree == null) return (tree);
-  if (typeof(tree) == "string") return (tree);
-  return (convert (new cons (reduce (tree.car), mapReduce (tree.cdr))));
-}
-
-// Reduce all the arguments in a list
-function mapReduce(tree) {
-  if (tree == null) return (tree);
-  if (tree.car == null) return (tree);
-  return (new cons (reduce (tree.car), mapReduce(tree.cdr )));
-}
-
-// If the list is of the form ((lambda var body) arg), do beta reduc.
-function convert(tree) {
-    if (isLet(tree)) {
-      return (sub(tree.cdr.car, tree.car.cdr.car.cdr, tree.car.cdr.cdr.car));}
-    else 
-      if (isConvertable(tree)) {
-        return (sub(tree.car.cdr.cdr.car, tree.car.cdr.car.cdr, tree.cdr.car));}
-      else return(tree);
-} 
-
-// Is of form ((let var arg) body)?
-function isLet(tree) {
-  if (tree == null) return (false);
-  if (!(isBindingForm(tree.car))) return (false);
-  if (tree.car.car.cdr != "let") return (false);
-  if (tree.cdr == null) return (false);
-  if (tree.cdr.car == null) return (false);
-  return(true);
-}  
-
-// Is of form ((lambda var body) arg)?
-function isConvertable(tree) {
-  if (tree == null) return (false);
-  if (!(isBindingForm(tree.car))) return (false);
-  if (tree.car.car.cdr != "lambda") return (false);
-  if (tree.cdr == null) return (false);
-  if (tree.cdr.car == null) return (false);
-  return(true);
-}  
-
-// Is of form (lambda var body)?
-function isBindingForm(tree) {
-  if (tree == null) return (false);
-  if (tree.car == null) return (false);
-  if (tree.car.car != null) return (false);
-  if ((tree.car.cdr != "lambda") 
-      && (tree.car.cdr != "let")
-      && (tree.car.cdr != "exists")
-      && (tree.car.cdr != "forall")
-      && (tree.car.cdr != "\u03BB")
-      && (tree.car.cdr != "\u2200")
-      && (tree.car.cdr != "\u2203")
-     )
-    return (false);
-  if (tree.car.cdr == null) return (false);
-  if (tree.cdr.car == null) return (false);
-  if (tree.cdr.car.car != null) return (false);
-  if (tree.cdr.cdr == null) return (false);
-  return (true);
-}
-
-function treeToString(tree) {
-  if (tree == null) return ("")
-  if (tree.car == null) return (tree.cdr)
-  if ((tree.car).car == null) 
-    return (treeToString(tree.car) + " " + treeToString(tree.cdr))
-  return ("(" + treeToString(tree.car) + ")" + treeToString(tree.cdr))
-}
-
-// use this instead of treeToString if you want to see the full structure
-function treeToStringRaw(tree) {
-  if (tree == null) return ("@")
-  if (typeof(tree) == "string") return (tree);
-  return ("(" + treeToStringRaw(tree.car) + "." + 
-                treeToStringRaw(tree.cdr) + ")")
-}
-
-// Make sure each paren will count as a separate token
-function stringToTree(input) {
-  input = input.replace(/let/g, " ( ( let ");
-  input = input.replace(/=/g, " ");
-  input = input.replace(/in/g, " ) ");
-  input = input.replace(/\(/g, " ( ");
-  input = input.replace(/\)/g, " ) ");
-  input = input.replace(/;.*\n/g," ");
-  input = input.replace(/\^/g, " ^ ");
-  input = input.replace(/[\\]/g, " lambda ");
-  input = input.replace(/\u03BB/g, "lambda");
-  return ((fold(input.split(/[ \f\n\r\t\v]+/))).car)
-}
-
-// Adjust spaces to print pretty
-function formatTree(tree) {
-  output = treeToString (tree);
-  output = output.replace(/^[ \f\n\r\t\v]+/, "");
-  output = output.replace(/[ \f\n\r\t\v]+$/, "");
-  output = output.replace(/[ \f\n\r\t\v]+\)/g, ")");
-  output = output.replace(/\)([^)(])/g, ") $1");
-  output = output.replace(/lambda/g, "\\");
-//  output = output.replace(/lambda/g, "\u03BB");
-//  output = output.replace(/exists/g, "\u2203");
-//  output = output.replace(/forall/g, "\u2200");
-  return (output)
-}
-
-function mytry(form) { 
-  i = 0;
-  form.result.value = formatTree(fixedPoint(stringToTree(form.input.value)));
+/*jslint bitwise: true,
+    eqeqeq: true,
+    immed: true,
+    newcap: true,
+    nomen: true,
+    onevar: true,
+    plusplus: true,
+    regexp: true,
+    rhino: true,
+    browser: false,
+    undef: true,
+    white: true,
+
+    evil: false,
+    regexp: false,
+    sub: true,
+    laxbreak: true,
+    onevar: false,
+    debug: true */
+
+
+// DeBruijn terms
+// substitution and translation algorithms from Chris Hankin, An Introduction to Lambda Calculi for Comptuer Scientists
+//
+function Db_free(variable) {
+    this.variable = variable;
+    this.subst = function (m, new_term) {
+        return this;
+    };
+    this.renumber = function (m, i) {
+        return this;
+    };
+    // we assume that other will have variable iff it's a Db_free
+    this.equal = function (other) {
+        return other.variable && this.variable.equal(other.variable);
+    };
+    this.contains = this.equal;
+}
+
+function Db_index(i) {
+    this.index = i;
+    this.subst = function (m, new_term) {
+        if (this.index < m) {
+            return this;
+        } else if (this.index > m) {
+            return new Db_index(this.index - 1);
+        } else {
+            return new_term.renumber(this.index, 1);
+        }
+    };
+    this.renumber = function (m, i) {
+        if (this.index < i) {
+            return this;
+        } else {
+            return new Db_index(this.index + m - 1);
+        }
+    };
+    // we assume that other will have index iff it's a Db_index
+    this.equal = function (other) {
+        return this.index === other.index;
+    };
+    this.contains = this.equal;
+}
+
+function Db_app(left, right) {
+    this.left = left;
+    this.right = right;
+    this.subst = function (m, new_term) {
+        return new Db_app(this.left.subst(m, new_term), this.right.subst(m, new_term));
+    };
+    this.renumber = function (m, i) {
+        return new Db_app(this.left.renumber(m, i), this.right.renumber(m, i));
+    };
+    // we assume that other will have left iff it's a Db_app
+    this.equal = function (other) {
+        return other.left && this.left.equal(other.left) && this.right.equal(other.right);
+    };
+    this.contains = function (other) {
+        if (other.left && this.left.equal(other.left) && this.right.equal(other.right)) {
+            return true;
+        } else {
+            return this.left.contains(other) || this.right.contains(other);
+        }
+    };
+}
+
+function Db_lam(body) {
+    this.body = body;
+    this.subst = function (m, new_term) {
+        return new Db_lam(this.body.subst(m + 1, new_term));
+    };
+    this.renumber = function (m, i) {
+        return new Db_lam(this.body.renumber(m, i + 1));
+    };
+    // we assume that other will have body iff it's a Db_lam
+    this.equal = function (other) {
+        return other.body && this.body.equal(other.body);
+    };
+    this.contains = function (other) {
+        if (other.body && this.body.equal(other.body)) {
+            return true;
+        } else {
+            return this.body.contains(other);
+        }
+    };
+}
+
+
+// lambda terms
+// substitution and normal-order evaluator based on Haskell version by Oleg Kisleyov
+// http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell
+//
+function Variable(name, tag) {
+    this.name = name;
+    this.tag = tag || 0;
+    this.to_string = function () {
+        // append count copies of str to accum
+//         function markup(accum, count) {
+//             if (count === 0) {
+//                 return accum;
+//             } else {
+//                 return markup(accum + "'", count - 1);
+//             }
+//         }
+//         return markup(this.name, this.tag);
+		var s = this.name;
+		for (var count = 0; count < this.tag; count += 1) {
+			s += "'";
+		}
+		return s;
+    };
+    this.equal = function (other) {
+        return (this.tag === other.tag) && (this.name === other.name);
+    };
+    // position of this in seq
+    this.position = function (seq) {
+        for (var i = 0; i < seq.length; i += 1) {
+            if (this.equal(seq[i])) {
+                return new Db_index(i + 1);
+            }
+        }
+        return new Db_free(this);
+    };
+}
+
+// if v occurs free_in term, returns Some v' where v' is the highest-tagged
+// variable with the same name as v occurring (free or bound) in term
+//
+function free_in(v, term) {
+    var res = term.has_free(v);
+    return res[0] && res[1];
+}
+
+function subst(v, new_term, expr) {
+    if (new_term.variable && new_term.variable.equal(v)) {
+        return expr;
+    } else {
+        return expr.subst(v, new_term);
+    }
 }
+
+function equal(expr1, expr2) {
+    return expr1.debruijn([]).equal(expr2.debruijn([]));
+}
+
+function contains(expr1, expr2) {
+    return expr1.debruijn([]).contains(expr2.debruijn([]));
+}
+
+
+function Lambda_var(variable) {
+    this.variable = variable;
+    this.debruijn = function (seq) {
+        return this.variable.position(seq);
+    };
+    this.to_string = function (as_atom) {
+        return this.variable.to_string();
+    };
+    this.has_free = function (v) {
+        if (v.name !== this.variable.name) {
+            return [false, v];
+        } else if (v.tag === this.variable.tag) {
+            return [true, v];
+        } else {
+            return [false, this.variable];
+        }
+    };
+    this.subst = function (v, new_term) {
+        if (this.variable.equal(v)) {
+            return new_term;
+        } else {
+            return this;
+        }
+    };
+    this.check_eta = function () {
+        return this;
+    };
+    this.eval_loop = function (stack, eta) {
+        function unwind(left, stack) {
+//             if (stack.length === 0) {
+//                 return left;
+//             } else {
+//                 var x = stack[0];
+//                 var xs = stack.slice(1);
+//                 return unwind(new Lambda_app(left, x.eval_loop([], eta)), xs);
+//             }
+			var res = left, x;
+			while (stack.length) {
+				x = stack.shift();
+				// res = new Lambda_app(res, x.eval_loop([], eta));
+				res = new Lambda_app(res, reduce(x, eta, false));
+			}
+			return res;
+        }
+        // return unwind(this, stack);
+		// trampoline to, args
+		return [null, unwind(this, stack)];
+    };
+    this.eval_cbv = function (aggressive) {
+        return this;
+    };
+}
+
+function Lambda_app(left, right) {
+    this.left = left;
+    this.right = right;
+    this.debruijn = function (seq) {
+        return new Db_app(this.left.debruijn(seq), this.right.debruijn(seq));
+    };
+    this.to_string = function (as_atom) {
+        var base;
+        if (this.left.left) {
+            base = this.left.to_string() + " " + this.right.to_string(true);
+        } else {
+            base = this.left.to_string(true) + " " + this.right.to_string(true);
+        }
+        if (as_atom) {
+            return "(" + base + ")";
+        } else {
+            return base;
+        }
+    };
+    this.has_free = function (v) {
+        var left_res = this.left.has_free(v);
+        var right_res = this.right.has_free(v);
+        var left_bool = left_res[0];
+        var right_bool = right_res[0];
+        var left_tag = left_res[1].tag;
+        var right_tag = right_res[1].tag;
+        var res;
+        if (left_tag > right_tag) {
+            res = left_res[1];
+        } else {
+            res = right_res[1];
+        }
+        return [left_bool || right_bool, res];
+    };
+    this.subst = function (v, new_term) {
+        return new Lambda_app(subst(v, new_term, this.left), subst(v, new_term, this.right));
+    };
+    this.check_eta = function () {
+        return this;
+    };
+    this.eval_loop = function (stack, eta) {
+        var new_stack = stack.slice(0);
+        new_stack.unshift(this.right);
+        // return this.left.eval_loop(new_stack, eta);
+		// trampoline to, args
+		return [this.left, new_stack, eta];
+    };
+    this.eval_cbv = function (aggressive) {
+        var left = this.left.eval_cbv(aggressive);
+        var right = this.right.eval_cbv(aggressive);
+        if (left.body) {
+            return subst(left.bound, right, left.body).eval_cbv(aggressive);
+        } else {
+            return new Lambda_app(left, right);
+        }
+    };
+}
+
+
+//     (* if v occurs free_in term, returns Some v' where v' is the highest-tagged
+//      * variable with the same name as v occurring (free or bound) in term *)
+
+
+function Lambda_lam(variable, body) {
+    this.bound = variable;
+    this.body = body;
+    this.debruijn = function (seq) {
+        var new_seq = seq.slice(0);
+        new_seq.unshift(this.bound);
+        return new Db_lam(this.body.debruijn(new_seq));
+    };
+    this.to_string = function (as_atom) {
+        var base = "\\" + this.to_string_funct();
+        if (as_atom) {
+            return "(" + base + ")";
+        } else {
+            return base;
+        }
+    };
+    this.to_string_funct = function () {
+        if (this.body.to_string_funct) {
+            return this.bound.to_string() + " " + this.body.to_string_funct();
+        } else {
+            return this.bound.to_string() + ". " + this.body.to_string();
+        }
+    };
+    this.has_free = function (v) {
+        if (this.bound.equal(v)) {
+            return [false, v];
+        } else {
+            return this.body.has_free(v);
+        }
+    };
+    this.subst = function (v, new_term) {
+        function bump_tag(v1, v2) {
+            var max;
+            if (v1.tag > v2.tag) {
+                max = v1.tag;
+            } else {
+                max = v2.tag;
+            }
+            return new Variable(v1.name, max + 1);
+        }
+        function bump_tag2(v1, v2) {
+            if (v1.name !== v2.name) {
+                return v1;
+            } else {
+                return bump_tag(v1, v2);
+            }
+        }
+        if (this.bound.equal(v)) {
+            return this;
+        } else {
+            var res = free_in(this.bound, new_term);
+            // if x is free in the inserted term new_term, a capture is possible
+            if (res) {
+                // this.bound is free in new_term, need to alpha-convert
+                var uniq_x = bump_tag2(bump_tag(this.bound, res), v);
+                var res2 = free_in(uniq_x, this.body);
+                if (res2) {
+                    uniq_x = bump_tag(uniq_x, res2);
+                }
+                var body2 = subst(this.bound, new Lambda_var(uniq_x), this.body);
+                return new Lambda_lam(uniq_x, subst(v, new_term, body2));
+            } else {
+                // this.bound not free in new_term, can substitute new_term for v without any captures
+                return new Lambda_lam(this.bound, subst(v, new_term, this.body));
+            }
+        }
+    };
+    this.check_eta = function () {
+        if (this.body.right && this.body.right.variable && this.bound.equal(this.body.right.variable) && !free_in(this.bound, this.body.left)) {
+            return this.body.left;
+        } else {
+            return this;
+        }
+    };
+    this.eval_loop = function (stack, eta) {
+        if (stack.length === 0) {
+			// var term = new Lambda_lam(this.bound, this.body.eval_loop([], eta));
+			var term = new Lambda_lam(this.bound, reduce(this.body, eta, false));
+			if (eta) {
+				return [null, term.check_eta()];
+			} else {
+				return [null, term];
+			}
+        } else {
+            var x = stack[0];
+            var xs = stack.slice(1);
+            // return subst(this.bound, x, this.body).eval_loop(xs, eta);
+			// trampoline to, args
+			return [subst(this.bound, x, this.body), xs, eta];
+        }
+    };
+    this.eval_cbv = function (aggressive) {
+        if (aggressive) {
+            return new Lambda_lam(this.bound, this.body.eval_cbv(aggressive));
+        } else {
+            return this;
+        }
+    };
+    this.to_int = function (sofar) {
+//         if (this.body.body && this.body.body.variable && this.body.bound.equal(this.body.body.variable)) {
+//             return 0 + sofar;
+//         } else if (this.body.variable && this.bound.equal(this.body.variable)) {
+//             return 1 + sofar;
+//         } else if (this.body.body && this.body.body.left && this.body.body.left.variable && this.bound.equal(this.body.body.left.variable)) {
+//             var new_int = new Lambda_lam(this.bound, new Lambda_lam(this.body.bound, this.body.body.right));
+//             return new_int.to_int(1 + sofar);
+//         } else {
+//             return "not a church numeral";
+//         }
+		var res = 0, s = this.bound, z, cursor;
+		if (this.body.variable && s.equal(this.body.variable)) {
+			return 1;
+		} else if (this.body.bound) {
+			z = this.body.bound;
+			cursor = this.body.body;
+			while (cursor.left && cursor.left.variable && s.equal(cursor.left.variable)) {
+				res += 1;
+				cursor = cursor.right;
+			}
+			if (cursor.variable && z.equal(cursor.variable)) {
+				return res;
+			}
+		}
+		return "not a church numeral";
+    };
+}
+
+
+
+///////////////////////////////////////////////////////////////////////////////////
+
+// cbv is false: use call-by-name
+// cbv is 1: use call-by-value, don't descend inside lambda
+// cbv is 2: applicative order
+function reduce(expr, eta, cbv) {
+    if (cbv) {
+        return expr.eval_cbv(cbv > 1);
+    } else {
+        // return expr.eval_loop([], eta);
+		var to_eval = expr, res = [[], eta];
+		while (to_eval !== null) {
+			res = to_eval.eval_loop.apply(to_eval, res);
+			to_eval = res.shift();
+		}
+		return res[0];
+    }
+}
+
+function make_var(name) {
+    return new Variable(name);
+}
+function make_app(aa, bb) {
+    return new Lambda_app(aa, bb);
+}
+function make_lam(a, aa) {
+    return new Lambda_lam(a, aa);
+}
+
+try {
+    if (console && console.debug) {
+        function print() {
+            console.debug.apply(this, arguments);
+        }
+    }
+} catch (e) {}
+
+/*
+let true = K in
+let false = \x y. y in
+let and = \l r. l r false in
+let or = \l r. l true r in
+let pair = \u v f. f u v in
+let triple = \u v w f. f u v w in
+let succ = \n s z. s (n s z) in
+let pred = \n s z. n (\u v. v (u s)) (K z) I in
+let ifzero = \n. n (\u v. v (u succ)) (K 0) (\n withp whenz. withp n) in
+let add = \m n. n succ m in
+let mul = \m n. n (\z. add m z) 0 in
+let mul = \m n s. m (n s) in
+let sub = (\mzero msucc mtail. \m n. n mtail (m msucc mzero) true) (pair 0 I) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let min = \m n. sub m (sub m n) in
+let max = \m n. add n (sub m n) in
+let lt = (\mzero msucc mtail. \n m. n mtail (m msucc mzero) true (\x. true) false) (pair 0 I) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let leq = (\mzero msucc mtail. \m n. n mtail (m msucc mzero) true (\x. false) true) (pair 0 I) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let eq = (\mzero msucc mtail. \m n. n mtail (m msucc mzero) true (\x. false) true) (pair 0 (K (pair 1 I))) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let divmod = (\mzero msucc mtail. \n divisor.
+               (\dhead. n (mtail dhead) (\sel. dhead (sel 0 0)))
+              (divisor msucc mzero (\a b c. c x))
+              (\d m a b c. pair d m) )
+          (triple succ (K 0) I)
+          (\d. triple I succ (K d))
+          (\dhead d. d (\dz mz df mf drest sel. drest dhead (sel (df dz) (mf mz)))) in
+let div = \n d. divmod n d true in
+let mod = \n d. divmod n d false in
+let Y = \f. (\y. f(y y)) (\y. f(y y)) in
+let Z = (\u f. f(u u f)) (\u f. f(u u f)) in
+let fact = \y. y (\f n. ifzero n (\p. mul n (f p)) 1) in
+fact Z 3
+*/
+
+
+
+// // Basic data structure, essentially a LISP/Scheme-like cons
+// // pre-terminal nodes are expected to be of the form new cons(null, "string")
+// function cons(car, cdr) {
+//   this.car = car;
+//   this.cdr = cdr;
+// }
+
+// // takes a stack of symbols, returns a pair: a tree and the remaining symbols
+// function parse(split) {
+//   if (split == null) return (new cons (null, null));
+//   if (split.length == 0) return (new cons (null, null));
+//   var token = split.shift();
+//   if (token == ")") return (new cons (null, split));
+//   var next = parse(split);
+//   if (token == "(") {
+//     var nextnext = parse(next.cdr);
+//     return (new cons ((new cons (next.car, nextnext.car)),
+//                       nextnext.cdr));
+//   }
+//   return (new cons ((new cons ((new cons (null, token)),
+//                                next.car)),
+//                     next.cdr))
+// }
+
+// // substitute arg in for v in tree
+// function sub(tree, v, arg) {
+//   if (tree == null) return (null);
+//   if (tree.car == null) if (tree.cdr == v) return (arg);
+//   if (tree.car == null) return (tree);
+
+//   // perform alpha reduction to prevent variable collision
+//   if (isBindingForm(tree)) 
+//     return (new cons (tree.car, 
+//                       sub(sub(tree.cdr,         // inner sub = alpha reduc.
+//                               tree.cdr.car.cdr, 
+//                               fresh(tree.cdr.car.cdr)),
+//                           v,
+//                           arg)));
+
+//   return (new cons ((sub (tree.car, v, arg)),
+//                     (sub (tree.cdr, v, arg))))
+// }
+
+// // Guaranteed unique for each call as long as string is not empty.
+// var i = 0;
+// function fresh(string) {
+//   i = i+1;
+//   if (typeof(string) != "string") return (string);
+//   return (new cons (null,  
+//                     string.substring(0,1) + (i).toString()));
+// }
+
+// // Keep reducing until there is no more change
+// function fixedPoint (tree) {
+//   var t2 = reduce(tree);
+//   if (treeToString(tree) == treeToString(t2)) return (tree);
+//   return (fixedPoint (t2));
+// }
+
+// // Reduce all the arguments, then try to do beta conversion on the whole
+// function reduce(tree) {
+//   if (tree == null) return (tree);
+//   if (typeof(tree) == "string") return (tree);
+//   return (convert (new cons (reduce (tree.car), mapReduce (tree.cdr))));
+// }
+
+// // Reduce all the arguments in a list
+// function mapReduce(tree) {
+//   if (tree == null) return (tree);
+//   if (tree.car == null) return (tree);
+//   return (new cons (reduce (tree.car), mapReduce(tree.cdr )));
+// }
+
+// // If the list is of the form ((lambda var body) arg), do beta reduc.
+// function convert(tree) {
+//     if (isLet(tree)) {
+//       return (sub(tree.cdr.car, tree.car.cdr.car.cdr, tree.car.cdr.cdr.car));}
+//     else 
+//       if (isConvertable(tree)) {
+//         return (sub(tree.car.cdr.cdr.car, tree.car.cdr.car.cdr, tree.cdr.car));}
+//       else return(tree);
+// } 
+
+// // Is of form ((let var arg) body)?
+// function isLet(tree) {
+//   if (tree == null) return (false);
+//   if (!(isBindingForm(tree.car))) return (false);
+//   if (tree.car.car.cdr != "let") return (false);
+//   if (tree.cdr == null) return (false);
+//   if (tree.cdr.car == null) return (false);
+//   return(true);
+// }  
+
+// // Is of form ((lambda var body) arg)?
+// function isConvertable(tree) {
+//   if (tree == null) return (false);
+//   if (!(isBindingForm(tree.car))) return (false);
+//   if (tree.car.car.cdr != "lambda") return (false);
+//   if (tree.cdr == null) return (false);
+//   if (tree.cdr.car == null) return (false);
+//   return(true);
+// }  
+
+// // Is of form (lambda var body)?
+// function isBindingForm(tree) {
+//   if (tree == null) return (false);
+//   if (tree.car == null) return (false);
+//   if (tree.car.car != null) return (false);
+//   if ((tree.car.cdr != "lambda") 
+//       && (tree.car.cdr != "let")
+//       && (tree.car.cdr != "exists")
+//       && (tree.car.cdr != "forall")
+//       && (tree.car.cdr != "\u03BB")
+//       && (tree.car.cdr != "\u2200")
+//       && (tree.car.cdr != "\u2203")
+//      )
+//     return (false);
+//   if (tree.car.cdr == null) return (false);
+//   if (tree.cdr.car == null) return (false);
+//   if (tree.cdr.car.car != null) return (false);
+//   if (tree.cdr.cdr == null) return (false);
+//   return (true);
+// }
+
+// function treeToString(tree) {
+//   if (tree == null) return ("")
+//   if (tree.car == null) return (tree.cdr)
+//   if ((tree.car).car == null) 
+//     return (treeToString(tree.car) + " " + treeToString(tree.cdr))
+//   return ("(" + treeToString(tree.car) + ")" + treeToString(tree.cdr))
+// }
+
+// // use this instead of treeToString if you want to see the full structure
+// function treeToStringRaw(tree) {
+//   if (tree == null) return ("@")
+//   if (typeof(tree) == "string") return (tree);
+//   return ("(" + treeToStringRaw(tree.car) + "." + 
+//                 treeToStringRaw(tree.cdr) + ")")
+// }
+
+// // Make sure each paren will count as a separate token
+// function stringToTree(input) {
+//   input = input.replace(/let/g, " ( ( let ");
+//   input = input.replace(/=/g, " ");
+//   input = input.replace(/in/g, " ) ");
+//   input = input.replace(/\(/g, " ( ");
+//   input = input.replace(/\)/g, " ) ");
+//   input = input.replace(/;.*\n/g," ");
+//   input = input.replace(/\^/g, " ^ ");
+//   input = input.replace(/[\\]/g, " lambda ");
+//   input = input.replace(/\u03BB/g, " lambda ");
+//   return ((parse(input.split(/[ \f\n\r\t\v]+/))).car)
+// }
+
+// // Adjust spaces to print pretty
+// function formatTree(tree) {
+//   output = treeToStringRaw (tree);
+//   output = output.replace(/^[ \f\n\r\t\v]+/, "");
+//   output = output.replace(/[ \f\n\r\t\v]+$/, "");
+//   output = output.replace(/[ \f\n\r\t\v]+\)/g, ")");
+//   output = output.replace(/\)([^)(])/g, ") $1");
+//   output = output.replace(/lambda/g, "\\");
+// //  output = output.replace(/lambda/g, "\u03BB");
+// //  output = output.replace(/exists/g, "\u2203");
+// //  output = output.replace(/forall/g, "\u2200");
+//   return (output)
+// }
+
+// function mytry(form) { 
+//   i = 0;
+//   form.result.value = formatTree((stringToTree(form.input.value)));
+//   // form.result.value = formatTree(fixedPoint(stringToTree(form.input.value)));
+// }