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+#lang sicp
+
+;; Chapter 1
+;; Building Abstractions with Procedures
+
+;; 1.2
+;; Procedures and the Processes They Generate
+
+#| 1.9 |#
+
+;; recursive process:
+#| (define (+ a b) |#
+#|   (if (= a 0) |#
+#|     b |#
+#|     (inc (+ (dec a) b)))) |#
+
+;; iterative process:
+#| (define (+ a b) |#
+#|   (if (= a 0) |#
+#|     b |#
+#|     (+ (dec a) (inc b)))) |#
+
+#| 1.10 |#
+
+(#%provide A)
+(define (A x y)
+  (cond
+    ((= y 0) 0)
+    ((= x 0) (* 2 y))
+    ((= y 1) 2)
+    (else
+      (A
+        (- x 1)
+        (A x (- y 1))))))
+
+#| (A 1 10) |#
+#| (A 0 (A 1 9)) |#
+#| (A 0 (A 0 (A 1 8))) |#
+#| (A 0 (A 0 (A 0 (A 1 7)))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 1 6))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 1 (A 1 5)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 32))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 32))))) |#
+#| (A 0 (A 0 (A 0 (A 0 64)))) |#
+#| (A 0 (A 0 (A 0 (* 2 64)))) |#
+#| (A 0 (A 0 (A 0 128))) |#
+#| (A 0 (A 0 (* 2 128))) |#
+#| (A 0 (A 0 256)) |#
+#| (A 0 (* 2 256)) |#
+#| (A 0 512) |#
+#| (* 2 512) |#
+#| 1024 |#
+
+#| (A 2 4) |#
+#| (A 1 (A 2 3)) |#
+#| (A 1 (A 1 (A 2 2))) |#
+#| (A 1 (A 1 (A 1 (A 2 1)))) |#
+#| (A 1 (A 1 (A 1 2))) |#
+#| (A 1 (A 1 (A 0 (A 1 1)))) |#
+#| (A 1 (A 1 (A 0 2))) |#
+#| (A 1 (A 1 (* 2 2))) |#
+#| (A 1 (A 1 4)) |#
+#| (A 1 (A 0 (A 1 3))) |#
+#| (A 1 (A 0 (A 0 (A 1 2)))) |#
+#| (A 1 (A 0 (A 0 (A 0 (A 1 1))))) |#
+#| (A 1 (A 0 (A 0 (A 0 2)))) |#
+#| (A 1 (A 0 (A 0 (* 2 2)))) |#
+#| (A 1 (A 0 (A 0 4))) |#
+#| (A 1 (A 0 (* 2 4))) |#
+#| (A 1 (A 0 8)) |#
+#| (A 1 (* 2 8)) |#
+#| (A 1 16) |#
+#| (A 0 (A 1 15)) |#
+#| (A 0 (A 0 (A 1 14))) |#
+#| (A 0 (A 0 (A 0 (A 1 13)))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 1 12))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 (A 1 5)))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16)))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16)))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 64)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 128))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 256)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 512))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 1024)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 2048))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (* 2 2048))))) |#
+#| (A 0 (A 0 (A 0 (A 0 4096)))) |#
+#| (A 0 (A 0 (A 0 (* 2 4096)))) |#
+#| (A 0 (A 0 (A 0 8182))) |#
+#| (A 0 (A 0 (* 2 8182))) |#
+#| (A 0 (A 0 16364)) |#
+#| (A 0 (* 2 16364)) |#
+#| (A 0 32768) |#
+#| (* 2 32768) |#
+#| 65536 |#
+
+
+#| (A 3 3) |#
+#| (A 2 (A 3 2)) |#
+#| (A 2 (A 2 (A 3 1))) |#
+#| (A 2 (A 2 2)) |#
+#| (A 2 (A 1 (A 2 1))) |#
+#| (A 2 (A 1 2)) |#
+#| (A 2 (A 0 (A 1 1))) |#
+#| (A 2 (A 0 2)) |#
+#| (A 2 (* 2 2)) |#
+#| (A 2 4) |#
+#| (A 1 (A 2 3)) |#
+#| (A 1 (A 1 (A 2 2))) |#
+#| (A 1 (A 1 (A 1 (A 2 1)))) |#
+#| (A 1 (A 1 (A 1 2))) |#
+#| (A 1 (A 1 (A 0 (A 1 1)))) |#
+#| (A 1 (A 1 (A 0 2))) |#
+#| (A 1 (A 1 (* 2 2))) |#
+#| (A 1 (A 1 4)) |#
+#| (A 1 (A 0 (A 1 3))) |#
+#| (A 1 (A 0 (A 0 (A 1 2)))) |#
+#| (A 1 (A 0 (A 0 (A 0 (A 1 1))))) |#
+#| (A 1 (A 0 (A 0 (A 0 2)))) |#
+#| (A 1 (A 0 (A 0 (* 2 2)))) |#
+#| (A 1 (A 0 (A 0 4))) |#
+#| (A 1 (A 0 (* 2 4))) |#
+#| (A 1 (A 0 8)) |#
+#| (A 1 (* 2 8)) |#
+#| (A 1 16) |#
+#| (A 0 (A 1 15)) |#
+#| (A 0 (A 0 (A 1 14))) |#
+#| (A 0 (A 0 (A 0 (A 1 13)))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 1 12))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 (A 1 5)))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8))))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16)))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16)))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 64)))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 128))))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 256)))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 512))))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 1024)))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (A 0 2048))))) |#
+#| (A 0 (A 0 (A 0 (A 0 (* 2 2048))))) |#
+#| (A 0 (A 0 (A 0 (A 0 4096)))) |#
+#| (A 0 (A 0 (A 0 (* 2 4096)))) |#
+#| (A 0 (A 0 (A 0 8182))) |#
+#| (A 0 (A 0 (* 2 8182))) |#
+#| (A 0 (A 0 16364)) |#
+#| (A 0 (* 2 16364)) |#
+#| (A 0 32768) |#
+#| (* 2 32768) |#
+#| 65536 |#
+
+;; f(n) = 2n
+(define (f n) (A 0 n))
+
+;; g(n) = 2^n
+(define (g n) (A 1 n))
+
+;; h(n) = 2^(2^... n times ...(2^2)...)
+(define (h n) (A 2 n))
+
+;; k(n) = 5n^2
+(define (k n) (* 5 n n))
+
+(#%provide count-change)
+(define (count-change amount)
+  (cc amount 5))
+
+(define (cc amount kinds-of-coins)
+  (cond
+    ((= amount 0) 1)
+    ((or (< amount 0) (= kinds-of-coins 0)) 0)
+    (else
+      (+
+        (cc amount (- kinds-of-coins 1))
+        (cc
+          (- amount (first-denomination kinds-of-coins))
+          kinds-of-coins)))))
+
+(define (first-denomination kinds-of-coins)
+  (cond
+    ((= kinds-of-coins 1) 1)
+    ((= kinds-of-coins 2) 5)
+    ((= kinds-of-coins 3) 10)
+    ((= kinds-of-coins 4) 25)
+    ((= kinds-of-coins 5) 50)))
+
+#| 1.11 |#
+
+(#%provide f-rec)
+(define (f-rec n)
+  (cond
+    ((< n 3) n)
+    (else
+      (+
+        (f-rec (- n 1))
+        (* 2 (f-rec (- n 2)))
+        (* 3 (f-rec (- n 3)))))))
+
+(#%provide f-iter)
+(define (f-iter n)
+  (f-iter- 2 1 0 n))
+
+(define (f-iter- a b c i)
+  (if (= i 0)
+    c
+    (f-iter-
+      (+ a (* 2 b) (* 3 c))
+      a
+      b
+      (- i 1))))
+
+#| 1.12 |#
+
+(#%provide pascal)
+(define (pascal row elem)
+  (cond
+    ((> elem row) 0)
+    ((= elem 0) 1)
+    (else
+      (+
+        (pascal (- row 1) (- elem 1))
+        (pascal (- row 1) elem)))))
+
+#| 1.13 |#
+
+;; phi = (1 + sqrt(5)) / 2
+;; psi = (1 - sqrt(5)) / 2
+
+;; note that:
+;; phi^2 = phi + 1, and
+;; psi^2 = psi + 1
+
+;; Proof by induction that Fib(n) = (phi^n - psi^n) / sqrt(5)
+
+;; Base cases:
+;; Fib(0) = 0 = (phi^0 - psi^0) / sqrt(5)
+;; Fib(1) = 1 = = (((1 + sqrt(5)) - (1 - sqrt(5))) / 2) / sqrt(5)
+;;        = (phi^1 - psi^1) / sqrt(5)
+
+;; Inductive case:
+
+;; Suppose
+;; Fib(n-2) = (phi^(n-2) - psi^(n-2)) / sqrt(5), and
+;; Fib(n-1) = (phi^(n-1) - psi^(n-1)) / sqrt(5), and
+
+;; Then
+;; Fib(n) = Fib(n-1) + Fib(n - 2)
+;;        = (phi^(n-1) - psi^(n-1) + phi^(n-2) - psi^(n-2)) / sqrt(5)
+;;        = (phi^(n-1) + phi^(n-2) - psi(^n-1) - psi^(n-2)) / sqrt(5)
+;;        = ((phi + 1)phi^(n-2) - (psi + 1)psi^(n-2)) / sqrt(5)
+;;        = ((phi^2)phi^(n-2) - (psi^2)psi^(n-2)) / sqrt(5)
+;;        = (phi^n - psi^n) / sqrt(5)
+
+;; For all n, psi^n / sqrt(5) < 1/2. So Fib(n) is the closest integer to phi^n / sqrt(5)
+
+
+#| 1.14 |#
+
+#| The process generated by (count-change 11): |#
+
+#| - cc 11 5 |#
+#| | - cc -39 5 |#
+#| | - cc 11 4 |#
+#|   | - cc -14 4 |#
+#|   | - cc 11 3 |#
+#|     | - cc 11 2 |#
+#|     | | - cc 11 1 |#
+#|     | | | - cc 11 0 |#
+#|     | | | - cc 10 1 |#
+#|     | |   | - cc 10 0 |#
+#|     | |   | - cc 9 1 |#
+#|     | |     | - cc 9 0 |#
+#|     | |     | - cc 8 1 |#
+#|     | |       | - cc 8 0 |#
+#|     | |       | - cc 7 1 |#
+#|     | |         | - cc 7 0 |#
+#|     | |         | - cc 6 1 |#
+#|     | |           | - cc 6 0 |#
+#|     | |           | - cc 5 1 |#
+#|     | |             | - cc 5 0 |#
+#|     | |             | - cc 4 1 |#
+#|     | |               | - cc 4 0 |#
+#|     | |               | - cc 3 1 |#
+#|     | |                 | - cc 3 0 |#
+#|     | |                 | - cc 2 1 |#
+#|     | |                   | - cc 2 0 |#
+#|     | |                   | - cc 1 1 |#
+#|     | |                     | - cc 1 0 |#
+#|     | |                     | - cc 0 1 |#
+#|     | | - cc 6 2 |#
+#|     |   | - cc 6 1 |#
+#|     |   | | - cc 6 0 |#
+#|     |   | | - cc 5 1 |#
+#|     |   |   | - cc 5 0 |#
+#|     |   |   | - cc 4 1 |#
+#|     |   |     | - cc 4 0 |#
+#|     |   |     | - cc 3 1 |#
+#|     |   |       | - cc 3 0 |#
+#|     |   |       | - cc 2 1 |#
+#|     |   |         | - cc 2 0 |#
+#|     |   |           - cc 1 1 |#
+#|     |   |           | - cc 1 0 |#
+#|     |   |           | - cc 0 1 |#
+#|     |   | - cc 1 2 |#
+#|     |     | - cc 1 1 |#
+#|     |     | | - cc 1 0 |#
+#|     |     | | - cc 0 1 |#
+#|     |     | - cc -4 2 |#
+#|     | - cc 1 3 |#
+#|       | - cc 1 2 |#
+#|       | | - cc 1 1 |#
+#|       | | | - cc 1 0 |#
+#|       | | | - cc 0 1 |#
+#|       | | - cc -4 2 |#
+#|       | - cc -9 3 |#
+
+#| count-change is Theta(n) in space (max depth of tree) |#
+#| count-change is Theta(e^n) in time (number of nodes in tree) |#
+
+#| 1.15 |#
+
+(#%provide sine)
+(define (sine ang)
+  (define (p x) (- (* 3 x) (* 4 (cube x))))
+  (if (not (> (abs ang) 0.1))
+    ang
+    (p (sine (/ ang 3.0)))))
+
+#| sine 12.15 |#
+#| p (sine 4.05) |#
+#| p (p (sine 1.35)) |#
+#| p (p (p (sine 0.45))) |#
+#| p (p (p (p (sine 0.15)))) |#
+#| p (p (p (p (p (sine 0.05))))) |#
+#| p (p (p (p (p 0.05)))) |#
+
+#| p is called 5 times |#
+
+#| The process generated by sine is recursive |#
+#| Order of growth in space: Theta(log_3(n)) |#
+#| Order of growth in time: Theta(log_3(n)) |#
+
+(#%provide expt-rec)
+(define (expt-rec b n)
+  (if (= n 0)
+    1
+    (* b (expt-rec b (- n 1)))))
+
+(#%provide expt-iter)
+(define (expt-iter b n)
+  (expt-iter- b n 1))
+
+(define (expt-iter- b i prod)
+  (if (= i 0)
+    prod
+    (expt-iter-
+      b
+      (- i 1)
+      (* b prod))))
+
+(define (even-? k) (= (remainder k 2) 0))
+
+(#%provide fast-expt-rec)
+(define (fast-expt-rec b n)
+  (cond
+    ((= n 0) 1)
+    ((even-? n) (square (fast-expt-rec b (/ n 2))))
+    (else (* b (fast-expt-rec b (- n 1))))))
+
+#| 1.16 |#
+
+(#%provide fast-expt-iter)
+(define (fast-expt-iter b n)
+  (fast-expt-iter- 1 b n))
+
+(define (fast-expt-iter- a b n)
+  (cond
+    ((= n 0) a)
+    ((even-? n) (fast-expt-iter- a (square b) (/ n 2)))
+    (else (fast-expt-iter- (* a b) b (- n 1)))))
+
+(#%provide mult)
+(define (mult a b)
+  (if (= b 0)
+    0
+    (+ a (mult a (- b 1)))))
+
+#| 1.17 |#
+
+(define (double x) (+ x x))
+(define (halve x) (/ x 2))
+
+(#%provide fast-times-rec)
+(define (fast-times-rec a b)
+  (cond
+    ((= b 0) 0)
+    ((even-? b) (double (fast-times-rec a (halve b))))
+    (else (+ a (fast-times-rec a (- b 1))))))
+
+#| 1.18 |#
+
+(#%provide fast-times-iter)
+(define (fast-times-iter a b)
+  (fast-times-iter- 0 a b))
+
+(define (fast-times-iter- res a b)
+  (cond
+    ((= b 0) res)
+    ((even-? b) (fast-times-iter- res (double a) (halve b)))
+    (else (fast-times-iter- (+ res a) a (- b 1)))))
+
+#| 1.19 |#
+
+;; T
+;; a <- a + b
+;; b <- a
+
+;; Tpq
+;; a <- bq + aq + ap
+;; b <- bp + aq
+
+;; Tpq^2
+;; a <- (bp + aq)q + (bq + aq + ap)q + (bq + aq + ap)p
+;; b <- (bp + aq)p + (bq + aq + ap)q
+
+;; a <- b(2pq + qq) + a(2pq + qq) + a(pp + qq)
+;; b <- b(pp + qq) + a(2qp + qq)
+
+;; p' = pp + qq
+;; q' = 2pq + qq
+
+(#%provide fib)
+(define (fib n)
+  (fib-iter 1 0 0 1 n))
+
+(define (fib-iter a b p q i)
+  (cond
+    ((= i 0) b)
+    ((even-? i)
+     (fib-iter
+       a
+       b
+       (+ (square p) (square q))
+       (+ (* 2 p q) (square q))
+       (/ i 2)))
+    (else
+      (fib-iter
+        (+ (* b q) (* a q) (* a p))
+        (+ (* b p) (* a q))
+        p
+        q
+        (- i 1)))))
+
+(#%provide fib-slow)
+(define (fib-slow n)
+  (cond
+    ((= n 0) 0)
+    ((= n 1) 1)
+    (else (+ (fib (- n 1)) (fib (- n 2))))))
+
+#| 1.20 |#
+
+(#%provide gcd-)
+(define (gcd- a b)
+  (if (= b 0)
+    a
+    (gcd- b (remainder a b))))
+
+;; Process generated by normal order evaluation
+
+#| (gcd- 206 40) |#
+#| (if (= 40 0) 206 (gcd- 40 (rem 206 40))) |#
+#| (gcd- 40 (rem 206 40)) |#
+#| (if (= (rem 206 40) 0) 40 (gcd- b (rem 40 (rem 206 40)))) ; one call |#
+#| (if (= 6 0) 40 (gcd- (rem 206 40) (rem 40 (rem 206 40)))) |#
+#| (gcd- (rem 206 40) (rem 40 (rem 206 40))) |#
+#| (if (= (rem 40 (rem 206 40)) 0) ... (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40))))) ; one call |#
+#| (if (= (rem 40 6) 0) ... (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40))))) ; one call |#
+#| (if (= 4 0) ... (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40))))) |#
+#| (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40)))) |#
+#| (if (= (rem (rem 206 40) (rem 40 (rem 206 40))) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
+#| (if (= (rem 6 (rem 40 (rem 206 40))) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
+#| (if (= (rem 6 (rem 40 6)) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
+#| (if (= (rem 6 4) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
+#| (if (= 2 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) |#
+#| (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) (rem (rem 40 (rem 206 40)) ...)) |#
+#| (if (= (rem (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) ...) ...) ; one call |#
+#| (if (= (rem (rem 40 6) (rem (rem 206 40) (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) ...) ...) ; one call |#
+#| (if (= (rem 4 (rem (rem 206 40) (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
+#| (if (= (rem 4 (rem 6 (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
+#| (if (= (rem 4 (rem 6 (rem 40 6))) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
+#| (if (= (rem 4 (rem 6 4)) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
+#| (if (= (rem 4 2) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
+#| (if (= 0 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) |#
+#| (rem (rem 206 40) (rem 40 (rem 206 40))) ; one call |#
+#| (rem 6 (rem 40 (rem 206 40))) ; one call |#
+#| (rem 6 (rem 40 6)) ; one call |#
+#| (rem 6 4) ; one call |#
+#| 2 |#
+
+;; 18 calls to remainder are performed
+
+;; Process generated by applicative order evaluation
+
+#| (gcd- 206 40) |#
+#| (if (= 40 0) 206 (gcd- 40 (remainder 206 40))) ; one call |#
+#| (gcd- 40 6) |#
+#| (gcd- 40 6) |#
+#| (if (= 6 0) 40 (gcd- 6 (remainder 40 6))) |#
+#| (gcd- 6 (remainder 40 6)) ; one call |#
+#| (gcd- 6 4) |#
+#| (if (= 4 0) 6 (gcd- 4 (remainder 6 4))) |#
+#| (gcd- 4 (remainder 6 4)) ; one call |#
+#| (gcd- 4 2) |#
+#| (if (= 2 0) 4 (gcd- 2 (remainder 4 2))) |#
+#| (gcd- 2 (remainder 4 2)) ; one call |#
+#| (gcd- 2 0) |#
+#| (if (= 0 0) 2 (gcd- 0 (remainder 2 0))) |#
+#| 2 |#
+
+;; 4 calls to remainder are performed
+
+(#%provide smallest-divisor)
+(define (smallest-divisor n)
+  (find-divisor n 2))
+
+(define (find-divisor n test-divisor)
+  (cond
+    ((> (square test-divisor) n) n)
+    ((divides? test-divisor n) test-divisor)
+    (else (find-divisor n (next test-divisor)))))
+
+(define (divides? a b)
+  (= (remainder b a) 0))
+
+(#%provide prime?)
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(define (expmod base expo m)
+  (cond
+    ((= expo 0) 1)
+    ((even-? expo)
+     (remainder (square (expmod base (/ expo 2) m)) m))
+    (else
+      (remainder
+        (* base (expmod base (- expo 1) m))
+        m))))
+
+(define (fermat-test n)
+  (define (try-it a)
+    (= (expmod a n n) a))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (fast-prime? n times)
+  (cond
+    ((= times 0) true)
+    ((fermat-test n) (fast-prime? n (- times 1)))
+    (else false)))
+
+#| 1.21 |#
+
+#| (smallest-divisor 199) |#
+#| (smallest-divisor 1999) |#
+#| (smallest-divisor 19999) |#
+
+(#%provide timed-prime-test)
+(define (timed-prime-test n)
+  (newline)
+  (display n)
+  (start-prime-test n (runtime)))
+
+(define (start-prime-test n start-time)
+  (if (fast-prime? n 100)
+    (report-prime (- (runtime) start-time))))
+
+(define (report-prime elapsed-time)
+(display " *** ")
+  (display elapsed-time))
+
+#| 1.22 |#
+
+(#%provide search-for-primes)
+(define (search-for-primes a b)
+  (cond
+    ((> a b) (newline))
+    (else
+      (timed-prime-test a)
+      (search-for-primes (+ a 1) b))))
+
+;; Smallest primes over 1000:
+;; 1009 3us
+;; 1013 3
+;; 1019 3
+
+;; Smallest primes over 10000:
+;; 10007 9us
+;; 10019 8
+;; 10037 9
+
+;; Smallest primes over 100000:
+;; 100003 22us
+;; 100019 21
+;; 100043 20
+
+;; Smallest primes over 1000000:
+;; 1000003 65us
+;; 1000033 65
+;; 1000037 65
+
+;; 9 / 3 = 3
+;; 21 / 9 = 2.33
+;; 65 / 21 = 3.095
+
+;; sqrt(10) = 3.16
+
+#| 1.23 |#
+
+(define (next n)
+  (if (= n 2) 3 (+ n 2)))
+
+;; Smallest primes over 1000:
+;; 1009 2us
+;; 1013 2
+;; 1019 2
+
+;; Smallest primes over 10000:
+;; 10007 5us
+;; 10019 4
+;; 10037 4
+
+;; Smallest primes over 100000:
+;; 100003 11us
+;; 100019 12
+;; 100043 11
+
+;; Smallest primes over 1000000:
+;; 1000003 34us
+;; 1000033 34
+;; 1000037 34
+
+;; 3 / 2 = 1.4
+;; 9 / 4 = 2.25
+;; 21 / 11 = 1.9
+;; 65 / 34 = 1.9
+
+#| 1.24 |#
+
+;; Using fast-prime?
+
+;; Smallest primes over 1000:
+;; 1009 .28ms
+;; 1013 .29
+;; 1019 .31
+
+;; Smallest primes over 10000:
+;; 10007 .37ms
+;; 10019 .36
+;; 10037 .40
+
+;; Smallest primes over 100000:
+;; 100003 .42ms
+;; 100019 .44
+;; 100043 .46
+
+;; Smallest primes over 1000000:
+;; 1000003 .50ms
+;; 1000033 .49
+;; 1000037 .51
+
+;; Scaling up by a factor of ten adds a constant amount of time
+
+#| 1.25 |#
+
+;; This version of expmod requires too much space to
+;; represent the intermediate result when using very
+;; large exponents
+
+(define (expmod-bad base expo m)
+  (remainder (fast-expt-iter base expo) m))
+
+#| 1.26 |#
+
+;; This version of expmod makes two recursive calls of
+;; size n / 2, making it Theta(n) in time rather than
+;; Theta(log(n))
+
+(define (expmod-slow base expo m)
+  (cond
+    ((= expo 0) 1)
+    ((even-? expo)
+     (remainder
+       (*
+         (remainder (expmod-slow base (/ expo 2) m)
+         (remainder (expmod-slow base (/ expo 2) m))))
+       m))
+    (else
+      (remainder
+        (*
+          base
+          (expmod-slow base (- expo 1) m))
+        m))))
+
+#| 1.27 |#
+
+(#%provide fermat-test-exhaustive)
+(define (fermat-test-exhaustive n)
+  (define (iter i res)
+    (define (try-it a) (= (expmod a n n) a))
+    (if (= i n)
+      res
+      (iter (+ i 1) (and res (try-it i)))))
+  (iter 1 true))
+
+#| (fermat-test-exhaustive 561) |#
+#| (fermat-test-exhaustive 1105) |#
+#| (fermat-test-exhaustive 1729) |#
+#| (fermat-test-exhaustive 2465) |#
+#| (fermat-test-exhaustive 2821) |#
+#| (fermat-test-exhaustive 6601) |#
+
+#| 1.28 |#
+
+(define (expmod-sig base expo m)
+  (define (square-sig x)
+    (define squared (remainder (square x) m))
+    (if
+      (and
+        (= squared 1)
+        (not (= x 1))
+        (not (= x expo)))
+      0
+      squared))
+  (cond
+    ((= expo 0) 1)
+    ((even-? expo)
+     (square-sig (expmod base (/ expo 2) m)))
+    (else
+      (remainder
+        (* base (expmod base (- expo 1) m))
+        m))))
+
+(define (miller-rabin-test n)
+  (define (try-it a)
+    (= (expmod a (- n 1) n) 1))
+  (try-it (+ 1 (random (- n 1)))))
+
+(#%provide mr-fast-prime?)
+(define (mr-fast-prime? n times)
+  (cond
+    ((= times 0) true)
+    ((miller-rabin-test n) (mr-fast-prime? n (- times 1)))
+    (else false)))
+
+(#%provide sum-integers-)
+(define (sum-integers- a b)
+  (if (> a b)
+    0
+    (+ a (sum-integers- (+ a 1) b))))
+
+(#%provide sum-cubes-)
+(define (sum-cubes- a b)
+  (if (> a b)
+    0
+    (+ (cube a) (sum-cubes- (+ a 1) b))))
+
+(#%provide pi-sum-)
+(define (pi-sum- a b)
+  (if (> a b)
+    0
+    (+ (/ 1.0 (* a (+ a 2))) (pi-sum- (+ a 4) b))))
+
+(#%provide sum)
+(define (sum term a next b)
+  (if (> a b)
+    0
+    (+
+      (term a)
+      (sum term (next a) next b))))
+
+(define (inc n) (+ n 1))
+
+(#%provide sum-cubes)
+(define (sum-cubes a b) (sum cube a inc b))
+
+(define (id x) x)
+
+(#%provide sum-integers)
+(define (sum-integers a b) (sum id a inc b))
+
+(#%provide pi-sum)
+(define (pi-sum a b)
+  (define (pi-term x) (/ 1.0 (* x (+ x 2))))
+  (define (pi-next x) (+ x 4))
+  (sum pi-term a pi-next b))
+
+(#%provide integral)
+(define (integral f a b dx)
+  (define (add-dx x) (+ x dx))
+  (* (sum f (+ a (/ dx 2.0)) add-dx b) dx))