1 (* Title: ex/Fib |
1 (* Title: HOL/NumberTheory/Fib.thy |
2 ID: $Id$ |
2 ID: $Id$ |
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
4 Copyright 1997 University of Cambridge |
4 Copyright 1997 University of Cambridge |
5 |
|
6 The Fibonacci function. Demonstrates the use of recdef. |
|
7 *) |
5 *) |
8 |
6 |
9 Fib = Primes + |
7 header {* The Fibonacci function *} |
10 |
8 |
11 consts fib :: "nat => nat" |
9 theory Fib = Primes: |
12 recdef fib "less_than" |
10 |
13 zero "fib 0 = 0" |
11 text {* |
14 one "fib 1 = 1" |
12 Fibonacci numbers: proofs of laws taken from: |
15 Suc_Suc "fib (Suc (Suc x)) = fib x + fib (Suc x)" |
13 R. L. Graham, D. E. Knuth, O. Patashnik. Concrete Mathematics. |
|
14 (Addison-Wesley, 1989) |
|
15 |
|
16 \bigskip |
|
17 *} |
|
18 |
|
19 consts fib :: "nat => nat" |
|
20 recdef fib less_than |
|
21 zero: "fib 0 = 0" |
|
22 one: "fib 1 = 1" |
|
23 Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)" |
|
24 |
|
25 text {* |
|
26 \medskip The difficulty in these proofs is to ensure that the |
|
27 induction hypotheses are applied before the definition of @{term |
|
28 fib}. Towards this end, the @{term fib} equations are not declared |
|
29 to the Simplifier and are applied very selectively at first. |
|
30 *} |
|
31 |
|
32 declare fib.Suc_Suc [simp del] |
|
33 |
|
34 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))" |
|
35 apply (rule fib.Suc_Suc) |
|
36 done |
|
37 |
|
38 |
|
39 text {* \medskip Concrete Mathematics, page 280 *} |
|
40 |
|
41 lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n" |
|
42 apply (induct n rule: fib.induct) |
|
43 prefer 3 |
|
44 txt {* simplify the LHS just enough to apply the induction hypotheses *} |
|
45 apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard]) |
|
46 apply (simp_all (no_asm_simp) add: fib.Suc_Suc add_mult_distrib add_mult_distrib2) |
|
47 done |
|
48 |
|
49 lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0" |
|
50 apply (induct n rule: fib.induct) |
|
51 apply (simp_all add: fib.Suc_Suc) |
|
52 done |
|
53 |
|
54 lemma [simp]: "0 < fib (Suc n)" |
|
55 apply (simp add: neq0_conv [symmetric]) |
|
56 done |
|
57 |
|
58 lemma fib_gr_0: "0 < n ==> 0 < fib n" |
|
59 apply (rule not0_implies_Suc [THEN exE]) |
|
60 apply auto |
|
61 done |
|
62 |
|
63 |
|
64 text {* |
|
65 \medskip Concrete Mathematics, page 278: Cassini's identity. It is |
|
66 much easier to prove using integers! |
|
67 *} |
|
68 |
|
69 lemma fib_Cassini: "int (fib (Suc (Suc n)) * fib n) = |
|
70 (if n mod #2 = 0 then int (fib (Suc n) * fib (Suc n)) - #1 |
|
71 else int (fib (Suc n) * fib (Suc n)) + #1)" |
|
72 apply (induct n rule: fib.induct) |
|
73 apply (simp add: fib.Suc_Suc) |
|
74 apply (simp add: fib.Suc_Suc mod_Suc) |
|
75 apply (simp add: fib.Suc_Suc |
|
76 add_mult_distrib add_mult_distrib2 mod_Suc zmult_int [symmetric] zmult_ac) |
|
77 done |
|
78 |
|
79 |
|
80 text {* \medskip Towards Law 6.111 of Concrete Mathematics *} |
|
81 |
|
82 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = 1" |
|
83 apply (induct n rule: fib.induct) |
|
84 prefer 3 |
|
85 apply (simp add: gcd_commute fib_Suc3) |
|
86 apply (simp_all add: fib.Suc_Suc) |
|
87 done |
|
88 |
|
89 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)" |
|
90 apply (simp (no_asm) add: gcd_commute [of "fib m"]) |
|
91 apply (case_tac "m = 0") |
|
92 apply simp |
|
93 apply (clarify dest!: not0_implies_Suc) |
|
94 apply (simp add: fib_add) |
|
95 apply (simp add: add_commute gcd_non_0) |
|
96 apply (simp add: gcd_non_0 [symmetric]) |
|
97 apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel) |
|
98 done |
|
99 |
|
100 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)" |
|
101 apply (rule gcd_fib_add [symmetric, THEN trans]) |
|
102 apply simp |
|
103 done |
|
104 |
|
105 lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)" |
|
106 apply (induct n rule: nat_less_induct) |
|
107 apply (subst mod_if) |
|
108 apply (simp add: gcd_fib_diff mod_geq not_less_iff_le diff_less) |
|
109 done |
|
110 |
|
111 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)" -- {* Law 6.111 *} |
|
112 apply (induct m n rule: gcd_induct) |
|
113 apply simp |
|
114 apply (simp add: gcd_non_0) |
|
115 apply (simp add: gcd_commute gcd_fib_mod) |
|
116 done |
|
117 |
|
118 lemma fib_mult_eq_setsum: |
|
119 "fib (Suc n) * fib n = setsum (\<lambda>k. fib k * fib k) (atMost n)" |
|
120 apply (induct n rule: fib.induct) |
|
121 apply (auto simp add: atMost_Suc fib.Suc_Suc) |
|
122 apply (simp add: add_mult_distrib add_mult_distrib2) |
|
123 done |
16 |
124 |
17 end |
125 end |