src/HOL/Old_Number_Theory/Fib.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 44821 a92f65e174cf child 57512 cc97b347b301 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Title:      HOL/Old_Number_Theory/Fib.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1997  University of Cambridge
4 *)
6 header {* The Fibonacci function *}
8 theory Fib
9 imports Primes
10 begin
12 text {*
13   Fibonacci numbers: proofs of laws taken from:
14   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
17   \bigskip
18 *}
20 fun fib :: "nat \<Rightarrow> nat"
21 where
22   "fib 0 = 0"
23 | "fib (Suc 0) = 1"
24 | fib_2: "fib (Suc (Suc n)) = fib n + fib (Suc n)"
26 text {*
27   \medskip The difficulty in these proofs is to ensure that the
28   induction hypotheses are applied before the definition of @{term
29   fib}.  Towards this end, the @{term fib} equations are not declared
30   to the Simplifier and are applied very selectively at first.
31 *}
33 text{*We disable @{text fib.fib_2fib_2} for simplification ...*}
34 declare fib_2 [simp del]
36 text{*...then prove a version that has a more restrictive pattern.*}
37 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
38   by (rule fib_2)
40 text {* \medskip Concrete Mathematics, page 280 *}
42 lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
43 proof (induct n rule: fib.induct)
44   case 1 show ?case by simp
45 next
46   case 2 show ?case  by (simp add: fib_2)
47 next
48   case 3 thus ?case by (simp add: fib_2 add_mult_distrib2)
49 qed
51 lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0"
52   apply (induct n rule: fib.induct)
53     apply (simp_all add: fib_2)
54   done
56 lemma fib_Suc_gr_0: "0 < fib (Suc n)"
57   by (insert fib_Suc_neq_0 [of n], simp)
59 lemma fib_gr_0: "0 < n ==> 0 < fib n"
60   by (case_tac n, auto simp add: fib_Suc_gr_0)
63 text {*
64   \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
65   much easier using integers, not natural numbers!
66 *}
68 lemma fib_Cassini_int:
69  "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 proof(induct n rule: fib.induct)
73   case 1 thus ?case by (simp add: fib_2)
74 next
75   case 2 thus ?case by (simp add: fib_2 mod_Suc)
76 next
77   case (3 x)
78   have "Suc 0 \<noteq> x mod 2 \<longrightarrow> x mod 2 = 0" by presburger
79   with "3.hyps" show ?case by (simp add: fib.simps add_mult_distrib add_mult_distrib2)
80 qed
82 text{*We now obtain a version for the natural numbers via the coercion
83    function @{term int}.*}
84 theorem fib_Cassini:
85  "fib (Suc (Suc n)) * fib n =
86   (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
87    else fib (Suc n) * fib (Suc n) + 1)"
88   apply (rule int_int_eq [THEN iffD1])
89   apply (simp add: fib_Cassini_int)
90   apply (subst of_nat_diff)
91    apply (insert fib_Suc_gr_0 [of n], simp_all)
92   done
95 text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
97 lemma gcd_fib_Suc_eq_1: "gcd (fib n) (fib (Suc n)) = Suc 0"
98   apply (induct n rule: fib.induct)
99     prefer 3
100     apply (simp add: gcd_commute fib_Suc3)
101    apply (simp_all add: fib_2)
102   done
104 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
105   apply (simp add: gcd_commute [of "fib m"])
106   apply (case_tac m)
107    apply simp
109   apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0])
110   apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
111   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
112   done
114 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
115   by (simp add: gcd_fib_add [symmetric, of _ "n-m"])
117 lemma gcd_fib_mod: "0 < m ==> gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
118 proof (induct n rule: less_induct)
119   case (less n)
120   from less.prems have pos_m: "0 < m" .
121   show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
122   proof (cases "m < n")
123     case True note m_n = True
124     then have m_n': "m \<le> n" by auto
125     with pos_m have pos_n: "0 < n" by auto
126     with pos_m m_n have diff: "n - m < n" by auto
127     have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
128     by (simp add: mod_if [of n]) (insert m_n, auto)
129     also have "\<dots> = gcd (fib m) (fib (n - m))" by (simp add: less.hyps diff pos_m)
130     also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff m_n')
131     finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
132   next
133     case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
134     by (cases "m = n") auto
135   qed
136 qed
138 lemma fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)"  -- {* Law 6.111 *}
139   apply (induct m n rule: gcd_induct)
140   apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
141   done
143 theorem fib_mult_eq_setsum:
144     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
145   apply (induct n rule: fib.induct)
146     apply (auto simp add: atMost_Suc fib_2)