src/HOL/NumberTheory/Fib.thy
 author wenzelm Fri Oct 05 21:52:39 2001 +0200 (2001-10-05) changeset 11701 3d51fbf81c17 parent 11468 02cd5d5bc497 child 11704 3c50a2cd6f00 permissions -rw-r--r--
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
"num" syntax (still with "#"), Numeral0, Numeral1;
1 (*  Title:      HOL/NumberTheory/Fib.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1997  University of Cambridge
5 *)
7 header {* The Fibonacci function *}
9 theory Fib = Primes:
11 text {*
12   Fibonacci numbers: proofs of laws taken from:
13   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
14   (Addison-Wesley, 1989)
16   \bigskip
17 *}
19 consts fib :: "nat => nat"
20 recdef fib  less_than
21   zero: "fib 0  = 0"
22   one:  "fib (Suc 0) = Suc 0"
23   Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
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 *}
32 declare fib.Suc_Suc [simp del]
34 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
35   apply (rule fib.Suc_Suc)
36   done
39 text {* \medskip Concrete Mathematics, page 280 *}
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
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
54 lemma [simp]: "0 < fib (Suc n)"
55   apply (simp add: neq0_conv [symmetric])
56   done
58 lemma fib_gr_0: "0 < n ==> 0 < fib n"
59   apply (rule not0_implies_Suc [THEN exE])
60    apply auto
61   done
64 text {*
65   \medskip Concrete Mathematics, page 278: Cassini's identity.  It is
66   much easier to prove using integers!
67 *}
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)) - Numeral1
71    else int (fib (Suc n) * fib (Suc n)) + Numeral1)"
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   apply (subgoal_tac "x mod # 2 < # 2", arith)
78   apply simp
79   done
82 text {* \medskip Towards Law 6.111 of Concrete Mathematics *}
84 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
85   apply (induct n rule: fib.induct)
86     prefer 3
87     apply (simp add: gcd_commute fib_Suc3)
88    apply (simp_all add: fib.Suc_Suc)
89   done
91 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
92   apply (simp (no_asm) add: gcd_commute [of "fib m"])
93   apply (case_tac "m = 0")
94    apply simp
95   apply (clarify dest!: not0_implies_Suc)
96   apply (simp add: fib_add)
97   apply (simp add: add_commute gcd_non_0)
98   apply (simp add: gcd_non_0 [symmetric])
99   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
100   done
102 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
103   apply (rule gcd_fib_add [symmetric, THEN trans])
104   apply simp
105   done
107 lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
108   apply (induct n rule: nat_less_induct)
109   apply (subst mod_if)
110   apply (simp add: gcd_fib_diff mod_geq not_less_iff_le diff_less)
111   done
113 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
114   apply (induct m n rule: gcd_induct)
115    apply simp
116   apply (simp add: gcd_non_0)
117   apply (simp add: gcd_commute gcd_fib_mod)
118   done
120 lemma fib_mult_eq_setsum:
121     "fib (Suc n) * fib n = setsum (\<lambda>k. fib k * fib k) (atMost n)"
122   apply (induct n rule: fib.induct)
123     apply (auto simp add: atMost_Suc fib.Suc_Suc)
124   apply (simp add: add_mult_distrib add_mult_distrib2)
125   done
127 end