15 begin |
15 begin |
16 |
16 |
17 |
17 |
18 subsection {* Main definitions *} |
18 subsection {* Main definitions *} |
19 |
19 |
20 class fib = |
20 fun fib :: "nat \<Rightarrow> nat" |
21 fixes fib :: "'a \<Rightarrow> 'a" |
|
22 |
|
23 |
|
24 (* definition for the natural numbers *) |
|
25 |
|
26 instantiation nat :: fib |
|
27 begin |
|
28 |
|
29 fun fib_nat :: "nat \<Rightarrow> nat" |
|
30 where |
21 where |
31 "fib_nat n = |
22 fib0: "fib 0 = 0" |
32 (if n = 0 then 0 else |
23 | fib1: "fib (Suc 0) = 1" |
33 (if n = 1 then 1 else |
24 | fib2: "fib (Suc (Suc n)) = fib (Suc n) + fib n" |
34 fib (n - 1) + fib (n - 2)))" |
|
35 |
|
36 instance .. |
|
37 |
|
38 end |
|
39 |
|
40 (* definition for the integers *) |
|
41 |
|
42 instantiation int :: fib |
|
43 begin |
|
44 |
|
45 definition fib_int :: "int \<Rightarrow> int" |
|
46 where "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)" |
|
47 |
|
48 instance .. |
|
49 |
|
50 end |
|
51 |
|
52 |
|
53 subsection {* Set up Transfer *} |
|
54 |
|
55 lemma transfer_nat_int_fib: |
|
56 "(x::int) >= 0 \<Longrightarrow> fib (nat x) = nat (fib x)" |
|
57 unfolding fib_int_def by auto |
|
58 |
|
59 lemma transfer_nat_int_fib_closure: |
|
60 "n >= (0::int) \<Longrightarrow> fib n >= 0" |
|
61 by (auto simp add: fib_int_def) |
|
62 |
|
63 declare transfer_morphism_nat_int[transfer add return: |
|
64 transfer_nat_int_fib transfer_nat_int_fib_closure] |
|
65 |
|
66 lemma transfer_int_nat_fib: "fib (int n) = int (fib n)" |
|
67 unfolding fib_int_def by auto |
|
68 |
|
69 lemma transfer_int_nat_fib_closure: "is_nat n \<Longrightarrow> fib n >= 0" |
|
70 unfolding fib_int_def by auto |
|
71 |
|
72 declare transfer_morphism_int_nat[transfer add return: |
|
73 transfer_int_nat_fib transfer_int_nat_fib_closure] |
|
74 |
|
75 |
25 |
76 subsection {* Fibonacci numbers *} |
26 subsection {* Fibonacci numbers *} |
77 |
27 |
78 lemma fib_0_nat [simp]: "fib (0::nat) = 0" |
28 lemma fib_1 [simp]: "fib (1::nat) = 1" |
79 by simp |
29 by (metis One_nat_def fib1) |
80 |
30 |
81 lemma fib_0_int [simp]: "fib (0::int) = 0" |
31 lemma fib_2 [simp]: "fib (2::nat) = 1" |
82 unfolding fib_int_def by simp |
32 using fib.simps(3) [of 0] |
|
33 by (simp add: numeral_2_eq_2) |
83 |
34 |
84 lemma fib_1_nat [simp]: "fib (1::nat) = 1" |
35 lemma fib_plus_2: "fib (n + 2) = fib (n + 1) + fib n" |
85 by simp |
36 by (metis Suc_eq_plus1 add_2_eq_Suc' fib.simps(3)) |
86 |
37 |
87 lemma fib_Suc_0_nat [simp]: "fib (Suc 0) = Suc 0" |
38 lemma fib_add: "fib (Suc (n+k)) = fib (Suc k) * fib (Suc n) + fib k * fib n" |
88 by simp |
39 by (induct n rule: fib.induct) (auto simp add: field_simps) |
89 |
40 |
90 lemma fib_1_int [simp]: "fib (1::int) = 1" |
41 lemma fib_neq_0_nat: "n > 0 \<Longrightarrow> fib n > 0" |
91 unfolding fib_int_def by simp |
42 by (induct n rule: fib.induct) (auto simp add: ) |
92 |
|
93 lemma fib_reduce_nat: "(n::nat) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)" |
|
94 by simp |
|
95 |
|
96 declare fib_nat.simps [simp del] |
|
97 |
|
98 lemma fib_reduce_int: "(n::int) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)" |
|
99 unfolding fib_int_def |
|
100 by (auto simp add: fib_reduce_nat nat_diff_distrib) |
|
101 |
|
102 lemma fib_neg_int [simp]: "(n::int) < 0 \<Longrightarrow> fib n = 0" |
|
103 unfolding fib_int_def by auto |
|
104 |
|
105 lemma fib_2_nat [simp]: "fib (2::nat) = 1" |
|
106 by (subst fib_reduce_nat, auto) |
|
107 |
|
108 lemma fib_2_int [simp]: "fib (2::int) = 1" |
|
109 by (subst fib_reduce_int, auto) |
|
110 |
|
111 lemma fib_plus_2_nat: "fib ((n::nat) + 2) = fib (n + 1) + fib n" |
|
112 by (subst fib_reduce_nat, auto simp add: One_nat_def) |
|
113 (* the need for One_nat_def is due to the natdiff_cancel_numerals |
|
114 procedure *) |
|
115 |
|
116 lemma fib_induct_nat: "P (0::nat) \<Longrightarrow> P (1::nat) \<Longrightarrow> |
|
117 (!!n. P n \<Longrightarrow> P (n + 1) \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n" |
|
118 apply (atomize, induct n rule: nat_less_induct) |
|
119 apply auto |
|
120 apply (case_tac "n = 0", force) |
|
121 apply (case_tac "n = 1", force) |
|
122 apply (subgoal_tac "n >= 2") |
|
123 apply (frule_tac x = "n - 1" in spec) |
|
124 apply (drule_tac x = "n - 2" in spec) |
|
125 apply (drule_tac x = "n - 2" in spec) |
|
126 apply auto |
|
127 apply (auto simp add: One_nat_def) (* again, natdiff_cancel *) |
|
128 done |
|
129 |
|
130 lemma fib_add_nat: "fib ((n::nat) + k + 1) = fib (k + 1) * fib (n + 1) + |
|
131 fib k * fib n" |
|
132 apply (induct n rule: fib_induct_nat) |
|
133 apply auto |
|
134 apply (subst fib_reduce_nat) |
|
135 apply (auto simp add: field_simps) |
|
136 apply (subst (1 3 5) fib_reduce_nat) |
|
137 apply (auto simp add: field_simps Suc_eq_plus1) |
|
138 (* hmmm. Why doesn't "n + (1 + (1 + k))" simplify to "n + k + 2"? *) |
|
139 apply (subgoal_tac "n + (k + 2) = n + (1 + (1 + k))") |
|
140 apply (erule ssubst) back back |
|
141 apply (erule ssubst) back |
|
142 apply auto |
|
143 done |
|
144 |
|
145 lemma fib_add'_nat: "fib (n + Suc k) = |
|
146 fib (Suc k) * fib (Suc n) + fib k * fib n" |
|
147 using fib_add_nat by (auto simp add: One_nat_def) |
|
148 |
|
149 |
|
150 (* transfer from nats to ints *) |
|
151 lemma fib_add_int: "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow> |
|
152 fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n " |
|
153 by (rule fib_add_nat [transferred]) |
|
154 |
|
155 lemma fib_neq_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n ~= 0" |
|
156 apply (induct n rule: fib_induct_nat) |
|
157 apply (auto simp add: fib_plus_2_nat) |
|
158 done |
|
159 |
|
160 lemma fib_gr_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n > 0" |
|
161 by (frule fib_neq_0_nat, simp) |
|
162 |
|
163 lemma fib_gr_0_int: "(n::int) > 0 \<Longrightarrow> fib n > 0" |
|
164 unfolding fib_int_def by (simp add: fib_gr_0_nat) |
|
165 |
43 |
166 text {* |
44 text {* |
167 \medskip Concrete Mathematics, page 278: Cassini's identity. The proof is |
45 \medskip Concrete Mathematics, page 278: Cassini's identity. The proof is |
168 much easier using integers, not natural numbers! |
46 much easier using integers, not natural numbers! |
169 *} |
47 *} |
170 |
48 |
171 lemma fib_Cassini_aux_int: "fib (int n + 2) * fib (int n) - |
49 lemma fib_Cassini_int: "int (fib (Suc (Suc n)) * fib n) - int((fib (Suc n))\<^sup>2) = - ((-1)^n)" |
172 (fib (int n + 1))\<^sup>2 = (-1)^(n + 1)" |
50 by (induction n rule: fib.induct) (auto simp add: field_simps power2_eq_square power_add) |
173 apply (induct n) |
|
174 apply (auto simp add: field_simps power2_eq_square fib_reduce_int power_add) |
|
175 done |
|
176 |
51 |
177 lemma fib_Cassini_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n - |
52 lemma fib_Cassini_nat: |
178 (fib (n + 1))\<^sup>2 = (-1)^(nat n + 1)" |
53 "fib (Suc (Suc n)) * fib n = |
179 by (insert fib_Cassini_aux_int [of "nat n"], auto) |
54 (if even n then (fib (Suc n))\<^sup>2 - 1 else (fib (Suc n))\<^sup>2 + 1)" |
180 |
55 using fib_Cassini_int [of n] by auto |
181 (* |
|
182 lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n = |
|
183 (fib (n + 1))\<^sup>2 + (-1)^(nat n + 1)" |
|
184 by (frule fib_Cassini_int, simp) |
|
185 *) |
|
186 |
|
187 lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib ((n::int) + 2) * fib n = |
|
188 (if even n then tsub ((fib (n + 1))\<^sup>2) 1 |
|
189 else (fib (n + 1))\<^sup>2 + 1)" |
|
190 apply (frule fib_Cassini_int, auto simp add: pos_int_even_equiv_nat_even) |
|
191 apply (subst tsub_eq) |
|
192 apply (insert fib_gr_0_int [of "n + 1"], force) |
|
193 apply auto |
|
194 done |
|
195 |
|
196 lemma fib_Cassini_nat: "fib ((n::nat) + 2) * fib n = |
|
197 (if even n then (fib (n + 1))\<^sup>2 - 1 |
|
198 else (fib (n + 1))\<^sup>2 + 1)" |
|
199 by (rule fib_Cassini'_int [transferred, of n], auto) |
|
200 |
56 |
201 |
57 |
202 text {* \medskip Toward Law 6.111 of Concrete Mathematics *} |
58 text {* \medskip Toward Law 6.111 of Concrete Mathematics *} |
203 |
59 |
204 lemma coprime_fib_plus_1_nat: "coprime (fib (n::nat)) (fib (n + 1))" |
60 lemma coprime_fib_Suc_nat: "coprime (fib (n::nat)) (fib (Suc n))" |
205 apply (induct n rule: fib_induct_nat) |
61 apply (induct n rule: fib.induct) |
206 apply auto |
62 apply auto |
207 apply (subst (2) fib_reduce_nat) |
63 apply (metis gcd_add1_nat nat_add_commute) |
208 apply (auto simp add: Suc_eq_plus1) (* again, natdiff_cancel *) |
|
209 apply (subst add_commute, auto) |
|
210 apply (subst gcd_commute_nat, auto simp add: field_simps) |
|
211 done |
64 done |
212 |
65 |
213 lemma coprime_fib_Suc_nat: "coprime (fib n) (fib (Suc n))" |
66 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)" |
214 using coprime_fib_plus_1_nat by (simp add: One_nat_def) |
|
215 |
|
216 lemma coprime_fib_plus_1_int: "n >= 0 \<Longrightarrow> coprime (fib (n::int)) (fib (n + 1))" |
|
217 by (erule coprime_fib_plus_1_nat [transferred]) |
|
218 |
|
219 lemma gcd_fib_add_nat: "gcd (fib (m::nat)) (fib (n + m)) = gcd (fib m) (fib n)" |
|
220 apply (simp add: gcd_commute_nat [of "fib m"]) |
67 apply (simp add: gcd_commute_nat [of "fib m"]) |
221 apply (rule cases_nat [of _ m]) |
68 apply (cases m) |
222 apply simp |
69 apply (auto simp add: fib_add) |
223 apply (subst add_assoc [symmetric]) |
|
224 apply (simp add: fib_add_nat) |
|
225 apply (subst gcd_commute_nat) |
70 apply (subst gcd_commute_nat) |
226 apply (subst mult_commute) |
71 apply (subst mult_commute) |
227 apply (subst gcd_add_mult_nat) |
72 apply (metis coprime_fib_Suc_nat gcd_add_mult_nat gcd_mult_cancel_nat gcd_nat.commute) |
228 apply (subst gcd_commute_nat) |
|
229 apply (rule gcd_mult_cancel_nat) |
|
230 apply (rule coprime_fib_plus_1_nat) |
|
231 done |
73 done |
232 |
74 |
233 lemma gcd_fib_add_int [rule_format]: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow> |
75 lemma gcd_fib_diff: "m \<le> n \<Longrightarrow> |
234 gcd (fib (m::int)) (fib (n + m)) = gcd (fib m) (fib n)" |
76 gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)" |
235 by (erule gcd_fib_add_nat [transferred]) |
77 by (simp add: gcd_fib_add [symmetric, of _ "n-m"]) |
236 |
78 |
237 lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow> |
79 lemma gcd_fib_mod: "0 < m \<Longrightarrow> |
238 gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)" |
|
239 by (simp add: gcd_fib_add_nat [symmetric, of _ "n-m"]) |
|
240 |
|
241 lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow> |
|
242 gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)" |
|
243 by (simp add: gcd_fib_add_int [symmetric, of _ "n-m"]) |
|
244 |
|
245 lemma gcd_fib_mod_nat: "0 < (m::nat) \<Longrightarrow> |
|
246 gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
80 gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
247 proof (induct n rule: less_induct) |
81 proof (induct n rule: less_induct) |
248 case (less n) |
82 case (less n) |
249 from less.prems have pos_m: "0 < m" . |
83 from less.prems have pos_m: "0 < m" . |
250 show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
84 show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
256 have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))" |
90 have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))" |
257 by (simp add: mod_if [of n]) (insert `m < n`, auto) |
91 by (simp add: mod_if [of n]) (insert `m < n`, auto) |
258 also have "\<dots> = gcd (fib m) (fib (n - m))" |
92 also have "\<dots> = gcd (fib m) (fib (n - m))" |
259 by (simp add: less.hyps diff pos_m) |
93 by (simp add: less.hyps diff pos_m) |
260 also have "\<dots> = gcd (fib m) (fib n)" |
94 also have "\<dots> = gcd (fib m) (fib n)" |
261 by (simp add: gcd_fib_diff_nat `m \<le> n`) |
95 by (simp add: gcd_fib_diff `m \<le> n`) |
262 finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" . |
96 finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" . |
263 next |
97 next |
264 case False |
98 case False |
265 then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
99 then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
266 by (cases "m = n") auto |
100 by (cases "m = n") auto |
267 qed |
101 qed |
268 qed |
102 qed |
269 |
103 |
270 lemma gcd_fib_mod_int: |
104 lemma fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)" |
271 assumes "0 < (m::int)" and "0 <= n" |
|
272 shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" |
|
273 apply (rule gcd_fib_mod_nat [transferred]) |
|
274 using assms apply auto |
|
275 done |
|
276 |
|
277 lemma fib_gcd_nat: "fib (gcd (m::nat) n) = gcd (fib m) (fib n)" |
|
278 -- {* Law 6.111 *} |
105 -- {* Law 6.111 *} |
279 apply (induct m n rule: gcd_nat_induct) |
106 by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod) |
280 apply (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod_nat) |
|
281 done |
|
282 |
|
283 lemma fib_gcd_int: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow> |
|
284 fib (gcd (m::int) n) = gcd (fib m) (fib n)" |
|
285 by (erule fib_gcd_nat [transferred]) |
|
286 |
|
287 lemma atMost_plus_one_nat: "{..(k::nat) + 1} = insert (k + 1) {..k}" |
|
288 by auto |
|
289 |
107 |
290 theorem fib_mult_eq_setsum_nat: |
108 theorem fib_mult_eq_setsum_nat: |
291 "fib ((n::nat) + 1) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)" |
|
292 apply (induct n) |
|
293 apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat field_simps) |
|
294 done |
|
295 |
|
296 theorem fib_mult_eq_setsum'_nat: |
|
297 "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)" |
109 "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)" |
298 using fib_mult_eq_setsum_nat by (simp add: One_nat_def) |
110 by (induct n rule: nat.induct) (auto simp add: field_simps) |
299 |
|
300 theorem fib_mult_eq_setsum_int [rule_format]: |
|
301 "n >= 0 \<Longrightarrow> fib ((n::int) + 1) * fib n = (\<Sum>k \<in> {0..n}. fib k * fib k)" |
|
302 by (erule fib_mult_eq_setsum_nat [transferred]) |
|
303 |
111 |
304 end |
112 end |
|
113 |