34 lemma fib_add: "fib (Suc (n+k)) = fib (Suc k) * fib (Suc n) + fib k * fib n" |
35 lemma fib_add: "fib (Suc (n+k)) = fib (Suc k) * fib (Suc n) + fib k * fib n" |
35 by (induct n rule: fib.induct) (auto simp add: field_simps) |
36 by (induct n rule: fib.induct) (auto simp add: field_simps) |
36 |
37 |
37 lemma fib_neq_0_nat: "n > 0 \<Longrightarrow> fib n > 0" |
38 lemma fib_neq_0_nat: "n > 0 \<Longrightarrow> fib n > 0" |
38 by (induct n rule: fib.induct) (auto simp add: ) |
39 by (induct n rule: fib.induct) (auto simp add: ) |
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40 |
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41 |
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42 subsection \<open>More efficient code\<close> |
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43 |
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44 text \<open> |
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45 The naive approach is very inefficient since the branching recursion leads to many |
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46 values of @{term fib} being computed multiple times. We can avoid this by ``remembering'' |
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47 the last two values in the sequence, yielding a tail-recursive version. |
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48 This is far from optimal (it takes roughly $O(n\cdot M(n))$ time where $M(n)$ is the |
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49 time required to multiply two $n$-bit integers), but much better than the naive version, |
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50 which is exponential. |
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51 \<close> |
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52 |
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53 fun gen_fib :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" where |
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54 "gen_fib a b 0 = a" |
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55 | "gen_fib a b (Suc 0) = b" |
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56 | "gen_fib a b (Suc (Suc n)) = gen_fib b (a + b) (Suc n)" |
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57 |
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58 lemma gen_fib_recurrence: "gen_fib a b (Suc (Suc n)) = gen_fib a b n + gen_fib a b (Suc n)" |
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59 by (induction a b n rule: gen_fib.induct) simp_all |
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60 |
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61 lemma gen_fib_fib: "gen_fib (fib n) (fib (Suc n)) m = fib (n + m)" |
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62 by (induction m rule: fib.induct) (simp_all del: gen_fib.simps(3) add: gen_fib_recurrence) |
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63 |
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64 lemma fib_conv_gen_fib: "fib n = gen_fib 0 1 n" |
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65 using gen_fib_fib[of 0 n] by simp |
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66 |
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67 declare fib_conv_gen_fib [code] |
39 |
68 |
40 |
69 |
41 subsection \<open>A Few Elementary Results\<close> |
70 subsection \<open>A Few Elementary Results\<close> |
42 |
71 |
43 text \<open> |
72 text \<open> |
102 |
131 |
103 theorem fib_mult_eq_sum_nat: "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)" |
132 theorem fib_mult_eq_sum_nat: "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)" |
104 by (induct n rule: nat.induct) (auto simp add: field_simps) |
133 by (induct n rule: nat.induct) (auto simp add: field_simps) |
105 |
134 |
106 |
135 |
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136 subsection \<open>Closed form\<close> |
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137 |
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138 lemma fib_closed_form: |
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139 defines "\<phi> \<equiv> (1 + sqrt 5) / (2::real)" and "\<psi> \<equiv> (1 - sqrt 5) / (2::real)" |
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140 shows "of_nat (fib n) = (\<phi> ^ n - \<psi> ^ n) / sqrt 5" |
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141 proof (induction n rule: fib.induct) |
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142 fix n :: nat |
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143 assume IH1: "of_nat (fib n) = (\<phi> ^ n - \<psi> ^ n) / sqrt 5" |
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144 assume IH2: "of_nat (fib (Suc n)) = (\<phi> ^ Suc n - \<psi> ^ Suc n) / sqrt 5" |
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145 have "of_nat (fib (Suc (Suc n))) = of_nat (fib (Suc n)) + of_nat (fib n)" by simp |
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146 also have "... = (\<phi>^n*(\<phi> + 1) - \<psi>^n*(\<psi> + 1)) / sqrt 5" |
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147 by (simp add: IH1 IH2 field_simps) |
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148 also have "\<phi> + 1 = \<phi>\<^sup>2" by (simp add: \<phi>_def field_simps power2_eq_square) |
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149 also have "\<psi> + 1 = \<psi>\<^sup>2" by (simp add: \<psi>_def field_simps power2_eq_square) |
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150 also have "\<phi>^n * \<phi>\<^sup>2 - \<psi>^n * \<psi>\<^sup>2 = \<phi> ^ Suc (Suc n) - \<psi> ^ Suc (Suc n)" |
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151 by (simp add: power2_eq_square) |
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152 finally show "of_nat (fib (Suc (Suc n))) = (\<phi> ^ Suc (Suc n) - \<psi> ^ Suc (Suc n)) / sqrt 5" . |
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153 qed (simp_all add: \<phi>_def \<psi>_def field_simps) |
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154 |
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155 lemma fib_closed_form': |
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156 defines "\<phi> \<equiv> (1 + sqrt 5) / (2 :: real)" and "\<psi> \<equiv> (1 - sqrt 5) / (2 :: real)" |
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157 assumes "n > 0" |
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158 shows "fib n = round (\<phi> ^ n / sqrt 5)" |
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159 proof (rule sym, rule round_unique') |
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160 have "\<bar>\<phi> ^ n / sqrt 5 - of_int (int (fib n))\<bar> = \<bar>\<psi>\<bar> ^ n / sqrt 5" |
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161 by (simp add: fib_closed_form[folded \<phi>_def \<psi>_def] field_simps power_abs) |
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162 also { |
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163 from assms have "\<bar>\<psi>\<bar>^n \<le> \<bar>\<psi>\<bar>^1" |
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164 by (intro power_decreasing) (simp_all add: algebra_simps real_le_lsqrt) |
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165 also have "... < sqrt 5 / 2" by (simp add: \<psi>_def field_simps) |
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166 finally have "\<bar>\<psi>\<bar>^n / sqrt 5 < 1/2" by (simp add: field_simps) |
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167 } |
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168 finally show "\<bar>\<phi> ^ n / sqrt 5 - of_int (int (fib n))\<bar> < 1/2" . |
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169 qed |
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170 |
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171 lemma fib_asymptotics: |
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172 defines "\<phi> \<equiv> (1 + sqrt 5) / (2 :: real)" |
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173 shows "(\<lambda>n. real (fib n) / (\<phi> ^ n / sqrt 5)) \<longlonglongrightarrow> 1" |
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174 proof - |
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175 define \<psi> where "\<psi> \<equiv> (1 - sqrt 5) / (2 :: real)" |
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176 have "\<phi> > 1" by (simp add: \<phi>_def) |
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177 hence A: "\<phi> \<noteq> 0" by auto |
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178 have "(\<lambda>n. (\<psi> / \<phi>) ^ n) \<longlonglongrightarrow> 0" |
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179 by (rule LIMSEQ_power_zero) (simp_all add: \<phi>_def \<psi>_def field_simps add_pos_pos) |
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180 hence "(\<lambda>n. 1 - (\<psi> / \<phi>) ^ n) \<longlonglongrightarrow> 1 - 0" by (intro tendsto_diff tendsto_const) |
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181 with A show ?thesis |
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182 by (simp add: divide_simps fib_closed_form [folded \<phi>_def \<psi>_def]) |
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183 qed |
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184 |
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185 |
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186 subsection \<open>Divide-and-Conquer recurrence\<close> |
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187 |
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188 text \<open> |
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189 The following divide-and-conquer recurrence allows for a more efficient computation |
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190 of Fibonacci numbers; however, it requires memoisation of values to be reasonably |
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191 efficient, cutting the number of values to be computed to logarithmically many instead of |
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192 linearly many. The vast majority of the computation time is then actually spent on the |
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193 multiplication, since the output number is exponential in the input number. |
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194 \<close> |
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195 |
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196 lemma fib_rec_odd: |
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197 defines "\<phi> \<equiv> (1 + sqrt 5) / (2 :: real)" and "\<psi> \<equiv> (1 - sqrt 5) / (2 :: real)" |
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198 shows "fib (Suc (2*n)) = fib n^2 + fib (Suc n)^2" |
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199 proof - |
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200 have "of_nat (fib n^2 + fib (Suc n)^2) = ((\<phi> ^ n - \<psi> ^ n)\<^sup>2 + (\<phi> * \<phi> ^ n - \<psi> * \<psi> ^ n)\<^sup>2)/5" |
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201 by (simp add: fib_closed_form[folded \<phi>_def \<psi>_def] field_simps power2_eq_square) |
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202 also have "(\<phi> ^ n - \<psi> ^ n)\<^sup>2 + (\<phi> * \<phi> ^ n - \<psi> * \<psi> ^ n)\<^sup>2 = |
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203 \<phi>^(2*n) + \<psi>^(2*n) - 2*(\<phi>*\<psi>)^n + \<phi>^(2*n+2) + \<psi>^(2*n+2) - 2*(\<phi>*\<psi>)^(n+1)" (is "_ = ?A") |
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204 by (simp add: power2_eq_square algebra_simps power_mult power_mult_distrib) |
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205 also have "\<phi> * \<psi> = -1" by (simp add: \<phi>_def \<psi>_def field_simps) |
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206 hence "?A = \<phi>^(2*n+1) * (\<phi> + inverse \<phi>) + \<psi>^(2*n+1) * (\<psi> + inverse \<psi>)" |
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207 by (auto simp: field_simps power2_eq_square) |
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208 also have "1 + sqrt 5 > 0" by (auto intro: add_pos_pos) |
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209 hence "\<phi> + inverse \<phi> = sqrt 5" by (simp add: \<phi>_def field_simps) |
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210 also have "\<psi> + inverse \<psi> = -sqrt 5" by (simp add: \<psi>_def field_simps) |
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211 also have "(\<phi> ^ (2*n+1) * sqrt 5 + \<psi> ^ (2*n+1)* - sqrt 5) / 5 = |
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212 (\<phi> ^ (2*n+1) - \<psi> ^ (2*n+1)) * (sqrt 5 / 5)" by (simp add: field_simps) |
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213 also have "sqrt 5 / 5 = inverse (sqrt 5)" by (simp add: field_simps) |
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214 also have "(\<phi> ^ (2*n+1) - \<psi> ^ (2*n+1)) * ... = of_nat (fib (Suc (2*n)))" |
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215 by (simp add: fib_closed_form[folded \<phi>_def \<psi>_def] divide_inverse) |
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216 finally show ?thesis by (simp only: of_nat_eq_iff) |
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217 qed |
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218 |
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219 lemma fib_rec_even: "fib (2*n) = (fib (n - 1) + fib (n + 1)) * fib n" |
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220 proof (induction n) |
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221 case (Suc n) |
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222 let ?rfib = "\<lambda>x. real (fib x)" |
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223 have "2 * (Suc n) = Suc (Suc (2*n))" by simp |
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224 also have "real (fib ...) = ?rfib n^2 + ?rfib (Suc n)^2 + (?rfib (n - 1) + ?rfib (n + 1)) * ?rfib n" |
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225 by (simp add: fib_rec_odd Suc) |
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226 also have "(?rfib (n - 1) + ?rfib (n + 1)) * ?rfib n = (2 * ?rfib (n + 1) - ?rfib n) * ?rfib n" |
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227 by (cases n) simp_all |
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228 also have "?rfib n^2 + ?rfib (Suc n)^2 + ... = (?rfib (Suc n) + 2 * ?rfib n) * ?rfib (Suc n)" |
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229 by (simp add: algebra_simps power2_eq_square) |
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230 also have "... = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))" by simp |
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231 finally show ?case by (simp only: of_nat_eq_iff) |
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232 qed simp |
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233 |
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234 lemma fib_rec_even': "fib (2*n) = (2*fib (n - 1) + fib n) * fib n" |
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235 by (subst fib_rec_even, cases n) simp_all |
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236 |
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237 lemma fib_rec: |
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238 "fib n = (if n = 0 then 0 else if n = 1 then 1 else |
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239 if even n then let n' = n div 2; fn = fib n' in (2 * fib (n' - 1) + fn) * fn |
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240 else let n' = n div 2 in fib n' ^ 2 + fib (Suc n') ^ 2)" |
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241 by (auto elim: evenE oddE simp: fib_rec_odd fib_rec_even' Let_def) |
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242 |
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243 |
107 subsection \<open>Fibonacci and Binomial Coefficients\<close> |
244 subsection \<open>Fibonacci and Binomial Coefficients\<close> |
108 |
245 |
109 lemma sum_drop_zero: "(\<Sum>k = 0..Suc n. if 0<k then (f (k - 1)) else 0) = (\<Sum>j = 0..n. f j)" |
246 lemma sum_drop_zero: "(\<Sum>k = 0..Suc n. if 0<k then (f (k - 1)) else 0) = (\<Sum>j = 0..n. f j)" |
110 by (induct n) auto |
247 by (induct n) auto |
111 |
248 |