src/HOL/Number_Theory/Fib.thy
changeset 32479 521cc9bf2958
parent 31952 40501bb2d57c
child 35644 d20cf282342e
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Number_Theory/Fib.thy	Tue Sep 01 15:39:33 2009 +0200
     1.3 @@ -0,0 +1,319 @@
     1.4 +(*  Title:      Fib.thy
     1.5 +    Authors:    Lawrence C. Paulson, Jeremy Avigad
     1.6 +
     1.7 +
     1.8 +Defines the fibonacci function.
     1.9 +
    1.10 +The original "Fib" is due to Lawrence C. Paulson, and was adapted by
    1.11 +Jeremy Avigad.
    1.12 +*)
    1.13 +
    1.14 +
    1.15 +header {* Fib *}
    1.16 +
    1.17 +theory Fib
    1.18 +imports Binomial
    1.19 +begin
    1.20 +
    1.21 +
    1.22 +subsection {* Main definitions *}
    1.23 +
    1.24 +class fib =
    1.25 +
    1.26 +fixes 
    1.27 +  fib :: "'a \<Rightarrow> 'a"
    1.28 +
    1.29 +
    1.30 +(* definition for the natural numbers *)
    1.31 +
    1.32 +instantiation nat :: fib
    1.33 +
    1.34 +begin 
    1.35 +
    1.36 +fun 
    1.37 +  fib_nat :: "nat \<Rightarrow> nat"
    1.38 +where
    1.39 +  "fib_nat n =
    1.40 +   (if n = 0 then 0 else
    1.41 +   (if n = 1 then 1 else
    1.42 +     fib (n - 1) + fib (n - 2)))"
    1.43 +
    1.44 +instance proof qed
    1.45 +
    1.46 +end
    1.47 +
    1.48 +(* definition for the integers *)
    1.49 +
    1.50 +instantiation int :: fib
    1.51 +
    1.52 +begin 
    1.53 +
    1.54 +definition
    1.55 +  fib_int :: "int \<Rightarrow> int"
    1.56 +where  
    1.57 +  "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)"
    1.58 +
    1.59 +instance proof qed
    1.60 +
    1.61 +end
    1.62 +
    1.63 +
    1.64 +subsection {* Set up Transfer *}
    1.65 +
    1.66 +
    1.67 +lemma transfer_nat_int_fib:
    1.68 +  "(x::int) >= 0 \<Longrightarrow> fib (nat x) = nat (fib x)"
    1.69 +  unfolding fib_int_def by auto
    1.70 +
    1.71 +lemma transfer_nat_int_fib_closure:
    1.72 +  "n >= (0::int) \<Longrightarrow> fib n >= 0"
    1.73 +  by (auto simp add: fib_int_def)
    1.74 +
    1.75 +declare TransferMorphism_nat_int[transfer add return: 
    1.76 +    transfer_nat_int_fib transfer_nat_int_fib_closure]
    1.77 +
    1.78 +lemma transfer_int_nat_fib:
    1.79 +  "fib (int n) = int (fib n)"
    1.80 +  unfolding fib_int_def by auto
    1.81 +
    1.82 +lemma transfer_int_nat_fib_closure:
    1.83 +  "is_nat n \<Longrightarrow> fib n >= 0"
    1.84 +  unfolding fib_int_def by auto
    1.85 +
    1.86 +declare TransferMorphism_int_nat[transfer add return: 
    1.87 +    transfer_int_nat_fib transfer_int_nat_fib_closure]
    1.88 +
    1.89 +
    1.90 +subsection {* Fibonacci numbers *}
    1.91 +
    1.92 +lemma fib_0_nat [simp]: "fib (0::nat) = 0"
    1.93 +  by simp
    1.94 +
    1.95 +lemma fib_0_int [simp]: "fib (0::int) = 0"
    1.96 +  unfolding fib_int_def by simp
    1.97 +
    1.98 +lemma fib_1_nat [simp]: "fib (1::nat) = 1"
    1.99 +  by simp
   1.100 +
   1.101 +lemma fib_Suc_0_nat [simp]: "fib (Suc 0) = Suc 0"
   1.102 +  by simp
   1.103 +
   1.104 +lemma fib_1_int [simp]: "fib (1::int) = 1"
   1.105 +  unfolding fib_int_def by simp
   1.106 +
   1.107 +lemma fib_reduce_nat: "(n::nat) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
   1.108 +  by simp
   1.109 +
   1.110 +declare fib_nat.simps [simp del]
   1.111 +
   1.112 +lemma fib_reduce_int: "(n::int) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
   1.113 +  unfolding fib_int_def
   1.114 +  by (auto simp add: fib_reduce_nat nat_diff_distrib)
   1.115 +
   1.116 +lemma fib_neg_int [simp]: "(n::int) < 0 \<Longrightarrow> fib n = 0"
   1.117 +  unfolding fib_int_def by auto
   1.118 +
   1.119 +lemma fib_2_nat [simp]: "fib (2::nat) = 1"
   1.120 +  by (subst fib_reduce_nat, auto)
   1.121 +
   1.122 +lemma fib_2_int [simp]: "fib (2::int) = 1"
   1.123 +  by (subst fib_reduce_int, auto)
   1.124 +
   1.125 +lemma fib_plus_2_nat: "fib ((n::nat) + 2) = fib (n + 1) + fib n"
   1.126 +  by (subst fib_reduce_nat, auto simp add: One_nat_def)
   1.127 +(* the need for One_nat_def is due to the natdiff_cancel_numerals
   1.128 +   procedure *)
   1.129 +
   1.130 +lemma fib_induct_nat: "P (0::nat) \<Longrightarrow> P (1::nat) \<Longrightarrow> 
   1.131 +    (!!n. P n \<Longrightarrow> P (n + 1) \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
   1.132 +  apply (atomize, induct n rule: nat_less_induct)
   1.133 +  apply auto
   1.134 +  apply (case_tac "n = 0", force)
   1.135 +  apply (case_tac "n = 1", force)
   1.136 +  apply (subgoal_tac "n >= 2")
   1.137 +  apply (frule_tac x = "n - 1" in spec)
   1.138 +  apply (drule_tac x = "n - 2" in spec)
   1.139 +  apply (drule_tac x = "n - 2" in spec)
   1.140 +  apply auto
   1.141 +  apply (auto simp add: One_nat_def) (* again, natdiff_cancel *)
   1.142 +done
   1.143 +
   1.144 +lemma fib_add_nat: "fib ((n::nat) + k + 1) = fib (k + 1) * fib (n + 1) + 
   1.145 +    fib k * fib n"
   1.146 +  apply (induct n rule: fib_induct_nat)
   1.147 +  apply auto
   1.148 +  apply (subst fib_reduce_nat)
   1.149 +  apply (auto simp add: ring_simps)
   1.150 +  apply (subst (1 3 5) fib_reduce_nat)
   1.151 +  apply (auto simp add: ring_simps Suc_eq_plus1)
   1.152 +(* hmmm. Why doesn't "n + (1 + (1 + k))" simplify to "n + k + 2"? *)
   1.153 +  apply (subgoal_tac "n + (k + 2) = n + (1 + (1 + k))")
   1.154 +  apply (erule ssubst) back back
   1.155 +  apply (erule ssubst) back 
   1.156 +  apply auto
   1.157 +done
   1.158 +
   1.159 +lemma fib_add'_nat: "fib (n + Suc k) = fib (Suc k) * fib (Suc n) + 
   1.160 +    fib k * fib n"
   1.161 +  using fib_add_nat by (auto simp add: One_nat_def)
   1.162 +
   1.163 +
   1.164 +(* transfer from nats to ints *)
   1.165 +lemma fib_add_int [rule_format]: "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow>
   1.166 +    fib (n + k + 1) = fib (k + 1) * fib (n + 1) + 
   1.167 +    fib k * fib n "
   1.168 +
   1.169 +  by (rule fib_add_nat [transferred])
   1.170 +
   1.171 +lemma fib_neq_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n ~= 0"
   1.172 +  apply (induct n rule: fib_induct_nat)
   1.173 +  apply (auto simp add: fib_plus_2_nat)
   1.174 +done
   1.175 +
   1.176 +lemma fib_gr_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n > 0"
   1.177 +  by (frule fib_neq_0_nat, simp)
   1.178 +
   1.179 +lemma fib_gr_0_int: "(n::int) > 0 \<Longrightarrow> fib n > 0"
   1.180 +  unfolding fib_int_def by (simp add: fib_gr_0_nat)
   1.181 +
   1.182 +text {*
   1.183 +  \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
   1.184 +  much easier using integers, not natural numbers!
   1.185 +*}
   1.186 +
   1.187 +lemma fib_Cassini_aux_int: "fib (int n + 2) * fib (int n) - 
   1.188 +    (fib (int n + 1))^2 = (-1)^(n + 1)"
   1.189 +  apply (induct n)
   1.190 +  apply (auto simp add: ring_simps power2_eq_square fib_reduce_int
   1.191 +      power_add)
   1.192 +done
   1.193 +
   1.194 +lemma fib_Cassini_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n - 
   1.195 +    (fib (n + 1))^2 = (-1)^(nat n + 1)"
   1.196 +  by (insert fib_Cassini_aux_int [of "nat n"], auto)
   1.197 +
   1.198 +(*
   1.199 +lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n = 
   1.200 +    (fib (n + 1))^2 + (-1)^(nat n + 1)"
   1.201 +  by (frule fib_Cassini_int, simp) 
   1.202 +*)
   1.203 +
   1.204 +lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib ((n::int) + 2) * fib n =
   1.205 +  (if even n then tsub ((fib (n + 1))^2) 1
   1.206 +   else (fib (n + 1))^2 + 1)"
   1.207 +  apply (frule fib_Cassini_int, auto simp add: pos_int_even_equiv_nat_even)
   1.208 +  apply (subst tsub_eq)
   1.209 +  apply (insert fib_gr_0_int [of "n + 1"], force)
   1.210 +  apply auto
   1.211 +done
   1.212 +
   1.213 +lemma fib_Cassini_nat: "fib ((n::nat) + 2) * fib n =
   1.214 +  (if even n then (fib (n + 1))^2 - 1
   1.215 +   else (fib (n + 1))^2 + 1)"
   1.216 +
   1.217 +  by (rule fib_Cassini'_int [transferred, of n], auto)
   1.218 +
   1.219 +
   1.220 +text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
   1.221 +
   1.222 +lemma coprime_fib_plus_1_nat: "coprime (fib (n::nat)) (fib (n + 1))"
   1.223 +  apply (induct n rule: fib_induct_nat)
   1.224 +  apply auto
   1.225 +  apply (subst (2) fib_reduce_nat)
   1.226 +  apply (auto simp add: Suc_eq_plus1) (* again, natdiff_cancel *)
   1.227 +  apply (subst add_commute, auto)
   1.228 +  apply (subst gcd_commute_nat, auto simp add: ring_simps)
   1.229 +done
   1.230 +
   1.231 +lemma coprime_fib_Suc_nat: "coprime (fib n) (fib (Suc n))"
   1.232 +  using coprime_fib_plus_1_nat by (simp add: One_nat_def)
   1.233 +
   1.234 +lemma coprime_fib_plus_1_int: 
   1.235 +    "n >= 0 \<Longrightarrow> coprime (fib (n::int)) (fib (n + 1))"
   1.236 +  by (erule coprime_fib_plus_1_nat [transferred])
   1.237 +
   1.238 +lemma gcd_fib_add_nat: "gcd (fib (m::nat)) (fib (n + m)) = gcd (fib m) (fib n)"
   1.239 +  apply (simp add: gcd_commute_nat [of "fib m"])
   1.240 +  apply (rule cases_nat [of _ m])
   1.241 +  apply simp
   1.242 +  apply (subst add_assoc [symmetric])
   1.243 +  apply (simp add: fib_add_nat)
   1.244 +  apply (subst gcd_commute_nat)
   1.245 +  apply (subst mult_commute)
   1.246 +  apply (subst gcd_add_mult_nat)
   1.247 +  apply (subst gcd_commute_nat)
   1.248 +  apply (rule gcd_mult_cancel_nat)
   1.249 +  apply (rule coprime_fib_plus_1_nat)
   1.250 +done
   1.251 +
   1.252 +lemma gcd_fib_add_int [rule_format]: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow> 
   1.253 +    gcd (fib (m::int)) (fib (n + m)) = gcd (fib m) (fib n)"
   1.254 +  by (erule gcd_fib_add_nat [transferred])
   1.255 +
   1.256 +lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow> 
   1.257 +    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
   1.258 +  by (simp add: gcd_fib_add_nat [symmetric, of _ "n-m"])
   1.259 +
   1.260 +lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow> 
   1.261 +    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
   1.262 +  by (simp add: gcd_fib_add_int [symmetric, of _ "n-m"])
   1.263 +
   1.264 +lemma gcd_fib_mod_nat: "0 < (m::nat) \<Longrightarrow> 
   1.265 +    gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
   1.266 +proof (induct n rule: less_induct)
   1.267 +  case (less n)
   1.268 +  from less.prems have pos_m: "0 < m" .
   1.269 +  show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
   1.270 +  proof (cases "m < n")
   1.271 +    case True note m_n = True
   1.272 +    then have m_n': "m \<le> n" by auto
   1.273 +    with pos_m have pos_n: "0 < n" by auto
   1.274 +    with pos_m m_n have diff: "n - m < n" by auto
   1.275 +    have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
   1.276 +    by (simp add: mod_if [of n]) (insert m_n, auto)
   1.277 +    also have "\<dots> = gcd (fib m)  (fib (n - m))" 
   1.278 +      by (simp add: less.hyps diff pos_m)
   1.279 +    also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff_nat m_n')
   1.280 +    finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
   1.281 +  next
   1.282 +    case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
   1.283 +    by (cases "m = n") auto
   1.284 +  qed
   1.285 +qed
   1.286 +
   1.287 +lemma gcd_fib_mod_int: 
   1.288 +  assumes "0 < (m::int)" and "0 <= n"
   1.289 +  shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
   1.290 +
   1.291 +  apply (rule gcd_fib_mod_nat [transferred])
   1.292 +  using prems apply auto
   1.293 +done
   1.294 +
   1.295 +lemma fib_gcd_nat: "fib (gcd (m::nat) n) = gcd (fib m) (fib n)"  
   1.296 +    -- {* Law 6.111 *}
   1.297 +  apply (induct m n rule: gcd_nat_induct)
   1.298 +  apply (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod_nat)
   1.299 +done
   1.300 +
   1.301 +lemma fib_gcd_int: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
   1.302 +    fib (gcd (m::int) n) = gcd (fib m) (fib n)"
   1.303 +  by (erule fib_gcd_nat [transferred])
   1.304 +
   1.305 +lemma atMost_plus_one_nat: "{..(k::nat) + 1} = insert (k + 1) {..k}" 
   1.306 +  by auto
   1.307 +
   1.308 +theorem fib_mult_eq_setsum_nat:
   1.309 +    "fib ((n::nat) + 1) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
   1.310 +  apply (induct n)
   1.311 +  apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat ring_simps)
   1.312 +done
   1.313 +
   1.314 +theorem fib_mult_eq_setsum'_nat:
   1.315 +    "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
   1.316 +  using fib_mult_eq_setsum_nat by (simp add: One_nat_def)
   1.317 +
   1.318 +theorem fib_mult_eq_setsum_int [rule_format]:
   1.319 +    "n >= 0 \<Longrightarrow> fib ((n::int) + 1) * fib n = (\<Sum>k \<in> {0..n}. fib k * fib k)"
   1.320 +  by (erule fib_mult_eq_setsum_nat [transferred])
   1.321 +
   1.322 +end