src/HOL/Old_Number_Theory/Fib.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63167 0909deb8059b child 64267 b9a1486e79be permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Old_Number_Theory/Fib.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1997  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section \<open>The Fibonacci function\<close>
```
```     7
```
```     8 theory Fib
```
```     9 imports Primes
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```    10 begin
```
```    11
```
```    12 text \<open>
```
```    13   Fibonacci numbers: proofs of laws taken from:
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```    14   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
```
```    15   (Addison-Wesley, 1989)
```
```    16
```
```    17   \bigskip
```
```    18 \<close>
```
```    19
```
```    20 fun fib :: "nat \<Rightarrow> nat"
```
```    21 where
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```    22   "fib 0 = 0"
```
```    23 | "fib (Suc 0) = 1"
```
```    24 | fib_2: "fib (Suc (Suc n)) = fib n + fib (Suc n)"
```
```    25
```
```    26 text \<open>
```
```    27   \medskip The difficulty in these proofs is to ensure that the
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```    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 \<close>
```
```    32
```
```    33 text\<open>We disable \<open>fib.fib_2fib_2\<close> for simplification ...\<close>
```
```    34 declare fib_2 [simp del]
```
```    35
```
```    36 text\<open>...then prove a version that has a more restrictive pattern.\<close>
```
```    37 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
```
```    38   by (rule fib_2)
```
```    39
```
```    40 text \<open>\medskip Concrete Mathematics, page 280\<close>
```
```    41
```
```    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
```
```    50
```
```    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
```
```    55
```
```    56 lemma fib_Suc_gr_0: "0 < fib (Suc n)"
```
```    57   by (insert fib_Suc_neq_0 [of n], simp)
```
```    58
```
```    59 lemma fib_gr_0: "0 < n ==> 0 < fib n"
```
```    60   by (case_tac n, auto simp add: fib_Suc_gr_0)
```
```    61
```
```    62
```
```    63 text \<open>
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```    64   \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
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```    65   much easier using integers, not natural numbers!
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```    66 \<close>
```
```    67
```
```    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
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```    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
```
```    81
```
```    82 text\<open>We now obtain a version for the natural numbers via the coercion
```
```    83    function @{term int}.\<close>
```
```    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 of_nat_eq_iff [where 'a = int, THEN iffD1])
```
```    89   using fib_Cassini_int apply (auto simp add: Suc_leI fib_Suc_gr_0 of_nat_diff)
```
```    90   done
```
```    91
```
```    92
```
```    93 text \<open>\medskip Toward Law 6.111 of Concrete Mathematics\<close>
```
```    94
```
```    95 lemma gcd_fib_Suc_eq_1: "gcd (fib n) (fib (Suc n)) = Suc 0"
```
```    96   apply (induct n rule: fib.induct)
```
```    97     prefer 3
```
```    98     apply (simp add: gcd_commute fib_Suc3)
```
```    99    apply (simp_all add: fib_2)
```
```   100   done
```
```   101
```
```   102 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
```
```   103   apply (simp add: gcd_commute [of "fib m"])
```
```   104   apply (case_tac m)
```
```   105    apply simp
```
```   106   apply (simp add: fib_add)
```
```   107   apply (simp add: add.commute gcd_non_0 [OF fib_Suc_gr_0])
```
```   108   apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
```
```   109   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
```
```   110   done
```
```   111
```
```   112 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
```
```   113   by (simp add: gcd_fib_add [symmetric, of _ "n-m"])
```
```   114
```
```   115 lemma gcd_fib_mod: "0 < m ==> gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
```
```   116 proof (induct n rule: less_induct)
```
```   117   case (less n)
```
```   118   from less.prems have pos_m: "0 < m" .
```
```   119   show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
```
```   120   proof (cases "m < n")
```
```   121     case True note m_n = True
```
```   122     then have m_n': "m \<le> n" by auto
```
```   123     with pos_m have pos_n: "0 < n" by auto
```
```   124     with pos_m m_n have diff: "n - m < n" by auto
```
```   125     have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
```
```   126     by (simp add: mod_if [of n]) (insert m_n, auto)
```
```   127     also have "\<dots> = gcd (fib m) (fib (n - m))" by (simp add: less.hyps diff pos_m)
```
```   128     also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff m_n')
```
```   129     finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
```
```   130   next
```
```   131     case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
```
```   132     by (cases "m = n") auto
```
```   133   qed
```
```   134 qed
```
```   135
```
```   136 lemma fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)"  \<comment> \<open>Law 6.111\<close>
```
```   137   apply (induct m n rule: gcd_induct)
```
```   138   apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
```
```   139   done
```
```   140
```
```   141 theorem fib_mult_eq_setsum:
```
```   142     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
```
```   143   apply (induct n rule: fib.induct)
```
```   144     apply (auto simp add: atMost_Suc fib_2)
```
```   145   apply (simp add: add_mult_distrib add_mult_distrib2)
```
```   146   done
```
```   147
```
```   148 end
```