src/HOL/NumberTheory/Fib.thy
 author paulson Fri Sep 14 15:27:12 2007 +0200 (2007-09-14) changeset 24573 5bbdc9b60648 parent 24549 c8cee92b06bc child 25222 78943ac46f6d permissions -rw-r--r--
tidied
```     1 (*  ID:         \$Id\$
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```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     3     Copyright   1997  University of Cambridge
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```     4 *)
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```     5
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```     6 header {* The Fibonacci function *}
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```     7
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```     8 theory Fib imports Primes begin
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```     9
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```    10 text {*
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```    11   Fibonacci numbers: proofs of laws taken from:
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```    12   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
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```    13   (Addison-Wesley, 1989)
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```    14
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```    15   \bigskip
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```    16 *}
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```    17
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```    18 fun fib :: "nat \<Rightarrow> nat"
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```    19 where
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```    20          "fib 0 = 0"
```
```    21 |        "fib (Suc 0) = 1"
```
```    22 | fib_2: "fib (Suc (Suc n)) = fib n + fib (Suc n)"
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```    23
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```    24 text {*
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```    25   \medskip The difficulty in these proofs is to ensure that the
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```    26   induction hypotheses are applied before the definition of @{term
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```    27   fib}.  Towards this end, the @{term fib} equations are not declared
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```    28   to the Simplifier and are applied very selectively at first.
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```    29 *}
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```    30
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```    31 text{*We disable @{text fib.fib_2fib_2} for simplification ...*}
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```    32 declare fib_2 [simp del]
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```    33
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```    34 text{*...then prove a version that has a more restrictive pattern.*}
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```    35 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
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```    36   by (rule fib_2)
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```    37
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```    38 text {* \medskip Concrete Mathematics, page 280 *}
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```    39
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```    40 lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
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```    41 proof (induct n rule: fib.induct)
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```    42   case 1 show ?case by simp
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```    43 next
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```    44   case 2 show ?case  by (simp add: fib_2)
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```    45 next
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```    46   case 3 thus ?case by (simp add: fib_2 add_mult_distrib2)
```
```    47 qed
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```    48
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```    49 lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0"
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```    50   apply (induct n rule: fib.induct)
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```    51     apply (simp_all add: fib_2)
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```    52   done
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```    53
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```    54 lemma fib_Suc_gr_0: "0 < fib (Suc n)"
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```    55   by (insert fib_Suc_neq_0 [of n], simp)
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```    56
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```    57 lemma fib_gr_0: "0 < n ==> 0 < fib n"
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```    58   by (case_tac n, auto simp add: fib_Suc_gr_0)
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```    59
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```    60
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```    61 text {*
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```    62   \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
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```    63   much easier using integers, not natural numbers!
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```    64 *}
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```    65
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```    66 lemma fib_Cassini_int:
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```    67  "int (fib (Suc (Suc n)) * fib n) =
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```    68   (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
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```    69    else int (fib (Suc n) * fib (Suc n)) + 1)"
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```    70 proof(induct n rule: fib.induct)
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```    71   case 1 thus ?case by (simp add: fib_2)
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```    72 next
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```    73   case 2 thus ?case by (simp add: fib_2 mod_Suc)
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```    74 next
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```    75   case (3 x)
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```    76   have "Suc 0 \<noteq> x mod 2 \<longrightarrow> x mod 2 = 0" by presburger
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```    77   with "3.hyps" show ?case by (simp add: fib_2 add_mult_distrib add_mult_distrib2)
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```    78 qed
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```    79
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```    80 text{*We now obtain a version for the natural numbers via the coercion
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```    81    function @{term int}.*}
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```    82 theorem fib_Cassini:
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```    83  "fib (Suc (Suc n)) * fib n =
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```    84   (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
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```    85    else fib (Suc n) * fib (Suc n) + 1)"
```
```    86   apply (rule int_int_eq [THEN iffD1])
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```    87   apply (simp add: fib_Cassini_int)
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```    88   apply (subst zdiff_int [symmetric])
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```    89    apply (insert fib_Suc_gr_0 [of n], simp_all)
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```    90   done
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```    91
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```    92
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```    93 text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
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```    94
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```    95 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
```
```    96   apply (induct n rule: fib.induct)
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```    97     prefer 3
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```    98     apply (simp add: gcd_commute fib_Suc3)
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```    99    apply (simp_all add: fib_2)
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```   100   done
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```   101
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```   102 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
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```   103   apply (simp add: gcd_commute [of "fib m"])
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```   104   apply (case_tac m)
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```   105    apply simp
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```   106   apply (simp add: fib_add)
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```   107   apply (simp add: add_commute gcd_non_0 [OF fib_Suc_gr_0])
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```   108   apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
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```   109   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
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```   110   done
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```   111
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```   112 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
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```   113   by (simp add: gcd_fib_add [symmetric, of _ "n-m"])
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```   114
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```   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)
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```   117   case (less n)
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```   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")
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```   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)" .
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```   130   next
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```   131     case False then show "gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
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```   132     by (cases "m = n") auto
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```   133   qed
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```   134 qed
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```   135
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```   136 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
```
```   137   apply (induct m n rule: gcd_induct)
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```   138   apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
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```   139   done
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```   140
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```   141 theorem fib_mult_eq_setsum:
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```   142     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
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```   143   apply (induct n rule: fib.induct)
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```   144     apply (auto simp add: atMost_Suc fib_2)
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```   145   apply (simp add: add_mult_distrib add_mult_distrib2)
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```   146   done
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```   147
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```   148 end
```