src/HOL/Number_Theory/Fib.thy
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```     1 (*  Title:      HOL/Number_Theory/Fib.thy
```
```     2     Author:     Lawrence C. Paulson
```
```     3     Author:     Jeremy Avigad
```
```     4
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```     5 Defines the fibonacci function.
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```     6
```
```     7 The original "Fib" is due to Lawrence C. Paulson, and was adapted by
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```     8 Jeremy Avigad.
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```     9 *)
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```    10
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```    11 section \<open>Fib\<close>
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```    12
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```    13 theory Fib
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```    14 imports Main "../GCD" "../Binomial"
```
```    15 begin
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```    16
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```    17
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```    18 subsection \<open>Fibonacci numbers\<close>
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```    19
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```    20 fun fib :: "nat \<Rightarrow> nat"
```
```    21 where
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```    22     fib0: "fib 0 = 0"
```
```    23   | fib1: "fib (Suc 0) = 1"
```
```    24   | fib2: "fib (Suc (Suc n)) = fib (Suc n) + fib n"
```
```    25
```
```    26 subsection \<open>Basic Properties\<close>
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```    27
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```    28 lemma fib_1 [simp]: "fib (1::nat) = 1"
```
```    29   by (metis One_nat_def fib1)
```
```    30
```
```    31 lemma fib_2 [simp]: "fib (2::nat) = 1"
```
```    32   using fib.simps(3) [of 0]
```
```    33   by (simp add: numeral_2_eq_2)
```
```    34
```
```    35 lemma fib_plus_2: "fib (n + 2) = fib (n + 1) + fib n"
```
```    36   by (metis Suc_eq_plus1 add_2_eq_Suc' fib.simps(3))
```
```    37
```
```    38 lemma fib_add: "fib (Suc (n+k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
```
```    39   by (induct n rule: fib.induct) (auto simp add: field_simps)
```
```    40
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```    41 lemma fib_neq_0_nat: "n > 0 \<Longrightarrow> fib n > 0"
```
```    42   by (induct n rule: fib.induct) (auto simp add: )
```
```    43
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```    44 subsection \<open>A Few Elementary Results\<close>
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```    45
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```    46 text \<open>
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```    47   \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
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```    48   much easier using integers, not natural numbers!
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```    49 \<close>
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```    50
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```    51 lemma fib_Cassini_int: "int (fib (Suc (Suc n)) * fib n) - int((fib (Suc n))\<^sup>2) = - ((-1)^n)"
```
```    52   by (induction n rule: fib.induct)  (auto simp add: field_simps power2_eq_square power_add)
```
```    53
```
```    54 lemma fib_Cassini_nat:
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```    55     "fib (Suc (Suc n)) * fib n =
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```    56        (if even n then (fib (Suc n))\<^sup>2 - 1 else (fib (Suc n))\<^sup>2 + 1)"
```
```    57 using fib_Cassini_int [of n] by auto
```
```    58
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```    59
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```    60 subsection \<open>Law 6.111 of Concrete Mathematics\<close>
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```    61
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```    62 lemma coprime_fib_Suc_nat: "coprime (fib (n::nat)) (fib (Suc n))"
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```    63   apply (induct n rule: fib.induct)
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```    64   apply auto
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```    65   apply (metis gcd_add1_nat add.commute)
```
```    66   done
```
```    67
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```    68 lemma gcd_fib_add: "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
```
```    69   apply (simp add: gcd_commute_nat [of "fib m"])
```
```    70   apply (cases m)
```
```    71   apply (auto simp add: fib_add)
```
```    72   apply (metis gcd_commute_nat mult.commute coprime_fib_Suc_nat gcd_add_mult_nat gcd_mult_cancel_nat gcd_nat.commute)
```
```    73   done
```
```    74
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```    75 lemma gcd_fib_diff: "m \<le> n \<Longrightarrow>
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```    76     gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
```
```    77   by (simp add: gcd_fib_add [symmetric, of _ "n-m"])
```
```    78
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```    79 lemma gcd_fib_mod: "0 < m \<Longrightarrow>
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```    80     gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
```
```    81 proof (induct n rule: less_induct)
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```    82   case (less n)
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```    83   from less.prems have pos_m: "0 < m" .
```
```    84   show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
```
```    85   proof (cases "m < n")
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```    86     case True
```
```    87     then have "m \<le> n" by auto
```
```    88     with pos_m have pos_n: "0 < n" by auto
```
```    89     with pos_m \<open>m < n\<close> have diff: "n - m < n" by auto
```
```    90     have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
```
```    91       by (simp add: mod_if [of n]) (insert \<open>m < n\<close>, auto)
```
```    92     also have "\<dots> = gcd (fib m)  (fib (n - m))"
```
```    93       by (simp add: less.hyps diff pos_m)
```
```    94     also have "\<dots> = gcd (fib m) (fib n)"
```
```    95       by (simp add: gcd_fib_diff \<open>m \<le> n\<close>)
```
```    96     finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
```
```    97   next
```
```    98     case False
```
```    99     then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
```
```   100       by (cases "m = n") auto
```
```   101   qed
```
```   102 qed
```
```   103
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```   104 lemma fib_gcd: "fib (gcd m n) = gcd (fib m) (fib n)"
```
```   105     -- \<open>Law 6.111\<close>
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```   106   by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod)
```
```   107
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```   108 theorem fib_mult_eq_setsum_nat:
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```   109     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
```
```   110   by (induct n rule: nat.induct) (auto simp add:  field_simps)
```
```   111
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```   112 subsection \<open>Fibonacci and Binomial Coefficients\<close>
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```   113
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```   114 lemma setsum_drop_zero: "(\<Sum>k = 0..Suc n. if 0<k then (f (k - 1)) else 0) = (\<Sum>j = 0..n. f j)"
```
```   115   by (induct n) auto
```
```   116
```
```   117 lemma setsum_choose_drop_zero:
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```   118     "(\<Sum>k = 0..Suc n. if k=0 then 0 else (Suc n - k) choose (k - 1)) = (\<Sum>j = 0..n. (n-j) choose j)"
```
```   119   by (rule trans [OF setsum.cong setsum_drop_zero]) auto
```
```   120
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```   121 lemma ne_diagonal_fib:
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```   122    "(\<Sum>k = 0..n. (n-k) choose k) = fib (Suc n)"
```
```   123 proof (induct n rule: fib.induct)
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```   124   case 1 show ?case by simp
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```   125 next
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```   126   case 2 show ?case by simp
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```   127 next
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```   128   case (3 n)
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```   129   have "(\<Sum>k = 0..Suc n. Suc (Suc n) - k choose k) =
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```   130         (\<Sum>k = 0..Suc n. (Suc n - k choose k) + (if k=0 then 0 else (Suc n - k choose (k - 1))))"
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```   131     by (rule setsum.cong) (simp_all add: choose_reduce_nat)
```
```   132   also have "... = (\<Sum>k = 0..Suc n. Suc n - k choose k) +
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```   133                    (\<Sum>k = 0..Suc n. if k=0 then 0 else (Suc n - k choose (k - 1)))"
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```   134     by (simp add: setsum.distrib)
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```   135   also have "... = (\<Sum>k = 0..Suc n. Suc n - k choose k) +
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```   136                    (\<Sum>j = 0..n. n - j choose j)"
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```   137     by (metis setsum_choose_drop_zero)
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```   138   finally show ?case using 3
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```   139     by simp
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```   140 qed
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```   141
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```   142 end
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```   143
```