src/HOL/NumberTheory/Fib.thy
 author nipkow Fri Jan 14 12:00:27 2005 +0100 (2005-01-14) changeset 15439 71c0f98e31f1 parent 15003 6145dd7538d7 child 15628 9f912f8fd2df permissions -rw-r--r--
```     1 (*  Title:      HOL/NumberTheory/Fib.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1997  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* The Fibonacci function *}
```
```     8
```
```     9 theory Fib = Primes:
```
```    10
```
```    11 text {*
```
```    12   Fibonacci numbers: proofs of laws taken from:
```
```    13   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
```
```    14   (Addison-Wesley, 1989)
```
```    15
```
```    16   \bigskip
```
```    17 *}
```
```    18
```
```    19 consts fib :: "nat => nat"
```
```    20 recdef fib  less_than
```
```    21   zero: "fib 0  = 0"
```
```    22   one:  "fib (Suc 0) = Suc 0"
```
```    23   Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
```
```    24
```
```    25 text {*
```
```    26   \medskip The difficulty in these proofs is to ensure that the
```
```    27   induction hypotheses are applied before the definition of @{term
```
```    28   fib}.  Towards this end, the @{term fib} equations are not declared
```
```    29   to the Simplifier and are applied very selectively at first.
```
```    30 *}
```
```    31
```
```    32 declare fib.Suc_Suc [simp del]
```
```    33
```
```    34 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
```
```    35   apply (rule fib.Suc_Suc)
```
```    36   done
```
```    37
```
```    38
```
```    39 text {* \medskip Concrete Mathematics, page 280 *}
```
```    40
```
```    41 lemma fib_add: "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
```
```    42   apply (induct n rule: fib.induct)
```
```    43     prefer 3
```
```    44     txt {* simplify the LHS just enough to apply the induction hypotheses *}
```
```    45     apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard])
```
```    46     apply (simp_all (no_asm_simp) add: fib.Suc_Suc add_mult_distrib add_mult_distrib2)
```
```    47     done
```
```    48
```
```    49 lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0"
```
```    50   apply (induct n rule: fib.induct)
```
```    51     apply (simp_all add: fib.Suc_Suc)
```
```    52   done
```
```    53
```
```    54 lemma [simp]: "0 < fib (Suc n)"
```
```    55   apply (simp add: neq0_conv [symmetric])
```
```    56   done
```
```    57
```
```    58 lemma fib_gr_0: "0 < n ==> 0 < fib n"
```
```    59   apply (rule not0_implies_Suc [THEN exE])
```
```    60    apply auto
```
```    61   done
```
```    62
```
```    63
```
```    64 text {*
```
```    65   \medskip Concrete Mathematics, page 278: Cassini's identity.  It is
```
```    66   much easier to prove using integers!
```
```    67 *}
```
```    68
```
```    69 lemma fib_Cassini:
```
```    70  "int (fib (Suc (Suc n)) * fib n) =
```
```    71   (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
```
```    72    else int (fib (Suc n) * fib (Suc n)) + 1)"
```
```    73   apply (induct n rule: fib.induct)
```
```    74     apply (simp add: fib.Suc_Suc)
```
```    75    apply (simp add: fib.Suc_Suc mod_Suc)
```
```    76   apply (simp add: fib.Suc_Suc
```
```    77     add_mult_distrib add_mult_distrib2 mod_Suc zmult_int [symmetric] zmult_ac)
```
```    78   apply (subgoal_tac "x mod 2 < 2", arith)
```
```    79   apply simp
```
```    80   done
```
```    81
```
```    82
```
```    83 text {* \medskip Towards Law 6.111 of Concrete Mathematics *}
```
```    84
```
```    85 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
```
```    86   apply (induct n rule: fib.induct)
```
```    87     prefer 3
```
```    88     apply (simp add: gcd_commute fib_Suc3)
```
```    89    apply (simp_all add: fib.Suc_Suc)
```
```    90   done
```
```    91
```
```    92 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
```
```    93   apply (simp (no_asm) add: gcd_commute [of "fib m"])
```
```    94   apply (case_tac "m = 0")
```
```    95    apply simp
```
```    96   apply (clarify dest!: not0_implies_Suc)
```
```    97   apply (simp add: fib_add)
```
```    98   apply (simp add: add_commute gcd_non_0)
```
```    99   apply (simp add: gcd_non_0 [symmetric])
```
```   100   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
```
```   101   done
```
```   102
```
```   103 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
```
```   104   apply (rule gcd_fib_add [symmetric, THEN trans])
```
```   105   apply simp
```
```   106   done
```
```   107
```
```   108 lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
```
```   109   apply (induct n rule: nat_less_induct)
```
```   110   apply (subst mod_if)
```
```   111   apply (simp add: gcd_fib_diff mod_geq not_less_iff_le)
```
```   112   done
```
```   113
```
```   114 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
```
```   115   apply (induct m n rule: gcd_induct)
```
```   116    apply simp
```
```   117   apply (simp add: gcd_non_0)
```
```   118   apply (simp add: gcd_commute gcd_fib_mod)
```
```   119   done
```
```   120
```
```   121 lemma fib_mult_eq_setsum:
```
```   122     "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
```
```   123   apply (induct n rule: fib.induct)
```
```   124     apply (auto simp add: atMost_Suc fib.Suc_Suc)
```
```   125   apply (simp add: add_mult_distrib add_mult_distrib2)
```
```   126   done
```
```   127
```
```   128 end
```