src/HOL/NumberTheory/Fib.thy
 author wenzelm Sat Oct 06 00:02:46 2001 +0200 (2001-10-06) changeset 11704 3c50a2cd6f00 parent 11701 3d51fbf81c17 child 11786 51ce34ef5113 permissions -rw-r--r--
* sane numerals (stage 2): plain "num" syntax (removed "#");
```     1 (*  Title:      HOL/NumberTheory/Fib.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1997  University of Cambridge
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```     5 *)
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```     6
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```     7 header {* The Fibonacci function *}
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```     8
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```     9 theory Fib = Primes:
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```    10
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```    11 text {*
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```    12   Fibonacci numbers: proofs of laws taken from:
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```    13   R. L. Graham, D. E. Knuth, O. Patashnik.  Concrete Mathematics.
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```    14   (Addison-Wesley, 1989)
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```    15
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```    16   \bigskip
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```    17 *}
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```    18
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```    19 consts fib :: "nat => nat"
```
```    20 recdef fib  less_than
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```    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
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```    25 text {*
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```    26   \medskip The difficulty in these proofs is to ensure that the
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```    27   induction hypotheses are applied before the definition of @{term
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```    28   fib}.  Towards this end, the @{term fib} equations are not declared
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```    29   to the Simplifier and are applied very selectively at first.
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```    30 *}
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```    31
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```    32 declare fib.Suc_Suc [simp del]
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```    33
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```    34 lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
```
```    35   apply (rule fib.Suc_Suc)
```
```    36   done
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```    37
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```    38
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```    39 text {* \medskip Concrete Mathematics, page 280 *}
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```    40
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```    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)
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```    43     prefer 3
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```    44     txt {* simplify the LHS just enough to apply the induction hypotheses *}
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```    45     apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard])
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```    46     apply (simp_all (no_asm_simp) add: fib.Suc_Suc add_mult_distrib add_mult_distrib2)
```
```    47     done
```
```    48
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```    49 lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0"
```
```    50   apply (induct n rule: fib.induct)
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```    51     apply (simp_all add: fib.Suc_Suc)
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```    52   done
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```    53
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```    54 lemma [simp]: "0 < fib (Suc n)"
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```    55   apply (simp add: neq0_conv [symmetric])
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```    56   done
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```    57
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```    58 lemma fib_gr_0: "0 < n ==> 0 < fib n"
```
```    59   apply (rule not0_implies_Suc [THEN exE])
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```    60    apply auto
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```    61   done
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```    62
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```    63
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```    64 text {*
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```    65   \medskip Concrete Mathematics, page 278: Cassini's identity.  It is
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```    66   much easier to prove using integers!
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```    67 *}
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```    68
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```    69 lemma fib_Cassini: "int (fib (Suc (Suc n)) * fib n) =
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```    70   (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - Numeral1
```
```    71    else int (fib (Suc n) * fib (Suc n)) + Numeral1)"
```
```    72   apply (induct n rule: fib.induct)
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```    73     apply (simp add: fib.Suc_Suc)
```
```    74    apply (simp add: fib.Suc_Suc mod_Suc)
```
```    75   apply (simp add: fib.Suc_Suc
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```    76     add_mult_distrib add_mult_distrib2 mod_Suc zmult_int [symmetric] zmult_ac)
```
```    77   apply (subgoal_tac "x mod 2 < 2", arith)
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```    78   apply simp
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```    79   done
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```    80
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```    81
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```    82 text {* \medskip Towards Law 6.111 of Concrete Mathematics *}
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```    83
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```    84 lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
```
```    85   apply (induct n rule: fib.induct)
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```    86     prefer 3
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```    87     apply (simp add: gcd_commute fib_Suc3)
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```    88    apply (simp_all add: fib.Suc_Suc)
```
```    89   done
```
```    90
```
```    91 lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
```
```    92   apply (simp (no_asm) add: gcd_commute [of "fib m"])
```
```    93   apply (case_tac "m = 0")
```
```    94    apply simp
```
```    95   apply (clarify dest!: not0_implies_Suc)
```
```    96   apply (simp add: fib_add)
```
```    97   apply (simp add: add_commute gcd_non_0)
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```    98   apply (simp add: gcd_non_0 [symmetric])
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```    99   apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
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```   100   done
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```   101
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```   102 lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
```
```   103   apply (rule gcd_fib_add [symmetric, THEN trans])
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```   104   apply simp
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```   105   done
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```   106
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```   107 lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
```
```   108   apply (induct n rule: nat_less_induct)
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```   109   apply (subst mod_if)
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```   110   apply (simp add: gcd_fib_diff mod_geq not_less_iff_le diff_less)
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```   111   done
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```   112
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```   113 lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
```
```   114   apply (induct m n rule: gcd_induct)
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```   115    apply simp
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```   116   apply (simp add: gcd_non_0)
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```   117   apply (simp add: gcd_commute gcd_fib_mod)
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```   118   done
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```   119
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```   120 lemma fib_mult_eq_setsum:
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```   121     "fib (Suc n) * fib n = setsum (\<lambda>k. fib k * fib k) (atMost n)"
```
```   122   apply (induct n rule: fib.induct)
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```   123     apply (auto simp add: atMost_Suc fib.Suc_Suc)
```
```   124   apply (simp add: add_mult_distrib add_mult_distrib2)
```
```   125   done
```
```   126
```
```   127 end
```