author paulson Fri Mar 25 16:20:57 2005 +0100 (2005-03-25) changeset 15628 9f912f8fd2df parent 15627 7a108ae4c798 child 15629 4066f01f1beb
tidied up
 src/HOL/Library/Primes.thy file | annotate | diff | revisions src/HOL/NumberTheory/Fib.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Library/Primes.thy	Fri Mar 25 14:14:01 2005 +0100
1.2 +++ b/src/HOL/Library/Primes.thy	Fri Mar 25 16:20:57 2005 +0100
1.3 @@ -210,11 +210,13 @@
1.4    done
1.5
1.6  lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
1.7 -  apply (rule gcd_commute [THEN trans])
1.10 -  apply (rule gcd_commute)
1.11 -  done
1.12 +proof -
1.13 +  have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute)
1.14 +  also have "... = gcd (n + m, m)" by (simp add: add_commute)
1.15 +  also have "... = gcd (n, m)" by simp
1.16 +  also have  "... = gcd (m, n)" by (rule gcd_commute)
1.17 +  finally show ?thesis .
1.18 +qed
1.19
1.20  lemma gcd_add2' [simp]: "gcd (m, n + m) = gcd (m, n)"
1.22 @@ -223,7 +225,7 @@
1.23
1.24  lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
1.25    apply (induct k)
1.28    done
1.29
1.30
1.31 @@ -235,8 +237,8 @@
1.32      apply (rule_tac n = k in relprime_dvd_mult)
1.35 -    apply (simp_all add: mult_commute gcd_dvd1 gcd_dvd2)
1.36 -  apply (blast intro: gcd_dvd1 dvd_trans)
1.37 +    apply (simp_all add: mult_commute)
1.38 +  apply (blast intro: dvd_trans)
1.39    done
1.40
1.41  end
```
```     2.1 --- a/src/HOL/NumberTheory/Fib.thy	Fri Mar 25 14:14:01 2005 +0100
2.2 +++ b/src/HOL/NumberTheory/Fib.thy	Fri Mar 25 16:20:57 2005 +0100
2.3 @@ -1,5 +1,4 @@
2.4 -(*  Title:      HOL/NumberTheory/Fib.thy
2.5 -    ID:         \$Id\$
2.6 +(*  ID:         \$Id\$
2.7      Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
2.8      Copyright   1997  University of Cambridge
2.9  *)
2.10 @@ -17,10 +16,10 @@
2.11  *}
2.12
2.13  consts fib :: "nat => nat"
2.14 -recdef fib  less_than
2.15 -  zero: "fib 0  = 0"
2.16 -  one:  "fib (Suc 0) = Suc 0"
2.17 -  Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
2.18 +recdef fib  "measure (\<lambda>x. x)"
2.19 +   zero:    "fib 0  = 0"
2.20 +   one:     "fib (Suc 0) = Suc 0"
2.21 +   Suc_Suc: "fib (Suc (Suc x)) = fib x + fib (Suc x)"
2.22
2.23  text {*
2.24    \medskip The difficulty in these proofs is to ensure that the
2.25 @@ -29,11 +28,12 @@
2.26    to the Simplifier and are applied very selectively at first.
2.27  *}
2.28
2.29 +text{*We disable @{text fib.Suc_Suc} for simplification ...*}
2.30  declare fib.Suc_Suc [simp del]
2.31
2.32 +text{*...then prove a version that has a more restrictive pattern.*}
2.33  lemma fib_Suc3: "fib (Suc (Suc (Suc n))) = fib (Suc n) + fib (Suc (Suc n))"
2.34 -  apply (rule fib.Suc_Suc)
2.35 -  done
2.36 +  by (rule fib.Suc_Suc)
2.37
2.38
2.39  text {* \medskip Concrete Mathematics, page 280 *}
2.40 @@ -42,45 +42,53 @@
2.41    apply (induct n rule: fib.induct)
2.42      prefer 3
2.43      txt {* simplify the LHS just enough to apply the induction hypotheses *}
2.44 -    apply (simp add: fib.Suc_Suc [of "Suc (m + n)", standard])
2.46 +    apply (simp add: fib_Suc3)
2.48      done
2.49
2.50 -lemma fib_Suc_neq_0 [simp]: "fib (Suc n) \<noteq> 0"
2.51 +lemma fib_Suc_neq_0: "fib (Suc n) \<noteq> 0"
2.52    apply (induct n rule: fib.induct)
2.54    done
2.55
2.56 -lemma [simp]: "0 < fib (Suc n)"
2.57 -  apply (simp add: neq0_conv [symmetric])
2.58 -  done
2.59 +lemma fib_Suc_gr_0: "0 < fib (Suc n)"
2.60 +  by (insert fib_Suc_neq_0 [of n], simp)
2.61
2.62  lemma fib_gr_0: "0 < n ==> 0 < fib n"
2.63 -  apply (rule not0_implies_Suc [THEN exE])
2.64 -   apply auto
2.65 -  done
2.66 +  by (case_tac n, auto simp add: fib_Suc_gr_0)
2.67
2.68
2.69  text {*
2.70 -  \medskip Concrete Mathematics, page 278: Cassini's identity.  It is
2.71 -  much easier to prove using integers!
2.72 +  \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
2.73 +  much easier using integers, not natural numbers!
2.74  *}
2.75
2.76 -lemma fib_Cassini:
2.77 +lemma fib_Cassini_int:
2.78   "int (fib (Suc (Suc n)) * fib n) =
2.79    (if n mod 2 = 0 then int (fib (Suc n) * fib (Suc n)) - 1
2.80     else int (fib (Suc n) * fib (Suc n)) + 1)"
2.81    apply (induct n rule: fib.induct)
2.83     apply (simp add: fib.Suc_Suc mod_Suc)
2.84 -  apply (simp add: fib.Suc_Suc
2.86 -  apply (subgoal_tac "x mod 2 < 2", arith)
2.87 -  apply simp
2.89 +                   mod_Suc zmult_int [symmetric])
2.90 +  apply presburger
2.91 +  done
2.92 +
2.93 +text{*We now obtain a version for the natural numbers via the coercion
2.94 +   function @{term int}.*}
2.95 +theorem fib_Cassini:
2.96 + "fib (Suc (Suc n)) * fib n =
2.97 +  (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
2.98 +   else fib (Suc n) * fib (Suc n) + 1)"
2.99 +  apply (rule int_int_eq [THEN iffD1])
2.100 +  apply (simp add: fib_Cassini_int)
2.101 +  apply (subst zdiff_int [symmetric])
2.102 +   apply (insert fib_Suc_gr_0 [of n], simp_all)
2.103    done
2.104
2.105
2.108
2.109  lemma gcd_fib_Suc_eq_1: "gcd (fib n, fib (Suc n)) = Suc 0"
2.110    apply (induct n rule: fib.induct)
2.111 @@ -90,35 +98,29 @@
2.112    done
2.113
2.114  lemma gcd_fib_add: "gcd (fib m, fib (n + m)) = gcd (fib m, fib n)"
2.115 -  apply (simp (no_asm) add: gcd_commute [of "fib m"])
2.116 -  apply (case_tac "m = 0")
2.117 -   apply simp
2.118 -  apply (clarify dest!: not0_implies_Suc)
2.119 +  apply (simp add: gcd_commute [of "fib m"])
2.120 +  apply (case_tac m)
2.121 +   apply simp
2.124 -  apply (simp add: gcd_non_0 [symmetric])
2.126 +  apply (simp add: gcd_non_0 [OF fib_Suc_gr_0, symmetric])
2.127    apply (simp add: gcd_fib_Suc_eq_1 gcd_mult_cancel)
2.128    done
2.129
2.130  lemma gcd_fib_diff: "m \<le> n ==> gcd (fib m, fib (n - m)) = gcd (fib m, fib n)"
2.131 -  apply (rule gcd_fib_add [symmetric, THEN trans])
2.132 -  apply simp
2.133 -  done
2.135
2.136  lemma gcd_fib_mod: "0 < m ==> gcd (fib m, fib (n mod m)) = gcd (fib m, fib n)"
2.137    apply (induct n rule: nat_less_induct)
2.138 -  apply (subst mod_if)
2.139 -  apply (simp add: gcd_fib_diff mod_geq not_less_iff_le)
2.140 +  apply (simp add: mod_if gcd_fib_diff mod_geq)
2.141    done
2.142
2.143  lemma fib_gcd: "fib (gcd (m, n)) = gcd (fib m, fib n)"  -- {* Law 6.111 *}
2.144    apply (induct m n rule: gcd_induct)
2.145 -   apply simp
2.146 -  apply (simp add: gcd_non_0)
2.147 -  apply (simp add: gcd_commute gcd_fib_mod)
2.148 +  apply (simp_all add: gcd_non_0 gcd_commute gcd_fib_mod)
2.149    done
2.150
2.151 -lemma fib_mult_eq_setsum:
2.152 +theorem fib_mult_eq_setsum:
2.153      "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
2.154    apply (induct n rule: fib.induct)
2.155      apply (auto simp add: atMost_Suc fib.Suc_Suc)
```