src/HOL/Number_Theory/Fib.thy
 changeset 32479 521cc9bf2958 parent 31952 40501bb2d57c child 35644 d20cf282342e
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Number_Theory/Fib.thy	Tue Sep 01 15:39:33 2009 +0200
@@ -0,0 +1,319 @@
+(*  Title:      Fib.thy
+    Authors:    Lawrence C. Paulson, Jeremy Avigad
+
+
+Defines the fibonacci function.
+
+The original "Fib" is due to Lawrence C. Paulson, and was adapted by
+*)
+
+
+
+theory Fib
+imports Binomial
+begin
+
+
+subsection {* Main definitions *}
+
+class fib =
+
+fixes
+  fib :: "'a \<Rightarrow> 'a"
+
+
+(* definition for the natural numbers *)
+
+instantiation nat :: fib
+
+begin
+
+fun
+  fib_nat :: "nat \<Rightarrow> nat"
+where
+  "fib_nat n =
+   (if n = 0 then 0 else
+   (if n = 1 then 1 else
+     fib (n - 1) + fib (n - 2)))"
+
+instance proof qed
+
+end
+
+(* definition for the integers *)
+
+instantiation int :: fib
+
+begin
+
+definition
+  fib_int :: "int \<Rightarrow> int"
+where
+  "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)"
+
+instance proof qed
+
+end
+
+
+subsection {* Set up Transfer *}
+
+
+lemma transfer_nat_int_fib:
+  "(x::int) >= 0 \<Longrightarrow> fib (nat x) = nat (fib x)"
+  unfolding fib_int_def by auto
+
+lemma transfer_nat_int_fib_closure:
+  "n >= (0::int) \<Longrightarrow> fib n >= 0"
+  by (auto simp add: fib_int_def)
+
+    transfer_nat_int_fib transfer_nat_int_fib_closure]
+
+lemma transfer_int_nat_fib:
+  "fib (int n) = int (fib n)"
+  unfolding fib_int_def by auto
+
+lemma transfer_int_nat_fib_closure:
+  "is_nat n \<Longrightarrow> fib n >= 0"
+  unfolding fib_int_def by auto
+
+    transfer_int_nat_fib transfer_int_nat_fib_closure]
+
+
+subsection {* Fibonacci numbers *}
+
+lemma fib_0_nat [simp]: "fib (0::nat) = 0"
+  by simp
+
+lemma fib_0_int [simp]: "fib (0::int) = 0"
+  unfolding fib_int_def by simp
+
+lemma fib_1_nat [simp]: "fib (1::nat) = 1"
+  by simp
+
+lemma fib_Suc_0_nat [simp]: "fib (Suc 0) = Suc 0"
+  by simp
+
+lemma fib_1_int [simp]: "fib (1::int) = 1"
+  unfolding fib_int_def by simp
+
+lemma fib_reduce_nat: "(n::nat) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
+  by simp
+
+declare fib_nat.simps [simp del]
+
+lemma fib_reduce_int: "(n::int) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
+  unfolding fib_int_def
+  by (auto simp add: fib_reduce_nat nat_diff_distrib)
+
+lemma fib_neg_int [simp]: "(n::int) < 0 \<Longrightarrow> fib n = 0"
+  unfolding fib_int_def by auto
+
+lemma fib_2_nat [simp]: "fib (2::nat) = 1"
+  by (subst fib_reduce_nat, auto)
+
+lemma fib_2_int [simp]: "fib (2::int) = 1"
+  by (subst fib_reduce_int, auto)
+
+lemma fib_plus_2_nat: "fib ((n::nat) + 2) = fib (n + 1) + fib n"
+  by (subst fib_reduce_nat, auto simp add: One_nat_def)
+(* the need for One_nat_def is due to the natdiff_cancel_numerals
+   procedure *)
+
+lemma fib_induct_nat: "P (0::nat) \<Longrightarrow> P (1::nat) \<Longrightarrow>
+    (!!n. P n \<Longrightarrow> P (n + 1) \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
+  apply (atomize, induct n rule: nat_less_induct)
+  apply auto
+  apply (case_tac "n = 0", force)
+  apply (case_tac "n = 1", force)
+  apply (subgoal_tac "n >= 2")
+  apply (frule_tac x = "n - 1" in spec)
+  apply (drule_tac x = "n - 2" in spec)
+  apply (drule_tac x = "n - 2" in spec)
+  apply auto
+  apply (auto simp add: One_nat_def) (* again, natdiff_cancel *)
+done
+
+lemma fib_add_nat: "fib ((n::nat) + k + 1) = fib (k + 1) * fib (n + 1) +
+    fib k * fib n"
+  apply (induct n rule: fib_induct_nat)
+  apply auto
+  apply (subst fib_reduce_nat)
+  apply (auto simp add: ring_simps)
+  apply (subst (1 3 5) fib_reduce_nat)
+  apply (auto simp add: ring_simps Suc_eq_plus1)
+(* hmmm. Why doesn't "n + (1 + (1 + k))" simplify to "n + k + 2"? *)
+  apply (subgoal_tac "n + (k + 2) = n + (1 + (1 + k))")
+  apply (erule ssubst) back back
+  apply (erule ssubst) back
+  apply auto
+done
+
+lemma fib_add'_nat: "fib (n + Suc k) = fib (Suc k) * fib (Suc n) +
+    fib k * fib n"
+
+
+(* transfer from nats to ints *)
+lemma fib_add_int [rule_format]: "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow>
+    fib (n + k + 1) = fib (k + 1) * fib (n + 1) +
+    fib k * fib n "
+
+
+lemma fib_neq_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n ~= 0"
+  apply (induct n rule: fib_induct_nat)
+  apply (auto simp add: fib_plus_2_nat)
+done
+
+lemma fib_gr_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n > 0"
+  by (frule fib_neq_0_nat, simp)
+
+lemma fib_gr_0_int: "(n::int) > 0 \<Longrightarrow> fib n > 0"
+  unfolding fib_int_def by (simp add: fib_gr_0_nat)
+
+text {*
+  \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
+  much easier using integers, not natural numbers!
+*}
+
+lemma fib_Cassini_aux_int: "fib (int n + 2) * fib (int n) -
+    (fib (int n + 1))^2 = (-1)^(n + 1)"
+  apply (induct n)
+  apply (auto simp add: ring_simps power2_eq_square fib_reduce_int
+done
+
+lemma fib_Cassini_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n -
+    (fib (n + 1))^2 = (-1)^(nat n + 1)"
+  by (insert fib_Cassini_aux_int [of "nat n"], auto)
+
+(*
+lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n =
+    (fib (n + 1))^2 + (-1)^(nat n + 1)"
+  by (frule fib_Cassini_int, simp)
+*)
+
+lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib ((n::int) + 2) * fib n =
+  (if even n then tsub ((fib (n + 1))^2) 1
+   else (fib (n + 1))^2 + 1)"
+  apply (frule fib_Cassini_int, auto simp add: pos_int_even_equiv_nat_even)
+  apply (subst tsub_eq)
+  apply (insert fib_gr_0_int [of "n + 1"], force)
+  apply auto
+done
+
+lemma fib_Cassini_nat: "fib ((n::nat) + 2) * fib n =
+  (if even n then (fib (n + 1))^2 - 1
+   else (fib (n + 1))^2 + 1)"
+
+  by (rule fib_Cassini'_int [transferred, of n], auto)
+
+
+
+lemma coprime_fib_plus_1_nat: "coprime (fib (n::nat)) (fib (n + 1))"
+  apply (induct n rule: fib_induct_nat)
+  apply auto
+  apply (subst (2) fib_reduce_nat)
+  apply (auto simp add: Suc_eq_plus1) (* again, natdiff_cancel *)
+  apply (subst gcd_commute_nat, auto simp add: ring_simps)
+done
+
+lemma coprime_fib_Suc_nat: "coprime (fib n) (fib (Suc n))"
+  using coprime_fib_plus_1_nat by (simp add: One_nat_def)
+
+lemma coprime_fib_plus_1_int:
+    "n >= 0 \<Longrightarrow> coprime (fib (n::int)) (fib (n + 1))"
+  by (erule coprime_fib_plus_1_nat [transferred])
+
+lemma gcd_fib_add_nat: "gcd (fib (m::nat)) (fib (n + m)) = gcd (fib m) (fib n)"
+  apply (simp add: gcd_commute_nat [of "fib m"])
+  apply (rule cases_nat [of _ m])
+  apply simp
+  apply (subst gcd_commute_nat)
+  apply (subst mult_commute)
+  apply (subst gcd_commute_nat)
+  apply (rule gcd_mult_cancel_nat)
+  apply (rule coprime_fib_plus_1_nat)
+done
+
+lemma gcd_fib_add_int [rule_format]: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
+    gcd (fib (m::int)) (fib (n + m)) = gcd (fib m) (fib n)"
+
+lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow>
+    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
+
+lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow>
+    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
+
+lemma gcd_fib_mod_nat: "0 < (m::nat) \<Longrightarrow>
+    gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
+proof (induct n rule: less_induct)
+  case (less n)
+  from less.prems have pos_m: "0 < m" .
+  show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
+  proof (cases "m < n")
+    case True note m_n = True
+    then have m_n': "m \<le> n" by auto
+    with pos_m have pos_n: "0 < n" by auto
+    with pos_m m_n have diff: "n - m < n" by auto
+    have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
+    by (simp add: mod_if [of n]) (insert m_n, auto)
+    also have "\<dots> = gcd (fib m)  (fib (n - m))"
+      by (simp add: less.hyps diff pos_m)
+    also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff_nat m_n')
+    finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
+  next
+    case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
+    by (cases "m = n") auto
+  qed
+qed
+
+lemma gcd_fib_mod_int:
+  assumes "0 < (m::int)" and "0 <= n"
+  shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
+
+  apply (rule gcd_fib_mod_nat [transferred])
+  using prems apply auto
+done
+
+lemma fib_gcd_nat: "fib (gcd (m::nat) n) = gcd (fib m) (fib n)"
+    -- {* Law 6.111 *}
+  apply (induct m n rule: gcd_nat_induct)
+  apply (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod_nat)
+done
+
+lemma fib_gcd_int: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
+    fib (gcd (m::int) n) = gcd (fib m) (fib n)"
+  by (erule fib_gcd_nat [transferred])
+
+lemma atMost_plus_one_nat: "{..(k::nat) + 1} = insert (k + 1) {..k}"
+  by auto
+
+theorem fib_mult_eq_setsum_nat:
+    "fib ((n::nat) + 1) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
+  apply (induct n)
+  apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat ring_simps)
+done
+
+theorem fib_mult_eq_setsum'_nat:
+    "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
+  using fib_mult_eq_setsum_nat by (simp add: One_nat_def)
+
+theorem fib_mult_eq_setsum_int [rule_format]:
+    "n >= 0 \<Longrightarrow> fib ((n::int) + 1) * fib n = (\<Sum>k \<in> {0..n}. fib k * fib k)"
+  by (erule fib_mult_eq_setsum_nat [transferred])
+
+end```