src/HOL/NewNumberTheory/Fib.thy

-(*  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
-Jeremy Avigad.
-*)
-
-
-header {* Fib *}
-
-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)
-
-declare TransferMorphism_nat_int[transfer add return: 
-    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
-
-declare TransferMorphism_int_nat[transfer add return: 
-    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"
-  using fib_add_nat by (auto simp add: One_nat_def)
-
-
-(* 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 "
-
-  by (rule fib_add_nat [transferred])
-
-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
-      power_add)
-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)
-
-
-text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
-
-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 add_commute, auto)
-  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 add_assoc [symmetric])
-  apply (simp add: fib_add_nat)
-  apply (subst gcd_commute_nat)
-  apply (subst mult_commute)
-  apply (subst gcd_add_mult_nat)
-  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)"
-  by (erule gcd_fib_add_nat [transferred])
-
-lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow> 
-    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
-  by (simp add: gcd_fib_add_nat [symmetric, of _ "n-m"])
-
-lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow> 
-    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
-  by (simp add: gcd_fib_add_int [symmetric, of _ "n-m"])
-
-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