src/HOL/Number_Theory/Fib.thy
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(*  Title:      Fib.thy
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    Authors:    Lawrence C. Paulson, Jeremy Avigad
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Defines the fibonacci function.
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The original "Fib" is due to Lawrence C. Paulson, and was adapted by
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Jeremy Avigad.
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*)
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header {* Fib *}
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theory Fib
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imports Binomial
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begin
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subsection {* Main definitions *}
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class fib =
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fixes 
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  fib :: "'a \<Rightarrow> 'a"
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(* definition for the natural numbers *)
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instantiation nat :: fib
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begin 
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fun 
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  fib_nat :: "nat \<Rightarrow> nat"
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where
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  "fib_nat n =
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   (if n = 0 then 0 else
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   (if n = 1 then 1 else
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     fib (n - 1) + fib (n - 2)))"
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instance proof qed
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end
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(* definition for the integers *)
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instantiation int :: fib
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begin 
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definition
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  fib_int :: "int \<Rightarrow> int"
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where  
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  "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)"
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instance proof qed
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end
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subsection {* Set up Transfer *}
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lemma transfer_nat_int_fib:
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  "(x::int) >= 0 \<Longrightarrow> fib (nat x) = nat (fib x)"
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  unfolding fib_int_def by auto
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lemma transfer_nat_int_fib_closure:
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  "n >= (0::int) \<Longrightarrow> fib n >= 0"
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  by (auto simp add: fib_int_def)
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declare transfer_morphism_nat_int[transfer add return: 
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    transfer_nat_int_fib transfer_nat_int_fib_closure]
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lemma transfer_int_nat_fib:
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  "fib (int n) = int (fib n)"
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  unfolding fib_int_def by auto
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lemma transfer_int_nat_fib_closure:
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  "is_nat n \<Longrightarrow> fib n >= 0"
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  unfolding fib_int_def by auto
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declare transfer_morphism_int_nat[transfer add return: 
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    transfer_int_nat_fib transfer_int_nat_fib_closure]
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subsection {* Fibonacci numbers *}
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lemma fib_0_nat [simp]: "fib (0::nat) = 0"
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  by simp
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lemma fib_0_int [simp]: "fib (0::int) = 0"
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  unfolding fib_int_def by simp
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lemma fib_1_nat [simp]: "fib (1::nat) = 1"
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  by simp
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lemma fib_Suc_0_nat [simp]: "fib (Suc 0) = Suc 0"
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  by simp
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lemma fib_1_int [simp]: "fib (1::int) = 1"
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  unfolding fib_int_def by simp
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lemma fib_reduce_nat: "(n::nat) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
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  by simp
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declare fib_nat.simps [simp del]
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lemma fib_reduce_int: "(n::int) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
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  unfolding fib_int_def
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  by (auto simp add: fib_reduce_nat nat_diff_distrib)
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lemma fib_neg_int [simp]: "(n::int) < 0 \<Longrightarrow> fib n = 0"
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  unfolding fib_int_def by auto
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lemma fib_2_nat [simp]: "fib (2::nat) = 1"
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  by (subst fib_reduce_nat, auto)
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lemma fib_2_int [simp]: "fib (2::int) = 1"
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  by (subst fib_reduce_int, auto)
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lemma fib_plus_2_nat: "fib ((n::nat) + 2) = fib (n + 1) + fib n"
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  by (subst fib_reduce_nat, auto simp add: One_nat_def)
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(* the need for One_nat_def is due to the natdiff_cancel_numerals
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   procedure *)
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lemma fib_induct_nat: "P (0::nat) \<Longrightarrow> P (1::nat) \<Longrightarrow> 
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    (!!n. P n \<Longrightarrow> P (n + 1) \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
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  apply (atomize, induct n rule: nat_less_induct)
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  apply auto
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  apply (case_tac "n = 0", force)
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  apply (case_tac "n = 1", force)
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  apply (subgoal_tac "n >= 2")
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  apply (frule_tac x = "n - 1" in spec)
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  apply (drule_tac x = "n - 2" in spec)
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  apply (drule_tac x = "n - 2" in spec)
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  apply auto
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  apply (auto simp add: One_nat_def) (* again, natdiff_cancel *)
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done
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lemma fib_add_nat: "fib ((n::nat) + k + 1) = fib (k + 1) * fib (n + 1) + 
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    fib k * fib n"
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  apply (induct n rule: fib_induct_nat)
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  apply auto
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  apply (subst fib_reduce_nat)
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  apply (auto simp add: field_simps)
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  apply (subst (1 3 5) fib_reduce_nat)
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  apply (auto simp add: field_simps Suc_eq_plus1)
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(* hmmm. Why doesn't "n + (1 + (1 + k))" simplify to "n + k + 2"? *)
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  apply (subgoal_tac "n + (k + 2) = n + (1 + (1 + k))")
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  apply (erule ssubst) back back
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  apply (erule ssubst) back 
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  apply auto
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done
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lemma fib_add'_nat: "fib (n + Suc k) = fib (Suc k) * fib (Suc n) + 
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    fib k * fib n"
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  using fib_add_nat by (auto simp add: One_nat_def)
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(* transfer from nats to ints *)
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lemma fib_add_int [rule_format]: "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow>
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    fib (n + k + 1) = fib (k + 1) * fib (n + 1) + 
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    fib k * fib n "
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  by (rule fib_add_nat [transferred])
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lemma fib_neq_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n ~= 0"
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  apply (induct n rule: fib_induct_nat)
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  apply (auto simp add: fib_plus_2_nat)
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done
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lemma fib_gr_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n > 0"
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  by (frule fib_neq_0_nat, simp)
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lemma fib_gr_0_int: "(n::int) > 0 \<Longrightarrow> fib n > 0"
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  unfolding fib_int_def by (simp add: fib_gr_0_nat)
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text {*
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  \medskip Concrete Mathematics, page 278: Cassini's identity.  The proof is
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  much easier using integers, not natural numbers!
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*}
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lemma fib_Cassini_aux_int: "fib (int n + 2) * fib (int n) - 
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    (fib (int n + 1))^2 = (-1)^(n + 1)"
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  apply (induct n)
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  apply (auto simp add: field_simps power2_eq_square fib_reduce_int
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      power_add)
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done
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lemma fib_Cassini_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n - 
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    (fib (n + 1))^2 = (-1)^(nat n + 1)"
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  by (insert fib_Cassini_aux_int [of "nat n"], auto)
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(*
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lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n = 
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    (fib (n + 1))^2 + (-1)^(nat n + 1)"
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  by (frule fib_Cassini_int, simp) 
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*)
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lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib ((n::int) + 2) * fib n =
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  (if even n then tsub ((fib (n + 1))^2) 1
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   else (fib (n + 1))^2 + 1)"
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  apply (frule fib_Cassini_int, auto simp add: pos_int_even_equiv_nat_even)
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  apply (subst tsub_eq)
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   206
  apply (insert fib_gr_0_int [of "n + 1"], force)
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  apply auto
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done
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lemma fib_Cassini_nat: "fib ((n::nat) + 2) * fib n =
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  (if even n then (fib (n + 1))^2 - 1
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   else (fib (n + 1))^2 + 1)"
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  by (rule fib_Cassini'_int [transferred, of n], auto)
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text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
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lemma coprime_fib_plus_1_nat: "coprime (fib (n::nat)) (fib (n + 1))"
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  apply (induct n rule: fib_induct_nat)
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  apply auto
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   222
  apply (subst (2) fib_reduce_nat)
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  apply (auto simp add: Suc_eq_plus1) (* again, natdiff_cancel *)
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  apply (subst add_commute, auto)
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   225
  apply (subst gcd_commute_nat, auto simp add: field_simps)
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done
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   228
lemma coprime_fib_Suc_nat: "coprime (fib n) (fib (Suc n))"
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   229
  using coprime_fib_plus_1_nat by (simp add: One_nat_def)
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   230
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   231
lemma coprime_fib_plus_1_int: 
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    "n >= 0 \<Longrightarrow> coprime (fib (n::int)) (fib (n + 1))"
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   233
  by (erule coprime_fib_plus_1_nat [transferred])
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   234
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   235
lemma gcd_fib_add_nat: "gcd (fib (m::nat)) (fib (n + m)) = gcd (fib m) (fib n)"
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   236
  apply (simp add: gcd_commute_nat [of "fib m"])
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   237
  apply (rule cases_nat [of _ m])
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   238
  apply simp
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   239
  apply (subst add_assoc [symmetric])
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   240
  apply (simp add: fib_add_nat)
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   241
  apply (subst gcd_commute_nat)
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   242
  apply (subst mult_commute)
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   243
  apply (subst gcd_add_mult_nat)
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   244
  apply (subst gcd_commute_nat)
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   245
  apply (rule gcd_mult_cancel_nat)
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   246
  apply (rule coprime_fib_plus_1_nat)
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   247
done
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   248
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   249
lemma gcd_fib_add_int [rule_format]: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow> 
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    gcd (fib (m::int)) (fib (n + m)) = gcd (fib m) (fib n)"
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   251
  by (erule gcd_fib_add_nat [transferred])
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   252
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   253
lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow> 
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    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
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   255
  by (simp add: gcd_fib_add_nat [symmetric, of _ "n-m"])
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   256
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   257
lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow> 
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   258
    gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
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   259
  by (simp add: gcd_fib_add_int [symmetric, of _ "n-m"])
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   260
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   261
lemma gcd_fib_mod_nat: "0 < (m::nat) \<Longrightarrow> 
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   262
    gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
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   263
proof (induct n rule: less_induct)
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   264
  case (less n)
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   265
  from less.prems have pos_m: "0 < m" .
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   266
  show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
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   267
  proof (cases "m < n")
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   268
    case True note m_n = True
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   269
    then have m_n': "m \<le> n" by auto
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   270
    with pos_m have pos_n: "0 < n" by auto
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   271
    with pos_m m_n have diff: "n - m < n" by auto
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   272
    have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
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   273
    by (simp add: mod_if [of n]) (insert m_n, auto)
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   274
    also have "\<dots> = gcd (fib m)  (fib (n - m))" 
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   275
      by (simp add: less.hyps diff pos_m)
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   276
    also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff_nat m_n')
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   277
    finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
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   278
  next
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   279
    case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
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   280
    by (cases "m = n") auto
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  qed
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qed
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   283
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   284
lemma gcd_fib_mod_int: 
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  assumes "0 < (m::int)" and "0 <= n"
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   286
  shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
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parents:
diff changeset
   287
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   288
  apply (rule gcd_fib_mod_nat [transferred])
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   289
  using prems apply auto
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   290
done
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   291
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   292
lemma fib_gcd_nat: "fib (gcd (m::nat) n) = gcd (fib m) (fib n)"  
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   293
    -- {* Law 6.111 *}
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   294
  apply (induct m n rule: gcd_nat_induct)
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diff changeset
   295
  apply (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod_nat)
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   296
done
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   297
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lemma fib_gcd_int: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
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    fib (gcd (m::int) n) = gcd (fib m) (fib n)"
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  by (erule fib_gcd_nat [transferred])
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lemma atMost_plus_one_nat: "{..(k::nat) + 1} = insert (k + 1) {..k}" 
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  by auto
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theorem fib_mult_eq_setsum_nat:
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    "fib ((n::nat) + 1) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
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  apply (induct n)
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  apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat field_simps)
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done
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theorem fib_mult_eq_setsum'_nat:
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    "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
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  using fib_mult_eq_setsum_nat by (simp add: One_nat_def)
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theorem fib_mult_eq_setsum_int [rule_format]:
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    "n >= 0 \<Longrightarrow> fib ((n::int) + 1) * fib n = (\<Sum>k \<in> {0..n}. fib k * fib k)"
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  by (erule fib_mult_eq_setsum_nat [transferred])
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end