theory FibInClass
imports Main
begin

(* Function package *)

fun fib :: "nat \<Rightarrow> nat" where
  "fib 0 = 0" |
  "fib (Suc 0) = 1" |
  "fib n = fib (n - 2) + fib (n - 1)"

primrec itfib :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" where
  "itfib f f' 0 = f" |
  "itfib f f' (Suc k) = itfib f' (f + f') k"



theorem "\<forall> n. itfib (fib n) (fib (n + 1)) k = fib (n + k)"
  apply (induct k)
  apply simp
  apply auto
  thm allE
  apply (erule_tac x="n+1" in allE)
  apply simp
  done

thm conjE
thm conjI
thm disjI1
thm disjI2
thm disjE

theorem "P \<and> Q \<longrightarrow> P"
proof -
  show "P \<and> Q \<longrightarrow> P"
  proof (rule impI)
    assume pq: "P \<and> Q"
    from pq show "P" by (rule conjunct1)
  qed
qed

(*
lemma L1[simp]: 
versus

lemma L1: 
then later
apply (auto simp: L1)
or
apply (simp add: L1)
*)

(* Example with custom termination measure *)

function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
  "sum i n = (if i > n then 0 else i + sum (i + 1) n)"
  by pat_completeness auto
termination sum
  apply (relation "measure (\<lambda>(i,n). (n + 1) - i)")
  apply auto
  done

end