theory MyFirst
imports Main
begin

datatype 'a list = Nil | Cons 'a "'a list"

fun app :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"app Nil ys = ys" |
"app (Cons x xs) ys = Cons x (app xs ys)"

fun rev :: "'a list \<Rightarrow> 'a list" where
"rev Nil = Nil" |
"rev (Cons x xs) = app (rev xs) (Cons x Nil)"

value "rev(Cons True (Cons False Nil))"

value "1 + (2::nat)"
value "1 + (2::int)"
value "1 - (2::nat)"
value "1 - (2::int)"

lemma app_Nil2 [simp]: "app xs Nil = xs"
apply(induction xs)
apply(auto)
done

lemma app_assoc [simp]: "app (app xs ys) zs = app xs (app ys zs)"
apply(induction xs)
apply(auto)
done

lemma rev_app [simp]: "rev(app xs ys) = app (rev ys) (rev xs)"
apply (induction xs)
apply (auto)
done

theorem rev_rev [simp]: "rev(rev xs) = xs"
apply (induction xs)
apply (auto)
done

fun add :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
"add 0 n = n" |
"add (Suc m) n = Suc(add m n)"

lemma add_02: "add m 0 = m"
apply(induction m)
apply(auto)
done

value "add 2 3"


(**commutative-associative**)
lemma add_04: "add m (add n k) = add (add m n) k"
apply(induct m)
apply(simp_all)
done

lemma add_zero: "add n 0 = n"
apply(induct n)
apply(auto)
done

lemma add_Suc: "add m (Suc n) = Suc (add m n)"
apply(induct m)
apply(metis add.simps(1))
apply(auto)
done

lemma add_comm: "add m n = add n m"
apply(induct m)
apply(simp add: add_zero)
apply(simp add: add_Suc)
done

lemma add_odd: "add m (add n k) = add k (add m n)"
apply(subst add_04)
apply(subst (2) add_comm)
apply(simp)
done


fun dub :: "nat \<Rightarrow> nat" where
"dub 0 = 0" |
"dub m = add m m"

lemma dub_01: "dub 0 = 0"
apply(induct)
apply(auto)
done

lemma dub_02: "dub m = add m m"
apply(induction m)
apply(auto)
done

value "dub 2"

fun trip :: "nat \<Rightarrow> nat" where
"trip 0 = 0" |
"trip m = add m (add m m)"

lemma trip_01: "trip 0 = 0"
apply(induct)
apply(auto)
done

lemma trip_02: "trip m = add m (add m m)"
apply(induction m)
apply(auto)
done

value "trip 1"
value "trip 2"

fun sum :: "nat \<Rightarrow> nat" where
  "sum 0 = 0"
| "sum (Suc n) = (Suc n) + sum n"

function sum1 :: "nat \<Rightarrow> nat" where
  "sum1 0 = 0"
| "n \<noteq> 0 \<Longrightarrow> sum1 n = n + sum1 (n - 1)"
apply(auto)
done

termination sum1
by (smt2 "termination" diff_less less_than_iff not_gr0 wf_less_than zero_neq_one)

lemma "sum n = sum1 n"
apply(induct n)
apply(auto)
done

lemma "sum n = (\<Sum>i \<le> n. i)"
apply(induct n)
apply(simp_all)
done

fun mull :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
"mull 0 0 = 0" |
"mull m 0 = 0" |
"mull m 1 = m" | 
(**"mull m (1::nat) = m" | **)
(**"mull m (suc(0)) = m" | **)
"mull m n = mull m (n-(1::nat))" 
apply(pat_completeness)
apply(auto)

done

  "mull 0 n = 0"
| "mull (Suc m) n = add n (mull m n)" 

lemma test: "mull m n = m * n"
sorry

fun poww :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
  "poww 0 n = 1"
| "poww (Suc m) n = mull n (poww m n)" 


"mull 0 0 = 0" |
"mull m 0 = 0" |
(**"mull m 1 = m" | **)
(**"mull m (1::nat) = m" | **)
(**"mull m (suc(0)) = m" | **)
"mull m n = mull m (n-(1::nat))" 

(**Define a function that counts the
number of occurrences of an element in a list **)
(**
fun count :: "'a\<Rightarrow>'a list\<Rightarrow>nat" where
"count  "
**)


(* prove n = n * (n + 1) div 2  *)