# Theory Termination

theory Termination
imports Multiset
```(* Title:       HOL/ex/Termination.thy
Author:      Lukas Bulwahn, TU Muenchen
Author:      Alexander Krauss, TU Muenchen
*)

section ‹Examples and regression tests for automated termination proofs›

theory Termination
imports Main "HOL-Library.Multiset"
begin

subsection ‹Manually giving termination relations using ‹relation› and
@{term measure}›

function sum :: "nat ⇒ nat ⇒ nat"
where
"sum i N = (if i > N then 0 else i + sum (Suc i) N)"
by pat_completeness auto

termination by (relation "measure (λ(i,N). N + 1 - i)") auto

function foo :: "nat ⇒ nat ⇒ nat"
where
"foo i N = (if i > N
then (if N = 0 then 0 else foo 0 (N - 1))
else i + foo (Suc i) N)"
by pat_completeness auto

termination by (relation "measures [λ(i, N). N, λ(i,N). N + 1 - i]") auto

subsection ‹‹lexicographic_order›: Trivial examples›

text ‹
The ‹fun› command uses the method ‹lexicographic_order› by default,
so it is not explicitly invoked.
›

fun identity :: "nat ⇒ nat"
where
"identity n = n"

fun yaSuc :: "nat ⇒ nat"
where
"yaSuc 0 = 0"
| "yaSuc (Suc n) = Suc (yaSuc n)"

subsection ‹Examples on natural numbers›

fun bin :: "(nat * nat) ⇒ nat"
where
"bin (0, 0) = 1"
| "bin (Suc n, 0) = 0"
| "bin (0, Suc m) = 0"
| "bin (Suc n, Suc m) = bin (n, m) + bin (Suc n, m)"

fun t :: "(nat * nat) ⇒ nat"
where
"t (0,n) = 0"
| "t (n,0) = 0"
| "t (Suc n, Suc m) = (if (n mod 2 = 0) then (t (Suc n, m)) else (t (n, Suc m)))"

fun k :: "(nat * nat) * (nat * nat) ⇒ nat"
where
"k ((0,0),(0,0)) = 0"
| "k ((Suc z, y), (u,v)) = k((z, y), (u, v))" (* z is descending *)
| "k ((0, Suc y), (u,v)) = k((1, y), (u, v))" (* y is descending *)
| "k ((0,0), (Suc u, v)) = k((1, 1), (u, v))" (* u is descending *)
| "k ((0,0), (0, Suc v)) = k((1,1), (1,v))"   (* v is descending *)

fun gcd2 :: "nat ⇒ nat ⇒ nat"
where
"gcd2 x 0 = x"
| "gcd2 0 y = y"
| "gcd2 (Suc x) (Suc y) = (if x < y then gcd2 (Suc x) (y - x)
else gcd2 (x - y) (Suc y))"

fun ack :: "(nat * nat) ⇒ nat"
where
"ack (0, m) = Suc m"
| "ack (Suc n, 0) = ack(n, 1)"
| "ack (Suc n, Suc m) = ack (n, ack (Suc n, m))"

fun greedy :: "nat * nat * nat * nat * nat => nat"
where
"greedy (Suc a, Suc b, Suc c, Suc d, Suc e) =
(if (a < 10) then greedy (Suc a, Suc b, c, d + 2, Suc e) else
(if (a < 20) then greedy (Suc a, b, Suc c, d, Suc e) else
(if (a < 30) then greedy (Suc a, b, Suc c, d, Suc e) else
(if (a < 40) then greedy (Suc a, b, Suc c, d, Suc e) else
(if (a < 50) then greedy (Suc a, b, Suc c, d, Suc e) else
(if (a < 60) then greedy (a, Suc b, Suc c, d, Suc e) else
(if (a < 70) then greedy (a, Suc b, Suc c, d, Suc e) else
(if (a < 80) then greedy (a, Suc b, Suc c, d, Suc e) else
(if (a < 90) then greedy (Suc a, Suc b, Suc c, d, e) else
greedy (Suc a, Suc b, Suc c, d, e))))))))))"
| "greedy (a, b, c, d, e) = 0"

fun blowup :: "nat => nat => nat => nat => nat => nat => nat => nat => nat => nat"
where
"blowup 0 0 0 0 0 0 0 0 0 = 0"
| "blowup 0 0 0 0 0 0 0 0 (Suc i) = Suc (blowup i i i i i i i i i)"
| "blowup 0 0 0 0 0 0 0 (Suc h) i = Suc (blowup h h h h h h h h i)"
| "blowup 0 0 0 0 0 0 (Suc g) h i = Suc (blowup g g g g g g g h i)"
| "blowup 0 0 0 0 0 (Suc f) g h i = Suc (blowup f f f f f f g h i)"
| "blowup 0 0 0 0 (Suc e) f g h i = Suc (blowup e e e e e f g h i)"
| "blowup 0 0 0 (Suc d) e f g h i = Suc (blowup d d d d e f g h i)"
| "blowup 0 0 (Suc c) d e f g h i = Suc (blowup c c c d e f g h i)"
| "blowup 0 (Suc b) c d e f g h i = Suc (blowup b b c d e f g h i)"
| "blowup (Suc a) b c d e f g h i = Suc (blowup a b c d e f g h i)"

subsection ‹Simple examples with other datatypes than nat, e.g. trees and lists›

datatype tree = Node | Branch tree tree

fun g_tree :: "tree * tree ⇒ tree"
where
"g_tree (Node, Node) = Node"
| "g_tree (Node, Branch a b) = Branch Node (g_tree (a,b))"
| "g_tree (Branch a b, Node) = Branch (g_tree (a,Node)) b"
| "g_tree (Branch a b, Branch c d) = Branch (g_tree (a,c)) (g_tree (b,d))"

fun acklist :: "'a list * 'a list ⇒ 'a list"
where
"acklist ([], m) = ((hd m)#m)"
|  "acklist (n#ns, []) = acklist (ns, [n])"
|  "acklist ((n#ns), (m#ms)) = acklist (ns, acklist ((n#ns), ms))"

subsection ‹Examples with mutual recursion›

fun evn od :: "nat ⇒ bool"
where
"evn 0 = True"
| "od 0 = False"
| "evn (Suc n) = od (Suc n)"
| "od (Suc n) = evn n"

fun sizechange_f :: "'a list => 'a list => 'a list" and
sizechange_g :: "'a list => 'a list => 'a list => 'a list"
where
"sizechange_f i x = (if i=[] then x else sizechange_g (tl i) x i)"
| "sizechange_g a b c = sizechange_f a (b @ c)"

fun
pedal :: "nat => nat => nat => nat"
and
coast :: "nat => nat => nat => nat"
where
"pedal 0 m c = c"
| "pedal n 0 c = c"
| "pedal n m c =
(if n < m then coast (n - 1) (m - 1) (c + m)
else pedal (n - 1) m (c + m))"

| "coast n m c =
(if n < m then coast n (m - 1) (c + n)
else pedal n m (c + n))"

subsection ‹Refined analysis: The ‹size_change› method›

text ‹Unsolvable for ‹lexicographic_order››

function fun1 :: "nat * nat ⇒ nat"
where
"fun1 (0,0) = 1"
| "fun1 (0, Suc b) = 0"
| "fun1 (Suc a, 0) = 0"
| "fun1 (Suc a, Suc b) = fun1 (b, a)"
by pat_completeness auto
termination by size_change

text ‹
‹lexicographic_order› can do the following, but it is much slower.
›

function
prod :: "nat => nat => nat => nat" and
eprod :: "nat => nat => nat => nat" and
oprod :: "nat => nat => nat => nat"
where
"prod x y z = (if y mod 2 = 0 then eprod x y z else oprod x y z)"
| "oprod x y z = eprod x (y - 1) (z+x)"
| "eprod x y z = (if y=0 then z else prod (2*x) (y div 2) z)"
by pat_completeness auto
termination by size_change

text ‹
Permutations of arguments:
›

function perm :: "nat ⇒ nat ⇒ nat ⇒ nat"
where
"perm m n r = (if r > 0 then perm m (r - 1) n
else if n > 0 then perm r (n - 1) m
else m)"
by auto
termination by size_change

text ‹
Artificial examples and regression tests:
›

function
fun2 :: "nat ⇒ nat ⇒ nat ⇒ nat"
where
"fun2 x y z =
(if x > 1000 ∧ z > 0 then
fun2 (min x y) y (z - 1)
else if y > 0 ∧ x > 100 then
fun2 x (y - 1) (2 * z)
else if z > 0 then
fun2 (min y (z - 1)) x x
else
0
)"
by pat_completeness auto
termination by size_change ― ‹requires Multiset›

definition negate :: "int ⇒ int"
where "negate i = - i"

function fun3 :: "int => nat"
where
"fun3 i =
(if i < 0 then fun3 (negate i)
else if i = 0 then 0
else fun3 (i - 1))"
by (pat_completeness) auto
termination
apply size_change