(* ID: $Id: NormalForm.thy,v 1.23 2007/10/25 11:52:01 haftmann Exp $ Authors: Klaus Aehlig, Tobias Nipkow *) header {* Test of normalization function *} theory NormalForm imports Main "~~/src/HOL/Real/Rational" begin lemma "True" by normalization lemma "x = x" by normalization lemma "p --> True" by normalization declare disj_assoc [code func] lemma "((P | Q) | R) = (P | (Q | R))" by normalization declare disj_assoc [code func del] lemma "0 + (n::nat) = n" by normalization lemma "0 + Suc n = Suc n" by normalization lemma "Suc n + Suc m = n + Suc (Suc m)" by normalization lemma "~((0::nat) < (0::nat))" by normalization datatype n = Z | S n consts add :: "n => n => n" add2 :: "n => n => n" mul :: "n => n => n" mul2 :: "n => n => n" exp :: "n => n => n" primrec "add Z = id" "add (S m) = S o add m" primrec "add2 Z n = n" "add2 (S m) n = S(add2 m n)" lemma [code]: "add2 (add2 n m) k = add2 n (add2 m k)" by(induct n) auto lemma [code]: "add2 n (S m) = S (add2 n m)" by(induct n) auto lemma [code]: "add2 n Z = n" by(induct n) auto lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization primrec "mul Z = (%n. Z)" "mul (S m) = (%n. add (mul m n) n)" primrec "mul2 Z n = Z" "mul2 (S m) n = add2 n (mul2 m n)" primrec "exp m Z = S Z" "exp m (S n) = mul (exp m n) m" lemma "mul2 (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization lemma "mul (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization lemma "split (%x y. x) (a, b) = a" by normalization lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization lemma "case Z of Z => True | S x => False" by normalization lemma "[] @ [] = []" by normalization normal_form "map f [x,y,z::'x] = [f x, f y, f z]" normal_form "[a, b, c] @ xs = a # b # c # xs" lemma "[] @ xs = xs" by normalization normal_form "map f [x,y,z::'x] = [f x, f y, f z]" normal_form "map (%f. f True) [id, g, Not] = [True, g True, False]" normal_form "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs" normal_form "rev [a, b, c] = [c, b, a]" normal_form "rev (a#b#cs) = rev cs @ [b, a]" normal_form "map (%F. F [a,b,c::'x]) (map map [f,g,h])" normal_form "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))" normal_form "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])" normal_form "map (%x. case x of None => False | Some y => True) [None, Some ()]" normal_form "case xs of [] => True | x#xs => False" normal_form "map (%x. case x of None => False | Some y => True) xs" normal_form "let x = y::'x in [x,x]" normal_form "Let y (%x. [x,x])" normal_form "case n of Z => True | S x => False" normal_form "(%(x,y). add x y) (S z,S z)" normal_form "filter (%x. x) ([True,False,x]@xs)" normal_form "filter Not ([True,False,x]@xs)" normal_form "[x,y,z] @ [a,b,c] = [x, y, z, a, b ,c]" normal_form "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" lemma "map (%x. case x of None => False | Some y => True) [None, Some ()] = [False, True]" by normalization lemma "last [a, b, c] = c" by normalization lemma "last ([a, b, c] @ xs) = (if null xs then c else last xs)" by normalization lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization lemma "(-4::int) * 2 = -8" by normalization lemma "abs ((-4::int) + 2 * 1) = 2" by normalization lemma "(2::int) + 3 = 5" by normalization lemma "(2::int) + 3 * (- 4) * (- 1) = 14" by normalization lemma "(2::int) + 3 * (- 4) * 1 + 0 = -10" by normalization lemma "(2::int) < 3" by normalization lemma "(2::int) <= 3" by normalization lemma "abs ((-4::int) + 2 * 1) = 2" by normalization lemma "4 - 42 * abs (3 + (-7::int)) = -164" by normalization lemma "(if (0::nat) ≤ (x::nat) then 0::nat else x) = 0" by normalization lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization lemma "max (Suc 0) 0 = Suc 0" by normalization lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization normal_form "Suc 0 ∈ set ms" normal_form "f" normal_form "f x" normal_form "(f o g) x" normal_form "(f o id) x" normal_form "λx. x" (* Church numerals: *) normal_form "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" normal_form "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" normal_form "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" end
lemma
True
lemma
x = x
lemma
p --> True
lemma
((P ∨ Q) ∨ R) = (P ∨ Q ∨ R)
lemma
0 + n = n
lemma
0 + Suc n = Suc n
lemma
Suc n + Suc m = n + Suc (Suc m)
lemma
¬ 0 < 0
lemma
add2 (add2 n m) k = add2 n (add2 m k)
lemma
add2 n (S m) = S (add2 n m)
lemma
add2 n Z = n
lemma
add2 (add2 n m) k = add2 n (add2 m k)
lemma
add2 (add2 (S n) (S m)) (S k) = S (S (S (add2 n (add2 m k))))
lemma
add2 (add2 (S n) (add2 (S m) Z)) (S k) = S (S (S (add2 n (add2 m k))))
lemma
mul2 (S (S (S (S (S Z))))) (S (S (S Z))) =
S (S (S (S (S (S (S (S (S (S (S (S (S (S (S Z))))))))))))))
lemma
mul (S (S (S (S (S Z))))) (S (S (S Z))) =
S (S (S (S (S (S (S (S (S (S (S (S (S (S (S Z))))))))))))))
lemma
exp (S (S Z)) (S (S (S (S Z)))) = exp (S (S (S (S Z)))) (S (S Z))
lemma
(let ((x, y), u, v) = ((Z, Z), Z, Z) in add (add x y) (add u v)) = Z
lemma
(λ(x, y). x) (a, b) = a
lemma
(λ((x, y), u, v). add (add x y) (add u v)) ((Z, Z), Z, Z) = Z
lemma
case Z of Z => True | S x => False
lemma
[] @ [] = []
lemma
[] @ xs = xs
lemma
map (option_case False (λy. True)) [None, Some ()] = [False, True]
lemma
last [a, b, c] = c
lemma
last ([a, b, c] @ xs) = (if null xs then c else last xs)
lemma
2 + 3 - 1 + - k * 2 = 4 + - k * 2
lemma
-4 * 2 = -8
lemma
¦-4 + 2 * 1¦ = 2
lemma
2 + 3 = 5
lemma
2 + 3 * - 4 * - 1 = 14
lemma
2 + 3 * - 4 * 1 + 0 = -10
lemma
2 < 3
lemma
2 ≤ 3
lemma
¦-4 + 2 * 1¦ = 2
lemma
4 - 42 * ¦3 + -7¦ = -164
lemma
(if 0 ≤ x then 0 else x) = 0
lemma
4 = Suc (Suc (Suc (Suc 0)))
lemma
nat 4 = Suc (Suc (Suc (Suc 0)))
lemma
[Suc 0, 0] = [Suc 0, 0]
lemma
max (Suc 0) 0 = Suc 0
lemma
42 / 1704 = 1 / 284 + 3 / 142