src/HOL/ex/Normalization_by_Evaluation.thy

no longer declare .psimps rules as [simp].

This regularly caused confusion (e.g., they show up in simp traces
when the regular simp rules are disabled). In the few places where the
rules are used, explicitly mentioning them actually clarifies the
proof text.

(*  Authors:  Klaus Aehlig, Tobias Nipkow *)

header {* Testing implementation of normalization by evaluation *}

theory Normalization_by_Evaluation
imports Complex_Main
begin

lemma "True" by normalization
lemma "p \<longrightarrow> True" by normalization
declare disj_assoc [code nbe]
lemma "((P | Q) | R) = (P | (Q | R))" by normalization
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

primrec add :: "n \<Rightarrow> n \<Rightarrow> n" where
   "add Z = id"
 | "add (S m) = S o add m"

primrec add2 :: "n \<Rightarrow> n \<Rightarrow> n" where
   "add2 Z n = n"
 | "add2 (S m) n = S(add2 m n)"

declare add2.simps [code]
lemma [code nbe]: "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 :: "n \<Rightarrow> n \<Rightarrow> n" where
   "mul Z = (%n. Z)"
 | "mul (S m) = (%n. add (mul m n) n)"

primrec mul2 :: "n \<Rightarrow> n \<Rightarrow> n" where
   "mul2 Z n = Z"
 | "mul2 (S m) n = add2 n (mul2 m n)"

primrec exp :: "n \<Rightarrow> n \<Rightarrow> n" where
   "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 \<Rightarrow> True | S x \<Rightarrow> False" by normalization

lemma "[] @ [] = []" by normalization
lemma "map f [x,y,z::'x] = [f x, f y, f z]" by normalization
lemma "[a, b, c] @ xs = a # b # c # xs" by normalization
lemma "[] @ xs = xs" by normalization
lemma "map (%f. f True) [id, g, Not] = [True, g True, False]" by normalization

lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs"
  by normalization rule+
lemma "rev [a, b, c] = [c, b, a]" by normalization
value [nbe] "rev (a#b#cs) = rev cs @ [b, a]"
value [nbe] "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
value [nbe] "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
value [nbe] "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" 
  by normalization
value [nbe] "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
value [nbe] "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
lemma "let x = y in [x, x] = [y, y]" by normalization
lemma "Let y (%x. [x,x]) = [y, y]" by normalization
value [nbe] "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization
value [nbe] "filter (%x. x) ([True,False,x]@xs)"
value [nbe] "filter Not ([True,False,x]@xs)"

lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b, c]" by normalization
lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" by normalization

lemma "last [a, b, c] = c" by normalization
lemma "last ([a, b, c] @ xs) = last (c # 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\<Colon>int)) = -164" by normalization
lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>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
value [nbe] "Suc 0 \<in> set ms"

lemma "f = f" by normalization
lemma "f x = f x" by normalization
lemma "(f o g) x = f (g x)" by normalization
lemma "(f o id) x = f x" by normalization
value [nbe] "(\<lambda>x. x)"

(* Church numerals: *)

value [nbe] "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
value [nbe] "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
value [nbe] "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"

(* handling of type classes in connection with equality *)

lemma "map f [x, y] = [f x, f y]" by normalization
lemma "(map f [x, y], w) = ([f x, f y], w)" by normalization
lemma "map f [x, y] = [f x \<Colon> 'a\<Colon>semigroup_add, f y]" by normalization
lemma "map f [x \<Colon> 'a\<Colon>semigroup_add, y] = [f x, f y]" by normalization
lemma "(map f [x \<Colon> 'a\<Colon>semigroup_add, y], w \<Colon> 'b\<Colon>finite) = ([f x, f y], w)" by normalization

end