theory Predicate_Compile_ex
imports Main Predicate_Compile_Alternative_Defs
begin
inductive even :: "nat \<Rightarrow> bool" and odd :: "nat \<Rightarrow> bool" where
"even 0"
| "even n \<Longrightarrow> odd (Suc n)"
| "odd n \<Longrightarrow> even (Suc n)"
code_pred (mode: [], [1]) [show_steps, inductify] even .
thm odd.equation
thm even.equation
(*
lemma unit: "unit_case f = (\<lambda>x. f)"
apply (rule ext)
apply (case_tac x)
apply (simp only: unit.cases)
done
lemma "even_1 (Suc (Suc n)) = even_1 n"
thm even.equation(2)
unfolding even.equation(1)[of "Suc (Suc n)"]
unfolding odd.equation
unfolding single_bind
apply simp
apply (simp add: unit)
unfolding bind_single
apply (rule refl)
done
*)
values "{x. even 2}"
values "{x. odd 2}"
values 10 "{n. even n}"
values 10 "{n. odd n}"
inductive append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"append [] xs xs"
| "append xs ys zs \<Longrightarrow> append (x # xs) ys (x # zs)"
code_pred (mode: [1, 2], [3], [2, 3], [1, 3], [1, 2, 3]) [inductify] append .
thm append.equation
code_pred [rpred] append .
thm append.equation
thm append.rpred_equation
values "{(ys, xs). append xs ys [0, Suc 0, 2]}"
values "{zs. append [0, Suc 0, 2] [17, 8] zs}"
values "{ys. append [0, Suc 0, 2] ys [0, Suc 0, 2, 17, 0, 5]}"
value [code] "Predicate.the (append_1_2 [0::int, 1, 2] [3, 4, 5])"
value [code] "Predicate.the (append_3 ([]::int list))"
subsection {* Tricky case with alternative rules *}
inductive append2
where
"append2 [] xs xs"
| "append2 xs ys zs \<Longrightarrow> append2 (x # xs) ys (x # zs)"
lemma append2_Nil: "append2 [] (xs::'b list) xs"
by (simp add: append2.intros(1))
lemmas [code_pred_intros] = append2_Nil append2.intros(2)
code_pred append2
proof -
case append2
from append2.cases[OF append2(1)] append2(2-3) show thesis by blast
qed
subsection {* Tricky cases with tuples *}
inductive tupled_append :: "'a list \<times> 'a list \<times> 'a list \<Rightarrow> bool"
where
"tupled_append ([], xs, xs)"
| "tupled_append (xs, ys, zs) \<Longrightarrow> tupled_append (x # xs, ys, x # zs)"
code_pred tupled_append .
thm tupled_append.equation
(*
TODO: values with tupled modes
values "{xs. tupled_append ([1,2,3], [4,5], xs)}"
*)
inductive tupled_append'
where
"tupled_append' ([], xs, xs)"
| "[| ys = fst (xa, y); x # zs = snd (xa, y);
tupled_append' (xs, ys, x # zs) |] ==> tupled_append' (x # xs, xa, y)"
ML {* aconv *}
code_pred tupled_append' .
thm tupled_append'.equation
(* TODO: Modeanalysis returns mode [2] ?? *)
inductive tupled_append'' :: "'a list \<times> 'a list \<times> 'a list \<Rightarrow> bool"
where
"tupled_append'' ([], xs, xs)"
| "ys = fst yszs ==> x # zs = snd yszs ==> tupled_append'' (xs, ys, x # zs) \<Longrightarrow> tupled_append'' (x # xs, yszs)"
thm tupled_append''.cases
code_pred [inductify] tupled_append'' .
thm tupled_append''.equation
(* TODO: Modeanalysis returns mode [2] ?? *)
inductive tupled_append''' :: "'a list \<times> 'a list \<times> 'a list \<Rightarrow> bool"
where
"tupled_append''' ([], xs, xs)"
| "yszs = (ys, zs) ==> tupled_append''' (xs, yszs) \<Longrightarrow> tupled_append''' (x # xs, ys, x # zs)"
code_pred [inductify] tupled_append''' .
thm tupled_append'''.equation
(* TODO: Missing a few modes *)
inductive map_ofP :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
where
"map_ofP ((a, b)#xs) a b"
| "map_ofP xs a b \<Longrightarrow> map_ofP (x#xs) a b"
code_pred (mode: [1], [1, 2], [1, 2, 3], [1, 3]) map_ofP .
thm map_ofP.equation
section {* reverse *}
inductive rev where
"rev [] []"
| "rev xs xs' ==> append xs' [x] ys ==> rev (x#xs) ys"
code_pred (mode: [1], [2], [1, 2]) rev .
thm rev.equation
values "{xs. rev [0, 1, 2, 3::nat] xs}"
inductive tupled_rev where
"tupled_rev ([], [])"
| "tupled_rev (xs, xs') \<Longrightarrow> tupled_append (xs', [x], ys) \<Longrightarrow> tupled_rev (x#xs, ys)"
code_pred tupled_rev .
thm tupled_rev.equation
inductive partition :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
for f where
"partition f [] [] []"
| "f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) (x # ys) zs"
| "\<not> f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) ys (x # zs)"
code_pred (mode: [1], [2, 3], [1, 2], [1, 3], [1, 2, 3]) partition .
inductive tupled_partition :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
for f where
"tupled_partition f ([], [], [])"
| "f x \<Longrightarrow> tupled_partition f (xs, ys, zs) \<Longrightarrow> tupled_partition f (x # xs, x # ys, zs)"
| "\<not> f x \<Longrightarrow> tupled_partition f (xs, ys, zs) \<Longrightarrow> tupled_partition f (x # xs, ys, x # zs)"
code_pred tupled_partition .
thm tupled_partition.equation
inductive member
for xs
where "x \<in> set xs ==> member xs x"
lemma [code_pred_intros]:
"member (x#xs') x"
by (auto intro: member.intros)
lemma [code_pred_intros]:
"member xs x ==> member (x'#xs) x"
by (auto intro: member.intros elim!: member.cases)
(* strange bug must be repaired! *)
(*
code_pred member sorry
*)
inductive is_even :: "nat \<Rightarrow> bool"
where
"n mod 2 = 0 \<Longrightarrow> is_even n"
code_pred is_even .
values 10 "{(ys, zs). partition is_even
[0, Suc 0, 2, 3, 4, 5, 6, 7] ys zs}"
values 10 "{zs. partition is_even zs [0, 2] [3, 5]}"
values 10 "{zs. partition is_even zs [0, 7] [3, 5]}"
lemma [code_pred_intros]:
"r a b \<Longrightarrow> tranclp r a b"
"r a b \<Longrightarrow> tranclp r b c \<Longrightarrow> tranclp r a c"
by auto
code_pred tranclp
proof -
case tranclp
from this converse_tranclpE[OF this(1)] show thesis by metis
qed
code_pred [inductify, rpred] tranclp .
thm tranclp.equation
thm tranclp.rpred_equation
inductive succ :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
"succ 0 1"
| "succ m n \<Longrightarrow> succ (Suc m) (Suc n)"
code_pred succ .
thm succ.equation
values 10 "{(m, n). succ n m}"
values "{m. succ 0 m}"
values "{m. succ m 0}"
(* FIXME: why does this not terminate? -- value chooses mode [] --> [1] and then starts enumerating all successors *)
(*
values 20 "{n. tranclp succ 10 n}"
values "{n. tranclp succ n 10}"
values 20 "{(n, m). tranclp succ n m}"
*)
subsection{* IMP *}
types
var = nat
state = "int list"
datatype com =
Skip |
Ass var "state => int" |
Seq com com |
IF "state => bool" com com |
While "state => bool" com
inductive exec :: "com => state => state => bool" where
"exec Skip s s" |
"exec (Ass x e) s (s[x := e(s)])" |
"exec c1 s1 s2 ==> exec c2 s2 s3 ==> exec (Seq c1 c2) s1 s3" |
"b s ==> exec c1 s t ==> exec (IF b c1 c2) s t" |
"~b s ==> exec c2 s t ==> exec (IF b c1 c2) s t" |
"~b s ==> exec (While b c) s s" |
"b s1 ==> exec c s1 s2 ==> exec (While b c) s2 s3 ==> exec (While b c) s1 s3"
code_pred exec .
values "{t. exec
(While (%s. s!0 > 0) (Seq (Ass 0 (%s. s!0 - 1)) (Ass 1 (%s. s!1 + 1))))
[3,5] t}"
inductive tupled_exec :: "(com \<times> state \<times> state) \<Rightarrow> bool" where
"tupled_exec (Skip, s, s)" |
"tupled_exec (Ass x e, s, s[x := e(s)])" |
"tupled_exec (c1, s1, s2) ==> tupled_exec (c2, s2, s3) ==> tupled_exec (Seq c1 c2, s1, s3)" |
"b s ==> tupled_exec (c1, s, t) ==> tupled_exec (IF b c1 c2, s, t)" |
"~b s ==> tupled_exec (c2, s, t) ==> tupled_exec (IF b c1 c2, s, t)" |
"~b s ==> tupled_exec (While b c, s, s)" |
"b s1 ==> tupled_exec (c, s1, s2) ==> tupled_exec (While b c, s2, s3) ==> tupled_exec (While b c, s1, s3)"
code_pred tupled_exec .
subsection{* CCS *}
text{* This example formalizes finite CCS processes without communication or
recursion. For simplicity, labels are natural numbers. *}
datatype proc = nil | pre nat proc | or proc proc | par proc proc
inductive step :: "proc \<Rightarrow> nat \<Rightarrow> proc \<Rightarrow> bool" where
"step (pre n p) n p" |
"step p1 a q \<Longrightarrow> step (or p1 p2) a q" |
"step p2 a q \<Longrightarrow> step (or p1 p2) a q" |
"step p1 a q \<Longrightarrow> step (par p1 p2) a (par q p2)" |
"step p2 a q \<Longrightarrow> step (par p1 p2) a (par p1 q)"
code_pred step .
inductive steps where
"steps p [] p" |
"step p a q \<Longrightarrow> steps q as r \<Longrightarrow> steps p (a#as) r"
code_pred steps .
values 5
"{as . steps (par (or (pre 0 nil) (pre 1 nil)) (pre 2 nil)) as (par nil nil)}"
(* FIXME
values 3 "{(a,q). step (par nil nil) a q}"
*)
inductive tupled_step :: "(proc \<times> nat \<times> proc) \<Rightarrow> bool"
where
"tupled_step (pre n p, n, p)" |
"tupled_step (p1, a, q) \<Longrightarrow> tupled_step (or p1 p2, a, q)" |
"tupled_step (p2, a, q) \<Longrightarrow> tupled_step (or p1 p2, a, q)" |
"tupled_step (p1, a, q) \<Longrightarrow> tupled_step (par p1 p2, a, par q p2)" |
"tupled_step (p2, a, q) \<Longrightarrow> tupled_step (par p1 p2, a, par p1 q)"
code_pred tupled_step .
thm tupled_step.equation
subsection {* divmod *}
inductive divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
"k < l \<Longrightarrow> divmod_rel k l 0 k"
| "k \<ge> l \<Longrightarrow> divmod_rel (k - l) l q r \<Longrightarrow> divmod_rel k l (Suc q) r"
code_pred divmod_rel ..
value [code] "Predicate.the (divmod_rel_1_2 1705 42)"
section {* Executing definitions *}
definition Min
where "Min s r x \<equiv> s x \<and> (\<forall>y. r x y \<longrightarrow> x = y)"
code_pred [inductify] Min .
subsection {* Examples with lists *}
(*
inductive filterP for Pa where
"(filterP::('a => bool) => 'a list => 'a list => bool) (Pa::'a => bool) [] []"
| "[| (res::'a list) = (y::'a) # (resa::'a list); (filterP::('a => bool) => 'a list => 'a list => bool) (Pa::'a => bool) (xt::'a list) resa; Pa y |]
==> filterP Pa (y # xt) res"
| "[| (filterP::('a => bool) => 'a list => 'a list => bool) (Pa::'a => bool) (xt::'a list) (res::'a list); ~ Pa (y::'a) |] ==> filterP Pa (y # xt) res"
*)
(*
code_pred (inductify_all) (rpred) filterP .
thm filterP.rpred_equation
*)
code_pred [inductify] lexord .
thm lexord.equation
lemma "(u, v) : lexord r ==> (x @ u, y @ v) : lexord r"
(*quickcheck[generator=pred_compile]*)
oops
lemmas [code_pred_def] = lexn_conv lex_conv lenlex_conv
code_pred [inductify] lexn .
thm lexn.equation
code_pred [inductify] lenlex .
thm lenlex.equation
code_pred [inductify, rpred] lenlex .
thm lenlex.rpred_equation
thm lists.intros
code_pred [inductify] lists .
thm lists.equation
section {* AVL Tree Example *}
datatype 'a tree = ET | MKT 'a "'a tree" "'a tree" nat
fun height :: "'a tree => nat" where
"height ET = 0"
| "height (MKT x l r h) = max (height l) (height r) + 1"
consts avl :: "'a tree => bool"
primrec
"avl ET = True"
"avl (MKT x l r h) = ((height l = height r \<or> height l = 1 + height r \<or> height r = 1+height l) \<and>
h = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
code_pred [inductify] avl .
thm avl.equation
fun set_of
where
"set_of ET = {}"
| "set_of (MKT n l r h) = insert n (set_of l \<union> set_of r)"
fun is_ord :: "nat tree => bool"
where
"is_ord ET = True"
| "is_ord (MKT n l r h) =
((\<forall>n' \<in> set_of l. n' < n) \<and> (\<forall>n' \<in> set_of r. n < n') \<and> is_ord l \<and> is_ord r)"
code_pred [inductify] set_of .
thm set_of.equation
text {* expected mode: [1], [1, 2] *}
(* FIXME *)
(*
code_pred (inductify_all) is_ord .
thm is_ord.equation
*)
section {* Definitions about Relations *}
code_pred [inductify] converse .
thm converse.equation
code_pred [inductify] Domain .
thm Domain.equation
section {* List functions *}
code_pred [inductify] length .
thm size_listP.equation
code_pred [inductify] concat .
thm concatP.equation
code_pred [inductify] hd .
code_pred [inductify] tl .
code_pred [inductify] last .
code_pred [inductify] butlast .
thm listsum.simps
(*code_pred [inductify] listsum .*)
code_pred [inductify] take .
code_pred [inductify] drop .
code_pred [inductify] zip .
code_pred [inductify] upt .
code_pred [inductify] remdups .
code_pred [inductify] remove1 .
code_pred [inductify] removeAll .
code_pred [inductify] distinct .
code_pred [inductify] replicate .
code_pred [inductify] splice .
code_pred [inductify] List.rev .
thm map.simps
code_pred [inductify, show_steps, show_intermediate_results] map .
code_pred [inductify] foldr .
code_pred [inductify] foldl .
code_pred [inductify] filter .
definition test where "test xs = filter (\<lambda>x. x = (1::nat)) xs"
(*
TODO: implement higher-order replacement correctly
code_pred [inductify] test .
thm testP.equation
code_pred [inductify, rpred] test .
*)
section {* Handling set operations *}
section {* Context Free Grammar *}
datatype alphabet = a | b
inductive_set S\<^isub>1 and A\<^isub>1 and B\<^isub>1 where
"[] \<in> S\<^isub>1"
| "w \<in> A\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1"
| "w \<in> B\<^isub>1 \<Longrightarrow> a # w \<in> S\<^isub>1"
| "w \<in> S\<^isub>1 \<Longrightarrow> a # w \<in> A\<^isub>1"
| "w \<in> S\<^isub>1 \<Longrightarrow> b # w \<in> S\<^isub>1"
| "\<lbrakk>v \<in> B\<^isub>1; v \<in> B\<^isub>1\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>1"
code_pred [inductify] S\<^isub>1p .
thm S\<^isub>1p.equation
(*
theorem S\<^isub>1_sound:
"w \<in> S\<^isub>1 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
quickcheck[generator=pred_compile]
oops
*)
inductive_set S\<^isub>2 and A\<^isub>2 and B\<^isub>2 where
"[] \<in> S\<^isub>2"
| "w \<in> A\<^isub>2 \<Longrightarrow> b # w \<in> S\<^isub>2"
| "w \<in> B\<^isub>2 \<Longrightarrow> a # w \<in> S\<^isub>2"
| "w \<in> S\<^isub>2 \<Longrightarrow> a # w \<in> A\<^isub>2"
| "w \<in> S\<^isub>2 \<Longrightarrow> b # w \<in> B\<^isub>2"
| "\<lbrakk>v \<in> B\<^isub>2; v \<in> B\<^isub>2\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>2"
code_pred [inductify, rpred] S\<^isub>2 .
thm S\<^isub>2.rpred_equation
thm A\<^isub>2.rpred_equation
thm B\<^isub>2.rpred_equation
theorem S\<^isub>2_sound:
"w \<in> S\<^isub>2 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
(*quickcheck[generator=SML]*)
(*quickcheck[generator=pred_compile, size=15, iterations=100]*)
oops
inductive_set S\<^isub>3 and A\<^isub>3 and B\<^isub>3 where
"[] \<in> S\<^isub>3"
| "w \<in> A\<^isub>3 \<Longrightarrow> b # w \<in> S\<^isub>3"
| "w \<in> B\<^isub>3 \<Longrightarrow> a # w \<in> S\<^isub>3"
| "w \<in> S\<^isub>3 \<Longrightarrow> a # w \<in> A\<^isub>3"
| "w \<in> S\<^isub>3 \<Longrightarrow> b # w \<in> B\<^isub>3"
| "\<lbrakk>v \<in> B\<^isub>3; w \<in> B\<^isub>3\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>3"
code_pred [inductify] S\<^isub>3 .
theorem S\<^isub>3_sound:
"w \<in> S\<^isub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
(*quickcheck[generator=pred_compile, size=10, iterations=1]*)
oops
lemma "\<not> (length w > 2) \<or> \<not> (length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b])"
(*quickcheck[size=10, generator = pred_compile]*)
oops
(*
inductive test
where
"length [x \<leftarrow> w. a = x] = length [x \<leftarrow> w. x = b] ==> test w"
ML {* @{term "[x \<leftarrow> w. x = a]"} *}
code_pred (inductify_all) test .
thm test.equation
*)
(*
theorem S\<^isub>3_complete:
"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. b = x] \<longrightarrow> w \<in> S\<^isub>3"
(*quickcheck[generator=SML]*)
quickcheck[generator=pred_compile, size=10, iterations=100]
oops
*)
inductive_set S\<^isub>4 and A\<^isub>4 and B\<^isub>4 where
"[] \<in> S\<^isub>4"
| "w \<in> A\<^isub>4 \<Longrightarrow> b # w \<in> S\<^isub>4"
| "w \<in> B\<^isub>4 \<Longrightarrow> a # w \<in> S\<^isub>4"
| "w \<in> S\<^isub>4 \<Longrightarrow> a # w \<in> A\<^isub>4"
| "\<lbrakk>v \<in> A\<^isub>4; w \<in> A\<^isub>4\<rbrakk> \<Longrightarrow> b # v @ w \<in> A\<^isub>4"
| "w \<in> S\<^isub>4 \<Longrightarrow> b # w \<in> B\<^isub>4"
| "\<lbrakk>v \<in> B\<^isub>4; w \<in> B\<^isub>4\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^isub>4"
(*
theorem S\<^isub>4_sound:
"w \<in> S\<^isub>4 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
quickcheck[generator = pred_compile, size=2, iterations=1]
oops
*)
theorem S\<^isub>4_complete:
"length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^isub>4"
(*quickcheck[generator = pred_compile, size=5, iterations=1]*)
oops
theorem S\<^isub>4_A\<^isub>4_B\<^isub>4_sound_and_complete:
"w \<in> S\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
"w \<in> A\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] + 1"
"w \<in> B\<^isub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = b] = length [x \<leftarrow> w. x = a] + 1"
(*quickcheck[generator = pred_compile, size=5, iterations=1]*)
oops
section {* Lambda *}
datatype type =
Atom nat
| Fun type type (infixr "\<Rightarrow>" 200)
datatype dB =
Var nat
| App dB dB (infixl "\<degree>" 200)
| Abs type dB
primrec
nth_el :: "'a list \<Rightarrow> nat \<Rightarrow> 'a option" ("_\<langle>_\<rangle>" [90, 0] 91)
where
"[]\<langle>i\<rangle> = None"
| "(x # xs)\<langle>i\<rangle> = (case i of 0 \<Rightarrow> Some x | Suc j \<Rightarrow> xs \<langle>j\<rangle>)"
inductive nth_el' :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> bool"
where
"nth_el' (x # xs) 0 x"
| "nth_el' xs i y \<Longrightarrow> nth_el' (x # xs) (Suc i) y"
inductive typing :: "type list \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile> _ : _" [50, 50, 50] 50)
where
Var [intro!]: "nth_el' env x T \<Longrightarrow> env \<turnstile> Var x : T"
| Abs [intro!]: "T # env \<turnstile> t : U \<Longrightarrow> env \<turnstile> Abs T t : (T \<Rightarrow> U)"
(* | App [intro!]: "env \<turnstile> s : T \<Rightarrow> U \<Longrightarrow> env \<turnstile> t : T \<Longrightarrow> env \<turnstile> (s \<degree> t) : U" *)
| App [intro!]: "env \<turnstile> s : U \<Rightarrow> T \<Longrightarrow> env \<turnstile> t : T \<Longrightarrow> env \<turnstile> (s \<degree> t) : U"
primrec
lift :: "[dB, nat] => dB"
where
"lift (Var i) k = (if i < k then Var i else Var (i + 1))"
| "lift (s \<degree> t) k = lift s k \<degree> lift t k"
| "lift (Abs T s) k = Abs T (lift s (k + 1))"
primrec
subst :: "[dB, dB, nat] => dB" ("_[_'/_]" [300, 0, 0] 300)
where
subst_Var: "(Var i)[s/k] =
(if k < i then Var (i - 1) else if i = k then s else Var i)"
| subst_App: "(t \<degree> u)[s/k] = t[s/k] \<degree> u[s/k]"
| subst_Abs: "(Abs T t)[s/k] = Abs T (t[lift s 0 / k+1])"
inductive beta :: "[dB, dB] => bool" (infixl "\<rightarrow>\<^sub>\<beta>" 50)
where
beta [simp, intro!]: "Abs T s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"
| appL [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"
| appR [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> u \<degree> s \<rightarrow>\<^sub>\<beta> u \<degree> t"
| abs [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> Abs T s \<rightarrow>\<^sub>\<beta> Abs T t"
lemma "Gamma \<turnstile> t : T \<Longrightarrow> t \<rightarrow>\<^sub>\<beta> t' \<Longrightarrow> Gamma \<turnstile> t' : T"
quickcheck[generator = pred_compile, size = 10, iterations = 1]
oops
lemma filter_eq_ConsD:
"filter P ys = x#xs \<Longrightarrow>
\<exists>us vs. ys = ts @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
(*quickcheck[generator = pred_compile]*)
oops
end