
diff -r a336af707767 -r 791ce568d40a src/HOL/Predicate_Compile_Examples/Predicate_Compile_Examples.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Predicate_Compile_Examples/Predicate_Compile_Examples.thy	Wed Mar 24 17:40:43 2010 +0100
@@ -0,0 +1,1278 @@
+theory Predicate_Compile_Examples
+imports "../ex/Predicate_Compile_Alternative_Defs"
+begin
+
+subsection {* Basic predicates *}
+
+inductive False' :: "bool"
+
+code_pred (expected_modes: bool) False' .
+code_pred [dseq] False' .
+code_pred [random_dseq] False' .
+
+values [expected "{}" pred] "{x. False'}"
+values [expected "{}" dseq 1] "{x. False'}"
+values [expected "{}" random_dseq 1, 1, 1] "{x. False'}"
+
+value "False'"
+
+
+inductive True' :: "bool"
+where
+  "True ==> True'"
+
+code_pred True' .
+code_pred [dseq] True' .
+code_pred [random_dseq] True' .
+
+thm True'.equation
+thm True'.dseq_equation
+thm True'.random_dseq_equation
+values [expected "{()}" ]"{x. True'}"
+values [expected "{}" dseq 0] "{x. True'}"
+values [expected "{()}" dseq 1] "{x. True'}"
+values [expected "{()}" dseq 2] "{x. True'}"
+values [expected "{}" random_dseq 1, 1, 0] "{x. True'}"
+values [expected "{}" random_dseq 1, 1, 1] "{x. True'}"
+values [expected "{()}" random_dseq 1, 1, 2] "{x. True'}"
+values [expected "{()}" random_dseq 1, 1, 3] "{x. True'}"
+
+inductive EmptySet :: "'a \<Rightarrow> bool"
+
+code_pred (expected_modes: o => bool, i => bool) EmptySet .
+
+definition EmptySet' :: "'a \<Rightarrow> bool"
+where "EmptySet' = {}"
+
+code_pred (expected_modes: o => bool, i => bool) [inductify] EmptySet' .
+
+inductive EmptyRel :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
+
+code_pred (expected_modes: o => o => bool, i => o => bool, o => i => bool, i => i => bool) EmptyRel .
+
+inductive EmptyClosure :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
+for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+
+code_pred
+  (expected_modes: (o => o => bool) => o => o => bool, (o => o => bool) => i => o => bool,
+         (o => o => bool) => o => i => bool, (o => o => bool) => i => i => bool,
+         (i => o => bool) => o => o => bool, (i => o => bool) => i => o => bool,
+         (i => o => bool) => o => i => bool, (i => o => bool) => i => i => bool,
+         (o => i => bool) => o => o => bool, (o => i => bool) => i => o => bool,
+         (o => i => bool) => o => i => bool, (o => i => bool) => i => i => bool,
+         (i => i => bool) => o => o => bool, (i => i => bool) => i => o => bool,
+         (i => i => bool) => o => i => bool, (i => i => bool) => i => i => bool)
+  EmptyClosure .
+
+thm EmptyClosure.equation
+
+(* TODO: inductive package is broken!
+inductive False'' :: "bool"
+where
+  "False \<Longrightarrow> False''"
+
+code_pred (expected_modes: []) False'' .
+
+inductive EmptySet'' :: "'a \<Rightarrow> bool"
+where
+  "False \<Longrightarrow> EmptySet'' x"
+
+code_pred (expected_modes: [1]) EmptySet'' .
+code_pred (expected_modes: [], [1]) [inductify] EmptySet'' .
+*)
+
+consts a' :: 'a
+
+inductive Fact :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
+"Fact a' a'"
+
+code_pred (expected_modes: o => o => bool, i => o => bool, o => i => bool, i => i => bool) Fact .
+
+inductive zerozero :: "nat * nat => bool"
+where
+  "zerozero (0, 0)"
+
+code_pred (expected_modes: i => bool, i * o => bool, o * i => bool, o => bool) zerozero .
+code_pred [dseq] zerozero .
+code_pred [random_dseq] zerozero .
+
+thm zerozero.equation
+thm zerozero.dseq_equation
+thm zerozero.random_dseq_equation
+
+text {* We expect the user to expand the tuples in the values command.
+The following values command is not supported. *}
+(*values "{x. zerozero x}" *)
+text {* Instead, the user must type *}
+values "{(x, y). zerozero (x, y)}"
+
+values [expected "{}" dseq 0] "{(x, y). zerozero (x, y)}"
+values [expected "{(0::nat, 0::nat)}" dseq 1] "{(x, y). zerozero (x, y)}"
+values [expected "{(0::nat, 0::nat)}" dseq 2] "{(x, y). zerozero (x, y)}"
+values [expected "{}" random_dseq 1, 1, 2] "{(x, y). zerozero (x, y)}"
+values [expected "{(0::nat, 0:: nat)}" random_dseq 1, 1, 3] "{(x, y). zerozero (x, y)}"
+
+inductive nested_tuples :: "((int * int) * int * int) => bool"
+where
+  "nested_tuples ((0, 1), 2, 3)"
+
+code_pred nested_tuples .
+
+inductive JamesBond :: "nat => int => code_numeral => bool"
+where
+  "JamesBond 0 0 7"
+
+code_pred JamesBond .
+
+values [expected "{(0::nat, 0::int , 7::code_numeral)}"] "{(a, b, c). JamesBond a b c}"
+values [expected "{(0::nat, 7::code_numeral, 0:: int)}"] "{(a, c, b). JamesBond a b c}"
+values [expected "{(0::int, 0::nat, 7::code_numeral)}"] "{(b, a, c). JamesBond a b c}"
+values [expected "{(0::int, 7::code_numeral, 0::nat)}"] "{(b, c, a). JamesBond a b c}"
+values [expected "{(7::code_numeral, 0::nat, 0::int)}"] "{(c, a, b). JamesBond a b c}"
+values [expected "{(7::code_numeral, 0::int, 0::nat)}"] "{(c, b, a). JamesBond a b c}"
+
+values [expected "{(7::code_numeral, 0::int)}"] "{(a, b). JamesBond 0 b a}"
+values [expected "{(7::code_numeral, 0::nat)}"] "{(c, a). JamesBond a 0 c}"
+values [expected "{(0::nat, 7::code_numeral)}"] "{(a, c). JamesBond a 0 c}"
+
+
+subsection {* Alternative Rules *}
+
+datatype char = C | D | E | F | G | H
+
+inductive is_C_or_D
+where
+  "(x = C) \<or> (x = D) ==> is_C_or_D x"
+
+code_pred (expected_modes: i => bool) is_C_or_D .
+thm is_C_or_D.equation
+
+inductive is_D_or_E
+where
+  "(x = D) \<or> (x = E) ==> is_D_or_E x"
+
+lemma [code_pred_intro]:
+  "is_D_or_E D"
+by (auto intro: is_D_or_E.intros)
+
+lemma [code_pred_intro]:
+  "is_D_or_E E"
+by (auto intro: is_D_or_E.intros)
+
+code_pred (expected_modes: o => bool, i => bool) is_D_or_E
+proof -
+  case is_D_or_E
+  from this(1) show thesis
+  proof
+    fix xa
+    assume x: "x = xa"
+    assume "xa = D \<or> xa = E"
+    from this show thesis
+    proof
+      assume "xa = D" from this x is_D_or_E(2) show thesis by simp
+    next
+      assume "xa = E" from this x is_D_or_E(3) show thesis by simp
+    qed
+  qed
+qed
+
+thm is_D_or_E.equation
+
+inductive is_F_or_G
+where
+  "x = F \<or> x = G ==> is_F_or_G x"
+
+lemma [code_pred_intro]:
+  "is_F_or_G F"
+by (auto intro: is_F_or_G.intros)
+
+lemma [code_pred_intro]:
+  "is_F_or_G G"
+by (auto intro: is_F_or_G.intros)
+
+inductive is_FGH
+where
+  "is_F_or_G x ==> is_FGH x"
+| "is_FGH H"
+
+text {* Compilation of is_FGH requires elimination rule for is_F_or_G *}
+
+code_pred (expected_modes: o => bool, i => bool) is_FGH
+proof -
+  case is_F_or_G
+  from this(1) show thesis
+  proof
+    fix xa
+    assume x: "x = xa"
+    assume "xa = F \<or> xa = G"
+    from this show thesis
+    proof
+      assume "xa = F"
+      from this x is_F_or_G(2) show thesis by simp
+    next
+      assume "xa = G"
+      from this x is_F_or_G(3) show thesis by simp
+    qed
+  qed
+qed
+
+subsection {* Preprocessor Inlining  *}
+
+definition "equals == (op =)"
+ 
+inductive zerozero' :: "nat * nat => bool" where
+  "equals (x, y) (0, 0) ==> zerozero' (x, y)"
+
+code_pred (expected_modes: i => bool) zerozero' .
+
+lemma zerozero'_eq: "zerozero' x == zerozero x"
+proof -
+  have "zerozero' = zerozero"
+    apply (auto simp add: mem_def)
+    apply (cases rule: zerozero'.cases)
+    apply (auto simp add: equals_def intro: zerozero.intros)
+    apply (cases rule: zerozero.cases)
+    apply (auto simp add: equals_def intro: zerozero'.intros)
+    done
+  from this show "zerozero' x == zerozero x" by auto
+qed
+
+declare zerozero'_eq [code_pred_inline]
+
+definition "zerozero'' x == zerozero' x"
+
+text {* if preprocessing fails, zerozero'' will not have all modes. *}
+
+code_pred (expected_modes: i * i => bool, i * o => bool, o * i => bool, o => bool) [inductify] zerozero'' .
+
+subsection {* Sets and Numerals *}
+
+definition
+  "one_or_two = {Suc 0, (Suc (Suc 0))}"
+
+code_pred [inductify] one_or_two .
+
+code_pred [dseq] one_or_two .
+code_pred [random_dseq] one_or_two .
+thm one_or_two.dseq_equation
+values [expected "{Suc 0::nat, 2::nat}"] "{x. one_or_two x}"
+values [random_dseq 0,0,10] 3 "{x. one_or_two x}"
+
+inductive one_or_two' :: "nat => bool"
+where
+  "one_or_two' 1"
+| "one_or_two' 2"
+
+code_pred one_or_two' .
+thm one_or_two'.equation
+
+values "{x. one_or_two' x}"
+
+definition one_or_two'':
+  "one_or_two'' == {1, (2::nat)}"
+
+code_pred [inductify] one_or_two'' .
+thm one_or_two''.equation
+
+values "{x. one_or_two'' x}"
+
+subsection {* even predicate *}
+
+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 (expected_modes: i => bool, o => bool) even .
+code_pred [dseq] even .
+code_pred [random_dseq] even .
+
+thm odd.equation
+thm even.equation
+thm odd.dseq_equation
+thm even.dseq_equation
+thm odd.random_dseq_equation
+thm even.random_dseq_equation
+
+values "{x. even 2}"
+values "{x. odd 2}"
+values 10 "{n. even n}"
+values 10 "{n. odd n}"
+values [expected "{}" dseq 2] "{x. even 6}"
+values [expected "{}" dseq 6] "{x. even 6}"
+values [expected "{()}" dseq 7] "{x. even 6}"
+values [dseq 2] "{x. odd 7}"
+values [dseq 6] "{x. odd 7}"
+values [dseq 7] "{x. odd 7}"
+values [expected "{()}" dseq 8] "{x. odd 7}"
+
+values [expected "{}" dseq 0] 8 "{x. even x}"
+values [expected "{0::nat}" dseq 1] 8 "{x. even x}"
+values [expected "{0::nat, 2}" dseq 3] 8 "{x. even x}"
+values [expected "{0::nat, 2}" dseq 4] 8 "{x. even x}"
+values [expected "{0::nat, 2, 4}" dseq 6] 8 "{x. even x}"
+
+values [random_dseq 1, 1, 0] 8 "{x. even x}"
+values [random_dseq 1, 1, 1] 8 "{x. even x}"
+values [random_dseq 1, 1, 2] 8 "{x. even x}"
+values [random_dseq 1, 1, 3] 8 "{x. even x}"
+values [random_dseq 1, 1, 6] 8 "{x. even x}"
+
+values [expected "{}" random_dseq 1, 1, 7] "{x. odd 7}"
+values [random_dseq 1, 1, 8] "{x. odd 7}"
+values [random_dseq 1, 1, 9] "{x. odd 7}"
+
+definition odd' where "odd' x == \<not> even x"
+
+code_pred (expected_modes: i => bool) [inductify] odd' .
+code_pred [dseq inductify] odd' .
+code_pred [random_dseq inductify] odd' .
+
+values [expected "{}" dseq 2] "{x. odd' 7}"
+values [expected "{()}" dseq 9] "{x. odd' 7}"
+values [expected "{}" dseq 2] "{x. odd' 8}"
+values [expected "{}" dseq 10] "{x. odd' 8}"
+
+
+inductive is_even :: "nat \<Rightarrow> bool"
+where
+  "n mod 2 = 0 \<Longrightarrow> is_even n"
+
+code_pred (expected_modes: i => bool) is_even .
+
+subsection {* append predicate *}
+
+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 (modes: i => i => o => bool as "concat", o => o => i => bool as "slice", o => i => i => bool as prefix,
+  i => o => i => bool as suffix, i => i => i => bool) append .
+code_pred [dseq] append .
+code_pred [random_dseq] append .
+
+thm append.equation
+thm append.dseq_equation
+thm append.random_dseq_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]}"
+
+values [expected "{}" dseq 0] 10 "{(xs, ys). append xs ys [1, 2, 3, 4, (5::nat)]}"
+values [expected "{(([]::nat list), [Suc 0, 2, 3, 4, (5::nat)])}" dseq 1] 10 "{(xs, ys). append xs ys [1, 2, 3, 4, (5::nat)]}"
+values [dseq 4] 10 "{(xs, ys). append xs ys [1, 2, 3, 4, (5::nat)]}"
+values [dseq 6] 10 "{(xs, ys). append xs ys [1, 2, 3, 4, (5::nat)]}"
+values [random_dseq 1, 1, 4] 10 "{(xs, ys). append xs ys [1, 2, 3, 4, (5::nat)]}"
+values [random_dseq 1, 1, 1] 10 "{(xs, ys, zs::int list). append xs ys zs}"
+values [random_dseq 1, 1, 3] 10 "{(xs, ys, zs::int list). append xs ys zs}"
+values [random_dseq 3, 1, 3] 10 "{(xs, ys, zs::int list). append xs ys zs}"
+values [random_dseq 1, 3, 3] 10 "{(xs, ys, zs::int list). append xs ys zs}"
+values [random_dseq 1, 1, 4] 10 "{(xs, ys, zs::int list). append xs ys zs}"
+
+value [code] "Predicate.the (concat [0::int, 1, 2] [3, 4, 5])"
+value [code] "Predicate.the (slice ([]::int list))"
+
+
+text {* 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_intro] = append2_Nil append2.intros(2)
+
+code_pred (expected_modes: i => i => o => bool, o => o => i => bool, o => i => i => bool,
+  i => o => i => bool, i => i => i => bool) append2
+proof -
+  case append2
+  from append2(1) show thesis
+  proof
+    fix xs
+    assume "xa = []" "xb = xs" "xc = xs"
+    from this append2(2) show thesis by simp
+  next
+    fix xs ys zs x
+    assume "xa = x # xs" "xb = ys" "xc = x # zs" "append2 xs ys zs"
+    from this append2(3) show thesis by fastsimp
+  qed
+qed
+
+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 (expected_modes: i * i * o => bool, o * o * i => bool, o * i * i => bool,
+  i * o * i => bool, i * i * i => bool) tupled_append .
+code_pred [random_dseq] tupled_append .
+thm tupled_append.equation
+
+values "{xs. tupled_append ([(1::nat), 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, zs) |] ==> tupled_append' (x # xs, xa, y)"
+
+code_pred (expected_modes: i * i * o => bool, o * o * i => bool, o * i * i => bool,
+  i * o * i => bool, i * i * i => bool) tupled_append' .
+thm tupled_append'.equation
+
+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, zs) \<Longrightarrow> tupled_append'' (x # xs, yszs)"
+
+code_pred (expected_modes: i * i * o => bool, o * o * i => bool, o * i * i => bool,
+  i * o * i => bool, i * i * i => bool) tupled_append'' .
+thm tupled_append''.equation
+
+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 (expected_modes: i * i * o => bool, o * o * i => bool, o * i * i => bool,
+  i * o * i => bool, i * i * i => bool) tupled_append''' .
+thm tupled_append'''.equation
+
+subsection {* map_ofP predicate *}
+
+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 (expected_modes: i => o => o => bool, i => i => o => bool, i => o => i => bool, i => i => i => bool) map_ofP .
+thm map_ofP.equation
+
+subsection {* filter predicate *}
+
+inductive filter1
+for P
+where
+  "filter1 P [] []"
+| "P x ==> filter1 P xs ys ==> filter1 P (x#xs) (x#ys)"
+| "\<not> P x ==> filter1 P xs ys ==> filter1 P (x#xs) ys"
+
+code_pred (expected_modes: (i => bool) => i => o => bool, (i => bool) => i => i => bool) filter1 .
+code_pred [dseq] filter1 .
+code_pred [random_dseq] filter1 .
+
+thm filter1.equation
+
+values [expected "{[0::nat, 2, 4]}"] "{xs. filter1 even [0, 1, 2, 3, 4] xs}"
+values [expected "{}" dseq 9] "{xs. filter1 even [0, 1, 2, 3, 4] xs}"
+values [expected "{[0::nat, 2, 4]}" dseq 10] "{xs. filter1 even [0, 1, 2, 3, 4] xs}"
+
+inductive filter2
+where
+  "filter2 P [] []"
+| "P x ==> filter2 P xs ys ==> filter2 P (x#xs) (x#ys)"
+| "\<not> P x ==> filter2 P xs ys ==> filter2 P (x#xs) ys"
+
+code_pred (expected_modes: (i => bool) => i => i => bool, (i => bool) => i => o => bool) filter2 .
+code_pred [dseq] filter2 .
+code_pred [random_dseq] filter2 .
+
+thm filter2.equation
+thm filter2.random_dseq_equation
+
+(*
+inductive filter3
+for P
+where
+  "List.filter P xs = ys ==> filter3 P xs ys"
+
+code_pred (expected_modes: (o => bool) => i => o => bool, (o => bool) => i => i => bool , (i => bool) => i => o => bool, (i => bool) => i => i => bool) [skip_proof] filter3 .
+
+code_pred [dseq] filter3 .
+thm filter3.dseq_equation
+*)
+(*
+inductive filter4
+where
+  "List.filter P xs = ys ==> filter4 P xs ys"
+
+code_pred (expected_modes: i => i => o => bool, i => i => i => bool) filter4 .
+(*code_pred [depth_limited] filter4 .*)
+(*code_pred [random] filter4 .*)
+*)
+subsection {* reverse predicate *}
+
+inductive rev where
+    "rev [] []"
+  | "rev xs xs' ==> append xs' [x] ys ==> rev (x#xs) ys"
+
+code_pred (expected_modes: i => o => bool, o => i => bool, i => i => bool) 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 (expected_modes: i * o => bool, o * i => bool, i * i => bool) tupled_rev .
+thm tupled_rev.equation
+
+subsection {* partition predicate *}
+
+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 (expected_modes: (i => bool) => i => o => o => bool, (i => bool) => o => i => i => bool,
+  (i => bool) => i => i => o => bool, (i => bool) => i => o => i => bool, (i => bool) => i => i => i => bool)
+  partition .
+code_pred [dseq] partition .
+code_pred [random_dseq] partition .
+
+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]}"
+
+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 (expected_modes: (i => bool) => i => bool, (i => bool) => (i * i * o) => bool, (i => bool) => (i * o * i) => bool,
+  (i => bool) => (o * i * i) => bool, (i => bool) => (i * o * o) => bool) tupled_partition .
+
+thm tupled_partition.equation
+
+lemma [code_pred_intro]:
+  "r a b \<Longrightarrow> tranclp r a b"
+  "r a b \<Longrightarrow> tranclp r b c \<Longrightarrow> tranclp r a c"
+  by auto
+
+subsection {* transitive predicate *}
+
+text {* Also look at the tabled transitive closure in the Library *}
+
+code_pred (modes: (i => o => bool) => i => i => bool, (i => o => bool) => i => o => bool as forwards_trancl,
+  (o => i => bool) => i => i => bool, (o => i => bool) => o => i => bool as backwards_trancl, (o => o => bool) => i => i => bool, (o => o => bool) => i => o => bool,
+  (o => o => bool) => o => i => bool, (o => o => bool) => o => o => bool) tranclp
+proof -
+  case tranclp
+  from this converse_tranclpE[OF this(1)] show thesis by metis
+qed
+
+
+code_pred [dseq] tranclp .
+code_pred [random_dseq] tranclp .
+thm tranclp.equation
+thm tranclp.random_dseq_equation
+
+inductive rtrancl' :: "'a => 'a => ('a => 'a => bool) => bool" 
+where
+  "rtrancl' x x r"
+| "r x y ==> rtrancl' y z r ==> rtrancl' x z r"
+
+code_pred [random_dseq] rtrancl' .
+
+thm rtrancl'.random_dseq_equation
+
+inductive rtrancl'' :: "('a * 'a * ('a \<Rightarrow> 'a \<Rightarrow> bool)) \<Rightarrow> bool"  
+where
+  "rtrancl'' (x, x, r)"
+| "r x y \<Longrightarrow> rtrancl'' (y, z, r) \<Longrightarrow> rtrancl'' (x, z, r)"
+
+code_pred rtrancl'' .
+
+inductive rtrancl''' :: "('a * ('a * 'a) * ('a * 'a => bool)) => bool" 
+where
+  "rtrancl''' (x, (x, x), r)"
+| "r (x, y) ==> rtrancl''' (y, (z, z), r) ==> rtrancl''' (x, (z, z), r)"
+
+code_pred rtrancl''' .
+
+
+inductive succ :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
+    "succ 0 1"
+  | "succ m n \<Longrightarrow> succ (Suc m) (Suc n)"
+
+code_pred (modes: i => i => bool, i => o => bool, o => i => bool, o => o => bool) succ .
+code_pred [random_dseq] succ .
+thm succ.equation
+thm succ.random_dseq_equation
+
+values 10 "{(m, n). succ n m}"
+values "{m. succ 0 m}"
+values "{m. succ m 0}"
+
+text {* values command needs mode annotation of the parameter succ
+to disambiguate which mode is to be chosen. *} 
+
+values [mode: i => o => bool] 20 "{n. tranclp succ 10 n}"
+values [mode: o => i => bool] 10 "{n. tranclp succ n 10}"
+values 20 "{(n, m). tranclp succ n m}"
+
+inductive example_graph :: "int => int => bool"
+where
+  "example_graph 0 1"
+| "example_graph 1 2"
+| "example_graph 1 3"
+| "example_graph 4 7"
+| "example_graph 4 5"
+| "example_graph 5 6"
+| "example_graph 7 6"
+| "example_graph 7 8"
+ 
+inductive not_reachable_in_example_graph :: "int => int => bool"
+where "\<not> (tranclp example_graph x y) ==> not_reachable_in_example_graph x y"
+
+code_pred (expected_modes: i => i => bool) not_reachable_in_example_graph .
+
+thm not_reachable_in_example_graph.equation
+thm tranclp.equation
+value "not_reachable_in_example_graph 0 3"
+value "not_reachable_in_example_graph 4 8"
+value "not_reachable_in_example_graph 5 6"
+text {* rtrancl compilation is strange! *}
+(*
+value "not_reachable_in_example_graph 0 4"
+value "not_reachable_in_example_graph 1 6"
+value "not_reachable_in_example_graph 8 4"*)
+
+code_pred [dseq] not_reachable_in_example_graph .
+
+values [dseq 6] "{x. tranclp example_graph 0 3}"
+
+values [dseq 0] "{x. not_reachable_in_example_graph 0 3}"
+values [dseq 0] "{x. not_reachable_in_example_graph 0 4}"
+values [dseq 20] "{x. not_reachable_in_example_graph 0 4}"
+values [dseq 6] "{x. not_reachable_in_example_graph 0 3}"
+values [dseq 3] "{x. not_reachable_in_example_graph 4 2}"
+values [dseq 6] "{x. not_reachable_in_example_graph 4 2}"
+
+
+inductive not_reachable_in_example_graph' :: "int => int => bool"
+where "\<not> (rtranclp example_graph x y) ==> not_reachable_in_example_graph' x y"
+
+code_pred not_reachable_in_example_graph' .
+
+value "not_reachable_in_example_graph' 0 3"
+(* value "not_reachable_in_example_graph' 0 5" would not terminate *)
+
+
+(*values [depth_limited 0] "{x. not_reachable_in_example_graph' 0 3}"*)
+(*values [depth_limited 3] "{x. not_reachable_in_example_graph' 0 3}"*) (* fails with undefined *)
+(*values [depth_limited 5] "{x. not_reachable_in_example_graph' 0 3}"*)
+(*values [depth_limited 1] "{x. not_reachable_in_example_graph' 0 4}"*)
+(*values [depth_limit = 4] "{x. not_reachable_in_example_graph' 0 4}"*) (* fails with undefined *)
+(*values [depth_limit = 20] "{x. not_reachable_in_example_graph' 0 4}"*) (* fails with undefined *)
+
+code_pred [dseq] not_reachable_in_example_graph' .
+
+(*thm not_reachable_in_example_graph'.dseq_equation*)
+
+(*values [dseq 0] "{x. not_reachable_in_example_graph' 0 3}"*)
+(*values [depth_limited 3] "{x. not_reachable_in_example_graph' 0 3}"*) (* fails with undefined *)
+(*values [depth_limited 5] "{x. not_reachable_in_example_graph' 0 3}"
+values [depth_limited 1] "{x. not_reachable_in_example_graph' 0 4}"*)
+(*values [depth_limit = 4] "{x. not_reachable_in_example_graph' 0 4}"*) (* fails with undefined *)
+(*values [depth_limit = 20] "{x. not_reachable_in_example_graph' 0 4}"*) (* fails with undefined *)
+
+
+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 .
+
+values "{s. tupled_exec (While (%s. s!0 > 0) (Seq (Ass 0 (%s. s!0 - 1)) (Ass 1 (%s. s!1 + 1))), [3, 5], s)}"
+
+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 3 
+ "{as . steps (par (or (pre 0 nil) (pre 1 nil)) (pre 2 nil)) as (par nil nil)}"
+
+values 5
+ "{as . steps (par (or (pre 0 nil) (pre 1 nil)) (pre 2 nil)) as (par nil nil)}"
+
+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 ..
+thm divmod_rel.equation
+value [code] "Predicate.the (divmod_rel_i_i_o_o 1705 42)"
+
+subsection {* Transforming predicate logic into logic programs *}
+
+subsection {* Transforming functions into logic programs *}
+definition
+  "case_f xs ys = (case (xs @ ys) of [] => [] | (x # xs) => xs)"
+
+code_pred [inductify] case_f .
+thm case_fP.equation
+thm case_fP.intros
+
+fun fold_map_idx where
+  "fold_map_idx f i y [] = (y, [])"
+| "fold_map_idx f i y (x # xs) =
+ (let (y', x') = f i y x; (y'', xs') = fold_map_idx f (Suc i) y' xs
+ in (y'', x' # xs'))"
+
+text {* mode analysis explores thousand modes - this is infeasible at the moment... *}
+(*code_pred [inductify, show_steps] fold_map_idx .*)
+
+subsection {* Minimum *}
+
+definition Min
+where "Min s r x \<equiv> s x \<and> (\<forall>y. r x y \<longrightarrow> x = y)"
+
+code_pred [inductify] Min .
+thm Min.equation
+
+subsection {* Lexicographic order *}
+
+declare lexord_def[code_pred_def]
+code_pred [inductify] lexord .
+code_pred [random_dseq inductify] lexord .
+
+thm lexord.equation
+thm lexord.random_dseq_equation
+
+inductive less_than_nat :: "nat * nat => bool"
+where
+  "less_than_nat (0, x)"
+| "less_than_nat (x, y) ==> less_than_nat (Suc x, Suc y)"
+ 
+code_pred less_than_nat .
+
+code_pred [dseq] less_than_nat .
+code_pred [random_dseq] less_than_nat .
+
+inductive test_lexord :: "nat list * nat list => bool"
+where
+  "lexord less_than_nat (xs, ys) ==> test_lexord (xs, ys)"
+
+code_pred test_lexord .
+code_pred [dseq] test_lexord .
+code_pred [random_dseq] test_lexord .
+thm test_lexord.dseq_equation
+thm test_lexord.random_dseq_equation
+
+values "{x. test_lexord ([1, 2, 3], [1, 2, 5])}"
+(*values [depth_limited 5] "{x. test_lexord ([1, 2, 3], [1, 2, 5])}"*)
+
+declare list.size(3,4)[code_pred_def]
+lemmas [code_pred_def] = lexn_conv lex_conv lenlex_conv
+(*
+code_pred [inductify] lexn .
+thm lexn.equation
+*)
+(*
+code_pred [random_dseq inductify] lexn .
+thm lexn.random_dseq_equation
+
+values [random_dseq 4, 4, 6] 100 "{(n, xs, ys::int list). lexn (%(x, y). x <= y) n (xs, ys)}"
+*)
+inductive has_length
+where
+  "has_length [] 0"
+| "has_length xs i ==> has_length (x # xs) (Suc i)" 
+
+lemma has_length:
+  "has_length xs n = (length xs = n)"
+proof (rule iffI)
+  assume "has_length xs n"
+  from this show "length xs = n"
+    by (rule has_length.induct) auto
+next
+  assume "length xs = n"
+  from this show "has_length xs n"
+    by (induct xs arbitrary: n) (auto intro: has_length.intros)
+qed
+
+lemma lexn_intros [code_pred_intro]:
+  "has_length xs i ==> has_length ys i ==> r (x, y) ==> lexn r (Suc i) (x # xs, y # ys)"
+  "lexn r i (xs, ys) ==> lexn r (Suc i) (x # xs, x # ys)"
+proof -
+  assume "has_length xs i" "has_length ys i" "r (x, y)"
+  from this has_length show "lexn r (Suc i) (x # xs, y # ys)"
+    unfolding lexn_conv Collect_def mem_def
+    by fastsimp
+next
+  assume "lexn r i (xs, ys)"
+  thm lexn_conv
+  from this show "lexn r (Suc i) (x#xs, x#ys)"
+    unfolding Collect_def mem_def lexn_conv
+    apply auto
+    apply (rule_tac x="x # xys" in exI)
+    by auto
+qed
+
+code_pred [random_dseq inductify] lexn
+proof -
+  fix r n xs ys
+  assume 1: "lexn r n (xs, ys)"
+  assume 2: "\<And>r' i x xs' y ys'. r = r' ==> n = Suc i ==> (xs, ys) = (x # xs', y # ys') ==> has_length xs' i ==> has_length ys' i ==> r' (x, y) ==> thesis"
+  assume 3: "\<And>r' i x xs' ys'. r = r' ==> n = Suc i ==> (xs, ys) = (x # xs', x # ys') ==> lexn r' i (xs', ys') ==> thesis"
+  from 1 2 3 show thesis
+    unfolding lexn_conv Collect_def mem_def
+    apply (auto simp add: has_length)
+    apply (case_tac xys)
+    apply auto
+    apply fastsimp
+    apply fastsimp done
+qed
+
+
+values [random_dseq 1, 2, 5] 10 "{(n, xs, ys::int list). lexn (%(x, y). x <= y) n (xs, ys)}"
+thm lenlex_conv
+thm lex_conv
+declare list.size(3,4)[code_pred_def]
+(*code_pred [inductify, show_steps, show_intermediate_results] length .*)
+setup {* Predicate_Compile_Data.ignore_consts [@{const_name Orderings.top_class.top}] *}
+code_pred [inductify] lex .
+thm lex.equation
+thm lex_def
+declare lenlex_conv[code_pred_def]
+code_pred [inductify] lenlex .
+thm lenlex.equation
+
+code_pred [random_dseq inductify] lenlex .
+thm lenlex.random_dseq_equation
+
+values [random_dseq 4, 2, 4] 100 "{(xs, ys::int list). lenlex (%(x, y). x <= y) (xs, ys)}"
+thm lists.intros
+
+code_pred [inductify] lists .
+thm lists.equation
+
+subsection {* AVL Tree *}
+
+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*)
+
+code_pred [random_dseq inductify] avl .
+thm avl.random_dseq_equation
+
+values [random_dseq 2, 1, 7] 5 "{t:: int tree. avl t}"
+
+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 (expected_modes: i => o => bool, i => i => bool) [inductify] set_of .
+thm set_of.equation
+
+code_pred (expected_modes: i => bool) [inductify] is_ord .
+thm is_ord_aux.equation
+thm is_ord.equation
+
+
+subsection {* Definitions about Relations *}
+term "converse"
+code_pred (modes:
+  (i * i => bool) => i * i => bool,
+  (i * o => bool) => o * i => bool,
+  (i * o => bool) => i * i => bool,
+  (o * i => bool) => i * o => bool,
+  (o * i => bool) => i * i => bool,
+  (o * o => bool) => o * o => bool,
+  (o * o => bool) => i * o => bool,
+  (o * o => bool) => o * i => bool,
+  (o * o => bool) => i * i => bool) [inductify] converse .
+
+thm converse.equation
+code_pred [inductify] rel_comp .
+thm rel_comp.equation
+code_pred [inductify] Image .
+thm Image.equation
+declare singleton_iff[code_pred_inline]
+declare Id_on_def[unfolded Bex_def UNION_def singleton_iff, code_pred_def]
+
+code_pred (expected_modes:
+  (o => bool) => o => bool,
+  (o => bool) => i * o => bool,
+  (o => bool) => o * i => bool,
+  (o => bool) => i => bool,
+  (i => bool) => i * o => bool,
+  (i => bool) => o * i => bool,
+  (i => bool) => i => bool) [inductify] Id_on .
+thm Id_on.equation
+thm Domain_def
+code_pred (modes:
+  (o * o => bool) => o => bool,
+  (o * o => bool) => i => bool,
+  (i * o => bool) => i => bool) [inductify] Domain .
+thm Domain.equation
+
+thm Range_def
+code_pred (modes:
+  (o * o => bool) => o => bool,
+  (o * o => bool) => i => bool,
+  (o * i => bool) => i => bool) [inductify] Range .
+thm Range.equation
+
+code_pred [inductify] Field .
+thm Field.equation
+
+thm refl_on_def
+code_pred [inductify] refl_on .
+thm refl_on.equation
+code_pred [inductify] total_on .
+thm total_on.equation
+code_pred [inductify] antisym .
+thm antisym.equation
+code_pred [inductify] trans .
+thm trans.equation
+code_pred [inductify] single_valued .
+thm single_valued.equation
+thm inv_image_def
+code_pred [inductify] inv_image .
+thm inv_image.equation
+
+subsection {* Inverting list functions *}
+
+(*code_pred [inductify] length .
+code_pred [random inductify] length .
+thm size_listP.equation
+thm size_listP.random_equation
+*)
+(*values [random] 1 "{xs. size_listP (xs::nat list) (5::nat)}"*)
+
+code_pred (expected_modes: i => o => bool, o => i => bool, i => i => bool) [inductify] List.concat .
+thm concatP.equation
+
+values "{ys. concatP [[1, 2], [3, (4::int)]] ys}"
+values "{ys. concatP [[1, 2], [3]] [1, 2, (3::nat)]}"
+
+code_pred [dseq inductify] List.concat .
+thm concatP.dseq_equation
+
+values [dseq 3] 3
+  "{xs. concatP xs ([0] :: nat list)}"
+
+values [dseq 5] 3
+  "{xs. concatP xs ([1] :: int list)}"
+
+values [dseq 5] 3
+  "{xs. concatP xs ([1] :: nat list)}"
+
+values [dseq 5] 3
+  "{xs. concatP xs [(1::int), 2]}"
+
+code_pred (expected_modes: i => o => bool, i => i => bool) [inductify] hd .
+thm hdP.equation
+values "{x. hdP [1, 2, (3::int)] x}"
+values "{(xs, x). hdP [1, 2, (3::int)] 1}"
+ 
+code_pred (expected_modes: i => o => bool, i => i => bool) [inductify] tl .
+thm tlP.equation
+values "{x. tlP [1, 2, (3::nat)] x}"
+values "{x. tlP [1, 2, (3::int)] [3]}"
+
+code_pred [inductify] last .
+thm lastP.equation
+
+code_pred [inductify] butlast .
+thm butlastP.equation
+
+code_pred [inductify] take .
+thm takeP.equation
+
+code_pred [inductify] drop .
+thm dropP.equation
+code_pred [inductify] zip .
+thm zipP.equation
+
+code_pred [inductify] upt .
+code_pred [inductify] remdups .
+thm remdupsP.equation
+code_pred [dseq inductify] remdups .
+values [dseq 4] 5 "{xs. remdupsP xs [1, (2::int)]}"
+
+code_pred [inductify] remove1 .
+thm remove1P.equation
+values "{xs. remove1P 1 xs [2, (3::int)]}"
+
+code_pred [inductify] removeAll .
+thm removeAllP.equation
+code_pred [dseq inductify] removeAll .
+
+values [dseq 4] 10 "{xs. removeAllP 1 xs [(2::nat)]}"
+
+code_pred [inductify] distinct .
+thm distinct.equation
+code_pred [inductify] replicate .
+thm replicateP.equation
+values 5 "{(n, xs). replicateP n (0::int) xs}"
+
+code_pred [inductify] splice .
+thm splice.simps
+thm spliceP.equation
+
+values "{xs. spliceP xs [1, 2, 3] [1, 1, 1, 2, 1, (3::nat)]}"
+
+code_pred [inductify] List.rev .
+code_pred [inductify] map .
+code_pred [inductify] foldr .
+code_pred [inductify] foldl .
+code_pred [inductify] filter .
+code_pred [random_dseq inductify] filter .
+
+subsection {* 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 .
+code_pred [random_dseq inductify] S\<^isub>1p .
+thm S\<^isub>1p.equation
+thm S\<^isub>1p.random_dseq_equation
+
+values [random_dseq 5, 5, 5] 5 "{x. S\<^isub>1p x}"
+
+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 [random_dseq inductify] S\<^isub>2p .
+thm S\<^isub>2p.random_dseq_equation
+thm A\<^isub>2p.random_dseq_equation
+thm B\<^isub>2p.random_dseq_equation
+
+values [random_dseq 5, 5, 5] 10 "{x. S\<^isub>2p x}"
+
+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>3p .
+thm S\<^isub>3p.equation
+
+values 10 "{x. S\<^isub>3p x}"
+
+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"
+
+code_pred (expected_modes: o => bool, i => bool) S\<^isub>4p .
+
+subsection {* 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"
+
+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"
+
+code_pred (expected_modes: i => i => o => bool, i => i => i => bool) typing .
+thm typing.equation
+
+code_pred (modes: i => i => bool,  i => o => bool as reduce') beta .
+thm beta.equation
+
+values "{x. App (Abs (Atom 0) (Var 0)) (Var 1) \<rightarrow>\<^sub>\<beta> x}"
+
+definition "reduce t = Predicate.the (reduce' t)"
+
+value "reduce (App (Abs (Atom 0) (Var 0)) (Var 1))"
+
+code_pred [dseq] typing .
+code_pred [random_dseq] typing .
+
+values [random_dseq 1,1,5] 10 "{(\<Gamma>, t, T). \<Gamma> \<turnstile> t : T}"
+
+subsection {* A minimal example of yet another semantics *}
+
+text {* thanks to Elke Salecker *}
+
+types
+  vname = nat
+  vvalue = int
+  var_assign = "vname \<Rightarrow> vvalue"  --"variable assignment"
+
+datatype ir_expr = 
+  IrConst vvalue
+| ObjAddr vname
+| Add ir_expr ir_expr
+
+datatype val =
+  IntVal  vvalue
+
+record  configuration =
+  Env :: var_assign
+
+inductive eval_var ::
+  "ir_expr \<Rightarrow> configuration \<Rightarrow> val \<Rightarrow> bool"
+where
+  irconst: "eval_var (IrConst i) conf (IntVal i)"
+| objaddr: "\<lbrakk> Env conf n = i \<rbrakk> \<Longrightarrow> eval_var (ObjAddr n) conf (IntVal i)"
+| plus: "\<lbrakk> eval_var l conf (IntVal vl); eval_var r conf (IntVal vr) \<rbrakk> \<Longrightarrow> eval_var (Add l r) conf (IntVal (vl+vr))"
+
+
+code_pred eval_var .
+thm eval_var.equation
+
+values "{val. eval_var (Add (IrConst 1) (IrConst 2)) (| Env = (\<lambda>x. 0)|) val}"
+
+end




