# Theory Predicate_Compile_Tests

```theory Predicate_Compile_Tests
imports "HOL-Library.Predicate_Compile_Alternative_Defs"
begin

declare [[values_timeout = 480.0]]

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 EmptyPred :: "'a ⇒ bool"

code_pred (expected_modes: o => bool, i => bool) EmptyPred .

definition EmptyPred' :: "'a ⇒ bool"
where "EmptyPred' = (λ x. False)"

code_pred (expected_modes: o => bool, i => bool) [inductify] EmptyPred' .

inductive EmptyRel :: "'a ⇒ 'b ⇒ bool"

code_pred (expected_modes: o => o => bool, i => o => bool, o => i => bool, i => i => bool) EmptyRel .

inductive EmptyClosure :: "('a ⇒ 'a ⇒ bool) ⇒ 'a ⇒ 'a ⇒ bool"
for r :: "'a ⇒ 'a ⇒ 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 ⟹ False''"

code_pred (expected_modes: bool) False'' .

inductive EmptySet'' :: "'a ⇒ bool"
where
"False ⟹ EmptySet'' x"

code_pred (expected_modes: i => bool, o => bool) [inductify] EmptySet'' .
*)

consts a' :: 'a

inductive Fact :: "'a ⇒ 'a ⇒ bool"
where
"Fact a' a'"

code_pred (expected_modes: o => o => bool, i => o => bool, o => i => bool, i => i => bool) Fact .

text ‹
The following example was derived from an predicate in the CakeML formalisation provided by Lars Hupel.
›
inductive predicate_where_argument_is_condition :: "bool ⇒ bool"
where
"ck ⟹ predicate_where_argument_is_condition ck"

code_pred predicate_where_argument_is_condition .

text ‹Other similar examples of this kind:›

inductive predicate_where_argument_is_in_equation :: "bool ⇒ bool"
where
"ck = True ⟹ predicate_where_argument_is_in_equation ck"

code_pred predicate_where_argument_is_in_equation .

inductive predicate_where_argument_is_condition_and_value :: "bool ⇒ bool"
where
"predicate_where_argument_is_condition_and_value ck ⟹ ck
⟹ predicate_where_argument_is_condition_and_value ck"

code_pred predicate_where_argument_is_condition_and_value .

inductive predicate_where_argument_is_neg_condition :: "bool ⇒ bool"
where
"¬ ck ⟹ predicate_where_argument_is_neg_condition ck"

code_pred predicate_where_argument_is_neg_condition .

inductive predicate_where_argument_is_neg_condition_and_in_equation :: "bool ⇒ bool"
where
"¬ ck ⟹ ck = False ⟹ predicate_where_argument_is_neg_condition_and_in_equation ck"

code_pred predicate_where_argument_is_neg_condition_and_in_equation .

text ‹Another related example that required slight adjustment of the proof procedure›

inductive if_as_predicate :: "bool ⇒ 'a ⇒ 'a ⇒ 'a ⇒ bool"
where
"condition ⟹ if_as_predicate condition then_value else_value then_value"
| "¬ condition ⟹ if_as_predicate condition then_value else_value else_value"

code_pred [show_proof_trace] if_as_predicate .

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 => natural => bool"
where
"JamesBond 0 0 7"

code_pred JamesBond .

values [expected "{(0::nat, 0::int , 7::natural)}"] "{(a, b, c). JamesBond a b c}"
values [expected "{(0::nat, 7::natural, 0:: int)}"] "{(a, c, b). JamesBond a b c}"
values [expected "{(0::int, 0::nat, 7::natural)}"] "{(b, a, c). JamesBond a b c}"
values [expected "{(0::int, 7::natural, 0::nat)}"] "{(b, c, a). JamesBond a b c}"
values [expected "{(7::natural, 0::nat, 0::int)}"] "{(c, a, b). JamesBond a b c}"
values [expected "{(7::natural, 0::int, 0::nat)}"] "{(c, b, a). JamesBond a b c}"

values [expected "{(7::natural, 0::int)}"] "{(a, b). JamesBond 0 b a}"
values [expected "{(7::natural, 0::nat)}"] "{(c, a). JamesBond a 0 c}"
values [expected "{(0::nat, 7::natural)}"] "{(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) ∨ (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) ∨ (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 is_D_or_E.prems show thesis
proof
fix xa
assume x: "x = xa"
assume "xa = D ∨ xa = E"
from this show thesis
proof
assume "xa = D" from this x is_D_or_E(1) show thesis by simp
next
assume "xa = E" from this x is_D_or_E(2) show thesis by simp
qed
qed
qed

thm is_D_or_E.equation

inductive is_F_or_G
where
"x = F ∨ 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 is_F_or_G.prems show thesis
proof
fix xa
assume x: "x = xa"
assume "xa = F ∨ xa = G"
from this show thesis
proof
assume "xa = F"
from this x is_F_or_G(1) show thesis by simp
next
assume "xa = G"
from this x is_F_or_G(2) show thesis by simp
qed
qed
qed

subsection ‹Named alternative rules›

inductive appending
where
nil: "appending [] ys ys"
| cons: "appending xs ys zs ⟹ appending (x#xs) ys (x#zs)"

lemma appending_alt_nil: "ys = zs ⟹ appending [] ys zs"
by (auto intro: appending.intros)

lemma appending_alt_cons: "xs' = x # xs ⟹ appending xs ys zs ⟹ zs' = x # zs ⟹ appending xs' ys zs'"
by (auto intro: appending.intros)

text ‹With code_pred_intro, we can give fact names to the alternative rules,
which are used for the code_pred command.›

declare appending_alt_nil[code_pred_intro alt_nil] appending_alt_cons[code_pred_intro alt_cons]

code_pred appending
proof -
case appending
from appending.prems show thesis
proof(cases)
case nil
from appending.alt_nil nil show thesis by auto
next
case cons
from appending.alt_cons cons show thesis by fastforce
qed
qed

inductive ya_even and ya_odd :: "nat => bool"
where
even_zero: "ya_even 0"
| odd_plus1: "ya_even x ==> ya_odd (x + 1)"
| even_plus1: "ya_odd x ==> ya_even (x + 1)"

declare even_zero[code_pred_intro even_0]

lemma [code_pred_intro odd_Suc]: "ya_even x ==> ya_odd (Suc x)"
by (auto simp only: Suc_eq_plus1 intro: ya_even_ya_odd.intros)

lemma [code_pred_intro even_Suc]:"ya_odd x ==> ya_even (Suc x)"
by (auto simp only: Suc_eq_plus1 intro: ya_even_ya_odd.intros)

code_pred ya_even
proof -
case ya_even
from ya_even.prems show thesis
proof (cases)
case even_zero
from even_zero ya_even.even_0 show thesis by simp
next
case even_plus1
from even_plus1 ya_even.even_Suc show thesis by simp
qed
next
case ya_odd
from ya_odd.prems show thesis
proof (cases)
case odd_plus1
from odd_plus1 ya_odd.odd_Suc show thesis by simp
qed
qed

subsection ‹Preprocessor Inlining›

definition "equals == (=)"

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 (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›
(*
inductive_set EmptySet :: "'a set"

code_pred (expected_modes: o => bool, i => bool) EmptySet .

definition EmptySet' :: "'a set"
where "EmptySet' = {}"

code_pred (expected_modes: o => bool, i => bool) [inductify] EmptySet' .
*)
subsection ‹Numerals›

definition
"one_or_two = (%x. x = Suc 0 ∨ ( x = 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 "{1::nat, 2}"] "{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'' == (%x. x = 1 ∨ x = (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 ⇒ bool" and odd :: "nat ⇒ bool" where
"even 0"
| "even n ⟹ odd (Suc n)"
| "odd n ⟹ 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 == ¬ 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 ⇒ bool"
where
"n mod 2 = 0 ⟹ is_even n"

code_pred (expected_modes: i => bool) is_even .

subsection ‹append predicate›

inductive append :: "'a list ⇒ 'a list ⇒ 'a list ⇒ bool" where
"append [] xs xs"
| "append xs ys zs ⟹ 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 (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), [1::nat, 2, 3, 4, 5])}" 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 "Predicate.the (concat [0::int, 1, 2] [3, 4, 5])"
value "Predicate.the (slice ([]::int list))"

text ‹tricky case with alternative rules›

inductive append2
where
"append2 [] xs xs"
| "append2 xs ys zs ⟹ append2 (x # xs) ys (x # zs)"

lemma append2_Nil: "append2 [] (xs::'b list) xs"

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.prems show thesis
proof
fix xs
assume "xa = []" "xb = xs" "xc = xs"
from this append2(1) 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(2) show thesis by fastforce
qed
qed

inductive tupled_append :: "'a list × 'a list × 'a list ⇒ bool"
where
"tupled_append ([], xs, xs)"
| "tupled_append (xs, ys, zs) ⟹ 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 (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 × 'a list × 'a list ⇒ bool"
where
"tupled_append'' ([], xs, xs)"
| "ys = fst yszs ==> x # zs = snd yszs ==> tupled_append'' (xs, ys, zs) ⟹ 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 × 'a list × 'a list ⇒ bool"
where
"tupled_append''' ([], xs, xs)"
| "yszs = (ys, zs) ==> tupled_append''' (xs, yszs) ⟹ 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 × 'b) list ⇒ 'a ⇒ 'b ⇒ bool"
where
"map_ofP ((a, b)#xs) a b"
| "map_ofP xs a b ⟹ 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)"
| "¬ 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)"
| "¬ 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 filter3 .
thm filter3.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') ⟹ tupled_append (xs', [x], ys) ⟹ 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 ⇒ bool) ⇒ 'a list ⇒ 'a list ⇒ 'a list ⇒ bool"
for f where
"partition f [] [] []"
| "f x ⟹ partition f xs ys zs ⟹ partition f (x # xs) (x # ys) zs"
| "¬ f x ⟹ partition f xs ys zs ⟹ 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 ⇒ bool) ⇒ ('a list × 'a list × 'a list) ⇒ bool"
for f where
"tupled_partition f ([], [], [])"
| "f x ⟹ tupled_partition f (xs, ys, zs) ⟹ tupled_partition f (x # xs, x # ys, zs)"
| "¬ f x ⟹ tupled_partition f (xs, ys, zs) ⟹ 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 ⟹ tranclp r a b"
"r a b ⟹ tranclp r b c ⟹ 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 tranclp.prems] 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 ⇒ 'a ⇒ bool)) ⇒ bool"
where
"rtrancl'' (x, x, r)"
| "r x y ⟹ rtrancl'' (y, z, r) ⟹ 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 ⇒ nat ⇒ bool" where
"succ 0 1"
| "succ m n ⟹ 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 "¬ (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 "¬ (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 ‹Free function variable›

inductive FF :: "nat => nat => bool"
where
"f x = y ==> FF x y"

code_pred FF .

subsection ‹IMP›

type_synonym var = nat
type_synonym state = "int list"

datatype com =
Skip |
Ass var "state => int" |
Seq com com |
IF "state => bool" com com |
While "state => bool" com

inductive tupled_exec :: "(com × state × state) ⇒ 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 tupled_step :: "(proc × nat × proc) ⇒ bool"
where
"tupled_step (pre n p, n, p)" |
"tupled_step (p1, a, q) ⟹ tupled_step (or p1 p2, a, q)" |
"tupled_step (p2, a, q) ⟹ tupled_step (or p1 p2, a, q)" |
"tupled_step (p1, a, q) ⟹ tupled_step (par p1 p2, a, par q p2)" |
"tupled_step (p2, a, q) ⟹ tupled_step (par p1 p2, a, par p1 q)"

code_pred tupled_step .
thm tupled_step.equation

subsection ‹divmod›

inductive divmod_rel :: "nat ⇒ nat ⇒ nat ⇒ nat ⇒ bool" where
"k < l ⟹ divmod_rel k l 0 k"
| "k ≥ l ⟹ divmod_rel (k - l) l q r ⟹ divmod_rel k l (Suc q) r"

code_pred divmod_rel .
thm divmod_rel.equation
value "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, skip_proof] case_f .
thm case_fP.equation

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'))"

code_pred [inductify] fold_map_idx .

subsection ‹Minimum›

definition Min
where "Min s r x ≡ s x ∧ (∀y. r x y ⟶ x = y)"

code_pred [inductify] Min .
thm Min.equation

subsection ‹Lexicographic order›
text ‹This example requires to handle the differences of sets and predicates in the predicate compiler,
or to have a copy of all definitions on predicates due to the set-predicate distinction.›

(*
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])}"*)

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 fastforce
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] lexn
proof -
fix r n xs ys
assume 1: "lexn r n (xs, ys)"
assume 2: "⋀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: "⋀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 (case_tac xys)
apply auto
apply fastforce
apply fastforce done
qed

values [random_dseq 1, 2, 5] 10 "{(n, xs, ys::int list). lexn (%(x, y). x <= y) n (xs, ys)}"

code_pred [inductify, skip_proof] lex .
thm lex.equation
thm lex_def
declare lenlex_conv[code_pred_def]
code_pred [inductify, skip_proof] 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"

primrec avl :: "'a tree => bool"
where
"avl ET = True"
| "avl (MKT x l r h) = ((height l = height r ∨ height l = 1 + height r ∨ height r = 1+height l) ∧
h = max (height l) (height r) + 1 ∧ avl l ∧ avl r)"
(*
code_pred [inductify] avl .
thm avl.equation*)

code_pred [new_random_dseq inductify] avl .
thm avl.new_random_dseq_equation
(* TODO: has highly non-deterministic execution time!

values [new_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 ∪ set_of r)"

fun is_ord :: "nat tree => bool"
where
"is_ord ET = True"
| "is_ord (MKT n l r h) =
((∀n' ∈ set_of l. n' < n) ∧ (∀n' ∈ set_of r. n < n') ∧ is_ord l ∧ 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
*)
(*
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] relcomp .
thm relcomp.equation
code_pred [inductify] Image .
thm Image.equation
declare singleton_iff[code_pred_inline]
declare Id_on_def[unfolded Bex_def UNION_eq 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_unfold
code_pred (modes:
(o * o => bool) => o => bool,
(o * o => bool) => i => bool,
(i * o => bool) => i => bool) [inductify] Domain .
thm Domain.equation

thm Domain_converse [symmetric]
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, skip_proof] size_list' .
code_pred [new_random_dseq inductify] size_list' .
thm size_list'P.equation
thm size_list'P.new_random_dseq_equation

values [new_random_dseq 2,3,10] 3 "{xs. size_list'P (xs::nat list) (5::nat)}"

code_pred (expected_modes: i => o => bool, o => i => bool, i => i => bool) [inductify, skip_proof] 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, skip_proof] last .
thm lastP.equation

code_pred [inductify, skip_proof] butlast .
thm butlastP.equation

code_pred [inductify, skip_proof] take .
thm takeP.equation

code_pred [inductify, skip_proof] drop .
thm dropP.equation
code_pred [inductify, skip_proof] zip .
thm zipP.equation

code_pred [inductify, skip_proof] upt .
(*
code_pred [inductify, skip_proof] remdups .
thm remdupsP.equation
code_pred [dseq inductify] remdups .
values [dseq 4] 5 "{xs. remdupsP xs [1, (2::int)]}"
*)
code_pred [inductify, skip_proof] remove1 .
thm remove1P.equation
values "{xs. remove1P 1 xs [2, (3::int)]}"

code_pred [inductify, skip_proof] 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, skip_proof] replicate .
thm replicateP.equation
values 5 "{(n, xs). replicateP n (0::int) xs}"

code_pred [inductify, skip_proof] splice .
thm splice.simps
thm spliceP.equation

values "{xs. spliceP xs [1, 2, 3] [1, 1, 1, 2, 1, (3::nat)]}"

code_pred [inductify, skip_proof] List.rev .
code_pred [inductify] map .
```