# Theory Nitpick

(*  Title:      HOL/Nitpick.thy
Author:     Jasmin Blanchette, TU Muenchen

Nitpick: Yet another counterexample generator for Isabelle/HOL.
*)

section ‹Nitpick: Yet Another Counterexample Generator for Isabelle/HOL›

theory Nitpick
imports Record GCD
keywords
"nitpick" :: diag and
"nitpick_params" :: thy_decl
begin

datatype (plugins only: extraction) (dead 'a, dead 'b) fun_box = FunBox "'a ⇒ 'b"
datatype (plugins only: extraction) (dead 'a, dead 'b) pair_box = PairBox 'a 'b
datatype (plugins only: extraction) (dead 'a) word = Word "'a set"

typedecl bisim_iterator
typedecl unsigned_bit
typedecl signed_bit

consts
unknown :: 'a
is_unknown :: "'a ⇒ bool"
bisim :: "bisim_iterator ⇒ 'a ⇒ 'a ⇒ bool"
bisim_iterator_max :: bisim_iterator
Quot :: "'a ⇒ 'b"
safe_The :: "('a ⇒ bool) ⇒ 'a"

text ‹
Alternative definitions.
›

lemma Ex1_unfold[nitpick_unfold]: "Ex1 P ≡ ∃x. {x. P x} = {x}"
apply (rule eq_reflection)
apply (rule iffI)
apply (erule exE)
apply (erule conjE)
apply (rule_tac x = x in exI)
apply (rule allI)
apply (rename_tac y)
apply (erule_tac x = y in allE)
by auto

lemma rtrancl_unfold[nitpick_unfold]: "r⇧* ≡ (r⇧+)⇧="
by (simp only: rtrancl_trancl_reflcl)

lemma rtranclp_unfold[nitpick_unfold]: "rtranclp r a b ≡ (a = b ∨ tranclp r a b)"
by (rule eq_reflection) (auto dest: rtranclpD)

lemma tranclp_unfold[nitpick_unfold]:
"tranclp r a b ≡ (a, b) ∈ trancl {(x, y). r x y}"

lemma [nitpick_simp]:
"of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
by (cases n) auto

definition prod :: "'a set ⇒ 'b set ⇒ ('a × 'b) set" where
"prod A B = {(a, b). a ∈ A ∧ b ∈ B}"

definition refl' :: "('a × 'a) set ⇒ bool" where
"refl' r ≡ ∀x. (x, x) ∈ r"

definition wf' :: "('a × 'a) set ⇒ bool" where
"wf' r ≡ acyclic r ∧ (finite r ∨ unknown)"

definition card' :: "'a set ⇒ nat" where
"card' A ≡ if finite A then length (SOME xs. set xs = A ∧ distinct xs) else 0"

definition sum' :: "('a ⇒ 'b::comm_monoid_add) ⇒ 'a set ⇒ 'b" where
"sum' f A ≡ if finite A then sum_list (map f (SOME xs. set xs = A ∧ distinct xs)) else 0"

inductive fold_graph' :: "('a ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a set ⇒ 'b ⇒ bool" where
"fold_graph' f z {} z" |
"⟦x ∈ A; fold_graph' f z (A - {x}) y⟧ ⟹ fold_graph' f z A (f x y)"

text ‹
The following lemmas are not strictly necessary but they help the
\textit{specialize} optimization.
›

lemma The_psimp[nitpick_psimp]: "P = (=) x ⟹ The P = x"
by auto

lemma Eps_psimp[nitpick_psimp]:
"⟦P x; ¬ P y; Eps P = y⟧ ⟹ Eps P = x"
apply (cases "P (Eps P)")
apply auto
apply (erule contrapos_np)
by (rule someI)

lemma case_unit_unfold[nitpick_unfold]:
"case_unit x u ≡ x"
apply (subgoal_tac "u = ()")
apply (simp only: unit.case)
by simp

declare unit.case[nitpick_simp del]

lemma case_nat_unfold[nitpick_unfold]:
"case_nat x f n ≡ if n = 0 then x else f (n - 1)"
apply (rule eq_reflection)
by (cases n) auto

declare nat.case[nitpick_simp del]

lemma size_list_simp[nitpick_simp]:
"size_list f xs = (if xs = [] then 0 else Suc (f (hd xs) + size_list f (tl xs)))"
"size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
by (cases xs) auto

text ‹
Auxiliary definitions used to provide an alternative representation for
‹rat› and ‹real›.
›

fun nat_gcd :: "nat ⇒ nat ⇒ nat" where
"nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"

declare nat_gcd.simps [simp del]

definition nat_lcm :: "nat ⇒ nat ⇒ nat" where
"nat_lcm x y = x * y div (nat_gcd x y)"

lemma gcd_eq_nitpick_gcd [nitpick_unfold]:
"gcd x y = Nitpick.nat_gcd x y"
by (induct x y rule: nat_gcd.induct)

lemma lcm_eq_nitpick_lcm [nitpick_unfold]:
"lcm x y = Nitpick.nat_lcm x y"
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)

definition Frac :: "int × int ⇒ bool" where
"Frac ≡ λ(a, b). b > 0 ∧ coprime a b"

consts
Abs_Frac :: "int × int ⇒ 'a"
Rep_Frac :: "'a ⇒ int × int"

definition zero_frac :: 'a where
"zero_frac ≡ Abs_Frac (0, 1)"

definition one_frac :: 'a where
"one_frac ≡ Abs_Frac (1, 1)"

definition num :: "'a ⇒ int" where
"num ≡ fst ∘ Rep_Frac"

definition denom :: "'a ⇒ int" where
"denom ≡ snd ∘ Rep_Frac"

function norm_frac :: "int ⇒ int ⇒ int × int" where
"norm_frac a b =
(if b < 0 then norm_frac (- a) (- b)
else if a = 0 ∨ b = 0 then (0, 1)
else let c = gcd a b in (a div c, b div c))"
by pat_completeness auto
termination by (relation "measure (λ(_, b). if b < 0 then 1 else 0)") auto

declare norm_frac.simps[simp del]

definition frac :: "int ⇒ int ⇒ 'a" where
"frac a b ≡ Abs_Frac (norm_frac a b)"

definition plus_frac :: "'a ⇒ 'a ⇒ 'a" where
[nitpick_simp]: "plus_frac q r = (let d = lcm (denom q) (denom r) in
frac (num q * (d div denom q) + num r * (d div denom r)) d)"

definition times_frac :: "'a ⇒ 'a ⇒ 'a" where
[nitpick_simp]: "times_frac q r = frac (num q * num r) (denom q * denom r)"

definition uminus_frac :: "'a ⇒ 'a" where
"uminus_frac q ≡ Abs_Frac (- num q, denom q)"

definition number_of_frac :: "int ⇒ 'a" where
"number_of_frac n ≡ Abs_Frac (n, 1)"

definition inverse_frac :: "'a ⇒ 'a" where
"inverse_frac q ≡ frac (denom q) (num q)"

definition less_frac :: "'a ⇒ 'a ⇒ bool" where
[nitpick_simp]: "less_frac q r ⟷ num (plus_frac q (uminus_frac r)) < 0"

definition less_eq_frac :: "'a ⇒ 'a ⇒ bool" where
[nitpick_simp]: "less_eq_frac q r ⟷ num (plus_frac q (uminus_frac r)) ≤ 0"

definition of_frac :: "'a ⇒ 'b::{inverse,ring_1}" where
"of_frac q ≡ of_int (num q) / of_int (denom q)"

axiomatization wf_wfrec :: "('a × 'a) set ⇒ (('a ⇒ 'b) ⇒ 'a ⇒ 'b) ⇒ 'a ⇒ 'b"

definition wf_wfrec' :: "('a × 'a) set ⇒ (('a ⇒ 'b) ⇒ 'a ⇒ 'b) ⇒ 'a ⇒ 'b" where
[nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"

definition wfrec' ::  "('a × 'a) set ⇒ (('a ⇒ 'b) ⇒ 'a ⇒ 'b) ⇒ 'a ⇒ 'b" where
"wfrec' R F x ≡ if wf R then wf_wfrec' R F x else THE y. wfrec_rel R (λf x. F (cut f R x) x) x y"

ML_file ‹Tools/Nitpick/kodkod.ML›
ML_file ‹Tools/Nitpick/kodkod_sat.ML›
ML_file ‹Tools/Nitpick/nitpick_util.ML›
ML_file ‹Tools/Nitpick/nitpick_hol.ML›
ML_file ‹Tools/Nitpick/nitpick_mono.ML›
ML_file ‹Tools/Nitpick/nitpick_preproc.ML›
ML_file ‹Tools/Nitpick/nitpick_scope.ML›
ML_file ‹Tools/Nitpick/nitpick_peephole.ML›
ML_file ‹Tools/Nitpick/nitpick_rep.ML›
ML_file ‹Tools/Nitpick/nitpick_nut.ML›
ML_file ‹Tools/Nitpick/nitpick_kodkod.ML›
ML_file ‹Tools/Nitpick/nitpick_model.ML›
ML_file ‹Tools/Nitpick/nitpick.ML›
ML_file ‹Tools/Nitpick/nitpick_commands.ML›
ML_file ‹Tools/Nitpick/nitpick_tests.ML›

setup ‹
Nitpick_HOL.register_ersatz_global
[(\<^const_name>‹card›, \<^const_name>‹card'›),
(\<^const_name>‹sum›, \<^const_name>‹sum'›),
(\<^const_name>‹fold_graph›, \<^const_name>‹fold_graph'›),
(\<^const_abbrev>‹wf›, \<^const_name>‹wf'›),
(\<^const_name>‹wf_wfrec›, \<^const_name>‹wf_wfrec'›),
(\<^const_name>‹wfrec›, \<^const_name>‹wfrec'›)]
›

hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The FunBox PairBox Word prod
refl' wf' card' sum' fold_graph' nat_gcd nat_lcm Frac Abs_Frac Rep_Frac
zero_frac one_frac num denom norm_frac frac plus_frac times_frac uminus_frac number_of_frac
inverse_frac less_frac less_eq_frac of_frac wf_wfrec wf_wfrec wfrec'

hide_type (open) bisim_iterator fun_box pair_box unsigned_bit signed_bit word

hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold prod_def refl'_def wf'_def
card'_def sum'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
size_list_simp nat_lcm_def Frac_def zero_frac_def one_frac_def
num_def denom_def frac_def plus_frac_def times_frac_def uminus_frac_def
number_of_frac_def inverse_frac_def less_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
wfrec'_def

end