Theory Nitpick

(*  Title:      HOL/Nitpick.thy
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2008, 2009, 2010

Nitpick: Yet another counterexample generator for Isabelle/HOL.

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

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

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

  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 (simp add: Ex1_def set_eq_iff)
  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}"
  by (simp add: trancl_def)

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:
  by simp

declare[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[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)
    (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)

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"

  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_namecard, const_namecard'),
     (const_namesum, const_namesum'),
     (const_namefold_graph, const_namefold_graph'),
     (const_abbrevwf, const_namewf'),
     (const_namewf_wfrec, const_namewf_wfrec'),
     (const_namewfrec, const_namewfrec')]

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