(*  Title:      HOL/Nitpick.thy

    Author:     Jasmin Blanchette, TU Muenchen

    Copyright   2008, 2009

Nitpick: Yet another counterexample generator for Isabelle/HOL.

*)

header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}

theory Nitpick

imports Map Quickcheck SAT

uses ("Tools/Nitpick/kodkod.ML")

     ("Tools/Nitpick/kodkod_sat.ML")

     ("Tools/Nitpick/nitpick_util.ML")

     ("Tools/Nitpick/nitpick_hol.ML")

     ("Tools/Nitpick/nitpick_mono.ML")

     ("Tools/Nitpick/nitpick_scope.ML")

     ("Tools/Nitpick/nitpick_peephole.ML")

     ("Tools/Nitpick/nitpick_rep.ML")

     ("Tools/Nitpick/nitpick_nut.ML")

     ("Tools/Nitpick/nitpick_kodkod.ML")

     ("Tools/Nitpick/nitpick_model.ML")

     ("Tools/Nitpick/nitpick.ML")

     ("Tools/Nitpick/nitpick_isar.ML")

     ("Tools/Nitpick/nitpick_tests.ML")

     ("Tools/Nitpick/minipick.ML")

begin

typedecl bisim_iterator

axiomatization unknown :: 'a

           and undefined_fast_The :: 'a

           and undefined_fast_Eps :: 'a

           and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"

           and bisim_iterator_max :: bisim_iterator

           and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"

datatype ('a, 'b) pair_box = PairBox 'a 'b

datatype ('a, 'b) fun_box = FunBox "'a \<Rightarrow> 'b"

text {*

Alternative definitions.

*}

lemma If_def [nitpick_def]:

"(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"

by (rule eq_reflection) (rule if_bool_eq_conj)

lemma Ex1_def [nitpick_def]:

"Ex1 P \<equiv> \<exists>x. P = {x}"

apply (rule eq_reflection)

apply (simp add: Ex1_def expand_set_eq)

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 simp: mem_def)

lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="

by simp

lemma rtranclp_def [nitpick_def]:

"rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"

by (rule eq_reflection) (auto dest: rtranclpD)

lemma tranclp_def [nitpick_def]:

"tranclp r a b \<equiv> trancl (split r) (a, b)"

by (simp add: trancl_def Collect_def mem_def)

definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

"refl' r \<equiv> \<forall>x. (x, x) \<in> r"

definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

"wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"

axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"

definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where

[nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"

definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where

"wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x

                else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"

definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where

"card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"

definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where

"setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"

inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where

"fold_graph' f z {} z" |

"\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"

text {*

The following lemmas are not strictly necessary but they help the

\textit{special\_level} optimization.

*}

lemma The_psimp [nitpick_psimp]:

"P = {x} \<Longrightarrow> The P = x"

by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)

lemma Eps_psimp [nitpick_psimp]:

"\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"

apply (case_tac "P (Eps P)")

 apply auto

apply (erule contrapos_np)

by (rule someI)

lemma unit_case_def [nitpick_def]:

"unit_case x u \<equiv> x"

apply (subgoal_tac "u = ()")

 apply (simp only: unit.cases)

by simp

declare unit.cases [nitpick_simp del]

lemma nat_case_def [nitpick_def]:

"nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"

apply (rule eq_reflection)

by (case_tac n) auto

declare nat.cases [nitpick_simp del]

lemmas [nitpick_def] = dvd_eq_mod_eq_0 [THEN eq_reflection]

lemma list_size_simp [nitpick_simp]:

"list_size f xs = (if xs = [] then 0

                   else Suc (f (hd xs) + list_size f (tl xs)))"

"size xs = (if xs = [] then 0 else Suc (size (tl xs)))"

by (case_tac xs) auto

text {*

Auxiliary definitions used to provide an alternative representation for

@{text rat} and @{text real}.

*}

function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

[simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"

by auto

termination

apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")

 apply auto

 apply (metis mod_less_divisor xt1(9))

by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))

definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

"nat_lcm x y = x * y div (nat_gcd x y)"

definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where

"int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"

definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where

"int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"

definition Frac :: "int \<times> int \<Rightarrow> bool" where

"Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"

axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"

           and Rep_Frac :: "'a \<Rightarrow> int \<times> int"

definition zero_frac :: 'a where

"zero_frac \<equiv> Abs_Frac (0, 1)"

definition one_frac :: 'a where

"one_frac \<equiv> Abs_Frac (1, 1)"

definition num :: "'a \<Rightarrow> int" where

"num \<equiv> fst o Rep_Frac"

definition denom :: "'a \<Rightarrow> int" where

"denom \<equiv> snd o Rep_Frac"

function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

[simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)

                              else if a = 0 \<or> b = 0 then (0, 1)

                              else let c = int_gcd a b in (a div c, b div c))"

by pat_completeness auto

termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto

definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where

"frac a b \<equiv> Abs_Frac (norm_frac a b)"

definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where

[nitpick_simp]:

"plus_frac q r = (let d = int_lcm (denom q) (denom r) in

                    frac (num q * (d div denom q) + num r * (d div denom r)) d)"

definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where

[nitpick_simp]:

"times_frac q r = frac (num q * num r) (denom q * denom r)"

definition uminus_frac :: "'a \<Rightarrow> 'a" where

"uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"

definition number_of_frac :: "int \<Rightarrow> 'a" where

"number_of_frac n \<equiv> Abs_Frac (n, 1)"

definition inverse_frac :: "'a \<Rightarrow> 'a" where

"inverse_frac q \<equiv> frac (denom q) (num q)"

definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where

[nitpick_simp]:

"less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"

definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where

"of_frac q \<equiv> of_int (num q) / of_int (denom q)"

(* While Nitpick normally avoids to unfold definitions for locales, it

   unfortunately needs to unfold them when dealing with the following built-in

   constants. A cleaner approach would be to change "Nitpick_HOL" and

   "Nitpick_Nits" so that they handle the unexpanded overloaded constants

   directly, but this is slightly more tricky to implement. *)

lemmas [nitpick_def] = div_int_inst.div_int div_int_inst.mod_int

    div_nat_inst.div_nat div_nat_inst.mod_nat lower_semilattice_fun_inst.inf_fun

    minus_fun_inst.minus_fun minus_int_inst.minus_int minus_nat_inst.minus_nat

    one_int_inst.one_int one_nat_inst.one_nat ord_fun_inst.less_eq_fun

    ord_int_inst.less_eq_int ord_int_inst.less_int ord_nat_inst.less_eq_nat

    ord_nat_inst.less_nat plus_int_inst.plus_int plus_nat_inst.plus_nat

    times_int_inst.times_int times_nat_inst.times_nat uminus_int_inst.uminus_int

    upper_semilattice_fun_inst.sup_fun zero_int_inst.zero_int

    zero_nat_inst.zero_nat

use "Tools/Nitpick/kodkod.ML"

use "Tools/Nitpick/kodkod_sat.ML"

use "Tools/Nitpick/nitpick_util.ML"

use "Tools/Nitpick/nitpick_hol.ML"

use "Tools/Nitpick/nitpick_mono.ML"

use "Tools/Nitpick/nitpick_scope.ML"

use "Tools/Nitpick/nitpick_peephole.ML"

use "Tools/Nitpick/nitpick_rep.ML"

use "Tools/Nitpick/nitpick_nut.ML"

use "Tools/Nitpick/nitpick_kodkod.ML"

use "Tools/Nitpick/nitpick_model.ML"

use "Tools/Nitpick/nitpick.ML"

use "Tools/Nitpick/nitpick_isar.ML"

use "Tools/Nitpick/nitpick_tests.ML"

use "Tools/Nitpick/minipick.ML"

setup {* Nitpick_Isar.setup *}

hide (open) const unknown undefined_fast_The undefined_fast_Eps bisim 

    bisim_iterator_max Tha refl' wf' wf_wfrec wf_wfrec' wfrec' card' setsum'

    fold_graph' nat_gcd nat_lcm int_gcd int_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_eq_frac of_frac

hide (open) type bisim_iterator pair_box fun_box

hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def

    wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def

    The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def

    nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def

    num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def

    uminus_frac_def number_of_frac_def inverse_frac_def less_eq_frac_def

    of_frac_def

end