src/HOL/Nitpick.thy
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use 'datatype_new' in 'Main'
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(*  Title:      HOL/Nitpick.thy
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2008, 2009, 2010
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Nitpick: Yet another counterexample generator for Isabelle/HOL.
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*)
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header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
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theory Nitpick
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imports Record
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keywords
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  "nitpick" :: diag and
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  "nitpick_params" :: thy_decl
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begin
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datatype_new (dead 'a, dead 'b) fun_box = FunBox "'a \<Rightarrow> 'b"
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datatype_new (dead 'a, dead 'b) pair_box = PairBox 'a 'b
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datatype_new (dead 'a) word = Word "'a set"
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typedecl bisim_iterator
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typedecl unsigned_bit
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typedecl signed_bit
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consts
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  unknown :: 'a
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  is_unknown :: "'a \<Rightarrow> bool"
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  bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
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  bisim_iterator_max :: bisim_iterator
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  Quot :: "'a \<Rightarrow> 'b"
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  safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
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text {*
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Alternative definitions.
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*}
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lemma Ex1_unfold[nitpick_unfold]: "Ex1 P \<equiv> \<exists>x. {x. P x} = {x}"
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  apply (rule eq_reflection)
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  apply (simp add: Ex1_def set_eq_iff)
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  apply (rule iffI)
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   apply (erule exE)
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   apply (erule conjE)
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   apply (rule_tac x = x in exI)
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   apply (rule allI)
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   apply (rename_tac y)
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   apply (erule_tac x = y in allE)
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  by auto
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lemma rtrancl_unfold[nitpick_unfold]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
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  by (simp only: rtrancl_trancl_reflcl)
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lemma rtranclp_unfold[nitpick_unfold]: "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
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  by (rule eq_reflection) (auto dest: rtranclpD)
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lemma tranclp_unfold[nitpick_unfold]:
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  "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
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  by (simp add: trancl_def)
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lemma [nitpick_simp]:
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  "of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
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  by (cases n) auto
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definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
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  "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
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definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
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  "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
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definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
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  "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
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definition card' :: "'a set \<Rightarrow> nat" where
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  "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
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definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
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  "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
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inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" where
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  "fold_graph' f z {} z" |
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  "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
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text {*
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The following lemmas are not strictly necessary but they help the
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\textit{specialize} optimization.
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*}
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lemma The_psimp[nitpick_psimp]: "P = (op =) x \<Longrightarrow> The P = x"
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  by auto
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lemma Eps_psimp[nitpick_psimp]:
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  "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
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  apply (cases "P (Eps P)")
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   apply auto
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  apply (erule contrapos_np)
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  by (rule someI)
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lemma case_unit_unfold[nitpick_unfold]:
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  "case_unit x u \<equiv> x"
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  apply (subgoal_tac "u = ()")
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   apply (simp only: unit.case)
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  by simp
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declare unit.case[nitpick_simp del]
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lemma case_nat_unfold[nitpick_unfold]:
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  "case_nat x f n \<equiv> if n = 0 then x else f (n - 1)"
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  apply (rule eq_reflection)
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  by (cases n) auto
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declare nat.case[nitpick_simp del]
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lemma size_list_simp[nitpick_simp]:
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  "size_list f xs = (if xs = [] then 0 else Suc (f (hd xs) + size_list f (tl xs)))"
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  "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
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  by (cases xs) auto
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text {*
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Auxiliary definitions used to provide an alternative representation for
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@{text rat} and @{text real}.
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*}
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function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
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  "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
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  by auto
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  termination
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  apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
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   apply auto
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   apply (metis mod_less_divisor xt1(9))
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  by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
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declare nat_gcd.simps[simp del]
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definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
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  "nat_lcm x y = x * y div (nat_gcd x y)"
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definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
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  "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
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definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
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  "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
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definition Frac :: "int \<times> int \<Rightarrow> bool" where
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  "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
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consts
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  Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
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  Rep_Frac :: "'a \<Rightarrow> int \<times> int"
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definition zero_frac :: 'a where
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  "zero_frac \<equiv> Abs_Frac (0, 1)"
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definition one_frac :: 'a where
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  "one_frac \<equiv> Abs_Frac (1, 1)"
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definition num :: "'a \<Rightarrow> int" where
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  "num \<equiv> fst o Rep_Frac"
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definition denom :: "'a \<Rightarrow> int" where
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  "denom \<equiv> snd o Rep_Frac"
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function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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  "norm_frac a b =
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    (if b < 0 then norm_frac (- a) (- b)
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     else if a = 0 \<or> b = 0 then (0, 1)
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     else let c = int_gcd a b in (a div c, b div c))"
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  by pat_completeness auto
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  termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
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declare norm_frac.simps[simp del]
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definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
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  "frac a b \<equiv> Abs_Frac (norm_frac a b)"
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definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  [nitpick_simp]: "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
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    frac (num q * (d div denom q) + num r * (d div denom r)) d)"
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definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  [nitpick_simp]: "times_frac q r = frac (num q * num r) (denom q * denom r)"
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definition uminus_frac :: "'a \<Rightarrow> 'a" where
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  "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
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definition number_of_frac :: "int \<Rightarrow> 'a" where
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  "number_of_frac n \<equiv> Abs_Frac (n, 1)"
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definition inverse_frac :: "'a \<Rightarrow> 'a" where
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  "inverse_frac q \<equiv> frac (denom q) (num q)"
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definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
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  [nitpick_simp]: "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
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definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
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  [nitpick_simp]: "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
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definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
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  "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
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axiomatization wf_wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
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definition wf_wfrec' :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
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  [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
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definition wfrec' ::  "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
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  "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x else THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y"
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ML_file "Tools/Nitpick/kodkod.ML"
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ML_file "Tools/Nitpick/kodkod_sat.ML"
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ML_file "Tools/Nitpick/nitpick_util.ML"
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ML_file "Tools/Nitpick/nitpick_hol.ML"
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ML_file "Tools/Nitpick/nitpick_mono.ML"
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ML_file "Tools/Nitpick/nitpick_preproc.ML"
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ML_file "Tools/Nitpick/nitpick_scope.ML"
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ML_file "Tools/Nitpick/nitpick_peephole.ML"
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ML_file "Tools/Nitpick/nitpick_rep.ML"
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ML_file "Tools/Nitpick/nitpick_nut.ML"
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ML_file "Tools/Nitpick/nitpick_kodkod.ML"
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ML_file "Tools/Nitpick/nitpick_model.ML"
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ML_file "Tools/Nitpick/nitpick.ML"
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ML_file "Tools/Nitpick/nitpick_commands.ML"
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ML_file "Tools/Nitpick/nitpick_tests.ML"
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setup {*
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  Nitpick_HOL.register_ersatz_global
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    [(@{const_name card}, @{const_name card'}),
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     (@{const_name setsum}, @{const_name setsum'}),
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     (@{const_name fold_graph}, @{const_name fold_graph'}),
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     (@{const_name wf}, @{const_name wf'}),
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     (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
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     (@{const_name wfrec}, @{const_name wfrec'})]
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*}
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hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The FunBox PairBox Word prod
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  refl' wf' card' setsum' fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac
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  zero_frac one_frac num denom norm_frac frac plus_frac times_frac uminus_frac number_of_frac
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  inverse_frac less_frac less_eq_frac of_frac wf_wfrec wf_wfrec wfrec'
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hide_type (open) bisim_iterator fun_box pair_box unsigned_bit signed_bit word
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hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold prod_def refl'_def wf'_def
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  card'_def setsum'_def fold_graph'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
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  size_list_simp nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
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  num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def uminus_frac_def
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  number_of_frac_def inverse_frac_def less_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
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  wfrec'_def
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end