src/HOL/Nitpick.thy

 keywords
   "nitpick" :: diag and
   "nitpick_params" :: thy_decl
 begin
 
+datatype ('a, 'b) fun_box = FunBox "'a \<Rightarrow> 'b"
+datatype ('a, 'b) pair_box = PairBox 'a 'b
+datatype 'a word = Word "'a set"
+
 typedecl bisim_iterator
-
-axiomatization unknown :: 'a
-           and is_unknown :: "'a \<Rightarrow> bool"
-           and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
-           and bisim_iterator_max :: bisim_iterator
-           and Quot :: "'a \<Rightarrow> 'b"
-           and safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
-
-datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
-datatype ('a, 'b) pair_box = PairBox 'a 'b
-
 typedecl unsigned_bit
 typedecl signed_bit
 
-datatype 'a word = Word "('a set)"
+consts
+  unknown :: 'a
+  is_unknown :: "'a \<Rightarrow> bool"
+  bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
+  bisim_iterator_max :: bisim_iterator
+  Quot :: "'a \<Rightarrow> 'b"
+  safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
 
 text {*
 Alternative definitions.
 *}
 
-lemma Ex1_unfold [nitpick_unfold]:
-"Ex1 P \<equiv> \<exists>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\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
+lemma Ex1_unfold[nitpick_unfold]: "Ex1 P \<equiv> \<exists>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\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
   by (simp only: rtrancl_trancl_reflcl)
 
-lemma rtranclp_unfold [nitpick_unfold]:
-"rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
-by (rule eq_reflection) (auto dest: rtranclpD)
-
-lemma tranclp_unfold [nitpick_unfold]:
-"tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
-by (simp add: trancl_def)
+lemma rtranclp_unfold[nitpick_unfold]: "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
+  by (rule eq_reflection) (auto dest: rtranclpD)
+
+lemma tranclp_unfold[nitpick_unfold]:
+  "tranclp r a b \<equiv> (a, b) \<in> 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
+  "of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
+  by (cases n) auto
 
 definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
-"prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
+  "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
 
 definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
-"refl' r \<equiv> \<forall>x. (x, x) \<in> r"
+  "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
 
 definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
-"wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
+  "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
 
 definition card' :: "'a set \<Rightarrow> nat" where
-"card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
+  "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
 
 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
-"setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
+  "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 set \<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)"
+  "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{specialize} optimization.
 *}
 
-lemma The_psimp [nitpick_psimp]:
-  "P = (op =) x \<Longrightarrow> The P = x"
+lemma The_psimp[nitpick_psimp]: "P = (op =) x \<Longrightarrow> The P = x"
   by auto
 
-lemma Eps_psimp [nitpick_psimp]:
-"\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> 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 \<equiv> 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 \<equiv> 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
+lemma Eps_psimp[nitpick_psimp]:
+  "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> 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 \<equiv> 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 \<equiv> 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
 @{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))
+  "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))
+
+declare nat_gcd.simps[simp del]
 
 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
-"nat_lcm x y = x * y div (nat_gcd x y)"
+  "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)))"
+  "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)))"
+  "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
+  "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
+
+consts
+  Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   Rep_Frac :: "'a \<Rightarrow> int \<times> int"
 
 definition zero_frac :: 'a where
-"zero_frac \<equiv> Abs_Frac (0, 1)"
+  "zero_frac \<equiv> Abs_Frac (0, 1)"
 
 definition one_frac :: 'a where
-"one_frac \<equiv> Abs_Frac (1, 1)"
+  "one_frac \<equiv> Abs_Frac (1, 1)"
 
 definition num :: "'a \<Rightarrow> int" where
-"num \<equiv> fst o Rep_Frac"
+  "num \<equiv> fst o Rep_Frac"
 
 definition denom :: "'a \<Rightarrow> int" where
-"denom \<equiv> snd o Rep_Frac"
+  "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
+  "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
+
+declare norm_frac.simps[simp del]
 
 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
-"frac a b \<equiv> Abs_Frac (norm_frac a b)"
+  "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)"
+  [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)"
+  [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)"
+  "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)"
+  "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)"
+  "inverse_frac q \<equiv> frac (denom q) (num q)"
 
 definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
-[nitpick_simp]:
-"less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
+  [nitpick_simp]: "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
 
 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"
+  [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)"
+  "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
 
 axiomatization wf_wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
 
 definition wf_wfrec' :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
-[nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
+  [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
 
 definition wfrec' ::  "('a \<times> 'a) set \<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 (cut f R x) x) x y"
+  "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"
 
 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"

      (@{const_name 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' 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_frac less_eq_frac of_frac wf_wfrec wf_wfrec
-    wfrec'
+hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The FunBox PairBox Word prod
+  refl' wf' 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_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 setsum'_def
-    fold_graph'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
-    size_list_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_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
-    wfrec'_def
+
+hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold prod_def refl'_def wf'_def
+  card'_def setsum'_def fold_graph'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
+  size_list_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_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
+  wfrec'_def
 
 end