
blanchet@33192: (*  Title:      HOL/Nitpick.thy
blanchet@33192:     Author:     Jasmin Blanchette, TU Muenchen
blanchet@35807:     Copyright   2008, 2009, 2010
blanchet@33192: 
blanchet@33192: Nitpick: Yet another counterexample generator for Isabelle/HOL.
blanchet@33192: *)
blanchet@33192: 
blanchet@33192: header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
blanchet@33192: 
blanchet@33192: theory Nitpick
blanchet@57231: imports BNF_FP_Base Map Record
blanchet@55539: keywords
blanchet@55539:   "nitpick" :: diag and
blanchet@55539:   "nitpick_params" :: thy_decl
blanchet@33192: begin
blanchet@33192: 
blanchet@33192: typedecl bisim_iterator
blanchet@33192: 
blanchet@33192: axiomatization unknown :: 'a
blanchet@34938:            and is_unknown :: "'a \<Rightarrow> bool"
blanchet@33192:            and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
blanchet@33192:            and bisim_iterator_max :: bisim_iterator
blanchet@34938:            and Quot :: "'a \<Rightarrow> 'b"
blanchet@35671:            and safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
blanchet@33192: 
blanchet@35665: datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
blanchet@33192: datatype ('a, 'b) pair_box = PairBox 'a 'b
blanchet@34124: 
blanchet@34124: typedecl unsigned_bit
blanchet@34124: typedecl signed_bit
blanchet@34124: 
blanchet@34124: datatype 'a word = Word "('a set)"
blanchet@33192: 
blanchet@33192: text {*
blanchet@33192: Alternative definitions.
blanchet@33192: *}
blanchet@33192: 
blanchet@54148: lemma Ex1_unfold [nitpick_unfold]:
haftmann@45970: "Ex1 P \<equiv> \<exists>x. {x. P x} = {x}"
blanchet@33192: apply (rule eq_reflection)
nipkow@39302: apply (simp add: Ex1_def set_eq_iff)
blanchet@33192: apply (rule iffI)
blanchet@33192:  apply (erule exE)
blanchet@33192:  apply (erule conjE)
blanchet@33192:  apply (rule_tac x = x in exI)
blanchet@33192:  apply (rule allI)
blanchet@33192:  apply (rename_tac y)
blanchet@33192:  apply (erule_tac x = y in allE)
haftmann@45970: by auto
blanchet@33192: 
blanchet@54148: lemma rtrancl_unfold [nitpick_unfold]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
haftmann@45140:   by (simp only: rtrancl_trancl_reflcl)
blanchet@33192: 
blanchet@54148: lemma rtranclp_unfold [nitpick_unfold]:
blanchet@33192: "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
blanchet@33192: by (rule eq_reflection) (auto dest: rtranclpD)
blanchet@33192: 
blanchet@54148: lemma tranclp_unfold [nitpick_unfold]:
haftmann@45970: "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
haftmann@45970: by (simp add: trancl_def)
blanchet@33192: 
blanchet@54148: lemma [nitpick_simp]:
blanchet@47909: "of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
wenzelm@47988: by (cases n) auto
blanchet@47909: 
blanchet@41046: definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
blanchet@41046: "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
blanchet@41046: 
haftmann@44278: definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
blanchet@33192: "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
blanchet@33192: 
haftmann@44278: definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
blanchet@33192: "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
blanchet@33192: 
haftmann@44278: definition card' :: "'a set \<Rightarrow> nat" where
blanchet@39365: "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
blanchet@33192: 
haftmann@44278: definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
blanchet@39365: "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
blanchet@33192: 
haftmann@44278: inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" where
blanchet@33192: "fold_graph' f z {} z" |
blanchet@33192: "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
blanchet@33192: 
blanchet@33192: text {*
blanchet@33192: The following lemmas are not strictly necessary but they help the
blanchet@47909: \textit{specialize} optimization.
blanchet@33192: *}
blanchet@33192: 
blanchet@54148: lemma The_psimp [nitpick_psimp]:
haftmann@45970:   "P = (op =) x \<Longrightarrow> The P = x"
haftmann@45970:   by auto
blanchet@33192: 
blanchet@54148: lemma Eps_psimp [nitpick_psimp]:
blanchet@33192: "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
wenzelm@47988: apply (cases "P (Eps P)")
blanchet@33192:  apply auto
blanchet@33192: apply (erule contrapos_np)
blanchet@33192: by (rule someI)
blanchet@33192: 
blanchet@55414: lemma case_unit_unfold [nitpick_unfold]:
blanchet@55414: "case_unit x u \<equiv> x"
blanchet@33192: apply (subgoal_tac "u = ()")
blanchet@55642:  apply (simp only: unit.case)
blanchet@33192: by simp
blanchet@33192: 
blanchet@55642: declare unit.case [nitpick_simp del]
blanchet@33556: 
blanchet@55415: lemma case_nat_unfold [nitpick_unfold]:
blanchet@55415: "case_nat x f n \<equiv> if n = 0 then x else f (n - 1)"
blanchet@33192: apply (rule eq_reflection)
wenzelm@47988: by (cases n) auto
blanchet@33192: 
blanchet@55642: declare nat.case [nitpick_simp del]
blanchet@33556: 
blanchet@56643: lemma size_list_simp [nitpick_simp]:
blanchet@56643: "size_list f xs = (if xs = [] then 0 else Suc (f (hd xs) + size_list f (tl xs)))"
blanchet@33192: "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
wenzelm@47988: by (cases xs) auto
blanchet@33192: 
blanchet@33192: text {*
blanchet@33192: Auxiliary definitions used to provide an alternative representation for
blanchet@33192: @{text rat} and @{text real}.
blanchet@33192: *}
blanchet@33192: 
blanchet@33192: function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
blanchet@33192: [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
blanchet@33192: by auto
blanchet@33192: termination
blanchet@33192: apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
blanchet@33192:  apply auto
blanchet@33192:  apply (metis mod_less_divisor xt1(9))
blanchet@33192: by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
blanchet@33192: 
blanchet@33192: definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
blanchet@33192: "nat_lcm x y = x * y div (nat_gcd x y)"
blanchet@33192: 
blanchet@33192: definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
blanchet@33192: "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
blanchet@33192: 
blanchet@33192: definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
blanchet@33192: "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
blanchet@33192: 
blanchet@33192: definition Frac :: "int \<times> int \<Rightarrow> bool" where
blanchet@33192: "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
blanchet@33192: 
blanchet@56643: axiomatization
blanchet@56643:   Abs_Frac :: "int \<times> int \<Rightarrow> 'a" and
blanchet@56643:   Rep_Frac :: "'a \<Rightarrow> int \<times> int"
blanchet@33192: 
blanchet@33192: definition zero_frac :: 'a where
blanchet@33192: "zero_frac \<equiv> Abs_Frac (0, 1)"
blanchet@33192: 
blanchet@33192: definition one_frac :: 'a where
blanchet@33192: "one_frac \<equiv> Abs_Frac (1, 1)"
blanchet@33192: 
blanchet@33192: definition num :: "'a \<Rightarrow> int" where
blanchet@33192: "num \<equiv> fst o Rep_Frac"
blanchet@33192: 
blanchet@33192: definition denom :: "'a \<Rightarrow> int" where
blanchet@33192: "denom \<equiv> snd o Rep_Frac"
blanchet@33192: 
blanchet@33192: function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
blanchet@33192: [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
blanchet@33192:                               else if a = 0 \<or> b = 0 then (0, 1)
blanchet@33192:                               else let c = int_gcd a b in (a div c, b div c))"
blanchet@33192: by pat_completeness auto
blanchet@33192: termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
blanchet@33192: 
blanchet@33192: definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
blanchet@33192: "frac a b \<equiv> Abs_Frac (norm_frac a b)"
blanchet@33192: 
blanchet@33192: definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
blanchet@33192: [nitpick_simp]:
blanchet@33192: "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
blanchet@33192:                     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
blanchet@33192: 
blanchet@33192: definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
blanchet@33192: [nitpick_simp]:
blanchet@33192: "times_frac q r = frac (num q * num r) (denom q * denom r)"
blanchet@33192: 
blanchet@33192: definition uminus_frac :: "'a \<Rightarrow> 'a" where
blanchet@33192: "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
blanchet@33192: 
blanchet@33192: definition number_of_frac :: "int \<Rightarrow> 'a" where
blanchet@33192: "number_of_frac n \<equiv> Abs_Frac (n, 1)"
blanchet@33192: 
blanchet@33192: definition inverse_frac :: "'a \<Rightarrow> 'a" where
blanchet@33192: "inverse_frac q \<equiv> frac (denom q) (num q)"
blanchet@33192: 
blanchet@37397: definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
blanchet@37397: [nitpick_simp]:
blanchet@37397: "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
blanchet@37397: 
blanchet@33192: definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
blanchet@33192: [nitpick_simp]:
blanchet@33192: "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
blanchet@33192: 
blanchet@33192: definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
blanchet@33192: "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
blanchet@33192: 
blanchet@55017: axiomatization wf_wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
blanchet@55017: 
blanchet@55017: definition wf_wfrec' :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
blanchet@55017: [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
blanchet@55017: 
blanchet@55017: definition wfrec' ::  "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
blanchet@55017: "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
blanchet@55017:                 else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
blanchet@55017: 
wenzelm@48891: ML_file "Tools/Nitpick/kodkod.ML"
wenzelm@48891: ML_file "Tools/Nitpick/kodkod_sat.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_util.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_hol.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_mono.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_preproc.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_scope.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_peephole.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_rep.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_nut.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_kodkod.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_model.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick.ML"
blanchet@55199: ML_file "Tools/Nitpick/nitpick_commands.ML"
wenzelm@48891: ML_file "Tools/Nitpick/nitpick_tests.ML"
blanchet@33192: 
krauss@44016: setup {*
krauss@44016:   Nitpick_HOL.register_ersatz_global
krauss@44016:     [(@{const_name card}, @{const_name card'}),
krauss@44016:      (@{const_name setsum}, @{const_name setsum'}),
krauss@44016:      (@{const_name fold_graph}, @{const_name fold_graph'}),
blanchet@55017:      (@{const_name wf}, @{const_name wf'}),
blanchet@55017:      (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
blanchet@55017:      (@{const_name wfrec}, @{const_name wfrec'})]
krauss@44016: *}
blanchet@33561: 
blanchet@39365: hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The
krauss@44013:     FunBox PairBox Word prod refl' wf' card' setsum'
blanchet@41052:     fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac
blanchet@41052:     one_frac num denom norm_frac frac plus_frac times_frac uminus_frac
blanchet@55017:     number_of_frac inverse_frac less_frac less_eq_frac of_frac wf_wfrec wf_wfrec
blanchet@55017:     wfrec'
blanchet@46324: hide_type (open) bisim_iterator fun_box pair_box unsigned_bit signed_bit word
blanchet@41797: hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold
krauss@44013:     prod_def refl'_def wf'_def card'_def setsum'_def
blanchet@55415:     fold_graph'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
blanchet@56643:     size_list_simp nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def
blanchet@41046:     zero_frac_def one_frac_def num_def denom_def norm_frac_def frac_def
blanchet@41046:     plus_frac_def times_frac_def uminus_frac_def number_of_frac_def
blanchet@55017:     inverse_frac_def less_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
blanchet@55017:     wfrec'_def
blanchet@33192: 
blanchet@33192: end




