
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
haftmann@38393: imports Map Quotient SAT Record
blanchet@33192: uses ("Tools/Nitpick/kodkod.ML")
blanchet@33192:      ("Tools/Nitpick/kodkod_sat.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_util.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_hol.ML")
blanchet@35070:      ("Tools/Nitpick/nitpick_preproc.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_mono.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_scope.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_peephole.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_rep.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_nut.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_kodkod.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_model.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_isar.ML")
blanchet@33192:      ("Tools/Nitpick/nitpick_tests.ML")
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 undefined_fast_The :: 'a
blanchet@33192:            and undefined_fast_Eps :: 'a
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@35671:            and safe_Eps :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
blanchet@33192: 
blanchet@35665: datatype ('a, 'b) fin_fun = FinFun "('a \<Rightarrow> 'b)"
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@36918: lemma If_def [nitpick_def, no_atp]:
blanchet@33192: "(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
blanchet@33192: by (rule eq_reflection) (rule if_bool_eq_conj)
blanchet@33192: 
blanchet@36918: lemma Ex1_def [nitpick_def, no_atp]:
blanchet@33192: "Ex1 P \<equiv> \<exists>x. P = {x}"
blanchet@33192: apply (rule eq_reflection)
blanchet@33192: apply (simp add: Ex1_def expand_set_eq)
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)
blanchet@33192: by (auto simp: mem_def)
blanchet@33192: 
blanchet@36918: lemma rtrancl_def [nitpick_def, no_atp]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
blanchet@33192: by simp
blanchet@33192: 
blanchet@36918: lemma rtranclp_def [nitpick_def, no_atp]:
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@36918: lemma tranclp_def [nitpick_def, no_atp]:
blanchet@33192: "tranclp r a b \<equiv> trancl (split r) (a, b)"
blanchet@33192: by (simp add: trancl_def Collect_def mem_def)
blanchet@33192: 
blanchet@33192: definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
blanchet@33192: "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
blanchet@33192: 
blanchet@33192: definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
blanchet@33192: "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
blanchet@33192: 
blanchet@33192: axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
blanchet@33192: 
blanchet@33192: definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
blanchet@33192: [nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
blanchet@33192: 
blanchet@33192: definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
blanchet@33192: "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
blanchet@33192:                 else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
blanchet@33192: 
blanchet@33192: definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
blanchet@35699: "card' A \<equiv> if finite A then length (safe_Eps (\<lambda>xs. set xs = A \<and> distinct xs)) else 0"
blanchet@33192: 
blanchet@33192: definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
blanchet@35699: "setsum' f A \<equiv> if finite A then listsum (map f (safe_Eps (\<lambda>xs. set xs = A \<and> distinct xs))) else 0"
blanchet@33192: 
blanchet@33192: inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<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@33192: \textit{special\_level} optimization.
blanchet@33192: *}
blanchet@33192: 
blanchet@36918: lemma The_psimp [nitpick_psimp, no_atp]:
blanchet@33192: "P = {x} \<Longrightarrow> The P = x"
blanchet@33192: by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
blanchet@33192: 
blanchet@36918: lemma Eps_psimp [nitpick_psimp, no_atp]:
blanchet@33192: "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
blanchet@33192: apply (case_tac "P (Eps P)")
blanchet@33192:  apply auto
blanchet@33192: apply (erule contrapos_np)
blanchet@33192: by (rule someI)
blanchet@33192: 
blanchet@36918: lemma unit_case_def [nitpick_def, no_atp]:
blanchet@33192: "unit_case x u \<equiv> x"
blanchet@33192: apply (subgoal_tac "u = ()")
blanchet@33192:  apply (simp only: unit.cases)
blanchet@33192: by simp
blanchet@33192: 
blanchet@33556: declare unit.cases [nitpick_simp del]
blanchet@33556: 
blanchet@36918: lemma nat_case_def [nitpick_def, no_atp]:
blanchet@33192: "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
blanchet@33192: apply (rule eq_reflection)
blanchet@33192: by (case_tac n) auto
blanchet@33192: 
blanchet@33556: declare nat.cases [nitpick_simp del]
blanchet@33556: 
blanchet@36918: lemma list_size_simp [nitpick_simp, no_atp]:
blanchet@33192: "list_size f xs = (if xs = [] then 0
blanchet@33192:                    else Suc (f (hd xs) + list_size f (tl xs)))"
blanchet@33192: "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
blanchet@33192: by (case_tac 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@33192: axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
blanchet@33192:            and 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@33192: use "Tools/Nitpick/kodkod.ML"
blanchet@33192: use "Tools/Nitpick/kodkod_sat.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_util.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_hol.ML"
blanchet@35665: use "Tools/Nitpick/nitpick_mono.ML"
blanchet@35070: use "Tools/Nitpick/nitpick_preproc.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_scope.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_peephole.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_rep.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_nut.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_kodkod.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_model.ML"
blanchet@33192: use "Tools/Nitpick/nitpick.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_isar.ML"
blanchet@33192: use "Tools/Nitpick/nitpick_tests.ML"
blanchet@33192: 
blanchet@33561: setup {* Nitpick_Isar.setup *}
blanchet@33561: 
wenzelm@36176: hide_const (open) unknown is_unknown undefined_fast_The undefined_fast_Eps bisim 
blanchet@35671:     bisim_iterator_max Quot safe_The safe_Eps FinFun FunBox PairBox Word refl'
blanchet@35671:     wf' wf_wfrec wf_wfrec' wfrec' card' setsum' fold_graph' nat_gcd nat_lcm
blanchet@35671:     int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac one_frac num denom
blanchet@35671:     norm_frac frac plus_frac times_frac uminus_frac number_of_frac inverse_frac
blanchet@37397:     less_frac less_eq_frac of_frac
wenzelm@36176: hide_type (open) bisim_iterator fin_fun fun_box pair_box unsigned_bit signed_bit
blanchet@35665:     word
blanchet@37704: hide_fact (open) If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
blanchet@37704:     wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
blanchet@37704:     The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def
blanchet@37704:     nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
blanchet@37704:     num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def
blanchet@37704:     uminus_frac_def number_of_frac_def inverse_frac_def less_frac_def
blanchet@37704:     less_eq_frac_def of_frac_def
blanchet@33192: 
blanchet@33192: end




