author | wenzelm |
Sat, 13 Mar 2010 14:44:47 +0100 | |
changeset 35743 | c506c029a082 |
parent 35699 | 9ed327529a44 |
child 35807 | e4d1b5cbd429 |
permissions | -rw-r--r-- |
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(* Title: HOL/Nitpick.thy |
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Author: Jasmin Blanchette, TU Muenchen |
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Copyright 2008, 2009 |
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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 Map Quotient SAT |
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uses ("Tools/Nitpick/kodkod.ML") |
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("Tools/Nitpick/kodkod_sat.ML") |
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("Tools/Nitpick/nitpick_util.ML") |
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("Tools/Nitpick/nitpick_hol.ML") |
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("Tools/Nitpick/nitpick_preproc.ML") |
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("Tools/Nitpick/nitpick_mono.ML") |
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("Tools/Nitpick/nitpick_scope.ML") |
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("Tools/Nitpick/nitpick_peephole.ML") |
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("Tools/Nitpick/nitpick_rep.ML") |
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("Tools/Nitpick/nitpick_nut.ML") |
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("Tools/Nitpick/nitpick_kodkod.ML") |
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("Tools/Nitpick/nitpick_model.ML") |
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("Tools/Nitpick/nitpick.ML") |
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("Tools/Nitpick/nitpick_isar.ML") |
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("Tools/Nitpick/nitpick_tests.ML") |
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("Tools/Nitpick/minipick.ML") |
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begin |
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typedecl bisim_iterator |
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axiomatization unknown :: 'a |
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and is_unknown :: "'a \<Rightarrow> bool" |
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and undefined_fast_The :: 'a |
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and undefined_fast_Eps :: 'a |
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and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" |
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and bisim_iterator_max :: bisim_iterator |
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and Quot :: "'a \<Rightarrow> 'b" |
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and safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" |
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and safe_Eps :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" |
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datatype ('a, 'b) fin_fun = FinFun "('a \<Rightarrow> 'b)" |
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datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)" |
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datatype ('a, 'b) pair_box = PairBox 'a 'b |
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typedecl unsigned_bit |
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typedecl signed_bit |
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datatype 'a word = Word "('a set)" |
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text {* |
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Alternative definitions. |
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*} |
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lemma If_def [nitpick_def]: |
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"(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)" |
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by (rule eq_reflection) (rule if_bool_eq_conj) |
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lemma Ex1_def [nitpick_def]: |
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"Ex1 P \<equiv> \<exists>x. P = {x}" |
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apply (rule eq_reflection) |
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apply (simp add: Ex1_def expand_set_eq) |
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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 simp: mem_def) |
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lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>=" |
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by simp |
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lemma rtranclp_def [nitpick_def]: |
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"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_def [nitpick_def]: |
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"tranclp r a b \<equiv> trancl (split r) (a, b)" |
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by (simp add: trancl_def Collect_def mem_def) |
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definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<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 \<Rightarrow> bool) \<Rightarrow> bool" where |
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"wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)" |
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axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" |
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definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<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 (Recdef.cut (wf_wfrec R F) R x) x" |
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definition wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<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 |
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else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y" |
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definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where |
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"card' A \<equiv> if finite A then length (safe_Eps (\<lambda>xs. set xs = A \<and> distinct xs)) else 0" |
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definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where |
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"setsum' f A \<equiv> if finite A then listsum (map f (safe_Eps (\<lambda>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 \<Rightarrow> bool) \<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{special\_level} optimization. |
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*} |
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lemma The_psimp [nitpick_psimp]: |
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"P = {x} \<Longrightarrow> The P = x" |
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by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def) |
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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 (case_tac "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 unit_case_def [nitpick_def]: |
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"unit_case x u \<equiv> x" |
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apply (subgoal_tac "u = ()") |
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apply (simp only: unit.cases) |
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by simp |
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declare unit.cases [nitpick_simp del] |
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lemma nat_case_def [nitpick_def]: |
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"nat_case 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 (case_tac n) auto |
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declare nat.cases [nitpick_simp del] |
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lemma list_size_simp [nitpick_simp]: |
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"list_size f xs = (if xs = [] then 0 |
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else Suc (f (hd xs) + list_size f (tl xs)))" |
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"size xs = (if xs = [] then 0 else Suc (size (tl xs)))" |
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by (case_tac 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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[simp del]: "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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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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axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a" |
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and 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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[simp del]: "norm_frac a b = (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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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]: |
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"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]: |
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"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_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where |
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[nitpick_simp]: |
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"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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use "Tools/Nitpick/kodkod.ML" |
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use "Tools/Nitpick/kodkod_sat.ML" |
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use "Tools/Nitpick/nitpick_util.ML" |
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use "Tools/Nitpick/nitpick_hol.ML" |
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use "Tools/Nitpick/nitpick_mono.ML" |
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use "Tools/Nitpick/nitpick_preproc.ML" |
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use "Tools/Nitpick/nitpick_scope.ML" |
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use "Tools/Nitpick/nitpick_peephole.ML" |
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use "Tools/Nitpick/nitpick_rep.ML" |
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use "Tools/Nitpick/nitpick_nut.ML" |
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use "Tools/Nitpick/nitpick_kodkod.ML" |
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use "Tools/Nitpick/nitpick_model.ML" |
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use "Tools/Nitpick/nitpick.ML" |
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use "Tools/Nitpick/nitpick_isar.ML" |
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use "Tools/Nitpick/nitpick_tests.ML" |
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use "Tools/Nitpick/minipick.ML" |
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setup {* Nitpick_Isar.setup *} |
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hide (open) const unknown is_unknown undefined_fast_The undefined_fast_Eps bisim |
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bisim_iterator_max Quot safe_The safe_Eps FinFun FunBox PairBox Word refl' |
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wf' wf_wfrec wf_wfrec' wfrec' card' setsum' fold_graph' nat_gcd nat_lcm |
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int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac one_frac num denom |
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norm_frac frac plus_frac times_frac uminus_frac number_of_frac inverse_frac |
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less_eq_frac of_frac |
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hide (open) type bisim_iterator fin_fun fun_box pair_box unsigned_bit signed_bit |
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word |
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hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def |
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wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def |
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The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def |
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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 |
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uminus_frac_def number_of_frac_def inverse_frac_def less_eq_frac_def |
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of_frac_def |
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end |