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Errorable.thy
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Errorable.thy
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(* Title: Safe OCL
Author: Denis Nikiforov, March 2019
Maintainer: Denis Nikiforov <denis.nikif at gmail.com>
License: LGPL
*)
section \<open>Errorable\<close>
theory Errorable
imports Main
begin
datatype 'a errorable = ErrorFree 'a | Errorable 'a
instantiation errorable :: (order) order
begin
fun less_errorable where
"ErrorFree \<tau> < ErrorFree \<sigma> = (\<tau> < \<sigma>)"
| "Errorable \<tau> < ErrorFree \<sigma> = False"
| "ErrorFree \<tau> < Errorable \<sigma> = (\<tau> \<le> \<sigma>)"
| "Errorable \<tau> < Errorable \<sigma> = (\<tau> < \<sigma>)"
fun less_eq_errorable where
"ErrorFree \<tau> \<le> ErrorFree \<sigma> = (\<tau> \<le> \<sigma>)"
| "Errorable \<tau> \<le> ErrorFree \<sigma> = False"
| "ErrorFree \<tau> \<le> Errorable \<sigma> = (\<tau> \<le> \<sigma>)"
| "Errorable \<tau> \<le> Errorable \<sigma> = (\<tau> \<le> \<sigma>)"
lemma less_le_not_le_errorable:
"\<tau> < \<sigma> \<longleftrightarrow> \<tau> \<le> \<sigma> \<and> \<not> \<sigma> \<le> \<tau>"
for \<tau> \<sigma> :: "'a errorable"
by (cases \<tau>; cases \<sigma>; auto)
lemma order_refl_errorable:
"\<tau> \<le> \<tau>"
for \<tau> :: "'a errorable"
by (cases \<tau>; auto)
lemma order_trans_errorable:
"\<tau> \<le> \<sigma> \<Longrightarrow> \<sigma> \<le> \<rho> \<Longrightarrow> \<tau> \<le> \<rho>"
for \<tau> \<sigma> \<rho> :: "'a errorable"
by (cases \<tau>; cases \<sigma>; cases \<rho>; auto)
lemma antisym_errorable:
"\<tau> \<le> \<sigma> \<Longrightarrow> \<sigma> \<le> \<tau> \<Longrightarrow> \<tau> = \<sigma>"
for \<tau> \<sigma> :: "'a errorable"
by (cases \<tau>; cases \<sigma>; auto)
instance
apply intro_classes
apply (simp add: less_le_not_le_errorable)
apply (simp add: order_refl_errorable)
using order_trans_errorable apply blast
by (simp add: antisym_errorable)
end
lemma less_x_ErrorFree_intro [intro]:
"\<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> < \<sigma> \<Longrightarrow> \<tau> < ErrorFree \<sigma>"
by simp
lemma less_eq_x_ErrorFree_intro [intro]:
"\<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> \<tau> \<le> ErrorFree \<sigma>"
by simp
lemma less_x_Errorable_intro [intro]:
"\<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> \<tau> < Errorable \<sigma>"
"\<tau> = Errorable \<rho> \<Longrightarrow> \<rho> < \<sigma> \<Longrightarrow> \<tau> < Errorable \<sigma>"
by simp_all
lemma less_eq_x_Errorable_intro [intro]:
"\<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> \<tau> \<le> Errorable \<sigma>"
"\<tau> = Errorable \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> \<tau> \<le> Errorable \<sigma>"
by simp_all
lemma less_x_ErrorFree [elim!]:
"\<tau> < ErrorFree \<sigma> \<Longrightarrow>
(\<And>\<rho>. \<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> < \<sigma> \<Longrightarrow> P) \<Longrightarrow> P"
by (erule less_errorable.elims; auto)
lemma less_eq_x_ErrorFree [elim!]:
"\<tau> \<le> ErrorFree \<sigma> \<Longrightarrow>
(\<And>\<rho>. \<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> P) \<Longrightarrow> P"
by (erule less_eq_errorable.elims; auto)
lemma less_x_Errorable [elim!]:
"\<tau> < Errorable \<sigma> \<Longrightarrow>
(\<And>\<rho>. \<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> P) \<Longrightarrow>
(\<And>\<rho>. \<tau> = Errorable \<rho> \<Longrightarrow> \<rho> < \<sigma> \<Longrightarrow> P) \<Longrightarrow> P"
by (erule less_errorable.elims; auto)
lemma less_eq_x_Errorable [elim!]:
"\<tau> \<le> Errorable \<sigma> \<Longrightarrow>
(\<And>\<rho>. \<tau> = ErrorFree \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> P) \<Longrightarrow>
(\<And>\<rho>. \<tau> = Errorable \<rho> \<Longrightarrow> \<rho> \<le> \<sigma> \<Longrightarrow> P) \<Longrightarrow> P"
by (erule less_eq_errorable.elims; auto)
lemma errorable_less_left_simps [simp]:
"ErrorFree \<rho> < \<sigma> = (\<exists>\<upsilon>.
\<sigma> = ErrorFree \<upsilon> \<and> \<rho> < \<upsilon> \<or>
\<sigma> = Errorable \<upsilon> \<and> \<rho> \<le> \<upsilon>)"
"Errorable \<rho> < \<sigma> = (\<exists>\<upsilon>.
\<sigma> = Errorable \<upsilon> \<and> \<rho> < \<upsilon>)"
by (induct \<sigma>; auto)+
lemma errorable_less_eq_left_simps [simp]:
"ErrorFree \<rho> \<le> \<sigma> = (\<exists>\<upsilon>.
\<sigma> = ErrorFree \<upsilon> \<and> \<rho> \<le> \<upsilon> \<or>
\<sigma> = Errorable \<upsilon> \<and> \<rho> \<le> \<upsilon>)"
"Errorable \<rho> \<le> \<sigma> = (\<exists>\<upsilon>.
\<sigma> = Errorable \<upsilon> \<and> \<rho> \<le> \<upsilon>)"
by (auto simp: dual_order.order_iff_strict)
lemma errorable_less_right_simps [simp]:
"\<tau> < ErrorFree \<upsilon> = (\<exists>\<rho>.
\<tau> = ErrorFree \<rho> \<and> \<rho> < \<upsilon>)"
"\<tau> < Errorable \<upsilon> = (\<exists>\<rho>.
\<tau> = ErrorFree \<rho> \<and> \<rho> \<le> \<upsilon> \<or>
\<tau> = Errorable \<rho> \<and> \<rho> < \<upsilon>)"
by auto
lemma errorable_less_eq_right_simps [simp]:
"\<tau> \<le> ErrorFree \<upsilon> = (\<exists>\<rho>. \<tau> = ErrorFree \<rho> \<and> \<rho> \<le> \<upsilon>)"
"\<tau> \<le> Errorable \<upsilon> = (\<exists>\<rho>. \<tau> = ErrorFree \<rho> \<and> \<rho> \<le> \<upsilon> \<or> \<tau> = Errorable \<rho> \<and> \<rho> \<le> \<upsilon>)"
by auto
notation sup (infixl "\<squnion>" 65)
instantiation errorable :: (semilattice_sup) semilattice_sup
begin
primrec sup_errorable where
"ErrorFree \<tau> \<squnion> \<sigma> = (case \<sigma>
of ErrorFree \<rho> \<Rightarrow> ErrorFree (\<tau> \<squnion> \<rho>)
| Errorable \<rho> \<Rightarrow> Errorable (\<tau> \<squnion> \<rho>))"
| "Errorable \<tau> \<squnion> \<sigma> = (case \<sigma>
of ErrorFree \<rho> \<Rightarrow> Errorable (\<tau> \<squnion> \<rho>)
| Errorable \<rho> \<Rightarrow> Errorable (\<tau> \<squnion> \<rho>))"
lemma sup_ge1_errorable:
"\<tau> \<le> \<tau> \<squnion> \<sigma>"
for \<tau> \<sigma> :: "'a errorable"
by (cases \<tau>; cases \<sigma>; simp)
lemma sup_commut_errorable:
"\<tau> \<squnion> \<sigma> = \<sigma> \<squnion> \<tau>"
for \<tau> \<sigma> :: "'a errorable"
by (cases \<tau>; cases \<sigma>; simp add: sup.commute)
lemma sup_least_errorable:
"\<tau> \<le> \<rho> \<Longrightarrow> \<sigma> \<le> \<rho> \<Longrightarrow> \<tau> \<squnion> \<sigma> \<le> \<rho>"
for \<tau> \<sigma> \<rho> :: "'a errorable"
by (cases \<tau>; cases \<sigma>; cases \<rho>; simp)
instance
apply intro_classes
apply (simp add: sup_ge1_errorable)
apply (simp add: sup_ge1_errorable sup_commut_errorable)
by (simp add: sup_least_errorable)
end
primrec unwrap_errorable_type where
"unwrap_errorable_type (ErrorFree \<tau>) = \<tau>"
| "unwrap_errorable_type (Errorable \<tau>) = \<tau>"
primrec errorable_type where
"errorable_type (ErrorFree \<tau>) = False"
| "errorable_type (Errorable \<tau>) = True"
abbreviation "error_free_type \<tau> \<equiv> \<not> errorable_type \<tau>"
primrec to_error_free_type where
"to_error_free_type (ErrorFree \<tau>) = (ErrorFree \<tau>)"
| "to_error_free_type (Errorable \<tau>) = (ErrorFree \<tau>)"
primrec to_errorable_type where
"to_errorable_type (ErrorFree \<tau>) = (Errorable \<tau>)"
| "to_errorable_type (Errorable \<tau>) = (Errorable \<tau>)"
lemma unwrap_errorable_type_ErrorFree [simp]:
"unwrap_errorable_type \<circ> ErrorFree = id"
using comp_def by auto
lemma unwrap_errorable_type_Errorable [simp]:
"unwrap_errorable_type \<circ> Errorable = id"
using comp_def by auto
end