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lib: none_top/none_bot for option predicates
Add theory None_Top_Bot with definitions, lemmas, and automation setup for predicates on option that map None to True/False, or more generally to top/bot. Can be used as e.g. "none_top tcb_at p s". Signed-off-by: Gerwin Klein <[email protected]>
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(* | ||
* Copyright 2024, Proofcraft Pty Ltd | ||
* | ||
* SPDX-License-Identifier: BSD-2-Clause | ||
*) | ||
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(* Predicates on option that map None to True/False, or more generally to top/bot. | ||
E.g. "none_top ((\<noteq>) 0) opt_ptr" or "none_top tcb_at p s". | ||
They are definitions, not abbreviations so that they don't participate in general | ||
option case splits (separate split rules are provided), and so that we can control | ||
simp/intro/elim setup a bit better. It should usually be unnecessary to unfold the | ||
definitions (e.g. see the all/ex rules + intro/dest/elim rules), but it is not | ||
harmful to do so. | ||
The main setup is for the general lattice case, followed by a section that spells | ||
out properties for the more common bool and function cases together with additional | ||
automation setup for these. | ||
*) | ||
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theory None_Top_Bot | ||
imports Monads.Fun_Pred_Syntax | ||
begin | ||
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definition none_top :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b::top" where | ||
"none_top \<equiv> case_option top" | ||
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definition none_bot :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b::bot" where | ||
"none_bot \<equiv> case_option bot" | ||
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section \<open>General lattice properties for @{const none_top}\<close> | ||
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lemma none_top_simps[simp]: | ||
"none_top f None = top" | ||
"none_top f (Some x) = f x" | ||
by (auto simp: none_top_def) | ||
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(* Mirrors option.splits *) | ||
lemma none_top_split: | ||
"P (none_top f opt) = ((opt = None \<longrightarrow> P top) \<and> (\<forall>x. opt = Some x \<longrightarrow> P (f x)))" | ||
by (cases opt) auto | ||
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lemma none_top_split_asm: | ||
"P (none_top f opt) = (\<not> (opt = None \<and> \<not> P top \<or> (\<exists>x. opt = Some x \<and> \<not> P (f x))))" | ||
by (cases opt) auto | ||
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lemmas none_top_splits = none_top_split none_top_split_asm | ||
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lemma none_top_if: | ||
"none_top f opt = (if opt = None then top else f (the opt))" | ||
by (cases opt) auto | ||
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(* General version of none_top_bool_all below *) | ||
lemma none_top_Inf: | ||
"none_top f opt = (Inf {f x |x. opt = Some x} :: 'a :: complete_lattice)" | ||
by (cases opt) auto | ||
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lemma none_top_top[simp]: | ||
"none_top top = top" | ||
by (rule ext) (simp split: none_top_splits) | ||
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section \<open>Bool/fun lemmas for @{const none_top}\<close> | ||
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lemma none_top_bool_all: | ||
"none_top f opt = (\<forall>x. opt = Some x \<longrightarrow> f x)" | ||
by (auto simp: none_top_Inf) | ||
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lemma none_top_bool_cases: | ||
"none_top f opt = (opt = None \<or> (\<exists>x. opt = Some x \<and> f x))" | ||
by (cases opt) auto | ||
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lemma none_top_boolI: | ||
"(\<And>x. opt = Some x \<Longrightarrow> f x) \<Longrightarrow> none_top f opt" | ||
by (simp add: none_top_bool_all) | ||
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lemma none_top_boolD: | ||
"none_top f opt \<Longrightarrow> opt = None \<or> (\<exists>x. opt = Some x \<and> f x)" | ||
by (cases opt) auto | ||
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lemma none_top_boolE: | ||
"\<lbrakk> none_top f opt; opt = None \<Longrightarrow> P; \<And>x. \<lbrakk> opt = Some x; f x \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" | ||
by (cases opt) auto | ||
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lemma none_top_False_bool_None[simp]: | ||
"none_top \<bottom> opt = (opt = None)" | ||
by (cases opt) auto | ||
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(* We don't usually use "bot" on bool -- only putting this here for completeness. *) | ||
lemma none_top_bot_None[intro!, simp]: | ||
"none_top bot opt = (opt = None)" | ||
by (cases opt) auto | ||
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lemma none_top_True_bool[intro!, simp]: | ||
"none_top \<top> opt" | ||
by (cases opt) auto | ||
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lemma none_top_fun_all: | ||
"none_top f opt s = (\<forall>x. opt = Some x \<longrightarrow> f x s)" | ||
by (cases opt) auto | ||
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lemma none_top_fun_cases: | ||
"none_top f opt s = (opt = None \<or> (\<exists>x. opt = Some x \<and> f x s))" | ||
by (cases opt) auto | ||
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lemma none_top_funI: | ||
"(\<And>x. opt = Some x \<Longrightarrow> f x s) \<Longrightarrow> none_top f opt s" | ||
by (simp add: none_top_fun_all) | ||
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lemma none_top_funD: | ||
"none_top f opt s \<Longrightarrow> opt = None \<or> (\<exists>x. opt = Some x \<and> f x s)" | ||
by (cases opt) auto | ||
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lemma none_top_funE: | ||
"\<lbrakk> none_top f opt s; opt = None \<Longrightarrow> P; \<And>x. \<lbrakk> opt = Some x; f x s \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" | ||
by (cases opt) auto | ||
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lemma none_top_False_fun_None[simp]: | ||
"none_top \<bottom>\<bottom> opt s = (opt = None)" | ||
by (cases opt) auto | ||
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lemma none_top_True_fun[intro!, simp]: | ||
"none_top \<top>\<top> opt s" | ||
by (cases opt) auto | ||
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section \<open>General lattice properties for @{const none_bot}\<close> | ||
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lemma none_bot_simps[simp]: | ||
"none_bot f None = bot" | ||
"none_bot f (Some x) = f x" | ||
by (auto simp: none_bot_def) | ||
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(* Mirrors option.splits *) | ||
lemma none_bot_split: | ||
"P (none_bot f opt) = ((opt = None \<longrightarrow> P bot) \<and> (\<forall>x. opt = Some x \<longrightarrow> P (f x)))" | ||
by (cases opt) auto | ||
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lemma none_bot_split_asm: | ||
"P (none_bot f opt) = (\<not> (opt = None \<and> \<not> P bot \<or> (\<exists>x. opt = Some x \<and> \<not> P (f x))))" | ||
by (cases opt) auto | ||
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lemmas none_bot_splits = none_bot_split none_bot_split_asm | ||
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(* General version of none_bot_bool_ex below *) | ||
lemma none_bot_Sup: | ||
"none_bot f opt = (Sup {f x |x. opt = Some x} :: 'a :: complete_lattice)" | ||
by (cases opt) auto | ||
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lemma none_bot_if: | ||
"none_bot f opt = (if opt = None then bot else f (the opt))" | ||
by (cases opt) auto | ||
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lemma none_bot_bot[simp]: | ||
"none_bot bot = bot" | ||
by (rule ext) (simp split: none_bot_splits) | ||
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section \<open>Bool/fun lemmas for @{const none_bot}\<close> | ||
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lemma none_bot_bool_ex: | ||
"none_bot f opt = (\<exists>x. opt = Some x \<and> f x)" | ||
by (auto simp: none_bot_Sup) | ||
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lemma none_bot_boolI: | ||
"\<exists>x. opt = Some x \<and> f x \<Longrightarrow> none_top f opt" | ||
by (auto simp: none_bot_bool_ex) | ||
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lemma none_bot_boolD: | ||
"none_bot f opt \<Longrightarrow> \<exists>x. opt = Some x \<and> f x" | ||
by (cases opt) auto | ||
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lemma none_bot_boolE: | ||
"\<lbrakk> none_bot f opt; \<And>x. \<lbrakk> opt = Some x; f x \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" | ||
by (cases opt) auto | ||
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(* As for none_top_bot, it would be unusual to see "top" for bool. Only adding the lemma here for | ||
completeness *) | ||
lemma none_bot_top_neq_None[simp]: | ||
"none_bot top opt = (opt \<noteq> None)" | ||
by (cases opt) auto | ||
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lemma none_bot_True_bool_neq_None[simp]: | ||
"none_bot \<top> opt = (opt \<noteq> None)" | ||
by (cases opt) auto | ||
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lemma none_bot_False_bool[simp]: | ||
"\<not>none_bot \<bottom> opt" | ||
by (cases opt) auto | ||
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lemma none_bot_fun_ex: | ||
"none_bot f opt s = (\<exists>x. opt = Some x \<and> f x s)" | ||
by (cases opt) auto | ||
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lemma none_bot_funI: | ||
"\<exists>x. opt = Some x \<and> f x s \<Longrightarrow> none_bot f opt s" | ||
by (simp add: none_bot_fun_ex) | ||
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lemma none_bot_funD: | ||
"none_bot f opt s \<Longrightarrow> \<exists>x. opt = Some x \<and> f x s" | ||
by (cases opt) auto | ||
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lemma none_bot_funE: | ||
"\<lbrakk> none_bot f opt s; \<And>x. \<lbrakk> opt = Some x; f x s \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" | ||
by (cases opt) auto | ||
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lemma none_bot_True_fun_neq_None[simp]: | ||
"none_bot \<top>\<top> opt s = (opt \<noteq> None)" | ||
by (cases opt) auto | ||
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lemma none_bot_False_fun[simp]: | ||
"\<not>none_bot \<bottom>\<bottom> opt s" | ||
by (cases opt) auto | ||
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section \<open>Automation setup and short-hand names\<close> | ||
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lemmas none_topI[intro!] = none_top_boolI none_top_funI | ||
lemmas none_topD = none_top_boolD none_top_funD | ||
lemmas none_topE = none_top_boolE none_top_funE | ||
lemmas none_top_all = none_top_bool_all none_top_fun_all | ||
lemmas none_top_case = none_top_bool_cases none_top_fun_cases | ||
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lemmas none_botI = none_bot_boolI none_bot_funI | ||
lemmas none_botD = none_bot_boolD none_bot_funD | ||
lemmas none_botE[elim!] = none_bot_boolE none_bot_funE | ||
lemmas none_bot_ex = none_bot_bool_ex none_bot_fun_ex | ||
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end |
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