(* Title: HOL/HOLCF/IOA/meta_theory/TL.thy
Author: Olaf Müller
*)
section \<open>A General Temporal Logic\<close>
theory TL
imports Pred Sequence
begin
default_sort type
type_synonym
'a temporal = "'a Seq predicate"
consts
suffix :: "'a Seq => 'a Seq => bool"
tsuffix :: "'a Seq => 'a Seq => bool"
validT :: "'a Seq predicate => bool"
unlift :: "'a lift => 'a"
Init :: "'a predicate => 'a temporal" ("\<langle>_\<rangle>" [0] 1000)
Box :: "'a temporal => 'a temporal" ("\<box>(_)" [80] 80)
Diamond :: "'a temporal => 'a temporal" ("\<diamond>(_)" [80] 80)
Next :: "'a temporal => 'a temporal"
Leadsto :: "'a temporal => 'a temporal => 'a temporal" (infixr "\<leadsto>" 22)
defs
unlift_def:
"unlift x == (case x of Def y => y)"
(* this means that for nil and UU the effect is unpredictable *)
Init_def:
"Init P s == (P (unlift (HD$s)))"
suffix_def:
"suffix s2 s == ? s1. (Finite s1 & s = s1 @@ s2)"
tsuffix_def:
"tsuffix s2 s == s2 ~= nil & s2 ~= UU & suffix s2 s"
Box_def:
"(\<box>P) s == ! s2. tsuffix s2 s --> P s2"
Next_def:
"(Next P) s == if (TL$s=UU | TL$s=nil) then (P s) else P (TL$s)"
Diamond_def:
"\<diamond>P == \<^bold>\<not> (\<box>(\<^bold>\<not> P))"
Leadsto_def:
"P \<leadsto> Q == (\<box>(P \<^bold>\<longrightarrow> (\<diamond>Q)))"
validT_def:
"validT P == ! s. s~=UU & s~=nil --> (s \<Turnstile> P)"
lemma simple: "\<box>\<diamond>(\<^bold>\<not> P) = (\<^bold>\<not> \<diamond>\<box>P)"
apply (rule ext)
apply (simp add: Diamond_def NOT_def Box_def)
done
lemma Boxnil: "nil \<Turnstile> \<box>P"
apply (simp add: satisfies_def Box_def tsuffix_def suffix_def nil_is_Conc)
done
lemma Diamondnil: "~(nil \<Turnstile> \<diamond>P)"
apply (simp add: Diamond_def satisfies_def NOT_def)
apply (cut_tac Boxnil)
apply (simp add: satisfies_def)
done
lemma Diamond_def2: "(\<diamond>F) s = (? s2. tsuffix s2 s & F s2)"
apply (simp add: Diamond_def NOT_def Box_def)
done
subsection "TLA Axiomatization by Merz"
lemma suffix_refl: "suffix s s"
apply (simp add: suffix_def)
apply (rule_tac x = "nil" in exI)
apply auto
done
lemma reflT: "s~=UU & s~=nil --> (s \<Turnstile> \<box>F \<^bold>\<longrightarrow> F)"
apply (simp add: satisfies_def IMPLIES_def Box_def)
apply (rule impI)+
apply (erule_tac x = "s" in allE)
apply (simp add: tsuffix_def suffix_refl)
done
lemma suffix_trans: "[| suffix y x ; suffix z y |] ==> suffix z x"
apply (simp add: suffix_def)
apply auto
apply (rule_tac x = "s1 @@ s1a" in exI)
apply auto
apply (simp (no_asm) add: Conc_assoc)
done
lemma transT: "s \<Turnstile> \<box>F \<^bold>\<longrightarrow> \<box>\<box>F"
apply (simp (no_asm) add: satisfies_def IMPLIES_def Box_def tsuffix_def)
apply auto
apply (drule suffix_trans)
apply assumption
apply (erule_tac x = "s2a" in allE)
apply auto
done
lemma normalT: "s \<Turnstile> \<box>(F \<^bold>\<longrightarrow> G) \<^bold>\<longrightarrow> \<box>F \<^bold>\<longrightarrow> \<box>G"
apply (simp (no_asm) add: satisfies_def IMPLIES_def Box_def)
done
subsection "TLA Rules by Lamport"
lemma STL1a: "validT P ==> validT (\<box>P)"
apply (simp add: validT_def satisfies_def Box_def tsuffix_def)
done
lemma STL1b: "valid P ==> validT (Init P)"
apply (simp add: valid_def validT_def satisfies_def Init_def)
done
lemma STL1: "valid P ==> validT (\<box>(Init P))"
apply (rule STL1a)
apply (erule STL1b)
done
(* Note that unlift and HD is not at all used !!! *)
lemma STL4: "valid (P \<^bold>\<longrightarrow> Q) ==> validT (\<box>(Init P) \<^bold>\<longrightarrow> \<box>(Init Q))"
apply (simp add: valid_def validT_def satisfies_def IMPLIES_def Box_def Init_def)
done
subsection "LTL Axioms by Manna/Pnueli"
lemma tsuffix_TL [rule_format (no_asm)]:
"s~=UU & s~=nil --> tsuffix s2 (TL$s) --> tsuffix s2 s"
apply (unfold tsuffix_def suffix_def)
apply auto
apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
apply (rule_tac x = "a\<leadsto>s1" in exI)
apply auto
done
lemmas tsuffix_TL2 = conjI [THEN tsuffix_TL]
declare split_if [split del]
lemma LTL1:
"s~=UU & s~=nil --> (s \<Turnstile> \<box>F \<^bold>\<longrightarrow> (F \<^bold>\<and> (Next (\<box>F))))"
apply (unfold Next_def satisfies_def NOT_def IMPLIES_def AND_def Box_def)
apply auto
(* \<box>F \<^bold>\<longrightarrow> F *)
apply (erule_tac x = "s" in allE)
apply (simp add: tsuffix_def suffix_refl)
(* \<box>F \<^bold>\<longrightarrow> Next \<box>F *)
apply (simp split add: split_if)
apply auto
apply (drule tsuffix_TL2)
apply assumption+
apply auto
done
declare split_if [split]
lemma LTL2a:
"s \<Turnstile> \<^bold>\<not> (Next F) \<^bold>\<longrightarrow> (Next (\<^bold>\<not> F))"
apply (unfold Next_def satisfies_def NOT_def IMPLIES_def)
apply simp
done
lemma LTL2b:
"s \<Turnstile> (Next (\<^bold>\<not> F)) \<^bold>\<longrightarrow> (\<^bold>\<not> (Next F))"
apply (unfold Next_def satisfies_def NOT_def IMPLIES_def)
apply simp
done
lemma LTL3:
"ex \<Turnstile> (Next (F \<^bold>\<longrightarrow> G)) \<^bold>\<longrightarrow> (Next F) \<^bold>\<longrightarrow> (Next G)"
apply (unfold Next_def satisfies_def NOT_def IMPLIES_def)
apply simp
done
lemma ModusPonens: "[| validT (P \<^bold>\<longrightarrow> Q); validT P |] ==> validT Q"
apply (simp add: validT_def satisfies_def IMPLIES_def)
done
end