(* Title: HOL/TLA/Init.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
Introduces type of temporal formulas. Defines interface between
temporal formulas and its "subformulas" (state predicates and
actions).
*)
theory Init
imports Action
begin
typedecl behavior
instance behavior :: world ..
type_synonym temporal = "behavior form"
consts
Initial :: "('w::world \<Rightarrow> bool) \<Rightarrow> temporal"
first_world :: "behavior \<Rightarrow> ('w::world)"
st1 :: "behavior \<Rightarrow> state"
st2 :: "behavior \<Rightarrow> state"
syntax
"_TEMP" :: "lift \<Rightarrow> 'a" ("(TEMP _)")
"_Init" :: "lift \<Rightarrow> lift" ("(Init _)"[40] 50)
translations
"TEMP F" => "(F::behavior \<Rightarrow> _)"
"_Init" == "CONST Initial"
"sigma \<Turnstile> Init F" <= "_Init F sigma"
defs
Init_def: "sigma \<Turnstile> Init F == (first_world sigma) \<Turnstile> F"
defs (overloaded)
fw_temp_def: "first_world == \<lambda>sigma. sigma"
fw_stp_def: "first_world == st1"
fw_act_def: "first_world == \<lambda>sigma. (st1 sigma, st2 sigma)"
lemma const_simps [int_rewrite, simp]:
"\<turnstile> (Init #True) = #True"
"\<turnstile> (Init #False) = #False"
by (auto simp: Init_def)
lemma Init_simps1 [int_rewrite]:
"\<And>F. \<turnstile> (Init \<not>F) = (\<not> Init F)"
"\<turnstile> (Init (P \<longrightarrow> Q)) = (Init P \<longrightarrow> Init Q)"
"\<turnstile> (Init (P \<and> Q)) = (Init P \<and> Init Q)"
"\<turnstile> (Init (P \<or> Q)) = (Init P \<or> Init Q)"
"\<turnstile> (Init (P = Q)) = ((Init P) = (Init Q))"
"\<turnstile> (Init (\<forall>x. F x)) = (\<forall>x. (Init F x))"
"\<turnstile> (Init (\<exists>x. F x)) = (\<exists>x. (Init F x))"
"\<turnstile> (Init (\<exists>!x. F x)) = (\<exists>!x. (Init F x))"
by (auto simp: Init_def)
lemma Init_stp_act: "\<turnstile> (Init $P) = (Init P)"
by (auto simp add: Init_def fw_act_def fw_stp_def)
lemmas Init_simps2 = Init_stp_act [int_rewrite] Init_simps1
lemmas Init_stp_act_rev = Init_stp_act [int_rewrite, symmetric]
lemma Init_temp: "\<turnstile> (Init F) = F"
by (auto simp add: Init_def fw_temp_def)
lemmas Init_simps = Init_temp [int_rewrite] Init_simps2
(* Trivial instances of the definitions that avoid introducing lambda expressions. *)
lemma Init_stp: "(sigma \<Turnstile> Init P) = P (st1 sigma)"
by (simp add: Init_def fw_stp_def)
lemma Init_act: "(sigma \<Turnstile> Init A) = A (st1 sigma, st2 sigma)"
by (simp add: Init_def fw_act_def)
lemmas Init_defs = Init_stp Init_act Init_temp [int_use]
end