Theory Init

theory Init
imports Action
(*  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
  first_world :: "behavior ⇒ ('w::world)"
  st1         :: "behavior ⇒ state"
  st2         :: "behavior ⇒ state"

definition Initial :: "('w::world ⇒ bool) ⇒ temporal"
  where Init_def: "Initial F sigma = F (first_world sigma)"

syntax
  "_TEMP"    :: "lift ⇒ 'a"                          ("(TEMP _)")
  "_Init"    :: "lift ⇒ lift"                        ("(Init _)"[40] 50)
translations
  "TEMP F"   => "(F::behavior ⇒ _)"
  "_Init"    == "CONST Initial"
  "sigma ⊨ Init F"  <= "_Init F sigma"

overloading
  fw_temp  "first_world :: behavior ⇒ behavior"
  fw_stp  "first_world :: behavior ⇒ state"
  fw_act  "first_world :: behavior ⇒ state × state"
begin

definition "first_world == λsigma. sigma"
definition "first_world == st1"
definition "first_world == λsigma. (st1 sigma, st2 sigma)"

end

lemma const_simps [int_rewrite, simp]:
  "⊢ (Init #True) = #True"
  "⊢ (Init #False) = #False"
  by (auto simp: Init_def)

lemma Init_simps1 [int_rewrite]:
  "⋀F. ⊢ (Init ¬F) = (¬ Init F)"
  "⊢ (Init (P ⟶ Q)) = (Init P ⟶ Init Q)"
  "⊢ (Init (P ∧ Q)) = (Init P ∧ Init Q)"
  "⊢ (Init (P ∨ Q)) = (Init P ∨ Init Q)"
  "⊢ (Init (P = Q)) = ((Init P) = (Init Q))"
  "⊢ (Init (∀x. F x)) = (∀x. (Init F x))"
  "⊢ (Init (∃x. F x)) = (∃x. (Init F x))"
  "⊢ (Init (∃!x. F x)) = (∃!x. (Init F x))"
  by (auto simp: Init_def)

lemma Init_stp_act: "⊢ (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: "⊢ (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 ⊨ Init P) = P (st1 sigma)"
  by (simp add: Init_def fw_stp_def)

lemma Init_act: "(sigma ⊨ 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