Theory TL (Isabelle2005: October 2005)

(*  Title:      HOLCF/IOA/meta_theory/TLS.thy
    ID:         $Id: TL.thy,v 1.10 2005/09/02 15:24:03 wenzelm Exp $
    Author:     Olaf Müller
*)

header {* A General Temporal Logic *}

theory TL
imports Pred Sequence
begin

defaultsort type

types
  '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"          ("<_>" [0] 1000)

Box          ::"'a temporal => 'a temporal"   ("[] (_)" [80] 80)
Diamond      ::"'a temporal => 'a temporal"   ("<> (_)" [80] 80)
Next         ::"'a temporal => 'a temporal"
Leadsto      ::"'a temporal => 'a temporal => 'a temporal"  (infixr "~>" 22)

syntax (xsymbols)
   "Box"        ::"'a temporal => 'a temporal"   ("\<box> (_)" [80] 80)
   "Diamond"    ::"'a temporal => 'a temporal"   ("\<diamond> (_)" [80] 80)
   "Leadsto"    ::"'a temporal => 'a temporal => 'a temporal"  (infixr "\<leadsto>" 22)

defs

unlift_def:
  "unlift x == (case x of
                 UU   => arbitrary
               | 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:
  "([] 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:
  "<> P == .~ ([] (.~ P))"

Leadsto_def:
   "P ~> Q == ([] (P .--> (<> Q)))"

validT_def:
  "validT P == ! s. s~=UU & s~=nil --> (s |= P)"

ML {* use_legacy_bindings (the_context ()) *}

end

theorem simple:

  [] <> (¬ P) = (¬ <> [] P)

theorem Boxnil:

  nil |= [] P

theorem Diamondnil:

  ¬ (nil |= <> P)

theorem Diamond_def2:

  (<> F) s = (∃s2. tsuffix s2 s ∧ F s2)

TLA Axiomatization by Merz

theorem suffix_refl:

  suffix s s

theorem reflT:

  s ≠ UU ∧ s ≠ nil --> (s |= [] F --> F)

theorem suffix_trans:

  [| suffix y x; suffix z y |] ==> suffix z x

theorem transT:

  s |= [] F --> [] [] F

theorem normalT:

  s |= [] (F --> G) --> [] F --> [] G

TLA Rules by Lamport

theorem STL1a:

  validT P ==> validT ([] P)

theorem STL1b:

  valid P ==> validT <P>

theorem STL1:

  valid P ==> validT ([] <P>)

theorem STL4:

  valid (P --> Q) ==> validT ([] <P> --> [] <Q>)

LTL Axioms by Manna/Pnueli

theorem tsuffix_TL:

  [| s ≠ UU ∧ s ≠ nil; tsuffix s2.0 (TL·s) |] ==> tsuffix s2.0 s

theorem LTL1:

  s ≠ UU ∧ s ≠ nil --> (s |= [] F --> F ∧ Next ([] F))

theorem LTL2a:

  s |= ¬ Next F --> Next (¬ F)

theorem LTL2b:

  s |= Next (¬ F) --> ¬ Next F

theorem LTL3:

  ex |= Next (F --> G) --> Next F --> Next G

theorem ModusPonens:

  [| validT (P --> Q); validT P |] ==> validT Q