Theory "Past_Temporal_Logic"

Parents Temporal_Logic

Signature

Constant	Type
InitPoint	:num -> bool
PALWAYS	:(num -> bool) -> num -> bool
PBEFORE	:(num -> bool) -> (num -> bool) -> num -> bool
PEVENTUAL	:(num -> bool) -> num -> bool
PNEXT	:(num -> bool) -> num -> bool
PSBEFORE	:(num -> bool) -> (num -> bool) -> num -> bool
PSNEXT	:(num -> bool) -> num -> bool
PSUNTIL	:(num -> bool) -> (num -> bool) -> num -> bool
PSWHEN	:(num -> bool) -> (num -> bool) -> num -> bool
PUNTIL	:(num -> bool) -> (num -> bool) -> num -> bool
PWHEN	:(num -> bool) -> (num -> bool) -> num -> bool

Definitions

PWHEN

⊢ ∀a b t0.
      (a PWHEN b) t0 ⇔
      (∀t. t ≤ t0 ⇒ ¬b t) ∨
      ∃delta. delta ≤ t0 ∧ a delta ∧ b delta ∧ ∀t. delta < t ∧ t ≤ t0 ⇒ ¬b t

PUNTIL

⊢ ∀a b t0.
      (a SINCE b) t0 ⇔
      (∀t. t ≤ t0 ⇒ a t) ∨
      ∃delta. delta ≤ t0 ∧ b delta ∧ ∀t. delta < t ∧ t ≤ t0 ⇒ a t ∧ ¬b t

PSWHEN

⊢ ∀a b t0.
      (a PSWHEN b) t0 ⇔
      ∃delta. delta ≤ t0 ∧ a delta ∧ b delta ∧ ∀t. delta < t ∧ t ≤ t0 ⇒ ¬b t

PSUNTIL

⊢ ∀a b t0.
      (a PSUNTIL b) t0 ⇔
      ∃delta. delta ≤ t0 ∧ b delta ∧ ∀t. delta < t ∧ t ≤ t0 ⇒ a t ∧ ¬b t

PSNEXT

⊢ ∀a t0. PSNEXT a t0 ⇔ 0 < t0 ∧ a (PRE t0)

PSBEFORE

⊢ ∀a b t0.
      (a PSBEFORE b) t0 ⇔
      ∃delta. delta ≤ t0 ∧ a delta ∧ ∀t. delta ≤ t ∧ t ≤ t0 ⇒ ¬b t

PNEXT

⊢ ∀a t0. PREV a t0 ⇔ (t0 = 0) ∨ a (PRE t0)

PEVENTUAL

⊢ ∀a t0. ONCE a t0 ⇔ ∃t. t ≤ t0 ∧ a t

PBEFORE

⊢ ∀a b t0.
      (a PBEFORE b) t0 ⇔
      (∀t. t ≤ t0 ⇒ ¬b t) ∨
      ∃delta. delta ≤ t0 ∧ a delta ∧ ∀t. delta ≤ t ∧ t ≤ t0 ⇒ ¬b t

PALWAYS

⊢ ∀a t0. PALWAYS a t0 ⇔ ∀t. t ≤ t0 ⇒ a t

InitPoint

⊢ InitPoint = (λt. t = 0)

Theorems

WHEN_EXPRESSIVE

⊢ (ALWAYS a = (λt. ((λt. F) WHEN (λt. ¬a t)) t)) ∧
  (EVENTUAL a = (λt. ¬((λt. F) WHEN a) t)) ∧
  (a SUNTIL b = (λt. ¬((λt. ¬b t) WHEN (λt. a t ⇒ b t)) t)) ∧
  (a UNTIL b = (λt. (b WHEN (λt. a t ⇒ b t)) t)) ∧
  (a SWHEN b = (λt. ¬((λt. ¬a t) WHEN b) t)) ∧
  (a BEFORE b = (λt. ((λt. ¬b t) WHEN (λt. a t ∨ b t)) t)) ∧
  (a SBEFORE b = (λt. ¬(b WHEN (λt. a t ∨ b t)) t))

UNTIL_EXPRESSIVE

⊢ (ALWAYS a = (λt. (a UNTIL (λt. F)) t)) ∧
  (EVENTUAL a = (λt. ¬((λt. ¬a t) UNTIL (λt. F)) t)) ∧
  (a SUNTIL b = (λt. ¬((λt. ¬b t) UNTIL (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a WHEN b = (λt. ((λt. ¬b t) UNTIL (λt. a t ∧ b t)) t)) ∧
  (a SWHEN b = (λt. ¬((λt. ¬a t ∨ ¬b t) UNTIL (λt. ¬a t ∧ b t)) t)) ∧
  (a BEFORE b = (λt. ((λt. ¬b t) UNTIL (λt. a t ∧ ¬b t)) t)) ∧
  (a SBEFORE b = (λt. ¬((λt. ¬a t) UNTIL b) t))

SWHEN_EXPRESSIVE

⊢ (ALWAYS a = (λt. ¬((λt. T) SWHEN (λt. ¬a t)) t)) ∧
  (EVENTUAL a = (λt. ((λt. T) SWHEN a) t)) ∧
  (a SUNTIL b = (λt. (b SWHEN (λt. a t ⇒ b t)) t)) ∧
  (a UNTIL b = (λt. ¬((λt. ¬b t) SWHEN (λt. a t ⇒ b t)) t)) ∧
  (a WHEN b = (λt. ¬((λt. ¬a t) SWHEN b) t)) ∧
  (a BEFORE b = (λt. ¬(b SWHEN (λt. a t ∨ b t)) t)) ∧
  (a SBEFORE b = (λt. ((λt. ¬b t) SWHEN (λt. a t ∨ b t)) t))

SUNTIL_EXPRESSIVE

⊢ (ALWAYS a = (λt. ¬((λt. T) SUNTIL (λt. ¬a t)) t)) ∧
  (EVENTUAL a = (λt. ((λt. T) SUNTIL a) t)) ∧
  (a UNTIL b = (λt. ¬((λt. ¬b t) SUNTIL (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a WHEN b = (λt. ¬((λt. ¬a t ∨ ¬b t) SUNTIL (λt. ¬a t ∧ b t)) t)) ∧
  (a BEFORE b = (λt. ¬((λt. ¬a t) SUNTIL b) t)) ∧
  (a SWHEN b = (λt. ((λt. ¬b t) SUNTIL (λt. a t ∧ b t)) t)) ∧
  (a SBEFORE b = (λt. ((λt. ¬b t) SUNTIL (λt. a t ∧ ¬b t)) t))

SOME_PAST_EVENT

⊢ ONCE b t ⇒
  ((a PWHEN b) t ⇔ (a PSWHEN b) t) ∧ ((a SINCE b) t ⇔ (a PSUNTIL b) t) ∧
  ((a PBEFORE b) t ⇔ (a PSBEFORE b) t) ∧ ((b PBEFORE a) t ⇔ (b PSBEFORE a) t)

SOME_FUTURE_EVENT

⊢ (EVENTUAL b t0 ⇔ ∀a. (a WHEN b) t0 ⇔ (a SWHEN b) t0) ∧
  (EVENTUAL b t0 ⇔ ∀a. (a UNTIL b) t0 ⇔ (a SUNTIL b) t0) ∧
  (EVENTUAL b t0 ⇔ ∀a. (a BEFORE b) t0 ⇔ (a SBEFORE b) t0)

SIMPLIFY

⊢ (NEXT (λt. F) = (λt. F)) ∧ (NEXT (λt. T) = (λt. T)) ∧
  (ALWAYS (λt. T) = (λt. T)) ∧ (ALWAYS (λt. F) = (λt. F)) ∧
  (EVENTUAL (λt. T) = (λt. T)) ∧ (EVENTUAL (λt. F) = (λt. F)) ∧
  ((λt. F) SUNTIL b = b) ∧ ((λt. T) SUNTIL b = EVENTUAL b) ∧
  (a SUNTIL (λt. F) = (λt. F)) ∧ (a SUNTIL (λt. T) = (λt. T)) ∧
  (a SUNTIL a = a) ∧ ((λt. F) UNTIL b = b) ∧ ((λt. T) UNTIL b = (λt. T)) ∧
  (a UNTIL (λt. F) = ALWAYS a) ∧ (a UNTIL (λt. T) = (λt. T)) ∧
  (a UNTIL a = a) ∧ ((λt. F) SWHEN b = (λt. F)) ∧
  ((λt. T) SWHEN b = EVENTUAL b) ∧ (a SWHEN (λt. F) = (λt. F)) ∧
  (a SWHEN (λt. T) = a) ∧ (a SWHEN a = EVENTUAL a) ∧
  ((λt. F) WHEN b = ALWAYS (λt. ¬b t)) ∧ ((λt. T) WHEN b = (λt. T)) ∧
  (a WHEN (λt. F) = (λt. T)) ∧ (a WHEN (λt. T) = a) ∧ (a WHEN a = (λt. T)) ∧
  ((λt. F) SBEFORE b = (λt. F)) ∧ ((λt. T) SBEFORE b = (λt. ¬b t)) ∧
  (a SBEFORE (λt. F) = EVENTUAL a) ∧ (a SBEFORE (λt. T) = (λt. F)) ∧
  (a SBEFORE a = (λt. F)) ∧ ((λt. F) BEFORE b = ALWAYS (λt. ¬b t)) ∧
  ((λt. T) BEFORE b = (λt. ¬b t)) ∧ (a BEFORE (λt. F) = (λt. T)) ∧
  (a BEFORE (λt. T) = (λt. F)) ∧ (a BEFORE a = ALWAYS (λt. ¬a t)) ∧
  (PREV (λt. F) = InitPoint) ∧ (PREV (λt. T) = (λt. T)) ∧
  (PSNEXT (λt. F) = (λt. F)) ∧ (PSNEXT (λt. T) = (λt. ¬InitPoint t)) ∧
  (PALWAYS (λt. T) = (λt. T)) ∧ (PALWAYS (λt. F) = (λt. F)) ∧
  (ONCE (λt. T) = (λt. T)) ∧ (ONCE (λt. F) = (λt. F)) ∧
  ((λt. F) PSUNTIL b = b) ∧ ((λt. T) PSUNTIL b = ONCE b) ∧
  (a PSUNTIL (λt. F) = (λt. F)) ∧ (a PSUNTIL (λt. T) = (λt. T)) ∧
  (a PSUNTIL a = a) ∧ ((λt. F) SINCE b = b) ∧ ((λt. T) SINCE b = (λt. T)) ∧
  (a SINCE (λt. F) = PALWAYS a) ∧ (a SINCE (λt. T) = (λt. T)) ∧
  (a SINCE a = a) ∧ ((λt. F) PSWHEN b = (λt. F)) ∧
  ((λt. T) PSWHEN b = ONCE b) ∧ (a PSWHEN (λt. F) = (λt. F)) ∧
  (a PSWHEN (λt. T) = a) ∧ (a PSWHEN a = ONCE a) ∧
  ((λt. F) PWHEN b = PALWAYS (λt. ¬b t)) ∧ ((λt. T) PWHEN b = (λt. T)) ∧
  (a PWHEN (λt. F) = (λt. T)) ∧ (a PWHEN (λt. T) = a) ∧
  (a PWHEN a = (λt. T)) ∧ ((λt. F) PSBEFORE b = (λt. F)) ∧
  ((λt. T) PSBEFORE b = (λt. ¬b t)) ∧ (a PSBEFORE (λt. F) = ONCE a) ∧
  (a PSBEFORE (λt. T) = (λt. F)) ∧ (a PSBEFORE a = (λt. F)) ∧
  ((λt. F) PBEFORE b = PALWAYS (λt. ¬b t)) ∧
  ((λt. T) PBEFORE b = (λt. ¬b t)) ∧ (a PBEFORE (λt. F) = (λt. T)) ∧
  (a PBEFORE (λt. T) = (λt. F)) ∧ (a PBEFORE a = PALWAYS (λt. ¬a t))

SEPARATE_SUNTIL_THM

⊢ (a SUNTIL (λt. b t ∨ c t) = (λt. (a SUNTIL b) t ∨ (a SUNTIL c) t)) ∧
  (a SUNTIL (λt. b t ∧ PREV c t) =
   (λt. b t ∧ PREV c t ∨ (a SUNTIL (λt. a t ∧ c t ∧ NEXT b t)) t)) ∧
  (a SUNTIL (λt. b t ∧ PSNEXT c t) =
   (λt. b t ∧ PSNEXT c t ∨ (a SUNTIL (λt. a t ∧ c t ∧ NEXT b t)) t)) ∧
  (a SUNTIL (λt. b t ∧ (c PSUNTIL d) t) =
   (λt.
        (c PSUNTIL d) t ∧ ((λt. a t ∧ NEXT c t) SUNTIL b) t ∨
        (a SUNTIL (λt. d t ∧ ((λt. a t ∧ NEXT c t) SUNTIL b) t)) t)) ∧
  (a SUNTIL (λt. b t ∧ (c PBEFORE d) t) =
   (λt.
        (c PBEFORE d) t ∧ ((λt. a t ∧ ¬NEXT d t) SUNTIL b) t ∨
        (a SUNTIL (λt. c t ∧ ¬d t ∧ ((λt. a t ∧ ¬NEXT d t) SUNTIL b) t)) t)) ∧
  ((λt. a t ∧ b t) SUNTIL c = (λt. (a SUNTIL c) t ∧ (b SUNTIL c) t)) ∧
  ((λt. a t ∨ PREV b t) SUNTIL c =
   (λt. c t ∨ (a t ∨ PREV b t) ∧ ((λt. b t ∨ NEXT a t) SUNTIL NEXT c) t)) ∧
  ((λt. a t ∨ PSNEXT b t) SUNTIL c =
   (λt. c t ∨ (a t ∨ PSNEXT b t) ∧ ((λt. b t ∨ NEXT a t) SUNTIL NEXT c) t)) ∧
  ((λt. a t ∨ (b PSUNTIL c) t) SUNTIL d =
   (λt.
        ((b PSUNTIL c) t ∨ ((λt. d t ∨ NEXT c t) BEFORE (λt. ¬a t ∧ ¬d t)) t) ∧
        ((λt. b t ∨ c t ∨ ((λt. d t ∨ NEXT c t) BEFORE (λt. ¬a t ∧ ¬d t)) t) SUNTIL
         d) t)) ∧
  ((λt. a t ∨ (b PBEFORE c) t) SUNTIL d =
   (λt.
        ((b PBEFORE c) t ∨ ((λt. d t ∨ NEXT b t) BEFORE (λt. ¬a t ∧ ¬d t)) t) ∧
        ((λt. ¬c t ∨ ((λt. d t ∨ NEXT b t) BEFORE (λt. ¬a t ∧ ¬d t)) t) SUNTIL
         d) t))

SEPARATE_PSUNTIL_THM

⊢ (a PSUNTIL (λt. b t ∨ c t) = (λt. (a PSUNTIL b) t ∨ (a PSUNTIL c) t)) ∧
  (a PSUNTIL (λt. b t ∧ NEXT c t) =
   (λt. b t ∧ NEXT c t ∨ (a PSUNTIL (λt. a t ∧ c t ∧ PSNEXT b t)) t)) ∧
  (a PSUNTIL (λt. b t ∧ (c SUNTIL d) t) =
   (λt.
        (c SUNTIL d) t ∧ ((λt. a t ∧ PREV c t) PSUNTIL b) t ∨
        (a PSUNTIL (λt. d t ∧ ((λt. a t ∧ PREV c t) PSUNTIL b) t)) t)) ∧
  (a PSUNTIL (λt. b t ∧ (c BEFORE d) t) =
   (λt.
        (c BEFORE d) t ∧ ((λt. a t ∧ ¬PREV d t) PSUNTIL b) t ∨
        (a PSUNTIL (λt. c t ∧ ¬d t ∧ ((λt. a t ∧ ¬PREV d t) PSUNTIL b) t)) t)) ∧
  ((λt. a t ∧ b t) PSUNTIL c = (λt. (a PSUNTIL c) t ∧ (b PSUNTIL c) t)) ∧
  ((λt. a t ∨ NEXT b t) PSUNTIL c =
   (λt. c t ∨ (a t ∨ NEXT b t) ∧ ((λt. b t ∨ PREV a t) PSUNTIL PSNEXT c) t)) ∧
  ((λt. a t ∨ (b SUNTIL c) t) PSUNTIL d =
   (λt.
        ((b SUNTIL c) t ∨ ((λt. d t ∨ PREV c t) PBEFORE (λt. ¬a t ∧ ¬d t)) t) ∧
        ((λt. b t ∨ c t ∨ ((λt. d t ∨ PREV c t) PBEFORE (λt. ¬a t ∧ ¬d t)) t) PSUNTIL
         d) t)) ∧
  ((λt. a t ∨ (b BEFORE c) t) PSUNTIL d =
   (λt.
        ((b BEFORE c) t ∨ ((λt. d t ∨ PSNEXT b t) PBEFORE (λt. ¬a t ∧ ¬d t)) t) ∧
        ((λt. ¬c t ∨ ((λt. d t ∨ PSNEXT b t) PBEFORE (λt. ¬a t ∧ ¬d t)) t) PSUNTIL
         d) t))

SEPARATE_PNEXT_THM

⊢ (PREV (λt. a t ∧ NEXT b t) = (λt. InitPoint t ∨ b t ∧ PREV a t)) ∧
  (PREV (λt. a t ∧ (b SUNTIL c) t) =
   (λt. PREV (λt. a t ∧ c t) t ∨ (b SUNTIL c) t ∧ PREV (λt. a t ∧ b t) t)) ∧
  (PREV (λt. a t ∧ (b BEFORE c) t) =
   (λt.
        PREV (λt. a t ∧ b t ∧ ¬c t) t ∨
        (b BEFORE c) t ∧ PREV (λt. a t ∧ ¬c t) t)) ∧
  (PREV (λt. a t ∨ NEXT b t) = (λt. b t ∨ PREV a t)) ∧
  (PREV (λt. a t ∨ (b SUNTIL c) t) =
   (λt. PREV (λt. a t ∨ c t) t ∨ (b SUNTIL c) t ∧ PREV b t)) ∧
  (PREV (λt. a t ∨ (b BEFORE c) t) =
   (λt. PREV (λt. a t ∨ ¬c t) t ∧ ((b BEFORE c) t ∨ PREV (λt. a t ∨ b t) t)))

SEPARATE_NEXT_THM

⊢ (NEXT (λt. a t ∧ PREV b t) = (λt. b t ∧ NEXT a t)) ∧
  (NEXT (λt. a t ∧ PSNEXT b t) = (λt. b t ∧ NEXT a t)) ∧
  (NEXT (λt. a t ∧ (b PSUNTIL c) t) =
   (λt. NEXT (λt. a t ∧ c t) t ∨ (b PSUNTIL c) t ∧ NEXT (λt. a t ∧ b t) t)) ∧
  (NEXT (λt. a t ∧ (b PBEFORE c) t) =
   (λt.
        NEXT (λt. a t ∧ b t ∧ ¬c t) t ∨
        (b PBEFORE c) t ∧ NEXT (λt. a t ∧ ¬c t) t)) ∧
  (NEXT (λt. a t ∨ PREV b t) = (λt. b t ∨ NEXT a t)) ∧
  (NEXT (λt. a t ∨ PSNEXT b t) = (λt. b t ∨ NEXT a t)) ∧
  (NEXT (λt. a t ∨ (b PSUNTIL c) t) =
   (λt. NEXT (λt. a t ∨ c t) t ∨ (b PSUNTIL c) t ∧ NEXT b t)) ∧
  (NEXT (λt. a t ∨ (b PBEFORE c) t) =
   (λt. NEXT (λt. a t ∨ ¬c t) t ∧ ((b PBEFORE c) t ∨ NEXT (λt. a t ∨ b t) t)))

SEPARATE_BEFORE_THM

⊢ (a BEFORE (λt. b t ∨ c t) = (λt. (a BEFORE b) t ∧ (a BEFORE c) t)) ∧
  ((λt. a t ∨ b t) BEFORE c = (λt. (a BEFORE c) t ∨ (b BEFORE c) t)) ∧
  (a BEFORE (λt. b t ∧ PREV c t) =
   (λt. ¬(b t ∧ PREV c t) ∧ (a t ∨ (NEXT a BEFORE (λt. c t ∧ NEXT b t)) t))) ∧
  (a BEFORE (λt. b t ∧ PSNEXT c t) =
   (λt. ¬(b t ∧ PSNEXT c t) ∧ (a t ∨ (NEXT a BEFORE (λt. c t ∧ NEXT b t)) t))) ∧
  (a BEFORE (λt. b t ∧ (c PSUNTIL d) t) =
   (λt.
        (((λt. ¬c t) PBEFORE d) t ∨ ((λt. a t ∨ ¬NEXT c t) BEFORE b) t) ∧
        (a BEFORE (λt. d t ∧ ((λt. ¬a t ∧ NEXT c t) SUNTIL b) t)) t)) ∧
  (a BEFORE (λt. b t ∧ (c PBEFORE d) t) =
   (λt.
        (((λt. ¬c t) PSUNTIL d) t ∨ ((λt. a t ∨ NEXT d t) BEFORE b) t) ∧
        (a BEFORE (λt. c t ∧ ¬d t ∧ ((λt. ¬a t ∧ ¬NEXT d t) SUNTIL b) t)) t)) ∧
  ((λt. a t ∧ PREV b t) BEFORE c =
   (λt. ¬c t ∧ a t ∧ PREV b t ∨ ¬c t ∧ ((λt. b t ∧ NEXT a t) BEFORE NEXT c) t)) ∧
  ((λt. a t ∧ PSNEXT b t) BEFORE c =
   (λt.
        ¬c t ∧ a t ∧ PSNEXT b t ∨
        ¬c t ∧ ((λt. b t ∧ NEXT a t) BEFORE NEXT c) t)) ∧
  ((λt. a t ∧ (b PBEFORE c) t) BEFORE d =
   (λt.
        (b PBEFORE c) t ∧ ((λt. ¬d t ∧ ¬NEXT c t) SUNTIL (λt. a t ∧ ¬d t)) t ∨
        ((λt. b t ∧ ¬c t ∧ ((λt. ¬d t ∧ ¬NEXT c t) SUNTIL (λt. a t ∧ ¬d t)) t) BEFORE
         d) t)) ∧
  ((λt. a t ∧ (b PSUNTIL c) t) BEFORE d =
   (λt.
        (b PSUNTIL c) t ∧ ((λt. ¬d t ∧ NEXT b t) SUNTIL (λt. a t ∧ ¬d t)) t ∨
        ((λt. c t ∧ ((λt. ¬d t ∧ NEXT b t) SUNTIL (λt. a t ∧ ¬d t)) t) BEFORE
         d) t))

SBEFORE_EXPRESSIVE

⊢ (ALWAYS a = (λt. ¬((λt. ¬a t) SBEFORE (λt. F)) t)) ∧
  (EVENTUAL a = (λt. (a SBEFORE (λt. F)) t)) ∧
  (a SUNTIL b = (λt. (b SBEFORE (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a UNTIL b = (λt. ¬((λt. ¬a t) SBEFORE b) t)) ∧
  (a SWHEN b = (λt. (b SBEFORE (λt. ¬a t ∧ b t)) t)) ∧
  (a WHEN b = (λt. ¬(b SBEFORE (λt. a t ∧ b t)) t)) ∧
  (a BEFORE b = (λt. ¬(b SBEFORE (λt. a t ∧ ¬b t)) t))

RECURSION

⊢ (ALWAYS a = (λt. a t ∧ NEXT (ALWAYS a) t)) ∧
  (EVENTUAL a = (λt. a t ∨ NEXT (EVENTUAL a) t)) ∧
  (a SUNTIL b = (λt. ¬b t ⇒ a t ∧ NEXT (a SUNTIL b) t)) ∧
  (a SWHEN b = (λt. if b t then a t else NEXT (a SWHEN b) t)) ∧
  (a SBEFORE b = (λt. ¬b t ∧ (a t ∨ NEXT (a SBEFORE b) t))) ∧
  (a UNTIL b = (λt. ¬b t ⇒ a t ∧ NEXT (a UNTIL b) t)) ∧
  (a WHEN b = (λt. if b t then a t else NEXT (a WHEN b) t)) ∧
  (a BEFORE b = (λt. ¬b t ∧ (a t ∨ NEXT (a BEFORE b) t))) ∧
  (PALWAYS a = (λt. a t ∧ PREV (PALWAYS a) t)) ∧
  (ONCE a = (λt. a t ∨ PSNEXT (ONCE a) t)) ∧
  (a PSUNTIL b = (λt. b t ∨ a t ∧ PSNEXT (a PSUNTIL b) t)) ∧
  (a PSWHEN b = (λt. a t ∧ b t ∨ ¬b t ∧ PSNEXT (a PSWHEN b) t)) ∧
  (a PSBEFORE b = (λt. ¬b t ∧ (a t ∨ PSNEXT (a PSBEFORE b) t))) ∧
  (a SINCE b = (λt. b t ∨ a t ∧ PREV (a SINCE b) t)) ∧
  (a PWHEN b = (λt. a t ∧ b t ∨ ¬b t ∧ PREV (a PWHEN b) t)) ∧
  (a PBEFORE b = (λt. ¬b t ∧ (a t ∨ PREV (a PBEFORE b) t)))

PWHEN_EXPRESSIVE

⊢ (PALWAYS a = (λt. ((λt. F) PWHEN (λt. ¬a t)) t)) ∧
  (ONCE a = (λt. ¬((λt. F) PWHEN a) t)) ∧
  (a PSUNTIL b = (λt. ¬((λt. ¬b t) PWHEN (λt. a t ⇒ b t)) t)) ∧
  (a SINCE b = (λt. (b PWHEN (λt. a t ⇒ b t)) t)) ∧
  (a PSWHEN b = (λt. ¬((λt. ¬a t) PWHEN b) t)) ∧
  (a PBEFORE b = (λt. ((λt. ¬b t) PWHEN (λt. a t ∨ b t)) t)) ∧
  (a PSBEFORE b = (λt. ¬(b PWHEN (λt. a t ∨ b t)) t))

PUNTIL_EXPRESSIVE

⊢ (PALWAYS a = (λt. (a SINCE (λt. F)) t)) ∧
  (ONCE a = (λt. ¬((λt. ¬a t) SINCE (λt. F)) t)) ∧
  (a PSUNTIL b = (λt. ¬((λt. ¬b t) SINCE (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a PWHEN b = (λt. ((λt. ¬b t) SINCE (λt. a t ∧ b t)) t)) ∧
  (a PSWHEN b = (λt. ¬((λt. ¬a t ∨ ¬b t) SINCE (λt. ¬a t ∧ b t)) t)) ∧
  (a PBEFORE b = (λt. ((λt. ¬b t) SINCE (λt. a t ∧ ¬b t)) t)) ∧
  (a PSBEFORE b = (λt. ¬((λt. ¬a t) SINCE b) t))

PSWHEN_EXPRESSIVE

⊢ (PALWAYS a = (λt. ¬((λt. T) PSWHEN (λt. ¬a t)) t)) ∧
  (ONCE a = (λt. ((λt. T) PSWHEN a) t)) ∧
  (a PSUNTIL b = (λt. (b PSWHEN (λt. a t ⇒ b t)) t)) ∧
  (a SINCE b = (λt. ¬((λt. ¬b t) PSWHEN (λt. a t ⇒ b t)) t)) ∧
  (a PWHEN b = (λt. ¬((λt. ¬a t) PSWHEN b) t)) ∧
  (a PBEFORE b = (λt. ¬(b PSWHEN (λt. a t ∨ b t)) t)) ∧
  (a PSBEFORE b = (λt. ((λt. ¬b t) PSWHEN (λt. a t ∨ b t)) t))

PSUNTIL_EXPRESSIVE

⊢ (PALWAYS a = (λt. ¬((λt. T) PSUNTIL (λt. ¬a t)) t)) ∧
  (ONCE a = (λt. ((λt. T) PSUNTIL a) t)) ∧
  (a SINCE b = (λt. ¬((λt. ¬b t) PSUNTIL (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a PWHEN b = (λt. ¬((λt. ¬a t ∨ ¬b t) PSUNTIL (λt. ¬a t ∧ b t)) t)) ∧
  (a PBEFORE b = (λt. ¬((λt. ¬a t) PSUNTIL b) t)) ∧
  (a PSWHEN b = (λt. ((λt. ¬b t) PSUNTIL (λt. a t ∧ b t)) t)) ∧
  (a PSBEFORE b = (λt. ((λt. ¬b t) PSUNTIL (λt. a t ∧ ¬b t)) t))

PSBEFORE_EXPRESSIVE

⊢ (PALWAYS a = (λt. ¬((λt. ¬a t) PSBEFORE (λt. F)) t)) ∧
  (ONCE a = (λt. (a PSBEFORE (λt. F)) t)) ∧
  (a PSUNTIL b = (λt. (b PSBEFORE (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a SINCE b = (λt. ¬((λt. ¬a t) PSBEFORE b) t)) ∧
  (a PSWHEN b = (λt. (b PSBEFORE (λt. ¬a t ∧ b t)) t)) ∧
  (a PWHEN b = (λt. ¬(b PSBEFORE (λt. a t ∧ b t)) t)) ∧
  (a PBEFORE b = (λt. ¬(b PSBEFORE (λt. a t ∧ ¬b t)) t))

PRENEX_NEXT_NORMAL_FORM

⊢ (¬NEXT a t ⇔ NEXT (λt. ¬a t) t) ∧
  (a t ∧ NEXT b t ⇔ NEXT (λt. PREV a t ∧ b t) t) ∧
  (a t ∨ NEXT b t ⇔ NEXT (λt. PREV a t ∨ b t) t) ∧
  (ALWAYS (NEXT a) = NEXT (ALWAYS a)) ∧
  (EVENTUAL (NEXT a) = NEXT (EVENTUAL a)) ∧
  (a SUNTIL NEXT b = NEXT (PREV a SUNTIL b)) ∧
  (a SWHEN NEXT b = NEXT (PREV a SWHEN b)) ∧
  (a SBEFORE NEXT b = NEXT (PREV a SBEFORE b)) ∧
  (a UNTIL NEXT b = NEXT (PREV a UNTIL b)) ∧
  (a WHEN NEXT b = NEXT (PREV a WHEN b)) ∧
  (a BEFORE NEXT b = NEXT (PREV a BEFORE b)) ∧
  (NEXT a SUNTIL b = NEXT (a SUNTIL PREV b)) ∧
  (NEXT a SWHEN b = NEXT (a SWHEN PREV b)) ∧
  (NEXT a SBEFORE b = NEXT (a SBEFORE PREV b)) ∧
  (NEXT a UNTIL b = NEXT (a UNTIL PREV b)) ∧
  (NEXT a WHEN b = NEXT (a WHEN PREV b)) ∧
  (NEXT a BEFORE b = NEXT (a BEFORE PREV b)) ∧
  (PREV (NEXT a) = (λt. InitPoint t ∨ a t)) ∧
  (PSNEXT (NEXT a) = (λt. ¬InitPoint t ∧ a t)) ∧
  (PALWAYS (NEXT a) = NEXT (PALWAYS (λt. InitPoint t ∨ a t))) ∧
  (ONCE (NEXT a) = NEXT (ONCE (λt. ¬InitPoint t ∧ a t))) ∧
  (a PSUNTIL NEXT b = NEXT (PREV a PSUNTIL (λt. ¬InitPoint t ∧ b t))) ∧
  (a PSWHEN NEXT b = NEXT (PREV a PSWHEN (λt. ¬InitPoint t ∧ b t))) ∧
  (a PSBEFORE NEXT b = NEXT (PSNEXT a PSBEFORE b)) ∧
  (a SINCE NEXT b = NEXT (PREV a SINCE b)) ∧
  (a PWHEN NEXT b = NEXT (PREV a PWHEN (λt. ¬InitPoint t ∧ b t))) ∧
  (a PBEFORE NEXT b = NEXT (PREV a PBEFORE (λt. ¬InitPoint t ∧ b t))) ∧
  (NEXT a PSUNTIL b = NEXT (a PSUNTIL PSNEXT b)) ∧
  (NEXT a PSWHEN b = NEXT (a PSWHEN PSNEXT b)) ∧
  (NEXT a PSBEFORE b = NEXT ((λt. ¬InitPoint t ∧ a t) PSBEFORE PREV b)) ∧
  (NEXT a SINCE b = NEXT ((λt. InitPoint t ∨ a t) SINCE PREV b)) ∧
  (NEXT a PWHEN b = NEXT (a PWHEN PSNEXT b)) ∧
  (NEXT a PBEFORE b = NEXT (a PBEFORE PSNEXT b))

PNEXT_INWARDS_NORMAL_FORM

⊢ (PREV (λt. ¬a t) = (λt. ¬PSNEXT a t)) ∧
  (PREV (λt. a t ∧ b t) = (λt. PREV a t ∧ PREV b t)) ∧
  (PREV (λt. a t ∨ b t) = (λt. PREV a t ∨ PREV b t)) ∧
  (PREV (NEXT a) = (λt. InitPoint t ∨ a t)) ∧
  (PREV (ALWAYS a) = (λt. InitPoint t ∨ ALWAYS (PREV a) t)) ∧
  (PREV (EVENTUAL a) = (λt. InitPoint t ∨ EVENTUAL (PREV a) t)) ∧
  (PREV (a SUNTIL b) = PREV a SUNTIL PREV b) ∧
  (PREV (a SWHEN b) = PREV a SWHEN PREV b) ∧
  (PREV (a SBEFORE b) = PREV a SBEFORE PSNEXT b) ∧
  (PREV (a UNTIL b) = PREV a UNTIL PREV b) ∧
  (PREV (a WHEN b) = PREV a WHEN PREV b) ∧
  (PREV (a BEFORE b) = PREV a BEFORE PSNEXT b) ∧
  (PREV (PALWAYS a) = PALWAYS (PREV a)) ∧
  (PREV (ONCE a) = (λt. InitPoint t ∨ ONCE (PSNEXT a) t)) ∧
  (PREV (a PSUNTIL b) = (λt. InitPoint t ∨ (PREV a PSUNTIL PSNEXT b) t)) ∧
  (PREV (a PSWHEN b) = (λt. InitPoint t ∨ (PREV a PSWHEN PSNEXT b) t)) ∧
  (PREV (a PSBEFORE b) = (λt. InitPoint t ∨ (PSNEXT a PSBEFORE PREV b) t)) ∧
  (PREV (a SINCE b) = PREV a SINCE PREV b) ∧
  (PREV (a PWHEN b) = PREV a PWHEN PREV b) ∧
  (PREV (a PBEFORE b) = PREV a PBEFORE PSNEXT b)

PBEFORE_EXPRESSIVE

⊢ (PALWAYS a = (λt. ((λt. F) PBEFORE (λt. ¬a t)) t)) ∧
  (ONCE a = (λt. ¬((λt. F) PBEFORE a) t)) ∧
  (a PSUNTIL b = (λt. ¬((λt. ¬a t) PBEFORE b) t)) ∧
  (a SINCE b = (λt. (b PBEFORE (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a PSWHEN b = (λt. ¬(b PBEFORE (λt. a t ∧ b t)) t)) ∧
  (a PWHEN b = (λt. ((λt. a t ∧ b t) PBEFORE (λt. ¬a t ∧ b t)) t)) ∧
  (a PSBEFORE b = (λt. ¬(b PBEFORE (λt. a t ∧ ¬b t)) t))

NO_PAST_EVENT

⊢ PALWAYS (λt. ¬b t) t ⇒
  ((a PWHEN b) t ⇔ T) ∧ ((a SINCE b) t ⇔ PALWAYS a t) ∧
  ((a PBEFORE b) t ⇔ T) ∧ ((b PBEFORE a) t ⇔ PALWAYS (λt. ¬a t) t) ∧
  ((a PSWHEN b) t ⇔ F) ∧ ((a PSUNTIL b) t ⇔ F) ∧
  ((a PSBEFORE b) t ⇔ ONCE a t) ∧ ((b PSBEFORE a) t ⇔ F)

NO_FUTURE_EVENT

⊢ ALWAYS (λt. ¬b t) t0 ⇒
  (∀a. (a WHEN b) t0 ⇔ T) ∧ (∀a. (a UNTIL b) t0 ⇔ ALWAYS a t0) ∧
  (∀a. (a BEFORE b) t0 ⇔ T) ∧ (∀a. (a SWHEN b) t0 ⇔ F) ∧
  (∀a. (a SUNTIL b) t0 ⇔ F) ∧ ∀a. (a SBEFORE b) t0 ⇔ EVENTUAL a t0

NEXT_INWARDS_NORMAL_FORM

⊢ (NEXT (λt. ¬a t) = (λt. ¬NEXT a t)) ∧
  (NEXT (λt. a t ∧ b t) = (λt. NEXT a t ∧ NEXT b t)) ∧
  (NEXT (λt. a t ∨ b t) = (λt. NEXT a t ∨ NEXT b t)) ∧
  (NEXT (ALWAYS a) = ALWAYS (NEXT a)) ∧
  (NEXT (EVENTUAL a) = EVENTUAL (NEXT a)) ∧
  (NEXT (a SUNTIL b) = NEXT a SUNTIL NEXT b) ∧
  (NEXT (a SWHEN b) = NEXT a SWHEN NEXT b) ∧
  (NEXT (a SBEFORE b) = NEXT a SBEFORE NEXT b) ∧
  (NEXT (a UNTIL b) = NEXT a UNTIL NEXT b) ∧
  (NEXT (a WHEN b) = NEXT a WHEN NEXT b) ∧
  (NEXT (a BEFORE b) = NEXT a BEFORE NEXT b) ∧ (NEXT (PREV a) = a) ∧
  (NEXT (PSNEXT a) = a) ∧ (NEXT (PALWAYS a) = (λt. PALWAYS a t ∧ NEXT a t)) ∧
  (NEXT (ONCE a) = (λt. ONCE a t ∨ NEXT a t)) ∧
  (NEXT (a PSUNTIL b) = (λt. NEXT b t ∨ NEXT a t ∧ (a PSUNTIL b) t)) ∧
  (NEXT (a PSWHEN b) = (λt. if NEXT b t then NEXT a t else (a PSWHEN b) t)) ∧
  (NEXT (a PSBEFORE b) = (λt. ¬NEXT b t ∧ (NEXT a t ∨ (a PSBEFORE b) t))) ∧
  (NEXT (a SINCE b) = (λt. NEXT b t ∨ NEXT a t ∧ (a SINCE b) t)) ∧
  (NEXT (a PWHEN b) = (λt. if NEXT b t then NEXT a t else (a PWHEN b) t)) ∧
  (NEXT (a PBEFORE b) = (λt. ¬NEXT b t ∧ (NEXT a t ∨ (a PBEFORE b) t)))

NEGATION_NORMAL_FORM

⊢ (¬NEXT a t ⇔ NEXT (λt. ¬a t) t) ∧ (¬ALWAYS a t ⇔ EVENTUAL (λt. ¬a t) t) ∧
  (¬EVENTUAL a t ⇔ ALWAYS (λt. ¬a t) t) ∧
  (¬(a WHEN b) t ⇔ ((λt. ¬a t) SWHEN b) t) ∧
  (¬(a UNTIL b) t ⇔ ((λt. ¬a t) SBEFORE b) t) ∧
  (¬(a BEFORE b) t ⇔ ((λt. ¬a t) SUNTIL b) t) ∧
  (¬(a SWHEN b) t ⇔ ((λt. ¬a t) WHEN b) t) ∧
  (¬(a SUNTIL b) t ⇔ ((λt. ¬a t) BEFORE b) t) ∧
  (¬(a SBEFORE b) t ⇔ ((λt. ¬a t) UNTIL b) t) ∧
  (¬PREV a t ⇔ PSNEXT (λt. ¬a t) t) ∧ (¬PSNEXT a t ⇔ PREV (λt. ¬a t) t) ∧
  (¬PALWAYS a t ⇔ ONCE (λt. ¬a t) t) ∧ (¬ONCE a t ⇔ PALWAYS (λt. ¬a t) t) ∧
  (¬(a PWHEN b) t ⇔ ((λt. ¬a t) PSWHEN b) t) ∧
  (¬(a SINCE b) t ⇔ ((λt. ¬a t) PSBEFORE b) t) ∧
  (¬(a PBEFORE b) t ⇔ ((λt. ¬a t) PSUNTIL b) t) ∧
  (¬(a PSWHEN b) t ⇔ ((λt. ¬a t) PWHEN b) t) ∧
  (¬(a PSUNTIL b) t ⇔ ((λt. ¬a t) PBEFORE b) t) ∧
  (¬(a PSBEFORE b) t ⇔ ((λt. ¬a t) SINCE b) t)

MORE_EVENT

⊢ (a WHEN b = (λt. a t ∧ b t) WHEN b) ∧
  (a UNTIL b = (λt. a t ∧ ¬b t) UNTIL b) ∧
  (a BEFORE b = (λt. a t ∧ ¬b t) BEFORE b) ∧
  (a SWHEN b = (λt. a t ∧ b t) SWHEN b) ∧
  (a SUNTIL b = (λt. a t ∧ ¬b t) SUNTIL b) ∧
  (a SBEFORE b = (λt. a t ∧ ¬b t) SBEFORE b) ∧
  (a PWHEN b = (λt. a t ∧ b t) PWHEN b) ∧
  (a SINCE b = (λt. a t ∧ ¬b t) SINCE b) ∧
  (a PBEFORE b = (λt. a t ∧ ¬b t) PBEFORE b) ∧
  (a PSWHEN b = (λt. a t ∧ b t) PSWHEN b) ∧
  (a PSUNTIL b = (λt. a t ∧ ¬b t) PSUNTIL b) ∧
  (a PSBEFORE b = (λt. a t ∧ ¬b t) PSBEFORE b)

INITIALISATION

⊢ (PREV a 0 ⇔ T) ∧ (PSNEXT a 0 ⇔ F) ∧ (PALWAYS a 0 ⇔ a 0) ∧ (ONCE a 0 ⇔ a 0) ∧
  ((a PSUNTIL b) 0 ⇔ b 0) ∧ ((a PSWHEN b) 0 ⇔ a 0 ∧ b 0) ∧
  ((a PSBEFORE b) 0 ⇔ a 0 ∧ ¬b 0) ∧ ((a SINCE b) 0 ⇔ a 0 ∨ b 0) ∧
  ((a PWHEN b) 0 ⇔ a 0 ∨ ¬b 0) ∧ ((a PBEFORE b) 0 ⇔ ¬b 0)

IMMEDIATE_EVENT

⊢ b t ⇒
  ((a WHEN b) t ⇔ a t) ∧ ((a UNTIL b) t ⇔ T) ∧ ((a BEFORE b) t ⇔ F) ∧
  ((b BEFORE a) t ⇔ ¬a t) ∧ ((a SWHEN b) t ⇔ a t) ∧ ((a SUNTIL b) t ⇔ T) ∧
  ((a SBEFORE b) t ⇔ F) ∧ ((b SBEFORE a) t ⇔ ¬a t) ∧ ((a PWHEN b) t ⇔ a t) ∧
  ((a SINCE b) t ⇔ T) ∧ ((a PBEFORE b) t ⇔ F) ∧ ((b PBEFORE a) t ⇔ ¬a t) ∧
  ((a PSWHEN b) t ⇔ a t) ∧ ((a PSUNTIL b) t ⇔ T) ∧ ((a PSBEFORE b) t ⇔ F) ∧
  ((b PSBEFORE a) t ⇔ ¬a t)

FIXPOINTS

⊢ ((y = (λt. a t ∧ NEXT y t)) ⇔ (y = ALWAYS a) ∨ (y = (λt. F))) ∧
  ((y = (λt. a t ∨ NEXT y t)) ⇔ (y = EVENTUAL a) ∨ (y = (λt. T))) ∧
  ((y = (λt. ¬b t ⇒ a t ∧ NEXT y t)) ⇔ (y = a UNTIL b) ∨ (y = a SUNTIL b)) ∧
  ((y = (λt. if b t then a t else NEXT y t)) ⇔
   (y = a WHEN b) ∨ (y = a SWHEN b)) ∧
  ((y = (λt. ¬b t ∧ (a t ∨ NEXT y t))) ⇔ (y = a BEFORE b) ∨ (y = a SBEFORE b)) ∧
  ((y = (λt. a t ∧ PREV y t)) ⇔ (y = PALWAYS a)) ∧
  ((y = (λt. a t ∨ PSNEXT y t)) ⇔ (y = ONCE a)) ∧
  ((y = (λt. b t ∨ a t ∧ PSNEXT y t)) ⇔ (y = a PSUNTIL b)) ∧
  ((y = (λt. a t ∧ b t ∨ ¬b t ∧ PSNEXT y t)) ⇔ (y = a PSWHEN b)) ∧
  ((y = (λt. ¬b t ∧ (a t ∨ PSNEXT y t))) ⇔ (y = a PSBEFORE b)) ∧
  ((y = (λt. b t ∨ a t ∧ PREV y t)) ⇔ (y = a SINCE b)) ∧
  ((y = (λt. a t ∧ b t ∨ ¬b t ∧ PREV y t)) ⇔ (y = a PWHEN b)) ∧
  ((y = (λt. ¬b t ∧ (a t ∨ PREV y t))) ⇔ (y = a PBEFORE b))

DISJUNCTIVE_NORMAL_FORM

⊢ (NEXT (λt. a t ∨ b t) = (λt. NEXT a t ∨ NEXT b t)) ∧
  (EVENTUAL (λt. a t ∨ b t) = (λt. EVENTUAL a t ∨ EVENTUAL b t)) ∧
  ((λt. a t ∨ b t) WHEN c = (λt. (a WHEN c) t ∨ (b WHEN c) t)) ∧
  ((λt. a t ∨ b t) SWHEN c = (λt. (a SWHEN c) t ∨ (b SWHEN c) t)) ∧
  (a UNTIL (λt. b t ∨ c t) = (λt. (a UNTIL b) t ∨ (a UNTIL c) t)) ∧
  (a SUNTIL (λt. b t ∨ c t) = (λt. (a SUNTIL b) t ∨ (a SUNTIL c) t)) ∧
  ((λt. a t ∨ b t) BEFORE c = (λt. (a BEFORE c) t ∨ (b BEFORE c) t)) ∧
  ((λt. a t ∨ b t) SBEFORE c = (λt. (a SBEFORE c) t ∨ (b SBEFORE c) t)) ∧
  (PREV (λt. a t ∨ b t) = (λt. PREV a t ∨ PREV b t)) ∧
  (ONCE (λt. a t ∨ b t) = (λt. ONCE a t ∨ ONCE b t)) ∧
  ((λt. a t ∨ b t) PWHEN c = (λt. (a PWHEN c) t ∨ (b PWHEN c) t)) ∧
  ((λt. a t ∨ b t) PSWHEN c = (λt. (a PSWHEN c) t ∨ (b PSWHEN c) t)) ∧
  (a SINCE (λt. b t ∨ c t) = (λt. (a SINCE b) t ∨ (a SINCE c) t)) ∧
  (a PSUNTIL (λt. b t ∨ c t) = (λt. (a PSUNTIL b) t ∨ (a PSUNTIL c) t)) ∧
  ((λt. a t ∨ b t) PBEFORE c = (λt. (a PBEFORE c) t ∨ (b PBEFORE c) t)) ∧
  ((λt. a t ∨ b t) PSBEFORE c = (λt. (a PSBEFORE c) t ∨ (b PSBEFORE c) t))

CONJUNCTIVE_NORMAL_FORM

⊢ (NEXT (λt. a t ∧ b t) = (λt. NEXT a t ∧ NEXT b t)) ∧
  (ALWAYS (λt. a t ∧ b t) = (λt. ALWAYS a t ∧ ALWAYS b t)) ∧
  ((λt. a t ∧ b t) WHEN c = (λt. (a WHEN c) t ∧ (b WHEN c) t)) ∧
  ((λt. a t ∧ b t) SWHEN c = (λt. (a SWHEN c) t ∧ (b SWHEN c) t)) ∧
  ((λt. a t ∧ b t) UNTIL c = (λt. (a UNTIL c) t ∧ (b UNTIL c) t)) ∧
  ((λt. a t ∧ b t) SUNTIL c = (λt. (a SUNTIL c) t ∧ (b SUNTIL c) t)) ∧
  (c BEFORE (λt. a t ∨ b t) = (λt. (c BEFORE a) t ∧ (c BEFORE b) t)) ∧
  (c SBEFORE (λt. a t ∨ b t) = (λt. (c SBEFORE a) t ∧ (c SBEFORE b) t)) ∧
  (PREV (λt. a t ∧ b t) = (λt. PREV a t ∧ PREV b t)) ∧
  (PSNEXT (λt. a t ∧ b t) = (λt. PSNEXT a t ∧ PSNEXT b t)) ∧
  (PALWAYS (λt. a t ∧ b t) = (λt. PALWAYS a t ∧ PALWAYS b t)) ∧
  ((λt. a t ∧ b t) PWHEN c = (λt. (a PWHEN c) t ∧ (b PWHEN c) t)) ∧
  ((λt. a t ∧ b t) PSWHEN c = (λt. (a PSWHEN c) t ∧ (b PSWHEN c) t)) ∧
  ((λt. a t ∧ b t) SINCE c = (λt. (a SINCE c) t ∧ (b SINCE c) t)) ∧
  ((λt. a t ∧ b t) PSUNTIL c = (λt. (a PSUNTIL c) t ∧ (b PSUNTIL c) t)) ∧
  (c PBEFORE (λt. a t ∨ b t) = (λt. (c PBEFORE a) t ∧ (c PBEFORE b) t)) ∧
  (c PSBEFORE (λt. a t ∨ b t) = (λt. (c PSBEFORE a) t ∧ (c PSBEFORE b) t))

BEFORE_EXPRESSIVE

⊢ (ALWAYS a = (λt. ((λt. F) BEFORE (λt. ¬a t)) t)) ∧
  (EVENTUAL a = (λt. ¬((λt. F) BEFORE a) t)) ∧
  (a SUNTIL b = (λt. ¬((λt. ¬a t) BEFORE b) t)) ∧
  (a UNTIL b = (λt. (b BEFORE (λt. ¬a t ∧ ¬b t)) t)) ∧
  (a SWHEN b = (λt. ¬(b BEFORE (λt. a t ∧ b t)) t)) ∧
  (a WHEN b = (λt. ((λt. a t ∧ b t) BEFORE (λt. ¬a t ∧ b t)) t)) ∧
  (a SBEFORE b = (λt. ¬(b BEFORE (λt. a t ∧ ¬b t)) t))