Theory "Temporal_Logic"

Parents list

Signature

Constant	Type
ALWAYS	:(num -> bool) -> num -> bool
BEFORE	:(num -> bool) -> (num -> bool) -> num -> bool
EVENTUAL	:(num -> bool) -> num -> bool
NEXT	:(num -> bool) -> num -> bool
SBEFORE	:(num -> bool) -> (num -> bool) -> num -> bool
SUNTIL	:(num -> bool) -> (num -> bool) -> num -> bool
SWHEN	:(num -> bool) -> (num -> bool) -> num -> bool
UNTIL	:(num -> bool) -> (num -> bool) -> num -> bool
UPTO	:num # num # (num -> bool) -> bool
WATCH	:(num -> bool) -> (num -> bool) -> num -> bool
WHEN	:(num -> bool) -> (num -> bool) -> num -> bool

Definitions

NEXT

|- ∀P. NEXT P = (λt. P (SUC t))

ALWAYS

|- ∀P t0. ALWAYS P t0 ⇔ ∀t. P (t + t0)

EVENTUAL

|- ∀P t0. EVENTUAL P t0 ⇔ ∃t. P (t + t0)

WATCH

|- ∀q b t0.
     (q WATCH b) t0 ⇔
     ∀t. (q t0 ⇔ F) ∧ (q (SUC (t + t0)) ⇔ q (t + t0) ∨ b (t + t0))

UPTO

|- ∀t0 t1 a. UPTO (t0,t1,a) ⇔ ∀t2. t0 ≤ t2 ∧ t2 < t1 ⇒ a t2

WHEN

|- ∀a b t0.
     (a WHEN b) t0 ⇔
     ∃q. (q WATCH b) t0 ∧ ∀t. q (t + t0) ∨ (b (t + t0) ⇒ a (t + t0))

UNTIL

|- ∀a b t0.
     (a UNTIL b) t0 ⇔
     ∃q. (q WATCH b) t0 ∧ ∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)

BEFORE

|- ∀a b t0.
     (a BEFORE b) t0 ⇔
     ∃q.
       (q WATCH b) t0 ∧
       ((∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)) ∨ ∀t. ¬b (t + t0))

SWHEN

|- ∀a b t0.
     (a SWHEN b) t0 ⇔
     ∃q. (q WATCH b) t0 ∧ ∃t. ¬q (t + t0) ∧ b (t + t0) ∧ a (t + t0)

SUNTIL

|- ∀a b t0.
     (a SUNTIL b) t0 ⇔
     ∃q.
       (q WATCH b) t0 ∧ (∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)) ∧
       ∃t. b (t + t0)

SBEFORE

|- ∀a b t0.
     (a SBEFORE b) t0 ⇔
     ∃q. (q WATCH b) t0 ∧ ∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)

Theorems

WATCH_EXISTS

|- ∀b t0. ∃q. (q WATCH b) t0

WELL_ORDER

|- (∃n. P n) ⇔ ∃m. P m ∧ ∀n. n < m ⇒ ¬P n

WELL_ORDER_UNIQUE

|- ∀m2 m1 P.
     (P m1 ∧ ∀n. n < m1 ⇒ ¬P n) ∧ P m2 ∧ (∀n. n < m2 ⇒ ¬P n) ⇒ (m1 = m2)

DELTA_CASES

|- (∃d. (∀t. t < d ⇒ ¬b (t + t0)) ∧ b (d + t0)) ∨ ∀d. ¬b (d + t0)

WHEN_IMP

|- (a WHEN b) t0 ⇔
   ∀q. (q WATCH b) t0 ⇒ ∀t. q (t + t0) ∨ (b (t + t0) ⇒ a (t + t0))

UNTIL_IMP

|- (a UNTIL b) t0 ⇔
   ∀q. (q WATCH b) t0 ⇒ ∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)

BEFORE_IMP

|- (a BEFORE b) t0 ⇔
   ∀q.
     (q WATCH b) t0 ⇒
     (∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)) ∨ ∀t. ¬b (t + t0)

SWHEN_IMP

|- (a SWHEN b) t0 ⇔
   ∀q. (q WATCH b) t0 ⇒ ∃t. ¬q (t + t0) ∧ b (t + t0) ∧ a (t + t0)

SUNTIL_IMP

|- (a SUNTIL b) t0 ⇔
   ∀q.
     (q WATCH b) t0 ⇒
     (∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)) ∧ ∃t. b (t + t0)

SBEFORE_IMP

|- (a SBEFORE b) t0 ⇔
   ∀q. (q WATCH b) t0 ⇒ ∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)

ALWAYS_SIGNAL

|- ALWAYS a t0 ⇔ ∀t. a (t + t0)

EVENTUAL_SIGNAL

|- EVENTUAL a t0 ⇔ ∃t. a (t + t0)

WATCH_SIGNAL

|- (q WATCH b) t0 ⇔
   ((∀t. ¬b (t + t0)) ⇒ ∀t. ¬q (t + t0)) ∧
   ∀d.
     b (d + t0) ∧ (∀t. t < d ⇒ ¬b (t + t0)) ⇒
     (∀t. t ≤ d ⇒ ¬q (t + t0)) ∧ ∀t. q (SUC (t + (d + t0)))

WHEN_SIGNAL

|- (a WHEN b) t0 ⇔
   ∀delta. (∀t. t < delta ⇒ ¬b (t + t0)) ∧ b (delta + t0) ⇒ a (delta + t0)

UNTIL_SIGNAL

|- (a UNTIL b) t0 ⇔
   ((∀t. ¬b (t + t0)) ⇒ ∀t. a (t + t0)) ∧
   ∀d. (∀t. t < d ⇒ ¬b (t + t0)) ∧ b (d + t0) ⇒ ∀t. t < d ⇒ a (t + t0)

BEFORE_SIGNAL

|- (a BEFORE b) t0 ⇔
   ∀delta.
     (∀t. t < delta ⇒ ¬b (t + t0)) ∧ b (delta + t0) ⇒
     ∃t. t < delta ∧ a (t + t0)

SWHEN_SIGNAL

|- (a SWHEN b) t0 ⇔
   ∃delta. (∀t. t < delta ⇒ ¬b (t + t0)) ∧ b (delta + t0) ∧ a (delta + t0)

SUNTIL_SIGNAL

|- (a SUNTIL b) t0 ⇔
   ∃delta. (∀t. t < delta ⇒ a (t + t0) ∧ ¬b (t + t0)) ∧ b (delta + t0)

SBEFORE_SIGNAL

|- (a SBEFORE b) t0 ⇔ ∃delta. a (delta + t0) ∧ ∀t. t ≤ delta ⇒ ¬b (t + t0)

NEXT_LINORD

|- NEXT a t0 ⇔ ∃t1. t0 < t1 ∧ (∀t3. t0 < t3 ⇒ t1 ≤ t3) ∧ a t1

ALWAYS_LINORD

|- ALWAYS a t0 ⇔ ∀t1. t0 ≤ t1 ⇒ a t1

EVENTUAL_LINORD

|- EVENTUAL a t0 ⇔ ∃t1. t0 ≤ t1 ∧ a t1

SUNTIL_LINORD

|- (a SUNTIL b) t0 ⇔ ∃t1. t0 ≤ t1 ∧ b t1 ∧ UPTO (t0,t1,a)

UNTIL_LINORD

|- (a UNTIL b) t0 ⇔ ∀t1. t0 ≤ t1 ∧ ¬b t1 ∧ UPTO (t0,t1,(λt. ¬b t)) ⇒ a t1

SBEFORE_LINORD

|- (a SBEFORE b) t0 ⇔ ∃t1. t0 ≤ t1 ∧ a t1 ∧ ¬b t1 ∧ UPTO (t0,t1,(λt. ¬b t))

BEFORE_LINORD

|- (a BEFORE b) t0 ⇔ ∀t1. t0 ≤ t1 ∧ UPTO (t0,t1,(λt. ¬a t)) ⇒ ¬b t1

SWHEN_LINORD

|- (a SWHEN b) t0 ⇔ ∃t1. t0 ≤ t1 ∧ a t1 ∧ b t1 ∧ UPTO (t0,t1,(λt. ¬b t))

WHEN_LINORD

|- (a WHEN b) t0 ⇔ ∀t1. t0 ≤ t1 ∧ b t1 ∧ UPTO (t0,t1,(λt. ¬b t)) ⇒ a t1

ALWAYS_AS_WHEN

|- ALWAYS a = (λt. F) WHEN (λt. ¬a t)

EVENTUAL_AS_WHEN

|- EVENTUAL a = (λt. ¬((λt. F) WHEN a) t)

UNTIL_AS_WHEN

|- a UNTIL b = b WHEN (λt. a t ⇒ b t)

BEFORE_AS_WHEN

|- a BEFORE b = (λt. ¬b t) WHEN (λt. a t ∨ b t)

SWHEN_AS_WHEN

|- a SWHEN b = (λt0. (a WHEN b) t0 ∧ EVENTUAL b t0)

SWHEN_AS_NOT_WHEN

|- (a SWHEN b) t0 ⇔ ¬((λt. ¬a t) WHEN b) t0

SUNTIL_AS_WHEN

|- a SUNTIL b = (λt. (b WHEN (λt. a t ⇒ b t)) t ∧ EVENTUAL b t)

SBEFORE_AS_WHEN

|- a SBEFORE b = (λt0. ((λt. ¬b t) WHEN (λt. a t ∨ b t)) t0 ∧ EVENTUAL a t0)

BEFORE_AS_WHEN_UNTIL

|- a BEFORE b = (λt. ((λt. ¬b t) UNTIL a) t ∧ ((λt. ¬b t) WHEN a) t)

BEFORE_HW

|- (a BEFORE b) t0 ⇔ ∃q. (q WATCH a) t0 ∧ ∀t. q (t + t0) ∨ ¬b (t + t0)

ALWAYS_AS_UNTIL

|- ALWAYS a = a UNTIL (λt. F)

EVENTUAL_AS_UNTIL

|- EVENTUAL a = (λt. ¬((λt. ¬a t) UNTIL (λt. F)) t)

WHEN_AS_UNTIL

|- a WHEN b = (λt. ¬b t) UNTIL (λt. a t ∧ b t)

BEFORE_AS_UNTIL

|- a BEFORE b = (λt0. ¬((λt. ¬a t) UNTIL b) t0 ∨ ALWAYS (λt. ¬b t) t0)

SWHEN_AS_UNTIL

|- a SWHEN b = (λt. ((λt. ¬b t) UNTIL (λt. a t ∧ b t)) t ∧ EVENTUAL b t)

SUNTIL_AS_UNTIL

|- a SUNTIL b = (λt0. (a UNTIL b) t0 ∧ EVENTUAL b t0)

SBEFORE_AS_UNTIL

|- a SBEFORE b = (λt0. ¬((λt. ¬a t) UNTIL b) t0)

ALWAYS_AS_BEFORE

|- ALWAYS b = (λt. F) BEFORE (λt. ¬b t)

EVENTUAL_AS_BEFORE

|- EVENTUAL b = (λt0. ¬((λt. F) BEFORE b) t0)

WHEN_AS_BEFORE

|- a WHEN b = (λt0. ¬(b BEFORE (λt. a t ∧ b t)) t0 ∨ ALWAYS (λt. ¬b t) t0)

UNTIL_AS_BEFORE

|- a UNTIL b = (λt0. ¬((λt. ¬a t) BEFORE b) t0 ∨ ALWAYS a t0)

SWHEN_AS_BEFORE

|- a SWHEN b = (λt0. ¬(b BEFORE (λt. a t ∧ b t)) t0)

SUNTIL_AS_BEFORE

|- a SUNTIL b = (λt0. ¬((λt. ¬a t) BEFORE b) t0)

SBEFORE_AS_BEFORE

|- a SBEFORE b = (λt0. (a BEFORE b) t0 ∧ EVENTUAL a t0)

WHEN_SWHEN_LEMMA

|- if ∀t1. ∃t2. b (t2 + t1) then ∀t0. (a WHEN b) t0 ⇔ (a SWHEN b) t0
   else ∃t1. ∀t2. (a WHEN b) (t2 + t1) ∧ ¬(a SWHEN b) (t2 + t1)

ALWAYS_AS_SWHEN

|- ALWAYS a = (λt. ¬((λt. T) SWHEN (λt. ¬a t)) t)

EVENTUAL_AS_SWHEN

|- EVENTUAL a = (λt. T) SWHEN a

WHEN_AS_SWHEN

|- a WHEN b = (λt. (a SWHEN b) t ∨ ALWAYS (λt. ¬b t) t)

WHEN_AS_NOT_SWHEN

|- (a WHEN b) t0 ⇔ ¬((λt. ¬a t) SWHEN b) t0

UNTIL_AS_SWHEN

|- a UNTIL b = (λt. (b SWHEN (λt. a t ⇒ b t)) t ∨ ALWAYS a t)

BEFORE_AS_SWHEN

|- a BEFORE b =
   (λt0. ((λt. ¬b t) SWHEN (λt. a t ∨ b t)) t0 ∨ ALWAYS (λt. ¬a t ∧ ¬b t) t0)

BEFORE_AS_NOT_SWHEN

|- a BEFORE b = (λt0. ¬(b SWHEN (λt. a t ∨ b t)) t0)

SUNTIL_AS_SWHEN

|- a SUNTIL b = b SWHEN (λt. a t ⇒ b t)

SBEFORE_AS_SWHEN

|- a SBEFORE b = (λt. ¬b t) SWHEN (λt. a t ∨ b t)

ALWAYS_AS_SUNTIL

|- ALWAYS a = (λt. ¬((λt. T) SUNTIL (λt. ¬a t)) t)

EVENTUAL_AS_SUNTIL

|- EVENTUAL a = (λt. T) SUNTIL a

WHEN_AS_SUNTIL

|- a WHEN b =
   (λt. ((λt. ¬b t) SUNTIL (λt. a t ∧ b t)) t ∨ ALWAYS (λt. ¬b t) t)

UNTIL_AS_SUNTIL

|- a UNTIL b = (λt. (a SUNTIL b) t ∨ ALWAYS a t)

BEFORE_AS_SUNTIL

|- a BEFORE b = (λt. ¬((λt. ¬a t) SUNTIL b) t)

SWHEN_AS_SUNTIL

|- a SWHEN b = (λt. ¬b t) SUNTIL (λt. a t ∧ b t)

SBEFORE_AS_SUNTIL

|- a SBEFORE b = (λt0. ¬((λt. ¬a t) SUNTIL b) t0 ∧ EVENTUAL a t0)

ALWAYS_AS_SBEFORE

|- ALWAYS b = (λt0. ¬((λt. ¬b t) SBEFORE (λt. F)) t0)

EVENTUAL_AS_SBEFORE

|- EVENTUAL b = b SBEFORE (λt. F)

WHEN_AS_SBEFORE

|- a WHEN b = (λt0. (b SBEFORE (λt. ¬a t ∧ b t)) t0 ∨ ALWAYS (λt. ¬b t) t0)

UNTIL_AS_SBEFORE

|- a UNTIL b = (λt0. ¬((λt. ¬a t) SBEFORE b) t0)

SWHEN_AS_SBEFORE

|- a SWHEN b = b SBEFORE (λt. ¬a t ∧ b t)

SUNTIL_AS_SBEFORE

|- a SUNTIL b = (λt0. ¬((λt. ¬a t) SBEFORE b) t0 ∧ EVENTUAL b t0)

BEFORE_AS_SBEFORE

|- a BEFORE b = (λt0. (a SBEFORE b) t0 ∨ ALWAYS (λt. ¬b t) t0)

WHEN_SIMP

|- ((λt. F) WHEN b = ALWAYS (λt. ¬b t)) ∧ ((λt. T) WHEN b = (λt. T)) ∧
   (a WHEN (λt. F) = (λt. T)) ∧ (a WHEN (λt. T) = (λt. a t)) ∧
   (a WHEN a = (λt. T))

UNTIL_SIMP

|- ((λt. F) UNTIL b = (λt. b t)) ∧ ((λt. T) UNTIL b = (λt. T)) ∧
   (a UNTIL (λt. F) = ALWAYS a) ∧ (a UNTIL (λt. T) = (λt. T)) ∧
   (a UNTIL a = (λt. a t))

BEFORE_SIMP

|- ((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))

SWHEN_SIMP

|- ((λt. F) SWHEN b = (λt. F)) ∧ ((λt. T) SWHEN b = EVENTUAL b) ∧
   (a SWHEN (λt. F) = (λt. F)) ∧ (a SWHEN (λt. T) = (λt. a t)) ∧
   (a SWHEN a = EVENTUAL a)

SUNTIL_SIMP

|- ((λt. F) SUNTIL b = (λt. b t)) ∧ ((λt. T) SUNTIL b = EVENTUAL b) ∧
   (a SUNTIL (λt. F) = (λt. F)) ∧ (a SUNTIL (λt. T) = (λt. T)) ∧
   (a SUNTIL a = (λt. a t))

SBEFORE_SIMP

|- ((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))

WHEN_EVENT

|- a WHEN b = (λt. a t ∧ b t) WHEN b

UNTIL_EVENT

|- a UNTIL b = (λt. a t ∧ ¬b t) UNTIL b

BEFORE_EVENT

|- a BEFORE b = (λt. a t ∧ ¬b t) BEFORE b

SWHEN_EVENT

|- a SWHEN b = (λt. a t ∧ b t) SWHEN b

SUNTIL_EVENT

|- a SUNTIL b = (λt. a t ∧ ¬b t) SUNTIL b

SBEFORE_EVENT

|- a SBEFORE b = (λt. a t ∧ ¬b t) SBEFORE b

IMMEDIATE_EVENT

|- b t0 ⇒
   (∀a. (a WHEN b) t0 ⇔ a t0) ∧ (∀a. (a UNTIL b) t0 ⇔ T) ∧
   (∀a. (a BEFORE b) t0 ⇔ F) ∧ (∀a. (a SWHEN b) t0 ⇔ a t0) ∧
   (∀a. (a SUNTIL b) t0 ⇔ T) ∧ ∀a. (a SBEFORE b) t0 ⇔ F

NO_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

SOME_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)

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)

NOT_NEXT

|- ∀P. NEXT (λt. ¬P t) = (λt. ¬NEXT P t)

AND_NEXT

|- ∀Q P. NEXT (λt. P t ∧ Q t) = (λt. NEXT P t ∧ NEXT Q t)

OR_NEXT

|- ∀Q P. NEXT (λt. P t ∨ Q t) = (λt. NEXT P t ∨ NEXT Q t)

IMP_NEXT

|- ∀Q P. NEXT (λt. P t ⇒ Q t) = (λt. NEXT P t ⇒ NEXT Q t)

EQUIV_NEXT

|- ∀Q P. NEXT (λt. P t ⇔ Q t) = (λt. NEXT P t ⇔ NEXT Q t)

ALWAYS_NEXT

|- ∀a. NEXT (ALWAYS a) = ALWAYS (NEXT a)

EVENTUAL_NEXT

|- ∀a. NEXT (EVENTUAL a) = EVENTUAL (NEXT a)

WHEN_NEXT

|- ∀a b. NEXT (a WHEN b) = NEXT a WHEN NEXT b

UNTIL_NEXT

|- ∀a b. NEXT (a UNTIL b) = NEXT a UNTIL NEXT b

BEFORE_NEXT

|- ∀a b. NEXT (a BEFORE b) = NEXT a BEFORE NEXT b

SWHEN_NEXT

|- ∀a b. NEXT (a SWHEN b) = NEXT a SWHEN NEXT b

SUNTIL_NEXT

|- ∀a b. NEXT (a SUNTIL b) = NEXT a SUNTIL NEXT b

SBEFORE_NEXT

|- ∀a b. NEXT (a SBEFORE b) = NEXT a SBEFORE NEXT b

ALWAYS_REC

|- ALWAYS P t0 ⇔ P t0 ∧ NEXT (ALWAYS P) t0

EVENTUAL_REC

|- EVENTUAL P t0 ⇔ P t0 ∨ NEXT (EVENTUAL P) t0

WATCH_REC

|- (q WATCH b) t0 ⇔
   ¬q t0 ∧ if b t0 then NEXT (ALWAYS q) t0 else NEXT (q WATCH b) t0

WHEN_REC

|- (a WHEN b) t0 ⇔ if b t0 then a t0 else NEXT (a WHEN b) t0

UNTIL_REC

|- (a UNTIL b) t0 ⇔ ¬b t0 ⇒ a t0 ∧ NEXT (a UNTIL b) t0

BEFORE_REC

|- (a BEFORE b) t0 ⇔ ¬b t0 ∧ (a t0 ∨ NEXT (a BEFORE b) t0)

SWHEN_REC

|- (a SWHEN b) t0 ⇔ if b t0 then a t0 else NEXT (a SWHEN b) t0

SUNTIL_REC

|- (a SUNTIL b) t0 ⇔ ¬b t0 ⇒ a t0 ∧ NEXT (a SUNTIL b) t0

SBEFORE_REC

|- (a SBEFORE b) t0 ⇔ ¬b t0 ∧ (a t0 ∨ NEXT (a SBEFORE b) t0)

ALWAYS_FIX

|- (y = (λt. a t ∧ y (t + 1))) ⇔ (y = ALWAYS a) ∨ (y = (λt. F))

EVENTUAL_FIX

|- (y = (λt. a t ∨ y (t + 1))) ⇔ (y = EVENTUAL a) ∨ (y = (λt. T))

WHEN_FIX

|- (y = (λt. if b t then a t else y (t + 1))) ⇔
   (y = a WHEN b) ∨ (y = a SWHEN b)

UNTIL_FIX

|- (y = (λt. ¬b t ⇒ a t ∧ y (t + 1))) ⇔ (y = a UNTIL b) ∨ (y = a SUNTIL b)

BEFORE_FIX

|- ∀y.
     (y = (λt. ¬b t ∧ (a t ∨ y (t + 1)))) ⇔
     (y = a BEFORE b) ∨ (y = a SBEFORE b)

WHEN_INVARIANT

|- (a WHEN b) t0 ⇔
   ∃J.
     J t0 ∧ (∀t. ¬b (t + t0) ∧ J (t + t0) ⇒ J (SUC (t + t0))) ∧
     ∀d. b (d + t0) ∧ J (d + t0) ⇒ a (d + t0)

UNTIL_INVARIANT

|- ∀t0.
     (a UNTIL b) t0 ⇔
     ∃J. J t0 ∧ ∀t. J (t + t0) ∧ ¬b (t + t0) ⇒ a (t + t0) ∧ J (SUC (t + t0))

BEFORE_INVARIANT

|- (a BEFORE b) t0 ⇔
   ∃J.
     J t0 ∧ (∀t. J (t + t0) ∧ ¬a (t + t0) ⇒ J (SUC (t + t0))) ∧
     ∀d. J (d + t0) ⇒ ¬b (d + t0)

ALWAYS_INVARIANT

|- ALWAYS a t0 ⇔ ∃J. J t0 ∧ ∀t. J (t + t0) ⇒ a (t + t0) ∧ J (t + (t0 + 1))

EVENTUAL_INVARIANT

|- EVENTUAL b t0 ⇔
   ∃J.
     0 < J t0 ∧ (∀t. J (SUC (t + t0)) < J (t + t0) ∨ (J (SUC (t + t0)) = 0)) ∧
     ∀t. 0 < J (t + t0) ∧ (J (SUC (t + t0)) = 0) ⇒ b (t + t0)

SWHEN_INVARIANT

|- (a SWHEN b) t0 ⇔
   (∃J1.
      J1 t0 ∧ (∀t. ¬b (t + t0) ∧ J1 (t + t0) ⇒ J1 (SUC (t + t0))) ∧
      ∀d. b (d + t0) ∧ J1 (d + t0) ⇒ a (d + t0)) ∧
   ∃J2.
     0 < J2 t0 ∧
     (∀t. J2 (SUC (t + t0)) < J2 (t + t0) ∨ (J2 (SUC (t + t0)) = 0)) ∧
     ∀t. 0 < J2 (t + t0) ∧ (J2 (SUC (t + t0)) = 0) ⇒ b (t + t0)

SUNTIL_INVARIANT

|- (a SUNTIL b) t0 ⇔
   (∃J1.
      J1 t0 ∧
      ∀t. J1 (t + t0) ∧ ¬b (t + t0) ⇒ a (t + t0) ∧ J1 (SUC (t + t0))) ∧
   ∃J2.
     0 < J2 t0 ∧
     (∀t. J2 (SUC (t + t0)) < J2 (t + t0) ∨ (J2 (SUC (t + t0)) = 0)) ∧
     ∀t. 0 < J2 (t + t0) ∧ (J2 (SUC (t + t0)) = 0) ⇒ b (t + t0)

SBEFORE_INVARIANT

|- (a SBEFORE b) t0 ⇔
   (∃J1.
      J1 t0 ∧ (∀t. J1 (t + t0) ∧ ¬a (t + t0) ⇒ J1 (SUC (t + t0))) ∧
      ∀d. J1 (d + t0) ⇒ ¬b (d + t0)) ∧
   ∃J2.
     0 < J2 t0 ∧
     (∀t. J2 (SUC (t + t0)) < J2 (t + t0) ∨ (J2 (SUC (t + t0)) = 0)) ∧
     ∀t. 0 < J2 (t + t0) ∧ (J2 (SUC (t + t0)) = 0) ⇒ a (t + t0)

ALWAYS_IDEM

|- ALWAYS a = ALWAYS (ALWAYS a)

EVENTUAL_IDEM

|- EVENTUAL a = EVENTUAL (EVENTUAL a)

WHEN_IDEM

|- a WHEN b = (a WHEN b) WHEN b

UNTIL_IDEM

|- a UNTIL b = (a UNTIL b) UNTIL b

BEFORE_IDEM

|- a BEFORE b = (a BEFORE b) BEFORE b

SWHEN_IDEM

|- a SWHEN b = (a SWHEN b) SWHEN b

SUNTIL_IDEM

|- a SUNTIL b = (a SUNTIL b) SUNTIL b

SBEFORE_IDEM

|- a SBEFORE b = (a SBEFORE b) SBEFORE b

NOT_ALWAYS

|- ¬ALWAYS a t0 ⇔ EVENTUAL (λt. ¬a t) t0

NOT_EVENTUAL

|- ¬EVENTUAL a t0 ⇔ ALWAYS (λt. ¬a t) t0

NOT_WHEN

|- ¬(a WHEN b) t0 ⇔ ((λt. ¬a t) SWHEN b) t0

NOT_UNTIL

|- ¬(a UNTIL b) t0 ⇔ ((λt. ¬a t) SBEFORE b) t0

NOT_BEFORE

|- ¬(a BEFORE b) t0 ⇔ ((λt. ¬a t) SUNTIL b) t0

NOT_SWHEN

|- ¬(a SWHEN b) t0 ⇔ ((λt. ¬a t) WHEN b) t0

NOT_SUNTIL

|- ¬(a SUNTIL b) t0 ⇔ ((λt. ¬a t) BEFORE b) t0

NOT_SBEFORE

|- ¬(a SBEFORE b) t0 ⇔ ((λt. ¬a t) UNTIL b) t0