Structure Temporal_LogicTheory
signature Temporal_LogicTheory =
sig
type thm = Thm.thm
(* Definitions *)
val ALWAYS : thm
val BEFORE : thm
val EVENTUAL : thm
val NEXT : thm
val SBEFORE : thm
val SUNTIL : thm
val SWHEN : thm
val UNTIL : thm
val UPTO : thm
val WATCH : thm
val WHEN : thm
(* Theorems *)
val ALWAYS_AS_BEFORE : thm
val ALWAYS_AS_SBEFORE : thm
val ALWAYS_AS_SUNTIL : thm
val ALWAYS_AS_SWHEN : thm
val ALWAYS_AS_UNTIL : thm
val ALWAYS_AS_WHEN : thm
val ALWAYS_FIX : thm
val ALWAYS_IDEM : thm
val ALWAYS_INVARIANT : thm
val ALWAYS_LINORD : thm
val ALWAYS_NEXT : thm
val ALWAYS_REC : thm
val ALWAYS_SIGNAL : thm
val AND_NEXT : thm
val BEFORE_AS_NOT_SWHEN : thm
val BEFORE_AS_SBEFORE : thm
val BEFORE_AS_SUNTIL : thm
val BEFORE_AS_SWHEN : thm
val BEFORE_AS_UNTIL : thm
val BEFORE_AS_WHEN : thm
val BEFORE_AS_WHEN_UNTIL : thm
val BEFORE_EVENT : thm
val BEFORE_FIX : thm
val BEFORE_HW : thm
val BEFORE_IDEM : thm
val BEFORE_IMP : thm
val BEFORE_INVARIANT : thm
val BEFORE_LINORD : thm
val BEFORE_NEXT : thm
val BEFORE_REC : thm
val BEFORE_SIGNAL : thm
val BEFORE_SIMP : thm
val DELTA_CASES : thm
val EQUIV_NEXT : thm
val EVENTUAL_AS_BEFORE : thm
val EVENTUAL_AS_SBEFORE : thm
val EVENTUAL_AS_SUNTIL : thm
val EVENTUAL_AS_SWHEN : thm
val EVENTUAL_AS_UNTIL : thm
val EVENTUAL_AS_WHEN : thm
val EVENTUAL_FIX : thm
val EVENTUAL_IDEM : thm
val EVENTUAL_INVARIANT : thm
val EVENTUAL_LINORD : thm
val EVENTUAL_NEXT : thm
val EVENTUAL_REC : thm
val EVENTUAL_SIGNAL : thm
val IMMEDIATE_EVENT : thm
val IMP_NEXT : thm
val MORE_EVENT : thm
val NEXT_LINORD : thm
val NOT_ALWAYS : thm
val NOT_BEFORE : thm
val NOT_EVENTUAL : thm
val NOT_NEXT : thm
val NOT_SBEFORE : thm
val NOT_SUNTIL : thm
val NOT_SWHEN : thm
val NOT_UNTIL : thm
val NOT_WHEN : thm
val NO_EVENT : thm
val OR_NEXT : thm
val SBEFORE_AS_BEFORE : thm
val SBEFORE_AS_SUNTIL : thm
val SBEFORE_AS_SWHEN : thm
val SBEFORE_AS_UNTIL : thm
val SBEFORE_AS_WHEN : thm
val SBEFORE_EVENT : thm
val SBEFORE_IDEM : thm
val SBEFORE_IMP : thm
val SBEFORE_INVARIANT : thm
val SBEFORE_LINORD : thm
val SBEFORE_NEXT : thm
val SBEFORE_REC : thm
val SBEFORE_SIGNAL : thm
val SBEFORE_SIMP : thm
val SOME_EVENT : thm
val SUNTIL_AS_BEFORE : thm
val SUNTIL_AS_SBEFORE : thm
val SUNTIL_AS_SWHEN : thm
val SUNTIL_AS_UNTIL : thm
val SUNTIL_AS_WHEN : thm
val SUNTIL_EVENT : thm
val SUNTIL_IDEM : thm
val SUNTIL_IMP : thm
val SUNTIL_INVARIANT : thm
val SUNTIL_LINORD : thm
val SUNTIL_NEXT : thm
val SUNTIL_REC : thm
val SUNTIL_SIGNAL : thm
val SUNTIL_SIMP : thm
val SWHEN_AS_BEFORE : thm
val SWHEN_AS_NOT_WHEN : thm
val SWHEN_AS_SBEFORE : thm
val SWHEN_AS_SUNTIL : thm
val SWHEN_AS_UNTIL : thm
val SWHEN_AS_WHEN : thm
val SWHEN_EVENT : thm
val SWHEN_IDEM : thm
val SWHEN_IMP : thm
val SWHEN_INVARIANT : thm
val SWHEN_LINORD : thm
val SWHEN_NEXT : thm
val SWHEN_REC : thm
val SWHEN_SIGNAL : thm
val SWHEN_SIMP : thm
val UNTIL_AS_BEFORE : thm
val UNTIL_AS_SBEFORE : thm
val UNTIL_AS_SUNTIL : thm
val UNTIL_AS_SWHEN : thm
val UNTIL_AS_WHEN : thm
val UNTIL_EVENT : thm
val UNTIL_FIX : thm
val UNTIL_IDEM : thm
val UNTIL_IMP : thm
val UNTIL_INVARIANT : thm
val UNTIL_LINORD : thm
val UNTIL_NEXT : thm
val UNTIL_REC : thm
val UNTIL_SIGNAL : thm
val UNTIL_SIMP : thm
val WATCH_EXISTS : thm
val WATCH_REC : thm
val WATCH_SIGNAL : thm
val WELL_ORDER : thm
val WELL_ORDER_UNIQUE : thm
val WHEN_AS_BEFORE : thm
val WHEN_AS_NOT_SWHEN : thm
val WHEN_AS_SBEFORE : thm
val WHEN_AS_SUNTIL : thm
val WHEN_AS_SWHEN : thm
val WHEN_AS_UNTIL : thm
val WHEN_EVENT : thm
val WHEN_FIX : thm
val WHEN_IDEM : thm
val WHEN_IMP : thm
val WHEN_INVARIANT : thm
val WHEN_LINORD : thm
val WHEN_NEXT : thm
val WHEN_REC : thm
val WHEN_SIGNAL : thm
val WHEN_SIMP : thm
val WHEN_SWHEN_LEMMA : thm
val Temporal_Logic_grammars : type_grammar.grammar * term_grammar.grammar
(*
[indexedLists] Parent theory of "Temporal_Logic"
[patternMatches] Parent theory of "Temporal_Logic"
[ALWAYS] Definition
⊢ ∀P t0. ALWAYS P t0 ⇔ ∀t. P (t + t0)
[BEFORE] Definition
⊢ ∀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))
[EVENTUAL] Definition
⊢ ∀P t0. EVENTUAL P t0 ⇔ ∃t. P (t + t0)
[NEXT] Definition
⊢ ∀P. NEXT P = (λt. P (SUC t))
[SBEFORE] Definition
⊢ ∀a b t0.
(a SBEFORE b) t0 ⇔
∃q. (q WATCH b) t0 ∧ ∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)
[SUNTIL] Definition
⊢ ∀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)
[SWHEN] Definition
⊢ ∀a b t0.
(a SWHEN b) t0 ⇔
∃q. (q WATCH b) t0 ∧ ∃t. ¬q (t + t0) ∧ b (t + t0) ∧ a (t + t0)
[UNTIL] Definition
⊢ ∀a b t0.
(a UNTIL b) t0 ⇔
∃q. (q WATCH b) t0 ∧ ∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)
[UPTO] Definition
⊢ ∀t0 t1 a. UPTO (t0,t1,a) ⇔ ∀t2. t0 ≤ t2 ∧ t2 < t1 ⇒ a t2
[WATCH] Definition
⊢ ∀q b t0.
(q WATCH b) t0 ⇔
∀t. (q t0 ⇔ F) ∧ (q (SUC (t + t0)) ⇔ q (t + t0) ∨ b (t + t0))
[WHEN] Definition
⊢ ∀a b t0.
(a WHEN b) t0 ⇔
∃q. (q WATCH b) t0 ∧ ∀t. q (t + t0) ∨ (b (t + t0) ⇒ a (t + t0))
[ALWAYS_AS_BEFORE] Theorem
⊢ ALWAYS b = (λt. F) BEFORE (λt. ¬b t)
[ALWAYS_AS_SBEFORE] Theorem
⊢ ALWAYS b = (λt0. ¬((λt. ¬b t) SBEFORE (λt. F)) t0)
[ALWAYS_AS_SUNTIL] Theorem
⊢ ALWAYS a = (λt. ¬((λt. T) SUNTIL (λt. ¬a t)) t)
[ALWAYS_AS_SWHEN] Theorem
⊢ ALWAYS a = (λt. ¬((λt. T) SWHEN (λt. ¬a t)) t)
[ALWAYS_AS_UNTIL] Theorem
⊢ ALWAYS a = a UNTIL (λt. F)
[ALWAYS_AS_WHEN] Theorem
⊢ ALWAYS a = (λt. F) WHEN (λt. ¬a t)
[ALWAYS_FIX] Theorem
⊢ (y = (λt. a t ∧ y (t + 1))) ⇔ (y = ALWAYS a) ∨ (y = (λt. F))
[ALWAYS_IDEM] Theorem
⊢ ALWAYS a = ALWAYS (ALWAYS a)
[ALWAYS_INVARIANT] Theorem
⊢ ALWAYS a t0 ⇔
∃J. J t0 ∧ ∀t. J (t + t0) ⇒ a (t + t0) ∧ J (t + (t0 + 1))
[ALWAYS_LINORD] Theorem
⊢ ALWAYS a t0 ⇔ ∀t1. t0 ≤ t1 ⇒ a t1
[ALWAYS_NEXT] Theorem
⊢ ∀a. NEXT (ALWAYS a) = ALWAYS (NEXT a)
[ALWAYS_REC] Theorem
⊢ ALWAYS P t0 ⇔ P t0 ∧ NEXT (ALWAYS P) t0
[ALWAYS_SIGNAL] Theorem
⊢ ALWAYS a t0 ⇔ ∀t. a (t + t0)
[AND_NEXT] Theorem
⊢ ∀Q P. NEXT (λt. P t ∧ Q t) = (λt. NEXT P t ∧ NEXT Q t)
[BEFORE_AS_NOT_SWHEN] Theorem
⊢ a BEFORE b = (λt0. ¬(b SWHEN (λt. a t ∨ b t)) t0)
[BEFORE_AS_SBEFORE] Theorem
⊢ a BEFORE b = (λt0. (a SBEFORE b) t0 ∨ ALWAYS (λt. ¬b t) t0)
[BEFORE_AS_SUNTIL] Theorem
⊢ a BEFORE b = (λt. ¬((λt. ¬a t) SUNTIL b) t)
[BEFORE_AS_SWHEN] Theorem
⊢ a BEFORE b =
(λt0.
((λt. ¬b t) SWHEN (λt. a t ∨ b t)) t0 ∨
ALWAYS (λt. ¬a t ∧ ¬b t) t0)
[BEFORE_AS_UNTIL] Theorem
⊢ a BEFORE b = (λt0. ¬((λt. ¬a t) UNTIL b) t0 ∨ ALWAYS (λt. ¬b t) t0)
[BEFORE_AS_WHEN] Theorem
⊢ a BEFORE b = (λt. ¬b t) WHEN (λt. a t ∨ b t)
[BEFORE_AS_WHEN_UNTIL] Theorem
⊢ a BEFORE b = (λt. ((λt. ¬b t) UNTIL a) t ∧ ((λt. ¬b t) WHEN a) t)
[BEFORE_EVENT] Theorem
⊢ a BEFORE b = (λt. a t ∧ ¬b t) BEFORE b
[BEFORE_FIX] Theorem
⊢ ∀y. (y = (λt. ¬b t ∧ (a t ∨ y (t + 1)))) ⇔
(y = a BEFORE b) ∨ (y = a SBEFORE b)
[BEFORE_HW] Theorem
⊢ (a BEFORE b) t0 ⇔ ∃q. (q WATCH a) t0 ∧ ∀t. q (t + t0) ∨ ¬b (t + t0)
[BEFORE_IDEM] Theorem
⊢ a BEFORE b = (a BEFORE b) BEFORE b
[BEFORE_IMP] Theorem
⊢ (a BEFORE b) t0 ⇔
∀q. (q WATCH b) t0 ⇒
(∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)) ∨ ∀t. ¬b (t + t0)
[BEFORE_INVARIANT] Theorem
⊢ (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)
[BEFORE_LINORD] Theorem
⊢ (a BEFORE b) t0 ⇔ ∀t1. t0 ≤ t1 ∧ UPTO (t0,t1,(λt. ¬a t)) ⇒ ¬b t1
[BEFORE_NEXT] Theorem
⊢ ∀a b. NEXT (a BEFORE b) = NEXT a BEFORE NEXT b
[BEFORE_REC] Theorem
⊢ (a BEFORE b) t0 ⇔ ¬b t0 ∧ (a t0 ∨ NEXT (a BEFORE b) t0)
[BEFORE_SIGNAL] Theorem
⊢ (a BEFORE b) t0 ⇔
∀delta.
(∀t. t < delta ⇒ ¬b (t + t0)) ∧ b (delta + t0) ⇒
∃t. t < delta ∧ a (t + t0)
[BEFORE_SIMP] Theorem
⊢ ((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))
[DELTA_CASES] Theorem
⊢ (∃d. (∀t. t < d ⇒ ¬b (t + t0)) ∧ b (d + t0)) ∨ ∀d. ¬b (d + t0)
[EQUIV_NEXT] Theorem
⊢ ∀Q P. NEXT (λt. P t ⇔ Q t) = (λt. NEXT P t ⇔ NEXT Q t)
[EVENTUAL_AS_BEFORE] Theorem
⊢ EVENTUAL b = (λt0. ¬((λt. F) BEFORE b) t0)
[EVENTUAL_AS_SBEFORE] Theorem
⊢ EVENTUAL b = b SBEFORE (λt. F)
[EVENTUAL_AS_SUNTIL] Theorem
⊢ EVENTUAL a = (λt. T) SUNTIL a
[EVENTUAL_AS_SWHEN] Theorem
⊢ EVENTUAL a = (λt. T) SWHEN a
[EVENTUAL_AS_UNTIL] Theorem
⊢ EVENTUAL a = (λt. ¬((λt. ¬a t) UNTIL (λt. F)) t)
[EVENTUAL_AS_WHEN] Theorem
⊢ EVENTUAL a = (λt. ¬((λt. F) WHEN a) t)
[EVENTUAL_FIX] Theorem
⊢ (y = (λt. a t ∨ y (t + 1))) ⇔ (y = EVENTUAL a) ∨ (y = (λt. T))
[EVENTUAL_IDEM] Theorem
⊢ EVENTUAL a = EVENTUAL (EVENTUAL a)
[EVENTUAL_INVARIANT] Theorem
⊢ 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)
[EVENTUAL_LINORD] Theorem
⊢ EVENTUAL a t0 ⇔ ∃t1. t0 ≤ t1 ∧ a t1
[EVENTUAL_NEXT] Theorem
⊢ ∀a. NEXT (EVENTUAL a) = EVENTUAL (NEXT a)
[EVENTUAL_REC] Theorem
⊢ EVENTUAL P t0 ⇔ P t0 ∨ NEXT (EVENTUAL P) t0
[EVENTUAL_SIGNAL] Theorem
⊢ EVENTUAL a t0 ⇔ ∃t. a (t + t0)
[IMMEDIATE_EVENT] Theorem
⊢ 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
[IMP_NEXT] Theorem
⊢ ∀Q P. NEXT (λt. P t ⇒ Q t) = (λt. NEXT P t ⇒ NEXT Q t)
[MORE_EVENT] Theorem
⊢ (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)
[NEXT_LINORD] Theorem
⊢ NEXT a t0 ⇔ ∃t1. t0 < t1 ∧ (∀t3. t0 < t3 ⇒ t1 ≤ t3) ∧ a t1
[NOT_ALWAYS] Theorem
⊢ ¬ALWAYS a t0 ⇔ EVENTUAL (λt. ¬a t) t0
[NOT_BEFORE] Theorem
⊢ ¬(a BEFORE b) t0 ⇔ ((λt. ¬a t) SUNTIL b) t0
[NOT_EVENTUAL] Theorem
⊢ ¬EVENTUAL a t0 ⇔ ALWAYS (λt. ¬a t) t0
[NOT_NEXT] Theorem
⊢ ∀P. NEXT (λt. ¬P t) = (λt. ¬NEXT P t)
[NOT_SBEFORE] Theorem
⊢ ¬(a SBEFORE b) t0 ⇔ ((λt. ¬a t) UNTIL b) t0
[NOT_SUNTIL] Theorem
⊢ ¬(a SUNTIL b) t0 ⇔ ((λt. ¬a t) BEFORE b) t0
[NOT_SWHEN] Theorem
⊢ ¬(a SWHEN b) t0 ⇔ ((λt. ¬a t) WHEN b) t0
[NOT_UNTIL] Theorem
⊢ ¬(a UNTIL b) t0 ⇔ ((λt. ¬a t) SBEFORE b) t0
[NOT_WHEN] Theorem
⊢ ¬(a WHEN b) t0 ⇔ ((λt. ¬a t) SWHEN b) t0
[NO_EVENT] Theorem
⊢ 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
[OR_NEXT] Theorem
⊢ ∀Q P. NEXT (λt. P t ∨ Q t) = (λt. NEXT P t ∨ NEXT Q t)
[SBEFORE_AS_BEFORE] Theorem
⊢ a SBEFORE b = (λt0. (a BEFORE b) t0 ∧ EVENTUAL a t0)
[SBEFORE_AS_SUNTIL] Theorem
⊢ a SBEFORE b = (λt0. ¬((λt. ¬a t) SUNTIL b) t0 ∧ EVENTUAL a t0)
[SBEFORE_AS_SWHEN] Theorem
⊢ a SBEFORE b = (λt. ¬b t) SWHEN (λt. a t ∨ b t)
[SBEFORE_AS_UNTIL] Theorem
⊢ a SBEFORE b = (λt0. ¬((λt. ¬a t) UNTIL b) t0)
[SBEFORE_AS_WHEN] Theorem
⊢ a SBEFORE b =
(λt0. ((λt. ¬b t) WHEN (λt. a t ∨ b t)) t0 ∧ EVENTUAL a t0)
[SBEFORE_EVENT] Theorem
⊢ a SBEFORE b = (λt. a t ∧ ¬b t) SBEFORE b
[SBEFORE_IDEM] Theorem
⊢ a SBEFORE b = (a SBEFORE b) SBEFORE b
[SBEFORE_IMP] Theorem
⊢ (a SBEFORE b) t0 ⇔
∀q. (q WATCH b) t0 ⇒ ∃t. ¬q (t + t0) ∧ ¬b (t + t0) ∧ a (t + t0)
[SBEFORE_INVARIANT] Theorem
⊢ (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)
[SBEFORE_LINORD] Theorem
⊢ (a SBEFORE b) t0 ⇔
∃t1. t0 ≤ t1 ∧ a t1 ∧ ¬b t1 ∧ UPTO (t0,t1,(λt. ¬b t))
[SBEFORE_NEXT] Theorem
⊢ ∀a b. NEXT (a SBEFORE b) = NEXT a SBEFORE NEXT b
[SBEFORE_REC] Theorem
⊢ (a SBEFORE b) t0 ⇔ ¬b t0 ∧ (a t0 ∨ NEXT (a SBEFORE b) t0)
[SBEFORE_SIGNAL] Theorem
⊢ (a SBEFORE b) t0 ⇔
∃delta. a (delta + t0) ∧ ∀t. t ≤ delta ⇒ ¬b (t + t0)
[SBEFORE_SIMP] Theorem
⊢ ((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))
[SOME_EVENT] Theorem
⊢ (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)
[SUNTIL_AS_BEFORE] Theorem
⊢ a SUNTIL b = (λt0. ¬((λt. ¬a t) BEFORE b) t0)
[SUNTIL_AS_SBEFORE] Theorem
⊢ a SUNTIL b = (λt0. ¬((λt. ¬a t) SBEFORE b) t0 ∧ EVENTUAL b t0)
[SUNTIL_AS_SWHEN] Theorem
⊢ a SUNTIL b = b SWHEN (λt. a t ⇒ b t)
[SUNTIL_AS_UNTIL] Theorem
⊢ a SUNTIL b = (λt0. (a UNTIL b) t0 ∧ EVENTUAL b t0)
[SUNTIL_AS_WHEN] Theorem
⊢ a SUNTIL b = (λt. (b WHEN (λt. a t ⇒ b t)) t ∧ EVENTUAL b t)
[SUNTIL_EVENT] Theorem
⊢ a SUNTIL b = (λt. a t ∧ ¬b t) SUNTIL b
[SUNTIL_IDEM] Theorem
⊢ a SUNTIL b = (a SUNTIL b) SUNTIL b
[SUNTIL_IMP] Theorem
⊢ (a SUNTIL b) t0 ⇔
∀q. (q WATCH b) t0 ⇒
(∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)) ∧ ∃t. b (t + t0)
[SUNTIL_INVARIANT] Theorem
⊢ (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)
[SUNTIL_LINORD] Theorem
⊢ (a SUNTIL b) t0 ⇔ ∃t1. t0 ≤ t1 ∧ b t1 ∧ UPTO (t0,t1,a)
[SUNTIL_NEXT] Theorem
⊢ ∀a b. NEXT (a SUNTIL b) = NEXT a SUNTIL NEXT b
[SUNTIL_REC] Theorem
⊢ (a SUNTIL b) t0 ⇔ ¬b t0 ⇒ a t0 ∧ NEXT (a SUNTIL b) t0
[SUNTIL_SIGNAL] Theorem
⊢ (a SUNTIL b) t0 ⇔
∃delta. (∀t. t < delta ⇒ a (t + t0) ∧ ¬b (t + t0)) ∧ b (delta + t0)
[SUNTIL_SIMP] Theorem
⊢ ((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))
[SWHEN_AS_BEFORE] Theorem
⊢ a SWHEN b = (λt0. ¬(b BEFORE (λt. a t ∧ b t)) t0)
[SWHEN_AS_NOT_WHEN] Theorem
⊢ (a SWHEN b) t0 ⇔ ¬((λt. ¬a t) WHEN b) t0
[SWHEN_AS_SBEFORE] Theorem
⊢ a SWHEN b = b SBEFORE (λt. ¬a t ∧ b t)
[SWHEN_AS_SUNTIL] Theorem
⊢ a SWHEN b = (λt. ¬b t) SUNTIL (λt. a t ∧ b t)
[SWHEN_AS_UNTIL] Theorem
⊢ a SWHEN b =
(λt. ((λt. ¬b t) UNTIL (λt. a t ∧ b t)) t ∧ EVENTUAL b t)
[SWHEN_AS_WHEN] Theorem
⊢ a SWHEN b = (λt0. (a WHEN b) t0 ∧ EVENTUAL b t0)
[SWHEN_EVENT] Theorem
⊢ a SWHEN b = (λt. a t ∧ b t) SWHEN b
[SWHEN_IDEM] Theorem
⊢ a SWHEN b = (a SWHEN b) SWHEN b
[SWHEN_IMP] Theorem
⊢ (a SWHEN b) t0 ⇔
∀q. (q WATCH b) t0 ⇒ ∃t. ¬q (t + t0) ∧ b (t + t0) ∧ a (t + t0)
[SWHEN_INVARIANT] Theorem
⊢ (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)
[SWHEN_LINORD] Theorem
⊢ (a SWHEN b) t0 ⇔
∃t1. t0 ≤ t1 ∧ a t1 ∧ b t1 ∧ UPTO (t0,t1,(λt. ¬b t))
[SWHEN_NEXT] Theorem
⊢ ∀a b. NEXT (a SWHEN b) = NEXT a SWHEN NEXT b
[SWHEN_REC] Theorem
⊢ (a SWHEN b) t0 ⇔ if b t0 then a t0 else NEXT (a SWHEN b) t0
[SWHEN_SIGNAL] Theorem
⊢ (a SWHEN b) t0 ⇔
∃delta.
(∀t. t < delta ⇒ ¬b (t + t0)) ∧ b (delta + t0) ∧ a (delta + t0)
[SWHEN_SIMP] Theorem
⊢ ((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)
[UNTIL_AS_BEFORE] Theorem
⊢ a UNTIL b = (λt0. ¬((λt. ¬a t) BEFORE b) t0 ∨ ALWAYS a t0)
[UNTIL_AS_SBEFORE] Theorem
⊢ a UNTIL b = (λt0. ¬((λt. ¬a t) SBEFORE b) t0)
[UNTIL_AS_SUNTIL] Theorem
⊢ a UNTIL b = (λt. (a SUNTIL b) t ∨ ALWAYS a t)
[UNTIL_AS_SWHEN] Theorem
⊢ a UNTIL b = (λt. (b SWHEN (λt. a t ⇒ b t)) t ∨ ALWAYS a t)
[UNTIL_AS_WHEN] Theorem
⊢ a UNTIL b = b WHEN (λt. a t ⇒ b t)
[UNTIL_EVENT] Theorem
⊢ a UNTIL b = (λt. a t ∧ ¬b t) UNTIL b
[UNTIL_FIX] Theorem
⊢ (y = (λt. ¬b t ⇒ a t ∧ y (t + 1))) ⇔
(y = a UNTIL b) ∨ (y = a SUNTIL b)
[UNTIL_IDEM] Theorem
⊢ a UNTIL b = (a UNTIL b) UNTIL b
[UNTIL_IMP] Theorem
⊢ (a UNTIL b) t0 ⇔
∀q. (q WATCH b) t0 ⇒ ∀t. q (t + t0) ∨ b (t + t0) ∨ a (t + t0)
[UNTIL_INVARIANT] Theorem
⊢ ∀t0.
(a UNTIL b) t0 ⇔
∃J. J t0 ∧
∀t. J (t + t0) ∧ ¬b (t + t0) ⇒ a (t + t0) ∧ J (SUC (t + t0))
[UNTIL_LINORD] Theorem
⊢ (a UNTIL b) t0 ⇔
∀t1. t0 ≤ t1 ∧ ¬b t1 ∧ UPTO (t0,t1,(λt. ¬b t)) ⇒ a t1
[UNTIL_NEXT] Theorem
⊢ ∀a b. NEXT (a UNTIL b) = NEXT a UNTIL NEXT b
[UNTIL_REC] Theorem
⊢ (a UNTIL b) t0 ⇔ ¬b t0 ⇒ a t0 ∧ NEXT (a UNTIL b) t0
[UNTIL_SIGNAL] Theorem
⊢ (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)
[UNTIL_SIMP] Theorem
⊢ ((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))
[WATCH_EXISTS] Theorem
⊢ ∀b t0. ∃q. (q WATCH b) t0
[WATCH_REC] Theorem
⊢ (q WATCH b) t0 ⇔
¬q t0 ∧ if b t0 then NEXT (ALWAYS q) t0 else NEXT (q WATCH b) t0
[WATCH_SIGNAL] Theorem
⊢ (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)))
[WELL_ORDER] Theorem
⊢ (∃n. P n) ⇔ ∃m. P m ∧ ∀n. n < m ⇒ ¬P n
[WELL_ORDER_UNIQUE] Theorem
⊢ ∀m2 m1 P.
(P m1 ∧ ∀n. n < m1 ⇒ ¬P n) ∧ P m2 ∧ (∀n. n < m2 ⇒ ¬P n) ⇒
(m1 = m2)
[WHEN_AS_BEFORE] Theorem
⊢ a WHEN b =
(λt0. ¬(b BEFORE (λt. a t ∧ b t)) t0 ∨ ALWAYS (λt. ¬b t) t0)
[WHEN_AS_NOT_SWHEN] Theorem
⊢ (a WHEN b) t0 ⇔ ¬((λt. ¬a t) SWHEN b) t0
[WHEN_AS_SBEFORE] Theorem
⊢ a WHEN b =
(λt0. (b SBEFORE (λt. ¬a t ∧ b t)) t0 ∨ ALWAYS (λt. ¬b t) t0)
[WHEN_AS_SUNTIL] Theorem
⊢ a WHEN b =
(λt. ((λt. ¬b t) SUNTIL (λt. a t ∧ b t)) t ∨ ALWAYS (λt. ¬b t) t)
[WHEN_AS_SWHEN] Theorem
⊢ a WHEN b = (λt. (a SWHEN b) t ∨ ALWAYS (λt. ¬b t) t)
[WHEN_AS_UNTIL] Theorem
⊢ a WHEN b = (λt. ¬b t) UNTIL (λt. a t ∧ b t)
[WHEN_EVENT] Theorem
⊢ a WHEN b = (λt. a t ∧ b t) WHEN b
[WHEN_FIX] Theorem
⊢ (y = (λt. if b t then a t else y (t + 1))) ⇔
(y = a WHEN b) ∨ (y = a SWHEN b)
[WHEN_IDEM] Theorem
⊢ a WHEN b = (a WHEN b) WHEN b
[WHEN_IMP] Theorem
⊢ (a WHEN b) t0 ⇔
∀q. (q WATCH b) t0 ⇒ ∀t. q (t + t0) ∨ (b (t + t0) ⇒ a (t + t0))
[WHEN_INVARIANT] Theorem
⊢ (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)
[WHEN_LINORD] Theorem
⊢ (a WHEN b) t0 ⇔
∀t1. t0 ≤ t1 ∧ b t1 ∧ UPTO (t0,t1,(λt. ¬b t)) ⇒ a t1
[WHEN_NEXT] Theorem
⊢ ∀a b. NEXT (a WHEN b) = NEXT a WHEN NEXT b
[WHEN_REC] Theorem
⊢ (a WHEN b) t0 ⇔ if b t0 then a t0 else NEXT (a WHEN b) t0
[WHEN_SIGNAL] Theorem
⊢ (a WHEN b) t0 ⇔
∀delta.
(∀t. t < delta ⇒ ¬b (t + t0)) ∧ b (delta + t0) ⇒ a (delta + t0)
[WHEN_SIMP] Theorem
⊢ ((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))
[WHEN_SWHEN_LEMMA] Theorem
⊢ 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)
*)
end
HOL 4, Trindemossen-1