# HG changeset patch
# User Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
# Date 1449833224 -32400
# Node ID a59bebe0265a51224f3993ae493c992090770b98
# Parent  26e64661b96951e3948df24645caf1b82fba07a0
Trying LTL

diff -r 26e64661b969 -r a59bebe0265a LTL/LTL.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/LTL/LTL.agda	Fri Dec 11 20:27:04 2015 +0900
@@ -0,0 +1,68 @@
+-- Principles of Model Checking
+
+open import Level
+open import Relation.Binary.PropositionalEquality
+open ≡-Reasoning
+
+module LTL where
+
+-- Syntax Definition (p227)
+
+data AP : Set where
+
+data LTL : Set  where
+  true     : LTL
+  ap       : AP -> LTL
+  _and_    : LTL -> LTL -> LTL
+  not      : LTL -> LTL
+  next     : LTL -> LTL
+  _until_  : LTL -> LTL -> LTL
+
+
+_or_ : LTL -> LTL -> LTL
+a or b = not ((not a) and (not b))
+
+_implies_ : LTL -> LTL -> LTL
+a implies b = (not a) or b
+
+_equiv_ : LTL -> LTL -> LTL
+a equiv b = (a implies b) and (b implies a)
+
+_xor_  : LTL -> LTL -> LTL
+a xor b = (a and (not b)) or (b and (not a))
+
+eventually : LTL -> LTL
+eventually a = true until a
+
+always : LTL -> LTL
+always a = not (eventually (not a))
+
+-- Equivalences (p244)
+
+next-duality-law : (a : LTL) -> not (next a) ≡ next (not a)
+next-duality-law true = begin
+  not (next true)
+  ≡⟨ {!!} ⟩
+  next (not true)
+  ∎
+next-duality-law (ap x) = begin
+  not (next (ap x))
+  ≡⟨ {!!} ⟩
+  next (not (ap x))
+  ∎
+next-duality-law (a and b) = begin
+  not (next (a and b))
+  ≡⟨ {!!} ⟩
+  next (not (a and b))
+  ∎
+next-duality-law (not a) = {!!}
+next-duality-law (next a) = {!!}
+next-duality-law (a until b) = {!!}
+
+eventually-duality-law : (a : LTL) -> not (eventually a) ≡ always (not a)
+eventually-duality-law true = {!!}
+eventually-duality-law (ap x) = {!!}
+eventually-duality-law (a and b) = {!!}
+eventually-duality-law (not a) = {!!}
+eventually-duality-law (next a) = {!!}
+eventually-duality-law (a until b) = {!!}
\ No newline at end of file