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TransitionSystems WIP

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Adam Chlipala 2016-02-08 18:14:11 -05:00
parent 3b82c0b2bd
commit 754784c286
2 changed files with 67 additions and 17 deletions
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40
Invariant.v
View file

@ -1,35 +1,41 @@
Require Import Relations.
Set Implicit Arguments.
Definition invariantFor {state} (initial : state -> Prop) (step : state -> state -> Prop) (invariant : state -> Prop) :=
forall s, initial s
-> forall s', step^* s s'
Record trsys state := {
Initial : state -> Prop;
Step : state -> state -> Prop
}.
Definition invariantFor {state} (sys : trsys state) (invariant : state -> Prop) :=
forall s, sys.(Initial) s
-> forall s', sys.(Step)^* s s'
-> invariant s'.
Theorem use_invariant : forall {state} (initial : state -> Prop)
(step : state -> state -> Prop) (invariant : state -> Prop) s s',
step^* s s'
-> initial s
-> invariantFor initial step invariant
Theorem use_invariant : forall {state} (sys : trsys state) (invariant : state -> Prop) s s',
sys.(Step)^* s s'
-> sys.(Initial) s
-> invariantFor sys invariant
-> invariant s'.
Proof.
firstorder.
Qed.
Theorem invariantFor_monotone : forall {state} (initial : state -> Prop)
(step : state -> state -> Prop) (invariant1 invariant2 : state -> Prop),
Theorem invariantFor_monotone : forall {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop),
(forall s, invariant1 s -> invariant2 s)
-> invariantFor initial step invariant1
-> invariantFor initial step invariant2.
-> invariantFor sys invariant1
-> invariantFor sys invariant2.
Proof.
unfold invariantFor; intuition eauto.
Qed.
Theorem invariant_induction : forall {state} (initial : state -> Prop)
(step : state -> state -> Prop) (invariant : state -> Prop),
(forall s, initial s -> invariant s)
-> (forall s, invariant s -> forall s', step s s' -> invariant s')
-> invariantFor initial step invariant.
Theorem invariant_induction : forall {state} (sys : trsys state)
(invariant : state -> Prop),
(forall s, sys.(Initial) s -> invariant s)
-> (forall s, invariant s -> forall s', sys.(Step) s s' -> invariant s')
-> invariantFor sys invariant.
Proof.
unfold invariantFor; intros.
assert (invariant s) by eauto.

44
TransitionSystems.v
View file

@ -110,6 +110,50 @@ Proof.
* both [trc] and [fact_step]. *)
Qed.
(* It will be useful to give state machines more first-class status, as
* *transition systems*, formalized by this record type. It has one type
* parameter, [state], which records the type of states. *)
Record trsys state := {
Initial : state -> Prop;
Step : state -> state -> Prop
}.
Definition invariantFor {state} (sys : trsys state) (invariant : state -> Prop) :=
forall s, sys.(Initial) s
-> forall s', sys.(Step)^* s s'
-> invariant s'.
Theorem use_invariant : forall {state} (sys : trsys state) (invariant : state -> Prop) s s',
sys.(Step)^* s s'
-> sys.(Initial) s
-> invariantFor sys invariant
-> invariant s'.
Proof.
firstorder.
Qed.
Theorem invariantFor_monotone : forall {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop),
(forall s, invariant1 s -> invariant2 s)
-> invariantFor sys invariant1
-> invariantFor sys invariant2.
Proof.
unfold invariantFor; intuition eauto.
Qed.
Theorem invariant_induction : forall {state} (sys : trsys state)
(invariant : state -> Prop),
(forall s, sys.(Initial) s -> invariant s)
-> (forall s, invariant s -> forall s', sys.(Step) s s' -> invariant s')
-> invariantFor sys invariant.
Proof.
unfold invariantFor; intros.
assert (invariant s) by eauto.
clear H1.
induction H2; eauto.
Qed.
(* To prove that our state machine is correct, we rely on the crucial technique
* of *invariants*. What is an invariant? Here's a general definition, in
* terms of an arbitrary *transition system* defined by a set of states,