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44 lines
1.2 KiB
Coq
44 lines
1.2 KiB
Coq
Require Import Relations.
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Set Implicit Arguments.
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Record trsys state := {
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Initial : state -> Prop;
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Step : state -> state -> Prop
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}.
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Definition invariantFor {state} (sys : trsys state) (invariant : state -> Prop) :=
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forall s, sys.(Initial) s
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-> forall s', sys.(Step)^* s s'
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-> invariant s'.
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Theorem use_invariant : forall {state} (sys : trsys state) (invariant : state -> Prop) s s',
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invariantFor sys invariant
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-> sys.(Step)^* s s'
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-> sys.(Initial) s
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-> invariant s'.
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Proof.
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firstorder.
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Qed.
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Theorem invariantFor_monotone : forall {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop),
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invariantFor sys invariant1
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-> (forall s, invariant1 s -> invariant2 s)
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-> invariantFor sys invariant2.
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Proof.
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unfold invariantFor; intuition eauto.
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Qed.
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Theorem invariant_induction : forall {state} (sys : trsys state)
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(invariant : state -> Prop),
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(forall s, sys.(Initial) s -> invariant s)
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-> (forall s, invariant s -> forall s', sys.(Step) s s' -> invariant s')
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-> invariantFor sys invariant.
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Proof.
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unfold invariantFor; intros.
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assert (invariant s) by eauto.
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clear H1.
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induction H2; eauto.
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Qed.
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