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98 lines
3 KiB
Coq
98 lines
3 KiB
Coq
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Require Import Invariant Sets.
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Definition oneStepClosure_current {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop) :=
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forall st, invariant1 st
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-> invariant2 st.
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Definition oneStepClosure_new {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop) :=
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forall st st', invariant1 st
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-> sys.(Step) st st'
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-> invariant2 st'.
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Definition oneStepClosure {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop) :=
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oneStepClosure_current sys invariant1 invariant2
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/\ oneStepClosure_new sys invariant1 invariant2.
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Theorem prove_oneStepClosure : forall state (sys : trsys state) (inv1 inv2 : state -> Prop),
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(forall st, inv1 st -> inv2 st)
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-> (forall st st', inv1 st -> sys.(Step) st st' -> inv2 st')
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-> oneStepClosure sys inv1 inv2.
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Proof.
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unfold oneStepClosure; tauto.
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Qed.
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Theorem oneStepClosure_done : forall state (sys : trsys state) (invariant : state -> Prop),
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(forall st, sys.(Initial) st -> invariant st)
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-> oneStepClosure sys invariant invariant
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-> invariantFor sys invariant.
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Proof.
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unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new.
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intuition eauto using invariant_induction.
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Qed.
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Inductive multiStepClosure {state} (sys : trsys state)
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: (state -> Prop) -> (state -> Prop) -> Prop :=
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| MscDone : forall inv,
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oneStepClosure sys inv inv
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-> multiStepClosure sys inv inv
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| MscStep : forall inv inv' inv'',
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oneStepClosure sys inv inv'
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-> multiStepClosure sys inv' inv''
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-> multiStepClosure sys inv inv''.
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Lemma multiStepClosure_ok' : forall state (sys : trsys state) (inv inv' : state -> Prop),
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multiStepClosure sys inv inv'
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-> (forall st, sys.(Initial) st -> inv st)
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-> invariantFor sys inv'.
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Proof.
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induction 1; simpl; intuition eauto using oneStepClosure_done.
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unfold oneStepClosure, oneStepClosure_current in *.
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intuition eauto.
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Qed.
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Theorem multiStepClosure_ok : forall state (sys : trsys state) (inv : state -> Prop),
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multiStepClosure sys sys.(Initial) inv
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-> invariantFor sys inv.
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Proof.
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eauto using multiStepClosure_ok'.
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Qed.
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Theorem oneStepClosure_empty : forall state (sys : trsys state),
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oneStepClosure sys (constant nil) (constant nil).
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Proof.
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unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; intuition.
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Qed.
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Theorem oneStepClosure_split : forall state (sys : trsys state) st sts (inv1 inv2 : state -> Prop),
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(forall st', sys.(Step) st st' -> inv1 st')
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-> oneStepClosure sys (constant sts) inv2
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-> oneStepClosure sys (constant (st :: sts)) ({st} \cup inv1 \cup inv2).
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Proof.
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unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; intuition.
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inversion H0; subst.
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unfold union; simpl; tauto.
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unfold union; simpl; eauto.
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unfold union in *; simpl in *.
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intuition (subst; eauto).
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Qed.
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Theorem singleton_in : forall {A} (x : A) rest,
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({x} \cup rest) x.
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Proof.
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unfold union; simpl; auto.
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Qed.
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Theorem singleton_in_other : forall {A} (x : A) (s1 s2 : set A),
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s2 x
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-> (s1 \cup s2) x.
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Proof.
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unfold union; simpl; auto.
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Qed.
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