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Factor the TypeOK inductive step out as LEMMA TypeOKStep #217

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Factor the TypeOK inductive step out as LEMMA TypeOKStep #217

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85 changes: 18 additions & 67 deletions specifications/ReadersWriters/ReadersWriters_proof.tla
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Original file line number Diff line number Diff line change
Expand Up @@ -42,10 +42,7 @@ LEMMA TailIsSeq ==
(***************************************************************************)
(* Type correctness. *)
(***************************************************************************)
THEOREM TypeCorrect == Spec => []TypeOK
<1>1. Init => TypeOK
BY DEF Init, TypeOK
<1>2. TypeOK /\ [Next]_vars => TypeOK'
LEMMA TypeOKStep == TypeOK /\ [Next]_vars => TypeOK'
<2>. SUFFICES ASSUME TypeOK, [Next]_vars PROVE TypeOK'
OBVIOUS
<2>. USE DEF TypeOK
Expand Down Expand Up @@ -97,66 +94,23 @@ THEOREM TypeCorrect == Spec => []TypeOK
<2>5. CASE UNCHANGED vars
BY <2>5 DEF vars
<2>. QED BY <2>1, <2>2, <2>3, <2>4, <2>5 DEF Next

THEOREM TypeCorrect == Spec => []TypeOK
<1>1. Init => TypeOK
BY DEF Init, TypeOK
<1>2. TypeOK /\ [Next]_vars => TypeOK'
BY TypeOKStep
<1>. QED BY <1>1, <1>2, PTL DEF Spec

(***************************************************************************)
(* The mutex / single-writer safety property. *)
(* Inductive together with TypeOK. *)
(* The mutex / single-writer safety property. Safety is inductive only *)
(* relative to TypeOK, which enters as a hypothesis here and is discharged *)
(* with TypeCorrect in the temporal step of THEOREM SafetyCorrect. *)
(***************************************************************************)
Inv == TypeOK /\ Safety

LEMMA SafetyStep == Inv /\ [Next]_vars => Inv'
<1>. SUFFICES ASSUME Inv, [Next]_vars PROVE Inv'
OBVIOUS
<1>. USE DEF Inv, TypeOK, Safety
<1>typok. TypeOK'
\* Re-derive the type-correctness step inline (same as in TypeCorrect).
<2>1. ASSUME NEW actor \in Actors, TryRead(actor)
PROVE TypeOK'
<3>. <<"read", actor>> \in {"read","write"} \X Actors
OBVIOUS
<3>. QED BY <2>1 DEF TryRead
<2>2. ASSUME NEW actor \in Actors, TryWrite(actor)
PROVE TypeOK'
<3>. <<"write", actor>> \in {"read","write"} \X Actors
OBVIOUS
<3>. QED BY <2>2 DEF TryWrite
<2>3. CASE ReadOrWrite
<3>. USE <2>3 DEF ReadOrWrite
<3>1. waiting # << >>
OBVIOUS
<3>2. Tail(waiting) \in Seq({"read","write"} \X Actors)
BY <3>1, TailIsSeq
<3>3. Head(waiting) \in {"read","write"} \X Actors
BY <3>1, HeadInSeqRange
<3>4. Head(waiting)[2] \in Actors
BY <3>3
<3>5. CASE Head(waiting)[1] = "read"
<4>. DEFINE actor == Head(waiting)[2]
<4>1. Read(actor)
BY <3>5
<4>. QED BY <3>2, <3>4, <4>1 DEF Read
<3>6. CASE Head(waiting)[1] = "write"
<4>. DEFINE actor == Head(waiting)[2]
<4>1. Write(actor)
BY <3>6
<4>. QED BY <3>2, <3>4, <4>1 DEF Write
<3>7. Head(waiting)[1] \in {"read","write"}
BY <3>3
<3>. QED BY <3>5, <3>6, <3>7
<2>4. ASSUME NEW actor \in readers \cup writers, StopActivity(actor)
PROVE TypeOK'
<3>1. CASE actor \in readers
BY <3>1, <2>4 DEF StopActivity
<3>2. CASE actor \notin readers
<4>1. actor \in writers
BY <3>2, <2>4
<4>. QED BY <3>2, <4>1, <2>4 DEF StopActivity
<3>. QED BY <3>1, <3>2
<2>5. CASE UNCHANGED vars
BY <2>5 DEF vars
<2>. QED BY <2>1, <2>2, <2>3, <2>4, <2>5 DEF Next, Stop
<1>safety. Safety'
LEMMA SafetyStep == TypeOK /\ Safety /\ [Next]_vars => Safety'
<2>. SUFFICES ASSUME TypeOK, Safety, [Next]_vars PROVE Safety'
OBVIOUS
<2>. USE DEF TypeOK, Safety
<2>1. ASSUME NEW actor \in Actors, TryRead(actor)
PROVE Safety'
BY <2>1 DEF TryRead
Expand Down Expand Up @@ -253,16 +207,13 @@ LEMMA SafetyStep == Inv /\ [Next]_vars => Inv'
<2>5. CASE UNCHANGED vars
BY <2>5 DEF vars
<2>. QED BY <2>1, <2>2, <2>3, <2>4, <2>5 DEF Next, Stop
<1>. QED BY <1>typok, <1>safety

THEOREM SafetyCorrect == Spec => []Safety
<1>1. Init => Inv
<1>1. Init => Safety
<2>1. Cardinality({}) = 0
BY FS_EmptySet
<2>. QED BY <2>1 DEF Init, Inv, TypeOK, Safety
<1>2. Inv /\ [Next]_vars => Inv'
<2>. QED BY <2>1 DEF Init, Safety
<1>2. TypeOK /\ Safety /\ [Next]_vars => Safety'
BY SafetyStep
<1>3. Inv => Safety
BY DEF Inv
<1>. QED BY <1>1, <1>2, <1>3, PTL DEF Spec
<1>. QED BY <1>1, <1>2, TypeCorrect, PTL DEF Spec
============================================================================
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