From 892b29f9259b9e49e6eea455134300a2104e7c87 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 07:41:46 -0700
Subject: [PATCH 01/14] DiningPhilosophers: rename APDiningPhilosophers to
 APDiningPhilosophersPCal

The Apalache wrapper instantiates DiningPhilosophersPCal, so give it
the same PCal suffix. Rename the module and its config, and update
manifest.json.

The non-PlusCal DiningPhilosophers gets no Apalache wrapper. Apalache
has nothing useful to check there: NoDeadlock uses ENABLED, which
Apalache does not support, and NoLivelock and NoStarvation are
liveness properties under SF fairness, which Apalache also rejects.
That leaves TypeOK, which tells us little.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 ...{APDiningPhilosophers.cfg => APDiningPhilosophersPCal.cfg} | 0
 ...{APDiningPhilosophers.tla => APDiningPhilosophersPCal.tla} | 2 +-
 specifications/DiningPhilosophers/manifest.json               | 4 ++--
 3 files changed, 3 insertions(+), 3 deletions(-)
 rename specifications/DiningPhilosophers/{APDiningPhilosophers.cfg => APDiningPhilosophersPCal.cfg} (100%)
 rename specifications/DiningPhilosophers/{APDiningPhilosophers.tla => APDiningPhilosophersPCal.tla} (90%)

diff --git a/specifications/DiningPhilosophers/APDiningPhilosophers.cfg b/specifications/DiningPhilosophers/APDiningPhilosophersPCal.cfg
similarity index 100%
rename from specifications/DiningPhilosophers/APDiningPhilosophers.cfg
rename to specifications/DiningPhilosophers/APDiningPhilosophersPCal.cfg
diff --git a/specifications/DiningPhilosophers/APDiningPhilosophers.tla b/specifications/DiningPhilosophers/APDiningPhilosophersPCal.tla
similarity index 90%
rename from specifications/DiningPhilosophers/APDiningPhilosophers.tla
rename to specifications/DiningPhilosophers/APDiningPhilosophersPCal.tla
index 43161f89..44997f25 100644
--- a/specifications/DiningPhilosophers/APDiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/APDiningPhilosophersPCal.tla
@@ -1,4 +1,4 @@
----- MODULE APDiningPhilosophers ----
+---- MODULE APDiningPhilosophersPCal ----
 (* Apalache type annotations for DiningPhilosophersPCal.tla, applied via
    INSTANCE so the original spec remains free of tool-specific
    idiosyncrasies. *)
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index 7491e783..44081ae5 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -9,11 +9,11 @@
   ],
   "modules": [
     {
-      "path": "specifications/DiningPhilosophers/APDiningPhilosophers.tla",
+      "path": "specifications/DiningPhilosophers/APDiningPhilosophersPCal.tla",
       "features": [],
       "models": [
         {
-          "path": "specifications/DiningPhilosophers/APDiningPhilosophers.cfg",
+          "path": "specifications/DiningPhilosophers/APDiningPhilosophersPCal.cfg",
           "runtime": "00:00:01",
           "mode": "symbolic",
           "result": "success"

From 615db5de3fa4010fdf2c99785ff4a54852f4f1ef Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:32 -0700
Subject: [PATCH 02/14] DP: remove the TLA+ spec to rebuild it in order

The footman solution belongs after taking both forks at once and before
the fork-order solutions. The commits on master already end with the
fork-order solutions, so the footman cannot go in between. Remove
DiningPhilosophers, its animation, and its proof, and add them back one
solution per commit, so that the history reads in order and the README
can link to each step.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 README.md                                     |   2 +-
 .../DiningPhilosophers/DiningPhilosophers.cfg |   4 -
 .../DiningPhilosophers/DiningPhilosophers.tla |  95 ---
 .../DiningPhilosophers_anim.cfg               |   6 -
 .../DiningPhilosophers_anim.gif               | Bin 60149 -> 0 bytes
 .../DiningPhilosophers_anim.tla               | 160 ----
 .../DiningPhilosophers_proof.tla              | 705 ------------------
 .../DiningPhilosophers/manifest.json          |  38 -
 8 files changed, 1 insertion(+), 1009 deletions(-)
 delete mode 100644 specifications/DiningPhilosophers/DiningPhilosophers.cfg
 delete mode 100644 specifications/DiningPhilosophers/DiningPhilosophers.tla
 delete mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg
 delete mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
 delete mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
 delete mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_proof.tla

diff --git a/README.md b/README.md
index e19baadc..7ad1609d 100644
--- a/README.md
+++ b/README.md
@@ -33,7 +33,7 @@ Here is a list of specs included in this repository which are validated by the C
 | [Boyer-Moore Majority Vote](specifications/Majority)                                                | Stephan Merz                                        |    ✔     |      ✔      |         |     ✔     |          |
 | [Proof x+x is Even](specifications/sums_even)                                                       | Martin Riener                                       |    ✔     |      ✔      |         |     ✔     |          |
 | [The N-Queens Puzzle](specifications/N-Queens)                                                      | Stephan Merz                                        |    ✔     |             |    ✔    |     ✔     |          |
-| [The Dining Philosophers Problem](specifications/DiningPhilosophers)                                | Jeff Hemphill, Markus Kuppe                         |    ✔     |      ✔      |   (✔)   |     ✔     |    ✔     |
+| [The Dining Philosophers Problem](specifications/DiningPhilosophers)                                | Jeff Hemphill, Markus Kuppe                         |    ✔     |             |   (✔)   |     ✔     |    ✔     |
 | [The Car Talk Puzzle](specifications/CarTalkPuzzle)                                                 | Leslie Lamport                                      |    ✔     |             |         |     ✔     |          |
 | [The Die Hard Problem](specifications/DieHard)                                                      | Leslie Lamport                                      |    ✔     |      ✔      |         |     ✔     |    ✔     |
 | [The Prisoners & Switches Puzzle](specifications/Prisoners)                                         | Leslie Lamport                                      |    ✔     |             |         |     ✔     |          |
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.cfg b/specifications/DiningPhilosophers/DiningPhilosophers.cfg
deleted file mode 100644
index c1a8d2d7..00000000
--- a/specifications/DiningPhilosophers/DiningPhilosophers.cfg
+++ /dev/null
@@ -1,4 +0,0 @@
-CONSTANTS N = 5
-SPECIFICATION Spec
-INVARIANT TypeOK NoDeadlock
-PROPERTY NoLivelock NoStarvation
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
deleted file mode 100644
index 32bd01b3..00000000
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ /dev/null
@@ -1,95 +0,0 @@
----------------------------- MODULE DiningPhilosophers ----------------------------
-
-EXTENDS Naturals, FiniteSets
-
-CONSTANT N \* Number of philosophers
-ASSUME NIsPosNat == N \in Nat \ {0}
-
-P == 1..N
-
-LeftFork(p) == p
-RightFork(p) == (p % N) + 1
-
-ASSUME ForksDistinct == \A p \in P : RightFork(p) \in P /\ RightFork(p) # LeftFork(p)
-ASSUME ForksCovered == {RightFork(p) : p \in P} = P
-
-VARIABLES philosophers, forks
-vars == <<philosophers, forks>>
-
-TypeOK ==
-    /\ philosophers \in [P -> SUBSET P]
-    /\ forks \in SUBSET P
-
-Init ==
-    \* each philosopher starts with no forks
-    /\ philosophers = [ p \in P |-> {} ]
-    \* all forks are available
-    /\ forks = 1..N
-
-TakeLeft(p) ==
-    \* Philosopher N grabs the right fork first.
-    /\ p = N => philosophers[p] = {RightFork(p)}
-    /\ LeftFork(p) \in forks
-    /\ forks' = forks \ {LeftFork(p)}
-    /\ philosophers' = 
-        [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
-
-TakeRight(p) ==
-    \* All other philosophers grab the left fork first.
-    /\ p # N => philosophers[p] = {LeftFork(p)}
-    /\ RightFork(p) \in forks 
-    /\ forks' = forks \ {RightFork(p)}
-    /\ philosophers' = 
-        [philosophers EXCEPT ![p] = philosophers[p] \cup {RightFork(p)}]
-
-Release(p) ==
-    /\ philosophers[p] = {LeftFork(p), RightFork(p)}
-    /\ forks' = forks \cup {LeftFork(p), RightFork(p)}
-    /\ philosophers' = [philosophers EXCEPT ![p] = {}]
-
-Next ==
-    \E p \in P:
-        \/ TakeLeft(p)
-        \/ TakeRight(p)
-        \/ Release(p)
-
-Spec ==
-    /\ Init
-    /\ [][Next]_vars
-    /\ \A p \in P :
-        SF_vars(TakeLeft(p) \/ TakeRight(p) \/ Release(p))
-
-IsEating(p) ==
-    Cardinality(philosophers[p]) = 2
-
-NoDeadlock ==
-    ENABLED Next
-
-NoLivelock ==
-    []<>(\E p \in P: IsEating(p))
-
-NoStarvation ==
-    \A p \in P: []<>IsEating(p)
-
-\* Proved in DiningPhilosophers_proof.
-THEOREM DeadlockFreedom == Spec => []NoDeadlock
-THEOREM LivelockFreedom == Spec => NoLivelock
-THEOREM StarvationFreedom == Spec => NoStarvation
-
------
-
-Alias ==
-    LET ThinkHungryEat(p) ==
-        IF IsEating(p) THEN "eating"
-        ELSE IF Cardinality(philosophers[p]) = 1 THEN "hungry"
-        ELSE "thinking" IN
-    [
-        philosophers |-> philosophers,
-        forks |-> forks,
-        eating |-> [ p \in DOMAIN philosophers |-> ThinkHungryEat(p) ]
-    ]
-
------------------------------------------------------------------------------
-
-
-=====
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg b/specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg
deleted file mode 100644
index 1a23796b..00000000
--- a/specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg
+++ /dev/null
@@ -1,6 +0,0 @@
-CONSTANT
-    N = 5
-SPECIFICATION 
-    Spec
-ALIAS 
-    AnimAlias
\ No newline at end of file
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
deleted file mode 100644
index d88a0f83cfd939b23e1fc82d2135e67aa376d882..0000000000000000000000000000000000000000
GIT binary patch
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diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
deleted file mode 100644
index cce05147..00000000
--- a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
+++ /dev/null
@@ -1,160 +0,0 @@
----- MODULE DiningPhilosophers_anim ----
-EXTENDS TLC, Integers, Sequences, SVG, DiningPhilosophers
-
-(***************************************************************************)
-(* Every seat is drawn once, in the frame of the philosopher at the top of *)
-(* the table, and then rotated about the table's center.  Philosophers are *)
-(* placed counter-clockwise so that, facing the table, LeftFork(p) is at   *)
-(* p's left hand and RightFork(p) at p's right hand.                       *)
-(***************************************************************************)
-
-\* Table radius.  Plates lie 25 inside the edge and need about 65 between
-\* neighbors, so the table grows with N.
-R == IF 25 + 11 * N > 80 THEN 25 + 11 * N ELSE 80
-
-\* The table's top edge is at y = 70, which all seat coordinates rely on.
-Cx == R + 70
-Cy == R + 70
-
-\* Angles are in tenths of a degree, in 0..7200 to avoid negative numbers.
-SeatAngle(p) == 3600 - (3600 * (p - 1)) \div N
-ForkAngle(f) == SeatAngle(f) + 1800 \div N   \* Fork f lies between seats f and f-1.
-
-Deg(t) == ToString(t \div 10) \o "." \o ToString(t % 10)
-
-RotateAboutCenter(t) ==
-    "rotate(" \o Deg(t) \o " " \o ToString(Cx) \o " " \o ToString(Cy) \o ")"
-
-\* Undo the rotation of the enclosing seat so that labels stay upright.
-Upright(t, x, y) ==
-    "rotate(" \o Deg(7200 - t) \o " " \o ToString(x) \o " " \o ToString(y) \o ")"
-
-Show(cond, elem) == IF cond THEN <<elem>> ELSE <<>>
-
-Label(x, y, t, str, size, color) ==
-    Text(x, y, str,
-        ("text-anchor" :> "middle" @@ "font-family" :> "sans-serif" @@
-         "font-size" :> ToString(size) @@ "fill" :> color @@
-         "transform" :> Upright(t, x, y - size \div 3)))
-
-ThinkingColor == "#9db4d0"
-HungryColor   == "#f2a541"
-EatingColor   == "#5cb85c"
-
-StateColor(p) ==
-    CASE IsEating(p)                      -> EatingColor
-      [] Cardinality(philosophers[p]) = 1 -> HungryColor
-      [] OTHER                            -> ThinkingColor
-
-(***************************************************************************)
-(* A fork centered at (x, y), rotated by a degrees.  Unrotated, its tines  *)
-(* point up.                                                               *)
-(***************************************************************************)
-ForkGlyph(x, y, a) ==
-    LET style(w) == ("stroke" :> "#5a5a5a" @@ "stroke-width" :> w @@
-                     "stroke-linecap" :> "round")
-    IN Group(<<
-        Line(0, 2, 0, 14, style("3")),
-        Line(-4, -2, 4, -2, style("1.5")),
-        Line(-4, -2, -4, -13, style("1.5")),
-        Line( 0, -2,  0, -13, style("1.5")),
-        Line( 4, -2,  4, -13, style("1.5"))
-       >>,
-       ("transform" :> "translate(" \o ToString(x) \o " " \o ToString(y) \o
-                       ") rotate(" \o ToString(a) \o ")"))
-
-Table ==
-    Circle(Cx, Cy, R, ("fill" :> "#deb887" @@ "stroke" :> "#a0764a" @@
-                       "stroke-width" :> "3"))
-
-\* A plate in front of philosopher p, with food on it while p eats.
-Plate(p) ==
-    Group(<<
-        Circle(Cx, 95, 15, ("fill" :> "white" @@ "stroke" :> "#999")),
-        Circle(Cx, 95, 10, ("fill" :> "none" @@ "stroke" :> "#ddd"))
-      >> \o
-      Show(IsEating(p), Circle(Cx, 95, 7, ("fill" :> "#e07b39"))),
-      ("transform" :> RotateAboutCenter(SeatAngle(p))))
-
-\* Philosopher p with an arm and a fork for each fork that p holds.
-Philosopher(p) ==
-    LET t     == SeatAngle(p)
-        arm(x1, x2) == Line(x1, 52, x2, 62,
-                            ("stroke" :> StateColor(p) @@ "stroke-width" :> "6" @@
-                             "stroke-linecap" :> "round"))
-        hand(x)     == Circle(x, 61, 4, ("fill" :> "#f6d7b0" @@ "stroke" :> "#555"))
-        left  == LeftFork(p) \in philosophers[p]
-        right == RightFork(p) \in philosophers[p]
-    IN Group(
-        Show(left,  arm(Cx + 10, Cx + 25)) \o
-        Show(right, arm(Cx - 10, Cx - 25)) \o
-        <<
-          Rect(Cx - 18, 44, 36, 24, ("rx" :> "12" @@ "fill" :> StateColor(p) @@ "stroke" :> "#555")),
-          Circle(Cx, 32, 12, ("fill" :> "#f6d7b0" @@ "stroke" :> "#555")),
-          Label(Cx, 36, t, ToString(p), 12, "#333")
-        >> \o
-        \* Held forks point towards the plate.
-        Show(left,  ForkGlyph(Cx + 22, 70, 200)) \o
-        Show(right, ForkGlyph(Cx - 22, 70, 160)) \o
-        Show(left,  hand(Cx + 25)) \o
-        Show(right, hand(Cx - 25)),
-        ("transform" :> RotateAboutCenter(t)))
-
-\* Fork f on the table; only its number remains while someone holds it.
-TableFork(f) ==
-    Group(
-      Show(f \in forks, ForkGlyph(Cx, 92, 180)) \o
-      <<Label(Cx, 114, ForkAngle(f), ToString(f), 9, "#7a5230")>>,
-      ("transform" :> RotateAboutCenter(ForkAngle(f))))
-
-Legend ==
-    LET x == 2 * Cx + 15
-        entry(i, color, str) ==
-            Group(<<
-                Rect(x, 20 + 22 * i, 14, 14, ("rx" :> "4" @@ "fill" :> color @@ "stroke" :> "#555")),
-                Text(x + 20, 32 + 22 * i, str,
-                     ("font-family" :> "sans-serif" @@ "font-size" :> "11" @@ "fill" :> "#333"))
-              >>, <<>>)
-    IN Group(<<
-        entry(0, ThinkingColor, "thinking"),
-        entry(1, HungryColor,   "hungry (1 fork)"),
-        entry(2, EatingColor,   "eating (2 forks)")
-      >>, <<>>)
-
-\* Scale the drawing, in hundredths, to 600 wide, which fits Spectacle's pane.
-ViewWidth == 600
-Scale == (100 * ViewWidth) \div (2 * Cx + 130)
-ViewHeight == (2 * Cy * Scale) \div 100
-
-ScaleStr ==
-    ToString(Scale \div 100) \o "." \o
-    (IF Scale % 100 < 10 THEN "0" ELSE "") \o ToString(Scale % 100)
-
-\* Spectacle has no TLCGet, so only AnimDoc, not AnimView, shows the step number.
-StepNumber ==
-    Text(ViewWidth - 10, ViewHeight - 10, "Step " \o ToString(TLCGet("level")),
-         ("text-anchor" :> "end" @@ "font-family" :> "monospace" @@
-          "font-size" :> "17" @@ "fill" :> "#666"))
-
-AnimView ==
-    Group(<<Table>> \o
-          [p \in P |-> Plate(p)] \o
-          [f \in P |-> TableFork(f)] \o
-          [p \in P |-> Philosopher(p)] \o
-          <<Legend>>,
-          ("transform" :> "scale(" \o ScaleStr \o " " \o ScaleStr \o ")"))
-
-AnimDoc == SVGDoc(Group(<<AnimView, StepNumber>>, <<>>), 0, 0, ViewWidth, ViewHeight, <<>>)
-
-\* Animation alias for generating SVG files with TLC.
-AnimAlias ==
-    [
-        philosophers |-> philosophers, forks |-> forks
-    ] @@
-    [ _anim |-> SVGSerialize(AnimDoc, "DiningPhilosophers_anim_", TLCGet("level")) ]
-
-\* Watch expression for the TLA+ Debugger; overwrites a single SVG file.
-AnimWatch ==
-    SVGSerialize(AnimDoc, "DiningPhilosophers_anim", 0)
-
-====
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla b/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
deleted file mode 100644
index 0286b8ec..00000000
--- a/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
+++ /dev/null
@@ -1,705 +0,0 @@
-------------------------- MODULE DiningPhilosophers_proof -------------------------
-(***************************************************************************)
-(* Proof that the solution in DiningPhilosophers is starvation free: every *)
-(* philosopher eats infinitely often.                                      *)
-(*                                                                         *)
-(* All philosophers but N take their left fork first, and N takes its      *)
-(* right fork first.  Either way, each philosopher first takes the lower-  *)
-(* numbered of its two forks (Dijkstra's resource hierarchy).              *)
-(*                                                                         *)
-(* A philosopher who holds fork f either eats, and then releases f, or     *)
-(* holds f as its low fork and waits for its high fork, whose number is    *)
-(* larger.  By induction from the largest fork down, every fork is free    *)
-(* infinitely often, so strong fairness lets every philosopher take its    *)
-(* low fork, then its high fork, and eat.                                  *)
-(***************************************************************************)
-EXTENDS DiningPhilosophers, TLAPS, NaturalsInduction, FiniteSetTheorems
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* Fork numbering.                                                         *)
-(***************************************************************************)
-
-LEMMA NAtLeastTwo == N \in Nat /\ N >= 2
-<1>1. N \in Nat /\ N # 0
-  BY NIsPosNat
-<1>2. N # 1
-  <2>. SUFFICES ASSUME N = 1 PROVE FALSE
-    OBVIOUS
-  <2>1. 1 \in P
-    BY <1>1 DEF P
-  <2>2. RightFork(1) # LeftFork(1)
-    BY <2>1, ForksDistinct
-  <2>. QED
-    BY <2>2 DEF RightFork, LeftFork
-<1>. QED
-  BY <1>1, <1>2
-
-LEMMA RightForkDef ==
-    \A q \in P : RightFork(q) = IF q = N THEN 1 ELSE q + 1
-BY NAtLeastTwo DEF P, RightFork
-
-\* The fork that a philosopher takes first, and the one it takes second.
-LowFork(p)  == IF p = N THEN RightFork(p) ELSE LeftFork(p)
-HighFork(p) == IF p = N THEN LeftFork(p) ELSE RightFork(p)
-
-LEMMA ForkFacts ==
-    \A q \in P :
-        /\ LowFork(q) \in P
-        /\ HighFork(q) \in P
-        /\ LowFork(q) < HighFork(q)
-        /\ {LowFork(q), HighFork(q)} = {LeftFork(q), RightFork(q)}
-BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork
-
-\* The philosopher whose right fork is f.
-Pred(f) == IF f = 1 THEN N ELSE f - 1
-
-LEMMA PredInP == \A f \in P : Pred(f) \in P
-BY NAtLeastTwo DEF P, Pred
-
-\* Fork f is only ever held by philosopher f or philosopher Pred(f).
-LEMMA Holders ==
-    \A f, q \in P : f \in {LowFork(q), HighFork(q)} => q = f \/ q = Pred(f)
-BY NAtLeastTwo, ForkFacts, RightForkDef DEF P, LeftFork, Pred
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* Inductive invariant.                                                    *)
-(***************************************************************************)
-
-Idle(p)   == philosophers[p] = {}
-Hungry(p) == philosophers[p] = {LowFork(p)}
-Eating(p) == philosophers[p] = {LowFork(p), HighFork(p)}
-
-Phases == \A p \in P : Idle(p) \/ Hungry(p) \/ Eating(p)
-
-\* A fork lies on the table iff no one holds it ...
-OnTable == \A f \in P : f \in forks <=> \A p \in P : f \notin philosophers[p]
-
-\* ... and no two philosophers hold the same fork.
-Exclusive == \A p, q \in P : p # q => philosophers[p] \cap philosophers[q] = {}
-
-Inv == TypeOK /\ Phases /\ OnTable /\ Exclusive
-
-\* TakeLeft and TakeRight, seen as taking the low fork, then the high one.
-TakeLow(p) ==
-    /\ philosophers[p] = {}
-    /\ LowFork(p) \in forks
-    /\ forks' = forks \ {LowFork(p)}
-    /\ philosophers' = [philosophers EXCEPT ![p] = {LowFork(p)}]
-
-TakeHigh(p) ==
-    /\ philosophers[p] = {LowFork(p)}
-    /\ HighFork(p) \in forks
-    /\ forks' = forks \ {HighFork(p)}
-    /\ philosophers' = [philosophers EXCEPT ![p] = {LowFork(p), HighFork(p)}]
-
-NextLH == \E p \in P : TakeLow(p) \/ TakeHigh(p) \/ Release(p)
-
-LEMMA TakeAgree ==
-    ASSUME Inv, NEW p \in P, TakeLeft(p) \/ TakeRight(p)
-    PROVE  TakeLow(p) \/ TakeHigh(p)
-<1>. USE DEF Inv, TypeOK, LowFork, HighFork
-<1>1. CASE TakeLeft(p) /\ p # N
-  <2>1. LeftFork(p) \in P /\ LeftFork(p) \notin philosophers[p]
-    BY <1>1 DEF TakeLeft, OnTable, LeftFork
-  <2>2. philosophers[p] = {}
-    BY <2>1, <1>1 DEF Phases, Idle, Hungry, Eating
-  <2>. QED
-    BY <1>1, <2>2 DEF TakeLeft, TakeLow
-<1>2. CASE TakeLeft(p) /\ p = N
-  BY <1>2 DEF TakeLeft, TakeHigh
-<1>3. CASE TakeRight(p) /\ p # N
-  BY <1>3 DEF TakeRight, TakeHigh
-<1>4. CASE TakeRight(p) /\ p = N
-  <2>1. RightFork(p) \in P /\ RightFork(p) \notin philosophers[p]
-    BY <1>4, ForksDistinct DEF TakeRight, OnTable
-  <2>2. philosophers[p] = {}
-    BY <2>1, <1>4 DEF Phases, Idle, Hungry, Eating
-  <2>. QED
-    BY <1>4, <2>2 DEF TakeRight, TakeLow
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4
-
-LEMMA NextAgree == ASSUME Inv, Next PROVE NextLH
-BY TakeAgree DEF Next, NextLH
-
-\* What each action does, stated pointwise.
-LEMMA TakeLowFrame ==
-    ASSUME TypeOK, NEW p \in P, TakeLow(p)
-    PROVE  /\ philosophers[p] = {} /\ LowFork(p) \in forks
-           /\ forks' = forks \ {LowFork(p)}
-           /\ philosophers'[p] = {LowFork(p)}
-           /\ \A r \in P : r # p => philosophers'[r] = philosophers[r]
-BY DEF TypeOK, TakeLow
-
-LEMMA TakeHighFrame ==
-    ASSUME TypeOK, NEW p \in P, TakeHigh(p)
-    PROVE  /\ philosophers[p] = {LowFork(p)} /\ HighFork(p) \in forks
-           /\ forks' = forks \ {HighFork(p)}
-           /\ philosophers'[p] = {LowFork(p), HighFork(p)}
-           /\ \A r \in P : r # p => philosophers'[r] = philosophers[r]
-BY DEF TypeOK, TakeHigh
-
-LEMMA ReleaseFrame ==
-    ASSUME TypeOK, NEW p \in P, Release(p)
-    PROVE  /\ philosophers[p] = {LowFork(p), HighFork(p)}
-           /\ forks' = forks \cup {LowFork(p), HighFork(p)}
-           /\ philosophers'[p] = {}
-           /\ \A r \in P : r # p => philosophers'[r] = philosophers[r]
-BY ForkFacts DEF TypeOK, Release
-
-THEOREM TypeOKInit == Init => TypeOK
-BY DEF Init, TypeOK, P
-
-THEOREM TypeOKNext == TypeOK /\ [Next]_vars => TypeOK'
-<1>. SUFFICES ASSUME TypeOK, [Next]_vars PROVE TypeOK'
-  OBVIOUS
-<1>. USE ForkFacts DEF TypeOK
-<1>1. ASSUME NEW p \in P, TakeLeft(p) PROVE TypeOK'
-  BY <1>1 DEF TakeLeft, LeftFork
-<1>2. ASSUME NEW p \in P, TakeRight(p) PROVE TypeOK'
-  BY <1>2, ForksDistinct DEF TakeRight
-<1>3. ASSUME NEW p \in P, Release(p) PROVE TypeOK'
-  BY <1>3, ReleaseFrame DEF Release
-<1>4. CASE UNCHANGED vars
-  BY <1>4 DEF vars
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4 DEF Next
-
-THEOREM PhasesNext == Inv /\ [Next]_vars => Phases'
-<1>. SUFFICES ASSUME Inv, [Next]_vars PROVE Phases'
-  OBVIOUS
-<1>. USE DEF Inv, Phases, Idle, Hungry, Eating
-<1>1. ASSUME NEW p \in P, TakeLow(p) PROVE Phases'
-  BY <1>1, TakeLowFrame
-<1>2. ASSUME NEW p \in P, TakeHigh(p) PROVE Phases'
-  BY <1>2, TakeHighFrame
-<1>3. ASSUME NEW p \in P, Release(p) PROVE Phases'
-  BY <1>3, ReleaseFrame
-<1>4. CASE UNCHANGED vars
-  BY <1>4 DEF vars
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, NextAgree DEF NextLH
-
-THEOREM OnTableNext == Inv /\ [Next]_vars => OnTable'
-<1>. SUFFICES ASSUME Inv, [Next]_vars, NEW g \in P
-              PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
-  BY DEF OnTable
-<1>. USE ForkFacts DEF Inv, TypeOK, OnTable
-<1>1. ASSUME NEW p \in P, TakeLow(p)
-      PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
-  <2>1. CASE g = LowFork(p)
-    BY <1>1, <2>1, TakeLowFrame
-  <2>2. CASE g # LowFork(p)
-    <3>1. \A r \in P : g \in philosophers'[r] <=> g \in philosophers[r]
-      BY <1>1, <2>2, TakeLowFrame
-    <3>. QED
-      BY <1>1, <2>2, <3>1, TakeLowFrame
-  <2>. QED
-    BY <2>1, <2>2
-<1>2. ASSUME NEW p \in P, TakeHigh(p)
-      PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
-  <2>1. CASE g = HighFork(p)
-    BY <1>2, <2>1, TakeHighFrame
-  <2>2. CASE g # HighFork(p)
-    <3>1. \A r \in P : g \in philosophers'[r] <=> g \in philosophers[r]
-      BY <1>2, <2>2, TakeHighFrame
-    <3>. QED
-      BY <1>2, <2>2, <3>1, TakeHighFrame
-  <2>. QED
-    BY <2>1, <2>2
-<1>3. ASSUME NEW p \in P, Release(p)
-      PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
-  <2>1. CASE g \in {LowFork(p), HighFork(p)}
-    BY <1>3, <2>1, ReleaseFrame DEF Exclusive
-  <2>2. CASE g \notin {LowFork(p), HighFork(p)}
-    <3>1. \A r \in P : g \in philosophers'[r] <=> g \in philosophers[r]
-      BY <1>3, <2>2, ReleaseFrame
-    <3>. QED
-      BY <1>3, <2>2, <3>1, ReleaseFrame
-  <2>. QED
-    BY <2>1, <2>2
-<1>4. CASE UNCHANGED vars
-  BY <1>4 DEF vars
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, NextAgree DEF NextLH
-
-THEOREM ExclusiveNext == Inv /\ [Next]_vars => Exclusive'
-<1>. SUFFICES ASSUME Inv, [Next]_vars,
-                     NEW r1 \in P, NEW r2 \in P, r1 # r2
-              PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
-  BY DEF Exclusive
-<1>. USE DEF Inv, TypeOK
-<1>1. ASSUME NEW p \in P, TakeLow(p)
-      PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
-  <2>1. \A r \in P : LowFork(p) \notin philosophers[r]
-    BY <1>1, TakeLowFrame DEF OnTable
-  <2>. QED
-    BY <1>1, <2>1, TakeLowFrame DEF Exclusive
-<1>2. ASSUME NEW p \in P, TakeHigh(p)
-      PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
-  <2>1. \A r \in P : HighFork(p) \notin philosophers[r]
-    BY <1>2, TakeHighFrame DEF OnTable
-  <2>2. \A r \in P : r # p => LowFork(p) \notin philosophers[r]
-    <3>. SUFFICES ASSUME NEW r \in P, r # p PROVE LowFork(p) \notin philosophers[r]
-      OBVIOUS
-    <3>1. philosophers[p] \cap philosophers[r] = {}
-      BY DEF Exclusive
-    <3>. QED
-      BY <1>2, <3>1, TakeHighFrame
-  <2>. QED
-    BY <1>2, <2>1, <2>2, TakeHighFrame DEF Exclusive
-<1>3. ASSUME NEW p \in P, Release(p)
-      PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
-  BY <1>3, ReleaseFrame DEF Exclusive
-<1>4. CASE UNCHANGED vars
-  BY <1>4 DEF vars, Exclusive
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, NextAgree DEF NextLH
-
-THEOREM InvInit == Init => Inv
-BY ForkFacts, TypeOKInit DEF Init, Inv, Phases, OnTable, Exclusive, Idle, P
-
-THEOREM InvNext == Inv /\ [Next]_vars => Inv'
-BY TypeOKNext, PhasesNext, OnTableNext, ExclusiveNext DEF Inv
-
-THEOREM InvSpec == Spec => []Inv
-BY InvInit, InvNext, PTL DEF Spec
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* The steps of a single philosopher.                                      *)
-(***************************************************************************)
-
-Act(p) == TakeLeft(p) \/ TakeRight(p) \/ Release(p)
-
-LEMMA SpecNext == Spec => [][Next]_vars
-BY DEF Spec
-
-LEMMA SpecFair == \A q \in P : Spec => SF_vars(Act(q))
-BY DEF Spec, Act
-
-\* A step either is a step of q or leaves q's forks alone.
-LEMMA NextSplit ==
-    ASSUME Inv, [Next]_vars, NEW q \in P
-    PROVE  <<Act(q)>>_vars \/ philosophers'[q] = philosophers[q]
-<1>1. CASE UNCHANGED vars
-  BY <1>1 DEF vars
-<1>2. ASSUME NEW p \in P, p # q, Act(p)
-      PROVE  philosophers'[q] = philosophers[q]
-  BY <1>2 DEF Act, TakeLeft, TakeRight, Release, Inv, TypeOK
-<1>3. CASE Act(q)
-  BY <1>3 DEF vars
-<1>. QED
-  BY <1>1, <1>2, <1>3 DEF Next, Act
-
-\* A step of q is a step of TakeLow, TakeHigh, or Release.
-LEMMA ActAgree ==
-    ASSUME Inv, NEW q \in P, Act(q)
-    PROVE  TakeLow(q) \/ TakeHigh(q) \/ Release(q)
-BY TakeAgree DEF Act
-
-LEMMA IdleStep ==
-    ASSUME Inv, NEW q \in P, Idle(q), <<Act(q)>>_vars
-    PROVE  Hungry(q)'
-BY ActAgree, ForkFacts DEF Inv, TypeOK, Idle, Hungry, TakeLow, TakeHigh, Release
-
-LEMMA HungryStep ==
-    ASSUME Inv, NEW q \in P, Hungry(q), <<Act(q)>>_vars
-    PROVE  Eating(q)'
-BY ActAgree, ForkFacts DEF Inv, TypeOK, Hungry, Eating, TakeLow, TakeHigh, Release
-
-LEMMA EatingStep ==
-    ASSUME Inv, NEW q \in P, Eating(q), <<Act(q)>>_vars
-    PROVE  \A f \in philosophers[q] : f \in forks'
-BY ActAgree, ForkFacts DEF Inv, TypeOK, Eating, TakeLow, TakeHigh, Release
-
-LEMMA IdleEnabled ==
-    ASSUME NEW q \in P
-    PROVE  Inv /\ Idle(q) /\ LowFork(q) \in forks => ENABLED <<Act(q)>>_vars
-<1>. SUFFICES ASSUME Inv, Idle(q), LowFork(q) \in forks
-              PROVE  ENABLED <<Act(q)>>_vars
-  OBVIOUS
-<1>1. philosophers \in [P -> SUBSET P] /\ philosophers[q] = {}
-  BY DEF Inv, TypeOK, Idle
-<1>2. CASE q # N
-  <2>1. LeftFork(q) \in forks
-    BY <1>2 DEF LowFork
-  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {LeftFork(q)}],
-          forks \ {LeftFork(q)}>> # <<philosophers, forks>>
-    BY <1>1
-  <2>3. ENABLED <<TakeLeft(q)>>_vars
-    BY <1>1, <1>2, <2>1, <2>2, ExpandENABLED DEF TakeLeft, vars
-  <2>. QED
-    BY <2>3, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-<1>3. CASE q = N
-  <2>1. RightFork(q) \in forks
-    BY <1>3 DEF LowFork
-  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {RightFork(q)}],
-          forks \ {RightFork(q)}>> # <<philosophers, forks>>
-    BY <1>1
-  <2>3. ENABLED <<TakeRight(q)>>_vars
-    BY <1>1, <1>3, <2>1, <2>2, ExpandENABLED DEF TakeRight, vars
-  <2>. QED
-    BY <2>3, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-<1>. QED
-  BY <1>2, <1>3
-
-LEMMA HungryEnabled ==
-    ASSUME NEW q \in P
-    PROVE  Inv /\ Hungry(q) /\ HighFork(q) \in forks => ENABLED <<Act(q)>>_vars
-<1>. SUFFICES ASSUME Inv, Hungry(q), HighFork(q) \in forks
-              PROVE  ENABLED <<Act(q)>>_vars
-  OBVIOUS
-<1>1. philosophers \in [P -> SUBSET P] /\ philosophers[q] = {LowFork(q)}
-  BY DEF Inv, TypeOK, Hungry
-<1>2. LeftFork(q) # RightFork(q)
-  BY ForksDistinct
-<1>3. CASE q # N
-  <2>1. philosophers[q] = {LeftFork(q)} /\ RightFork(q) \in forks
-    BY <1>1, <1>3 DEF LowFork, HighFork
-  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {RightFork(q)}],
-          forks \ {RightFork(q)}>> # <<philosophers, forks>>
-    BY <1>1, <1>2, <2>1
-  <2>3. (q # N => philosophers[q] = {LeftFork(q)}) /\ RightFork(q) \in forks
-    BY <2>1
-  <2>4. ENABLED <<TakeRight(q)>>_vars
-    BY <2>2, <2>3, ExpandENABLED DEF TakeRight, vars
-  <2>. QED
-    BY <2>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-<1>4. CASE q = N
-  <2>1. philosophers[q] = {RightFork(q)} /\ LeftFork(q) \in forks
-    BY <1>1, <1>4 DEF LowFork, HighFork
-  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {LeftFork(q)}],
-          forks \ {LeftFork(q)}>> # <<philosophers, forks>>
-    BY <1>1, <1>2, <2>1
-  <2>3. (q = N => philosophers[q] = {RightFork(q)}) /\ LeftFork(q) \in forks
-    BY <2>1
-  <2>4. ENABLED <<TakeLeft(q)>>_vars
-    BY <2>2, <2>3, ExpandENABLED DEF TakeLeft, vars
-  <2>. QED
-    BY <2>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-<1>. QED
-  BY <1>3, <1>4
-
-LEMMA EatingEnabled ==
-    ASSUME NEW q \in P
-    PROVE  Inv /\ Eating(q) => ENABLED <<Act(q)>>_vars
-<1>. SUFFICES ASSUME Inv, Eating(q)
-              PROVE  ENABLED <<Act(q)>>_vars
-  OBVIOUS
-<1>1. philosophers \in [P -> SUBSET P]
-      /\ philosophers[q] = {LeftFork(q), RightFork(q)}
-  BY ForkFacts DEF Inv, TypeOK, Eating
-<1>2. {LeftFork(q), RightFork(q)} # {}
-  OBVIOUS
-<1>3. <<[philosophers EXCEPT ![q] = {}], forks \cup {LeftFork(q), RightFork(q)}>>
-        # <<philosophers, forks>>
-  BY <1>1, <1>2
-<1>4. ENABLED <<Release(q)>>_vars
-  BY <1>1, <1>3, ExpandENABLED DEF Release, vars
-<1>. QED
-  BY <1>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* Progress of a single philosopher.                                       *)
-(***************************************************************************)
-
-FreeIO(f) == []<>(f \in forks)
-
-HeldEat(f, q) == Eating(q) /\ f \in philosophers[q]
-
-\* Named so that the lemmas below can be instantiated: backends other than
-\* PTL only see temporal formulas through opaque operators.
-IdleToHungry(q)   == Idle(q) ~> Hungry(q)
-HungryToEating(q) == Hungry(q) ~> Eating(q)
-EatingFrees(f, q) == HeldEat(f, q) ~> (f \in forks)
-HungryFrees(f, q) == (Hungry(q) /\ f = LowFork(q)) ~> (f \in forks)
-
-LEMMA IdleHungry ==
-    ASSUME NEW q \in P
-    PROVE  Spec /\ FreeIO(LowFork(q)) => IdleToHungry(q)
-<1>1. Inv /\ Idle(q) /\ [Next]_vars => Idle(q)' \/ Hungry(q)'
-  <2>. SUFFICES ASSUME Inv, Idle(q), [Next]_vars PROVE Idle(q)' \/ Hungry(q)'
-    OBVIOUS
-  <2>1. CASE philosophers'[q] = philosophers[q]
-    BY <2>1 DEF Idle
-  <2>2. CASE <<Act(q)>>_vars
-    BY <2>2, IdleStep
-  <2>. QED
-    BY <2>1, <2>2, NextSplit
-<1>2. Inv /\ Idle(q) /\ <<Act(q)>>_vars => Hungry(q)'
-  BY IdleStep
-<1>3. Inv /\ Idle(q) /\ LowFork(q) \in forks => ENABLED <<Act(q)>>_vars
-  BY IdleEnabled
-<1>4. Spec => SF_vars(Act(q))
-  BY SpecFair
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF FreeIO, IdleToHungry
-
-LEMMA HungryEat ==
-    ASSUME NEW q \in P
-    PROVE  Spec /\ FreeIO(HighFork(q)) => HungryToEating(q)
-<1>1. Inv /\ Hungry(q) /\ [Next]_vars => Hungry(q)' \/ Eating(q)'
-  <2>. SUFFICES ASSUME Inv, Hungry(q), [Next]_vars PROVE Hungry(q)' \/ Eating(q)'
-    OBVIOUS
-  <2>1. CASE philosophers'[q] = philosophers[q]
-    BY <2>1 DEF Hungry
-  <2>2. CASE <<Act(q)>>_vars
-    BY <2>2, HungryStep
-  <2>. QED
-    BY <2>1, <2>2, NextSplit
-<1>2. Inv /\ Hungry(q) /\ <<Act(q)>>_vars => Eating(q)'
-  BY HungryStep
-<1>3. Inv /\ Hungry(q) /\ HighFork(q) \in forks => ENABLED <<Act(q)>>_vars
-  BY HungryEnabled
-<1>4. Spec => SF_vars(Act(q))
-  BY SpecFair
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF FreeIO, HungryToEating
-
-LEMMA EatRelease ==
-    ASSUME NEW f \in P, NEW q \in P
-    PROVE  Spec => EatingFrees(f, q)
-<1>1. Inv /\ HeldEat(f, q) /\ [Next]_vars => HeldEat(f, q)' \/ (f \in forks)'
-  <2>. SUFFICES ASSUME Inv, HeldEat(f, q), [Next]_vars
-                PROVE  HeldEat(f, q)' \/ (f \in forks)'
-    OBVIOUS
-  <2>1. CASE philosophers'[q] = philosophers[q]
-    BY <2>1 DEF HeldEat, Eating
-  <2>2. CASE <<Act(q)>>_vars
-    BY <2>2, EatingStep DEF HeldEat
-  <2>. QED
-    BY <2>1, <2>2, NextSplit
-<1>2. Inv /\ HeldEat(f, q) /\ <<Act(q)>>_vars => (f \in forks)'
-  BY EatingStep DEF HeldEat
-<1>3. Inv /\ HeldEat(f, q) => ENABLED <<Act(q)>>_vars
-  BY EatingEnabled DEF HeldEat
-<1>4. Spec => SF_vars(Act(q))
-  BY SpecFair
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF EatingFrees
-
-\* A hungry philosopher who holds f as its low fork eventually frees f,
-\* provided that all forks larger than f are free infinitely often.
-LEMMA HungryRelease ==
-    ASSUME NEW f \in P, NEW q \in P
-    PROVE  Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, q)
-<1>1. CASE f = LowFork(q)
-  <2>1. HighFork(q) \in P /\ HighFork(q) > f
-    BY <1>1, ForkFacts
-  <2>2. (\A g \in P : g > f => FreeIO(g)) => FreeIO(HighFork(q))
-    BY <2>1
-  <2>3. Spec /\ FreeIO(HighFork(q)) => HungryToEating(q)
-    BY HungryEat
-  <2>4. Spec => EatingFrees(f, q)
-    BY EatRelease
-  <2>5. Eating(q) => HeldEat(f, q)
-    BY <1>1 DEF HeldEat, Eating
-  <2>6. (Hungry(q) /\ f = LowFork(q)) => Hungry(q)
-    OBVIOUS
-  <2>. QED
-    BY <2>2, <2>3, <2>4, <2>5, <2>6, PTL
-       DEF HungryToEating, EatingFrees, HungryFrees
-<1>2. CASE f # LowFork(q)
-  <2>1. ~(Hungry(q) /\ f = LowFork(q))
-    BY <1>2
-  <2>. QED
-    BY <2>1, PTL DEF HungryFrees
-<1>. QED
-  BY <1>1, <1>2
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* Every fork is free infinitely often.                                    *)
-(***************************************************************************)
-
-LEMMA HeldBy ==
-    ASSUME NEW f \in P
-    PROVE  Inv /\ ~(f \in forks) =>
-              \/ HeldEat(f, f) \/ HeldEat(f, Pred(f))
-              \/ Hungry(f) /\ f = LowFork(f)
-              \/ Hungry(Pred(f)) /\ f = LowFork(Pred(f))
-<1>. SUFFICES ASSUME Inv, ~(f \in forks)
-              PROVE  \/ HeldEat(f, f) \/ HeldEat(f, Pred(f))
-                     \/ Hungry(f) /\ f = LowFork(f)
-                     \/ Hungry(Pred(f)) /\ f = LowFork(Pred(f))
-  OBVIOUS
-<1>1. PICK q \in P : f \in philosophers[q]
-  BY DEF Inv, OnTable
-<1>2. f \in {LowFork(q), HighFork(q)}
-  BY <1>1 DEF Inv, Phases, Idle, Hungry, Eating
-<1>3. q = f \/ q = Pred(f)
-  BY <1>2, Holders
-<1>4. HeldEat(f, q) \/ (Hungry(q) /\ f = LowFork(q))
-  BY <1>1 DEF Inv, Phases, Idle, Hungry, Eating, HeldEat
-<1>. QED
-  BY <1>3, <1>4
-
-LEMMA FreeStep ==
-    ASSUME NEW f \in P
-    PROVE  Spec /\ (\A g \in P : g > f => FreeIO(g)) => FreeIO(f)
-<1>1. Pred(f) \in P
-  BY PredInP
-<1>2. Inv /\ ~(f \in forks) =>
-          \/ HeldEat(f, f) \/ HeldEat(f, Pred(f))
-          \/ Hungry(f) /\ f = LowFork(f)
-          \/ Hungry(Pred(f)) /\ f = LowFork(Pred(f))
-  BY HeldBy
-<1>3. Spec => EatingFrees(f, f)
-  BY EatRelease
-<1>4. Spec => EatingFrees(f, Pred(f))
-  BY <1>1, EatRelease
-<1>5. Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, f)
-  BY HungryRelease
-<1>6. Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, Pred(f))
-  BY <1>1, HungryRelease
-<1>. QED
-  BY <1>2, <1>3, <1>4, <1>5, <1>6, InvSpec, PTL
-     DEF FreeIO, EatingFrees, HungryFrees
-
-THEOREM AllFree == \A f \in P : Spec => FreeIO(f)
-<1>. DEFINE Q(m) == Spec => \A g \in P : g + m > N => FreeIO(g)
-<1>1. Q(0)
-  BY NAtLeastTwo DEF P
-<1>2. \A m \in Nat : Q(m) => Q(m + 1)
-  <2>. SUFFICES ASSUME NEW m \in Nat, Q(m)
-                PROVE  Q(m + 1)
-    OBVIOUS
-  <2>1. ASSUME NEW g \in P, g + (m + 1) > N
-        PROVE  Spec => FreeIO(g)
-    <3>1. CASE g + m > N
-      BY <3>1
-    <3>2. CASE ~(g + m > N)
-      <4>1. \A h \in P : h > g => h + m > N
-        BY <2>1, <3>2, NAtLeastTwo DEF P
-      <4>2. Spec => \A h \in P : h > g => FreeIO(h)
-        BY <4>1
-      <4>. QED
-        BY <4>2, FreeStep
-    <3>. QED
-      BY <3>1, <3>2
-  <2>. QED
-    BY <2>1
-<1>. HIDE DEF Q
-<1>3. \A m \in Nat : Q(m)
-  BY <1>1, <1>2, NatInduction
-<1>4. Q(N)
-  BY <1>3, NAtLeastTwo
-<1>. QED
-  BY <1>4, NAtLeastTwo DEF Q, P
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* Starvation freedom.                                                     *)
-(***************************************************************************)
-
-LEMMA EatingIsEating == \A p \in P : Eating(p) => IsEating(p)
-<1>. SUFFICES ASSUME NEW p \in P, Eating(p) PROVE IsEating(p)
-  OBVIOUS
-<1>1. LowFork(p) # HighFork(p)
-  BY ForkFacts
-<1>2. IsFiniteSet({LowFork(p)}) /\ Cardinality({LowFork(p)}) = 1
-  BY FS_Singleton
-<1>3. Cardinality({LowFork(p)} \cup {HighFork(p)}) = 2
-  BY <1>1, <1>2, FS_AddElement
-<1>. QED
-  BY <1>3 DEF Eating, IsEating
-
-THEOREM StarvationFreedom
-<1>. SUFFICES ASSUME NEW p \in P PROVE Spec => []<>IsEating(p)
-  BY DEF NoStarvation
-<1>1. Spec => FreeIO(LowFork(p))
-  BY AllFree, ForkFacts
-<1>2. Spec => FreeIO(HighFork(p))
-  BY AllFree, ForkFacts
-<1>3. Spec /\ FreeIO(LowFork(p)) => IdleToHungry(p)
-  BY IdleHungry
-<1>4. Spec /\ FreeIO(HighFork(p)) => HungryToEating(p)
-  BY HungryEat
-<1>5. Inv => Idle(p) \/ Hungry(p) \/ Eating(p)
-  BY DEF Inv, Phases
-<1>6. Eating(p) => IsEating(p)
-  BY EatingIsEating
-<1>. QED
-  BY <1>1, <1>2, <1>3, <1>4, <1>5, <1>6, InvSpec, PTL
-     DEF IdleToHungry, HungryToEating
-
-EatsIO(p) == []<>IsEating(p)
-
-THEOREM StarvationImpliesNoLivelock == NoStarvation => NoLivelock
-<1>1. 1 \in P
-  BY NIsPosNat DEF P
-<1>2. NoStarvation => \A p \in P : EatsIO(p)
-  BY DEF NoStarvation, EatsIO
-<1>3. NoStarvation => EatsIO(1)
-  BY <1>1, <1>2
-<1>4. IsEating(1) => \E p \in P : IsEating(p)
-  BY <1>1
-<1>. QED
-  BY <1>3, <1>4, PTL DEF EatsIO, NoLivelock
-
-THEOREM LivelockFreedom
-BY StarvationFreedom, StarvationImpliesNoLivelock
-
------------------------------------------------------------------------------
-(***************************************************************************)
-(* Deadlock freedom.  If no one eats, fork N is free, because it is the    *)
-(* high fork of both philosophers who can hold it.  So philosopher N-1 or  *)
-(* N can take it if hungry; if both are idle, fork N-1 is free too.        *)
-(***************************************************************************)
-
-LEMMA ActEnabledNext ==
-    ASSUME NEW q \in P
-    PROVE  (ENABLED <<Act(q)>>_vars) => ENABLED Next
-BY ExpandENABLED DEF Act, Next, TakeLeft, TakeRight, Release, vars
-
-LEMMA TopForks ==
-    /\ N - 1 \in P /\ N \in P
-    /\ LowFork(N - 1) = N - 1 /\ HighFork(N - 1) = N
-    /\ LowFork(N) = 1 /\ HighFork(N) = N
-    /\ Pred(N) = N - 1 /\ 1 # N /\ N - 1 # N
-    /\ Pred(N - 1) = N \/ LowFork(Pred(N - 1)) # N - 1
-BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork, Pred
-
-THEOREM InvNoDeadlock == Inv => NoDeadlock
-<1>. SUFFICES ASSUME Inv PROVE ENABLED Next
-  BY DEF NoDeadlock
-<1>. DEFINE M == N - 1
-<1>0. TopForks
-  BY TopForks
-<1>1. CASE \E q \in P : Eating(q)
-  BY <1>1, EatingEnabled, ActEnabledNext
-<1>2. CASE \A q \in P : ~Eating(q)
-  <2>1. ~HeldEat(N, N) /\ ~HeldEat(N, Pred(N))
-        /\ ~HeldEat(M, M) /\ ~HeldEat(M, Pred(M))
-    BY <1>0, <1>2, PredInP DEF HeldEat
-  <2>2. N \in forks
-    <3>1. N # LowFork(N) /\ N # LowFork(Pred(N))
-      BY <1>0
-    <3>. QED
-      BY <1>0, <2>1, <3>1, HeldBy
-  <2>3. CASE Hungry(M)
-    BY <1>0, <2>2, <2>3, HungryEnabled, ActEnabledNext
-  <2>4. CASE Hungry(N)
-    BY <1>0, <2>2, <2>4, HungryEnabled, ActEnabledNext
-  <2>5. CASE ~Hungry(M) /\ ~Hungry(N)
-    <3>1. Idle(M) /\ Idle(N)
-      BY <1>0, <1>2, <2>5 DEF Inv, Phases
-    <3>2. ~Hungry(Pred(M)) \/ LowFork(Pred(M)) # M
-      BY <1>0, <2>5
-    <3>3. M \in forks
-      BY <1>0, <2>1, <2>5, <3>2, HeldBy
-    <3>. QED
-      BY <1>0, <3>1, <3>3, IdleEnabled, ActEnabledNext
-  <2>. QED
-    BY <2>3, <2>4, <2>5
-<1>. QED
-  BY <1>1, <1>2
-
-THEOREM DeadlockFreedom
-BY InvNoDeadlock, InvSpec, PTL
-
-=============================================================================
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index 44081ae5..ad6bb467 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -20,21 +20,6 @@
         }
       ]
     },
-    {
-      "path": "specifications/DiningPhilosophers/DiningPhilosophers.tla",
-      "features": [],
-      "models": [
-        {
-          "path": "specifications/DiningPhilosophers/DiningPhilosophers.cfg",
-          "runtime": "00:00:01",
-          "mode": "exhaustive search",
-          "result": "success",
-          "distinctStates": 70,
-          "totalStates": 220,
-          "stateDepth": 6
-        }
-      ]
-    },
     {
       "path": "specifications/DiningPhilosophers/DiningPhilosophersPCal.tla",
       "features": [
@@ -51,29 +36,6 @@
           "stateDepth": 29
         }
       ]
-    },
-    {
-      "path": "specifications/DiningPhilosophers/DiningPhilosophers_anim.tla",
-      "features": [],
-      "models": [
-        {
-          "path": "specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg",
-          "runtime": "00:00:01",
-          "mode": "exhaustive search",
-          "result": "success",
-          "distinctStates": 70,
-          "totalStates": 220,
-          "stateDepth": 6
-        }
-      ]
-    },
-    {
-      "path": "specifications/DiningPhilosophers/DiningPhilosophers_proof.tla",
-      "features": [],
-      "models": [],
-      "proof": {
-        "maxRuntimeMinutes": 3
-      }
     }
   ]
 }
\ No newline at end of file

From 8c5d5f7a3671e9f39ffb08867ff6abf83cc7b973 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:32 -0700
Subject: [PATCH 03/14] DP: add a README that walks through the solutions

The folder holds two independent solutions, and the TLA+ one is best
read as a series of commits. Add the README before the series, so that
each of the following commits adds its step to the table. A commit
cannot link to itself, so each commit links the step before it.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 specifications/DiningPhilosophers/README.md | 35 +++++++++++++++++++++
 1 file changed, 35 insertions(+)
 create mode 100644 specifications/DiningPhilosophers/README.md

diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
new file mode 100644
index 00000000..8af4cfa2
--- /dev/null
+++ b/specifications/DiningPhilosophers/README.md
@@ -0,0 +1,35 @@
+# Dining Philosophers
+
+This folder contains specifications of the
+[Dining Philosophers problem](https://en.wikipedia.org/wiki/Dining_philosophers_problem).
+
+## From naive to correct (TLA+)
+
+The git history builds a TLA+ spec of the forks, one solution per
+commit. The table lists, for each commit, the properties for which TLC
+finds a counterexample:
+
+| Step | Solution | Known as | Coordination | Deadlock | Livelock | Starvation | Spectacle |
+|---|---|---|---|---|---|---|---|
+
+This repository has specs of two other solutions:
+
+- **Chandy-Misra:** philosophers hand clean forks to their neighbors on
+  request. See [`DiningPhilosophersPCal.tla`](DiningPhilosophersPCal.tla)
+  [below](#chandy-misra-pluscal).
+- **Bakery:** hungry philosophers take a ticket, and the lowest ticket eats
+  first. The ticket order gives a fair, first-come-first-served solution.
+  Lamport's Bakery algorithm and its Boulangerie variant are in
+  [`Bakery-Boulangerie`](../Bakery-Boulangerie). They solve mutual exclusion,
+  not dining, but the ticket idea carries over.
+
+## Chandy-Misra (PlusCal)
+
+[`DiningPhilosophersPCal.tla`](DiningPhilosophersPCal.tla) is a PlusCal
+algorithm of the
+[Chandy-Misra solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Chandy/Misra_solution).
+Unlike in Dijkstra's formulation, philosophers here talk to each other: they
+hand clean forks to their neighbors on request. It is based on the exercise
+at [learntla.com](https://learntla.com/temporal-logic/operators/).
+[`APDiningPhilosophersPCal.tla`](APDiningPhilosophersPCal.tla) adds the type
+annotations that Apalache needs.

From 89da1aa3e41b33fdb00062fa0a0a57ef4494e211 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 04/14] DP: add a pure TLA+ spec with an animation

The PlusCal spec fixes one solution in code, so it cannot show why the
naive solutions fail. DiningPhilosophers models the forks directly:
philosophers take their left and right forks in any order and release
both after eating. Strong fairness of each philosopher makes
NoStarvation counterexamples show actual starvation. NoDeadlock and
NoLivelock tell whether the table as a whole makes progress. The
ASSUMEs reject fork layouts in which a fork is outside 1..N, a
philosopher's two forks are the same, or a fork is no one's right fork.

TLC finds the deadlock where every philosopher holds one fork.

DiningPhilosophers_anim draws the table, plates, forks, and
philosophers for any N, colored by state, and writes one SVG per state
with SVGSerialize. The spec and animation come from Spectacle:
https://github.com/will62794/spectacle

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.cfg |   4 +
 .../DiningPhilosophers/DiningPhilosophers.tla |  86 ++++++++++
 .../DiningPhilosophers_anim.cfg               |   6 +
 .../DiningPhilosophers_anim.gif               | Bin 0 -> 28577 bytes
 .../DiningPhilosophers_anim.tla               | 160 ++++++++++++++++++
 specifications/DiningPhilosophers/README.md   |  16 +-
 .../DiningPhilosophers/manifest.json          |  24 +++
 7 files changed, 292 insertions(+), 4 deletions(-)
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers.cfg
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers.tla
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_anim.tla

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.cfg b/specifications/DiningPhilosophers/DiningPhilosophers.cfg
new file mode 100644
index 00000000..c1a8d2d7
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.cfg
@@ -0,0 +1,4 @@
+CONSTANTS N = 5
+SPECIFICATION Spec
+INVARIANT TypeOK NoDeadlock
+PROPERTY NoLivelock NoStarvation
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
new file mode 100644
index 00000000..263bb3f5
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -0,0 +1,86 @@
+---------------------------- MODULE DiningPhilosophers ----------------------------
+
+EXTENDS Naturals, FiniteSets
+
+CONSTANT N \* Number of philosophers
+ASSUME NIsPosNat == N \in Nat \ {0}
+
+P == 1..N
+
+LeftFork(p) == p
+RightFork(p) == (p % N) + 1
+
+ASSUME ForksDistinct == \A p \in P : RightFork(p) \in P /\ RightFork(p) # LeftFork(p)
+ASSUME ForksCovered == {RightFork(p) : p \in P} = P
+
+VARIABLES philosophers, forks
+vars == <<philosophers, forks>>
+
+TypeOK ==
+    /\ philosophers \in [P -> SUBSET P]
+    /\ forks \in SUBSET P
+
+Init ==
+    \* each philosopher starts with no forks
+    /\ philosophers = [ p \in P |-> {} ]
+    \* all forks are available
+    /\ forks = 1..N
+
+TakeLeft(p) ==
+    /\ LeftFork(p) \in forks
+    /\ forks' = forks \ {LeftFork(p)}
+    /\ philosophers' = 
+        [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
+
+TakeRight(p) ==
+    /\ RightFork(p) \in forks 
+    /\ forks' = forks \ {RightFork(p)}
+    /\ philosophers' = 
+        [philosophers EXCEPT ![p] = philosophers[p] \cup {RightFork(p)}]
+
+Release(p) ==
+    /\ philosophers[p] = {LeftFork(p), RightFork(p)}
+    /\ forks' = forks \cup {LeftFork(p), RightFork(p)}
+    /\ philosophers' = [philosophers EXCEPT ![p] = {}]
+
+Next ==
+    \E p \in P:
+        \/ TakeLeft(p)
+        \/ TakeRight(p)
+        \/ Release(p)
+
+Spec ==
+    /\ Init
+    /\ [][Next]_vars
+    /\ \A p \in P :
+        SF_vars(TakeLeft(p) \/ TakeRight(p) \/ Release(p))
+
+IsEating(p) ==
+    Cardinality(philosophers[p]) = 2
+
+NoDeadlock ==
+    ENABLED Next
+
+NoLivelock ==
+    []<>(\E p \in P: IsEating(p))
+
+NoStarvation ==
+    \A p \in P: []<>IsEating(p)
+
+-----
+
+Alias ==
+    LET ThinkHungryEat(p) ==
+        IF IsEating(p) THEN "eating"
+        ELSE IF Cardinality(philosophers[p]) = 1 THEN "hungry"
+        ELSE "thinking" IN
+    [
+        philosophers |-> philosophers,
+        forks |-> forks,
+        eating |-> [ p \in DOMAIN philosophers |-> ThinkHungryEat(p) ]
+    ]
+
+-----------------------------------------------------------------------------
+
+
+=====
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg b/specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg
new file mode 100644
index 00000000..1a23796b
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg
@@ -0,0 +1,6 @@
+CONSTANT
+    N = 5
+SPECIFICATION 
+    Spec
+ALIAS 
+    AnimAlias
\ No newline at end of file
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
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diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
new file mode 100644
index 00000000..cce05147
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
@@ -0,0 +1,160 @@
+---- MODULE DiningPhilosophers_anim ----
+EXTENDS TLC, Integers, Sequences, SVG, DiningPhilosophers
+
+(***************************************************************************)
+(* Every seat is drawn once, in the frame of the philosopher at the top of *)
+(* the table, and then rotated about the table's center.  Philosophers are *)
+(* placed counter-clockwise so that, facing the table, LeftFork(p) is at   *)
+(* p's left hand and RightFork(p) at p's right hand.                       *)
+(***************************************************************************)
+
+\* Table radius.  Plates lie 25 inside the edge and need about 65 between
+\* neighbors, so the table grows with N.
+R == IF 25 + 11 * N > 80 THEN 25 + 11 * N ELSE 80
+
+\* The table's top edge is at y = 70, which all seat coordinates rely on.
+Cx == R + 70
+Cy == R + 70
+
+\* Angles are in tenths of a degree, in 0..7200 to avoid negative numbers.
+SeatAngle(p) == 3600 - (3600 * (p - 1)) \div N
+ForkAngle(f) == SeatAngle(f) + 1800 \div N   \* Fork f lies between seats f and f-1.
+
+Deg(t) == ToString(t \div 10) \o "." \o ToString(t % 10)
+
+RotateAboutCenter(t) ==
+    "rotate(" \o Deg(t) \o " " \o ToString(Cx) \o " " \o ToString(Cy) \o ")"
+
+\* Undo the rotation of the enclosing seat so that labels stay upright.
+Upright(t, x, y) ==
+    "rotate(" \o Deg(7200 - t) \o " " \o ToString(x) \o " " \o ToString(y) \o ")"
+
+Show(cond, elem) == IF cond THEN <<elem>> ELSE <<>>
+
+Label(x, y, t, str, size, color) ==
+    Text(x, y, str,
+        ("text-anchor" :> "middle" @@ "font-family" :> "sans-serif" @@
+         "font-size" :> ToString(size) @@ "fill" :> color @@
+         "transform" :> Upright(t, x, y - size \div 3)))
+
+ThinkingColor == "#9db4d0"
+HungryColor   == "#f2a541"
+EatingColor   == "#5cb85c"
+
+StateColor(p) ==
+    CASE IsEating(p)                      -> EatingColor
+      [] Cardinality(philosophers[p]) = 1 -> HungryColor
+      [] OTHER                            -> ThinkingColor
+
+(***************************************************************************)
+(* A fork centered at (x, y), rotated by a degrees.  Unrotated, its tines  *)
+(* point up.                                                               *)
+(***************************************************************************)
+ForkGlyph(x, y, a) ==
+    LET style(w) == ("stroke" :> "#5a5a5a" @@ "stroke-width" :> w @@
+                     "stroke-linecap" :> "round")
+    IN Group(<<
+        Line(0, 2, 0, 14, style("3")),
+        Line(-4, -2, 4, -2, style("1.5")),
+        Line(-4, -2, -4, -13, style("1.5")),
+        Line( 0, -2,  0, -13, style("1.5")),
+        Line( 4, -2,  4, -13, style("1.5"))
+       >>,
+       ("transform" :> "translate(" \o ToString(x) \o " " \o ToString(y) \o
+                       ") rotate(" \o ToString(a) \o ")"))
+
+Table ==
+    Circle(Cx, Cy, R, ("fill" :> "#deb887" @@ "stroke" :> "#a0764a" @@
+                       "stroke-width" :> "3"))
+
+\* A plate in front of philosopher p, with food on it while p eats.
+Plate(p) ==
+    Group(<<
+        Circle(Cx, 95, 15, ("fill" :> "white" @@ "stroke" :> "#999")),
+        Circle(Cx, 95, 10, ("fill" :> "none" @@ "stroke" :> "#ddd"))
+      >> \o
+      Show(IsEating(p), Circle(Cx, 95, 7, ("fill" :> "#e07b39"))),
+      ("transform" :> RotateAboutCenter(SeatAngle(p))))
+
+\* Philosopher p with an arm and a fork for each fork that p holds.
+Philosopher(p) ==
+    LET t     == SeatAngle(p)
+        arm(x1, x2) == Line(x1, 52, x2, 62,
+                            ("stroke" :> StateColor(p) @@ "stroke-width" :> "6" @@
+                             "stroke-linecap" :> "round"))
+        hand(x)     == Circle(x, 61, 4, ("fill" :> "#f6d7b0" @@ "stroke" :> "#555"))
+        left  == LeftFork(p) \in philosophers[p]
+        right == RightFork(p) \in philosophers[p]
+    IN Group(
+        Show(left,  arm(Cx + 10, Cx + 25)) \o
+        Show(right, arm(Cx - 10, Cx - 25)) \o
+        <<
+          Rect(Cx - 18, 44, 36, 24, ("rx" :> "12" @@ "fill" :> StateColor(p) @@ "stroke" :> "#555")),
+          Circle(Cx, 32, 12, ("fill" :> "#f6d7b0" @@ "stroke" :> "#555")),
+          Label(Cx, 36, t, ToString(p), 12, "#333")
+        >> \o
+        \* Held forks point towards the plate.
+        Show(left,  ForkGlyph(Cx + 22, 70, 200)) \o
+        Show(right, ForkGlyph(Cx - 22, 70, 160)) \o
+        Show(left,  hand(Cx + 25)) \o
+        Show(right, hand(Cx - 25)),
+        ("transform" :> RotateAboutCenter(t)))
+
+\* Fork f on the table; only its number remains while someone holds it.
+TableFork(f) ==
+    Group(
+      Show(f \in forks, ForkGlyph(Cx, 92, 180)) \o
+      <<Label(Cx, 114, ForkAngle(f), ToString(f), 9, "#7a5230")>>,
+      ("transform" :> RotateAboutCenter(ForkAngle(f))))
+
+Legend ==
+    LET x == 2 * Cx + 15
+        entry(i, color, str) ==
+            Group(<<
+                Rect(x, 20 + 22 * i, 14, 14, ("rx" :> "4" @@ "fill" :> color @@ "stroke" :> "#555")),
+                Text(x + 20, 32 + 22 * i, str,
+                     ("font-family" :> "sans-serif" @@ "font-size" :> "11" @@ "fill" :> "#333"))
+              >>, <<>>)
+    IN Group(<<
+        entry(0, ThinkingColor, "thinking"),
+        entry(1, HungryColor,   "hungry (1 fork)"),
+        entry(2, EatingColor,   "eating (2 forks)")
+      >>, <<>>)
+
+\* Scale the drawing, in hundredths, to 600 wide, which fits Spectacle's pane.
+ViewWidth == 600
+Scale == (100 * ViewWidth) \div (2 * Cx + 130)
+ViewHeight == (2 * Cy * Scale) \div 100
+
+ScaleStr ==
+    ToString(Scale \div 100) \o "." \o
+    (IF Scale % 100 < 10 THEN "0" ELSE "") \o ToString(Scale % 100)
+
+\* Spectacle has no TLCGet, so only AnimDoc, not AnimView, shows the step number.
+StepNumber ==
+    Text(ViewWidth - 10, ViewHeight - 10, "Step " \o ToString(TLCGet("level")),
+         ("text-anchor" :> "end" @@ "font-family" :> "monospace" @@
+          "font-size" :> "17" @@ "fill" :> "#666"))
+
+AnimView ==
+    Group(<<Table>> \o
+          [p \in P |-> Plate(p)] \o
+          [f \in P |-> TableFork(f)] \o
+          [p \in P |-> Philosopher(p)] \o
+          <<Legend>>,
+          ("transform" :> "scale(" \o ScaleStr \o " " \o ScaleStr \o ")"))
+
+AnimDoc == SVGDoc(Group(<<AnimView, StepNumber>>, <<>>), 0, 0, ViewWidth, ViewHeight, <<>>)
+
+\* Animation alias for generating SVG files with TLC.
+AnimAlias ==
+    [
+        philosophers |-> philosophers, forks |-> forks
+    ] @@
+    [ _anim |-> SVGSerialize(AnimDoc, "DiningPhilosophers_anim_", TLCGet("level")) ]
+
+\* Watch expression for the TLA+ Debugger; overwrites a single SVG file.
+AnimWatch ==
+    SVGSerialize(AnimDoc, "DiningPhilosophers_anim", 0)
+
+====
diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 8af4cfa2..57d818b5 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -1,16 +1,18 @@
 # Dining Philosophers
 
-This folder contains specifications of the
+This folder contains two independent specifications of the
 [Dining Philosophers problem](https://en.wikipedia.org/wiki/Dining_philosophers_problem).
 
 ## From naive to correct (TLA+)
 
-The git history builds a TLA+ spec of the forks, one solution per
-commit. The table lists, for each commit, the properties for which TLC
-finds a counterexample:
+[`DiningPhilosophers.tla`](DiningPhilosophers.tla) models the forks directly.
+
+The git history shows how the spec got there. The table lists, for each
+commit, the properties for which TLC finds a counterexample:
 
 | Step | Solution | Known as | Coordination | Deadlock | Livelock | Starvation | Spectacle |
 |---|---|---|---|---|---|---|---|
+| 1 | Take the forks in any order | Unordered fork acquisition | none | yes | yes | yes |  |
 
 This repository has specs of two other solutions:
 
@@ -23,6 +25,12 @@ This repository has specs of two other solutions:
   [`Bakery-Boulangerie`](../Bakery-Boulangerie). They solve mutual exclusion,
   not dining, but the ticket idea carries over.
 
+[`DiningPhilosophers_anim.tla`](DiningPhilosophers_anim.tla) draws the table,
+forks, and philosophers for any `N` and writes one SVG per state. The
+animation comes from [Spectacle](https://github.com/will62794/spectacle).
+
+![Animation of DiningPhilosophers](DiningPhilosophers_anim.gif)
+
 ## Chandy-Misra (PlusCal)
 
 [`DiningPhilosophersPCal.tla`](DiningPhilosophersPCal.tla) is a PlusCal
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index ad6bb467..17bd0824 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -20,6 +20,18 @@
         }
       ]
     },
+    {
+      "path": "specifications/DiningPhilosophers/DiningPhilosophers.tla",
+      "features": [],
+      "models": [
+        {
+          "path": "specifications/DiningPhilosophers/DiningPhilosophers.cfg",
+          "runtime": "00:00:01",
+          "mode": "exhaustive search",
+          "result": "safety failure"
+        }
+      ]
+    },
     {
       "path": "specifications/DiningPhilosophers/DiningPhilosophersPCal.tla",
       "features": [
@@ -36,6 +48,18 @@
           "stateDepth": 29
         }
       ]
+    },
+    {
+      "path": "specifications/DiningPhilosophers/DiningPhilosophers_anim.tla",
+      "features": [],
+      "models": [
+        {
+          "path": "specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg",
+          "runtime": "00:00:01",
+          "mode": "exhaustive search",
+          "result": "deadlock failure"
+        }
+      ]
     }
   ]
 }
\ No newline at end of file

From bf33e1bb350d46fc55904002e5de3807466de573 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 05/14] DP: take the left fork before the right fork

TakeRight had no ordering guard, so philosophers could grab either
fork first. Require the left fork to be held before taking the right
one, which models the naive left-then-right algorithm. TLC finds the
classic deadlock where every philosopher holds only the left fork.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |   1 +
 .../DiningPhilosophers_anim.gif               | Bin 28577 -> 28487 bytes
 specifications/DiningPhilosophers/README.md   |   3 ++-
 3 files changed, 3 insertions(+), 1 deletion(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index 263bb3f5..cb26d800 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -33,6 +33,7 @@ TakeLeft(p) ==
         [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
 
 TakeRight(p) ==
+    /\ LeftFork(p) \in philosophers[p]
     /\ RightFork(p) \in forks 
     /\ forks' = forks \ {RightFork(p)}
     /\ philosophers' = 
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
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diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 57d818b5..16df8484 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -12,7 +12,8 @@ commit, the properties for which TLC finds a counterexample:
 
 | Step | Solution | Known as | Coordination | Deadlock | Livelock | Starvation | Spectacle |
 |---|---|---|---|---|---|---|---|
-| 1 | Take the forks in any order | Unordered fork acquisition | none | yes | yes | yes |  |
+| 1 | [Take the forks in any order](https://github.com/tlaplus/examples/commit/89da1aa3e41b33fdb00062fa0a0a57ef4494e211) | Unordered fork acquisition | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F89da1aa3e41b33fdb00062fa0a0a57ef4494e211%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 2 | Take the left fork before the right fork | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes |  |
 
 This repository has specs of two other solutions:
 

From 3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 06/14] DP: put the left fork back when the right fork is taken

The naive left-then-right algorithm deadlocks once every philosopher
holds a left fork. Let a philosopher who cannot get the right fork put
the left fork back and retry. This removes the deadlock, but TLC still
finds NoStarvation violations: philosophers can livelock by taking and
returning their left forks, or starve while a neighbor keeps eating.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |   9 ++++++++-
 .../DiningPhilosophers_anim.gif               | Bin 28487 -> 76564 bytes
 specifications/DiningPhilosophers/README.md   |   3 ++-
 .../DiningPhilosophers/manifest.json          |   7 +++++--
 4 files changed, 15 insertions(+), 4 deletions(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index cb26d800..7af434cf 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -39,6 +39,12 @@ TakeRight(p) ==
     /\ philosophers' = 
         [philosophers EXCEPT ![p] = philosophers[p] \cup {RightFork(p)}]
 
+PutBack(p) ==
+    /\ philosophers[p] = {LeftFork(p)}
+    /\ RightFork(p) \notin forks
+    /\ forks' = forks \cup {LeftFork(p)}
+    /\ philosophers' = [philosophers EXCEPT ![p] = {}]
+
 Release(p) ==
     /\ philosophers[p] = {LeftFork(p), RightFork(p)}
     /\ forks' = forks \cup {LeftFork(p), RightFork(p)}
@@ -48,13 +54,14 @@ Next ==
     \E p \in P:
         \/ TakeLeft(p)
         \/ TakeRight(p)
+        \/ PutBack(p)
         \/ Release(p)
 
 Spec ==
     /\ Init
     /\ [][Next]_vars
     /\ \A p \in P :
-        SF_vars(TakeLeft(p) \/ TakeRight(p) \/ Release(p))
+        SF_vars(TakeLeft(p) \/ TakeRight(p) \/ PutBack(p) \/ Release(p))
 
 IsEating(p) ==
     Cardinality(philosophers[p]) = 2
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
index b6a0d35ca1f1d9599f0ce718665ff6f2268fabd2..2c00bc2130f5054832e184d5e6a75e6279f2b5c7 100644
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diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 16df8484..d3c04025 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -13,7 +13,8 @@ commit, the properties for which TLC finds a counterexample:
 | Step | Solution | Known as | Coordination | Deadlock | Livelock | Starvation | Spectacle |
 |---|---|---|---|---|---|---|---|
 | 1 | [Take the forks in any order](https://github.com/tlaplus/examples/commit/89da1aa3e41b33fdb00062fa0a0a57ef4494e211) | Unordered fork acquisition | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F89da1aa3e41b33fdb00062fa0a0a57ef4494e211%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 2 | Take the left fork before the right fork | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes |  |
+| 2 | [Take the left fork before the right fork](https://github.com/tlaplus/examples/commit/bf33e1bb350d46fc55904002e5de3807466de573) | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fbf33e1bb350d46fc55904002e5de3807466de573%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 3 | Put the left fork back if the right fork is taken | Back-off and retry | none | no | yes | yes |  |
 
 This repository has specs of two other solutions:
 
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index 17bd0824..7a4d40f5 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -28,7 +28,7 @@
           "path": "specifications/DiningPhilosophers/DiningPhilosophers.cfg",
           "runtime": "00:00:01",
           "mode": "exhaustive search",
-          "result": "safety failure"
+          "result": "liveness failure"
         }
       ]
     },
@@ -57,7 +57,10 @@
           "path": "specifications/DiningPhilosophers/DiningPhilosophers_anim.cfg",
           "runtime": "00:00:01",
           "mode": "exhaustive search",
-          "result": "deadlock failure"
+          "result": "success",
+          "distinctStates": 82,
+          "totalStates": 351,
+          "stateDepth": 6
         }
       ]
     }

From de09177ff01f1c821ce17a0243286a27d605eba5 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 07/14] DP: take both forks at once

Philosophers who put the left fork back can livelock. Replace TakeLeft,
TakeRight, and PutBack with TakeBoth, which takes both forks in one
step and only when both are free (Dijkstra's solution, as in
Tanenbaum's monitor version). No philosopher ever holds a single fork,
so there is no deadlock or livelock. TLC still finds a NoStarvation
violation: a philosopher's two neighbors can eat in overlapping turns,
so the philosopher's forks are never both free, and not even strong
fairness forces it to eat.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |  28 ++++--------------
 .../DiningPhilosophers_anim.gif               | Bin 76564 -> 87441 bytes
 specifications/DiningPhilosophers/README.md   |   3 +-
 .../DiningPhilosophers/manifest.json          |   6 ++--
 4 files changed, 11 insertions(+), 26 deletions(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index 7af434cf..3b3aaee6 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -26,24 +26,10 @@ Init ==
     \* all forks are available
     /\ forks = 1..N
 
-TakeLeft(p) ==
-    /\ LeftFork(p) \in forks
-    /\ forks' = forks \ {LeftFork(p)}
-    /\ philosophers' = 
-        [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
-
-TakeRight(p) ==
-    /\ LeftFork(p) \in philosophers[p]
-    /\ RightFork(p) \in forks 
-    /\ forks' = forks \ {RightFork(p)}
-    /\ philosophers' = 
-        [philosophers EXCEPT ![p] = philosophers[p] \cup {RightFork(p)}]
-
-PutBack(p) ==
-    /\ philosophers[p] = {LeftFork(p)}
-    /\ RightFork(p) \notin forks
-    /\ forks' = forks \cup {LeftFork(p)}
-    /\ philosophers' = [philosophers EXCEPT ![p] = {}]
+TakeBoth(p) ==
+    /\ {LeftFork(p), RightFork(p)} \subseteq forks
+    /\ forks' = forks \ {LeftFork(p), RightFork(p)}
+    /\ philosophers' = [philosophers EXCEPT ![p] = {LeftFork(p), RightFork(p)}]
 
 Release(p) ==
     /\ philosophers[p] = {LeftFork(p), RightFork(p)}
@@ -52,16 +38,14 @@ Release(p) ==
 
 Next ==
     \E p \in P:
-        \/ TakeLeft(p)
-        \/ TakeRight(p)
-        \/ PutBack(p)
+        \/ TakeBoth(p)
         \/ Release(p)
 
 Spec ==
     /\ Init
     /\ [][Next]_vars
     /\ \A p \in P :
-        SF_vars(TakeLeft(p) \/ TakeRight(p) \/ PutBack(p) \/ Release(p))
+        SF_vars(TakeBoth(p) \/ Release(p))
 
 IsEating(p) ==
     Cardinality(philosophers[p]) = 2
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
index 2c00bc2130f5054832e184d5e6a75e6279f2b5c7..e77d7bdefa4577a2e53f42d457f1e9a12f6047a6 100644
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diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index d3c04025..257b415e 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -14,7 +14,8 @@ commit, the properties for which TLC finds a counterexample:
 |---|---|---|---|---|---|---|---|
 | 1 | [Take the forks in any order](https://github.com/tlaplus/examples/commit/89da1aa3e41b33fdb00062fa0a0a57ef4494e211) | Unordered fork acquisition | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F89da1aa3e41b33fdb00062fa0a0a57ef4494e211%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 2 | [Take the left fork before the right fork](https://github.com/tlaplus/examples/commit/bf33e1bb350d46fc55904002e5de3807466de573) | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fbf33e1bb350d46fc55904002e5de3807466de573%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 3 | Put the left fork back if the right fork is taken | Back-off and retry | none | no | yes | yes |  |
+| 3 | [Put the left fork back if the right fork is taken](https://github.com/tlaplus/examples/commit/3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa) | Back-off and retry | none | no | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 4 | Take both forks at once | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes |  |
 
 This repository has specs of two other solutions:
 
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index 7a4d40f5..ad17735e 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -58,9 +58,9 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 82,
-          "totalStates": 351,
-          "stateDepth": 6
+          "distinctStates": 11,
+          "totalStates": 31,
+          "stateDepth": 3
         }
       ]
     }

From 173c60f8d1375217f61b901094b0dd36acd967c2 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 08/14] DP: let a philosopher who ate wait for its neighbors

Taking both forks at once lets a philosopher's neighbors starve it by
eating in overlapping turns. Make a philosopher who releases its forks
wait until both neighbors have eaten before it takes them again. The
neighbors of a hungry philosopher thus cannot eat twice while it waits.
Philosophers who wait for each other never form a cycle: of two
neighbors, the one who ate last waits for the other. TLC finds no
violation of NoDeadlock, NoLivelock, or NoStarvation for N = 2 to 7,
even under weak fairness.

The new variable waits relates only neighbors. The solution promises
that every philosopher eats eventually, but not that hungry philosophers
eat in the order in which they got hungry.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |  16 ++++++++++++++--
 .../DiningPhilosophers_anim.gif               | Bin 87441 -> 44359 bytes
 specifications/DiningPhilosophers/README.md   |   3 ++-
 .../DiningPhilosophers/manifest.json          |  11 +++++++----
 4 files changed, 23 insertions(+), 7 deletions(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index 3b3aaee6..12471f7b 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -13,28 +13,39 @@ RightFork(p) == (p % N) + 1
 ASSUME ForksDistinct == \A p \in P : RightFork(p) \in P /\ RightFork(p) # LeftFork(p)
 ASSUME ForksCovered == {RightFork(p) : p \in P} = P
 
-VARIABLES philosophers, forks
-vars == <<philosophers, forks>>
+Neighbors(p) ==
+    {q \in P \ {p} : {LeftFork(q), RightFork(q)} \cap {LeftFork(p), RightFork(p)} # {}}
+
+VARIABLES philosophers, forks, waits
+vars == <<philosophers, forks, waits>>
 
 TypeOK ==
     /\ philosophers \in [P -> SUBSET P]
     /\ forks \in SUBSET P
+    /\ waits \in [P -> SUBSET P]
 
 Init ==
     \* each philosopher starts with no forks
     /\ philosophers = [ p \in P |-> {} ]
     \* all forks are available
     /\ forks = 1..N
+    \* no philosopher has eaten yet
+    /\ waits = [ p \in P |-> {} ]
 
 TakeBoth(p) ==
+    \* p has let its neighbors eat since it last ate.
+    /\ waits[p] = {}
     /\ {LeftFork(p), RightFork(p)} \subseteq forks
     /\ forks' = forks \ {LeftFork(p), RightFork(p)}
     /\ philosophers' = [philosophers EXCEPT ![p] = {LeftFork(p), RightFork(p)}]
+    /\ waits' = [q \in P |-> waits[q] \ {p}]
 
 Release(p) ==
     /\ philosophers[p] = {LeftFork(p), RightFork(p)}
     /\ forks' = forks \cup {LeftFork(p), RightFork(p)}
     /\ philosophers' = [philosophers EXCEPT ![p] = {}]
+    \* p waits until both neighbors have eaten.
+    /\ waits' = [waits EXCEPT ![p] = Neighbors(p)]
 
 Next ==
     \E p \in P:
@@ -69,6 +80,7 @@ Alias ==
     [
         philosophers |-> philosophers,
         forks |-> forks,
+        waits |-> waits,
         eating |-> [ p \in DOMAIN philosophers |-> ThinkHungryEat(p) ]
     ]
 
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
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diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 257b415e..b38b7cb7 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -15,7 +15,8 @@ commit, the properties for which TLC finds a counterexample:
 | 1 | [Take the forks in any order](https://github.com/tlaplus/examples/commit/89da1aa3e41b33fdb00062fa0a0a57ef4494e211) | Unordered fork acquisition | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F89da1aa3e41b33fdb00062fa0a0a57ef4494e211%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 2 | [Take the left fork before the right fork](https://github.com/tlaplus/examples/commit/bf33e1bb350d46fc55904002e5de3807466de573) | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fbf33e1bb350d46fc55904002e5de3807466de573%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 3 | [Put the left fork back if the right fork is taken](https://github.com/tlaplus/examples/commit/3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa) | Back-off and retry | none | no | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 4 | Take both forks at once | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes |  |
+| 4 | [Take both forks at once](https://github.com/tlaplus/examples/commit/de09177ff01f1c821ce17a0243286a27d605eba5) | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fde09177ff01f1c821ce17a0243286a27d605eba5%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 5 | Let a philosopher who ate wait for its neighbors | [Priority to waiting neighbors](https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD310.html) | neighbors + atomic fork pair | no | no | no |  |
 
 This repository has specs of two other solutions:
 
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index ad17735e..fd9ae14b 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -28,7 +28,10 @@
           "path": "specifications/DiningPhilosophers/DiningPhilosophers.cfg",
           "runtime": "00:00:01",
           "mode": "exhaustive search",
-          "result": "liveness failure"
+          "result": "success",
+          "distinctStates": 161,
+          "totalStates": 331,
+          "stateDepth": 11
         }
       ]
     },
@@ -58,9 +61,9 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 11,
-          "totalStates": 31,
-          "stateDepth": 3
+          "distinctStates": 161,
+          "totalStates": 331,
+          "stateDepth": 11
         }
       ]
     }

From 637ef15fc210e5785bb3f9a68769392ca36601d3 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 09/14] DP: let a waiter serve philosophers in order

Waiting for the neighbors makes every philosopher eat eventually, but
hungry philosophers do not eat in the order in which they got hungry.
Replace waits with a waiter who keeps a queue of hungry philosophers and
lets a philosopher take both forks once no neighbor asked before it.
Neighbors thus eat first come, first served, and philosophers who share
no fork do not wait for each other. TLC finds no violation of
NoDeadlock, NoLivelock, or NoStarvation for N = 2 to 7, even under weak
fairness.

The queue holds every hungry philosopher, so the waiter is global state.

DiningPhilosophers_stats.tla measures with TLC how many philosophers
eat at once, on average. The README now shows the average per step.

TLC's -generate does not choose uniformly among the philosophers who
can act: https://github.com/tlaplus/tlaplus/issues/1485. Keeping Next
one action reduces the bias, so TLC's averages come close to the exact
ones computed by PRISM.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |  34 +++++++++++-------
 .../DiningPhilosophers_anim.gif               | Bin 44359 -> 40185 bytes
 .../DiningPhilosophers_stats.cfg              |   5 +++
 .../DiningPhilosophers_stats.tla              |  33 +++++++++++++++++
 specifications/DiningPhilosophers/README.md   |  25 +++++++++----
 .../DiningPhilosophers/manifest.json          |  25 +++++++++----
 6 files changed, 96 insertions(+), 26 deletions(-)
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_stats.cfg
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_stats.tla

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index 12471f7b..f6964c30 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -1,6 +1,6 @@
 ---------------------------- MODULE DiningPhilosophers ----------------------------
 
-EXTENDS Naturals, FiniteSets
+EXTENDS Naturals, FiniteSets, Sequences
 
 CONSTANT N \* Number of philosophers
 ASSUME NIsPosNat == N \in Nat \ {0}
@@ -16,39 +16,47 @@ ASSUME ForksCovered == {RightFork(p) : p \in P} = P
 Neighbors(p) ==
     {q \in P \ {p} : {LeftFork(q), RightFork(q)} \cap {LeftFork(p), RightFork(p)} # {}}
 
-VARIABLES philosophers, forks, waits
-vars == <<philosophers, forks, waits>>
+VARIABLES philosophers, forks, queue
+vars == <<philosophers, forks, queue>>
 
 TypeOK ==
     /\ philosophers \in [P -> SUBSET P]
     /\ forks \in SUBSET P
-    /\ waits \in [P -> SUBSET P]
+    /\ queue \in Seq(P)
 
 Init ==
     \* each philosopher starts with no forks
     /\ philosophers = [ p \in P |-> {} ]
     \* all forks are available
     /\ forks = 1..N
-    \* no philosopher has eaten yet
-    /\ waits = [ p \in P |-> {} ]
+    \* the waiter's queue of hungry philosophers is empty
+    /\ queue = <<>>
+
+Request(p) ==
+    /\ philosophers[p] = {}
+    /\ \A i \in 1..Len(queue) : queue[i] # p
+    /\ queue' = Append(queue, p)
+    /\ UNCHANGED <<philosophers, forks>>
 
 TakeBoth(p) ==
-    \* p has let its neighbors eat since it last ate.
-    /\ waits[p] = {}
+    \* The waiter serves p once no neighbor of p asked before p.
+    /\ \E i \in 1..Len(queue) :
+        /\ queue[i] = p
+        /\ \A j \in 1..(i - 1) : queue[j] \notin Neighbors(p)
+        /\ queue' = SubSeq(queue, 1, i - 1) \o SubSeq(queue, i + 1, Len(queue))
     /\ {LeftFork(p), RightFork(p)} \subseteq forks
     /\ forks' = forks \ {LeftFork(p), RightFork(p)}
     /\ philosophers' = [philosophers EXCEPT ![p] = {LeftFork(p), RightFork(p)}]
-    /\ waits' = [q \in P |-> waits[q] \ {p}]
 
 Release(p) ==
     /\ philosophers[p] = {LeftFork(p), RightFork(p)}
     /\ forks' = forks \cup {LeftFork(p), RightFork(p)}
     /\ philosophers' = [philosophers EXCEPT ![p] = {}]
-    \* p waits until both neighbors have eaten.
-    /\ waits' = [waits EXCEPT ![p] = Neighbors(p)]
+    /\ UNCHANGED queue
 
 Next ==
     \E p \in P:
+        \/ Request(p)
         \/ TakeBoth(p)
         \/ Release(p)
 
@@ -56,7 +64,7 @@ Spec ==
     /\ Init
     /\ [][Next]_vars
     /\ \A p \in P :
-        SF_vars(TakeBoth(p) \/ Release(p))
+        SF_vars(Request(p) \/ TakeBoth(p) \/ Release(p))
 
 IsEating(p) ==
     Cardinality(philosophers[p]) = 2
@@ -80,7 +88,7 @@ Alias ==
     [
         philosophers |-> philosophers,
         forks |-> forks,
-        waits |-> waits,
+        queue |-> queue,
         eating |-> [ p \in DOMAIN philosophers |-> ThinkHungryEat(p) ]
     ]
 
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
index 8240c82b2d4a59ac7d6287da99054525e6598661..5de6ab090fe79394714756239c492c5cf72ace65 100644
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diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_stats.cfg b/specifications/DiningPhilosophers/DiningPhilosophers_stats.cfg
new file mode 100644
index 00000000..40b251a0
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_stats.cfg
@@ -0,0 +1,5 @@
+CONSTANT N = 5
+INIT Init
+NEXT StatsNext
+CONSTRAINT Stats
+POSTCONDITION PrintStats
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_stats.tla b/specifications/DiningPhilosophers/DiningPhilosophers_stats.tla
new file mode 100644
index 00000000..bb5fda25
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_stats.tla
@@ -0,0 +1,33 @@
+------------------------ MODULE DiningPhilosophers_stats ------------------------
+(***************************************************************************)
+(* Estimate the average number of eating philosophers per state.  TLC      *)
+(* generates random behaviors of DiningPhilosophers and sums over their    *)
+(* states.  The spec does not model time, so states stand in for it.  A    *)
+(* solution with fewer steps per meal thus has more eating states.         *)
+(***************************************************************************)
+EXTENDS DiningPhilosophers, TLC
+
+ASSUME
+    /\ TLCGet("config").mode = "generate"
+    \* TLC's registers are local to a worker.
+    /\ TLCGet("config").worker = 1
+    \* Register 1 sums the eating philosophers, register 2 counts the states.
+    /\ TLCSet(1, 0)
+    /\ TLCSet(2, 0)
+
+\* The conjunct keeps TLC from splitting Next into subactions, which reduces
+\* the bias of -generate: https://github.com/tlaplus/tlaplus/issues/1485
+StatsNext == Next /\ TRUE
+
+Stats ==
+    /\ TLCSet(1, TLCGet(1) + Cardinality({p \in P : IsEating(p)}))
+    /\ TLCSet(2, TLCGet(2) + 1)
+
+\* The average is eating / states.
+PrintStats ==
+    PrintT([eating |-> TLCGet(1), states |-> TLCGet(2)])
+=============================================================================
+
+Check out the commit of a solution, then run:
+
+$ tlc DiningPhilosophers_stats -generate num=1000 -depth 1000
diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index b38b7cb7..6e221adc 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -10,13 +10,14 @@ This folder contains two independent specifications of the
 The git history shows how the spec got there. The table lists, for each
 commit, the properties for which TLC finds a counterexample:
 
-| Step | Solution | Known as | Coordination | Deadlock | Livelock | Starvation | Spectacle |
-|---|---|---|---|---|---|---|---|
-| 1 | [Take the forks in any order](https://github.com/tlaplus/examples/commit/89da1aa3e41b33fdb00062fa0a0a57ef4494e211) | Unordered fork acquisition | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F89da1aa3e41b33fdb00062fa0a0a57ef4494e211%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 2 | [Take the left fork before the right fork](https://github.com/tlaplus/examples/commit/bf33e1bb350d46fc55904002e5de3807466de573) | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fbf33e1bb350d46fc55904002e5de3807466de573%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 3 | [Put the left fork back if the right fork is taken](https://github.com/tlaplus/examples/commit/3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa) | Back-off and retry | none | no | yes | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 4 | [Take both forks at once](https://github.com/tlaplus/examples/commit/de09177ff01f1c821ce17a0243286a27d605eba5) | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fde09177ff01f1c821ce17a0243286a27d605eba5%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 5 | Let a philosopher who ate wait for its neighbors | [Priority to waiting neighbors](https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD310.html) | neighbors + atomic fork pair | no | no | no |  |
+| Step | Solution | Known as | Coordination | Deadlock | Livelock | Starvation | Eating[^eating] | Spectacle |
+|---|---|---|---|---|---|---|---|---|
+| 1 | [Take the forks in any order](https://github.com/tlaplus/examples/commit/89da1aa3e41b33fdb00062fa0a0a57ef4494e211) | Unordered fork acquisition | none | yes | yes | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F89da1aa3e41b33fdb00062fa0a0a57ef4494e211%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 2 | [Take the left fork before the right fork](https://github.com/tlaplus/examples/commit/bf33e1bb350d46fc55904002e5de3807466de573) | [Naive (symmetric) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Problems) | none | yes | yes | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fbf33e1bb350d46fc55904002e5de3807466de573%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 3 | [Put the left fork back if the right fork is taken](https://github.com/tlaplus/examples/commit/3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa) | Back-off and retry | none | no | yes | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 4 | [Take both forks at once](https://github.com/tlaplus/examples/commit/de09177ff01f1c821ce17a0243286a27d605eba5) | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fde09177ff01f1c821ce17a0243286a27d605eba5%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 5 | [Let a philosopher who ate wait for its neighbors](https://github.com/tlaplus/examples/commit/173c60f8d1375217f61b901094b0dd36acd967c2) | [Priority to waiting neighbors](https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD310.html) | neighbors + atomic fork pair | no | no | no | —[^step5] | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F173c60f8d1375217f61b901094b0dd36acd967c2%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 6 | Let a waiter serve philosophers in order | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 |  |
 
 This repository has specs of two other solutions:
 
@@ -35,6 +36,16 @@ animation comes from [Spectacle](https://github.com/will62794/spectacle).
 
 ![Animation of DiningPhilosophers](DiningPhilosophers_anim.gif)
 
+[^eating]: Average number of philosophers eating per state, for `N = 5`. The
+    spec does not model time, so states stand in for it. In each step, a
+    random philosopher who can act takes one step. TLC's averages come close
+    to the exact ones computed by [PRISM](https://www.prismmodelchecker.org).
+
+[^step5]: Not measured. A meal takes three steps in the other solutions but
+    two with priority to waiting neighbors, because philosophers get hungry
+    without a step. More of its states are thus eating states, so its average
+    is not comparable.
+
 ## Chandy-Misra (PlusCal)
 
 [`DiningPhilosophersPCal.tla`](DiningPhilosophersPCal.tla) is a PlusCal
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index fd9ae14b..774f118e 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -29,9 +29,9 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 161,
-          "totalStates": 331,
-          "stateDepth": 11
+          "distinctStates": 731,
+          "totalStates": 1951,
+          "stateDepth": 8
         }
       ]
     },
@@ -61,9 +61,22 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 161,
-          "totalStates": 331,
-          "stateDepth": 11
+          "distinctStates": 731,
+          "totalStates": 1951,
+          "stateDepth": 8
+        }
+      ]
+    },
+    {
+      "path": "specifications/DiningPhilosophers/DiningPhilosophers_stats.tla",
+      "features": [],
+      "models": [
+        {
+          "path": "specifications/DiningPhilosophers/DiningPhilosophers_stats.cfg",
+          "runtime": "unknown",
+          "mode": "generate",
+          "result": "unknown",
+          "workers": 1
         }
       ]
     }

From 20efef8f7b1b790d3643284d7c96434453580581 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 10/14] DP: let a footman seat at most N-1 philosophers

The waiter keeps a queue of every hungry philosopher, and TakeBoth takes
two forks in one step, which no lock on a single fork provides. Go
back to TakeLeft and TakeRight, with every philosopher taking the left
fork first, and add a footman who seats at most N-1 philosophers, as in
Hoare's CSP book. Not every philosopher can hold one fork, so the
circular wait cannot close. TLC finds no violation of NoDeadlock,
NoLivelock, or NoStarvation for N = 2 to 7.

The footman needs strong fairness: under weak fairness, TLC finds a
NoStarvation violation in which a philosopher never gets a seat. Strong
fairness plays the part of a fair semaphore. The footman's guard reads
the state of every philosopher, which makes the footman global state.

The animation shows the number of seated philosophers and fades a
philosopher who waits for a seat.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |  53 ++++++++----------
 .../DiningPhilosophers_anim.gif               | Bin 40185 -> 96557 bytes
 .../DiningPhilosophers_anim.tla               |  16 +++++-
 specifications/DiningPhilosophers/README.md   |   3 +-
 .../DiningPhilosophers/manifest.json          |  12 ++--
 5 files changed, 45 insertions(+), 39 deletions(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index f6964c30..a289dbb5 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -1,6 +1,6 @@
 ---------------------------- MODULE DiningPhilosophers ----------------------------
 
-EXTENDS Naturals, FiniteSets, Sequences
+EXTENDS Naturals, FiniteSets
 
 CONSTANT N \* Number of philosophers
 ASSUME NIsPosNat == N \in Nat \ {0}
@@ -13,58 +13,52 @@ RightFork(p) == (p % N) + 1
 ASSUME ForksDistinct == \A p \in P : RightFork(p) \in P /\ RightFork(p) # LeftFork(p)
 ASSUME ForksCovered == {RightFork(p) : p \in P} = P
 
-Neighbors(p) ==
-    {q \in P \ {p} : {LeftFork(q), RightFork(q)} \cap {LeftFork(p), RightFork(p)} # {}}
-
-VARIABLES philosophers, forks, queue
-vars == <<philosophers, forks, queue>>
+VARIABLES philosophers, forks
+vars == <<philosophers, forks>>
 
 TypeOK ==
     /\ philosophers \in [P -> SUBSET P]
     /\ forks \in SUBSET P
-    /\ queue \in Seq(P)
 
 Init ==
     \* each philosopher starts with no forks
     /\ philosophers = [ p \in P |-> {} ]
     \* all forks are available
     /\ forks = 1..N
-    \* the waiter's queue of hungry philosophers is empty
-    /\ queue = <<>>
-
-Request(p) ==
-    /\ philosophers[p] = {}
-    /\ \A i \in 1..Len(queue) : queue[i] # p
-    /\ queue' = Append(queue, p)
-    /\ UNCHANGED <<philosophers, forks>>
-
-TakeBoth(p) ==
-    \* The waiter serves p once no neighbor of p asked before p.
-    /\ \E i \in 1..Len(queue) :
-        /\ queue[i] = p
-        /\ \A j \in 1..(i - 1) : queue[j] \notin Neighbors(p)
-        /\ queue' = SubSeq(queue, 1, i - 1) \o SubSeq(queue, i + 1, Len(queue))
-    /\ {LeftFork(p), RightFork(p)} \subseteq forks
-    /\ forks' = forks \ {LeftFork(p), RightFork(p)}
-    /\ philosophers' = [philosophers EXCEPT ![p] = {LeftFork(p), RightFork(p)}]
+
+TakeLeft(p) ==
+    \* The footman seats at most N-1 philosophers.  This conjunct reads
+    \* every philosopher's state: the footman is essentially global state.
+    /\ Cardinality({q \in P : philosophers[q] # {}}) < N - 1
+    /\ LeftFork(p) \in forks
+    /\ forks' = forks \ {LeftFork(p)}
+    /\ philosophers' = 
+        [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
+
+TakeRight(p) ==
+    \* Philosophers grab the left fork first.
+    /\ philosophers[p] = {LeftFork(p)}
+    /\ RightFork(p) \in forks 
+    /\ forks' = forks \ {RightFork(p)}
+    /\ philosophers' = 
+        [philosophers EXCEPT ![p] = philosophers[p] \cup {RightFork(p)}]
 
 Release(p) ==
     /\ philosophers[p] = {LeftFork(p), RightFork(p)}
     /\ forks' = forks \cup {LeftFork(p), RightFork(p)}
     /\ philosophers' = [philosophers EXCEPT ![p] = {}]
-    /\ UNCHANGED queue
 
 Next ==
     \E p \in P:
-        \/ Request(p)
-        \/ TakeBoth(p)
+        \/ TakeLeft(p)
+        \/ TakeRight(p)
         \/ Release(p)
 
 Spec ==
     /\ Init
     /\ [][Next]_vars
     /\ \A p \in P :
-        SF_vars(Request(p) \/ TakeBoth(p) \/ Release(p))
+        SF_vars(TakeLeft(p) \/ TakeRight(p) \/ Release(p))
 
 IsEating(p) ==
     Cardinality(philosophers[p]) = 2
@@ -88,7 +82,6 @@ Alias ==
     [
         philosophers |-> philosophers,
         forks |-> forks,
-        queue |-> queue,
         eating |-> [ p \in DOMAIN philosophers |-> ThinkHungryEat(p) ]
     ]
 
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
index 5de6ab090fe79394714756239c492c5cf72ace65..628a1982b08a83e958321aa3d59c3cced5ce38c9 100644
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diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
index cce05147..3a6d2e17 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
@@ -46,6 +46,12 @@ StateColor(p) ==
       [] Cardinality(philosophers[p]) = 1 -> HungryColor
       [] OTHER                            -> ThinkingColor
 
+\* Philosophers who hold a fork are seated by the footman.
+Seated == Cardinality({q \in P : philosophers[q] # {}})
+
+\* A thinking philosopher waits for the footman while all N-1 seats are taken.
+WaitsForSeat(p) == philosophers[p] = {} /\ Seated = N - 1
+
 (***************************************************************************)
 (* A fork centered at (x, y), rotated by a degrees.  Unrotated, its tines  *)
 (* point up.                                                               *)
@@ -98,7 +104,8 @@ Philosopher(p) ==
         Show(right, ForkGlyph(Cx - 22, 70, 160)) \o
         Show(left,  hand(Cx + 25)) \o
         Show(right, hand(Cx - 25)),
-        ("transform" :> RotateAboutCenter(t)))
+        ("transform" :> RotateAboutCenter(t) @@
+         "opacity" :> IF WaitsForSeat(p) THEN "0.35" ELSE "1"))
 
 \* Fork f on the table; only its number remains while someone holds it.
 TableFork(f) ==
@@ -118,7 +125,12 @@ Legend ==
     IN Group(<<
         entry(0, ThinkingColor, "thinking"),
         entry(1, HungryColor,   "hungry (1 fork)"),
-        entry(2, EatingColor,   "eating (2 forks)")
+        entry(2, EatingColor,   "eating (2 forks)"),
+        Text(x, 32 + 22 * 3, "seated: " \o ToString(Seated) \o "/" \o ToString(N - 1),
+             ("font-family" :> "sans-serif" @@ "font-size" :> "11" @@
+              "fill" :> IF Seated = N - 1 THEN "#c0392b" ELSE "#333")),
+        Text(x, 32 + 22 * 4, "faded: no free seat",
+             ("font-family" :> "sans-serif" @@ "font-size" :> "11" @@ "fill" :> "#333"))
       >>, <<>>)
 
 \* Scale the drawing, in hundredths, to 600 wide, which fits Spectacle's pane.
diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 6e221adc..601e8456 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -17,7 +17,8 @@ commit, the properties for which TLC finds a counterexample:
 | 3 | [Put the left fork back if the right fork is taken](https://github.com/tlaplus/examples/commit/3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa) | Back-off and retry | none | no | yes | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F3f171e1ec2db66c4ffa87727e8ed1a74f40c46fa%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 4 | [Take both forks at once](https://github.com/tlaplus/examples/commit/de09177ff01f1c821ce17a0243286a27d605eba5) | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fde09177ff01f1c821ce17a0243286a27d605eba5%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 5 | [Let a philosopher who ate wait for its neighbors](https://github.com/tlaplus/examples/commit/173c60f8d1375217f61b901094b0dd36acd967c2) | [Priority to waiting neighbors](https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD310.html) | neighbors + atomic fork pair | no | no | no | —[^step5] | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F173c60f8d1375217f61b901094b0dd36acd967c2%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 6 | Let a waiter serve philosophers in order | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 |  |
+| 6 | [Let a waiter serve philosophers in order](https://github.com/tlaplus/examples/commit/637ef15fc210e5785bb3f9a68769392ca36601d3) | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F637ef15fc210e5785bb3f9a68769392ca36601d3%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 7 | Let a footman seat at most `N-1` philosophers | [Footman (butler) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Limiting_the_number_of_diners_in_the_table) | global counter | no | no | no | 0.42 |  |
 
 This repository has specs of two other solutions:
 
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index 774f118e..80ad233e 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -29,9 +29,9 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 731,
-          "totalStates": 1951,
-          "stateDepth": 8
+          "distinctStates": 81,
+          "totalStates": 261,
+          "stateDepth": 6
         }
       ]
     },
@@ -61,9 +61,9 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 731,
-          "totalStates": 1951,
-          "stateDepth": 8
+          "distinctStates": 81,
+          "totalStates": 261,
+          "stateDepth": 6
         }
       ]
     },

From 048764145a337a362be7bcb3140c158ed06d35f6 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:33 -0700
Subject: [PATCH 11/14] DP: let philosopher N take its right fork first

The footman reads the state of every philosopher. Replace it with a
rule that each philosopher checks alone, the order in which it takes
its forks: all but philosopher N take the left fork
first, and N takes the right fork first. Each philosopher thus takes
the lower-numbered of its two forks first (Dijkstra's resource
hierarchy), which breaks the circular wait. TLC finds no violation of
NoDeadlock, NoLivelock, or NoStarvation: under strong fairness, a
philosopher who holds its first fork eventually gets the second,
because that fork becomes free again and again.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |   9 ++++-----
 .../DiningPhilosophers_anim.gif               | Bin 96557 -> 60149 bytes
 .../DiningPhilosophers_anim.tla               |  16 ++--------------
 specifications/DiningPhilosophers/README.md   |   3 ++-
 .../DiningPhilosophers/manifest.json          |   8 ++++----
 5 files changed, 12 insertions(+), 24 deletions(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index a289dbb5..76463d8b 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -27,17 +27,16 @@ Init ==
     /\ forks = 1..N
 
 TakeLeft(p) ==
-    \* The footman seats at most N-1 philosophers.  This conjunct reads
-    \* every philosopher's state: the footman is essentially global state.
-    /\ Cardinality({q \in P : philosophers[q] # {}}) < N - 1
+    \* Philosopher N grabs the right fork first.
+    /\ p = N => philosophers[p] = {RightFork(p)}
     /\ LeftFork(p) \in forks
     /\ forks' = forks \ {LeftFork(p)}
     /\ philosophers' = 
         [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
 
 TakeRight(p) ==
-    \* Philosophers grab the left fork first.
-    /\ philosophers[p] = {LeftFork(p)}
+    \* All other philosophers grab the left fork first.
+    /\ p # N => philosophers[p] = {LeftFork(p)}
     /\ RightFork(p) \in forks 
     /\ forks' = forks \ {RightFork(p)}
     /\ philosophers' = 
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
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diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
index 3a6d2e17..cce05147 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_anim.tla
@@ -46,12 +46,6 @@ StateColor(p) ==
       [] Cardinality(philosophers[p]) = 1 -> HungryColor
       [] OTHER                            -> ThinkingColor
 
-\* Philosophers who hold a fork are seated by the footman.
-Seated == Cardinality({q \in P : philosophers[q] # {}})
-
-\* A thinking philosopher waits for the footman while all N-1 seats are taken.
-WaitsForSeat(p) == philosophers[p] = {} /\ Seated = N - 1
-
 (***************************************************************************)
 (* A fork centered at (x, y), rotated by a degrees.  Unrotated, its tines  *)
 (* point up.                                                               *)
@@ -104,8 +98,7 @@ Philosopher(p) ==
         Show(right, ForkGlyph(Cx - 22, 70, 160)) \o
         Show(left,  hand(Cx + 25)) \o
         Show(right, hand(Cx - 25)),
-        ("transform" :> RotateAboutCenter(t) @@
-         "opacity" :> IF WaitsForSeat(p) THEN "0.35" ELSE "1"))
+        ("transform" :> RotateAboutCenter(t)))
 
 \* Fork f on the table; only its number remains while someone holds it.
 TableFork(f) ==
@@ -125,12 +118,7 @@ Legend ==
     IN Group(<<
         entry(0, ThinkingColor, "thinking"),
         entry(1, HungryColor,   "hungry (1 fork)"),
-        entry(2, EatingColor,   "eating (2 forks)"),
-        Text(x, 32 + 22 * 3, "seated: " \o ToString(Seated) \o "/" \o ToString(N - 1),
-             ("font-family" :> "sans-serif" @@ "font-size" :> "11" @@
-              "fill" :> IF Seated = N - 1 THEN "#c0392b" ELSE "#333")),
-        Text(x, 32 + 22 * 4, "faded: no free seat",
-             ("font-family" :> "sans-serif" @@ "font-size" :> "11" @@ "fill" :> "#333"))
+        entry(2, EatingColor,   "eating (2 forks)")
       >>, <<>>)
 
 \* Scale the drawing, in hundredths, to 600 wide, which fits Spectacle's pane.
diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 601e8456..b9cbf369 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -18,7 +18,8 @@ commit, the properties for which TLC finds a counterexample:
 | 4 | [Take both forks at once](https://github.com/tlaplus/examples/commit/de09177ff01f1c821ce17a0243286a27d605eba5) | [Dijkstra's semaphore solution (Tanenbaum's monitor)](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Dijkstra's_solution) | atomic fork pair | no | no | yes | | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2Fde09177ff01f1c821ce17a0243286a27d605eba5%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 5 | [Let a philosopher who ate wait for its neighbors](https://github.com/tlaplus/examples/commit/173c60f8d1375217f61b901094b0dd36acd967c2) | [Priority to waiting neighbors](https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD310.html) | neighbors + atomic fork pair | no | no | no | —[^step5] | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F173c60f8d1375217f61b901094b0dd36acd967c2%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 6 | [Let a waiter serve philosophers in order](https://github.com/tlaplus/examples/commit/637ef15fc210e5785bb3f9a68769392ca36601d3) | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F637ef15fc210e5785bb3f9a68769392ca36601d3%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 7 | Let a footman seat at most `N-1` philosophers | [Footman (butler) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Limiting_the_number_of_diners_in_the_table) | global counter | no | no | no | 0.42 |  |
+| 7 | [Let a footman seat at most `N-1` philosophers](https://github.com/tlaplus/examples/commit/20efef8f7b1b790d3643284d7c96434453580581) | [Footman (butler) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Limiting_the_number_of_diners_in_the_table) | global counter | no | no | no | 0.42 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F20efef8f7b1b790d3643284d7c96434453580581%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 8 | Let philosopher `N` take its right fork first | [Resource hierarchy (asymmetric) solution](https://doi.org/10.1147/sj.72.0074) | fork order (local) | no | no | no | 0.55 |  |
 
 This repository has specs of two other solutions:
 
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index 80ad233e..f7a65e1a 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -29,8 +29,8 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 81,
-          "totalStates": 261,
+          "distinctStates": 70,
+          "totalStates": 220,
           "stateDepth": 6
         }
       ]
@@ -61,8 +61,8 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 81,
-          "totalStates": 261,
+          "distinctStates": 70,
+          "totalStates": 220,
           "stateDepth": 6
         }
       ]

From 57a50b9008e41e108b270b97ef804929c7c300dd Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:34 -0700
Subject: [PATCH 12/14] DP: prove starvation freedom with TLAPS

TLC checks NoStarvation only for N = 5. The proof shows Spec =>
NoStarvation for every N. Forks are freed infinitely often, by
induction from the highest fork down: a hungry philosopher holds its
low fork and waits only for a higher one. Strong fairness then lets
each philosopher take both forks.

Check with: tlapm --strict DiningPhilosophers_proof.tla

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 README.md                                     |   2 +-
 .../DiningPhilosophers/DiningPhilosophers.tla |   5 +
 .../DiningPhilosophers_proof.tla              | 705 ++++++++++++++++++
 specifications/DiningPhilosophers/README.md   |  10 +-
 .../DiningPhilosophers/manifest.json          |   8 +
 5 files changed, 728 insertions(+), 2 deletions(-)
 create mode 100644 specifications/DiningPhilosophers/DiningPhilosophers_proof.tla

diff --git a/README.md b/README.md
index 7ad1609d..e19baadc 100644
--- a/README.md
+++ b/README.md
@@ -33,7 +33,7 @@ Here is a list of specs included in this repository which are validated by the C
 | [Boyer-Moore Majority Vote](specifications/Majority)                                                | Stephan Merz                                        |    ✔     |      ✔      |         |     ✔     |          |
 | [Proof x+x is Even](specifications/sums_even)                                                       | Martin Riener                                       |    ✔     |      ✔      |         |     ✔     |          |
 | [The N-Queens Puzzle](specifications/N-Queens)                                                      | Stephan Merz                                        |    ✔     |             |    ✔    |     ✔     |          |
-| [The Dining Philosophers Problem](specifications/DiningPhilosophers)                                | Jeff Hemphill, Markus Kuppe                         |    ✔     |             |   (✔)   |     ✔     |    ✔     |
+| [The Dining Philosophers Problem](specifications/DiningPhilosophers)                                | Jeff Hemphill, Markus Kuppe                         |    ✔     |      ✔      |   (✔)   |     ✔     |    ✔     |
 | [The Car Talk Puzzle](specifications/CarTalkPuzzle)                                                 | Leslie Lamport                                      |    ✔     |             |         |     ✔     |          |
 | [The Die Hard Problem](specifications/DieHard)                                                      | Leslie Lamport                                      |    ✔     |      ✔      |         |     ✔     |    ✔     |
 | [The Prisoners & Switches Puzzle](specifications/Prisoners)                                         | Leslie Lamport                                      |    ✔     |             |         |     ✔     |          |
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index 76463d8b..32bd01b3 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -71,6 +71,11 @@ NoLivelock ==
 NoStarvation ==
     \A p \in P: []<>IsEating(p)
 
+\* Proved in DiningPhilosophers_proof.
+THEOREM DeadlockFreedom == Spec => []NoDeadlock
+THEOREM LivelockFreedom == Spec => NoLivelock
+THEOREM StarvationFreedom == Spec => NoStarvation
+
 -----
 
 Alias ==
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla b/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
new file mode 100644
index 00000000..0286b8ec
--- /dev/null
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
@@ -0,0 +1,705 @@
+------------------------- MODULE DiningPhilosophers_proof -------------------------
+(***************************************************************************)
+(* Proof that the solution in DiningPhilosophers is starvation free: every *)
+(* philosopher eats infinitely often.                                      *)
+(*                                                                         *)
+(* All philosophers but N take their left fork first, and N takes its      *)
+(* right fork first.  Either way, each philosopher first takes the lower-  *)
+(* numbered of its two forks (Dijkstra's resource hierarchy).              *)
+(*                                                                         *)
+(* A philosopher who holds fork f either eats, and then releases f, or     *)
+(* holds f as its low fork and waits for its high fork, whose number is    *)
+(* larger.  By induction from the largest fork down, every fork is free    *)
+(* infinitely often, so strong fairness lets every philosopher take its    *)
+(* low fork, then its high fork, and eat.                                  *)
+(***************************************************************************)
+EXTENDS DiningPhilosophers, TLAPS, NaturalsInduction, FiniteSetTheorems
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* Fork numbering.                                                         *)
+(***************************************************************************)
+
+LEMMA NAtLeastTwo == N \in Nat /\ N >= 2
+<1>1. N \in Nat /\ N # 0
+  BY NIsPosNat
+<1>2. N # 1
+  <2>. SUFFICES ASSUME N = 1 PROVE FALSE
+    OBVIOUS
+  <2>1. 1 \in P
+    BY <1>1 DEF P
+  <2>2. RightFork(1) # LeftFork(1)
+    BY <2>1, ForksDistinct
+  <2>. QED
+    BY <2>2 DEF RightFork, LeftFork
+<1>. QED
+  BY <1>1, <1>2
+
+LEMMA RightForkDef ==
+    \A q \in P : RightFork(q) = IF q = N THEN 1 ELSE q + 1
+BY NAtLeastTwo DEF P, RightFork
+
+\* The fork that a philosopher takes first, and the one it takes second.
+LowFork(p)  == IF p = N THEN RightFork(p) ELSE LeftFork(p)
+HighFork(p) == IF p = N THEN LeftFork(p) ELSE RightFork(p)
+
+LEMMA ForkFacts ==
+    \A q \in P :
+        /\ LowFork(q) \in P
+        /\ HighFork(q) \in P
+        /\ LowFork(q) < HighFork(q)
+        /\ {LowFork(q), HighFork(q)} = {LeftFork(q), RightFork(q)}
+BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork
+
+\* The philosopher whose right fork is f.
+Pred(f) == IF f = 1 THEN N ELSE f - 1
+
+LEMMA PredInP == \A f \in P : Pred(f) \in P
+BY NAtLeastTwo DEF P, Pred
+
+\* Fork f is only ever held by philosopher f or philosopher Pred(f).
+LEMMA Holders ==
+    \A f, q \in P : f \in {LowFork(q), HighFork(q)} => q = f \/ q = Pred(f)
+BY NAtLeastTwo, ForkFacts, RightForkDef DEF P, LeftFork, Pred
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* Inductive invariant.                                                    *)
+(***************************************************************************)
+
+Idle(p)   == philosophers[p] = {}
+Hungry(p) == philosophers[p] = {LowFork(p)}
+Eating(p) == philosophers[p] = {LowFork(p), HighFork(p)}
+
+Phases == \A p \in P : Idle(p) \/ Hungry(p) \/ Eating(p)
+
+\* A fork lies on the table iff no one holds it ...
+OnTable == \A f \in P : f \in forks <=> \A p \in P : f \notin philosophers[p]
+
+\* ... and no two philosophers hold the same fork.
+Exclusive == \A p, q \in P : p # q => philosophers[p] \cap philosophers[q] = {}
+
+Inv == TypeOK /\ Phases /\ OnTable /\ Exclusive
+
+\* TakeLeft and TakeRight, seen as taking the low fork, then the high one.
+TakeLow(p) ==
+    /\ philosophers[p] = {}
+    /\ LowFork(p) \in forks
+    /\ forks' = forks \ {LowFork(p)}
+    /\ philosophers' = [philosophers EXCEPT ![p] = {LowFork(p)}]
+
+TakeHigh(p) ==
+    /\ philosophers[p] = {LowFork(p)}
+    /\ HighFork(p) \in forks
+    /\ forks' = forks \ {HighFork(p)}
+    /\ philosophers' = [philosophers EXCEPT ![p] = {LowFork(p), HighFork(p)}]
+
+NextLH == \E p \in P : TakeLow(p) \/ TakeHigh(p) \/ Release(p)
+
+LEMMA TakeAgree ==
+    ASSUME Inv, NEW p \in P, TakeLeft(p) \/ TakeRight(p)
+    PROVE  TakeLow(p) \/ TakeHigh(p)
+<1>. USE DEF Inv, TypeOK, LowFork, HighFork
+<1>1. CASE TakeLeft(p) /\ p # N
+  <2>1. LeftFork(p) \in P /\ LeftFork(p) \notin philosophers[p]
+    BY <1>1 DEF TakeLeft, OnTable, LeftFork
+  <2>2. philosophers[p] = {}
+    BY <2>1, <1>1 DEF Phases, Idle, Hungry, Eating
+  <2>. QED
+    BY <1>1, <2>2 DEF TakeLeft, TakeLow
+<1>2. CASE TakeLeft(p) /\ p = N
+  BY <1>2 DEF TakeLeft, TakeHigh
+<1>3. CASE TakeRight(p) /\ p # N
+  BY <1>3 DEF TakeRight, TakeHigh
+<1>4. CASE TakeRight(p) /\ p = N
+  <2>1. RightFork(p) \in P /\ RightFork(p) \notin philosophers[p]
+    BY <1>4, ForksDistinct DEF TakeRight, OnTable
+  <2>2. philosophers[p] = {}
+    BY <2>1, <1>4 DEF Phases, Idle, Hungry, Eating
+  <2>. QED
+    BY <1>4, <2>2 DEF TakeRight, TakeLow
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4
+
+LEMMA NextAgree == ASSUME Inv, Next PROVE NextLH
+BY TakeAgree DEF Next, NextLH
+
+\* What each action does, stated pointwise.
+LEMMA TakeLowFrame ==
+    ASSUME TypeOK, NEW p \in P, TakeLow(p)
+    PROVE  /\ philosophers[p] = {} /\ LowFork(p) \in forks
+           /\ forks' = forks \ {LowFork(p)}
+           /\ philosophers'[p] = {LowFork(p)}
+           /\ \A r \in P : r # p => philosophers'[r] = philosophers[r]
+BY DEF TypeOK, TakeLow
+
+LEMMA TakeHighFrame ==
+    ASSUME TypeOK, NEW p \in P, TakeHigh(p)
+    PROVE  /\ philosophers[p] = {LowFork(p)} /\ HighFork(p) \in forks
+           /\ forks' = forks \ {HighFork(p)}
+           /\ philosophers'[p] = {LowFork(p), HighFork(p)}
+           /\ \A r \in P : r # p => philosophers'[r] = philosophers[r]
+BY DEF TypeOK, TakeHigh
+
+LEMMA ReleaseFrame ==
+    ASSUME TypeOK, NEW p \in P, Release(p)
+    PROVE  /\ philosophers[p] = {LowFork(p), HighFork(p)}
+           /\ forks' = forks \cup {LowFork(p), HighFork(p)}
+           /\ philosophers'[p] = {}
+           /\ \A r \in P : r # p => philosophers'[r] = philosophers[r]
+BY ForkFacts DEF TypeOK, Release
+
+THEOREM TypeOKInit == Init => TypeOK
+BY DEF Init, TypeOK, P
+
+THEOREM TypeOKNext == TypeOK /\ [Next]_vars => TypeOK'
+<1>. SUFFICES ASSUME TypeOK, [Next]_vars PROVE TypeOK'
+  OBVIOUS
+<1>. USE ForkFacts DEF TypeOK
+<1>1. ASSUME NEW p \in P, TakeLeft(p) PROVE TypeOK'
+  BY <1>1 DEF TakeLeft, LeftFork
+<1>2. ASSUME NEW p \in P, TakeRight(p) PROVE TypeOK'
+  BY <1>2, ForksDistinct DEF TakeRight
+<1>3. ASSUME NEW p \in P, Release(p) PROVE TypeOK'
+  BY <1>3, ReleaseFrame DEF Release
+<1>4. CASE UNCHANGED vars
+  BY <1>4 DEF vars
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4 DEF Next
+
+THEOREM PhasesNext == Inv /\ [Next]_vars => Phases'
+<1>. SUFFICES ASSUME Inv, [Next]_vars PROVE Phases'
+  OBVIOUS
+<1>. USE DEF Inv, Phases, Idle, Hungry, Eating
+<1>1. ASSUME NEW p \in P, TakeLow(p) PROVE Phases'
+  BY <1>1, TakeLowFrame
+<1>2. ASSUME NEW p \in P, TakeHigh(p) PROVE Phases'
+  BY <1>2, TakeHighFrame
+<1>3. ASSUME NEW p \in P, Release(p) PROVE Phases'
+  BY <1>3, ReleaseFrame
+<1>4. CASE UNCHANGED vars
+  BY <1>4 DEF vars
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, NextAgree DEF NextLH
+
+THEOREM OnTableNext == Inv /\ [Next]_vars => OnTable'
+<1>. SUFFICES ASSUME Inv, [Next]_vars, NEW g \in P
+              PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
+  BY DEF OnTable
+<1>. USE ForkFacts DEF Inv, TypeOK, OnTable
+<1>1. ASSUME NEW p \in P, TakeLow(p)
+      PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
+  <2>1. CASE g = LowFork(p)
+    BY <1>1, <2>1, TakeLowFrame
+  <2>2. CASE g # LowFork(p)
+    <3>1. \A r \in P : g \in philosophers'[r] <=> g \in philosophers[r]
+      BY <1>1, <2>2, TakeLowFrame
+    <3>. QED
+      BY <1>1, <2>2, <3>1, TakeLowFrame
+  <2>. QED
+    BY <2>1, <2>2
+<1>2. ASSUME NEW p \in P, TakeHigh(p)
+      PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
+  <2>1. CASE g = HighFork(p)
+    BY <1>2, <2>1, TakeHighFrame
+  <2>2. CASE g # HighFork(p)
+    <3>1. \A r \in P : g \in philosophers'[r] <=> g \in philosophers[r]
+      BY <1>2, <2>2, TakeHighFrame
+    <3>. QED
+      BY <1>2, <2>2, <3>1, TakeHighFrame
+  <2>. QED
+    BY <2>1, <2>2
+<1>3. ASSUME NEW p \in P, Release(p)
+      PROVE  g \in forks' <=> \A r \in P : g \notin philosophers'[r]
+  <2>1. CASE g \in {LowFork(p), HighFork(p)}
+    BY <1>3, <2>1, ReleaseFrame DEF Exclusive
+  <2>2. CASE g \notin {LowFork(p), HighFork(p)}
+    <3>1. \A r \in P : g \in philosophers'[r] <=> g \in philosophers[r]
+      BY <1>3, <2>2, ReleaseFrame
+    <3>. QED
+      BY <1>3, <2>2, <3>1, ReleaseFrame
+  <2>. QED
+    BY <2>1, <2>2
+<1>4. CASE UNCHANGED vars
+  BY <1>4 DEF vars
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, NextAgree DEF NextLH
+
+THEOREM ExclusiveNext == Inv /\ [Next]_vars => Exclusive'
+<1>. SUFFICES ASSUME Inv, [Next]_vars,
+                     NEW r1 \in P, NEW r2 \in P, r1 # r2
+              PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
+  BY DEF Exclusive
+<1>. USE DEF Inv, TypeOK
+<1>1. ASSUME NEW p \in P, TakeLow(p)
+      PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
+  <2>1. \A r \in P : LowFork(p) \notin philosophers[r]
+    BY <1>1, TakeLowFrame DEF OnTable
+  <2>. QED
+    BY <1>1, <2>1, TakeLowFrame DEF Exclusive
+<1>2. ASSUME NEW p \in P, TakeHigh(p)
+      PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
+  <2>1. \A r \in P : HighFork(p) \notin philosophers[r]
+    BY <1>2, TakeHighFrame DEF OnTable
+  <2>2. \A r \in P : r # p => LowFork(p) \notin philosophers[r]
+    <3>. SUFFICES ASSUME NEW r \in P, r # p PROVE LowFork(p) \notin philosophers[r]
+      OBVIOUS
+    <3>1. philosophers[p] \cap philosophers[r] = {}
+      BY DEF Exclusive
+    <3>. QED
+      BY <1>2, <3>1, TakeHighFrame
+  <2>. QED
+    BY <1>2, <2>1, <2>2, TakeHighFrame DEF Exclusive
+<1>3. ASSUME NEW p \in P, Release(p)
+      PROVE  philosophers'[r1] \cap philosophers'[r2] = {}
+  BY <1>3, ReleaseFrame DEF Exclusive
+<1>4. CASE UNCHANGED vars
+  BY <1>4 DEF vars, Exclusive
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, NextAgree DEF NextLH
+
+THEOREM InvInit == Init => Inv
+BY ForkFacts, TypeOKInit DEF Init, Inv, Phases, OnTable, Exclusive, Idle, P
+
+THEOREM InvNext == Inv /\ [Next]_vars => Inv'
+BY TypeOKNext, PhasesNext, OnTableNext, ExclusiveNext DEF Inv
+
+THEOREM InvSpec == Spec => []Inv
+BY InvInit, InvNext, PTL DEF Spec
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* The steps of a single philosopher.                                      *)
+(***************************************************************************)
+
+Act(p) == TakeLeft(p) \/ TakeRight(p) \/ Release(p)
+
+LEMMA SpecNext == Spec => [][Next]_vars
+BY DEF Spec
+
+LEMMA SpecFair == \A q \in P : Spec => SF_vars(Act(q))
+BY DEF Spec, Act
+
+\* A step either is a step of q or leaves q's forks alone.
+LEMMA NextSplit ==
+    ASSUME Inv, [Next]_vars, NEW q \in P
+    PROVE  <<Act(q)>>_vars \/ philosophers'[q] = philosophers[q]
+<1>1. CASE UNCHANGED vars
+  BY <1>1 DEF vars
+<1>2. ASSUME NEW p \in P, p # q, Act(p)
+      PROVE  philosophers'[q] = philosophers[q]
+  BY <1>2 DEF Act, TakeLeft, TakeRight, Release, Inv, TypeOK
+<1>3. CASE Act(q)
+  BY <1>3 DEF vars
+<1>. QED
+  BY <1>1, <1>2, <1>3 DEF Next, Act
+
+\* A step of q is a step of TakeLow, TakeHigh, or Release.
+LEMMA ActAgree ==
+    ASSUME Inv, NEW q \in P, Act(q)
+    PROVE  TakeLow(q) \/ TakeHigh(q) \/ Release(q)
+BY TakeAgree DEF Act
+
+LEMMA IdleStep ==
+    ASSUME Inv, NEW q \in P, Idle(q), <<Act(q)>>_vars
+    PROVE  Hungry(q)'
+BY ActAgree, ForkFacts DEF Inv, TypeOK, Idle, Hungry, TakeLow, TakeHigh, Release
+
+LEMMA HungryStep ==
+    ASSUME Inv, NEW q \in P, Hungry(q), <<Act(q)>>_vars
+    PROVE  Eating(q)'
+BY ActAgree, ForkFacts DEF Inv, TypeOK, Hungry, Eating, TakeLow, TakeHigh, Release
+
+LEMMA EatingStep ==
+    ASSUME Inv, NEW q \in P, Eating(q), <<Act(q)>>_vars
+    PROVE  \A f \in philosophers[q] : f \in forks'
+BY ActAgree, ForkFacts DEF Inv, TypeOK, Eating, TakeLow, TakeHigh, Release
+
+LEMMA IdleEnabled ==
+    ASSUME NEW q \in P
+    PROVE  Inv /\ Idle(q) /\ LowFork(q) \in forks => ENABLED <<Act(q)>>_vars
+<1>. SUFFICES ASSUME Inv, Idle(q), LowFork(q) \in forks
+              PROVE  ENABLED <<Act(q)>>_vars
+  OBVIOUS
+<1>1. philosophers \in [P -> SUBSET P] /\ philosophers[q] = {}
+  BY DEF Inv, TypeOK, Idle
+<1>2. CASE q # N
+  <2>1. LeftFork(q) \in forks
+    BY <1>2 DEF LowFork
+  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {LeftFork(q)}],
+          forks \ {LeftFork(q)}>> # <<philosophers, forks>>
+    BY <1>1
+  <2>3. ENABLED <<TakeLeft(q)>>_vars
+    BY <1>1, <1>2, <2>1, <2>2, ExpandENABLED DEF TakeLeft, vars
+  <2>. QED
+    BY <2>3, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
+<1>3. CASE q = N
+  <2>1. RightFork(q) \in forks
+    BY <1>3 DEF LowFork
+  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {RightFork(q)}],
+          forks \ {RightFork(q)}>> # <<philosophers, forks>>
+    BY <1>1
+  <2>3. ENABLED <<TakeRight(q)>>_vars
+    BY <1>1, <1>3, <2>1, <2>2, ExpandENABLED DEF TakeRight, vars
+  <2>. QED
+    BY <2>3, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
+<1>. QED
+  BY <1>2, <1>3
+
+LEMMA HungryEnabled ==
+    ASSUME NEW q \in P
+    PROVE  Inv /\ Hungry(q) /\ HighFork(q) \in forks => ENABLED <<Act(q)>>_vars
+<1>. SUFFICES ASSUME Inv, Hungry(q), HighFork(q) \in forks
+              PROVE  ENABLED <<Act(q)>>_vars
+  OBVIOUS
+<1>1. philosophers \in [P -> SUBSET P] /\ philosophers[q] = {LowFork(q)}
+  BY DEF Inv, TypeOK, Hungry
+<1>2. LeftFork(q) # RightFork(q)
+  BY ForksDistinct
+<1>3. CASE q # N
+  <2>1. philosophers[q] = {LeftFork(q)} /\ RightFork(q) \in forks
+    BY <1>1, <1>3 DEF LowFork, HighFork
+  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {RightFork(q)}],
+          forks \ {RightFork(q)}>> # <<philosophers, forks>>
+    BY <1>1, <1>2, <2>1
+  <2>3. (q # N => philosophers[q] = {LeftFork(q)}) /\ RightFork(q) \in forks
+    BY <2>1
+  <2>4. ENABLED <<TakeRight(q)>>_vars
+    BY <2>2, <2>3, ExpandENABLED DEF TakeRight, vars
+  <2>. QED
+    BY <2>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
+<1>4. CASE q = N
+  <2>1. philosophers[q] = {RightFork(q)} /\ LeftFork(q) \in forks
+    BY <1>1, <1>4 DEF LowFork, HighFork
+  <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {LeftFork(q)}],
+          forks \ {LeftFork(q)}>> # <<philosophers, forks>>
+    BY <1>1, <1>2, <2>1
+  <2>3. (q = N => philosophers[q] = {RightFork(q)}) /\ LeftFork(q) \in forks
+    BY <2>1
+  <2>4. ENABLED <<TakeLeft(q)>>_vars
+    BY <2>2, <2>3, ExpandENABLED DEF TakeLeft, vars
+  <2>. QED
+    BY <2>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
+<1>. QED
+  BY <1>3, <1>4
+
+LEMMA EatingEnabled ==
+    ASSUME NEW q \in P
+    PROVE  Inv /\ Eating(q) => ENABLED <<Act(q)>>_vars
+<1>. SUFFICES ASSUME Inv, Eating(q)
+              PROVE  ENABLED <<Act(q)>>_vars
+  OBVIOUS
+<1>1. philosophers \in [P -> SUBSET P]
+      /\ philosophers[q] = {LeftFork(q), RightFork(q)}
+  BY ForkFacts DEF Inv, TypeOK, Eating
+<1>2. {LeftFork(q), RightFork(q)} # {}
+  OBVIOUS
+<1>3. <<[philosophers EXCEPT ![q] = {}], forks \cup {LeftFork(q), RightFork(q)}>>
+        # <<philosophers, forks>>
+  BY <1>1, <1>2
+<1>4. ENABLED <<Release(q)>>_vars
+  BY <1>1, <1>3, ExpandENABLED DEF Release, vars
+<1>. QED
+  BY <1>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* Progress of a single philosopher.                                       *)
+(***************************************************************************)
+
+FreeIO(f) == []<>(f \in forks)
+
+HeldEat(f, q) == Eating(q) /\ f \in philosophers[q]
+
+\* Named so that the lemmas below can be instantiated: backends other than
+\* PTL only see temporal formulas through opaque operators.
+IdleToHungry(q)   == Idle(q) ~> Hungry(q)
+HungryToEating(q) == Hungry(q) ~> Eating(q)
+EatingFrees(f, q) == HeldEat(f, q) ~> (f \in forks)
+HungryFrees(f, q) == (Hungry(q) /\ f = LowFork(q)) ~> (f \in forks)
+
+LEMMA IdleHungry ==
+    ASSUME NEW q \in P
+    PROVE  Spec /\ FreeIO(LowFork(q)) => IdleToHungry(q)
+<1>1. Inv /\ Idle(q) /\ [Next]_vars => Idle(q)' \/ Hungry(q)'
+  <2>. SUFFICES ASSUME Inv, Idle(q), [Next]_vars PROVE Idle(q)' \/ Hungry(q)'
+    OBVIOUS
+  <2>1. CASE philosophers'[q] = philosophers[q]
+    BY <2>1 DEF Idle
+  <2>2. CASE <<Act(q)>>_vars
+    BY <2>2, IdleStep
+  <2>. QED
+    BY <2>1, <2>2, NextSplit
+<1>2. Inv /\ Idle(q) /\ <<Act(q)>>_vars => Hungry(q)'
+  BY IdleStep
+<1>3. Inv /\ Idle(q) /\ LowFork(q) \in forks => ENABLED <<Act(q)>>_vars
+  BY IdleEnabled
+<1>4. Spec => SF_vars(Act(q))
+  BY SpecFair
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF FreeIO, IdleToHungry
+
+LEMMA HungryEat ==
+    ASSUME NEW q \in P
+    PROVE  Spec /\ FreeIO(HighFork(q)) => HungryToEating(q)
+<1>1. Inv /\ Hungry(q) /\ [Next]_vars => Hungry(q)' \/ Eating(q)'
+  <2>. SUFFICES ASSUME Inv, Hungry(q), [Next]_vars PROVE Hungry(q)' \/ Eating(q)'
+    OBVIOUS
+  <2>1. CASE philosophers'[q] = philosophers[q]
+    BY <2>1 DEF Hungry
+  <2>2. CASE <<Act(q)>>_vars
+    BY <2>2, HungryStep
+  <2>. QED
+    BY <2>1, <2>2, NextSplit
+<1>2. Inv /\ Hungry(q) /\ <<Act(q)>>_vars => Eating(q)'
+  BY HungryStep
+<1>3. Inv /\ Hungry(q) /\ HighFork(q) \in forks => ENABLED <<Act(q)>>_vars
+  BY HungryEnabled
+<1>4. Spec => SF_vars(Act(q))
+  BY SpecFair
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF FreeIO, HungryToEating
+
+LEMMA EatRelease ==
+    ASSUME NEW f \in P, NEW q \in P
+    PROVE  Spec => EatingFrees(f, q)
+<1>1. Inv /\ HeldEat(f, q) /\ [Next]_vars => HeldEat(f, q)' \/ (f \in forks)'
+  <2>. SUFFICES ASSUME Inv, HeldEat(f, q), [Next]_vars
+                PROVE  HeldEat(f, q)' \/ (f \in forks)'
+    OBVIOUS
+  <2>1. CASE philosophers'[q] = philosophers[q]
+    BY <2>1 DEF HeldEat, Eating
+  <2>2. CASE <<Act(q)>>_vars
+    BY <2>2, EatingStep DEF HeldEat
+  <2>. QED
+    BY <2>1, <2>2, NextSplit
+<1>2. Inv /\ HeldEat(f, q) /\ <<Act(q)>>_vars => (f \in forks)'
+  BY EatingStep DEF HeldEat
+<1>3. Inv /\ HeldEat(f, q) => ENABLED <<Act(q)>>_vars
+  BY EatingEnabled DEF HeldEat
+<1>4. Spec => SF_vars(Act(q))
+  BY SpecFair
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF EatingFrees
+
+\* A hungry philosopher who holds f as its low fork eventually frees f,
+\* provided that all forks larger than f are free infinitely often.
+LEMMA HungryRelease ==
+    ASSUME NEW f \in P, NEW q \in P
+    PROVE  Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, q)
+<1>1. CASE f = LowFork(q)
+  <2>1. HighFork(q) \in P /\ HighFork(q) > f
+    BY <1>1, ForkFacts
+  <2>2. (\A g \in P : g > f => FreeIO(g)) => FreeIO(HighFork(q))
+    BY <2>1
+  <2>3. Spec /\ FreeIO(HighFork(q)) => HungryToEating(q)
+    BY HungryEat
+  <2>4. Spec => EatingFrees(f, q)
+    BY EatRelease
+  <2>5. Eating(q) => HeldEat(f, q)
+    BY <1>1 DEF HeldEat, Eating
+  <2>6. (Hungry(q) /\ f = LowFork(q)) => Hungry(q)
+    OBVIOUS
+  <2>. QED
+    BY <2>2, <2>3, <2>4, <2>5, <2>6, PTL
+       DEF HungryToEating, EatingFrees, HungryFrees
+<1>2. CASE f # LowFork(q)
+  <2>1. ~(Hungry(q) /\ f = LowFork(q))
+    BY <1>2
+  <2>. QED
+    BY <2>1, PTL DEF HungryFrees
+<1>. QED
+  BY <1>1, <1>2
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* Every fork is free infinitely often.                                    *)
+(***************************************************************************)
+
+LEMMA HeldBy ==
+    ASSUME NEW f \in P
+    PROVE  Inv /\ ~(f \in forks) =>
+              \/ HeldEat(f, f) \/ HeldEat(f, Pred(f))
+              \/ Hungry(f) /\ f = LowFork(f)
+              \/ Hungry(Pred(f)) /\ f = LowFork(Pred(f))
+<1>. SUFFICES ASSUME Inv, ~(f \in forks)
+              PROVE  \/ HeldEat(f, f) \/ HeldEat(f, Pred(f))
+                     \/ Hungry(f) /\ f = LowFork(f)
+                     \/ Hungry(Pred(f)) /\ f = LowFork(Pred(f))
+  OBVIOUS
+<1>1. PICK q \in P : f \in philosophers[q]
+  BY DEF Inv, OnTable
+<1>2. f \in {LowFork(q), HighFork(q)}
+  BY <1>1 DEF Inv, Phases, Idle, Hungry, Eating
+<1>3. q = f \/ q = Pred(f)
+  BY <1>2, Holders
+<1>4. HeldEat(f, q) \/ (Hungry(q) /\ f = LowFork(q))
+  BY <1>1 DEF Inv, Phases, Idle, Hungry, Eating, HeldEat
+<1>. QED
+  BY <1>3, <1>4
+
+LEMMA FreeStep ==
+    ASSUME NEW f \in P
+    PROVE  Spec /\ (\A g \in P : g > f => FreeIO(g)) => FreeIO(f)
+<1>1. Pred(f) \in P
+  BY PredInP
+<1>2. Inv /\ ~(f \in forks) =>
+          \/ HeldEat(f, f) \/ HeldEat(f, Pred(f))
+          \/ Hungry(f) /\ f = LowFork(f)
+          \/ Hungry(Pred(f)) /\ f = LowFork(Pred(f))
+  BY HeldBy
+<1>3. Spec => EatingFrees(f, f)
+  BY EatRelease
+<1>4. Spec => EatingFrees(f, Pred(f))
+  BY <1>1, EatRelease
+<1>5. Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, f)
+  BY HungryRelease
+<1>6. Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, Pred(f))
+  BY <1>1, HungryRelease
+<1>. QED
+  BY <1>2, <1>3, <1>4, <1>5, <1>6, InvSpec, PTL
+     DEF FreeIO, EatingFrees, HungryFrees
+
+THEOREM AllFree == \A f \in P : Spec => FreeIO(f)
+<1>. DEFINE Q(m) == Spec => \A g \in P : g + m > N => FreeIO(g)
+<1>1. Q(0)
+  BY NAtLeastTwo DEF P
+<1>2. \A m \in Nat : Q(m) => Q(m + 1)
+  <2>. SUFFICES ASSUME NEW m \in Nat, Q(m)
+                PROVE  Q(m + 1)
+    OBVIOUS
+  <2>1. ASSUME NEW g \in P, g + (m + 1) > N
+        PROVE  Spec => FreeIO(g)
+    <3>1. CASE g + m > N
+      BY <3>1
+    <3>2. CASE ~(g + m > N)
+      <4>1. \A h \in P : h > g => h + m > N
+        BY <2>1, <3>2, NAtLeastTwo DEF P
+      <4>2. Spec => \A h \in P : h > g => FreeIO(h)
+        BY <4>1
+      <4>. QED
+        BY <4>2, FreeStep
+    <3>. QED
+      BY <3>1, <3>2
+  <2>. QED
+    BY <2>1
+<1>. HIDE DEF Q
+<1>3. \A m \in Nat : Q(m)
+  BY <1>1, <1>2, NatInduction
+<1>4. Q(N)
+  BY <1>3, NAtLeastTwo
+<1>. QED
+  BY <1>4, NAtLeastTwo DEF Q, P
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* Starvation freedom.                                                     *)
+(***************************************************************************)
+
+LEMMA EatingIsEating == \A p \in P : Eating(p) => IsEating(p)
+<1>. SUFFICES ASSUME NEW p \in P, Eating(p) PROVE IsEating(p)
+  OBVIOUS
+<1>1. LowFork(p) # HighFork(p)
+  BY ForkFacts
+<1>2. IsFiniteSet({LowFork(p)}) /\ Cardinality({LowFork(p)}) = 1
+  BY FS_Singleton
+<1>3. Cardinality({LowFork(p)} \cup {HighFork(p)}) = 2
+  BY <1>1, <1>2, FS_AddElement
+<1>. QED
+  BY <1>3 DEF Eating, IsEating
+
+THEOREM StarvationFreedom
+<1>. SUFFICES ASSUME NEW p \in P PROVE Spec => []<>IsEating(p)
+  BY DEF NoStarvation
+<1>1. Spec => FreeIO(LowFork(p))
+  BY AllFree, ForkFacts
+<1>2. Spec => FreeIO(HighFork(p))
+  BY AllFree, ForkFacts
+<1>3. Spec /\ FreeIO(LowFork(p)) => IdleToHungry(p)
+  BY IdleHungry
+<1>4. Spec /\ FreeIO(HighFork(p)) => HungryToEating(p)
+  BY HungryEat
+<1>5. Inv => Idle(p) \/ Hungry(p) \/ Eating(p)
+  BY DEF Inv, Phases
+<1>6. Eating(p) => IsEating(p)
+  BY EatingIsEating
+<1>. QED
+  BY <1>1, <1>2, <1>3, <1>4, <1>5, <1>6, InvSpec, PTL
+     DEF IdleToHungry, HungryToEating
+
+EatsIO(p) == []<>IsEating(p)
+
+THEOREM StarvationImpliesNoLivelock == NoStarvation => NoLivelock
+<1>1. 1 \in P
+  BY NIsPosNat DEF P
+<1>2. NoStarvation => \A p \in P : EatsIO(p)
+  BY DEF NoStarvation, EatsIO
+<1>3. NoStarvation => EatsIO(1)
+  BY <1>1, <1>2
+<1>4. IsEating(1) => \E p \in P : IsEating(p)
+  BY <1>1
+<1>. QED
+  BY <1>3, <1>4, PTL DEF EatsIO, NoLivelock
+
+THEOREM LivelockFreedom
+BY StarvationFreedom, StarvationImpliesNoLivelock
+
+-----------------------------------------------------------------------------
+(***************************************************************************)
+(* Deadlock freedom.  If no one eats, fork N is free, because it is the    *)
+(* high fork of both philosophers who can hold it.  So philosopher N-1 or  *)
+(* N can take it if hungry; if both are idle, fork N-1 is free too.        *)
+(***************************************************************************)
+
+LEMMA ActEnabledNext ==
+    ASSUME NEW q \in P
+    PROVE  (ENABLED <<Act(q)>>_vars) => ENABLED Next
+BY ExpandENABLED DEF Act, Next, TakeLeft, TakeRight, Release, vars
+
+LEMMA TopForks ==
+    /\ N - 1 \in P /\ N \in P
+    /\ LowFork(N - 1) = N - 1 /\ HighFork(N - 1) = N
+    /\ LowFork(N) = 1 /\ HighFork(N) = N
+    /\ Pred(N) = N - 1 /\ 1 # N /\ N - 1 # N
+    /\ Pred(N - 1) = N \/ LowFork(Pred(N - 1)) # N - 1
+BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork, Pred
+
+THEOREM InvNoDeadlock == Inv => NoDeadlock
+<1>. SUFFICES ASSUME Inv PROVE ENABLED Next
+  BY DEF NoDeadlock
+<1>. DEFINE M == N - 1
+<1>0. TopForks
+  BY TopForks
+<1>1. CASE \E q \in P : Eating(q)
+  BY <1>1, EatingEnabled, ActEnabledNext
+<1>2. CASE \A q \in P : ~Eating(q)
+  <2>1. ~HeldEat(N, N) /\ ~HeldEat(N, Pred(N))
+        /\ ~HeldEat(M, M) /\ ~HeldEat(M, Pred(M))
+    BY <1>0, <1>2, PredInP DEF HeldEat
+  <2>2. N \in forks
+    <3>1. N # LowFork(N) /\ N # LowFork(Pred(N))
+      BY <1>0
+    <3>. QED
+      BY <1>0, <2>1, <3>1, HeldBy
+  <2>3. CASE Hungry(M)
+    BY <1>0, <2>2, <2>3, HungryEnabled, ActEnabledNext
+  <2>4. CASE Hungry(N)
+    BY <1>0, <2>2, <2>4, HungryEnabled, ActEnabledNext
+  <2>5. CASE ~Hungry(M) /\ ~Hungry(N)
+    <3>1. Idle(M) /\ Idle(N)
+      BY <1>0, <1>2, <2>5 DEF Inv, Phases
+    <3>2. ~Hungry(Pred(M)) \/ LowFork(Pred(M)) # M
+      BY <1>0, <2>5
+    <3>3. M \in forks
+      BY <1>0, <2>1, <2>5, <3>2, HeldBy
+    <3>. QED
+      BY <1>0, <3>1, <3>3, IdleEnabled, ActEnabledNext
+  <2>. QED
+    BY <2>3, <2>4, <2>5
+<1>. QED
+  BY <1>1, <1>2
+
+THEOREM DeadlockFreedom
+BY InvNoDeadlock, InvSpec, PTL
+
+=============================================================================
diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index b9cbf369..3b96a9fd 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -19,7 +19,7 @@ commit, the properties for which TLC finds a counterexample:
 | 5 | [Let a philosopher who ate wait for its neighbors](https://github.com/tlaplus/examples/commit/173c60f8d1375217f61b901094b0dd36acd967c2) | [Priority to waiting neighbors](https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD310.html) | neighbors + atomic fork pair | no | no | no | —[^step5] | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F173c60f8d1375217f61b901094b0dd36acd967c2%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 6 | [Let a waiter serve philosophers in order](https://github.com/tlaplus/examples/commit/637ef15fc210e5785bb3f9a68769392ca36601d3) | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F637ef15fc210e5785bb3f9a68769392ca36601d3%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 7 | [Let a footman seat at most `N-1` philosophers](https://github.com/tlaplus/examples/commit/20efef8f7b1b790d3643284d7c96434453580581) | [Footman (butler) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Limiting_the_number_of_diners_in_the_table) | global counter | no | no | no | 0.42 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F20efef8f7b1b790d3643284d7c96434453580581%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 8 | Let philosopher `N` take its right fork first | [Resource hierarchy (asymmetric) solution](https://doi.org/10.1147/sj.72.0074) | fork order (local) | no | no | no | 0.55 |  |
+| 8 | [Let philosopher `N` take its right fork first](https://github.com/tlaplus/examples/commit/048764145a337a362be7bcb3140c158ed06d35f6) | [Resource hierarchy (asymmetric) solution](https://doi.org/10.1147/sj.72.0074) | fork order (local) | no | no | no | 0.55 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F048764145a337a362be7bcb3140c158ed06d35f6%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 
 This repository has specs of two other solutions:
 
@@ -32,6 +32,14 @@ This repository has specs of two other solutions:
   [`Bakery-Boulangerie`](../Bakery-Boulangerie). They solve mutual exclusion,
   not dining, but the ticket idea carries over.
 
+TLC checks the spec only for `N = 5`.
+[`DiningPhilosophers_proof.tla`](DiningPhilosophers_proof.tla) proves with
+TLAPS that the spec is free of deadlock, livelock, and starvation for every `N`:
+
+```
+tlapm --strict DiningPhilosophers_proof.tla
+```
+
 [`DiningPhilosophers_anim.tla`](DiningPhilosophers_anim.tla) draws the table,
 forks, and philosophers for any `N` and writes one SVG per state. The
 animation comes from [Spectacle](https://github.com/will62794/spectacle).
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index f7a65e1a..ec0ff55e 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -79,6 +79,14 @@
           "workers": 1
         }
       ]
+    },
+    {
+      "path": "specifications/DiningPhilosophers/DiningPhilosophers_proof.tla",
+      "features": [],
+      "models": [],
+      "proof": {
+        "maxRuntimeMinutes": 3
+      }
     }
   ]
 }
\ No newline at end of file

From 0f6e467d33ee55cdd23936e72b1d1931ef05eb5f Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:34 -0700
Subject: [PATCH 13/14] DP: let even philosophers take their right fork first

If only philosopher N takes its right fork first, philosophers 1 to N-1
can wait for each other in one long chain. Alternating the fork order
between odd and even philosophers keeps these chains short, so more
philosophers can eat at once. TLC finds no violation for N = 2 to 7.
The animation shows pairs of philosophers eating at once.

The proof now ranks the forks in place of their numbers: odd forks but
fork 1, then fork 1, then the even forks.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 .../DiningPhilosophers/DiningPhilosophers.tla |   8 +-
 .../DiningPhilosophers_anim.gif               | Bin 60149 -> 59561 bytes
 .../DiningPhilosophers_proof.tla              | 172 ++++++++++--------
 specifications/DiningPhilosophers/README.md   |   1 +
 .../DiningPhilosophers/manifest.json          |   8 +-
 5 files changed, 108 insertions(+), 81 deletions(-)

diff --git a/specifications/DiningPhilosophers/DiningPhilosophers.tla b/specifications/DiningPhilosophers/DiningPhilosophers.tla
index 32bd01b3..2078bc05 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers.tla
@@ -27,16 +27,16 @@ Init ==
     /\ forks = 1..N
 
 TakeLeft(p) ==
-    \* Philosopher N grabs the right fork first.
-    /\ p = N => philosophers[p] = {RightFork(p)}
+    \* Even philosophers grab the right fork first.
+    /\ p % 2 = 0 => philosophers[p] = {RightFork(p)}
     /\ LeftFork(p) \in forks
     /\ forks' = forks \ {LeftFork(p)}
     /\ philosophers' = 
         [philosophers EXCEPT ![p] = philosophers[p] \cup {LeftFork(p)}]
 
 TakeRight(p) ==
-    \* All other philosophers grab the left fork first.
-    /\ p # N => philosophers[p] = {LeftFork(p)}
+    \* Odd philosophers grab the left fork first.
+    /\ p % 2 = 1 => philosophers[p] = {LeftFork(p)}
     /\ RightFork(p) \in forks 
     /\ forks' = forks \ {RightFork(p)}
     /\ philosophers' = 
diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif b/specifications/DiningPhilosophers/DiningPhilosophers_anim.gif
index d88a0f83cfd939b23e1fc82d2135e67aa376d882..ecd4ea83753c212fdb2b7fc4ff87a788be1cd6bd 100644
GIT binary patch
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diff --git a/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla b/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
index 0286b8ec..2856ec3e 100644
--- a/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
+++ b/specifications/DiningPhilosophers/DiningPhilosophers_proof.tla
@@ -3,13 +3,14 @@
 (* Proof that the solution in DiningPhilosophers is starvation free: every *)
 (* philosopher eats infinitely often.                                      *)
 (*                                                                         *)
-(* All philosophers but N take their left fork first, and N takes its      *)
-(* right fork first.  Either way, each philosopher first takes the lower-  *)
-(* numbered of its two forks (Dijkstra's resource hierarchy).              *)
+(* Odd philosophers take their left fork first, and even philosophers      *)
+(* take their right fork first.  Either way, each philosopher first takes  *)
+(* the lower-ranked of its two forks, where odd forks but fork 1 rank      *)
+(* lowest, then fork 1, then the even forks (a resource hierarchy).        *)
 (*                                                                         *)
 (* A philosopher who holds fork f either eats, and then releases f, or     *)
-(* holds f as its low fork and waits for its high fork, whose number is    *)
-(* larger.  By induction from the largest fork down, every fork is free    *)
+(* holds f as its low fork and waits for its high fork, whose rank is      *)
+(* higher.  By induction from the highest rank down, every fork is free    *)
 (* infinitely often, so strong fairness lets every philosopher take its    *)
 (* low fork, then its high fork, and eat.                                  *)
 (***************************************************************************)
@@ -40,16 +41,25 @@ LEMMA RightForkDef ==
 BY NAtLeastTwo DEF P, RightFork
 
 \* The fork that a philosopher takes first, and the one it takes second.
-LowFork(p)  == IF p = N THEN RightFork(p) ELSE LeftFork(p)
-HighFork(p) == IF p = N THEN LeftFork(p) ELSE RightFork(p)
+LowFork(p)  == IF p % 2 = 0 THEN RightFork(p) ELSE LeftFork(p)
+HighFork(p) == IF p % 2 = 0 THEN LeftFork(p) ELSE RightFork(p)
+
+Rank(f) == IF f % 2 = 0 THEN 2 ELSE IF f = 1 THEN 1 ELSE 0
+
+LEMMA RankType == \A f \in P : Rank(f) \in 0..2
+BY DEF Rank
 
 LEMMA ForkFacts ==
     \A q \in P :
         /\ LowFork(q) \in P
         /\ HighFork(q) \in P
-        /\ LowFork(q) < HighFork(q)
+        /\ LowFork(q) # HighFork(q)
+        /\ Rank(LowFork(q)) < Rank(HighFork(q))
         /\ {LowFork(q), HighFork(q)} = {LeftFork(q), RightFork(q)}
-BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork
+BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork, Rank
+
+LEMMA Parity == \A q \in P : q % 2 = 0 \/ q % 2 = 1
+BY DEF P
 
 \* The philosopher whose right fork is f.
 Pred(f) == IF f = 1 THEN N ELSE f - 1
@@ -100,18 +110,18 @@ LEMMA TakeAgree ==
     ASSUME Inv, NEW p \in P, TakeLeft(p) \/ TakeRight(p)
     PROVE  TakeLow(p) \/ TakeHigh(p)
 <1>. USE DEF Inv, TypeOK, LowFork, HighFork
-<1>1. CASE TakeLeft(p) /\ p # N
+<1>1. CASE TakeLeft(p) /\ p % 2 = 1
   <2>1. LeftFork(p) \in P /\ LeftFork(p) \notin philosophers[p]
     BY <1>1 DEF TakeLeft, OnTable, LeftFork
   <2>2. philosophers[p] = {}
     BY <2>1, <1>1 DEF Phases, Idle, Hungry, Eating
   <2>. QED
     BY <1>1, <2>2 DEF TakeLeft, TakeLow
-<1>2. CASE TakeLeft(p) /\ p = N
+<1>2. CASE TakeLeft(p) /\ p % 2 = 0
   BY <1>2 DEF TakeLeft, TakeHigh
-<1>3. CASE TakeRight(p) /\ p # N
+<1>3. CASE TakeRight(p) /\ p % 2 = 1
   BY <1>3 DEF TakeRight, TakeHigh
-<1>4. CASE TakeRight(p) /\ p = N
+<1>4. CASE TakeRight(p) /\ p % 2 = 0
   <2>1. RightFork(p) \in P /\ RightFork(p) \notin philosophers[p]
     BY <1>4, ForksDistinct DEF TakeRight, OnTable
   <2>2. philosophers[p] = {}
@@ -119,7 +129,7 @@ LEMMA TakeAgree ==
   <2>. QED
     BY <1>4, <2>2 DEF TakeRight, TakeLow
 <1>. QED
-  BY <1>1, <1>2, <1>3, <1>4
+  BY <1>1, <1>2, <1>3, <1>4, Parity
 
 LEMMA NextAgree == ASSUME Inv, Next PROVE NextLH
 BY TakeAgree DEF Next, NextLH
@@ -323,7 +333,7 @@ LEMMA IdleEnabled ==
   OBVIOUS
 <1>1. philosophers \in [P -> SUBSET P] /\ philosophers[q] = {}
   BY DEF Inv, TypeOK, Idle
-<1>2. CASE q # N
+<1>2. CASE q % 2 = 1
   <2>1. LeftFork(q) \in forks
     BY <1>2 DEF LowFork
   <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {LeftFork(q)}],
@@ -333,7 +343,7 @@ LEMMA IdleEnabled ==
     BY <1>1, <1>2, <2>1, <2>2, ExpandENABLED DEF TakeLeft, vars
   <2>. QED
     BY <2>3, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-<1>3. CASE q = N
+<1>3. CASE q % 2 = 0
   <2>1. RightFork(q) \in forks
     BY <1>3 DEF LowFork
   <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {RightFork(q)}],
@@ -344,7 +354,7 @@ LEMMA IdleEnabled ==
   <2>. QED
     BY <2>3, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
 <1>. QED
-  BY <1>2, <1>3
+  BY <1>2, <1>3, Parity
 
 LEMMA HungryEnabled ==
     ASSUME NEW q \in P
@@ -356,32 +366,32 @@ LEMMA HungryEnabled ==
   BY DEF Inv, TypeOK, Hungry
 <1>2. LeftFork(q) # RightFork(q)
   BY ForksDistinct
-<1>3. CASE q # N
+<1>3. CASE q % 2 = 1
   <2>1. philosophers[q] = {LeftFork(q)} /\ RightFork(q) \in forks
     BY <1>1, <1>3 DEF LowFork, HighFork
   <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {RightFork(q)}],
           forks \ {RightFork(q)}>> # <<philosophers, forks>>
     BY <1>1, <1>2, <2>1
-  <2>3. (q # N => philosophers[q] = {LeftFork(q)}) /\ RightFork(q) \in forks
+  <2>3. (q % 2 = 1 => philosophers[q] = {LeftFork(q)}) /\ RightFork(q) \in forks
     BY <2>1
   <2>4. ENABLED <<TakeRight(q)>>_vars
     BY <2>2, <2>3, ExpandENABLED DEF TakeRight, vars
   <2>. QED
     BY <2>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
-<1>4. CASE q = N
+<1>4. CASE q % 2 = 0
   <2>1. philosophers[q] = {RightFork(q)} /\ LeftFork(q) \in forks
     BY <1>1, <1>4 DEF LowFork, HighFork
   <2>2. <<[philosophers EXCEPT ![q] = philosophers[q] \cup {LeftFork(q)}],
           forks \ {LeftFork(q)}>> # <<philosophers, forks>>
     BY <1>1, <1>2, <2>1
-  <2>3. (q = N => philosophers[q] = {RightFork(q)}) /\ LeftFork(q) \in forks
+  <2>3. (q % 2 = 0 => philosophers[q] = {RightFork(q)}) /\ LeftFork(q) \in forks
     BY <2>1
   <2>4. ENABLED <<TakeLeft(q)>>_vars
     BY <2>2, <2>3, ExpandENABLED DEF TakeLeft, vars
   <2>. QED
     BY <2>4, ExpandENABLED DEF Act, TakeLeft, TakeRight, Release, vars
 <1>. QED
-  BY <1>3, <1>4
+  BY <1>3, <1>4, Parity
 
 LEMMA EatingEnabled ==
     ASSUME NEW q \in P
@@ -483,14 +493,14 @@ LEMMA EatRelease ==
   BY <1>1, <1>2, <1>3, <1>4, InvSpec, SpecNext, PTL DEF EatingFrees
 
 \* A hungry philosopher who holds f as its low fork eventually frees f,
-\* provided that all forks larger than f are free infinitely often.
+\* provided that all forks ranked above f are free infinitely often.
 LEMMA HungryRelease ==
     ASSUME NEW f \in P, NEW q \in P
-    PROVE  Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, q)
+    PROVE  Spec /\ (\A g \in P : Rank(g) > Rank(f) => FreeIO(g)) => HungryFrees(f, q)
 <1>1. CASE f = LowFork(q)
-  <2>1. HighFork(q) \in P /\ HighFork(q) > f
+  <2>1. HighFork(q) \in P /\ Rank(HighFork(q)) > Rank(f)
     BY <1>1, ForkFacts
-  <2>2. (\A g \in P : g > f => FreeIO(g)) => FreeIO(HighFork(q))
+  <2>2. (\A g \in P : Rank(g) > Rank(f) => FreeIO(g)) => FreeIO(HighFork(q))
     BY <2>1
   <2>3. Spec /\ FreeIO(HighFork(q)) => HungryToEating(q)
     BY HungryEat
@@ -540,7 +550,7 @@ LEMMA HeldBy ==
 
 LEMMA FreeStep ==
     ASSUME NEW f \in P
-    PROVE  Spec /\ (\A g \in P : g > f => FreeIO(g)) => FreeIO(f)
+    PROVE  Spec /\ (\A g \in P : Rank(g) > Rank(f) => FreeIO(g)) => FreeIO(f)
 <1>1. Pred(f) \in P
   BY PredInP
 <1>2. Inv /\ ~(f \in forks) =>
@@ -552,30 +562,30 @@ LEMMA FreeStep ==
   BY EatRelease
 <1>4. Spec => EatingFrees(f, Pred(f))
   BY <1>1, EatRelease
-<1>5. Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, f)
+<1>5. Spec /\ (\A g \in P : Rank(g) > Rank(f) => FreeIO(g)) => HungryFrees(f, f)
   BY HungryRelease
-<1>6. Spec /\ (\A g \in P : g > f => FreeIO(g)) => HungryFrees(f, Pred(f))
+<1>6. Spec /\ (\A g \in P : Rank(g) > Rank(f) => FreeIO(g)) => HungryFrees(f, Pred(f))
   BY <1>1, HungryRelease
 <1>. QED
   BY <1>2, <1>3, <1>4, <1>5, <1>6, InvSpec, PTL
      DEF FreeIO, EatingFrees, HungryFrees
 
 THEOREM AllFree == \A f \in P : Spec => FreeIO(f)
-<1>. DEFINE Q(m) == Spec => \A g \in P : g + m > N => FreeIO(g)
+<1>. DEFINE Q(m) == Spec => \A g \in P : Rank(g) + m > 2 => FreeIO(g)
 <1>1. Q(0)
-  BY NAtLeastTwo DEF P
+  BY RankType
 <1>2. \A m \in Nat : Q(m) => Q(m + 1)
   <2>. SUFFICES ASSUME NEW m \in Nat, Q(m)
                 PROVE  Q(m + 1)
     OBVIOUS
-  <2>1. ASSUME NEW g \in P, g + (m + 1) > N
+  <2>1. ASSUME NEW g \in P, Rank(g) + (m + 1) > 2
         PROVE  Spec => FreeIO(g)
-    <3>1. CASE g + m > N
+    <3>1. CASE Rank(g) + m > 2
       BY <3>1
-    <3>2. CASE ~(g + m > N)
-      <4>1. \A h \in P : h > g => h + m > N
-        BY <2>1, <3>2, NAtLeastTwo DEF P
-      <4>2. Spec => \A h \in P : h > g => FreeIO(h)
+    <3>2. CASE ~(Rank(g) + m > 2)
+      <4>1. \A h \in P : Rank(h) > Rank(g) => Rank(h) + m > 2
+        BY <2>1, <3>2, RankType
+      <4>2. Spec => \A h \in P : Rank(h) > Rank(g) => FreeIO(h)
         BY <4>1
       <4>. QED
         BY <4>2, FreeStep
@@ -586,10 +596,10 @@ THEOREM AllFree == \A f \in P : Spec => FreeIO(f)
 <1>. HIDE DEF Q
 <1>3. \A m \in Nat : Q(m)
   BY <1>1, <1>2, NatInduction
-<1>4. Q(N)
-  BY <1>3, NAtLeastTwo
+<1>4. Q(3)
+  BY <1>3
 <1>. QED
-  BY <1>4, NAtLeastTwo DEF Q, P
+  BY <1>4, RankType DEF Q
 
 -----------------------------------------------------------------------------
 (***************************************************************************)
@@ -646,9 +656,11 @@ BY StarvationFreedom, StarvationImpliesNoLivelock
 
 -----------------------------------------------------------------------------
 (***************************************************************************)
-(* Deadlock freedom.  If no one eats, fork N is free, because it is the    *)
-(* high fork of both philosophers who can hold it.  So philosopher N-1 or  *)
-(* N can take it if hungry; if both are idle, fork N-1 is free too.        *)
+(* Deadlock freedom.  If no one eats, every even fork is free, because it  *)
+(* is the high fork of both philosophers who can hold it.  So a hungry     *)
+(* philosopher whose high fork is even can take it.  Otherwise philosopher *)
+(* 1 is idle, and fork 1 is free: philosopher N could hold it only as its  *)
+(* low fork, and then N's high fork would be even.                         *)
 (***************************************************************************)
 
 LEMMA ActEnabledNext ==
@@ -656,46 +668,60 @@ LEMMA ActEnabledNext ==
     PROVE  (ENABLED <<Act(q)>>_vars) => ENABLED Next
 BY ExpandENABLED DEF Act, Next, TakeLeft, TakeRight, Release, vars
 
-LEMMA TopForks ==
-    /\ N - 1 \in P /\ N \in P
-    /\ LowFork(N - 1) = N - 1 /\ HighFork(N - 1) = N
-    /\ LowFork(N) = 1 /\ HighFork(N) = N
-    /\ Pred(N) = N - 1 /\ 1 # N /\ N - 1 # N
-    /\ Pred(N - 1) = N \/ LowFork(Pred(N - 1)) # N - 1
-BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork, Pred
+LEMMA EvenFree ==
+    ASSUME Inv, \A q \in P : ~Eating(q), NEW f \in P, Rank(f) = 2
+    PROVE  f \in forks
+<1>. SUFFICES ASSUME ~(f \in forks) PROVE FALSE
+  OBVIOUS
+<1>1. PICK q \in P : f \in philosophers[q]
+  BY DEF Inv, OnTable
+<1>2. f = LowFork(q)
+  BY <1>1 DEF Inv, Phases, Idle, Hungry
+<1>. QED
+  BY <1>2, ForkFacts, RankType
+
+LEMMA OneForks ==
+    /\ 1 \in P /\ 2 \in P /\ N \in P
+    /\ LowFork(1) = 1 /\ HighFork(1) = 2 /\ Rank(2) = 2
+    /\ Pred(1) = N /\ N # 1
+    /\ LowFork(N) = 1 => Rank(HighFork(N)) = 2
+BY NAtLeastTwo, RightForkDef DEF P, LowFork, HighFork, LeftFork, Pred, Rank
 
 THEOREM InvNoDeadlock == Inv => NoDeadlock
 <1>. SUFFICES ASSUME Inv PROVE ENABLED Next
   BY DEF NoDeadlock
-<1>. DEFINE M == N - 1
-<1>0. TopForks
-  BY TopForks
+<1>0. OneForks
+  BY OneForks
 <1>1. CASE \E q \in P : Eating(q)
   BY <1>1, EatingEnabled, ActEnabledNext
 <1>2. CASE \A q \in P : ~Eating(q)
-  <2>1. ~HeldEat(N, N) /\ ~HeldEat(N, Pred(N))
-        /\ ~HeldEat(M, M) /\ ~HeldEat(M, Pred(M))
-    BY <1>0, <1>2, PredInP DEF HeldEat
-  <2>2. N \in forks
-    <3>1. N # LowFork(N) /\ N # LowFork(Pred(N))
-      BY <1>0
+  <2>1. CASE \E q \in P : Hungry(q) /\ Rank(HighFork(q)) = 2
+    <3>1. PICK q \in P : Hungry(q) /\ Rank(HighFork(q)) = 2
+      BY <2>1
+    <3>2. HighFork(q) \in forks
+      BY <1>2, <3>1, ForkFacts, EvenFree
     <3>. QED
-      BY <1>0, <2>1, <3>1, HeldBy
-  <2>3. CASE Hungry(M)
-    BY <1>0, <2>2, <2>3, HungryEnabled, ActEnabledNext
-  <2>4. CASE Hungry(N)
-    BY <1>0, <2>2, <2>4, HungryEnabled, ActEnabledNext
-  <2>5. CASE ~Hungry(M) /\ ~Hungry(N)
-    <3>1. Idle(M) /\ Idle(N)
-      BY <1>0, <1>2, <2>5 DEF Inv, Phases
-    <3>2. ~Hungry(Pred(M)) \/ LowFork(Pred(M)) # M
-      BY <1>0, <2>5
-    <3>3. M \in forks
-      BY <1>0, <2>1, <2>5, <3>2, HeldBy
+      BY <3>1, <3>2, HungryEnabled, ActEnabledNext
+  <2>2. CASE \A q \in P : ~(Hungry(q) /\ Rank(HighFork(q)) = 2)
+    <3>1. Idle(1)
+      BY <1>0, <1>2, <2>2 DEF Inv, Phases
+    <3>2. 1 \in forks
+      <4>. SUFFICES ASSUME ~(1 \in forks) PROVE FALSE
+        OBVIOUS
+      <4>1. PICK r \in P : 1 \in philosophers[r]
+        BY <1>0 DEF Inv, OnTable
+      <4>2. Hungry(r) /\ 1 = LowFork(r)
+        BY <1>2, <4>1 DEF Inv, Phases, Idle, Hungry
+      <4>3. r = 1 \/ r = N
+        BY <1>0, <4>2, Holders
+      <4>4. r # 1
+        BY <3>1, <4>1 DEF Idle
+      <4>. QED
+        BY <1>0, <2>2, <4>2, <4>3, <4>4
     <3>. QED
-      BY <1>0, <3>1, <3>3, IdleEnabled, ActEnabledNext
+      BY <1>0, <3>1, <3>2, IdleEnabled, ActEnabledNext
   <2>. QED
-    BY <2>3, <2>4, <2>5
+    BY <2>1, <2>2
 <1>. QED
   BY <1>1, <1>2
 
diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 3b96a9fd..90d9e567 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -20,6 +20,7 @@ commit, the properties for which TLC finds a counterexample:
 | 6 | [Let a waiter serve philosophers in order](https://github.com/tlaplus/examples/commit/637ef15fc210e5785bb3f9a68769392ca36601d3) | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F637ef15fc210e5785bb3f9a68769392ca36601d3%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 7 | [Let a footman seat at most `N-1` philosophers](https://github.com/tlaplus/examples/commit/20efef8f7b1b790d3643284d7c96434453580581) | [Footman (butler) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Limiting_the_number_of_diners_in_the_table) | global counter | no | no | no | 0.42 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F20efef8f7b1b790d3643284d7c96434453580581%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 8 | [Let philosopher `N` take its right fork first](https://github.com/tlaplus/examples/commit/048764145a337a362be7bcb3140c158ed06d35f6) | [Resource hierarchy (asymmetric) solution](https://doi.org/10.1147/sj.72.0074) | fork order (local) | no | no | no | 0.55 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F048764145a337a362be7bcb3140c158ed06d35f6%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
+| 9 | Let even philosophers take their right fork first | Odd/even (alternating) solution | fork order (local) | no | no | no | 0.72 |  |
 
 This repository has specs of two other solutions:
 
diff --git a/specifications/DiningPhilosophers/manifest.json b/specifications/DiningPhilosophers/manifest.json
index ec0ff55e..46546f4b 100644
--- a/specifications/DiningPhilosophers/manifest.json
+++ b/specifications/DiningPhilosophers/manifest.json
@@ -29,8 +29,8 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 70,
-          "totalStates": 220,
+          "distinctStates": 58,
+          "totalStates": 174,
           "stateDepth": 6
         }
       ]
@@ -61,8 +61,8 @@
           "runtime": "00:00:01",
           "mode": "exhaustive search",
           "result": "success",
-          "distinctStates": 70,
-          "totalStates": 220,
+          "distinctStates": 58,
+          "totalStates": 174,
           "stateDepth": 6
         }
       ]

From 1868d4c2401410409e4361ac9d37342de5ca97f9 Mon Sep 17 00:00:00 2001
From: Markus Alexander Kuppe <github.com@lemmster.de>
Date: Wed, 21 Oct 2026 11:29:34 -0700
Subject: [PATCH 14/14] DP: link the last step in the README

A commit cannot link to itself, so the odd/even step is the only one
in the table without a link.

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Markus Alexander Kuppe <github.com@lemmster.de>
---
 specifications/DiningPhilosophers/README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specifications/DiningPhilosophers/README.md b/specifications/DiningPhilosophers/README.md
index 90d9e567..da4a518e 100644
--- a/specifications/DiningPhilosophers/README.md
+++ b/specifications/DiningPhilosophers/README.md
@@ -20,7 +20,7 @@ commit, the properties for which TLC finds a counterexample:
 | 6 | [Let a waiter serve philosophers in order](https://github.com/tlaplus/examples/commit/637ef15fc210e5785bb3f9a68769392ca36601d3) | Waiter (arbitrator) with a queue | global queue | no | no | no | 0.73 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F637ef15fc210e5785bb3f9a68769392ca36601d3%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 7 | [Let a footman seat at most `N-1` philosophers](https://github.com/tlaplus/examples/commit/20efef8f7b1b790d3643284d7c96434453580581) | [Footman (butler) solution](https://en.wikipedia.org/wiki/Dining_philosophers_problem#Limiting_the_number_of_diners_in_the_table) | global counter | no | no | no | 0.42 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F20efef8f7b1b790d3643284d7c96434453580581%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 | 8 | [Let philosopher `N` take its right fork first](https://github.com/tlaplus/examples/commit/048764145a337a362be7bcb3140c158ed06d35f6) | [Resource hierarchy (asymmetric) solution](https://doi.org/10.1147/sj.72.0074) | fork order (local) | no | no | no | 0.55 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F048764145a337a362be7bcb3140c158ed06d35f6%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
-| 9 | Let even philosophers take their right fork first | Odd/even (alternating) solution | fork order (local) | no | no | no | 0.72 |  |
+| 9 | [Let even philosophers take their right fork first](https://github.com/tlaplus/examples/commit/0f6e467d33ee55cdd23936e72b1d1931ef05eb5f) | Odd/even (alternating) solution | fork order (local) | no | no | no | 0.72 | [open](https://will62794.github.io/spectacle/#!/home?specpath=https%3A%2F%2Fraw.githubusercontent.com%2Ftlaplus%2Fexamples%2F0f6e467d33ee55cdd23936e72b1d1931ef05eb5f%2Fspecifications%2FDiningPhilosophers%2FDiningPhilosophers.tla&constants%5BN%5D=5) |
 
 This repository has specs of two other solutions: