// Petri Net Reachability - Full Example
//
// This demonstrates the core ideas from the original geolog design document:
// modeling Petri net reachability using geometric logic with the chase algorithm.
//
// Original design: loose_thoughts/2025-12-12_12:10.md
//
// Key concepts:
// - PetriNet: places, transitions, input/output arcs
// - Marking: assignment of tokens to places (parameterized theory)
// - Trace: sequence of transition firings connecting markings
// - Reachability: computed via chase algorithm

// ============================================================
// THEORY: PetriNet
// ============================================================

theory PetriNet {
  P : Sort;      // Places
  T : Sort;      // Transitions
  In : Sort;     // Input arcs (place -> transition)
  Out : Sort;    // Output arcs (transition -> place)

  // Arc structure
  in/place : In -> P;
  in/trans : In -> T;
  out/trans : Out -> T;
  out/place : Out -> P;
}

// ============================================================
// THEORY: Marking (parameterized)
// A marking assigns tokens to places in a specific net
// ============================================================

theory (N : PetriNet instance) Marking {
  Token : Sort;
  of : Token -> N/P;
}

// ============================================================
// THEORY: PlaceReachability
// Simplified reachability at the place level
// ============================================================

theory PlaceReachability {
  P : Sort;
  T : Sort;

  // Which transition connects which places
  // Fires(t, from, to) means transition t can move a token from 'from' to 'to'
  Fires : [trans: T, from: P, to: P] -> Prop;

  // Reachability relation (transitive closure)
  CanReach : [from: P, to: P] -> Prop;

  // Reflexivity: every place can reach itself
  ax/refl : forall p : P.
    |- [from: p, to: p] CanReach;

  // Transition firing creates reachability
  ax/fire : forall t : T, x : P, y : P.
    [trans: t, from: x, to: y] Fires |- [from: x, to: y] CanReach;

  // Transitivity: reachability composes
  ax/trans : forall x : P, y : P, z : P.
    [from: x, to: y] CanReach, [from: y, to: z] CanReach |- [from: x, to: z] CanReach;
}

// ============================================================
// INSTANCE: SimpleNet
// A -> B -> C with bidirectional A <-> B
//
//   (A) <--[ba]-- (B) --[bc]--> (C)
//    |             ^
//    +---[ab]------+
// ============================================================

// Uses chase to derive CanReach from axioms (reflexivity, fire, transitivity)
instance SimpleNet : PlaceReachability = chase {
  // Places
  A : P;
  B : P;
  C : P;

  // Transitions
  ab : T;   // A -> B
  ba : T;   // B -> A
  bc : T;   // B -> C

  // Firing relations
  [trans: ab, from: A, to: B] Fires;
  [trans: ba, from: B, to: A] Fires;
  [trans: bc, from: B, to: C] Fires;
}

// ============================================================
// INSTANCE: MutexNet
// Two processes competing for a mutex
//
//   idle1 --[enter1]--> crit1 --[exit1]--> idle1
//              ^                    |
//              |       mutex        |
//              |                    v
//   idle2 --[enter2]--> crit2 --[exit2]--> idle2
// ============================================================

// Uses chase to derive reachability relation
instance MutexNet : PlaceReachability = chase {
  // Places
  idle1 : P;
  crit1 : P;
  idle2 : P;
  crit2 : P;
  mutex : P;

  // Transitions
  enter1 : T;
  exit1 : T;
  enter2 : T;
  exit2 : T;

  // Process 1 acquires mutex: idle1 + mutex -> crit1
  // (simplified: we track place-level, not token-level)
  [trans: enter1, from: idle1, to: crit1] Fires;
  [trans: enter1, from: mutex, to: crit1] Fires;

  // Process 1 releases mutex: crit1 -> idle1 + mutex
  [trans: exit1, from: crit1, to: idle1] Fires;
  [trans: exit1, from: crit1, to: mutex] Fires;

  // Process 2 acquires mutex: idle2 + mutex -> crit2
  [trans: enter2, from: idle2, to: crit2] Fires;
  [trans: enter2, from: mutex, to: crit2] Fires;

  // Process 2 releases mutex: crit2 -> idle2 + mutex
  [trans: exit2, from: crit2, to: idle2] Fires;
  [trans: exit2, from: crit2, to: mutex] Fires;
}

// ============================================================
// INSTANCE: ProducerConsumerNet
// Producer creates items, consumer processes them
//
//   ready --[produce]--> buffer --[consume]--> done
// ============================================================

// Uses chase to derive reachability relation
instance ProducerConsumerNet : PlaceReachability = chase {
  // Places
  ready : P;
  buffer : P;
  done : P;

  // Transitions
  produce : T;
  consume : T;

  // Produce: ready -> buffer
  [trans: produce, from: ready, to: buffer] Fires;

  // Consume: buffer -> done
  [trans: consume, from: buffer, to: done] Fires;
}