// Test Trace theory with axioms using product codomains

theory PetriNet {
  P : Sort;
  T : Sort;
  in : Sort;
  out : Sort;
  in/src : in -> P;
  in/tgt : in -> T;
  out/src : out -> T;
  out/tgt : out -> P;
}

// Trace theory with axioms
theory (N : PetriNet instance) Trace {
  F : Sort;
  F/of : F -> N/T;

  W : Sort;
  W/src : W -> [firing : F, arc : N/out];
  W/tgt : W -> [firing : F, arc : N/in];

  input_terminal : Sort;
  output_terminal : Sort;
  input_terminal/of : input_terminal -> N/P;
  output_terminal/of : output_terminal -> N/P;
  input_terminal/tgt : input_terminal -> [firing : F, arc : N/in];
  output_terminal/src : output_terminal -> [firing : F, arc : N/out];

  // Axiom: wires are injective on source
  // forall w1, w2 : W. w1 W/src = w2 W/src |- w1 = w2;
  // (Commented out - requires product codomain equality in premises)

  // Axiom: every arc endpoint must be wired or terminated
  // forall f : F, arc : N/out. arc N/out/src = f F/of |-
  //   (exists w : W. w W/src = [firing: f, arc: arc]) \/
  //   (exists o : output_terminal. o output_terminal/src = [firing: f, arc: arc]);
  // (Commented out - requires product codomain values in conclusions)
}

// Simple net for testing
instance SimpleNet : PetriNet = {
  A : P;
  B : P;
  t : T;
  arc_in : in;
  arc_in in/src = A;
  arc_in in/tgt = t;
  arc_out : out;
  arc_out out/src = t;
  arc_out out/tgt = B;
}

// Test that the basic theory without axioms still works
instance SimpleTrace : SimpleNet Trace = {
  f1 : F;
  f1 F/of = SimpleNet/t;

  it : input_terminal;
  it input_terminal/of = SimpleNet/A;
  it input_terminal/tgt = [firing: f1, arc: SimpleNet/arc_in];

  ot : output_terminal;
  ot output_terminal/of = SimpleNet/B;
  ot output_terminal/src = [firing: f1, arc: SimpleNet/arc_out];
}