# Concrete Evidence of the Imposition Theory Bug

## Summary

The bug occurs in the evaluation of necessity formulas involving imposition. Specifically:
- `¬(A ⊡ B)` is TRUE at all worlds (verified by checking each world individually)
- But `□¬(A ⊡ B)` evaluates to FALSE (has falsifiers)
- This violates the basic semantics of the necessity operator

## Evidence from PROBE2 Output

From running `./dev_cli.py src/model_checker/theory_lib/imposition/debug.py`:

```
3.  |\Box \neg (A \imposition B)| = < ∅, {□} >  (False in b)
      |\neg (A \imposition B)| = < {□}, ∅ >  (True in a)
      |\neg (A \imposition B)| = < {□}, ∅ >  (True in b)
```

This shows:
1. `¬(A ⊡ B)` has verifiers {□} at world a → TRUE at a
2. `¬(A ⊡ B)` has verifiers {□} at world b → TRUE at b
3. `□¬(A ⊡ B)` has falsifiers {□} → FALSE

## The Problem

The necessity operator (`□`) should check if its argument is true at all worlds. The implementation does this by:

1. Creating a formula: `ForAll(u, is_world(u) → true_at(argument, u))`
2. Evaluating this formula using Z3's `evaluate()` function
3. If true, returning verifiers = {□}; if false, returning falsifiers = {□}

## Root Cause

The issue is NOT that the model is incomplete. The Z3 model is fully constructed when `find_verifiers_and_falsifiers` is called. Instead, the problem is:

1. **Complex Nested Functions**: The expanded ForAll formula references multiple Z3 functions:
   - `verify(state, sentence)`
   - `falsify(state, sentence)` 
   - `extended_verify(state, sentence, world)`
   - `extended_falsify(state, sentence, world)`
   - `imposition(state, world, outcome)`
   - `true_at(sentence, world)`

2. **Z3 Function Interpretations**: These functions use Z3's function interpretation mechanism, which includes default "else" values for unconstrained inputs.

3. **Evaluation Discrepancy**: When Z3 evaluates the large expanded formula (with all quantifiers expanded), it may use these default values or other internal evaluation strategies that produce different results than checking each world individually.

## Proof the Model is Complete

1. The model is created by Z3's solver before any propositions are built
2. `find_verifiers_and_falsifiers` is called during proposition initialization
3. It accesses the model via: `evaluate = leftarg.proposition.model_structure.z3_model.evaluate`
4. Individual world evaluations work correctly (as shown in the output)

## Why Custom ForAll/Exists Don't Help

The code already uses custom `ForAll` and `Exists` functions from `utils.py` that explicitly expand quantifiers. However, the expanded formula still contains complex nested function calls that Z3 evaluates differently than expected.

## Implications

This bug affects any formula involving:
- Modal operators (□, ◇) applied to imposition formulas
- Nested evaluation contexts where the truth of complex formulas must be determined

The bug does NOT affect:
- Simple imposition formulas without modal operators
- Direct evaluation at specific worlds

## Next Steps

The solution requires changing how `find_verifiers_and_falsifiers` works for complex formulas, potentially:
1. Using a different evaluation strategy that doesn't rely on Z3's `evaluate()`
2. Ensuring all function values are fully constrained (no reliance on defaults)
3. Computing verifiers/falsifiers differently for modal operators