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README.md

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+---
+license: mit
+tags:
+- formal-verification
+- coq
+- threshold-logic
+- neuromorphic
+- multi-layer
+- equivalence
+---
+
+# tiny-BiImplies-prover
+
+Formally verified biconditional gate (if and only if). Two-layer threshold network computing logical equivalence with 100% accuracy.
+
+## Architecture
+
+| Component | Value |
+|-----------|-------|
+| Inputs | 2 |
+| Outputs | 1 |
+| Neurons | 3 (2 hidden, 1 output) |
+| Layers | 2 |
+| Parameters | 9 |
+| **Layer 1, Neuron 1 (NOR)** | |
+| Weights | [-1, -1] |
+| Bias | 0 |
+| **Layer 1, Neuron 2 (AND)** | |
+| Weights | [1, 1] |
+| Bias | -2 |
+| **Layer 2 (OR)** | |
+| Weights | [1, 1] |
+| Bias | -1 |
+| Activation | Heaviside step (all layers) |
+
+## Key Properties
+
+- 100% accuracy (4/4 inputs correct)
+- Coq-proven correctness
+- Minimal 2-layer architecture
+- Integer weights
+- Equivalence relation (reflexive, symmetric, transitive)
+- Identical to XNOR
+
+## Usage
+
+```python
+import torch
+from safetensors.torch import load_file
+
+weights = load_file('biimplies.safetensors')
+
+def biimplies_gate(x, y):
+    inputs = torch.tensor([float(x), float(y)])
+
+    # Layer 1: NOR and AND
+    nor_sum = (inputs * weights['layer1.neuron1.weight']).sum() + weights['layer1.neuron1.bias']
+    nor_out = int(nor_sum >= 0)
+
+    and_sum = (inputs * weights['layer1.neuron2.weight']).sum() + weights['layer1.neuron2.bias']
+    and_out = int(and_sum >= 0)
+
+    # Layer 2: OR
+    layer1_outs = torch.tensor([float(nor_out), float(and_out)])
+    or_sum = (layer1_outs * weights['layer2.weight']).sum() + weights['layer2.bias']
+    return int(or_sum >= 0)
+
+# Test
+print(biimplies_gate(0, 0))  # 1 (both false, equivalent)
+print(biimplies_gate(0, 1))  # 0 (different, not equivalent)
+print(biimplies_gate(1, 0))  # 0 (different, not equivalent)
+print(biimplies_gate(1, 1))  # 1 (both true, equivalent)
+```
+
+## Verification
+
+**Coq Theorem**:
+```coq
+Theorem biimplies_correct : forall x y, biimplies_circuit x y = eqb x y.
+```
+
+Proven axiom-free with properties:
+- Reflexivity (x ↔ x)
+- Symmetry (x ↔ y → y ↔ x)
+- Transitivity (x ↔ y ∧ y ↔ z → x ↔ z)
+- Full equivalence relation
+
+Full proof: [coq-circuits/Boolean/BiImplies.v](https://github.com/CharlesCNorton/coq-circuits)
+
+## Circuit Operation
+
+BiImplies outputs true when inputs are equal.
+
+BiImplies(x,y) = OR(NOR(x,y), AND(x,y)) = (x ↔ y)
+
+## Citation
+
+```bibtex
+@software{tiny_biimplies_prover_2025,
+  title={tiny-BiImplies-prover: Formally Verified Biconditional Gate},
+  author={Norton, Charles},
+  url={https://huggingface.co/phanerozoic/tiny-BiImplies-prover},
+  year={2025}
+}
+```