tiny-Implies-verified

Formally verified implication gate (material conditional). Single threshold neuron computing logical implication with 100% accuracy.

Architecture

Component	Value
Inputs	2
Outputs	1
Neurons	1
Parameters	3
Weights	[-1, 1]
Bias	0
Activation	Heaviside step

Key Properties

• 100% accuracy (4/4 inputs correct)
• Coq-proven correctness
• Single threshold neuron
• Integer weights
• Reflexive: Implies(x,x) = true
• Ex falso quodlibet: Implies(false,y) = true

Usage

import torch
from safetensors.torch import load_file

weights = load_file('implies.safetensors')

def implies_gate(x, y):
    # Heaviside: weighted_sum + bias >= 0
    inputs = torch.tensor([float(x), float(y)])
    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
    return int(weighted_sum >= 0)

# Test
print(implies_gate(0, 0))  # 1 (false implies false)
print(implies_gate(0, 1))  # 1 (false implies true)
print(implies_gate(1, 0))  # 0 (true does not imply false)
print(implies_gate(1, 1))  # 1 (true implies true)

Verification

Coq Theorem:

Theorem implies_correct : forall x y, implies_circuit x y = implb x y.

Proven axiom-free with properties:

• Reflexivity (x → x)
• Ex falso quodlibet (⊥ → y)
• Modus ponens structure

Full proof: coq-circuits/Boolean/Implies.v

Circuit Operation

Input combination produces weighted sum:

• (0,0): 0*(-1) + 0*1 + 0 = 0 >= 0 → 1
• (0,1): 0*(-1) + 1*1 + 0 = 1 >= 0 → 1
• (1,0): 1*(-1) + 0*1 + 0 = -1 < 0 → 0
• (1,1): 1*(-1) + 1*1 + 0 = 0 >= 0 → 1

Fails only when antecedent is true and consequent is false.

Citation

@software{tiny_implies_prover_2025,
  title={tiny-Implies-verified: Formally Verified Implication Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-Implies-verified},
  year={2025}
}