Editing Proof Theory (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Modeling 'Inference Rules' (Automated Deduction):'''
<syntaxhighlight lang="python">
def apply_modus_ponens(known_facts, if_then_rule):
    """
    Modus Ponens: If A is true, and (A -> B) is true, then B is true.
    """
    a, b = if_then_rule # Rule is [A, B] representing A -> B
    
    if a in known_facts:
        print(f"Step: Since {a} is known, we derive {b}.")
        return b
    return None

# Axioms: 'It is raining', 'If raining -> Wet'
facts = {"Raining"}
rule = ["Raining", "Wet"]

new_fact = apply_modus_ponens(facts, rule)
# This simple 'Replacement' logic is how computer 
# solvers prove complex math.
</syntaxhighlight>

; Proof Theory Concepts
: '''Transfinite Induction''' → A method of proof that allows us to reason about infinite sets larger than the counting numbers.
: '''Ordinal Analysis''' → A way to measure the "Strength" of a mathematical system by assigning it a specific infinite number.
: '''Proof Compression''' → The study of how to make a proof as short as possible (important for cryptography and blockchain).
: '''Linear Logic''' → A type of logic where "Assumptions" are treated as resources that are "used up" when they are used in a proof.
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