Foundation Primitive Recognition Calculus Trace Logic Trace Logic Certificate

The trace logic certificate

A trace, in this framework, is a discrete record of events that can be extended with more events. The certificate named trace_logic_certificate is a machine-checked theorem in the framework's library of formal proofs. It establishes that a certain collection of logical operations, defined on properties of traces, behaves correctly. The properties in question are stable: if a property holds for a trace, it continues to hold for any extension of that trace. This stability is the central idea, and the certificate proves that the standard logical connectives, when defined with this stability in mind, work as expected.

The library defines a trace predicate as a property of traces that is stable under extension. It then defines the logical operations: truth, falsehood, conjunction, disjunction, implication, negation, and universal and existential quantification. Each definition is crafted so that the result is again a stable trace predicate. For example, conjunction of two stable predicates is stable because if both hold on a trace, they both hold on any extension. Implication is defined in a way that makes it stable by construction: it holds on a trace if the consequent holds on every future extension where the antecedent holds. The certificate proves the basic rules of inference for these operations, such as introduction and elimination rules for conjunction, disjunction, implication, negation, and quantifiers. It also proves that any stable predicate persists along trace extensions, which is the defining property of the system.

The certificate is a single theorem, but its content is the collection of these proofs. It shows that the first pass of a trace-based logic is internally consistent. The framework's goal is to build physical structure from the cost of recognition events, and this certificate is a foundational step: it establishes the logical language that can be used to make statements about traces. It does not, however, claim to have built any physical structure. It does not derive any constants, forces, or dimensions. It does not claim that this logic is complete, meaning that every true statement about traces is provable within it. It does not claim that the definitions of the logical connectives are the only possible ones, or that they match any particular philosophical notion of truth. The certificate is a proof of internal consistency for a specific, deliberately chosen set of definitions.

The practical consequence is that the framework can now reason about traces with a machine-checked logical foundation. Any future theorem that uses these trace predicates can rely on the fact that the basic logical rules are sound. This is a necessary, but not sufficient, step for the larger project. The certificate itself is a small piece of the overall framework, but it is a piece that is verified to be correct. It is the difference between building on sand and building on a checked foundation.

theorem trace_logic_certificate : TraceLogicCertificate where
  proposition_surface := ⟨TracePredicate.top⟩
  truth_intro := TracePredicate.top_intro
  conjunction_intro := by
    intro P Q T hP hQ
    exact TracePredicate.and_intro hP hQ
  conjunction_left := by
    intro P Q T h
    exact TracePredicate.and_left h
  conjunction_right := by
    intro P Q T h
    exact TracePredicate.and_right h
  disjunction_left := by
    intro P Q T hP
    exact TracePredicate.or_inl hP
  disjunction_right := by
    intro P Q T hQ
    exact TracePredicate.or_inr hQ
  implication_elim := by
    intro P Q T U himp hTU hP
    exact TracePredicate.imp_elim himp hTU hP
  negation_elim := by
    intro P T U hn hTU hP
    exact TracePredicate.not_elim hn hTU hP
  universal_intro := by
    intro α P T h
    exact TracePredicate.all_intro h
  universal_elim := by
    intro α P T h a
    exact TracePredicate.all_elim h a
  existential_intro := by
    intro α P T a h
    exact TracePredicate.exists_intro a h
  persistence := by
    intro P T U hTU hP
    exact TracePredicate.persists hTU hP
  strength_tag := rfl

/-- A proposition in the first PRC logic pass is a predicate on finite traces
that persists under trace extension. -/
structure TracePredicate where
  holds : Trace → Prop
  stable :
    ∀ {T U : Trace}, Trace.Extends T U → holds T → holds U

theorem persists {P : TracePredicate} {T U : Trace}
    (hTU : Trace.Extends T U) (hP : P.holds T) :
    P.holds U :=
  P.stable hTU hP

What this page does not claim

The certificate does not derive any physical constants or dimensions. The certificate does not prove the logic is complete for all statements about traces. The certificate does not claim the chosen definitions of the logical connectives are the only possible ones.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/TraceLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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