Foundation Primitive Recognition Calculus Same Diff Same Trans

The transitivity law

In ordinary mathematics, transitivity is the property that if A equals B and B equals C, then A equals C. The Recognition Science declaration same_trans states this exact property for a specific kind of judgment: the recognition ledger, a discrete record of events in which objects are compared at a given trace, or moment of observation. The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified step by step by a computer.

The theorem reads: for any admissible trace judgment surface, at any trace, if the ledger records that object a is the same as object b, and that b is the same as c, then it records that a is the same as c. This is not an assumption the framework makes freely; it is a consequence of deeper axioms. The admissibility field of a trace judgment surface already requires reflexivity (a is the same as itself), symmetry (if a is the same as b, then b is the same as a), and transitivity. The declaration same_trans simply extracts the transitivity requirement into a standalone theorem, making it available for later proofs.

The practical effect is that the ledger's notion of sameness behaves like a proper equivalence relation, not a loose similarity. If two objects are interchangeable in one context, and one of them is interchangeable with a third, then the first and third are interchangeable too. This is what allows substitution: if a predicate respects sameness, and a is the same as b, then any property holding of a also holds of b. The framework models this with a separate theorem called substitute.

What same_trans does not claim is that sameness is the same as literal identity in the ambient type theory. The ledger's sameness is a judgment at a trace, a comparison the recognition process makes, not a statement about the underlying objects themselves. The framework is careful to distinguish object-level equality from the ledger's witnessed sameness. The declaration also does not claim that difference is transitive; the framework treats sameness and difference as separate relations, with only an exclusivity rule stating they cannot both hold for the same ordered pair.

/-- Transitivity of SameT, extracted from the admissibility field. -/
theorem same_trans (J : TraceJudgment) (T : Trace) {a b c : Endpoint}
    (hab : J.same T a b) (hbc : J.same T b c) :
    J.same T a c :=
  J.same_trans_proof T hab hbc

/-- Reflexivity of SameT, extracted from the admissibility field. -/
theorem same_refl (J : TraceJudgment) (T : Trace) (a : Endpoint) :
    J.same T a a :=
  J.same_refl_proof T a

/-- Symmetry of SameT, extracted from the admissibility field. -/
theorem same_symm (J : TraceJudgment) (T : Trace) {a b : Endpoint}
    (h : J.same T a b) :
    J.same T b a :=
  J.same_symm_proof T h

/-- Transitivity of SameT, extracted from the admissibility field. -/
theorem same_trans (J : TraceJudgment) (T : Trace) {a b c : Endpoint}
    (hab : J.same T a b) (hbc : J.same T b c) :
    J.same T a c :=
  J.same_trans_proof T hab hbc

/-- K2.10 and R7. Substitution for contexts that respect SameT. -/
theorem substitute
    (J : TraceJudgment) (T : Trace) (P : Endpoint → Prop)
    (hP : RespectsSame J T P) {a b : Endpoint}
    (hsame : J.same T a b) (ha : P a) :
    P b :=
  hP hsame ha

What this page does not claim

same_trans does not claim that sameness is identical to the ambient type theory's equality. same_trans does not claim that difference is transitive. same_trans does not claim that the ledger's sameness applies outside a given trace.

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/SameDiff.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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