Action Path Space Fixed Endpoints Refl

The reflexivity lemma

In the calculus of variations, a path is a continuous curve between two points, and a common problem is to find the curve that minimizes some quantity, such as energy or time. The Recognition Science framework builds a formal version of this setting in its machine-checked library of formal theorems. It defines an admissible path, a continuous, strictly positive function on a closed interval, and a cost, a measure of how expensive a value is. The framework then defines the action, the integral of the cost along a path, and asks which admissible path minimizes it.

Within this setup, the declaration fixedEndpoints_refl establishes a property so basic it is almost invisible: every admissible path shares its endpoints with itself. The formal statement says that for any path γ, the relation fixedEndpoints γ γ holds, meaning γ evaluated at the start of the interval equals itself, and γ evaluated at the end equals itself. The proof is immediate from the definition, two applications of reflexivity of equality. This is not a deep theorem about the calculus of variations; it is a foundational brick in the formal structure, the reflexivity half of the fact that sharing endpoints is an equivalence relation.

The lemma matters because it is one of three properties that make fixedEndpoints an equivalence relation. The library also proves symmetry (if γ₁ shares endpoints with γ₂, then γ₂ shares them with γ₁) and transitivity (if γ₁ shares with γ₂ and γ₂ with γ₃, then γ₁ shares with γ₃). Together, these three lemmas allow the framework to treat paths as equivalent when they have the same endpoints, a step needed for the variational argument that follows. The reflexivity lemma is the entry point: without it, the relation would not be an equivalence, and the path space would not have the structure the framework needs.

In Recognition Science, this declaration is part of a larger project to derive physical laws from the cost function J(x) = (x + 1/x)/2 - 1. The path space module sets up the variational stage for a least-action principle derived from this cost. The reflexivity lemma itself does not prove that any path minimizes the action, nor does it establish the existence of a minimizer. It only records the trivial fact that a path is its own endpoint-mate. What it enables is the formal machinery of equivalence classes, which the framework uses to reason about paths that share boundaries.

lemma fixedEndpoints_refl {a b : ℝ} (γ : AdmissiblePath a b) :
    fixedEndpoints γ γ := And.intro rfl rfl

lemma fixedEndpoints_refl {a b : ℝ} (γ : AdmissiblePath a b) :
    fixedEndpoints γ γ := And.intro rfl rfl

lemma fixedEndpoints_symm {a b : ℝ} {γ₁ γ₂ : AdmissiblePath a b}
    (h : fixedEndpoints γ₁ γ₂) : fixedEndpoints γ₂ γ₁ :=
  ⟨h.1.symm, h.2.symm⟩

lemma fixedEndpoints_trans {a b : ℝ} {γ₁ γ₂ γ₃ : AdmissiblePath a b}
    (h₁ : fixedEndpoints γ₁ γ₂) (h₂ : fixedEndpoints γ₂ γ₃) :
    fixedEndpoints γ₁ γ₃ := ⟨h₁.1.trans h₂.1, h₁.2.trans h₂.2⟩

What this page does not claim

The reflexivity lemma does not prove that any path minimizes the action. The reflexivity lemma does not establish the existence of a minimizer. The reflexivity lemma does not state that all paths with the same endpoints are equal.

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

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