actual_prescribed_path_reversal_and_survival

Plain-English statement

Independent science d9850090b15210e29ecd9b932f02c358d2c9e786. Original six analytic assumptions and literal actual path definitions. On 0<=s<=T away from finitely many reversed event times, barGamma(s)=R Gamma(T-s). Both actual scalar rates are interval integrable and their survival integrals agree. Actual state reversal, path interpolation, harmonic flow, phase reversal and prescribed-density integrability discharge all private premises. Endpoints, n=0,T=0,rank0,zero normals/rates included. No genuine-jump pointwise equality, weighted density/sector, invariant probability, Markovness, hypocoercivity, errors, costs, main results or composition.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem actual_prescribed_path_reversal_and_survival
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ) * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ) * ‖v‖ ^ 2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) : actual_prescribed_path_reversal_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_prescribed_path_reversal_statement
  intro c Φ S Rec Hist Δ Prefix m zAt K Path Ψ Γ R rate n T
  obtain ⟨rho, hrho, h99⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedStateReversal.actual_prescribed_state_reversal_and_rate_product
      hα hαβ hV hH hη hβη n T
  refine ⟨rho, hrho, ?_⟩
  intro y r z t A gamma barGamma
  let grid : Fin (n+2) → ℝ := Fin.snoc (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) (T : ℝ)
  let barGrid : Fin (n+2) → ℝ := Fin.snoc (Fin.cons (0 : ℝ) (rho t).val) (T : ℝ)
  let post := zAt y r z n T t
  let barPost := zAt y r (R A) n T (rho t)
  have hpair : ∀ j : Fin (n+1), barPost j =
      R (Φ y r (grid j.rev.succ-grid j.rev.castSucc) (post j.rev)) :=
    fun j => ((h99 y r z t).1 j).1
  have hbl (j : Fin (n+1)) : barGrid j.castSucc = (T : ℝ) - grid j.rev.succ := by
    induction j using Fin.cases with
    | zero => simp [grid, barGrid]
    | succ i => simp only [grid, barGrid, Fin.rev_succ, Fin.succ_castSucc,
        Fin.snoc_castSucc, Fin.cons_succ, hrho]
  have hbr (j : Fin (n+1)) : barGrid j.succ = (T : ℝ) - grid j.rev.castSucc := by
    induction j using Fin.lastCases with
    | last => simp [grid, barGrid]
    | cast i => simp only [grid, barGrid, Fin.rev_castSucc, Fin.succ_castSucc,
        Fin.snoc_castSucc, Fin.cons_succ, hrho]
  have hgrid : Monotone grid := by
    have htT : ∀ j : Fin (n+1), (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) j ≤ (T : ℝ) := by
      intro j
      refine Fin.cases ?_ (fun i => ?_) j
      · simpa using T.coe_nonneg
      · exact (t.property.2 i).le
    intro i j hij
    induction i using Fin.lastCases with
    | last =>
      have hj : j = Fin.last (n+1) := le_antisymm (Fin.le_last j) hij
      subst j
      exact le_rfl
    | cast i =>
      induction j using Fin.lastCases with
      | last => simpa [grid] using htT i
      | cast j => simpa [grid] using t.property.1.monotone (show i ≤ j from hij)
  obtain ⟨Z, _, _, _, _, hlaw⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath.actual_finite_count_path_law
      hα hαβ hV hH hη hβη
  have harc (w : E × E) (u : Prefix n T) (s : ℝ) (hs0 : 0 ≤ s) (hsT : s ≤ (T : ℝ))
      (j : Fin (n+1)) (hj : (Fin.cons (0 : ℝ) (u.val : Fin n → ℝ) : Fin (n+1) → ℝ) j ≤ s)
      (hn : ∀ k : Fin (n+1), j < k → s < (Fin.cons (0 : ℝ) (u.val : Fin n → ℝ) : Fin (n+1) → ℝ) k) :
      Γ y r w n T u ⟨min (Real.toNNReal s) T, min_le_right _ _⟩ =
        Φ y r (s - (Fin.cons (0 : ℝ) (u.val : Fin n → ℝ) : Fin (n+1) → ℝ) j) (zAt y r w n T u j) := by
    have hclip : min (Real.toNNReal s) T = Real.toNNReal s :=
      min_eq_left (by exact_mod_cast (Real.toNNReal_le_iff_le_coe).2 hsT)
    have hcoe : ((min (Real.toNNReal s) T : ℝ≥0) : ℝ) = s := by
      rw [hclip, Real.coe_toNNReal s hs0]
    have hp := ((hlaw n T).2.2.2 y r w).2.2.2.2.1
      u ⟨min (Real.toNNReal s) T, min_le_right _ _⟩ j
      (by simpa only [hcoe] using hj) (by simpa only [hcoe] using hn)
    change Γ y r w n T u ⟨min (Real.toNNReal s) T, min_le_right _ _⟩ =
      Φ y r (((min (Real.toNNReal s) T : ℝ≥0) : ℝ) - (Fin.cons (0 : ℝ) (u.val : Fin n → ℝ) : Fin (n+1) → ℝ) j)
        (zAt y r w n T u j) at hp
    simpa only [hcoe] using hp
  obtain ⟨_, _, _, hgroup, _⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHarmonicFlow.actual_harmonic_flow_laws
      hα hαβ hV hH hη hβη
  obtain ⟨_, _, _, _, _, _, _, _, hR, hrev, _⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseReversalVolume.actual_phase_reversal_volume_laws
      hα hαβ hV hH hη hβη
  have hoff : ∀ s : ℝ, 0 ≤ s → s ≤ (T : ℝ) →
      (∀ i : Fin n, s ≠ (rho t).val i) → barGamma s = R (gamma ((T : ℝ)-s)) := by
    have hp := path_reversal n (T : ℝ) grid barGrid hgrid
      (by simp [barGrid]) (by simp [barGrid]) hbl hbr (Φ y r) R hR
      (hgroup y r) (hrev y r) post barPost gamma barGamma hpair
      (fun s hs0 hsT j hj hn => by
        simpa only [gamma, grid, post, Fin.snoc_castSucc] using
          harc z t s hs0 hsT j (by simpa only [grid, Fin.snoc_castSucc] using hj)
            (by simpa only [grid, Fin.snoc_castSucc] using hn))
      (fun s hs0 hsT j hj hn => by
        simpa only [barGamma, barGrid, barPost, Fin.snoc_castSucc] using
          harc (R A) (rho t) s hs0 hsT j (by simpa only [barGrid, Fin.snoc_castSucc] using hj)
            (by simpa only [barGrid, Fin.snoc_castSucc] using hn))
    intro s hs0 hsT hn
    exact hp s hs0 hsT (by simpa only [barGrid, Fin.snoc_castSucc, Fin.cons_succ] using hn)
  have hi : IntervalIntegrable (fun s => rate r (barGamma s)) volume 0 (T : ℝ) :=
    (AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathDensity.actual_prescribed_path_density_rewrite
      hα hαβ hV hH hη hβη y r (R A) n T (rho t)).1
  have hs := scalar_reflection (T : ℝ) T.coe_nonneg
    (fun s => rate r (R (gamma s))) (fun s => rate r (barGamma s))
    (Set.range (rho t).val) (Set.finite_range _) (by
      intro s hs hn
      exact congrArg (rate r) (hoff s hs.1 hs.2 (fun i he => hn ⟨i, he.symm⟩))) hi
  exact ⟨hoff, hi, hs⟩

#print axioms actual_prescribed_path_reversal_and_survival

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathReversal

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPrescribedPathReversal.lean:156published source at ed709a7f320d

Proof architecture

Actual finite PBPS path reversal and survival substitution; no SALD claim

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Density statements retain normalization and absolute-continuity prerequisites.