actual_prescribed_gibbs_survival_balance

Plain-English statement

- Actual finite prescribed target-energy and Gibbs/survival identity under the six source hypotheses.

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.

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

How to read the exact Lean declaration

Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.

NameWhat reusable mathematical fact is being created?
ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
PropositionAfter the colon, translate the Lean expression back into a paper statement.
Proof actionsAfter by, ask what each tactic does to the mathematical goal—not only what syntax it uses.

Syntax used on this page

This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.

Source voice and ASTIS voice stay separate

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

theorem actual_prescribed_gibbs_survival_balance
    {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_gibbs_survival_statement hα hαβ hV hH hη hβη := by
  unfold actual_prescribed_gibbs_survival_statement
  intro c Φ S Rec Hist Δ Prefix m zAt K Path Ψ Γ R rate U y r z n T t gamma A Iplus Iminus
  let a : Fin (n+1) → ℝ := Fin.cons 0 t.val
  let grid : Fin (n+2) → ℝ := Fin.snoc a (T : ℝ)
  let u : Fin (n+2) → ℝ := Fin.snoc (fun j => U y (zAt y r z n T t j)) (U y A)
  have hflow := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHarmonicFlow.actual_harmonic_flow_laws
    hα hαβ hV hH hη hβη
  have hzero : ∀ q : E × E, Φ y r 0 q = q := hflow.2.2.1 y r
  have hb := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBounceRate.actual_bounce_rate_energy_laws
    hα hαβ hV hH hη hβη
  have hsU (q : E × E) : U y (S r q) = U y q := by
    have hn := (hb.2.2.2.2.1 (gradient V q.1 - gradient V r) q.2).2.1
    change V q.1 + ‖q.1 - y‖ ^ 2 / (2 * η) + _ / 2 = _
    rw [show ‖(S r q).2‖ = ‖q.2‖ from hn]
  have hRate : Continuous (fun q : E × (E × E) => rate q.1 q.2) := hb.2.1
  have hPhi : Continuous (fun q : (E × E) × ℝ × (E × E) => Φ q.1.1 q.1.2 q.2.1 q.2.2) := hflow.1
  have ham : StrictMono a := t.property.1
  have haT : ∀ j, a j ≤ (T : ℝ) := by
    intro j
    refine Fin.cases ?_ (fun i => ?_) j
    · simpa [a] using T.coe_nonneg
    · exact (t.property.2 i).le
  have hgrid : Monotone grid := by
    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 haT i
      | cast j => simpa [grid] using ham.monotone (show i ≤ j from hij)
  have hg0 : grid 0 = 0 := by simp [grid, a]
  have hgT : grid (Fin.last (n+1)) = (T : ℝ) := by simp [grid]
  obtain ⟨Z, _, _, _, _, hlaw⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath.actual_finite_count_path_law
      hα hαβ hV hH hη hβη
  have harc := ((hlaw n T).2.2.2 y r z).2.2.2.2.1
  have heq (j : Fin (n+1)) : EqOn gamma
      (fun s => Φ y r (s - grid j.castSucc) (zAt y r z n T t j))
      (Ioo (grid j.castSucc) (grid j.succ)) := by
    intro s hs
    have hs0 : 0 ≤ s := by
      have := hgrid (show (0 : Fin (n+2)) ≤ j.castSucc from Fin.zero_le _)
      rw [hg0] at this
      exact this.trans hs.1.le
    have hsT : s ≤ (T : ℝ) := by
      have := hgrid (Fin.le_last j.succ)
      rw [hgT] at this
      exact hs.2.le.trans this
    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 hclipcoe : ((min (Real.toNNReal s) T : ℝ≥0) : ℝ) = s := by
      rw [hclip, Real.coe_toNNReal s hs0]
    have hstart : a j ≤ s := by simpa [grid] using hs.1.le
    have hafter : ∀ k : Fin (n+1), j < k → s < a k := by
      intro k hjk
      have hidx : j.succ ≤ k.castSucc := by exact_mod_cast hjk
      exact hs.2.trans_le (by simpa [grid] using hgrid hidx)
    have hh := harc t ⟨min (Real.toNNReal s) T, min_le_right _ _⟩ j
      (by simpa only [hclipcoe] using hstart)
      (by simpa only [hclipcoe] using hafter)
    change gamma s = Φ y r (((min (Real.toNNReal s) T : ℝ≥0) : ℝ) - a j)
      (zAt y r z n T t j) at hh
    simpa only [hclipcoe, grid, Fin.snoc_castSucc] using hh
  obtain ⟨rho, _, _, _, _, hend, _⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualEndpointTimeReversal.actual_endpoint_time_reversal_laws
      hα hαβ hV hH hη hβη n T
  have hterminal : A = Φ y r ((T : ℝ) - a (Fin.last n)) (zAt y r z n T t (Fin.last n)) :=
    hend y r z t
  have hnext (i : Fin n) : zAt y r z n T t i.succ =
      S r (Φ y r (t.val i - a i.castSucc) (zAt y r z n T t i.castSucc)) := by
    simp only [zAt, Fin.induction_succ]
    rfl
  have hEnd (j : Fin (n+1)) :
      U y (Φ y r (grid j.succ - grid j.castSucc) (zAt y r z n T t j)) = u j.succ := by
    induction j using Fin.lastCases with
    | last =>
      simpa only [Fin.succ_last, grid, u, Fin.snoc_last, Fin.snoc_castSucc] using
        (congrArg (U y) hterminal).symm
    | cast i =>
      simp only [Fin.succ_castSucc, u, Fin.snoc_castSucc]
      rw [hnext]
      simpa only [grid, Fin.succ_castSucc, Fin.snoc_castSucc, a, Fin.cons_succ] using
        (hsU (Φ y r (t.val i - a i.castSucc) (zAt y r z n T t i.castSucc))).symm
  have hstep (j : Fin (n+1)) :
      IntervalIntegrable (fun s => rate r (gamma s)) volume (grid j.castSucc) (grid j.succ) ∧
      IntervalIntegrable (fun s => rate r (R (gamma s))) volume (grid j.castSucc) (grid j.succ) ∧
      (∫ s in grid j.castSucc..grid j.succ, rate r (gamma s)) -
        (∫ s in grid j.castSucc..grid j.succ, rate r (R (gamma s))) = u j.succ - u j.castSucc := by
    let q : ℝ → E × E := fun s => Φ y r (s - grid j.castSucc) (zAt y r z n T t j)
    have harg : Continuous (fun s : ℝ => ((y,r),(s - grid j.castSucc),zAt y r z n T t j)) :=
      continuous_const.prodMk ((continuous_id.sub continuous_const).prodMk continuous_const)
    have hq : Continuous q := hPhi.comp harg
    have hp : Continuous (fun s : ℝ => rate r (q s)) :=
      hRate.comp ((continuous_const (y := r)).prodMk hq)
    have hm : Continuous (fun s : ℝ => rate r (R (q s))) :=
      hRate.comp ((continuous_const (y := r)).prodMk (hq.fst.prodMk hq.snd.neg))
    have hle := hgrid (Fin.castSucc_le_succ j)
    have hip : IntervalIntegrable (fun s => rate r (q s)) volume (grid j.castSucc) (grid j.succ) := hp.intervalIntegrable _ _
    have him : IntervalIntegrable (fun s => rate r (R (q s))) volume (grid j.castSucc) (grid j.succ) := hm.intervalIntegrable _ _
    have hep : EqOn (fun s => rate r (gamma s)) (fun s => rate r (q s)) (Ioo (grid j.castSucc) (grid j.succ)) :=
      fun s hs => congrArg (rate r) (heq j hs)
    have hem : EqOn (fun s => rate r (R (gamma s))) (fun s => rate r (R (q s))) (Ioo (grid j.castSucc) (grid j.succ)) :=
      fun s hs => congrArg (fun w => rate r (R w)) (heq j hs)
    refine ⟨hip.congr_uIoo (by simpa only [uIoo_of_le hle] using hep.symm),
      him.congr_uIoo (by simpa only [uIoo_of_le hle] using hem.symm), ?_⟩
    rw [intervalIntegral.integral_congr_Ioo_of_le hle hep,
      intervalIntegral.integral_congr_Ioo_of_le hle hem, ← intervalIntegral.integral_sub hip him]
    have hd (s : ℝ) : HasDerivAt (fun w => U y (q w))
        (rate r (q s) - rate r (R (q s))) s := by
      have hh : HasDerivAt (fun w => U y (Φ y r w (zAt y r z n T t j)))
          (rate r (q s) - rate r (R (q s))) (s - grid j.castSucc) :=
        (actual_arc_energy hα hαβ hV hH hη hβη y r (zAt y r z n T t j)).1 (s - grid j.castSucc)
      simpa only [Function.comp_def, id_eq, one_smul] using
        hh.scomp s ((hasDerivAt_id s).sub_const (grid j.castSucc))
    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun s _ => hd s) (hip.sub him)]
    change U y (Φ y r (grid j.succ - grid j.castSucc) (zAt y r z n T t j)) -
      U y (Φ y r (grid j.castSucc - grid j.castSucc) (zAt y r z n T t j)) = _
    rw [hEnd, sub_self, hzero]
    simp only [u, Fin.snoc_castSucc]
  have hf := finite_energy_balance (n+1) grid u
    (fun s => rate r (gamma s)) (fun s => rate r (R (gamma s)))
    (fun j => (hstep j).1) (fun j => (hstep j).2.1) (fun j => (hstep j).2.2)
  have hi : IntervalIntegrable (fun s => rate r (gamma s)) volume 0 (T : ℝ) := by
    simpa only [hg0, hgT] using hf.1
  have hj : IntervalIntegrable (fun s => rate r (R (gamma s))) volume 0 (T : ℝ) := by
    simpa only [hg0, hgT] using hf.2.1
  have henergy : U y A - U y z = Iplus - Iminus := by
    simpa [u, zAt, hg0, hgT] using hf.2.2
  refine ⟨hi, hj, henergy, ?_⟩
  rw [← Real.exp_add, ← Real.exp_add]
  congr 1
  linarith


end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedGibbsSurvival
#print axioms AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedGibbsSurvival.actual_prescribed_gibbs_survival_balance

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPrescribedGibbsSurvival.lean:182published source at ed709a7f320d

Proof architecture

Actual finite PBPS target-energy and survival balance; 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.
A Gibbs expression is not automatically a probability law or an invariant law.