import Mathlib

/-! # Fluid Limit Stability

This module provides the mathematical foundation for stability analysis of
large-scale session-typed systems via fluid approximations. Fluid limits
replace discrete stochastic systems with continuous deterministic flows,
making stability analysis tractable while preserving essential dynamics.

## Physical Intuition

Consider water flowing through a pipe network. At a microscopic level, water
consists of individual molecules bouncing chaotically. But at a macroscopic
level, we can describe flow rates and pressures without tracking molecules.
The "fluid limit" is this macroscopic view: it averages over microscopic
fluctuations to reveal the deterministic backbone of the system.

The stability criterion is intuitive: if the fluid level in each tank is
dropping over time, eventually all tanks will empty. Mathematically, this
becomes an "energy dissipation" argument: if the squared norm ||x||² of the
system state decreases with each step, the system must converge to zero.

## Canonical Mathematical Formulation

Consider a queueing network with queue lengths Q(t). The **fluid model** is
obtained by scaling:

  Q̄(t) = lim_{n→∞} Q(nt) / n

Under this limit, the random dynamics become deterministic, satisfying a
differential equation:

  dQ̄/dt = λ - μ(Q̄)

where λ is the arrival rate and μ(Q̄) is the service rate (possibly state-dependent).

The system is **fluid stable** if for any initial condition Q̄(0), the fluid
solution satisfies Q̄(t) → 0 as t → ∞. A sufficient condition is the existence
of an **energy function** (Lyapunov function) E : ℝⁿ → ℝ₊ such that:

  dE/dt ≤ -κ · ||Q̄||

for some κ > 0 whenever Q̄ ≠ 0.

## Application to Telltale

In session-typed systems, fluid stability applies to:

1. **Buffer accumulation**: Do buffers grow without bound, or do they drain?
   The fluid model tracks the deterministic component of buffer dynamics.

2. **Multi-session systems**: With many concurrent sessions, individual
   randomness averages out, and fluid analysis captures aggregate behavior.

3. **Capacity planning**: Fluid models identify the stability region: the
   set of arrival rates for which the system remains stable.

The `energy` function ||x||² (squared Euclidean norm) captures total "backlog
pressure" in the system. The `FluidDescentWitness` certifies that this energy
dissipates at rate κ · |x|, guaranteeing eventual drainage.

For Coherence, the fluid perspective reveals that buffer boundedness follows
from structural properties of the protocol (balanced send/receive rates)
rather than fine-grained timing.

## Key Properties

- **Quadratic energy**: The energy function E(x) = x² is the simplest choice,
  measuring total "stress" in the system.

- **Dissipation rate**: The descent condition E(n+1) ≤ E(n) - κ|x(n)| ensures
  energy drains at a rate proportional to the state magnitude.

- **Stability implies termination**: Energy dissipation implies the system
  reaches E = 0 in finite time, corresponding to session completion.

## References

- Dai, J.G. (1995). On positive Harris recurrence of multiclass queueing
  networks: a unified approach via fluid limit models.
- Bramson, M. (2008). Stability of Queueing Networks.
- See also `Runtime/Proofs/WeightedMeasure.lean` for protocol stability analysis.
-/

/-
The Problem. This module needs a concise statement of its theorem-family boundary and reusable assumptions.
Solution Structure. Introduce the core model/contracts first, then derive canonical lemmas and API wrappers from those contracts.
-/

/-! ## Core Development -/

namespace Classical
namespace FluidLimitStability

/-- The energy function: squared state value.

    In the Telltale setting:
    - x(n) is the buffer occupancy (or aggregate backlog) at step n
    - energy x n = x(n)² measures the "pressure" in the system

    Stability is equivalent to energy approaching zero. -/
def energy (x : Nat → Real) (n : Nat) : Real :=
  (x n) ^ 2

/-- A `FluidDescentWitness` certifies that energy dissipates.

    The witness provides:
    - κ: the dissipation rate (must be ≥ 0)
    - descent: proof that E(n+1) ≤ E(n) - κ|x(n)|

    In the Telltale setting, this witnesses that buffer pressure decreases
    at a rate proportional to buffer occupancy, ensuring eventual drainage. -/
structure FluidDescentWitness (x : Nat → Real) where
  κ : Real
  kappa_nonneg : 0 ≤ κ
  descent : ∀ n, energy x (n + 1) ≤ energy x n - κ * |x n|

/-- Energy is non-increasing when dissipation holds.

    For protocols: if the descent condition holds, energy can only decrease.
    Combined with non-negativity of energy, this gives eventual convergence. -/
theorem nonincreasing_energy {x : Nat → Real} (h : FluidDescentWitness x) :
    ∀ n, energy x (n + 1) ≤ energy x n := by
  intro n
  exact le_trans (h.descent n) (sub_le_self _ (mul_nonneg h.kappa_nonneg (abs_nonneg _)))

/-- Each step dissipates at least κ|x(n)| of energy.

    For protocols: the "slack" in each step is at least κ times the current
    backlog. This means large backlogs drain faster, preventing runaway growth. -/
theorem one_step_dissipation {x : Nat → Real} (h : FluidDescentWitness x) (n : Nat) :
    energy x n - energy x (n + 1) ≥ h.κ * |x n| := by
  have hstep := h.descent n
  linarith

end FluidLimitStability
end Classical
