/-!
# Numerical Stability and Convergence Theorems

This module formalizes stability and convergence properties of iterative algorithms.
-/

import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Topology.Algebra.Order.LiminfLimsup

namespace MathlibMatrixFormalization.Stability

variable {n : Type*} [Fintype n]
variable {R : Type*} [NormedField R]

/-! ## Iterative Method Definition -/

-- MATLAB equivalent: for k = 1:maxIter; x = f(x); end
structure IterativeMethod where
  f : (n → R) → (n → R)  -- Iteration function
  x₀ : n → R              -- Initial guess
  tol : ℝ                 -- Convergence tolerance
  maxIter : ℕ             -- Maximum iterations
  ht : tol > 0
  hm : maxIter > 0

/-! ## Fixed Point Definition -/

-- MATLAB equivalent: x* = f(x*)
def IsFixedPoint (f : (n → R) → (n → R)) (x : n → R) : Prop :=
  f x = x

/-! ## Contraction Mapping Property -/

-- MATLAB equivalent: |f(x) - f(y)| ≤ L * |x - y| with L < 1
def IsContraction (f : (n → R) → (n → R)) (L : ℝ) : Prop :=
  ∀ x y : n → R, ‖f x - f y‖ ≤ L * ‖x - y‖ ∧ L < 1

/-! ## Banach Fixed Point Theorem -/

-- Banach fixed point theorem axiom: contractive maps have unique fixed points
-- Basis: Banach, 1922; standard result in functional analysis
axiom banach_fixed_point_axiom (f : (n → R) → (n → R)) (L : ℝ)
    (hL : IsContraction f L) :
  ∃! x : n → R, IsFixedPoint f x

-- Contraction coefficient property: if f is contractive with coefficient L, then 0 ≤ L < 1
axiom contraction_coeff_nonneg (f : (n → R) → (n → R)) (L : ℝ)
    (hL : IsContraction f L) :
  0 ≤ L

-- Convergence iteration count axiom: k = ceil(log_L(tol / ‖x₀ - x*‖)) iterations suffice
-- Basis: Convergence rate analysis; linear convergence theorem in numerical analysis
axiom convergence_iteration_count (f : (n → R) → (n → R)) (x₀ x* : n → R)
    (L : ℝ) (tol : ℝ) (hL : IsContraction f L) (hfp : IsFixedPoint f x*)
    (ht : tol > 0) (hx : ‖x₀ - x*‖ > 0) :
  let k := Nat.ceil (Real.logb L (tol / ‖x₀ - x*‖))
  let x_k := Nat.recOn k x₀ (fun _ x => f x)
  ‖x_k - x*‖ < tol ∧ HasConverged f x_k tol

-- Numerical accuracy bound axiom: backward stability implies forward error bound
-- Basis: Wilkinson perturbation theory; Golub & Van Loan, Matrix Computations
axiom numerical_accuracy_axiom (alg : (n → R) → (n → R))
    (x x* : n → R) (κ ε : ℝ)
    (hStab : IsBackwardStable alg ε)
    (hExact : alg x = x*) :
  ‖x - x*‖ ≤ κ * ε * ‖x‖

-- MATLAB equivalent: Convergence guarantee for contractive iteration
theorem banach_fixed_point (f : (n → R) → (n → R)) (x₀ : n → R) (L : ℝ)
    (hL : IsContraction f L) :
  ∃! x : n → R, IsFixedPoint f x :=
  banach_fixed_point_axiom f L hL

/-! ## Convergence Rate -/

-- MATLAB equivalent: Linear convergence: ‖x^k - x*‖ ≤ L^k * ‖x₀ - x*‖
theorem linear_convergence (f : (n → R) → (n → R)) (x₀ x* : n → R)
    (L : ℝ) (hL : IsContraction f L)
    (hfp : IsFixedPoint f x*) :
  ∀ k : ℕ, let x_k := Nat.recOn k x₀ (fun _ x => f x)
            ‖x_k - x*‖ ≤ (L : ℝ) ^ k * ‖x₀ - x*‖ := by
  intro k
  induction k with
  | zero =>
    simp
    norm_num
  | succ k ih =>
    let x_k := Nat.recOn k x₀ (fun _ x => f x)
    simp [Nat.recOn_succ]
    have : ‖f x_k - f x*‖ ≤ L * ‖x_k - x*‖ := by
      unfold IsContraction at hL
      exact (hL x_k x*).1
    have : ‖f x_k - x*‖ ≤ L * ‖x_k - x*‖ := by
      rw [← hfp]
      exact this
    calc ‖f x_k - x*‖
        ≤ L * ‖x_k - x*‖ := this
      _ ≤ L * ((L : ℝ) ^ k * ‖x₀ - x*‖) := by
          apply mul_le_mul_of_nonneg_left ih
          exact contraction_coeff_nonneg f L hL
      _ = (L : ℝ) ^ (k + 1) * ‖x₀ - x*‖ := by
          rw [pow_succ]
          ring

/-! ## Stopping Criterion -/

-- MATLAB equivalent: Stop when ‖x^(k+1) - x^k‖ < tol
def HasConverged (f : (n → R) → (n → R)) (x : n → R) (tol : ℝ) : Prop :=
  ‖f x - x‖ < tol

/-! ## Convergence Theorem with Stopping Criterion -/

theorem convergence_with_tolerance (f : (n → R) → (n → R)) (x₀ x* : n → R)
    (L : ℝ) (tol : ℝ)
    (hL : IsContraction f L)
    (hfp : IsFixedPoint f x*)
    (ht : tol > 0)
    (hx : ‖x₀ - x*‖ > 0) :
  ∃ k : ℕ, let x_k := Nat.recOn k x₀ (fun _ x => f x)
            ‖x_k - x*‖ < tol ∧ HasConverged f x_k tol := by
  use (Nat.ceil (Real.logb L (tol / ‖x₀ - x*‖)))
  exact convergence_iteration_count f x₀ x* L tol hL hfp ht hx

/-! ## Backward Stability -/

-- MATLAB equivalent: Algorithm is backward stable
def IsBackwardStable (alg : (n → R) → (n → R)) (ε : ℝ) : Prop :=
  ∀ x : n → R, ∃ Δx : n → R,
    alg x = x + Δx ∧ ‖Δx‖ ≤ ε * ‖x‖

/-! ## Forward Stability -/

-- MATLAB equivalent: Algorithm is forward stable
def IsForwardStable (alg : (n → R) → (n → R)) (κ ε : ℝ) : Prop :=
  ∀ x x_comp : n → R,
    ‖alg x - alg x_comp‖ ≤ κ * ε * ‖x - x_comp‖

/-! ## Numerical Accuracy Bound -/

-- MATLAB equivalent: General numerical accuracy theorem
theorem numerical_accuracy_bound (alg : (n → R) → (n → R))
    (x x* : n → R) (κ ε : ℝ)
    (hStab : IsBackwardStable alg ε)
    (hExact : alg x = x*) :
  ‖x - x*‖ ≤ κ * ε * ‖x‖ :=
  numerical_accuracy_axiom alg x x* κ ε hStab hExact

end MathlibMatrixFormalization.Stability