import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Probability.Kernel.Composition.Prod
import Mathlib.Probability.Kernel.Composition.CompProd
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Lean.Elab.Tactic.Location
import Lean.Meta.DecLevel
import Lean.Meta.Transform
import Lean.Util.Recognizers
import Lean.Meta.Tactic.Replace
import Lean.Meta.Tactic.Rewrite
import Mathlib.Probability.Kernel.Deterministic

/-! # Standalone extraction for `finisherKernelUnlift`
Definitions are copied verbatim; theorem proofs are replaced by `sorry`.
Auto-generated by Referee. -/

set_option quotPrecheck false

-- Namespace stubs (so later `open`s resolve).
namespace Finset
end Finset
namespace Lean
end Lean
namespace Lean.Meta
end Lean.Meta
namespace ProbabilityTheory
end ProbabilityTheory
namespace ProbabilityTheory.Kernel
end ProbabilityTheory.Kernel

-- ═══ ForMathlib.MeasureTheory.Order.Lattice ═══
section
open Finset
variable {α δ : Type*} [MeasurableSpace δ] [SemilatticeInf α] {m : MeasurableSpace α} [MeasurableInf₂ α]
attribute [to_dual existing] MeasurableInf₂
end

-- ═══ Tactic.EqLift.Tactic.Kernel.Utils ═══
section
public meta section
open Lean Meta ProbabilityTheory Elab Term

/-- Extract `(X, Y, u, v)` from an expression of type `Kernel X Y`. -/
def getTypesFromKernel (κ : Expr) : MetaM (Expr × Expr × Level × Level) := do
  let κType ← inferType κ
  match κType.getAppFn with
  | Expr.const ``Kernel univs =>
    let args := κType.getAppArgs
    if args.size < 2 then
      throwError "Kernel type with insufficient arguments: {κType}."
    let X := args[0]!
    let Y := args[1]!
    let xLevel := univs[0]!
    let yLevel := univs[1]!
    return (X, Y, xLevel, yLevel)
  | _ => throwError "Expected a kernel type, got: {κType}."

/-- Build a measurable equivalence for `e` into universe `maxLvl` (recursive on products). -/
partial def constructMeasurableEquiv (e : Expr) (eLevel maxLvl : Level) : MetaM Expr := do
  let ewhnf ← whnf e
  match ewhnf.getAppFn with
  | Expr.const ``PUnit _ | Expr.const ``Unit _ =>
    mkAppOptM' (Expr.const `MeasurableEquiv.punit [maxLvl, eLevel]) #[]
  | Expr.const ``Prod univs =>
    let args := ewhnf.getAppArgs
    let X := args[0]!
    let Y := args[1]!
    let xLevel := univs[0]!
    let yLevel := univs[1]!
    let ex ← constructMeasurableEquiv X xLevel maxLvl
    let ey ← constructMeasurableEquiv Y yLevel maxLvl
    let res ← mkAppOptM' (Expr.const ``MeasurableEquiv.prodCongr [maxLvl, xLevel, maxLvl, yLevel])
      #[none, none, none, none, none, none, none, none, ex, ey]
    return res
  | _ => mkAppOptM' (Expr.const ``MeasurableEquiv.ulift [eLevel, maxLvl]) #[e, none]

end
end

-- ═══ Tactic.EqLift.Tactic.Kernel.KernelUnlift ═══
section
public meta section
open Lean Meta Parser.Tactic ProbabilityTheory ProbabilityTheory.Kernel

/-- Constructs the finisher proof that concludes the unlifting after rewriting the equalities. -/
def finisherKernelUnlift (_ _ lhs rhs : Expr) (maxLvl : Level) : MetaM Expr := do
  let (X, Y, xLvl, yLvl) ← getTypesFromKernel lhs
  let ex ← constructMeasurableEquiv X xLvl maxLvl
  let ey ← constructMeasurableEquiv Y yLvl maxLvl
  let lift_congr_expr ← mkAppM ``lift_congr #[ex, ey, lhs, rhs]
  mkAppM ``Iff.symm #[lift_congr_expr]

end
end