openAI-math-dataset / lean /OAI /Probability /LogConcave /OraclePrograms /TerminalMeanEquivariant.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.OraclePrograms.Expression | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling.SeedCompiler | |
| open OracleCompiler OracleCompiler.Expression | |
| lemma terminalMean_equivariant {d : ℕ} (r σ : ℝ) (Δ : Point d) : | |
| (terminalMean d r σ).Equivariant | |
| (moveSeed (fun x => x+r • Δ) (terminalShift 1) Δ) id := by | |
| apply Program.oneQuery_equivariant | |
| · intro z | |
| exact terminalQuery_invariant _ _ _ _ | |
| · intro z a | |
| simp [moveSeed,shiftSlots,terminalShift] | |
| lemma terminalSample_equivariant {d : ℕ} (r η : ℝ) (Δ : Point d) : | |
| (terminalSample d r η).Equivariant | |
| (moveSeed (fun x => x+r • Δ) (terminalShift 2) Δ) (fun y => y-Δ) := by | |
| apply Program.oneQuery_equivariant | |
| · intro z | |
| exact terminalQuery_invariant _ _ _ _ | |
| · intro z a | |
| simp [moveSeed,shiftSlots,terminalShift] | |
| module | |
| lemma terminalSample_reserved {d : ℕ} (r η : ℝ) (Δ : Point d) : | |
| (terminalSample d r η).Equivariant | |
| (moveSeed id (fun i => if i=2 then 1 else 0) Δ) (fun y => y+(η/2) • Δ) := by | |
| apply Program.oneQuery_equivariant | |
| · intro z | |
| simp [moveSeed,shiftSlots] | |
| · intro z a | |
| simp [moveSeed,shiftSlots,smul_add] | |
| abel | |
| end LogConcaveSampling.SeedCompiler | |
| end | |
| end | |
| end | |
| end OAI | |