openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /FieldReindex.lean
| import OAI.MathematicalPhysics.NavierStokes.ShearFlows.FieldExpressions | |
| namespace OAI | |
| /-! Coordinate changes in finite field expressions. This moves the planar | |
| processor's periodic phase from its third spatial coordinate to time. -/ | |
| namespace ForcedComputation | |
| open ShearFlows | |
| noncomputable def reindexPoint (σ : Fin 4 → Fin 4) (y : SpaceTime) : SpaceTime := | |
| (timeSpaceCoord (σ 0) y, fun j => timeSpaceCoord (σ j.succ) y) | |
| theorem timeSpaceCoord_reindex (σ : Fin 4 → Fin 4) (j : Fin 4) (y : SpaceTime) : | |
| timeSpaceCoord j (reindexPoint σ y) = timeSpaceCoord (σ j) y := by | |
| fin_cases j <;> rfl | |
| def _root_.OAI.ShearFlows.FieldExpr.reindex (σ : Fin 4 → Fin 4) : FieldExpr → FieldExpr | |
| | .const r => .const r | |
| | .atom j p => .atom (σ j) p | |
| | .add e f => .add (e.reindex σ) (f.reindex σ) | |
| | .mul e f => .mul (e.reindex σ) (f.reindex σ) | |
| theorem _root_.OAI.ShearFlows.FieldExpr.valid_reindex {e : FieldExpr} (he : e.Valid) (σ : Fin 4 → Fin 4) : | |
| (e.reindex σ).Valid := by | |
| induction e with | |
| | const => trivial | |
| | atom j p => exact he | |
| | add e f ih₁ ih₂ => exact ⟨ih₁ he.1, ih₂ he.2⟩ | |
| | mul e f ih₁ ih₂ => exact ⟨ih₁ he.1, ih₂ he.2⟩ | |
| theorem _root_.OAI.ShearFlows.FieldExpr.val_reindex (e : FieldExpr) (σ : Fin 4 → Fin 4) (y : SpaceTime) : | |
| (e.reindex σ).val y = e.val (reindexPoint σ y) := by | |
| induction e with | |
| | const => rfl | |
| | atom j p => simp only [FieldExpr.reindex, FieldExpr.val, timeSpaceCoord_reindex] | |
| | add e f ih₁ ih₂ => simp only [FieldExpr.reindex, FieldExpr.val, ih₁, ih₂] | |
| | mul e f ih₁ ih₂ => simp only [FieldExpr.reindex, FieldExpr.val, ih₁, ih₂] | |
| def timeHeightSwap : Fin 4 → Fin 4 := ![3, 1, 2, 0] | |
| theorem reindexPoint_timeHeightSwap (y : SpaceTime) : | |
| reindexPoint timeHeightSwap y = (y.2 2, atHeight (horizontal y.2) y.1) := by | |
| apply Prod.ext | |
| · rfl | |
| · ext j | |
| fin_cases j <;> rfl | |
| theorem _root_.OAI.ShearFlows.FieldExpr.val_timeHeightSwap (e : FieldExpr) (y : SpaceTime) : | |
| (e.reindex timeHeightSwap).val y = e.val (y.2 2, atHeight (horizontal y.2) y.1) := by | |
| rw [FieldExpr.val_reindex, reindexPoint_timeHeightSwap] | |
| end ForcedComputation | |
| end OAI | |