openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Local /LocalConditionUnusedSupport.lean
| import OAI.Combinatorics.BalancedRyser.Local.LocalConditionFibers | |
| namespace OAI | |
| namespace Balanced.Local.ConditionSpec | |
| noncomputable section | |
| open scoped Classical | |
| open LocalFiberBounds | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| variable {b κ : ℕ} | |
| theorem indexedThrough_eq_of_unused_eq (a : DirectionAssignment F b κ) | |
| (M N : MergedPairs F) | |
| (h : ∀ d, d ∉ FourEvents.usedDirections a → M d = N d) | |
| (p : Point F) : | |
| indexedThrough M (unusedDirections a) p = | |
| indexedThrough N (unusedDirections a) p := by | |
| ext ds | |
| by_cases hd : ds.1 ∉ FourEvents.usedDirections a | |
| · simp [indexedThrough, unusedDirections, hd, constituent, h ds.1 hd] | |
| · simp [indexedThrough, unusedDirections, hd] | |
| theorem pointHolds_iff_of_unused_eq (a : DirectionAssignment F b κ) | |
| (p : Point F) (M N : MergedPairs F) | |
| (h : ∀ d, d ∉ FourEvents.usedDirections a → M d = N d) | |
| (x : Position b → Point F) (v : Position b) (z : Code b κ) : | |
| PointHolds a p M x v z ↔ PointHolds a p N x v z := by | |
| rcases z with z | xy | w | cw | |
| · simp only [PointHolds] | |
| apply exists_congr | |
| intro d | |
| by_cases hd : d ∉ FourEvents.usedDirections a | |
| · simp only [hd, constituent, h d hd] | |
| · simp only [hd, false_and] | |
| · rfl | |
| · simp only [PointHolds] | |
| apply exists_congr | |
| intro d | |
| by_cases hd : d ∉ FourEvents.usedDirections a | |
| · simp only [hd, mergedPoints, constituent, h d hd] | |
| · simp only [hd, false_and] | |
| · rfl | |
| theorem holds_iff_pointHolds_of_unused_eq (P : EqualityPattern b κ) | |
| (a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) | |
| (M N : MergedPairs F) | |
| (h : ∀ d, d ∉ FourEvents.usedDirections a → M d = N d) | |
| (v : Position b) (z : Code b κ) : | |
| Holds P a bits p M v z ↔ | |
| PointHolds a p N (prescribedPoint P a bits M) v z := by | |
| rw [holds_eq_pointHolds] | |
| exact pointHolds_iff_of_unused_eq a p M N h _ v z | |
| end | |
| end Balanced.Local.ConditionSpec | |
| end OAI | |