| import OAI.Computability.LoopMatching.Model | |
| namespace OAI | |
| /-! Polynomial-time approximation of weighted matchings with singleton loops. -/ | |
| open scoped BigOperators | |
| namespace LoopMatching | |
| theorem loopVertices_card_le {n : ℕ} (M : PartialMatching n) : | |
| (loopVertices M).card ≤ n := by | |
| classical | |
| simpa using Finset.card_le_card (Finset.subset_univ (loopVertices M)) | |
| theorem countWithLoops_eq_zero_of_lt (G : Input) {j : ℕ} (hj : G.n < j) : | |
| countWithLoops G j = 0 := by | |
| classical | |
| unfold countWithLoops | |
| apply Finset.sum_eq_zero | |
| intro M hM | |
| have hne : (loopVertices M).card ≠ j := by | |
| intro heq | |
| have hle := loopVertices_card_le M | |
| rw [heq] at hle | |
| exact (Nat.not_le_of_gt hj) hle | |
| simp only [hne, ↓reduceIte] | |
| theorem count_eq_sum_countWithLoops (G : Input) : | |
| count G = ∑ j ∈ Finset.range (G.n + 1), countWithLoops G j := by | |
| classical | |
| unfold count countWithLoops | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro M hM | |
| symm | |
| calc | |
| _ = (if (loopVertices M).card = (loopVertices M).card then weight G M else 0) := by | |
| apply Finset.sum_eq_single (loopVertices M).card | |
| · intro j hj hne | |
| simp only [Ne.symm hne, ↓reduceIte] | |
| · intro hnot | |
| exact (hnot (Finset.mem_range.mpr | |
| (Nat.lt_succ_of_le (loopVertices_card_le M)))).elim | |
| _ = weight G M := ite_eq_left rfl | |
| theorem weight_eq_zero_of_loopMultiplicity_eq_zero (G : Input) | |
| (M : PartialMatching G.n) (v : Fin G.n) | |
| (hv : v ∈ loopVertices M) (hzero : loopMultiplicity G v = 0) : | |
| weight G M = 0 := by | |
| classical | |
| unfold weight | |
| rw [Finset.prod_eq_zero hv hzero, mul_zero] | |
| theorem weight_eq_zero_of_multiplicity_eq_zero (G : Input) | |
| (M : PartialMatching G.n) (e : BinaryMatching.Pair G.n) | |
| (he : e ∈ M.val) (hzero : BinaryMatching.multiplicity G.ordinary e = 0) : | |
| weight G M = 0 := by | |
| classical | |
| unfold weight | |
| rw [Finset.prod_eq_zero he hzero, zero_mul] | |
| theorem count_eq_zero_of_forall_weight_eq_zero (G : Input) | |
| (hzero : ∀ M : PartialMatching G.n, weight G M = 0) : count G = 0 := by | |
| classical | |
| unfold count | |
| exact Finset.sum_eq_zero (fun M _ => hzero M) | |
| end LoopMatching | |
| end OAI | |