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8df23f4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Fintype.Prod
import OAI.Computability.BinPacking.Inventory.InventoryScores
namespace OAI
namespace BinPackingGap.InventoryData
open scoped BigOperators
variable (D : InventoryData)
theorem card_jobCopy_edge_of_endpoint (v : D.Vertex) (e : D.Edge) (s : Fin 2)
(hs : D.graph.endpoint e s = v) :
(Finset.univ.filter (fun j : D.JobCopy v => (D.jobPosition j).1 = e)).card =
D.d * D.R := by
classical
let f : Fin D.R × Fin D.d → D.JobCopy v :=
fun rc => ((⟨(e, s), hs⟩, rc.1), rc.2)
have hf : Function.Injective f := by
intro a b hab
apply Prod.ext
· exact congrArg (fun j : D.JobCopy v => j.1.2) hab
· exact congrArg (fun j : D.JobCopy v => j.2) hab
have hfilter :
Finset.univ.filter (fun j : D.JobCopy v => (D.jobPosition j).1 = e) =
Finset.univ.image f := by
ext j
simp only [Finset.mem_filter, Finset.mem_univ, true_and]
constructor
· intro hj
apply Finset.mem_image.mpr
refine ⟨(j.1.2, j.2), Finset.mem_univ _, ?_⟩
rcases j with ⟨⟨⟨⟨e', s'⟩, hv⟩, r⟩, c⟩
change e' = e at hj
subst e'
have hslot : s = s' := D.graph.endpoint_injective e (hs.trans hv.symm)
subst s'
rfl
· intro hj
obtain ⟨rc, _, rfl⟩ := Finset.mem_image.mp hj
rfl
rw [hfilter, Finset.card_image_of_injective _ hf, Finset.card_univ]
simp [Nat.mul_comm]
theorem card_jobCopy_edge (v : D.Vertex) (e : D.Edge) :
(Finset.univ.filter (fun j : D.JobCopy v => (D.jobPosition j).1 = e)).card =
if D.graph.left e = v ∨ D.graph.right e = v then D.d * D.R else 0 := by
classical
by_cases hv : D.graph.left e = v ∨ D.graph.right e = v
· rw [ite_eq_left hv]
rcases hv with hl | hr
· exact D.card_jobCopy_edge_of_endpoint v e 0 (by simpa using hl)
· exact D.card_jobCopy_edge_of_endpoint v e 1 (by simpa using hr)
· rw [ite_eq_right hv]
apply Finset.card_eq_zero.mpr
apply Finset.eq_empty_iff_forall_notMem.mpr
intro j hj
apply hv
have he : j.1.1.val.1 = e := (Finset.mem_filter.mp hj).2
have hep : D.graph.endpoint e j.1.1.val.2 = v := by
rw [← he]
exact j.1.1.property
have hslot : j.1.1.val.2 = 0 ∨ j.1.1.val.2 = 1 := by omega
rcases hslot with hzero | hone
· left
simpa only [hzero, GraphInput.endpoint_zero] using hep
· right
simpa only [hone, GraphInput.endpoint_one] using hep
theorem sum_card_jobCopy_edge (e : D.Edge) :
(∑ v : D.Vertex,
(Finset.univ.filter (fun j : D.JobCopy v => (D.jobPosition j).1 = e)).card) =
2 * D.d * D.R := by
classical
calc
_ =
(Finset.univ.filter (fun j : D.JobCopy (D.graph.left e) =>
(D.jobPosition j).1 = e)).card +
(Finset.univ.filter (fun j : D.JobCopy (D.graph.right e) =>
(D.jobPosition j).1 = e)).card := by
refine Finset.sum_eq_add_of_mem (D.graph.left e) (D.graph.right e)
(Finset.mem_univ _) (Finset.mem_univ _) (D.graph.left_ne_right e) ?_
intro v _ hv
rw [D.card_jobCopy_edge]
simp [Ne.symm hv.1, Ne.symm hv.2]
_ = D.d * D.R + D.d * D.R := by simp [D.card_jobCopy_edge]
_ = 2 * D.d * D.R := by simp [two_mul, Nat.add_mul]
end BinPackingGap.InventoryData
end OAI
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