| import Lean.Elab.Tactic.Omega | |
| import Mathlib.Algebra.BigOperators.Group.Finset.Basic | |
| import Mathlib.Data.Finset.Card | |
| import Mathlib.Data.Fintype.Basic | |
| namespace OAI | |
| namespace BinPackingGap | |
| namespace FinitePrefixConservation | |
| variable {α β : Type*} [DecidableEq α] [DecidableEq β] | |
| theorem remaining_eq_bad (keys good : Finset α) (resources : Finset β) | |
| (f : α → β) (hgood : good ⊆ keys) | |
| (hmap : ∀ x ∈ good, f x ∈ resources) | |
| (hinj : Set.InjOn f (good : Set α)) | |
| (hcard : resources.card = keys.card) : | |
| (resources \ good.image f).card = (keys \ good).card := by | |
| have himage : good.image f ⊆ resources := Finset.image_subset_iff.mpr hmap | |
| rw [Finset.card_sdiff_of_subset himage, Finset.card_sdiff_of_subset hgood, | |
| Finset.card_image_of_injOn hinj, hcard] | |
| theorem remaining_le_bad (keys good : Finset α) (resources remaining : Finset β) | |
| (f : α → β) (hgood : good ⊆ keys) | |
| (hmap : ∀ x ∈ good, f x ∈ resources) | |
| (hinj : Set.InjOn f (good : Set α)) | |
| (hcard : resources.card = keys.card) | |
| (hremaining : remaining ⊆ resources \ good.image f) : | |
| remaining.card ≤ (keys \ good).card := by | |
| calc | |
| remaining.card ≤ (resources \ good.image f).card := Finset.card_le_card hremaining | |
| _ = (keys \ good).card := remaining_eq_bad keys good resources f hgood hmap hinj hcard | |
| end FinitePrefixConservation | |
| namespace LabelConservation | |
| open scoped BigOperators | |
| variable {n : ℕ} | |
| variable {T A R D : Type*} | |
| variable [Fintype T] [Fintype A] [Fintype R] [Fintype D] | |
| variable [DecidableEq A] [DecidableEq R] [DecidableEq D] | |
| def outsideItems (f : T → A) : Finset A := Finset.univ \ Finset.univ.image f | |
| def prefixItems (label : A → Fin n) (h : ℕ) : Finset A := | |
| Finset.univ.filter fun x => (label x).val < h | |
| def prefixOutside (label : A → Fin n) (f : T → A) (h : ℕ) : Finset A := | |
| (outsideItems f).filter fun x => (label x).val < h | |
| def exports (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) (h : ℕ) : Finset T := | |
| Finset.univ.filter fun t => | |
| (labelR (role t)).val < h ∧ ¬ (labelA (anchor t)).val < h | |
| def labelDefects (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) (v : Fin n) : Finset T := | |
| Finset.univ.filter fun t => labelA (anchor t) = v ∧ labelR (role t) ≠ v | |
| omit [DecidableEq A] in | |
| @[simp] theorem prefixItems_zero (label : A → Fin n) : prefixItems label 0 = ∅ := by | |
| simp [prefixItems] | |
| @[simp] theorem prefixOutside_zero (label : A → Fin n) (f : T → A) : | |
| prefixOutside label f 0 = ∅ := by | |
| simp [prefixOutside] | |
| omit [Fintype A] [Fintype R] [DecidableEq A] [DecidableEq R] in | |
| @[simp] theorem exports_zero (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) : exports labelA labelR anchor role 0 = ∅ := by | |
| simp [exports] | |
| omit [DecidableEq A] [DecidableEq R] in | |
| theorem prefix_inventory_eq (labelA : A → Fin n) (labelR : R → Fin n) | |
| (hInv : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun r => labelR r = v).card) (h : ℕ) : | |
| (prefixItems labelA h).card = (prefixItems labelR h).card := by | |
| let labels : Finset (Fin n) := Finset.univ.filter fun v => v.val < h | |
| have hA := Finset.sum_card_fiberwise_eq_card_filter (Finset.univ : Finset A) | |
| labels labelA | |
| have hR := Finset.sum_card_fiberwise_eq_card_filter (Finset.univ : Finset R) | |
| labels labelR | |
| have hsum : (∑ v ∈ labels, (Finset.univ.filter fun a => labelA a = v).card) = | |
| ∑ v ∈ labels, (Finset.univ.filter fun r => labelR r = v).card := by | |
| exact Finset.sum_congr rfl fun v _ => hInv v | |
| simpa only [labels, Finset.mem_filter, Finset.mem_univ, true_and, prefixItems] | |
| using hA.symm.trans (hsum.trans hR) | |
| theorem outside_add_selected_eq (label : A → Fin n) (f : T → A) | |
| (hf : Function.Injective f) (h : ℕ) : | |
| (prefixOutside label f h).card + (prefixItems (label ∘ f) h).card = | |
| (prefixItems label h).card := by | |
| have hsplit := Finset.card_sdiff_add_card_inter | |
| (prefixItems label h) (Finset.univ.image f) | |
| have hout : prefixItems label h \ Finset.univ.image f = prefixOutside label f h := by | |
| ext x | |
| simp [prefixItems, prefixOutside, outsideItems, and_comm] | |
| have hin : prefixItems label h ∩ Finset.univ.image f = | |
| (Finset.univ.image f).filter (fun x => (label x).val < h) := by | |
| ext x | |
| simp [prefixItems, and_comm] | |
| rw [hout, hin, Finset.filter_image, Finset.card_image_of_injective _ hf] at hsplit | |
| simpa only [prefixItems, Function.comp_apply] using hsplit | |
| omit [Fintype A] [Fintype R] [DecidableEq A] [DecidableEq R] in | |
| theorem selected_add_exports_eq (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) | |
| (hle : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) (h : ℕ) : | |
| (prefixItems (labelA ∘ anchor) h).card + (exports labelA labelR anchor role h).card = | |
| (prefixItems (labelR ∘ role) h).card := by | |
| have hsplit := Finset.card_filter_add_card_filter_not | |
| (s := prefixItems (labelR ∘ role) h) (fun t => (labelA (anchor t)).val < h) | |
| have hin : (prefixItems (labelR ∘ role) h).filter | |
| (fun t => (labelA (anchor t)).val < h) = prefixItems (labelA ∘ anchor) h := by | |
| ext t | |
| simp only [Finset.mem_filter, prefixItems, Finset.mem_univ, true_and, | |
| Function.comp_apply] | |
| exact ⟨fun ht => ht.2, fun ht => ⟨lt_of_le_of_lt (hle t) ht, ht⟩⟩ | |
| have hout : (prefixItems (labelR ∘ role) h).filter | |
| (fun t => ¬ (labelA (anchor t)).val < h) = exports labelA labelR anchor role h := by | |
| ext t | |
| simp [prefixItems, exports] | |
| rw [hin, hout] at hsplit | |
| exact hsplit | |
| theorem exports_add_outsideRole_eq_outsideAnchor | |
| (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) | |
| (hAnchor : Function.Injective anchor) (hRole : Function.Injective role) | |
| (hle : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) | |
| (hInv : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun r => labelR r = v).card) (h : ℕ) : | |
| (exports labelA labelR anchor role h).card + (prefixOutside labelR role h).card = | |
| (prefixOutside labelA anchor h).card := by | |
| have hA := outside_add_selected_eq labelA anchor hAnchor h | |
| have hR := outside_add_selected_eq labelR role hRole h | |
| have hE := selected_add_exports_eq labelA labelR anchor role hle h | |
| have hI := prefix_inventory_eq labelA labelR hInv h | |
| omega | |
| theorem exports_eq_outside_sub | |
| (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) | |
| (hAnchor : Function.Injective anchor) (hRole : Function.Injective role) | |
| (hle : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) | |
| (hInv : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun r => labelR r = v).card) (h : ℕ) : | |
| ((exports labelA labelR anchor role h).card : ℤ) = | |
| (prefixOutside labelA anchor h).card - (prefixOutside labelR role h).card := by | |
| have hcon := exports_add_outsideRole_eq_outsideAnchor | |
| labelA labelR anchor role hAnchor hRole hle hInv h | |
| omega | |
| theorem exports_card_le_outside | |
| (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) | |
| (hAnchor : Function.Injective anchor) (hRole : Function.Injective role) | |
| (hle : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) | |
| (hInv : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun r => labelR r = v).card) (h : ℕ) : | |
| (exports labelA labelR anchor role h).card ≤ (outsideItems anchor).card := by | |
| have hcon := exports_add_outsideRole_eq_outsideAnchor | |
| labelA labelR anchor role hAnchor hRole hle hInv h | |
| have hout : (prefixOutside labelA anchor h).card ≤ (outsideItems anchor).card := | |
| Finset.card_le_card (Finset.filter_subset _ _) | |
| omega | |
| omit [Fintype A] [Fintype R] [DecidableEq A] [DecidableEq R] in | |
| theorem labelDefects_subset_exports | |
| (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) | |
| (hle : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) (v : Fin n) : | |
| labelDefects labelA labelR anchor role v ⊆ exports labelA labelR anchor role v.val := by | |
| intro t ht | |
| obtain ⟨ha, hr⟩ := (Finset.mem_filter.mp ht).2 | |
| have hl := hle t | |
| rw [ha] at hl | |
| have hn : (labelR (role t)).val ≠ v.val := fun he => hr (Fin.ext he) | |
| apply Finset.mem_filter.mpr | |
| refine ⟨Finset.mem_univ _, ?_, ?_⟩ | |
| · omega | |
| · simp [ha] | |
| theorem labelDefects_card_le_outside | |
| (labelA : A → Fin n) (labelR : R → Fin n) | |
| (anchor : T → A) (role : T → R) | |
| (hAnchor : Function.Injective anchor) (hRole : Function.Injective role) | |
| (hle : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) | |
| (hInv : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun r => labelR r = v).card) (v : Fin n) : | |
| (labelDefects labelA labelR anchor role v).card ≤ (outsideItems anchor).card := by | |
| exact (Finset.card_le_card (labelDefects_subset_exports labelA labelR anchor role hle v)).trans | |
| (exports_card_le_outside labelA labelR anchor role hAnchor hRole hle hInv v.val) | |
| def badAnchors (labelA : A → Fin n) (labelR : R → Fin n) (labelD : D → Fin n) | |
| (anchor : T → A) (role : T → R) (localRole : T → D) (v : Fin n) : Finset A := | |
| ((outsideItems anchor).filter fun a => labelA a = v) ∪ | |
| (labelDefects labelA labelR anchor role v).image anchor ∪ | |
| (labelDefects labelA labelD anchor localRole v).image anchor | |
| omit [Fintype R] [Fintype D] [DecidableEq R] [DecidableEq D] in | |
| theorem mem_badAnchors_iff | |
| (labelA : A → Fin n) (labelR : R → Fin n) (labelD : D → Fin n) | |
| (anchor : T → A) (role : T → R) (localRole : T → D) | |
| (hAnchor : Function.Injective anchor) (v : Fin n) (a : A) : | |
| a ∈ badAnchors labelA labelR labelD anchor role localRole v ↔ | |
| labelA a = v ∧ | |
| ¬ ∃ t, anchor t = a ∧ labelR (role t) = v ∧ labelD (localRole t) = v := by | |
| constructor | |
| · intro hbad | |
| rcases Finset.mem_union.mp hbad with hleft | hd | |
| · rcases Finset.mem_union.mp hleft with ho | hr | |
| · obtain ⟨hoa, hv⟩ := Finset.mem_filter.mp ho | |
| refine ⟨hv, ?_⟩ | |
| rintro ⟨t, hat, _, _⟩ | |
| exact (Finset.mem_sdiff.mp hoa).2 | |
| (Finset.mem_image.mpr ⟨t, Finset.mem_univ _, hat⟩) | |
| · obtain ⟨t, ht, hat⟩ := Finset.mem_image.mp hr | |
| obtain ⟨hva, hvr⟩ := (Finset.mem_filter.mp ht).2 | |
| refine ⟨by simpa only [hat] using hva, ?_⟩ | |
| rintro ⟨u, hua, hur, _⟩ | |
| have htu : t = u := hAnchor (hat.trans hua.symm) | |
| exact hvr (by simpa only [htu] using hur) | |
| · obtain ⟨t, ht, hat⟩ := Finset.mem_image.mp hd | |
| obtain ⟨hva, hvd⟩ := (Finset.mem_filter.mp ht).2 | |
| refine ⟨by simpa only [hat] using hva, ?_⟩ | |
| rintro ⟨u, hua, _, hud⟩ | |
| have htu : t = u := hAnchor (hat.trans hua.symm) | |
| exact hvd (by simpa only [htu] using hud) | |
| · rintro ⟨hv, hnot⟩ | |
| by_cases ha : a ∈ Finset.univ.image anchor | |
| · obtain ⟨t, _, hat⟩ := Finset.mem_image.mp ha | |
| have htv : labelA (anchor t) = v := by simpa only [hat] using hv | |
| by_cases hr : labelR (role t) = v | |
| · have hd : labelD (localRole t) ≠ v := fun hd => hnot ⟨t, hat, hr, hd⟩ | |
| apply Finset.mem_union_right | |
| exact Finset.mem_image.mpr ⟨t, Finset.mem_filter.mpr | |
| ⟨Finset.mem_univ _, htv, hd⟩, hat⟩ | |
| · apply Finset.mem_union_left | |
| apply Finset.mem_union_right | |
| exact Finset.mem_image.mpr ⟨t, Finset.mem_filter.mpr | |
| ⟨Finset.mem_univ _, htv, hr⟩, hat⟩ | |
| · apply Finset.mem_union_left | |
| apply Finset.mem_union_left | |
| exact Finset.mem_filter.mpr | |
| ⟨Finset.mem_sdiff.mpr ⟨Finset.mem_univ _, ha⟩, hv⟩ | |
| theorem badAnchors_card_le_three_mul | |
| (labelA : A → Fin n) (labelR : R → Fin n) (labelD : D → Fin n) | |
| (anchor : T → A) (role : T → R) (localRole : T → D) | |
| (hAnchor : Function.Injective anchor) (hRole : Function.Injective role) | |
| (hLocal : Function.Injective localRole) | |
| (hleR : ∀ t, (labelR (role t)).val ≤ (labelA (anchor t)).val) | |
| (hleD : ∀ t, (labelD (localRole t)).val ≤ (labelA (anchor t)).val) | |
| (hInvR : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun r => labelR r = v).card) | |
| (hInvD : ∀ v, (Finset.univ.filter fun a => labelA a = v).card = | |
| (Finset.univ.filter fun d => labelD d = v).card) | |
| (K : ℕ) (hOutside : (outsideItems anchor).card ≤ K) (v : Fin n) : | |
| (badAnchors labelA labelR labelD anchor role localRole v).card ≤ 3 * K := by | |
| have hR := labelDefects_card_le_outside labelA labelR anchor role hAnchor hRole hleR hInvR v | |
| have hD := labelDefects_card_le_outside labelA labelD anchor localRole | |
| hAnchor hLocal hleD hInvD v | |
| have hO : ((outsideItems anchor).filter fun a => labelA a = v).card ≤ | |
| (outsideItems anchor).card := Finset.card_le_card (Finset.filter_subset _ _) | |
| have hRi := Finset.card_image_le (s := labelDefects labelA labelR anchor role v) (f := anchor) | |
| have hDi := Finset.card_image_le (s := labelDefects labelA labelD anchor localRole v) | |
| (f := anchor) | |
| have hU := Finset.card_union_le | |
| ((outsideItems anchor).filter fun a => labelA a = v) | |
| ((labelDefects labelA labelR anchor role v).image anchor) | |
| have hV := Finset.card_union_le | |
| (((outsideItems anchor).filter fun a => labelA a = v) ∪ | |
| (labelDefects labelA labelR anchor role v).image anchor) | |
| ((labelDefects labelA labelD anchor localRole v).image anchor) | |
| unfold badAnchors | |
| omega | |
| end LabelConservation | |
| end BinPackingGap | |
| end OAI | |