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import Mathlib.Data.List.Nodup
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching
def crossRecord (n : ℕ) (r : LoopRecord) (d : ℕ) : BinaryMatching.Record :=
⟨r.vertex, n + d, r.multiplicity⟩
def crossRecords (n j : ℕ) (rs : List LoopRecord) : List BinaryMatching.Record :=
rs.flatMap fun r => (List.range j).map (crossRecord n r)
lemma mem_crossRecords {n j : ℕ} {rs : List LoopRecord} {e : BinaryMatching.Record} :
e ∈ crossRecords n j rs ↔ ∃ r ∈ rs, ∃ d < j, crossRecord n r d = e := by
simp [crossRecords]
lemma crossRecords_length (n j : ℕ) (rs : List LoopRecord) :
(crossRecords n j rs).length = rs.length * j := by
induction rs with
| nil => simp [crossRecords]
| cons r rs ih =>
change (((List.range j).map (crossRecord n r)) ++ crossRecords n j rs).length =
(rs.length + 1) * j
rw [List.length_append, List.length_map, List.length_range, ih]
simp [Nat.add_mul, Nat.add_comm]
lemma crossRecords_unique (n j : ℕ) (rs : List LoopRecord)
(h : (rs.map LoopRecord.vertex).Nodup) :
((crossRecords n j rs).map (fun e => (e.left,e.right))).Nodup := by
induction rs with
| nil => simp [crossRecords]
| cons r rs ih =>
obtain ⟨hn,ht⟩ := List.nodup_cons.mp h
simp only [crossRecords, List.flatMap_cons, List.map_append, List.map_map]
apply List.nodup_append.mpr
refine ⟨?_, ih ht, ?_⟩
· apply (List.nodup_range (n := j)).map
intro a b hab
have hh := congrArg Prod.snd hab
dsimp [Function.comp_def, crossRecord] at hh
omega
· intro a ha b hb hab
obtain ⟨d,hd,rfl⟩ := List.mem_map.mp ha
obtain ⟨e,he,rfl⟩ := List.mem_map.mp hb
obtain ⟨s,hs,c,hc,rfl⟩ := mem_crossRecords.mp he
have hv := congrArg Prod.fst hab
dsimp [Function.comp_def, crossRecord] at hv
exact hn (List.mem_map.mpr ⟨s,hs,hv.symm⟩)
def expand (G : Input) (j : ℕ) : BinaryMatching.Input where
n := G.n + j
records := G.ordinary.records ++ crossRecords G.n j G.loops
valid := by
intro e he
rcases List.mem_append.mp he with he | he
· obtain ⟨hlt,hbound⟩ := G.ordinary.valid e he
change e.right < G.n at hbound
exact ⟨hlt,by omega⟩
· obtain ⟨r,hr,d,hd,rfl⟩ := mem_crossRecords.mp he
have hrn := G.valid r hr
change r.vertex < G.n at hrn
dsimp [crossRecord]
constructor <;> omega
unique := by
rw [List.map_append]
apply List.nodup_append.mpr
refine ⟨G.ordinary.unique, crossRecords_unique _ _ _ G.unique, ?_⟩
intro a ha b hb hab
obtain ⟨e,he,rfl⟩ := List.mem_map.mp ha
obtain ⟨f,hf,rfl⟩ := List.mem_map.mp hb
obtain ⟨r,hr,d,hd,rfl⟩ := mem_crossRecords.mp hf
have hn := (G.ordinary.valid e he).2
change e.right < G.n at hn
have hh := congrArg Prod.snd hab
dsimp [crossRecord] at hh
omega
@[simp] lemma expand_n (G : Input) (j : ℕ) : (expand G j).n = G.n+j := rfl
@[simp] lemma expand_records_length (G : Input) (j : ℕ) :
(expand G j).records.length = G.ordinary.records.length + G.loops.length*j := by
simp [expand, crossRecords_length]
lemma expand_records_length_le (G : Input) {j : ℕ} (hj : j ≤ G.n) :
(expand G j).records.length ≤ G.ordinary.records.length + G.loops.length*G.n := by
rw [expand_records_length]
exact Nat.add_le_add_left (Nat.mul_le_mul_left _ hj) _
def oldPair (G : Input) (j : ℕ) (e : BinaryMatching.Pair G.n) :
BinaryMatching.Pair (expand G j).n :=
⟨(⟨e.val.1.val,by have := e.val.1.isLt; simp only [expand_n]; omega⟩,
⟨e.val.2.val,by have := e.val.2.isLt; simp only [expand_n]; omega⟩),e.property⟩
def crossPair (G : Input) (j : ℕ) (v : Fin G.n) (d : Fin j) :
BinaryMatching.Pair (expand G j).n :=
⟨(⟨v.val,by have := v.isLt; simp only [expand_n]; omega⟩,
⟨G.n+d.val,by have := d.isLt; simp only [expand_n]; omega⟩),by
change v.val < G.n+d.val
have := v.isLt
omega⟩
def dummyPair (G : Input) (j : ℕ) (e : BinaryMatching.Pair j) :
BinaryMatching.Pair (expand G j).n :=
⟨(⟨G.n+e.val.1.val,by have := e.val.1.isLt; simp only [expand_n]; omega⟩,
⟨G.n+e.val.2.val,by have := e.val.2.isLt; simp only [expand_n]; omega⟩),by
change G.n+e.val.1.val < G.n+e.val.2.val
have := e.property
exact Nat.add_lt_add_left this _⟩
private lemma find_some_unique {α : Type*} (p : α → Bool) (xs : List α) (a : α)
(ha : a ∈ xs) (hp : p a = true) (hu : ∀ b ∈ xs, p b = true → b=a) :
xs.find? p = some a := by
cases hf : xs.find? p with
| none => exact False.elim ((List.find?_eq_none.mp hf a ha) hp)
| some b =>
have hb := hu b (List.mem_of_find?_eq_some hf) (List.find?_some hf)
exact congrArg some hb
private lemma find_cross_block (n j i d : ℕ) (r : LoopRecord) (hd : d<j) :
((List.range j).map (crossRecord n r)).find?
(fun e => e.left=i && e.right=n+d) =
if r.vertex=i then some (crossRecord n r d) else none := by
by_cases hi : r.vertex=i
· rw [ite_eq_left hi]
apply find_some_unique
· exact List.mem_map.mpr ⟨d,List.mem_range.mpr hd,rfl⟩
· simp [crossRecord,hi]
· intro b hb hp
obtain ⟨c,hc,rfl⟩ := List.mem_map.mp hb
have he : n+c=n+d := by simpa [crossRecord,hi] using hp
have : c=d := by omega
subst c
rfl
· rw [ite_eq_right hi]
apply List.find?_eq_none.mpr
intro b hb
obtain ⟨c,hc,rfl⟩ := List.mem_map.mp hb
simp [crossRecord,hi]
private lemma find_crossRecords (n j i d : ℕ) (rs : List LoopRecord) (hd : d<j) :
(crossRecords n j rs).find? (fun e => e.left=i && e.right=n+d) =
(rs.find? (fun r => r.vertex == i)).map (fun r => crossRecord n r d) := by
induction rs with
| nil => simp [crossRecords]
| cons r rs ih =>
change ((((List.range j).map (crossRecord n r)) ++ crossRecords n j rs).find?
(fun e => e.left=i && e.right=n+d)) =
((r :: rs).find? (fun r => r.vertex == i)).map (fun r => crossRecord n r d)
rw [List.find?_append, find_cross_block n j i d r hd]
by_cases hi : r.vertex=i
· simp [hi]
· simp [hi, ih]
lemma multiplicity_expand_old (G : Input) (j : ℕ) (e : BinaryMatching.Pair G.n) :
BinaryMatching.multiplicity (expand G j) (oldPair G j e) =
BinaryMatching.multiplicity G.ordinary e := by
have hn : (crossRecords G.n j G.loops).find?
(fun r => r.left=e.val.1.val && r.right=e.val.2.val) = none := by
apply List.find?_eq_none.mpr
intro r hr
obtain ⟨s,hs,d,hd,rfl⟩ := mem_crossRecords.mp hr
have he := e.val.2.isLt
have hne : G.n+d ≠ e.val.2.val := by omega
simp [crossRecord,hne]
change (match (G.ordinary.records ++ crossRecords G.n j G.loops).find?
(fun r => r.left=e.val.1.val && r.right=e.val.2.val) with
| none => 0 | some r => r.multiplicity) =
(match G.ordinary.records.find?
(fun r => r.left=e.val.1.val && r.right=e.val.2.val) with
| none => 0 | some r => r.multiplicity)
rw [List.find?_append, hn]
cases G.ordinary.records.find? (fun r => r.left=e.val.1.val && r.right=e.val.2.val) <;> rfl
lemma multiplicity_expand_cross (G : Input) (j : ℕ) (v : Fin G.n) (d : Fin j) :
BinaryMatching.multiplicity (expand G j) (crossPair G j v d) = loopMultiplicity G v := by
have hn : G.ordinary.records.find?
(fun r => r.left=v.val && r.right=G.n+d.val) = none := by
apply List.find?_eq_none.mpr
intro r hr
have hb := (G.ordinary.valid r hr).2
change r.right < G.n at hb
have hne : r.right ≠ G.n+d.val := by omega
simp [hne]
change (match (G.ordinary.records ++ crossRecords G.n j G.loops).find?
(fun r => r.left=v.val && r.right=G.n+d.val) with
| none => 0 | some r => r.multiplicity) = loopMultiplicity G v
rw [List.find?_append, hn]
change (match (crossRecords G.n j G.loops).find?
(fun r => r.left=v.val && r.right=G.n+d.val) with
| none => 0 | some r => r.multiplicity) = _
rw [find_crossRecords G.n j v.val d.val G.loops d.isLt]
unfold loopMultiplicity
cases G.loops.find? (fun r => r.vertex == v.val) <;> rfl
lemma multiplicity_expand_dummy (G : Input) (j : ℕ) (e : BinaryMatching.Pair j) :
BinaryMatching.multiplicity (expand G j) (dummyPair G j e) = 0 := by
have hn : (expand G j).records.find?
(fun r => r.left=G.n+e.val.1.val && r.right=G.n+e.val.2.val) = none := by
apply List.find?_eq_none.mpr
intro r hr
have hb : r.left < G.n := by
rcases List.mem_append.mp hr with hr | hr
· obtain ⟨hlt,hbound⟩ := G.ordinary.valid r hr
change r.right < G.n at hbound
omega
· obtain ⟨s,hs,d,hd,rfl⟩ := mem_crossRecords.mp hr
exact G.valid s hs
have hne : r.left ≠ G.n+e.val.1.val := by omega
simp [hne]
change (match (expand G j).records.find?
(fun r => r.left=G.n+e.val.1.val && r.right=G.n+e.val.2.val) with
| none => 0 | some r => r.multiplicity) = 0
rw [hn]
end LoopMatching
end OAI
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