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97c3405 | 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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.Combinatorics.ContingencyTables.Machines.LowerCapacityProgram
import OAI.Combinatorics.MatchingCount.Machines.TreeFiniteOps
namespace OAI
/-! # The executed feasibility test with one extra lower cell bound -/
namespace ContingencyTables.CellLowerTestProgram
open MatchingFPRAS.TreeTyped Polynomial
abbrev Context := NetworkVertexCode.Input×(ℕ×ℕ)
abbrev Input := Context×ℕ
def shifted (z : Input) : NetworkVertexCode.Input :=
((ListSubtractProgram.subtractAt ((z.1.1.1.1,z.1.2.1),z.2),
ListSubtractProgram.subtractAt ((z.1.1.1.2,z.1.2.2),z.2)),
LowerCapacityProgram.matrix ((z.1.1.2,z.1.2),z.2))
def rowInputRealizer : Realizer (fun z : Input=>((z.1.1.1.1,z.1.2.1),z.2)) :=
pair (pair (composition (composition (composition first first) first) first)
(composition (composition first second) first)) second
def columnInputRealizer : Realizer (fun z : Input=>((z.1.1.1.2,z.1.2.2),z.2)) :=
pair (pair (composition (composition (composition first first) first) second)
(composition (composition first second) second)) second
def capacityInputRealizer : Realizer (fun z : Input=>((z.1.1.2,z.1.2),z.2)) :=
pair (pair (composition (composition first first) second) (composition first second)) second
lemma polynomial_rowInput : PolynomialTime rowInputRealizer :=
polynomial_pair (polynomial_pair
(polynomial_composition (polynomial_composition
(polynomial_composition polynomial_first polynomial_first) polynomial_first) polynomial_first)
(polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_first))
polynomial_second
lemma polynomial_columnInput : PolynomialTime columnInputRealizer :=
polynomial_pair (polynomial_pair
(polynomial_composition (polynomial_composition
(polynomial_composition polynomial_first polynomial_first) polynomial_first) polynomial_second)
(polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_second))
polynomial_second
lemma polynomial_capacityInput : PolynomialTime capacityInputRealizer :=
polynomial_pair (polynomial_pair
(polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_second)
(polynomial_composition polynomial_first polynomial_second)) polynomial_second
def shiftedRealizer : Realizer shifted :=
pair (pair
(composition (f := fun z : Input=>((z.1.1.1.1,z.1.2.1),z.2))
(g := ListSubtractProgram.subtractAt) rowInputRealizer ListSubtractProgram.subtractAtRealizer)
(composition (f := fun z : Input=>((z.1.1.1.2,z.1.2.2),z.2))
(g := ListSubtractProgram.subtractAt) columnInputRealizer ListSubtractProgram.subtractAtRealizer))
(composition (f := fun z : Input=>((z.1.1.2,z.1.2),z.2))
(g := LowerCapacityProgram.matrix) capacityInputRealizer LowerCapacityProgram.matrixRealizer)
lemma polynomial_shifted : PolynomialTime shiftedRealizer :=
polynomial_pair (polynomial_pair
(polynomial_composition polynomial_rowInput ListSubtractProgram.polynomial_subtractAt)
(polynomial_composition polynomial_columnInput ListSubtractProgram.polynomial_subtractAt))
(polynomial_composition polynomial_capacityInput LowerCapacityProgram.polynomial_matrix)
def guards (z : Input) : Bool :=
decide (z.2≤z.1.1.1.1.getD z.1.2.1 0) &&
(decide (z.2≤z.1.1.1.2.getD z.1.2.2 0) &&
decide (z.2≤ResidualGraphProgram.capacityEntry (z.1.1.2,z.1.2)))
def guardsRealizer : Realizer guards := by
let rowBound : Realizer (fun z : Input=>z.1.1.1.1.getD z.1.2.1 0) :=
composition (composition rowInputRealizer first) (getDFast 0)
let columnBound : Realizer (fun z : Input=>z.1.1.1.2.getD z.1.2.2 0) :=
composition (composition columnInputRealizer first) (getDFast 0)
let capacityBound : Realizer (fun z : Input=>ResidualGraphProgram.capacityEntry (z.1.1.2,z.1.2)) :=
composition (composition capacityInputRealizer first) ResidualGraphProgram.capacityEntryRealizer
exact composition (pair (composition (pair second rowBound) leNat)
(composition (pair (composition (pair second columnBound) leNat)
(composition (pair second capacityBound) leNat)) boolAnd)) boolAnd
lemma polynomial_guards : PolynomialTime guardsRealizer := by
unfold guardsRealizer
apply polynomial_composition
· apply polynomial_pair
· exact polynomial_composition (polynomial_pair polynomial_second
(polynomial_composition (polynomial_composition polynomial_rowInput polynomial_first)
(polynomial_getDFast 0))) polynomial_leNat
· apply polynomial_composition
· apply polynomial_pair
· exact polynomial_composition (polynomial_pair polynomial_second
(polynomial_composition (polynomial_composition polynomial_columnInput polynomial_first)
(polynomial_getDFast 0))) polynomial_leNat
· exact polynomial_composition (polynomial_pair polynomial_second
(polynomial_composition (polynomial_composition polynomial_capacityInput polynomial_first)
ResidualGraphProgram.polynomial_capacityEntry)) polynomial_leNat
· exact polynomial_boolAnd
· exact polynomial_boolAnd
def test (z : Input) : Bool := guards z && NetworkFeasibilityProgram.test (shifted z)
def testRealizer : Realizer test :=
composition (pair guardsRealizer
(composition (f := shifted) (g := NetworkFeasibilityProgram.test)
shiftedRealizer NetworkFeasibilityProgram.testRealizer)) boolAnd
lemma polynomial_test : PolynomialTime testRealizer :=
polynomial_composition (polynomial_pair polynomial_guards
(polynomial_composition polynomial_shifted NetworkFeasibilityProgram.polynomial_test))
polynomial_boolAnd
lemma shifted_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n) (t : ℕ) :
shifted ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=
NetworkVertexCode.data (BoundedCellShift.margin r i t)
(BoundedCellShift.margin c j t) (BoundedCellShift.lower b i j t) := by
unfold shifted NetworkVertexCode.data
rw [ListSubtractProgram.subtractAt_ofFn,ListSubtractProgram.subtractAt_ofFn,
LowerCapacityProgram.matrix_code]
lemma guards_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n) (t : ℕ) :
guards ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=true ↔
t≤r i ∧ t≤c j ∧ t≤b i j := by
dsimp only [guards, NetworkVertexCode.data]
rw [ResidualGraphProgram.capacityEntry_code]
simp only [List.getD_eq_getElem?_getD,List.getElem?_ofFn,
i.isLt,j.isLt,dite_true,Option.getD_some,Fin.eta,Bool.and_eq_true]
constructor
· rintro ⟨hr,hc,hb⟩
exact ⟨_root_.of_decide_eq_true hr,_root_.of_decide_eq_true hc,_root_.of_decide_eq_true hb⟩
· rintro ⟨hr,hc,hb⟩
exact ⟨_root_.decide_eq_true hr,_root_.decide_eq_true hc,_root_.decide_eq_true hb⟩
theorem test_correct {m n : ℕ} (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1)
(r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n) (t : ℕ) :
test ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=true ↔
∃ X,HasMargins X r c ∧ (∀ u v,X u v≤b u v) ∧ t≤X i j := by
rw [test,Bool.and_eq_true,guards_code,shifted_code,NetworkFeasibilityProgram.test_correct hEK]
simpa only [and_assoc] using (BoundedCellShift.feasible_lower_iff r c b i j t).symm
end ContingencyTables.CellLowerTestProgram
end OAI
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