File size: 3,567 Bytes
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 | import OAI.Combinatorics.ContingencyTables.Sampling.ProfileExchangeProgram
import OAI.Combinatorics.ContingencyTables.Machines.ListWordProgram
import OAI.Combinatorics.ContingencyTables.Dense.DenseAxisProposal
namespace OAI
/-! # Bounded binary coordinate moves in the dense grid -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms
abbrev AxisData := ((ℕ×List ℕ)×ℕ)×Bool
def axisListStep (z : AxisData) : Option (List ℕ) :=
let K := z.1.1.1
let xs := z.1.1.2
let i := z.1.2
if i<xs.length then
if z.2 then
if xs.getD i 0+1<K then some (listExchange (xs,(i,xs.length))) else none
else
if 0<xs.getD i 0 then some (listExchange (xs,(xs.length,i))) else none
else none
def axisListStepRealizer : Realizer axisListStep := by
let K : Realizer (fun z : AxisData => z.1.1.1) :=
composition (composition first first) first
let X : Realizer (fun z : AxisData => z.1.1.2) :=
composition (composition first first) second
let I : Realizer (fun z : AxisData => z.1.2) := composition first second
let L := composition X listLength
let V := composition (pair X I) (getDFast 0)
let up := composition (composition (pair X (pair I L)) listExchangeRealizer) TreeTyped.some
let down := composition (composition (pair X (pair L I)) listExchangeRealizer) TreeTyped.some
exact (choose (composition (pair I L) less)
(choose second (choose (composition (pair (composition V successor) K) less)
up (constant none))
(choose (composition (pair (constant 0) V) less) down (constant none)))
(constant none)).congr (by
intro z
simp only [Function.comp_apply,decide_eq_true_eq,axisListStep])
/-- An invalid or boundary move terminates immediately. Its index is
never used as an iteration count. -/
theorem polynomial_axisListStep : PolynomialTime axisListStepRealizer := by
have hK : PolynomialTime (composition
(composition (first (A := (ℕ×List ℕ)×ℕ) (B := Bool)) first) first) :=
polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_first
have hX : PolynomialTime (composition
(composition (first (A := (ℕ×List ℕ)×ℕ) (B := Bool)) first) second) :=
polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_second
have hI : PolynomialTime (composition
(first (A := (ℕ×List ℕ)×ℕ) (B := Bool)) second) :=
polynomial_composition polynomial_first polynomial_second
have hL := polynomial_composition hX polynomial_listLength
have hV := polynomial_composition (polynomial_pair hX hI) (polynomial_getDFast 0)
have hup := polynomial_composition (polynomial_composition
(polynomial_pair hX (polynomial_pair hI hL)) polynomial_listExchange) polynomial_some
have hdown := polynomial_composition (polynomial_composition
(polynomial_pair hX (polynomial_pair hL hI)) polynomial_listExchange) polynomial_some
unfold axisListStepRealizer
exact polynomial_congr _ (polynomial_choose
(polynomial_composition (polynomial_pair hI hL) polynomial_less)
(polynomial_choose polynomial_second
(polynomial_choose (polynomial_composition (polynomial_pair
(polynomial_composition hV polynomial_successor) hK) polynomial_less)
hup (polynomial_constant none))
(polynomial_choose (polynomial_composition (polynomial_pair (polynomial_constant 0) hV)
polynomial_less) hdown (polynomial_constant none))) (polynomial_constant none))
end ContingencyTables.DensePrograms
end OAI
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