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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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionMetropolisProgram
import OAI.Combinatorics.ContingencyTables.Dense.DenseWordProgram
namespace OAI
/-! # Bounded slicing of one flat companion transition word -/
namespace ContingencyTables.CompanionMetropolisWordProgram
open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms
/-- The fixed frame, the acceptance precision, current code, and fresh word. -/
abbrev Input := ((CompanionMetropolisProgram.Frame × ℕ) × List ℕ) × List Bool
def proposalWidth (z : Input) : ℕ := Nat.clog 2 ((z.1.2.length/2)^2)
def proposalWidthRealizer : Realizer proposalWidth := by
let p : Realizer (fun z : Input => z.1.2.length/2) :=
composition (f := fun z : Input => (z.1.2.length,2))
(g := fun ab : ℕ × ℕ => ab.1/ab.2)
(pair (composition (composition first second) listLength) (constant 2)) quotient
exact composition (f := fun z : Input => (z.1.2.length/2)^2) (g := Nat.clog 2)
((composition (pair p p) multiply).congr (fun z => (pow_two _).symm)) ceilLogTwoRealizer
lemma polynomial_proposalWidth : PolynomialTime proposalWidthRealizer := by
unfold proposalWidthRealizer
apply polynomial_composition _ polynomial_ceilLogTwo
apply polynomial_congr
apply polynomial_composition _ polynomial_multiply
apply polynomial_pair
all_goals exact (polynomial_composition
(polynomial_pair (polynomial_composition (polynomial_composition polynomial_first polynomial_second)
polynomial_listLength) (polynomial_constant 2)) polynomial_quotient)
def context (z : Input) : RootedEstimatorLayoutProgram.Context := z.1.1.1.1.1.1
def contextRealizer : Realizer context :=
composition (composition (composition (composition (composition first first) first) first) first) first
lemma polynomial_context : PolynomialTime contextRealizer := by
unfold contextRealizer
exact polynomial_composition (polynomial_composition (polynomial_composition
(polynomial_composition (polynomial_composition polynomial_first polynomial_first)
polynomial_first) polynomial_first) polynomial_first) polynomial_first
def queryWidth (z : Input) : ℕ := z.1.1.1.1.1.2*RootedEstimatorLayoutProgram.wordWidth (context z)
def queryWidthRealizer : Realizer queryWidth :=
composition (f := fun z : Input =>
(z.1.1.1.1.1.2,RootedEstimatorLayoutProgram.wordWidth (context z)))
(g := fun ab : ℕ × ℕ => ab.1*ab.2)
(pair (composition (composition (composition (composition (composition first first) first) first) first) second)
(composition (f := context) (g := RootedEstimatorLayoutProgram.wordWidth)
contextRealizer RootedEstimatorLayoutProgram.wordWidthRealizer)) multiply
lemma polynomial_queryWidth : PolynomialTime queryWidthRealizer :=
polynomial_composition (polynomial_pair (by tree_poly)
(polynomial_composition polynomial_context RootedEstimatorLayoutProgram.polynomial_wordWidth)) polynomial_multiply
attribute [local irreducible] context queryWidth proposalWidth
/-- Offsets are computed in binary; the physical `take`/`drop` calls remain
bounded by the supplied word on every malformed input. -/
def firstOffset (z : Input) : ℕ := 1+proposalWidth z
def firstOffsetRealizer : Realizer firstOffset :=
composition (f := fun z : Input => (1,proposalWidth z)) (g := fun p : ℕ × ℕ => p.1+p.2)
(pair (constant 1) proposalWidthRealizer) TreeTyped.add
lemma polynomial_firstOffset : PolynomialTime firstOffsetRealizer :=
polynomial_composition (polynomial_pair (polynomial_constant 1) polynomial_proposalWidth) polynomial_add
def secondOffset (z : Input) : ℕ := firstOffset z+queryWidth z
attribute [local irreducible] firstOffset
def secondOffsetRealizer : Realizer secondOffset :=
composition (f := fun z : Input => (firstOffset z,queryWidth z)) (g := fun p : ℕ × ℕ => p.1+p.2)
(pair firstOffsetRealizer queryWidthRealizer) TreeTyped.add
lemma polynomial_secondOffset : PolynomialTime secondOffsetRealizer :=
polynomial_composition (polynomial_pair polynomial_firstOffset polynomial_queryWidth) polynomial_add
def coinOffset (z : Input) : ℕ := secondOffset z+queryWidth z
attribute [local irreducible] secondOffset
def coinOffsetRealizer : Realizer coinOffset :=
composition (f := fun z : Input => (secondOffset z,queryWidth z)) (g := fun p : ℕ × ℕ => p.1+p.2)
(pair secondOffsetRealizer queryWidthRealizer) TreeTyped.add
lemma polynomial_coinOffset : PolynomialTime coinOffsetRealizer :=
polynomial_composition (polynomial_pair polynomial_secondOffset polynomial_queryWidth) polynomial_add
private def slice (start count : Input → ℕ) (z : Input) : List Bool :=
(z.2.drop (start z)).take (count z)
private def sliceRealizer {start count : Input → ℕ}
(S : Realizer start) (C : Realizer count) : Realizer (slice start count) :=
composition (pair (composition (pair second S) dropFast) C) takeFast
private lemma polynomial_slice {start count : Input → ℕ}
(S : Realizer start) (C : Realizer count) (hS : PolynomialTime S) (hC : PolynomialTime C) :
PolynomialTime (sliceRealizer S C) :=
polynomial_composition (polynomial_pair (polynomial_composition
(polynomial_pair polynomial_second hS) polynomial_dropFast) hC) polynomial_takeFast
def fields (z : Input) : CompanionMetropolisProgram.Words :=
(z.2.getD 0 false,
(slice (fun _ => 1) proposalWidth z,
((slice firstOffset queryWidth z,slice secondOffset queryWidth z),
slice coinOffset (fun z => z.1.1.2) z)))
def fieldsRealizer : Realizer fields :=
pair (composition (pair second (constant 0)) (getDFast false))
(pair (sliceRealizer (constant 1) proposalWidthRealizer)
(pair (pair (sliceRealizer firstOffsetRealizer queryWidthRealizer)
(sliceRealizer secondOffsetRealizer queryWidthRealizer))
(sliceRealizer coinOffsetRealizer (composition (composition first first) second))))
lemma polynomial_fields : PolynomialTime fieldsRealizer :=
polynomial_pair (polynomial_composition (polynomial_pair polynomial_second (polynomial_constant 0))
(polynomial_getDFast false))
(polynomial_pair (polynomial_slice _ _ (polynomial_constant 1) polynomial_proposalWidth)
(polynomial_pair (polynomial_pair
(polynomial_slice _ _ polynomial_firstOffset polynomial_queryWidth)
(polynomial_slice _ _ polynomial_secondOffset polynomial_queryWidth))
(polynomial_slice _ _ polynomial_coinOffset (by tree_poly))))
def stepInput (z : Input) : CompanionMetropolisProgram.Input := ((z.1.1.1,z.1.2),fields z)
def stepInputRealizer : Realizer stepInput :=
pair (pair (composition (composition first first) first) (composition first second)) fieldsRealizer
lemma polynomial_stepInput : PolynomialTime stepInputRealizer :=
polynomial_pair (polynomial_pair (by tree_poly) (polynomial_composition polynomial_first polynomial_second))
polynomial_fields
attribute [local irreducible] stepInput
def output (z : Input) : List ℕ := CompanionMetropolisProgram.output (stepInput z)
def outputRealizer : Realizer output :=
composition (f := stepInput) (g := CompanionMetropolisProgram.output)
stepInputRealizer CompanionMetropolisProgram.outputRealizer
theorem polynomial_output : PolynomialTime outputRealizer :=
polynomial_composition polynomial_stepInput CompanionMetropolisProgram.polynomial_output
end ContingencyTables.CompanionMetropolisWordProgram
end OAI
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