4dd182c554
This PR removes the deprecated aliases `Int.div := Int.tdiv` and `Int.mod := Int.tmod`. Later we will rename `Int.ediv` to `Int.div` and `Int.emod` to `Int.mod`.
396 lines
13 KiB
Text
396 lines
13 KiB
Text
/-
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Linear Diophantine equation solver
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Author: Marc Huisinga
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-/
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import Lean.Data.HashMap
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open Lean
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namespace Int
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def roundedDiv (a b : Int) : Int := Id.run <| do
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if b = 0 then
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return 0
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let mut div := a / b
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let rest := a % b
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-- This determines how we should adjust the divisor.
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-- The extra logic is to preserve tie-breaking behavior from
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-- a time when div used T-rounding
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if a ≥ 0 then
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if 2*rest ≥ b.natAbs then
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div := div + (if b ≥ 0 then 1 else -1)
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else
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if 2*rest > b.natAbs then
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div := div + (if b >= 0 then 1 else -1)
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return div
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def mod' (a b : Int) : Int :=
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a - b*(a.roundedDiv b)
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end Int
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namespace Lean.AssocList
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def map (f : α → β → δ) : AssocList α β → AssocList α δ
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| AssocList.nil => AssocList.nil
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| AssocList.cons k v t => AssocList.cons k (f k v) (map f t)
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def filter (p : α → β → Bool) : AssocList α β → AssocList α β
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| AssocList.nil => AssocList.nil
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| AssocList.cons k v t =>
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if p k v then
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AssocList.cons k v (filter p t)
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else
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filter p t
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end Lean.AssocList
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namespace Lean.HashMap
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variable [BEq α] [Hashable α]
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@[inline] protected def forIn {δ : Type w} {m : Type w → Type w'} [Monad m]
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(as : HashMap α β) (init : δ) (f : (α × β) → δ → m (ForInStep δ)) : m δ := do
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forIn as.val.buckets.val init fun bucket acc => do
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let (done, v) ← bucket.forIn (false, acc) fun v (_, acc) => do
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let r ← f v acc
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match r with
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| ForInStep.done r' =>
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return ForInStep.done (true, r')
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| ForInStep.yield r' =>
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return ForInStep.yield (false, r')
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if done then
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return ForInStep.done v
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else
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return ForInStep.yield v
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instance : ForIn m (HashMap α β) (α × β) where
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forIn := HashMap.forIn
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def modify! [Inhabited β] (xs : HashMap α β) (k : α) (f : β → β) : HashMap α β :=
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let v := xs.find! k
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xs.erase k |>.insert k (f v)
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def any (xs : HashMap α β) (p : α → β → Bool) : Bool := Id.run <| do
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for (k, v) in xs do
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if p k v then
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return true
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return false
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def mapValsM [Monad m] (f : β → m γ) (xs : HashMap α β) : m (HashMap α γ) :=
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mkHashMap (capacity := xs.size) |> xs.foldM fun acc k v => return acc.insert k (←f v)
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def mapVals (f : β → γ) (xs : HashMap α β) : HashMap α γ :=
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mkHashMap (capacity := xs.size) |> xs.fold fun acc k v => acc.insert k (f v)
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def fastMapVals (f : α → β → β) (xs : HashMap α β) : HashMap α β :=
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let size := xs.val.size
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let buckets := xs.val.buckets.val.map (·.map f)
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⟨⟨size, ⟨buckets, sorry⟩⟩, sorry⟩
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def filter (p : α → β → Bool) (xs : HashMap α β) : HashMap α β :=
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let buckets := xs.val.buckets.val.map (·.filter p)
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let size := buckets.foldl (fun acc bucket => bucket.foldl (fun acc _ _ => acc + 1) acc) 0
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⟨⟨size, ⟨buckets, sorry⟩⟩, sorry⟩
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def getAny? (x : HashMap α β) : Option (α × β) := Id.run <| do
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for bucket in x.val.buckets.val do
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for (k, v) in bucket do
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return some (k, v)
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return none
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end Lean.HashMap
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structure Equation where
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id : Nat
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coeffs : HashMap Nat Int
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const : Int
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deriving Inhabited
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def gcd (coeffs : HashMap Nat Int) : Nat :=
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let coeffs := coeffs.mapVals (·.natAbs)
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let coeffsContent := coeffs.toArray
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match coeffsContent with
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| #[] => panic! "Cannot calculate GCD of empty list of coefficients"
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| #[(i, x)] => x
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| coeffsContent =>
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coeffsContent[0]!.2.gcd coeffsContent[1]!.2
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|> coeffs.fold fun acc k v => acc.gcd v
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namespace Equation
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def preprocess? (e : Equation) : Option Equation := Id.run <| do
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let gcd : Int := gcd e.coeffs
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if e.const % gcd ≠ 0 then
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return none
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return some { e with
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coeffs := e.coeffs.fastMapVals fun _ coeff => coeff / gcd
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const := e.const / gcd }
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def subst (fromEq toEq : Equation) (varIdx : Nat) : Equation := Id.run <| do
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-- varIdx ≡ k
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-- fromEq ≡ sₖxₖ + ∑ i ∈ V_fromEq\{k}. aᵢxᵢ = A
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-- ⇔ xₖ = sₖA - ∑ i ∈ V_fromEq\{k}. sₖaᵢxᵢ
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-- toEq ≡ bₖxₖ + ∑ i ∈ V_toEq\{k}. bᵢxᵢ = B
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-- ⇝ B = bₖ(sₖA - ∑ i ∈ V_fromEq\{k}. sₖaᵢxᵢ) + ∑ i ∈ V_toEq\{k}. bᵢxᵢ
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-- = bₖsₖA - ∑ i ∈ V_fromEq\{k}. sₖbₖaᵢxᵢ + ∑ i ∈ V_toEq\{k}. bᵢxᵢ
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-- = bₖsₖA + ∑ i ∈ V_fromEq\V_toEq. -sₖbₖaᵢxᵢ
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-- + ∑ i ∈ V_toEq\{k} ∩ V_fromEq. (bᵢ - sₖbₖaᵢ)xᵢ
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-- + ∑ i ∈ V_toEq\V_fromEq. bᵢxᵢ
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-- ⇔ B - bₖsₖA = + ∑ i ∈ X. -sₖbₖaᵢxᵢ
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-- + ∑ i ∈ Y. (bᵢ - sₖbₖaᵢ)xᵢ
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-- + ∑ i ∈ Z. bᵢxᵢ
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-- with X, Y, Z defined as above, X ∪ Y ∪ Z = (V_fromEq ∪ V_toEq)\{k}
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-- and X, Y, Z pairwise disjoint
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let A := fromEq.const
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let B := toEq.const
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let V_fromEq := fromEq.coeffs
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let V_toEq := toEq.coeffs
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let k := varIdx
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let sₖ := V_fromEq.find! k
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let bₖ := V_toEq.find! k
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let mut V_toEq := V_toEq.fastMapVals fun i bᵢ =>
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match V_fromEq.find? i with
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| none =>
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bᵢ
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| some aᵢ =>
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bᵢ - sₖ*bₖ*aᵢ
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for (i, aᵢ) in V_fromEq do
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if ¬V_toEq.contains i then
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V_toEq := V_toEq.insert i (-sₖ*bₖ*aᵢ)
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V_toEq := V_toEq.filter fun i bᵢ => i ≠ k ∧ bᵢ ≠ 0
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let B' := B - bₖ*sₖ*A
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{ toEq with coeffs := V_toEq, const := B' }
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def normalize (e : Equation) : Equation := Id.run <| do
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if e.coeffs.size ≠ 1 then
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return e
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let (i, c) := e.coeffs.getAny?.get!
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return { e with
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coeffs := e.coeffs.insert i 1
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const := Int.ediv e.const c }
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def invert (e : Equation) : Equation :=
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{ e with
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coeffs := e.coeffs.fastMapVals fun _ coeff => (-1)*coeff
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const := (-1)*e.const }
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def reorganizeFor (e : Equation) (varIdx : Nat) : Equation := Id.run <| do
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let singletonCoeff := e.coeffs.find! varIdx
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let mut e := { e with coeffs := e.coeffs.fastMapVals fun _ coeff => (-1)*coeff }
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if singletonCoeff = -1 then
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e := e.invert
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{ e with coeffs := e.coeffs.erase varIdx }
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def findSingleton? (e : Equation) : Option (Nat × Int) := Id.run <| do
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for (i, coeff) in e.coeffs do
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if coeff = 1 ∨ coeff = -1 then
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return some (i, coeff)
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return none
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def findAbsMinimumCoeff? (e : Equation) : Option (Nat × Int) := Id.run <| do
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let mut r? : Option (Nat × Int) := none
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for (i, coeff) in e.coeffs do
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match r? with
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| none =>
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r? := some (i, coeff)
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| some (i', coeff') =>
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if coeff.natAbs < coeff'.natAbs then
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r? := some (i, coeff)
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return r?
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end Equation
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structure Problem where
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equations : HashMap Nat Equation
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solvedEquations : HashMap Nat Equation
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nEquations : Nat
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nVars : Nat
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deriving Inhabited
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def preprocess? (eqs : HashMap Nat Equation) : Option (HashMap Nat Equation) :=
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eqs.mapValsM (·.preprocess?)
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def eliminateSingleton (p : Problem) (singletonEq : Equation) (varIdx : Nat) : Problem := Id.run <| do
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let mut eqsWithVarIdx : Array Nat := #[]
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for (id, eq) in p.equations do
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if eq.coeffs.contains varIdx then
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eqsWithVarIdx := eqsWithVarIdx.push id
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let mut equations := p.equations
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for id in eqsWithVarIdx do
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if id == singletonEq.id then
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continue
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equations := equations.modify! id fun eq => singletonEq.subst eq varIdx |>.normalize
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equations := equations.erase singletonEq.id
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let solvedEquations := p.solvedEquations.insert varIdx <| singletonEq.reorganizeFor varIdx
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return { p with
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equations := equations
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solvedEquations := solvedEquations }
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partial def eliminateSingletons (p : Problem) : Problem := Id.run <| do
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let mut r? : Option (Equation × Nat) := none
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for (id, eq) in p.equations do
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match eq.findSingleton? with
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| none =>
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continue
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| some (varIdx, coeff) =>
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r? := some (eq, varIdx)
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match r? with
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| none =>
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return p
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| some (eq, varIdx) =>
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let p := eliminateSingleton p eq varIdx
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return eliminateSingletons p
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def addAuxEquation (p : Problem) : Problem := Id.run <| do
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let mut E? : Option Equation := none
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let mut k? : Option Nat := none
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let mut aₖ? : Option Int := none
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for (_, eq) in p.equations do
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match eq.findAbsMinimumCoeff?, aₖ? with
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| none, _ => continue
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| some (k', aₖ'), none =>
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E? := some eq
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k? := some k'
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aₖ? := some aₖ'
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| some (k', aₖ'), some aₖ =>
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if aₖ'.natAbs < aₖ.natAbs then
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E? := some eq
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k? := some k'
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aₖ? := some aₖ'
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let mut E := E?.get!
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let k := k?.get!
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let mut aₖ := aₖ?.get!
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if aₖ < 0 then
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aₖ := -aₖ
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E := E.invert
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let m := aₖ + 1
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let σIdx := p.nVars
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let newEqCoeffs := E.coeffs.fastMapVals (fun _ coeff => coeff.mod' m)
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|>.insert σIdx (-m)
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|>.filter (fun _ coeff => coeff ≠ 0)
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let newEqConst := E.const.mod' m
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let newEq : Equation := ⟨p.nEquations, newEqCoeffs, newEqConst⟩
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let E'coeffs := E.coeffs.filter (fun i _ => i ≠ k)
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|>.fastMapVals (fun _ aᵢ => aᵢ.roundedDiv m + aᵢ.mod' m)
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|>.insert σIdx (-aₖ)
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|>.filter (fun _ coeff => coeff ≠ 0)
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let c := E.const
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let E'const := c.roundedDiv m + c.mod' m
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let E' := { E with coeffs := E'coeffs, const := E'const }.normalize
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let equations' := p.equations.insert E'.id E' |>.insert newEq.id newEq
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let p' : Problem := { p with
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equations := equations'
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nVars := p.nVars + 1
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nEquations := p.nEquations + 1 }
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return eliminateSingleton p' newEq k
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inductive Solution
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| unsat
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| sat (assignment : Array Int)
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deriving Inhabited
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partial def readSolution? (p : Problem) : Option Solution := Id.run <| do
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if p.equations.any (fun _ eq => eq.coeffs.size ≠ 0) then
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return none
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if p.equations.any (fun _ eq => eq.const ≠ 0) then
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return some Solution.unsat
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let mut assignment : Array (Option Int) := mkArray p.nVars none
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for i in [0:p.nVars] do
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assignment := readSolution i assignment
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return Solution.sat <| assignment.map (·.get!)
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where
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readSolution (varIdx : Nat) (assignment : Array (Option Int)) : Array (Option Int) := Id.run <| do
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match p.solvedEquations.find? varIdx with
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| none =>
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return assignment.set! varIdx (some 0)
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| some eq =>
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let mut assignment := assignment
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let mut r := eq.const
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for (i, coeff) in eq.coeffs do
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if assignment[i]!.isNone then
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assignment := readSolution i assignment
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r := r + coeff*assignment[i]!.get!
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return assignment.set! varIdx (some r)
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partial def solveProblem' (p : Problem) : Solution := Id.run <| do
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match readSolution? p with
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| some solution => return solution
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| none =>
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let p := eliminateSingletons p
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match readSolution? p with
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| some solution => return solution
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| none =>
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let p := addAuxEquation p
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return solveProblem' p
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def isSatAssignment (p : Problem) (assignment : Array Int) : Bool :=
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¬ p.equations.any fun _ (eq : Equation) => Id.run <| do
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let mut r := 0
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for (i, coeff) in eq.coeffs do
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r := r + coeff*assignment[i]!
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return r ≠ eq.const
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def solveProblem (p : Problem) : Solution :=
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let nVars := p.nVars
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match solveProblem' p with
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| Solution.unsat =>
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Solution.unsat
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| Solution.sat assignment =>
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let assignment' := assignment.extract 0 nVars
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if isSatAssignment p assignment' then
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Solution.sat assignment'
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else
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Solution.unsat
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def error (msg : String) : IO α :=
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throw <| IO.userError s!"Error: {msg}."
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def Array.ithVal (xs : Array String) (i : Nat) (name : String) : IO Int := do
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let some unparsed := xs.get? i
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| error s!"Missing {name}"
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let some parsed := String.toInt? unparsed
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| error s!"Invalid {name}: `{unparsed}`"
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return parsed
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def main (args : List String) : IO UInt32 := do
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let some path := args.head?
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| error "Usage: liasolver <input file>"
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let lines ← IO.FS.lines path <&> Array.filter (¬·.isEmpty)
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let some headerLine := lines.get? 0
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| error "No header line"
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let header := headerLine.splitOn.toArray
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let nEquations ← header.ithVal 0 "amount of equations"
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let nVars ← header.ithVal 1 "amount of variables"
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let mut equations : HashMap Nat Equation := mkHashMap
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for line in lines[1:] do
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let elems := line.splitOn.toArray
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let nTerms ← elems.ithVal 0 "amount of equation terms"
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let 0 ← elems.ithVal (elems.size - 1) "end of line symbol"
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| error "Non-zero end of line symbol"
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let const ← elems.ithVal (elems.size - 2) "constant value"
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let mut coeffs := mkHashMap
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for i in [1:elems.size-2:2] do
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let coeff ← elems.ithVal i "coefficient"
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let varIdx ← elems.ithVal (i + 1) "variable index"
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if varIdx < 1 then
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error "Invalid variable index"
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let varIdx := varIdx.toNat - 1
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if coeff ≠ 0 then
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coeffs := coeffs.insert varIdx coeff
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if coeffs.size ≠ 0 then
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equations := equations.insert equations.size ⟨equations.size, coeffs, const⟩
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match preprocess? equations with
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| none =>
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IO.println "UNSAT"
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| some equations' =>
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let problem : Problem := ⟨equations', mkHashMap, equations'.size, nVars.natAbs⟩
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match solveProblem problem with
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| Solution.unsat =>
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IO.println "UNSAT"
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| Solution.sat assignment =>
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IO.println "SAT"
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IO.println <| String.intercalate " " <| assignment.toList.map toString
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return 0
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