@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.instToExprLinearCombo : ToExpr Omega.LinearCombo

@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.instToExprConstraint : ToExpr Omega.Constraint

@[reducible, inline]

abbrev Lean.Elab.Tactic.Omega.Proof : Type

A delayed proof that a constraint is satisfied at the atoms.

inductive Lean.Elab.Tactic.Omega.Justification : Omega.Constraint → Omega.Coeffs → Type

Our internal representation of an argument "justifying" that a constraint holds on some coefficients. We'll use this to construct the proof term once a contradiction is found.

• assumption (s : Omega.Constraint) (x : Omega.Coeffs) (i : Nat) : Justification s x

  Problem.assumptions[i] generates a proof that s.sat' coeffs atoms

• tidy {s : Omega.Constraint} {c : Omega.Coeffs} (j : Justification s c) : Justification (Omega.tidyConstraint s c) (Omega.tidyCoeffs s c)

  The result of tidy on another Justification.

• combine {s t : Omega.Constraint} {c : Omega.Coeffs} (j : Justification s c) (k : Justification t c) : Justification (s.combine t) c

  The result of combine on two Justifications.

• combo {s t : Omega.Constraint} {x y : Omega.Coeffs} (a : Int) (j : Justification s x) (b : Int) (k : Justification t y) : Justification (Omega.Constraint.combo a s b t) (Omega.Coeffs.combo a x b y)

  A linear combo of two Justifications.

• bmod (m : Nat) (r : Int) (i : Nat) {x : Omega.Coeffs} (j : Justification (Omega.Constraint.exact r) x) : Justification (Omega.Constraint.exact (r.bmod m)) (Omega.bmod_coeffs m i x)

  The justification for the constraint constructed using "balanced mod" while eliminating an equality.

def Lean.Elab.Tactic.Omega.Justification.tidy? {s : Omega.Constraint} {c : Omega.Coeffs} (j : Justification s c) : Option ((s' : Omega.Constraint) × (c' : Omega.Coeffs) × Justification s' c')

Wrapping for Justification.tidy when tidy? is nonempty.

def Lean.Elab.Tactic.Omega.Justification.toString {s : Omega.Constraint} {x : Omega.Coeffs} : Justification s x → String

Print a Justification as an indented tree structure.

@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.Justification.instToString {s : Omega.Constraint} {x : Omega.Coeffs} : ToString (Justification s x)

def Lean.Elab.Tactic.Omega.Justification.tidyProof (s : Omega.Constraint) (x : Omega.Coeffs) (v prf : Expr) : Expr

Construct the proof term associated to a tidy step.

def Lean.Elab.Tactic.Omega.Justification.combineProof (s t : Omega.Constraint) (x : Omega.Coeffs) (v ps pt : Expr) : Expr

Construct the proof term associated to a combine step.

def Lean.Elab.Tactic.Omega.Justification.comboProof (s t : Omega.Constraint) (a : Int) (x : Omega.Coeffs) (b : Int) (y : Omega.Coeffs) (v px py : Expr) : Expr

Construct the proof term associated to a combo step.

def Lean.Elab.Tactic.Omega.Justification.bmodProof (m : Nat) (r : Int) (i : Nat) (x : Omega.Coeffs) (v w : Expr) : MetaM Expr

Construct the proof term associated to a bmod step.

def Lean.Elab.Tactic.Omega.Justification.proof {s : Omega.Constraint} {c : Omega.Coeffs} (v : Expr) (assumptions : Array Proof) : Justification s c → Proof

Constructs a proof that s.sat' c v = true

structure Lean.Elab.Tactic.Omega.Fact : Type

A Justification bundled together with its parameters.

• coeffs : Omega.Coeffs

  The coefficients of a constraint.

• constraint : Omega.Constraint

  The constraint.

• justification : Justification self.constraint self.coeffs

  The justification of a derived fact.

@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.Fact.instToString : ToString Fact

def Lean.Elab.Tactic.Omega.Fact.tidy (f : Fact) : Fact

tidy, implemented on Fact.

def Lean.Elab.Tactic.Omega.Fact.combo (a : Int) (f : Fact) (b : Int) (g : Fact) : Fact

combo, implemented on Fact.

structure Lean.Elab.Tactic.Omega.Problem : Type

A omega problem.

This is a hybrid structure: it contains both Expr objects giving proofs of the "ground" assumptions (or rather continuations which will produce the proofs when needed) and internal representations of the linear constraints that we manipulate in the algorithm.

While the algorithm is running we do not synthesize any new Expr proofs: proof extraction happens only once we've found a contradiction.

• assumptions : Array Proof

  The ground assumptions that the algorithm starts from.

• numVars : Nat

  The number of variables in the problem.

• constraints : Std.HashMap Omega.Coeffs Fact

  The current constraints, indexed by their coefficients.

• equalities : Std.HashSet Omega.Coeffs

  The coefficients for which constraints contains an exact constraint (i.e. an equality).

• eliminations : List (Fact × Nat × Int)

  Equations that have already been used to eliminate variables, along with the variable which was removed, and its coefficient (either 1 or -1). The earlier elements are more recent, so if these are being reapplied it is essential to use List.foldr.

• possible : Bool

  Whether the problem is possible.

• proveFalse? : Option Proof

  If the problem is impossible, then proveFalse? will contain a proof of False.

• proveFalse?_spec : (self.possible || self.proveFalse?.isSome) = true

  Invariant between possible and proveFalse?.

• explanation? : Thunk String

  If we have found a contradiction, explanation? will contain a human readable account of the derivation.

def Lean.Elab.Tactic.Omega.Problem.isEmpty (p : Problem) : Bool

Check if a problem has no constraints.

@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.Problem.instToString : ToString Problem

def Lean.Elab.Tactic.Omega.Problem.proveFalse {s : Omega.Constraint} {x : Omega.Coeffs} (j : Justification s x) (assumptions : Array Proof) : Proof

Takes a proof that s.sat' x v for some s such that s.isImpossible, and constructs a proof of False.

def Lean.Elab.Tactic.Omega.Problem.insertConstraint (p : Problem) : Fact → Problem

Insert a constraint into the problem, without checking if there is already a constraint for these coefficients.

def Lean.Elab.Tactic.Omega.Problem.addConstraint (p : Problem) : Fact → Problem

Add a constraint into the problem, combining it with any existing constraints for the same coefficients.

def Lean.Elab.Tactic.Omega.Problem.selectEquality (p : Problem) : Option (Omega.Coeffs × Nat)

Walk through the equalities, finding either the first equality with minimal coefficient ±1, or otherwise the equality with minimal (r.minNatAbs, r.maxNatAbs) (ordered lexicographically).

Returns the coefficients of the equality, along with the value of minNatAbs.

Although we don't need to run a termination proof here, it's nevertheless important that we use this ordering so the algorithm terminates in practice!

def Lean.Elab.Tactic.Omega.Problem.replayEliminations (p : Problem) (f : Fact) : Fact

If we have already solved some equalities, apply those to some new Fact.

def Lean.Elab.Tactic.Omega.Problem.solveEasyEquality (p : Problem) (c : Omega.Coeffs) : Problem

Solve an "easy" equality, i.e. one with a coefficient that is ±1.

After solving, the variable will have been eliminated from all constraints.

def Lean.Elab.Tactic.Omega.Problem.dealWithHardEquality (p : Problem) (c : Omega.Coeffs) : OmegaM Problem

We deal with a hard equality by introducing a new easy equality.

After solving the easy equality, the minimum lexicographic value of (c.minNatAbs, c.maxNatAbs) will have been reduced.

def Lean.Elab.Tactic.Omega.Problem.solveEquality (p : Problem) (c : Omega.Coeffs) (m : Nat) : OmegaM Problem

Solve an equality, by deciding whether it is easy (has a ±1 coefficient) or hard, and delegating to the appropriate function.

partial def Lean.Elab.Tactic.Omega.Problem.solveEqualities (p : Problem) : OmegaM Problem

Recursively solve all equalities.

def Lean.Elab.Tactic.Omega.Problem.addInequality_proof (c : Int) (x : Omega.Coeffs) (p : Proof) : Proof

Constructing the proof term for addInequality.

def Lean.Elab.Tactic.Omega.Problem.addEquality_proof (c : Int) (x : Omega.Coeffs) (p : Proof) : Proof

Constructing the proof term for addEquality.

def Lean.Elab.Tactic.Omega.Problem.addInequality (p : Problem) (const : Int) (coeffs : Omega.Coeffs) (prf? : Option Proof) : Problem

Helper function for adding an inequality of the form const + Coeffs.dot coeffs atoms ≥ 0 to a Problem.

(This is only used while initializing a Problem. During elimination we use addConstraint.)

def Lean.Elab.Tactic.Omega.Problem.addEquality (p : Problem) (const : Int) (coeffs : Omega.Coeffs) (prf? : Option Proof) : Problem

Helper function for adding an equality of the form const + Coeffs.dot coeffs atoms = 0 to a Problem.

(This is only used while initializing a Problem. During elimination we use addConstraint.)

def Lean.Elab.Tactic.Omega.Problem.addInequalities (p : Problem) (ineqs : List (Int × Omega.Coeffs × Option Proof)) : Problem

Folding addInequality over a list.

def Lean.Elab.Tactic.Omega.Problem.addEqualities (p : Problem) (eqs : List (Int × Omega.Coeffs × Option Proof)) : Problem

Folding addEquality over a list.

structure Lean.Elab.Tactic.Omega.Problem.FourierMotzkinData : Type

Representation of the data required to run Fourier-Motzkin elimination on a variable.

• var : Nat

  Which variable is being eliminated.

• irrelevant : List Fact

  The "irrelevant" facts which do not involve the target variable.

• lowerBounds : List (Fact × Int)

  The facts which give a lower bound on the target variable, and the coefficient of the target variable in each.

• upperBounds : List (Fact × Int)

  The facts which give an upper bound on the target variable, and the coefficient of the target variable in each.

• lowerExact : Bool

  Whether the elimination would be exact, because all of the lower bound coefficients are ±1.

• upperExact : Bool

  Whether the elimination would be exact, because all of the upper bound coefficients are ±1.

@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.Problem.instInhabitedFourierMotzkinData : Inhabited FourierMotzkinData

def Lean.Elab.Tactic.Omega.Problem.instInhabitedFourierMotzkinData.default : FourierMotzkinData

@[implicit_reducible]

instance Lean.Elab.Tactic.Omega.Problem.instToStringFourierMotzkinData : ToString FourierMotzkinData

def Lean.Elab.Tactic.Omega.Problem.FourierMotzkinData.isEmpty (d : FourierMotzkinData) : Bool

Is a Fourier-Motzkin elimination empty (i.e. there are no relevant constraints).

def Lean.Elab.Tactic.Omega.Problem.FourierMotzkinData.size (d : FourierMotzkinData) : Nat

The number of new constraints that would be introduced by Fourier-Motzkin elimination.

def Lean.Elab.Tactic.Omega.Problem.FourierMotzkinData.exact (d : FourierMotzkinData) : Bool

Is the Fourier-Motzkin elimination known to be exact?

def Lean.Elab.Tactic.Omega.Problem.fourierMotzkinData (p : Problem) : Array FourierMotzkinData

Prepare the Fourier-Motzkin elimination data for each variable.

def Lean.Elab.Tactic.Omega.Problem.fourierMotzkinSelect (data : Array FourierMotzkinData) : MetaM FourierMotzkinData

Decides which variable to run Fourier-Motzkin elimination on, and returns the associated data.

We prefer eliminations which introduce no new inequalities, or otherwise exact eliminations, and break ties by the number of new inequalities introduced.

def Lean.Elab.Tactic.Omega.Problem.fourierMotzkin (p : Problem) : MetaM Problem

Run Fourier-Motzkin elimination on one variable.

partial def Lean.Elab.Tactic.Omega.Problem.runOmega (p : Problem) : OmegaM Problem

Run the omega algorithm (for now without dark and grey shadows!) on a problem.

partial def Lean.Elab.Tactic.Omega.Problem.elimination (p : Problem) : OmegaM Problem

As for runOmega, but assuming the first round of solving equalities has already happened.