Frontend to the omega tactic.

See Lean.Elab.Tactic.Omega for an overview of the tactic.

def Lean.Elab.Tactic.Omega.elabOmegaConfig : Lean.Syntax → Lean.Elab.TermElabM Lean.Meta.Omega.OmegaConfig

Allow elaboration of OmegaConfig arguments to tactics.

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def Lean.Elab.Tactic.Omega.succ? (e : Lean.Expr) : Option Lean.Expr

Match on the two defeq expressions for successor: n+1, n.succ.

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

A partially processed omega context.

We have:

• a Problem representing the integer linear constraints extracted so far, and their proofs
• the unprocessed facts : List Expr taken from the local context,
• the unprocessed disjunctions : List Expr, which will only be split one at a time if we can't otherwise find a contradiction.

We begin with facts := ← getLocalHyps and problem := .trivial, and progressively process the facts.

As we process the facts, we may generate additional facts (e.g. about coercions and integer divisions). To avoid duplicates, we maintain a HashSet of previously processed facts.

• problem : Lean.Elab.Tactic.Omega.Problem

  An integer linear arithmetic problem.

• facts : List Lean.Expr

  Pending facts which have not been processed yet.

• disjunctions : List Lean.Expr

  Pending disjunctions, which we will case split one at a time if we can't get a contradiction.

• processedFacts : Std.HashSet Lean.Expr

  Facts which have already been processed; we keep these to avoid duplicates.

def Lean.Elab.Tactic.Omega.mkEvalRflProof (e : Lean.Expr) (lc : Lean.Omega.LinearCombo) : Lean.Elab.Tactic.Omega.OmegaM Lean.Expr

Construct the rfl proof that lc.eval atoms = e.

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def Lean.Elab.Tactic.Omega.mkCoordinateEvalAtomsEq (e : Lean.Expr) (n : Nat) : Lean.Elab.Tactic.Omega.OmegaM Lean.Expr

If e : Expr is the n-th atom, construct the proof that e = (coordinate n).eval atoms.

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def Lean.Elab.Tactic.Omega.mkAtomLinearCombo (e : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Omega.LinearCombo × Lean.Elab.Tactic.Omega.OmegaM Lean.Expr × Std.HashSet Lean.Expr)

Construct the linear combination (and its associated proof and new facts) for an atom.

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partial def Lean.Elab.Tactic.Omega.asLinearCombo (e : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Omega.LinearCombo × Lean.Elab.Tactic.Omega.OmegaM Lean.Expr × Std.HashSet Lean.Expr)

Wrapper for asLinearComboImpl, using a cache for previously visited expressions.

Gives a small (10%) speedup in testing. I tried using a pointer based cache, but there was never enough subexpression sharing to make it effective.

partial def Lean.Elab.Tactic.Omega.asLinearComboImpl (e : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Omega.LinearCombo × Lean.Elab.Tactic.Omega.OmegaM Lean.Expr × Std.HashSet Lean.Expr)

Translates an expression into a LinearCombo. Also returns:

• a proof that this linear combo evaluated at the atoms is equal to the original expression
• a list of new facts which should be recorded:
  • for each new atom a of the form ((x : Nat) : Int), the fact that 0 ≤ a
  • for each new atom a of the form x / k, for k a positive numeral, the facts that k * a ≤ x < (k + 1) * a
  • for each new atom of the form ((a - b : Nat) : Int), the fact: b ≤ a ∧ ((a - b : Nat) : Int) = a - b ∨ a < b ∧ ((a - b : Nat) : Int) = 0

We also transform the expression as we descend into it:

• pushing coercions: ↑(x + y), ↑(x * y), ↑(x / k), ↑(x % k), ↑k
• unfolding emod: x % k → x - x / k

partial def Lean.Elab.Tactic.Omega.asLinearComboImpl.rewrite (lhs : Lean.Expr) (rw : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Omega.LinearCombo × Lean.Elab.Tactic.Omega.OmegaM Lean.Expr × Std.HashSet Lean.Expr)

Apply a rewrite rule to an expression, and interpret the result as a LinearCombo. (We're not rewriting any subexpressions here, just the top level, for efficiency.)

partial def Lean.Elab.Tactic.Omega.asLinearComboImpl.handleNatCast (e : Lean.Expr) (i : Lean.Expr) (n : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Omega.LinearCombo × Lean.Elab.Tactic.Omega.OmegaM Lean.Expr × Std.HashSet Lean.Expr)

partial def Lean.Elab.Tactic.Omega.asLinearComboImpl.handleFinVal (e : Lean.Expr) (i : Lean.Expr) (n : Lean.Expr) (x : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Omega.LinearCombo × Lean.Elab.Tactic.Omega.OmegaM Lean.Expr × Std.HashSet Lean.Expr)

def Lean.Elab.Tactic.Omega.MetaProblem.trivial : Lean.Elab.Tactic.Omega.MetaProblem

The trivial MetaProblem, with no facts to process and a trivial Problem.

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instance Lean.Elab.Tactic.Omega.MetaProblem.instInhabited : Inhabited Lean.Elab.Tactic.Omega.MetaProblem

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• Lean.Elab.Tactic.Omega.MetaProblem.instInhabited = { default := Lean.Elab.Tactic.Omega.MetaProblem.trivial }

def Lean.Elab.Tactic.Omega.MetaProblem.addIntEquality (p : Lean.Elab.Tactic.Omega.MetaProblem) (h : Lean.Expr) (x : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM Lean.Elab.Tactic.Omega.MetaProblem

Add an integer equality to the Problem.

We solve equalities as they are discovered, as this often results in an earlier contradiction.

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def Lean.Elab.Tactic.Omega.MetaProblem.addIntInequality (p : Lean.Elab.Tactic.Omega.MetaProblem) (h : Lean.Expr) (y : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM Lean.Elab.Tactic.Omega.MetaProblem

Add an integer inequality to the Problem.

We solve equalities as they are discovered, as this often results in an earlier contradiction.

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def Lean.Elab.Tactic.Omega.MetaProblem.pushNot (h : Lean.Expr) (P : Lean.Expr) : Lean.MetaM (Option Lean.Expr)

Given a fact h with type ¬ P, return a more useful fact obtained by pushing the negation.

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partial def Lean.Elab.Tactic.Omega.MetaProblem.addFact (p : Lean.Elab.Tactic.Omega.MetaProblem) (h : Lean.Expr) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Elab.Tactic.Omega.MetaProblem × Nat)

Parse an Expr and extract facts, also returning the number of new facts found.

partial def Lean.Elab.Tactic.Omega.MetaProblem.processFacts (p : Lean.Elab.Tactic.Omega.MetaProblem) : Lean.Elab.Tactic.Omega.OmegaM (Lean.Elab.Tactic.Omega.MetaProblem × Nat)

Process all the facts in a MetaProblem, returning the new problem, and the number of new facts.

This is partial because new facts may be generated along the way.

def Lean.Elab.Tactic.Omega.cases₂ (mvarId : Lean.MVarId) (p : Lean.Expr) (hName : optParam Lake.Name `h) : Lean.MetaM ((Lean.MVarId × Lean.FVarId) × Lean.MVarId × Lean.FVarId)

Given p : P ∨ Q (or any inductive type with two one-argument constructors), split the goal into two subgoals: one containing the hypothesis h : P and another containing h : Q.

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def Lean.Elab.Tactic.Omega.formatErrorMessage (p : Lean.Elab.Tactic.Omega.Problem) : Lean.Elab.Tactic.Omega.OmegaM Lean.MessageData

Helpful error message when omega cannot find a solution

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def Lean.Elab.Tactic.Omega.formatErrorMessage.varNameOf (i : Nat) : String

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def Lean.Elab.Tactic.Omega.formatErrorMessage.inScope (s : String) : Lean.MetaM Bool

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def Lean.Elab.Tactic.Omega.formatErrorMessage.varNames (mask : Array Bool) : Lean.MetaM (Array String)

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def Lean.Elab.Tactic.Omega.formatErrorMessage.prettyConstraints (names : Array String) (constraints : Std.HashMap Lean.Omega.Coeffs Lean.Elab.Tactic.Omega.Fact) : String

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def Lean.Elab.Tactic.Omega.formatErrorMessage.prettyConstraint (e : String) : Lean.Omega.Constraint → String

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• Lean.Elab.Tactic.Omega.formatErrorMessage.prettyConstraint e { lowerBound := none, upperBound := none } = toString "" ++ toString e ++ toString " is unconstrained"
• Lean.Elab.Tactic.Omega.formatErrorMessage.prettyConstraint e { lowerBound := none, upperBound := some y } = toString "" ++ toString e ++ toString " ≤ " ++ toString y ++ toString ""
• Lean.Elab.Tactic.Omega.formatErrorMessage.prettyConstraint e { lowerBound := some x_1, upperBound := none } = toString "" ++ toString e ++ toString " ≥ " ++ toString x_1 ++ toString ""

def Lean.Elab.Tactic.Omega.formatErrorMessage.prettyCoeffs (names : Array String) (coeffs : Lean.Omega.Coeffs) : String

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def Lean.Elab.Tactic.Omega.formatErrorMessage.mentioned (atoms : Array Lean.Expr) (constraints : Std.HashMap Lean.Omega.Coeffs Lean.Elab.Tactic.Omega.Fact) : Lean.MetaM (Array Bool)

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def Lean.Elab.Tactic.Omega.formatErrorMessage.prettyAtoms (names : Array String) (atoms : Array Lean.Expr) (mask : Array Bool) : Lean.MessageData

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partial def Lean.Elab.Tactic.Omega.splitDisjunction (m : Lean.Elab.Tactic.Omega.MetaProblem) (g : Lean.MVarId) : Lean.Elab.Tactic.Omega.OmegaM Unit

Split a disjunction in a MetaProblem, and if we find a new usable fact call omegaImpl in both branches.

partial def Lean.Elab.Tactic.Omega.omegaImpl (m : Lean.Elab.Tactic.Omega.MetaProblem) (g : Lean.MVarId) : Lean.Elab.Tactic.Omega.OmegaM Unit

Implementation of the omega algorithm, and handling disjunctions.

def Lean.Elab.Tactic.Omega.omega (facts : List Lean.Expr) (g : Lean.MVarId) (cfg : optParam Lean.Meta.Omega.OmegaConfig { splitDisjunctions := true, splitNatSub := true, splitNatAbs := true, splitMinMax := true }) : Lean.MetaM Unit

Given a collection of facts, try prove False using the omega algorithm, and close the goal using that.

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def Lean.Elab.Tactic.Omega.omegaTactic (cfg : Lean.Meta.Omega.OmegaConfig) : Lean.Elab.Tactic.TacticM Unit

The omega tactic, for resolving integer and natural linear arithmetic problems.

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def Lean.Elab.Tactic.Omega.omegaDefault : Lean.Elab.Tactic.TacticM Unit

The omega tactic, for resolving integer and natural linear arithmetic problems. This TacticM Unit frontend with default configuration can be used as an Aesop rule, for example via the tactic call aesop (add 50% tactic Lean.Omega.omegaDefault).

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• Lean.Elab.Tactic.Omega.omegaDefault = Lean.Elab.Tactic.Omega.omegaTactic { splitDisjunctions := true, splitNatSub := true, splitNatAbs := true, splitMinMax := true }

def Lean.Elab.Tactic.Omega.evalOmega : Lean.Elab.Tactic.Tactic

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opaque Lean.Elab.Tactic.Omega.bvOmegaSimpExtension : Lean.Meta.SimpExtension