Frontend to the omega tactic. #

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

Allow elaboration of OmegaConfig arguments to tactics.

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

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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.

• An integer linear arithmetic problem.

• facts :

Pending facts which have not been processed yet.

• disjunctions :

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

• processedFacts :

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

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

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

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

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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.

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 % kx - x / k

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.)

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

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

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

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 : ) :

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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Helpful error message when omega cannot find a solution

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

Implementation of the omega algorithm, and handling disjunctions.

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

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

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

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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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