Linarith preprocessing

This file contains methods used to preprocess inputs to linarith.

In particular, linarith works over comparisons of the form t R 0, where R ∈ {<,≤,=}. It assumes that expressions in t have integer coefficients and that the type of t has well-behaved subtraction.

Implementation details

A GlobalPreprocessor is a function List Expr → TacticM (List Expr). Users can add custom preprocessing steps by adding them to the LinarithConfig object. Linarith.defaultPreprocessors is the main list, and generally none of these should be skipped unless you know what you're doing.

Preprocessing

def Linarith.splitConjunctions : Linarith.Preprocessor

Processor that recursively replaces P ∧ Q hypotheses with the pair P and Q.

Equations

• Linarith.splitConjunctions = { name := "split conjunctions", transform := Linarith.splitConjunctions.aux }

partial def Linarith.splitConjunctions.aux (proof : Lean.Expr) : Lean.MetaM (List Lean.Expr)

Implementation of the splitConjunctions preprocessor.

def Linarith.filterComparisons : Linarith.Preprocessor

Removes any expressions that are not proofs of inequalities, equalities, or negations thereof.

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def Linarith.filterComparisons.aux (e : Lean.Expr) : Lean.MetaM Bool

Implementation of the filterComparisons preprocessor.

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def Linarith.flipNegatedComparison (prf : Lean.Expr) (e : Lean.Expr) : Lean.MetaM (Option Lean.Expr)

If prf is a proof of ¬ e, where e is a comparison, flipNegatedComparison prf e flips the comparison in e and returns a proof. For example, if prf : ¬ a < b, flipNegatedComparison prf q(a < b) returns a proof of a ≥ b.

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def Linarith.removeNegations : Linarith.Preprocessor

Replaces proofs of negations of comparisons with proofs of the reversed comparisons. For example, a proof of ¬ a < b will become a proof of a ≥ b.

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partial def Linarith.isNatProp (e : Lean.Expr) : Bool

isNatProp tp is true iff tp is an inequality or equality between natural numbers or the negation thereof.

def Linarith.isNatCoe (e : Lean.Expr) : Option (Lean.Expr × Lean.Expr)

If e is of the form ((n : ℕ) : C), isNatCoe e returns ⟨n, C⟩.

Equations

• Linarith.isNatCoe e = match Lean.Expr.getAppFnArgs e with | (`Nat.cast, #[target, head, n]) => some (n, target) | x => none

partial def Linarith.getNatComparisons (e : Lean.Expr) : List (Lean.Expr × Lean.Expr)

getNatComparisons e returns a list of all subexpressions of e of the form ((t : ℕ) : C).

def Linarith.mk_natCast_nonneg_prf (p : Lean.Expr × Lean.Expr) : Lean.MetaM (Option Lean.Expr)

If e : ℕ, returns a proof of 0 ≤ (e : C).

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def Linarith.Expr.Ord : Ord Lean.Expr

Ordering on Expr.

Equations

• Linarith.Expr.Ord = { compare := fun (a b : Lean.Expr) => if Lean.Expr.lt a b = true then Ordering.lt else if Lean.Expr.equal a b = true then Ordering.eq else Ordering.gt }

def Linarith.natToInt : Linarith.GlobalBranchingPreprocessor

If h is an equality or inequality between natural numbers, natToInt lifts this inequality to the integers. It also adds the facts that the integers involved are nonnegative. To avoid adding the same nonnegativity facts many times, it is a global preprocessor.

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def Linarith.mkNonstrictIntProof (pf : Lean.Expr) : Lean.MetaM (Option Lean.Expr)

If pf is a proof of a strict inequality (a : ℤ) < b, mkNonstrictIntProof pf returns a proof of a + 1 ≤ b, and similarly if pf proves a negated weak inequality.

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def Linarith.strengthenStrictInt : Linarith.Preprocessor

strengthenStrictInt h turns a proof h of a strict integer inequality t1 < t2 into a proof of t1 ≤ t2 + 1.

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def Linarith.rearrangeComparison (e : Lean.Expr) : Lean.MetaM (Option Lean.Expr)

rearrangeComparison e takes a proof e of an equality, inequality, or negation thereof, and turns it into a proof of a comparison _ R 0, where R ∈ {=, ≤, <}.

Equations

• Linarith.rearrangeComparison e = do let __do_lift ← Lean.Meta.inferType e let __do_lift ← Lean.instantiateMVars __do_lift Linarith.rearrangeComparison.aux e __do_lift

partial def Linarith.rearrangeComparison.aux (proof : Lean.Expr) (e : Lean.Expr) : Lean.MetaM (Option Lean.Expr)

Implementation of rearrangeComparison, after type inference.

def Linarith.compWithZero : Linarith.Preprocessor

compWithZero h takes a proof h of an equality, inequality, or negation thereof, and turns it into a proof of a comparison _ R 0, where R ∈ {=, ≤, <}.

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theorem Linarith.without_one_mul {M : Type u_1} [MulOneClass M] {a : M} {b : M} (h : 1 * a = b) : a = b

def Linarith.normalizeDenominatorsLHS (h : Lean.Expr) (lhs : Lean.Expr) : Lean.MetaM Lean.Expr

normalizeDenominatorsLHS h lhs assumes that h is a proof of lhs R 0. It creates a proof of lhs' R 0, where all numeric division in lhs has been cancelled.

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def Linarith.cancelDenoms : Linarith.Preprocessor

cancelDenoms pf assumes pf is a proof of t R 0. If t contains the division symbol /, it tries to scale t to cancel out division by numerals.

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partial def Linarith.findSquares (s : Lean.HashSet (Lean.Expr × Bool)) (e : Lean.Expr) : Lean.MetaM (Lean.HashSet (Lean.Expr × Bool))

findSquares s e collects all terms of the form a ^ 2 and a * a that appear in e and adds them to the set s. A pair (a, true) is added to s when a^2 appears in e, and (a, false) is added to s when a*a appears in e.

def Linarith.nlinarithExtras : Linarith.GlobalPreprocessor

nlinarithExtras is the preprocessor corresponding to the nlinarith tactic.

• For every term t such that t^2 or t*t appears in the input, adds a proof of t^2 ≥ 0 or t*t ≥ 0.
• For every pair of comparisons t1 R1 0 and t2 R2 0, adds a proof of t1*t2 R 0.

This preprocessor is typically run last, after all inputs have been canonized.

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partial def Linarith.removeNe_aux : Lean.MVarId → List Lean.Expr → Lean.MetaM (List Linarith.Branch)

removeNe_aux case splits on any proof h : a ≠ b in the input, turning it into a < b ∨ a > b. This produces 2^n branches when there are n such hypotheses in the input.

def Linarith.removeNe : Linarith.GlobalBranchingPreprocessor

removeNe case splits on any proof h : a ≠ b in the input, turning it into a < b ∨ a > b, by calling linarith.removeNe_aux. This produces 2^n branches when there are n such hypotheses in the input.

Equations

• Linarith.removeNe = { name := "removeNe", transform := Linarith.removeNe_aux }

def Linarith.defaultPreprocessors : List Linarith.GlobalBranchingPreprocessor

The default list of preprocessors, in the order they should typically run.

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def Linarith.preprocess (pps : List Linarith.GlobalBranchingPreprocessor) (g : Lean.MVarId) (l : List Lean.Expr) : Lean.MetaM (List Linarith.Branch)

preprocess pps l takes a list l of proofs of propositions. It maps each preprocessor pp ∈ pps over this list. The preprocessors are run sequentially: each receives the output of the previous one. Note that a preprocessor may produce multiple or no expressions from each input expression, so the size of the list may change.

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