# Documentation

Mathlib.Tactic.Linarith.Preprocessing

# 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 #

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

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Implementation of the splitConjunctions preprocessor.

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

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Implementation of the filterComparisons preprocessor.

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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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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) :

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

If e is of the form ((n : ℕ) : ℤ), isNatIntCoe e returns n : ℕ.

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

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

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Ordering on Expr.

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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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isStrictIntComparison tp is true iff tp is a strict inequality between integers or the negation of a weak inequality between integers.

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

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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 ∈ {=, ≤, <}.

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Implementation of rearrangeComparison, after type inference.

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} [] {a : M} {b : M} (h : 1 * a = b) :
a = b

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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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 : ) (e : Lean.Expr) :

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.

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 :

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.

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.

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

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def Linarith.preprocess (g : Lean.MVarId) (l : ) :

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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