Datatypes for linarith

Some of the data structures here are used in multiple parts of the tactic. We split them into their own file.

This file also contains a few convenient auxiliary functions.

def Linarith.linarithGetProofsMessage (l : List Lean.Expr) : Lean.MetaM Lean.MessageData

A shorthand for getting the types of a list of proofs terms, to trace.

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def Linarith.linarithTraceProofs {α : Type} [Lean.ToMessageData α] (s : α) (l : List Lean.Expr) : Lean.MetaM Unit

A shorthand for tracing the types of a list of proof terms when the trace.linarith option is set to true.

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Linear expressions

@[reducible, inline]

abbrev Linarith.Linexp : Type

A linear expression is a list of pairs of variable indices and coefficients, representing the sum of the products of each coefficient with its corresponding variable.

Some functions on Linexp assume that n : Nat occurs at most once as the first element of a pair, and that the list is sorted in decreasing order of the first argument. This is not enforced by the type but the operations here preserve it.

Equations

• Linarith.Linexp = List (ℕ × ℤ)

partial def Linarith.Linexp.add : Linarith.Linexp → Linarith.Linexp → Linarith.Linexp

Add two Linexps together componentwise. Preserves sorting and uniqueness of the first argument.

def Linarith.Linexp.scale (c : ℤ) (l : Linarith.Linexp) : Linarith.Linexp

l.scale c scales the values in l by c without modifying the order or keys.

Equations

• Linarith.Linexp.scale c l = if c = 0 then [] else if c = 1 then l else List.map (fun (x : ℕ × ℤ) => match x with | (n, z) => (n, z * c)) l

def Linarith.Linexp.get (n : ℕ) : Linarith.Linexp → Option ℤ

l.get n returns the value in l associated with key n, if it exists, and none otherwise. This function assumes that l is sorted in decreasing order of the first argument, that is, it will return none as soon as it finds a key smaller than n.

Equations

• Linarith.Linexp.get n [] = none
• Linarith.Linexp.get n ((a, b) :: t) = if a < n then none else if a = n then some b else Linarith.Linexp.get n t

def Linarith.Linexp.contains (n : ℕ) : Linarith.Linexp → Bool

l.contains n is true iff n is the first element of a pair in l.

Equations

• Linarith.Linexp.contains n = Option.isSome ∘ Linarith.Linexp.get n

def Linarith.Linexp.zfind (n : ℕ) (l : Linarith.Linexp) : ℤ

l.zfind n returns the value associated with key n if there is one, and 0 otherwise.

Equations

• Linarith.Linexp.zfind n l = match Linarith.Linexp.get n l with | none => 0 | some v => v

def Linarith.Linexp.vars (l : Linarith.Linexp) : List ℕ

l.vars returns the list of variables that occur in l.

Equations

• l.vars = List.map Prod.fst l

def Linarith.Linexp.cmp : Linarith.Linexp → Linarith.Linexp → Ordering

Defines a lex ordering on Linexp. This function is performance critical.

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• Linarith.Linexp.cmp [] [] = Ordering.eq
• Linarith.Linexp.cmp [] x✝ = Ordering.lt
• x✝.cmp [] = Ordering.gt

Comparisons with 0

structure Linarith.Comp : Type

The main datatype for FM elimination. Variables are represented by natural numbers, each of which has an integer coefficient. Index 0 is reserved for constants, i.e. coeffs.find 0 is the coefficient of 1. The represented term is coeffs.sum (fun ⟨k, v⟩ ↦ v * Var[k]). str determines the strength of the comparison -- is it < 0, ≤ 0, or = 0?

• str : Mathlib.Ineq

  The strength of the comparison, <, ≤, or =.

• coeffs : Linarith.Linexp

  The coefficients of the comparison, stored as list of pairs (i, a), where i is the index of a recorded atom, and a is the coefficient.

instance Linarith.instInhabitedComp : Inhabited Linarith.Comp

Equations

• Linarith.instInhabitedComp = { default := { str := default, coeffs := default } }

instance Linarith.instReprComp : Repr Linarith.Comp

Equations

• Linarith.instReprComp = { reprPrec := Linarith.reprComp✝ }

def Linarith.Comp.vars : Linarith.Comp → List ℕ

c.vars returns the list of variables that appear in the linear expression contained in c.

Equations

• Linarith.Comp.vars = Linarith.Linexp.vars ∘ Linarith.Comp.coeffs

def Linarith.Comp.coeffOf (c : Linarith.Comp) (a : ℕ) : ℤ

c.coeffOf a projects the coefficient of variable a out of c.

Equations

• c.coeffOf a = Linarith.Linexp.zfind a c.coeffs

def Linarith.Comp.scale (c : Linarith.Comp) (n : ℕ) : Linarith.Comp

c.scale n scales the coefficients of c by n.

Equations

• c.scale n = { str := c.str, coeffs := Linarith.Linexp.scale (↑n) c.coeffs }

def Linarith.Comp.add (c1 c2 : Linarith.Comp) : Linarith.Comp

Comp.add c1 c2 adds the expressions represented by c1 and c2. The coefficient of variable a in c1.add c2 is the sum of the coefficients of a in c1 and c2.

Equations

• c1.add c2 = { str := c1.str.max c2.str, coeffs := c1.coeffs.add c2.coeffs }

def Linarith.Comp.cmp : Linarith.Comp → Linarith.Comp → Ordering

Comp has a lex order. First the ineqs are compared, then the coeffs.

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def Linarith.Comp.isContr (c : Linarith.Comp) : Bool

A Comp represents a contradiction if its expression has no coefficients and its strength is <, that is, it represents the fact 0 < 0.

Equations

• c.isContr = (List.isEmpty c.coeffs && decide (c.str = Mathlib.Ineq.lt))

instance Linarith.Comp.ToFormat : Std.ToFormat Linarith.Comp

Equations

• Linarith.Comp.ToFormat = { format := fun (p : Linarith.Comp) => Std.format p.coeffs ++ Std.Format.text (toString p.str) ++ Std.Format.text "0" }

Parsing into linear form

Control

structure Linarith.PreprocessorBase : Type

Metadata about preprocessors, for trace output.

• name : Lean.Name

  The name of the preprocessor, populated automatically, to create linkable trace messages.

• description : String

  The description of the preprocessor.

structure Linarith.Preprocessorextends Linarith.PreprocessorBase : Type

A preprocessor transforms a proof of a proposition into a proof of a different proposition. The return type is List Expr, since some preprocessing steps may create multiple new hypotheses, and some may remove a hypothesis from the list. A "no-op" preprocessor should return its input as a singleton list.

• name : Lean.Name
• description : String
• transform : Lean.Expr → Lean.MetaM (List Lean.Expr)

  Replace a hypothesis by a list of hypotheses. These expressions are the proof terms.

structure Linarith.GlobalPreprocessorextends Linarith.PreprocessorBase : Type

Some preprocessors need to examine the full list of hypotheses instead of working item by item. As with Preprocessor, the input to a GlobalPreprocessor is replaced by, not added to, its output.

• name : Lean.Name
• description : String
• transform : List Lean.Expr → Lean.MetaM (List Lean.Expr)

  Replace the collection of all hypotheses with new hypotheses. These expressions are proof terms.

def Linarith.Branch : Type

Some preprocessors perform branching case splits. A Branch is used to track one of these case splits. The first component, an MVarId, is the goal corresponding to this branch of the split, given as a metavariable. The List Expr component is the list of hypotheses for linarith in this branch.

Equations

• Linarith.Branch = (Lean.MVarId × List Lean.Expr)

structure Linarith.GlobalBranchingPreprocessorextends Linarith.PreprocessorBase : Type

Some preprocessors perform branching case splits. A GlobalBranchingPreprocessor produces a list of branches to run. Each branch is independent, so hypotheses that appear in multiple branches should be duplicated. The preprocessor is responsible for making sure that each branch contains the correct goal metavariable.

• name : Lean.Name
• description : String
• transform : Lean.MVarId → List Lean.Expr → Lean.MetaM (List Linarith.Branch)

  Given a goal, and a list of hypotheses, produce a list of pairs (consisting of a goal and list of hypotheses).

def Linarith.Preprocessor.globalize (pp : Linarith.Preprocessor) : Linarith.GlobalPreprocessor

A Preprocessor lifts to a GlobalPreprocessor by folding it over the input list.

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def Linarith.GlobalPreprocessor.branching (pp : Linarith.GlobalPreprocessor) : Linarith.GlobalBranchingPreprocessor

A GlobalPreprocessor lifts to a GlobalBranchingPreprocessor by producing only one branch.

Equations

• pp.branching = { toPreprocessorBase := pp.toPreprocessorBase, transform := fun (g : Lean.MVarId) (l : List Lean.Expr) => do let __do_lift ← pp.transform l pure [(g, __do_lift)] }

def Linarith.GlobalBranchingPreprocessor.process (pp : Linarith.GlobalBranchingPreprocessor) (g : Lean.MVarId) (l : List Lean.Expr) : Lean.MetaM (List Linarith.Branch)

process pp l runs pp.transform on l and returns the result, tracing the result if trace.linarith is on.

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instance Linarith.PreprocessorToGlobalBranchingPreprocessor : Coe Linarith.Preprocessor Linarith.GlobalBranchingPreprocessor

Equations

• Linarith.PreprocessorToGlobalBranchingPreprocessor = { coe := Linarith.GlobalPreprocessor.branching ∘ Linarith.Preprocessor.globalize }

instance Linarith.GlobalPreprocessorToGlobalBranchingPreprocessor : Coe Linarith.GlobalPreprocessor Linarith.GlobalBranchingPreprocessor

Equations

• Linarith.GlobalPreprocessorToGlobalBranchingPreprocessor = { coe := Linarith.GlobalPreprocessor.branching }

structure Linarith.CertificateOracle : Type

A CertificateOracle provides a function produceCertificate : List Comp → Nat → MetaM (HashMap Nat Nat).

The default CertificateOracle used by linarith is Linarith.CertificateOracle.simplexAlgorithmSparse. Linarith.CertificateOracle.simplexAlgorithmDense and Linarith.CertificateOracle.fourierMotzkin are also available (though the Fourier-Motzkin oracle has some bugs).

• produceCertificate (hyps : List Linarith.Comp) (max_var : ℕ) : Lean.MetaM (Std.HashMap ℕ ℕ)

  produceCertificate hyps max_var tries to derive a contradiction from the comparisons in hyps by eliminating all variables ≤ max_var. If successful, it returns a map coeff : Nat → Nat as a certificate. This map represents that we can find a contradiction by taking the sum ∑ (coeff i) * hyps[i].

Auxiliary functions

These functions are used by multiple modules, so we put them here for accessibility.

def Linarith.parseCompAndExpr (e : Lean.Expr) : Lean.MetaM (Mathlib.Ineq × Lean.Expr)

parseCompAndExpr e checks if e is of the form t < 0, t ≤ 0, or t = 0. If it is, it returns the comparison along with t.

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def Linarith.mkSingleCompZeroOf (c : ℕ) (h : Lean.Expr) : Lean.MetaM (Mathlib.Ineq × Lean.Expr)

mkSingleCompZeroOf c h assumes that h is a proof of t R 0. It produces a pair (R', h'), where h' is a proof of c*t R' 0. Typically R and R' will be the same, except when c = 0, in which case R' is =. If c = 1, h' is the same as h -- specifically, it does not change the type to 1*t R 0.

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