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.
A shorthand for tracing the types of a list of proof terms
when the trace.linarith
option is set to true.
Equations
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Linear expressions #
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.
Add two Linexp
s together componentwise.
Preserves sorting and uniqueness of the first argument.
l.scale c
scales the values in l
by c
without modifying the order or keys.
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
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
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
l.vars
returns the list of variables that occur in l
.
Equations
- Linarith.Linexp.vars l = List.map Prod.fst l
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
- Linarith.Linexp.cmp x [] = Ordering.gt
Inequalities #
- eq: Linarith.Ineq
- le: Linarith.Ineq
- lt: Linarith.Ineq
The three-element type Ineq
is used to represent the strength of a comparison between
terms.
Instances For
Equations
- Linarith.instDecidableEqIneq x y = if h : Linarith.Ineq.toCtorIdx x = Linarith.Ineq.toCtorIdx y then isTrue (_ : x = y) else isFalse (_ : x = y → False)
Equations
- Linarith.instInhabitedIneq = { default := Linarith.Ineq.eq }
Equations
- Linarith.instReprIneq = { reprPrec := Linarith.reprIneq✝ }
max R1 R2
computes the strength of the sum of two inequalities. If t1 R1 0
and t2 R2 0
,
then t1 + t2 (max R1 R2) 0
.
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Ineq
is ordered eq < le < lt
.
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Prints an Ineq
as the corresponding infix symbol.
Equations
- Linarith.Ineq.toString x = match x with | Linarith.Ineq.eq => "=" | Linarith.Ineq.le => "≤" | Linarith.Ineq.lt => "<"
Finds the name of a multiplicative lemma corresponding to an inequality strength.
Equations
- Linarith.Ineq.toConstMulName x = match x with | Linarith.Ineq.lt => `Linarith.mul_neg | Linarith.Ineq.le => `Linarith.mul_nonpos | Linarith.Ineq.eq => `Linarith.mul_eq
Equations
- Linarith.Ineq.instToStringIneq = { toString := Linarith.Ineq.toString }
Equations
- Linarith.Ineq.instToFormatIneq = { format := fun i => Std.Format.text (Linarith.Ineq.toString i) }
Comparisons with 0 #
The strength of the comparison,
<
,≤≤
, or=
.str : Linarith.IneqThe coefficients of the comparison, stored as list of pairs
(i, a)
, wherei
is the index of a recorded atom, anda
is the coefficient.coeffs : Linarith.Linexp
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 (λ ⟨k, v⟩, v * Var[k])
.
str determines the strength of the comparison -- is it < 0, ≤ 0, or = 0?
Instances For
Equations
- Linarith.instInhabitedComp = { default := { str := default, coeffs := default } }
Equations
- Linarith.instReprComp = { reprPrec := Linarith.reprComp✝ }
c.vars
returns the list of variables that appear in the linear expression contained in c
.
c.coeffOf a
projects the coefficient of variable a
out of c
.
Equations
- Linarith.Comp.coeffOf c a = Linarith.Linexp.zfind a c.coeffs
c.scale n
scales the coefficients of c
by n
.
Equations
- Linarith.Comp.scale c n = { str := c.str, coeffs := Linarith.Linexp.scale (↑n) c.coeffs }
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
- Linarith.Comp.add c1 c2 = { str := Linarith.Ineq.max c1.str c2.str, coeffs := Linarith.Linexp.add c1.coeffs c2.coeffs }
Comp
has a lex order. First the ineq
s are compared, then the coeff
s.
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A Comp
represents a contradiction if its expression has no coefficients and its strength is <,
that is, it represents the fact 0 < 0
.
Equations
- Linarith.Comp.isContr c = (List.isEmpty c.coeffs && decide (c.str = Linarith.Ineq.lt))
Equations
- Linarith.Comp.ToFormat = { format := fun p => Std.format p.coeffs ++ Std.Format.text (toString p.str) ++ Std.Format.text "0" }
Parsing into linear form #
Control #
The name of the preprocessor, used in trace output.
name : StringReplace a hypothesis by a list of hypotheses. These expressions are the proof terms.
transform : Lean.Expr → Lean.MetaM (List Lean.Expr)
A preprocessor transforms a proof of a proposition into a proof of a different propositon.
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.
Instances For
The name of the global preprocessor, used in trace output.
name : StringReplace the collection of all hypotheses with new hypotheses. These expressions are proof terms.
transform : List Lean.Expr → Lean.MetaM (List Lean.Expr)
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.
Instances For
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
The name of the global branching preprocessor, used in trace output.
name : StringGiven a goal, and a list of hypotheses, produce a list of pairs (consisting of a goal and list of hypotheses).
transform : Lean.MVarId → List Lean.Expr → Lean.MetaM (List Linarith.Branch)
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.
Instances For
A Preprocessor
lifts to a GlobalPreprocessor
by folding it over the input list.
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A GlobalPreprocessor
lifts to a GlobalBranchingPreprocessor
by producing only one branch.
Equations
- Linarith.GlobalPreprocessor.branching pp = { name := pp.name, transform := fun g l => do let __do_lift ← Linarith.GlobalPreprocessor.transform pp l pure [(g, __do_lift)] }
process pp l
runs pp.transform
on l
and returns the result,
tracing the result if trace.linarith
is on.
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A CertificateOracle
is a function
produceCertificate : List Comp → Nat → MetaM (HashMap Nat Nat)→ Nat → MetaM (HashMap Nat Nat)→ MetaM (HashMap Nat Nat)
.
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→ Nat
as a certificate.
This map represents that we can find a contradiction by taking the sum ∑ (coeff i) * hyps[i]∑ (coeff i) * hyps[i]
.
The default CertificateOracle
used by linarith
is
Linarith.FourierMotzkin.produceCertificate
.
Equations
Discharger to prove that a candidate linear combination of hypothesis is zero.
discharger : Lean.Elab.Tactic.TacticM UnitProve goals which are not linear comparisons by first calling
exfalso
.exfalso : BoolTransparency mode for identifying atomic expressions in comparisons.
transparency : Lean.Meta.TransparencyModeSplit conjunctions in hypotheses.
split_hypotheses : BoolSplit
≠≠
in hypotheses, by branching in cases<
and>
.split_ne : BoolOverride the list of preprocessors.
preprocessors : Option (List Linarith.GlobalBranchingPreprocessor)Specify an oracle for identifying candidate contradictions. The only implementation here is Fourier-Motzkin elimination.
oracle : Option Linarith.CertificateOracle
A configuration object for linarith
.
Instances For
cfg.updateReducibility reduce_default
will change the transparency setting of cfg
to
default
if reduce_default
is true. In this case, it also sets the discharger to ring!
,
since this is typically needed when using stronger unification.
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Auxiliary functions #
These functions are used by multiple modules, so we put them here for accessibility.
getRelSides e
returns the left and right hand sides of e
if e
is a comparison,
and fails otherwise.
This function is more naturally in the Option
monad, but it is convenient to put in MetaM
for compositionality.
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parseCompAndExpr e
checks if e
is of the form t < 0
, t ≤ 0≤ 0
, or t = 0
.
If it is, it returns the comparison along with t
.
Equations
- One or more equations did not get rendered due to their size.
helper function for error message
Equations
- Linarith.parseCompAndExpr.throwNotZero z = Lean.throwError (Lean.toMessageData "invalid comparison, rhs not zero: " ++ Lean.toMessageData z ++ Lean.toMessageData "")
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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