Documentation

Mathlib.Testing.SlimCheck.Testable

Testable Class #

Testable propositions have a procedure that can generate counter-examples together with a proof that they invalidate the proposition.

This is a port of the Haskell QuickCheck library.

Creating Customized Instances #

The type classes Testable, SampleableExt and Shrinkable are the means by which SlimCheck creates samples and tests them. For instance, the proposition ∀ i j : ℕ, i ≤ j has a Testable instance because is sampleable and i ≤ j is decidable. Once SlimCheck finds the Testable instance, it can start using the instance to repeatedly creating samples and checking whether they satisfy the property. Once it has found a counter-example it will then use a Shrinkable instance to reduce the example. This allows the user to create new instances and apply SlimCheck to new situations.

What do I do if I'm testing a property about my newly defined type? #

Let us consider a type made for a new formalization:

structure MyType where
  x : ℕ
  y : ℕ
  h : x ≤ y
  deriving Repr

How do we test a property about MyType? For instance, let us consider Testable.check <| ∀ a b : MyType, a.y ≤ b.x → a.x ≤ b.y. Writing this property as is will give us an error because we do not have an instance of Shrinkable MyType and SampleableExt MyType. We can define one as follows:

instance : Shrinkable MyType where
  shrink := fun ⟨x,y,h⟩ ↦
    let proxy := Shrinkable.shrink (x, y - x)
    proxy.map (fun ⟨⟨fst, snd⟩, ha⟩ ↦ ⟨⟨fst, fst + snd, sorry⟩, sorry⟩)

instance : SampleableExt MyType :=
  SampleableExt.mkSelfContained do
    let x ← SampleableExt.interpSample Nat
    let xyDiff ← SampleableExt.interpSample Nat
    pure <| ⟨x, x + xyDiff, sorry⟩

Again, we take advantage of the fact that other types have useful Shrinkable implementations, in this case Prod. Note that the second proof is heavily based on WellFoundedRelation since it's used for termination so the first step you want to take is almost always to simp_wf in order to get through the WellFoundedRelation.

Main definitions #

Tags #

random testing

References #

inductive SlimCheck.TestResult (p : Prop) :

Result of trying to disprove p The constructors are:

  • success : (Unit ⊕' p) → TestResult p succeed when we find another example satisfying p In success h, h is an optional proof of the proposition. Without the proof, all we know is that we found one example where p holds. With a proof, the one test was sufficient to prove that p holds and we do not need to keep finding examples.
  • gaveUp : ℕ → TestResult p give up when a well-formed example cannot be generated. gaveUp n tells us that n invalid examples were tried. Above 100, we give up on the proposition and report that we did not find a way to properly test it.
  • failure : ¬ p → (List String) → ℕ → TestResult p a counter-example to p; the strings specify values for the relevant variables. failure h vs n also carries a proof that p does not hold. This way, we can guarantee that there will be no false positive. The last component, n, is the number of times that the counter-example was shrunk.
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    Configuration for testing a property.

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      Allow elaboration of Configuration arguments to tactics.

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        PrintableProp p allows one to print a proposition so that SlimCheck can indicate how values relate to each other. It's basically a poor man's delaborator.

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          • SlimCheck.instPrintableProp = { printProp := "⋯" }

          Testable p uses random examples to try to disprove p.

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                • SlimCheck.TestResult.instToString = { toString := SlimCheck.TestResult.toString }
                def SlimCheck.TestResult.combine {p : Prop} {q : Prop} :
                Unit ⊕' (pq)Unit ⊕' pUnit ⊕' q

                Applicative combinator proof carrying test results.

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                  Combine the test result for properties p and q to create a test for their conjunction.

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                    Combine the test result for properties p and q to create a test for their disjunction.

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                      def SlimCheck.TestResult.imp {p : Prop} {q : Prop} (h : qp✝) (r : SlimCheck.TestResult p✝) (p : optParam (Unit ⊕' (p✝q)) (PSum.inl ())) :

                      If q → p, then ¬ p → ¬ q which means that testing p can allow us to find counter-examples to q.

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                        Test q by testing p and proving the equivalence between the two.

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                          def SlimCheck.TestResult.addInfo {p : Prop} {q : Prop} (x : String) (h : qp✝) (r : SlimCheck.TestResult p✝) (p : optParam (Unit ⊕' (p✝q)) (PSum.inl ())) :

                          When we assign a value to a universally quantified variable, we record that value using this function so that our counter-examples can be informative.

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                            def SlimCheck.TestResult.addVarInfo {p : Prop} {q : Prop} {γ : Type u_1} [Repr γ] (var : String) (x : γ) (h : qp✝) (r : SlimCheck.TestResult p✝) (p : optParam (Unit ⊕' (p✝q)) (PSum.inl ())) :

                            Add some formatting to the information recorded by addInfo.

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                                A configuration with all the trace options enabled, useful for debugging.

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                                  A dbgTrace with special formatting

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                                    instance SlimCheck.Testable.decGuardTestable {p : Prop} {var : String} [SlimCheck.PrintableProp p] [Decidable p] {β : pProp} [(h : p) → SlimCheck.Testable (β h)] :
                                    SlimCheck.Testable (SlimCheck.NamedBinder var (∀ (h : p), β h))
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                                    Format the counter-examples found in a test failure.

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                                      Increase the number of shrinking steps in a test result.

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                                        instance SlimCheck.Testable.instInhabitedOptionTOfPure {α : Type u} {m : Type u → Type u_1} [Pure m] :
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                                        • SlimCheck.Testable.instInhabitedOptionTOfPure = { default := pure none }

                                        Shrink a counter-example x by using Shrinkable.shrink x, picking the first candidate that falsifies a property and recursively shrinking that one. The process is guaranteed to terminate because shrink x produces a proof that all the values it produces are smaller (according to SizeOf) than x.

                                        Once a property fails to hold on an example, look for smaller counter-examples to show the user.

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                                          instance SlimCheck.Testable.varTestable {var : String} {α : Sort u_1} [SlimCheck.SampleableExt α] {β : αProp} [(x : α) → SlimCheck.Testable (β x)] :
                                          SlimCheck.Testable (SlimCheck.NamedBinder var (∀ (x : α), β x))

                                          Test a universal property by creating a sample of the right type and instantiating the bound variable with it.

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                                          instance SlimCheck.Testable.propVarTestable {var : String} {β : PropProp} [(b : Bool) → SlimCheck.Testable (β (b = true))] :

                                          Test a universal property about propositions

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                                          instance SlimCheck.Testable.subtypeVarTestable {var : String} {α : Sort u_1} {p : αProp} {β : αProp} [(x : α) → SlimCheck.PrintableProp (p x)] [(x : α) → SlimCheck.Testable (β x)] [SlimCheck.SampleableExt (Subtype p)] {var' : String} :
                                          SlimCheck.Testable (SlimCheck.NamedBinder var (∀ (x : α), SlimCheck.NamedBinder var' (p xβ x)))
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                                          instance SlimCheck.Eq.printableProp {α : Type u_1} [Repr α] {x : α} {y : α} :
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                                          instance SlimCheck.Ne.printableProp {α : Type u_1} [Repr α] {x : α} {y : α} :
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                                          instance SlimCheck.LE.printableProp {α : Type u_1} [Repr α] [LE α] {x : α} {y : α} :
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                                          instance SlimCheck.LT.printableProp {α : Type u_1} [Repr α] [LT α] {x : α} {y : α} :
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                                          • SlimCheck.Bool.printableProp = { printProp := if b = true then "true" else "false" }
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                                          Execute cmd and repeat every time the result is gave_up (at most n times).

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                                            Count the number of times the test procedure gave up.

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                                              Try n times to find a counter-example for p.

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                                                def SlimCheck.Testable.runSuite (p : Prop) [SlimCheck.Testable p] (cfg : optParam SlimCheck.Configuration { numInst := 100, maxSize := 100, numRetries := 10, traceDiscarded := false, traceSuccesses := false, traceShrink := false, traceShrinkCandidates := false, randomSeed := none, quiet := false }) :

                                                Try to find a counter-example of p.

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                                                  def SlimCheck.Testable.checkIO (p : Prop) [SlimCheck.Testable p] (cfg : optParam SlimCheck.Configuration { numInst := 100, maxSize := 100, numRetries := 10, traceDiscarded := false, traceSuccesses := false, traceShrink := false, traceShrinkCandidates := false, randomSeed := none, quiet := false }) :

                                                  Run a test suite for p in BaseIO using the global RNG in stdGenRef.

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                                                    Traverse the syntax of a proposition to find universal quantifiers quantifiers and add NamedBinder annotations next to them.

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                                                    DecorationsOf p is used as a hint to mk_decorations to specify that the goal should be satisfied with a proposition equivalent to p with added annotations.

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                                                      In a goal of the shape DecorationsOf p, mk_decoration examines the syntax of p and adds NamedBinder around universal quantifications to improve error messages. This tool can be used in the declaration of a function as follows:

                                                      def foo (p : Prop) (p' : Decorations.DecorationsOf p := by mk_decorations) [Testable p'] : ...
                                                      

                                                      p is the parameter given by the user, p' is a definitionally equivalent proposition where the quantifiers are annotated with NamedBinder.

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                                                        def SlimCheck.Testable.check (p : Prop) (cfg : optParam SlimCheck.Configuration { numInst := 100, maxSize := 100, numRetries := 10, traceDiscarded := false, traceSuccesses := false, traceShrink := false, traceShrinkCandidates := false, randomSeed := none, quiet := false }) (p' : autoParam (SlimCheck.Decorations.DecorationsOf p) _auto✝) [SlimCheck.Testable p'] :

                                                        Run a test suite for p and throw an exception if p does not hold.

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