Packages

•  package root

  KeYmaera X is a theorem prover for differential dynamic logic (dL), a logic for specifying and verifying properties of hybrid systems with mixed discrete and continuous dynamics.

  KeYmaera X: An aXiomatic Tactical Theorem Prover

  KeYmaera X is a theorem prover for differential dynamic logic (dL), a logic for specifying and verifying properties of hybrid systems with mixed discrete and continuous dynamics. Reasoning about complicated hybrid systems requires support for sophisticated proof techniques, efficient computation, and a user interface that crystallizes salient properties of the system. KeYmaera X allows users to specify custom proof search techniques as tactics, execute tactics in parallel, and interface with partial proofs via an extensible user interface.

  http://keymaeraX.org/

  Concrete syntax for input language Differential Dynamic Logic

  Package Structure

  Main documentation entry points for KeYmaera X API:

  • edu.cmu.cs.ls.keymaerax.core - KeYmaera X kernel, proof certificates, main data structures
    • Expression - Differential dynamic logic expressions: Term, Formula, Program
    • Sequent - Sequents of formulas
    • Provable - Proof certificates transformed by rules/axioms
    • Rule - Proof rules as well as USubstOne for (one-pass) uniform substitutions and renaming.
    • StaticSemantics - Static semantics with free and bound variable analysis
    • Instead of constructing dL expressions from constructors, it is easier to use the KeYmaeraXParser.
  • edu.cmu.cs.ls.keymaerax.parser - Parser and pretty printer with concrete syntax and notation for differential dynamic logic.
    • Concrete syntax for input language Differential Dynamic Logic
    • KeYmaeraXPrettyPrinter - Pretty printer producing concrete KeYmaera X syntax
    • KeYmaeraXParser - Parser reading concrete KeYmaera X syntax
    • KeYmaeraXArchiveParser - Parser reading KeYmaera X model and proof archive .kyx files
    • DLParser - Combinator parser reading concrete KeYmaera X syntax
    • DLArchiveParser - Combinator parser reading KeYmaera X model and proof archive .kyx files
  • edu.cmu.cs.ls.keymaerax.infrastruct - Prover infrastructure outside the kernel
    • UnificationMatch - Unification algorithm
    • RenUSubst - Renaming Uniform Substitution quickly combining kernel's renaming and substitution.
    • Context - Representation for contexts of formulas in which they occur.
    • Augmentors - Augmenting formula and expression data structures with additional functionality
    • ExpressionTraversal - Generic traversal functionality for expressions
  • edu.cmu.cs.ls.keymaerax.bellerophon - Bellerophon tactic language and tactic interpreter
    • BelleExpr - Tactic language expressions
    • SequentialInterpreter - Sequential tactic interpreter for Bellerophon tactics
  • edu.cmu.cs.ls.keymaerax.btactics - Bellerophon tactic library for conducting proofs.
    • TactixLibrary - Main KeYmaera X tactic library including many proof tactics.
    • HilbertCalculus - Hilbert Calculus for differential dynamic logic
    • SequentCalculus - Sequent Calculus for propositional and first-order logic
    • HybridProgramCalculus - Hybrid Program Calculus for differential dynamic logic
    • DifferentialEquationCalculus - Differential Equation Calculus for differential dynamic logic
    • UnifyUSCalculus - Unification-based uniform substitution calculus underlying the other calculi
    • [edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.ForwardTactic ForwardTactic] - Forward tactic framework for conducting proofs from premises to conclusions
  • edu.cmu.cs.ls.keymaerax.lemma - Lemma mechanism
    • Lemma - Lemmas are Provables stored under a name, e.g., in files.
    • LemmaDB - Lemma database stored in files or database etc.
  • edu.cmu.cs.ls.keymaerax.tools.qe - Real arithmetic back-end solvers
    • MathematicaQETool - Mathematica interface for real arithmetic.
    • Z3QETool - Z3 interface for real arithmetic.
  • edu.cmu.cs.ls.keymaerax.tools.ext - Extended back-ends for noncritical ODE solving, counterexamples, algebra, simplifiers, etc.
    • Mathematica - Mathematica interface for ODE solving, algebra, simplification, invariant generation, etc.
    • Z3 - Z3 interface for real arithmetic including simplifiers.

  Entry Points

  Additional entry points and usage points for KeYmaera X API:

  • edu.cmu.cs.ls.keymaerax.launcher.KeYmaeraX - Command-line launcher for KeYmaera X supports command-line argument -help to obtain usage information
  • edu.cmu.cs.ls.keymaerax.btactics.AxIndex - Axiom indexing data structures with keys and recursors for canonical proof strategies.
  • edu.cmu.cs.ls.keymaerax.btactics.DerivationInfo - Meta-information on all derivation steps (axioms, derived axioms, proof rules, tactics) with user-interface info.
  • edu.cmu.cs.ls.keymaerax.bellerophon.UIIndex - Index determining which canonical reasoning steps to display on the KeYmaera X User Interface.
  • edu.cmu.cs.ls.keymaerax.btactics.Ax - Registry for derived axioms and axiomatic proof rules that are proved from the core.

  References

  Full references on KeYmaera X are provided at http://keymaeraX.org/. The main references are the following:

  1. André Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219-265, 2017.

  2. Nathan Fulton, Stefan Mitsch, Jan-David Quesel, Marcus Völp and André Platzer. KeYmaera X: An axiomatic tactical theorem prover for hybrid systems. In Amy P. Felty and Aart Middeldorp, editors, International Conference on Automated Deduction, CADE'15, Berlin, Germany, Proceedings, volume 9195 of LNCS, pp. 527-538. Springer, 2015.

  3. André Platzer. Logical Foundations of Cyber-Physical Systems. Springer, 2018. Videos

  Definition Classes

  root

•  package edu

  Definition Classes

  root

•  package cmu

  Definition Classes

  edu

•  package cs

  Definition Classes

  cmu

•  package ls

  Definition Classes

  cs

•  package keymaerax

  Definition Classes

  ls

•  package btactics

  Tactic library in the Bellerophon tactic language.

  Tactic library in the Bellerophon tactic language.

  • edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary Main tactic library consisting of:
    • HilbertCalculus Hilbert Calculus for differential dynamic logic
    • SequentCalculus Sequent Calculus for propositional and first-order logic
    • HybridProgramCalculus Hybrid Program Calculus for differential dynamic logic
    • DifferentialEquationCalculus Differential Equation Calculus for differential dynamic logic
    • UnifyUSCalculus Unification-based Uniform Substitution Calculus
  • Tactic tools
    • AxIndex]: Axiom Indexing data structures for canonical proof strategies.
    • DerivationInfo: Meta-information for derivation steps such as axioms for user interface etc.

  All tactics are implemented in the Bellerophon tactic language, including its dependent tactics, which ultimately produce edu.cmu.cs.ls.keymaerax.core.Provable proof certificates by the Bellerophon interpreter. The Provables that tactics produce can be extracted, for example, with edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary.proveBy().

  Proof Styles

  KeYmaera X supports many different proof styles, including flexible combinations of the following styles:

  1. Explicit proof certificates directly program the proof rules from the core.

  2. Explicit proofs use tactics to describe a proof directly mentioning all or most proof steps.

  3. Proof by search relies mainly on proof search with occasional additional guidance.

  4. Proof by pointing points out facts and where to use them.

  5. Proof by congruence is based on equivalence or equality or implicational rewriting within a context.

  6. Proof by chase is based on chasing away operators at an indicated position.

  Explicit Proof Certificates

  The most explicit types of proofs can be constructed directly using the edu.cmu.cs.ls.keymaerax.core.Provable certificates in KeYmaera X's kernel without using any tactics. Also see edu.cmu.cs.ls.keymaerax.core.

  import edu.cmu.cs.ls.keymaerax.core._
  // explicit proof certificate construction of |- !!p() <-> p()
  val proof = (Provable.startProof(
    "!!p() <-> p()".asFormula)
    (EquivRight(SuccPos(0)), 0)
    // right branch
      (NotRight(SuccPos(0)), 1)
      (NotLeft(AntePos(1)), 1)
      (Close(AntePos(0),SuccPos(0)), 1)
    // left branch
      (NotLeft(AntePos(0)), 0)
      (NotRight(SuccPos(1)), 0)
      (Close(AntePos(0),SuccPos(0)), 0)
  )

  Explicit Proofs

  Explicit proofs construct similarly explicit proof steps, just with explicit tactics from TactixLibrary: The only actual difference is the order of the branches, which is fixed to be from left to right in tactic branching. Tactics also use more elegant signed integers to refer to antecedent positions (negative) or succedent positions (positive).

  import TactixLibrary._
  // Explicit proof tactic for |- !!p() <-> p()
  val proof = TactixLibrary.proveBy("!!p() <-> p()".asFormula,
     equivR(1) & <(
       (notL(-1) &
         notR(2) &
         closeId)
       ,
       (notR(1) &
         notL(-2) &
         closeId)
       )
   )

  Proof by Search

  Proof by search primarily relies on proof search procedures to conduct a proof. That gives very short proofs but, of course, they are not always entirely informative about how the proof worked. It is easiest to see in simple situations where propositional logic proof search will find a proof but works well in more general situations, too.

  import TactixLibrary._
  // Proof by search of |- (p() & q()) & r() <-> p() & (q() & r())
  val proof = TactixLibrary.proveBy("(p() & q()) & r() <-> p() & (q() & r())".asFormula,
     prop
  )

  Common tactics for proof by search include edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary.prop(), edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary.auto() and the like.

  Proof by Pointing

  Proof by pointing emphasizes the facts to use and is implicit about the details on how to use them exactly. Proof by pointing works by pointing to a position in the sequent and using a given fact at that position. For example, for proving

  ⟨v:=2*v+1;⟩v!=0 <-> 2*v+1!=0

  it is enough to point to the highlighted position using the Ax.diamond axiom fact ![a;]!p(||) <-> ⟨a;⟩p(||) at the highlighted position to reduce the proof to a proof of

  ![v:=2*v+1;]!(v!=0) <-> 2*v+1!=0

  which is, in turn, easy to prove by pointing to the highlighted position using the Ax.assignbAxiom axiom [x:=t();]p(x) <-> p(t()) at the highlighted position to reduce the proof to

  !!(2*v+1!=0) <-> 2*v+1!=0

  Finally, using double negation !!p() <-> p() at the highlighted position yields the following

  2*v+1!=0 <-> 2*v+1!=0

  which easily proves by reflexivity p() <-> p().

  Proof by pointing matches the highlighted position against the highlighted position in the fact and does what it takes to use that knowledge. There are multiple variations of proof by pointing in edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.useAt and edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.byUS, which are also imported into edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary. The above proof by pointing implements directly in KeYmaera X:

  import TactixLibrary._
  // Proof by pointing of  |- <v:=2*v+1;>v!=0 <-> 2*v+1!=0
  val proof = TactixLibrary.proveBy("<v:=2*v+1;>q(v) <-> q(2*v+1)".asFormula,
    // use Ax.diamond axiom backwards at the indicated position on
    // |- __<v:=2*v+1;>q(v)__ <-> q(2*v+1)
    useExpansionAt(Ax.diamond)(1, 0::Nil) &
    // use Ax.assignbAxiom axiom forward at the indicated position on
    // |- !__[v:=2*v+1;]!q(v)__ <-> q(2*v+1)
    useAt(Ax.assignbAxiom(1, 0::0::Nil) &
    // use double negation at the indicated position on
    // |- __!!q(2*v+1)__ <-> q(2*v+1)
    useAt(Ax.doubleNegation)(1, 0::Nil) &
    // close by (an instance of) reflexivity |- p() <-> p()
    // |- q(2*v+1) <-> q(2*v+1)
    byUS(Ax.equivReflexive)
  )

  Another example is:

  import TactixLibrary._
  // Proof by pointing of  |- <a;++b;>p(x) <-> (<a;>p(x) | <b;>p(x))
  val proof = TactixLibrary.proveBy("<a;++b;>p(x) <-> (<a;>p(x) | <b;>p(x))".asFormula,
    // use Ax.diamond axiom backwards at the indicated position on
    // |- __<a;++b;>p(x)__  <->  <a;>p(x) | <b;>p(x)
    useExpansionAt(Ax.diamond)(1, 0::Nil) &
    // use Ax.choiceb axiom forward at the indicated position on
    // |- !__[a;++b;]!p(x)__  <->  <a;>p(x) | <b;>p(x)
    useAt(Ax.choiceb)(1, 0::0::Nil) &
    // use Ax.diamond axiom forward at the indicated position on
    // |- !([a;]!p(x) & [b;]!p(x))  <->  __<a;>p(x)__ | <b;>p(x)
    useExpansionAt(Ax.diamond)(1, 1::0::Nil) &
    // use Ax.diamond axiom forward at the indicated position on
    // |- !([a;]!p(x) & [b;]!p(x))  <->  ![a;]!p(x) | __<b;>p(x)__
    useExpansionAt(Ax.diamond)(1, 1::1::Nil) &
    // use propositional logic to show
    // |- !([a;]!p(x) & [b;]!p(x))  <->  ![a;]!p(x) | ![b;]!p(x)
    prop
  )

  edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary.stepAt also uses proof by pointing but figures out the appropriate fact to use on its own. Here's a similar proof:

  import TactixLibrary._
  // Proof by pointing with steps of  |- ⟨a++b⟩p(x) <-> (⟨a⟩p(x) | ⟨b⟩p(x))
  val proof = TactixLibrary.proveBy("p(x) <-> (p(x) | p(x))".asFormula,
    // use Ax.diamond axiom backwards at the indicated position on
    // |- __⟨a++b⟩p(x)__  <->  ⟨a⟩p(x) | ⟨b⟩p(x)
    useExpansionAt(Ax.diamond)(1, 0::Nil) &
    // |- !__[a;++b;]!p(x)__  <->  ⟨a⟩p(x) | ⟨b⟩p(x)
    // step Ax.choiceb axiom forward at the indicated position
    stepAt(1, 0::0::Nil) &
    // |- __!([a;]!p(x) & [b;]!p(x))__  <-> ⟨a⟩p(x) | ⟨b⟩p(x)
    // step deMorgan forward at the indicated position
    stepAt(1, 0::Nil) &
    // |- __![a;]!p(x)__ | ![b;]!p(x)  <-> ⟨a⟩p(x) | ⟨b⟩p(x)
    // step Ax.diamond forward at the indicated position
    stepAt(1, 0::0::Nil) &
    // |- ⟨a⟩p(x) | __![b;]!p(x)__  <-> ⟨a⟩p(x) | ⟨b⟩p(x)
    // step Ax.diamond forward at the indicated position
    stepAt(1, 0::1::Nil) &
    // |- ⟨a⟩p(x) | ⟨b⟩p(x)  <-> ⟨a⟩p(x) | ⟨b⟩p(x)
    byUS(Ax.equivReflexive)
  )

  Likewise, for proving

  x>5 |- !([x:=x+1; ++ x:=0;]x>=6) | x<2

  it is enough to point to the highlighted position

  x>5 |- !([x:=x+1; ++ x:=0;]x>=6) | x<2

  and using the Ax.choiceb axiom fact [a;++b;]p(||) <-> [a;]p(||) & [b;]p(||) to reduce the proof to a proof of

  x>5 |- !([x:=x+1;]x>6 & [x:=0;]x>=6) | x<2

  which is, in turn, easy to prove by pointing to the assignments using Ax.assignbAxiom axioms and ultimately asking propositional logic.

  More proofs by pointing are shown in edu.cmu.cs.ls.keymaerax.btactics.Ax source code.

  Proof by Congruence

  Proof by congruence is based on equivalence or equality or implicational rewriting within a context. This proof style can make quite quick inferences leading to significant progress using the CE, CQ, CT congruence proof rules or combinations thereof.

  import TactixLibrary._
  // |- x*(x+1)>=0 -> [y:=0;x:=__x^2+x__;]x>=y
  val proof = TactixLibrary.proveBy("x*(x+1)>=0 -> [y:=0;x:=x^2+x;]x>=y".asFormula,
    CEat(proveBy("x*(x+1)=x^2+x".asFormula, QE)) (1, 1::0::1::1::Nil) &
    // |- x*(x+1)>=0 -> [y:=0;x:=__x*(x+1)__;]x>=y by CE/CQ using x*(x+1)=x^2+x at the indicated position
    // step uses top-level operator [;]
    stepAt(1, 1::Nil) &
    // |- x*(x+1)>=0 -> [y:=0;][x:=x*(x+1);]x>=y
    // step uses top-level operator [:=]
    stepAt(1, 1::Nil) &
    // |- x*(x+1)>=0 -> [x:=x*(x+1);]x>=0
    // step uses top-level [:=]
    stepAt(1, 1::Nil) &
    // |- x*(x+1)>=0 -> x*(x+1)>=0
    prop
  )

  Proof by congruence can also make use of a fact in multiple places at once by defining an appropriate context C:

  import TactixLibrary._
  val C = Context("x<5 & ⎵ -> [{x' = 5*x & ⎵}](⎵ & x>=1)".asFormula)
  // |- x<5 & __x^2<4__ -> [{x' = 5*x & __x^2<4__}](__x^2<4__ & x>=1)
  val proof = TactixLibrary.proveBy("x<5 & x^2<4 -> [{x' = 5*x & x^2<4}](x^2<4 & x>=1)".asFormula,
    CEat(proveBy("-2x^2<4".asFormula, QE), C) (1))
  )
  // |- x<5 & (__-2 [{x' = 5*x & __-2=1) by CE
  println(proof.subgoals)

  Proof by Chase

  Proof by chase chases the expression at the indicated position forward until it is chased away or can't be chased further without critical choices. The canonical examples use edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.chase() to chase away differential forms:

  import TactixLibrary._
  val proof = TactixLibrary.proveBy("[{x'=22}](2*x+x*y>=5)'".asFormula,
   // chase the differential prime away in the postcondition
   chase(1, 1 :: Nil)
   // |- [{x'=22}]2*x'+(x'*y+x*y')>=0
  )
  // Remaining subgoals: |- [{x'=22}]2*x'+(x'*y+x*y')>=0
  println(proof.subgoals)

  import TactixLibrary._
  val proof = TactixLibrary.proveBy("[{x'=22}](2*x+x*y>=5)' <-> [{x'=22}]2*x'+(x'*y+x*y')>=0".asFormula,
    // chase the differential prime away in the left postcondition
    chase(1, 0:: 1 :: Nil) &
    // |- [{x'=22}]2*x'+(x'*y+x*y')>=0 <-> [{x'=22}]2*x'+(x'*y+x*y')>=0
    byUS(Ax.equivReflexive)
  )

  Yet edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.chase() is also useful to chase away other operators, say, modalities:

  import TactixLibrary._
  // proof by chase of |- [?x>0;x:=x+1;x:=2*x; ++ ?x=0;x:=1;]x>=1
  val proof = TactixLibrary.proveBy(
    "[?x>0;x:=x+1;x:=2*x; ++ ?x=0;x:=1;]x>=1".asFormula,
    // chase the box in the succedent away
    chase(1,Nil) &
    // |- (x>0->2*(x+1)>=1)&(x=0->1>=1)
    QE
  )

  Additional Mechanisms

  • Tactic framework for Hilbert-style Forward Tactics: Tactics are functions Provable=>Provable
    • edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.ForwardTactic: Forward Hilbert-style tactics Provable=>Provable
    • edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.ForwardPositionTactic: Positional forward Hilbert-style tactics Position=>(Provable=>Provable)
    • edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus: Forward Hilbert-style tactic combinators.

  Definition Classes

  keymaerax

  To do

  Expand descriptions

  See also

  Andre Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219-266, 2017.

  edu.cmu.cs.ls.keymaerax.btactics.TactixLibrary

  edu.cmu.cs.ls.keymaerax.btactics.HilbertCalculus

  edu.cmu.cs.ls.keymaerax.btactics.SequentCalculus

  edu.cmu.cs.ls.keymaerax.btactics.HybridProgramCalculus

  edu.cmu.cs.ls.keymaerax.btactics.DifferentialEquationCalculus

  edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus

•  package arithmetic

  Definition Classes

  btactics

•  package cexsearch

  Definition Classes

  btactics

•  package coasterx

  Definition Classes

  btactics

•  package components

  Definition Classes

  btactics

•  package dRL

  Definition Classes

  btactics

•  package helpers

  Helper functions for implementing tactics.

  Helper functions for implementing tactics. Note that these are **not** themselves tactics.

  Definition Classes

  btactics

  See also

  helpers.DifferentialHelper Differential Equation-specific program helpers.

  helpers.ProgramHelpers Hybrid Program helpers that are not specific to Differential Equations.

•  package macros

  Definition Classes

  btactics

•  package robustify

  Definition Classes

  btactics

• AnonymousLemmas
• Approximator
• ArithmeticLibrary
• ArithmeticSimplification
• Ax
• AxIndex
• AxiomaticODESolver
• BelleLabels
• BelleREPL
• Bifurcation
• Case
• ConfigurableGenerator
• DebuggingTactics
• DefaultTacticIndex
• DerivationInfoRegistry
• Derive
• DifferentialDecisionProcedures
• DifferentialEquationCalculus
• DifferentialSaturation
• FOQuantifierTactics
• FixedGenerator
• Generator
• HilbertCalculus
• HybridProgramCalculus
• Idioms
• ImplicitAx
• Integrator
• IntegratorODESolverTool
• IntervalArithmeticV2
• InvGenTool
• InvariantGenerator
• InvariantProvers
• IsabelleSyntax
• Kaisar
• MathematicaToolProvider
• ModelPlex
• ModelPlexConjecture
• ModelPlexTrait
• MonomialOrders
• MultiToolProvider
• NanoTimer
• NoTimer
• NoneToolProvider
• ODEInvariance
• ODELiveness
• ODEStability
• PolynomialArith
• PolynomialArithV2
• PolynomialArithV2Helpers
• PolynomialRing
• PreferredToolProvider
• RicattiEquation
• RicattiSystem
• SOSSolve
• SequentCalculus
• Simplifier
• SimplifierV2
• SimplifierV3
• SwitchedSystems
• TacticFactory
• TacticHelper
• TacticIndex
• TactixInit
• TactixLibrary
• TaylorModelArith
• TaylorModelOptions
• TaylorModelTactics
• TimeStepOptions
• Timer
• ToolProvider
• Transitivity
• TwoThreeTreePolynomialRing
• UnifyUSCalculus
• WolframEngineToolProvider
• WolframScriptToolProvider
• WolframToolProvider
• Z3ToolProvider

edu.cmu.cs.ls.keymaerax.btactics

PolynomialArithV2 