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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.

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:

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 speculative

Definition Classes: arithmetic

ArithmeticSpeculativeSimplification

edu.cmu.cs.ls.keymaerax.btactics.arithmetic.speculative

ArithmeticSpeculativeSimplification

object ArithmeticSpeculativeSimplification

Tactics for simplifying arithmetic sub-goals.

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Definition Classes
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def autoMonotonicityTransform: BelleExpr
Auto-transforms formulas from upper/lower bounds to concrete value (e.g., t<=ep, transform d>=A*ep to d>=A*t.
def clone(): AnyRef

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protected[java.lang]
Definition Classes
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@native() @throws( ... )
final def eq(arg0: AnyRef): Boolean

Definition Classes
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def equals(arg0: Any): Boolean

Definition Classes
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def exhaustiveAbsSplit: BelleExpr
Splits absolute value functions to create more, but hopefully simpler, goals.
def finalize(): Unit

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final def getClass(): Class[_]

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@native()
def hashCode(): Int

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@native()
def hideInconsistentSigns: BelleExpr
def hideNonmatchingBounds: BelleExpr
Hides formulas with non-matching bounds.
final def isInstanceOf[T0]: Boolean

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final def ne(arg0: AnyRef): Boolean

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final def notify(): Unit

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@native()
def orIntro(finish: BelleExpr): BelleExpr
Uses the disjunction introduction proof rule to prove a disjunctions by proving any 1 of the disjuncts.
lazy val proveOrRefuteAbs: BelleExpr
Proves abs by trying to find contradictions; falls back to QE if contradictions fail
lazy val speculativeQE: BelleExpr
Tries decreasingly aggressive strategies of hiding formulas before QE, until finally falling back to full QE if none of the simplifications work out.
Tries decreasingly aggressive strategies of hiding formulas before QE, until finally falling back to full QE if none of the simplifications work out.

Annotations
@Tactic( x$1 , x$2 , x$6 , x$3 , x$4 , x$7 , x$8 , x$5 , x$9 , x$10 , x$11 , x$12 )
final def synchronized[T0](arg0: ⇒ T0): T0

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Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

Proof Styles

Explicit Proof Certificates

Explicit Proofs

Proof by Search

Proof by Pointing

Proof by Congruence

Proof by Chase

Additional Mechanisms

ArithmeticSpeculativeSimplification

object ArithmeticSpeculativeSimplification

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Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

Proof Styles

Explicit Proof Certificates

Explicit Proofs

Proof by Search

Proof by Pointing

Proof by Congruence

Proof by Chase

Additional Mechanisms

ArithmeticSpeculativeSimplification 

object ArithmeticSpeculativeSimplification

Value Members

Inherited from AnyRef

Inherited from Any

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ArithmeticSpeculativeSimplification