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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 infrastruct

The infrastruct package provides central prover infrastructure such as unification and augmented formula analysis.

The infrastruct package provides central prover infrastructure such as unification and augmented formula analysis. This infrastructure is of basic interest and useful independently of any particular tactic language etc.

Main functionality:

Position with PosInExpr positioning within subformulas.
Context Convenience representation of formulas used as contexts that provide ways of substituting expressions in by conceptually splitting and managing the context around a position in a formula.
RenUSubst Renaming Uniform Substitutions that quickly combine and ultimately reduce to uniform renaming and uniform substitution of the kernel.
UnificationMatch single-sided matching unification algorithms.
Augmentors: Implicit convenience additions of helper functions to formulas, terms, programs, sequents that are useful for analysis and transformation.
ExpressionTraversal generic functionality for traversing expressions for analysis or transformation.

Definition Classes: keymaerax
See also: edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus

AntePosition

Augmentors

BaseMatcher

Context

DependencyAnalysis

DirectUSubstAboveURen

ExpressionTraversal

FastURenAboveUSubst

FastUSubstAboveURen

FormulaTools

FreshUnificationMatch

InsistentMatcher

LinearMatcher

Matcher

MultiRename

NonSubstUnificationMatch

PosInExpr

Position

ProvableHelper

RenUSubst

RenUSubstBase

RestrictedBiDiUnificationMatch

SchematicComposedUnificationMatch

SchematicUnificationMatch

StaticSemanticsTools

Statistics

SubstitutionHelper

SuccPosition

TopAntePosition

TopPosition

TopSuccPosition

TreeForm

URenSubstitutionPair

USubstAboveURen

USubstRenChurch

USubstRenOne

UnificationMatch

UnificationMatchBase

UnificationMatchUSubstAboveURen

UnificationTools

UniformMatcher

UniformMatching

edu.cmu.cs.ls.keymaerax.infrastruct

SubstitutionHelper

Companion class SubstitutionHelper

object SubstitutionHelper

Created by smitsch on 2/19/15.

To do: generalize to replacing formula by formula, too.

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def replaceFn(fn: Function, fml: Formula, subst: Map[Term, Variable]): Formula
Replaces any function application fn(...) in fml per subst.
def replaceFree[T <: Expression](expr: T)(what: Term, repl: Term): T
Return the result of replacing all free occurrences of what in expression expr by repl.
def replaceFree(prg: Program)(what: Term, repl: Term): Program
Return the result of replacing all free occurrences of what in program prg by repl.
def replaceFree(seq: Sequent)(what: Term, repl: Term): Sequent
Return the result of replacing all free occurrences of what in sequent seq by repl.
def replaceFree(t: Term)(what: Term, repl: Term): Term
Return the result of replacing all free occurrences of what in formula f by repl.
def replaceFree(f: Formula)(what: Term, repl: Term): Formula
Return the result of replacing all free occurrences of what in term t by repl.
def replacesFree[T <: Expression](expr: T)(replaces: (Term) ⇒ Option[Term]): T
Return the result of replacing all free occurrences of what in expression expr by repl whenever replaces(what) = Some(repl).
def replacesFree(prg: Program)(replaces: (Term) ⇒ Option[Term]): Program
Return the result of replacing all free occurrences of what in program prg by repl whenever replaces(what) = Some(repl).
def replacesFree(seq: Sequent)(replaces: (Term) ⇒ Option[Term]): Sequent
Return the result of replacing all free occurrences of what in sequent seq by repl whenever replaces(what) = Some(repl).
def replacesFree(t: Term)(replaces: (Term) ⇒ Option[Term]): Term
Return the result of replacing all free occurrences of what in formula f by repl whenever replaces(what) = Some(repl).
def replacesFree(f: Formula)(replaces: (Term) ⇒ Option[Term]): Formula
Return the result of replacing all free occurrences of what in term t by repl whenever replaces(what) = Some(repl).
final def synchronized[T0](arg0: ⇒ T0): T0

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Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

SubstitutionHelper

Companion class SubstitutionHelper

object SubstitutionHelper

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KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

SubstitutionHelper 

Companion class SubstitutionHelper

object SubstitutionHelper

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SubstitutionHelper