case class Minus(left: Term, right: Term) extends RBinaryCompositeTerm with Product with Serializable
 binary subtraction of reals.
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def
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final
def
##(): Int
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final
def
==(arg0: Any): Boolean
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final
def
asInstanceOf[T0]: T0
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def
clone(): AnyRef
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def
eq(arg0: AnyRef): Boolean
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finalize(): scala.Unit
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final
def
getClass(): Class[_]
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final
def
isInstanceOf[T0]: Boolean
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final
val
kind: Kind
What kind of an expression this is, e.g., TermKind, FormulaKind, ProgramKind.
What kind of an expression this is, e.g., TermKind, FormulaKind, ProgramKind.
 Definition Classes
 Term → Expression

val
left: Term
The left child of this binary composite expression
The left child of this binary composite expression
 Definition Classes
 Minus → RBinaryCompositeTerm → BinaryCompositeTerm → BinaryComposite

final
def
ne(arg0: AnyRef): Boolean
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final
def
notify(): scala.Unit
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final
def
notifyAll(): scala.Unit
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def
prettyString: String
Prettyprinted string representing this expression
Prettyprinted string representing this expression
 Definition Classes
 Expression

def
reapply: (Term, Term) ⇒ Minus
Create a term of this constructor but with the give left and right arguments instead.
Create a term of this constructor but with the give left and right arguments instead. (copy)
 Definition Classes
 Minus → BinaryCompositeTerm
Times(Number(7), Variable("v")).reapply(Variable("a"), Number(99)) == Times(Variable("a"), Number(99))
 Note
Convenience method not used in the soundnesscritical core but simplifies homogeneous data processing.
Example: 
val
right: Term
The right child of this binary composite expression
The right child of this binary composite expression
 Definition Classes
 Minus → RBinaryCompositeTerm → BinaryCompositeTerm → BinaryComposite

final
val
sort: Sort
The sort of this expression, e.g., Real, Bool.
 Definition Classes
 RTerm → Expression

final
def
synchronized[T0](arg0: ⇒ T0): T0
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def
toString(): String
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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 structuresExpression
 Differential dynamic logic expressions:Term
,Formula
,Program
Sequent
 Sequents of formulasProvable
 Proof certificates transformed by rules/axiomsRule
 Proof rules as well asUSubstOne
for (onepass) uniform substitutions and renaming.StaticSemantics
 Static semantics with free and bound variable analysisKeYmaeraXParser
.edu.cmu.cs.ls.keymaerax.parser
 Parser and pretty printer with concrete syntax and notation for differential dynamic logic.KeYmaeraXPrettyPrinter
 Pretty printer producing concrete KeYmaera X syntaxKeYmaeraXParser
 Parser reading concrete KeYmaera X syntaxKeYmaeraXArchiveParser
 Parser reading KeYmaera X model and proof archive.kyx
filesDLParser
 Combinator parser reading concrete KeYmaera X syntaxDLArchiveParser
 Combinator parser reading KeYmaera X model and proof archive.kyx
filesedu.cmu.cs.ls.keymaerax.infrastruct
 Prover infrastructure outside the kernelUnificationMatch
 Unification algorithmRenUSubst
 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 functionalityExpressionTraversal
 Generic traversal functionality for expressionsedu.cmu.cs.ls.keymaerax.bellerophon
 Bellerophon tactic language and tactic interpreterBelleExpr
 Tactic language expressionsSequentialInterpreter
 Sequential tactic interpreter for Bellerophon tacticsedu.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 logicSequentCalculus
 Sequent Calculus for propositional and firstorder logicHybridProgramCalculus
 Hybrid Program Calculus for differential dynamic logicDifferentialEquationCalculus
 Differential Equation Calculus for differential dynamic logicUnifyUSCalculus
 Unificationbased 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 conclusionsedu.cmu.cs.ls.keymaerax.lemma
 Lemma mechanismLemma
 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 backend solversMathematicaQETool
 Mathematica interface for real arithmetic.Z3QETool
 Z3 interface for real arithmetic.edu.cmu.cs.ls.keymaerax.tools.ext
 Extended backends 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
 Commandline launcher for KeYmaera X supports commandline argumenthelp
to obtain usage informationedu.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
 Metainformation on all derivation steps (axioms, derived axioms, proof rules, tactics) with userinterface 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. 219265, 2017.
2. Nathan Fulton, Stefan Mitsch, JanDavid 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. 527538. Springer, 2015.
3. André Platzer. Logical Foundations of CyberPhysical Systems. Springer, 2018. Videos