trait BaseMatcher extends Matcher with Logging
Basic matcher infrastructure that simply forwards all unification functionality to BaseMatcher.unify(). Only apply() functions wrap exceptions in context while unify() functions simply pass it upwards.
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- BaseMatcher
- Logging
- LazyLogging
- LoggerHolder
- Matcher
- Function2
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Type Members
-
type
Subst = RenUSubst
The (generalized) substitutions used for unification purposes
-
type
SubstRepl = (Expression, Expression)
A (generalized) substitution pair, which is either like a SubstitutionPair for uniform substitution or a pair of Variable for uniform renaming.
A (generalized) substitution pair, which is either like a SubstitutionPair for uniform substitution or a pair of Variable for uniform renaming.
- Definition Classes
- Matcher
- See also
SubstitutionPair
Abstract Value Members
-
abstract
def
unify(shape: Sequent, input: Sequent): List[SubstRepl]
unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Attributes
- protected
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
abstract
def
unify(shape: Program, input: Program): List[SubstRepl]
unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Attributes
- protected
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
abstract
def
unify(shape: Formula, input: Formula): List[SubstRepl]
unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Attributes
- protected
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
abstract
def
unify(shape: Term, input: Term): List[SubstRepl]
unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Attributes
- protected
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
abstract
def
unifyODE(shape: DifferentialProgram, input: DifferentialProgram): List[SubstRepl]
unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.unify(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Attributes
- protected
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
Concrete Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
Subst(subs: List[SubstRepl]): Subst
Create a (generalized) substitution from the given representation
subs
.Create a (generalized) substitution from the given representation
subs
.- Attributes
- protected
- Definition Classes
- Matcher
-
def
SubstRepl(what: Expression, repl: Expression): SubstRepl
Create a (generalized) substitution pair.
Create a (generalized) substitution pair.
- Attributes
- protected
- Definition Classes
- Matcher
- See also
SubstitutionPair
-
def
apply(e1: Sequent, e2: Sequent): Subst
apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Definition Classes
- BaseMatcher → Matcher
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
def
apply(e1: DifferentialProgram, e2: DifferentialProgram): Subst
apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Definition Classes
- BaseMatcher → Matcher
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
def
apply(e1: Program, e2: Program): Subst
apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Definition Classes
- BaseMatcher → Matcher
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
def
apply(e1: Formula, e2: Formula): Subst
apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Definition Classes
- BaseMatcher → Matcher
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
def
apply(e1: Term, e2: Term): Subst
apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Definition Classes
- BaseMatcher → Matcher
- Exceptions thrown
UnificationException
ifinput
cannot be matched against the patternshape
.
-
def
apply(e1: Expression, e2: Expression): Subst
apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.apply(shape, input) matches
input
against the patternshape
to find a uniform substitution\result
such that\result(shape)==input
.- Definition Classes
- BaseMatcher → Matcher → Function2
-
final
def
asInstanceOf[T0]: T0
- Definition Classes
- Any
-
def
clone(): AnyRef
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
-
def
curried: (Expression) ⇒ (Expression) ⇒ RenUSubst
- Definition Classes
- Function2
- Annotations
- @unspecialized()
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
equals(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
finalize(): Unit
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( classOf[java.lang.Throwable] )
-
final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
def
hashCode(): Int
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
val
id: List[SubstRepl]
Identity substitution
{}
that does not change anything.Identity substitution
{}
that does not change anything.- Attributes
- protected
- Definition Classes
- Matcher
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
def
join(s: List[SubstRepl], t: List[SubstRepl]): List[SubstRepl]
Union of renaming substitution representations:
join(s, t)
gives the representation ofs
performed together witht
, if compatible.Union of renaming substitution representations:
join(s, t)
gives the representation ofs
performed together witht
, if compatible.s \cup t = {p(.)~>F | (p(.)~>F) \in s} ++ {(p(.)~>F) | (p(.)~>F) \in t} if s and t are compatible, so do not map the same p(.) or f(.) or a to different replacements
- returns
a substitution that has the same effect as applying both substitutions
s
andt
simultaneously. Similar to returnings++t
but skipping duplicates and checking compatibility in passing.
- Attributes
- protected
-
lazy val
logger: Logger
- Attributes
- protected
- Definition Classes
- LazyLogging → LoggerHolder
-
final
val
loggerName: String
- Attributes
- protected
- Definition Classes
- LoggerHolder
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
final
def
notify(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
final
def
notifyAll(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toString(): String
- Definition Classes
- Function2 → AnyRef → Any
-
def
tupled: ((Expression, Expression)) ⇒ RenUSubst
- Definition Classes
- Function2
- Annotations
- @unspecialized()
-
def
unifiable(shape: Sequent, input: Sequent): Option[Subst]
unifiable(shape, input) Compute some unifier matching
input
against the patternshape
if unifiable else Noneunifiable(shape, input) Compute some unifier matching
input
against the patternshape
if unifiable else None- Definition Classes
- BaseMatcher → Matcher
-
def
unifiable(shape: Expression, input: Expression): Option[Subst]
unifiable(shape, input) Compute some unifier matching
input
against the patternshape
if unifiable else Noneunifiable(shape, input) Compute some unifier matching
input
against the patternshape
if unifiable else None- Definition Classes
- BaseMatcher → Matcher
-
def
unifier(e1: Sequent, e2: Sequent, us: List[SubstRepl]): Subst
Create the unifier
us
(which was formed for e1 and e2, just for the record).Create the unifier
us
(which was formed for e1 and e2, just for the record).- Attributes
- protected
-
def
unifier(e1: Expression, e2: Expression, us: List[SubstRepl]): Subst
Create the unifier
us
(which was formed for e1 and e2, just for the record).Create the unifier
us
(which was formed for e1 and e2, just for the record).- Attributes
- protected
-
def
unifier(shape: Expression, input: Expression): List[SubstRepl]
Construct the base unifier that forces
shape
andinput
to unify unless equal alreadyConstruct the base unifier that forces
shape
andinput
to unify unless equal already- returns
{shape~>input} unless shape=input in which case it's {}.
- Attributes
- protected
-
def
unify(shape: Expression, input: Expression): List[SubstRepl]
- Attributes
- protected
-
def
ununifiable(shape: Sequent, input: Sequent): Nothing
Indicates that
input
cannot be matched against the patternshape
by raising a UnificationException.Indicates that
input
cannot be matched against the patternshape
by raising a UnificationException.- Attributes
- protected
- Exceptions thrown
UnificationException
always.
-
def
ununifiable(shape: Expression, input: Expression): Nothing
Indicates that
input
cannot be matched against the patternshape
by raising a UnificationException.Indicates that
input
cannot be matched against the patternshape
by raising a UnificationException.- Attributes
- protected
- Exceptions thrown
UnificationException
always.
-
final
def
wait(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
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 (one-pass) 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 first-order logicHybridProgramCalculus
- Hybrid Program Calculus for differential dynamic logicDifferentialEquationCalculus
- Differential Equation Calculus for differential dynamic logicUnifyUSCalculus
- 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 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 back-end solversMathematicaQETool
- 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 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
- 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