sealed trait Stack[+A] extends AnyRef
Stack with top on the right. For example the stack Bottom :+ a3 :+ a2 +: a1 has element a1 on the top, then a2 as the top of the tail.
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- Stack
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Abstract Value Members
-
abstract
def
drop(n: Int): Stack[A]
Select all elements except the top n elements of this stack, or empty if there are not that many.
-
abstract
def
isEmpty: Boolean
Whether this stack is empty
-
abstract
def
tail: Stack[A]
Tail of this stack, i.e.
Tail of this stack, i.e. all but top element, or error if empty.
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abstract
def
take(n: Int): Stack[A]
Select only the top n elements of this stack, or less if there are not that many.
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abstract
def
top: A
Top element of this stack or error if empty.
Concrete Value Members
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final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
++[B >: A](pushStack: Stack[B]): Stack[B]
S++T result of pushing the whole stack T as is on top of the stack S.
-
def
:+[B >: A](push: B): Stack[B]
S:+b result of pushing b on top of the stack S.
-
final
def
==(arg0: Any): Boolean
- Definition Classes
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-
final
def
asInstanceOf[T0]: T0
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-
def
clone(): AnyRef
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- @native() @throws( ... )
-
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]
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- @throws( classOf[java.lang.Throwable] )
-
def
find(p: (A) ⇒ Boolean): Option[A]
Find the top-most element satisfying p, if any
-
def
fold[B](z: B)(f: (B, A) ⇒ B): B
Fold the elements of this stack by f starting with z at the Bottom.
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final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
def
hashCode(): Int
- Definition Classes
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- @native()
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
def
length: Int
Number of elements on this stack
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
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-
final
def
notify(): Unit
- Definition Classes
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-
final
def
notifyAll(): Unit
- Definition Classes
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- @native()
-
def
split[B >: A](p: (B) ⇒ Boolean): (Stack[B], Stack[B])
Splits the stack at first occurrence of an element next s.t.
Splits the stack at first occurrence of an element next s.t. p(next) = true; p is placed in the later stack.
(A B C D E).split(C) = ( (A,B), (C, D, E) )
Example: -
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toList: List[A]
Convert stack A,B,C where top = A to the list A :: B :: C :: Nil
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def
toString(): String
- Definition Classes
- Stack → AnyRef → Any
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final
def
wait(): Unit
- Definition Classes
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- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
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- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): Unit
- Definition Classes
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- 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