object AxiomaticODESolver
An Axiomatic ODE solver. Current limitations: - No support for explicit-form diamond ODEs/box ODEs in context: <{x'=0*x+1}>P, ![{x'=0*x+1}]P
- See also
Page 25 in http://arxiv.org/abs/1503.01981 for a high-level sketch.
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val
addTimeVar: DependentPositionTactic
Rewrites [{c}]p to [{c, t'=1}]p.
Rewrites [{c}]p to [{c, t'=1}]p. The positional argument should point to the location of c, NOT the location of the box. This tactic should work at any top-level position and also in any context.
- Note
If we want an initial value for time (kyxtime:=0) then this is the place to add that functionality.
- def alist(ode: DifferentialProgram): Option[List[AtomicODE]]
- def apply(): DependentPositionTactic
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asInstanceOf[T0]: T0
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- @native() @throws( ... )
- def commAxiomInst(dom: Formula, post: Formula, ode1: DifferentialProgram, ode2: DifferentialProgram): Provable
- def commSubst(dom: Formula, post: Formula, ode1: DifferentialProgram, ode2: DifferentialProgram): USubst
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def
cutAndProveFml(cut: Formula, contextSize: Int = 0): DependentPositionTactic
Augment ODE with formula
cut
, consider context of sizecontextSize
when proving with DI. - def cutInSoln(odeSize: Int, diffArg: Term = Variable("kyxtime")): DependentPositionTactic
- def dfs(ode: DifferentialProgram): Option[List[Variable]]
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def
isSolved(v: Variable, system: ODESystem): Boolean
- v
A variable occuring in the odes program.
- system
An ode system.
- returns
true if the program does not already contain an = constraint (a.k.a. sol'n) for v in the evolution domain.
- def makeCanonical(ode: DifferentialProgram, ord: List[Variable], dom: Formula, post: Formula, pos: Position): BelleExpr
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- def odeSolverPreconds(ord: List[Variable]): DependentPositionTactic
- def ofAtoms(atoms: List[AtomicODE]): DifferentialProgram
- def selectionSort(dom: Formula, post: Formula, ode: DifferentialProgram, goal: List[Variable], pos: Position): BelleExpr
- def simplifyPostCondition(odeSize: Int): DependentPositionTactic
- def sortAxiomInst(dom: Formula, post: Formula, context: DifferentialProgram, ode1: DifferentialProgram, ode2: DifferentialProgram): Provable
- def sortSubst(dom: Formula, post: Formula, context: DifferentialProgram, ode1: DifferentialProgram, ode2: DifferentialProgram): USubst
- def splitODEAt(ode: DifferentialProgram, v: Variable): (List[AtomicODE], List[AtomicODE], List[AtomicODE])
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synchronized[T0](arg0: ⇒ T0): T0
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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 (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