object ArithmeticSimplification
Tactics for simplifying arithmetic sub-goals.
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def
hideFactsAbout(xs: List[Variable]): InputTactic
Version of hideFactsAbout that hides all formulas mentioning free variable TODO: would be nice to allow multiple variable hide on the UI somehow?
Version of hideFactsAbout that hides all formulas mentioning free variable TODO: would be nice to allow multiple variable hide on the UI somehow?
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the variable to hide
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- @Tactic( x$13 , x$17 , x$18 , x$14 , x$15 , x$19 , x$20 , x$16 , x$21 , x$22 , x$23 , x$24 )
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def
hideFactsAbout(irrelevant: String*): BelleExpr
Simplifies arithmetic by removing all formulas (both antecedent and succedent) that mention any of the irrelevant names.
Simplifies arithmetic by removing all formulas (both antecedent and succedent) that mention any of the irrelevant names.
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Same as smartHide except does both the succedent and the antecedent, and assumes that a list of irrelevant names is already available.
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def
irrelevantIndices(fmls: IndexedSeq[Formula], baseline: IndexedSeq[Formula]): Seq[Int]
Computes indices that 'seem' irrelevant for proving the formulas in baseline.
Computes indices that 'seem' irrelevant for proving the formulas in baseline. The result is sorted in descending order (optimized for subsequent hiding).
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keepFactsAbout(xs: List[Variable]): InputTactic
Version of hideFactsAbout that hides all formulas mentioning free variable TODO: would be nice to allow multiple variable hide on the UI somehow?
Version of hideFactsAbout that hides all formulas mentioning free variable TODO: would be nice to allow multiple variable hide on the UI somehow?
- xs
the variable to hide
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- @Tactic( x$25 , x$29 , x$30 , x$26 , x$27 , x$31 , x$32 , x$28 , x$33 , x$34 , x$35 , x$36 )
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- lazy val smartCoHideAt: DependentPositionTactic
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smartHide: BuiltInTactic
Simplifies arithmetic by removing formulas that are **probably** irrelevant to the current sub-goal.
Simplifies arithmetic by removing formulas that are **probably** irrelevant to the current sub-goal. Does not necessarily retain validity???
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- @Tactic( x$1 , x$5 , x$6 , x$2 , x$3 , x$7 , x$8 , x$4 , x$9 , x$10 , x$11 , x$12 )
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lazy val
smartSuccHide: DependentTactic
Simplifies arithmetic by removing formulas that are **probably** unprovable from the current facts.
Simplifies arithmetic by removing formulas that are **probably** unprovable from the current facts. Does not necessarily retain validity???
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def
transformEquality(equality: Formula): DependentPositionWithAppliedInputTactic
Transforms the formula at position by replacing all free occurrences of equality.left with equality.right
Transforms the formula at position by replacing all free occurrences of equality.left with equality.right
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- @Tactic( x$37 , x$41 , x$42 , x$38 , x$39 , x$43 , x$44 , x$40 , x$45 , x$46 , x$47 , x$48 )
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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