Documentation

LeanCert.Tactic.IntervalAuto.Multivariate

Multivariate Bound Tactic #

The multivariate_bound tactic proves bounds on multivariate expressions.

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def LeanCert.Tactic.Auto.multivariateBoundCore (maxIters : ℕ) (tolerance : ℚ) (useMonotonicity : Bool) (taylorDepth : ℕ) :
Lean.Elab.Tactic.TacticM Unit

The main multivariate_bound tactic implementation

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    def LeanCert.Tactic.Auto.multivariateBoundCore.extractRatBound (bound : Lean.Expr) :
    Lean.Elab.Tactic.TacticM Lean.Expr

    Extract rational bound from possible coercion (reusing logic from intervalBoundCore)

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      def LeanCert.Tactic.Auto.multivariateBoundCore.getVarExprs (vars : Array VarIntervalInfo) :
      Lean.Elab.Tactic.TacticM (Array Lean.Expr)

      Fetch local variable expressions in the order of VarIntervalInfo.

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        def LeanCert.Tactic.Auto.multivariateBoundCore.mkEnvExpr (varsListExpr : Lean.Expr) :
        Lean.Elab.Tactic.TacticM Lean.Expr

        Build an environment function ρ from a list of variables.

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          def LeanCert.Tactic.Auto.multivariateBoundCore.proveMultivariateLe (goal : Lean.MVarId) (vars : Array VarIntervalInfo) (func bound : Lean.Expr) (maxIters : ℕ) (tolerance : ℚ) (useMonotonicity : Bool) (taylorDepth : ℕ) :
          Lean.Elab.Tactic.TacticM Unit

          Prove ∀ x₁ ∈ I₁, ..., ∀ xₙ ∈ Iₙ, f(x) ≤ c using verify_global_upper_bound

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            def LeanCert.Tactic.Auto.multivariateBoundCore.proveMultivariateGe (goal : Lean.MVarId) (vars : Array VarIntervalInfo) (func bound : Lean.Expr) (maxIters : ℕ) (tolerance : ℚ) (useMonotonicity : Bool) (taylorDepth : ℕ) :
            Lean.Elab.Tactic.TacticM Unit

            Prove ∀ x₁ ∈ I₁, ..., ∀ xₙ ∈ Iₙ, c ≤ f(x) using verify_global_lower_bound

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              def LeanCert.Tactic.Auto.tacticMultivariate_bound_ :
              Lean.ParserDescr

              The multivariate_bound tactic.

              Automatically proves bounds on multivariate expressions using global optimization.

              Usage:

              • multivariate_bound - uses defaults (1000 iterations, tolerance 1/1000, Taylor depth 10)
              • multivariate_bound 2000 - uses 2000 iterations

              Supports goals of the form:

              • ∀ x ∈ I, ∀ y ∈ J, f(x,y) ≤ c
              • ∀ x ∈ I, ∀ y ∈ J, c ≤ f(x,y)
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