`BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.kinetic_inner` (v : L.D) : (inner ℂ (v : F) (L.kinetic v : F) : ℂ) = (((1 / 2 : ℝ) * ∑ i : Fin 3, ‖(L.P i v : F)‖ ^ 2 :

Proved

BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.kinetic_inner

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNavierStokesLagrangianEsa.

BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.kinetic_inner (v : L.D) : (inner ℂ (v : F) (L.kinetic v : F) : ℂ) = (((1 / 2 : ℝ) * ∑ i : Fin 3, ‖(L.P i v : F)‖ ^ 2 : ℝ) : ℂ)

Formalization note: Lean 4 identifier BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.kinetic_inner.

Preamble

-- Generated from ChapterNavierStokesLagrangianEsa.lean — theorem BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.kinetic_inner
import Mathlib
import Definitions.Def_ChapterNavierStokesLagrangianEsa
import Definitions.Def_ChapterF7
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
open BookProof.ChapterF7
open BookProof.NavierStokesFlow
open BookProof.NavierStokesFlow.LagrangianEsa




open FullEsa

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable (L : LagrangianFullData F)

Formal statement

theorem BookProof.NavierStokesFlow.LagrangianEsa.LagrangianFullData.kinetic_inner (v : L.D) :
    (inner ℂ (v : F) (L.kinetic v : F) : ℂ)
      = (((1 / 2 : ℝ) * ∑ i : Fin 3, ‖(L.P i v : F)‖ ^ 2 : ℝ) : ℂ) := by sorry

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