`BookProof.ChapterGravityPolymomentum.hamiltonian_momentum_eq_velocity` (he : e ≠ 0) (Scal E : Matrix (Fin 4) (Fin 4) ℝ) (P trE R : ℝ) (hS : Scal = (2 * e) • S) (hP : P = -4 * e *

Proved

BookProof.ChapterGravityPolymomentum.hamiltonian_momentum_eq_velocity

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGravityPolymomentum.

BookProof.ChapterGravityPolymomentum.hamiltonian_momentum_eq_velocity (he : e ≠ 0) (Scal E : Matrix (Fin 4) (Fin 4) ℝ) (P trE R : ℝ) (hS : Scal = (2 * e) • S) (hP : P = -4 * e * T) : (1 / (16 * e)) * contract Scal Scal - (1 / (24 * e)) * P ^ 2 + (1 / 2) * contract Scal E + (1 / 3) * P * trE - e * R = e * ((1 / 4) * contract S S - (2 / 3) * T ^ 2 + contract S E - (4 / 3) * T * trE - R)

Formalization note: Lean 4 identifier BookProof.ChapterGravityPolymomentum.hamiltonian_momentum_eq_velocity.

Preamble

-- Generated from ChapterGravityPolymomentum.lean — theorem BookProof.ChapterGravityPolymomentum.hamiltonian_momentum_eq_velocity
import Definitions.Def_ChapterGravityProjector
import Mathlib
import Definitions.Def_ChapterGravityPolymomentum
open BookProof.ChapterGravityPolymomentum



open Matrix
open scoped BigOperators
open BookProof.ChapterGravityProjector

variable {e T : ℝ} {S Tc : Matrix (Fin 4) (Fin 4) ℝ} {u v : Fin 4 → ℝ}

Formal statement

theorem BookProof.ChapterGravityPolymomentum.hamiltonian_momentum_eq_velocity (he : e ≠ 0) (Scal E : Matrix (Fin 4) (Fin 4) ℝ)
    (P trE R : ℝ) (hS : Scal = (2 * e) • S) (hP : P = -4 * e * T) :
    (1 / (16 * e)) * contract Scal Scal - (1 / (24 * e)) * P ^ 2
        + (1 / 2) * contract Scal E + (1 / 3) * P * trE - e * R
      = e * ((1 / 4) * contract S S - (2 / 3) * T ^ 2 + contract S E
        - (4 / 3) * T * trE - R) := by sorry

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