PoissonBracket.lean`

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The canonical Poisson bracket on a polynomial algebra in $2n$ variables

For a commutative ring R and a finite type σ, we equip MvPolynomial (σ ⊕ σ) R with its canonical Poisson bracket $$\{f, g\} = \sum_{i} \left(\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial y_i} - \frac{\partial f}{\partial y_i}\frac{\partial g}{\partial x_i}\right),$$ where $x_i$ (resp. $y_i$) denotes the variable indexed by Sum.inl i (resp. Sum.inr i). This is the polynomial algebra underlying the n-th canonical Poisson algebra $P_n(R)$ when σ = Fin n.

Classical sources index the same object by $2n$ variables $X_1, \dots, X_{2n}$ with bracket ${f, g} = \sum_{i=1}^n (\partial f/\partial X_i \cdot \partial g/\partial X_{i+n}

\partial f/\partial X_{i+n} \cdot \partial g/\partial X_i)$; here Sum.inl i plays the role of $X_i$ and Sum.inr i the role of $X_{i+n}$ (see e.g. AvdE07, Notations 1).

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def MvPolynomial.poissonBracket Documented declaration: poissonBracket line 49 source
Source documentation
The canonical Poisson bracket on MvPolynomial (σ ⊕ σ) R, given by ${f, g} = \sum_{i} (\partial_{x_i} f \cdot \partial_{y_i} g
- \partial_{y_i} f \cdot \partial_{x_i} g)$ where $x_i$ (resp. $y_i$) is the variable indexed by Sum.inl i (resp. Sum.inr i).
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, MvPolynomial, Sum
Proof uses: Finset.sum, Finsupp, Nat, Finset.univ, HSub.hSub, HMul.hMul, Derivation, MvPolynomial.pderiv, Sum.inl, Sum.inr
29 instances, coercions and auxiliary constants left out.
```
noncomputable def poissonBracket (f g : MvPolynomial (σ ⊕ σ) R) : MvPolynomial (σ ⊕ σ) R :=
…
```
theorem MvPolynomial.poissonBracket_self line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, MvPolynomial, Sum, Eq, MvPolynomial.poissonBracket, OfNat.ofNat, Finsupp, Nat
Proof uses: Finset.sum_eq_zero, Finset.univ, HSub.hSub, HMul.hMul, Derivation, MvPolynomial.pderiv, Sum.inl, Sum.inr, Membership.mem, Finset, Eq.mpr, id, congrArg, mul_comm, sub_self, Eq.refl
49 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem MvPolynomial.poissonBracket_swap line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, MvPolynomial, Sum, Eq, MvPolynomial.poissonBracket, Neg.neg, Finsupp, Nat
Proof uses: Eq.mpr, Finset.sum, Finset.univ, HSub.hSub, HMul.hMul, Derivation, MvPolynomial.pderiv, Sum.inl, Sum.inr, id, congrArg, Eq.symm, Finset.sum_neg_distrib, Finset.sum_congr, rfl, Finset, Membership.mem, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, HPow.hPow and 36 more
64 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem MvPolynomial.poissonBracket_X_inl_inl line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, Eq, MvPolynomial, Sum, MvPolynomial.poissonBracket, MvPolynomial.X, Sum.inl, OfNat.ofNat, Finsupp, Nat
Proof uses: Finset.sum_eq_zero, Finset.univ, HSub.hSub, HMul.hMul, Derivation, MvPolynomial.pderiv, Sum.inr, Membership.mem, Finset, Eq.mpr, id, congrArg, MvPolynomial.pderiv_X_of_ne, Ne, of_eq_true, Not, Eq.trans, False, True, eq_false' and 8 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem MvPolynomial.poissonBracket_X_inr_inr line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, Eq, MvPolynomial, Sum, MvPolynomial.poissonBracket, MvPolynomial.X, Sum.inr, OfNat.ofNat, Finsupp, Nat
Proof uses: Finset.sum_eq_zero, Finset.univ, HSub.hSub, HMul.hMul, Derivation, MvPolynomial.pderiv, Sum.inl, Membership.mem, Finset, Eq.mpr, id, congrArg, MvPolynomial.pderiv_X_of_ne, Ne, of_eq_true, Not, Eq.trans, False, True, eq_false' and 8 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem MvPolynomial.poissonBracket_X_inl_inr line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, DecidableEq, Eq, MvPolynomial, Sum, MvPolynomial.poissonBracket, MvPolynomial.X, Sum.inl, Sum.inr, ite, OfNat.ofNat, Finsupp, Nat
Proof uses: Eq.mpr, Finset.sum, Finset.univ, HSub.hSub, HMul.hMul, Derivation, MvPolynomial.pderiv, id, congrArg, Finset.sum_eq_single, Membership.mem, Finset, Ne, MvPolynomial.pderiv_X_of_ne, Not, Sum.inl.injEq, Ne.symm, of_eq_true, Eq.trans, False and 27 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
theorem MvPolynomial.poissonBracket_X_inr_inl line 96 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, DecidableEq, Eq, MvPolynomial, Sum, MvPolynomial.poissonBracket, MvPolynomial.X, Sum.inr, Sum.inl, ite, Neg.neg, Finsupp, Nat, OfNat.ofNat
Proof uses: Eq.mpr, id, congrArg, MvPolynomial.poissonBracket_swap, MvPolynomial.poissonBracket_X_inl_inr, dite, congrFun', if_pos, of_eq_true, eq_self, Not, if_neg, False.elim, Eq.mp, False, Eq.trans, True, eq_false, not_false_eq_true, not_true_eq_false and 1 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.