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Compatible rewriting of noncommutative polynomials for proving operator identities C.Chenavier C.Hofstadler C.G.Raab G.Regensburger Johannes Kepler Universitt, Linz, Austria 45th ISSAC Kalamata, Greece, July 20-23, 2020 Chenavier,

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Compatible rewriting of noncommutative polynomials for proving operator identities

C.Chenavier C.Hofstadler C.G.Raab G.Regensburger

Johannes Kepler Universität, Linz, Austria

45th ISSAC

Kalamata, Greece, July 20-23, 2020

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 1 / 14

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SLIDE 2

Proving operator identities

Objective: formally prove operator identities ⊲ operators are expressible in terms of basic operators ⊲ "forgetting" the analytic meaning by replacing basic operators by symbols Prove new identities establish equalities in suitable algebraic structures, e.g., ⊲ linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras ⊲ integro-diff. systems with smooth unknown functions tensor algebras ⊲ other systems with mixed operations Ore algebras/extensions, tensor rings Prove algebraic equalities use rewriting theory ⊲ e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . ⊲ simplify a syntactic expression into an equivalent one, e.g., ∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 2 / 14

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SLIDE 3

Proving operator identities

Objective: formally prove operator identities ⊲ operators are expressible in terms of basic operators ⊲ "forgetting" the analytic meaning by replacing basic operators by symbols Prove new identities establish equalities in suitable algebraic structures, e.g., ⊲ linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras ⊲ integro-diff. systems with smooth unknown functions tensor algebras ⊲ other systems with mixed operations Ore algebras/extensions, tensor rings Prove algebraic equalities use rewriting theory ⊲ e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . ⊲ simplify a syntactic expression into an equivalent one, e.g., ∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 2 / 14

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SLIDE 4

Proving operator identities

Objective: formally prove operator identities ⊲ operators are expressible in terms of basic operators ⊲ "forgetting" the analytic meaning by replacing basic operators by symbols Prove new identities establish equalities in suitable algebraic structures, e.g., ⊲ linear P.D.E. ’s with constant/polynomial coeff. polynomial/Weyl algebras ⊲ integro-diff. systems with smooth unknown functions tensor algebras ⊲ other systems with mixed operations Ore algebras/extensions, tensor rings Prove algebraic equalities use rewriting theory ⊲ e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . ⊲ simplify a syntactic expression into an equivalent one, e.g., ∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 2 / 14

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SLIDE 5

Proving operator identities

Objective: formally prove operator identities ⊲ operators are expressible in terms of basic operators ⊲ "forgetting" the analytic meaning by replacing basic operators by symbols Prove new identities establish equalities in suitable algebraic structures, e.g., ⊲ linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras ⊲ integro-diff. systems with smooth unknown functions tensor algebras ⊲ other systems with mixed operations Ore algebras/extensions, tensor rings Prove algebraic equalities use rewriting theory ⊲ e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . ⊲ simplify a syntactic expression into an equivalent one, e.g., ∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 2 / 14

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SLIDE 6

Proving operator identities

Objective: formally prove operator identities ⊲ operators are expressible in terms of basic operators ⊲ "forgetting" the analytic meaning by replacing basic operators by symbols Prove new identities establish equalities in suitable algebraic structures, e.g., ⊲ linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras ⊲ integro-diff. systems with smooth unknown functions tensor algebras ⊲ other systems with mixed operations Ore algebras/extensions, tensor rings Prove algebraic equalities use rewriting theory ⊲ e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . ⊲ simplify a syntactic expression into an equivalent one, e.g., ∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 2 / 14

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Additional task

Take compatibility conditions into account Ck ⊲ multiplication is not defined everywhere, e.g., matrices ⊲ composition depends on domains and codomains e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Existing method: based on quiver representation (Hossein Poor, R., R., arXiv:1910.06165) ⊲ requires to work with "uniformly compatible" polynomials

Our contributions

Theoretical part: extend the quiver approach to prove more identities ➔ based on Q-consequences Algorithmic part: compute Q-consequences using rewriting ➔ restrictions on the computations with G.B.

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 3 / 14

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Additional task

Take compatibility conditions into account Ck ⊲ multiplication is not defined everywhere, e.g., matrices ⊲ composition depends on domains and codomains e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Existing method: based on quiver representation (Hossein Poor, R., R., arXiv:1910.06165) ⊲ requires to work with "uniformly compatible" polynomials

Our contributions

Theoretical part: extend the quiver approach to prove more identities ➔ based on Q-consequences Algorithmic part: compute Q-consequences using rewriting ➔ restrictions on the computations with G.B.

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 3 / 14

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Additional task

Take compatibility conditions into account Ck ⊲ multiplication is not defined everywhere, e.g., matrices ⊲ composition depends on domains and codomains e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Existing method: based on quiver representation (Hossein Poor, R., R., arXiv:1910.06165) ⊲ requires to work with "uniformly compatible" polynomials

Our contributions

Theoretical part: extend the quiver approach to prove more identities ➔ based on Q-consequences Algorithmic part: compute Q-consequences using rewriting ➔ restrictions on the computations with G.B.

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 3 / 14

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Additional task

Take compatibility conditions into account Ck ⊲ multiplication is not defined everywhere, e.g., matrices ⊲ composition depends on domains and codomains e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Existing method: based on quiver representation (Hossein Poor, R., R., arXiv:1910.06165) ⊲ requires to work with "uniformly compatible" polynomials

Our contributions

Theoretical part: extend the quiver approach to prove more identities ➔ based on Q-consequences Algorithmic part: compute Q-consequences using rewriting ➔ restrictions on the computations with G.B.

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 3 / 14

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SLIDE 11

Additional task

Take compatibility conditions into account Ck ⊲ multiplication is not defined everywhere, e.g., matrices ⊲ composition depends on domains and codomains e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Existing method: based on quiver representation (Hossein Poor, R., R., arXiv:1910.06165) ⊲ requires to work with "uniformly compatible" polynomials

Our contributions

Theoretical part: extend the quiver approach to prove more identities ➔ based on Q-consequences Algorithmic part: compute Q-consequences using rewriting ➔ restrictions on the computations with G.B.

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 3 / 14

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Proof of operator identities

Given: basic operators satisfying identities, e.g., ∂(f ) := f ′,

  • (f ) :=

x

x0

f (t)dt, Eval(f ) := f (x0) are s.t.

  • ∂ = Id − Eval,

∂ ◦ = Id i.e., ∀f :

x

x0 f ′(t)dt = f (x) − f (x0),

x

x0 f (t)dt

′

= f (x) Objective: prove new identities using symbolic methods, e.g., Eval ◦ = 0, i.e., ∀f :

x0

x0 f (t)dt = 0,

follows from

  • − Eval ◦

=

  • ∂ ◦

=

  • Chenavier, Hofstadler, Raab, Regensburger

Rewriting and operator identities July 20-22, 2020 4 / 14

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Proof of operator identities

Given: basic operators satisfying identities, e.g., ∂(f ) := f ′,

  • (f ) :=

x

x0

f (t)dt, Eval(f ) := f (x0) are s.t.

  • ∂ = Id − Eval,

∂ ◦ = Id i.e., ∀f :

x

x0 f ′(t)dt = f (x) − f (x0),

x

x0 f (t)dt

′

= f (x) Objective: prove new identities using symbolic methods, e.g., Eval ◦ = 0, i.e., ∀f :

x0

x0 f (t)dt = 0,

follows from

  • − Eval ◦

=

  • ∂ ◦

=

  • Chenavier, Hofstadler, Raab, Regensburger

Rewriting and operator identities July 20-22, 2020 4 / 14

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SLIDE 14

Proof of operator identities

Given: basic operators satisfying identities, e.g., ∂(f ) := f ′,

  • (f ) :=

x

x0

f (t)dt, Eval(f ) := f (x0) are s.t.

  • ∂ = Id − Eval,

∂ ◦ = Id i.e., ∀f :

x

x0 f ′(t)dt = f (x) − f (x0),

x

x0 f (t)dt

′

= f (x) Objective: prove new identities using symbolic methods, e.g., Eval ◦ = 0, i.e., ∀f :

x0

x0 f (t)dt = 0,

follows from

  • − Eval ◦

=

  • ∂ ◦

=

  • Chenavier, Hofstadler, Raab, Regensburger

Rewriting and operator identities July 20-22, 2020 4 / 14

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Formal computations with noncommutative polynomials

Example: Ck ∂,

  • ,

Eval and

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 Polynomial translation: Ck Kd, i, e ∋ id − 1 + e, di − 1 New identity: Ck ei = (id − 1 + e)i − i(di − 1) Additionally: check compatiblity of cofactor decomposition with domains and codomains ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 5 / 14

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Formal computations with noncommutative polynomials

Example: Ck ∂,

  • ,

Eval and

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 Polynomial translation: Ck Kd, i, e ∋ id − 1 + e, di − 1 New identity: Ck ei = (id − 1 + e)i − i(di − 1) Additionally: check compatiblity of cofactor decomposition with domains and codomains ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 5 / 14

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Formal computations with noncommutative polynomials

Example: Ck ∂,

  • ,

Eval and

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 Polynomial translation: Ck Kd, i, e ∋ id − 1 + e, di − 1 New identity: Ck ei = (id − 1 + e)i − i(di − 1) Additionally: check compatiblity of cofactor decomposition with domains and codomains ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 5 / 14

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SLIDE 18

Formal computations with noncommutative polynomials

Example: Ck ∂,

  • ,

Eval and

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 Polynomial translation: Ck Kd, i, e ∋ id − 1 + e, di − 1 New identity: Ck ei = (id − 1 + e)i − i(di − 1) Additionally: check compatiblity of cofactor decomposition with domains and codomains ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 5 / 14

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Quivers represented by operators

  • e

d i

Def.: consider a labelled quiver Q (one letter may label multiple edges) ⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

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Quivers represented by operators

  • e

d i

  • e

d i

Def.: consider a labelled quiver Q (one letter may label multiple edges) ⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

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SLIDE 21

Quivers represented by operators

  • e

d i

Def.: consider a labelled quiver Q (one letter may label multiple edges) ⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

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SLIDE 22

Quivers represented by operators

  • e

d i

Def.: consider a labelled quiver Q (one letter may label multiple edges) ⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

slide-23
SLIDE 23

Quivers represented by operators

  • e

d i Ck+1(I) Ck(I) Eval ∂

  • Def.: consider a labelled quiver Q (one letter may label multiple edges) with a representation

⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

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SLIDE 24

Quivers represented by operators

  • e

d i C∞(I) C∞(I) Eval ∂

  • Def.: consider a labelled quiver Q (one letter may label multiple edges) with a representation

⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

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SLIDE 25

Quivers represented by operators

  • e

d i Ck+1(I) Ck(I) Eval ∂

  • Def.: consider a labelled quiver Q (one letter may label multiple edges) with a representation

⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

slide-26
SLIDE 26

Quivers represented by operators

  • e

d i Ck+1(I) Ck(I) Eval ∂

  • Def.: consider a labelled quiver Q (one letter may label multiple edges) with a representation

⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition ⊲ a realization of f ∈ KX is an image of f by the representation

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

slide-27
SLIDE 27

Quivers represented by operators

  • e

d i Ck+1(I) Ck(I) Eval ∂

  • Def.: consider a labelled quiver Q (one letter may label multiple edges) with a representation

⊲ f ∈ KX is a Q-consequence of F ⊆ KX if it admits a compatible decomposition ⊲ a realization of f ∈ KX is an image of f by the representation

  • Ex. of a Q-csq.: ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i

each monomial labels a path •

∗

→ •

Theorem

If all elements of realizations of F are zero and if f is a Q-consequence of F, then all realizations

  • f f are zero

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 6 / 14

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Illustrating example

Consider the inhomogeneous linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (1) Assumption: (1) can be factored into the 1st order equations y′(x) − B2(x)y(x) = z(x) and z′(x) − B1(x)z(x) = r(x) General solution: given by y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt (2) where Hi(x) is s.t. H′

i (x) − Bi(x)Hi(x) = 0 and Hi(x)−1 exists

Illustration of the theorem: formally prove that (2) is a solution of (1)

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 7 / 14

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SLIDE 29

Illustrating example

Consider the inhomogeneous linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (1) Assumption: (1) can be factored into the 1st order equations y′(x) − B2(x)y(x) = z(x) and z′(x) − B1(x)z(x) = r(x) General solution: given by y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt (2) where Hi(x) is s.t. H′

i (x) − Bi(x)Hi(x) = 0 and Hi(x)−1 exists

Illustration of the theorem: formally prove that (2) is a solution of (1)

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 7 / 14

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SLIDE 30

Illustrating example

Assumptions: prove that y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is solution of

  • (∂ − B1) ◦ (∂ − B2)

(y(x)) = r(x) where Hi is s.t. H′

i (x) − Bi(x)Hi(x) = 0 and Hi(x)−1 exists

Algebraic part: X := {h1, h2, b1, b2, ˜ h1, ˜ h2, i, d}, F := {f1, . . . , f5} ⊂ KX, where f1 := dh1 − h1d − b1h1, f2 := dh2 − h2d − b2h2, f3 := h1˜ h1 − 1, f4 := h2˜ h2 − 1, f5 := di − 1 Objective: prove that f is a Q-consequence of F, where f := (d − b1)(d − b2)h2i˜ h2h1i˜ h1 − 1

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 8 / 14

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SLIDE 31

First method for proving Q-consequences

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • Fact: F := {f1, . . . , f5}

⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) proves that f is a Q-consequence of F

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 9 / 14

slide-32
SLIDE 32

First method for proving Q-consequences

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) proves that f is a Q-consequence of F

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 9 / 14

slide-33
SLIDE 33

First method for proving Q-consequences

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) proves that f is a Q-consequence of F

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 9 / 14

slide-34
SLIDE 34

First method for proving Q-consequences

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) proves that f is a Q-consequence of F

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 9 / 14

slide-35
SLIDE 35

First method for proving Q-consequences

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) proves that f is a Q-consequence of F

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 9 / 14

slide-36
SLIDE 36

First method for proving Q-consequences

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) proves that f is a Q-consequence of F

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 9 / 14

slide-37
SLIDE 37

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

slide-38
SLIDE 38

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

slide-39
SLIDE 39

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

slide-40
SLIDE 40

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

slide-41
SLIDE 41

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

slide-42
SLIDE 42

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

slide-43
SLIDE 43

Second method for proving Q-consequences

Restrict to rew. steps s.t.

We only use "valid" computations Compatible rewriting rules

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible (1st method involved invalid computations) ⊲ h2d → dh2 − b2h2 is compatible

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 10 / 14

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SLIDE 44

Theorem

Let Q be a quiver labelled by X, let F ⊂ KX and let f ∈ KX. Assume that each rew. rule is Q-compatible and f

∗

→F 0. Then, f is compatible with Q ⇔ f is a Q-consequence

Summary of the 2nd method for proving Q-consequences

Using the Theorem: ⊲ representation(s) of the quiver map any polynomial to the operator(s) it represents ⊲ elements of F polynomial expressions of known operator identities ⊲ f polynomial expression of the identity we wish to prove ⊲ f

∗

→F 0 with compatible rew. rules only the identity is proven

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 11 / 14

slide-45
SLIDE 45

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-46
SLIDE 46

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-47
SLIDE 47

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-48
SLIDE 48

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-49
SLIDE 49

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-50
SLIDE 50

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-51
SLIDE 51

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-52
SLIDE 52

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f ⊲ for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1 ⊲ problem: f does not rewrite into 0

  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 12 / 14

slide-53
SLIDE 53

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ F : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where ⊲ A0, A1 are functions of class Ck Ck ⊲ r is a function of class Ck Ck If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a function of class Ck+2 solution of (4) Ck

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 13 / 14

slide-54
SLIDE 54

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ F : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where ⊲ A0, A1 are functions of class Ck Ck ⊲ r is a function of class Ck Ck If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a function of class Ck+2 solution of (4) Ck

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 13 / 14

slide-55
SLIDE 55

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ F : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where ⊲ A0, A1 are analytic functions Ck ⊲ r is an analytic function Ck If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is an analytic function solution of (4) Ck

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 13 / 14

slide-56
SLIDE 56

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ F : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where ⊲ A0, A1 are n × n matrices of functions of class Ck Ck ⊲ r is a vector of n functions of class Ck Ck If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a vector of n functions of class Ck+2 solution of (4) Ck

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 13 / 14

slide-57
SLIDE 57

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ F : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where ⊲ A0, A1 are · · · Ck ⊲ r is a · · · Ck If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a · · · solution of (4) Ck

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 13 / 14

slide-58
SLIDE 58

Summary

Our contributions Ck ⊲ we develop an approach based on Q-consequences to formally prove identities ⊲ we provided a method for computing Q-consequences using rewriting Implementation: Ck ⊲ Mathematica package OperatorGB (by Clemens Hofstadler) ⊲ link: http://gregensburger.com/softw/OperatorGB/

THANK YOU FOR YOUR ATTENTION!

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 14 / 14

slide-59
SLIDE 59

Summary

Our contributions Ck ⊲ we develop an approach based on Q-consequences to formally prove identities ⊲ we provided a method for computing Q-consequences using rewriting Implementation: Ck ⊲ Mathematica package OperatorGB (by Clemens Hofstadler) ⊲ link: http://gregensburger.com/softw/OperatorGB/

THANK YOU FOR YOUR ATTENTION!

Chenavier, Hofstadler, Raab, Regensburger Rewriting and operator identities July 20-22, 2020 14 / 14