Folding polynomial is unique
Polynomial.FoldingPolynomial.folding_polynomial_is_unique'
Project documentation
Alternative uniqueness theorem for the folding polynomial. The only difference is the h_x condition which in this theorem is only and inequality. Handy in practice since degreeX is defined as a supremum so inequality is much easier to prove for it.
Exact Lean statement
theorem folding_polynomial_is_unique' {q f : Polynomial F} {Q : Polynomial (Polynomial F)}
(h : (Q.map (Polynomial.compRingHom q)).eval Polynomial.X = f)
(h_x : degreeX Q ≤ f.natDegree / q.natDegree)
(h_y : natDegreeY Q < q.natDegree) :
Q = foldingPolynomial q fFormal artifact
Lean source
theorem folding_polynomial_is_unique' {q f : Polynomial F} {Q : Polynomial (Polynomial F)} (h : (Q.map (Polynomial.compRingHom q)).eval Polynomial.X = f) (h_x : degreeX Q ≤ f.natDegree / q.natDegree) (h_y : natDegreeY Q < q.natDegree) : Q = foldingPolynomial q f := by by_cases hq_const : q.degree ≤ 0 · rw [Polynomial.eq_C_of_degree_le_zero hq_const] at h h_y ⊢ aesop · apply folding_polynomial_is_unique h (by have h_deg : f.natDegree ≤ degreeX Q * q.natDegree + q.natDegree - 1 := by have h_deg : Polynomial.natDegree (Polynomial.eval Polynomial.X (Polynomial.map q.compRingHom Q)) ≤ degreeX Q * q.natDegree + q.natDegree - 1 := by have := folded_poly_degree_bound (Nat.lt_succ_self _ : degreeX Q < degreeX Q + 1) h_y exact Nat.le_sub_one_of_lt (by linarith) generalize_proofs at * aesop exact le_antisymm h_x <| Nat.le_of_lt_succ (Nat.div_lt_of_lt_mul <| by linarith [Nat.sub_add_cancel ( show 1 ≤ degreeX Q * q.natDegree + q.natDegree from Nat.succ_le_iff.mpr <| by nlinarith [show q.natDegree > 0 from Polynomial.natDegree_pos_iff_degree_pos.mpr <| lt_of_not_ge hq_const])])) h_y- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Polynomial/FoldingPolynomial.lean:814-845
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