Folded poly degree bound
Polynomial.FoldingPolynomial.folded_poly_degree_bound
Plain-language statement
If we fold a polynomial using a folding polynomial Q with appropriate degree bounds in each variable we get a univariate polynomial with a degree bound.
Exact Lean statement
lemma folded_poly_degree_bound {Q : F[X][Y]} {q : F[X]} {t : ℕ}
(h_x : degreeX Q < t)
(h_y : natDegreeY Q < q.natDegree) :
((Q.map (Polynomial.compRingHom q)).eval X).natDegree < t * q.natDegreeFormal artifact
Lean source
lemma folded_poly_degree_bound {Q : F[X][Y]} {q : F[X]} {t : ℕ} (h_x : degreeX Q < t) (h_y : natDegreeY Q < q.natDegree) : ((Q.map (Polynomial.compRingHom q)).eval X).natDegree < t * q.natDegree := by have h : Q = foldingPolynomial q ((Q.map (Polynomial.compRingHom q)).eval X) := by apply folding_polynomial_is_unique · aesop · by_cases hq : q = 0 · aesop · rw [Polynomial.eval_map, Polynomial.eval₂_eq_sum_range, Polynomial.natDegree_sum_eq_of_disjoint] · apply le_antisymm <;> simp_all +decide only [degreeX, coe_compRingHom, Finset.sup_le_iff, mem_support_iff, ne_eq] · intro n hn apply Nat.le_div_iff_mul_le (Nat.pos_of_ne_zero (ne_of_gt (Nat.pos_of_ne_zero (by aesop)))) |>.2 · apply le_trans _ (Finset.le_sup (f := fun i ↦ Polynomial.natDegree (Polynomial.comp (Q.coeff i) q * Polynomial.X ^ i)) (Finset.mem_range.mpr (Nat.lt_succ_of_le (Polynomial.le_natDegree_of_ne_zero hn)))) rw [Polynomial.natDegree_mul'] <;> simp +decide only [ monic_X_pow, Monic.leadingCoeff, mul_one, ne_eq, leadingCoeff_eq_zero, natDegree_comp, natDegree_pow, natDegree_X, mul_one, le_add_iff_nonneg_right, zero_le] have h_comp_nonzero : Polynomial.natDegree (Polynomial.comp (Q.coeff n) q) = Polynomial.natDegree (Q.coeff n) * Polynomial.natDegree q := by rw [Polynomial.natDegree_comp] by_contra h_comp_zero have h_deg_zero : Polynomial.natDegree (Polynomial.comp (Q.coeff n) q) = 0 := by rw [h_comp_zero, Polynomial.natDegree_zero] simp_all +decide cases h_comp_nonzero <;> simp_all +decide [Polynomial.natDegree_eq_zero_iff_degree_le_zero] rw [ Polynomial.eq_C_of_degree_le_zero ‹Polynomial.degree (Q.coeff n) ≤ 0›] at hn h_comp_zero aesop · rw [Nat.div_le_iff_le_mul_add_pred] <;> norm_num · intro b hb have h_deg : Polynomial.natDegree (Polynomial.comp (Q.coeff b) q) ≤ Polynomial.natDegree q * Polynomial.natDegree (Q.coeff b) := by rw [Polynomial.natDegree_comp, mul_comm] by_cases h : Polynomial.comp (Q.coeff b) q = 0 <;> simp_all +decide only [ natDegree_zero, zero_le, zero_mul, monic_X_pow, Monic.leadingCoeff, mul_one, ne_eq, leadingCoeff_eq_zero, not_false_eq_true, natDegree_mul', natDegree_pow, natDegree_X, ge_iff_le] apply add_le_add (le_trans h_deg (Nat.mul_le_mul_left _ (Finset.le_sup (f := fun n ↦ Polynomial.natDegree (Q.coeff n)) (by aesop)))) exact Nat.le_sub_one_of_lt (lt_of_lt_of_le (Nat.lt_succ_of_le hb) (Nat.succ_le_of_lt (lt_of_le_of_lt (Polynomial.le_natDegree_of_mem_supp _ (by aesop)) h_y))) · exact Nat.pos_of_ne_zero (by aesop) · intro i hi j hj hij simp_all +decide only [Finset.mem_range, Order.lt_add_one_iff, coe_compRingHom, ne_eq, mul_eq_zero, pow_eq_zero_iff', X_ne_zero, false_and, or_false, Set.mem_setOf_eq, Function.comp_apply, monic_X_pow, Monic.leadingCoeff, mul_one, leadingCoeff_eq_zero, not_false_eq_true, natDegree_mul', natDegree_comp, natDegree_pow, natDegree_X] by_contra h_contra exact hij (by nlinarith [show Polynomial.natDegree (Q.coeff i) = Polynomial.natDegree (Q.coeff j) by nlinarith [show i < q.natDegree from lt_of_le_of_lt (Polynomial.le_natDegree_of_ne_zero (by aesop)) h_y, show j < q.natDegree from lt_of_le_of_lt (Polynomial.le_natDegree_of_ne_zero (by aesop)) h_y]]) · aesop contrapose! h_x rw [h, folding_polynomial_deg_x] exact Nat.le_div_iff_mul_le (Nat.pos_of_ne_zero (by rintro h; simp_all +singlePass)) |>.2 h_x- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Polynomial/FoldingPolynomial.lean:712-807
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