H tilde equiv H tilde
BCIKS20AppendixA.H_tilde_equiv_H_tilde'
Plain-language statement
The polynomial H_tilde' agrees with the monicization H_tilde after embedding into Polynomial (RatFunc F).
Exact Lean statement
lemma H_tilde_equiv_H_tilde' (H : F[X][Y]) : (H_tilde' H).map univPolyHom = H_tilde H
Formal artifact
Lean source
lemma H_tilde_equiv_H_tilde' (H : F[X][Y]) : (H_tilde' H).map univPolyHom = H_tilde H := by classical by_cases hdeg : H.natDegree = 0 · simp only [H_tilde', hdeg, ↓reduceIte, map_C] have hconst : H = Polynomial.C (H.coeff 0) := Polynomial.eq_C_of_natDegree_le_zero (by omega) rw [hconst, H_tilde] simp · have hH_ne : H ≠ 0 := by intro hzero apply hdeg simp [hzero] have hw_ne_zero : univPolyHom H.leadingCoeff ≠ 0 := by apply IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors rw [mem_nonZeroDivisors_iff_ne_zero] exact Polynomial.leadingCoeff_ne_zero.mpr hH_ne have hd : 0 < H.natDegree := Nat.pos_of_ne_zero hdeg have hEval : Polynomial.eval₂ (RingHom.comp Polynomial.C univPolyHom) (Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree)) H = ∑ i ∈ Finset.range (H.natDegree + 1), Polynomial.C (univPolyHom (H.coeff i)) * (Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree)) ^ i := by simpa using (Polynomial.eval₂_eq_sum_range (p := H) (f := RingHom.comp Polynomial.C univPolyHom) (x := Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree))) simp only [H_tilde', hdeg, ↓reduceIte, coeff_natDegree, map_mul, map_pow, Polynomial.map_add, Polynomial.map_pow, map_X] rw [H_tilde, hEval, Finset.sum_range_succ, mul_add, Finset.mul_sum, Polynomial.map_sum] have hsum : ∑ i ∈ Finset.range H.natDegree, ((RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree) ^ (H.natDegree - 1)) * (Polynomial.C (univPolyHom (H.coeff i)) * (Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree)) ^ i) = ∑ i ∈ Finset.range H.natDegree, Polynomial.map univPolyHom (Polynomial.C (H.coeff i) * Polynomial.C H.leadingCoeff ^ (H.natDegree - 1 - i) * Polynomial.X ^ i) := by refine Finset.sum_congr rfl ?_ intro i hi simpa [Polynomial.coeff_natDegree, map_mul, map_pow] using monicize_term (univPolyHom H.leadingCoeff) (univPolyHom (H.coeff i)) i H.natDegree hw_ne_zero (Finset.mem_range.mp hi) have hlead : ((RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree) ^ (H.natDegree - 1)) * (Polynomial.C (univPolyHom (H.coeff H.natDegree)) * (Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree)) ^ H.natDegree) = Polynomial.X ^ H.natDegree := by simpa [Polynomial.coeff_natDegree] using monicize_leading_term (univPolyHom H.leadingCoeff) H.natDegree hw_ne_zero hd rw [hlead] calc Polynomial.X ^ H.natDegree + ∑ i ∈ Finset.range H.natDegree, Polynomial.map univPolyHom (Polynomial.C (H.coeff i) * Polynomial.C H.leadingCoeff ^ (H.natDegree - 1 - i) * Polynomial.X ^ i) = Polynomial.X ^ H.natDegree + ∑ i ∈ Finset.range H.natDegree, (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree) ^ (H.natDegree - 1) * (Polynomial.C (univPolyHom (H.coeff i)) * (Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree)) ^ i) := by exact congrArg (fun p => Polynomial.X ^ H.natDegree + p) hsum.symm _ = ∑ i ∈ Finset.range H.natDegree, (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree) ^ (H.natDegree - 1) * (Polynomial.C (univPolyHom (H.coeff i)) * (Polynomial.X / (RingHom.comp Polynomial.C univPolyHom) ((fun i => H.coeff i) H.natDegree)) ^ i) + Polynomial.X ^ H.natDegree := by rw [add_comm]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Polynomial/RationalFunctions.lean:162-245
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Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.