Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
| Mathlib/Data/Real/GoldenRatio.lean | 64 | 66 | theorem goldConj_mul_gold : Ο * Ο = -1 := by |
rw [mul_comm]
exact gold_mul_goldConj
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
| Mathlib/Data/Real/GoldenRatio.lean | 70 | 72 | theorem gold_add_goldConj : Ο + Ο = 1 := by |
rw [goldenRatio, goldenConj]
ring
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
| Mathlib/Data/Real/GoldenRatio.lean | 75 | 76 | theorem one_sub_goldConj : 1 - Ο = Ο := by |
linarith [gold_add_goldConj]
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
| Mathlib/Data/Real/GoldenRatio.lean | 79 | 80 | theorem one_sub_gold : 1 - Ο = Ο := by |
linarith [gold_add_goldConj]
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
| Mathlib/Data/Real/GoldenRatio.lean | 84 | 84 | theorem gold_sub_goldConj : Ο - Ο = β5 := by | ring
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
| Mathlib/Data/Real/GoldenRatio.lean | 87 | 88 | theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by |
rw [goldenRatio]; ring_nf; norm_num; ring
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
| Mathlib/Data/Real/GoldenRatio.lean | 91 | 94 | theorem gold_sq : Ο ^ 2 = Ο + 1 := by |
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
| Mathlib/Data/Real/GoldenRatio.lean | 98 | 101 | theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by |
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
| Mathlib/Data/Real/GoldenRatio.lean | 112 | 114 | theorem one_lt_gold : 1 < Ο := by |
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
| Mathlib/Data/Real/GoldenRatio.lean | 117 | 119 | theorem gold_lt_two : Ο < 2 := by | calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
theorem gold_lt_two : Ο < 2 := by calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
| Mathlib/Data/Real/GoldenRatio.lean | 121 | 122 | theorem goldConj_neg : Ο < 0 := by |
linarith [one_sub_goldConj, one_lt_gold]
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
theorem gold_lt_two : Ο < 2 := by calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
theorem goldConj_neg : Ο < 0 := by
linarith [one_sub_goldConj, one_lt_gold]
#align gold_conj_neg goldConj_neg
theorem goldConj_ne_zero : Ο β 0 :=
ne_of_lt goldConj_neg
#align gold_conj_ne_zero goldConj_ne_zero
| Mathlib/Data/Real/GoldenRatio.lean | 129 | 131 | theorem neg_one_lt_goldConj : -1 < Ο := by |
rw [neg_lt, β inv_gold]
exact inv_lt_one one_lt_gold
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
theorem gold_lt_two : Ο < 2 := by calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
theorem goldConj_neg : Ο < 0 := by
linarith [one_sub_goldConj, one_lt_gold]
#align gold_conj_neg goldConj_neg
theorem goldConj_ne_zero : Ο β 0 :=
ne_of_lt goldConj_neg
#align gold_conj_ne_zero goldConj_ne_zero
theorem neg_one_lt_goldConj : -1 < Ο := by
rw [neg_lt, β inv_gold]
exact inv_lt_one one_lt_gold
#align neg_one_lt_gold_conj neg_one_lt_goldConj
| Mathlib/Data/Real/GoldenRatio.lean | 140 | 146 | theorem gold_irrational : Irrational Ο := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
theorem gold_lt_two : Ο < 2 := by calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
theorem goldConj_neg : Ο < 0 := by
linarith [one_sub_goldConj, one_lt_gold]
#align gold_conj_neg goldConj_neg
theorem goldConj_ne_zero : Ο β 0 :=
ne_of_lt goldConj_neg
#align gold_conj_ne_zero goldConj_ne_zero
theorem neg_one_lt_goldConj : -1 < Ο := by
rw [neg_lt, β inv_gold]
exact inv_lt_one one_lt_gold
#align neg_one_lt_gold_conj neg_one_lt_goldConj
theorem gold_irrational : Irrational Ο := by
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
#align gold_irrational gold_irrational
| Mathlib/Data/Real/GoldenRatio.lean | 150 | 156 | theorem goldConj_irrational : Irrational Ο := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_sub 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
theorem gold_lt_two : Ο < 2 := by calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
theorem goldConj_neg : Ο < 0 := by
linarith [one_sub_goldConj, one_lt_gold]
#align gold_conj_neg goldConj_neg
theorem goldConj_ne_zero : Ο β 0 :=
ne_of_lt goldConj_neg
#align gold_conj_ne_zero goldConj_ne_zero
theorem neg_one_lt_goldConj : -1 < Ο := by
rw [neg_lt, β inv_gold]
exact inv_lt_one one_lt_gold
#align neg_one_lt_gold_conj neg_one_lt_goldConj
theorem gold_irrational : Irrational Ο := by
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
#align gold_irrational gold_irrational
theorem goldConj_irrational : Irrational Ο := by
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_sub 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
#align gold_conj_irrational goldConj_irrational
section Fibrec
variable {Ξ± : Type*} [CommSemiring Ξ±]
def fibRec : LinearRecurrence Ξ± where
order := 2
coeffs := ![1, 1]
#align fib_rec fibRec
section Poly
open Polynomial
| Mathlib/Data/Real/GoldenRatio.lean | 178 | 181 | theorem fibRec_charPoly_eq {Ξ² : Type*} [CommRing Ξ²] :
fibRec.charPoly = X ^ 2 - (X + (1 : Ξ²[X])) := by |
rw [fibRec, LinearRecurrence.charPoly]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ', β smul_X_eq_monomial]
| 1,817 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open Polynomial
abbrev goldenRatio : β := (1 + β5) / 2
#align golden_ratio goldenRatio
abbrev goldenConj : β := (1 - β5) / 2
#align golden_conj goldenConj
@[inherit_doc goldenRatio] scoped[goldenRatio] notation "Ο" => goldenRatio
@[inherit_doc goldenConj] scoped[goldenRatio] notation "Ο" => goldenConj
open Real goldenRatio
theorem inv_gold : Οβ»ΒΉ = -Ο := by
have : 1 + β5 β 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
#align inv_gold inv_gold
theorem inv_goldConj : Οβ»ΒΉ = -Ο := by
rw [inv_eq_iff_eq_inv, β neg_inv, β neg_eq_iff_eq_neg]
exact inv_gold.symm
#align inv_gold_conj inv_goldConj
@[simp]
theorem gold_mul_goldConj : Ο * Ο = -1 := by
field_simp
rw [β sq_sub_sq]
norm_num
#align gold_mul_gold_conj gold_mul_goldConj
@[simp]
theorem goldConj_mul_gold : Ο * Ο = -1 := by
rw [mul_comm]
exact gold_mul_goldConj
#align gold_conj_mul_gold goldConj_mul_gold
@[simp]
theorem gold_add_goldConj : Ο + Ο = 1 := by
rw [goldenRatio, goldenConj]
ring
#align gold_add_gold_conj gold_add_goldConj
theorem one_sub_goldConj : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold_conj one_sub_goldConj
theorem one_sub_gold : 1 - Ο = Ο := by
linarith [gold_add_goldConj]
#align one_sub_gold one_sub_gold
@[simp]
theorem gold_sub_goldConj : Ο - Ο = β5 := by ring
#align gold_sub_gold_conj gold_sub_goldConj
theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by
rw [goldenRatio]; ring_nf; norm_num; ring
@[simp 1200]
theorem gold_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenRatio, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_sq gold_sq
@[simp 1200]
theorem goldConj_sq : Ο ^ 2 = Ο + 1 := by
rw [goldenConj, β sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
#align gold_conj_sq goldConj_sq
theorem gold_pos : 0 < Ο :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
#align gold_pos gold_pos
theorem gold_ne_zero : Ο β 0 :=
ne_of_gt gold_pos
#align gold_ne_zero gold_ne_zero
theorem one_lt_gold : 1 < Ο := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [β sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
#align one_lt_gold one_lt_gold
theorem gold_lt_two : Ο < 2 := by calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
theorem goldConj_neg : Ο < 0 := by
linarith [one_sub_goldConj, one_lt_gold]
#align gold_conj_neg goldConj_neg
theorem goldConj_ne_zero : Ο β 0 :=
ne_of_lt goldConj_neg
#align gold_conj_ne_zero goldConj_ne_zero
theorem neg_one_lt_goldConj : -1 < Ο := by
rw [neg_lt, β inv_gold]
exact inv_lt_one one_lt_gold
#align neg_one_lt_gold_conj neg_one_lt_goldConj
theorem gold_irrational : Irrational Ο := by
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
#align gold_irrational gold_irrational
theorem goldConj_irrational : Irrational Ο := by
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_sub 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
#align gold_conj_irrational goldConj_irrational
section Fibrec
variable {Ξ± : Type*} [CommSemiring Ξ±]
def fibRec : LinearRecurrence Ξ± where
order := 2
coeffs := ![1, 1]
#align fib_rec fibRec
| Mathlib/Data/Real/GoldenRatio.lean | 187 | 192 | theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : β β Ξ±) := by |
rw [fibRec]
intro n
simp only
rw [Nat.fib_add_two, add_comm]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ']
| 1,817 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
def PowerSeries (R : Type*) :=
MvPowerSeries Unit R
#align power_series PowerSeries
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section
-- Porting note: not available in Lean 4
-- local reducible PowerSeries
scoped notation:9000 R "β¦Xβ§" => PowerSeries R
instance [Inhabited R] : Inhabited Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Zero R] : Zero Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddMonoid R] : AddMonoid Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddGroup R] : AddGroup Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommMonoid R] : AddCommMonoid Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommGroup R] : AddCommGroup Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Semiring R] : Semiring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [CommSemiring R] : CommSemiring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Ring R] : Ring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [CommRing R] : CommRing Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Nontrivial R] : Nontrivial Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R Aβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]
[IsScalarTower R S A] : IsScalarTower R S Aβ¦Xβ§ :=
Pi.isScalarTower
instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R Aβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
end
section Semiring
variable (R) [Semiring R]
def coeff (n : β) : Rβ¦Xβ§ ββ[R] R :=
MvPowerSeries.coeff R (single () n)
#align power_series.coeff PowerSeries.coeff
def monomial (n : β) : R ββ[R] Rβ¦Xβ§ :=
MvPowerSeries.monomial R (single () n)
#align power_series.monomial PowerSeries.monomial
variable {R}
| Mathlib/RingTheory/PowerSeries/Basic.lean | 150 | 151 | theorem coeff_def {s : Unit ββ β} {n : β} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by |
erw [coeff, β h, β Finsupp.unique_single s]
| 1,818 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
def PowerSeries (R : Type*) :=
MvPowerSeries Unit R
#align power_series PowerSeries
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section
-- Porting note: not available in Lean 4
-- local reducible PowerSeries
scoped notation:9000 R "β¦Xβ§" => PowerSeries R
instance [Inhabited R] : Inhabited Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Zero R] : Zero Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddMonoid R] : AddMonoid Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddGroup R] : AddGroup Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommMonoid R] : AddCommMonoid Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommGroup R] : AddCommGroup Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Semiring R] : Semiring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [CommSemiring R] : CommSemiring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Ring R] : Ring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [CommRing R] : CommRing Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Nontrivial R] : Nontrivial Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R Aβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]
[IsScalarTower R S A] : IsScalarTower R S Aβ¦Xβ§ :=
Pi.isScalarTower
instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R Aβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
end
section Semiring
variable (R) [Semiring R]
def coeff (n : β) : Rβ¦Xβ§ ββ[R] R :=
MvPowerSeries.coeff R (single () n)
#align power_series.coeff PowerSeries.coeff
def monomial (n : β) : R ββ[R] Rβ¦Xβ§ :=
MvPowerSeries.monomial R (single () n)
#align power_series.monomial PowerSeries.monomial
variable {R}
theorem coeff_def {s : Unit ββ β} {n : β} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by
erw [coeff, β h, β Finsupp.unique_single s]
#align power_series.coeff_def PowerSeries.coeff_def
@[ext]
theorem ext {Ο Ο : Rβ¦Xβ§} (h : β n, coeff R n Ο = coeff R n Ο) : Ο = Ο :=
MvPowerSeries.ext fun n => by
rw [β coeff_def]
Β· apply h
rfl
#align power_series.ext PowerSeries.ext
theorem ext_iff {Ο Ο : Rβ¦Xβ§} : Ο = Ο β β n, coeff R n Ο = coeff R n Ο :=
β¨fun h n => congr_arg (coeff R n) h, extβ©
#align power_series.ext_iff PowerSeries.ext_iff
instance [Subsingleton R] : Subsingleton Rβ¦Xβ§ := by
simp only [subsingleton_iff, ext_iff]
exact fun _ _ _ β¦ (subsingleton_iff).mp (by infer_instance) _ _
def mk {R} (f : β β R) : Rβ¦Xβ§ := fun s => f (s ())
#align power_series.mk PowerSeries.mk
@[simp]
theorem coeff_mk (n : β) (f : β β R) : coeff R n (mk f) = f n :=
congr_arg f Finsupp.single_eq_same
#align power_series.coeff_mk PowerSeries.coeff_mk
| Mathlib/RingTheory/PowerSeries/Basic.lean | 181 | 184 | theorem coeff_monomial (m n : β) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=
calc
coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _
_ = if m = n then a else 0 := by | simp only [Finsupp.unique_single_eq_iff]
| 1,818 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
def PowerSeries (R : Type*) :=
MvPowerSeries Unit R
#align power_series PowerSeries
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section
-- Porting note: not available in Lean 4
-- local reducible PowerSeries
scoped notation:9000 R "β¦Xβ§" => PowerSeries R
instance [Inhabited R] : Inhabited Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Zero R] : Zero Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddMonoid R] : AddMonoid Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddGroup R] : AddGroup Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommMonoid R] : AddCommMonoid Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [AddCommGroup R] : AddCommGroup Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Semiring R] : Semiring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [CommSemiring R] : CommSemiring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Ring R] : Ring Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [CommRing R] : CommRing Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance [Nontrivial R] : Nontrivial Rβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R Aβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]
[IsScalarTower R S A] : IsScalarTower R S Aβ¦Xβ§ :=
Pi.isScalarTower
instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R Aβ¦Xβ§ := by
dsimp only [PowerSeries]
infer_instance
end
section Semiring
variable (R) [Semiring R]
def coeff (n : β) : Rβ¦Xβ§ ββ[R] R :=
MvPowerSeries.coeff R (single () n)
#align power_series.coeff PowerSeries.coeff
def monomial (n : β) : R ββ[R] Rβ¦Xβ§ :=
MvPowerSeries.monomial R (single () n)
#align power_series.monomial PowerSeries.monomial
variable {R}
theorem coeff_def {s : Unit ββ β} {n : β} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by
erw [coeff, β h, β Finsupp.unique_single s]
#align power_series.coeff_def PowerSeries.coeff_def
@[ext]
theorem ext {Ο Ο : Rβ¦Xβ§} (h : β n, coeff R n Ο = coeff R n Ο) : Ο = Ο :=
MvPowerSeries.ext fun n => by
rw [β coeff_def]
Β· apply h
rfl
#align power_series.ext PowerSeries.ext
theorem ext_iff {Ο Ο : Rβ¦Xβ§} : Ο = Ο β β n, coeff R n Ο = coeff R n Ο :=
β¨fun h n => congr_arg (coeff R n) h, extβ©
#align power_series.ext_iff PowerSeries.ext_iff
instance [Subsingleton R] : Subsingleton Rβ¦Xβ§ := by
simp only [subsingleton_iff, ext_iff]
exact fun _ _ _ β¦ (subsingleton_iff).mp (by infer_instance) _ _
def mk {R} (f : β β R) : Rβ¦Xβ§ := fun s => f (s ())
#align power_series.mk PowerSeries.mk
@[simp]
theorem coeff_mk (n : β) (f : β β R) : coeff R n (mk f) = f n :=
congr_arg f Finsupp.single_eq_same
#align power_series.coeff_mk PowerSeries.coeff_mk
theorem coeff_monomial (m n : β) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=
calc
coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _
_ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff]
#align power_series.coeff_monomial PowerSeries.coeff_monomial
theorem monomial_eq_mk (n : β) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 :=
ext fun m => by rw [coeff_monomial, coeff_mk]
#align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk
@[simp]
theorem coeff_monomial_same (n : β) (a : R) : coeff R n (monomial R n a) = a :=
MvPowerSeries.coeff_monomial_same _ _
#align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same
@[simp]
theorem coeff_comp_monomial (n : β) : (coeff R n).comp (monomial R n) = LinearMap.id :=
LinearMap.ext <| coeff_monomial_same n
#align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial
variable (R)
def constantCoeff : Rβ¦Xβ§ β+* R :=
MvPowerSeries.constantCoeff Unit R
#align power_series.constant_coeff PowerSeries.constantCoeff
def C : R β+* Rβ¦Xβ§ :=
MvPowerSeries.C Unit R
set_option linter.uppercaseLean3 false in
#align power_series.C PowerSeries.C
variable {R}
def X : Rβ¦Xβ§ :=
MvPowerSeries.X ()
set_option linter.uppercaseLean3 false in
#align power_series.X PowerSeries.X
theorem commute_X (Ο : Rβ¦Xβ§) : Commute Ο X :=
MvPowerSeries.commute_X _ _
set_option linter.uppercaseLean3 false in
#align power_series.commute_X PowerSeries.commute_X
@[simp]
| Mathlib/RingTheory/PowerSeries/Basic.lean | 229 | 231 | theorem coeff_zero_eq_constantCoeff : β(coeff R 0) = constantCoeff R := by |
rw [coeff, Finsupp.single_zero]
rfl
| 1,818 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section Trunc
variable [Semiring R]
open Finset Nat
def trunc (n : β) (Ο : Rβ¦Xβ§) : R[X] :=
β m β Ico 0 n, Polynomial.monomial m (coeff R m Ο)
#align power_series.trunc PowerSeries.trunc
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 44 | 46 | theorem coeff_trunc (m) (n) (Ο : Rβ¦Xβ§) :
(trunc n Ο).coeff m = if m < n then coeff R m Ο else 0 := by |
simp [trunc, Polynomial.coeff_sum, Polynomial.coeff_monomial, Nat.lt_succ_iff]
| 1,819 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section Trunc
variable [Semiring R]
open Finset Nat
def trunc (n : β) (Ο : Rβ¦Xβ§) : R[X] :=
β m β Ico 0 n, Polynomial.monomial m (coeff R m Ο)
#align power_series.trunc PowerSeries.trunc
theorem coeff_trunc (m) (n) (Ο : Rβ¦Xβ§) :
(trunc n Ο).coeff m = if m < n then coeff R m Ο else 0 := by
simp [trunc, Polynomial.coeff_sum, Polynomial.coeff_monomial, Nat.lt_succ_iff]
#align power_series.coeff_trunc PowerSeries.coeff_trunc
@[simp]
theorem trunc_zero (n) : trunc n (0 : Rβ¦Xβ§) = 0 :=
Polynomial.ext fun m => by
rw [coeff_trunc, LinearMap.map_zero, Polynomial.coeff_zero]
split_ifs <;> rfl
#align power_series.trunc_zero PowerSeries.trunc_zero
@[simp]
theorem trunc_one (n) : trunc (n + 1) (1 : Rβ¦Xβ§) = 1 :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_one, Polynomial.coeff_one]
split_ifs with h _ h'
Β· rfl
Β· rfl
Β· subst h'; simp at h
Β· rfl
#align power_series.trunc_one PowerSeries.trunc_one
@[simp]
theorem trunc_C (n) (a : R) : trunc (n + 1) (C R a) = Polynomial.C a :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_C, Polynomial.coeff_C]
split_ifs with H <;> first |rfl|try simp_all
set_option linter.uppercaseLean3 false in
#align power_series.trunc_C PowerSeries.trunc_C
@[simp]
theorem trunc_add (n) (Ο Ο : Rβ¦Xβ§) : trunc n (Ο + Ο) = trunc n Ο + trunc n Ο :=
Polynomial.ext fun m => by
simp only [coeff_trunc, AddMonoidHom.map_add, Polynomial.coeff_add]
split_ifs with H
Β· rfl
Β· rw [zero_add]
#align power_series.trunc_add PowerSeries.trunc_add
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 84 | 86 | theorem trunc_succ (f : Rβ¦Xβ§) (n : β) :
trunc n.succ f = trunc n f + Polynomial.monomial n (coeff R n f) := by |
rw [trunc, Ico_zero_eq_range, sum_range_succ, trunc, Ico_zero_eq_range]
| 1,819 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section Trunc
variable [Semiring R]
open Finset Nat
def trunc (n : β) (Ο : Rβ¦Xβ§) : R[X] :=
β m β Ico 0 n, Polynomial.monomial m (coeff R m Ο)
#align power_series.trunc PowerSeries.trunc
theorem coeff_trunc (m) (n) (Ο : Rβ¦Xβ§) :
(trunc n Ο).coeff m = if m < n then coeff R m Ο else 0 := by
simp [trunc, Polynomial.coeff_sum, Polynomial.coeff_monomial, Nat.lt_succ_iff]
#align power_series.coeff_trunc PowerSeries.coeff_trunc
@[simp]
theorem trunc_zero (n) : trunc n (0 : Rβ¦Xβ§) = 0 :=
Polynomial.ext fun m => by
rw [coeff_trunc, LinearMap.map_zero, Polynomial.coeff_zero]
split_ifs <;> rfl
#align power_series.trunc_zero PowerSeries.trunc_zero
@[simp]
theorem trunc_one (n) : trunc (n + 1) (1 : Rβ¦Xβ§) = 1 :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_one, Polynomial.coeff_one]
split_ifs with h _ h'
Β· rfl
Β· rfl
Β· subst h'; simp at h
Β· rfl
#align power_series.trunc_one PowerSeries.trunc_one
@[simp]
theorem trunc_C (n) (a : R) : trunc (n + 1) (C R a) = Polynomial.C a :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_C, Polynomial.coeff_C]
split_ifs with H <;> first |rfl|try simp_all
set_option linter.uppercaseLean3 false in
#align power_series.trunc_C PowerSeries.trunc_C
@[simp]
theorem trunc_add (n) (Ο Ο : Rβ¦Xβ§) : trunc n (Ο + Ο) = trunc n Ο + trunc n Ο :=
Polynomial.ext fun m => by
simp only [coeff_trunc, AddMonoidHom.map_add, Polynomial.coeff_add]
split_ifs with H
Β· rfl
Β· rw [zero_add]
#align power_series.trunc_add PowerSeries.trunc_add
theorem trunc_succ (f : Rβ¦Xβ§) (n : β) :
trunc n.succ f = trunc n f + Polynomial.monomial n (coeff R n f) := by
rw [trunc, Ico_zero_eq_range, sum_range_succ, trunc, Ico_zero_eq_range]
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 88 | 95 | theorem natDegree_trunc_lt (f : Rβ¦Xβ§) (n) : (trunc (n + 1) f).natDegree < n + 1 := by |
rw [Nat.lt_succ_iff, natDegree_le_iff_coeff_eq_zero]
intros
rw [coeff_trunc]
split_ifs with h
Β· rw [lt_succ, β not_lt] at h
contradiction
Β· rfl
| 1,819 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section Trunc
variable [Semiring R]
open Finset Nat
def trunc (n : β) (Ο : Rβ¦Xβ§) : R[X] :=
β m β Ico 0 n, Polynomial.monomial m (coeff R m Ο)
#align power_series.trunc PowerSeries.trunc
theorem coeff_trunc (m) (n) (Ο : Rβ¦Xβ§) :
(trunc n Ο).coeff m = if m < n then coeff R m Ο else 0 := by
simp [trunc, Polynomial.coeff_sum, Polynomial.coeff_monomial, Nat.lt_succ_iff]
#align power_series.coeff_trunc PowerSeries.coeff_trunc
@[simp]
theorem trunc_zero (n) : trunc n (0 : Rβ¦Xβ§) = 0 :=
Polynomial.ext fun m => by
rw [coeff_trunc, LinearMap.map_zero, Polynomial.coeff_zero]
split_ifs <;> rfl
#align power_series.trunc_zero PowerSeries.trunc_zero
@[simp]
theorem trunc_one (n) : trunc (n + 1) (1 : Rβ¦Xβ§) = 1 :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_one, Polynomial.coeff_one]
split_ifs with h _ h'
Β· rfl
Β· rfl
Β· subst h'; simp at h
Β· rfl
#align power_series.trunc_one PowerSeries.trunc_one
@[simp]
theorem trunc_C (n) (a : R) : trunc (n + 1) (C R a) = Polynomial.C a :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_C, Polynomial.coeff_C]
split_ifs with H <;> first |rfl|try simp_all
set_option linter.uppercaseLean3 false in
#align power_series.trunc_C PowerSeries.trunc_C
@[simp]
theorem trunc_add (n) (Ο Ο : Rβ¦Xβ§) : trunc n (Ο + Ο) = trunc n Ο + trunc n Ο :=
Polynomial.ext fun m => by
simp only [coeff_trunc, AddMonoidHom.map_add, Polynomial.coeff_add]
split_ifs with H
Β· rfl
Β· rw [zero_add]
#align power_series.trunc_add PowerSeries.trunc_add
theorem trunc_succ (f : Rβ¦Xβ§) (n : β) :
trunc n.succ f = trunc n f + Polynomial.monomial n (coeff R n f) := by
rw [trunc, Ico_zero_eq_range, sum_range_succ, trunc, Ico_zero_eq_range]
theorem natDegree_trunc_lt (f : Rβ¦Xβ§) (n) : (trunc (n + 1) f).natDegree < n + 1 := by
rw [Nat.lt_succ_iff, natDegree_le_iff_coeff_eq_zero]
intros
rw [coeff_trunc]
split_ifs with h
Β· rw [lt_succ, β not_lt] at h
contradiction
Β· rfl
@[simp] lemma trunc_zero' {f : Rβ¦Xβ§} : trunc 0 f = 0 := rfl
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 99 | 106 | theorem degree_trunc_lt (f : Rβ¦Xβ§) (n) : (trunc n f).degree < n := by |
rw [degree_lt_iff_coeff_zero]
intros
rw [coeff_trunc]
split_ifs with h
Β· rw [β not_le] at h
contradiction
Β· rfl
| 1,819 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section Trunc
variable [Semiring R]
open Finset Nat
def trunc (n : β) (Ο : Rβ¦Xβ§) : R[X] :=
β m β Ico 0 n, Polynomial.monomial m (coeff R m Ο)
#align power_series.trunc PowerSeries.trunc
theorem coeff_trunc (m) (n) (Ο : Rβ¦Xβ§) :
(trunc n Ο).coeff m = if m < n then coeff R m Ο else 0 := by
simp [trunc, Polynomial.coeff_sum, Polynomial.coeff_monomial, Nat.lt_succ_iff]
#align power_series.coeff_trunc PowerSeries.coeff_trunc
@[simp]
theorem trunc_zero (n) : trunc n (0 : Rβ¦Xβ§) = 0 :=
Polynomial.ext fun m => by
rw [coeff_trunc, LinearMap.map_zero, Polynomial.coeff_zero]
split_ifs <;> rfl
#align power_series.trunc_zero PowerSeries.trunc_zero
@[simp]
theorem trunc_one (n) : trunc (n + 1) (1 : Rβ¦Xβ§) = 1 :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_one, Polynomial.coeff_one]
split_ifs with h _ h'
Β· rfl
Β· rfl
Β· subst h'; simp at h
Β· rfl
#align power_series.trunc_one PowerSeries.trunc_one
@[simp]
theorem trunc_C (n) (a : R) : trunc (n + 1) (C R a) = Polynomial.C a :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_C, Polynomial.coeff_C]
split_ifs with H <;> first |rfl|try simp_all
set_option linter.uppercaseLean3 false in
#align power_series.trunc_C PowerSeries.trunc_C
@[simp]
theorem trunc_add (n) (Ο Ο : Rβ¦Xβ§) : trunc n (Ο + Ο) = trunc n Ο + trunc n Ο :=
Polynomial.ext fun m => by
simp only [coeff_trunc, AddMonoidHom.map_add, Polynomial.coeff_add]
split_ifs with H
Β· rfl
Β· rw [zero_add]
#align power_series.trunc_add PowerSeries.trunc_add
theorem trunc_succ (f : Rβ¦Xβ§) (n : β) :
trunc n.succ f = trunc n f + Polynomial.monomial n (coeff R n f) := by
rw [trunc, Ico_zero_eq_range, sum_range_succ, trunc, Ico_zero_eq_range]
theorem natDegree_trunc_lt (f : Rβ¦Xβ§) (n) : (trunc (n + 1) f).natDegree < n + 1 := by
rw [Nat.lt_succ_iff, natDegree_le_iff_coeff_eq_zero]
intros
rw [coeff_trunc]
split_ifs with h
Β· rw [lt_succ, β not_lt] at h
contradiction
Β· rfl
@[simp] lemma trunc_zero' {f : Rβ¦Xβ§} : trunc 0 f = 0 := rfl
theorem degree_trunc_lt (f : Rβ¦Xβ§) (n) : (trunc n f).degree < n := by
rw [degree_lt_iff_coeff_zero]
intros
rw [coeff_trunc]
split_ifs with h
Β· rw [β not_le] at h
contradiction
Β· rfl
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 108 | 120 | theorem evalβ_trunc_eq_sum_range {S : Type*} [Semiring S] (s : S) (G : R β+* S) (n) (f : Rβ¦Xβ§) :
(trunc n f).evalβ G s = β i β range n, G (coeff R i f) * s ^ i := by |
cases n with
| zero =>
rw [trunc_zero', range_zero, sum_empty, evalβ_zero]
| succ n =>
have := natDegree_trunc_lt f n
rw [evalβ_eq_sum_range' (hn := this)]
apply sum_congr rfl
intro _ h
rw [mem_range] at h
congr
rw [coeff_trunc, if_pos h]
| 1,819 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Ring
variable {R S : Type*} [Ring R] [Ring S]
def invUnitsSub (u : RΛ£) : PowerSeries R :=
mk fun n => 1 /β u ^ (n + 1)
#align power_series.inv_units_sub PowerSeries.invUnitsSub
@[simp]
theorem coeff_invUnitsSub (u : RΛ£) (n : β) : coeff R n (invUnitsSub u) = 1 /β u ^ (n + 1) :=
coeff_mk _ _
#align power_series.coeff_inv_units_sub PowerSeries.coeff_invUnitsSub
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 47 | 48 | theorem constantCoeff_invUnitsSub (u : RΛ£) : constantCoeff R (invUnitsSub u) = 1 /β u := by |
rw [β coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Ring
variable {R S : Type*} [Ring R] [Ring S]
def invUnitsSub (u : RΛ£) : PowerSeries R :=
mk fun n => 1 /β u ^ (n + 1)
#align power_series.inv_units_sub PowerSeries.invUnitsSub
@[simp]
theorem coeff_invUnitsSub (u : RΛ£) (n : β) : coeff R n (invUnitsSub u) = 1 /β u ^ (n + 1) :=
coeff_mk _ _
#align power_series.coeff_inv_units_sub PowerSeries.coeff_invUnitsSub
@[simp]
theorem constantCoeff_invUnitsSub (u : RΛ£) : constantCoeff R (invUnitsSub u) = 1 /β u := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]
#align power_series.constant_coeff_inv_units_sub PowerSeries.constantCoeff_invUnitsSub
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 52 | 55 | theorem invUnitsSub_mul_X (u : RΛ£) : invUnitsSub u * X = invUnitsSub u * C R u - 1 := by |
ext (_ | n)
Β· simp
Β· simp [n.succ_ne_zero, pow_succ']
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Ring
variable {R S : Type*} [Ring R] [Ring S]
def invUnitsSub (u : RΛ£) : PowerSeries R :=
mk fun n => 1 /β u ^ (n + 1)
#align power_series.inv_units_sub PowerSeries.invUnitsSub
@[simp]
theorem coeff_invUnitsSub (u : RΛ£) (n : β) : coeff R n (invUnitsSub u) = 1 /β u ^ (n + 1) :=
coeff_mk _ _
#align power_series.coeff_inv_units_sub PowerSeries.coeff_invUnitsSub
@[simp]
theorem constantCoeff_invUnitsSub (u : RΛ£) : constantCoeff R (invUnitsSub u) = 1 /β u := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]
#align power_series.constant_coeff_inv_units_sub PowerSeries.constantCoeff_invUnitsSub
@[simp]
theorem invUnitsSub_mul_X (u : RΛ£) : invUnitsSub u * X = invUnitsSub u * C R u - 1 := by
ext (_ | n)
Β· simp
Β· simp [n.succ_ne_zero, pow_succ']
set_option linter.uppercaseLean3 false in
#align power_series.inv_units_sub_mul_X PowerSeries.invUnitsSub_mul_X
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 60 | 61 | theorem invUnitsSub_mul_sub (u : RΛ£) : invUnitsSub u * (C R u - X) = 1 := by |
simp [mul_sub, sub_sub_cancel]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Ring
variable {R S : Type*} [Ring R] [Ring S]
def invUnitsSub (u : RΛ£) : PowerSeries R :=
mk fun n => 1 /β u ^ (n + 1)
#align power_series.inv_units_sub PowerSeries.invUnitsSub
@[simp]
theorem coeff_invUnitsSub (u : RΛ£) (n : β) : coeff R n (invUnitsSub u) = 1 /β u ^ (n + 1) :=
coeff_mk _ _
#align power_series.coeff_inv_units_sub PowerSeries.coeff_invUnitsSub
@[simp]
theorem constantCoeff_invUnitsSub (u : RΛ£) : constantCoeff R (invUnitsSub u) = 1 /β u := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]
#align power_series.constant_coeff_inv_units_sub PowerSeries.constantCoeff_invUnitsSub
@[simp]
theorem invUnitsSub_mul_X (u : RΛ£) : invUnitsSub u * X = invUnitsSub u * C R u - 1 := by
ext (_ | n)
Β· simp
Β· simp [n.succ_ne_zero, pow_succ']
set_option linter.uppercaseLean3 false in
#align power_series.inv_units_sub_mul_X PowerSeries.invUnitsSub_mul_X
@[simp]
theorem invUnitsSub_mul_sub (u : RΛ£) : invUnitsSub u * (C R u - X) = 1 := by
simp [mul_sub, sub_sub_cancel]
#align power_series.inv_units_sub_mul_sub PowerSeries.invUnitsSub_mul_sub
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 64 | 68 | theorem map_invUnitsSub (f : R β+* S) (u : RΛ£) :
map f (invUnitsSub u) = invUnitsSub (Units.map (f : R β* S) u) := by |
ext
simp only [β map_pow, coeff_map, coeff_invUnitsSub, one_divp]
rfl
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section invOneSubPow
variable {S : Type*} [CommRing S] (d : β)
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 84 | 89 | theorem mk_one_mul_one_sub_eq_one : (mk 1 : Sβ¦Xβ§) * (1 - X) = 1 := by |
rw [mul_comm, ext_iff]
intro n
cases n with
| zero => simp
| succ n => simp [sub_mul]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section invOneSubPow
variable {S : Type*} [CommRing S] (d : β)
theorem mk_one_mul_one_sub_eq_one : (mk 1 : Sβ¦Xβ§) * (1 - X) = 1 := by
rw [mul_comm, ext_iff]
intro n
cases n with
| zero => simp
| succ n => simp [sub_mul]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 96 | 106 | theorem mk_one_pow_eq_mk_choose_add :
(mk 1 : Sβ¦Xβ§) ^ (d + 1) = (mk fun n => Nat.choose (d + n) d : Sβ¦Xβ§) := by |
induction d with
| zero => ext; simp
| succ d hd =>
ext n
rw [pow_add, hd, pow_one, mul_comm, coeff_mul]
simp_rw [coeff_mk, Pi.one_apply, one_mul]
norm_cast
rw [Finset.sum_antidiagonal_choose_add, β Nat.choose_succ_succ, Nat.succ_eq_add_one,
add_right_comm]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 174 | 176 | theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by |
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 181 | 182 | theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by |
rw [sin, coeff_mk, if_pos (even_bit0 n)]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by
rw [sin, coeff_mk, if_pos (even_bit0 n)]
#align power_series.coeff_sin_bit0 PowerSeries.coeff_sin_bit0
set_option linter.deprecated false in
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 187 | 189 | theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by |
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by
rw [sin, coeff_mk, if_pos (even_bit0 n)]
#align power_series.coeff_sin_bit0 PowerSeries.coeff_sin_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_sin_bit1 PowerSeries.coeff_sin_bit1
set_option linter.deprecated false in
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 194 | 196 | theorem coeff_cos_bit0 : coeff A (bit0 n) (cos A) = (-1) ^ n * coeff A (bit0 n) (exp A) := by |
rw [cos, coeff_mk, if_pos (even_bit0 n), Nat.bit0_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by
rw [sin, coeff_mk, if_pos (even_bit0 n)]
#align power_series.coeff_sin_bit0 PowerSeries.coeff_sin_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_sin_bit1 PowerSeries.coeff_sin_bit1
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit0 : coeff A (bit0 n) (cos A) = (-1) ^ n * coeff A (bit0 n) (exp A) := by
rw [cos, coeff_mk, if_pos (even_bit0 n), Nat.bit0_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_cos_bit0 PowerSeries.coeff_cos_bit0
set_option linter.deprecated false in
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 201 | 202 | theorem coeff_cos_bit1 : coeff A (bit1 n) (cos A) = 0 := by |
rw [cos, coeff_mk, if_neg n.not_even_bit1]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by
rw [sin, coeff_mk, if_pos (even_bit0 n)]
#align power_series.coeff_sin_bit0 PowerSeries.coeff_sin_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_sin_bit1 PowerSeries.coeff_sin_bit1
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit0 : coeff A (bit0 n) (cos A) = (-1) ^ n * coeff A (bit0 n) (exp A) := by
rw [cos, coeff_mk, if_pos (even_bit0 n), Nat.bit0_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_cos_bit0 PowerSeries.coeff_cos_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit1 : coeff A (bit1 n) (cos A) = 0 := by
rw [cos, coeff_mk, if_neg n.not_even_bit1]
#align power_series.coeff_cos_bit1 PowerSeries.coeff_cos_bit1
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 206 | 208 | theorem map_exp : map (f : A β+* A') (exp A) = exp A' := by |
ext
simp
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by
rw [sin, coeff_mk, if_pos (even_bit0 n)]
#align power_series.coeff_sin_bit0 PowerSeries.coeff_sin_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_sin_bit1 PowerSeries.coeff_sin_bit1
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit0 : coeff A (bit0 n) (cos A) = (-1) ^ n * coeff A (bit0 n) (exp A) := by
rw [cos, coeff_mk, if_pos (even_bit0 n), Nat.bit0_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_cos_bit0 PowerSeries.coeff_cos_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit1 : coeff A (bit1 n) (cos A) = 0 := by
rw [cos, coeff_mk, if_neg n.not_even_bit1]
#align power_series.coeff_cos_bit1 PowerSeries.coeff_cos_bit1
@[simp]
theorem map_exp : map (f : A β+* A') (exp A) = exp A' := by
ext
simp
#align power_series.map_exp PowerSeries.map_exp
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 212 | 214 | theorem map_sin : map f (sin A) = sin A' := by |
ext
simp [sin, apply_ite f]
| 1,820 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerSeries
section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra β A] [Algebra β A']
open Nat
def exp : PowerSeries A :=
mk fun n => algebraMap β A (1 / n !)
#align power_series.exp PowerSeries.exp
def sin : PowerSeries A :=
mk fun n => if Even n then 0 else algebraMap β A ((-1) ^ (n / 2) / n !)
#align power_series.sin PowerSeries.sin
def cos : PowerSeries A :=
mk fun n => if Even n then algebraMap β A ((-1) ^ (n / 2) / n !) else 0
#align power_series.cos PowerSeries.cos
variable {A A'} [Ring A] [Ring A'] [Algebra β A] [Algebra β A'] (n : β) (f : A β+* A')
@[simp]
theorem coeff_exp : coeff A n (exp A) = algebraMap β A (1 / n !) :=
coeff_mk _ _
#align power_series.coeff_exp PowerSeries.coeff_exp
@[simp]
theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by
rw [β coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
#align power_series.constant_coeff_exp PowerSeries.constantCoeff_exp
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 := by
rw [sin, coeff_mk, if_pos (even_bit0 n)]
#align power_series.coeff_sin_bit0 PowerSeries.coeff_sin_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_sin_bit1 PowerSeries.coeff_sin_bit1
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit0 : coeff A (bit0 n) (cos A) = (-1) ^ n * coeff A (bit0 n) (exp A) := by
rw [cos, coeff_mk, if_pos (even_bit0 n), Nat.bit0_div_two, β mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
#align power_series.coeff_cos_bit0 PowerSeries.coeff_cos_bit0
set_option linter.deprecated false in
@[simp]
theorem coeff_cos_bit1 : coeff A (bit1 n) (cos A) = 0 := by
rw [cos, coeff_mk, if_neg n.not_even_bit1]
#align power_series.coeff_cos_bit1 PowerSeries.coeff_cos_bit1
@[simp]
theorem map_exp : map (f : A β+* A') (exp A) = exp A' := by
ext
simp
#align power_series.map_exp PowerSeries.map_exp
@[simp]
theorem map_sin : map f (sin A) = sin A' := by
ext
simp [sin, apply_ite f]
#align power_series.map_sin PowerSeries.map_sin
@[simp]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 218 | 220 | theorem map_cos : map f (cos A) = cos A' := by |
ext
simp [cos, apply_ite f]
| 1,820 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
| Mathlib/RingTheory/PowerSeries/Order.lean | 47 | 51 | theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by |
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
| Mathlib/RingTheory/PowerSeries/Order.lean | 68 | 75 | theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by |
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
| Mathlib/RingTheory/PowerSeries/Order.lean | 80 | 84 | theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by |
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
| Mathlib/RingTheory/PowerSeries/Order.lean | 89 | 94 | theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by |
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
#align power_series.order_le PowerSeries.order_le
| Mathlib/RingTheory/PowerSeries/Order.lean | 99 | 101 | theorem coeff_of_lt_order (n : β) (h : βn < order Ο) : coeff R n Ο = 0 := by |
contrapose! h
exact order_le _ h
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
#align power_series.order_le PowerSeries.order_le
theorem coeff_of_lt_order (n : β) (h : βn < order Ο) : coeff R n Ο = 0 := by
contrapose! h
exact order_le _ h
#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order
@[simp]
theorem order_eq_top {Ο : Rβ¦Xβ§} : Ο.order = β€ β Ο = 0 :=
PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left
#align power_series.order_eq_top PowerSeries.order_eq_top
| Mathlib/RingTheory/PowerSeries/Order.lean | 112 | 116 | theorem nat_le_order (Ο : Rβ¦Xβ§) (n : β) (h : β i < n, coeff R i Ο = 0) : βn β€ order Ο := by |
by_contra H; rw [not_le] at H
have : (order Ο).Dom := PartENat.dom_of_le_natCast H.le
rw [β PartENat.natCast_get this, PartENat.coe_lt_coe] at H
exact coeff_order this (h _ H)
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
#align power_series.order_le PowerSeries.order_le
theorem coeff_of_lt_order (n : β) (h : βn < order Ο) : coeff R n Ο = 0 := by
contrapose! h
exact order_le _ h
#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order
@[simp]
theorem order_eq_top {Ο : Rβ¦Xβ§} : Ο.order = β€ β Ο = 0 :=
PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left
#align power_series.order_eq_top PowerSeries.order_eq_top
theorem nat_le_order (Ο : Rβ¦Xβ§) (n : β) (h : β i < n, coeff R i Ο = 0) : βn β€ order Ο := by
by_contra H; rw [not_le] at H
have : (order Ο).Dom := PartENat.dom_of_le_natCast H.le
rw [β PartENat.natCast_get this, PartENat.coe_lt_coe] at H
exact coeff_order this (h _ H)
#align power_series.nat_le_order PowerSeries.nat_le_order
| Mathlib/RingTheory/PowerSeries/Order.lean | 121 | 129 | theorem le_order (Ο : Rβ¦Xβ§) (n : PartENat) (h : β i : β, βi < n β coeff R i Ο = 0) :
n β€ order Ο := by |
induction n using PartENat.casesOn
Β· show _ β€ _
rw [top_le_iff, order_eq_top]
ext i
exact h _ (PartENat.natCast_lt_top i)
Β· apply nat_le_order
simpa only [PartENat.coe_lt_coe] using h
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
#align power_series.order_le PowerSeries.order_le
theorem coeff_of_lt_order (n : β) (h : βn < order Ο) : coeff R n Ο = 0 := by
contrapose! h
exact order_le _ h
#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order
@[simp]
theorem order_eq_top {Ο : Rβ¦Xβ§} : Ο.order = β€ β Ο = 0 :=
PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left
#align power_series.order_eq_top PowerSeries.order_eq_top
theorem nat_le_order (Ο : Rβ¦Xβ§) (n : β) (h : β i < n, coeff R i Ο = 0) : βn β€ order Ο := by
by_contra H; rw [not_le] at H
have : (order Ο).Dom := PartENat.dom_of_le_natCast H.le
rw [β PartENat.natCast_get this, PartENat.coe_lt_coe] at H
exact coeff_order this (h _ H)
#align power_series.nat_le_order PowerSeries.nat_le_order
theorem le_order (Ο : Rβ¦Xβ§) (n : PartENat) (h : β i : β, βi < n β coeff R i Ο = 0) :
n β€ order Ο := by
induction n using PartENat.casesOn
Β· show _ β€ _
rw [top_le_iff, order_eq_top]
ext i
exact h _ (PartENat.natCast_lt_top i)
Β· apply nat_le_order
simpa only [PartENat.coe_lt_coe] using h
#align power_series.le_order PowerSeries.le_order
| Mathlib/RingTheory/PowerSeries/Order.lean | 134 | 139 | theorem order_eq_nat {Ο : Rβ¦Xβ§} {n : β} :
order Ο = n β coeff R n Ο β 0 β§ β i, i < n β coeff R i Ο = 0 := by |
classical
rcases eq_or_ne Ο 0 with (rfl | hΟ)
Β· simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm
simp [order, dif_neg hΟ, Nat.find_eq_iff]
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
#align power_series.order_le PowerSeries.order_le
theorem coeff_of_lt_order (n : β) (h : βn < order Ο) : coeff R n Ο = 0 := by
contrapose! h
exact order_le _ h
#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order
@[simp]
theorem order_eq_top {Ο : Rβ¦Xβ§} : Ο.order = β€ β Ο = 0 :=
PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left
#align power_series.order_eq_top PowerSeries.order_eq_top
theorem nat_le_order (Ο : Rβ¦Xβ§) (n : β) (h : β i < n, coeff R i Ο = 0) : βn β€ order Ο := by
by_contra H; rw [not_le] at H
have : (order Ο).Dom := PartENat.dom_of_le_natCast H.le
rw [β PartENat.natCast_get this, PartENat.coe_lt_coe] at H
exact coeff_order this (h _ H)
#align power_series.nat_le_order PowerSeries.nat_le_order
theorem le_order (Ο : Rβ¦Xβ§) (n : PartENat) (h : β i : β, βi < n β coeff R i Ο = 0) :
n β€ order Ο := by
induction n using PartENat.casesOn
Β· show _ β€ _
rw [top_le_iff, order_eq_top]
ext i
exact h _ (PartENat.natCast_lt_top i)
Β· apply nat_le_order
simpa only [PartENat.coe_lt_coe] using h
#align power_series.le_order PowerSeries.le_order
theorem order_eq_nat {Ο : Rβ¦Xβ§} {n : β} :
order Ο = n β coeff R n Ο β 0 β§ β i, i < n β coeff R i Ο = 0 := by
classical
rcases eq_or_ne Ο 0 with (rfl | hΟ)
Β· simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm
simp [order, dif_neg hΟ, Nat.find_eq_iff]
#align power_series.order_eq_nat PowerSeries.order_eq_nat
| Mathlib/RingTheory/PowerSeries/Order.lean | 144 | 157 | theorem order_eq {Ο : Rβ¦Xβ§} {n : PartENat} :
order Ο = n β (β i : β, βi = n β coeff R i Ο β 0) β§ β i : β, βi < n β coeff R i Ο = 0 := by |
induction n using PartENat.casesOn
Β· rw [order_eq_top]
constructor
Β· rintro rfl
constructor <;> intros
Β· exfalso
exact PartENat.natCast_ne_top βΉ_βΊ βΉ_βΊ
Β· exact (coeff _ _).map_zero
Β· rintro β¨_hβ, hββ©
ext i
exact hβ i (PartENat.natCast_lt_top i)
Β· simpa [PartENat.natCast_inj] using order_eq_nat
| 1,821 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_antidiagonal)
namespace PowerSeries
open Finsupp (single)
variable {R : Type*}
section OrderBasic
open multiplicity
variable [Semiring R] {Ο : Rβ¦Xβ§}
theorem exists_coeff_ne_zero_iff_ne_zero : (β n : β, coeff R n Ο β 0) β Ο β 0 := by
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero
def order (Ο : Rβ¦Xβ§) : PartENat :=
letI := Classical.decEq R
letI := Classical.decEq Rβ¦Xβ§
if h : Ο = 0 then β€ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)
#align power_series.order PowerSeries.order
@[simp]
theorem order_zero : order (0 : Rβ¦Xβ§) = β€ :=
dif_pos rfl
#align power_series.order_zero PowerSeries.order_zero
theorem order_finite_iff_ne_zero : (order Ο).Dom β Ο β 0 := by
simp only [order]
constructor
Β· split_ifs with h <;> intro H
Β· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
Β· exact h
Β· intro h
simp [h]
#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero
theorem coeff_order (h : (order Ο).Dom) : coeff R (Ο.order.get h) Ο β 0 := by
classical
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']
generalize_proofs h
exact Nat.find_spec h
#align power_series.coeff_order PowerSeries.coeff_order
theorem order_le (n : β) (h : coeff R n Ο β 0) : order Ο β€ n := by
classical
rw [order, dif_neg]
Β· simp only [PartENat.coe_le_coe]
exact Nat.find_le h
Β· exact exists_coeff_ne_zero_iff_ne_zero.mp β¨n, hβ©
#align power_series.order_le PowerSeries.order_le
theorem coeff_of_lt_order (n : β) (h : βn < order Ο) : coeff R n Ο = 0 := by
contrapose! h
exact order_le _ h
#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order
@[simp]
theorem order_eq_top {Ο : Rβ¦Xβ§} : Ο.order = β€ β Ο = 0 :=
PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left
#align power_series.order_eq_top PowerSeries.order_eq_top
theorem nat_le_order (Ο : Rβ¦Xβ§) (n : β) (h : β i < n, coeff R i Ο = 0) : βn β€ order Ο := by
by_contra H; rw [not_le] at H
have : (order Ο).Dom := PartENat.dom_of_le_natCast H.le
rw [β PartENat.natCast_get this, PartENat.coe_lt_coe] at H
exact coeff_order this (h _ H)
#align power_series.nat_le_order PowerSeries.nat_le_order
theorem le_order (Ο : Rβ¦Xβ§) (n : PartENat) (h : β i : β, βi < n β coeff R i Ο = 0) :
n β€ order Ο := by
induction n using PartENat.casesOn
Β· show _ β€ _
rw [top_le_iff, order_eq_top]
ext i
exact h _ (PartENat.natCast_lt_top i)
Β· apply nat_le_order
simpa only [PartENat.coe_lt_coe] using h
#align power_series.le_order PowerSeries.le_order
theorem order_eq_nat {Ο : Rβ¦Xβ§} {n : β} :
order Ο = n β coeff R n Ο β 0 β§ β i, i < n β coeff R i Ο = 0 := by
classical
rcases eq_or_ne Ο 0 with (rfl | hΟ)
Β· simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm
simp [order, dif_neg hΟ, Nat.find_eq_iff]
#align power_series.order_eq_nat PowerSeries.order_eq_nat
theorem order_eq {Ο : Rβ¦Xβ§} {n : PartENat} :
order Ο = n β (β i : β, βi = n β coeff R i Ο β 0) β§ β i : β, βi < n β coeff R i Ο = 0 := by
induction n using PartENat.casesOn
Β· rw [order_eq_top]
constructor
Β· rintro rfl
constructor <;> intros
Β· exfalso
exact PartENat.natCast_ne_top βΉ_βΊ βΉ_βΊ
Β· exact (coeff _ _).map_zero
Β· rintro β¨_hβ, hββ©
ext i
exact hβ i (PartENat.natCast_lt_top i)
Β· simpa [PartENat.natCast_inj] using order_eq_nat
#align power_series.order_eq PowerSeries.order_eq
| Mathlib/RingTheory/PowerSeries/Order.lean | 162 | 164 | theorem le_order_add (Ο Ο : Rβ¦Xβ§) : min (order Ο) (order Ο) β€ order (Ο + Ο) := by |
refine le_order _ _ ?_
simp (config := { contextual := true }) [coeff_of_lt_order]
| 1,821 |
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
open Pointwise Polynomial
noncomputable section
variable {Ξ : Type*} {R : Type*}
namespace HahnSeries
section Semiring
variable [Semiring R]
@[simps]
def toPowerSeries : HahnSeries β R β+* PowerSeries R where
toFun f := PowerSeries.mk f.coeff
invFun f := β¨fun n => PowerSeries.coeff R n f, (Nat.lt_wfRel.wf.isWF _).isPWOβ©
left_inv f := by
ext
simp
right_inv f := by
ext
simp
map_add' f g := by
ext
simp
map_mul' f g := by
ext n
simp only [PowerSeries.coeff_mul, PowerSeries.coeff_mk, mul_coeff, isPWO_support]
classical
refine (sum_filter_ne_zero _).symm.trans <| (sum_congr ?_ fun _ _ β¦ rfl).trans <|
sum_filter_ne_zero _
ext m
simp only [mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter,
mem_support]
rintro h
rw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]
#align hahn_series.to_power_series HahnSeries.toPowerSeries
theorem coeff_toPowerSeries {f : HahnSeries β R} {n : β} :
PowerSeries.coeff R n (toPowerSeries f) = f.coeff n :=
PowerSeries.coeff_mk _ _
#align hahn_series.coeff_to_power_series HahnSeries.coeff_toPowerSeries
theorem coeff_toPowerSeries_symm {f : PowerSeries R} {n : β} :
(HahnSeries.toPowerSeries.symm f).coeff n = PowerSeries.coeff R n f :=
rfl
#align hahn_series.coeff_to_power_series_symm HahnSeries.coeff_toPowerSeries_symm
variable (Ξ R) [StrictOrderedSemiring Ξ]
def ofPowerSeries : PowerSeries R β+* HahnSeries Ξ R :=
(HahnSeries.embDomainRingHom (Nat.castAddMonoidHom Ξ) Nat.strictMono_cast.injective fun _ _ =>
Nat.cast_le).comp
(RingEquiv.toRingHom toPowerSeries.symm)
#align hahn_series.of_power_series HahnSeries.ofPowerSeries
variable {Ξ} {R}
theorem ofPowerSeries_injective : Function.Injective (ofPowerSeries Ξ R) :=
embDomain_injective.comp toPowerSeries.symm.injective
#align hahn_series.of_power_series_injective HahnSeries.ofPowerSeries_injective
theorem ofPowerSeries_apply (x : PowerSeries R) :
ofPowerSeries Ξ R x =
HahnSeries.embDomain
β¨β¨((β) : β β Ξ), Nat.strictMono_cast.injectiveβ©, by
simp only [Function.Embedding.coeFn_mk]
exact Nat.cast_leβ©
(toPowerSeries.symm x) :=
rfl
#align hahn_series.of_power_series_apply HahnSeries.ofPowerSeries_apply
| Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 112 | 113 | theorem ofPowerSeries_apply_coeff (x : PowerSeries R) (n : β) :
(ofPowerSeries Ξ R x).coeff n = PowerSeries.coeff R n x := by | simp [ofPowerSeries_apply]
| 1,822 |
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
open Pointwise Polynomial
noncomputable section
variable {Ξ : Type*} {R : Type*}
namespace HahnSeries
section Semiring
variable [Semiring R]
@[simps]
def toPowerSeries : HahnSeries β R β+* PowerSeries R where
toFun f := PowerSeries.mk f.coeff
invFun f := β¨fun n => PowerSeries.coeff R n f, (Nat.lt_wfRel.wf.isWF _).isPWOβ©
left_inv f := by
ext
simp
right_inv f := by
ext
simp
map_add' f g := by
ext
simp
map_mul' f g := by
ext n
simp only [PowerSeries.coeff_mul, PowerSeries.coeff_mk, mul_coeff, isPWO_support]
classical
refine (sum_filter_ne_zero _).symm.trans <| (sum_congr ?_ fun _ _ β¦ rfl).trans <|
sum_filter_ne_zero _
ext m
simp only [mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter,
mem_support]
rintro h
rw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]
#align hahn_series.to_power_series HahnSeries.toPowerSeries
theorem coeff_toPowerSeries {f : HahnSeries β R} {n : β} :
PowerSeries.coeff R n (toPowerSeries f) = f.coeff n :=
PowerSeries.coeff_mk _ _
#align hahn_series.coeff_to_power_series HahnSeries.coeff_toPowerSeries
theorem coeff_toPowerSeries_symm {f : PowerSeries R} {n : β} :
(HahnSeries.toPowerSeries.symm f).coeff n = PowerSeries.coeff R n f :=
rfl
#align hahn_series.coeff_to_power_series_symm HahnSeries.coeff_toPowerSeries_symm
variable (Ξ R) [StrictOrderedSemiring Ξ]
def ofPowerSeries : PowerSeries R β+* HahnSeries Ξ R :=
(HahnSeries.embDomainRingHom (Nat.castAddMonoidHom Ξ) Nat.strictMono_cast.injective fun _ _ =>
Nat.cast_le).comp
(RingEquiv.toRingHom toPowerSeries.symm)
#align hahn_series.of_power_series HahnSeries.ofPowerSeries
variable {Ξ} {R}
theorem ofPowerSeries_injective : Function.Injective (ofPowerSeries Ξ R) :=
embDomain_injective.comp toPowerSeries.symm.injective
#align hahn_series.of_power_series_injective HahnSeries.ofPowerSeries_injective
theorem ofPowerSeries_apply (x : PowerSeries R) :
ofPowerSeries Ξ R x =
HahnSeries.embDomain
β¨β¨((β) : β β Ξ), Nat.strictMono_cast.injectiveβ©, by
simp only [Function.Embedding.coeFn_mk]
exact Nat.cast_leβ©
(toPowerSeries.symm x) :=
rfl
#align hahn_series.of_power_series_apply HahnSeries.ofPowerSeries_apply
theorem ofPowerSeries_apply_coeff (x : PowerSeries R) (n : β) :
(ofPowerSeries Ξ R x).coeff n = PowerSeries.coeff R n x := by simp [ofPowerSeries_apply]
#align hahn_series.of_power_series_apply_coeff HahnSeries.ofPowerSeries_apply_coeff
@[simp]
| Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 117 | 128 | theorem ofPowerSeries_C (r : R) : ofPowerSeries Ξ R (PowerSeries.C R r) = HahnSeries.C r := by |
ext n
simp only [ofPowerSeries_apply, C, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, ne_eq,
single_coeff]
split_ifs with hn
Β· subst hn
convert @embDomain_coeff β R _ _ Ξ _ _ _ 0 <;> simp
Β· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_C]
intro
simp (config := { contextual := true }) [Ne.symm hn]
| 1,822 |
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
open Pointwise Polynomial
noncomputable section
variable {Ξ : Type*} {R : Type*}
namespace HahnSeries
section Semiring
variable [Semiring R]
@[simps]
def toPowerSeries : HahnSeries β R β+* PowerSeries R where
toFun f := PowerSeries.mk f.coeff
invFun f := β¨fun n => PowerSeries.coeff R n f, (Nat.lt_wfRel.wf.isWF _).isPWOβ©
left_inv f := by
ext
simp
right_inv f := by
ext
simp
map_add' f g := by
ext
simp
map_mul' f g := by
ext n
simp only [PowerSeries.coeff_mul, PowerSeries.coeff_mk, mul_coeff, isPWO_support]
classical
refine (sum_filter_ne_zero _).symm.trans <| (sum_congr ?_ fun _ _ β¦ rfl).trans <|
sum_filter_ne_zero _
ext m
simp only [mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter,
mem_support]
rintro h
rw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]
#align hahn_series.to_power_series HahnSeries.toPowerSeries
theorem coeff_toPowerSeries {f : HahnSeries β R} {n : β} :
PowerSeries.coeff R n (toPowerSeries f) = f.coeff n :=
PowerSeries.coeff_mk _ _
#align hahn_series.coeff_to_power_series HahnSeries.coeff_toPowerSeries
theorem coeff_toPowerSeries_symm {f : PowerSeries R} {n : β} :
(HahnSeries.toPowerSeries.symm f).coeff n = PowerSeries.coeff R n f :=
rfl
#align hahn_series.coeff_to_power_series_symm HahnSeries.coeff_toPowerSeries_symm
variable (Ξ R) [StrictOrderedSemiring Ξ]
def ofPowerSeries : PowerSeries R β+* HahnSeries Ξ R :=
(HahnSeries.embDomainRingHom (Nat.castAddMonoidHom Ξ) Nat.strictMono_cast.injective fun _ _ =>
Nat.cast_le).comp
(RingEquiv.toRingHom toPowerSeries.symm)
#align hahn_series.of_power_series HahnSeries.ofPowerSeries
variable {Ξ} {R}
theorem ofPowerSeries_injective : Function.Injective (ofPowerSeries Ξ R) :=
embDomain_injective.comp toPowerSeries.symm.injective
#align hahn_series.of_power_series_injective HahnSeries.ofPowerSeries_injective
theorem ofPowerSeries_apply (x : PowerSeries R) :
ofPowerSeries Ξ R x =
HahnSeries.embDomain
β¨β¨((β) : β β Ξ), Nat.strictMono_cast.injectiveβ©, by
simp only [Function.Embedding.coeFn_mk]
exact Nat.cast_leβ©
(toPowerSeries.symm x) :=
rfl
#align hahn_series.of_power_series_apply HahnSeries.ofPowerSeries_apply
theorem ofPowerSeries_apply_coeff (x : PowerSeries R) (n : β) :
(ofPowerSeries Ξ R x).coeff n = PowerSeries.coeff R n x := by simp [ofPowerSeries_apply]
#align hahn_series.of_power_series_apply_coeff HahnSeries.ofPowerSeries_apply_coeff
@[simp]
theorem ofPowerSeries_C (r : R) : ofPowerSeries Ξ R (PowerSeries.C R r) = HahnSeries.C r := by
ext n
simp only [ofPowerSeries_apply, C, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, ne_eq,
single_coeff]
split_ifs with hn
Β· subst hn
convert @embDomain_coeff β R _ _ Ξ _ _ _ 0 <;> simp
Β· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_C]
intro
simp (config := { contextual := true }) [Ne.symm hn]
#align hahn_series.of_power_series_C HahnSeries.ofPowerSeries_C
@[simp]
| Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 132 | 142 | theorem ofPowerSeries_X : ofPowerSeries Ξ R PowerSeries.X = single 1 1 := by |
ext n
simp only [single_coeff, ofPowerSeries_apply, RingHom.coe_mk]
split_ifs with hn
Β· rw [hn]
convert @embDomain_coeff β R _ _ Ξ _ _ _ 1 <;> simp
Β· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_X]
intro
simp (config := { contextual := true }) [Ne.symm hn]
| 1,822 |
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
open Pointwise Polynomial
noncomputable section
variable {Ξ : Type*} {R : Type*}
namespace HahnSeries
section Semiring
variable [Semiring R]
@[simps]
def toPowerSeries : HahnSeries β R β+* PowerSeries R where
toFun f := PowerSeries.mk f.coeff
invFun f := β¨fun n => PowerSeries.coeff R n f, (Nat.lt_wfRel.wf.isWF _).isPWOβ©
left_inv f := by
ext
simp
right_inv f := by
ext
simp
map_add' f g := by
ext
simp
map_mul' f g := by
ext n
simp only [PowerSeries.coeff_mul, PowerSeries.coeff_mk, mul_coeff, isPWO_support]
classical
refine (sum_filter_ne_zero _).symm.trans <| (sum_congr ?_ fun _ _ β¦ rfl).trans <|
sum_filter_ne_zero _
ext m
simp only [mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter,
mem_support]
rintro h
rw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]
#align hahn_series.to_power_series HahnSeries.toPowerSeries
theorem coeff_toPowerSeries {f : HahnSeries β R} {n : β} :
PowerSeries.coeff R n (toPowerSeries f) = f.coeff n :=
PowerSeries.coeff_mk _ _
#align hahn_series.coeff_to_power_series HahnSeries.coeff_toPowerSeries
theorem coeff_toPowerSeries_symm {f : PowerSeries R} {n : β} :
(HahnSeries.toPowerSeries.symm f).coeff n = PowerSeries.coeff R n f :=
rfl
#align hahn_series.coeff_to_power_series_symm HahnSeries.coeff_toPowerSeries_symm
variable (Ξ R) [StrictOrderedSemiring Ξ]
def ofPowerSeries : PowerSeries R β+* HahnSeries Ξ R :=
(HahnSeries.embDomainRingHom (Nat.castAddMonoidHom Ξ) Nat.strictMono_cast.injective fun _ _ =>
Nat.cast_le).comp
(RingEquiv.toRingHom toPowerSeries.symm)
#align hahn_series.of_power_series HahnSeries.ofPowerSeries
variable {Ξ} {R}
theorem ofPowerSeries_injective : Function.Injective (ofPowerSeries Ξ R) :=
embDomain_injective.comp toPowerSeries.symm.injective
#align hahn_series.of_power_series_injective HahnSeries.ofPowerSeries_injective
theorem ofPowerSeries_apply (x : PowerSeries R) :
ofPowerSeries Ξ R x =
HahnSeries.embDomain
β¨β¨((β) : β β Ξ), Nat.strictMono_cast.injectiveβ©, by
simp only [Function.Embedding.coeFn_mk]
exact Nat.cast_leβ©
(toPowerSeries.symm x) :=
rfl
#align hahn_series.of_power_series_apply HahnSeries.ofPowerSeries_apply
theorem ofPowerSeries_apply_coeff (x : PowerSeries R) (n : β) :
(ofPowerSeries Ξ R x).coeff n = PowerSeries.coeff R n x := by simp [ofPowerSeries_apply]
#align hahn_series.of_power_series_apply_coeff HahnSeries.ofPowerSeries_apply_coeff
@[simp]
theorem ofPowerSeries_C (r : R) : ofPowerSeries Ξ R (PowerSeries.C R r) = HahnSeries.C r := by
ext n
simp only [ofPowerSeries_apply, C, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, ne_eq,
single_coeff]
split_ifs with hn
Β· subst hn
convert @embDomain_coeff β R _ _ Ξ _ _ _ 0 <;> simp
Β· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_C]
intro
simp (config := { contextual := true }) [Ne.symm hn]
#align hahn_series.of_power_series_C HahnSeries.ofPowerSeries_C
@[simp]
theorem ofPowerSeries_X : ofPowerSeries Ξ R PowerSeries.X = single 1 1 := by
ext n
simp only [single_coeff, ofPowerSeries_apply, RingHom.coe_mk]
split_ifs with hn
Β· rw [hn]
convert @embDomain_coeff β R _ _ Ξ _ _ _ 1 <;> simp
Β· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_X]
intro
simp (config := { contextual := true }) [Ne.symm hn]
#align hahn_series.of_power_series_X HahnSeries.ofPowerSeries_X
| Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 145 | 147 | theorem ofPowerSeries_X_pow {R} [Semiring R] (n : β) :
ofPowerSeries Ξ R (PowerSeries.X ^ n) = single (n : Ξ) 1 := by |
simp
| 1,822 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Function AddMonoidAlgebra Finsupp
noncomputable section
variable {R : Type*}
abbrev LaurentPolynomial (R : Type*) [Semiring R] :=
AddMonoidAlgebra R β€
#align laurent_polynomial LaurentPolynomial
@[nolint docBlame]
scoped[LaurentPolynomial] notation:9000 R "[T;Tβ»ΒΉ]" => LaurentPolynomial R
open LaurentPolynomial
-- Porting note: `ext` no longer applies `Finsupp.ext` automatically
@[ext]
theorem LaurentPolynomial.ext [Semiring R] {p q : R[T;Tβ»ΒΉ]} (h : β a, p a = q a) : p = q :=
Finsupp.ext h
def Polynomial.toLaurent [Semiring R] : R[X] β+* R[T;Tβ»ΒΉ] :=
(mapDomainRingHom R Int.ofNatHom).comp (toFinsuppIso R)
#align polynomial.to_laurent Polynomial.toLaurent
theorem Polynomial.toLaurent_apply [Semiring R] (p : R[X]) :
toLaurent p = p.toFinsupp.mapDomain (β) :=
rfl
#align polynomial.to_laurent_apply Polynomial.toLaurent_apply
def Polynomial.toLaurentAlg [CommSemiring R] : R[X] ββ[R] R[T;Tβ»ΒΉ] :=
(mapDomainAlgHom R R Int.ofNatHom).comp (toFinsuppIsoAlg R).toAlgHom
#align polynomial.to_laurent_alg Polynomial.toLaurentAlg
@[simp] lemma Polynomial.coe_toLaurentAlg [CommSemiring R] :
(toLaurentAlg : R[X] β R[T;Tβ»ΒΉ]) = toLaurent :=
rfl
theorem Polynomial.toLaurentAlg_apply [CommSemiring R] (f : R[X]) : toLaurentAlg f = toLaurent f :=
rfl
#align polynomial.to_laurent_alg_apply Polynomial.toLaurentAlg_apply
namespace LaurentPolynomial
section Semiring
variable [Semiring R]
theorem single_zero_one_eq_one : (Finsupp.single 0 1 : R[T;Tβ»ΒΉ]) = (1 : R[T;Tβ»ΒΉ]) :=
rfl
#align laurent_polynomial.single_zero_one_eq_one LaurentPolynomial.single_zero_one_eq_one
def C : R β+* R[T;Tβ»ΒΉ] :=
singleZeroRingHom
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C LaurentPolynomial.C
theorem algebraMap_apply {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :
algebraMap R (LaurentPolynomial A) r = C (algebraMap R A r) :=
rfl
#align laurent_polynomial.algebra_map_apply LaurentPolynomial.algebraMap_apply
theorem C_eq_algebraMap {R : Type*} [CommSemiring R] (r : R) : C r = algebraMap R R[T;Tβ»ΒΉ] r :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C_eq_algebra_map LaurentPolynomial.C_eq_algebraMap
theorem single_eq_C (r : R) : Finsupp.single 0 r = C r := rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.single_eq_C LaurentPolynomial.single_eq_C
@[simp] lemma C_apply (t : R) (n : β€) : C t n = if n = 0 then t else 0 := by
rw [β single_eq_C, Finsupp.single_apply]; aesop
def T (n : β€) : R[T;Tβ»ΒΉ] :=
Finsupp.single n 1
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T LaurentPolynomial.T
@[simp] lemma T_apply (m n : β€) : (T n : R[T;Tβ»ΒΉ]) m = if n = m then 1 else 0 :=
Finsupp.single_apply
@[simp]
theorem T_zero : (T 0 : R[T;Tβ»ΒΉ]) = 1 :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_zero LaurentPolynomial.T_zero
| Mathlib/Algebra/Polynomial/Laurent.lean | 185 | 187 | theorem T_add (m n : β€) : (T (m + n) : R[T;Tβ»ΒΉ]) = T m * T n := by |
-- Porting note: was `convert single_mul_single.symm`
simp [T, single_mul_single]
| 1,823 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Function AddMonoidAlgebra Finsupp
noncomputable section
variable {R : Type*}
abbrev LaurentPolynomial (R : Type*) [Semiring R] :=
AddMonoidAlgebra R β€
#align laurent_polynomial LaurentPolynomial
@[nolint docBlame]
scoped[LaurentPolynomial] notation:9000 R "[T;Tβ»ΒΉ]" => LaurentPolynomial R
open LaurentPolynomial
-- Porting note: `ext` no longer applies `Finsupp.ext` automatically
@[ext]
theorem LaurentPolynomial.ext [Semiring R] {p q : R[T;Tβ»ΒΉ]} (h : β a, p a = q a) : p = q :=
Finsupp.ext h
def Polynomial.toLaurent [Semiring R] : R[X] β+* R[T;Tβ»ΒΉ] :=
(mapDomainRingHom R Int.ofNatHom).comp (toFinsuppIso R)
#align polynomial.to_laurent Polynomial.toLaurent
theorem Polynomial.toLaurent_apply [Semiring R] (p : R[X]) :
toLaurent p = p.toFinsupp.mapDomain (β) :=
rfl
#align polynomial.to_laurent_apply Polynomial.toLaurent_apply
def Polynomial.toLaurentAlg [CommSemiring R] : R[X] ββ[R] R[T;Tβ»ΒΉ] :=
(mapDomainAlgHom R R Int.ofNatHom).comp (toFinsuppIsoAlg R).toAlgHom
#align polynomial.to_laurent_alg Polynomial.toLaurentAlg
@[simp] lemma Polynomial.coe_toLaurentAlg [CommSemiring R] :
(toLaurentAlg : R[X] β R[T;Tβ»ΒΉ]) = toLaurent :=
rfl
theorem Polynomial.toLaurentAlg_apply [CommSemiring R] (f : R[X]) : toLaurentAlg f = toLaurent f :=
rfl
#align polynomial.to_laurent_alg_apply Polynomial.toLaurentAlg_apply
namespace LaurentPolynomial
section Semiring
variable [Semiring R]
theorem single_zero_one_eq_one : (Finsupp.single 0 1 : R[T;Tβ»ΒΉ]) = (1 : R[T;Tβ»ΒΉ]) :=
rfl
#align laurent_polynomial.single_zero_one_eq_one LaurentPolynomial.single_zero_one_eq_one
def C : R β+* R[T;Tβ»ΒΉ] :=
singleZeroRingHom
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C LaurentPolynomial.C
theorem algebraMap_apply {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :
algebraMap R (LaurentPolynomial A) r = C (algebraMap R A r) :=
rfl
#align laurent_polynomial.algebra_map_apply LaurentPolynomial.algebraMap_apply
theorem C_eq_algebraMap {R : Type*} [CommSemiring R] (r : R) : C r = algebraMap R R[T;Tβ»ΒΉ] r :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C_eq_algebra_map LaurentPolynomial.C_eq_algebraMap
theorem single_eq_C (r : R) : Finsupp.single 0 r = C r := rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.single_eq_C LaurentPolynomial.single_eq_C
@[simp] lemma C_apply (t : R) (n : β€) : C t n = if n = 0 then t else 0 := by
rw [β single_eq_C, Finsupp.single_apply]; aesop
def T (n : β€) : R[T;Tβ»ΒΉ] :=
Finsupp.single n 1
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T LaurentPolynomial.T
@[simp] lemma T_apply (m n : β€) : (T n : R[T;Tβ»ΒΉ]) m = if n = m then 1 else 0 :=
Finsupp.single_apply
@[simp]
theorem T_zero : (T 0 : R[T;Tβ»ΒΉ]) = 1 :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_zero LaurentPolynomial.T_zero
theorem T_add (m n : β€) : (T (m + n) : R[T;Tβ»ΒΉ]) = T m * T n := by
-- Porting note: was `convert single_mul_single.symm`
simp [T, single_mul_single]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_add LaurentPolynomial.T_add
| Mathlib/Algebra/Polynomial/Laurent.lean | 191 | 191 | theorem T_sub (m n : β€) : (T (m - n) : R[T;Tβ»ΒΉ]) = T m * T (-n) := by | rw [β T_add, sub_eq_add_neg]
| 1,823 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Function AddMonoidAlgebra Finsupp
noncomputable section
variable {R : Type*}
abbrev LaurentPolynomial (R : Type*) [Semiring R] :=
AddMonoidAlgebra R β€
#align laurent_polynomial LaurentPolynomial
@[nolint docBlame]
scoped[LaurentPolynomial] notation:9000 R "[T;Tβ»ΒΉ]" => LaurentPolynomial R
open LaurentPolynomial
-- Porting note: `ext` no longer applies `Finsupp.ext` automatically
@[ext]
theorem LaurentPolynomial.ext [Semiring R] {p q : R[T;Tβ»ΒΉ]} (h : β a, p a = q a) : p = q :=
Finsupp.ext h
def Polynomial.toLaurent [Semiring R] : R[X] β+* R[T;Tβ»ΒΉ] :=
(mapDomainRingHom R Int.ofNatHom).comp (toFinsuppIso R)
#align polynomial.to_laurent Polynomial.toLaurent
theorem Polynomial.toLaurent_apply [Semiring R] (p : R[X]) :
toLaurent p = p.toFinsupp.mapDomain (β) :=
rfl
#align polynomial.to_laurent_apply Polynomial.toLaurent_apply
def Polynomial.toLaurentAlg [CommSemiring R] : R[X] ββ[R] R[T;Tβ»ΒΉ] :=
(mapDomainAlgHom R R Int.ofNatHom).comp (toFinsuppIsoAlg R).toAlgHom
#align polynomial.to_laurent_alg Polynomial.toLaurentAlg
@[simp] lemma Polynomial.coe_toLaurentAlg [CommSemiring R] :
(toLaurentAlg : R[X] β R[T;Tβ»ΒΉ]) = toLaurent :=
rfl
theorem Polynomial.toLaurentAlg_apply [CommSemiring R] (f : R[X]) : toLaurentAlg f = toLaurent f :=
rfl
#align polynomial.to_laurent_alg_apply Polynomial.toLaurentAlg_apply
namespace LaurentPolynomial
section Semiring
variable [Semiring R]
theorem single_zero_one_eq_one : (Finsupp.single 0 1 : R[T;Tβ»ΒΉ]) = (1 : R[T;Tβ»ΒΉ]) :=
rfl
#align laurent_polynomial.single_zero_one_eq_one LaurentPolynomial.single_zero_one_eq_one
def C : R β+* R[T;Tβ»ΒΉ] :=
singleZeroRingHom
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C LaurentPolynomial.C
theorem algebraMap_apply {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :
algebraMap R (LaurentPolynomial A) r = C (algebraMap R A r) :=
rfl
#align laurent_polynomial.algebra_map_apply LaurentPolynomial.algebraMap_apply
theorem C_eq_algebraMap {R : Type*} [CommSemiring R] (r : R) : C r = algebraMap R R[T;Tβ»ΒΉ] r :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C_eq_algebra_map LaurentPolynomial.C_eq_algebraMap
theorem single_eq_C (r : R) : Finsupp.single 0 r = C r := rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.single_eq_C LaurentPolynomial.single_eq_C
@[simp] lemma C_apply (t : R) (n : β€) : C t n = if n = 0 then t else 0 := by
rw [β single_eq_C, Finsupp.single_apply]; aesop
def T (n : β€) : R[T;Tβ»ΒΉ] :=
Finsupp.single n 1
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T LaurentPolynomial.T
@[simp] lemma T_apply (m n : β€) : (T n : R[T;Tβ»ΒΉ]) m = if n = m then 1 else 0 :=
Finsupp.single_apply
@[simp]
theorem T_zero : (T 0 : R[T;Tβ»ΒΉ]) = 1 :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_zero LaurentPolynomial.T_zero
theorem T_add (m n : β€) : (T (m + n) : R[T;Tβ»ΒΉ]) = T m * T n := by
-- Porting note: was `convert single_mul_single.symm`
simp [T, single_mul_single]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_add LaurentPolynomial.T_add
theorem T_sub (m n : β€) : (T (m - n) : R[T;Tβ»ΒΉ]) = T m * T (-n) := by rw [β T_add, sub_eq_add_neg]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_sub LaurentPolynomial.T_sub
@[simp]
| Mathlib/Algebra/Polynomial/Laurent.lean | 196 | 197 | theorem T_pow (m : β€) (n : β) : (T m ^ n : R[T;Tβ»ΒΉ]) = T (n * m) := by |
rw [T, T, single_pow n, one_pow, nsmul_eq_mul]
| 1,823 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Function AddMonoidAlgebra Finsupp
noncomputable section
variable {R : Type*}
abbrev LaurentPolynomial (R : Type*) [Semiring R] :=
AddMonoidAlgebra R β€
#align laurent_polynomial LaurentPolynomial
@[nolint docBlame]
scoped[LaurentPolynomial] notation:9000 R "[T;Tβ»ΒΉ]" => LaurentPolynomial R
open LaurentPolynomial
-- Porting note: `ext` no longer applies `Finsupp.ext` automatically
@[ext]
theorem LaurentPolynomial.ext [Semiring R] {p q : R[T;Tβ»ΒΉ]} (h : β a, p a = q a) : p = q :=
Finsupp.ext h
def Polynomial.toLaurent [Semiring R] : R[X] β+* R[T;Tβ»ΒΉ] :=
(mapDomainRingHom R Int.ofNatHom).comp (toFinsuppIso R)
#align polynomial.to_laurent Polynomial.toLaurent
theorem Polynomial.toLaurent_apply [Semiring R] (p : R[X]) :
toLaurent p = p.toFinsupp.mapDomain (β) :=
rfl
#align polynomial.to_laurent_apply Polynomial.toLaurent_apply
def Polynomial.toLaurentAlg [CommSemiring R] : R[X] ββ[R] R[T;Tβ»ΒΉ] :=
(mapDomainAlgHom R R Int.ofNatHom).comp (toFinsuppIsoAlg R).toAlgHom
#align polynomial.to_laurent_alg Polynomial.toLaurentAlg
@[simp] lemma Polynomial.coe_toLaurentAlg [CommSemiring R] :
(toLaurentAlg : R[X] β R[T;Tβ»ΒΉ]) = toLaurent :=
rfl
theorem Polynomial.toLaurentAlg_apply [CommSemiring R] (f : R[X]) : toLaurentAlg f = toLaurent f :=
rfl
#align polynomial.to_laurent_alg_apply Polynomial.toLaurentAlg_apply
namespace LaurentPolynomial
section Semiring
variable [Semiring R]
theorem single_zero_one_eq_one : (Finsupp.single 0 1 : R[T;Tβ»ΒΉ]) = (1 : R[T;Tβ»ΒΉ]) :=
rfl
#align laurent_polynomial.single_zero_one_eq_one LaurentPolynomial.single_zero_one_eq_one
def C : R β+* R[T;Tβ»ΒΉ] :=
singleZeroRingHom
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C LaurentPolynomial.C
theorem algebraMap_apply {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :
algebraMap R (LaurentPolynomial A) r = C (algebraMap R A r) :=
rfl
#align laurent_polynomial.algebra_map_apply LaurentPolynomial.algebraMap_apply
theorem C_eq_algebraMap {R : Type*} [CommSemiring R] (r : R) : C r = algebraMap R R[T;Tβ»ΒΉ] r :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C_eq_algebra_map LaurentPolynomial.C_eq_algebraMap
theorem single_eq_C (r : R) : Finsupp.single 0 r = C r := rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.single_eq_C LaurentPolynomial.single_eq_C
@[simp] lemma C_apply (t : R) (n : β€) : C t n = if n = 0 then t else 0 := by
rw [β single_eq_C, Finsupp.single_apply]; aesop
def T (n : β€) : R[T;Tβ»ΒΉ] :=
Finsupp.single n 1
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T LaurentPolynomial.T
@[simp] lemma T_apply (m n : β€) : (T n : R[T;Tβ»ΒΉ]) m = if n = m then 1 else 0 :=
Finsupp.single_apply
@[simp]
theorem T_zero : (T 0 : R[T;Tβ»ΒΉ]) = 1 :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_zero LaurentPolynomial.T_zero
theorem T_add (m n : β€) : (T (m + n) : R[T;Tβ»ΒΉ]) = T m * T n := by
-- Porting note: was `convert single_mul_single.symm`
simp [T, single_mul_single]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_add LaurentPolynomial.T_add
theorem T_sub (m n : β€) : (T (m - n) : R[T;Tβ»ΒΉ]) = T m * T (-n) := by rw [β T_add, sub_eq_add_neg]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_sub LaurentPolynomial.T_sub
@[simp]
theorem T_pow (m : β€) (n : β) : (T m ^ n : R[T;Tβ»ΒΉ]) = T (n * m) := by
rw [T, T, single_pow n, one_pow, nsmul_eq_mul]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_pow LaurentPolynomial.T_pow
@[simp]
| Mathlib/Algebra/Polynomial/Laurent.lean | 203 | 204 | theorem mul_T_assoc (f : R[T;Tβ»ΒΉ]) (m n : β€) : f * T m * T n = f * T (m + n) := by |
simp [β T_add, mul_assoc]
| 1,823 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Function AddMonoidAlgebra Finsupp
noncomputable section
variable {R : Type*}
abbrev LaurentPolynomial (R : Type*) [Semiring R] :=
AddMonoidAlgebra R β€
#align laurent_polynomial LaurentPolynomial
@[nolint docBlame]
scoped[LaurentPolynomial] notation:9000 R "[T;Tβ»ΒΉ]" => LaurentPolynomial R
open LaurentPolynomial
-- Porting note: `ext` no longer applies `Finsupp.ext` automatically
@[ext]
theorem LaurentPolynomial.ext [Semiring R] {p q : R[T;Tβ»ΒΉ]} (h : β a, p a = q a) : p = q :=
Finsupp.ext h
def Polynomial.toLaurent [Semiring R] : R[X] β+* R[T;Tβ»ΒΉ] :=
(mapDomainRingHom R Int.ofNatHom).comp (toFinsuppIso R)
#align polynomial.to_laurent Polynomial.toLaurent
theorem Polynomial.toLaurent_apply [Semiring R] (p : R[X]) :
toLaurent p = p.toFinsupp.mapDomain (β) :=
rfl
#align polynomial.to_laurent_apply Polynomial.toLaurent_apply
def Polynomial.toLaurentAlg [CommSemiring R] : R[X] ββ[R] R[T;Tβ»ΒΉ] :=
(mapDomainAlgHom R R Int.ofNatHom).comp (toFinsuppIsoAlg R).toAlgHom
#align polynomial.to_laurent_alg Polynomial.toLaurentAlg
@[simp] lemma Polynomial.coe_toLaurentAlg [CommSemiring R] :
(toLaurentAlg : R[X] β R[T;Tβ»ΒΉ]) = toLaurent :=
rfl
theorem Polynomial.toLaurentAlg_apply [CommSemiring R] (f : R[X]) : toLaurentAlg f = toLaurent f :=
rfl
#align polynomial.to_laurent_alg_apply Polynomial.toLaurentAlg_apply
namespace LaurentPolynomial
section Semiring
variable [Semiring R]
theorem single_zero_one_eq_one : (Finsupp.single 0 1 : R[T;Tβ»ΒΉ]) = (1 : R[T;Tβ»ΒΉ]) :=
rfl
#align laurent_polynomial.single_zero_one_eq_one LaurentPolynomial.single_zero_one_eq_one
def C : R β+* R[T;Tβ»ΒΉ] :=
singleZeroRingHom
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C LaurentPolynomial.C
theorem algebraMap_apply {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :
algebraMap R (LaurentPolynomial A) r = C (algebraMap R A r) :=
rfl
#align laurent_polynomial.algebra_map_apply LaurentPolynomial.algebraMap_apply
theorem C_eq_algebraMap {R : Type*} [CommSemiring R] (r : R) : C r = algebraMap R R[T;Tβ»ΒΉ] r :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.C_eq_algebra_map LaurentPolynomial.C_eq_algebraMap
theorem single_eq_C (r : R) : Finsupp.single 0 r = C r := rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.single_eq_C LaurentPolynomial.single_eq_C
@[simp] lemma C_apply (t : R) (n : β€) : C t n = if n = 0 then t else 0 := by
rw [β single_eq_C, Finsupp.single_apply]; aesop
def T (n : β€) : R[T;Tβ»ΒΉ] :=
Finsupp.single n 1
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T LaurentPolynomial.T
@[simp] lemma T_apply (m n : β€) : (T n : R[T;Tβ»ΒΉ]) m = if n = m then 1 else 0 :=
Finsupp.single_apply
@[simp]
theorem T_zero : (T 0 : R[T;Tβ»ΒΉ]) = 1 :=
rfl
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_zero LaurentPolynomial.T_zero
theorem T_add (m n : β€) : (T (m + n) : R[T;Tβ»ΒΉ]) = T m * T n := by
-- Porting note: was `convert single_mul_single.symm`
simp [T, single_mul_single]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_add LaurentPolynomial.T_add
theorem T_sub (m n : β€) : (T (m - n) : R[T;Tβ»ΒΉ]) = T m * T (-n) := by rw [β T_add, sub_eq_add_neg]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_sub LaurentPolynomial.T_sub
@[simp]
theorem T_pow (m : β€) (n : β) : (T m ^ n : R[T;Tβ»ΒΉ]) = T (n * m) := by
rw [T, T, single_pow n, one_pow, nsmul_eq_mul]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.T_pow LaurentPolynomial.T_pow
@[simp]
theorem mul_T_assoc (f : R[T;Tβ»ΒΉ]) (m n : β€) : f * T m * T n = f * T (m + n) := by
simp [β T_add, mul_assoc]
set_option linter.uppercaseLean3 false in
#align laurent_polynomial.mul_T_assoc LaurentPolynomial.mul_T_assoc
@[simp]
| Mathlib/Algebra/Polynomial/Laurent.lean | 209 | 212 | theorem single_eq_C_mul_T (r : R) (n : β€) :
(Finsupp.single n r : R[T;Tβ»ΒΉ]) = (C r * T n : R[T;Tβ»ΒΉ]) := by |
-- Porting note: was `convert single_mul_single.symm`
simp [C, T, single_mul_single]
| 1,823 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
variable (M : Type*) [Monoid M]
open Polynomial
namespace Polynomial
variable (R : Type*) [Semiring R]
variable {M}
-- Porting note: changed `(Β· β’ Β·) m` to `HSMul.hSMul m`
| Mathlib/Algebra/Polynomial/GroupRingAction.lean | 31 | 39 | theorem smul_eq_map [MulSemiringAction M R] (m : M) :
HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m) := by |
suffices DistribMulAction.toAddMonoidHom R[X] m =
(mapRingHom (MulSemiringAction.toRingHom M R m)).toAddMonoidHom by
ext1 r
exact DFunLike.congr_fun this r
ext n r : 2
change m β’ monomial n r = map (MulSemiringAction.toRingHom M R m) (monomial n r)
rw [Polynomial.map_monomial, Polynomial.smul_monomial, MulSemiringAction.toRingHom_apply]
| 1,824 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
variable (M : Type*) [Monoid M]
open Polynomial
namespace Polynomial
variable (R : Type*) [Semiring R]
variable {M}
-- Porting note: changed `(Β· β’ Β·) m` to `HSMul.hSMul m`
theorem smul_eq_map [MulSemiringAction M R] (m : M) :
HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m) := by
suffices DistribMulAction.toAddMonoidHom R[X] m =
(mapRingHom (MulSemiringAction.toRingHom M R m)).toAddMonoidHom by
ext1 r
exact DFunLike.congr_fun this r
ext n r : 2
change m β’ monomial n r = map (MulSemiringAction.toRingHom M R m) (monomial n r)
rw [Polynomial.map_monomial, Polynomial.smul_monomial, MulSemiringAction.toRingHom_apply]
#align polynomial.smul_eq_map Polynomial.smul_eq_map
variable (M)
noncomputable instance [MulSemiringAction M R] : MulSemiringAction M R[X] :=
{ Polynomial.distribMulAction with
smul_one := fun m β¦
smul_eq_map R m βΈ Polynomial.map_one (MulSemiringAction.toRingHom M R m)
smul_mul := fun m _ _ β¦
smul_eq_map R m βΈ Polynomial.map_mul (MulSemiringAction.toRingHom M R m) }
variable {M R}
variable [MulSemiringAction M R]
@[simp]
theorem smul_X (m : M) : (m β’ X : R[X]) = X :=
(smul_eq_map R m).symm βΈ map_X _
set_option linter.uppercaseLean3 false in
#align polynomial.smul_X Polynomial.smul_X
variable (S : Type*) [CommSemiring S] [MulSemiringAction M S]
theorem smul_eval_smul (m : M) (f : S[X]) (x : S) : (m β’ f).eval (m β’ x) = m β’ f.eval x :=
Polynomial.induction_on f (fun r β¦ by rw [smul_C, eval_C, eval_C])
(fun f g ihf ihg β¦ by rw [smul_add, eval_add, ihf, ihg, eval_add, smul_add]) fun n r _ β¦ by
rw [smul_mul', smul_pow', smul_C, smul_X, eval_mul, eval_C, eval_pow, eval_X, eval_mul, eval_C,
eval_pow, eval_X, smul_mul', smul_pow']
#align polynomial.smul_eval_smul Polynomial.smul_eval_smul
variable (G : Type*) [Group G]
| Mathlib/Algebra/Polynomial/GroupRingAction.lean | 71 | 73 | theorem eval_smul' [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
f.eval (g β’ x) = g β’ (gβ»ΒΉ β’ f).eval x := by |
rw [β smul_eval_smul, smul_inv_smul]
| 1,824 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
variable (M : Type*) [Monoid M]
open Polynomial
namespace Polynomial
variable (R : Type*) [Semiring R]
variable {M}
-- Porting note: changed `(Β· β’ Β·) m` to `HSMul.hSMul m`
theorem smul_eq_map [MulSemiringAction M R] (m : M) :
HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m) := by
suffices DistribMulAction.toAddMonoidHom R[X] m =
(mapRingHom (MulSemiringAction.toRingHom M R m)).toAddMonoidHom by
ext1 r
exact DFunLike.congr_fun this r
ext n r : 2
change m β’ monomial n r = map (MulSemiringAction.toRingHom M R m) (monomial n r)
rw [Polynomial.map_monomial, Polynomial.smul_monomial, MulSemiringAction.toRingHom_apply]
#align polynomial.smul_eq_map Polynomial.smul_eq_map
variable (M)
noncomputable instance [MulSemiringAction M R] : MulSemiringAction M R[X] :=
{ Polynomial.distribMulAction with
smul_one := fun m β¦
smul_eq_map R m βΈ Polynomial.map_one (MulSemiringAction.toRingHom M R m)
smul_mul := fun m _ _ β¦
smul_eq_map R m βΈ Polynomial.map_mul (MulSemiringAction.toRingHom M R m) }
variable {M R}
variable [MulSemiringAction M R]
@[simp]
theorem smul_X (m : M) : (m β’ X : R[X]) = X :=
(smul_eq_map R m).symm βΈ map_X _
set_option linter.uppercaseLean3 false in
#align polynomial.smul_X Polynomial.smul_X
variable (S : Type*) [CommSemiring S] [MulSemiringAction M S]
theorem smul_eval_smul (m : M) (f : S[X]) (x : S) : (m β’ f).eval (m β’ x) = m β’ f.eval x :=
Polynomial.induction_on f (fun r β¦ by rw [smul_C, eval_C, eval_C])
(fun f g ihf ihg β¦ by rw [smul_add, eval_add, ihf, ihg, eval_add, smul_add]) fun n r _ β¦ by
rw [smul_mul', smul_pow', smul_C, smul_X, eval_mul, eval_C, eval_pow, eval_X, eval_mul, eval_C,
eval_pow, eval_X, smul_mul', smul_pow']
#align polynomial.smul_eval_smul Polynomial.smul_eval_smul
variable (G : Type*) [Group G]
theorem eval_smul' [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
f.eval (g β’ x) = g β’ (gβ»ΒΉ β’ f).eval x := by
rw [β smul_eval_smul, smul_inv_smul]
#align polynomial.eval_smul' Polynomial.eval_smul'
| Mathlib/Algebra/Polynomial/GroupRingAction.lean | 76 | 78 | theorem smul_eval [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
(g β’ f).eval x = g β’ f.eval (gβ»ΒΉ β’ x) := by |
rw [β smul_eval_smul, smul_inv_smul]
| 1,824 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
| Mathlib/Algebra/Polynomial/Smeval.lean | 54 | 54 | theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by | rw [smeval_def]
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
| Mathlib/Algebra/Polynomial/Smeval.lean | 57 | 58 | theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
| Mathlib/Algebra/Polynomial/Smeval.lean | 61 | 63 | theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
| Mathlib/Algebra/Polynomial/Smeval.lean | 65 | 67 | theorem eval_eq_smeval : p.eval r = p.smeval r := by |
rw [eval_eq_sum, smeval_eq_sum]
rfl
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
theorem eval_eq_smeval : p.eval r = p.smeval r := by
rw [eval_eq_sum, smeval_eq_sum]
rfl
| Mathlib/Algebra/Polynomial/Smeval.lean | 69 | 74 | theorem evalβ_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R β+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.evalβ f x = p.smeval x := by |
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, evalβ_eq_sum]
rfl
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
theorem eval_eq_smeval : p.eval r = p.smeval r := by
rw [eval_eq_sum, smeval_eq_sum]
rfl
theorem evalβ_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R β+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.evalβ f x = p.smeval x := by
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, evalβ_eq_sum]
rfl
variable (R)
@[simp]
| Mathlib/Algebra/Polynomial/Smeval.lean | 79 | 80 | theorem smeval_zero : (0 : R[X]).smeval x = 0 := by |
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
theorem eval_eq_smeval : p.eval r = p.smeval r := by
rw [eval_eq_sum, smeval_eq_sum]
rfl
theorem evalβ_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R β+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.evalβ f x = p.smeval x := by
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, evalβ_eq_sum]
rfl
variable (R)
@[simp]
theorem smeval_zero : (0 : R[X]).smeval x = 0 := by
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
@[simp]
| Mathlib/Algebra/Polynomial/Smeval.lean | 83 | 85 | theorem smeval_one : (1 : R[X]).smeval x = 1 β’ x ^ 0 := by |
rw [β C_1, smeval_C]
simp only [Nat.cast_one, one_smul]
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
theorem eval_eq_smeval : p.eval r = p.smeval r := by
rw [eval_eq_sum, smeval_eq_sum]
rfl
theorem evalβ_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R β+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.evalβ f x = p.smeval x := by
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, evalβ_eq_sum]
rfl
variable (R)
@[simp]
theorem smeval_zero : (0 : R[X]).smeval x = 0 := by
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
@[simp]
theorem smeval_one : (1 : R[X]).smeval x = 1 β’ x ^ 0 := by
rw [β C_1, smeval_C]
simp only [Nat.cast_one, one_smul]
@[simp]
| Mathlib/Algebra/Polynomial/Smeval.lean | 88 | 90 | theorem smeval_X :
(X : R[X]).smeval x = x ^ 1 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_X_index, one_smul]
| 1,825 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[MulActionWithZero R S] (x : S)
def smul_pow : β β R β S := fun n r => r β’ x^n
irreducible_def smeval : S := p.sum (smul_pow x)
theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
@[simp]
theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
@[simp]
theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
theorem eval_eq_smeval : p.eval r = p.smeval r := by
rw [eval_eq_sum, smeval_eq_sum]
rfl
theorem evalβ_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R β+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.evalβ f x = p.smeval x := by
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, evalβ_eq_sum]
rfl
variable (R)
@[simp]
theorem smeval_zero : (0 : R[X]).smeval x = 0 := by
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
@[simp]
theorem smeval_one : (1 : R[X]).smeval x = 1 β’ x ^ 0 := by
rw [β C_1, smeval_C]
simp only [Nat.cast_one, one_smul]
@[simp]
theorem smeval_X :
(X : R[X]).smeval x = x ^ 1 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_X_index, one_smul]
@[simp]
| Mathlib/Algebra/Polynomial/Smeval.lean | 93 | 95 | theorem smeval_X_pow {n : β} :
(X ^ n : R[X]).smeval x = x ^ n := by |
simp only [smeval_eq_sum, smul_pow, X_pow_eq_monomial, zero_smul, sum_monomial_index, one_smul]
| 1,825 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 37 | 39 | theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by |
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 42 | 44 | theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by |
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 48 | 50 | theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by |
ext
simp
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by
ext
simp
#align polynomial.zero_scale_roots Polynomial.zero_scaleRoots
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 53 | 59 | theorem scaleRoots_ne_zero {p : R[X]} (hp : p β 0) (s : R) : scaleRoots p s β 0 := by |
intro h
have : p.coeff p.natDegree β 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] β β β R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by
ext
simp
#align polynomial.zero_scale_roots Polynomial.zero_scaleRoots
theorem scaleRoots_ne_zero {p : R[X]} (hp : p β 0) (s : R) : scaleRoots p s β 0 := by
intro h
have : p.coeff p.natDegree β 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] β β β R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
#align polynomial.scale_roots_ne_zero Polynomial.scaleRoots_ne_zero
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 62 | 64 | theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support β€ p.support := by |
intro
simpa using left_ne_zero_of_mul
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by
ext
simp
#align polynomial.zero_scale_roots Polynomial.zero_scaleRoots
theorem scaleRoots_ne_zero {p : R[X]} (hp : p β 0) (s : R) : scaleRoots p s β 0 := by
intro h
have : p.coeff p.natDegree β 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] β β β R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
#align polynomial.scale_roots_ne_zero Polynomial.scaleRoots_ne_zero
theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support β€ p.support := by
intro
simpa using left_ne_zero_of_mul
#align polynomial.support_scale_roots_le Polynomial.support_scaleRoots_le
theorem support_scaleRoots_eq (p : R[X]) {s : R} (hs : s β nonZeroDivisors R) :
(scaleRoots p s).support = p.support :=
le_antisymm (support_scaleRoots_le p s)
(by intro i
simp only [coeff_scaleRoots, Polynomial.mem_support_iff]
intro p_ne_zero ps_zero
have := pow_mem hs (p.natDegree - i) _ ps_zero
contradiction)
#align polynomial.support_scale_roots_eq Polynomial.support_scaleRoots_eq
@[simp]
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 78 | 86 | theorem degree_scaleRoots (p : R[X]) {s : R} : degree (scaleRoots p s) = degree p := by |
haveI := Classical.propDecidable
by_cases hp : p = 0
Β· rw [hp, zero_scaleRoots]
refine le_antisymm (Finset.sup_mono (support_scaleRoots_le p s)) (degree_le_degree ?_)
rw [coeff_scaleRoots_natDegree]
intro h
have := leadingCoeff_eq_zero.mp h
contradiction
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by
ext
simp
#align polynomial.zero_scale_roots Polynomial.zero_scaleRoots
theorem scaleRoots_ne_zero {p : R[X]} (hp : p β 0) (s : R) : scaleRoots p s β 0 := by
intro h
have : p.coeff p.natDegree β 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] β β β R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
#align polynomial.scale_roots_ne_zero Polynomial.scaleRoots_ne_zero
theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support β€ p.support := by
intro
simpa using left_ne_zero_of_mul
#align polynomial.support_scale_roots_le Polynomial.support_scaleRoots_le
theorem support_scaleRoots_eq (p : R[X]) {s : R} (hs : s β nonZeroDivisors R) :
(scaleRoots p s).support = p.support :=
le_antisymm (support_scaleRoots_le p s)
(by intro i
simp only [coeff_scaleRoots, Polynomial.mem_support_iff]
intro p_ne_zero ps_zero
have := pow_mem hs (p.natDegree - i) _ ps_zero
contradiction)
#align polynomial.support_scale_roots_eq Polynomial.support_scaleRoots_eq
@[simp]
theorem degree_scaleRoots (p : R[X]) {s : R} : degree (scaleRoots p s) = degree p := by
haveI := Classical.propDecidable
by_cases hp : p = 0
Β· rw [hp, zero_scaleRoots]
refine le_antisymm (Finset.sup_mono (support_scaleRoots_le p s)) (degree_le_degree ?_)
rw [coeff_scaleRoots_natDegree]
intro h
have := leadingCoeff_eq_zero.mp h
contradiction
#align polynomial.degree_scale_roots Polynomial.degree_scaleRoots
@[simp]
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 90 | 91 | theorem natDegree_scaleRoots (p : R[X]) (s : R) : natDegree (scaleRoots p s) = natDegree p := by |
simp only [natDegree, degree_scaleRoots]
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by
ext
simp
#align polynomial.zero_scale_roots Polynomial.zero_scaleRoots
theorem scaleRoots_ne_zero {p : R[X]} (hp : p β 0) (s : R) : scaleRoots p s β 0 := by
intro h
have : p.coeff p.natDegree β 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] β β β R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
#align polynomial.scale_roots_ne_zero Polynomial.scaleRoots_ne_zero
theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support β€ p.support := by
intro
simpa using left_ne_zero_of_mul
#align polynomial.support_scale_roots_le Polynomial.support_scaleRoots_le
theorem support_scaleRoots_eq (p : R[X]) {s : R} (hs : s β nonZeroDivisors R) :
(scaleRoots p s).support = p.support :=
le_antisymm (support_scaleRoots_le p s)
(by intro i
simp only [coeff_scaleRoots, Polynomial.mem_support_iff]
intro p_ne_zero ps_zero
have := pow_mem hs (p.natDegree - i) _ ps_zero
contradiction)
#align polynomial.support_scale_roots_eq Polynomial.support_scaleRoots_eq
@[simp]
theorem degree_scaleRoots (p : R[X]) {s : R} : degree (scaleRoots p s) = degree p := by
haveI := Classical.propDecidable
by_cases hp : p = 0
Β· rw [hp, zero_scaleRoots]
refine le_antisymm (Finset.sup_mono (support_scaleRoots_le p s)) (degree_le_degree ?_)
rw [coeff_scaleRoots_natDegree]
intro h
have := leadingCoeff_eq_zero.mp h
contradiction
#align polynomial.degree_scale_roots Polynomial.degree_scaleRoots
@[simp]
theorem natDegree_scaleRoots (p : R[X]) (s : R) : natDegree (scaleRoots p s) = natDegree p := by
simp only [natDegree, degree_scaleRoots]
#align polynomial.nat_degree_scale_roots Polynomial.natDegree_scaleRoots
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 94 | 95 | theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) β Monic p := by |
simp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]
| 1,826 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R S A K : Type*}
namespace Polynomial
open Polynomial
section Semiring
variable [Semiring R] [Semiring S]
noncomputable def scaleRoots (p : R[X]) (s : R) : R[X] :=
β i β p.support, monomial i (p.coeff i * s ^ (p.natDegree - i))
#align polynomial.scale_roots Polynomial.scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : β) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
#align polynomial.coeff_scale_roots Polynomial.coeff_scaleRoots
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
(scaleRoots p s).coeff p.natDegree = p.leadingCoeff := by
rw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]
#align polynomial.coeff_scale_roots_nat_degree Polynomial.coeff_scaleRoots_natDegree
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by
ext
simp
#align polynomial.zero_scale_roots Polynomial.zero_scaleRoots
theorem scaleRoots_ne_zero {p : R[X]} (hp : p β 0) (s : R) : scaleRoots p s β 0 := by
intro h
have : p.coeff p.natDegree β 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] β β β R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
#align polynomial.scale_roots_ne_zero Polynomial.scaleRoots_ne_zero
theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support β€ p.support := by
intro
simpa using left_ne_zero_of_mul
#align polynomial.support_scale_roots_le Polynomial.support_scaleRoots_le
theorem support_scaleRoots_eq (p : R[X]) {s : R} (hs : s β nonZeroDivisors R) :
(scaleRoots p s).support = p.support :=
le_antisymm (support_scaleRoots_le p s)
(by intro i
simp only [coeff_scaleRoots, Polynomial.mem_support_iff]
intro p_ne_zero ps_zero
have := pow_mem hs (p.natDegree - i) _ ps_zero
contradiction)
#align polynomial.support_scale_roots_eq Polynomial.support_scaleRoots_eq
@[simp]
theorem degree_scaleRoots (p : R[X]) {s : R} : degree (scaleRoots p s) = degree p := by
haveI := Classical.propDecidable
by_cases hp : p = 0
Β· rw [hp, zero_scaleRoots]
refine le_antisymm (Finset.sup_mono (support_scaleRoots_le p s)) (degree_le_degree ?_)
rw [coeff_scaleRoots_natDegree]
intro h
have := leadingCoeff_eq_zero.mp h
contradiction
#align polynomial.degree_scale_roots Polynomial.degree_scaleRoots
@[simp]
theorem natDegree_scaleRoots (p : R[X]) (s : R) : natDegree (scaleRoots p s) = natDegree p := by
simp only [natDegree, degree_scaleRoots]
#align polynomial.nat_degree_scale_roots Polynomial.natDegree_scaleRoots
theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) β Monic p := by
simp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]
#align polynomial.monic_scale_roots_iff Polynomial.monic_scaleRoots_iff
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 98 | 101 | theorem map_scaleRoots (p : R[X]) (x : R) (f : R β+* S) (h : f p.leadingCoeff β 0) :
(p.scaleRoots x).map f = (p.map f).scaleRoots (f x) := by |
ext
simp [Polynomial.natDegree_map_of_leadingCoeff_ne_zero _ h]
| 1,826 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section CommRing
variable [CommRing R] {p q : R[X]}
section
variable [Semiring S]
theorem natDegree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S}
(hz : aeval z p = 0) (inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.natDegree :=
natDegree_pos_of_evalβ_root hp (algebraMap R S) hz inj
#align polynomial.nat_degree_pos_of_aeval_root Polynomial.natDegree_pos_of_aeval_root
theorem degree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S} (hz : aeval z p = 0)
(inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.degree :=
natDegree_pos_iff_degree_pos.mp (natDegree_pos_of_aeval_root hp hz inj)
#align polynomial.degree_pos_of_aeval_root Polynomial.degree_pos_of_aeval_root
| Mathlib/Algebra/Polynomial/RingDivision.lean | 50 | 55 | theorem modByMonic_eq_of_dvd_sub (hq : q.Monic) {pβ pβ : R[X]} (h : q β£ pβ - pβ) :
pβ %β q = pβ %β q := by |
nontriviality R
obtain β¨f, sub_eqβ© := h
refine (div_modByMonic_unique (pβ /β q + f) _ hq β¨?_, degree_modByMonic_lt _ hqβ©).2
rw [sub_eq_iff_eq_add.mp sub_eq, mul_add, β add_assoc, modByMonic_add_div _ hq, add_comm]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section CommRing
variable [CommRing R] {p q : R[X]}
section
variable [Semiring S]
theorem natDegree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S}
(hz : aeval z p = 0) (inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.natDegree :=
natDegree_pos_of_evalβ_root hp (algebraMap R S) hz inj
#align polynomial.nat_degree_pos_of_aeval_root Polynomial.natDegree_pos_of_aeval_root
theorem degree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S} (hz : aeval z p = 0)
(inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.degree :=
natDegree_pos_iff_degree_pos.mp (natDegree_pos_of_aeval_root hp hz inj)
#align polynomial.degree_pos_of_aeval_root Polynomial.degree_pos_of_aeval_root
theorem modByMonic_eq_of_dvd_sub (hq : q.Monic) {pβ pβ : R[X]} (h : q β£ pβ - pβ) :
pβ %β q = pβ %β q := by
nontriviality R
obtain β¨f, sub_eqβ© := h
refine (div_modByMonic_unique (pβ /β q + f) _ hq β¨?_, degree_modByMonic_lt _ hqβ©).2
rw [sub_eq_iff_eq_add.mp sub_eq, mul_add, β add_assoc, modByMonic_add_div _ hq, add_comm]
#align polynomial.mod_by_monic_eq_of_dvd_sub Polynomial.modByMonic_eq_of_dvd_sub
| Mathlib/Algebra/Polynomial/RingDivision.lean | 58 | 69 | theorem add_modByMonic (pβ pβ : R[X]) : (pβ + pβ) %β q = pβ %β q + pβ %β q := by |
by_cases hq : q.Monic
Β· cases' subsingleton_or_nontrivial R with hR hR
Β· simp only [eq_iff_true_of_subsingleton]
Β· exact
(div_modByMonic_unique (pβ /β q + pβ /β q) _ hq
β¨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div _ hq, β add_assoc,
add_comm (q * _), modByMonic_add_div _ hq],
(degree_add_le _ _).trans_lt
(max_lt (degree_modByMonic_lt _ hq) (degree_modByMonic_lt _ hq))β©).2
Β· simp_rw [modByMonic_eq_of_not_monic _ hq]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section CommRing
variable [CommRing R] {p q : R[X]}
section
variable [Semiring S]
theorem natDegree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S}
(hz : aeval z p = 0) (inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.natDegree :=
natDegree_pos_of_evalβ_root hp (algebraMap R S) hz inj
#align polynomial.nat_degree_pos_of_aeval_root Polynomial.natDegree_pos_of_aeval_root
theorem degree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S} (hz : aeval z p = 0)
(inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.degree :=
natDegree_pos_iff_degree_pos.mp (natDegree_pos_of_aeval_root hp hz inj)
#align polynomial.degree_pos_of_aeval_root Polynomial.degree_pos_of_aeval_root
theorem modByMonic_eq_of_dvd_sub (hq : q.Monic) {pβ pβ : R[X]} (h : q β£ pβ - pβ) :
pβ %β q = pβ %β q := by
nontriviality R
obtain β¨f, sub_eqβ© := h
refine (div_modByMonic_unique (pβ /β q + f) _ hq β¨?_, degree_modByMonic_lt _ hqβ©).2
rw [sub_eq_iff_eq_add.mp sub_eq, mul_add, β add_assoc, modByMonic_add_div _ hq, add_comm]
#align polynomial.mod_by_monic_eq_of_dvd_sub Polynomial.modByMonic_eq_of_dvd_sub
theorem add_modByMonic (pβ pβ : R[X]) : (pβ + pβ) %β q = pβ %β q + pβ %β q := by
by_cases hq : q.Monic
Β· cases' subsingleton_or_nontrivial R with hR hR
Β· simp only [eq_iff_true_of_subsingleton]
Β· exact
(div_modByMonic_unique (pβ /β q + pβ /β q) _ hq
β¨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div _ hq, β add_assoc,
add_comm (q * _), modByMonic_add_div _ hq],
(degree_add_le _ _).trans_lt
(max_lt (degree_modByMonic_lt _ hq) (degree_modByMonic_lt _ hq))β©).2
Β· simp_rw [modByMonic_eq_of_not_monic _ hq]
#align polynomial.add_mod_by_monic Polynomial.add_modByMonic
| Mathlib/Algebra/Polynomial/RingDivision.lean | 72 | 80 | theorem smul_modByMonic (c : R) (p : R[X]) : c β’ p %β q = c β’ (p %β q) := by |
by_cases hq : q.Monic
Β· cases' subsingleton_or_nontrivial R with hR hR
Β· simp only [eq_iff_true_of_subsingleton]
Β· exact
(div_modByMonic_unique (c β’ (p /β q)) (c β’ (p %β q)) hq
β¨by rw [mul_smul_comm, β smul_add, modByMonic_add_div p hq],
(degree_smul_le _ _).trans_lt (degree_modByMonic_lt _ hq)β©).2
Β· simp_rw [modByMonic_eq_of_not_monic _ hq]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section CommRing
variable [CommRing R] {p q : R[X]}
section
variable [Semiring S]
theorem natDegree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S}
(hz : aeval z p = 0) (inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.natDegree :=
natDegree_pos_of_evalβ_root hp (algebraMap R S) hz inj
#align polynomial.nat_degree_pos_of_aeval_root Polynomial.natDegree_pos_of_aeval_root
theorem degree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p β 0) {z : S} (hz : aeval z p = 0)
(inj : β x : R, algebraMap R S x = 0 β x = 0) : 0 < p.degree :=
natDegree_pos_iff_degree_pos.mp (natDegree_pos_of_aeval_root hp hz inj)
#align polynomial.degree_pos_of_aeval_root Polynomial.degree_pos_of_aeval_root
theorem modByMonic_eq_of_dvd_sub (hq : q.Monic) {pβ pβ : R[X]} (h : q β£ pβ - pβ) :
pβ %β q = pβ %β q := by
nontriviality R
obtain β¨f, sub_eqβ© := h
refine (div_modByMonic_unique (pβ /β q + f) _ hq β¨?_, degree_modByMonic_lt _ hqβ©).2
rw [sub_eq_iff_eq_add.mp sub_eq, mul_add, β add_assoc, modByMonic_add_div _ hq, add_comm]
#align polynomial.mod_by_monic_eq_of_dvd_sub Polynomial.modByMonic_eq_of_dvd_sub
theorem add_modByMonic (pβ pβ : R[X]) : (pβ + pβ) %β q = pβ %β q + pβ %β q := by
by_cases hq : q.Monic
Β· cases' subsingleton_or_nontrivial R with hR hR
Β· simp only [eq_iff_true_of_subsingleton]
Β· exact
(div_modByMonic_unique (pβ /β q + pβ /β q) _ hq
β¨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div _ hq, β add_assoc,
add_comm (q * _), modByMonic_add_div _ hq],
(degree_add_le _ _).trans_lt
(max_lt (degree_modByMonic_lt _ hq) (degree_modByMonic_lt _ hq))β©).2
Β· simp_rw [modByMonic_eq_of_not_monic _ hq]
#align polynomial.add_mod_by_monic Polynomial.add_modByMonic
theorem smul_modByMonic (c : R) (p : R[X]) : c β’ p %β q = c β’ (p %β q) := by
by_cases hq : q.Monic
Β· cases' subsingleton_or_nontrivial R with hR hR
Β· simp only [eq_iff_true_of_subsingleton]
Β· exact
(div_modByMonic_unique (c β’ (p /β q)) (c β’ (p %β q)) hq
β¨by rw [mul_smul_comm, β smul_add, modByMonic_add_div p hq],
(degree_smul_le _ _).trans_lt (degree_modByMonic_lt _ hq)β©).2
Β· simp_rw [modByMonic_eq_of_not_monic _ hq]
#align polynomial.smul_mod_by_monic Polynomial.smul_modByMonic
@[simps]
def modByMonicHom (q : R[X]) : R[X] ββ[R] R[X] where
toFun p := p %β q
map_add' := add_modByMonic
map_smul' := smul_modByMonic
#align polynomial.mod_by_monic_hom Polynomial.modByMonicHom
theorem neg_modByMonic (p mod : R[X]) : (-p) %β mod = - (p %β mod) :=
(modByMonicHom mod).map_neg p
theorem sub_modByMonic (a b mod : R[X]) : (a - b) %β mod = a %β mod - b %β mod :=
(modByMonicHom mod).map_sub a b
end
section
variable [Ring S]
| Mathlib/Algebra/Polynomial/RingDivision.lean | 103 | 107 | theorem aeval_modByMonic_eq_self_of_root [Algebra R S] {p q : R[X]} (hq : q.Monic) {x : S}
(hx : aeval x q = 0) : aeval x (p %β q) = aeval x p := by |
--`evalβ_modByMonic_eq_self_of_root` doesn't work here as it needs commutativity
rw [modByMonic_eq_sub_mul_div p hq, _root_.map_sub, _root_.map_mul, hx, zero_mul,
sub_zero]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
| Mathlib/Algebra/Polynomial/RingDivision.lean | 124 | 126 | theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by |
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
| Mathlib/Algebra/Polynomial/RingDivision.lean | 129 | 136 | theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by |
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
| Mathlib/Algebra/Polynomial/RingDivision.lean | 140 | 145 | theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by |
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
| Mathlib/Algebra/Polynomial/RingDivision.lean | 148 | 153 | theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by |
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
| Mathlib/Algebra/Polynomial/RingDivision.lean | 156 | 158 | theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by |
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
| Mathlib/Algebra/Polynomial/RingDivision.lean | 161 | 163 | theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by |
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
#align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd
| Mathlib/Algebra/Polynomial/RingDivision.lean | 166 | 169 | theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by |
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
#align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd
theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_degree_lt Polynomial.eq_zero_of_dvd_of_degree_lt
| Mathlib/Algebra/Polynomial/RingDivision.lean | 172 | 175 | theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (hβ : p β£ q) (hβ : natDegree q < natDegree p) :
q = 0 := by |
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (natDegree_le_of_dvd hβ hc)
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
#align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd
theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_degree_lt Polynomial.eq_zero_of_dvd_of_degree_lt
theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (hβ : p β£ q) (hβ : natDegree q < natDegree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (natDegree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_nat_degree_lt Polynomial.eq_zero_of_dvd_of_natDegree_lt
| Mathlib/Algebra/Polynomial/RingDivision.lean | 178 | 180 | theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q β 0) (hl : q.degree < p.degree) : Β¬p β£ q := by |
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl)
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
#align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd
theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_degree_lt Polynomial.eq_zero_of_dvd_of_degree_lt
theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (hβ : p β£ q) (hβ : natDegree q < natDegree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (natDegree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_nat_degree_lt Polynomial.eq_zero_of_dvd_of_natDegree_lt
theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q β 0) (hl : q.degree < p.degree) : Β¬p β£ q := by
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl)
#align polynomial.not_dvd_of_degree_lt Polynomial.not_dvd_of_degree_lt
| Mathlib/Algebra/Polynomial/RingDivision.lean | 183 | 186 | theorem not_dvd_of_natDegree_lt {p q : R[X]} (h0 : q β 0) (hl : q.natDegree < p.natDegree) :
Β¬p β£ q := by |
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_natDegree_lt hcontra hl)
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
#align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd
theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_degree_lt Polynomial.eq_zero_of_dvd_of_degree_lt
theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (hβ : p β£ q) (hβ : natDegree q < natDegree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (natDegree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_nat_degree_lt Polynomial.eq_zero_of_dvd_of_natDegree_lt
theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q β 0) (hl : q.degree < p.degree) : Β¬p β£ q := by
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl)
#align polynomial.not_dvd_of_degree_lt Polynomial.not_dvd_of_degree_lt
theorem not_dvd_of_natDegree_lt {p q : R[X]} (h0 : q β 0) (hl : q.natDegree < p.natDegree) :
Β¬p β£ q := by
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_natDegree_lt hcontra hl)
#align polynomial.not_dvd_of_nat_degree_lt Polynomial.not_dvd_of_natDegree_lt
| Mathlib/Algebra/Polynomial/RingDivision.lean | 190 | 195 | theorem natDegree_sub_eq_of_prod_eq {pβ pβ qβ qβ : R[X]} (hpβ : pβ β 0) (hqβ : qβ β 0)
(hpβ : pβ β 0) (hqβ : qβ β 0) (h_eq : pβ * qβ = pβ * qβ) :
(pβ.natDegree : β€) - qβ.natDegree = (pβ.natDegree : β€) - qβ.natDegree := by |
rw [sub_eq_sub_iff_add_eq_add]
norm_cast
rw [β natDegree_mul hpβ hqβ, β natDegree_mul hpβ hqβ, h_eq]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [Semiring R] [NoZeroDivisors R] {p q : R[X]}
instance : NoZeroDivisors R[X] where
eq_zero_or_eq_zero_of_mul_eq_zero h := by
rw [β leadingCoeff_eq_zero, β leadingCoeff_eq_zero]
refine eq_zero_or_eq_zero_of_mul_eq_zero ?_
rw [β leadingCoeff_zero, β leadingCoeff_mul, h]
theorem natDegree_mul (hp : p β 0) (hq : q β 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [β Nat.cast_inj (R := WithBot β), β degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, β degree_eq_natDegree hp, β degree_eq_natDegree hq, degree_mul]
#align polynomial.nat_degree_mul Polynomial.natDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0
Β· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
Β· rw [hq, mul_zero, trailingDegree_zero, add_top]
Β· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
apply WithTop.coe_add
#align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul
@[simp]
theorem natDegree_pow (p : R[X]) (n : β) : natDegree (p ^ n) = n * natDegree p := by
classical
obtain rfl | hp := eq_or_ne p 0
Β· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [β leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
#align polynomial.nat_degree_pow Polynomial.natDegree_pow
theorem degree_le_mul_left (p : R[X]) (hq : q β 0) : degree p β€ degree (p * q) := by
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
#align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left
theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : p.natDegree β€ q.natDegree := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
#align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd
theorem degree_le_of_dvd {p q : R[X]} (h1 : p β£ q) (h2 : q β 0) : degree p β€ degree q := by
rcases h1 with β¨q, rflβ©; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
#align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd
theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_degree_lt Polynomial.eq_zero_of_dvd_of_degree_lt
theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (hβ : p β£ q) (hβ : natDegree q < natDegree p) :
q = 0 := by
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (natDegree_le_of_dvd hβ hc)
#align polynomial.eq_zero_of_dvd_of_nat_degree_lt Polynomial.eq_zero_of_dvd_of_natDegree_lt
theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q β 0) (hl : q.degree < p.degree) : Β¬p β£ q := by
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl)
#align polynomial.not_dvd_of_degree_lt Polynomial.not_dvd_of_degree_lt
theorem not_dvd_of_natDegree_lt {p q : R[X]} (h0 : q β 0) (hl : q.natDegree < p.natDegree) :
Β¬p β£ q := by
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_natDegree_lt hcontra hl)
#align polynomial.not_dvd_of_nat_degree_lt Polynomial.not_dvd_of_natDegree_lt
theorem natDegree_sub_eq_of_prod_eq {pβ pβ qβ qβ : R[X]} (hpβ : pβ β 0) (hqβ : qβ β 0)
(hpβ : pβ β 0) (hqβ : qβ β 0) (h_eq : pβ * qβ = pβ * qβ) :
(pβ.natDegree : β€) - qβ.natDegree = (pβ.natDegree : β€) - qβ.natDegree := by
rw [sub_eq_sub_iff_add_eq_add]
norm_cast
rw [β natDegree_mul hpβ hqβ, β natDegree_mul hpβ hqβ, h_eq]
#align polynomial.nat_degree_sub_eq_of_prod_eq Polynomial.natDegree_sub_eq_of_prod_eq
| Mathlib/Algebra/Polynomial/RingDivision.lean | 198 | 203 | theorem natDegree_eq_zero_of_isUnit (h : IsUnit p) : natDegree p = 0 := by |
nontriviality R
obtain β¨q, hqβ© := h.exists_right_inv
have := natDegree_mul (left_ne_zero_of_mul_eq_one hq) (right_ne_zero_of_mul_eq_one hq)
rw [hq, natDegree_one, eq_comm, add_eq_zero_iff] at this
exact this.1
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [CommSemiring R] [NoZeroDivisors R] {p q : R[X]}
| Mathlib/Algebra/Polynomial/RingDivision.lean | 245 | 256 | theorem irreducible_of_monic (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β f g : R[X], f.Monic β g.Monic β f * g = p β f = 1 β¨ g = 1 := by |
refine
β¨fun h f g hf hg hp => (h.2 f g hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
β¨hp1 β hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)β©β©
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, mul_comm, β hfg, β Monic]
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, β hfg, β Monic]
Β· rw [mul_mul_mul_comm, β C_mul, β leadingCoeff_mul, β hfg, hp.leadingCoeff, C_1, mul_one,
mul_comm, β hfg]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [CommSemiring R] [NoZeroDivisors R] {p q : R[X]}
theorem irreducible_of_monic (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β f g : R[X], f.Monic β g.Monic β f * g = p β f = 1 β¨ g = 1 := by
refine
β¨fun h f g hf hg hp => (h.2 f g hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
β¨hp1 β hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)β©β©
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, mul_comm, β hfg, β Monic]
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, β hfg, β Monic]
Β· rw [mul_mul_mul_comm, β C_mul, β leadingCoeff_mul, β hfg, hp.leadingCoeff, C_1, mul_one,
mul_comm, β hfg]
#align polynomial.irreducible_of_monic Polynomial.irreducible_of_monic
| Mathlib/Algebra/Polynomial/RingDivision.lean | 259 | 265 | theorem Monic.irreducible_iff_natDegree (hp : p.Monic) :
Irreducible p β
p β 1 β§ β f g : R[X], f.Monic β g.Monic β f * g = p β f.natDegree = 0 β¨ g.natDegree = 0 := by |
by_cases hp1 : p = 1; Β· simp [hp1]
rw [irreducible_of_monic hp hp1, and_iff_right hp1]
refine forallβ_congr fun a b ha hb => ?_
rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [CommSemiring R] [NoZeroDivisors R] {p q : R[X]}
theorem irreducible_of_monic (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β f g : R[X], f.Monic β g.Monic β f * g = p β f = 1 β¨ g = 1 := by
refine
β¨fun h f g hf hg hp => (h.2 f g hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
β¨hp1 β hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)β©β©
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, mul_comm, β hfg, β Monic]
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, β hfg, β Monic]
Β· rw [mul_mul_mul_comm, β C_mul, β leadingCoeff_mul, β hfg, hp.leadingCoeff, C_1, mul_one,
mul_comm, β hfg]
#align polynomial.irreducible_of_monic Polynomial.irreducible_of_monic
theorem Monic.irreducible_iff_natDegree (hp : p.Monic) :
Irreducible p β
p β 1 β§ β f g : R[X], f.Monic β g.Monic β f * g = p β f.natDegree = 0 β¨ g.natDegree = 0 := by
by_cases hp1 : p = 1; Β· simp [hp1]
rw [irreducible_of_monic hp hp1, and_iff_right hp1]
refine forallβ_congr fun a b ha hb => ?_
rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]
#align polynomial.monic.irreducible_iff_nat_degree Polynomial.Monic.irreducible_iff_natDegree
| Mathlib/Algebra/Polynomial/RingDivision.lean | 268 | 279 | theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p β p β 1 β§
β f g : R[X], f.Monic β g.Monic β f * g = p β g.natDegree β Ioc 0 (p.natDegree / 2) := by |
simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
apply and_congr_right'
constructor <;> intro h f g hf hg he <;> subst he
Β· rw [hf.natDegree_mul hg, add_le_add_iff_right]
exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
Β· simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
contrapose! h
obtain hl | hl := le_total f.natDegree g.natDegree
Β· exact β¨g, f, hg, hf, mul_comm g f, h.1, add_le_add_left hl _β©
Β· exact β¨f, g, hf, hg, rfl, h.2, add_le_add_right hl _β©
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [CommSemiring R] [NoZeroDivisors R] {p q : R[X]}
theorem irreducible_of_monic (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β f g : R[X], f.Monic β g.Monic β f * g = p β f = 1 β¨ g = 1 := by
refine
β¨fun h f g hf hg hp => (h.2 f g hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
β¨hp1 β hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)β©β©
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, mul_comm, β hfg, β Monic]
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, β hfg, β Monic]
Β· rw [mul_mul_mul_comm, β C_mul, β leadingCoeff_mul, β hfg, hp.leadingCoeff, C_1, mul_one,
mul_comm, β hfg]
#align polynomial.irreducible_of_monic Polynomial.irreducible_of_monic
theorem Monic.irreducible_iff_natDegree (hp : p.Monic) :
Irreducible p β
p β 1 β§ β f g : R[X], f.Monic β g.Monic β f * g = p β f.natDegree = 0 β¨ g.natDegree = 0 := by
by_cases hp1 : p = 1; Β· simp [hp1]
rw [irreducible_of_monic hp hp1, and_iff_right hp1]
refine forallβ_congr fun a b ha hb => ?_
rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]
#align polynomial.monic.irreducible_iff_nat_degree Polynomial.Monic.irreducible_iff_natDegree
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p β p β 1 β§
β f g : R[X], f.Monic β g.Monic β f * g = p β g.natDegree β Ioc 0 (p.natDegree / 2) := by
simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
apply and_congr_right'
constructor <;> intro h f g hf hg he <;> subst he
Β· rw [hf.natDegree_mul hg, add_le_add_iff_right]
exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
Β· simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
contrapose! h
obtain hl | hl := le_total f.natDegree g.natDegree
Β· exact β¨g, f, hg, hf, mul_comm g f, h.1, add_le_add_left hl _β©
Β· exact β¨f, g, hf, hg, rfl, h.2, add_le_add_right hl _β©
#align polynomial.monic.irreducible_iff_nat_degree' Polynomial.Monic.irreducible_iff_natDegree'
| Mathlib/Algebra/Polynomial/RingDivision.lean | 284 | 291 | theorem Monic.irreducible_iff_lt_natDegree_lt {p : R[X]} (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β q, Monic q β natDegree q β Finset.Ioc 0 (natDegree p / 2) β Β¬ q β£ p := by |
rw [hp.irreducible_iff_natDegree', and_iff_right hp1]
constructor
Β· rintro h g hg hdg β¨f, rflβ©
exact h f g (hg.of_mul_monic_left hp) hg (mul_comm f g) hdg
Β· rintro h f g - hg rfl hdg
exact h g hg hdg (dvd_mul_left g f)
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [CommSemiring R] [NoZeroDivisors R] {p q : R[X]}
theorem irreducible_of_monic (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β f g : R[X], f.Monic β g.Monic β f * g = p β f = 1 β¨ g = 1 := by
refine
β¨fun h f g hf hg hp => (h.2 f g hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
β¨hp1 β hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)β©β©
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, mul_comm, β hfg, β Monic]
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, β hfg, β Monic]
Β· rw [mul_mul_mul_comm, β C_mul, β leadingCoeff_mul, β hfg, hp.leadingCoeff, C_1, mul_one,
mul_comm, β hfg]
#align polynomial.irreducible_of_monic Polynomial.irreducible_of_monic
theorem Monic.irreducible_iff_natDegree (hp : p.Monic) :
Irreducible p β
p β 1 β§ β f g : R[X], f.Monic β g.Monic β f * g = p β f.natDegree = 0 β¨ g.natDegree = 0 := by
by_cases hp1 : p = 1; Β· simp [hp1]
rw [irreducible_of_monic hp hp1, and_iff_right hp1]
refine forallβ_congr fun a b ha hb => ?_
rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]
#align polynomial.monic.irreducible_iff_nat_degree Polynomial.Monic.irreducible_iff_natDegree
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p β p β 1 β§
β f g : R[X], f.Monic β g.Monic β f * g = p β g.natDegree β Ioc 0 (p.natDegree / 2) := by
simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
apply and_congr_right'
constructor <;> intro h f g hf hg he <;> subst he
Β· rw [hf.natDegree_mul hg, add_le_add_iff_right]
exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
Β· simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
contrapose! h
obtain hl | hl := le_total f.natDegree g.natDegree
Β· exact β¨g, f, hg, hf, mul_comm g f, h.1, add_le_add_left hl _β©
Β· exact β¨f, g, hf, hg, rfl, h.2, add_le_add_right hl _β©
#align polynomial.monic.irreducible_iff_nat_degree' Polynomial.Monic.irreducible_iff_natDegree'
theorem Monic.irreducible_iff_lt_natDegree_lt {p : R[X]} (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β q, Monic q β natDegree q β Finset.Ioc 0 (natDegree p / 2) β Β¬ q β£ p := by
rw [hp.irreducible_iff_natDegree', and_iff_right hp1]
constructor
Β· rintro h g hg hdg β¨f, rflβ©
exact h f g (hg.of_mul_monic_left hp) hg (mul_comm f g) hdg
Β· rintro h f g - hg rfl hdg
exact h g hg hdg (dvd_mul_left g f)
| Mathlib/Algebra/Polynomial/RingDivision.lean | 293 | 316 | theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
Β¬Irreducible p β β cβ cβ, p.coeff 0 = cβ * cβ β§ p.coeff 1 = cβ + cβ := by |
cases subsingleton_or_nontrivial R
Β· simp [natDegree_of_subsingleton] at hnd
rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
Β· push_neg
constructor
Β· rintro β¨a, b, ha, hb, rfl, hdbβ©
simp only [zero_lt_two, Nat.div_self, ge_iff_le,
Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
have hda := hnd
rw [ha.natDegree_mul hb, hdb] at hda
use a.coeff 0, b.coeff 0, mul_coeff_zero a b
simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
Β· rintro β¨cβ, cβ, hmul, haddβ©
refine
β¨X + C cβ, X + C cβ, monic_X_add_C _, monic_X_add_C _, ?_, ?_β©
Β· rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
Finset.sum_range_one, β hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
ring
Β· rw [mem_Ioc, natDegree_X_add_C _]
simp
Β· rintro rfl
simp [natDegree_one] at hnd
| 1,827 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : β}
section NoZeroDivisors
variable [CommSemiring R] [NoZeroDivisors R] {p q : R[X]}
theorem irreducible_of_monic (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β f g : R[X], f.Monic β g.Monic β f * g = p β f = 1 β¨ g = 1 := by
refine
β¨fun h f g hf hg hp => (h.2 f g hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
β¨hp1 β hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)β©β©
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, mul_comm, β hfg, β Monic]
Β· rwa [Monic, leadingCoeff_mul, leadingCoeff_C, β leadingCoeff_mul, β hfg, β Monic]
Β· rw [mul_mul_mul_comm, β C_mul, β leadingCoeff_mul, β hfg, hp.leadingCoeff, C_1, mul_one,
mul_comm, β hfg]
#align polynomial.irreducible_of_monic Polynomial.irreducible_of_monic
theorem Monic.irreducible_iff_natDegree (hp : p.Monic) :
Irreducible p β
p β 1 β§ β f g : R[X], f.Monic β g.Monic β f * g = p β f.natDegree = 0 β¨ g.natDegree = 0 := by
by_cases hp1 : p = 1; Β· simp [hp1]
rw [irreducible_of_monic hp hp1, and_iff_right hp1]
refine forallβ_congr fun a b ha hb => ?_
rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]
#align polynomial.monic.irreducible_iff_nat_degree Polynomial.Monic.irreducible_iff_natDegree
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p β p β 1 β§
β f g : R[X], f.Monic β g.Monic β f * g = p β g.natDegree β Ioc 0 (p.natDegree / 2) := by
simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
apply and_congr_right'
constructor <;> intro h f g hf hg he <;> subst he
Β· rw [hf.natDegree_mul hg, add_le_add_iff_right]
exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
Β· simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
contrapose! h
obtain hl | hl := le_total f.natDegree g.natDegree
Β· exact β¨g, f, hg, hf, mul_comm g f, h.1, add_le_add_left hl _β©
Β· exact β¨f, g, hf, hg, rfl, h.2, add_le_add_right hl _β©
#align polynomial.monic.irreducible_iff_nat_degree' Polynomial.Monic.irreducible_iff_natDegree'
theorem Monic.irreducible_iff_lt_natDegree_lt {p : R[X]} (hp : p.Monic) (hp1 : p β 1) :
Irreducible p β β q, Monic q β natDegree q β Finset.Ioc 0 (natDegree p / 2) β Β¬ q β£ p := by
rw [hp.irreducible_iff_natDegree', and_iff_right hp1]
constructor
Β· rintro h g hg hdg β¨f, rflβ©
exact h f g (hg.of_mul_monic_left hp) hg (mul_comm f g) hdg
Β· rintro h f g - hg rfl hdg
exact h g hg hdg (dvd_mul_left g f)
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
Β¬Irreducible p β β cβ cβ, p.coeff 0 = cβ * cβ β§ p.coeff 1 = cβ + cβ := by
cases subsingleton_or_nontrivial R
Β· simp [natDegree_of_subsingleton] at hnd
rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
Β· push_neg
constructor
Β· rintro β¨a, b, ha, hb, rfl, hdbβ©
simp only [zero_lt_two, Nat.div_self, ge_iff_le,
Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
have hda := hnd
rw [ha.natDegree_mul hb, hdb] at hda
use a.coeff 0, b.coeff 0, mul_coeff_zero a b
simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
Β· rintro β¨cβ, cβ, hmul, haddβ©
refine
β¨X + C cβ, X + C cβ, monic_X_add_C _, monic_X_add_C _, ?_, ?_β©
Β· rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
Finset.sum_range_one, β hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
ring
Β· rw [mem_Ioc, natDegree_X_add_C _]
simp
Β· rintro rfl
simp [natDegree_one] at hnd
#align polynomial.monic.not_irreducible_iff_exists_add_mul_eq_coeff Polynomial.Monic.not_irreducible_iff_exists_add_mul_eq_coeff
| Mathlib/Algebra/Polynomial/RingDivision.lean | 319 | 320 | theorem root_mul : IsRoot (p * q) a β IsRoot p a β¨ IsRoot q a := by |
simp_rw [IsRoot, eval_mul, mul_eq_zero]
| 1,827 |
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