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[Merged by Bors] - feat(Algebra/CharP/{Basic|ExpChar}): add sum_pow_{char|expChar}_pow #9799

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[Merged by Bors] - feat(Algebra/CharP/{Basic|ExpChar}): add sum_pow_{char|expChar}_pow #9799

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add `sum_pow_{char|expChar}_pow`
acmepjz Jan 17, 2024
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acmepjz Jan 17, 2024
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Merge branch 'master' into acmepjz_sum_pow_char_pow
acmepjz Jan 17, 2024
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19 changes: 19 additions & 0 deletions Mathlib/Algebra/CharP/Basic.lean
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Expand Up @@ -442,6 +442,25 @@ theorem sum_pow_char {ι : Type*} (s : Finset ι) (f : ι → R) :
(frobenius R p).map_sum _ _
#align sum_pow_char sum_pow_char

variable (n : ℕ)

theorem list_sum_pow_char_pow (l : List R) : l.sum ^ p ^ n = (l.map (· ^ p ^ n : R → R)).sum := by
induction n
case zero => simp_rw [pow_zero, pow_one, List.map_id']
case succ n ih => simp_rw [pow_succ', pow_mul, ih, list_sum_pow_char, List.map_map]; rfl

theorem multiset_sum_pow_char_pow (s : Multiset R) :
s.sum ^ p ^ n = (s.map (· ^ p ^ n : R → R)).sum := by
induction n
case zero => simp_rw [pow_zero, pow_one, Multiset.map_id']
case succ n ih => simp_rw [pow_succ', pow_mul, ih, multiset_sum_pow_char, Multiset.map_map]; rfl

theorem sum_pow_char_pow {ι : Type*} (s : Finset ι) (f : ι → R) :
(∑ i in s, f i) ^ p ^ n = ∑ i in s, f i ^ p ^ n := by
induction n
case zero => simp_rw [pow_zero, pow_one]
case succ n ih => simp_rw [pow_succ', pow_mul, ih, sum_pow_char]

end CommSemiring

section CommRing
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44 changes: 44 additions & 0 deletions Mathlib/Algebra/CharP/ExpChar.lean
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Expand Up @@ -243,3 +243,47 @@ theorem ExpChar.neg_one_pow_expChar_pow [Ring R] (q n : ℕ) [hR : ExpChar R q]
cases' hR with _ _ hprime _
· simp only [one_pow, pow_one]
haveI := Fact.mk hprime; exact CharP.neg_one_pow_char_pow R q n

section BigOperators

open BigOperators

variable {R}

variable [CommSemiring R] (q : ℕ) [hR : ExpChar R q] (n : ℕ)

theorem list_sum_pow_expChar (l : List R) : l.sum ^ q = (l.map (· ^ q : R → R)).sum := by
cases hR
case zero => simp_rw [pow_one, List.map_id']
case prime hprime _ => haveI := Fact.mk hprime; exact list_sum_pow_char q l

theorem multiset_sum_pow_expChar (s : Multiset R) : s.sum ^ q = (s.map (· ^ q : R → R)).sum := by
cases hR
case zero => simp_rw [pow_one, Multiset.map_id']
case prime hprime _ => haveI := Fact.mk hprime; exact multiset_sum_pow_char q s

theorem sum_pow_expChar {ι : Type*} (s : Finset ι) (f : ι → R) :
(∑ i in s, f i) ^ q = ∑ i in s, f i ^ q := by
cases hR
case zero => simp_rw [pow_one]
case prime hprime _ => haveI := Fact.mk hprime; exact sum_pow_char q s f

theorem list_sum_pow_expChar_pow (l : List R) :
l.sum ^ q ^ n = (l.map (· ^ q ^ n : R → R)).sum := by
cases hR
case zero => simp_rw [one_pow, pow_one, List.map_id']
case prime hprime _ => haveI := Fact.mk hprime; exact list_sum_pow_char_pow q n l

theorem multiset_sum_pow_expChar_pow (s : Multiset R) :
s.sum ^ q ^ n = (s.map (· ^ q ^ n : R → R)).sum := by
cases hR
case zero => simp_rw [one_pow, pow_one, Multiset.map_id']
case prime hprime _ => haveI := Fact.mk hprime; exact multiset_sum_pow_char_pow q n s

theorem sum_pow_expChar_pow {ι : Type*} (s : Finset ι) (f : ι → R) :
(∑ i in s, f i) ^ q ^ n = ∑ i in s, f i ^ q ^ n := by
cases hR
case zero => simp_rw [one_pow, pow_one]
case prime hprime _ => haveI := Fact.mk hprime; exact sum_pow_char_pow q n s f

end BigOperators
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