Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Multiset_Lemmas.jsonl

{"state":"α : Type u_1\ninst✝ : CommMonoid α\ns : Multiset α\na✝ : α\nl : List α\na : α\nh : a ∈ ⟦l⟧\n⊢ a ∣ prod ⟦l⟧","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by ","nextTactic":"simpa using List.dvd_prod h","declUpToTactic":"theorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.18_0.kGoXX9k6n37QCh9","decl":"theorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod "}
{"state":"α : Type u_1\ninst✝ : CanonicallyOrderedCommMonoid α\nm : Multiset α\nl : List α\n⊢ prod ⟦l⟧ = 1 ↔ ∀ x ∈ ⟦l⟧, x = 1","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by ","nextTactic":"simpa using List.prod_eq_one_iff l","declUpToTactic":"@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.22_0.kGoXX9k6n37QCh9","decl":"@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) "}
{"state":"α : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\n⊢ 0 < Multiset.prod m ↔ ∀ x ∈ m, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  ","nextTactic":"rcases m with ⟨l⟩","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
{"state":"case mk\nα : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\nl : List R\n⊢ 0 < Multiset.prod (Quot.mk Setoid.r l) ↔ ∀ x ∈ Quot.mk Setoid.r l, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  ","nextTactic":"rw [Multiset.quot_mk_to_coe'']","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
{"state":"case mk\nα : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\nl : List R\n⊢ 0 < Multiset.prod ↑l ↔ ∀ x ∈ ↑l, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  ","nextTactic":"rw [Multiset.coe_prod]","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
{"state":"case mk\nα : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\nl : List R\n⊢ 0 < List.prod l ↔ ∀ x ∈ ↑l, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  ","nextTactic":"exact CanonicallyOrderedCommSemiring.list_prod_pos","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
{"state":"α : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\nh : ∀ b ∈ s, Commute a b\n⊢ Commute a (sum s)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  ","nextTactic":"induction s using Quotient.inductionOn","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}
{"state":"case h\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (sum ⟦a✝⟧)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  ","nextTactic":"rw [quot_mk_to_coe]","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}
{"state":"case h\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (sum ↑a✝)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  ","nextTactic":"rw [coe_sum]","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}
{"state":"case h\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (List.sum a✝)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  rw [coe_sum]\n  ","nextTactic":"exact Commute.list_sum_right _ _ h","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  rw [coe_sum]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}