Loogle!

Result

Found 13817 declarations mentioning CommSemiring. Of these, only the first 200 are shown.

• CommSemiring 📋 Mathlib.Algebra.Ring.Defs
  (R : Type u) : Type u
• CommRing.toCommSemiring 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [s : CommRing α] : CommSemiring α
• CommSemiring.toCommMonoid 📋 Mathlib.Algebra.Ring.Defs
  {R : Type u} [self : CommSemiring R] : CommMonoid R
• CommSemiring.toCommMonoidWithZero 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] : CommMonoidWithZero α
• CommSemiring.toNonAssocCommSemiring 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] : NonAssocCommSemiring α
• CommSemiring.toNonUnitalCommSemiring 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] : NonUnitalCommSemiring α
• CommSemiring.toSemiring 📋 Mathlib.Algebra.Ring.Defs
  {R : Type u} [self : CommSemiring R] : Semiring R
• IsMulCommutative.instCommSemiring 📋 Mathlib.Algebra.Ring.Defs
  {R : Type v} [Semiring R] [IsMulCommutative R] : CommSemiring R
• IsUnital.toCommSemiring 📋 Mathlib.Algebra.Ring.Defs
  {A : Type u_1} [NonUnitalCommSemiring A] [IsUnital A] : CommSemiring A
• CommSemiring.mk 📋 Mathlib.Algebra.Ring.Defs
  {R : Type u} [toSemiring : Semiring R] (mul_comm : ∀ (a b : R), a * b = b * a) : CommSemiring R
• CommSemiring.mul_comm 📋 Mathlib.Algebra.Ring.Defs
  {R : Type u} [self : CommSemiring R] (a b : R) : a * b = b * a
• add_mul_self_eq 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] (a b : α) : (a + b) * (a + b) = a * a + 2 * a * b + b * b
• add_pow_two 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] (a b : α) : (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2
• add_sq 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] (a b : α) : (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2
• add_sq' 📋 Mathlib.Algebra.Ring.Defs
  {α : Type u} [CommSemiring α] (a b : α) : (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b
• Int.instCommSemiring 📋 Mathlib.Algebra.Ring.Int.Defs
  : CommSemiring ℤ
• Nat.instCommSemiring 📋 Mathlib.Algebra.Ring.Nat
  : CommSemiring ℕ
• Function.Injective.commSemiring 📋 Mathlib.Algebra.Ring.InjSurj
  {R : Type u_1} {S : Type u_2} (f : S → R) (hf : Function.Injective f) [Add S] [Mul S] [Zero S] [One S] [SMul ℕ S] [Pow S ℕ] [NatCast S] [CommSemiring R] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : S), f (x + y) = f x + f y) (mul : ∀ (x y : S), f (x * y) = f x * f y) (nsmul : ∀ (n : ℕ) (x : S), f (n • x) = n • f x) (npow : ∀ (x : S) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) : CommSemiring S
• Function.Surjective.commSemiring 📋 Mathlib.Algebra.Ring.InjSurj
  {R : Type u_1} {S : Type u_2} (f : R → S) (hf : Function.Surjective f) [Add S] [Mul S] [Zero S] [One S] [SMul ℕ S] [Pow S ℕ] [NatCast S] [CommSemiring R] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : R), f (x + y) = f x + f y) (mul : ∀ (x y : R), f (x * y) = f x * f y) (nsmul : ∀ (n : ℕ) (x : R), f (n • x) = n • f x) (npow : ∀ (x : R) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) : CommSemiring S
• Units.divp_add_divp 📋 Mathlib.Algebra.Ring.Units
  {α : Type u} [CommSemiring α] (a b : α) (u₁ u₂ : αˣ) : a /ₚ u₁ + b /ₚ u₂ = (a * ↑u₂ + ↑u₁ * b) /ₚ (u₁ * u₂)
• Semifield.toCommSemiring 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : CommSemiring K
• Semifield.mk 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [toCommSemiring : CommSemiring K] [toInv : Inv K] [toDiv : Div K] [toZPow : ZPow K] (div_eq_mul_inv : ∀ (a b : K), a / b = a * b⁻¹ := by intros; rfl) (zpow_zero' : ∀ (a : K), a ^ 0 = 1 := by intros; rfl) (zpow_succ' : ∀ (n : ℕ) (a : K), a ^ ↑n.succ = a ^ ↑n * a := by intros; rfl) (zpow_neg' : ∀ (n : ℕ) (a : K), a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹ := by intros; rfl) [toNontrivial : Nontrivial K] (inv_zero : 0⁻¹ = 0) (mul_inv_cancel : ∀ (a : K), a ≠ 0 → a * a⁻¹ = 1) [toNNRatCast : NNRatCast K] (nnratCast_def : ∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den := by intros; rfl) (nnqsmul : ℚ≥0 → K → K) (nnqsmul_def : ∀ (q : ℚ≥0) (a : K), nnqsmul q a = ↑q * a := by intros; rfl) : Semifield K
• four_mul_le_pow_two_add 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] (a b : R) : 4 * a * b ≤ (a + b) ^ 2
• four_mul_le_sq_add 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] (a b : R) : 4 * a * b ≤ (a + b) ^ 2
• max_mul_mul_le_max_mul_max 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] {a d : R} [PosMulMono R] [MulPosMono R] (b c : R) (ha : 0 ≤ a) (hd : 0 ≤ d) : max (a * b) (d * c) ≤ max a c * max d b
• two_mul_le_add_pow_two 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] (a b : R) : 2 * a * b ≤ a ^ 2 + b ^ 2
• two_mul_le_add_sq 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] (a b : R) : 2 * a * b ≤ a ^ 2 + b ^ 2
• two_mul_le_add_of_sq_eq_mul 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R] [PosMulStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] {a b r : R} (ha : 0 ≤ a) (hb : 0 ≤ b) (ht : r ^ 2 = a * b) : 2 * r ≤ a + b
• two_mul_le_add_of_sq_le_mul 📋 Mathlib.Algebra.Order.Ring.Unbundled.Basic
  {R : Type u} [CommSemiring R] [LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R] [PosMulStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] {a b r : R} (ha : 0 ≤ a) (hb : 0 ≤ b) (ht : r ^ 2 ≤ a * b) : 2 * r ≤ a + b
• CommSemiring.toGrindCommSemiring 📋 Mathlib.Algebra.Ring.GrindInstances
  (α : Type u_1) [s : CommSemiring α] : Lean.Grind.CommSemiring α
• Nat.cast_tsub 📋 Mathlib.Data.Nat.Cast.Order.Ring
  {α : Type u_2} [CommSemiring α] [PartialOrder α] [IsOrderedRing α] [CanonicallyOrderedAdd α] [Sub α] [OrderedSub α] [AddLeftReflectLE α] (m n : ℕ) : ↑(m - n) = ↑m - ↑n
• Lex.instCommSemiring 📋 Mathlib.Algebra.Order.Ring.Synonym
  {R : Type u_1} [CommSemiring R] : CommSemiring (Lex R)
• OrderDual.instCommSemiring 📋 Mathlib.Algebra.Order.Ring.Synonym
  {R : Type u_1} [CommSemiring R] : CommSemiring Rᵒᵈ
• Nonneg.commMonoidWithZero 📋 Mathlib.Algebra.Order.Nonneg.Basic
  {α : Type u_1} [CommSemiring α] [PartialOrder α] [ZeroLEOneClass α] [AddLeftMono α] [PosMulMono α] : CommMonoidWithZero (Nonneg α)
• Nonneg.commSemiring 📋 Mathlib.Algebra.Order.Nonneg.Basic
  {α : Type u_1} [CommSemiring α] [PartialOrder α] [ZeroLEOneClass α] [AddLeftMono α] [PosMulMono α] : CommSemiring (Nonneg α)
• Rat.commSemiring 📋 Mathlib.Algebra.Ring.Rat
  : CommSemiring ℚ
• instCommSemiringNNRat 📋 Mathlib.Data.NNRat.Defs
  : CommSemiring ℚ≥0
• Mathlib.Tactic.Ring.Common.pow_one 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) : a ^ 1 = a
• Mathlib.Tactic.Ring.Common.pow_one_cast 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) : a ^ Nat.rawCast 1 = a
• Mathlib.Tactic.Ring.Common.add_pf_add_zero 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) : a + 0 = a
• Mathlib.Tactic.Ring.Common.add_pf_zero_add 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (b : R) : 0 + b = b
• Mathlib.Tactic.Ring.Common.mul_one 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) : a * Nat.rawCast 1 = a
• Mathlib.Tactic.Ring.Common.one_mul 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) : Nat.rawCast 1 * a = a
• Mathlib.Tactic.Ring.Common.pow_one_cast_of_isNat 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) (b : ℕ) (hb : Mathlib.Meta.NormNum.IsNat b 1) : a ^ b = a
• Mathlib.Tactic.Ring.Common.one_pow 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a : R} (b : ℕ) (ha : Mathlib.Meta.NormNum.IsNat a 1) : a ^ b = a
• Mathlib.Tactic.Ring.Common.mul_zero 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) : a * 0 = 0
• Mathlib.Tactic.Ring.Common.zero_mul 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (b : R) : 0 * b = 0
• Mathlib.Tactic.Ring.Common.add_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a a' b b' c : R} : a = a' → b = b' → a' + b' = c → a + b = c
• Mathlib.Tactic.Ring.Common.mul_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a a' b b' c : R} : a = a' → b = b' → a' * b' = c → a * b = c
• Mathlib.Tactic.Ring.Common.zero_pow 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {b : ℕ} : 0 < b → 0 ^ b = 0
• Mathlib.Tactic.Ring.Common.pow_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a a' c : R} {b b' : ℕ} : a = a' → b = b' → a' ^ b' = c → a ^ b = c
• Mathlib.Tactic.Ring.Common.toProd_pf 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a a' : R} (p : a = a') {e : ℕ} (hone : Nat.rawCast 1 = e) : a = a' ^ e * Nat.rawCast 1
• Mathlib.Tactic.Ring.Common.pow_bit0 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b c : R} {k : ℕ} : a ^ k = b → b * b = c → a ^ Nat.mul 2 k = c
• Mathlib.Tactic.Ring.Common.nsmul_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {b b' c : R} {a a' : ℕ} : a = a' → b = b' → a' • b' = c → a • b = c
• Mathlib.Tactic.Ring.Common.pow_zero 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) {e : R} (h : Nat.rawCast 1 = e) : a ^ 0 = e + 0
• Mathlib.Tactic.Ring.Common.add_pf_add_gt 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b₂ c : R} (b₁ : R) : a + b₂ = c → a + (b₁ + b₂) = b₁ + c
• Mathlib.Tactic.Ring.Common.add_pf_add_lt 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₂ b c : R} (a₁ : R) : a₂ + b = c → a₁ + a₂ + b = a₁ + c
• Mathlib.Tactic.Ring.Common.smul_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} {α : Type u_3} [CommSemiring α] [SMul R α] {r : R} {a b t c : α} : a = b → (∀ (x : α), r • x = t * x) → t * b = c → r • a = c
• Mathlib.Tactic.Ring.Common.inv_add 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {b₁ b₂ : R} {a₁ a₂ : ℕ} : ↑a₁ = b₁ → ↑a₂ = b₂ → ↑(a₁ + a₂) = b₁ + b₂
• Mathlib.Tactic.Ring.Common.pow_nat 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a : R} {b c k : ℕ} {d e : R} : b = c * k → a ^ c = d → d ^ k = e → a ^ b = e
• Mathlib.Tactic.Ring.Common.atom_pf 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {b : R} (a : R) {e : ℕ} (hone : Nat.rawCast 1 = e) (hb : a ^ e * Nat.rawCast 1 = b) : a = b + 0
• Mathlib.Tactic.Ring.Common.pow_bit1 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b c : R} {k : ℕ} {d : R} : a ^ k = b → b * b = c → c * a = d → a ^ (Nat.mul 2 k).add 1 = d
• Mathlib.Tactic.Ring.Common.atom_pf' 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a a' b : R} (p : a = a') {e : ℕ} (hone : Nat.rawCast 1 = e) (hb : a' ^ e * Nat.rawCast 1 = b) : a = b + 0
• Mathlib.Tactic.Ring.Common.add_mul 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₁ a₂ b c₁ c₂ d : R} : a₁ * b = c₁ → a₂ * b = c₂ → c₁ + c₂ = d → (a₁ + a₂) * b = d
• Mathlib.Tactic.Ring.Common.pow_add 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a c₁ c₂ : R} {b₁ b₂ : ℕ} {d : R} : a ^ b₁ = c₁ → a ^ b₂ = c₂ → c₁ * c₂ = d → a ^ (b₁ + b₂) = d
• Mathlib.Tactic.Ring.Common.pow_atom 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) (b : ℕ) {e : R} (h : a ^ b * Nat.rawCast 1 = e) : a ^ b = e + 0
• Mathlib.Tactic.Ring.Common.pow_prod_atom 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] (a : R) (b : ℕ) {e : R} (h : (a + 0) ^ b * Nat.rawCast 1 = e) : a ^ b = e
• Mathlib.Tactic.Ring.Common.single_pow 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a c : R} {b : ℕ} : a ^ b = c → (a + 0) ^ b = c + 0
• Mathlib.Tactic.Ring.Common.add_pf_add_overlap_zero 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₁ a₂ b₁ b₂ c : R} (h : Mathlib.Meta.NormNum.IsNat (a₁ + b₁) 0) (h₄ : a₂ + b₂ = c) : a₁ + a₂ + (b₁ + b₂) = c
• Mathlib.Tactic.Ring.Common.add_pf_add_overlap 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₁ a₂ b₁ b₂ c₁ c₂ : R} : a₁ + b₁ = c₁ → a₂ + b₂ = c₂ → a₁ + a₂ + (b₁ + b₂) = c₁ + c₂
• Mathlib.Tactic.Ring.Common.mul_pf_left 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₃ b c : R} (a₁ : R) (a₂ : ℕ) : a₃ * b = c → a₁ ^ a₂ * a₃ * b = a₁ ^ a₂ * c
• Mathlib.Tactic.Ring.Common.mul_pf_right 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b₃ c : R} (b₁ : R) (b₂ : ℕ) : a * b₃ = c → a * (b₁ ^ b₂ * b₃) = b₁ ^ b₂ * c
• Mathlib.Tactic.Ring.Common.mul_add 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b₁ b₂ c₁ c₂ d : R} : a * b₁ = c₁ → a * b₂ = c₂ → c₁ + 0 + c₂ = d → a * (b₁ + b₂) = d
• Mathlib.Tactic.Ring.Common.add_overlap_pf_zero 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b : R} (x : R) (e : ℕ) : Mathlib.Meta.NormNum.IsNat (a + b) 0 → Mathlib.Meta.NormNum.IsNat (x ^ e * a + x ^ e * b) 0
• Mathlib.Tactic.Ring.Common.mul_pow 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₂ c₂ : R} {ea₁ b c₁ : ℕ} {xa₁ : R} : ea₁ * b = c₁ → a₂ ^ b = c₂ → (xa₁ ^ ea₁ * a₂) ^ b = xa₁ ^ c₁ * c₂
• Mathlib.Tactic.Ring.Common.add_overlap_pf 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a b c : R} (x : R) (e : ℕ) (pq_pf : a + b = c) : x ^ e * a + x ^ e * b = x ^ e * c
• Mathlib.Tactic.Ring.Common.mul_pp_pf_overlap 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₂ b₂ c : R} {ea eb e : ℕ} (x : R) : ea + eb = e → a₂ * b₂ = c → x ^ ea * a₂ * (x ^ eb * b₂) = x ^ e * c
• Mathlib.Tactic.Ring.Common.mul_pow_mul 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_1} [CommSemiring R] {a₂ c₂ : R} {ea₁ b c₁ : ℕ} {xa₁ c₃ d : R} : ea₁ * b = c₁ → a₂ ^ b = c₂ → xa₁ ^ c₁ * Nat.rawCast 1 = c₃ → c₃ * c₂ = d → (xa₁ ^ ea₁ * a₂) ^ b = d
• Mathlib.Tactic.Ring.cast_zero 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] {a : R} : Mathlib.Meta.NormNum.IsNat a 0 → a = 0
• Mathlib.Tactic.Ring.natCast_nat 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] (n : ℕ) : ↑n.rawCast = n.rawCast
• Mathlib.Tactic.Ring.natCast_zero 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] : ↑0 = 0
• Mathlib.Tactic.Ring.cast_pos 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] {a : R} {n : ℕ} : Mathlib.Meta.NormNum.IsNat a n → a = n.rawCast + 0
• Mathlib.Tactic.Ring.Nat.smul_eq_mul 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] {n n' : ℕ} {r : R} (hr : ↑n = r) (hn : n' = n) (a : R) : n' • a = r * a
• Mathlib.Tactic.Ring.natCast_add 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] {b₁ b₂ : R} {a₁ a₂ : ℕ} : ↑a₁ = b₁ → ↑a₂ = b₂ → ↑(a₁ + a₂) = b₁ + b₂
• Mathlib.Tactic.Ring.natCast_mul 📋 Mathlib.Tactic.Ring.Basic
  {R : Type u_1} [CommSemiring R] {b₁ b₃ : R} {a₁ a₃ : ℕ} (a₂ : ℕ) : ↑a₁ = b₁ → ↑a₃ = b₃ → ↑(a₁ ^ a₂ * a₃) = b₁ ^ a₂ * b₃
• CanonicallyOrderedAdd.toIsOrderedRing 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] : IsOrderedRing R
• CanonicallyOrderedAdd.toIsOrderedMonoid 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] : IsOrderedMonoid R
• CanonicallyOrderedAdd.pow_pos 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [IsReduced R] {a : R} (ha : 0 < a) (n : ℕ) : 0 < a ^ n
• CanonicallyOrderedAdd.mul_lt_mul_of_lt_of_lt 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [PosMulStrictMono R] {a b c d : R} (hab : a < b) (hcd : c < d) : a * c < b * d
• CanonicallyOrderedAdd.mul_pos 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [NoZeroDivisors R] {a b : R} : 0 < a * b ↔ 0 < a ∧ 0 < b
• mul_self_tsub_mul_self 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [Std.Total fun x1 x2 => x1 ≤ x2] [AddLeftReflectLE R] (a b : R) : a * a - b * b = (a + b) * (a - b)
• sq_tsub_sq 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [Std.Total fun x1 x2 => x1 ≤ x2] [AddLeftReflectLE R] (a b : R) : a ^ 2 - b ^ 2 = (a + b) * (a - b)
• mul_self_tsub_one 📋 Mathlib.Algebra.Order.Ring.Canonical
  {R : Type u} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [Sub R] [OrderedSub R] [Std.Total fun x1 x2 => x1 ≤ x2] [AddLeftReflectLE R] (a : R) : a * a - 1 = (a + 1) * (a - 1)
• Nonneg.linearOrderedCommMonoidWithZero 📋 Mathlib.Algebra.Order.Nonneg.Ring
  {α : Type u_1} [CommSemiring α] [LinearOrder α] [IsStrictOrderedRing α] : LinearOrderedCommMonoidWithZero (Nonneg α)
• Positive.commMonoid 📋 Mathlib.Algebra.Order.Positive.Ring
  {R : Type u_2} [CommSemiring R] [PartialOrder R] [IsStrictOrderedRing R] : CommMonoid { x // 0 < x }
• Positive.isOrderedMonoid 📋 Mathlib.Algebra.Order.Positive.Ring
  {R : Type u_2} [CommSemiring R] [PartialOrder R] [IsStrictOrderedRing R] : IsOrderedMonoid { x // 0 < x }
• Positive.isOrderedCancelMonoid 📋 Mathlib.Algebra.Order.Positive.Ring
  {R : Type u_2} [CommSemiring R] [LinearOrder R] [IsStrictOrderedRing R] : IsOrderedCancelMonoid { x // 0 < x }
• Mathlib.Tactic.RingNF.nat_rawCast_0 📋 Mathlib.Tactic.Ring.RingNF
  {R : Type u_1} [CommSemiring R] : Nat.rawCast 0 = 0
• Mathlib.Tactic.RingNF.nat_rawCast_1 📋 Mathlib.Tactic.Ring.RingNF
  {R : Type u_1} [CommSemiring R] : Nat.rawCast 1 = 1
• Mathlib.Tactic.RingNF.nat_rawCast_2 📋 Mathlib.Tactic.Ring.RingNF
  {R : Type u_1} [CommSemiring R] {n : ℕ} [n.AtLeastTwo] : n.rawCast = OfNat.ofNat n
• Mathlib.Tactic.RingNF.add_assoc_rev 📋 Mathlib.Tactic.Ring.RingNF
  {R : Type u_1} [CommSemiring R] (a b c : R) : a + (b + c) = a + b + c
• Mathlib.Tactic.RingNF.mul_assoc_rev 📋 Mathlib.Tactic.Ring.RingNF
  {R : Type u_1} [CommSemiring R] (a b c : R) : a * (b * c) = a * b * c
• pow_add_mul_le_add_pow 📋 Mathlib.Algebra.Order.Ring.Pow
  {R : Type u_1} [CommSemiring R] [LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a b : R} (ha : 0 ≤ a) (H : 0 ≤ 2 * a + b) (n : ℕ) : a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n
• pow_add_mul_le_add_pow_of_sq_nonneg 📋 Mathlib.Algebra.Order.Ring.Pow
  {R : Type u_1} [CommSemiring R] [PartialOrder R] [IsOrderedRing R] {a b : R} (ha : 0 ≤ a) (Hsq : 0 ≤ b ^ 2) (Hsq' : 0 ≤ (a + b) ^ 2) (H : 0 ≤ 2 * a + b) (n : ℕ) : a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n
• Nonneg.instMulArchimedean 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {R : Type u_3} [CommSemiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] [ExistsAddOfLE R] : MulArchimedean (Nonneg R)
• WithBot.instCommSemiring 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [Subsingleton (AddUnits α)] [NoZeroDivisors α] [Nontrivial α] : CommSemiring (WithBot α)
• WithTop.instCommSemiring 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [Subsingleton (AddUnits α)] [NoZeroDivisors α] [Nontrivial α] : CommSemiring (WithTop α)
• WithTop.instIsOrderedRing 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [PartialOrder α] [CanonicallyOrderedAdd α] [NoZeroDivisors α] [Nontrivial α] : IsOrderedRing (WithTop α)
• WithBot.instIsOrderedRing 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [PartialOrder α] [IsOrderedRing α] [CanonicallyOrderedAdd α] [NoZeroDivisors α] [Nontrivial α] : IsOrderedRing (WithBot α)
• WithTop.pow_right_strictMono 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [PartialOrder α] [OrderBot α] [CanonicallyOrderedAdd α] [PosMulStrictMono α] [NoZeroDivisors α] [Nontrivial α] {n : ℕ} : n ≠ 0 → StrictMono fun a => a ^ n
• WithTop.mul_lt_mul 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [PartialOrder α] [OrderBot α] [CanonicallyOrderedAdd α] [PosMulStrictMono α] {a₁ a₂ b₁ b₂ : WithTop α} (ha : a₁ < a₂) (hb : b₁ < b₂) : a₁ * b₁ < a₂ * b₂
• WithTop.pow_lt_pow_left 📋 Mathlib.Algebra.Order.Ring.WithTop
  {α : Type u_1} [DecidableEq α] [CommSemiring α] [PartialOrder α] [OrderBot α] [CanonicallyOrderedAdd α] [PosMulStrictMono α] [NoZeroDivisors α] [Nontrivial α] {a b : WithTop α} (hab : a < b) {n : ℕ} (hn : n ≠ 0) : a ^ n < b ^ n
• instCommSemiringENat 📋 Mathlib.Data.ENat.Monoid
  : CommSemiring ℕ∞
• RingHom.ENatMap 📋 Mathlib.Data.ENat.Monoid
  {S : Type u_1} [CommSemiring S] [PartialOrder S] [CanonicallyOrderedAdd S] [DecidableEq S] [Nontrivial S] (f : ℕ →+* S) (hf : Function.Injective ⇑f) : ℕ∞ →+* WithTop S
• RingHom.ENatMap_apply 📋 Mathlib.Data.ENat.Monoid
  {S : Type u_1} [CommSemiring S] [PartialOrder S] [CanonicallyOrderedAdd S] [DecidableEq S] [Nontrivial S] (f : ℕ →+* S) (hf : Function.Injective ⇑f) : ⇑(f.ENatMap hf) = (↑(f.toMonoidWithZeroHom.ENatMap hf)).toFun
• Cardinal.commSemiring 📋 Mathlib.SetTheory.Cardinal.Order
  : CommSemiring Cardinal.{u}
• MulActionHom.instCommSemiring 📋 Mathlib.GroupTheory.GroupAction.Hom
  {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [SMul M X] [Monoid N] [CommSemiring Y] [MulSemiringAction N Y] : CommSemiring (X →ₑ[σ] Y)
• IsLinearMap.isLinearMap_smul 📋 Mathlib.Algebra.Module.LinearMap.Defs
  {R : Type u_14} {M : Type u_15} [CommSemiring R] [AddCommMonoid M] [Module R M] (c : R) : IsLinearMap R fun z => c • z
• Algebra 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] : Type (max u v)
• Algebra.id 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) [CommSemiring R] : Algebra R R
• Algebra.cast 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] : R → A
• Algebra.toSMul 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} [self : Algebra R A] : SMul R A
• algebraMap.coeHTCT 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] : CoeHTCT R A
• Algebra.subsingleton 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] [Subsingleton R] : Subsingleton A
• Algebra.toModule 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} {x✝ : CommSemiring R} {x✝¹ : Semiring A} [Algebra R A] : Module R A
• Algebra.algebraMap 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) (A : Type v) {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} [self : Algebra R A] : R →+* A
• RingHom.toAlgebra 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (i : R →+* S) : Algebra R S
• Algebra.compHom 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {S : Type v} (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R A] (f : S →+* R) : Algebra S A
• Algebra.algebraMap_self 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} [CommSemiring R] : algebraMap R R = RingHom.id R
• IsScalarTower.right 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] : IsScalarTower R A A
• toAlgebra_algebraMap 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] [Algebra R S] : (algebraMap R S).toAlgebra = inst✝
• algebraMap.coe_zero 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] : ↑0 = 0
• algebraMap.coe_natCast 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : ℕ) : ↑↑a = ↑a
• RingHom.algebraMap_toAlgebra 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (i : R →+* S) : algebraMap R S = i
• Algebra.algebraMap_self_apply 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} [CommSemiring R] (x : R) : (algebraMap R R) x = x
• Algebra.linearMap 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) (A : Type w) [CommSemiring R] [Semiring A] [Algebra R A] : R →ₗ[R] A
• algebraMap.coe_one 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] : ↑1 = 1
• RingHom.smulOneHom_eq_algebraMap 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] : RingHom.smulOneHom = algebraMap R A
• Algebra.commute_algebraMap_left 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A) : Commute ((algebraMap R A) r) x
• Algebra.commute_algebraMap_right 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A) : Commute x ((algebraMap R A) r)
• algebraMap.coe_add 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a b : R) : ↑(a + b) = ↑a + ↑b
• algebraMap.coe_mul 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a b : R) : ↑(a * b) = ↑a * ↑b
• algebraMap.coe_pow 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : R) (n : ℕ) : ↑(a ^ n) = ↑a ^ n
• algebraMap.coe_smul 📋 Mathlib.Algebra.Algebra.Defs
  {A : Type u_1} {B : Type u_2} (a : A) (b : B) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [SMul A C] [IsScalarTower A B C] : ↑(a • b) = a • ↑b
• Algebra.mul_smul_comm 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (s : R) (x y : A) : x * s • y = s • (x * y)
• Algebra.smul_mul_assoc 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x y : A) : r • x * y = r • (x * y)
• Algebra.linearMap_self 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) [CommSemiring R] : Algebra.linearMap R R = LinearMap.id
• Algebra.algebraMap_eq_smul_one 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) : (algebraMap R A) r = r • 1
• Algebra.algebraMap_eq_smul_one' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] : ⇑(algebraMap R A) = fun r => r • 1
• Algebra.smul_def 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A) : r • x = (algebraMap R A) r * x
• Algebra.smul_def' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} [self : Algebra R A] (r : R) (x : A) : r • x = (algebraMap R A) r * x
• Algebra.compHom_algebraMap_eq 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {S : Type v} (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R A] (f : S →+* R) : algebraMap S A = (algebraMap R A).comp f
• Algebra.algebra_ext 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] (P Q : Algebra R A) (h : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r) : P = Q
• Algebra.algebra_ext_iff 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {P Q : Algebra R A} : P = Q ↔ ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r
• RingHom.smul_toAlgebra 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (i : R →+* S) (r : R) (s : S) : have x := i.toAlgebra; r • s = i r * s
• Algebra.compHom_smul_def 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {S : Type v} (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R A] (f : S →+* R) (s : S) (x : A) : s • x = f s • x
• Algebra.commutes 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A) : (algebraMap R A) r * x = x * (algebraMap R A) r
• Algebra.commutes' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} [self : Algebra R A] (r : R) (x : A) : (algebraMap R A) r * x = x * (algebraMap R A) r
• RingHom.toAlgebra' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] (i : R →+* S) (h : ∀ (c : R) (x : S), i c * x = x * i c) : Algebra R S
• algebraMap.coe_smul' 📋 Mathlib.Algebra.Algebra.Defs
  {A : Type u_1} {B : Type u_2} (a : A) (b : B) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [Monoid A] [MulDistribMulAction A C] [SMulDistribClass A B C] : ↑(a • b) = a • ↑b
• algebraMap.smul 📋 Mathlib.Algebra.Algebra.Defs
  {A : Type u_1} {B : Type u_2} (a : A) (b : B) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [SMul A C] [IsScalarTower A B C] : (algebraMap B C) (a • b) = a • (algebraMap B C) b
• RingHom.algebraMap_toAlgebra' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] (i : R →+* S) (h : ∀ (c : R) (x : S), i c * x = x * i c) : algebraMap R S = i
• Algebra.coe_linearMap 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) (A : Type w) [CommSemiring R] [Semiring A] [Algebra R A] : ⇑(Algebra.linearMap R A) = ⇑(algebraMap R A)
• Algebra.left_comm 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (r : R) (y : A) : x * ((algebraMap R A) r * y) = (algebraMap R A) r * (x * y)
• Algebra.right_comm 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (r : R) (y : A) : x * (algebraMap R A) r * y = x * y * (algebraMap R A) r
• Algebra.linearMap_apply 📋 Mathlib.Algebra.Algebra.Defs
  (R : Type u) (A : Type w) [CommSemiring R] [Semiring A] [Algebra R A] (r : R) : (Algebra.linearMap R A) r = (algebraMap R A) r
• Algebra.compHom_algebraMap_apply 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {S : Type v} (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R A] (f : S →+* R) (s : S) : (algebraMap S A) s = (algebraMap R A) (f s)
• smul_algebraMap 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] {α : Type u_1} [Monoid α] [MulDistribMulAction α A] [SMulCommClass α R A] (a : α) (r : R) : a • (algebraMap R A) r = (algebraMap R A) r
• algebraMap.smul' 📋 Mathlib.Algebra.Algebra.Defs
  {A : Type u_1} {B : Type u_2} (a : A) (b : B) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [Monoid A] [MulDistribMulAction A C] [SMulDistribClass A B C] : (algebraMap B C) (a • b) = a • (algebraMap B C) b
• Algebra.mk 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [toSMul : SMul R A] (algebraMap : R →+* A) (commutes' : ∀ (r : R) (x : A), algebraMap r * x = x * algebraMap r) (smul_def' : ∀ (r : R) (x : A), r • x = algebraMap r * x) : Algebra R A
• RingHom.smul_toAlgebra' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] (i : R →+* S) (h : ∀ (c : R) (x : S), i c * x = x * i c) (r : R) (s : S) : have x := i.toAlgebra' h; r • s = i r * s
• IsUnital.toAlgebra 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [IsUnital A] : Algebra R A
• Algebra.ofModule 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Module R A] (h₁ : ∀ (r : R) (x y : A), r • x * y = r • (x * y)) (h₂ : ∀ (r : R) (x y : A), x * r • y = r • (x * y)) : Algebra R A
• Algebra.ofModule' 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Module R A] (h₁ : ∀ (r : R) (x : A), r • 1 * x = r • x) (h₂ : ∀ (r : R) (x : A), x * r • 1 = r • x) : Algebra R A
• AddOpposite.instCommSemiring 📋 Mathlib.Algebra.Ring.Opposite
  {R : Type u_1} [CommSemiring R] : CommSemiring Rᵃᵒᵖ
• MulOpposite.instCommSemiring 📋 Mathlib.Algebra.Ring.Opposite
  {R : Type u_1} [CommSemiring R] : CommSemiring Rᵐᵒᵖ
• Pi.commSemiring 📋 Mathlib.Algebra.Ring.Pi
  {I : Type u} {f : I → Type v} [(i : I) → CommSemiring (f i)] : CommSemiring ((i : I) → f i)
• Module.End.smulLeft_eq 📋 Mathlib.Algebra.Module.LinearMap.End
  {M : Type u_4} [AddCommMonoid M] {R : Type u_9} [CommSemiring R] [Module R M] (α : R) (hα : α ∈ Set.center R := by simp) : Module.End.smulLeft α hα = α • LinearMap.id
• LinearMap.applyₗ 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {M : Type u_4} {M₂ : Type u_6} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : M →ₗ[R] (M →ₗ[R] M₂) →ₗ[R] M₂
• LinearMap.smulRightₗ 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {M : Type u_4} {M₂ : Type u_6} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : (M₂ →ₗ[R] R) →ₗ[R] M →ₗ[R] M₂ →ₗ[R] M
• LinearMap.applyₗ_apply_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {M : Type u_4} {M₂ : Type u_6} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (v : M) (f : M →ₗ[R] M₂) : (LinearMap.applyₗ v) f = f v
• LinearMap.smulRightₗ_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {M : Type u_4} {M₂ : Type u_6} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : M₂ →ₗ[R] R) (x : M) : (LinearMap.smulRightₗ f) x = f.smulRight x
• LinearMap.smulRightₗ_apply_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {M : Type u_4} {M₂ : Type u_6} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : M₂ →ₗ[R] R) (x : M) (y : M₂) : ((LinearMap.smulRightₗ f) x) y = f y • x
• LinearEquiv.smulOfUnit 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_1} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (a : Rˣ) : M ≃ₗ[R] M
• LinearEquiv.conj 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') : Module.End R₁' M₁' ≃ₛₗ[σ₁'₂'] Module.End R₂' M₂'
• LinearEquiv.congrRight 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_1} {M : Type u_5} {M₂ : Type u_7} {M₃ : Type u_8} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M₂ ≃ₗ[R] M₃) : (M →ₗ[R] M₂) ≃ₗ[R] M →ₗ[R] M₃
• LinearEquiv.arrowCongr 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁ : Type u_9} {R₂ : Type u_10} {R₁' : Type u_12} {R₂' : Type u_13} {M₁ : Type u_17} {M₂ : Type u_18} {M₁' : Type u_20} {M₂' : Type u_21} [Semiring R₁] [Semiring R₂] [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁ M₁] [Module R₂ M₂] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₂₂' : R₂ →+* R₂'} {σ₂₁' : R₂ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [RingHomCompTriple σ₂₁ σ₁₂' σ₂₂'] [RingHomCompTriple σ₂₂' σ₂'₁' σ₂₁'] [RingHomCompTriple σ₁₂ σ₂₁' σ₁₁'] (e₁ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₂ : M₁' ≃ₛₗ[σ₁'₂'] M₂') : (M₁ →ₛₗ[σ₁₁'] M₁') ≃ₛₗ[σ₁'₂'] M₂ →ₛₗ[σ₂₂'] M₂'
• LinearEquiv.conj_id 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') : e.conj LinearMap.id = LinearMap.id
• LinearEquiv.conj_refl 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_1} {M : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : Module.End R M) : (LinearEquiv.refl R M).conj f = f
• LinearMap.piApply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_1} {M : Type u_5} {V : M → Type u_9} [CommSemiring R] [(x : M) → AddCommMonoid (V x)] [(x : M) → Module R (V x)] : ((x : M) → V x →ₗ[R] R) →ₗ[R] ((x : M) → V x) →ₗ[R] M → R
• LinearEquiv.conj_trans 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {R₃' : Type u_14} {M₁' : Type u_20} {M₂' : Type u_21} {M₃' : Type u_22} [CommSemiring R₁'] [CommSemiring R₂'] [CommSemiring R₃'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [AddCommMonoid M₃'] [Module R₁' M₁'] [Module R₂' M₂'] [Module R₃' M₃'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₂'₃' : R₂' →+* R₃'} {σ₃'₂' : R₃' →+* R₂'} {σ₁'₃' : R₁' →+* R₃'} {σ₃'₁' : R₃' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomInvPair σ₂'₃' σ₃'₂'] [RingHomInvPair σ₃'₂' σ₂'₃'] [RingHomInvPair σ₁'₃' σ₃'₁'] [RingHomInvPair σ₃'₁' σ₁'₃'] [RingHomCompTriple σ₁'₂' σ₂'₃' σ₁'₃'] [RingHomCompTriple σ₃'₂' σ₂'₁' σ₃'₁'] (e₁ : M₁' ≃ₛₗ[σ₁'₂'] M₂') (e₂ : M₂' ≃ₛₗ[σ₂'₃'] M₃') : e₁.conj.trans e₂.conj = (e₁.trans e₂).conj
• LinearEquiv.conj_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : Module.End R₁' M₁') : e.conj f = (↑e ∘ₛₗ f) ∘ₛₗ ↑e.symm
• LinearEquiv.symm_conj_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : Module.End R₂' M₂') : e.symm.conj f = (↑e.symm ∘ₛₗ f) ∘ₛₗ ↑e
• LinearEquiv.conj_apply_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : Module.End R₁' M₁') (x : M₂') : (e.conj f) x = e (f (e.symm x))
• LinearEquiv.arrowCongr_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁ : Type u_9} {R₂ : Type u_10} {R₁' : Type u_12} {R₂' : Type u_13} {M₁ : Type u_17} {M₂ : Type u_18} {M₁' : Type u_20} {M₂' : Type u_21} [Semiring R₁] [Semiring R₂] [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁ M₁] [Module R₂ M₂] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₂₂' : R₂ →+* R₂'} {σ₂₁' : R₂ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [RingHomCompTriple σ₂₁ σ₁₂' σ₂₂'] [RingHomCompTriple σ₂₂' σ₂'₁' σ₂₁'] [RingHomCompTriple σ₁₂ σ₂₁' σ₁₁'] (e₁ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₂ : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : M₁ →ₛₗ[σ₁₁'] M₁') (x : M₂) : ((e₁.arrowCongr e₂) f) x = e₂ (f (e₁.symm x))
• LinearEquiv.arrowCongr_symm_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁ : Type u_9} {R₂ : Type u_10} {R₁' : Type u_12} {R₂' : Type u_13} {M₁ : Type u_17} {M₂ : Type u_18} {M₁' : Type u_20} {M₂' : Type u_21} [Semiring R₁] [Semiring R₂] [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁ M₁] [Module R₂ M₂] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₂₂' : R₂ →+* R₂'} {σ₂₁' : R₂ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [RingHomCompTriple σ₂₁ σ₁₂' σ₂₂'] [RingHomCompTriple σ₂₂' σ₂'₁' σ₂₁'] [RingHomCompTriple σ₁₂ σ₂₁' σ₁₁'] (e₁ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₂ : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : M₂ →ₛₗ[σ₂₂'] M₂') (x : M₁) : ((e₁.arrowCongr e₂).symm f) x = e₂.symm (f (e₁ x))
• LinearEquiv.arrowCongr_trans 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁ : Type u_9} {R₂ : Type u_10} {R₃ : Type u_11} {R₁' : Type u_12} {R₂' : Type u_13} {R₃' : Type u_14} {M₁ : Type u_17} {M₂ : Type u_18} {M₃ : Type u_19} {M₁' : Type u_20} {M₂' : Type u_21} {M₃' : Type u_22} [Semiring R₁] [Semiring R₂] [Semiring R₃] [CommSemiring R₁'] [CommSemiring R₂'] [CommSemiring R₃'] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [AddCommMonoid M₃'] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [Module R₁' M₁'] [Module R₂' M₂'] [Module R₃' M₃'] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} {σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₂'₃' : R₂' →+* R₃'} {σ₃'₂' : R₃' →+* R₂'} {σ₁'₃' : R₁' →+* R₃'} {σ₃'₁' : R₃' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₂₂' : R₂ →+* R₂'} {σ₃₃' : R₃ →+* R₃'} {σ₂₁' : R₂ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} {σ₃₂' : R₃ →+* R₂'} {σ₂₃' : R₂ →+* R₃'} {σ₃₁' : R₃ →+* R₁'} {σ₁₃' : R₁ →+* R₃'} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomInvPair σ₂₃ σ₃₂] [RingHomInvPair σ₃₂ σ₂₃] [RingHomInvPair σ₂'₃' σ₃'₂'] [RingHomInvPair σ₃'₂' σ₂'₃'] [RingHomInvPair σ₁₃ σ₃₁] [RingHomInvPair σ₃₁ σ₁₃] [RingHomInvPair σ₁'₃' σ₃'₁'] [RingHomInvPair σ₃'₁' σ₁'₃'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [RingHomCompTriple σ₂₁ σ₁₂' σ₂₂'] [RingHomCompTriple σ₂₂' σ₂'₁' σ₂₁'] [RingHomCompTriple σ₁₂ σ₂₁' σ₁₁'] [RingHomCompTriple σ₁₁' σ₁'₃' σ₁₃'] [RingHomCompTriple σ₃₁ σ₁₃' σ₃₃'] [RingHomCompTriple σ₃₃' σ₃'₁' σ₃₁'] [RingHomCompTriple σ₁₃ σ₃₁' σ₁₁'] [RingHomCompTriple σ₂₂' σ₂'₃' σ₂₃'] [RingHomCompTriple σ₃₂ σ₂₃' σ₃₃'] [RingHomCompTriple σ₃₃' σ₃'₂' σ₃₂'] [RingHomCompTriple σ₂₃ σ₃₂' σ₂₂'] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] [RingHomCompTriple σ₁'₂' σ₂'₃' σ₁'₃'] [RingHomCompTriple σ₃'₂' σ₂'₁' σ₃'₁'] (e₁ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₁' : M₁' ≃ₛₗ[σ₁'₂'] M₂') (e₂ : M₂ ≃ₛₗ[σ₂₃] M₃) (e₂' : M₂' ≃ₛₗ[σ₂'₃'] M₃') : (e₁.arrowCongr e₁').trans (e₂.arrowCongr e₂') = (e₁.trans e₂).arrowCongr (e₁'.trans e₂')
• LinearEquiv.conj_conj_symm 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : Module.End R₂' M₂') : e.conj (e.symm.conj f) = f
• LinearEquiv.conj_symm_conj 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R₁' : Type u_12} {R₂' : Type u_13} {M₁' : Type u_20} {M₂' : Type u_21} [CommSemiring R₁'] [CommSemiring R₂'] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] (e : M₁' ≃ₛₗ[σ₁'₂'] M₂') (f : Module.End R₁' M₁') : e.symm.conj (e.conj f) = f
• LinearMap.piApply_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_1} {M : Type u_5} {V : M → Type u_9} [CommSemiring R] [(x : M) → AddCommMonoid (V x)] [(x : M) → Module R (V x)] (e : (x : M) → V x →ₗ[R] R) (s : (x : M) → V x) : (LinearMap.piApply e) s = fun x => (e x) (s x)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c