Loogle!
Result
Found 59 definitions mentioning Function.Injective and Prod. Of these, 26 match your pattern(s).
- Function.Injective2.uncurry Mathlib.Logic.Function.Basic
∀ {α : Type u_4} {β : Type u_5} {γ : Type u_6} {f : α → β → γ}, Function.Injective2 f → Function.Injective (Function.uncurry f) - Prod.fst_injective Mathlib.Data.Prod.Basic
∀ {α : Type u_1} {β : Type u_2} [inst : Subsingleton β], Function.Injective Prod.fst - Prod.snd_injective Mathlib.Data.Prod.Basic
∀ {α : Type u_1} {β : Type u_2} [inst : Subsingleton α], Function.Injective Prod.snd - Prod.swap_injective Mathlib.Data.Prod.Basic
∀ {α : Type u_1} {β : Type u_2}, Function.Injective Prod.swap - Function.Injective.Prod_map Mathlib.Data.Prod.Basic
∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ} {g : β → δ}, Function.Injective f → Function.Injective g → Function.Injective (Prod.map f g) - Prod.map_injective Mathlib.Data.Prod.Basic
∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : Nonempty α] [inst : Nonempty β] {f : α → γ} {g : β → δ}, Function.Injective (Prod.map f g) ↔ Function.Injective f ∧ Function.Injective g - Equiv.Perm.sumCongrHom_injective Mathlib.GroupTheory.Perm.Basic
∀ {α : Type u_1} {β : Type u_2}, Function.Injective ⇑(Equiv.Perm.sumCongrHom α β) - Equiv.Perm.subtypeCongrHom_injective Mathlib.GroupTheory.Perm.Basic
∀ {α : Type u} (p : α → Prop) [inst : DecidablePred p], Function.Injective ⇑(Equiv.Perm.subtypeCongrHom p) - AddSubgroup.add_injective_of_disjoint Mathlib.Algebra.Group.Subgroup.Basic
∀ {G : Type u_1} [inst : AddGroup G] {H₁ H₂ : AddSubgroup G}, Disjoint H₁ H₂ → Function.Injective fun g => ↑g.1 + ↑g.2 - Subgroup.mul_injective_of_disjoint Mathlib.Algebra.Group.Subgroup.Basic
∀ {G : Type u_1} [inst : Group G] {H₁ H₂ : Subgroup G}, Disjoint H₁ H₂ → Function.Injective fun g => ↑g.1 * ↑g.2 - LinearMap.tailing Mathlib.LinearAlgebra.Prod
{R : Type u} → {M : Type v} → [inst : Ring R] → {N : Type u_3} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (f : M × N →ₗ[R] M) → Function.Injective ⇑f → ℕ → Submodule R M - LinearMap.tailings Mathlib.LinearAlgebra.Prod
{R : Type u} → {M : Type v} → [inst : Ring R] → {N : Type u_3} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (f : M × N →ₗ[R] M) → Function.Injective ⇑f → ℕ → Submodule R M - LinearMap.tailings_zero Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f), f.tailings i 0 = f.tailing i 0 - LinearMap.tunnel Mathlib.LinearAlgebra.Prod
{R : Type u} → {M : Type v} → [inst : Ring R] → {N : Type u_3} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (f : M × N →ₗ[R] M) → Function.Injective ⇑f → ℕ →o (Submodule R M)ᵒᵈ - LinearMap.tailings_disjoint_tailing Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ), Disjoint (f.tailings i n) (f.tailing i (n + 1)) - LinearMap.tunnel' Mathlib.LinearAlgebra.Prod
{R : Type u} → {M : Type v} → [inst : Ring R] → {N : Type u_3} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (f : M × N →ₗ[R] M) → Function.Injective ⇑f → ℕ → (K : Submodule R M) × ↥K ≃ₗ[R] M - LinearMap.tailingLinearEquiv Mathlib.LinearAlgebra.Prod
{R : Type u} → {M : Type v} → [inst : Ring R] → {N : Type u_3} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (f : M × N →ₗ[R] M) → (i : Function.Injective ⇑f) → (n : ℕ) → ↥(f.tailing i n) ≃ₗ[R] N - LinearMap.tailings_succ Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ), f.tailings i (n + 1) = f.tailings i n ⊔ f.tailing i (n + 1) - LinearMap.tunnelAux_injective Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M), Function.Injective ⇑f → ∀ (Kφ : (K : Submodule R M) × ↥K ≃ₗ[R] M), Function.Injective ⇑(f.tunnelAux Kφ) - LinearMap.tailing_disjoint_tunnel_succ Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ), Disjoint (f.tailing i n) (OrderDual.ofDual ((f.tunnel i) (n + 1))) - LinearMap.tailing_le_tunnel Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ), f.tailing i n ≤ OrderDual.ofDual ((f.tunnel i) n) - LinearMap.tailings_disjoint_tunnel Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ), Disjoint (f.tailings i n) (OrderDual.ofDual ((f.tunnel i) (n + 1))) - LinearMap.tailing_sup_tunnel_succ_le_tunnel Mathlib.LinearAlgebra.Prod
∀ {R : Type u} {M : Type v} [inst : Ring R] {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ), f.tailing i n ⊔ OrderDual.ofDual ((f.tunnel i) (n + 1)) ≤ OrderDual.ofDual ((f.tunnel i) n) - IsNoetherian.equivPUnitOfProdInjective Mathlib.RingTheory.Noetherian
{R : Type u_1} → {M : Type u_2} → {N : Type w} → [inst : Ring R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → [inst_5 : IsNoetherian R M] → (f : M × N →ₗ[R] M) → Function.Injective ⇑f → N ≃ₗ[R] PUnit.{w + 1} - Unitization.splitMul_injective Mathlib.Analysis.NormedSpace.Unitization
∀ (𝕜 : Type u_1) (A : Type u_2) [inst : NontriviallyNormedField 𝕜] [inst_1 : NonUnitalNormedRing A] [inst_2 : NormedSpace 𝕜 A] [inst_3 : IsScalarTower 𝕜 A A] [inst_4 : SMulCommClass 𝕜 A A] [inst_5 : RegularNormedAlgebra 𝕜 A], Function.Injective ⇑(Unitization.splitMul 𝕜 A) - Unitization.splitMul_injective_of_clm_mul_injective Mathlib.Analysis.NormedSpace.Unitization
∀ {𝕜 : Type u_1} {A : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NonUnitalNormedRing A] [inst_2 : NormedSpace 𝕜 A] [inst_3 : IsScalarTower 𝕜 A A] [inst_4 : SMulCommClass 𝕜 A A], Function.Injective ⇑(ContinuousLinearMap.mul 𝕜 A) → Function.Injective ⇑(Unitization.splitMul 𝕜 A)
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.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
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 findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
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, _ * _, _ ^ _, |- _ < _ → _
woould 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 currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision d874bdf