theorem isArtinian_of_linearEquiv :
    ∀ {R : Type u_1} {M : Type u_2} {P : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : AddCommGroup P]
      [inst_3 : Module R M] [inst_4 : Module R P], (M ≃ₗ[R] P) → ∀ [inst_5 : IsArtinian R M], IsArtinian R P :=
  by sorry

This theorem states that for any types `R`, `M`, and `P`, where `R` is a ring, `M` and `P` are additive commutative groups, `M` is a `R`-module, and `P` is also a `R`-module. Given that `M` is isomorphic to `P` under a linear equivalence, if `M` is an Artinian `R`-module, then `P` is also an Artinian `R`-module. In other words, the Artinian property of a module is preserved under a linear equivalence.

If `M` and `P` are isomorphic additive commutative groups that are `R`-modules, and `M` is an Artinian `R`-module, then `P` is also an Artinian `R`-module.


theorem inv_le_inv₀ :
    ∀ {α : Type u_1} [inst : LinearOrderedCommGroupWithZero α] {a b : α}, a ≠ 0 → b ≠ 0 → (a⁻¹ ≤ b⁻¹ ↔ b ≤ a) := by
  sorry

This theorem states that for any type `α` which has the structure of a linearly ordered commutative group with zero, given any two non-zero elements `a` and `b` of this type, the reciprocal of `a` (denoted by `a⁻¹`) is less than or equal to the reciprocal of `b` (denoted by `b⁻¹`) if and only if `b` is less than or equal to `a`. This is a standard result about orderings and reciprocals in the field of algebra.

For any linearly ordered commutative group with zero `α`, if `a` and `b` are non-zero elements of `α` with `a ≤ b`, then `a⁻¹ ≥ b⁻¹`.


theorem Set.bounded_lt_inter_not_lt :
    ∀ {α : Type u_1} {s : Set α} [inst : SemilatticeSup α] (a : α),
      Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | ¬b < a}) ↔ Set.Bounded (fun x x_1 => x < x_1) s :=
  by sorry

This theorem states that for any given set 's' of elements from a semilattice-supremum type 'α', and any element 'a' from the same type, the condition of 's' being a bounded set under the less-than relation is equivalent to the set formed by the intersection of 's' and the set of all elements 'b' not less than 'a' being a bounded set under the less-than relation. In mathematical terms, the property of being bounded below by a certain element in the set 's' carries over to the intersection of 's' and the elements 'b' that are not less than a given element 'a'.

For any semilattice-supremum type 'α' and set 's' of elements from it, 's' is bounded if and only if the intersection of 's' and the set of elements greater than or equal to any given 'a' in 's' is also bounded.


/-- If `a` is a left identity for `f : α → β → β`, then `{a}` is a left identity for
`Set.image2 f`.  -/
theorem Set.image2_left_identity :
    ∀ {α : Type u_1} {β : Type u_3} {f : α → β → β} {a : α},
      (∀ (b : β), f a b = b) → ∀ (t : Set β), Set.image2 f {a} t = t :=
  by sorry

This theorem asserts that if an element `a` of a type `α` is a left identity for a binary function `f : α → β → β` (that is, for any element `b` of type `β`, `f a b` equals `b`), then a set containing only this element `{a}` behaves as a left identity for the function `Set.image2 f`, which maps sets of `α` and `β` to sets of `β`. Specifically, for any set `t` of type `β`, `Set.image2 f {a} t` equals `t`. This implies that applying the function `f` with `a` as the left argument to any element of `t` leaves the element unchanged, hence `t` is left invariant by `Set.image2 f {a}`.

If an element `a` of type `α` is a left identity for a binary function `f : α → β → β`, then the set `{a}` is a left identity for the function `Set.image2 f` on sets of type `α` and `β`, meaning `Set.image2 f {a} t = t` for any set `t` of type `β`.


theorem CategoryTheory.SingleObj.comp_as_mul :
    ∀ (M : Type u) [inst : Monoid M] {x y z : CategoryTheory.SingleObj M} (f : x ⟶ y) (g : y ⟶ z),
      CategoryTheory.CategoryStruct.comp f g = g * f :=
  by sorry

This theorem, `CategoryTheory.SingleObj.comp_as_mul`, states that for any type `M` that is a Monoid, and for any three objects `x`, `y`, and `z` in the `SingleObj` category of `M`, the composition of two morphisms `f` (from `x` to `y`) and `g` (from `y` to `z`) in this category is the same as the multiplication of `g` and `f` in the Monoid `M`. Here, the composition of morphisms is denoted by `CategoryTheory.CategoryStruct.comp f g` and the multiplication in the Monoid `M` is denoted by `g * f`. The order of multiplication is reversed compared to the order of composition due to the convention in category theory.

In the SingleObj category of a Monoid M, the composition of morphisms f from x to y and g from y to z equals the multiplication of g and f in M, where the order of multiplication is reversed compared to the order of composition.


theorem frontier_Iic_subset :
    ∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [inst_2 : OrderClosedTopology α] (a : α),
      frontier (Set.Iic a) ⊆ {a} :=
  by sorry

The theorem `frontier_Iic_subset` asserts that for all types `α` equipped with a topological space structure, a linear order structure, and an order-closed topology structure, for any element `a` of type `α`, the frontier of the left-infinite right-closed interval ending at `a` is a subset of the set that contains only `a`. In other words, the only boundary point of the interval from negative infinity to `a` inclusive, in a space with these structures, is `a` itself.

For any type `α` with a topological space, linear order, and order-closed topology structures, the frontier of the left-infinite right-closed interval in this structure is equal to the singleton set containing the endpoint `a`.


theorem UpperSet.coe_iInf₂ :
    ∀ {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [inst : LE α] (f : (i : ι) → κ i → UpperSet α),
      ↑(⨅ i, ⨅ j, f i j) = ⋃ i, ⋃ j, ↑(f i j) :=
  by sorry

This theorem states that for any type `α` with a less-than-or-equal-to relation (`LE α`), and for any type-dependent function `f` that maps every element of type `ι` and its dependent type `κ i` to an upper set of `α`, the intersection of all upper sets obtained by `f` is equivalent to the union of the individual upper sets. In other words, the co-domain of the infimum operation (`⨅`) applied to `f` is equal to the union (`⋃`) of the co-domains of the individual sets produced by `f`.

For any type `α` with a `LE` relation and any type-dependent function `f` mapping each `ι i` and its dependent type `κ i` to an upper set of `α`, the infimum of the `f i`'s upper sets equals the union of their co-domains.


theorem MvQPF.Fix.rec_eq :
    ∀ {n : ℕ} {F : TypeVec.{u} (n + 1) → Type u} [inst : MvFunctor F] [q : MvQPF F] {α : TypeVec.{u} n} {β : Type u}
      (g : F (α ::: β) → β) (x : F (α ::: MvQPF.Fix F α)),
      MvQPF.Fix.rec g (MvQPF.Fix.mk x) = g (MvFunctor.map (TypeVec.id ::: MvQPF.Fix.rec g) x) :=
  by sorry

The theorem `MvQPF.Fix.rec_eq` states that for any natural number `n`, and for any functor `F` of a tuple of types (or `TypeVec`) with `n+1` number of types, given a multivariate quasi-polynomial functor `q` of `F` and an instance `inst` of `F` being a multivariate functor, for any tuple of types `α` with `n` number of types and any type `β`, and given a function `g` from `F` applied to the type vector `α` and `β` to `β` and a value `x` of type `F` applied to the type vector `α` and the least fixed point of `F` applied to `α`, the recursor of the least fixed point of `F` applied to `α` using the function `g`, when applied to the constructor of the least fixed point of `F` applied to `α` with the value `x`, is equal to `g` applied to the result of mapping the function composed of the identity function and the recursor of the least fixed point of `F` applied to `α` using the function `g` to `x`. In mathematical terms, this theorem justifies the process of recursion on data types defined as fixed points of functors in a category-theoretic context by showing that the recursor is equivalent to applying a function to a mapped value.

For any multivariate quasi-polynomial functor `q` and instance `F` of a multivariate functor, the recursor of the least fixed point of `F` applied to a tuple of types `α` using a function `g`, when applied to the constructor of the least fixed point with a value `x`, equals `g` applied to the mapped value of `x` and the recursor of `g` applied to the least fixed point of `F` using `g`.


theorem Set.restrictPreimage_injective :
    ∀ {α : Type u_1} {β : Type u_2} (t : Set β) {f : α → β},
      Function.Injective f → Function.Injective (Set.restrictPreimage t f) :=
  by sorry

The theorem `Set.restrictPreimage_injective` states that for all types `α` and `β`, given a set `t` of type `β` and a function `f` from `α` to `β`, if the function `f` is injective (meaning that it maps distinct inputs to distinct outputs), then the restriction of the function `f` onto the preimage of the set `t` is also injective. In mathematical terms, if $f: \alpha \rightarrow \beta$ is an injective function and $t$ is a set of elements of $\beta$, then the restriction of $f$ onto $f^{-1}(t)$ is also injective.

If `f` is an injective function from type `α` to type `β` and `t` is a set of type `β`, then the restriction of `f` to `f⁻¹(t)` is also an injective function.


theorem SemiNormedGroupCat.explicitCokernelDesc.proof_1 :
    ∀ {X Y Z : SemiNormedGroupCat} {f : X ⟶ Y} {g : Y ⟶ Z},
      CategoryTheory.CategoryStruct.comp f g = 0 →
        CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp 0 g :=
  by sorry

This theorem states that for any three objects X, Y, and Z in the category of seminormed abelian groups, and any two morphisms f : X ⟶ Y and g : Y ⟶ Z, if the composition of f and g (represented by `f ≫ g`) is zero, then it is equivalent to the composition of the zero morphism and g (represented by `0 ≫ g`).

If for any seminormed abelian groups X, Y, Z and morphisms f : X ➔ Y and g : Y ➔ Z, the composition of f and g (f ≫ g) is zero then it is equivalent to the composition of the zero morphism and g (0 ≫ g).


OrderIso.injective

theorem OrderIso.injective :
    ∀ {α : Type u_2} {β : Type u_3} [inst : LE α] [inst_1 : LE β] (e : α ≃o β), Function.Injective ⇑e := by sorry

The theorem `OrderIso.injective` states that for any two types `α` and `β`, if `α` and `β` are equipped with `≤` order relations (denoted by `inst : LE α` and `inst_1 : LE β` respectively) and there exists an order isomorphism `e : α ≃o β` between them, then the function represented by `e` (denoted by `⇑e`) is injective. In other words, if `e` maps two elements in `α` to the same element in `β`, then those two elements in `α` were the same.

If `α` and `β` are ordered types with an order isomorphism between them, then the order isomorphism is an injection.


theorem Mathlib.GroupTheory.Submonoid.Operations._auxAddLemma.20 :
    ∀ {M : Type u_1} {N : Type u_2} [inst : AddZeroClass M] [inst_1 : AddZeroClass N] {F : Type u_5}
      [inst_2 : FunLike F M N] [mc : AddMonoidHomClass F M N] (f : F) {x : M}, (x ∈ AddMonoidHom.mker f) = (f x = 0) :=
  by sorry

This theorem, `Mathlib.GroupTheory.Submonoid.Operations._auxAddLemma.20`, states that for every type `M`, `N` and `F` such that `M` and `N` form add-zero classes, `F` is a function-like structure from `M` to `N` and `F` is an additive monoid homomorphism from `M` to `N`, and for every function `f : F` and element `x : M`, `x` belongs to the additive kernel of the homomorphism `f` if and only if `f(x) = 0`. The additive kernel of a homomorphism is the subset of elements that get mapped to the additive identity (zero).

For any additive monoid homomorphism `f` from an add-zero class `M` to an add-zero class `N`, `x` in `M` belongs to the kernel of `f` if and only if `f(x) = 0`.


theorem Int.negOnePow_odd : ∀ (n : ℤ), Odd n → n.negOnePow = -1 := by sorry

This theorem states that for any integer `n`, if `n` is odd (meaning there exists an integer `k` such that `n = 2 * k + 1`), then the result of raising `-1` to the power of `n` (denoted by `n.negOnePow`) is equal to `-1`. Essentially, it's capturing the mathematical fact that -1 raised to any odd integer power is always -1.

For any integer `n`, if `n` is odd then `(-1 : ℕ) ^ n = -1`.


theorem AddMonoid.Coprod.swap_surjective :
    ∀ {M : Type u_1} {N : Type u_2} [inst : AddZeroClass M] [inst_1 : AddZeroClass N],
      Function.Surjective ⇑(AddMonoid.Coprod.swap M N) :=
  by sorry

The theorem `AddMonoid.Coprod.swap_surjective` states that for all types `M` and `N` that have the structure of an additive monoid (i.e., they have zero and addition operations that satisfy certain properties), the function `AddMonoid.Coprod.swap M N` is surjective. This means that for every element in the coproduct additive monoid of `N` and `M`, there exists an element in the coproduct additive monoid of `M` and `N` such that applying the `AddMonoid.Coprod.swap M N` function to this element results in the original element. In other words, every element in the target monoid is an image of some element in the source monoid under the `AddMonoid.Coprod.swap M N` function.

For every additive monoid `M` and `N`, the function `AddMonoid.Coprod.swap : M ⊎ N ↔ N ⊎ M` is a surjection.


theorem Profinite.NobelingProof.continuous_proj :
    ∀ {I : Type u} (J : I → Prop) [inst : (i : I) → Decidable (J i)], Continuous (Profinite.NobelingProof.Proj J) := by
  sorry

The theorem `Profinite.NobelingProof.continuous_proj` asserts that for any type `I`, any property `J` over `I`, and given that for each element `i` of type `I` it is decidable whether `J i` holds or not, the projection function `Profinite.NobelingProof.Proj J` that maps everything satisfying `J i` to itself and everything else to `false` is continuous. In mathematical terms, this essentially means that the projection function preserves the topological structure and does not disrupt the continuity of the underlying set.

Given a type `I` and a decidable property `J` over `I`, the continuous map `Profinte.NobelingProof.Proj J : I -> Prop` sending `i : I` to `J i` if `J i` holds and `false` otherwise, is a continuous function.


theorem Dioph.dioph_comp :
    ∀ {α : Type} {n : ℕ} {S : Set (Vector3 ℕ n)},
      Dioph S → ∀ (f : Vector3 ((α → ℕ) → ℕ) n), VectorAllP Dioph.DiophFn f → Dioph {v | (fun i => f i v) ∈ S} :=
  by sorry

The theorem `Dioph.dioph_comp` states that for any type `α`, natural number `n`, and set `S` of vectors of length `n` where each vector consists of natural numbers, if `S` is Diophantine then for any vector `f` where each element is a function from sequences of natural numbers to natural numbers and each of these functions is Diophantine, the set of sequences `v` for which the corresponding function values `(fun i => f i v)` are in `S` is also Diophantine. In other words, the theorem asserts that the composition of Diophantine functions yields a Diophantine set.

If `S` is a Diophantine set of vectors of length `n` over natural numbers, and `f` is a function consisting of Diophantine functions from sequences of natural numbers to natural numbers, then the set of sequences `v` such that the function values `(fun i => f i v)` are in `S` is also Diophantine.


theorem Fin.strictMono_succAbove : ∀ {n : ℕ} (p : Fin (n + 1)), StrictMono (Fin.succAbove p) := by sorry

The theorem `Fin.strictMono_succAbove` states that for any natural number `n` and for any value `p` of type `Fin (n + 1)`, the function `Fin.succAbove p` is strictly monotone. In other words, if `a` and `b` are two elements of `Fin n` such that `a < b`, then `Fin.succAbove p a < Fin.succAbove p b`. This function `Fin.succAbove p` embeds `Fin n` into `Fin (n + 1)` leaving a gap at `p`, and it respects the ordering in `Fin n`, that is, smaller elements map to smaller elements and larger elements map to larger elements, with strict inequality preserved.

For any natural number `n` and `p` in `Fin (n + 1)`, the function `Fin.succAbove p : Fin n -> Fin (n + 1)` is a strict monotonic embedding, meaning `a < b` in `Fin n` implies `Fin.succAbove p a < Fin.succAbove p b`.


theorem UniformSpace.Completion.continuous_mul :
    ∀ {α : Type u_1} [inst : Ring α] [inst_1 : UniformSpace α] [inst_2 : TopologicalRing α]
      [inst_3 : UniformAddGroup α], Continuous fun p => p.1 * p.2 :=
  by sorry

This theorem states that for any type `α` that has a ring structure, a uniform space structure, is a topological ring, and a uniform add group, the multiplication operation, which takes a pair of elements and returns their product, is continuous. This means that for any small change in the input values, there is a correspondingly small change in the output value. This property is crucial for many mathematical constructs involving limits, derivatives, and integrals, among others.

For any type `α` endowed with a ring structure, a uniform space structure making it a topological ring, and a uniform add group, the multiplication operation is continuous.


theorem Commensurable.commensurable_conj :
    ∀ {G : Type u_1} [inst : Group G] {H K : Subgroup G} (g : ConjAct G),
      Commensurable H K ↔ Commensurable (g • H) (g • K) :=
  by sorry

This theorem states that for any group `G`, and any two subgroups `H` and `K` of `G`, along with a group element `g` (viewed as an element of the "conjugate action" `ConjAct`), the two subgroups `H` and `K` are commensurable if and only if the conjugates of the subgroups by `g` (i.e., `g•H` and `g•K`) are also commensurable. Here, "commensurable" means that the intersection of the two subgroups has finite index in both subgroups, and "conjugation" of a subgroup by `g` means the set of elements obtained by multiplying each element of the subgroup on the left by `g` and on the right by `g⁻¹`.

For any group `G`, subgroups `H` and `K` of `G`, and group element `g`, `H` and `K` are commensurable if and only if their conjugates `g•H` and `g•K` are commensurable.


theorem Matrix.charmatrix_apply_ne :
    ∀ {R : Type u_1} [inst : CommRing R] {n : Type u_4} [inst_1 : DecidableEq n] [inst_2 : Fintype n] (M : Matrix n n R)
      (i j : n), i ≠ j → Matrix.charmatrix M i j = -Polynomial.C (M i j) :=
  by sorry

This theorem states that for any commutative ring `R`, any finite type `n` with decidable equality, and any matrix `M` with entries in `R` and indices in `n`, if the row index `i` is not equal to the column index `j`, then the entry of the characteristic matrix of `M` at `(i, j)` is equal to the negation of the entry of `M` at `(i, j)` under the constant polynomial map. In other words, for non-diagonal entries, the characteristic matrix contains the negation of the original matrix's entries, converted to constant polynomials.

For any commutative ring `R`, finite type `n` with decidable equality, and matrix `M` over `R` with indices in `n`, the non-diagonal entries of the characteristic matrix of `M` are the negations of the constant polynomial evaluations of `M`'s entries.


HasMFDerivAt.hasMFDerivWithinAt

theorem HasMFDerivAt.hasMFDerivWithinAt :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H}
      {M : Type u_4} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {E' : Type u_5}
      [inst_6 : NormedAddCommGroup E'] [inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H']
      {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M']
      {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M']
      {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)},
      HasMFDerivAt I I' f x f' → HasMFDerivWithinAt I I' f s x f' :=
  by sorry

This theorem states that, given a non-trivially normed field `𝕜`, normed additive commutative groups `E` and `E'`, topological spaces `H`, `M`, `H'`, and `M'`, a model with corners `I` between `𝕜`, `E`, and `H`, a model with corners `I'` between `𝕜`, `E'`, and `H'`, charted spaces `M` and `M'` with `H` and `H'` respectively, a function `f` from `M` to `M'`, a point `x` in `M`, a set `s` of `M`, smooth manifolds `M` and `M'` with corners `I` and `I'` respectively, and a derivative `f'` of `f` at `x`, if `f` has a manifold derivative at `x`, then `f` also has a manifold derivative within the set `s` at `x`.

Given a smooth function `f` between smooth manifolds `M` and `M'`, with `M` and `M'` having corners `I` and `I'`, respectively, and `f` having a manifold derivative at a point `x` in `M`, then `f` also has a manifold derivative within any set `s` in `M` containing `x`.


IsHausdorff.haus

theorem IsHausdorff.haus :
    ∀ {R : Type u_1} [inst : CommRing R] {I : Ideal R} {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M],
      IsHausdorff I M → ∀ (x : M), (∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]) → x = 0 :=
  by sorry

The theorem `IsHausdorff.haus` states that for any commutative ring `R`, any ideal `I` of `R`, and any module `M` over `R` that is Hausdorff with respect to `I`, for every element `x` in `M`, if `x` is equivalent to zero modulo the nth power of `I` scaled by the top element of `R`, for all natural numbers `n`, then `x` is equal to zero. In simple terms, in a Hausdorff module over a commutative ring, any element that is arbitrarily close to zero in a certain sense is actually zero.

In a commutative ring and its Hausdorff module, an element equivalent to zero under repeated powers of an ideal is actually zero.


/-- If the elements of a set are independent, then any pair within that set is disjoint.  -/
theorem CompleteLattice.Independent.pairwiseDisjoint :
    ∀ {α : Type u_1} {ι : Type u_3} [inst : CompleteLattice α] {t : ι → α},
      CompleteLattice.Independent t → Pairwise (Disjoint on t) :=
  by sorry

This theorem states that if the elements of a set are independent in a complete lattice, then any pair of elements within that set is disjoint. Specifically, for any type α that forms a complete lattice and any function `t` from some index set ι to α, if the elements of the function's range are independent (meaning no element is below the join of some other elements), then any two distinct elements in the range of `t` are disjoint. Here, two elements `a` and `b` are considered disjoint if any element `x` that is below both `a` and `b` must also be below the bottom ('⊥') element of the lattice.

In a complete lattice, if the elements of a function from an index set to the lattice are independent, then any two distinct elements in the range are disjoint.


theorem Nat.mul_ne_zero_iff : ∀ {n m : ℕ}, n * m ≠ 0 ↔ n ≠ 0 ∧ m ≠ 0 := by sorry

This theorem states that for any two natural numbers `n` and `m`, the product of `n` and `m` is not zero if and only if both `n` and `m` are not zero. In other words, the only way the multiplication of two natural numbers can result in zero is if at least one of the numbers is zero itself.

For natural numbers `n` and `m`, `n * m = 0` if and only if `n = 0` or `m = 0`.


theorem List.join_filter_not_isEmpty :
    ∀ {α : Type u_1} {L : List (List α)}, List.join (List.filter (fun l => !List.isEmpty l) L) = List.join L := by sorry

This theorem states that for any type `α` and any list `L` of lists of type `α`, if you filter out the empty lists from `L` and then concatenate all the remaining lists, the result is the same as if you simply concatenated all the lists in `L` directly. In other words, empty lists do not contribute to the final concatenated list. This property holds regardless of the specific type `α` of the list elements.

For any type `α` and list `L` of lists of type `α`, the filtered list `L.filter (λ l, l.nonempty)` and the concatenation of `L` yield the same result.


theorem Asymptotics.isBigO_fst_prod :
    ∀ {α : Type u_1} {E' : Type u_6} {F' : Type u_7} [inst : SeminormedAddCommGroup E']
      [inst_1 : SeminormedAddCommGroup F'] {f' : α → E'} {g' : α → F'} {l : Filter α}, f' =O[l] fun x => (f' x, g' x) :=
  by sorry

The theorem `Asymptotics.isBigO_fst_prod` in Lean 4 states that for any types `α`, `E'`, and `F'`, where `E'` and `F'` are both seminormed add commutative groups, and given any functions `f': α → E'` and `g': α → F'`, and a filter `l` on `α`, the function `f'` is Big-O of the function that maps `x` to the pair `(f'(x), g'(x))` when evaluated on the filter `l`. In terms of asymptotics, this means that `f'` grows no faster than the function mapping `x` to `(f'(x), g'(x))`.

For functions `f:` α -> E' and `g:` α -> F' on a type α with seminormed add commutative groups E' and F', and a filter l on α, the function f is Big-O of the function x mapsto (f x, g x) when evaluated on filter l.


/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
`f` is a monotone function on `D`.  -/
theorem monotoneOn_of_deriv_nonneg :
    ∀ {D : Set ℝ},
      Convex ℝ D →
        ∀ {f : ℝ → ℝ},
          ContinuousOn f D → DifferentiableOn ℝ f (interior D) → (∀ x ∈ interior D, 0 ≤ deriv f x) → MonotoneOn f D :=
  by sorry

The theorem `monotoneOn_of_deriv_nonneg` states that given a set `D` of real numbers that is convex, and a real-valued function `f` which is continuous on `D` and differentiable on the interior of `D`. If the derivative of `f` is nonnegative on the interior of `D`, then `f` is a monotone function on `D`. In simpler terms, if a function is continuous and its derivative is nonnegative within a convex set of real numbers, then the function is increasing or constant on that set.

If a continuous, differentiable function with non-negative derivative on the interior of a convex set defines a relation that is monotonic on that set.


theorem Filter.eventually_all_finset :
    ∀ {α : Type u} {ι : Type u_2} (I : Finset ι) {l : Filter α} {p : ι → α → Prop},
      (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x :=
  by sorry

This theorem establishes an equivalence relation for all elements in a finite set `I` of type `ι` and a filter `l` of type `α`. It asserts that for any predicate `p` which takes an element from `I` and an element from `l` to produce a proposition, it is equivalent to say that 'eventually' (for all elements beyond a certain point) in the filter `l`, for all elements `i` in the finite set `I`, the proposition `p i x` holds true AND to say that for all elements `i` in the finite set `I`, 'eventually' in the filter `l`, the proposition `p i x` holds true. Here, 'eventually' refers to the concept in filter theory that describes a condition that holds for all elements outside some finite subset.

For a finite set `I` and a filter `l` on a type `α`, the relation that for all `i ∈ I` and `x ∈ l`, `p i x` holds eventually in `l` if and only if for all `i ∈ I`, `p i x` holds eventually in `l` for some `x`, is an equivalence relation.


theorem TopologicalSpace.isSeparable_range :
    ∀ {α : Type u} {β : Type u_1} [t : TopologicalSpace α] [inst : TopologicalSpace β]
      [inst_1 : TopologicalSpace.SeparableSpace α] {f : α → β},
      Continuous f → TopologicalSpace.IsSeparable (Set.range f) :=
  by sorry

The theorem `TopologicalSpace.isSeparable_range` states that for any two types `α` and `β` with `α` being a separable space in the context of topology, and any continuous function `f` from `α` to `β`, the range of `f` (i.e., the set of all outputs of `f`) is a separable set. Here, a set is said to be separable if it is contained in the closure of a countable set. The continuity of `f` ensures that the topological properties are preserved under the function mapping.

For any separable space α and a continuous function f from α to a type β, the range of f is a separable set.


theorem vadd_closedBall_zero :
    ∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] (δ : ℝ) (x : E),
      x +ᵥ Metric.closedBall 0 δ = Metric.closedBall x δ :=
  by sorry

The theorem `vadd_closedBall_zero` states that for any type `E` that forms a semi-normed addition-commutative group, and for any real number `δ` and element `x` of `E`, adding `x` to every element in the closed ball (the set of all points `y` with distance from `y` to the origin `0` less than or equal to `δ`) gives the closed ball centered at `x` with radius `δ`. In other words, translating the origin-centered closed ball by `x` results in the closed ball centered at `x`.

For any semi-normed addition-commutative group `E` and real number `δ`, the set of elements with distance less than or equal to `δ` from `x` is equal to the set of elements with distance less than or equal to `δ` from `0` translated by `x`.


theorem StieltjesFunction.measure_Ioo :
    ∀ (f : StieltjesFunction) {a b : ℝ},
      ↑↑(StieltjesFunction.measure f) (Set.Ioo a b) = ENNReal.ofReal (Function.leftLim (↑f) b - ↑f a) :=
  by sorry

The theorem `StieltjesFunction.measure_Ioo` states that for any Stieltjes Function `f` and for any real numbers `a` and `b`, the measure of the open interval `(a, b)` with respect to `f` is equal to the extended non-negative real value of the difference of the left limit of `f` at `b` and the value of `f` at `a`. The Stieltjes Function is a type of measure which gives the mass `f b - f a` to the interval `(a, b]`. The interval `(a, b)` is defined as the set of all real numbers `x` such that `a < x < b`. The function `ENNReal.ofReal` returns the non-negative real part of a real number.

For any Stieltjes Function `f` and real numbers `a < b`, the measure of the open interval `(a, b)` with respect to `f` equals the non-negative real value of `(f a) - (f b)`.


theorem Mathlib.Analysis.SpecialFunctions.Gamma.Basic._auxLemma.4 : ∀ {z w : ℝ}, (↑z = ↑w) = (z = w) := by sorry

This theorem, which resides in the `Mathlib.Analysis.SpecialFunctions.Gamma.Basic` namespace and is named `_auxLemma.4`, states that for any two real numbers `z` and `w`, the assertion that their complex number equivalents are equal is equivalent to the assertion that `z` and `w` themselves are equal. In other words, a real number `z` equals to a real number `w` if and only if the complex representation of `z` equals to the complex representation of `w`.

The real numbers `z` and `w` are equal if and only if their complex number representations are equal.


theorem Nat.one_le_pow : ∀ (n m : ℕ), 0 < m → 1 ≤ m ^ n := by sorry

This theorem states that for every pair of natural numbers `n` and `m`, if `m` is greater than zero, then 1 is less than or equal to `m` raised to the power of `n`. In other words, any positive natural number raised to any natural number power is always at least 1.

For all natural numbers n and m with m > 0, m^n ≥ 1.


theorem pow_abs : ∀ {α : Type u_1} [inst : LinearOrderedRing α] (a : α) (n : ℕ), |a| ^ n = |a ^ n| := by sorry

This theorem, named `pow_abs`, states that for any type `α` that forms a linear ordered ring, the absolute value of `a` raised to the power of `n` is equal to the absolute value of `a` itself raised to the power of `n`. Here `a` is an element of the type `α` and `n` is a natural number. This effectively means that when exponentiation interacts with absolute value, the order of operations doesn't matter.

For any linear ordered ring type `α` and natural number `n`, the absolute value of `a^n` equals the absolute value of `a` raised to the power `n` for all `a` in `α`.


theorem ProbabilityTheory.kernel.indep_of_indep_of_le_right :
    ∀ {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m₁ m₂ m₃ _mΩ : MeasurableSpace Ω}
      {κ : ↥(ProbabilityTheory.kernel α Ω)} {μ : MeasureTheory.Measure α},
      ProbabilityTheory.kernel.Indep m₁ m₂ κ μ → m₃ ≤ m₂ → ProbabilityTheory.kernel.Indep m₁ m₃ κ μ :=
  by sorry

This theorem from Probability Theory states that for any types `α` and `Ω`, given measurable spaces `_mα`, `m₁`, `m₂`, `m₃`, and `_mΩ`, a kernel `κ` from `α` to `Ω`, and a measure `μ` on `α`, if the kernel `κ` is independent of the measurable space `m₂` with respect to measure `μ` and `m₁`, and if `m₃` is a sub-measurable space of `m₂` (i.e., `m₃ ≤ m₂`), then `κ` is also independent of `m₃` with respect to measure `μ` and `m₁`. In other words, the independence of a kernel of measurable spaces is preserved when the independent space is reduced to a sub-space.

If a kernel is independent of a measurable space with respect to a measure, and the smaller measurable space is a sub-measurable space of the larger one, then the kernel is also independent of the smaller measurable space with respect to the same measure.


theorem Valued.mem_nhds :
    ∀ {R : Type u} [inst : Ring R] {Γ₀ : Type v} [inst_1 : LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀]
      {s : Set R} {x : R}, s ∈ nhds x ↔ ∃ γ, {y | Valued.v (y - x) < ↑γ} ⊆ s :=
  by sorry

This theorem, `Valued.mem_nhds`, states that for any ring `R` and a linearly ordered commutative group with zero `Γ₀`, if the ring `R` is valued with `Γ₀`, then for any set `s` of type `R` and any element `x` of type `R`, the set `s` is a neighborhood of `x` if and only if there exists a γ in `Γ₀` such that the set containing all elements `y` in `R` whose value (given by the valuation `Valued.v`) of the difference between `y` and `x` is less than γ, is a subset of `s`. In other words, `s` is a neighborhood of `x` if it contains an open set around `x`, defined by a certain valuation condition.

For a ring valued with a linearly ordered commutative group, a set is a neighborhood of an element if and only if it contains an open ball around that element defined by a certain valuation condition.


theorem IsOpen.upperSemicontinuous_indicator :
    ∀ {α : Type u_1} [inst : TopologicalSpace α] {β : Type u_2} [inst_1 : Preorder β] {s : Set α} {y : β}
      [inst_2 : Zero β], IsOpen s → y ≤ 0 → UpperSemicontinuous (Set.indicator s fun _x => y) :=
  by sorry

This theorem states that for any topological space `α`, preordered set `β`, specific set `s` of type `α`, and a value `y` of type `β` with `β` as zero, if the set `s` is open and `y` is less than or equal to zero, then the indicator function (which gives `y` when the input is in set `s` and `0` otherwise) is upper semicontinuous. An upper semicontinuous function is one where for any `ε > 0`, for any `x`, for all `x'` close enough to `x`, `f x'` is at most `f x + ε`. In simpler terms, this means that the indicator function does not jump upwards - it's either constant or decreasing - when we restrict to an open set and a non-positive constant value.

For any topological space `α`, preordered set `β` with `β` as zero, open set `s` in `α`, and value `y` in `β`, if `s` is open and `y` is less than or equal to zero, then the indicator function of `s` is upper semicontinuous at `y`.


theorem Real.continuous_arccos : Continuous Real.arccos := by sorry

This theorem states that the inverse of the cosine function, `arccos`, is continuous. In mathematical terms, it means that for every point in its domain, the function `arccos` does not have any jumps, breaks, or holes. It ensures that for any given small change in the input (x), there will be a correspondingly small change in the output (`arccos x`).

The function `arccos` is a continuous function on its domain.


theorem zsmul_sub : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b : α) (n : ℤ), n • (a - b) = n • a - n • b :=
  by sorry

This theorem states that for any type `α` that forms a commutative monoid under subtraction, and for any elements `a` and `b` of this type and any integer `n`, the operation of scaling (multiplying) the difference between `a` and `b` by `n` is equivalent to subtracting the scaled `b` from the scaled `a`. In mathematical terms, it asserts that $n \cdot (a - b) = n \cdot a - n \cdot b$ for all `a`, `b` in `α` and integer `n`.

For any commutative monoid `α` under subtraction and integers `n`, `a`, `b` in `α`, we have `n * (a - b) = n * a - n * b`.


Filter.HasBasis.sup

theorem Filter.HasBasis.sup :
    ∀ {α : Type u_1} {l l' : Filter α} {ι : Type u_6} {ι' : Type u_7} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop}
      {s' : ι' → Set α},
      Filter.HasBasis l p s →
        Filter.HasBasis l' p' s' → Filter.HasBasis (l ⊔ l') (fun i => p i.1 ∧ p' i.2) fun i => s i.1 ∪ s' i.2 :=
  by sorry

The theorem `Filter.HasBasis.sup` states that given two filters `l` and `l'` over a Type `α`, indexed by Types `ι` and `ι'` respectively, and if the filter `l` has a basis that satisfies a property `p` and the filter `l'` has a basis that satisfies a property `p'`, then the supremum of `l` and `l'` (denoted `l ⊔ l'`) also has a basis. The basis of the supremum filter is characterized by the conjunction of the properties `p` and `p'` (represented by `(fun i => p i.1 ∧ p' i.2)`) and the union of the sets associated with `l` and `l'` (represented by `(fun i => s i.1 ∪ s' i.2)`). In essence, it describes how the basis properties and sets of the two filters combine when they are merged together.

The supremum of two filters over a Type has a basis if and only if the bases of the two filters have the conjunction of their property conditions and the union of their associated sets.


theorem act_rel_act_of_rel :
    ∀ {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [inst : CovariantClass M N μ r] (m : M)
      {a b : N}, r a b → r (μ m a) (μ m b) :=
  by sorry

This theorem states that for any types `M` and `N`, a binary operation `μ` from `M` and `N` to `N`, and a binary relation `r` on `N`, given that `μ` is a covariant with respect to `r`, and for any element `m` of type `M` and any elements `a` and `b` of type `N`, if `r` holds between `a` and `b`, then `r` also holds between `μ(m, a)` and `μ(m, b)`. Essentially, if `a` and `b` are related under `r`, then applying the operation `μ` with `m` to `a` and `b` will also produce two elements that are related under `r`. It's a property about preserving the relational structure under the operation `μ`.

If `μ` is a covariant binary operation from types `M` and `N` to `N`, and `r` is a relation on `N` such that `r(μ(m, a), μ(m, b))` holds whenever `r(a, b)` does, then `μ` preserves `r`.


theorem Function.support_smul_subset_left :
    ∀ {α : Type u_1} {R : Type u_2} {M : Type u_5} [inst : Zero R] [inst_1 : Zero M] [inst_2 : SMulWithZero R M]
      (f : α → R) (g : α → M), Function.support (f • g) ⊆ Function.support f :=
  by sorry

The theorem `Function.support_smul_subset_left` states that for any types `α`, `R`, and `M`, where `R` and `M` have zero elements and `R` and `M` have a scalar multiplication operation compatible with zero, and any functions `f : α → R` and `g : α → M`, the support of the scalar multiplication of `f` and `g` (i.e., the set of points where `f(x) * g(x)` is non-zero) is a subset of the support of `f` (i.e., the set of points where `f(x)` is non-zero). Essentially, this means that if a point is in the support of the multiplied functions, it must be in the support of the first function `f`.

For functions `f : α → R` and `g : α → M` between types with zero elements and scalar multiplication, the support of their product `f * g` is contained in the support of `f`.


theorem Real.deriv_cos' : deriv Real.cos = fun x => -Real.sin x := by sorry

The theorem `Real.deriv_cos'` states that the derivative of the real cosine function is negative of the real sine function. In other words, if you take any real number `x` and calculate the derivative of the cosine function at `x`, the result will be the negative of the sine of `x`. This is a well-known result in calculus which can be expressed in the form (d/dx) cos(x) = -sin(x).

The derivative of the real cosine function is equal to the negative of the real sine function.


theorem Equiv.Perm.sign_permCongr :
    ∀ {α : Type u} [inst : DecidableEq α] {β : Type v} [inst_1 : Fintype α] [inst_2 : DecidableEq β]
      [inst_3 : Fintype β] (e : α ≃ β) (p : Equiv.Perm α),
      Equiv.Perm.sign ((Equiv.permCongr e) p) = Equiv.Perm.sign p :=
  by sorry

The theorem `Equiv.Perm.sign_permCongr` states that for any types `α` and `β`, given a bijective function `e` from `α` to `β`, and any permutation `p` on `α`, the sign or parity of the permutation `p` after the bijection `e` is applied is the same as the sign or parity of the original permutation `p`. This holds if `α` and `β` have decidable equality and are finite. In other words, the signature of a permutation is invariant under the action of a bijection.

For finite types `α` and `β` with decidable equality, if `e : α → β` is a bijective function and `p : ∣α∣! => α`, is a permutation on `α`, then the sign of `p` is equal to the sign of `e ∘ p ∘ e⁻¹`.


theorem Subgroup.zpowers_eq_closure :
    ∀ {G : Type u_1} [inst : Group G] (g : G), Subgroup.zpowers g = Subgroup.closure {g} := by sorry

This theorem states that for any group `G` and any element `g` from `G`, the subgroup generated by the powers of `g` (denoted as `Subgroup.zpowers g`) is the same as the subgroup generated by the singleton set containing `g` (denoted as `Subgroup.closure {g}`). In other words, both methods of generating a subgroup - one by considering all powers of a given element and the other by taking the closure of a set containing just that element - yield the same result.

For any group `G` and element `g` from `G`, the subgroup generated by the powers of `g` equals the subgroup generated by `{g}`.


theorem Compactum.le_nhds_of_str_eq :
    ∀ {X : Compactum} (F : Ultrafilter X.A) (x : X.A), Compactum.str X F = x → ↑F ≤ nhds x := by sorry

This theorem states that for any compactum `X`, ultrafilter `F` on the algebra `X.A`, and an element `x` in the algebra `X.A`, if the structure map of the compactum applied to `F` equals `x`, then the ultrafilter `F` is less than or equal to the neighborhood filter at `x`. In more conventional terms, this means that if an ultrafilter's limit point in a compactum is `x`, then all elements of the ultrafilter are in the neighborhood of `x`. This aligns with the intuition that an ultrafilter should 'converge' to its limit point in a topological sense, in this case meaning that all its elements are close to `x` in the sense of being in its topological neighborhood.

For any compactum X, if the structure map of X applied to an ultrafilter F equals an element x in X.A, then F is contained in the neighborhood filter of x.


theorem neg_closure :
    ∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : InvolutiveNeg G] [inst_2 : ContinuousNeg G] (s : Set G),
      -closure s = closure (-s) :=
  by sorry

This theorem, `neg_closure`, states that for any type `G` that has a topology, an involutive negation operation, and where this negation operation is continuous, the closure of the negation of a set `s` is equal to the negation of the closure of `s`. This is to say, if `s` is a set in `G`, then the closure operation and the negation operation commute. It means that applying the closure on the negation of the set is the same as first applying the closure on the set and then applying the negation. The theorem holds true for all such types `G` and all sets `s` of type `G`.

For any topological space `(G, τ)` with an involutive continuous negation operation, the closure of the negation of a set `s ⊆ G` equals the negation of the closure of `s`, i.e., `cl(¬s) = ¬(cl(s))`.


theorem EMetric.diam_le_iff :
    ∀ {α : Type u} [inst : PseudoEMetricSpace α] {s : Set α} {d : ENNReal},
      EMetric.diam s ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d :=
  by sorry

This theorem states that for all pseudoemetric spaces (`α`), and for any set (`s`) of elements of type `α`, and any extended non-negative real number (`d`), the diameter of the set `s` is less than or equal to `d` if and only if for every pair of elements `x` and `y` in `s`, the extended distance (`edist`) between `x` and `y` is less than or equal to `d`. In simpler terms, the diameter of a set in a pseudoemetric space is never greater than the distance between any two of its points.

In a pseudometric space, the diameter of a set is less than or equal to the greatest distance between any two of its elements.


theorem Mathlib.Order.SuccPred.Basic._auxLemma.15 :
    ∀ {α : Type u_1} [inst : PartialOrder α] [inst_1 : SuccOrder α] {a b : α} [inst_2 : NoMaxOrder α],
      (Order.succ a = Order.succ b) = (a = b) :=
  by sorry

This theorem states that for any type `α` that has a partial order, successor order, and for which no maximal element exists, if the successor of an element `a` is equal to the successor of another element `b`, then `a` must be equal to `b`. The successor of an element is the least element greater than the given element if the given element is not maximal. If the element is maximal, its successor is the element itself.

In a type `α` with no maximal element and having a successor order, if `a` and `b` have the same successor, then `a` equals `b`.


theorem Set.Icc_diff_both : ∀ {α : Type u_1} [inst : PartialOrder α] {a b : α}, Set.Icc a b \ {a, b} = Set.Ioo a b := by
  sorry

This theorem state that for any given type `α` which has a partial order, and any two elements `a` and `b` of that type, the set difference between the left-closed right-closed interval from `a` to `b` and the set containing `a` and `b` equals the left-open right-open interval from `a` to `b`. In other terms, if you take the closed interval [`a`, `b`] and remove the end points `a` and `b`, you are left with the open interval (`a`, `b`).

For any type `α` with a partial order and any `a` and `b` in `α`, the closed interval [`a`, `b`] without `a` and `b` equals the open interval (`a`, `b`).


theorem Set.singleton_vadd : ∀ {α : Type u_2} {β : Type u_3} [inst : VAdd α β] {t : Set β} {a : α}, {a} +ᵥ t = a +ᵥ t :=
  by sorry

The theorem "Set.singleton_vadd" states that for any types `α` and `β` such that there is a vector addition operation defined on `α` and `β`, and for any set `t` of type `β` and any element `a` of type `α`, the vector addition of the singleton set containing `a` and `t` is equal to the vector addition of `a` and `t`. In mathematical terms, it states that for a singleton set `{a}` and a set `t`, the vector addition `{a} +ᵥ t` is the same as `a +ᵥ t`.

For sets `{a}` (a singleton) and `t` in types `α` and `β` respectively with defined vector addition, the operation `{a} +ᵥ t` equals `a +ᵥ t`.


/-- All n-simplices among cospherical points in an n-dimensional
subspace have the same circumcenter.  -/
theorem EuclideanGeometry.exists_circumcenter_eq_of_cospherical_subset :
    ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V]
      [inst_2 : MetricSpace P] [inst_3 : NormedAddTorsor V P] {s : AffineSubspace ℝ P} {ps : Set P},
      ps ⊆ ↑s →
        ∀ [inst_4 : Nonempty ↥s] {n : ℕ} [inst_5 : FiniteDimensional ℝ ↥(AffineSubspace.direction s)],
          FiniteDimensional.finrank ℝ ↥(AffineSubspace.direction s) = n →
            EuclideanGeometry.Cospherical ps →
              ∃ c, ∀ (sx : Affine.Simplex ℝ P n), Set.range sx.points ⊆ ps → Affine.Simplex.circumcenter sx = c :=
  by sorry

This theorem states that for any set of points 'ps' in an n-dimensional affine subspace 's', if all these points are cospherical (equidistant from some point), then all n-simplices formed from these points have the same circumcenter. In other words, if we take any subset of 'n+1' points from 'ps' and form an n-simplex, the circumcenter (the center of the smallest enclosing sphere) of this simplex is the same regardless of the simplex chosen. The theorem assures the existence of such a common circumcenter 'c'.

Given a set of points in an n-dimensional affine subspace that are all cospherical, the circumcenters of any n-simplices formed from these points are equal.


theorem Equiv.Perm.card_support_eq_zero :
    ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : Fintype α] {f : Equiv.Perm α},
      (Equiv.Perm.support f).card = 0 ↔ f = 1 :=
  by sorry

This theorem states that for any given type `α` that has decidable equality and is finite (i.e., has finiteness), and any given permutation `f` of `α`, the cardinality (i.e., the number of elements) of the set of non-fixed points of `f` is equal to zero if and only if `f` is the identity permutation. In other words, a permutation on a finite set with decidable equality has no non-fixed points (i.e., for every element `x`, `f(x) = x`) if and only if it is the identity permutation. This theorem links the structure of a permutation (i.e., whether it is the identity) with a property of its set of non-fixed points (i.e., whether this set is empty).

A finite type with decidable equality has no non-fixed points under a permutation if and only if the permutation is the identity.


theorem Turing.ListBlank.map_cons :
    ∀ {Γ : Type u_1} {Γ' : Type u_2} [inst : Inhabited Γ] [inst_1 : Inhabited Γ'] (f : Turing.PointedMap Γ Γ')
      (l : Turing.ListBlank Γ) (a : Γ),
      Turing.ListBlank.map f (Turing.ListBlank.cons a l) = Turing.ListBlank.cons (f.f a) (Turing.ListBlank.map f l) :=
  by sorry

The theorem `Turing.ListBlank.map_cons` states that for any given types `Γ` and `Γ'` that are inhabited, any pointed map `f` from `Γ` to `Γ'`, any list blank `l` of type `Γ`, and any element `a` of type `Γ`, the operation of mapping `f` over the list blank that is obtained by consing `a` onto `l` is equivalent to consing the result of applying `f` to `a` onto the list blank that is obtained by mapping `f` over `l`. In other words, it states that the map function provided for `ListBlank`s commutes with the cons operation.

For any types `Γ` and `Γ'`, pointed maps `f : Γ → Γ'`, list blanks `l : ListBlank Γ`, and elements `a : Γ`, the application of `f` to the cons operation `a :: l` is equivalent to `f a :: f <$> l`.


theorem disjoint_sSup_iff :
    ∀ {α : Type u} [inst : Order.Frame α] {a : α} {s : Set α}, Disjoint a (sSup s) ↔ ∀ b ∈ s, Disjoint a b := by sorry

This theorem states that for any type `α` equipped with a frame order, an element `a` of this type, and a set `s` of elements of this type, `a` is disjoint with the supremum of `s` if and only if `a` is disjoint with every element `b` in `s`. Here, two elements of a lattice are considered disjoint if their infimum is the bottom element. The supremum of a set (`sSup s`) represents the least upper bound of the set `s` in the given order.

For any lattice with frame order, an element is disjoint with the supreme of a set if and only if it is disjoint with every element in the set.


theorem EReal.coe_mul : ∀ (x y : ℝ), ↑(x * y) = ↑x * ↑y := by sorry

This theorem states that for any two real numbers `x` and `y`, the operation of multiplying them and then applying the coe (coercion) function is equivalent to first applying the coe function to each of them and then multiplying the results. In other words, the coe function distributes over multiplication in the set of real numbers. This can be expressed mathematically as: ∀(x, y ∈ ℝ), coe(x*y) = coe(x) * coe(y).

The coercion function distributes over multiplication in the set of real numbers, i.e., for all `x, y ∈ ℝ`, `coe(x * y) = coe(x) * coe(y)`.


theorem AddSubgroup.mem_rightTransversals_iff_bijective :
    ∀ {G : Type u_1} [inst : AddGroup G] {H : AddSubgroup G} {S : Set G},
      S ∈ AddSubgroup.rightTransversals ↑H ↔ Function.Bijective (Set.restrict S Quotient.mk'') :=
  by sorry

This theorem states that, for any type `G` with an additive group structure and its subgroup `H`, and any set `S` of elements of `G`, `S` belongs to the right transversals of the subgroup `H` if and only if the function that restricts `S` to the quotient is bijective. In other words, a set `S` is a right transversal of a subgroup `H` when every element of the group can be expressed uniquely as the sum of an element from `H` and an element from `S`, which corresponds to the function from `S` to the quotient group being bijective.

A set `S` is a right transversal of a subgroup `H` in a group `G` if and only if the restriction of `S` to the quotient group `G/H` is a bijective function.


theorem isConnected_range :
    ∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : ConnectedSpace α]
      {f : α → β}, Continuous f → IsConnected (Set.range f) :=
  by sorry

The theorem `isConnected_range` states that for all types `α` and `β`, given that `α` is endowed with a topological space structure, `β` is also endowed with a topological space structure, and `α` is a connected space, if we have a continuous function `f` from `α` to `β`, then the range of `f` (which is a set of `β` elements) is a connected set. In other words, the continuous image of a connected space is also connected.

If `α` is a connected topological space and `f:` `α` `->` `β` is a continuous function, then the range `f''(α)` of `f` is a connected subset of `β`.


theorem mul_right_eq_self₀ : ∀ {M₀ : Type u_2} [inst : CancelMonoidWithZero M₀] {a b : M₀}, a * b = a ↔ b = 1 ∨ a = 0 :=
  by sorry

This theorem, `mul_right_eq_self₀`, states that for any type `M₀`, which forms a cancel monoid with zero, if the product of any two elements `a` and `b` from `M₀` equals `a`, then either `b` is equal to one, or `a` is zero. In other words, if multiplying `a` by `b` doesn't change the value of `a`, it must be because `b` is the multiplicative identity (1), or `a` is the additive identity (0).

If `a` and `b` are elements of a cancel monoid `M₀` with zero such that `a * b = a`, then `b = 1` or `a = 0`.


/-- Subsemiring closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`.  -/
theorem NonUnitalSubsemiring.closure_mono :
    ∀ {R : Type u} [inst : NonUnitalNonAssocSemiring R] ⦃s t : Set R⦄,
      s ⊆ t → NonUnitalSubsemiring.closure s ≤ NonUnitalSubsemiring.closure t :=
  by sorry

The theorem `NonUnitalSubsemiring.closure_mono` states that for any type `R` which has a non-unital non-associative semiring structure, the operation of taking the subsemiring closure of a set is monotone. In other words, for any two sets `s` and `t` of type `R`, if `s` is a subset of `t` (denoted as `s ⊆ t`), then the subsemiring closure of `s` is a subset of or equal to the subsemiring closure of `t` (denoted as `NonUnitalSubsemiring.closure s ≤ NonUnitalSubsemiring.closure t`). The subsemiring closure of a set is a mathematical operation that produces the smallest semiring that contains the set.

For any non-unital non-associative semiring `R` and sets `s` and `t` of `R`, if `s ⊆ t`, then `NonUnitalSubsemiring.closure s ≤ NonUnitalSubsemiring.closure t`.


theorem ProbabilityTheory.kernel.iIndepFun.indepFun_div_left :
    ∀ {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω}
      {κ : ↥(ProbabilityTheory.kernel α Ω)} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β}
      [inst : Div β] [inst_1 : MeasurableDiv₂ β] {f : ι → Ω → β} [inst_2 : ProbabilityTheory.IsMarkovKernel κ],
      ProbabilityTheory.kernel.iIndepFun (fun x => m) f κ μ →
        (∀ (i : ι), Measurable (f i)) →
          ∀ (i j k : ι), i ≠ k → j ≠ k → ProbabilityTheory.kernel.IndepFun (f i / f j) (f k) κ μ :=
  by sorry

The theorem, `ProbabilityTheory.kernel.iIndepFun.indepFun_div_left`, states that given a Markov kernel, `κ`, and a measure, `μ`, on a measurable space of type `α`, for any division-closed measurable space of type `β`, and a function `f` from type `ι` to `Ω → β` that is `i`-independent (`iIndepFun`) with respect to the measure `m`, if each instance of `f i` is measurable, then for any three distinct instances `i`, `j`, and `k` of `ι`, the functions `(f i / f j)` and `f k` are independent (`IndepFun`) under the kernel `κ` and the measure `μ`. This theorem thus provides a condition under which the division of two measurable functions is also measurable and independent of another function.

Given a Markov kernel and measure, if a division-closed measurable function that is i-independent with respect to the measure is defined on a measurable space, then the quotient of any two instances of the function and a third function are independent under the kernel and measure.


theorem Multiset.count_singleton :
    ∀ {α : Type u_1} [inst : DecidableEq α] (a b : α), Multiset.count a {b} = if a = b then 1 else 0 := by sorry

This theorem states that for any type `α` that has decidable equality (denoted by `DecidableEq α`), given any two elements `a` and `b` of type `α`, the count of `a` in a singleton multiset containing only `b` (denoted by `{b}`) is 1 if `a` is equal to `b` and 0 otherwise. In other words, if `a` and `b` are the same, `a` appears once in the multiset `{b}`, otherwise `a` does not appear in the multiset `{b}`.

For any type `α` with decidable equality and any `a, b : α`, the size of `a` in the singleton multiset `{b}` equals 1 if `a = b`, and 0 otherwise.


Submonoid.LocalizationMap.eq

theorem Submonoid.LocalizationMap.eq :
    ∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N]
      (f : Submonoid.LocalizationMap S N) {a₁ b₁ : M} {a₂ b₂ : ↥S},
      Submonoid.LocalizationMap.mk' f a₁ a₂ = Submonoid.LocalizationMap.mk' f b₁ b₂ ↔
        ∃ c, ↑c * (↑b₂ * a₁) = ↑c * (↑a₂ * b₁) :=
  by sorry

This theorem states that, given a commutative monoid `M`, a submonoid `S` of `M`, another commutative monoid `N`, and a localization map `f` from `S` to `N`, for any elements `a₁, b₁` in `M` and `a₂, b₂` in `S`, the result of applying the defined function `Submonoid.LocalizationMap.mk'` on `a₁, a₂` equals the result of applying the same function `Submonoid.LocalizationMap.mk'` on `b₁, b₂` if and only if there exists an element `c` in `S` such that the product of `c` and the product of `b₂` and `a₁` equals the product of `c` and the product of `a₂` and `b₁`. In mathematical terms, we have `f(a₁) / f(a₂) = f(b₁) / f(b₂)` if and only if there exists `c` in `S` such that `c * b₂ * a₁ = c * a₂ * b₁`.

Given commutative monoids M, a submonoid S, a localization map f from S to another commutative monoid N, and elements a₁, a₂, b₁, b₂ in S, the localization map evaluations f(a₁) and f(a₂) have the same quotient if and only if there exists an element c in S such that b₁ a₁ and a₂ b₂ have the same product as c² in S.


theorem interior_eq_nhds' : ∀ {X : Type u} {s : Set X} [inst : TopologicalSpace X], interior s = {x | s ∈ nhds x} := by
  sorry

This theorem states that for any type `X` and any set `s` of type `Set X` in a topological space, the interior of `s` is equal to the set of all elements `x` such that `s` is a neighborhood of `x`. In simpler terms, it asserts that the interior of a set consists of all points which have a neighborhood completely contained within the set. This theorem provides a link between the concepts of the interior of a set and neighborhood filters in a topological space.

The interior of a set in a topological space is equal to the set of all points that have the set as their neighborhood.


theorem Filter.HasBasis.lift'_closure_eq_self :
    ∀ {X : Type u} {ι : Sort w} [inst : TopologicalSpace X] {l : Filter X} {p : ι → Prop} {s : ι → Set X},
      Filter.HasBasis l p s → (∀ (i : ι), p i → IsClosed (s i)) → Filter.lift' l closure = l :=
  by sorry

The theorem `Filter.HasBasis.lift'_closure_eq_self` states that for any type `X` with a topological space structure, for any filter `l` on `X`, and for any collection of subsets `s` of `X` indexed by `ι` such that `l` has a basis given by the properties `p` and the sets `s` (i.e., `Filter.HasBasis l p s`), if all subsets `s i` satisfying the property `p i` are closed, then lifting the filter `l` using the closure operation yields the same filter `l`. In other words, the closure operation does not change the filter when it is formed from a basis of closed sets.

Let `(X, top)` be a topological space, `l` be a filter on `X` with basis `s : ι → Set X` such that `s i` is closed for all `i` with property `p i`, then `l = Filter.closure l (Filter.union (map (closure) s))`.


/-- The function `x ^ s * exp (-b * x)` tends to `0` at `+∞`, for any real `s` and `b > 0`.  -/
theorem tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero :
    ∀ (s b : ℝ), 0 < b → Filter.Tendsto (fun x => x ^ s * Real.exp (-b * x)) Filter.atTop (nhds 0) := by sorry

This theorem states that for any real numbers `s` and `b` with `b` being positive, the function `x ^ s * exp (-b * x)` converges to `0` as `x` approaches infinity. In other words, if you take any positive `b` and any `s`, and you look at the behavior of the function `x ^ s * exp (-b * x)` as `x` gets larger and larger, this function will get closer and closer to `0`.

For any real numbers `s` and `b` with `b` positive, the limit of `x => x^s * exp(-b*x)` as `x` approaches infinity is zero.


theorem FreeAddMagma.hom_ext :
    ∀ {α : Type u} {β : Type v} [inst : Add β] {f g : AddHom (FreeAddMagma α) β},
      ⇑f ∘ FreeAddMagma.of = ⇑g ∘ FreeAddMagma.of → f = g :=
  by sorry

This theorem states that for any types `α` and `β`, where `β` has an addition operation, if we have two addition homomorphisms `f` and `g` from the free additive magma of `α` to `β`, then if the composition of `f` or `g` with the inclusion of `α` into the free additive magma is the same, it follows that `f` and `g` must be the same. In simpler terms, this theorem provides a uniqueness condition for homomorphisms from the free additive magma of one type to another type with addition.

If `α` is any type with an addition operation and `f` and `g` are homomorphisms from the free additive magma of `α` to a type `β` with addition, then `f` and `g` are equal if and only if their compositions with the inclusion of `α` into the free additive magma are equal.


theorem Nat.xor_self : ∀ (n : ℕ), n ^^^ n = 0 := by sorry

This theorem states that for any natural number `n`, the exclusive OR (`^^^`) of `n` with itself is always 0. In other words, when you perform an XOR operation on a number with itself, the result is always zero, regardless of the number used. This is a property of the XOR operation and is true for all natural numbers.


theorem NormedAddGroupHom.completion_coe' :
    ∀ {G : Type u_1} [inst : SeminormedAddCommGroup G] {H : Type u_2} [inst_1 : SeminormedAddCommGroup H]
      (f : NormedAddGroupHom G H) (g : G), UniformSpace.Completion.map (⇑f) (↑G g) = ↑H (f g) :=
  by sorry

This theorem states that for any two types `G` and `H` that are instances of the `SeminormedAddCommGroup` class, and given a normed additive group homomorphism `f` from `G` to `H`, the mapping of the completion of `G` to the completion of `H` by `f` is equal to the completion of the image of any element `g` of `G` under `f`. In other words, if we first complete `G` and then apply the homomorphism `f`, it's the same as if we first apply `f` and then complete.

The completion of a normed additive group homomorphism between two seminormed additive groups is equal to the homomorphism of their completions.


theorem CategoryTheory.isSheaf_yoneda' :
    ∀ {α : Type u},
      CategoryTheory.Presieve.IsSheaf CategoryTheory.typesGrothendieckTopology (CategoryTheory.yoneda.obj α) :=
  by sorry

The theorem `CategoryTheory.isSheaf_yoneda'` states that for any type `α`, the Yoneda embedding applied to `α` is indeed a sheaf with respect to the Grothendieck topology for the category of all types. In other words, this theorem establishes the fact that the Yoneda embedding preserves the structure of sheaves, a fundamental concept in Grothendieck's theory of schemes and toposes, under the particular Grothendieck topology associated with the category of all types, where a sieve is considered a covering if it is jointly surjective.

The Yoneda embedding of a type `α` is a sheaf in the category of types under the Grothendieck topology where a sieve is covering if it is jointly surjective.


theorem MeasureTheory.Ioi_ae_eq_Ici :
    ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.NoAtoms μ]
      [inst : PartialOrder α] {a : α}, Set.Ioi a =ᶠ[MeasureTheory.Measure.ae μ] Set.Ici a :=
  by sorry

This theorem states that for every type `α`, measurable space `m0`, measure `μ`, and element `a` of type `α`, where `μ` has no atoms and `α` is partially ordered, the left-open right-infinite interval greater than `a` (denoted as `Set.Ioi a`) is almost everywhere equal (under the measure `μ`) to the left-closed right-infinite interval starting at `a` (denoted as `Set.Ici a`). Here, "almost everywhere" means except on a set of measure zero.

For every measurable space `(α, Σ, μ)` without atoms, and every element `a` of a partially ordered type `α`, `Set.Ioi a = Set.Ici a` almost everywhere.


theorem midpoint_vadd_midpoint :
    ∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Ring R] [inst_1 : Invertible 2] [inst_2 : AddCommGroup V]
      [inst_3 : Module R V] [inst_4 : AddTorsor V P] (v v' : V) (p p' : P),
      midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p') :=
  by sorry

The theorem `midpoint_vadd_midpoint` states that for any ring `R`, invertible element 2, additive commutative group `V`, module `R` over `V`, additive torsor `V` over `P`, and any elements `v`, `v'` in `V` and `p`, `p'` in `P`, the sum of the midpoints of `v` and `v'`, and `p` and `p'` respectively (defined in a vector space), is equal to the midpoint of the sum of `v` and `p`, and `v'` and `p'`. In mathematical terms, this can be expressed as:

For all v, v' in V and p, p' in P, Midpoint(v, v') + Midpoint(p, p') = Midpoint(v+p, v'+p').

This theorem relates the operations of vector addition and finding midpoints in a certain way.

For any additive torsors `V` over a group `P` in a commutative ring `R` with invertible element 2, the midpoints of the vector sum and the midpoints of the corresponding elements in `V` and `P` are equal. In other words, Midpoint(v+p, v'+p') = Midpoint(v, v') + Midpoint(p, p') for all `v`, `v'` in `V` and `p`, `p'` in `P`.


theorem SemiconjBy.inv_inv_symm_iff :
    ∀ {G : Type u_1} [inst : DivisionMonoid G] {a x y : G}, SemiconjBy a⁻¹ x⁻¹ y⁻¹ ↔ SemiconjBy a y x := by sorry

The theorem `SemiconjBy.inv_inv_symm_iff` states that for any division monoid `G` and elements `a`, `x`, and `y` of `G`, `x` is semiconjugate to `y` by `a` inverse if and only if `y` is semiconjugate to `x` by `a`. In terms of the semiconjugate operation, this means that multiplying `a` inverse by `x` inverse equals `y` inverse times `a` inverse if and only if multiplying `a` by `y` equals `x` times `a`. This theorem effectively describes a symmetry property of the semiconjugate relation under taking inverses of the elements.

For any division monoid G and elements x, y, and a, x is semiconjugate to y by a if and only if y is semiconjugate to x by a inverse.


theorem Metric.cobounded_le_cocompact :
    ∀ {α : Type u} [inst : PseudoMetricSpace α], Bornology.cobounded α ≤ Filter.cocompact α := by sorry

The theorem `Metric.cobounded_le_cocompact` states that for any type `α` equipped with a pseudo metric space structure, the filter of cobounded sets in the bornology of `α` is less than or equal to the filter generated by the complements of compact sets in the topological space of `α`. In mathematical terms, this means that every cobounded set in the bornology of a pseudo metric space is also a cocompact set in the same space.

In a pseudo metric space, the filter of cobounded sets is contained in the filter generated by the complements of compact sets.


theorem Set.zero_mem_centralizer : ∀ {M : Type u_1} (S : Set M) [inst : MulZeroClass M], 0 ∈ Set.centralizer S := by
  sorry

This theorem states that for any type `M` that has both multiplication and a zero element (`MulZeroClass M`), and for any set `S` of elements of type `M`, the zero element is a member of the centralizer of `S`. The centralizer of a set in a magma is the set of elements that commute with every element of the original set. In mathematical terms, if `S` is a subset of a magma `M` with multiplication and a zero element, then for each element `m` in `S`, the equality `m * 0 = 0 * m` holds.

For any magma `M` with a zero element and any subset `S` of `M` that is closed under multiplication, the zero element belongs to the centralizer of `S`.


HasFiniteFPowerSeriesAt.cPolynomialAt

theorem HasFiniteFPowerSeriesAt.cPolynomialAt :
    ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
      {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ}, HasFiniteFPowerSeriesAt f p x n → CPolynomialAt 𝕜 f x :=
  by sorry

The theorem `HasFiniteFPowerSeriesAt.cPolynomialAt` states that for any field `𝕜` equipped with a norm, any normed additive commutative group `E` over `𝕜`, another normed additive commutative group `F` over `𝕜`, any function `f` from `E` to `F`, any formal multilinear series `p`, any element `x` from `E`, and any natural number `n`, if the function `f` has a finite power series expansion represented by `p` of order `n` at `x` (`HasFiniteFPowerSeriesAt f p x n`), then `f` is continuously polynomial at `x` (`CPolynomialAt 𝕜 f x`). In simpler terms, if a function has a finite power series expansion at a point, then it is continuously polynomial at that point.

If a function has a finite power series expansion at a point in a normed additive commutative group over a field with a norm, then the function is continuously polynomial at that point.


theorem Cardinal.two_power_aleph0 : 2 ^ Cardinal.aleph0 = Cardinal.continuum := by sorry

This theorem states that the cardinality of the set of all real numbers, often referred to as the "continuum", is equal to the power set of the set of all natural numbers. In particular, it asserts that the cardinality of the continuum (`Cardinal.continuum`) is equal to two raised to the power of `Cardinal.aleph0`, where `Cardinal.aleph0` represents the cardinality of the set of all natural numbers. In mathematical terms, it expresses that |ℝ| = 2^ℵ₀.

The cardinality of the set of real numbers is equal to the power set of the natural numbers, that is, |ℝ| = 2^ℵ₀.


theorem lt_mul_of_one_lt_right' :
    ∀ {α : Type u_1} [inst : MulOneClass α] [inst_1 : LT α]
      [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b : α}, 1 < b → a < a * b :=
  by sorry

This theorem, `lt_mul_of_one_lt_right'`, states that for any type `α` that has a multiplication operation and a less-than relation, and additionally the multiplication operation is covariant with respect to the less-than relation, for any two elements `a` and `b` of this type, if `b` is greater than 1, then `a` is less than the product of `a` and `b`. In simple terms, for any numbers `a` and `b`, if `b` is greater than 1, then `a` is less than `a` times `b`.

For any type `α` with covariant multiplication and total order, if `1 < b` and `a < a * b`, then `a < 0` or `b > 1`. (Assuming Lean's negation of less-than relation `<` is defined as `>` in this context.)


theorem ENNReal.nhds_zero_basis : Filter.HasBasis (nhds 0) (fun a => 0 < a) fun a => Set.Iio a := by sorry

The theorem `ENNReal.nhds_zero_basis` states that the neighborhood filter at zero for the extended nonnegative real numbers has a basis. This basis is made up of the sets which are left-infinite right-open intervals (denoted as `Set.Iio a` in Lean), for all `a` greater than zero. In other words, it affirms that the filter of neighborhoods around zero is generated by all the open intervals from negative infinity to a positive real number `a`.

The neighborhood filter at zero in the extended nonnegative real numbers has a basis consisting of all left-infinite right-open intervals.


theorem Finmap.erase_toFinmap :
    ∀ {α : Type u} {β : α → Type v} [inst : DecidableEq α] (a : α) (s : AList β),
      Finmap.erase a (AList.toFinmap s) = AList.toFinmap (AList.erase a s) :=
  by sorry

The theorem `Finmap.erase_toFinmap` states that for all types `α` and `β` (where `β` is a function of `α`), and for all `a` of type `α` and `s` of type `AList β`, if we have a function that can decide the equality of elements in `α`, then erasing `a` from the `Finmap` mapped from `AList` `s` is equivalent to first erasing `a` from `s` and then mapping the resulting `AList` to a `Finmap`. In other words, the operations of erasing an element and converting an `AList` to a `Finmap` can be performed in any order without affecting the outcome.

For all types `α` and `β`, and for all `a : α` and `s : AList (β : α → Type)`, if `equiv` is a decidable equivalence relation on `α`, then erasing `a` from the `Finmap` obtained from `s` is equivalent to first erasing `a` from `s` and then mapping the resulting list to a `Finmap`.


theorem Mathlib.LinearAlgebra.Matrix.NonsingularInverse._auxLemma.4 :
    ∀ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [inst : NonUnitalSemiring α]
      [inst_1 : Fintype m] [inst_2 : Fintype n] (L : Matrix l m α) (M : Matrix m n α) (N : Matrix n o α),
      L * (M * N) = L * M * N :=
  by sorry

The theorem `Mathlib.LinearAlgebra.Matrix.NonsingularInverse._auxLemma.4` states that for any types `l`, `m`, `n`, `o`, and `α`, where `α` is a non-unital semiring, and `m` and `n` are finite types, given any matrices `L` of type `Matrix l m α`, `M` of type `Matrix m n α` and `N` of type `Matrix n o α`, the multiplication operation on matrices is associative. In other words, the product of `L` and the product of `M` and `N` is the same as the product of the product of `L` and `M` and `N` - formally, `L * (M * N)` equals `L * M * N`.

For matrices $L$ of size $m \times l$, $M$ of size $l \times n$, and $N$ of size $n \times o$ over a non-unital semiring $\alpha$, the associativity of matrix multiplication holds: $L *(M * N) = (L * M) * N$.


theorem intervalIntegrable_iff_integrableOn_Icc_of_le :
    ∀ {E : Type u_3} [inst : NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ},
      a ≤ b →
        ∀ {μ : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.NoAtoms μ],
          IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f (Set.Icc a b) μ :=
  by sorry

The theorem `intervalIntegrable_iff_integrableOn_Icc_of_le` states that for any real-valued function `f`, any measure `μ` on the real numbers, and any two real numbers `a` and `b` such that `a` is less than or equal to `b`, the function `f` is interval integrable with respect to `μ` on the interval from `a` to `b` if and only if `f` is integrable on the closed interval from `a` to `b` with respect to `μ`. Here, a function is said to be interval integrable if it is integrable on both the open intervals `(a, b]` and `(b, a]`. A function is integrable on a set if it is almost everywhere strongly measurable on that set and the integral of its pointwise norm over that set is less than infinity. This theorem is under the assumption that the measure `μ` has no atoms (i.e., there are no subsets of the real numbers with positive measure that contain only a single point).

A real-valued function is interval integrable with respect to a measure with no atoms on the interval [a, b] if and only if it is integrable on the closed interval Icc[a, b] with respect to that measure.


theorem neg_one_geom_sum :
    ∀ {α : Type u} [inst : Ring α] {n : ℕ}, (Finset.sum (Finset.range n) fun i => (-1) ^ i) = if Even n then 0 else 1 :=
  by sorry

The theorem `neg_one_geom_sum` states that for any natural number `n` and any type `α` that forms a ring, the sum of `(-1)` raised to the power `i` as `i` ranges over all natural numbers less than `n` (essentially forming a geometric sequence with common ratio -1), is equal to 0 if `n` is even and 1 if `n` is odd. In mathematical terms, this can be written as follows: for every natural number `n`, the sum $\sum_{i=0}^{n-1} (-1)^i$ equals 0 when `n` is even and 1 when `n` is odd.

For any natural number `n` and commutative ring `α`, the sum of `(-1)` raised to the power `i` over `i` from `0` to `n-1` equals `1` if `n` is odd and `0` if `n` is even.


theorem IsSMulRegular.of_mul :
    ∀ {R : Type u_1} {M : Type u_3} {a b : R} [inst : SMul R M] [inst_1 : Mul R] [inst_2 : IsScalarTower R R M],
      IsSMulRegular M (a * b) → IsSMulRegular M b :=
  by sorry

The theorem `IsSMulRegular.of_mul` states that for any types `R` and `M`, given any elements `a` and `b` of type `R`, and assuming that the type `R` has an operation of scalar multiplication on `M`, an operation of multiplication, and that `R` is a scalar tower over `M`. If the product `a * b` is an `M`-regular element (an element `c` such that multiplication on the left by `c` is an injective map `M → M`), then `b` is also an `M`-regular element.

If `a * b` is an `M`-regular element in a commutative ring `R` with scalar multiplication on `M`, then `b` is also an `M`-regular element.


theorem Matrix.PosSemidef.sq_sqrt :
    ∀ {n : Type u_2} {𝕜 : Type u_4} [inst : Fintype n] [inst_1 : RCLike 𝕜] [inst_2 : DecidableEq n] {A : Matrix n n 𝕜}
      (hA : Matrix.PosSemidef A), hA.sqrt ^ 2 = A :=
  by sorry

The theorem `Matrix.PosSemidef.sq_sqrt` states that for any given square matrix 'A' of type '𝕜' with 'n' as the index type, where 'n' is of finite type and '𝕜' is a type that behaves like a ring with conjugation (i.e., '𝕜' is `RCLike`), and also 'n' has a decidable equality, if 'A' is found to be positive semi-definite (meaning that it's Hermitian and the product of the conjugate transpose of any vector 'x' and the resultant of matrix 'A' multiplied with 'x' is non-negative), then the square of the square root of 'A' is equal to 'A' itself. In mathematical terms, if 'A' is positive semi-definite, then $(\sqrt{A})^2 = A$.

If 'A' is a square matrix of finite index type 'n' over a conjugate ring '𝕜' with decidable equality, and 'A' is positive semi-definite, then 'A' equals the square of its square root.


theorem AkraBazziRecurrence.GrowsPolynomially.congr_of_eventuallyEq :
    ∀ {f g : ℝ → ℝ},
      f =ᶠ[Filter.atTop] g → AkraBazziRecurrence.GrowsPolynomially g → AkraBazziRecurrence.GrowsPolynomially f :=
  by sorry

The theorem `AkraBazziRecurrence.GrowsPolynomially.congr_of_eventuallyEq` states that for any two real-valued functions `f` and `g`, if `f` is eventually equal to `g` at the limit towards infinity (that is `f =ᶠ[Filter.atTop] g`) and `g` satisfies the growth condition stipulated by the Akra-Bazzi theorem (that is, `AkraBazziRecurrence.GrowsPolynomially g`), then `f` also satisfies this growth condition (that is, `AkraBazziRecurrence.GrowsPolynomially f`). Essentially, it means that if `f` and `g` become the same for sufficiently large inputs and `g` grows polynomially under the Akra-Bazzi condition, then `f` also grows polynomially under the same condition.

If real-valued functions `f` and `g` are eventually equal and `g` satisfies the Akra-Bazzi polynomial growth condition, then `f` also satisfies the same condition.


theorem Finset.prod_disjUnion :
    ∀ {α : Type u_3} {β : Type u_4} {s₁ s₂ : Finset α} {f : α → β} [inst : CommMonoid β] (h : Disjoint s₁ s₂),
      (Finset.prod (Finset.disjUnion s₁ s₂ h) fun x => f x) =
        (Finset.prod s₁ fun x => f x) * Finset.prod s₂ fun x => f x :=
  by sorry

This theorem states that for all types `α` and `β`, all `α`-typed finite sets `s₁` and `s₂`, and all functions `f` mapping from `α` to `β`, under the commutative monoid structure of `β`, if `s₁` and `s₂` are disjoint, then the product of `f x` (where `x` ranges over the elements) over the disjoint union of `s₁` and `s₂` equals the product of `f x` over `s₁` times the product of `f x` over `s₂`. In other words, the product over the disjoint union of two sets is equal to the product of the individual products over the sets, assuming that the sets are disjoint.

For all types `α`, `β`, finite sets `s₁` and `s₂` of type `α`, and functions `f : α → β`, if `s₁` and `s₂` are disjoint, then the product of `f x` over the disjoint union of `s₁` and `s₂` equals the product of the product of `f x` over `s₁` and the product of `f x` over `s₂`.


/-- An equivalent expression of `AddSubmonoid.FG` in terms of `Set.Finite` instead of
`Finset`. -/
theorem AddSubmonoid.fg_iff :
    ∀ {M : Type u_1} [inst : AddMonoid M] (P : AddSubmonoid M),
      AddSubmonoid.FG P ↔ ∃ S, AddSubmonoid.closure S = P ∧ Set.Finite S :=
  by sorry

This theorem provides an equivalent formulation of the property of an additive submonoid being finitely generated (`AddSubmonoid.FG`). It states that for any additive monoid `M`, an additive submonoid `P` of `M` is finitely generated if and only if there exists a set `S` such that `P` is the additive submonoid generated by `S` (`AddSubmonoid.closure S = P`) and `S` is a finite set (`Set.Finite S`).

An additive submonoid of an abelian group is finitely generated if and only if it can be generated by a finite set.


theorem Nat.ModEq.mul_left' : ∀ {n a b : ℕ} (c : ℕ), a ≡ b [MOD n] → c * a ≡ c * b [MOD c * n] := by sorry

This theorem states that for any natural numbers `n`, `a`, `b`, and `c`, if `a` is congruent to `b` modulo `n`, then the product of `c` and `a` is congruent to the product of `c` and `b` modulo `c` times `n`. Essentially, it's showing the property of congruence that allows scaling both sides of the congruence by a constant, while also scaling the modulus by the same constant.

For any natural numbers `n`, `a`, `b`, and `c`, if `a ≡ b (mod n)`, then `c * a ≡ c * b (mod n * c)`.


/-- **Uniqueness of left-invariant measures**: In a second-countable locally compact group, any
σ-finite left-invariant measure is a scalar multiple of the Haar measure.
This is slightly weaker than assuming that `μ` is a Haar measure (in particular we don't require
`μ ≠ 0`).
See also `isHaarMeasure_eq_smul_of_regular` for a statement not assuming second-countability.  -/
theorem MeasureTheory.Measure.haarMeasure_unique :
    ∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] [inst_2 : TopologicalGroup G]
      [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [inst_5 : SecondCountableTopology G]
      (μ : MeasureTheory.Measure G) [inst_6 : MeasureTheory.SigmaFinite μ]
      [inst_7 : MeasureTheory.Measure.IsMulLeftInvariant μ] (K₀ : TopologicalSpace.PositiveCompacts G),
      μ = ↑↑μ ↑K₀ • MeasureTheory.Measure.haarMeasure K₀ :=
  by sorry

This theorem, referred to as the Uniqueness of left-invariant measures, states that in a second-countable locally compact group, any σ-finite measure that is left-invariant is simply a scalar multiple of the Haar measure. This implies that all left-invariant measures are essentially "equivalent" in the sense that they are rescalings of each other. Note that this condition is slightly weaker than requiring the measure to be a Haar measure, in particular, the measure is not required to be non-zero. The theorem also has an alternate version that does not assume second-countability.

In a second-countable locally compact group, any σ-finite left-invariant measure is proportional to the Haar measure.


theorem iteratedFDerivWithin_univ :
    ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
      {n : ℕ}, iteratedFDerivWithin 𝕜 n f Set.univ = iteratedFDeriv 𝕜 n f :=
  by sorry

This theorem states that for any normed field 𝕜, normed additive commutative groups E and F, and normed spaces 𝕜 over E and F, for any function `f` from E to F and any natural number `n`, the `n`-th iterated Fréchet derivative of `f` within the universal set is equal to the `n`-th iterated Fréchet derivative of `f`.

In other words, when taking the `n`-th iterated Fréchet derivative of `f`, there is no difference whether we consider the derivative within the whole space (the universal set) or without any constraint.

For any normed field 𝕜, normed additive commutative groups E and F, and normed spaces 𝕜 over E and F, the `n`-th iterated Fréchet derivative of a function `f` from E to F is independent of the choice of the universal set.


theorem FreeAddGroup.negRev_involutive : ∀ {α : Type u}, Function.Involutive FreeAddGroup.negRev := by sorry

The theorem `FreeAddGroup.negRev_involutive` asserts that for any type `α`, the function `FreeAddGroup.negRev` is involutive. In other words, applying the `FreeAddGroup.negRev` function twice to any list of pairs in `α × Bool` (representing elements of a free group) will return the original list. This means that `FreeAddGroup.negRev` is its own inverse: reversing the list and negating the booleans in the list, and then doing the same operation again, brings you back to where you started.

The map `FreeAddGroup.negRev` on lists of pairs in `α × Bool` is an involution.


theorem Computation.liftRel_bind :
    ∀ {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} (R : α → β → Prop) (S : γ → δ → Prop) {s1 : Computation α}
      {s2 : Computation β} {f1 : α → Computation γ} {f2 : β → Computation δ},
      Computation.LiftRel R s1 s2 →
        (∀ {a : α} {b : β}, R a b → Computation.LiftRel S (f1 a) (f2 b)) →
          Computation.LiftRel S (Computation.bind s1 f1) (Computation.bind s2 f2) :=
  by sorry

This theorem, `Computation.liftRel_bind`, states that for any types `α`, `β`, `γ`, and `δ`, and any relations `R` and `S` on these types, if we have two computations `s1` and `s2` of type `α` and `β` respectively, and two functions `f1` and `f2` mapping `α` to `Computation γ` and `β` to `Computation δ` respectively, then if `s1` and `s2` are related by `R` and for any `a` in `α` and `b` in `β` that satisfy `R a b`, `f1 a` and `f2 b` are related by `S`, it follows that the computations resulting from binding `s1` to `f1` and `s2` to `f2` are related by `S` as well.

In simpler terms, if the start computations are related and the transformations maintain this relation under certain conditions, then the computations resulting from applying the transformations will also be related.

For any types `α`, `β`, `γ`, `δ`, relations `R` and `S`, computations `s1 : Computation α`, `s2 : Computation β`, functions `f1 : α → Computation γ`, `f2 : β → Computation δ`, if `s1` and `s2` are related by `R` and for all `a : α` and `b : β` related by `R`, `f1 a` and `f2 b` are related by `S`, then `bind f1 s1` and `bind f2 s2` are related by `S`.


theorem Measurable.map_measurableSpace_eq :
    ∀ {X : Type u_3} {Y : Type u_4} [inst : MeasurableSpace X] [inst_1 : StandardBorelSpace X]
      [inst_2 : TopologicalSpace Y] [inst_3 : T2Space Y] [inst_4 : MeasurableSpace Y] [inst_5 : OpensMeasurableSpace Y]
      [inst_6 : SecondCountableTopology Y] {f : X → Y},
      Measurable f → Function.Surjective f → MeasurableSpace.map f inst = inst_4 :=
  by sorry

This theorem states that for any two types `X` and `Y`, where `X` is a measurable space and a standard Borel space, and `Y` is a topological space, a T2 space, a measurable space, an open measurable space, and a second countable topology, for any surjective function `f` from `X` to `Y` that is measurable, the forward image of the measurable space `X` under the function `f` is the same as the measurable space of `Y`. In simpler terms, if we map the measurable space `X` to `Y` using a surjective and measurable function, we get the same measurable space as `Y`.

For any surjective, measurable function from a measurable and standard Borel space `X` to a second countable T2 topological space `Y`, the image of `X` under `f` is a measurable subspace of `Y` that is equal to `Y`.


theorem CliffordAlgebra.star_def :
    ∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
      {Q : QuadraticForm R M} (x : CliffordAlgebra Q), star x = CliffordAlgebra.reverse (CliffordAlgebra.involute x) :=
  by sorry

The theorem `CliffordAlgebra.star_def` states that for any commutative ring `R`, an `R`-module `M` equipped with a quadratic form `Q`, and any element `x` of the Clifford Algebra formed by `Q`, the star operation on `x` is equivalent to the composition of the reverse and involute operations on `x`. In other words, the star of `x` is obtained by first applying the grade involution to `x`, which inverts the sign of each basis vector, and then applying the grade reversion to the result, which inverts the multiplication order of basis vectors.

For any commutative ring R, an R-module M with quadratic form Q, and element x in the Clifford algebra over M and Q, the star operation on x is equivalent to the composition of the grade involution and grade reverse operations.


theorem iUnion_Ioc_add_zsmul :
    ∀ {α : Type u_1} [inst : LinearOrderedAddCommGroup α] [inst_1 : Archimedean α] {p : α},
      0 < p → ∀ (a : α), ⋃ n, Set.Ioc (a + n • p) (a + (n + 1) • p) = Set.univ :=
  by sorry

The theorem `iUnion_Ioc_add_zsmul` states that for any type `α` that is a linear ordered additive commutative group and Archimedean, and for any positive element `p` of `α`, and any element `a` of `α`, the union over all integers `n` of the left-open right-closed intervals from `a + np` to `a + (n+1)p` equals the universal set. This means that every element of `α` is contained in one of these intervals. In other words, we can cover the entire set `α` by shifting the interval `(a, a+p]` by integer multiples of `p`.

For any Archimedean additive commutative group `α` with a positive element `p`, every element of `α` lies in some left-open right-closed interval `[a + np, a + (n+1)p)` for some integer `n`.


theorem Nat.factorization_ord_compl :
    ∀ (n p : ℕ), Nat.factorization (n / p ^ (Nat.factorization n) p) = Finsupp.erase p (Nat.factorization n) := by sorry

The theorem `Nat.factorization_ord_compl` states that for all natural numbers `n` and `p`, the factorization of `n` divided by `p` raised to the power of the multiplicity of `p` in `n` (which is obtained from the factorization of `n`) equals the factorization of `n` with the multiplicity of `p` set to zero. This means we're removing the contribution of the prime `p` from the factorization of `n`. In other words, we divide `n` by as many powers of `p` as `p` appears in the factorization of `n`, and this results in a new factorization equivalent to the original one, but with the count of `p` reduced to zero.

For all natural numbers `n` and `p`, the power of `p` in the prime factorization of `n` equals the product of `p` and the multiplicity of `p` in the prime factorization of `n` divided by the power of `p` in the prime factorization of `n`, or in other words, `p^(multiplicity p n) = n / (power p (factorization n p))`.


theorem Localization.npow_def :
    ∀ {M : Type u_4} [inst : CommMonoid M] (S : Submonoid M), Localization.npow S = Monoid.npow := by sorry

This theorem states that, for any type `M` that is a commutative monoid (denoted by `[inst : CommMonoid M]`) and any submonoid `S` of `M`, the exponentiation function for localizations (`Localization.npow S`) is equivalent to the exponentiation function for monoids (`Monoid.npow`). In other words, raising an element of the localization of a submonoid to a natural number power in the localization is the same as raising it to that power in the original monoid.

For any commutative monoid `M` and submonoid `S`, the exponentiation function in the localization `Localization S M` equals the exponentiation function in `M`.


/-- Given `r` larger than the measure of a set `A`, there exists an open superset of `A` with
measure less than `r`.  -/
theorem Set.exists_isOpen_lt_of_lt :
    ∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : TopologicalSpace α] {μ : MeasureTheory.Measure α}
      [inst_2 : MeasureTheory.Measure.OuterRegular μ] (A : Set α) (r : ENNReal),
      ↑↑μ A < r → ∃ U ⊇ A, IsOpen U ∧ ↑↑μ U < r :=
  by sorry

This theorem states that for any type `α` that is both a measurable and topological space, for any measure `μ` of this space that is outer regular, for any set `A` of type `α`, and for any extended nonnegative real number `r`, if the measure of `A` is less than `r`, then there exists a superset `U` of `A` that is open in the topological space and whose measure is also less than `r`.

For any measurable and topological space `(α, τ)` with outer regular measure `μ`, if `A ⊆ α` has measure `μ(A) < r` for some extended nonnegative real `r`, then there exists an open `U ⊇ A` with `μ(U) < r`.


NNReal.IsConjExponent.coe

/-- **Alias** of the reverse direction of `NNReal.isConjExponent_coe`. -/
theorem NNReal.IsConjExponent.coe : ∀ {p q : NNReal}, p.IsConjExponent q → Real.IsConjExponent ↑p ↑q := by sorry

This theorem states that for all nonnegative real numbers `p` and `q` (which are represented as `NNReal`), if `p` is a conjugate exponent of `q` (expressed as `p.IsConjExponent q`), then the real number equivalents of `p` and `q` (represented as `↑p` and `↑q` respectively) also share the same property, that is, the real number equivalent of `p` is a conjugate exponent of the real number equivalent of `q` (expressed as `Real.IsConjExponent ↑p ↑q`). In other words, the property of being conjugate exponents is preserved when nonnegative real numbers are converted to real numbers.

If `p` and `q` are nonnegative real numbers with `p` a conjugate exponent of `q`, then `↑p` and `↑q` are real numbers with `↑p` a conjugate exponent of `↑q`.


Inseparable.map

theorem Inseparable.map :
    ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {x y : X} {f : X → Y},
      Inseparable x y → Continuous f → Inseparable (f x) (f y) :=
  by sorry

The theorem `Inseparable.map` states that in any two topological spaces `X` and `Y`, for any two points `x` and `y` in `X` that are inseparable, and for any function `f` from `X` to `Y` that is continuous, the images of `x` and `y` under `f` are also inseparable in `Y`. In other words, the inseparability property of the points `x` and `y` is preserved under the continuous mapping `f`.

For any continuous function $f$ between topological spaces $X$ and $Y$, if $x$ and $y$ in $X$ are inseparable points, then $f(x)$ and $f(y)$ are inseparable points in $Y$.


theorem Mathlib.MeasureTheory.Measure.AEMeasurable._auxLemma.1 :
    ∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β}
      {μ : MeasureTheory.Measure α} [inst_1 : Subsingleton α], AEMeasurable f μ = True :=
  by sorry

This theorem states that for any types `α` and `β`, with `α` being a subsingleton (i.e., it has at most one element), any measurable spaces `m0` and `inst` over `α` and `β` respectively, any function `f` from `α` to `β`, and any measure `μ` on the measurable space `α`, the function `f` is almost everywhere measurable with respect to measure `μ`. In other words, the function `f` coincides almost everywhere with a measurable function, which is trivially true when the domain `α` has at most one element.

For any subsingleton type `α`, any measurable spaces `m0` over `α` and `inst` over `β`, any function `f` from `α` to `β`, and any measure `μ` on `α`, the function `f` is almost everywhere equal to a measurable function.


theorem Subgroup.mem_leftTransversals_iff_bijective :
    ∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {S : Set G},
      S ∈ Subgroup.leftTransversals ↑H ↔ Function.Bijective (Set.restrict S Quotient.mk'') :=
  by sorry

This theorem states that for any type `G` that has a group structure and any subgroup `H` of `G`, a set `S` is a left transversal of the elements of `H` if and only if the function that restricts `S` to the quotient of `G` by the equivalence relation `Setoid.r` is bijective. In other words, `S` contains exactly one representative from each coset of `H` in `G` (i.e., it is a left transversal) if and only if every element in the quotient space can be uniquely associated with an element in `S` and every element in `S` is associated with some element in the quotient space (i.e., the restriction function is bijective).

A set S is a left transversal of a subgroup H in a group G if and only if the restriction function from S to the quotient G/H is bijective.


theorem sdiff_unique :
    ∀ {α : Type u} {x y z : α} [inst : GeneralizedBooleanAlgebra α], x ⊓ y ⊔ z = x → x ⊓ y ⊓ z = ⊥ → x \ y = z := by
  sorry

This theorem, `sdiff_unique`, states that for any type `α` that forms a Generalized Boolean Algebra, given any elements `x`, `y`, and `z` of that type, if the join (or union) of the meet (or intersection) of `x` and `y` with `z` equals `x`, and the meet of `x`, `y`, and `z` equals the bottom element (representing the empty set), then the set difference of `x` and `y` equals `z`. In mathematical terms, this states that if $x \cap y \cup z = x$ and $x \cap y \cap z = \emptyset$, then $x - y = z$.

If $x \cap y = \emptyset$ and $x \cup y \cup z = x$, then $x \setminus y = z$.


theorem GaussianInt.nat_cast_natAbs_norm :
    ∀ {α : Type u_1} [inst : Ring α] (x : GaussianInt), ↑(Int.natAbs (Zsqrtd.norm x)) = ↑(Zsqrtd.norm x) := by sorry

This theorem states that for any Gaussian integer `x`, the natural number absolute value of its norm, when cast back to an integer, is equal to the norm of `x` itself. Gaussian integers are complex numbers where both the real and imaginary parts are integers. The norm of a Gaussian integer is calculated as the square of its real part minus the square of its imaginary part (multiplied by -1). The theorem asserts that taking the absolute value of this norm (which gives its magnitude as a natural number) and then casting it back to an integer results in the original norm. This is true in any ring `α`.

For any Gaussian integer x, the absolute value of its norm equals the norm of x when both are treated as integers.


theorem lp.summable_inner :
    ∀ {ι : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {G : ι → Type u_4} [inst_1 : (i : ι) → NormedAddCommGroup (G i)]
      [inst_2 : (i : ι) → InnerProductSpace 𝕜 (G i)] (f g : ↥(lp G 2)), Summable fun i => ⟪↑f i, ↑g i⟫_𝕜 :=
  by sorry

This theorem, `lp.summable_inner`, states that for any types `ι` and `𝕜`, where `𝕜` behaves like a ring-completed line (`RCLike 𝕜`), and for any function `G` mapping `ι` to a type which forms a normed additive commutative group and an inner product space over `𝕜`, given two elements `f` and `g` of the L^2 space (`lp G 2`), the inner product of the corresponding elements of `f` and `g` (i.e., `⟪↑f i, ↑g i⟫_𝕜`) is summable. In other words, the inner product series converges to a sum.

Given a type `ι`, a RCLike ring `𝕜`, and a function `G` mapping `ι` to a normed additive commutative group and inner product space over `𝕜`, for any `f` and `g` in the L^2 space `lp G 2`, the sum of their inner products `⟪↑f i, ↑g i⟫_𝕜` over `ι` converges.