theorem Finset.image₂_swap :
    ∀ {α : Type u_1} {β : Type u_3} {γ : Type u_5} [inst : DecidableEq γ] (f : α → β → γ) (s : Finset α) (t : Finset β),
      Finset.image₂ f s t = Finset.image₂ (fun a b => f b a) t s :=
  by sorry

The theorem `Finset.image₂_swap` states that for any types `α`, `β`, and `γ` and for any binary function `f` from `α` and `β` to `γ`, and for any finite sets `s` of type `α` and `t` of type `β`, applying the image of `f` over `s` and `t` is the same as applying the image of the swapped version of `f` over `t` and `s`. In other words, the order of the sets in the image operation does not affect the result when the arguments of the function are swapped. This operation is equivalent in natural language to swapping the order of the two sets in the Cartesian product and then applying the swapped function.

For any finite sets $s$ of type $\alpha$, $t$ of type $\beta$, and binary function $f$ from $\alpha$ and $\beta$ to $\gamma$, $f(s \times t) = f(t \times s)$.


theorem ProbabilityTheory.kernel.indep_limsup_atTop_self :
    ∀ {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {m0 : MeasurableSpace Ω}
      {κ : ↥(ProbabilityTheory.kernel α Ω)} {μα : MeasureTheory.Measure α} [inst : ProbabilityTheory.IsMarkovKernel κ]
      {s : ι → MeasurableSpace Ω} [inst : SemilatticeSup ι] [inst_1 : NoMaxOrder ι] [inst_2 : Nonempty ι],
      (∀ (n : ι), s n ≤ m0) →
        ProbabilityTheory.kernel.iIndep s κ μα →
          ProbabilityTheory.kernel.Indep (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) κ μα :=
  by sorry

This theorem, `ProbabilityTheory.kernel.indep_limsup_atTop_self`, states that for any types `α`, `Ω`, and `ι`, with associated measurable spaces `_mα` and `m0`, and any probability kernel `κ` from `α` to `Ω`, and a measure `μα` on `α`, if `κ` is a Markov kernel, and `s` is a function from `ι` to the measurable space `Ω`, with `ι` being a semilattice sup (it has an associative, commutative, and idempotent binary operation) and having no maximum order, and also, `ι` is nonempty, if every `s n` is less than or equal to `m0`, and the family of measurable space structures `s` is independent with respect to `κ` and `μα`, then, the `limsup` of `s` as `ι` goes to infinity in both arguments of the `Indep` predicate (which states a certain form of independence) is also independent with respect to `κ` and `μα`. This theorem essentially shows that the independence of the family of measurable space structures `s` is preserved under the limit superior operation as `ι` goes to infinity.

If `κ` is a Markov kernel from measurable spaces `α` to `Ω`, `μα` is a measure on `α`, `ι` is a semilattice sup without maximum, `s` is a family of measurable functions from `ι` to `Ω` with `s n ≤ m0` for all `n`, and `s` is independent of `κ` and `μα` for all `n`, then `limsup s` is independent of `κ` and `μα`.


/-- See also `pow_left_strictMonoOn`.  -/
theorem pow_left_strictMono :
    ∀ {M : Type u_3} [inst : Monoid M] [inst_1 : Preorder M]
      [inst_2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1]
      [inst_3 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ},
      n ≠ 0 → StrictMono fun x => x ^ n :=
  by sorry

The theorem `pow_left_strictMono` states that for any type `M` that forms a monoid and has a preorder structure, if the operations of multiplication and comparison are covariant, then for any non-zero natural number `n`, the function that takes `x` to the `n`th power of `x` is strictly monotone. In other words, if `a` and `b` are elements of `M` such that `a < b`, then `a^n < b^n`.

For any monoid `M` with covariant multiplication and preorder structure, the function from `M` to itself that maps each element to its `n`th power is strictly monotone on the positive elements.


theorem Polynomial.C_eq_int_cast : ∀ {R : Type u} [inst : Ring R] (n : ℤ), Polynomial.C ↑n = ↑n := by sorry

This theorem states that for any ring `R` and any integer `n`, the constant polynomial of `n` cast into `R` is equal to `n` cast into `R`. Essentially, the operation of creating a constant polynomial doesn't change the value of the integer when it is cast into the ring.

For any ring `R` and integer `n`, the constant polynomial `x => n` over `R` is equal to the numeric constant `n` when evaluated at any element of `R`.


theorem Fin.predAbove_castSucc_of_le : ∀ {n : ℕ} (p i : Fin n), i ≤ p → Fin.predAbove p (Fin.castSucc i) = i := by sorry

This theorem states that for any natural number `n` and for any two elements `p` and `i` of the finite set `Fin n` (which contains exactly `n` elements), if `i` is less than or equal to `p`, then if you first cast `i` to the set `Fin(n+1)` using `Fin.castSucc` and then apply the `Fin.predAbove` function on `p` and the result, you will end up with `i` again. This means that, under these conditions, `Fin.predAbove p (Fin.castSucc i)` is essentially a no-op.

For any natural number `n` and elements `i` and `p` of `Fin n`, if `i ≤ p`, then `Fin.predAbove (Fin.castSucc i) p = i`.


theorem List.rel_of_sorted_cons :
    ∀ {α : Type uu} {r : α → α → Prop} {a : α} {l : List α}, List.Sorted r (a :: l) → ∀ b ∈ l, r a b := by sorry

This theorem states that for any type `α`, and any binary relation `r` on `α`, if we have a sorted list `l` under the relation `r` such that some element `a` is the first element of the list, then for any element `b` contained in `l`, the relation `r` holds between `a` and `b`. In other words, if a list is sorted according to a certain relation, the first element of the list will relate to all other elements in the list via this relation.

If `α` is a type, `r` is a binary relation on `α`, and `l` is a list of `α` that is sorted under `r` with `a` as its first element, then `r(a, b)` holds for all `b` in `l`.


theorem SequentiallyComplete.setSeq_mem :
    ∀ {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → Set (α × α)}
      (U_mem : ∀ (n : ℕ), U n ∈ uniformity α) (n : ℕ), SequentiallyComplete.setSeq hf U_mem n ∈ f :=
  by sorry

This theorem states that for any type `α` that has a uniform space structure, any Cauchy filter `f` on `α`, and any sequence `U` of sets of ordered pairs from `α` such that each `U n` is in the uniformity of `α`, the set sequence created by `SequentiallyComplete.setSeq` applied to the filter `f`, the sequence `U`, and any natural number `n` is a member of the filter `f`. In other words, it guarantees that the sets we construct through the `SequentiallyComplete.setSeq` function from a Cauchy filter and a sequence of entourages always belong to the original filter.

For any Cauchy filter on a uniform space and any sequence of entourages in its uniformity, the sequence of sets generated by applying `SequentiallyComplete.setSeq` belongs to the Cauchy filter.


theorem le_kstar : ∀ {α : Type u_1} [inst : KleeneAlgebra α] {a : α}, a ≤ KStar.kstar a := by sorry

The theorem `le_kstar` states that for any type `α` that forms a Kleene algebra, and for any element `a` of that type, `a` is less than or equal to `KStar.kstar a`. Here, "less than or equal to" is interpreted in the sense of the partial order defined on the Kleene algebra. The function `KStar.kstar` represents the Kleene star operation, which is a unary operation in the algebra.

For any Kleene algebra type `α` and element `a` in `α`, `a ≤ KStar.kstar a`.


MeasurableSet.preimage

theorem MeasurableSet.preimage :
    ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {t : Set β},
      MeasurableSet t → Measurable f → MeasurableSet (f ⁻¹' t) :=
  by sorry

The theorem `MeasurableSet.preimage` states that for any two types `α` and `β`, a function `f` from `α` to `β`, and a set `t` of type `β`, if `t` is a measurable set and the function `f` is measurable, then the preimage of `t` under `f` is also a measurable set. In other words, it asserts that the preimage of a measurable set through a measurable function retains the property of measurability. Here, measurability of a set is defined in the context of a given measure space, and a function is considered measurable if the preimage of every measurable set is measurable.

If `f` is a measurable function and `t` is a measurable subset of `β`, then the preimage `f⁻¹(t)` is a measurable subset of `α`.


theorem Asymptotics.IsLittleO.right_isBigO_add :
    ∀ {α : Type u_1} {E' : Type u_6} [inst : SeminormedAddCommGroup E'] {l : Filter α} {f₁ f₂ : α → E'},
      f₁ =o[l] f₂ → f₂ =O[l] fun x => f₁ x + f₂ x :=
  by sorry

This theorem states that for all types `α` and `E'`, with `E'` being a seminormed additive commutative group, and for all filters `l` and functions `f₁` and `f₂` from `α` to `E'`, if `f₁` is little-o of `f₂` at filter `l` (meaning that `f₁` grows asymptotically slower than `f₂` as they approach the limit points of the filter), then `f₂` is Big-O of `f₁ + f₂` at filter `l` (meaning that the growth rate of `f₂` is bounded above by the growth rate of `f₁ + f₂`). In short, it expresses a certain relative growth rate condition between two functions in the context of asymptotic analysis.

If `f₁` is little-o of `f₂` at filter `l` in the seminormed additive commutative group `E'` over type `α`, then `f₂` is Big-O of `f₁ + f₂` at filter `l`.


theorem Finset.left_mem_uIcc :
    ∀ {α : Type u_2} [inst : Lattice α] [inst_1 : LocallyFiniteOrder α] {a b : α}, a ∈ Finset.uIcc a b := by sorry

The theorem `Finset.left_mem_uIcc` states that for any given type `α` that forms a lattice and is locally finite ordered, and for any elements `a` and `b` of type `α`, the element `a` is always a member of the set `Finset.uIcc a b`. In other words, in the set of all elements between `a` and `b` inclusive (defined in terms of the infimum and supremum of `a` and `b`), `a` is always included.

For any locally finite ordered type `α` and elements `a`, `b` of `α`, `a` is an element of the upward closed interval `Finset.uIcc a b`.


theorem eventually_mem_nhdsWithin :
    ∀ {α : Type u_1} [inst : TopologicalSpace α] {a : α} {s : Set α}, ∀ᶠ (x : α) in nhdsWithin a s, x ∈ s := by sorry

This theorem, `eventually_mem_nhdsWithin`, states that for any type `α` that has a topological space structure, any point `a` of type `α`, and any set `s` of type `α`, eventually, for all points `x` in the neighborhood within `s` of `a` (represented as `nhdsWithin a s`), `x` belongs to set `s`. In other words, it guarantees that in the vicinity of any point `a` within a set `s` in a topological space, all the points are eventually elements of `s`.

For any topological space `(α, τ)` and point `a` in `α` with `s ∈ τ`, eventually all points in the neighborhood `nhdsWithin a s` belong to `s`.


theorem MeasureTheory.IsFundamentalDomain.measure_eq_tsum :
    ∀ {G : Type u_1} {α : Type u_3} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : MeasurableSpace α] {s : Set α}
      {μ : MeasureTheory.Measure α} [inst_3 : MeasurableSpace G] [inst_4 : MeasurableSMul G α]
      [inst_5 : MeasureTheory.SMulInvariantMeasure G α μ] [inst_6 : Countable G],
      MeasureTheory.IsFundamentalDomain G s μ → ∀ (t : Set α), ↑↑μ t = ∑' (g : G), ↑↑μ (g • t ∩ s) :=
  by sorry

This theorem states that for any given types 'G' and 'α', where 'G' is a group with a multiplication action on 'α', and 'α' is equipped with a measurable space structure, along with a set 's' of type 'α' and a measure 'μ' on 'α'. Further, 'G' is also equipped with a measurable space structure and a measurable scalar multiplication operation, and 'μ' is assumed to be a scalar multiplication-invariant measure on 'α' under the action of 'G', with 'G' being countable. Under these conditions, the theorem asserts that if 's' is a fundamental domain for the action of 'G' on 'α' with respect to the measure 'μ', then for any set 't' of type 'α', the measure of 't' is equal to the sum of the measures of the intersections of 't' acted on by elements 'g' of 'G' and the set 's'. The sum is over all elements 'g' in 'G'.

If 'G' is a countable group acting measurably on measurable space ('α', μ) with 's' as a fundamental domain, then for any measurable set 't', the measure of 't' equals the sum of measures of intersections 't ∩ g(s)' over all elements 'g' in 'G'.


theorem Fin.insertNth_last' :
    ∀ {n : ℕ} {β : Type v} (x : β) (p : Fin n → β), Fin.insertNth (Fin.last n) x p = Fin.snoc p x := by sorry

This theorem, `Fin.insertNth_last'`, states that for any natural number `n` and any type `β`, if you have an element `x` of type `β` and a function `p` that maps each element of `Fin n` (finite set of elements from 0 to n-1) to `β`, then inserting `x` at the last position of the tuple (using `Fin.insertNth`) with `Fin.last n` (which gives the greatest value in `Fin (n+1)`) is equivalent to adding `x` at the end of the `n`-tuple to get an `n+1`-tuple (using `Fin.snoc`). In essence, it's saying that you can insert an element at the end of a tuple either by using `Fin.insertNth` with `Fin.last` or by using `Fin.snoc`, and these two methods are equivalent.

For any natural number `n` and type `β`, the application of `Fin.snoc` to an `n`-tuple `ts` with an element `x` is equivalent to the application of `Fin.insertNth` to the same `n`-tuple `ts` and index `Fin.last n` with element `x`.


theorem Associates.factors_one :
    ∀ {α : Type u_1} [inst : CancelCommMonoidWithZero α] [inst_1 : UniqueFactorizationMonoid α] [inst_2 : Nontrivial α],
      Associates.factors 1 = 0 :=
  by sorry

This theorem, `Associates.factors_one`, states that for any type `α` that forms a `CancelCommMonoidWithZero` and a `UniqueFactorizationMonoid` and is nontrivial, the multiset of irreducible factors of the associate of `1` (which is itself) is `0`. In other words, `1` has no irreducible factors, which aligns with the mathematical concept that `1` is not considered a prime or irreducible number.

In a nontrivial `CancelCommMonoidWithZero` and `UniqueFactorizationMonoid` type `α`, the set of irreducible factors of `1` is empty.


theorem ProbabilityTheory.gaussianReal_zero_var :
    ∀ (μ : ℝ), ProbabilityTheory.gaussianReal μ 0 = MeasureTheory.Measure.dirac μ := by sorry

The theorem `ProbabilityTheory.gaussianReal_zero_var` states that for all real numbers `μ`, the Gaussian distribution with mean `μ` and variance `0` is equivalent to the dirac measure at `μ`. In other words, a Gaussian distribution with zero variance behaves like a point mass located at the mean. This is consistent with the principle that a zero-variance Gaussian distribution is a distribution where every point except the mean has almost zero probability.

For all real numbers `μ`, the Gaussian distribution with mean `μ` and variance `0` is equal to the Dirac measure at `μ`.


Function.IsPeriodicPt.isFixedPt

theorem Function.IsPeriodicPt.isFixedPt :
    ∀ {α : Type u_1} {f : α → α} {x : α} {n : ℕ}, Function.IsPeriodicPt f n x → Function.IsFixedPt f^[n] x := by sorry

The theorem `Function.IsPeriodicPt.isFixedPt` states that for any type `α`, function `f` from `α` to `α`, element `x` of type `α`, and natural number `n`, if `x` is a periodic point of `f` with period `n` (i.e., `f` applied `n` times to `x` returns `x`), then `x` is also a fixed point of `f` applied `n` times (i.e., `f` applied `n` times to `x` still gives `x`). Thus, the periodicity of a point for a function implies its fixed point property when the function is iterated the same number of times as the period.

If `x` is a periodic point of period `n` for function `f`: i.e., `f^n(x) = x`, then `x` is a fixed point of `f` raised to the power `n`: i.e., `f^n(x) = x`.


theorem embedding_sigmaMk :
    ∀ {ι : Type u_5} {σ : ι → Type u_7} [inst : (i : ι) → TopologicalSpace (σ i)] {i : ι}, Embedding (Sigma.mk i) := by
  sorry

This theorem, `embedding_sigmaMk`, states that for any type `ι`, any function `σ` from `ι` to some type, and for any index `i` of type `ι`, the function `Sigma.mk i` is an embedding. Here, each `σ i` is equipped with a topology, making `σ i` a topological space for each `i` in `ι`. In this context, an embedding is a function that is both injective (or one-to-one) and continuous, with its image also being a closed set. The function `Sigma.mk i` creates a sigma type, essentially a tagged union, with the tag being `i`.

For any type `ι`, function `σ : ι → X` with each `σ i` a topological space, and index `i : ι`, the function `Σ.mk i : σ i → Σ i × *, where Σ i is the sigma type over `ι` with tag `i`, is an injective and continuous embedding.


theorem Order.Ideal.bot_mem : ∀ {P : Type u_1} [inst : LE P] [inst_1 : OrderBot P] (s : Order.Ideal P), ⊥ ∈ s := by
  sorry

This theorem, `Order.Ideal.bot_mem`, states that for any given type `P` that has an order (`LE P`) and a least element (`OrderBot P`), the least element (denoted by `⊥`) is an element of any order ideal `s` of `P`. In other words, in any ordered set with a bottom (or minimum) element, this bottom element is included in all of its order ideals.

For any type `P` with order `LE P` and least element `⊥`, `⊥` is an element of every order ideal `s` of `P`.


theorem dist_lineMap_left :
    ∀ {V : Type u_2} {P : Type u_3} [inst : SeminormedAddCommGroup V] [inst_1 : PseudoMetricSpace P]
      [inst_2 : NormedAddTorsor V P] {𝕜 : Type u_6} [inst_3 : NormedField 𝕜] [inst_4 : NormedSpace 𝕜 V] (p₁ p₂ : P)
      (c : 𝕜), dist ((AffineMap.lineMap p₁ p₂) c) p₁ = ‖c‖ * dist p₁ p₂ :=
  by sorry

The theorem `dist_lineMap_left` states that for all points `p₁` and `p₂` in a pseudometric space `P` that is also a normed add-torsor over a seminormed add-commutative group `V`, and for all scalars `c` in a normed field `𝕜` that is also a normed space over `V`, the distance from the point mapped from `c` by the affine map from `𝕜` to `P` sending `0` to `p₁` and `1` to `p₂` to `p₁` is equal to the product of the norm of `c` and the distance from `p₁` to `p₂`. In mathematical terms, we have $dist( AffineMap.lineMap(p₁, p₂)(c), p₁) = ‖c‖ * dist(p₁, p₂)$.

The distance between the point in a normed add-torsor over a seminormed add-commutative group that is the image of a scalar under the affine map from the scalar field to the group, sending 0 to a point p₁ and 1 to another point p₂, and p₁ is equal to the product of the norm of the scalar and the distance between p₁ and p₂.


theorem ExteriorAlgebra.lift.proof_1 :
    ∀ (R : Type u_3) [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {A : Type u_1}
      [inst_3 : Semiring A] [inst_4 : Algebra R A] (a : M →ₗ[R] A),
      (∀ (m : M), a m * a m = 0) ↔ ∀ (a_1 : M), a a_1 * a a_1 = (algebraMap R A) 0 :=
  by sorry

The theorem `ExteriorAlgebra.lift.proof_1` states that for any commutative ring `R`, additive commutative group `M`, module `M` over `R`, semiring `A`, and an algebra `A` over `R`, given a linear map `a` from `M` to `A`, the condition that the square of the image of any element of `M` under `a` equals zero is equivalent to the condition that the square of the image of any element of `M` under `a` equals the image of zero from `R` under the algebra map to `A`.

In simpler terms, this means that if for a linear map, the square of the result of mapping any element is always zero, it's the same as saying the square of the result of mapping any element is always the image of zero in the algebra structure.

For a linear map from an additive commutative group to an algebra over a commutative ring, the condition that the image of any element squared is zero is equivalent to the image of any element squared being the image of zero under the algebra map.


theorem Real.hasStrictDerivAt_arsinh : ∀ (x : ℝ), HasStrictDerivAt Real.arsinh (Real.sqrt (1 + x ^ 2))⁻¹ x := by sorry

The theorem `Real.hasStrictDerivAt_arsinh` states that for any real number `x`, the function `arsinh` has a strict derivative at the point `x`. Specifically, the derivative of `arsinh` at `x` is the reciprocal of the square root of `1 + x^2`. In mathematical terms, this can be written as: for all `x` in ℝ, the derivative of the `arsinh` function at `x` is `(1 + x^2)^{-0.5}`.

The theorem `Real.hasStrictDerivAt_arsinh` asserts that the derivative of `arsinh(x)` at `x` is `(1 + x^2)^(-0.5)`.


theorem CategoryTheory.Limits.sequentialFunctor_map :
    ∀ (J : Type u_2) [inst : Countable J] [inst_1 : Preorder J] [inst_2 : CategoryTheory.IsCofiltered J],
      Antitone (CategoryTheory.Limits.sequentialFunctor_obj J) :=
  by sorry

This theorem states that for any type `J` that satisfies the conditions of being countable, having a preorder, and being cofiltered in the category theory sense, the function `CategoryTheory.Limits.sequentialFunctor_obj J` is antitone. An antitone function is one where if `a ≤ b`, then `f b ≤ f a`. Specifically, in this context, `CategoryTheory.Limits.sequentialFunctor_obj J` maps natural numbers to elements of `J` in an antitone fashion.

For any countable, preordered and cocofiltered type `J`, the function `CategoryTheory.Limits.sequentialFunctor_obj J` is an antitone function from natural numbers to elements of `J`.


theorem CategoryTheory.Limits.π_comp_cokernelComparison :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      {X Y : C} (f : X ⟶ Y) {D : Type u₂} [inst_2 : CategoryTheory.Category.{v₂, u₂} D]
      [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] (G : CategoryTheory.Functor C D)
      [inst_4 : CategoryTheory.Functor.PreservesZeroMorphisms G] [inst_5 : CategoryTheory.Limits.HasCokernel f]
      [inst_6 : CategoryTheory.Limits.HasCokernel (G.map f)],
      CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.π (G.map f))
          (CategoryTheory.Limits.cokernelComparison f G) =
        G.map (CategoryTheory.Limits.cokernel.π f) :=
  by sorry

This theorem, `CategoryTheory.Limits.π_comp_cokernelComparison`, states that for any categories `C` and `D`, where `C` and `D` have zero morphisms, and for any functor `G` from `C` to `D` that preserves zero morphisms, if a morphism `f` from `X` to `Y` in `C` has a cokernel and the same morphism `f` mapped by `G` also has a cokernel in `D`, then the composition of the map from the target of the mapped `f` to its cokernel and the cokernel comparison of `f` and `G` is equal to the map of the cokernel projection of `f` under `G`. In terms of category theory, this is about the commutativity of the cokernel in the presence of a functor that preserves zero morphisms.

For functors G between categories C and D preserving zero morphisms, if morphism f in C has a cokernel and G(f) also has a cokernel in D, then the composition of the map from the cokernel of G(f) to its cokernel and the cokernel comparison of f and G is equal to the map of the cokernel projection of f under G.


theorem NonUnitalSubalgebra.unitization_range :
    ∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
      [inst_3 : SetLike S A] [hSA : NonUnitalSubsemiringClass S A] [hSRA : SMulMemClass S R A] (s : S),
      AlgHom.range (NonUnitalSubalgebra.unitization s) = Algebra.adjoin R ↑s :=
  by sorry

The theorem `NonUnitalSubalgebra.unitization_range` states that for any type `R` with commutative semiring structure, type `A` with semiring structure and algebra structure over `R`, type `S` that behaves like a set of elements from `A` with non-unital subsemiring structure and scalar multiplication behavior, and any `s` of type `S`, the range of the algebra homomorphism obtained by unitizing the non-unital subalgebra `s` is the same as the subalgebra generated by the algebraic adjunction of `s` over `R`. In simple terms, unitizing a non-unital subalgebra and then taking its range gives us the same result as simply adjoining the elements of the subalgebra to `R`.

For any commutative semiring `R`, semiring and algebra `A` over `R`, non-unital subsemiring `S` of `A` with scalar multiplication behavior, and `s` in `S`, the range of the algebra homomorphism obtained by unitizing `s` equals the subalgebra generated by `s` and `R`.


theorem BoundedContinuousFunction.coe_smul :
    ∀ {α : Type u} {β : Type v} {𝕜 : Type u_2} [inst : PseudoMetricSpace 𝕜] [inst_1 : TopologicalSpace α]
      [inst_2 : PseudoMetricSpace β] [inst_3 : Zero 𝕜] [inst_4 : Zero β] [inst_5 : SMul 𝕜 β] [inst_6 : BoundedSMul 𝕜 β]
      (c : 𝕜) (f : BoundedContinuousFunction α β), ⇑(c • f) = fun x => c • f x :=
  by sorry

This theorem states that for any types `α`, `β`, and `𝕜`, where `𝕜` is a pseudometric space, `α` is a topological space, and `β` is a pseudometric space with zero and scalar multiplication properties, and where `𝕜` and `β` can be scalars of a bounded scalar multiplication, the function of a scalar `c` multiplied by a bounded continuous function `f` (denoted as `c • f`) applied to an element `x` is equal to the scalar `c` multiplied by the application of `f` to `x` (expressed as `c • f x`). In simpler terms, it describes a property of scalar multiplication in the context of bounded continuous functions.

For any bounded continuous function `f` from a topological space `α` to a pseudometric space `β` with scalar multiplication, and for any scalar `c` and point `x` in `α`, the scalar multiplication `c • (f x)` equals the application of `c • f` to `x`.


theorem Polynomial.monomial_add :
    ∀ {R : Type u} [inst : Semiring R] (n : ℕ) (r s : R),
      (Polynomial.monomial n) (r + s) = (Polynomial.monomial n) r + (Polynomial.monomial n) s :=
  by sorry

The theorem `Polynomial.monomial_add` states that for any semiring `R`, any natural number `n`, and any elements `r` and `s` of `R`, the monomial generated by the sum `(r + s)` with exponent `n` is equal to the sum of the monomials generated by `r` and `s` separately with the same exponent `n`. In other words, the operation of creating a monomial with exponent `n` distributes over the addition of elements in the semiring. This can be written symbolically as `a * X^n + b * X^n = (a + b) * X^n` where `a` and `b` are elements in the semiring `R` and `X^n` represents a monomial of degree `n`.

For any semiring `R` and natural number `n`, the sum of monomials `(r * X^n + s * X^n)` equals `(r + s) * X^n`, where `r` and `s` are elements of `R`.


Ordnode.Valid'.node'

theorem Ordnode.Valid'.node' :
    ∀ {α : Type u_1} [inst : Preorder α] {l : Ordnode α} {x : α} {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},
      Ordnode.Valid' o₁ l ↑x →
        Ordnode.Valid' (↑x) r o₂ →
          Ordnode.BalancedSz (Ordnode.size l) (Ordnode.size r) → Ordnode.Valid' o₁ (Ordnode.node' l x r) o₂ :=
  by sorry

This theorem, `Ordnode.Valid'.node'`, asserts that for any type `α` which has a preorder relation, given a left Ordnode `l`, a value `x` of type `α`, a right Ordnode `r`, and two optional types `o₁` and `o₂` (which can be thought of as possible maximum and minimum bounds), if `l` is a valid Ordnode with respect to `o₁` and `x`, and `r` is a valid Ordnode with respect to `x` and `o₂`, and if the sizes of `l` and `r` satisfy the `Ordnode.BalancedSz` condition (meaning that either their sum is less than or equal to 1, or each one is less than or equal to `Ordnode.delta` times the other), then the new Ordnode created by `Ordnode.node'` with `l`, `x`, and `r` is also a valid Ordnode with respect to `o₁` and `o₂`. This essentially states that constructing a new Ordnode with correct size information from valid child nodes gives you a valid parent node, assuming that the sizes of the child nodes satisfy the balance condition.

If `α` has a preorder relation, `l` is a valid Ordnode with respect to optional `o₁` and `x`, `r` is a valid Ordnode with respect to `x` and `o₂`, and their sizes satisfy the `Ordnode.BalancedSz` condition, then the new Ordnode created by `Ordnode.node'` with `l`, `x`, and `r` is also a valid Ordnode with respect to `o₁` and `o₂`.


theorem Mathlib.GroupTheory.GroupAction.FixedPoints._auxAddLemma.2 :
    ∀ {M : Type u} [inst : AddMonoid M] {α : Type v} [inst_1 : AddAction M α] {m : M} {a : α},
      (a ∈ AddAction.fixedBy α m) = (m +ᵥ a = a) :=
  by sorry

The theorem `Mathlib.GroupTheory.GroupAction.FixedPoints._auxAddLemma.2` states that for any type `M` with an additive monoid structure, any type `α` with an `M`-additive action, any element `m` of `M`, and any element `a` of `α`, `a` is in the set of elements fixed by `m` (i.e., `a` is in `AddAction.fixedBy α m`) if and only if `m` added to `a` via the additive action (`m +ᵥ a`) equals `a` itself. In essence, the theorem formalizes the definition of fixed points under an additive action of a monoid.

For any additive monoid M, type α with an M-additive action, element m of M, and α-element a, a is a fixed point of the action by m if and only if m + a = a.


theorem Btw.btw.cyclic_left : ∀ {α : Type u_1} [inst : CircularPreorder α] {a b c : α}, btw a b c → btw b c a := by
  sorry

This theorem, `Btw.btw.cyclic_left`, states that for any type `α` that has a circular preorder (a relation that is reflexive and transitive), and for any three elements `a`, `b`, and `c` of this type, if `a` is between `b` and `c` (denoted as `btw a b c`), then `b` is between `c` and `a` (denoted as `btw b c a`). This is essentially a property of circularity where the betweenness relation is cyclic.

For any type with a circular preorder, if `a` is between `b` and `c`, then `b` is between `c` and `a` (i.e., the betweenness relation is reflexively cyclic).


theorem MvPolynomial.degrees_def :
    ∀ {R : Type u} {σ : Type u_1} [inst : CommSemiring R] [inst_1 : DecidableEq σ] (p : MvPolynomial σ R),
      MvPolynomial.degrees p = Finset.sup (MvPolynomial.support p) fun s => Finsupp.toMultiset s :=
  by sorry

The theorem `MvPolynomial.degrees_def` states that for any commutative semiring `R`, any type `σ`, and any instance of decidable equality on `σ`, for any multivariate polynomial `p` with coefficients in `R` and variables indexed by `σ`, the degrees of `p` (defined as the maximal degrees of each variable, expressed as a multiset) is equal to the supremum (greatest element) of the multiset conversion of the finite set of all monomials `m : σ →₀ ℕ` in `p` that have non-zero coefficients. This multiset is obtained by taking each monomial in the polynomial's support and converting it to a multiset where each variable appears a number of times equal to its exponent in the monomial.

The degrees of a multivariate polynomial with coefficients in a commutative semiring and decidable equality on its index type is equal to the multiset of monomial degrees in the polynomial, where each monomial is counted by its highest variable exponent.


theorem List.support_formPerm_le' :
    ∀ {α : Type u_1} [inst : DecidableEq α] (l : List α), {x | (List.formPerm l) x ≠ x} ≤ ↑(List.toFinset l) := by sorry

This theorem is stating that for any list `l` of elements of type `α`, where `α` has decidable equality, the set of elements that are not fixed by the permutation formed by `l` (i.e., those `x` for which applying the permutation `List.formPerm l` to `x` does not result in `x`) is a subset of the finset obtained by removing duplicates from `l`. In other words, any element that the permutation `List.formPerm l` moves is an element of the list `l`.

For any list `l` of distinct elements of type `α` with decidable equality, the set of elements not fixed by the permutation `List.formPerm l` is a subset of the list `l` without duplicates.


theorem Finset.inter_subset_inter :
    ∀ {α : Type u_1} [inst : DecidableEq α] {x y s t : Finset α}, x ⊆ y → s ⊆ t → x ∩ s ⊆ y ∩ t := by sorry

This theorem states that for any type `α` equipped with a decidable equality operation, given four finite sets `x`, `y`, `s`, and `t` of this type, if `x` is a subset of `y` and `s` is a subset of `t`, then the intersection of `x` and `s` is a subset of the intersection of `y` and `t`. This is a generalization of the principle from set theory that intersecting with a subset yields a subset of the original intersection.

For any type `α` with decidable equality, if `x` is a subset of `y` and `s` is a subset of `t`, then the intersection of `x` and `s` is a subset of the intersection of `y` and `t`.


theorem Nat.lt_size_self : ∀ (n : ℕ), n < 2 ^ Nat.size n := by sorry

This theorem states that, for any natural number 'n', 'n' is less than 2 raised to the power of the size of 'n'. Here, the size of 'n' refers to the length of its binary representation. In other words, any natural number is less than 2 to the power of the number of bits in its binary form.

For all natural numbers n, the binary representation of n has length at most the log2 base 2 of n + 1, so 2^(log2 base 2 of n) ≤ n.


theorem ProbabilityTheory.set_lintegral_condCDF :
    ∀ {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ))
      [inst : MeasureTheory.IsFiniteMeasure ρ] (x : ℝ) {s : Set α},
      MeasurableSet s →
        ∫⁻ (a : α) in s, ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) x) ∂MeasureTheory.Measure.fst ρ =
          ↑↑ρ (s ×ˢ Set.Iic x) :=
  by sorry

This theorem, named `ProbabilityTheory.set_lintegral_condCDF`, states that for any given measure `ρ` on the product space of some type `α` and the real numbers (`α × ℝ`), and for any real number `x` and a measurable set `s` of type `α`, the integral over `s` with respect to the marginal measure on `α` obtained from `ρ`, of the function that assigns to each element `a` in `s` the extended non-negative real number obtained from the conditional cumulative distribution function (CDF) at `a` and `x`, is equal to the measure `ρ` of the Cartesian product of `s` and the left-closed right-infinite interval of all real numbers less than or equal to `x`. This theorem connects the Lebesgue integral of the conditional CDF and the measure of a set in the product space.

For measure `ρ` on `α × ℝ`, and measurable set `s` in `α`, the integral of the conditional CDF with respect to the marginal measure on `α` over `s` equals the measure of `s` times the length of the interval `[-\infty, x]` for any real number `x`.


List.IsRotated.trans

theorem List.IsRotated.trans : ∀ {α : Type u} {l l' l'' : List α}, l ~r l' → l' ~r l'' → l ~r l'' := by sorry

This theorem states that for any type `α`, and any three lists of type `α` named `l`, `l'`, and `l''`, if `l` is a rotation of `l'` and `l'` is a rotation of `l''`, then `l` is a rotation of `l''`. Here, a rotation is a transformation where we take an element from one end of the list and put it on the other end. The "~r" symbol is used to indicate a rotation relation between two lists.

If `l` is a rotation of `l'` and `l'` is a rotation of `l''`, then `l` is a rotation of `l''` (in the context of lists `l`, `l'`, and `l''` of type `α`).


theorem List.Perm.subperm_right :
    ∀ {α : Type u_1} {l₁ l₂ l : List α}, List.Perm l₁ l₂ → (List.Subperm l₁ l ↔ List.Subperm l₂ l) := by sorry

This theorem states that for any three lists `l₁`, `l₂`, and `l` of any type `α`, if `l₁` is a permutation of `l₂`, then `l₁` is a sublist of a permutation of `l` if and only if `l₂` is a sublist of a permutation of `l`. This is essentially saying that the property of being a sublist of a permutation of another list is preserved under permutation of the lists.

For any lists `l₁`, `l₂`, and `l` of type `α`, if `l₁` is a permutation of `l₂`, then `l₁` is a sublist of some permutation of `l` if and only if `l₂` is a sublist of some permutation of `l`.


theorem AddCircle.measure_univ : ∀ (T : ℝ) [hT : Fact (0 < T)], ↑↑MeasureTheory.volume Set.univ = ENNReal.ofReal T := by
  sorry

This theorem, `AddCircle.measure_univ`, states that for any real number `T` for which there is a fact proving that it's positive (`0 < T`), the measure of the universal set, which includes all elements of a certain type, is equal to the nonnegative extended real number representation of `T`. In other words, when measuring the size of the universal set under the `MeasureTheory.volume`, it will always be equivalent to `T` if `T` is nonnegative, or to `0` if `T` is negative.

For any positive real number `T`, the measure of the universal set under `MeasureTheory.volume` is equal to the nonnegative extended real number `T`.


theorem lt_add_of_nonneg_of_lt :
    ∀ {α : Type u_1} [inst : AddZeroClass α] [inst_1 : Preorder α]
      [inst_2 : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α},
      0 ≤ a → b < c → b < a + c :=
  by sorry

The theorem `lt_add_of_nonneg_of_lt` states that for any type `α` which has an addition operation, zero element (is an `AddZeroClass`), and has a preorder relation (is a `Preorder`), and the addition operation is covariant (satisfies the `CovariantClass` property), if `a`, `b`, and `c` are elements of `α` such that `a` is nonnegative (i.e., `0 ≤ a`) and `b` is less than `c` (i.e., `b < c`), then `b` is also less than the sum of `a` and `c` (i.e., `b < a + c`). This is a statement about the order-preserving nature of addition when one of the added values is nonnegative.

If `α` is a type with Addition, Zero, Preorder, and Covariant properties, and `0 ≤ a` and `b < c` in `α`, then `b < a + c`.


theorem Decidable.not_iff_not : ∀ {a b : Prop} [inst : Decidable a] [inst : Decidable b], (¬a ↔ ¬b) ↔ (a ↔ b) := by
  sorry

The theorem `Decidable.not_iff_not` states that for any two propositions `a` and `b`, given that `a` and `b` are decidable (i.e., either `a` or `¬a` is true, and either `b` or `¬b` is true), the equivalence of the negations of `a` and `b` (i.e., `¬a` is equivalent to `¬b`) if and only if `a` is equivalent to `b`. In mathematical terms, for all propositions `a` and `b`, `(¬a ↔ ¬b) ↔ (a ↔ b)`.

The negations of propositions `a` and `b` are equivalent if and only if `a` and `b` are equivalent, given that `a` and `b` are decidable.


theorem eventually_nhds_nhdsWithin :
    ∀ {α : Type u_1} [inst : TopologicalSpace α] {a : α} {s : Set α} {p : α → Prop},
      (∀ᶠ (y : α) in nhds a, ∀ᶠ (x : α) in nhdsWithin y s, p x) ↔ ∀ᶠ (x : α) in nhdsWithin a s, p x :=
  by sorry

This theorem, `eventually_nhds_nhdsWithin`, states that for any topological space `α`, a point `a` in that space, a set `s` of `α`, and a property `p` that elements of `α` might satisfy, the following two conditions are equivalent:

1. For any point `y` in the neighborhood of `a`, eventually for every point `x` in the intersection of `s` and the neighborhood of `y`, `p` holds for `x`.
2. Eventually for every point `x` in the intersection of `s` and the neighborhood of `a`, `p` holds for `x`.

Here, "eventually" is used in the sense of filter theory: a property holds eventually in a filter if the property is true within some element of the filter, and a neighborhood of a point is an open set containing that point.

For any topological space, point, set, and property, the two conditions are equivalent: the property holds eventually for all points in the intersection of the set and the neighborhood of any point in the neighborhood of the given point, if and only if the property holds eventually for all points in the intersection of the set and the neighborhood of the given point.


theorem Padic.rat_dense' :
    ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (q : ℚ_[p]) {ε : ℚ}, 0 < ε → ∃ r, padicNormE (q - ↑r) < ε := by sorry

This theorem, `Padic.rat_dense'`, states that for any prime number `p`, and for any element `q` in the `p`-adic numbers ℚ_[p] and any positive rational number `ε`, there exists a rational number `r` such that the `p`-adic norm of the difference between `q` and `r` is less than `ε`. In other words, the rational numbers are densely embedded within the `p`-adic numbers to any desired level of precision `ε`.

For any prime number `p` and positive rational number `ε`, the `p`-adic numbers ℚ_[p] contain a rational number `r` with `p`-adic norm of the difference between `q` and `r` less than `ε` for any given `q` in ℚ_[p].


theorem CategoryTheory.ShortComplex.Exact.mono_fromOpcycles :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
      {S : CategoryTheory.ShortComplex C},
      CategoryTheory.ShortComplex.Exact S →
        ∀ [inst_2 : CategoryTheory.ShortComplex.HasRightHomology S],
          CategoryTheory.Mono (CategoryTheory.ShortComplex.fromOpcycles S) :=
  by sorry

This theorem states that for any category `C` that is preadditive, and for any short complex `S` in `C`, if `S` is exact and has right homology, then the morphism from the opposite cycles of `S` is a monomorphism. In the context of category theory, a category is preadditive if its hom-sets are equipped with an abelian group structure, a short complex is a sequence of objects and morphisms with certain properties, and a monomorphism is a morphism that is left-cancellable.

In a preadditive category `C`, if a short exact and right-homologous complex `S` has a monomorphic opposite cycles morphism.


theorem Mathlib.LinearAlgebra.Dual._auxLemma.11 :
    ∀ {R : Type u} {M : Type v} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
      {W : Submodule R M} (φ : Module.Dual R M), (φ ∈ Submodule.dualAnnihilator W) = ∀ w ∈ W, φ w = 0 :=
  by sorry

This theorem states that for any commutative semiring `R`, any additive commutative monoid `M` serving as an `R`-module, and any submodule `W` of `M`, a linear map `φ` from `M` to `R` (identified as a member of the dual space of `M`) is in the dual annihilator of `W` if and only if it maps every element of `W` to zero. In simpler terms, it states that a linear map is in the dual annihilator of a submodule when it annihilates every element of that submodule.

A linear map from an `R`-module to `R` belongs to the dual annihilator of a submodule if and only if it maps every submodule element to zero.


theorem Matrix.trace_mul_comm :
    ∀ {m : Type u_2} {n : Type u_3} {R : Type u_6} [inst : Fintype m] [inst_1 : Fintype n] [inst_2 : AddCommMonoid R]
      [inst_3 : CommSemigroup R] (A : Matrix m n R) (B : Matrix n m R), Matrix.trace (A * B) = Matrix.trace (B * A) :=
  by sorry

This theorem states that for any types `m` and `n`, and a type `R` with structures of a finite type, an addition-commutative monoid and a commutative semigroup, given two matrices `A` and `B` of types `Matrix m n R` and `Matrix n m R` respectively, the trace of the product `A * B` is equal to the trace of the product `B * A`. In other words, the trace of the product of two matrices is commutative. This mirrors the familiar property from linear algebra where the trace of the product of two matrices is invariant under cyclic permutations.

For any types `m`, `n`, and matrices `A` of size `m x n` and `B` of size `n x m` over a commutative semigroup `R` with finite type and additive commutativity, the trace of their product `A * B` equals the trace of `B * A`.


theorem List.sublists_append :
    ∀ {α : Type u} (l₁ l₂ : List α),
      List.sublists (l₁ ++ l₂) = do
        let x ← List.sublists l₂
        List.map (fun x_1 => x_1 ++ x) (List.sublists l₁) :=
  by sorry

The theorem `List.sublists_append` states that for any two lists `l₁` and `l₂` of the same type `α`, the list of all sublists of the concatenation of `l₁` and `l₂` is equivalent to the list generated by taking each sublist `x` of `l₂`, and then mapping each sublist of `l₁` to the concatenation of that sublist with `x`. This theorem describes a property of how the sublists of concatenated lists relate to the sublists of the original lists.

The set of sublists of the concatenation of lists `l₁` and `l₂` is equal to the image of the set of sublists of `l₁` under the function that maps each sublist to its concatenation with `l₂`.


theorem PiNat.apply_firstDiff_ne :
    ∀ {E : ℕ → Type u_1} {x y : (n : ℕ) → E n}, x ≠ y → x (PiNat.firstDiff x y) ≠ y (PiNat.firstDiff x y) := by sorry

This theorem states that for any function `E` that maps natural numbers to a certain type, and any two functions `x` and `y` (that are both of type `E`), if `x` and `y` are not equal, then the values of `x` and `y` at the first index at which they differ (given by `PiNat.firstDiff x y`) are also not equal. In other words, it asserts that if two such functions are not the same, they must differ at least at one index.

For any function `E` mapping natural numbers to a type and two distinct functions `x` and `y` of type `E`, `PiNat.firstDiff x y` is also distinct.


theorem MeasureTheory.Measure.restrict_restrict₀ :
    ∀ {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},
      MeasureTheory.NullMeasurableSet s (μ.restrict t) → (μ.restrict t).restrict s = μ.restrict (s ∩ t) :=
  by sorry

This theorem is about the restriction of measures in measure theory. It states that for any type `α` equipped with a `MeasurableSpace` structure, any measure `μ` on `α`, and any two sets `s` and `t` of type `α`, if `s` is a Null Measurable Set with respect to the restriction of `μ` to `t`, then the restriction of the restriction of `μ` to `t` to `s` is equal to the restriction of `μ` to the intersection of `s` and `t`. In other words, restricting the measure twice (first on `t` then on `s`) is the same as restricting it once on the intersection of `s` and `t`, provided `s` is a Null Measurable Set under the restricted measure.

If `μ` is a measure on a measurable space `(α, Σ)` and `s` is a Null Measurable Set in `t ∈ Σ`, then the restriction of `μ` to `s` under `t` equals the restriction of `μ` to `s ∩ t`.


theorem PerfectClosure.mk_eq_iff :
    ∀ (K : Type u) [inst : CommRing K] (p : ℕ) [inst_1 : Fact (Nat.Prime p)] [inst_2 : CharP K p] (x y : ℕ × K),
      PerfectClosure.mk K p x = PerfectClosure.mk K p y ↔
        ∃ z, (⇑(frobenius K p))^[y.1 + z] x.2 = (⇑(frobenius K p))^[x.1 + z] y.2 :=
  by sorry

The theorem `PerfectClosure.mk_eq_iff` states that for any commutative ring `K`, prime number `p`, and pairs of natural number and element of `K` denoted as `x` and `y`, the equality of `PerfectClosure.mk K p x` and `PerfectClosure.mk K p y` is equivalent to the existence of a natural number `z` such that applying the Frobenius map `p` times to `x.2` incremented by `y.1` and `z` times is equal to applying the Frobenius map `p` times to `y.2` incremented by `x.1` and `z` times.

In mathematical terms, this means that two elements in the perfect closure of a commutative ring are equal if and only if there exists a natural number such that the p-th power Frobenius map applied to the second element of the first pair, raised to the power of the sum of the first element of the second pair and the natural number, is equal to the p-th power Frobenius map applied to the second element of the second pair, raised to the sum of the first element of the first pair and the natural number.

For any commutative ring K and prime number p, two elements in the perfect closure of K with respect to p are equal if and only if their second components differ by a natural number z such that (Frobenius^p(k.2 + z))^(p+1) = (Frobenius^p(l.2 + z))^(p+1), where k and l are the given pairs and Frobenius denotes the p-th power map.


theorem MeasureTheory.Integrable.integral_prod_left :
    ∀ {α : Type u_1} {β : Type u_3} {E : Type u_6} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β]
      {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [inst_2 : NormedAddCommGroup E]
      [inst_3 : MeasureTheory.SigmaFinite ν] [inst_4 : NormedSpace ℝ E] ⦃f : α × β → E⦄,
      MeasureTheory.Integrable f (MeasureTheory.Measure.prod μ ν) →
        MeasureTheory.Integrable (fun x => ∫ (y : β), f (x, y) ∂ν) μ :=
  by sorry

The theorem `MeasureTheory.Integrable.integral_prod_left` states that for any two types `α` and `β` with respective measures `μ` and `ν`, and a function `f` from the Cartesian product of `α` and `β` to another type `E` (where 'E' is a Normed Additive Commutative Group and a Normed Space over the Reals), if `f` is integrable with respect to the product measure of `μ` and `ν`, then the function that maps each element of `α` to the integral of `f` over `β` (with the element of `α` fixed) is also integrable with respect to measure `μ`. In other words, it allows us to integrate the function `f` with respect to one variable first, ensuring the result is still an integrable function of the remaining variable.

Given measurable spaces `(α, μ)` and `(β, ν)`, a function `f : α × β → E` (where `E` is a real Normed Additive Commutative Group) that is μν-integrable, the function `g : α → E` defined by `g(a) = ∫β f(a, b) dν` is μ-integrable.


theorem Real.deriv_sin : deriv Real.sin = Real.cos := by sorry

This theorem states that the derivative of the real sine function equals the real cosine function. In mathematical terms, if `f(x) = sin(x)` for a real number `x`, then the derivative of `f` with respect to `x`, denoted as `f'(x)`, is equal to `cos(x)`. This fact is a well-known result in calculus.

The derivative of the real sine function is equal to the real cosine function. (i.e., sin'(x) = cos(x) for all x in R)


theorem Cardinal.lift_two_power :
    ∀ (a : Cardinal.{u_1}), Cardinal.lift.{v, u_1} (2 ^ a) = 2 ^ Cardinal.lift.{v, u_1} a := by sorry

This theorem states that for any cardinal number `a` in a given universe `u_1`, lifting the cardinality of the set `{0, 1}` raised to the power `a` to a bigger universe is equivalent to the set `{0, 1}` raised to the power of the lifted cardinality of `a`. In other words, the operation of lifting the universe of a cardinal number commutes with the operation of taking a power of 2. This asserts the consistency of cardinal arithmetic across different universe levels.

For any cardinal number `a` in a universe `u_1`, the cardinality of `{0, 1}^a` in a bigger universe is equal to the cardinality of the lifted cardinality of `{0, 1}^a` in `u_1`.


/-- We shorten this from `measure_preimage_mul_left`, since left invariant is the preferred option
for measures in this formalization.  -/
theorem MeasureTheory.measure_preimage_mul :
    ∀ {G : Type u_2} [inst : MeasurableSpace G] [inst_1 : Group G] [inst_2 : MeasurableMul G]
      (μ : MeasureTheory.Measure G) [inst_3 : MeasureTheory.Measure.IsMulLeftInvariant μ] (g : G) (A : Set G),
      ↑↑μ ((fun h => g * h) ⁻¹' A) = ↑↑μ A :=
  by sorry

This theorem, referred to as `MeasureTheory.measure_preimage_mul`, states that for any type `G` possessing a `MeasurableSpace` structure, a `Group` structure, and a `MeasurableMul` structure, any `MeasureTheory.Measure` `μ` that is `MulLeftInvariant` and any element `g` of `G` and set `A` of `G`, the measure of the preimage of `A` under the function that multiplies `g` to the left (`fun h => g * h`) is equal to the measure of `A`. In other words, the left multiplication by `g` does not change the measure of the set `A` which is a property often needed in measure theory as applied to groups.

For any measurable space `(G, Σ)` with a group and MeasurableMul structure, and any MulLeftInvariant measure `μ` on `(G, Σ)`, left multiplication by any group element `g` preserves the measure: `μ(g⁻¹(A)) = μ(A)` for all sets `A` in `Σ`.


theorem AddLECancellable.tsub_lt_iff_left :
    ∀ {α : Type u_1} [inst : AddCommSemigroup α] [inst_1 : PartialOrder α] [inst_2 : ExistsAddOfLE α]
      [inst_3 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [inst_4 : Sub α] [inst_5 : OrderedSub α]
      {a b c : α}, AddLECancellable b → b ≤ a → (a - b < c ↔ a < b + c) :=
  by sorry

The theorem `AddLECancellable.tsub_lt_iff_left` applies to an additive commutative semi-group that is also a partially ordered set on which subtraction is defined in a specific way. Given elements `a`, `b`, and `c` of this set, if `b` is add-left-cancellable and `b` is less than or equal to `a`, then the theorem states that `a` minus `b` is less than `c` if and only if `a` is less than `b + c`. Here, `b` is said to be add-left-cancellable if for any two elements `x` and `y` of the set, `b + x` being less than or equal to `b + y` implies `x` is less than or equal to `y`.

Given an additive commutative semi-group with subtraction that is also a partially ordered set, if `b` is add-left-cancellable and `b <= a`, then `a - b < c` if and only if `a < b + c`.


theorem smul_algebra_smul_comm :
    ∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {M : Type u_3}
      [inst_3 : AddCommMonoid M] [inst_4 : Module A M] [inst_5 : Module R M] [inst_6 : IsScalarTower R A M] (r : R)
      (a : A) (m : M), a • r • m = r • a • m :=
  by sorry

This theorem states that for any commutative semiring `R`, any semiring `A` that is also an `R`-algebra, and any additively commutative monoid `M` that is an `A`-module and an `R`-module (with `R`, `A` and `M` forming a scalar tower), scalar multiplication is commutative with respect to `R` and `A`. In other words, for any elements `r` in `R`, `a` in `A`, and `m` in `M`, the operations of scalar multiplication `a • r • m` and `r • a • m` are equal.

For any commutative semiring `R`, `R-algebra `A`, `A-module and `R-module `M` forming a scalar tower, the scalar multiplication of elements in `A` by elements in `R` commutes with the scalar multiplication of `M` by both `A` and `R`.


theorem le_div_iff_mul_le :
    ∀ {α : Type u} [inst : Group α] [inst_1 : LE α]
      [inst_2 : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α},
      a ≤ c / b ↔ a * b ≤ c :=
  by sorry

This theorem, `le_div_iff_mul_le`, is a statement about elements `a`, `b`, and `c` in a group `α` which is also equipped with a less than or equal to relation (`LE`). The theorem states that `a` is less than or equal to the division of `c` by `b` if and only if the product of `a` and `b` is less than or equal to `c`. The group `α` needs to have a certain property, specifically, it should be a `CovariantClass`. This implies that for any two elements `x` and `x_1` in the group, the swapped multiplication operation (`x * x_1` swapped to `x_1 * x`) should preserve the less than or equal to relation. In other words, if `x ≤ x_1`, then `x * x_1 ≤ x_1 * x`.

In a CovariantClass group equipped with a less than or equal to relation, `a ≤ b` if and only if `a * b ≤ c`, for some `c` in the group.


theorem Set.inv_mem_center₀ :
    ∀ {M : Type u_1} [inst : GroupWithZero M] {a : M}, a ∈ Set.center M → a⁻¹ ∈ Set.center M := by sorry

This theorem states that for any type `M` with a `GroupWithZero` structure, if an element `a` belongs to the center of `M` (i.e., it commutes with every other element in `M`), then its inverse `a⁻¹` also belongs to the center of `M`. In other words, if `a` is a central element in a group with zero, so is its inverse.

If `a` is a central element in a group with zero `M`, then its inverse `a⁻¹` is also a central element in `M`.


theorem NonUnitalStarSubalgebra.toNonUnitalSubalgebra_injective :
    ∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]
      [inst_3 : Star A], Function.Injective NonUnitalStarSubalgebra.toNonUnitalSubalgebra :=
  by sorry

This theorem states that for any commutative semiring `R` and any non-unital, non-associative semiring `A` that is also a module over `R` and posseses a star operation, the function `NonUnitalStarSubalgebra.toNonUnitalSubalgebra` is injective. In other words, if `A` and `B` are two non-unital star subalgebras of `R` such that their corresponding non-unital subalgebras are equal (as determined by `toNonUnitalSubalgebra`), then `A` and `B` were already equal.

Given commutative semirings `R`, a non-unital, non-associative semiring `A` over `R` with a star operation, if `toNonUnitalSubalgebra R A = toNonUnitalSubalgebra R B` for non-unital subalgebras `A` and `B`, then `A = B`.


theorem ENNReal.continuousAt_mul_const : ∀ {a b : ENNReal}, a ≠ ⊤ ∨ b ≠ 0 → ContinuousAt (fun x => x * a) b := by sorry

This theorem states that for any two extended nonnegative real numbers `a` and `b`, if `a` is different from positive infinity or `b` is different from zero, then the function that multiplies `x` by `a` is continuous at `b`. In terms of mathematics, if we have two numbers `a` and `b` from the set [0, ∞], and if `a` is any number other than ∞ or `b` is any number other than 0, then the function `f(x) = xa` is continuous at `x = b`.

For any `a in [0, ∞)` different from positive infinity and `b in (0, ∞]`, the function `f(x) = x * a` is continuous at `x = b`.


/-- Closure of a subsemigroup `S` equals `S`.  -/
theorem Subsemigroup.closure_eq : ∀ {M : Type u_1} [inst : Mul M] (S : Subsemigroup M), Subsemigroup.closure ↑S = S :=
  by sorry

The theorem `Subsemigroup.closure_eq` states that for any type `M` that has a multiplication operation and for any subsemigroup `S` of `M`, the closure of `S` is equal to `S` itself. Here, the closure of a set in a semigroup is the smallest subsemigroup that contains it. This means that applying the closure operation to a set that is already a subsemigroup does not modify the set.

For any subsemigroup `S` of a type `M` with multiplication operation, `S` is equal to its closure in `M`.


theorem Finset.exists_nat_subset_range : ∀ (s : Finset ℕ), ∃ n, s ⊆ Finset.range n := by sorry

The theorem `Finset.exists_nat_subset_range` states that for every finite set `s` of natural numbers, there exists a natural number `n` such that `s` is a subset of the set of natural numbers less than `n`. In other words, you can always find a number `n` such that all elements of the finite set `s` are less than `n`.

For every finite set of natural numbers, there exists a natural number that is an upper bound for the set.


theorem Mathlib.Analysis.SpecialFunctions.Gaussian._auxLemma.1 : ∀ (x : ℝ), x ^ 2 = x ^ 2 := by sorry

This theorem, `Mathlib.Analysis.SpecialFunctions.Gaussian._auxLemma.1`, states that for all real numbers `x`, the square of `x` is equal to the square of `x`. In mathematical terms, it is asserting that for every real number `x`, $x^2 = x^2$. This is an example of a reflexive property, which is true for all real numbers.


theorem Submonoid.eq_bot_iff_forall :
    ∀ {M : Type u_1} [inst : MulOneClass M] (S : Submonoid M), S = ⊥ ↔ ∀ x ∈ S, x = 1 := by sorry

This theorem states that for any type `M` having a multiplicative identity (i.e., an instance of `MulOneClass`), a submonoid `S` is equal to the trivial submonoid (denoted as `⊥` in Lean) if and only if all elements `x` in `S` are equal to the multiplicative identity (`1`). In other words, a submonoid is trivial if it contains only the multiplicative identity element.

A submonoid of a type with a multiplicative identity is the trivial submonoid if and only if all its elements equal the multiplicative identity.


theorem Bool.and_assoc : ∀ (a b c : Bool), (a && b && c) = (a && (b && c)) := by sorry

This theorem states that the logical AND operation (denoted by `&&`) is associative for boolean values. That is, for any three boolean values `a`, `b`, and `c`, the result of `a AND b` ANDed with `c` is the same as the result of `a` ANDed with the result of `b AND c`. In mathematical terms, this can be written as `(a ∧ b) ∧ c = a ∧ (b ∧ c)` where `∧` denotes the logical AND operation.

The logical AND operation is associative for boolean values, that is, `(a && b) && c = a && (b && c)` for all boolean values `a`, `b`, and `c`.


theorem inv_eq_iff_mul_eq_one : ∀ {G : Type u_3} [inst : Group G] {a b : G}, a⁻¹ = b ↔ a * b = 1 := by sorry

This theorem states that for any type `G` that has a group structure, and any two elements `a` and `b` of this group, the inverse of `a` is equal to `b` if and only if the product of `a` and `b` is the identity element of the group. In mathematical terms, this is saying that for all $a, b \in G$, we have $a^{-1} = b$ if and only if $a*b = 1$, where $*$ denotes the group operation, $1$ is the identity element, and $a^{-1}$ is the inverse of $a$.

For any group G and elements a, b in G, a^(-1) = b if and only if a * b = 1.


/-- A sort of triangle inequality.  -/
theorem Set.uIcc_subset_uIcc_union_uIcc :
    ∀ {α : Type u_1} [inst : LinearOrder α] {a b c : α}, Set.uIcc a c ⊆ Set.uIcc a b ∪ Set.uIcc b c := by sorry

The theorem titled "A sort of triangle inequality" states the following: For any type 'α' that has a linear order, and for any three elements 'a', 'b', and 'c' of type 'α', the set of elements that lie between 'a' and 'c' inclusive is a subset of the union of the set of elements that lie between 'a' and 'b' inclusive and the set of elements that lie between 'b' and 'c' inclusive. In other words, it is stating a property similar to the triangle inequality but for sets, where the set from 'a' to 'c' is contained in the combined set from 'a' to 'b' and 'b' to 'c'.

For any type 'α' with a linear order, the set of elements between 'a' and 'c' (inclusive) is contained in the union of the sets between 'a' and 'b' (inclusive) and between 'b' and 'c' (inclusive).


theorem MeasureTheory.Content.le_outerMeasure_compacts :
    ∀ {G : Type w} [inst : TopologicalSpace G] (μ : MeasureTheory.Content G) [inst_1 : R1Space G]
      (K : TopologicalSpace.Compacts G), (fun s => ↑(μ.toFun s)) K ≤ ↑(MeasureTheory.Content.outerMeasure μ) ↑K :=
  by sorry

This theorem states that for any topological space `G`, given a measure content `μ` and a compact set `K` within that space, the measure of `K` under `μ` is less than or equal to the outer measure of `K` with respect to the measure extended from `μ`. This holds under the additional assumption that `G` is a one-dimensional real vector space (`R1Space G`). This extends the measure content, which is defined only on compact sets, to an outer measure, which is defined on all subsets of the topological space.

For any one-dimensional real topological space `G` with measure content `μ` and compact set `K`, the content `μ(K)` is less than or equal to the outer measure `μ*` of `K` in `G`. (i.e., `μ(K) ≤ μ*(K)`)


theorem Complex.dist_eq_re_im : ∀ (z w : ℂ), dist z w = Real.sqrt ((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2) := by sorry

This theorem states that for any two complex numbers `z` and `w`, the distance between them is equal to the square root of the sum of the squares of their real and imaginary part differences. In other words, if you represent the complex numbers `z` and `w` in the complex plane, the geometric distance between them is calculated using Pythagoras' theorem: the square root of ((real part of z - real part of w) squared + (imaginary part of z - imaginary part of w) squared). This is the complex number version of the Euclidean distance formula.

The theorem asserts that the magnitude of the complex difference between two complex numbers z and w is equal to the square root of the sum of the squares of the real and imaginary part differences: |z - w| = sqrt((Re(z) - Re(w))^2 + (Im(z) - Im(w))^2).


theorem geom_sum_succ :
    ∀ {α : Type u} [inst : Semiring α] {x : α} {n : ℕ},
      (Finset.sum (Finset.range (n + 1)) fun i => x ^ i) = (x * Finset.sum (Finset.range n) fun i => x ^ i) + 1 :=
  by sorry

This theorem is called `geom_sum_succ` and it states that for any semiring `α`, any element `x` of `α`, and any natural number `n`, the sum of `x` to the power `i` for all `i` in the range from `0` to `n + 1` (inclusive of `0` but exclusive of `n + 1`) is equal to `x` times the sum of `x` to the power `i` for all `i` in the range from `0` to `n` (inclusive of `0` but exclusive of `n`), plus `1`. In mathematical notation, this theorem states that $\sum_{i=0}^{n} x^i = x \sum_{i=0}^{n-1} x^i + 1$ for all `x` in a semiring and all natural numbers `n`.

For any semiring `α` and natural number `n`, the sum of the elements `x` raised to the power `i` for `i` from `0` to `n` is equal to `x` times the sum of `x` raised to the power `i` for `i` from `0` to `n-1`, plus `1`. In mathematical notation: $\sum\_{i=0}^n x^i = x \sum\_{i=0}^{n-1} x^i + 1$ for all `x` in a semiring.


theorem Real.ennnorm_eq_ofReal : ∀ {r : ℝ}, 0 ≤ r → ↑‖r‖₊ = ENNReal.ofReal r := by sorry

This theorem states that for every real number `r` that is nonnegative (i.e., `r` is greater than or equal to 0), the nonnegative extended real number norm of `r`, denoted as `↑‖r‖₊`, is equal to the nonnegative extended real number obtained by applying the `ENNReal.ofReal` function to `r`. In other words, if you have a nonnegative real number `r`, its norm in the extended nonnegative real number domain is the same as if you just convert `r` to an extended nonnegative real number using the `ENNReal.ofReal` function.

For all non-negative real numbers `r`, `↑‖r‖₊ = ENNReal.ofReal r`.


theorem sdiff_top : ∀ {α : Type u_2} [inst : CoheytingAlgebra α] (a : α), a \ ⊤ = ⊥ := by sorry

This theorem states that for any element `a` of a given type `α` in a Coheyting Algebra, the set difference of `a` and the top element `⊤` is always the bottom element `⊥`. In other words, when we subtract the largest possible element from any element in a Coheyting Algebra, we always get the smallest possible element.

In a Cohecting Algebra, for all elements `a` in `α`, the set difference `a \ ⊤` equals the bottom element `⊥`.


Filter.HasBasis.inf'

theorem Filter.HasBasis.inf' :
    ∀ {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l l' : Filter α} {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.fst ∧ p' i.snd) fun i => s i.fst ∩ s' i.snd :=
  by sorry

The theorem `Filter.HasBasis.inf'` states that for any types `α`, `ι`, and `ι'`, and any two filters `l` and `l'` over `α`, indexed by `ι` and `ι'` respectively, along with index-based properties `p` and `p'` and index-based sets `s` and `s'` respectively, if `l` has a basis characterized by predicate `p` and sets `s`, and `l'` has a basis characterized by predicate `p'` and sets `s'`, then the infimum (greatest lower bound) of `l` and `l'` also has a basis. This basis is characterized by the predicate that corresponds to the conjunction of `p` and `p'`, and the set that corresponds to the intersection of `s` and `s'`.

If filters `l` and `l'` over type `α`, indexed by `ι` and `ι'` respectively, have bases characterized by predicates `p` and `p'` and sets `s` and `s'` respectively, then their infimum has a basis characterized by the conjunction of `p` and `p'` and the intersection of `s` and `s'`.


/-- For any `x : M ⧸ S` and any `0 < ε`, there is `m : M` such that `mk' S m = x`
and `‖m‖ < ‖x‖ + ε`.  -/
theorem norm_mk_lt :
    ∀ {M : Type u_1} [inst : SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S) {ε : ℝ},
      0 < ε → ∃ m, (QuotientAddGroup.mk' S) m = x ∧ ‖m‖ < ‖x‖ + ε :=
  by sorry

The theorem `norm_mk_lt` states that for any element `x` of the quotient group `M/S` of a given seminormed additve commutative group `M` by an additive subgroup `S`, and for any real number `ε` greater than zero, there exists an element `m` of `M` such that the image of `m` under the additive group homomorphism from `M` to `M/S` is `x` and the norm of `m` is less than the norm of `x` plus `ε`. This theorem essentially states that we can find an approximation of `x` in `M` that is arbitrarily close in norm.

For any seminormed additive commutative group `M` with subgroup `S`, and for any element `x` in `M/S` and real `ε > 0`, there exists an element `m` in `M` such that `norm m < norm x + ε` and `x = m % S`, where `%` denotes the quotient group operation.


theorem Mathlib.Data.Multiset.Basic._auxLemma.4 :
    ∀ {α : Type u_1} (a : α) {s t : Multiset α}, (a ::ₘ s = a ::ₘ t) = (s = t) := by sorry

This theorem states that for any type `α`, and any given element `a` of that type, as well as any two multisets `s` and `t` of the same type, the condition that prepending `a` to `s` results in the same multiset as prepending `a` to `t` is equivalent to `s` and `t` being the same multiset. Here, the operator `::ₘ` represents the operation of prepending an element to a multiset.

For any type `α` and elements `a` of type `α`, the multisets `s` and `t` of type `set α` are equal if and only if prepending `a` to `s` results in the same multiset as prepending `a` to `t`, denoted by `(a :: s) = (a :: t)`.


theorem MeasureTheory.OuterMeasure.boundedBy.proof_1 :
    ∀ {α : Type u_1} (m : Set α → ENNReal), ⨆ (_ : Set.Nonempty ∅), m ∅ = 0 := by sorry

The theorem `MeasureTheory.OuterMeasure.boundedBy.proof_1` states that for any function `m` from a set of elements of arbitrary type `α` to the extended nonnegative real numbers (usually denoted by `[0, ∞]`), the supremum of `m ∅` over the condition that the empty set is nonempty (which is never true) is equal to zero. In other words, in the context of measure theory, the measure of the empty set is always zero, regardless of how the measure function is defined.

The measure of the empty set, under any measure function, is equal to 0.


theorem List.reverse_prefix : ∀ {α : Type u_1} {l₁ l₂ : List α}, List.reverse l₁ <+: List.reverse l₂ ↔ l₁ <:+ l₂ := by
  sorry

This theorem, `List.reverse_prefix`, states that for all types `α` and two lists of type `α`, `l₁` and `l₂`, the reverse of `l₁` is a prefix of the reverse of `l₂` if and only if `l₁` is a suffix of `l₂`. Here, a prefix of a list is a sublist that appears at the start of the list, and a suffix of a list is a sublist that appears at the end of the list.

For all types α and lists l₁ and l₂ of length m and n with m ≤ n, respectively, the reverses of l₁ and l₂ are equal as suffixes if and only if l₁ is a suffix of l₂.


/-- In principle this could be an instance, but in practice it causes performance issues.  -/
theorem Subsingleton.of_oneHomClass :
    ∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : One M] [inst_1 : One N] [inst_2 : FunLike F M N]
      [inst_3 : Subsingleton M] [inst : OneHomClass F M N], Subsingleton F :=
  by sorry

The theorem `Subsingleton.of_oneHomClass` states that for all types `M`, `N`, and `F`, if `M` is a subsingleton type (i.e., it has at most one element), `M` and `N` have an identity element (`One M` and `One N`), `F` injectively maps `M` to `N` (`FunLike F M N`), and `F` is a homomorphism that preserves the identity element (`OneHomClass F M N`), then `F` is also a subsingleton type. In other words, if `M` has at most one element and `F` is a structure-preserving map from `M` to `N`, then there is at most one such map `F`.

If `M` is a subsingleton type with at most one element, `F` is a function from `M` to another type `N` that is injective, preserves identity elements, and is a homomorphism, then `N` is also a subsingleton type with at most one element.


theorem Subgroup.zpowers_eq_bot : ∀ {G : Type u_1} [inst : Group G] {g : G}, Subgroup.zpowers g = ⊥ ↔ g = 1 := by sorry

This theorem states that for any group `G` and any element `g` of `G`, the subgroup generated by `g` (denoted by `Subgroup.zpowers g`) is the trivial subgroup (denoted by `⊥`) if and only if `g` is the identity element of the group (`g = 1`). In other words, the subgroup generated by an element in a group is trivial if and only if that element is the identity element.

The subgroup generated by an element in a group is the trivial subgroup if and only if that element is the identity.


theorem nhds_sInf : ∀ {α : Type u} {s : Set (TopologicalSpace α)} {a : α}, nhds a = ⨅ t ∈ s, nhds a := by sorry

This theorem, denoted as `nhds_sInf`, states that for any type `α`, any set `s` of topological spaces on `α`, and any element `a` of type `α`, the neighborhood filter at `a` is equal to the infimum of the neighborhood filters at `a` for all topological spaces in `s`. In mathematical terms, the neighborhood filter at any point `a` in a given topological space is the same as the greatest lower bound (infimum) of the neighborhood filters at point `a` across all topological spaces within a specified set.

For any type `α`, any set `s` of topological spaces on `α`, and any `a` in `α`, the neighborhood filter of `a` is the infimum of the neighborhood filters of `a` in all topologies in `s`.


theorem AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty :
    ∀ {A : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} A] [inst_1 : CategoryTheory.Abelian A]
      {X : CategoryTheory.SimplicialObject A} (n : ℕ),
      CategoryTheory.Subobject.Factors (AlgebraicTopology.NormalizedMooreComplex.objX X n)
        (AlgebraicTopology.DoldKan.PInfty.f n) :=
  by sorry

The theorem `AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty` states that for any category `A` (with type `Type u_1`) which also has an Abelian structure, and any simplicial object `X` in `A`, the `PInfty.f` map of the Dold-Kan correspondence `PInfty` factors through the object `n` of the Normalized Moore Complex of `X`. Here, `n` is any natural number. This theorem is about the relationship between the Dold-Kan correspondence and the Normalized Moore Complex in the context of algebraic topology.

For any Abelian category `A` and simplicial object `X` in `A`, the `PInfty.f` map of the Dold-Kan correspondence `PInfty` factors through the object `n` of the Normalized Moore Complex of `X` for all natural numbers `n`.


Set.Nonempty.mul

theorem Set.Nonempty.mul :
    ∀ {α : Type u_2} [inst : Mul α] {s t : Set α}, Set.Nonempty s → Set.Nonempty t → Set.Nonempty (s * t) := by sorry

The theorem `Set.Nonempty.mul` states that for any two sets `s` and `t` of elements from a type `α` that supports multiplication, if both `s` and `t` are nonempty (i.e., each contains at least one element), then the set `s * t` is also nonempty. Here, `s * t` represents the set obtained by multiplying all possible pairs of elements where the first element is from `s` and the second element is from `t`.

If sets `s` and `t` over type `α` are nonempty, then their Cartesian product `s × t` is nonempty.


/-- Note that the inequality can be strict; e.g., the inclusion of an Abelian subalgebra of a
simple algebra.  -/
theorem LieIdeal.map_bracket_le :
    ∀ {R : Type u} {L : Type v} {L' : Type w₂} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
      [inst_3 : LieRing L'] [inst_4 : LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') {I₁ I₂ : LieIdeal R L},
      LieIdeal.map f ⁅I₁, I₂⁆ ≤ ⁅LieIdeal.map f I₁, LieIdeal.map f I₂⁆ :=
  by sorry

This theorem states that for any commutative ring `R`, Lie rings `L` and `L'`, and Lie algebras `L` and `L'` over `R`, given a Lie algebra homomorphism `f` from `L` to `L'` and any two Lie Ideals `I₁` and `I₂` of `L`, the image under `f` of the Lie bracket of `I₁` and `I₂` is a subset of the Lie bracket of the images under `f` of `I₁` and `I₂`. In other words, the mapping of the Lie bracket of two ideals through a homomorphism is always contained within the Lie bracket of the mapped ideals. Note, however, that this containment can be strict. For example, this can be the case when we consider the inclusion of an Abelian subalgebra in a simple algebra.

For any commutative ring `R`, given Lie algebra homomorphisms `f: L → L'` between Lie algebras `L` and `L'` over `R`, and Lie ideals `I₁` and `I₂` in `L`, we have `f[{[x, y] | x ∈ I₁, y ∈ I₂}] ⊆ {[f(x), f(y)] | x ∈ I₁, y ∈ I₂}`.


/-- If `C` is cofiltered or empty, and we have a functor `L : C ⥤ D` with a right adjoint,
then `D` is cofiltered or empty.
 -/
theorem CategoryTheory.IsCofilteredOrEmpty.of_left_adjoint :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.IsCofilteredOrEmpty C]
      {D : Type u₁} [inst_2 : CategoryTheory.Category.{v₁, u₁} D] {L : CategoryTheory.Functor C D}
      {R : CategoryTheory.Functor D C}, (L ⊣ R) → CategoryTheory.IsCofilteredOrEmpty D :=
  by sorry

The theorem `CategoryTheory.IsCofilteredOrEmpty.of_left_adjoint` states that if we have a category `C` that is either cofiltered or empty, and a functor `L` from `C` to another category `D` which has a right adjoint, then `D` is also either cofiltered or empty. Here, "cofiltered" refers to a property of a category that generalizes the notion of a directed set from order theory.

If `C` is a cofiltered or empty category and `L : C -> D` has a right adjoint, then `D` is also cofiltered or empty.


theorem PMF.tsum_coe_indicator_ne_top :
    ∀ {α : Type u_1} (p : PMF α) (s : Set α), ∑' (a : α), Set.indicator s (⇑p) a ≠ ⊤ := by sorry

The theorem `PMF.tsum_coe_indicator_ne_top` states that for any given probability mass function `p` and any set `s` of the same type `α`, the infinite sum of the `Set.indicator` function applied to `s` and `p` for all elements in `α` is never equal to infinity. The `Set.indicator` function here returns the value of `p` for an element if the element is in the set `s`, and `0` otherwise.

For any probability mass function `p` and set `s` of the same type, the sum of `Set.indicator` functions of `s` and `p` over all elements of the type is finite.


theorem Subtype.ne_of_val_ne : ∀ {α : Type u} {p : α → Prop} {a1 a2 : { x // p x }}, ↑a1 ≠ ↑a2 → a1 ≠ a2 := by sorry

This theorem, named 'Subtype.ne_of_val_ne', states that for any type 'α' and any predicate 'p' on 'α', if we have two elements 'a1' and 'a2' of the subtype of 'α' satisfying 'p', then if their underlying values in 'α' are not equal (i.e., `↑a1 ≠ ↑a2`), the original elements 'a1' and 'a2' are also not equal. In other words, distinct elements in the original type lead to distinct elements in the subtype.

For any type α and predicate p, if a1 and a2 are distinct elements in the subtype {x : α | p x}, then ↑a1 and ↑a2 are distinct in α.


Set.Finite.isBounded

theorem Set.Finite.isBounded :
    ∀ {α : Type u_2} [inst : Bornology α] {s : Set α}, Set.Finite s → Bornology.IsBounded s := by sorry

This theorem states that for any type `α` that has a bornology structure, and for any set `s` of elements of that type, if the set `s` is finite, then it is also bounded with respect to the bornology on `α`. In other words, finite sets are always bounded in any bornology.

For any type `α` with a bornology structure and any finite set `s` of elements from `α`, `s` is bounded in the bornology on `α`.


/-- If `a` semiconjugates a unit `x` to a unit `y`, then it semiconjugates `x⁻¹` to `y⁻¹`.  -/
theorem SemiconjBy.units_inv_right :
    ∀ {M : Type u_1} [inst : Monoid M] {a : M} {x y : Mˣ}, SemiconjBy a ↑x ↑y → SemiconjBy a ↑x⁻¹ ↑y⁻¹ := by sorry

The theorem `SemiconjBy.units_inv_right` states that for any type `M` that has a monoid structure, given elements `a`, `x`, and `y` from the group of units of `M`, if `a` semiconjugates `x` to `y` (i.e., if `a * x = y * a`), then `a` also semiconjugates the inverse of `x` to the inverse of `y` (i.e., `a * x⁻¹ = y⁻¹ * a`). Here, `x⁻¹` and `y⁻¹` denote the inverses of `x` and `y` in the group of units.

If `a` semiconjugates `x` to `y` in the group of units of a monoid `M`, then `a` semiconjugates the inverses of `x` and `y`, i.e., `a * x⁻¹ = y⁻¹ * a`.


theorem Mathlib.Algebra.Homology.ShortComplex.Exact._auxLemma.1 :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      (X : C), CategoryTheory.Limits.IsZero X = (CategoryTheory.CategoryStruct.id X = 0) :=
  by sorry

This theorem, from the Mathematics Library in Algebra and Homology, states that for any type `C` which forms a category (with the category theory structure) and has zero morphisms, and for any object `X` in this category, the object `X` is considered as a "zero object" (i.e., both initial and terminal object, or an object having certain properties with respect to morphisms) if and only if the identity morphism on `X` is the zero morphism.

A type `C` in a category with zero morphisms is the initial and terminal object if and only if its identity morphism is the zero morphism.


theorem extendFrom_eq :
    ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [inst_2 : T2Space Y]
      {A : Set X} {f : X → Y} {x : X} {y : Y},
      x ∈ closure A → Filter.Tendsto f (nhdsWithin x A) (nhds y) → extendFrom A f x = y :=
  by sorry

The theorem `extendFrom_eq` states that for any types `X` and `Y` endowed with topological space structures (where `Y` is also a T2 space, also known as a Hausdorff space), given a set `A` of `X`, a function `f` from `X` to `Y`, and elements `x` of `X` and `y` of `Y`, if `x` belongs to the closure of set `A` and the function `f` tends to `y` at `x` within the set `A` (as formalized by the `Filter.Tendsto` predicate), then the function `extendFrom A f`, which extends the function `f` from the set `A`, evaluates to `y` at `x`. In other words, the extended function at `x` is the limit of the original function `f` as we approach `x` within the set `A`, if such a limit exists at `y`.

If a function `f` from a set `A` in a T2 topological space `X` to a topological space `Y`, with closure `cl(A)`, tends to `y` at every limit point `x` in `cl(A)`, then `extendFrom A f` evaluates to `y` at `x`.


theorem PartialHomeomorph.symm_trans_self :
    ∀ {X : Type u_1} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
      (e : PartialHomeomorph X Y),
      PartialHomeomorph.trans (PartialHomeomorph.symm e) e ≈ PartialHomeomorph.ofSet e.target ⋯ :=
  by sorry

This theorem states that for any partial homeomorphism `e` from a topological space `X` to another topological space `Y`, the composition of the inverse of `e` (as given by `PartialHomeomorph.symm e`) followed by `e` itself is equivalent to the identity partial homeomorphism on the set `e.target`. This identity partial homeomorphism is given by `PartialHomeomorph.ofSet e.target`, where `e.target` is the target set of the partial homeomorphism `e`. In essence, this theorem highlights an important property of partial homeomorphisms: namely, that they preserve the topological structure when inverted and then composed with the original homeomorphism.

For any partial homeomorphism `e` from topological space `X` to `Y`, `PartialHomeomorph.symm e ∘ e = PartialHomeomorph.ofSet e.target`.


HasDerivWithinAt.congr

theorem HasDerivWithinAt.congr :
    ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
      [inst_2 : NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜},
      HasDerivWithinAt f f' s x → (∀ x ∈ s, f₁ x = f x) → f₁ x = f x → HasDerivWithinAt f₁ f' s x :=
  by sorry

The theorem `HasDerivWithinAt.congr` states that for any types `𝕜` (with `NontriviallyNormedField 𝕜`) and `F` (with `NormedAddCommGroup F` and `NormedSpace 𝕜 F`), given two functions `f` and `f₁` from `𝕜` to `F`, a derivative `f'` of type `F`, a point `x` of type `𝕜`, and a set `s` of points of type `𝕜`, if the function `f` has the derivative `f'` at the point `x` within the set `s`, and for all `x` in `s`, `f₁ x` equals `f x`, and `f₁ x` equals `f x`, then the function `f₁` also has the derivative `f'` at the point `x` within the set `s`. In other words, if two functions agree on a subset and have the same point value at a given point, then having a derivative at that point for one implies the same for the other.

If two real-valued functions agree on a set and have the same value at a point, and one of them has a derivative at that point within the set, then the other function also has the derivative of the same value at that point within the set.


/-- If the second of three points is weakly between the other two, the oriented angle at the
first point is zero.  -/
theorem Wbtw.oangle₂₁₃_eq_zero :
    ∀ {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] [hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)]
      [inst_4 : Module.Oriented ℝ V (Fin 2)] {p₁ p₂ p₃ : P}, Wbtw ℝ p₁ p₂ p₃ → EuclideanGeometry.oangle p₂ p₁ p₃ = 0 :=
  by sorry

This theorem states that for a vector space `V` with an inner product over the real numbers ℝ, that is associated to a metric space `P` and is of finite rank 2, if a point `p₂` is weakly between two other points `p₁` and `p₃` (i.e., `p₂` lies on the line segment joining `p₁` and `p₃`), then the oriented angle formed at the first point `p₁` by the three points is zero. This means the three points are co-linear with `p₂` in between `p₁` and `p₃`. The theorem is applicable under the condition that the vector space is oriented.

For an oriented finite-rank 2 real inner-product vector space with associated metric space, if point `p₂` lies on the line segment between `p₁` and `p₃`, then the oriented angle formed at `p₁` by the three points is zero, implying they are collinear.


theorem MeasureTheory.withDensity_mul₀ :
    ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ENNReal},
      AEMeasurable f μ → AEMeasurable g μ → μ.withDensity (f * g) = (μ.withDensity f).withDensity g :=
  by sorry

This theorem states that for any type `α`, given a measurable space `m0` of `α`, a measure `μ` on the measurable space, and two functions `f` and `g` mapping from `α` to the extended nonnegative real numbers (usually denoted [0, ∞]), if `f` and `g` are almost everywhere measurable with respect to measure `μ`, then the measure `μ` with density `f * g` is equal to the measure with density `g` derived from the measure `μ` with density `f`. In mathematical terms, it says that for almost every measurable functions `f` and `g`, the measure with density `f * g` is the same as composing the measure with density `f` and then with density `g`.

For measurable functions `f` and `g` mapping from a measurable space `(α, m0)` to the extended nonnegative real numbers, if `f` and `g` are almost everywhere equal almost everywhere with respect to measure `μ`, then the measure `μ` with density `f * g` equals the measure obtained by composing the measures with densities `f` and `g`: `μ {| α | f * g} = (μ {| α | f}) {| α | g}`.


DifferentiableWithinAt.pow

theorem DifferentiableWithinAt.pow :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [inst_3 : NormedRing 𝔸] [inst_4 : NormedAlgebra 𝕜 𝔸]
      {a : E → 𝔸}, DifferentiableWithinAt 𝕜 a s x → ∀ (n : ℕ), DifferentiableWithinAt 𝕜 (fun x => a x ^ n) s x :=
  by sorry

The theorem `DifferentiableWithinAt.pow` states that for any nontrivially normed field `𝕜`, normed addative commutative group `E`, normed ring `𝔸`, normed algebra of `𝕜` and `𝔸`, and set `s` of `E`, if a function `a` from `E` to `𝔸` is differentiable at a point `x` within `s`, then the function obtained by raising `a(x)` to the power `n` (for any natural number `n`), is also differentiable at `x` within `s`. The theorem represents the property that the power function preserves differentiability under certain conditions.

If `a: E -> 𝔸` is differentiable at `x` in the normed additive commutative group `E`, then the function `b(y) = a(y)^n` is differentiable at `x` in `E` for any natural number `n`.


theorem MeasureTheory.dominatedFinMeasAdditive_weightedSMul :
    ∀ {α : Type u_1} {F : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {x : MeasurableSpace α}
      (μ : MeasureTheory.Measure α), MeasureTheory.DominatedFinMeasAdditive μ (MeasureTheory.weightedSMul μ) 1 :=
  by sorry

The theorem `MeasureTheory.dominatedFinMeasAdditive_weightedSMul` states that for any measurable space `α` and any normed additive commutative group `F` that is a normed space over the real numbers, given a measure `μ` on `α`, the function `weightedSMul μ` is `DominatedFinMeasAdditive` with respect to `μ` and constant 1. In other words, the `weightedSMul` function, which scales elements of `F` by the real measure of a given set, respects the addition of measurable sets and the norm of the result for any measurable set is less than or equal to the measure of the set itself.

For any measurable space (α, Σ), normed additive commutative group (F, +) as a real normed space, and measure μ on Σ, the weighted scaling function weightedSMul μ : Σ → F is dominated finitely additive with respect to μ and constant 1.


theorem Real.volume_Ioi : ∀ {a : ℝ}, ↑↑MeasureTheory.volume (Set.Ioi a) = ⊤ := by sorry

The theorem `Real.volume_Ioi` states that for any real number `a`, the Lebesgue measure (or "volume") of the set of all real numbers greater than `a` (denoted by `Set.Ioi a`), is infinite. This is symbolized in Lean 4 by `⊤`. In other words, the size of any interval that begins at a real number and extends infinitely to the right is always considered to be infinite.

The Lebesgue measure of the set of real numbers strictly greater than a real number `a` is infinite.


theorem Fin.addCases_right :
    ∀ {m n : ℕ} {motive : Fin (m + n) → Sort u_1} {left : (i : Fin m) → motive (Fin.castAdd n i)}
      {right : (i : Fin n) → motive (Fin.natAdd m i)} (i : Fin n), Fin.addCases left right (Fin.natAdd m i) = right i :=
  by sorry

The theorem `Fin.addCases_right` states that for any two natural numbers `m` and `n`, any motive (a function or property) depending on a finite ordinal of size `m + n`, and any definitions of this motive for finite ordinals smaller than `m` and `n` respectively, if we have a finite ordinal `i` less than `n`, then applying the function `Fin.addCases` with the motive, the definitions, and `i` after adding `m` to it through `Fin.natAdd`, will yield the same result as directly applying the second definition (`right`) to `i`. In other words, `Fin.addCases` correctly applies the second definition when the input is in the range of the second definition.

For any natural numbers `m` and `n`, and any motive defined for finite ordinals less than `m + n`, applying `Fin.addCases` to the motive with definitions for `m` and `n` yields the same result as directly applying the definition for `n` to any finite ordinal less than `n`.


theorem Mathlib.Analysis.Normed.Group.AddCircle._auxLemma.10 :
    ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, (y ∈ Metric.closedBall x ε) = (dist y x ≤ ε) := by
  sorry

This theorem, `Mathlib.Analysis.Normed.Group.AddCircle._auxLemma.10`, asserts that for any pseudometric space `α`, any two elements `x` and `y` in this space, and any real number `ε`, the proposition that `y` belongs to the closed ball with center `x` and radius `ε` (i.e., `y ∈ Metric.closedBall x ε`) is equivalent to the statement that the distance between `y` and `x` is less than or equal to `ε` (i.e., `dist y x ≤ ε`). In other words, a point `y` is in the closed ball centered at `x` with radius `ε` in a pseudometric space if and only if the distance from `y` to `x` is not greater than `ε`.

In a pseudometric space, an element belongs to the closed ball centered at another element with a given radius if and only if the distance between them is less than or equal to the radius.


theorem Set.not_subsingleton_iff : ∀ {α : Type u} {s : Set α}, ¬Set.Subsingleton s ↔ Set.Nontrivial s := by sorry

The theorem `Set.not_subsingleton_iff` states that for every set `s` of a type `α`, the set `s` is not a subsingleton if and only if `s` is nontrivial. In mathematical terms, a set is defined as nontrivial (has at least two distinct elements) if and only if it is not a subsingleton (has at most one element). This theorem is providing a characterization of nontrivial sets in terms of the negation of being a subsingleton set.

A set is nontrivial if and only if it is not a subsingleton.


/-- A set `s` is not bounded below if and only if for each `x` there exists `y ∈ s` that is less
than `x`. A version for preorders is called `not_bddBelow_iff'`.  -/
theorem not_bddBelow_iff :
    ∀ {α : Type u_1} [inst : LinearOrder α] {s : Set α}, ¬BddBelow s ↔ ∀ (x : α), ∃ y ∈ s, y < x := by sorry

The theorem `not_bddBelow_iff` describes a condition for a set `s` of a certain type `α` under a linear order to not be bounded below. The theorem states that a set `s` is not bounded below if and only if for every element `x` of type `α`, there exists an element `y` in the set `s` such that `y` is less than `x`. This theorem provides a characterization for sets that lack a lower bound in the context of linear orders.

A set of type `α` under a linear order is not bounded below if and only if for every `x` in `α`, there exists `y` in the set such that `y` is strictly less than `x`.


theorem CategoryTheory.Functor.CommShift.isoAdd.proof_1 :
    ∀ {A : Type u_1} [inst : AddMonoid A] {a b : A}, a + b = a + b := by sorry

This theorem states that for any type 'A' that forms an addition monoid, and for any two elements 'a' and 'b' of type 'A', the sum of 'a' and 'b' is equal to the sum of 'a' and 'b'. Essentially, it's simply asserting that the addition operation is well-defined and consistent in such a context. It's a fundamental property of addition in any additive monoid, which includes structures like integers, real numbers, matrices, and more.

For any additive monoid type 'A' and elements 'a' and 'b' of type 'A', the operation + is commutative, i.e., a + b = b + a.


theorem map_finset_sup :
    ∀ {F : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Type u_5} [inst : SemilatticeSup α] [inst_1 : OrderBot α]
      [inst_2 : SemilatticeSup β] [inst_3 : OrderBot β] [inst_4 : FunLike F α β] [inst_5 : SupBotHomClass F α β] (f : F)
      (s : Finset ι) (g : ι → α), f (Finset.sup s g) = Finset.sup s (⇑f ∘ g) :=
  by sorry

The theorem `map_finset_sup` states that for all types `F`, `α`, `β`, and `ι`, given instances of `SemilatticeSup` and `OrderBot` for `α` and `β`, an injective function-like instance `FunLike F α β`, and a instance of `SupBotHomClass F α β` (which indicates that `F` is a function that preserves suprema and bottom elements), along with a function `f` of type `F`, a finite set `s` of type `Finset ι`, and a function `g` from `ι` to `α`, the supremum of the mapping of `f` over the set `s` after applying `g` (`f (Finset.sup s g)`) is the same as the supremum of the set `s` after applying the composition of `f` and `g` (`Finset.sup s (⇑f ∘ g)`).

In other words, the function `f` can be "pushed through" the `sup` operation, demonstrating a distributive property of `f` over the `sup` operation for finite sets.

For any type `F`, `α`, `β`, and `ι`, given instances of `SemilatticeSup` and `OrderBot` for `α` and `β`, an injective `FunLike F α β`, and a `SupBotHomClass F α β`, the application of `f ∘ g` to the supremum of a finite set `s` is equal to the supremum of the image of `f` over the set `s` after applying `g`.


theorem Associated.gcd_eq_right :
    ∀ {α : Type u_1} [inst : CancelCommMonoidWithZero α] [inst_1 : NormalizedGCDMonoid α] {m n : α},
      Associated m n → ∀ (k : α), gcd k m = gcd k n :=
  by sorry

The theorem `Associated.gcd_eq_right` states that for any type `α` that is a `CancelCommMonoidWithZero` and a `NormalizedGCDMonoid`, and for any elements `m` and `n` of `α` such that `m` is associated with `n`, the greatest common divisor (gcd) of `k` and `m` is equal to the gcd of `k` and `n` for any `k` in `α`. In other words, if `m` and `n` are associated in such a monoid, then computing the gcd by pairing either `m` or `n` with a given third element `k` yields the same result.

If `α` is a `CancelCommMonoidWithZero` and a `NormalizedGCDMonoid` where `m` and `n` are associated elements, then `gcd m k = gcd n k`.


theorem lt_div_iff_mul_lt :
    ∀ {α : Type u} [inst : Group α] [inst_1 : LT α]
      [inst_2 : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α},
      a < c / b ↔ a * b < c :=
  by sorry

The theorem `lt_div_iff_mul_lt` states that for any type `α` that has a group structure and a less-than (`<`) relation, and any elements `a`, `b`, `c` of `α`, where the multiplication operation on `α` is covariant with respect to the less-than relation, the inequality `a < c / b` is equivalent to `a * b < c`. Essentially, it relates division and multiplication in the context of inequalities, allowing us to rewrite or simplify expressions involving such inequalities.

For any type `α` with a group structure and a covariant multiplication with respect to the `<` relation, `a < c / b` if and only if `a * b < c`.


theorem sdiff_sdiff_comm :
    ∀ {α : Type u_2} [inst : GeneralizedCoheytingAlgebra α] {a b c : α}, (a \ b) \ c = (a \ c) \ b := by sorry

This theorem states that for any type `α` that forms a Generalized Coheyting Algebra, and for any three elements `a`, `b`, and `c` of type `α`, the operation of subtracting `b` from `a` and then subtracting `c` from the result is equal to the operation of subtracting `c` from `a` and then subtracting `b` from the result. This can be written in mathematical notation as `(a - b) - c = (a - c) - b`.

For any type `α` that is a Generalized Cohen algebra and for all `a, b, c` in `α`, `(a - b) - c = (a - c) - b`.


theorem Real.volume_Ico : ∀ {a b : ℝ}, ↑↑MeasureTheory.volume (Set.Ico a b) = ENNReal.ofReal (b - a) := by sorry

The `Real.volume_Ico` theorem states that for any pair of real numbers `a` and `b`, the measure (or "volume") of the left-closed right-open interval from `a` to `b` is equal to the extended non-negative real number representation of the difference `(b - a)`. In simpler terms, the length of the interval from `a` to `b` (where `a` is included and `b` is excluded) is equal to the absolute difference between `b` and `a`. This is subject to the condition that if the difference `(b - a)` is negative, it's considered as `0`.

The volume of the left-closed right-open real interval [a, b) is equal to the absolute difference b - a.


theorem PartENat.le_of_lt_add_one : ∀ {x y : PartENat}, x < y + 1 → x ≤ y := by sorry

This theorem states that for any two elements `x` and `y` of the type `PartENat` (which represents natural numbers with an additional element representing infinity), if `x` is less than `y + 1`, then `x` is also less than or equal to `y`. This is a commonly understood property in the domain of natural numbers and extends appropriately to this extended number system that includes infinity.

For any `x` and `y` in `PartENat`, if `x < y + 1`, then `x ≤ y`.