theorem Set.iInter₂_mono :
    ∀ {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s t : (i : ι) → κ i → Set α},
      (∀ (i : ι) (j : κ i), s i j ⊆ t i j) → ⋂ i, ⋂ j, s i j ⊆ ⋂ i, ⋂ j, t i j :=
  by sorry

The theorem `Set.iInter₂_mono` states that for any type `α`, and for any sorts `ι` and `κ` indexed by `ι`, given two families `(s : (i : ι) → κ i → Set α)` and `(t : (i : ι) → κ i → Set α)` of sets indexed by `ι` and `κ i`, if every set in the family `s` is a subset of the corresponding set in the family `t` (i.e., for every index `i` of type `ι` and `j` of type `κ i`, `s i j` is a subset of `t i j`), then the intersection of all sets in the family `s` is a subset of the intersection of all sets in the family `t`. In mathematical terms, if ∀(i : ι)(j : κ i), s i j ⊆ t i j, then ⋂ i, ⋂ j, s i j ⊆ ⋂ i, ⋂ j, t i j.

If for every index `i` of type `ι` and `j` of type `κ i`, the set `s i j` is a subset of `t i j`, then the intersection of all sets `s i j` in the family indexed by `ι` and `κ i` is a subset of the intersection of all sets `t i j`.


theorem Monotone.comp_antitone :
    ∀ {α : Type u} {β : Type v} {γ : Type w} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : Preorder γ] {g : β → γ}
      {f : α → β}, Monotone g → Antitone f → Antitone (g ∘ f) :=
  by sorry

The theorem `Monotone.comp_antitone` states that for any three types `α`, `β`, and `γ`, which have the preorder relation, and any two functions `g : β → γ` and `f : α → β`, if `g` is a monotone function and `f` is an antitone function, then the composition of `g` and `f` (i.e., `g ∘ f`) is an antitone function. In other words, if the function `g` preserves order and the function `f` reverses order, then their composition also reverses order.

If `f : α → β` is antitone and `g : β → γ` is monotone, then their composition `g ∘ f : α → γ` is antitone.


theorem Mathlib.ModelTheory.Semantics._auxLemma.7 :
    ∀ {L : FirstOrder.Language} {M : Type w} [inst : FirstOrder.Language.Structure L M] {α : Type u'} {l : ℕ}
      {φ ψ : FirstOrder.Language.BoundedFormula L α l} {v : α → M} {xs : Fin l → M},
      FirstOrder.Language.BoundedFormula.Realize (φ ⊓ ψ) v xs =
        (FirstOrder.Language.BoundedFormula.Realize φ v xs ∧ FirstOrder.Language.BoundedFormula.Realize ψ v xs) :=
  by sorry

This theorem is about the semantics of first-order logic in the context of Model Theory. Specifically, it states that for any first-order language `L`, a type `M` which is a structure of `L`, a type `α`, a natural number `l`, and two bounded formulas `φ` and `ψ` of `L` with `α` and `l`, the realization of the conjunction of `φ` and `ψ` with respect to a function `v` mapping `α` to `M` and a function `xs` mapping `Fin l` to `M` is equal to the conjunction of the realization of `φ` and the realization of `ψ` with respect to the same `v` and `xs`. In other words, the realization operation respects the logical conjunction operation in the sense that realizing a conjunction of two formulas is the same as realizing each formula individually and then taking the logical conjunction of the results.

For any first-order language L, structure M of L, type α, natural number l, bounded formulas φ and ψ of L with α and l, and functions v mapping α to M and xs mapping Fin l to M, the realization of φ ∧ ψ with respect to v and xs equals the realization of φ with respect to v and xs conjuncted with the realization of ψ with respect to v and xs.


theorem CategoryTheory.NatTrans.id_app :
    ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂}
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C),
      (CategoryTheory.CategoryStruct.id F).app X = CategoryTheory.CategoryStruct.id (F.obj X) :=
  by sorry

The theorem `CategoryTheory.NatTrans.id_app` states that for any objects `C` and `D` in the categories `u₁` and `u₂` respectively, and any functor `F` from `C` to `D` and object `X` in `C`, the application of the identity natural transformation on `F` at `X` is equal to the identity morphism on the object `F.obj X` in `D`. In other words, applying the identity natural transformation to a functor and then applying the result to an object is the same as first applying the functor to the object and then applying the identity morphism.

For any functors F : C -> D between categories C and D, and object X in C, the identity natural transformation on F at X equals the identity morphism on F(X) in D.


theorem zpow_add_one₀ :
    ∀ {G₀ : Type u_1} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → ∀ (n : ℤ), a ^ (n + 1) = a ^ n * a := by sorry

This theorem states that for any non-zero element 'a' of a type 'G₀' that forms a group with zero, and for any integer 'n', the 'n+1'-th power of 'a' is equal to the 'n'-th power of 'a' multiplied by 'a'. Essentially, this is the power rule for multiplication applied to groups with zero.

For any group element `a` in `G₀` different from zero and any integer `n`, the `(n + 1)`-th power of `a` equals the `n`-th power of `a` multiplied by `a`. In Lean notation: `(a ^ (n + 1)) = a ^ n * a`.


theorem Mathlib.Data.Finset.Sups._auxLemma.28 :
    ∀ {α : Type u_2} [inst : BooleanAlgebra α] {s : Finset α} {a : α}, (a ∈ Finset.compls s) = (aᶜ ∈ s) := by sorry

The theorem `Mathlib.Data.Finset.Sups._auxLemma.28` states that for any type `α` that forms a Boolean algebra, for any finite set `s` of `α`, and for any element `a` of `α`, `a` is an element of the complement of `s` if and only if the complement of `a` is an element of `s`. Here the complement operation is as per the Boolean algebra. This theorem establishes a relationship between an element and its complement in terms of their membership in a set and its complement.

For any Boolean algebra `α`, and finite set `s` from `α`, an element `a` is in the complement of `s` if and only if the complement of `a` is in `s`.


theorem lt_min_iff : ∀ {α : Type u} [inst : LinearOrder α] {a b c : α}, a < min b c ↔ a < b ∧ a < c := by sorry

This theorem states that for all types `α` that have a linear order, and for all instances of this type `a`, `b`, and `c`, `a` is less than the minimum of `b` and `c` if and only if `a` is less than `b` and `a` is less than `c`. In other words, an element is less than the minimum of two other elements if and only if it is less than both of these elements.

For all types with a linear order and elements `a`, `b`, `c`, if `a` is less than the minimum of `b` and `c`, then `a` is less than both `b` and `c`.


theorem MeasureTheory.condexp_of_not_le :
    ∀ {α : Type u_1} {F' : Type u_3} [inst : NormedAddCommGroup F'] [inst_1 : NormedSpace ℝ F']
      [inst_2 : CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'},
      ¬m ≤ m0 → MeasureTheory.condexp m μ f = 0 :=
  by sorry

This theorem is about the conditional expectation of a function in measure theory. It says that for any types `α` and `F'`, where `F'` is a normed addition commutative group, a real normed space, and a complete space, given two measurable spaces `m` and `m0` on `α`, a measure `μ` on `α`, and a function `f` from `α` to `F'`, if `m` is not a sub-σ-algebra of `m0`, then the conditional expectation of `f` with respect to the measure space `m` and measure `μ` is equal to 0. This is in line with the definition of the conditional expectation in Lean, which is defined to be 0 if `m` is not a sub-σ-algebra of `m0`.

If `μ` is a measure on the measurable spaces `(α, m)` and `(α, m0)`, `f : α → F'` is a measurable function, and `m` is not a sub-σ-algebra of `m0`, then the conditional expectation of `f` with respect to `(α, m, μ)` is equal to the zero function.


theorem MvPolynomial.rename_id :
    ∀ {σ : Type u_1} {R : Type u_4} [inst : CommSemiring R] (p : MvPolynomial σ R), (MvPolynomial.rename id) p = p := by
  sorry

This theorem states that for any multivariate polynomial `p` with variables indexed by the set `σ` and coefficients in a commutative semiring `R`, renaming the variables in `p` by the identity function results in the same polynomial `p`. The identity function here maps each variable to itself. In other words, if we do not change the names of the variables in a multivariate polynomial, the polynomial remains unchanged.

For any multivariate polynomial over a commutative semiring, renaming variables by the identity function leaves the polynomial unchanged.


ContinuousWithinAt.continuousAt

theorem ContinuousWithinAt.continuousAt :
    ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {s : Set α}
      {x : α}, ContinuousWithinAt f s x → s ∈ nhds x → ContinuousAt f x :=
  by sorry

This theorem states that for any given types `α` and `β` that have topological space structures, if a function `f` from `α` to `β` is continuous at a point `x` within a subset `s` of `α` (i.e., the value of `f` at `x` approaches the value of `f` at `x₀` as `x` tends to `x₀` within `s`), and if `s` is a neighborhood of `x` (i.e., `s` contains an open set around `x`), then `f` is continuous at `x` (i.e., the value of `f` at `x` approaches the value of `f` at `x₀` as `x` tends to `x₀` without restriction).

If a continuous function between topological spaces `f : α → β` is continuous at a point `x` in a neighborhood `s ⊆ α` of `x`, then `f` is continuous at `x`.


theorem PowerSeries.coeff_mul :
    ∀ {R : Type u_1} [inst : Semiring R] (n : ℕ) (φ ψ : PowerSeries R),
      (PowerSeries.coeff R n) (φ * ψ) =
        Finset.sum (Finset.antidiagonal n) fun p => (PowerSeries.coeff R p.1) φ * (PowerSeries.coeff R p.2) ψ :=
  by sorry

This theorem, `PowerSeries.coeff_mul`, states that for any semiring `R`, and any natural number `n` and power series `φ` and `ψ` over `R`, the `n`th coefficient of the product of `φ` and `ψ` is equal to the sum, over all pairs `(i, j)` such that `i + j = n`, of the product of the `i`th coefficient of `φ` and the `j`th coefficient of `ψ`. This can be seen as a statement about how multiplication works in the ring of power series: it reflects the pattern of combining terms that we see when multiplying out expressions like `(a + b)(c + d)`.

For any semiring `R`, natural number `n`, and power series `φ` and `ψ` over `R`, the `n`th coefficient of their product `φ * ψ` is equal to the sum, over all `i + j = n`, of the product of the `i`th coefficient of `φ` and the `j`th coefficient of `ψ`.


theorem zpow_mem :
    ∀ {M : Type u_5} {S : Type u_6} [inst : DivInvMonoid M] [inst_1 : SetLike S M] [hSM : SubgroupClass S M] {K : S}
      {x : M}, x ∈ K → ∀ (n : ℤ), x ^ n ∈ K :=
  by sorry

This theorem states that for any type `M` with a division and inversion operation structure (`DivInvMonoid`) and any type `S` that behaves like a set of `M` and forms a subgroup (`SubgroupClass`), given a subgroup `K` of type `S` and an element `x` in `M` that belongs to `K`, then the integer power (`n : ℤ`) of `x` (denoted by `x ^ n`) also belongs to `K`. This theorem essentially means that the power of any element in a subgroup remains in the subgroup, which is a fundamental property of mathematical groups.

If `M` is a `DivInvMonoid` type with subgroup `S` (as a `SubgroupClass`), then for any `x` in `M` in `S` and `n` in `ℤ`, `x ^ n` is also in `S`.


theorem PiNat.dist_eq_of_ne :
    ∀ {E : ℕ → Type u_1} {x y : (n : ℕ) → E n}, x ≠ y → dist x y = (1 / 2) ^ PiNat.firstDiff x y := by sorry

This theorem states that for any function `E` from natural numbers to some type `u_1`, and for any two functions `x` and `y` from natural numbers to `E`, if `x` and `y` are not equal, then the distance between `x` and `y` (using the distance metric in the product space `Π n, E n`) is equal to `(1 / 2)` raised to the power of the first index at which `x` and `y` differ (`PiNat.firstDiff x y`). Here, `PiNat.firstDiff x y` is defined as the first natural number `n` for which `x(n)` and `y(n)` are distinct.

For any function `E : Nat → Type_, and functions `x, y : Nat → E`, if `x ≠ y`, then the distance between `x` and `y` in the product space `Π n, E n` is equal to `(1 / 2) ^ (PiNat.firstDiff x y)`.


theorem List.mem_iff_get : ∀ {α : Type u_1} {a : α} {l : List α}, a ∈ l ↔ ∃ n, List.get l n = a := by sorry

This theorem states that for any given type `α`, an element `a` of type `α`, and a list `l` of elements of type `α`, `a` is a member of `l` if and only if there exists an index `n` such that the `n`-th element of the list `l` is `a`. Essentially, this theorem connects list membership with the existence of a specific index where the element can be found in the list.

For any type `α`, list `l` of length `n` in `α`, and element `a` in `α`, `a ∈ l` if and only if there exists an index `i` in `{0, 1, ..., n-1}` such that `l(i) = a`.


theorem ascPochhammer_map :
    ∀ {S : Type u} [inst : Semiring S] {T : Type v} [inst_1 : Semiring T] (f : S →+* T) (n : ℕ),
      Polynomial.map f (ascPochhammer S n) = ascPochhammer T n :=
  by sorry

The theorem `ascPochhammer_map` states that for any two types `S` and `T`, both having a `Semiring` structure, any ring homomorphism `f` from `S` to `T`, and any natural number `n`, the action of mapping the ascending Pochhammer polynomial of `S` and `n` across the ring homomorphism `f` is equivalent to the ascending Pochhammer polynomial of `T` and `n`. In other words, the ring homomorphism `f` commutes with the ascending Pochhammer polynomial operation.

Given types `S` and `T` with `Semiring` structures, a ring homomorphism `f` from `S` to `T`, and a natural number `n`, the ascending Pochhammer polynomials `(x + 1)^n` in `S` and `f((x + 1)^n)` in `T` are equal.


theorem ZeroHom.map_zero :
    ∀ {M : Type u_4} {N : Type u_5} [inst : Zero M] [inst_1 : Zero N] (f : ZeroHom M N), f 0 = 0 := by sorry

This theorem states that, for any two types `M` and `N` that have a zero element (denoted by the `Zero M` and `Zero N` instances respectively), any `ZeroHom` (zero-preserving function) `f` from `M` to `N` will map the zero element of `M` to the zero element of `N`. In mathematical terms, given a zero-preserving function `f`, we have `f(0) = 0`.

If `f` is a zero-preserving function from type `M` to type `N` with zero elements `Zero M` and `Zero N`, then `f (Zero M) = Zero N`.


theorem CategoryTheory.ShortComplex.Exact.liftFromProjective_comp :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Abelian C]
      {S : CategoryTheory.ShortComplex C} (hS : CategoryTheory.ShortComplex.Exact S) {P : C} (f : P ⟶ S.X₂)
      [inst_2 : CategoryTheory.Projective P] (hf : CategoryTheory.CategoryStruct.comp f S.g = 0),
      CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.Exact.liftFromProjective hS f hf) S.f = f :=
  by sorry

This theorem is about the category theory in mathematics. It states that for any category `C` that is abelian, any short complex `S` that is exact, and any morphism `f` from a projective object `P` to the second object of the short complex `S`, if the composition of `f` with the morphism `g` from the short complex `S` is zero, then the composition of the lift from the projective `P` to `S` via `f` and the morphism `f` from the short complex `S` is equal to `f`.

In other words, we can imagine the objects and morphisms as forming a diagram, and this theorem is saying that if a certain square in this diagram commutes (i.e., the two different paths from `P` to `S.X₂` via `S.g` and `S.f` are equal), then a corresponding triangle also commutes.

In an abelian category `C`, if a short exact sequence `S` has a projective object `P` and a morphism `f : P -> S.X₁` such that `S.g . f = 0`, then the lift of `f` to `S` is `f`.


theorem LaurentPolynomial.single_eq_C_mul_T :
    ∀ {R : Type u_1} [inst : Semiring R] (r : R) (n : ℤ),
      (fun₀ | n => r) = LaurentPolynomial.C r * LaurentPolynomial.T n :=
  by sorry

This theorem states that for any semiring `R`, and for any element `r` of `R` and integer `n`, the function mapping `n` to `r` is equivalent to the multiplication of the constant Laurent polynomial of `r` and the Laurent polynomial generated by `T` raised to the power `n`. In mathematical terms, this can be written as f(n) = r = C(r) * T^n, where f is a function, C is the constant Laurent polynomial, and T is the Laurent polynomial. This establishes a relationship between the definition of a constant Laurent polynomial, a Laurent polynomial generated by `T`, and a mapping from an integer to a constant of the semiring.

For any semiring `R`, and any constant `r` in `R` and integer `n`, the function `r : ℤ → R` is equal to the constant Laurent polynomial `C(r)` multiplied by `T^n`, where `T` is a Laurent polynomial.


theorem Primrec.nat_rec' :
    ∀ {α : Type u_1} {β : Type u_2} [inst : Primcodable α] [inst_1 : Primcodable β] {f : α → ℕ} {g : α → β}
      {h : α → ℕ × β → β},
      Primrec f → Primrec g → Primrec₂ h → Primrec fun a => Nat.rec (g a) (fun n IH => h a (n, IH)) (f a) :=
  by sorry

The theorem `Primrec.nat_rec'` asserts that for any types `α` and `β` that are primitive codable, and for any functions `f: α → ℕ`, `g: α → β`, `h: α → ℕ × β → β`, if `f`, `g`, and `h` are all primitive recursive functions, then the function that takes an input `a` from type `α` and applies the natural number recursion method on `f(a)`, starting from `g(a)` and with recursion function `h(a, (n, IH))`, is also a primitive recursive function. Here, `Nat.rec (g a) (fun n IH => h a (n, IH)) (f a)` is a way to represent recursion in the natural number `f(a)`, where `g(a)` is the base case and `h(a, (n, IH))` is the recursive step applied to each preceding result `IH` and its index `n`.

If `f:` α -> Nat, `g:` α -> β, and `h:` α -> Nat x β are primitive recursive functions, then the function `Nat.rec (g a) (fun n IH => h a (n, IH)) (f a)` is also primitive recursive.


theorem CategoryTheory.ShortComplex.LeftHomologyData.wi_assoc :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      {S : CategoryTheory.ShortComplex C} (self : CategoryTheory.ShortComplex.LeftHomologyData S) {Z : C}
      (h : S.X₃ ⟶ Z),
      CategoryTheory.CategoryStruct.comp self.i (CategoryTheory.CategoryStruct.comp S.g h) =
        CategoryTheory.CategoryStruct.comp 0 h :=
  by sorry

This theorem belongs to the category theory and it is about the composition of morphisms in a category. Specifically, it states that for any category `C` with zero morphisms, given a short complex `S` in `C` and a `LeftHomologyData` instance for `S`, the composition of the morphism `i` from the `LeftHomologyData` with the composition of the morphism `g` from the short complex `S` and any morphism `h` from `S.X₃` to any object `Z` in `C` is equal to the composition of the zero morphism with `h`. This is an important property that characterizes how morphisms interact with the zero morphism in category theory.

For any short exact sequence S in a category C with zero morphisms and a given LeftHomologyData instance, the composition of i from the LeftHomologyData, g from S, and any morphism h from S.X₃ to Z equals the composition of the zero morphism with h.


theorem Stream'.Seq.nil_append : ∀ {α : Type u} (s : Stream'.Seq α), Stream'.Seq.append Stream'.Seq.nil s = s := by
  sorry

This theorem states that for all sequences `s` of any type `α`, appending the empty sequence to `s` results in `s` itself. In other words, it states that the operation of appending an empty sequence to any given sequence has no effect, and the original sequence remains unchanged. This mirrors the behavior we would expect in standard list or string concatenation where an element concatenated with an empty element doesn't change the original element.

For all sequences `s` of type `α`, appending the empty sequence to `s` yields `s`.


/-- **Alias** of the forward direction of `isOfFinOrder_iff_pow_eq_one`. -/
theorem IsOfFinOrder.exists_pow_eq_one :
    ∀ {G : Type u_1} [inst : Monoid G] {x : G}, IsOfFinOrder x → ∃ n, 0 < n ∧ x ^ n = 1 := by sorry

This theorem, **Alias of the forward direction of `isOfFinOrder_iff_pow_eq_one`**, states that for any type `G` which forms a monoid, and any element `x` of `G` that satisfies the predicate `IsOfFinOrder`, there exists a natural number `n` greater than 0 such that raising `x` to the power of `n` equals the identity element of the monoid. In other words, if `x` has finite order in the monoid, then there's a positive integer `n` such that `x` to the power `n` is the identity.

For any monoid `G` and element `x` of finite order in `G`, there exists a positive integer `n` such that `x^n = 1_G`, where `1_G` is the identity element of `G`.


theorem Subgroup.map_mono :
    ∀ {G : Type u_1} [inst : Group G] {N : Type u_5} [inst_1 : Group N] {f : G →* N} {K K' : Subgroup G},
      K ≤ K' → Subgroup.map f K ≤ Subgroup.map f K' :=
  by sorry

The theorem `Subgroup.map_mono` states that for any two subgroups `K` and `K'` of a group `G`, and a monoid homomorphism `f` from `G` to another group `N`, if `K` is a subset of `K'` (i.e., `K` is less than or equal to `K'`), then the image of `K` under `f` is also a subset of the image of `K'` under `f`. In other words, the mapping of subgroups through a monoid homomorphism preserves the subset relation.

If `K` is a subgroup of `K'` in a group `G` and `f` is a monoid homomorphism from `G` to a group `N`, then `f(K)` is a subgroup of `f(K')`.


/-- For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓤 β` as a
filter basis.  -/
theorem UniformFun.hasBasis_nhds :
    ∀ (α : Type u_1) (β : Type u_2) [inst : UniformSpace β] (f : UniformFun α β),
      Filter.HasBasis (nhds f) (fun V => V ∈ uniformity β) fun V => {g | (f, g) ∈ UniformFun.gen α β V} :=
  by sorry

The theorem named `UniformFun.hasBasis_nhds` states that for every uniform space `β` and every function `f` from one type `α` to `β`, the neighborhood filter of `f` (denoted by `𝓝 f`) is generated by a basis consisting of sets of the form `{g | ∀ x, (f x, g x) ∈ V}`, where `V` is drawn from the uniformity of `β` (denoted by `𝓤 β`). In other words, these sets form a filter basis for the neighborhood filter of `f`, capturing the idea that `f` and `g` are "close" if their values are "close" for every input.

The neighborhood filter of a function from type `α` to uniform space `β` is generated by sets of the form `{g : β | ∀ x : α, (f x, g x) ∈ V}`, where `V` is from the uniformity of `β`.


theorem Mathlib.Computability.TuringMachine._auxLemma.8 :
    ∀ {α : Sort u_1} {p : α → Prop} {a' : α}, (∀ (a : α), a' = a → p a) = p a' := by sorry

This theorem, from the Turing Machine section of the Mathlib library in Lean, states that for any type 'α' and any predicate 'p' on this type, and for any element 'a' of type 'α', the proposition that 'p' is true for all elements 'a' that are equal to 'a', is equivalent to the proposition that 'p' is true for 'a'. In simpler terms, it means that if a predicate is true for all elements equal to a given element, then it is true for that given element.

For any type 'α' and predicate 'p' on 'α', the equality 'a = x' implies 'p(a) <-> ∀(y : α), a = y -> p(y)' (where '<->' denotes logical equivalence).


theorem nonZeroDivisors_le_comap_nonZeroDivisors_of_injective :
    ∀ {M : Type u_2} {M' : Type u_3} {F : Type u_7} [inst : MonoidWithZero M] [inst_1 : MonoidWithZero M']
      [inst_2 : FunLike F M M'] [inst_3 : NoZeroDivisors M'] [inst_4 : MonoidWithZeroHomClass F M M'] (f : F),
      Function.Injective ⇑f → nonZeroDivisors M ≤ Submonoid.comap f (nonZeroDivisors M') :=
  by sorry

This theorem states that for all structures `M` and `M'` of type `Type`, where `M` and `M'` are monoids with zero, and for all structures `F` of type `Type` that behave functionally from `M` to `M'` (i.e., `F` is a monoid homomorphism from `M` to `M'`), if `f` is an injective function of type `F` and `M'` has no zero divisors, then the submonoid of non-zero divisors of `M` is a subset of or equal to the preimage of the submonoid of non-zero divisors of `M'` under the function `f`. In other words, if two elements of `M` have the same image in `M'` under `f`, and `M'` does not have zero divisors, then those two elements must be the same in `M`.

If `M` and `M'` are monoids with zero, `F : M → M'` is a monoid homomorphism, `f : F → M'` is an injection, and `M'` has no zero divisors, then the preimage of the submonoid of non-zero divisors of `M'` under `f` is contained in the submonoid of non-zero divisors of `M`.


/-- A subring is closed under multiplication.  -/
theorem Subring.mul_mem : ∀ {R : Type u} [inst : Ring R] (s : Subring R) {x y : R}, x ∈ s → y ∈ s → x * y ∈ s := by
  sorry

This theorem states that a subring is closed under multiplication. In other words, for any ring `R`, and any subring `s` of `R`, if `x` and `y` are any elements of `s`, then the product `x * y` is also an element of `s`. This property is one of the defining characteristics of a subring in ring theory, a branch of abstract algebra.

For any ring `R` and subring `s` of `R`, the product of any two elements in `s` belongs to `s`.


theorem MeasureTheory.AEEqFun.coeFn_sup :
    ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α}
      [inst_1 : TopologicalSpace β] [inst_2 : SemilatticeSup β] [inst_3 : ContinuousSup β] (f g : α →ₘ[μ] β),
      ↑(f ⊔ g) =ᶠ[MeasureTheory.Measure.ae μ] fun x => ↑f x ⊔ ↑g x :=
  by sorry

The theorem `MeasureTheory.AEEqFun.coeFn_sup` states that for any two almost everywhere equal functions `f` and `g` from a measurable space `α` to a topological space `β` equipped with a semilattice structure and a continuous supremum operation, and for any measure `μ` on `α`, the supremum of `f` and `g` (denoted by `f ⊔ g`) is also an almost everywhere equal function. Moreover, this supremum function is almost everywhere equal to the function that sends each point `x` in `α` to the supremum of the values of `f` and `g` at `x` (denoted by `↑f x ⊔ ↑g x`). The equality is almost everywhere with respect to the measure `μ`, meaning that the set of points in `α` where the two functions do not agree has measure zero.

For measurable functions $f$ and $g$ from a measurable space to a topological semilattice with a continuous supremum operation, if $f$ and $g$ are almost everywhere equal with respect to some measure, then their supremum is almost everywhere equal to the pointwise supremum, and this equality holds almost everywhere with respect to the measure.


/-- Shows that `IntFractPair.stream` has the sequence property, that is once we return `none` at
position `n`, we also return `none` at `n + 1`.
 -/
theorem GeneralizedContinuedFraction.IntFractPair.stream_isSeq :
    ∀ {K : Type u_1} [inst : LinearOrderedField K] [inst_1 : FloorRing K] (v : K),
      Stream'.IsSeq (GeneralizedContinuedFraction.IntFractPair.stream v) :=
  by sorry

This theorem states that for any linearly ordered field `K` that also has the structure of a floor ring, the function `GeneralizedContinuedFraction.IntFractPair.stream` applied to any element `v` of `K`, produces a sequence in the sense defined by `Stream'.IsSeq`. That is, if for some natural number `n`, the `n`-th element of the stream is `none`, then the `(n + 1)`-th element is also `none`. In other words, once the stream returns `none`, it will continue to do so for all subsequent indices.

For any linearly ordered field `K` that is also a floor ring, the sequence produced by `GeneralizedContinuedFraction.IntFractPair.stream` on any element `v` of `K` is a stream in the sense of `Stream'.IsSeq`, meaning that if the `n`-th element is `none`, then all subsequent elements are also `none`.


theorem Filter.eventually_prod_iff :
    ∀ {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} {p : α × β → Prop},
      (∀ᶠ (x : α × β) in f ×ˢ g, p x) ↔
        ∃ pa, (∀ᶠ (x : α) in f, pa x) ∧ ∃ pb, (∀ᶠ (y : β) in g, pb y) ∧ ∀ {x : α}, pa x → ∀ {y : β}, pb y → p (x, y) :=
  by sorry

This theorem states that for any two types `α` and `β`, any two filters `f` and `g` on these types respectively, and any property `p` of pairs of elements from `α` and `β`, the property `p` holds eventually (i.e., outside of a finite number of exceptions) for pairs `(x, y)` drawn from the filters `f` and `g` if and only if there exist properties `pa` and `pb` such that `pa` holds eventually for elements `x` drawn from `f`, `pb` holds eventually for elements `y` drawn from `g`, and for any `x` that satisfies `pa` and any `y` that satisfies `pb`, the pair `(x, y)` satisfies `p`.

For filters `f` on type `α` and `g` on type `β`, and any property `p` of pairs from `α × β`, `p` holds eventually for pairs drawn from `f` and `g` if and only if there exist properties `pa` and `pb` such that `pa` holds eventually for elements of `f` and `pb` holds eventually for elements of `g`, and for all `x` satisfying `pa` and `y` satisfying `pb`, `(x, y)` satisfies `p`.


HasFDerivWithinAt.neg

theorem HasFDerivWithinAt.neg :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
      {f' : E →L[𝕜] F} {x : E} {s : Set E}, HasFDerivWithinAt f f' s x → HasFDerivWithinAt (fun x => -f x) (-f') s x :=
  by sorry

This theorem states that for any nontrivially normed field `𝕜`, normed additive commutative groups `E` and `F`, and normed spaces `E` over `𝕜` and `F` over `𝕜`, if a function `f : E → F` has a Frechet derivative `f'` at a point `x` within a set `s` of `E`, then the negation function `(fun x => -f x)` also has a Frechet derivative which is the negation of `f'` within the same set `s` at the same point `x`. In the language of calculus, if a function has a derivative at a point within a set, then the derivative of the negative of the function at that point is the negative of the original derivative.

If `f : E → F` has a Frechet derivative `f'` at `x` in `s` in a nontrivially normed field `𝕜`, then `-f` has a Frechet derivative `-f'` at `x` in `s`.


theorem Subsingleton.eq_univ_of_nonempty :
    ∀ {α : Type u_1} [inst : Subsingleton α] {s : Set α}, Set.Nonempty s → s = Set.univ := by sorry

The theorem `Subsingleton.eq_univ_of_nonempty` states that for any type `α` that has the `Subsingleton` property (meaning that there can be at most one element of that type), any nonempty set `s` of that type must be equivalent to the universal set on `α`. In other words, if a type can have at most one element and we have a set of that type which contains at least one element, then this set must contain every element of the type, which means that it is equal to the universal set on `α`.

If `α` is a type with at most one element and `s : set α` is nonempty, then `s` equals the universal set on `α`.


theorem uniformContinuous_subtype_val :
    ∀ {α : Type ua} {p : α → Prop} [inst : UniformSpace α], UniformContinuous Subtype.val := by sorry

The theorem `uniformContinuous_subtype_val` states that for any type `α` equipped with a uniform space structure and any predicate `p` on `α`, the function `Subtype.val` is uniformly continuous. In other words, when the inputs of `Subtype.val` (which are elements of a subtype defined by `p`) are sufficiently close to each other, their images (which are elements of the base type `α`) are also close, regardless of where these inputs are located in the subtype.

For any uniform space `(α, d)` and subtype `s : α → Prop` with predicate `p := ⦃x : α⦄. s x`, the function `Subtype.val : s → α` is uniformly continuous.


theorem not_injective_infinite_finite :
    ∀ {α : Sort u_4} {β : Sort u_5} [inst : Infinite α] [inst : Finite β] (f : α → β), ¬Function.Injective f := by sorry

The theorem `not_injective_infinite_finite` states that for any two types `α` and `β`, if `α` is infinite and `β` is finite, then any function `f` from `α` to `β` cannot be injective. In other words, there does not exist a function that maps distinct elements of an infinite set to distinct elements of a finite set.

If `α` is infinite and `β` is finite, then no function from `α` to `β` can be injective.


theorem EReal.coe_strictMono : StrictMono Real.toEReal := by sorry

The theorem `EReal.coe_strictMono` asserts that the function `Real.toEReal`, which is the canonical inclusion from the real numbers to the extended real numbers, is strictly monotone. In other words, if `a` and `b` are real numbers with `a < b`, then `Real.toEReal a < Real.toEReal b`.

The canonical inclusion `Real.toEReal` from the real numbers to the extended real numbers is a strictly increasing function.


theorem Subgroup.normalCore_le : ∀ {G : Type u_1} [inst : Group G] (H : Subgroup G), Subgroup.normalCore H ≤ H := by
  sorry

This theorem states that for any group `G` and any subgroup `H` of `G`, the normal core of `H` is a subgroup of `H`. In other words, every element of the normal core of `H` is also an element of `H`. The normal core of a subgroup `H` is defined as the largest normal subgroup of `G` that is contained in `H`. Here, a normal subgroup is a subgroup where each element `a` satisfies the property that for any element `b` from the group, the element resulting from the operation `b * a * b⁻¹` (where `*` denotes group multiplication and `b⁻¹` denotes the inverse of `b`) is also in `H`.

The normal core of a subgroup is a subgroup containing only its normal elements.


/-- In an R₁ space, the filters `coclosedCompact` and `cocompact` are equal.  -/
theorem Filter.coclosedCompact_eq_cocompact :
    ∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : R1Space X], Filter.coclosedCompact X = Filter.cocompact X :=
  by sorry

This theorem states that in any R₁ space, which is a type of topological space, the filters `coclosedCompact` and `cocompact` are equivalent. The `coclosedCompact` filter is generated by the complements of closed compact sets, while the `cocompact` filter is generated by the complements of compact sets. In simpler terms, it says that in an R₁ space, taking the complements of all compact sets or the complements of all closed compact sets generate the same filters.

In an R₁ space, the filters generated by the complements of compact sets and the complements of closed compact sets are equivalent.


theorem Mathlib.MeasureTheory.Integral.IntegrableOn._auxLemma.2 :
    ∀ {α : Type u_1} {β : Type u_2} [inst : NormedAddCommGroup β] {x : MeasurableSpace α} {f : α → β},
      MeasureTheory.Integrable f 0 = True :=
  by sorry

This theorem states that for any types `α` and `β`, where `β` is a normed additive commutative group, any measurable function `f` from `α` to `β` is integrable with respect to the zero measure. In mathematical terms, the theorem says that $\int_{0}^{} |f(a)| da$ is finite for every function `f` from `α` to `β`, where `α` is any type and `β` is a normed additive commutative group.

Every measurable function from a type `α` to a normed additive commutative group `β` has finite integral of absolute value.


theorem Set.Finite.isClosed_biUnion :
    ∀ {X : Type u} {α : Type u_1} [inst : TopologicalSpace X] {s : Set α} {f : α → Set X},
      Set.Finite s → (∀ i ∈ s, IsClosed (f i)) → IsClosed (⋃ i ∈ s, f i) :=
  by sorry

This theorem states that for any type `X` equipped with a topology (i.e., `X` is a topological space) and any type `α`, if `s` is a finite set of elements of type `α` and `f` is a function from `α` to the set of subsets of `X`, then if every subset `f(i)` where `i` is an element of `s` is closed, the union of all such subsets `f(i)` for `i` in `s` is also closed. In simpler terms, it says that the union of a finite collection of closed sets in a topological space is itself a closed set.

In a topological space, the union of a finite number of closed sets is closed.


Set.EqOn.mono

theorem Set.EqOn.mono :
    ∀ {α : Type u_1} {β : Type u_2} {s₁ s₂ : Set α} {f₁ f₂ : α → β}, s₁ ⊆ s₂ → Set.EqOn f₁ f₂ s₂ → Set.EqOn f₁ f₂ s₁ :=
  by sorry

The theorem `Set.EqOn.mono` states that for any two types `α` and `β`, two sets `s₁` and `s₂` of type `α`, and two functions `f₁` and `f₂` from `α` to `β`, if `s₁` is a subset of `s₂` and the functions `f₁` and `f₂` are equal on `s₂` (meaning that for each element `x` in `s₂`, `f₁(x)` is equal to `f₂(x)`), then `f₁` and `f₂` are also equal on `s₁`. In other words, the equality of two functions on a larger set implies their equality on any subset of that set.

If `s₁ ⊆ s₂`, `f₁ : α → β`, `f₂ : α → β`, and `∀ x ∈ s₂, f₁ x = f₂ x`, then `∀ x ∈ s₁, f₁ x = f₂ x`.


/-- `∨` distributes over `∧` (on the left).  -/
theorem or_and_left : ∀ {a b c : Prop}, a ∨ b ∧ c ↔ (a ∨ b) ∧ (a ∨ c) := by sorry

This theorem states that the logical "OR" operation is distributive over the logical "AND" operation when "OR" is on the left. In other words, for any three propositions `a`, `b`, and `c`, the proposition "`a` OR (`b` AND `c`)" is equivalent to the proposition "(`a` OR `b`) AND (`a` OR `c`)".

The theorem asserts that `(p ∨ (q ∧ r)) ≡ (p ∨ q) ∧ (p ∨ r)` for all propositions `p`, `q`, and `r`.


theorem not_isMax_of_lt : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, a < b → ¬IsMax a := by sorry

The theorem named `not_isMax_of_lt` states that for any preorder set `α` and any two elements `a` and `b` of this set, if `a` is strictly less than `b`, then `a` cannot be a maximal element of the set. In other words, there exists another element `b` that is strictly greater than `a`, thus `a` can't be the greatest element in the set `α`.

If `α` is a preorder set and `a` is strictly less than `b` in `α`, then `a` cannot be a maximal element of `α`.


/-- `g` converges to `f.smallSets` if for all `s ∈ f`, eventually we have `g x ⊆ s`.  -/
theorem Filter.tendsto_smallSets_iff :
    ∀ {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {f : α → Set β},
      Filter.Tendsto f la (Filter.smallSets lb) ↔ ∀ t ∈ lb, ∀ᶠ (x : α) in la, f x ⊆ t :=
  by sorry

This theorem states that a function `g` (denoted as `f` in the theorem) from a type `α` to a set of type `β` converges to the 'small sets' of a filter `lb` (i.e., the filter containing all powersets of members of `lb`), with respect to a filter `la` on `α`, if and only if for every set `t` in `lb`, there is an 'eventually' condition in `la` such that for all `x` of type `α`, `g(x)` is a subset of `t`. Here, 'eventually' refers to the property that the condition holds for all `x` in the filter `la` from some point forward.

A function `g` from type `α` to sets of type `β` converges to the filter `lb` with respect to filter `la` if and only if for every set `t` in `lb`, there exists an eventually true condition in `la` such that for all `x` in `α`, `g(x)` is a subset of `t`.


Primrec.eq

theorem Primrec.eq : ∀ {α : Type u_1} [inst : Primcodable α] [inst_1 : DecidableEq α], PrimrecRel Eq := by sorry

The theorem `Primrec.eq` asserts that for any type `α` that is primcodable and has decidable equality, the equality relation `Eq` is a primitive recursive relation. In other words, it's stating that for all `α`, if `α` is a type that can be encoded in a primitive way and can decide whether two elements are equal or not, then the standard equality function from `α` to `α` to `Prop` can be determined in a primitive recursive manner. This implies that the function `decide ∘ Eq : α → α → Bool` is primitive recursive.

For any primcodable and decidably equal type α, the standard equality relation Eq : α → α → Prop is a primitive recursive relation.


theorem Set.Iic_subset_Iio : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, Set.Iic a ⊆ Set.Iio b ↔ a < b := by sorry

The theorem `Set.Iic_subset_Iio` states that for any type `α` with a preorder relation, and any two elements `a` and `b` of this type, the set of all elements less than or equal to `a` (`Set.Iic a`) is a subset of the set of all elements strictly less than `b` (`Set.Iio b`) if and only if `a` is strictly less than `b`. In other words, all elements less than or equal to `a` are also strictly less than `b` if and only if `a` is strictly less than `b`.

For any preordered type `α` and elements `a` and `b` of `α`, `Set.Iic a ⊆ Set.Iio b` if and only if `a` is strictly less than `b`.


theorem Mathlib.Data.Finset.Basic._auxLemma.47 :
    ∀ {α : Sort u_1} {p : α → Prop} {a' : α}, (∀ (a : α), a = a' → p a) = p a' := by sorry

This is a theorem about properties in the context of mathematical logic and set theory. Specifically, it states that for any type `α`, and any property `p` that a member of `α` might have, and a particular member `a'` of `α`, if we know that `p` holds for any member `a` of `α` if `a` is equal to `a'`, then it is equivalent to say `p` holds specifically for `a'`. In other words, if the property is true for all elements identical to `a'`, then it is equivalent to saying the property is true for `a'`.

For any type `α` and property `p` on `α`, if for all `a` in `α`, `p(a)` if and only if `a = a'`, then `p(a')` is equivalent to `∀x ∈ α, x = a' �plies p(x)`.


IsLocalization.surj

/-- The `algebraMap` is surjective  -/
theorem IsLocalization.surj :
    ∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) {S : Type u_2} [inst_1 : CommSemiring S]
      [inst_2 : Algebra R S] [inst_3 : IsLocalization M S] (z : S),
      ∃ x, z * (algebraMap R S) ↑x.2 = (algebraMap R S) x.1 :=
  by sorry

The theorem `IsLocalization.surj` states that for any commutative semiring `R`, any submonoid `M` of `R`, any commutative semiring `S`, and any algebra structure from `R` to `S`, if `S` is a localization of `R` at `M`, then for every element `z` in `S`, there exists an element `x` such that `z` multiplied by the image of the second component of `x` under the algebra map from `R` to `S` equals the image of the first component of `x` under the same algebra map. In other words, every element `z` of `S` is in the image of the multiplication by some element of `M` of the algebra map from `R` to `S`.

If `S` is a localization of commutative semiring `R` at submonoid `M` via an algebra structure from `R` to `S`, then for every `z` in `S`, there exists `x` in `R` such that `z = x * m * s`, where `m` is in `M` and `s` is the image of `x` under the algebra map from `R` to `S`.


theorem MeasureTheory.integral_dirac :
    ∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [hE : CompleteSpace E]
      [inst_2 : MeasurableSpace α] [inst_3 : MeasurableSingletonClass α] (f : α → E) (a : α),
      ∫ (x : α), f x ∂MeasureTheory.Measure.dirac a = f a :=
  by sorry

The theorem `MeasureTheory.integral_dirac` states that for any normed add commutative group `E`, which is also a normed space over the real numbers ℝ and is complete, and for any measurable singleton space `α`, if we have a function `f : α → E` and an element `a` in `α`, then the integral of `f` with respect to the dirac measure at `a` is simply the value of the function at `a`. In other words, the integration of a function over a dirac measure at a point is just the evaluation of the function at that point.

For any complete normed add commutative group `E` over ℝ, any measurable singleton space `α`, and function `f : α → E` with an element `a` in `α`, the integral of `f` with respect to the dirac measure at `a` equals `f a`.


theorem Nat.lt_add_of_pos_left : ∀ {k n : ℕ}, 0 < k → n < k + n := by sorry

This theorem, `Nat.lt_add_of_pos_left`, states that for any two natural numbers `k` and `n`, if `k` is greater than zero, then `n` is less than the sum of `k` and `n`. In other words, if you have a positive natural number `k`, adding it to any other natural number `n` will result in a number that is strictly larger than `n`.

For all natural numbers `k` and `n`, if `k > 0` then `n < k + n`.


theorem BoxIntegral.BoxAdditiveMap.sum_partition_boxes :
    ∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} {I : BoxIntegral.Box ι}
      (f : BoxIntegral.BoxAdditiveMap ι M I₀),
      ↑I ≤ I₀ →
        ∀ {π : BoxIntegral.Prepartition I},
          BoxIntegral.Prepartition.IsPartition π → (Finset.sum π.boxes fun J => f J) = f I :=
  by sorry

This theorem states that for any type `ι`, and `M` which is an additive commutative monoid, given a 'top-attached' box `I₀` of type `ι` and a box `I` of the same type, and a box-additive map `f` from `I₀` to `M`, if `I` is less than or equal to `I₀`, then for any prepartition `π` of `I` which is also a partition, the sum of `f` applied to each box in `π` is equal to `f` applied to `I`. In other words, the value of the box-additive map on the entire box `I` is equal to the sum of its values on the smaller boxes of the partition.

For any additive commutative monoid M, type ι, top-attached box I₀ of type ι, box I with I ≤ I₀, and box-additive map f from I₀ to M, if π is a partition of I that is also a prepartition, then ∑(π)(i) (f i) = f (∑(ι) i), where ∑(π)(i) denotes the sum of boxes in partition π that map to i under the canonical bijection from I to π.


/-- A monotone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsInf` to the `Filter.liminf` of the image if the function is continuous at the `limsInf`
(and the filter is bounded from above and below).  -/
theorem Monotone.map_limsInf_of_continuousAt :
    ∀ {R : Type u_4} {S : Type u_5} [inst : ConditionallyCompleteLinearOrder R] [inst_1 : TopologicalSpace R]
      [inst_2 : OrderTopology R] [inst_3 : ConditionallyCompleteLinearOrder S] [inst_4 : TopologicalSpace S]
      [inst_5 : OrderTopology S] {F : Filter R} [inst_6 : Filter.NeBot F] {f : R → S},
      Monotone f →
        ContinuousAt f (Filter.limsInf F) →
          autoParam (Filter.IsBounded (fun x x_1 => x ≤ x_1) F) _auto✝ →
            autoParam (Filter.IsBounded (fun x x_1 => x ≥ x_1) F) _auto✝¹ → f (Filter.limsInf F) = Filter.liminf f F :=
  by sorry

The theorem `Monotone.map_limsInf_of_continuousAt` states that for any monotone function `f` mapping between two conditionally complete linear ordered spaces `R` and `S`, if `f` is continuous at the `limsInf` of a specific filter `F` in `R` and `F` is both upper and lower bounded (these are automatic parameters), then the image of the `limsInf` of `F` under `f` equals the `liminf` of the image of `F` under `f`. This illustrates the preservation of liminf under the action of a continuous monotone function.

If `f` is a monotone, continuous at the limit of an upper and lower bounded filter in a conditionally complete linear ordered space `R`, then `f` preserves the liminf, that is, `liminf (map f F) = f (limsInf F)`.


/-- See also `Nat.card_subgroup`. -/
theorem Fintype.card_zmultiples :
    ∀ {G : Type u_1} [inst : AddGroup G] [inst_1 : Fintype G] {x : G},
      Fintype.card ↥(AddSubgroup.zmultiples x) = addOrderOf x :=
  by sorry

This theorem states that for any additive group `G` that is also a finite type, and for any element `x` of `G`, the cardinality of the subgroup generated by `x` (also known as `zmultiples` of `x`) is equal to the order of the element `x`. Here, the order of `x` (defined by `addOrderOf x`) is the smallest positive integer `n` such that `n` times `x` equals to the identity element of the group (0 in the case of additive groups), or is defined to be `0` if `x` has infinite order.

For any finite additive group G and its element x, the cardinality of the subgroup generated by x equals the order of x.


theorem StrictMono.comp_strictMonoOn :
    ∀ {α : Type u} {β : Type v} {γ : Type w} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : Preorder γ] {g : β → γ}
      {f : α → β} {s : Set α}, StrictMono g → StrictMonoOn f s → StrictMonoOn (g ∘ f) s :=
  by sorry

The theorem `StrictMono.comp_strictMonoOn` states that for any types `α`, `β`, and `γ` that have a pre-order relation, if `g` is a strictly monotonic function from `β` to `γ`, and `f` is a strictly monotonic function on a set `s` of `α`, then the composition function `(g ∘ f)` is also strictly monotonic on the same set `s`. In other words, the composition of a strictly monotonic function and a function that is strictly monotonic on a certain set inherits the strict monotonicity on this set.

If `f` is strictly monotonic on a set `s` of `α` and `g` is a strictly monotonic function from `β` to `γ`, then the composition `(g ∘ f)` is strictly monotonic on `s`.


/-- If each partial application of `f` is injective, and images of `s` under those partial
applications are disjoint (but not necessarily distinct!), then the size of `t` divides the size of
`finset.image₂ f s t`.  -/
theorem Finset.card_dvd_card_image₂_right :
    ∀ {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},
      (∀ a ∈ s, Function.Injective (f a)) →
        Set.PairwiseDisjoint ((fun a => Finset.image (f a) t) '' ↑s) id → t.card ∣ (Finset.image₂ f s t).card :=
  by sorry

The theorem `Finset.card_dvd_card_image₂_right` states that for any types `α`, `β`, `γ` and a function `f : α → β → γ`, and finite sets `s : Finset α` and `t : Finset β`, given that each partial application of the function `f` with elements in `s` is injective, and the images of `s` under those partial applications are pairwise disjoint, then the size of `t` divides the size of the image of the binary function `f` applied to `s` and `t`. In simpler terms, if we apply `f` to each pair of elements from `s` and `t` and the results do not overlap, then the number of elements in `t` will be a divisor of the total number of outcomes.

If `f : α → β → γ`, `s : Finset α`, `t : Finset β` satisfy that `f` restricted to pairs of elements from `s` and `t` is injective and the images are disjoint, then `#t` divides `#(image (λ x, λ y, f x y) s)`.


RelEmbedding.isWellOrder

theorem RelEmbedding.isWellOrder :
    ∀ {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop},
      r ↪r s → ∀ [inst : IsWellOrder β s], IsWellOrder α r :=
  by sorry

The theorem `RelEmbedding.isWellOrder` states that for any two types `α` and `β` with relations `r` and `s` respectively, if `r` is embeddable into `s` (denoted by `r ↪r s`), then given `s` is a well-order relation on `β` (captured by `IsWellOrder β s`), it implies `r` is a well-order relation on `α` (represented by `IsWellOrder α r`). In essence, it means that the well-orderedness is preserved under the relative embedding from `r` to `s`.

If relation `r` on type `α` is embeddable into relation `s` on type `β` and `s` is a well-order relation on `β`, then `r` is a well-order relation on `α`.


theorem Submodule.mem_inf :
    ∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
      {p q : Submodule R M} {x : M}, x ∈ p ⊓ q ↔ x ∈ p ∧ x ∈ q :=
  by sorry

This theorem states that for any semiring `R`, additive commutative monoid `M`, and `R`-module `M`, given two submodules `p` and `q` of `M` and an element `x` of `M`, `x` is in the intersection of `p` and `q` if and only if `x` is in `p` and `x` is in `q`. In other words, the intersection of two submodules is defined as the set of elements that belong to both submodules.

For any semiring `R`, additive commutative monoid `M`, and `R`-module `M`, the intersection of submodules `p` and `q` of `M` is equal to the set of elements `x` in `M` such that `x` belongs to both `p` and `q` (i.e., `x` is in `p` and `x` is in `q`).


theorem List.nodup_reverse : ∀ {α : Type u} {l : List α}, List.Nodup (List.reverse l) ↔ List.Nodup l := by sorry

This theorem states that for any list `l` of elements of any type `α`, the list `l` has no duplicates if and only if the reverse of list `l` has no duplicates. This implies that reversing a list does not change its property of having no duplicate elements.

For any list `l` of elements from type `α`, the list `l` has no duplicates if and only if its reverse `rev l` has no duplicates.


theorem IsOpenMap.nhds_le :
    ∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
      IsOpenMap f → ∀ (x : X), nhds (f x) ≤ Filter.map f (nhds x) :=
  by sorry

The theorem `IsOpenMap.nhds_le` states that for any types `X` and `Y`, and any function `f` from `X` to `Y`, if `f` is an open map (i.e., the image of any open set in `X` under `f` is an open set in `Y`), then for any element `x` in `X`, the neighborhood filter of `f(x)` in `Y` is less than or equal to the filter obtained by mapping the neighborhood filter of `x` in `X` through the function `f`. This means that every neighborhood of `f(x)` contains the image under `f` of some neighborhood of `x`.

For any open map f from X to Y and any x in X, the neighborhood filter of f(x) in Y is contained in the filter obtained by mapping neighborhoods of x in X through f.


theorem Polynomial.coeff_zero : ∀ {R : Type u} [inst : Semiring R] (n : ℕ), Polynomial.coeff 0 n = 0 := by sorry

This theorem states that for any semiring `R` and natural number `n`, the coefficient of `X^n` in the zero polynomial (which is a polynomial with all coefficients equal to zero) is zero. In more mathematical terms, if `p(X)` represents the zero polynomial in `R[X]`, the theorem says that the coefficient of `X^n` in `p(X)` is zero for any `n ∈ ℕ`.

For any semiring `R` and natural number `n`, the coefficient of `X^n` in the zero polynomial `p(X)` over `R` is equal to zero.


theorem mul_sub_left_distrib :
    ∀ {α : Type u} [inst : NonUnitalNonAssocRing α] (a b c : α), a * (b - c) = a * b - a * c := by sorry

This theorem, named `mul_sub_left_distrib`, states that for all types `α` that have a structure of a non-unital, non-associative ring (denoted by `NonUnitalNonAssocRing α`), the standard property of left distributivity of multiplication over subtraction holds. In more formal mathematical terms, for any three elements `a`, `b` and `c` of type `α`, the product of `a` and the difference of `b` and `c` is equal to the difference of the product of `a` and `b` and the product of `a` and `c`. This can be represented in LaTeX math as: \(a \cdot (b - c) = a \cdot b - a \cdot c\).

For any non-unital, non-associative ring type `α`, the left distributivity of multiplication over subtraction holds: `a * (b - c) = a * b - a * c`.


theorem Cardinal.mul_eq_left : ∀ {a b : Cardinal.{u_1}}, Cardinal.aleph0 ≤ a → b ≤ a → b ≠ 0 → a * b = a := by sorry

The theorem `Cardinal.mul_eq_left` states that for any two cardinal numbers `a` and `b`, if the cardinal number `a` is greater than or equal to `ℵ₀` (aleph null, the smallest infinite cardinal) and `b` is less than or equal to `a` and `b` is not zero, then the product of `a` and `b` is equal to `a`. This theorem captures an important property of cardinal arithmetic at infinity, where the multiplication of an infinite cardinal number with another cardinal that is not zero and is less than or equal to the infinite cardinal, results in the original infinite cardinal number.

For any cardinal numbers `a` greater than or equal to `ℵ₀` and `b` less than or equal to `a` with `b` not zero, `a * b = a`.


theorem ENNReal.coe_one : ↑1 = 1 := by sorry

This theorem states that the coercion of the integer 1 to the extended non-negative real number type (`ENNReal`) results in the same number, 1. In other words, when 1 is interpreted as an extended non-negative real number, it still remains 1.

The theorem asserts that the coercion of the integer 1 to the type of extended non-negative real numbers is the identity function.


theorem finsum_def' :
    ∀ {M : Type u_7} {α : Sort u_8} [inst : AddCommMonoid M] (f : α → M),
      finsum f =
        if h : Set.Finite (Function.support (f ∘ PLift.down)) then Finset.sum (Set.Finite.toFinset h) fun i => f i.down
        else 0 :=
  by sorry

This theorem states that for any function `f` from a given sort `α` to a type `M` that forms an additive commutative monoid, the finite sum of `f x` for `x` in the support of `f` is defined as follows: if the support of the function `f`, when considering `x` as a lifted proposition, is a finite set, then the sum is the total of `f x.down` for `x` in the finite set equivalent to the support of `f`; otherwise, if the set is not finite, the sum is defined to be zero. Here, the `support` of a function is the set of points where the function is non-zero, `PLift.down` is a function that extracts a value from a lifted proposition, and `Finset.sum` sums up all the values of a function at points in a finite set.

For any additive commutative monoid-valued function `f` on a sort `α`, if the support of `f` is finite, then the sum of `f x` over the support is equal to the sum of `f x.down`; otherwise, the sum is zero.


theorem CategoryTheory.MorphismProperty.subset_isoClosure :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P : CategoryTheory.MorphismProperty C),
      P ⊆ CategoryTheory.MorphismProperty.isoClosure P :=
  by sorry

This theorem states that for any category `C`, and for any property `P` of morphisms in `C`, the property `P` is a subset of its closure by isomorphisms. This means that any morphism that satisfies property `P` also satisfies the property obtained by taking the closure of `P` by isomorphisms. The closure by isomorphisms of a property `P` consists of those morphisms that can be obtained by taking a morphism satisfying `P` and precomposing or postcomposing it with an isomorphism.

For any property `P` of morphisms in a category `C`, the morphisms with property `P` are contained in the closure of `P` under isomorphisms.


/-- Constant zero function has sum `0` -/
theorem hasSum_zero :
    ∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α], HasSum (fun x => 0) 0 := by
  sorry

This theorem states that for any types `α` and `β`, where `α` is a commutative monoid under addition and is also equipped with a topology, the sum of the constant zero function (which maps every element of type `β` to `0` in `α`) is `0`. In other words, if you add up all the zeros produced by this function, the result is zero.

For any commutative monoid `α` with topology and function `zero : β → α`, the sum of `zero` is equal to the additive identity `0` in `α`.


theorem sub_toIcoDiv_zsmul_mem_Ico :
    ∀ {α : Type u_1} [inst : LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α),
      b - toIcoDiv hp a b • p ∈ Set.Ico a (a + p) :=
  by sorry

This theorem states that for any given type `α` which is a linear ordered additive commutative group and satisfies the Archimedean property, and for any positive `p` of type `α`, and any `a` and `b` of type `α`, the result of subtracting `p` times the unique integer (which is given by the function `toIcoDiv`) from `b` lies in the semi-open interval from `a` to `a + p`. In other words, after we subtract `p` multiplied by this unique integer from `b`, the resulting value will always be greater than or equal to `a` and less than `a + p`.

For any Archimedean linear ordered additive commutative group `α`, positive `p` in `α`, and `a, b` in `α`, the result of subtracting `p` times the unique integer `n` with `b = a + p * n` is in the semi-open interval `(a, a + p)`.


theorem FirstOrder.Language.Hom.range_eq_map :
    ∀ {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [inst : FirstOrder.Language.Structure L M]
      [inst_1 : FirstOrder.Language.Structure L N] (f : FirstOrder.Language.Hom L M N),
      FirstOrder.Language.Hom.range f = FirstOrder.Language.Substructure.map f ⊤ :=
  by sorry

This theorem states that for any first-order language `L`, and any types `M` and `N` with structures of `L`, the range of a homomorphism `f : M → N` is equivalent to the image of the entire structure `M` (denoted by the top element `⊤`) under the homomorphism `f`. In other words, applying `f` to all elements of `M` gives us the same set as the range of `f`.

For any first-order language L, any types M and N with structures, and any homomorphism f : M -> N, the range of f is equal to the image of the entire structure M under f (denoted by f(⊤)).


theorem Monotone.le_map_iSup₂ :
    ∀ {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_7} [inst : CompleteLattice α]
      [inst_1 : CompleteLattice β] {f : α → β},
      Monotone f → ∀ (s : (i : ι) → κ i → α), ⨆ i, ⨆ j, f (s i j) ≤ f (⨆ i, ⨆ j, s i j) :=
  by sorry

The theorem `Monotone.le_map_iSup₂` states that for any two types `α` and `β`, where both `α` and `β` are complete lattices, any function `f` from `α` to `β`, and any family of sets `s` indexed by two sorts `ι` and `κ`, if the function `f` is monotone, then the supremum (denoted by ⨆) of the function `f` applied to the elements of the sets in the family `s` is less than or equal to the function `f` applied to the supremum of the elements of the sets in the family `s`. In other words, the function `f` preserves the order of the suprema.

For any complete lattices α and β, monotone function f : α → β, and families s : ι → Set κ with supremum s.sup, f (⨆ x in s : α) ≤ ⨆ x in s : α → β. (f(x))


theorem finprod_one : ∀ {M : Type u_2} {α : Sort u_4} [inst : CommMonoid M], (finprod fun x => 1) = 1 := by sorry

The theorem `finprod_one` states that for any type `M` with a commutative monoid structure and any type `α`, the product, as determined by the `finprod` function, of the constant function that always yields 1 (i.e., `(fun x => 1)`) is 1. This is akin to stating that the product of 1, taken over any number of elements (possibly infinite), remains 1 in the context of a commutative monoid.

For any commutative monoid `M` and type `α`, the constant function `(fun x => 1)` has a product of 1 according to the `finprod` function.


theorem CategoryTheory.Limits.coprod.map_id_id :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C}
      [inst_1 : CategoryTheory.Limits.HasBinaryCoproduct X Y],
      CategoryTheory.Limits.coprod.map (CategoryTheory.CategoryStruct.id X) (CategoryTheory.CategoryStruct.id Y) =
        CategoryTheory.CategoryStruct.id (X ⨿ Y) :=
  by sorry

The theorem `CategoryTheory.Limits.coprod.map_id_id` states that for any category `C` and any two objects `X` and `Y` in `C` that have a binary coproduct, the mapping from the binary coproduct of `X` and `Y` to itself using the identity morphisms of `X` and `Y` is the same as the identity morphism of the binary coproduct of `X` and `Y`. In other words, in the category theory context, mapping a binary coproduct to itself with identity morphisms doesn't change the binary coproduct, which is a fundamental property of identity mappings.

Given a category `C` and objects `X` and `Y` with a binary coproduct, the identity mappings of `X` and `Y` to their coproduct are equal to the coproduct of the identity mappings of `X` and `Y`.


theorem LieSubmodule.Quotient.is_quotient_mk :
    ∀ {R : Type u} {L : Type v} {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : AddCommGroup M]
      [inst_3 : Module R M] [inst_4 : LieRingModule L M] {N : LieSubmodule R L M} (m : M),
      Quotient.mk'' m = LieSubmodule.Quotient.mk m :=
  by sorry

This theorem states that for any commutative ring `R`, any Lie ring `L`, and any additive commutative group `M` that is also a module over `R` and a Lie ring module over `L`, given any Lie submodule `N` of `M`, the Quotient of any element `m` of `M` by the equivalence relation on `M` induced by `N` using the `Quotient.mk''` function is the same as the result of applying the `LieSubmodule.Quotient.mk` function to `m`. In other words, the `Quotient.mk''` function and `LieSubmodule.Quotient.mk` function are equivalent for Lie submodules.

For any commutative ring `R`, Lie ring `L`, additive commutative group `M` that is both an `R`-module and a Lie ring module over `L`, and any Lie submodule `N` of `M`, the Quotient of any `m` in `M` by the equivalence relation induced by `N` is equivalent to the Lie submodule quotient of `m` by `N`.


theorem Nat.sub_add_comm : ∀ {n m k : ℕ}, k ≤ n → n + m - k = n - k + m := by sorry

This theorem, `Nat.sub_add_comm`, states that for any three natural numbers `n`, `m`, and `k`, if `k` is less than or equal to `n`, then adding `m` to `n` and then subtracting `k` is the same as subtracting `k` from `n` first and then adding `m`. In mathematical notation, this is stating that if $k \leq n$, then $n + m - k = n - k + m$.

For all natural numbers $n$, $m$, and $k$ with $k \leq n$, we have $n + m = n + m - k + k = n - k + m$.


theorem mdifferentiableAt_const :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)
      {M : Type u_4} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [inst_6 : SmoothManifoldWithCorners I M]
      {E' : Type u_5} [inst_7 : NormedAddCommGroup E'] [inst_8 : NormedSpace 𝕜 E'] {H' : Type u_6}
      [inst_9 : TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [inst_10 : TopologicalSpace M']
      [inst_11 : ChartedSpace H' M'] [inst_12 : SmoothManifoldWithCorners I' M'] {x : M} {c : M'},
      MDifferentiableAt I I' (fun x => c) x :=
  by sorry

This theorem states that for any types 𝕜, E, H, M, E', H', and M', where 𝕜 is a non-trivially normed field, E and E' are normed add commutative groups, H, M, H', and M' are topological spaces, with M and M' also charted spaces H and H' respectively, and given the model with corners I for 𝕜, E and H, I' for 𝕜, E' and H', if these satisfy the condition of a smooth manifold with corners, then for any point x in M and constant c in M', the function that maps every point in M to the constant c is differentiable at x.

For any smooth manifold with corners (𝕜, E, H, I) and (𝕜, E', H', I'), where 𝕜 is a non-trivially normed field, E and E' are normed add commutative groups, H, H' are topological spaces, and M, M' are charted subspaces of H, H' respectively, the constant function from M to M' is differentiable at every point in M.


/-- The neighborhood filter of a compact set is disjoint with a filter `l` if and only if the
neighborhood filter of each point of this set is disjoint with `l`.  -/
theorem IsCompact.disjoint_nhdsSet_left :
    ∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X} {l : Filter X},
      IsCompact s → (Disjoint (nhdsSet s) l ↔ ∀ x ∈ s, Disjoint (nhds x) l) :=
  by sorry

This theorem states that for a given compact set 's' in a topological space 'X' and a filter 'l', the neighborhood filter of the set 's' is disjoint with the filter 'l' if and only if the neighborhood filter of each point in the set 's' is disjoint with the filter 'l'. In other words, there is no overlap between the collection of neighborhoods of the set 's' and the filter 'l' if and only if there is no overlap between the collection of neighborhoods of each individual point in 's' and the filter 'l'.

For a compact set $S$ in a topological space $X$ and a filter $l$, the neighborhood filter of $S$ is disjoint with $l$ if and only if the neighborhood filter of each point in $S$ is disjoint with $l$.


theorem Function.IsPeriodicPt.minimalPeriod_pos :
    ∀ {α : Type u_1} {f : α → α} {x : α} {n : ℕ},
      0 < n → Function.IsPeriodicPt f n x → 0 < Function.minimalPeriod f x :=
  by sorry

This theorem states that for any type `α`, a function `f` from `α` to `α`, any element `x` of type `α`, and any natural number `n`, if `n` is greater than 0 and `x` is a periodic point of `f` with a period of `n` (i.e., applying function `f` `n` times to `x` gives us `x` again), then the minimal period of `x` under the function `f` is also greater than 0. The minimal period is the smallest positive integer `n` such that `f` applied `n` times to `x` returns `x`, or 0 if no such integer exists.

If `x` is a periodic point of period `n` (`n` > 0) for function `f` from type `α` to `α`, then the minimal period of `x` under `f` is also a positive integer.


theorem nhdsWithin_le_of_mem :
    ∀ {α : Type u_1} [inst : TopologicalSpace α] {a : α} {s t : Set α},
      s ∈ nhdsWithin a t → nhdsWithin a t ≤ nhdsWithin a s :=
  by sorry

This theorem states that for any type `α` that has a topological space structure, and any elements `a` of type `α`, and for any sets `s` and `t` of type `α`; if `s` is a member of the "neighborhood within" filter of `a` and `t`, then the "neighborhood within" filter of `a` and `t` is less than or equal to the "neighborhood within" filter of `a` and `s`. In mathematical terms, if `s` is in the intersection of `t` and a neighborhood of `a`, then every set that is in the intersection of `t` and a neighborhood of `a` is also in the intersection of `s` and a neighborhood of `a`.

If `a` is an element of a topological space `α`, `s` and `t` are subsets of `α` such that `s` is a neighborhood of `a` containing `t`, then for any neighborhood `U` of `a`, `s ∩ U` is a subset of `t ∩ U`.


theorem inv_zpow : ∀ {α : Type u_1} [inst : DivisionMonoid α] (a : α) (n : ℤ), a⁻¹ ^ n = (a ^ n)⁻¹ := by sorry

This theorem states that for any type `α` that forms a division monoid, and for any element `a` of type `α` and any integer `n`, the inverse of `a` raised to the power of `n` is equal to the inverse of `a` raised to the power of `n`. In mathematical terms, this can be written as $(a^{-1})^n = (a^n)^{-1}$ for all `a` in the division monoid and all integers `n`.

For all division monoids `α`, and for all `a` in `α` and integer `n`, $(a^{-1})^n = (a^n)^{-1}$.


theorem Mathlib.Topology.Algebra.GroupCompletion._auxLemma.3 :
    ∀ {α : Type u_3} [inst : UniformSpace α] [inst_1 : AddGroup α] [inst_2 : UniformAddGroup α] (a b : α),
      ↑α a + ↑α b = ↑α (a + b) :=
  by sorry

This theorem, from the Mathlib.Topology.Algebra.GroupCompletion module, states that for any type `α` that has a Uniform Space structure, an Addition Group structure, and a Uniform Addition Group structure, the sum of the embeddings of two elements `a` and `b` from `α` is equal to the embedding of the sum of these two elements. In simpler terms, this is stating that the operation of embedding preserves the addition operation in the space `α`.

For any type `α` with a Uniform Space, Addition Group, and Uniform Addition Group structure, the embedding function maps the sum of any two elements to the sum of their embeddings.


theorem Complex.exp_antiperiodic : Function.Antiperiodic Complex.exp (↑Real.pi * Complex.I) := by sorry

This theorem states that the complex exponential function is "antiperiodic" with an antiperiod of "pi times the imaginary unit". In other words, for any complex number z, the value of the complex exponential function at (z + pi * i) is the negative of its value at z. This property is similar to periodicity, but involves a sign change in addition to the function repeating its values.

The complex exponential function is a complex-valued function with a negative conjugate symmetry, specifically, exp(z) = -exp(z + πi) for all complex numbers z.


theorem Class.coe_mem : ∀ {x : ZFSet} {A : Class}, ↑x ∈ A ↔ A x := by sorry

The theorem `Class.coe_mem` states that for any set `x` from the ZFC universe of sets (`ZFSet`) and any `Class` `A`, the set `x` belongs to the class `A` if and only if the class `A` contains the set `x`. This theorem establishes the equivalence between the membership of a set in a class and the class containing the set.

For any set `x` in the ZFC universe and class `A`, `x` belongs to `A` if and only if `A` contains `x`.


theorem Set.OrdConnected.apply_wcovBy_apply_iff :
    ∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] {a b : α} (f : α ↪o β),
      Set.OrdConnected (Set.range ⇑f) → (f a ⩿ f b ↔ a ⩿ b) :=
  by sorry

This theorem states that, for any two types `α` and `β` with preorder structures, and for any function `f` from `α` to `β` which is an order-embedding, if the range of `f` is order-connected, then the relation of being covered by (`⩿`) between the images of two elements `a` and `b` under `f` is equivalent to the relation of being covered by between `a` and `b` themselves. In simpler terms, it means if `f(a)` is covered by `f(b)` then `a` is also covered by `b` and vice versa, assuming that the range of the function `f` is order-connected.

If `f : α → β` is an order-embedding from preordered types `α` and `β`, and the range of `f` is order-connected, then for all `a, b ∈ α`, `a ⩿ b` if and only if `f a ⩿ f b`.


theorem ZMod.int_cast_zmod_eq_zero_iff_dvd : ∀ (a : ℤ) (b : ℕ), ↑a = 0 ↔ ↑b ∣ a := by sorry

This theorem states that for any integer `a` and natural number `b`, the cast of `a` to the integers modulo `b` (using `zmod`) equals zero if and only if `b` divides `a`. In other words, if `b` is a factor of `a`, then `a` modulo `b` is zero, and conversely, if `a` modulo `b` is zero, then `b` is a factor of `a`.

For any integers `a` and natural number `b`, `a % b = 0` if and only if `b` divides `a`.


theorem Metric.nonempty_ball :
    ∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, Set.Nonempty (Metric.ball x ε) ↔ 0 < ε := by sorry

The theorem `Metric.nonempty_ball` states that in any pseudometric space, a ball of radius `ε` centered at a point `x` is not empty if and only if `ε` is greater than zero. In other words, there is at least one point `y` such that the distance from `y` to `x` is less than `ε`, if and only if `ε` is positive. Here, `dist y x < ε` defines the metric ball, and `Set.Nonempty (Metric.ball x ε)` means the set of all such points `y` is nonempty.

In a pseudometric space, the metric ball centered at a point with positive radius is nonempty.


theorem OrderIso.lt_iff_lt :
    ∀ {α : Type u_2} {β : Type u_3} [inst : Preorder α] [inst_1 : Preorder β] (e : α ≃o β) {x y : α},
      e x < e y ↔ x < y :=
  by sorry

This theorem states that for any two types `α` and `β` that have a preorder (a binary relation that is reflexive and transitive), if there is an order-preserving isomorphism `e` from `α` to `β`, then for any two elements `x` and `y` in `α`, `x` is less than `y` if and only if `e(x)` is less than `e(y)`. In other words, the order-preserving isomorphism preserves the less-than relation.

Given types `α` and `β` with a preorder, an order-preserving isomorphism `e` from `α` to `β` implies that `x ≤ y` in `α` if and only if `e(x) ≤ e(y)` in `β`.


theorem Set.range_inl_union_range_inr :
    ∀ {α : Type u_1} {β : Type u_2}, Set.range Sum.inl ∪ Set.range Sum.inr = Set.univ := by sorry

This theorem states that for any two types `α` and `β`, the union of the ranges of the functions `Sum.inl` and `Sum.inr`, which separately map each type into the sum type `α ⊕ β`, equals the universal set over the sum type `α ⊕ β`. In other words, every element in the sum type can be obtained either from an element of `α` using `Sum.inl` or from an element of `β` using `Sum.inr`.

For any types `α` and `β`, the ranges of `Sum.inl` from `α` and `Sum.inr` from `β` cover the entire sum type `α ⊕ β`.


theorem and_congr_left' : ∀ {a b c : Prop}, (a ↔ b) → (a ∧ c ↔ b ∧ c) := by sorry

This theorem states that for any three propositions `a`, `b`, and `c`, if `a` is logically equivalent to `b` (denoted as `a ↔ b`), then the logical conjunction of `a` and `c` (denoted as `a ∧ c`) is logically equivalent to the logical conjunction of `b` and `c` (denoted as `b ∧ c`). In other words, if `a` and `b` can be substituted for each other in a logical context, then `a` and `b` can also be substituted for each other when both are conjoined with the same third proposition, `c`.


theorem Real.sqrt_inv : ∀ (x : ℝ), Real.sqrt x⁻¹ = (Real.sqrt x)⁻¹ := by sorry

This theorem states that for every real number x, the square root of the reciprocal of x is equal to the reciprocal of the square root of x. In mathematical notation, it's saying that for all x in the real numbers, √(1/x) = 1/√x. Note that the theorem applies only to real numbers.


IsPathConnected.joinedIn

theorem IsPathConnected.joinedIn :
    ∀ {X : Type u_1} [inst : TopologicalSpace X] {F : Set X}, IsPathConnected F → ∀ x ∈ F, ∀ y ∈ F, JoinedIn F x y := by
  sorry

The theorem `IsPathConnected.joinedIn` states that for any topological space `X` and any subset `F` of `X`, if `F` is path-connected, then for any two points `x` and `y` in `F`, there exists a path within `F` that connects `x` and `y`. In the context of topology, a path between two points is essentially a continuous function from the unit interval into the topological space that starts at one point and ends at the other. In other words, if a set is path-connected, then any two points in the set can be joined by a path that stays entirely within the set.

If `F` is a path-connected subset of a topological space `X`, then for any `x, y ∈ F`, there exists a path `p : I → F` with `p(0) = x` and `p(1) = y`.


theorem Finset.disjoint_singleton_right : ∀ {α : Type u_1} {s : Finset α} {a : α}, Disjoint s {a} ↔ a ∉ s := by sorry

The theorem `Finset.disjoint_singleton_right` states that for any set `s` of elements of a certain type `α` and any single element `a` of the same type, `s` and `{a}` are disjoint (in the sense that their greatest lower bound, or infimum, is the least element in the order) if and only if `a` is not an element of `s`. In other words, the singleton set `{a}` is disjoint from the set `s` precisely when `a` does not belong to `s`.

The theorem `Finset.disjoint_right_singleton` asserts that for any set `s` of type `α` and element `a` of type `α`, the singleton set `{a}` is disjoint from `s` if and only if `a` is not in `s`.


theorem Set.nonempty_Ici : ∀ {α : Type u_1} [inst : Preorder α] {a : α}, Set.Nonempty (Set.Ici a) := by sorry

The theorem `Set.nonempty_Ici` states that for every type `α` that has a preorder structure, and for any element `a` of this type `α`, the left-closed right-infinite interval `Set.Ici a` is nonempty. In terms of mathematics, for any value `a`, there always exists at least one element `x` such that `a` is less than or equal to `x` (i.e., `a ≤ x`) within the given set.

For any preordered type `α` and element `a` in `α`, the left-closed right-infinite interval `Ici a` is nonempty, containing some element `x` with `a ≤ x`.


theorem LinearIndependent.ne_zero :
    ∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
      [inst_2 : Module R M] [inst_3 : Nontrivial R] (i : ι), LinearIndependent R v → v i ≠ 0 :=
  by sorry

The theorem `LinearIndependent.ne_zero` states that for any type `ι`, semiring `R`, additively commutative monoid `M`, function `v` mapping from `ι` to `M`, module structure on `R` and `M`, and nontrivial structure on `R`, if `v` is a family of vectors that is defined to be linearly independent over `R`, then each individual vector in the family (represented by `v i`) is not equal to zero. In other words, no vector in a linearly independent family can be the zero vector.

Given a type ι, a semiring R, an additively commutative monoid M with a module structure over R, a nontrivial R-module M, and a linearly independent family {vi : M | i ∈ ι} over R, then vi ≠ 0 for all i ∈ ι.


theorem Filter.tendsto_add_atTop_nat : ∀ (k : ℕ), Filter.Tendsto (fun a => a + k) Filter.atTop Filter.atTop := by sorry

This theorem, `Filter.tendsto_add_atTop_nat`, states that for any natural number `k`, the function which adds `k` to its input tends towards infinity along with its input. In other words, if we take any sequence of natural numbers that goes to infinity and add `k` to every number in that sequence, the resulting sequence also goes to infinity. In mathematical terms, this can be expressed as $\lim_{n \to \infty} (n+k) = \infty$ for all natural numbers `k`.

For all natural numbers `k`, the sequence `x => x + k` tends to infinity along with any sequence `x` that goes to infinity. (Mathematically, for all increasing sequences `{x_n}`, `lim (n : ℕ) (x_n + k) = ∞`. In Lean, this theorem is named `Filter.tendsto_add_atTop_nat`.)


theorem Set.nonempty_Iic : ∀ {α : Type u_1} [inst : Preorder α] {a : α}, Set.Nonempty (Set.Iic a) := by sorry

This theorem states that for any type `α` with a given preordering, and for any element `a` of this type `α`, the left-infinite right-closed interval `Set.Iic a` is not empty. In simpler terms, it says that there always exists an element in the set of all elements less than or equal to `a`.

For any type `α` with a preorder and any element `a` in `α`, the left-infinite right-closed interval `Set.Iic a` is non-empty.


theorem Filter.forall_mem_nonempty_iff_neBot :
    ∀ {α : Type u} {f : Filter α}, (∀ s ∈ f, Set.Nonempty s) ↔ Filter.NeBot f := by sorry

This theorem states that for any type 'α' and any filter 'f' on that type, the filter 'f' is not the bottom filter (`Filter.NeBot f`) if and only if every set 's' in the filter 'f' is not empty (`∀ s ∈ f, Set.Nonempty s`). Here, a bottom filter refers to the smallest filter in the filter lattice structure that contains all filters, analogous to the role of zero in a number system. The 'Set.Nonempty s' property indicates that there exists at least one element in the set 's'.

A filter on a type is not the bottom filter if and only if it contains only non-empty sets.


theorem map_add_left_nhds_zero :
    ∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : TopologicalAddGroup G] (x : G),
      Filter.map (fun x_1 => x + x_1) (nhds 0) = nhds x :=
  by sorry

The theorem `map_add_left_nhds_zero` states that for any type `G` that has a topological space structure, an additive group structure, and a topological additive group structure, we have for any element `x` of `G`, the filter obtained by mapping `(fun x_1 => x + x_1)` to the neighborhood filter of `0` is equal to the neighborhood filter of `x`. In other words, applying the mapping function that adds `x` to every element to the neighborhood filter of zero gives the neighborhood filter of `x`. This is an important property in topological additive groups.

For any topological additive group (G, +, 0), the mapping (x ↦ x + 0) preserves neighborhood filters, i.e., the neighborhood filter of 0 in G maps to the neighborhood filter of x.


theorem Finset.prod_apply :
    ∀ {α : Type u_1} {β : α → Type u_2} {γ : Type u_3} [inst : (a : α) → CommMonoid (β a)] (a : α) (s : Finset γ)
      (g : γ → (a : α) → β a), Finset.prod s (fun c => g c) a = Finset.prod s fun c => g c a :=
  by sorry

The theorem `Finset.prod_apply` states that for all types `α`, `β` which is a function of `α`, and `γ`, along with a commutative monoid structure on `β a` for every `a` in `α`, a specific `a` in `α`, a finite set `s` of `γ`, and a function `g` mapping from `γ` to `β a`, the product of the function `g` over the set `s` applied to `a` is equal to the product over the set `s` of the function `g c a`.

In other words, when computing the product of the function `g` over all elements in the finite set `s` and then applying this to `a`, it's the same as computing the product over the set `s` of the function `g` applied to each element `c` and `a`.

For any commutative monoid `β` over type `α`, function `g` from `γ` to `β a`, and finite set `s` of `γ`, the product of `g` over `s` and applied to `a` equals the product over `s` of `g a` applied to each element in `s`.


theorem Filter.Iic_mem_atBot : ∀ {α : Type u_3} [inst : Preorder α] (a : α), Set.Iic a ∈ Filter.atBot := by sorry

This theorem states that for any type `α` with a preorder relationship, and for any element `a` of type `α`, the set of all elements `x` such that `x ≤ a` (`Set.Iic a`) is a member of the filter representing the limit towards negative infinity (`Filter.atBot`). Essentially, this theorem confirms that the set of all elements less than or equal to a given element is a valid down-set in the filter representing the concept of "going to negative infinity".

For any type `α` with a preorder relation and any element `a` in `α`, the down-set `{x : α | x ≤ a}` belongs to the filter `Filter.atBot` of elements approaching the bottom element.


theorem List.mem_sublists' : ∀ {α : Type u} {s t : List α}, s ∈ List.sublists' t ↔ List.Sublist s t := by sorry

This theorem states that for any lists `s` and `t` of any type `α`, `s` is a member of the list of all non-contiguous sublists of `t` (`List.sublists' t`) if and only if `s` is a sublist of `t` (`List.Sublist s t`). In other words, it establishes the equivalence between the membership of `s` in the list of all non-contiguous sublists of `t` and the sublist relationship of `s` to `t`.

The theorem asserts that a list `s` is a non-contiguous sublist of list `t` if and only if `s` is a sublist of `t`.


theorem Cardinal.toPartENat_apply_of_aleph0_le :
    ∀ {c : Cardinal.{u_1}}, Cardinal.aleph0 ≤ c → Cardinal.toPartENat c = ⊤ := by sorry

The theorem `Cardinal.toPartENat_apply_of_aleph0_le` states that for any cardinal number `c`, if the cardinality of `c` is greater than or equal to the cardinality of the set of natural numbers (ℵ₀), then the function `Cardinal.toPartENat` applied to `c` returns the greatest possible element (⊤). In other words, the function `Cardinal.toPartENat` maps any infinite cardinal (cardinal at least as large as the set of natural numbers) to the top element, indicating that the cardinality of the set is infinite.

For any infinite cardinal number `c`, `Cardinal.toPartENat c = ⊤`.


/-- A variant of `modEq_iff_dvd` with `Nat` divisibility  -/
theorem Nat.modEq_iff_dvd' : ∀ {n a b : ℕ}, a ≤ b → (a ≡ b [MOD n] ↔ n ∣ b - a) := by sorry

This theorem states that for any natural numbers `n`, `a`, and `b`, where `a` is less than or equal to `b`, `a` is equivalent to `b` modulo `n` if and only if `n` divides the difference of `b` and `a`. In other words, `a` and `b` have the same remainder when divided by `n` if and only if `b - a` is a multiple of `n`. This is a variant of the theorem `modEq_iff_dvd` with natural number divisibility.

For natural numbers `n`, `a`, and `b`, `a` is congruent to `b` modulo `n` if and only if `n` divides the difference `b - a`.


theorem List.pmap_eq_map :
    ∀ {α : Type u} {β : Type v} (p : α → Prop) (f : α → β) (l : List α) (H : ∀ a ∈ l, p a),
      List.pmap (fun a x => f a) l H = List.map f l :=
  by sorry

The theorem `List.pmap_eq_map` states that for any types `α` and `β`, a predicate `p` on `α`, a function `f` from `α` to `β`, and a list `l` of type `α` with the property that the predicate `p` holds for every element in the list `l`, the `pmap` function of the list `l`, which applies the function `f` to each element in `l` (using a proof that the predicate `p` holds for each element), is equal to the `map` function of the list `l`, where `f` is applied to each element. In other words, if we know that every element of a list satisfies a certain property, then applying a function to each element of the list while simultaneously proving the property is equivalent to simply applying the function to each element of the list.

For any types `α` and `β`, if `l` is a list of `α` such that `p(x)` holds for all `x` in `l`, then `List.pmap f l` equals `List.map f l`, where `p` is a predicate on `α` and `f` is a function from `α` to `β`.


theorem monotone_closure : ∀ (X : Type u_3) [inst : TopologicalSpace X], Monotone closure := by sorry

The theorem `monotone_closure` states that for any type `X` equipped with a topology (i.e., `X` is a topological space), the function `closure` is monotone. In other words, for any two sets `a` and `b` in `X`, if `a` is a subset of `b`, then the closure of `a` is also a subset of the closure of `b`. This essentially means that taking the closure of a set preserves the subset relationship.

The closure operation on a topological space is monotone, that is, for any sets `a` and `b` such that `a` is a subset of `b`, we have `closure(a)` is a subset of `closure(b)`.


theorem LowerSet.coe_iSup :
    ∀ {α : Type u_1} {ι : Sort u_4} [inst : LE α] (f : ι → LowerSet α), ↑(⨆ i, f i) = ⋃ i, ↑(f i) := by sorry

This theorem states that for any type `α`, any sort `ι`, and any function `f` from `ι` to the LowerSet of `α`, the supremum (`⨆ i, f i`) of the `f` expressed in terms of `ι`, when coerced (`↑`), equals the union (`⋃ i, ↑(f i)`) of the individual coerced values of `f i`. In simpler terms, in the context of order theory, this theorem asserts that the supremum of a collection of lower sets is equivalent to the union of those individual lower sets.

For any type `α` and sort `ι`, the supremum of a function from `ι` to the LowerSet of `α` equals the union of the images of that function.


theorem WithBot.unbot'_le_iff :
    ∀ {α : Type u_1} [inst : LE α] {a : WithBot α} {b c : α}, (a = ⊥ → b ≤ c) → (WithBot.unbot' b a ≤ c ↔ a ≤ ↑c) := by
  sorry

This theorem states that for any type `α` that has a less-than-or-equal-to (`≤`) relation, and for any elements `a` of the type `WithBot α` and `b` and `c` of type `α`, if `b` is less than or equal to `c` when `a` is bottom (`⊥`), then `WithBot.unbot' b a` being less than or equal to `c` is equivalent to `a` being less than or equal to the co-domain of `c`. In other words, it shows the equivalence of these two conditions under the provided assumptions.

For any type `α` with a `≤` relation, and for any `a ∈ WithBot α`, `b`, `c ∈ α`, if `a = ⊥` implies `b ≤ c` then `WithBot.unbot' b a ≤ c` if and only if `a ≤ co-domain of c`.


theorem Sum.inr_ne_inl : ∀ {β : Type u_1} {b : β} {α : Type u_2} {a : α}, Sum.inr b ≠ Sum.inl a := by sorry

This theorem states that for any types `α` and `β`, and for any elements `a` of type `α` and `b` of type `β`, the sum type operation `Sum.inr b` (injecting `b` into the sum type from the right) is not equal to the sum type operation `Sum.inl a` (injecting `a` into the sum type from the left). In other words, it declares that elements from different summands of a sum type are distinct.

For all types `α` and `β` and elements `a : α`, `b : β`, `Sum.inr b` and `Sum.inl a` are distinct summand injections of `Sum α β`.


theorem Mathlib.CategoryTheory.Sites.Sieves._auxLemma.4 :
    ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X : C} (self : CategoryTheory.Sieve X) {Y Z : C}
      {f : Y ⟶ X}, self.arrows f → ∀ (g : Z ⟶ Y), self.1 (CategoryTheory.CategoryStruct.comp g f) = True :=
  by sorry

This theorem states that for any category `C`, and any object `X` in this category, let `self` be a sieve on `X`. If an arrow `f` from some object `Y` to `X` is in this sieve, then for any arrow `g` from any object `Z` to `Y`, the composition of `g` and `f` (`g ≫ f`) also belongs to the sieve `self`. This establishes that sieves in a category are stable under precomposition.

For any category C and object X, if a sieve on X contains an arrow from Y to X, then it also contains the composition of any arrow from Z to Y with that arrow.


/-- Embedding `i : Fin n` into `Fin (n + 1)` with a hole around `p : Fin (n + 1)`
never results in `p` itself  -/
theorem Fin.succAbove_ne : ∀ {n : ℕ} (p : Fin (n + 1)) (i : Fin n), Fin.succAbove p i ≠ p := by sorry

The theorem `Fin.succAbove_ne` states that for any natural number `n`, and any elements `p` of the finite set of size `n+1` and `i` of the finite set of size `n`, embedding `i` into the larger set with a hole around `p` using the `Fin.succAbove` function never results in `p` itself. In other words, the process of embedding maintains the distinctiveness of `p` and the embedded `i`.

For all natural numbers n, and all i in Fin (n), and p in Fin (n+1), p is not in the image of i under the Fin.succAbove function.


theorem Polynomial.natDegree_pow_le :
    ∀ {R : Type u} [inst : Semiring R] {p : Polynomial R} {n : ℕ},
      Polynomial.natDegree (p ^ n) ≤ n * Polynomial.natDegree p :=
  by sorry

The theorem `Polynomial.natDegree_pow_le` states that for any type `R` that is a semiring, for any polynomial `p` of type `R`, and for any natural number `n`, the natural degree of the polynomial `p` raised to the power of `n` is less than or equal to `n` times the natural degree of `p`. In mathematical terms, if we denote the natural degree of a polynomial `p` as `deg(p)`, this theorem asserts that `deg(p^n) ≤ n * deg(p)` for any `n` in natural numbers.

For any semiring R and polynomial p of type R, the natural degree of p raised to the power n is less than or equal to n times the natural degree of p. Equivalently, deg(p^n) ≤ n * deg(p) for all natural numbers n.


theorem Nat.dvd_zero : ∀ (a : ℕ), a ∣ 0 := by sorry

This theorem states that any natural number 'a' divides zero. In other words, for any given natural number, there exists another natural number such that when the former multiplies the latter, the result is zero. This is a standard property of division in mathematics represented in Lean 4.

There exists a natural number that makes any natural number 'a' such that 'a * x = 0' holds in Lean 4. (This is the negative of the multiplicative identity for 'a', equivalent to 'a' dividing '0' in the context of Lean 4's natural numbers.)


theorem borel_eq_top_of_discrete :
    ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : DiscreteTopology α], borel α = ⊤ := by sorry

This theorem states that for any type `α` equipped with a topological space structure and a discrete topology, the borel measurable space generated from the topological space is equal to the maximal (or top) measurable space, denoted by '⊤'. In other words, when the topology on `α` is discrete, every subset of `α` is both open and measurable, thus the borel measurable space is as large as it could possibly be.

For any topological space `α` with a discrete topology, its Borel sigma-algebra equals the maximal measurable space.


theorem Real.sin_le_one : ∀ (x : ℝ), Real.sin x ≤ 1 := by sorry

This theorem states that for all real numbers `x`, the sine of `x` is less than or equal to 1. In other words, it asserts the well-known mathematical fact that the sine function always returns a value between -1 and 1 for any real number input.


theorem SetTheory.PGame.LF.not_ge : ∀ {x y : SetTheory.PGame}, SetTheory.PGame.LF x y → ¬y ≤ x := by sorry

This theorem states that for any two pre-games `x` and `y` in set theory, if the "less or fuzzy" relation holds between `x` and `y` (i.e., `SetTheory.PGame.LF x y`), then `y` is not less than or equal to `x` (denoted as `¬y ≤ x`). The less or fuzzy relation is defined such that if `0` is not less than or equal to `x` (`0 ⧏ x`), then the Left player can win `x` as the first player. Therefore, this theorem is essentially asserting that if the Left player has a winning strategy for `x` as the first player, then `y` cannot be less than or equal to `x`.

For any pre-games x and y in set theory, if the Left player has a winning strategy as the first player in game x then y is strictly greater than x (i.e., y > x).


theorem Asymptotics.IsEquivalent.add_isLittleO :
    ∀ {α : Type u_1} {β : Type u_2} [inst : NormedAddCommGroup β] {u v w : α → β} {l : Filter α},
      Asymptotics.IsEquivalent l u v → w =o[l] v → Asymptotics.IsEquivalent l (u + w) v :=
  by sorry

The theorem `Asymptotics.IsEquivalent.add_isLittleO` states that for any types `α` and `β`, where `β` is a normed additive commutative group, and for any functions `u`, `v`, and `w` from `α` to `β`, and any filter `l` on `α`, if `u` and `v` are asymptotically equivalent along `l` and `w` is little-o of `v` along `l`, then the function `u` added to `w` is also asymptotically equivalent to `v` along `l`. In mathematical terms, if `u(x) - v(x) = o(v(x))` as `x` approaches along `l`, and `w(x) = o(v(x))` as `x` approaches along `l`, then `(u(x) + w(x)) - v(x) = o(v(x))` as `x` approaches along `l`.

If `u`, `v`, and `w` are functions from a normed additive commutative group `α` to another normed additive commutative group `β`, and `u` is asymptotically equivalent to `v` along a filter `l` on `α` with `w = o(v)` along `l`, then `u + w` is asymptotically equivalent to `v` along `l`.


NormedAddGroupHom.NormNoninc.comp

theorem NormedAddGroupHom.NormNoninc.comp :
    ∀ {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [inst : SeminormedAddCommGroup V₁]
      [inst_1 : SeminormedAddCommGroup V₂] [inst_2 : SeminormedAddCommGroup V₃] {g : NormedAddGroupHom V₂ V₃}
      {f : NormedAddGroupHom V₁ V₂},
      NormedAddGroupHom.NormNoninc g →
        NormedAddGroupHom.NormNoninc f → NormedAddGroupHom.NormNoninc (NormedAddGroupHom.comp g f) :=
  by sorry

This theorem states that if we have two normed add group homomorphisms `g` and `f` between three seminormed add commutative groups `V₁`, `V₂`, and `V₃`, then if both `g` and `f` are norm-nonincreasing (meaning that the norm of each image is less than or equal to the norm of the corresponding input), the composition of `g` and `f` (denoted by `NormedAddGroupHom.comp g f`) is also norm-nonincreasing.

In symbols, if for all `v` in `V₁` and `w` in `V₂`, we have `‖f v‖ ≤ ‖v‖` and `‖g w‖ ≤ ‖w‖`, then for all `v` in `V₁`, we also have `‖(NormedAddGroupHom.comp g f) v‖ ≤ ‖v‖`. This is a property about preserving norms when we have the composition of two norm-nonincreasing normed add group homomorphisms.

If `g` and `f` are norm-nonincreasing normed add group homomorphisms between seminormed add commutative groups `V₁`, `V₂`, and `V₃`, then the composition `NormedAddGroupHom.comp g f` is also a norm-nonincreasing homomorphism.


theorem exists_and_right : ∀ {α : Sort u_1} {p : α → Prop} {b : Prop}, (∃ x, p x ∧ b) ↔ (∃ x, p x) ∧ b := by sorry

This theorem, `exists_and_right`, states that for any type `α` and any predicate `p` of type `α → Prop` and any proposition `b`, the existence of an element of type `α` such that both `p` holds for that element and `b` is true is equivalent to the conjunction of the existence of an element of type `α` for which `p` holds and `b` being true. In other words, it equates the existence of an element satisfying a property and another independent proposition being true, with the existence of an element satisfying the property and the other proposition being true separately.

For any type `α`, predicate `p` of type `α → Prop`, and proposition `b`, the existence of an element of type `α` satisfying `p` and `b` is equivalent to the existence of an element of type `α` satisfying `p` and the truth of `b` separately.


theorem MvPolynomial.degrees_C :
    ∀ {R : Type u} {σ : Type u_1} [inst : CommSemiring R] (a : R), MvPolynomial.degrees (MvPolynomial.C a) = 0 := by
  sorry

This theorem states that for any type `R` and `σ`, with `R` being a commutative semiring, and for any element `a` of `R`, the multiset of degrees of the constant polynomial with value `a` is zero. In other words, a constant polynomial has no variables to have degrees of, so the multiset of degrees is empty, which is represented by `0` in this context.

For any commutative semiring `R` and its element `a`, the multiset of degrees of the constant polynomial `λ x : R, a` is the empty multiset.