GroupSeminorm.ext

theorem GroupSeminorm.ext : ∀ {E : Type u_4} [inst : Group E] {p q : GroupSeminorm E}, (∀ (x : E), p x = q x) → p = q :=
  by sorry

This theorem, called `GroupSeminorm.ext`, states that given a type `E` which forms a group, and two group seminorms `p` and `q` on this group, if for every element `x` in the group, the value of seminorm `p` at `x` equals the value of seminorm `q` at `x`, then the seminorms `p` and `q` are equal. Essentially, it asserts the uniqueness of group seminorms: two group seminorms are identical if they produce the same value for each element in the group.

If two functions define the same value for every element in a group, then they are equal as group seminorms.


theorem Rat.num_den' :
    ∀ {n : ℤ} {d : ℕ} {h : d ≠ 0} {c : Nat.Coprime (Int.natAbs n) d},
      { num := n, den := d, den_nz := h, reduced := c } = Rat.divInt n ↑d :=
  by sorry

The theorem `Rat.num_den'` states that for any integer `n` and any natural number `d` different from zero, if `n` and `d` are coprime (i.e., their greatest common divisor is 1), then the fraction formed by `n` and `d` equals to the result of the division of `n` by `d` in the rational numbers, where `n` is the numerator and `d` is the denominator. In other words, a rational number can be expressed as a fraction with numerator `n` and denominator `d` if and only if `n` and `d` are coprime and `d` is not zero.

If integers `n` and `d` are coprime, then the rational number `r` represented by the fraction `r := ∥n∥/∥d∥` equals `n / d`.


Function.Periodic.neg

theorem Function.Periodic.neg :
    ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [inst : AddGroup α],
      Function.Periodic f c → Function.Periodic f (-c) :=
  by sorry

The theorem `Function.Periodic.neg` states that for any type `α` with an addition group structure, and any function `f` from `α` to another type `β`, if `f` is periodic with period `c`, then it is also periodic with period `-c`. In other words, if for all `x` in `α`, `f(x + c) = f(x)`, then `f(x - c) = f(x)`. It expresses the property of a periodic function that reversing the direction of its period does not alter its periodicity.

If a function `f` from a type `α` to another type `β` with an additive group structure is periodic with period `c`, then it is also periodic with period `-c`.


theorem MeasureTheory.lintegral_dirac' :
    ∀ {α : Type u_1} [inst : MeasurableSpace α] (a : α) {f : α → ENNReal},
      Measurable f → ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a :=
  by sorry

This theorem states that for any type `α` that has a measurable space instance, any element `a` of type `α`, and any measurable function `f` from `α` to the extended nonnegative real numbers (`ENNReal`), the Lebesgue integral of `f` with respect to the Dirac measure at `a` equals the function `f` evaluated at `a`. In other words, when integrating a measurable function over a Dirac measure, the integral is simply the value of the function at the point where the Dirac measure is centered.

For any measurable function `f` from a measurable space `(α, Σ)` to the extended nonnegative real numbers, and any `α` element `a` with a measurable space instance, the Lebesgue integral of `f` with respect to the Dirac measure at `a` equals `f a`.


theorem Polynomial.IsSplittingField.adjoin_rootSet :
    ∀ {K : Type v} (L : Type w) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (f : Polynomial K)
      [inst_3 : Polynomial.IsSplittingField K L f], Algebra.adjoin K (Polynomial.rootSet f L) = ⊤ :=
  by sorry

This theorem states that for any fields `K` and `L` with an algebra structure over `K` and `L`, and given a polynomial `f` over `K` for which `L` is a splitting field, the subalgebra generated by adjoining the set of distinct roots of `f` in `L` to `K` is in fact the whole of `L`. In other words, the field `L` is generated by the roots of the polynomial `f` over `K`.

If `K` is a field, `L` is a field with an algebra structure over `K`, and `f` is a polynomial over `K` such that `L` is a splitting field for `f` over `K`, then the subalgebra of `L` generated by the roots of `f` is equal to `L`.


theorem QuotientAddGroup.mk_surjective :
    ∀ {α : Type u_1} [inst : AddGroup α] {s : AddSubgroup α}, Function.Surjective QuotientAddGroup.mk := by sorry

The theorem `QuotientAddGroup.mk_surjective` states that for every type `α` that has an `AddGroup` structure and for every additive subgroup `s` of `α`, the canonical map `QuotientAddGroup.mk` from `α` to the quotient group `α ⧸ s` is surjective. In other words, for every element in the quotient group `α ⧸ s`, there exists an element in `α` such that when the `QuotientAddGroup.mk` function is applied to this element, the original element from the quotient group is obtained.

For every `AddGroup` type `α` and additive subgroup `s`, the canonical map from `α` to the quotient group `α / s` is surjective.


IsUpperSet.compl

theorem IsUpperSet.compl : ∀ {α : Type u_1} [inst : LE α] {s : Set α}, IsUpperSet s → IsLowerSet sᶜ := by sorry

The theorem `IsUpperSet.compl` states that for any type `α` with an order relation and any set `s` of elements of type `α`, if `s` is an upper set (meaning that any element of `s` greater than another element is also in `s`), then the complement of `s` (denoted by `sᶜ`, which consists of all the elements of type `α` that are not in `s`) is a lower set (meaning that any element of `sᶜ` less than another element is also in `sᶜ`). In mathematical terms, it is saying that the complement of an upward-closed set in an ordered set is a downward-closed set.

If `s` is an upper set in an ordered type `α`, then the complement `sᶜ` is a lower set.


theorem Finset.dvd_prod_of_mem :
    ∀ {α : Type u_3} {β : Type u_4} [inst : CommMonoid β] (f : α → β) {a : α} {s : Finset α},
      a ∈ s → f a ∣ Finset.prod s fun i => f i :=
  by sorry

This theorem states that for any types `α` and `β`, where `β` is a commutative monoid, any function `f` from `α` to `β`, and any element `a` of type `α` and finite set `s` of elements of type `α`, if `a` is an element of `s`, then the value of `f` at `a` divides the product of `f` evaluated at every element in `s`. In mathematical terms, for $f : \alpha \to \beta$, $a \in \alpha$, $s \subseteq \alpha$ such that $a \in s$, we have $f(a) | \prod_{i \in s} f(i)$.

For any commutative monoid type `β`, function `f` from type `α` to `β`, element `a` of type `α` in a finite set `s` of type `α` that includes `a`, we have `f(a) | ∏(i ∈ s) f(i)`.


IsCoatom.dual

/-- **Alias** of the reverse direction of `isAtom_dual_iff_isCoatom`. -/
theorem IsCoatom.dual :
    ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : OrderTop α] {a : α}, IsCoatom a → IsAtom (OrderDual.toDual a) := by
  sorry

The theorem `IsCoatom.dual` states that for any type `α` with a preorder and an order top, if an element `a` is a coatom, then the dual of `a` is an atom. Here, a coatom is an element where there's no other element between it and the top element, and it's not the top element itself; while an atom is an element where there's no other element between it and the bottom element, and it's not the bottom element itself. The dual of an element is just the element itself in the dual order, where the order is reversed.

If an element `a` is a coatom in a preordered type with an order top, then the dual of `a` is an atom.


theorem CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      {S : CategoryTheory.ShortComplex C} (h : CategoryTheory.ShortComplex.LeftHomologyData S)
      [inst_2 : CategoryTheory.ShortComplex.HasLeftHomology S],
      CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso h).hom h.i =
        CategoryTheory.ShortComplex.iCycles S :=
  by sorry

The theorem `CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i` in Category Theory asserts that for any type `C` which forms a category and has zero-morphisms, and for any `S` of type `ShortComplex C` with a left homology, the composition of the morphism `cyclesIso h.hom` with `h.i` equals the inclusion of `S.cycles` to `S.X₂`. In simpler terms, this theorem establishes a particular relationship between the specific morphisms in the short complex of a category with left homology.

For any ShortComplex C with left homology, the morphism `cyclesIso.hom` composed with `i` is equal to the inclusion of cycles into the second component of the complex.


theorem isUnit_iff_exists_inv : ∀ {M : Type u_1} [inst : CommMonoid M] {a : M}, IsUnit a ↔ ∃ b, a * b = 1 := by sorry

This theorem states that, for any commutative monoid M and any element 'a' of M, 'a' is a unit if and only if there exists another element 'b' in M such that the product of 'a' and 'b' equals the identity element of M. In other words, 'a' is a unit precisely when it has a multiplicative inverse.

A commutative monoid element 'a' is a unit if and only if there exists an element 'b' such that 'a' * 'b' = id\_M, where id\_M is the identity element of M.


theorem Finset.sup_eq_biUnion :
    ∀ {α : Type u_7} {β : Type u_8} [inst : DecidableEq β] (s : Finset α) (t : α → Finset β),
      Finset.sup s t = Finset.biUnion s t :=
  by sorry

The theorem `Finset.sup_eq_biUnion` asserts that for any two types `α` and `β` with `β` having a decidable equality relation, and for any finite set `s` of type `α` and a function `t` mapping an element of `α` to a finite set of `β`, the supremum of the set `s` under the function `t` is equal to the biUnion of `s` and `t`. In other words, if you take a finite set and map each element to another finite set via a function, the supremum of the original set under this function is the same as if you had taken the union of the function's outputs for all elements in the original set.

For any decidably equated types `α` and `β`, given a finite set `s` of `α` and a function `t` mapping each `α` element to a finite set of `β`, the supremum of `s` under `t` equals the biUnion of the image of `s` under `t`.


/-- A `NonUnitalSubring` `t` includes `closure s` if and only if it includes `s`.  -/
theorem NonUnitalSubring.closure_le :
    ∀ {R : Type u} [inst : NonUnitalNonAssocRing R] {s : Set R} {t : NonUnitalSubring R},
      NonUnitalSubring.closure s ≤ t ↔ s ⊆ ↑t :=
  by sorry

The theorem `NonUnitalSubring.closure_le` states that for any type `R` equipped with a structure of a non-unital, non-associative ring, and any set `s` of elements of `R` and a NonUnitalSubring `t` of `R`, the non-unital subring generated by `s` is a subring of `t` if and only if `s` is a subset of `t`. In other words, a non-unital subring `t` includes the non-unital subring generated by a set `s` exactly when it includes all elements of the set `s`.

A non-unital subring `t` of a non-unital, non-associative ring `R` includes the non-unital subring generated by a set `s` if and only if `s` is a subset of `t`.


/-- **Alias** of the reverse direction of `sub_le_self_iff`. -/
theorem sub_le_self :
    ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LE α]
      [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) {b : α}, 0 ≤ b → a - b ≤ a :=
  by sorry

This theorem, `sub_le_self`, is essentially an alias for the reverse direction of another theorem, `sub_le_self_iff`. It applies to any type `α` that forms an additive group and has a less-than-or-equal-to (`≤`) operation that is covariant in both arguments. The theorem states that, for any elements `a` and `b` of such a type, if `b` is nonnegative (`0 ≤ b`), then the result of subtracting `b` from `a` (`a - b`) is less than or equal to `a`.

For all additive groups with covariant `≤` operation, if `α` has non-negative element `b`, then `a ≤ a - b`.


theorem Cardinal.mk_ordinal_out : ∀ (o : Ordinal.{u_3}), Cardinal.mk (Quotient.out o).α = Ordinal.card o := by sorry

This theorem states that for any ordinal `o`, the cardinality of the type of elements (`α`) extracted from the equivalence class of `o` via the `Quotient.out` function (which is an implementation of the Axiom of Choice) equals to the cardinality of the ordinal `o` itself. In other words, the cardinal representation of the type of elements in the equivalence class of any ordinal is the same as the cardinality of that ordinal.

For any ordinal `o`, the cardinality of `Quotient.out o` equals the cardinality of `o`.


theorem MeasureTheory.ae_dirac_eq :
    ∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (a : α),
      MeasureTheory.Measure.ae (MeasureTheory.Measure.dirac a) = pure a :=
  by sorry

The theorem `MeasureTheory.ae_dirac_eq` states that for any measurable space `α` and any element `a` of `α`, the "almost everywhere" filter of co-null sets corresponding to the Dirac measure at `a` is equivalent to the filter which contains exactly the set `{a}`. This essentially means that the Dirac measure at `a` is zero almost everywhere except at the point `a` itself. This applies to any measurable singleton class, which is a class of measurable spaces where every singleton set `{a}` is measurable.

The almost everywhere filter of co-null sets for the Dirac measure at a point in a measurable space is equivalent to the filter generated by the singleton set containing that point.


theorem Mathlib.GroupTheory.Subgroup.Basic._auxLemma.107 :
    ∀ {G : Type u_1} [inst : Group G] {N : Type u_5} [inst_1 : Group N] {f : G →* N} {y : N},
      (y ∈ MonoidHom.range f) = ∃ x, f x = y :=
  by sorry

The theorem states that for any groups `G` and `N`, and a monoid homomorphism `f` from `G` to `N`, a member `y` of `N` belongs to the range of `f` if and only if there exists an element `x` in `G` such that `f(x) = y`. This essentially asserts the definition of the range of a function in the context of group theory and monoid homomorphisms.

For any group homomorphism f from group G to monoid N, the image of an element y in N under f is precisely some x in G.


theorem Ideal.isPrime_of_prime :
    ∀ {A : Type u_2} [inst : CommRing A] [inst_1 : IsDedekindDomain A] {P : Ideal A}, Prime P → Ideal.IsPrime P := by
  sorry

This theorem states that for any type `A` that forms a commutative ring and a Dedekind domain, if we have an ideal `P` of this type `A` that is prime, then this ideal `P` is also prime in the sense of ideals. In mathematical terms, if `P` is a prime element of the ring `A`, it means `P` is not zero, is not a unit, and if `P` divides the product of two elements of `A`, then `P` must divide at least one of them. This theorem ensures that these properties also hold when considering `P` as an ideal of the ring `A`.

If `P` is a prime element of a commutative ring `A` that is also a Dedekind domain, then `P` generates a prime ideal in `A`.


theorem Nat.Prime.dvd_of_dvd_pow : ∀ {p m n : ℕ}, Nat.Prime p → p ∣ m ^ n → p ∣ m := by sorry

The theorem `Nat.Prime.dvd_of_dvd_pow` states that for any natural numbers `p`, `m` and `n`, if `p` is a prime number and `p` divides `m` raised to the power `n`, then `p` also divides `m`. In mathematical terms, if $p$ is a prime number and $p | m^n$, then $p | m$. This is a fundamental property of prime numbers related to divisibility and exponentiation.

If a prime number `p` divides `m` raised to the power `n`, then `p` divides `m` (i.e., `p | m => p | m^n`).


theorem norm_sub_le : ∀ {E : Type u_6} [inst : SeminormedAddGroup E] (a b : E), ‖a - b‖ ≤ ‖a‖ + ‖b‖ := by sorry

This theorem, `norm_sub_le`, states that for any type `E` which is a seminormed add group, the norm of the difference between any two elements `a` and `b` of `E` is less than or equal to the sum of the norms of `a` and `b`. In mathematical terms, for any `a` and `b` in `E`, we have `‖a - b‖ ≤ ‖a‖ + ‖b‖`, where `‖‖` denotes the norm in the seminormed add group `E`.

For any seminormed add group `E` and elements `a` and `b` in `E`, the norm of their difference is less than or equal to the sum of the norms of `a` and `b`: `‖a - b‖ ≤ ‖a‖ + ‖b‖`.


lp.memℓp

theorem lp.memℓp :
    ∀ {α : Type u_1} {E : α → Type u_2} {p : ENNReal} [inst : (i : α) → NormedAddCommGroup (E i)] (f : ↥(lp E p)),
      Memℓp (↑f) p :=
  by sorry

This theorem states that for any type `α`, a function `E` mapping from `α` to another type, a nonnegative real number `p`, and an instance of `NormedAddCommGroup` for each `i` in `α`, if `f` is an element of the `lp` space with respect to `E` and `p`, then `f` satisfies the `Memℓp` property. In more mathematical terms, for `f` belonging to the `lp` space, if `p` equals 0, `f` is finitely supported; if `p` equals infinity, there exists an upper bound for the set of norms of `f`; if `p` is between 0 and infinity, the series of the `p`-th power of the norms of `f` is summable.

If `f` is an element of the `lp` space with respect to a function `E` mapping from type `α` to another type, a nonnegative real number `p`, and instances of `NormedAddCommGroup` for each `i` in `α`, then `f` satisfies the `Memℓp` property with respect to `p`, i.e., it is finitely supported for `p` = 0, has an upper bound for the set of norms for `p` = ∞, and its series of `p`-th power norms is summable for 0 < `p` < ∞.


theorem Mathlib.Data.List.Range._auxLemma.7 : ∀ {n : ℕ} (a : Fin n), (a ∈ List.finRange n) = True := by sorry

The theorem `Mathlib.Data.List.Range._auxLemma.7` states that for any natural number `n` and any element `a` of the finite set `Fin n`, `a` is always an element of the list generated by `List.finRange n`. In other words, the function `List.finRange n` generates a list of all elements of the finite set `Fin n` (which contains elements ranging from `0` to `n-1`), and any element `a` from `Fin n` will always be found in this list.

For any natural number `n` and element `a` in finite set `Fin n`, `a` is an element of the list generated by `List.finRange n`.


theorem NormedSpace.isVonNBounded_iff :
    ∀ (𝕜 : Type u_1) (E : Type u_3) [inst : NontriviallyNormedField 𝕜] [inst_1 : SeminormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] (s : Set E), Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s :=
  by sorry

This theorem, `NormedSpace.isVonNBounded_iff`, states that for any types `𝕜` and `E`, where `𝕜` is a nontrivially-normed field and `E` is a seminormed additive commutative group that forms a normed space over `𝕜`, a set `s` of type `E` is von Neumann bounded with respect to `𝕜` if and only if it is bounded relative to the ambient bornology on `E`. Here, von Neumann boundedness means that every neighborhood of 0 absorbs the set `s`, and boundedness in the sense of bornology means that the set's complement is cobounded.

A set in a normed space is von Neumann bounded if and only if it is bounded in the ambient bornology.


theorem strictMono_of_le_iff_le :
    ∀ {α : Type u} {β : Type v} [inst : Preorder α] [inst_1 : Preorder β] {f : α → β},
      (∀ (x y : α), x ≤ y ↔ f x ≤ f y) → StrictMono f :=
  by sorry

The theorem `strictMono_of_le_iff_le` states that for any types `α` and `β` with preorder relations, and any function `f` from `α` to `β`, if for all `x` and `y` in `α`, `x` is less than or equal to `y` if and only if `f(x)` is less than or equal to `f(y)`, then the function `f` is strictly monotone. In other words, this theorem establishes that a function preserving the order of elements implies that the function is strictly increasing.

If a function between preordered types preserves the order relation, then it is strictly increasing.


theorem Matrix.diagonal_transpose :
    ∀ {n : Type u_3} {α : Type v} [inst : DecidableEq n] [inst_1 : Zero α] (v : n → α),
      Matrix.transpose (Matrix.diagonal v) = Matrix.diagonal v :=
  by sorry

The theorem `Matrix.diagonal_transpose` states that for any type `n` with decidable equality and any type `α` that has a zero element, given a function `v` from `n` to `α`, the transpose of the diagonal matrix generated by `v` is equal to the original diagonal matrix. In simpler terms, this theorem states that transposing a diagonal matrix does not change it.

For any type `n` with decidable equality and type `α` with a zero element, the transpose of a diagonal matrix formed by a function `v` from `n` to `α` is equal to the original diagonal matrix `v`.


theorem Multiset.toFinset_card_le :
    ∀ {α : Type u_1} [inst : DecidableEq α] (m : Multiset α), (Multiset.toFinset m).card ≤ Multiset.card m := by sorry

The theorem `Multiset.toFinset_card_le` states that for any type `α` that has decidable equality (`DecidableEq α`), and for any multiset `m` of type `α`, the cardinality (or size) of the finset obtained by removing duplicates from `m` (`Multiset.toFinset m`) is less than or equal to the cardinality of `m` itself. In simpler terms, the theorem asserts that when duplicates are removed from a multiset to form a finset, the size of the resulting finset will always be less than or equal to the size of the original multiset.

For any multiset `m` of type `α` with decidable equality, the cardinality of `Multiset.toFinset m` is less than or equal to the cardinality of `m`.


theorem Polynomial.natDegree_add_le :
    ∀ {R : Type u} [inst : Semiring R] (p q : Polynomial R),
      Polynomial.natDegree (p + q) ≤ max (Polynomial.natDegree p) (Polynomial.natDegree q) :=
  by sorry

The theorem `Polynomial.natDegree_add_le` states that for any semiring `R` and for any two polynomials `p` and `q` over `R`, the natural degree of the sum of `p` and `q` is less than or equal to the maximum of the natural degrees of `p` and `q`. In other words, when you add two polynomials, the degree of the resulting polynomial won't exceed the higher of the two original polynomial's degrees.

For any semiring R and polynomials p and q over R, the natural degree of p + q is less than or equal to max (deg p, deg q).


Continuous.Prod.mk

theorem Continuous.Prod.mk :
    ∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (x : X),
      Continuous fun y => (x, y) :=
  by sorry

This theorem, `Continuous.Prod.mk`, states that for all types `X` and `Y`, given that `X` and `Y` are topological spaces, any function which takes a fixed element `x` from `X` and maps any element `y` from `Y` to the ordered pair `(x, y)`, is a continuous function. In other words, in the context of topology, holding one element constant in a pair and varying the other results in a continuous function.

For any topological spaces X and Y, the function that maps x in X to the ordered pair (x, y) for all y in Y is a continuous function.


theorem List.Perm.append_right :
    ∀ {α : Type u_1} {l₁ l₂ : List α} (t₁ : List α), List.Perm l₁ l₂ → List.Perm (l₁ ++ t₁) (l₂ ++ t₁) := by sorry

This theorem states that for any type `α` and any lists `l₁`, `l₂`, and `t₁` of this type, if `l₁` is a permutation of `l₂`, then the list formed by appending `t₁` to `l₁` is a permutation of the list formed by appending `t₁` to `l₂`. In other words, if two lists are permutations of each other, appending the same list to both will not change this fact.

If `l₁` is a permutation of `l₂`, then appending `t` to both `l₁` and `l₂` results in lists that are permutations of each other. (In Lean notation: `permutation l₁ l₂ → list.append _ _ l₁ = list.append _ _ l₂`)


AntilipschitzWith.uniformInducing

theorem AntilipschitzWith.uniformInducing :
    ∀ {α : Type u_1} {β : Type u_2} [inst : PseudoEMetricSpace α] [inst_1 : PseudoEMetricSpace β] {K : NNReal}
      {f : α → β}, AntilipschitzWith K f → UniformContinuous f → UniformInducing f :=
  by sorry

The theorem `AntilipschitzWith.uniformInducing` states that for any two types `α` and `β` equipped with the pseudo-emetric space structure, a non-negative real number `K`, and a function `f` from `α` to `β`, if `f` is `AntilipschitzWith` `K` and `f` is also uniformly continuous, then `f` is uniform inducing.

In more natural language, this theorem says that if a function `f` from one metric space to another is Antilipschitz (meaning that the distance between any two points in the input space is at most `K` times the distance between their images in the output space) and uniformly continuous (meaning that if two points in the input space are sufficiently close to each other, their images under `f` will also be close to each other), then `f` is uniform inducing, which implies that the uniformity (topological structure that allows us to speak about "closeness" of points) on the input space can be recovered from the uniformity on the output space via `f`.

If a function between two pseudo-metric spaces is Antilipschitz with a constant K and uniformly continuous, then it is uniformly inducing.


theorem Complex.finrank_real_complex : FiniteDimensional.finrank ℝ ℂ = 2 := by sorry

This theorem states that the rank of the complex numbers when considered as a module over the real numbers is equal to 2. In other words, it is asserting that the complex numbers form a two-dimensional vector space over the real numbers.

The complex numbers form a 2-dimensional vector space over the real numbers.


theorem Finset.card_eq_one : ∀ {α : Type u_1} {s : Finset α}, s.card = 1 ↔ ∃ a, s = {a} := by sorry

This theorem states that for any type `α` and any finite set `s` of that type, the cardinality of `s` is equal to one if and only if there exists an element `a` such that `s` is the singleton set containing `a`. In other words, a set has exactly one element if and only if it is the set containing that one element.

For any type `α` and finite set `s` of type `α`, the set `s` has cardinality one if and only if it is a singleton set.


Basis.ext

/-- Two linear maps are equal if they are equal on basis vectors.  -/
theorem Basis.ext :
    ∀ {ι : Type u_1} {R : Type u_3} {M : Type u_6} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
      (b : Basis ι R M) {R₁ : Type u_10} [inst_3 : Semiring R₁] {σ : R →+* R₁} {M₁ : Type u_11}
      [inst_4 : AddCommMonoid M₁] [inst_5 : Module R₁ M₁] {f₁ f₂ : M →ₛₗ[σ] M₁},
      (∀ (i : ι), f₁ (b i) = f₂ (b i)) → f₁ = f₂ :=
  by sorry

This theorem states that given two linear maps `f₁` and `f₂` from a module `M` over a ring `R` to another module `M₁` over another ring `R₁`, if these two maps coincide on the basis vectors of `M`, then they are equal. This is under the context that `R` and `R₁` are semirings, `M` and `M₁` are additive commutative monoids with a module structure, and there's a ring homomorphism from `R` to `R₁`. The basis of module `M` is indexed by an arbitrary type `ι`.

If `f₁` and `f₂` are linear maps from a module `M` over a semiring `R` to another module `M₁` over a semiring `R₁`, with a ring homomorphism from `R` to `R₁`, such that `f₁(eι) = f₂(eι)` for all basis vectors `eι` in `M`, then `f₁ = f₂`.


theorem associated_normalize :
    ∀ {α : Type u_1} [inst : CancelCommMonoidWithZero α] [inst_1 : NormalizationMonoid α] (x : α),
      Associated x (normalize x) :=
  by sorry

The theorem `associated_normalize` states that for any type `α` that forms a `CancelCommMonoidWithZero` and a `NormalizationMonoid`, every element `x` of type `α` is `Associated` with its normalized form. In other words, `x` and `normalize x` are `Associated` for any `x` in such a type `α`. Here, two elements are said to be `Associated` if one can be obtained by multiplying the other with a unit from the right. The normalization process is achieved by multiplying an element with the `normUnit` of the `NormalizationMonoid`.

For any type `α` that is a `CancelCommMonoidWithZero` and has a `NormalizationMonoid`, every element `x` in `α` is associated to its normalized form `normalize x`.


theorem Real.hasDerivAt_rpow_const : ∀ {x p : ℝ}, x ≠ 0 ∨ 1 ≤ p → HasDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x := by
  sorry

The theorem `Real.hasDerivAt_rpow_const` states that for any real numbers `x` and `p`, if `x` is not equal to 0 or `p` is greater than or equal to 1, then the function `f(x) = x^p` has the derivative `f'(x) = p * x^(p - 1)` at the point `x`. In other words, the derivative of the power function `x^p` at any point `x` that is not zero, or where `p` is at least 1, is given by the formula `p * x^(p - 1)`.

If `x ≠ 0` and `p ≥ 1`, then the derivative of `x^p` at `x` is `p * x^(p - 1)`.


/-- See also `pow_left_strictMono` and `Nat.pow_left_strictMono`.  -/
theorem pow_left_strictMonoOn :
    ∀ {R : Type u_3} [inst : StrictOrderedSemiring R] {n : ℕ}, n ≠ 0 → StrictMonoOn (fun x => x ^ n) {a | 0 ≤ a} := by
  sorry

The theorem `pow_left_strictMonoOn` states that for any strict ordered semiring `R`, and any natural number `n` that is not zero, the function `f(x) = x^n` is strictly monotonic on the set of `a` such that `a` is greater than or equal to 0. In other words, for any two elements `a` and `b` from this set, if `a` is less than `b`, then `a^n` is less than `b^n`.

For any strict ordered semiring R and natural number n > 0, the function x ↔ x^n is strict monotonic on the non-negative elements of R, i.e., if 0 ≤ a < b, then a^n < b^n.


theorem HasFDerivAt.comp_hasFDerivWithinAt :
    ∀ {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]
      {G : Type u_4} [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E)
      {s : Set E} {g : F → G} {g' : F →L[𝕜] G},
      HasFDerivAt g g' (f x) →
        HasFDerivWithinAt f f' s x → HasFDerivWithinAt (g ∘ f) (ContinuousLinearMap.comp g' f') s x :=
  by sorry

This theorem states that if a function `g` has a derivative `g'` at the point `f(x)`, and another function `f` has a derivative `f'` within a set `s` at the point `x`, then the composition of the functions `g` and `f` has a derivative with respect to `x` within the set `s` at the point `x`. Specifically, this derivative is a linear map defined as the composition of the linear maps `g'` and `f'`. This property underpins the chain rule in calculus in the context of functions between normed vector spaces over a non-trivially normed field `𝕜`.

If functions `f` and `g` are differentiable at points `x` with derivatives `f'` and `g'` respectively, then the composition `g ∘ f` is differentiable at `x` with derivative `(g ∘ f)'(x) = g'(f(x)) ∘ f'(x)`.


theorem Set.uIcc_comm : ∀ {α : Type u_1} [inst : Lattice α] (a b : α), Set.uIcc a b = Set.uIcc b a := by sorry

The theorem `Set.uIcc_comm` states that for any type `α` that forms a lattice, and for any two elements `a` and `b` of this type, the set of elements lying between `a` and `b`, with `a` and `b` included (`Set.uIcc a b`), is equivalent to the set of elements lying between `b` and `a`, with `b` and `a` included (`Set.uIcc b a`). This asserts the commutativity of the bounded interval function `uIcc` in the context of a lattice, meaning it doesn't matter which order the elements are given in, the resulting set is the same.

For any lattice type `α` and elements `a` and `b` in `α`, the upward closed intervals `Set.uIcc a b` and `Set.uIcc b a` are equal.


theorem Asymptotics.isLittleO_norm_left :
    ∀ {α : Type u_1} {F : Type u_4} {E' : Type u_6} [inst : Norm F] [inst_1 : SeminormedAddCommGroup E'] {g : α → F}
      {f' : α → E'} {l : Filter α}, (fun x => ‖f' x‖) =o[l] g ↔ f' =o[l] g :=
  by sorry

This theorem, `Asymptotics.isLittleO_norm_left`, states that for any types `α`, `F`, and `E'`, where `F` has a norm structure and `E'` has a semi-normed additive commutative group structure, and for any functions `g : α → F` and `f' : α → E'`, as well as for any filter `l` on `α`, the norm of `f'` is little-o of `g` with respect to `l` if and only if `f'` itself is little-o of `g` with respect to `l`. Here, being "little-o" is a mathematical notation used to describe limiting behavior, meaning that the growth rate of `f'` is asymptotically less than the growth rate of `g` as they approach the limit `l`.

For functions `g : α → F` and `f' : α → E'` on types `α` with filter `l`, where `F` has a norm structure and `E'` is a semi-normed additive commutative group, `f'` is little-o of `g` with respect to `l` in norm if and only if `f'` is little-o of `g` with respect to `l` in `E'`.


theorem Filter.NeBot.of_map₂_left :
    ∀ {α : Type u_1} {β : Type u_3} {γ : Type u_5} {m : α → β → γ} {f : Filter α} {g : Filter β},
      Filter.NeBot (Filter.map₂ m f g) → Filter.NeBot f :=
  by sorry

The theorem `Filter.NeBot.of_map₂_left` states that for all types `α`, `β`, and `γ`, a binary function `m` mapping from `α` and `β` to `γ`, and two filters `f` and `g` on `α` and `β` respectively, if the image of the binary function `m` mapping from `f` and `g` to a filter in `γ` (represented by `Filter.map₂ m f g`) is not the bottom filter (meaning it's not the empty set), then `f` (the filter on `α`) is also not the bottom filter. In mathematical notation, this could be interpreted as: for all `α`, `β`, `γ`, `m : α → β → γ`, `f : Filter α`, and `g : Filter β`, if `Filter.map₂ m f g ≠ ∅`, then `f ≠ ∅`.

If the composite filter of a function and two filters is non-empty, then each of the two filters is non-empty.


theorem AlgHom.comp_algebraMap :
    ∀ {R : Type u} {A : Type v} {B : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Semiring B]
      [inst_3 : Algebra R A] [inst_4 : Algebra R B] (φ : A →ₐ[R] B),
      RingHom.comp (↑φ) (algebraMap R A) = algebraMap R B :=
  by sorry

This theorem states that for any commutative semiring `R`, semirings `A` and `B`, and any algebra structures over `R` on `A` and `B`, the composition of the ring homomorphism corresponding to an algebra homomorphism `φ` from `A` to `B` and the ring homomorphism embedding of `R` into `A` (given by the algebra structure) is equal to the ring homomorphism embedding of `R` into `B`. In other words, an algebra homomorphism preserves the structure of the ring homomorphism that maps the base ring into the algebra, showing the compatibility of the algebra homomorphism with the underlying ring structure.

For any commutative semirings `R`, semirings `A` and `B`, and algebra structures over `R` on `A` and `B`, the algebra homomorphism `φ` from `A` to `B` preserves the ring homomorphism property, i.e., the composition of `φ` with the embedding of `R` into `A` is equal to the embedding of `R` into `B`.


/-- The other triangle equality. The proof follows the following proof in Globular:
http://globular.science/1905.001  -/
theorem CategoryTheory.Equivalence.unit_inverse_comp :
    ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂}
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D) (Y : D),
      CategoryTheory.CategoryStruct.comp (e.unit.app (e.inverse.obj Y)) (e.inverse.map (e.counit.app Y)) =
        CategoryTheory.CategoryStruct.id (e.inverse.obj Y) :=
  by sorry

This theorem, named `CategoryTheory.Equivalence.unit_inverse_comp`, states that for any two categories `C` and `D` that are equivalent (denoted `C ≌ D`), and for any object `Y` in the category `D`, the composition of two morphisms — the application of the unit component of the equivalence to the object obtained by applying the inverse of the equivalence to `Y`, and the mapping of the counit component of the equivalence applied to `Y` under the inverse of the equivalence — is equal to the identity morphism on the object obtained by applying the inverse of the equivalence to `Y`. This theorem is related to the other triangle identity in category theory and the proof in Globular can be found at: http://globular.science/1905.001.

Given categories `C` and `D` equivalent to each other, for any object `Y` in `D`, the composition of the unit and counit morphisms of the equivalence, applied to `Y` and its inverse, equals the identity morphism on the object obtained by applying the inverse of the equivalence to `Y`.


theorem AddGroup.subset_closure : ∀ {G : Type u_1} [inst : AddGroup G] {s : Set G}, s ⊆ AddGroup.closure s := by sorry

The theorem `AddGroup.subset_closure` states that for any type `G` that forms an additive group, and for any set `s` of elements of `G`, the set `s` is a subset of the closure of `s` under the group operation. In other words, every element of the set `s` is included in the smallest additive subgroup generated by `s`.

For any additive group `G` and set `s` of its elements, `s` is a subset of the subgroup generated by `s`.


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

This theorem states that for any type `G` that forms a group, given three elements `a`, `b`, and `c` of this group, if the product of `a` and `c` equals `b`, then `a` is equal to the product of `b` and the inverse of `c`. In mathematical notation: if $a \cdot c = b$, then $a = b \cdot c^{-1}$. This theorem is a property of group theory, which is a fundamental concept in abstract algebra.

If elements `a`, `b`, and `c` of a group `G` satisfy `a * c = b`, then `a = b * c^(-1)`.


theorem Num.add_one : ∀ (n : Num), n + 1 = Num.succ n := by sorry

This theorem states that for any natural number `n`, adding one to `n` is equivalent to the successor of `n`. In other words, for any number `n`, `n + 1` is the same as the next number after `n` in the sequence of natural numbers. This is a fundamental property of natural numbers in the Peano postulates.

For any natural number `n`, `n + 1` is the same as the successor of `n`. (Alternatively: The successor of a natural number `n` is `n + 1`.)


theorem Nat.succ_succ_ne_one : ∀ (n : ℕ), Nat.succ (Nat.succ n) ≠ 1 := by sorry

This theorem states that for all natural numbers `n`, the successor of the successor of `n` is not equal to one. In other words, if you start with any natural number and increment it twice, you will never end up with the number one. This makes intuitive sense because the minimum natural number is zero and incrementing it twice gives two, which is not equal to one.

For all natural numbers `n`, `(S (S n)) ≠ 1`, where `S` denotes the successor function.


theorem Nat.sub_le_sub_right : ∀ {n m : ℕ}, n ≤ m → ∀ (k : ℕ), n - k ≤ m - k := by sorry

This theorem states that for any three natural numbers `n`, `m`, and `k`, if `n` is less than or equal to `m`, then the difference between `n` and `k` is also less than or equal to the difference between `m` and `k`. In mathematical terms, if `n ≤ m`, then `n - k ≤ m - k` for any `k`. This is a fundamental property of subtraction in the natural numbers.

For all natural numbers `n`, `m`, and `k`, if `n ≤ m`, then `n - k ≤ m - k`.


theorem fderivWithin_univ :
    ∀ {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},
      fderivWithin 𝕜 f Set.univ = fderiv 𝕜 f :=
  by sorry

This theorem, `fderivWithin_univ`, states that for any type `𝕜` with a non-trivial normed field structure, type `E` with a normed group and a normed space over `𝕜` structure, and type `F` with the same structure as `E`, for any function `f` from `E` to `F`, the derivative of `f` within the universal set is identical to the derivative of `f`. This essentially establishes that taking the derivative of a function anywhere in the universal set is the same as taking the derivative without any set restriction.

For any normed field `𝕜`, normed groups and normed spaces `E` and `F` over `𝕜`, and a function `f` from `E` to `F`, the derivative of `f` within the universal set equals the derivative of `f` without any set restriction.


theorem Nat.not_prime_one : ¬Nat.Prime 1 := by sorry

The theorem `Nat.not_prime_one` states that the number 1 is not a prime number. In the context of natural numbers, a prime number is defined as a number greater than or equal to 2 whose only divisors are itself and 1. Therefore, 1 does not meet the criteria to be considered prime.

The number 1 is not a prime number in the natural numbers.


theorem Set.smul_mem_smul_set_iff₀ :
    ∀ {α : Type u_2} {β : Type u_3} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α},
      a ≠ 0 → ∀ (A : Set β) (x : β), a • x ∈ a • A ↔ x ∈ A :=
  by sorry

This theorem states that for any arbitrary types `α` and `β`, where `α` is a group with zero and `β` is an `α`-module, and given any non-zero element `a` from `α`, any set `A` of elements of type `β`, and any element `x` from `β`, the scaled element `a • x` is an element of the scaled set `a • A` if and only if `x` is an element of `A`. This theorem establishes a correspondence between an element's membership in a set and that of its scaled version in the scaled set, under the condition that the scaling factor is non-zero.

For any group `α` with zero, `α`-module `β`, and non-zero `a` in `α`, an element `x` in `β` belongs to the scaled set `a • A` if and only if `x` is in `A`.


theorem Mathlib.Algebra.Order.Monoid.Lemmas._auxLemma.27 :
    ∀ {α : Type u_1} [inst : AddZeroClass α] [inst_1 : LT α]
      [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1]
      [inst_3 : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α}, (a + b < a) = (b < 0) :=
  by sorry

This theorem states that, for any type `α` which has an additive zero element and a less-than relation, and is both covariant and contravariant for the addition operation and the less-than relation, for any elements `a` and `b` of this type, the statement "the sum of `a` and `b` is less than `a`" is equivalent to the statement "`b` is less than zero". That is, for given `a` and `b`, `a + b < a` if and only if `b < 0`.

For any additive type `α` with a zero element and a negative element `−`, the equality `a + b < a` holds if and only if `b < −a`.


theorem CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff :
    ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      {S₁ S₂ : CategoryTheory.ShortComplex C} [inst_2 : CategoryTheory.ShortComplex.HasHomology S₁]
      [inst_3 : CategoryTheory.ShortComplex.HasHomology S₂] {φ : S₁ ⟶ S₂}
      {h₁ : CategoryTheory.ShortComplex.LeftHomologyData S₁} {h₂ : CategoryTheory.ShortComplex.LeftHomologyData S₂}
      (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂),
      CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso γ.φH :=
  by sorry

This theorem states that in the context of a category `C` equipped with zero morphisms, given two short complexes `S₁` and `S₂` that have homologies, a morphism `φ` from `S₁` to `S₂`, and left homology data `h₁` for `S₁` and `h₂` for `S₂`, then a left homology map data `γ` for `φ`, `h₁`, and `h₂` exists if and only if `φ` is a quasi-isomorphism in the short complex category if and only if the associated `φH` in the left homology map data `γ` is an isomorphism in the category `C`.

A morphism `φ` between two short complexes `S₁` and `S₂` in a category `C` equipped with zero morphisms with homologies and given left homology data `h₁` for `S₁` and `h₂` for `S₂`, has an isomorphism `γ(φ)` in the category of left homology data if and only if `φ` is a quasi-isomorphism and `γ(φ)` is an isomorphism in the category `C`.


theorem lt_div_iff' : ∀ {α : Type u_2} [inst : LinearOrderedSemifield α] {a b c : α}, 0 < c → (a < b / c ↔ c * a < b) :=
  by sorry

This theorem, `lt_div_iff'`, states that for any type `α` that is a linearly ordered semifield, and for any three elements `a`, `b`, and `c` of type `α`, if `c` is greater than zero, then `a` is less than `b` divided by `c` if and only if `c` multiplied by `a` is less than `b`. In other words, it establishes an equivalence between the comparison of `a` with the quotient `b/c` and the comparison of the product `c*a` with `b`, given the condition that `c` is positive.

For any linearly ordered semifield type `α` and elements `a`, `b`, and `c` with `c > 0`, `a < b / c` if and only if `c * a < b`.


Polynomial.Separable.map

theorem Polynomial.Separable.map :
    ∀ {R : Type u} [inst : CommSemiring R] {S : Type v} [inst_1 : CommSemiring S] {p : Polynomial R},
      Polynomial.Separable p → ∀ {f : R →+* S}, Polynomial.Separable (Polynomial.map f p) :=
  by sorry

The theorem `Polynomial.Separable.map` states that for any commutative semirings `R` and `S`, any polynomial `p` from `R`, and any ring homomorphism `f` from `R` to `S`, if the polynomial `p` is separable over `R`, then the mapped polynomial `Polynomial.map f p` is also separable over `S`. Here, a polynomial is considered separable if it is coprime with its derivative.

If `p` is a separable polynomial over a commutative semiring `R`, then `Polynomial.map f p` is a separable polynomial over the image semiring `f(R)` under a ring homomorphism `f`.


theorem AffineMap.lineMap_apply_zero :
    ∀ {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V1] [inst_2 : Module k V1]
      [inst_3 : AddTorsor V1 P1] (p₀ p₁ : P1), (AffineMap.lineMap p₀ p₁) 0 = p₀ :=
  by sorry

This theorem states that for any type `k` forming a ring, type `V1` forming an additive commutative group, and an affine space `P1` formed over the vector space `V1` and the field `k`, the affine map `lineMap`, which maps `0` to a point `p₀` and `1` to a point `p₁`, when applied to `0`, always yields `p₀`. In other words, the `lineMap` function associates the scalar `0` with the initial point of the line segment in the affine space.

For any ring `k`, additive commutative group `V1`, and affine space `P1` over `V1` and `k`, the affine map `lineMap` from `k` to `P1` satisfies `lineMap 0 = p₀`.


theorem SimpleGraph.Hom.map_adj :
    ∀ {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {v w : V},
      G.Adj v w → G'.Adj (f v) (f w) :=
  by sorry

This theorem, `SimpleGraph.Hom.map_adj`, states that for any two types `V` and `W`, and any two simple graphs `G` of type `V` and `G'` of type `W`, if there is a graph homomorphism `f` from `G` to `G'`, then for any two vertices `v` and `w` in `G`, if `v` and `w` are adjacent in `G` (i.e., there is an edge between them), then the images of `v` and `w` under `f` (i.e., `f v` and `f w`) are also adjacent in `G'`. In short, the adjacency relationship between two vertices is preserved under a graph homomorphism.

Given simple graphs G of type V and G' of type W, and a graph homomorphism f from G to G', if v and w are adjacent vertices in G, then f(v) and f(w) are adjacent vertices in G'.


theorem Polynomial.coeff_C_succ :
    ∀ {R : Type u} [inst : Semiring R] {r : R} {n : ℕ}, Polynomial.coeff (Polynomial.C r) (n + 1) = 0 := by sorry

The theorem `Polynomial.coeff_C_succ` states that for any semiring `R`, any element `r` of `R`, and any natural number `n`, the coefficient of the term `X^(n+1)` in the constant polynomial `r` is `0`. In other words, it confirms that the constant polynomial `r` in the semiring `R` does not have any terms of degree greater than `0`, by ensuring that all coefficients of higher degree terms are `0`.

The theorem `Polynomial.coeff_C_succ` asserts that the coefficient of the term `X^(n+1)` in the constant polynomial `r` over a semiring `R` is equal to `0` for any natural number `n`.


ClosedEmbedding.measurableEmbedding

theorem ClosedEmbedding.measurableEmbedding :
    ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : MeasurableSpace α] [inst_2 : BorelSpace α]
      [inst_3 : TopologicalSpace β] [inst_4 : MeasurableSpace β] [inst_5 : BorelSpace β] {f : α → β},
      ClosedEmbedding f → MeasurableEmbedding f :=
  by sorry

This theorem states that for every function `f` mapping from a type `α` to a type `β`, if `f` is a closed embedding (i.e., a homeomorphism onto its image that also maps closed sets to closed sets), then `f` is a measurable embedding. This holds under the assumptions that `α` and `β` are topological spaces with measurable spaces and their corresponding Borel sigma algebras. In other words, closed embeddings respect not only the topological structure but also the measurable structure of the spaces involved.

A closed embedding between topological spaces `α` and `β` that are also measurable spaces preserves the measurable structure, that is, it is a measurable function.


/-- The edist to a set depends continuously on the point  -/
theorem EMetric.continuous_infEdist :
    ∀ {α : Type u} [inst : PseudoEMetricSpace α] {s : Set α}, Continuous fun x => EMetric.infEdist x s := by sorry

The theorem `EMetric.continuous_infEdist` states that for any type 'α' that is a pseudo-emetric space and any set 's' of type 'α', the function that maps a point 'x' to the minimum extended distance (or edist) from 'x' to the set 's' is continuous. In simpler terms, it says that the smallest distance from any given point to a particular set varies smoothly as you move the point around.

The function that maps a point in a pseudo-metric space to the minimum extended distance from that point to a given set is continuous.


theorem List.bind_singleton : ∀ {α : Type u} {β : Type v} (f : α → List β) (x : α), List.bind [x] f = f x := by sorry

The theorem `List.bind_singleton` states that for any types `α` and `β`, for any function `f` from type `α` to a list of type `β`, and for any element `x` of type `α`, the bind operation of the list monad on the singleton list containing `x` with function `f` is equal to the result of the function `f` applied to `x`. In other words, if we apply the function `f` to each element of the singleton list `[x]` to get a list of lists, and then concatenate them all together, the result will be the same as just applying `f` to `x`.

For all types `α` and `β`, and for every function `f : α → list β`, the application of the bind operation `>>=` to the singleton list `[x]` with `f` equals the application of `f` to `x`, i.e., `[x] >>= f = f x`.


theorem Nat.one_add : ∀ (n : ℕ), 1 + n = Nat.succ n := by sorry

This theorem states that for any natural number `n`, the sum of `1` and `n` is equal to the successor of `n`. In more mathematical terms, the theorem is saying that for all natural numbers `n`, 1 + n = n + 1, where "succ" denotes the successor function (i.e., the function that gives the next natural number). This corresponds to the intuitive concept of adding one to a number to get the next number.

For all natural numbers `n`, `1 + n = succ(n)`.


theorem CategoryTheory.eqToHom_refl :
    ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] (X : C) (p : X = X),
      CategoryTheory.eqToHom p = CategoryTheory.CategoryStruct.id X :=
  by sorry

This theorem states that for any category `C` and any object `X` in that category, the morphism obtained by applying the function `CategoryTheory.eqToHom` to the reflexive equality `p` (i.e., `X = X`) is equal to the identity morphism on `X`. In other words, in any given category, the morphism created by asserting the identity of an object with itself is indeed the identity morphism for that object.

For any category `C` and object `X` in `C`, the morphism `CategoryTheory.eqToHom X X` equals the identity morphism `id X`.


theorem Finite.bddAbove_range :
    ∀ {α : Type u_1} {β : Type u_2} [inst : Finite β] [inst : Nonempty α] [inst : Preorder α]
      [inst_1 : IsDirected α fun x x_1 => x ≤ x_1] (f : β → α), BddAbove (Set.range f) :=
  by sorry

The theorem `Finite.bddAbove_range` states that for any two types `α` and `β` with `α` being a nonempty type with a preorder (a binary relation that is reflexive and transitive) and being directed (any two elements in `α` have an upper bound), and `β` being finite, for any function `f` from `β` to `α`, the range of `f` (the set of all possible outputs of `f`) is bounded above. In other words, there exists an element in `α` that is greater than or equal to every element in the range of `f`.

For any nonempty type `α` with a preorder and directed structure, and any finite type `β`, if `f` is a function from `β` to `α`, then the range of `f` has an upper bound in `α`.


theorem LinearIsometryEquiv.symm_apply_apply :
    ∀ {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]
      {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [inst_2 : RingHomInvPair σ₁₂ σ₂₁] [inst_3 : RingHomInvPair σ₂₁ σ₁₂]
      [inst_4 : SeminormedAddCommGroup E] [inst_5 : SeminormedAddCommGroup E₂] [inst_6 : Module R E]
      [inst_7 : Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (x : E), (LinearIsometryEquiv.symm e) (e x) = x :=
  by sorry

This theorem states that for any semiring `R`, another semiring `R₂`, types `E` and `E₂` representing seminormed add commutative groups, and two ring homomorphisms `σ₁₂` and `σ₂₁` that form a pair of inverse homomorphisms between `R` and `R₂`, if `e` is a linear isometry equivalence between `E` and `E₂` with respect to `σ₁₂`, then applying the inverse of this equivalence to the result of applying the equivalence to any element `x` from `E` returns the original element `x`. In simpler terms, the theorem states that the inverse operation of a linear isometry equivalence undoes the effect of the original operation.

Given semirings `R` and `R₂`, seminormed add commutative groups `E` and `E₂`, ring homomorphisms σ₁₂ and σ₂₁ forming a pair of inverse homomorphisms between `R` and `R₂`, and a linear isometry equivalence `e` between `E` and `E₂`, for all `x` in `E`, `e⁻¹(e(x)) = x`.


theorem forall_imp : ∀ {α : Sort u_1} {p q : α → Prop}, (∀ (a : α), p a → q a) → (∀ (a : α), p a) → ∀ (a : α), q a := by
  sorry

This theorem states that for any type `α` and any two predicates `p` and `q` on `α`, if for every element `a` of `α`, `p a` implies `q a`, and if `p a` holds for all `a` in `α`, then `q a` also holds for all `a` in `α`. In the language of mathematical logic, this is the principle of modus ponens applied universally: if "`p` implies `q`" is a universal truth and `p` is universally true, then `q` must also be universally true. Here, `p` and `q` could be any properties or conditions applicable to elements of the type `α`.

If `p` implies `q` and `p` holds for all elements of type `α`, then `q` holds for all elements of type `α`. (Modus ponens for universally quantified statements)


theorem Polynomial.coeff_mul_X :
    ∀ {R : Type u} [inst : Semiring R] (p : Polynomial R) (n : ℕ),
      Polynomial.coeff (p * Polynomial.X) (n + 1) = Polynomial.coeff p n :=
  by sorry

This theorem states that for any semiring `R` and for any polynomial `p` over this semiring, the coefficient of `X^(n + 1)` in the product of `p` and `X` (the polynomial variable) is the same as the coefficient of `X^n` in the polynomial `p`. In other words, multiplying a polynomial by `X` shifts all its coefficients one degree higher.

For any semiring `R` and polynomial `p(X)` over `R`, the coefficient of `X^(n+1)` in `p(X) * X` equals the coefficient of `X^n` in `p(X)`.


theorem WithTop.add_top : ∀ {α : Type u} [inst : Add α] (a : WithTop α), a + ⊤ = ⊤ := by sorry

The theorem `WithTop.add_top` states that for any given type `α` that has an addition operation (denoted by `Add α`) and any instance `a` of the type `WithTop α` (which is type `α` plus an extra top element `⊤`), the sum of `a` and `⊤` is always equal to `⊤`. Essentially, `⊤` behaves like an absorbing element for the addition operation in this context.

For any type `α` with an addition operation and instance `WithTop α` having top element `⊤`, the addition of any element `a ∈ α` with `⊤` results in `⊤`. In mathematical notation, `a + ⊤ = ⊤` for all `a ∈ α`.


theorem Filter.neBot_of_le : ∀ {α : Type u} {f g : Filter α} [hf : Filter.NeBot f], f ≤ g → Filter.NeBot g := by sorry

This theorem states that for any type `α`, and for any two filters `f` and `g` on that type, if `f` is not the bottom filter (i.e., `f` is not the filter that includes all subsets of `α`), and if `f` is less than or equal to `g` (in the filter ordering), then `g` is also not the bottom filter. In other words, the theorem asserts that a filter which is greater than or equal to a non-bottom filter cannot be the bottom filter itself.

If `α` is a type, `f` and `g` are filters on `α` with `f` non-bottom and `f ≤ g`, then `g` is also non-bottom.


theorem Polynomial.aeval_map_algebraMap :
    ∀ {R : Type u_1} (A : Type u_2) {B : Type u_3} [inst : CommSemiring R] [inst_1 : CommSemiring A]
      [inst_2 : Semiring B] [inst_3 : Algebra R A] [inst_4 : Algebra A B] [inst_5 : Algebra R B]
      [inst_6 : IsScalarTower R A B] (x : B) (p : Polynomial R),
      (Polynomial.aeval x) (Polynomial.map (algebraMap R A) p) = (Polynomial.aeval x) p :=
  by sorry

This theorem, `Polynomial.aeval_map_algebraMap`, states that for any commutative semirings `R` and `A`, a semiring `B`, and given the algebra structures over them, if there is a scalar tower of `R`, `A`, and `B`, then for any element `x` of `B` and any polynomial `p` with coefficients in `R`, applying the algebra homomorphism `Polynomial.aeval x` to the polynomial obtained by mapping `p` across the ring homomorphism `algebraMap R A` is exactly the same as applying `Polynomial.aeval x` directly to `p`. This means that the process of mapping the polynomial `p` across the algebra map does not change the result when the polynomial is evaluated at `x` in the algebra structure.

For any commutative semirings R, A, and B, and given algebra structures over them with a scalar tower, the application of the algebra homomorphism Polynomial.aeval to a polynomial obtained by mapping across the ring homomorphism algebraMap is equal to directly applying Polynomial.aeval to the polynomial.


IsBoundedLinearMap.contDiff

/-- Unbundled bounded linear functions are `C^∞`.
 -/
theorem IsBoundedLinearMap.contDiff :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
      {n : ℕ∞}, IsBoundedLinearMap 𝕜 f → ContDiff 𝕜 n f :=
  by sorry

This theorem states that for any unbundled bounded linear map `f` from a normed additive commutative group `E` over a non-trivially normed field `𝕜` to another normed additive commutative group `F` over the same field, the map `f` is infinitely differentiable (`C^∞`). In more technical terms, if `f` is a bounded linear map, then `f` is continuously differentiable any number of times (`n`), where `n` can even be infinite (`ℕ∞`).

A bounded linear map between two normed additive commutative groups over a non-trivially normed field is infinitely differentiable.


theorem DirectSum.of_zero_mul :
    ∀ {ι : Type u_1} [inst : DecidableEq ι] (A : ι → Type u_2) [inst_1 : AddZeroClass ι]
      [inst_2 : (i : ι) → AddCommMonoid (A i)] [inst_3 : DirectSum.GNonUnitalNonAssocSemiring A] (a b : A 0),
      (DirectSum.of A 0) (a * b) = (DirectSum.of A 0) a * (DirectSum.of A 0) b :=
  by sorry

This theorem states that for any type `ι` with decidable equality, a function `A` mapping from `ι` to another type, assuming `ι` is an additive zero class and `A i` is an additive commutative monoid for each `i` in `ι`, and `A` forms a generalized non-unital non-associative semiring, then for any two elements `a` and `b` of `A 0`, the value of `DirectSum.of A 0` applied to the product of `a` and `b` is the same as the product of `DirectSum.of A 0` applied to `a` and `DirectSum.of A 0` applied to `b`. In simpler terms, this theorem confirms the distributive property of multiplication over addition for `DirectSum.of A 0`.

For any additive zero class ι with decidable equality, and a function A from ι to an additive commutative monoid such that A forms a generalized non-unital non-associative semiring, the distribution law holds for DirectSum.of A 0, that is, DirectSum.of A 0 (A i j) = A i (DirectSum.of A 0 j) * A i (DirectSum.of A 0 k) for all i, j, k in ι.


theorem mul_left_injective₀ :
    ∀ {M₀ : Type u_1} [inst : Mul M₀] [inst_1 : Zero M₀] [inst_2 : IsRightCancelMulZero M₀] {b : M₀},
      b ≠ 0 → Function.Injective fun a => a * b :=
  by sorry

The theorem `mul_left_injective₀` states that for any type `M₀` equipped with a multiplication operation, a zero element, and a property that right multiplication by zero is cancelable, for any non-zero element `b` of type `M₀`, the function that multiplies an element `a` of `M₀` by `b` is injective. In other words, in such a mathematical structure, if `a * b = c * b` then `a = c` whenever `b` is not zero.

For any type `M₀` with multiplication, zero, and right zero-cancelation, the function `(× b) : M₀ → M₀` is injective, where `b` is a non-zero element.


theorem isLeast_Ici : ∀ {α : Type u} [inst : Preorder α] {a : α}, IsLeast (Set.Ici a) a := by sorry

This theorem states that for any given type `α` that has a preorder defined on it, and any element `a` of type `α`, `a` is a least element of the set of all elements `x` such that `a` is less than or equal to `x`. In mathematical terms, given a set `S` defined as {x | a ≤ x}, `a` is both a member of `S` and a lower bound for `S` under the given preorder.

For any type `α` with a preorder and any element `a` of `α`, `a` is the least element of the set `{x : α | a ≤ x}` under the given preorder.


/-- An induction principle for additive closure membership. If `p`
holds for `0` and all elements of `k`, and is preserved under addition and inverses, then `p`
holds for all elements of the additive closure of `k`. -/
theorem AddSubgroup.closure_induction :
    ∀ {G : Type u_1} [inst : AddGroup G] {k : Set G} {p : G → Prop} {x : G},
      x ∈ AddSubgroup.closure k →
        (∀ x ∈ k, p x) → p 0 → (∀ (x y : G), p x → p y → p (x + y)) → (∀ (x : G), p x → p (-x)) → p x :=
  by sorry

The theorem `AddSubgroup.closure_induction` states the following induction principle for membership in an additive closure: Given a type `G` with an additive group structure, a set `k` of type `G`, and a property `p` from `G` to `Prop`, for any element `x` of `G`, if `x` belongs to the additive closure of `k`, and if property `p` holds for all elements of `k`, for `0`, is preserved under addition (i.e., if `p` holds for `x` and `y`, then it also holds for `x + y`), and is preserved under taking additive inverses (i.e., if `p` holds for `x`, then it also holds for `-x`), then `p` holds for `x`. This principle can be used to prove properties of all elements of an additive closure, given that these properties hold for the generating set and are preserved under the group operations.

Given an additive group `(G, +)` and a subset `k` of `G`, if `x` is in the additive closure of `k`, and `p(y)` holds for all `y` in `k` and is closed under addition and additive inverses, then `p(x)` holds.


theorem Real.tan_zero : Real.tan 0 = 0 := by sorry

This theorem states that the real tangent function, when evaluated at zero, is equal to zero. In mathematical terms, it can be written as $\tan(0) = 0$. It expresses a well-known property of the tangent function from trigonometry in the context of real numbers.

The real tangent function equals zero at zero: $\tan(0) = 0$.


theorem Multiset.eq_of_le_of_card_le :
    ∀ {α : Type u_1} {s t : Multiset α}, s ≤ t → Multiset.card t ≤ Multiset.card s → s = t := by sorry

This theorem states that for any two multisets `s` and `t` of any type `α`, if `s` is a sub-multiset of `t` (denoted by `s ≤ t`) and the cardinality of `t` (i.e., the total number of elements in `t`, including duplicates) is less than or equal to the cardinality of `s`, then `s` is equal to `t`. This essentially means that if a multiset is a subset of another and they have the same cardinality, they must be the same multiset.

If two multisets `s` and `t` of type `α` satisfy `s ≤ t` and the cardinality of `t` is less than or equal to that of `s`, then `s = t`.


theorem MonoidHom.id_comp :
    ∀ {M : Type u_4} {N : Type u_5} [inst : MulOneClass M] [inst_1 : MulOneClass N] (f : M →* N),
      MonoidHom.comp (MonoidHom.id N) f = f :=
  by sorry

The theorem `MonoidHom.id_comp` states that for any monoid morphism `f` from a monoid `M` to another monoid `N`, the composition of `f` with the identity map on `N` is just `f` itself. Here, the monoids `M` and `N` can be any types equipped with a multiplication operation and a distinguished "one" element, and `f` is any function that preserves these structures (i.e., a monoid homomorphism). This is an instance of the general principle in mathematics that composing any function with the identity function leaves the original function unchanged.

For any monoid homomorphism `f` from monoid `M` to monoid `N`, `f ∘ id_N = f`.


theorem star_neg : ∀ {R : Type u} [inst : AddGroup R] [inst_1 : StarAddMonoid R] (r : R), star (-r) = -star r := by
  sorry

This theorem, named `star_neg`, states that for any type `R` that forms an add group and a star add monoid, the star operation applied to the negation of an element `r` of `R` equals the negation of the star of `r`. In more familiar terms, this corresponds to the mathematical property that the star operation and negation commute in some sort of way. In the language of algebra, if you have a certain structure (like a group with an additional operation, here denoted as `star`), the way negation and the `star` operation interact is specified by this theorem. The star of the negative of `r` is the same as the negative of the star of `r`.

For any add group and star add monoid `R`, the star of the additive negation of an element `r` equals the additive negation of the star of `r`: `-(star r) = star (-r)`.


theorem LieRing.of_associative_ring_bracket : ∀ {A : Type v} [inst : Ring A] (x y : A), ⁅x, y⁆ = x * y - y * x := by
  sorry

This theorem states that for any type `A` that has a ring structure, the Lie bracket of two elements `x` and `y` from `A` is defined as the difference of the product of `x` and `y` in one order and the product in the reverse order. In other words, the Lie bracket `[x, y]` is equal to `x * y - y * x`. This is a definition often used in the theory of Lie algebras, which are algebraic structures that generalize certain properties of associative algebras.

For any ring `A`, the Lie bracket `[x, y]` of elements `x` and `y` in `A` is defined as their commutator, i.e., `x * y - y * x`.


theorem List.count_eq_one_of_mem :
    ∀ {α : Type u} [inst : DecidableEq α] {a : α} {l : List α}, List.Nodup l → a ∈ l → List.count a l = 1 := by sorry

This theorem states that for any type `α` with a decidable equality relation and any element `a` of this type, if `a` is a member of a list `l` of elements of type `α` and this list `l` has no duplicates (therefore each element appears at most once in the list), then the number of occurrences of `a` in `l` is exactly 1. In other words, if we have a list with no duplicates and an element is in that list, it must appear exactly once.

For any type `α` with decidable equality and any list `l` of distinct elements of type `α`, the size of the subset of `l` equal to an element `a` is equal to 1 if and only if `a` is in `l`.


theorem csSup_of_not_bddAbove :
    ∀ {α : Type u_1} [inst : ConditionallyCompleteLinearOrder α] {s : Set α}, ¬BddAbove s → sSup s = sSup ∅ := by sorry

This theorem states that for any type `α` that is a conditionally complete linear order and for any set `s` of type `α`, if `s` is not bounded above, then the supremum (least upper bound) of `s` is equal to the supremum of the empty set.

In mathematical terms, if there does not exist an upper bound for the set `s`, the least upper bound of `s` is equal to the least upper bound of the empty set. This theorem is a statement about the properties of boundedness and the supremum in a conditionally complete linear order.

In a conditionally complete linear order, if a set has no upper bound, then its supremum equals the supremum of the empty set.


theorem ENNReal.two_ne_top : 2 ≠ ⊤ := by sorry

This theorem states that the number 2 is not equal to infinity (denoted by ⊤) in the extended non-negative real numbers, abbreviated as ENNReal. In other words, in the context of the non-negative real numbers expanded to include infinity, 2 is finite and is distinct from the concept of infinity.

The theorem asserts that 2 is distinct from infinity in the extended non-negative real numbers.


Convex.ordConnected

/-- **Alias** of the forward direction of `convex_iff_ordConnected`. -/
theorem Convex.ordConnected :
    ∀ {𝕜 : Type u_1} [inst : LinearOrderedField 𝕜] {s : Set 𝕜}, Convex 𝕜 s → Set.OrdConnected s := by sorry

The theorem `Convex.ordConnected` states that for any type `𝕜` which is a `LinearOrderedField`, and any set `s` of type `Set 𝕜`, if `s` is convex then `s` is also order-connected. Here, a set is said to be convex if for any element `x` in the set, the set is star-convex at `x`. A set is said to be order-connected if for any two elements in the set, all elements between them are also in the set.

If `𝕜` is a LinearOrderedField and `s` is a convex subset of `Set 𝕜`, then `s` is order-connected.


theorem Polynomial.degree_mul_le :
    ∀ {R : Type u} [inst : Semiring R] (p q : Polynomial R),
      Polynomial.degree (p * q) ≤ Polynomial.degree p + Polynomial.degree q :=
  by sorry

The theorem `Polynomial.degree_mul_le` states that for any type `R` that is a semiring, and for any two polynomials `p` and `q` over that semiring, the degree of the product of `p` and `q` is less than or equal to the sum of the degrees of `p` and `q`. In other words, the highest power of `X` that appears in the polynomial resulting from the multiplication of `p` and `q` cannot exceed the sum of the highest powers of `X` that appear in `p` and `q` respectively.

For any semiring R and polynomials p and q over R, the degree of the product p q is less than or equal to the sum of the degrees of p and q.


theorem IsLocalization.lift_eq :
    ∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} {S : Type u_2} [inst_1 : CommSemiring S]
      [inst_2 : Algebra R S] {P : Type u_3} [inst_3 : CommSemiring P] [inst_4 : IsLocalization M S] {g : R →+* P}
      (hg : ∀ (y : ↥M), IsUnit (g ↑y)) (x : R), (IsLocalization.lift hg) ((algebraMap R S) x) = g x :=
  by sorry

The theorem `IsLocalization.lift_eq` states that for any commutative semirings `R`, `S`, and `P`, a submonoid `M` of `R`, and an algebra structure on `R` and `S`, if `S` is a localization of `R` at `M`, and `g` is a ring homomorphism from `R` to `P` such that each element of `M` is a unit under `g`, then for every element `x` of `R`, applying the localization lift `IsLocalization.lift` to the algebra map of `x` from `R` to `S` yields the same result as applying `g` directly to `x`. In other words, the diagram commutes. The localization lift is a way of extending a map defined on `R` to a map defined on the localization of `R` at `M`.

If `R` is a commutative semiring, `M` a submonoid of `R`, `S` a localization of `R` at `M`, `g` a ring homomorphism from `R` to a ring `P` such that each element of `M` is a unit under `g`, then for all `x` in `R`, `IsLocalization.lift(algebraMap x) = g x`.


theorem Ideal.radical_isRadical :
    ∀ {R : Type u} [inst : CommSemiring R] (I : Ideal R), Ideal.IsRadical (Ideal.radical I) := by sorry

This theorem states that for any commutative semiring `R` and any ideal `I` in `R`, the radical of `I` is a radical ideal. In other words, within the context of a commutative semiring, the radical of any given ideal always contains its own radical. This radical of an ideal consists of all the elements `r` such that `r` to the power of `n` is in the ideal for some `n`.

In a commutative semiring, the radical of an ideal is an ideal that contains all elements equal to some power of elements in the ideal.


/-- To prove a property of a sum, it suffices to prove that
the property is additive and holds on summands. -/
theorem Finset.sum_induction :
    ∀ {α : Type u_3} {s : Finset α} {M : Type u_6} [inst : AddCommMonoid M] (f : α → M) (p : M → Prop),
      (∀ (a b : M), p a → p b → p (a + b)) → p 0 → (∀ x ∈ s, p (f x)) → p (Finset.sum s fun x => f x) :=
  by sorry

This theorem states that to prove a certain property of a sum over a finite set, it is sufficient to fulfill three conditions. The first condition is that this property is additive; that is, if the property holds for two elements, it also holds for their sum. The second condition is that the property holds for the additive identity (zero). The third condition is that the property holds for each element in the finite set, after it is mapped by a function. If these conditions are met, then the property holds for the sum over the finite set, where each element in the set is mapped by the function before summing.

If a property is additive, holds for the additive identity, and holds for each element mapped by a function in a finite set, then it holds for the sum of those elements mapped by the function.


theorem ProbabilityTheory.gaussianReal_of_var_ne_zero :
    ∀ (μ : ℝ) {v : NNReal},
      v ≠ 0 →
        ProbabilityTheory.gaussianReal μ v = MeasureTheory.volume.withDensity (ProbabilityTheory.gaussianPDF μ v) :=
  by sorry

This theorem asserts that for any real number `μ` and any nonnegative real number `v` different from zero, the Gaussian distribution on real numbers with mean `μ` and variance `v` is equal to the measure that is obtained by applying the `withDensity` operation on the probability density function (pdf) of the Gaussian distribution with mean `μ` and variance `v`. In simpler terms, it shows that when the variance `v` is not zero, the Gaussian distribution is defined by the density function associated with that Gaussian distribution.

For any real number `μ` and nonzero real number `v`, the Gaussian distribution with mean `μ` and variance `v` is equal to the measure obtained by applying `withDensity` to its probability density function.


theorem Polynomial.degree_map_le :
    ∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S) (p : Polynomial R),
      Polynomial.degree (Polynomial.map f p) ≤ Polynomial.degree p :=
  by sorry

The theorem `Polynomial.degree_map_le` states that for any semirings `R` and `S`, and for any ring homomorphism `f` from `R` to `S`, the degree of the polynomial obtained by mapping the polynomial `p` over `R` using `f` is less than or equal to the degree of the original polynomial `p`. This means that applying a ring homomorphism to a polynomial doesn't increase its degree.

For any semirings R and S, and ring homomorphism f from R to S, the degree of the polynomial image of p under f is less than or equal to the degree of p.


theorem Complex.arg_real_mul : ∀ (x : ℂ) {r : ℝ}, 0 < r → Complex.arg (↑r * x) = Complex.arg x := by sorry

The theorem `Complex.arg_real_mul` states that for any complex number `x` and any real number `r` greater than zero, the argument of the product of `x` and `r` (when `r` is considered as a complex number with zero imaginary part) is the same as the argument of `x`. In other words, multiplying a complex number by a positive real number doesn't change its argument. Here, the argument of a complex number is a real number in the range (-π, π] satisfying certain trigonometric conditions.

For any complex number x and positive real number r, the argument of x \* r equals the argument of x.


theorem Asymptotics.IsBigO.exists_pos :
    ∀ {α : Type u_1} {E : Type u_3} {F' : Type u_7} [inst : Norm E] [inst_1 : SeminormedAddCommGroup F'] {f : α → E}
      {g' : α → F'} {l : Filter α}, f =O[l] g' → ∃ c > 0, Asymptotics.IsBigOWith c l f g' :=
  by sorry

The theorem `Asymptotics.IsBigO.exists_pos` states that for any types `α`, `E`, `F'`, where `E` has a norm and `F'` is a semi-normed add commutative group, and for any functions `f : α → E` and `g' : α → F'`, and any filter `l` on `α`, if `f` is of the same order of magnitude as `g'` at `l` (denoted as `f =O[l] g'`), then there exists a positive real number `c` such that `f` is 'big O' of `g'` with constant `c` at `l`. In other words, `f` is asymptotically bounded by `c * g'` near the filter `l`.

If a function `f` is of the same order of magnitude as `g'` at a filter `l` on type `α`, then there exists a constant `c` such that `f(x) ≤ c * g'(x)` for all `x` in `α` near `l`.


theorem Polynomial.eval₂_at_apply :
    ∀ {R : Type u} [inst : Semiring R] {p : Polynomial R} {S : Type u_1} [inst_1 : Semiring S] (f : R →+* S) (r : R),
      Polynomial.eval₂ f (f r) p = f (Polynomial.eval r p) :=
  by sorry

The theorem `Polynomial.eval₂_at_apply` states that for any semiring `R`, any polynomial `p` from `R`, any semiring `S`, any ring homomorphism `f` from `R` to `S`, and any element `r` of `R`, evaluating `p` at `r` (under the ring homomorphism `f`) is the same as applying `f` to the result of evaluating `p` at `r`. In terms of mathematical notation, this theorem states that for all `f : R →+* S` and `r ∈ R`, we have `f(p(r)) = p(f(r))`.

For any semiring homomorphism $f$ and element $r$ in a semiring, we have $f(p(r)) = p(f(r))$ for all polynomials $p$ over the semiring.


theorem Finset.range_eq_Ico : Finset.range = Finset.Ico 0 := by sorry

This theorem states that the function `Finset.range`, which generates a set of natural numbers less than a given number `n`, is equivalent to the function `Finset.Ico` when the latter is invoked with `0` as the start of the range. In other words, it asserts that generating a set of natural numbers less than `n` is the same as generating a set of numbers in the interval `[0, n)` (including `0` but excluding `n`).

The `Finset.range` function generating natural numbers less than `n` is equivalent to the `Finset.Ico` function with start `0` generating numbers in the interval `[0, n)`.


MonoidWithZeroHom.ext

theorem MonoidWithZeroHom.ext :
    ∀ {α : Type u_2} {β : Type u_3} [inst : MulZeroOneClass α] [inst_1 : MulZeroOneClass β] ⦃f g : α →*₀ β⦄,
      (∀ (x : α), f x = g x) → f = g :=
  by sorry

This theorem, `MonoidWithZeroHom.ext`, states that for any two types `α` and `β`, where both types have a multiplication, zero, and one operation (i.e., they are instances of the `MulZeroOneClass`), if you have two monoid homomorphisms (`f` and `g`) from `α` to `β` such that for every element `x` in `α`, `f(x)` equals `g(x)`, then `f` and `g` must be the same monoid homomorphism. In other words, two monoid homomorphisms between two `MulZeroOneClass` types are equal if they map every element to the same value.

If `f` and `g` are monoid homomorphisms from a `MulZeroOneClass` type `α` to another `MulZeroOneClass` type `β` such that `f(x) = g(x)` for all `x` in `α`, then `f` and `g` are equal.


theorem Ideal.radical_eq_sInf :
    ∀ {R : Type u} [inst : CommSemiring R] (I : Ideal R), Ideal.radical I = sInf {J | I ≤ J ∧ Ideal.IsPrime J} := by
  sorry

This theorem states that for any commutative semiring `R` and any ideal `I` of `R`, the radical of the ideal `I` is equal to the infimum of the set of all prime ideals `J` of `R` that contain `I`. In other words, the radical of `I` is the lowest bound of all prime ideals that contain `I`. In terms of prime ideals, the radical of an ideal can be seen as the "intersection" of all prime ideals containing it.

The radical of an ideal in a commutative semiring is the infimum of prime ideals containing it.


theorem CompleteLattice.wellFounded_characterisations :
    ∀ (α : Type u_2) [inst : CompleteLattice α],
      List.TFAE
        [WellFounded fun x x_1 => x > x_1, CompleteLattice.IsSupFiniteCompact α, CompleteLattice.IsSupClosedCompact α,
          ∀ (k : α), CompleteLattice.IsCompactElement k] :=
  by sorry

This theorem states that for any type `α` that forms a complete lattice, the following four properties are equivalent:

1. The relation `x > x_1` is well-founded, i.e., there are no infinitely decreasing chains.
2. The complete lattice has the sup-finite compactness property, i.e., any subset has a finite subset with the same supremum.
3. The complete lattice has the sup-closed compactness property, i.e., any non-empty subset that is closed under taking suprema contains its own supremum.
4. Every element `k` of the complete lattice is a compact element, i.e., for any set with supremum above `k`, there exists a finite subset also with supremum above `k`.

In other words, for a complete lattice, these four properties - well-foundedness of the greater-than relation, sup-finite compactness, sup-closed compactness, and compactness of each element – either all hold or all fail together.

A complete lattice is well-founded if and only if it satisfies the sup-finite compactness, sup-closed compactness, and compactness properties for every element.


theorem CategoryTheory.ShortComplex.zero_τ₃ :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      (S₁ S₂ : CategoryTheory.ShortComplex C), 0.τ₃ = 0 :=
  by sorry

This theorem states that in the context of Category Theory, given any two short complexes `S₁` and `S₂` in a category `C` that has zero morphisms, the `τ₃` function of the zero object is equal to zero.

In the category with zero morphisms, the `τ₃` function of the zero object equals zero for any two short complexes `S₁` and `S₂`.


theorem CategoryTheory.ShortComplex.zero_τ₂ :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      (S₁ S₂ : CategoryTheory.ShortComplex C), 0.τ₂ = 0 :=
  by sorry

This theorem states that in the context of category theory, given any category `C` which has zero morphisms, and any two short complexes `S₁` and `S₂` in `C`, the second boundary map `τ₂` of the zero object is always equal to zero.

In any category with zero morphisms, the second boundary map of any short complex at the zero object is equal to the identity morphism, but since the category has zero morphisms, this identity is effectively equal to the zero morphism.


theorem CategoryTheory.ShortComplex.zero_τ₁ :
    ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      (S₁ S₂ : CategoryTheory.ShortComplex C), 0.τ₁ = 0 :=
  by sorry

This theorem states that for any category `C` with zero morphisms and any two short complexes `S₁` and `S₂` in `C`, the τ₁ morphism of the zero object is equal to zero. Here, `CategoryTheory.ShortComplex` refers to a short exact sequence in the category `C`, `τ₁` is a specific morphism within such a sequence, and `0` represents the zero object in the category.

For any short complexes $S\_1$ and $S\_2$ in a category $C$ with zero morphisms, the morphism $\tau\_1$ in their respective short exact sequences is equal to the zero morphism.


theorem Set.range_const : ∀ {α : Type u_1} {ι : Sort u_4} [inst : Nonempty ι] {c : α}, (Set.range fun x => c) = {c} :=
  by sorry

The theorem `Set.range_const` states that for any type `α` and for any type or sort `ι` that is nonempty, the range of a function that maps each element of `ι` to a constant `c` in `α` is a set containing only the element `c`. In other words, when a function always returns the same value, the range of this function is a set with that single value.

For any constant function from a nonempty type or sort to a type, its range is a set containing only that constant.


theorem Asymptotics.isLittleO_const_left :
    ∀ {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [inst : NormedAddCommGroup E'']
      [inst_1 : NormedAddCommGroup F''] {g'' : α → F''} {l : Filter α} {c : E''},
      (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Filter.Tendsto (norm ∘ g'') l Filter.atTop :=
  by sorry

The theorem `Asymptotics.isLittleO_const_left` states that for any types `α`, `E''`, and `F''` and for any function `g''` from `α` to `F''`, and any filter `l` on `α`, and any constant `c` of type `E''`, the function that is constantly equal to `c` is little-o of `g''` with respect to the filter `l` if and only if `c` is zero or the norm of `g''` tends to infinity with respect to the filter `l`. Here, a function `f` being little-o of `g` (denoted `f =o[l] g`) essentially means that `f` grows no faster than `g` near the points determined by the filter `l`.

For any types `α`, `E''`, and `F''`, function `g''` from `α` to `F''`, filter `l` on `α`, and constant `c` of type `E''`, the constant function equal to `c` is little-o of `g''` with respect to `l` if and only if `c` is zero or the norm of `g''` approaches infinity at points in the filter `l`.


/-- Two points are collinear.  -/
theorem collinear_pair :
    ∀ (k : Type u_1) {V : Type u_2} {P : Type u_3} [inst : DivisionRing k] [inst_1 : AddCommGroup V]
      [inst_2 : Module k V] [inst_3 : AddTorsor V P] (p₁ p₂ : P), Collinear k {p₁, p₂} :=
  by sorry

The theorem `collinear_pair` states that, for any given division ring `k`, commutative additive group `V`, and additive torsor `P`, along with any two points `p₁` and `p₂` from `P`, these two points are collinear. Here, being collinear means that the vector span of the set containing these two points, when considered as a module over the division ring, has a rank less than or equal to `1`.

Given a division ring `k`, a commutative additive group `V`, and an additive torsor `P` over `V`, any two points `p₁, p₂ ∈ P` are collinear, that is, their vector span over `k` has rank at most 1.


theorem Relation.TransGen.to_reflTransGen :
    ∀ {α : Type u_1} {r : α → α → Prop} {a b : α}, Relation.TransGen r a b → Relation.ReflTransGen r a b := by sorry

This theorem states that for any type `α` and for any binary relation `r` on `α`, if the transitive closure of `r` relates two elements `a` and `b` of `α`, then the reflexive transitive closure of `r` also relates `a` and `b`. In mathematical terms, this theorem is saying that if `a` is related to `b` through a series of steps where each step is defined by the relation `r`, then `a` is also related to `b` in a series of steps defined by `r` where we can also stay at the same element.

If `r` is a binary relation on type `α` and `a` and `b` are related in the transitive closure of `r`, then `a` and `b` are also related in the reflexive transitive closure of `r`.


theorem Nat.mul_div_assoc : ∀ {k n : ℕ} (m : ℕ), k ∣ n → m * n / k = m * (n / k) := by sorry

This theorem states that for any natural numbers `k`, `n`, and `m`, if `k` divides `n`, then the result of first multiplying `m` and `n` and then dividing by `k` is equal to the result of first dividing `n` by `k` and then multiplying by `m`. In mathematical notation, this would be represented as: if $k | n$, then $m \cdot \frac{n}{k} = \frac{m \cdot n}{k}$.

If `k` divides `n` in Lean, then `m * (n / k)` equals `(m * n) / k`.


theorem sub_lt_comm :
    ∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : LT α]
      [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}, a - b < c ↔ a - c < b :=
  by sorry

This theorem, `sub_lt_comm`, states that for any type `α` that has an additive commutative group structure and a less than operation (`<`), and that satisfies the covariant class properties with respect to addition and the less than operation, the inequality `a - b < c` holds if and only if `a - c < b` holds for any elements `a`, `b`, `c` in `α`. In other words, it says that subtracting `b` from `a` and testing whether this is less than `c` is equivalent to subtracting `c` from `a` and testing whether this is less than `b`.

For any additive commutative group `α` with a covariant `<` relation, `a - b < c` if and only if `a - c < b`.


theorem orderOf_injective :
    ∀ {G : Type u_1} [inst : Monoid G] {H : Type u_6} [inst_1 : Monoid H] (f : G →* H),
      Function.Injective ⇑f → ∀ (x : G), orderOf (f x) = orderOf x :=
  by sorry

The theorem `orderOf_injective` states that for any two types `G` and `H` that are both monoids, if there exists a monoid homomorphism `f` from `G` to `H` that is injective, then for any element `x` of `G`, the order of `f(x)` in `H` is equal to the order of `x` in `G`.

In other words, an injective monoid homomorphism preserves the order of elements. Here, the order of an element is the smallest positive integer `n` such that `x^n = 1`, or `0` if no such `n` exists (meaning that `x` is of infinite order).

If `f` is an injective monoid homomorphism from monoid `G` to monoid `H`, then for all `x` in `G`, the order of `x` in `G` equals the order of `f(x)` in `H`.


theorem IsCompact.exists_isLeast :
    ∀ {α : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [inst_2 : ClosedIicTopology α] {s : Set α},
      IsCompact s → Set.Nonempty s → ∃ x, IsLeast s x :=
  by sorry

This theorem states that for any set `s` of a given type `α`, where `α` has a linear order, topological space, and closed lower semi-continuous topology, if the set `s` is compact and nonempty, then there exists an element `x` in `s` that is the least element of the set `s`. In other words, any compact and nonempty set in such a space will always have a minimum element.

If `α` is a type with a linear order, topology, and closed lower semi-continuous topology, then any compact and nonempty subset of `α` has a least element.


theorem embedding_uLift_down : ∀ {X : Type u} [inst : TopologicalSpace X], Embedding ULift.down := by sorry

This theorem states that for every type `X` which has an instance of a Topological Space, the function `ULift.down` is an embedding. In mathematical terms, the function `ULift.down` is a homeomorphism from the topological space `ULift X` onto its image under `ULift.down`, which is a subset of `X`. An embedding is a function that preserves the topological structure, meaning the function and its inverse are both continuous. Here, `ULift.down` is such a function that extracts a value from an uplifted type `ULift α`, maintaining the topological structure.

The function `ULift.down` is a homeomorphism between the topological space `ULift X` and its image in `X`.


IsGLB.dual

theorem IsGLB.dual :
    ∀ {α : Type u} [inst : Preorder α] {s : Set α} {a : α},
      IsGLB s a → IsLUB (⇑OrderDual.ofDual ⁻¹' s) (OrderDual.toDual a) :=
  by sorry

The theorem `IsGLB.dual` states that for any type `α` with a pre-order relation, any set `s` of elements of type `α`, and any element `a` of type `α`, if `a` is a greatest lower bound of the set `s`, then the dual of `a` is the least upper bound of the dual of the set `s`. The `OrderDual.ofDual` function is used to map elements from the original order to the dual order and `OrderDual.toDual` is used to map `a` to its dual. Essentially, it is stating that taking the dual of a pre-order reverses the order of the bounds.

For any pre-order type `α` and set `s` of elements, if `a` is the greatest lower bound of `s`, then the dual of `a` is the least upper bound of the dual of `s`.


theorem Polynomial.support_monomial :
    ∀ {R : Type u} [inst : Semiring R] (n : ℕ) {a : R}, a ≠ 0 → Polynomial.support ((Polynomial.monomial n) a) = {n} :=
  by sorry

This theorem states that for any type `R` forming a semiring, any natural number `n`, and any element `a` of `R` that is not zero, the support of the monomial `a * X^n` (denoted as `Polynomial.monomial n a`) is the singleton set containing `n`. The support of a polynomial is the set of all exponents `n` for which the coefficient of `X^n` is non-zero. In other words, for a non-zero coefficient `a` and an exponent `n`, the non-zero terms of the monomial `a * X^n` only exist at the `n`th power of `X`.

For any semiring `R`, given a non-zero element `a` in `R` and a natural number `n`, the support of the monomial `a * X^n` in the polynomial ring over `R` is the singleton set `{n}`.


theorem measurable_norm :
    ∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : NormedAddCommGroup α] [inst_2 : OpensMeasurableSpace α],
      Measurable norm :=
  by sorry

The theorem `measurable_norm` states that for any type `α` that is a measurable space, a normed additive commutative group, and for which all open sets are measurable, the norm function (which assigns each element in `α` its mathematical norm) is a measurable function. This means that for every measurable set in the codomain, the preimage of this set under the norm function is also a measurable set in `α`.

If `α` is a measurable space and normed additive commutative group where all open sets are measurable, then the norm function is a measurable function.


ContinuousOn.div

theorem ContinuousOn.div :
    ∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀]
      [inst_2 : HasContinuousInv₀ G₀] [inst_3 : ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α]
      {s : Set α}, ContinuousOn f s → ContinuousOn g s → (∀ x ∈ s, g x ≠ 0) → ContinuousOn (f / g) s :=
  by sorry

The theorem `ContinuousOn.div` states that for any type `α`, and `G₀` which is a group with zero and a topological space that has continuous inversion and multiplication, if we have two functions `f` and `g` from `α` to `G₀` where `g` is never zero on a set `s`, and if both `f` and `g` are continuous on `s`, then the function that maps each element of `s` to the quotient of `f` and `g` at that point is also continuous on `s`. In other words, the division of two continuous functions is continuous, provided the denominator function is never zero.

If `f` and `g` are continuous functions from a set `s` to a group `G₀` with zero and topology having continuous inversion and multiplication, where `g` is never zero on `s`, then the function that maps each point in `s` to the quotient of `f` and `g` at that point is continuous on `s`.


/-- The direction of `⊤` is the whole module as a submodule.  -/
theorem AffineSubspace.direction_top :
    ∀ (k : Type u_1) (V : Type u_2) (P : Type u_3) [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
      [S : AddTorsor V P], AffineSubspace.direction ⊤ = ⊤ :=
  by sorry

The theorem `AffineSubspace.direction_top` states that for any ring `k`, additive commutative group `V`, and the torsor `P` over `V`, the direction of the top (`⊤`) affine subspace is the entire module as a submodule. In other words, the set of all pairwise differences of points in the top affine subspace, also known as its direction, covers the whole vector space `V`.

For any additive commutative group `V` with a ring structure `k`, and a torsor `P` over `V`, the direction of the top affine subspace `P` is the entire module `V` as a submodule of `k≃V`.


theorem continuous_projIcc :
    ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α} {h : a ≤ b} [inst_1 : TopologicalSpace α]
      [inst_2 : OrderTopology α], Continuous (Set.projIcc a b h) :=
  by sorry

This theorem states that for any type `α` that has a linear order and a topology which is an order topology, the projection of `α` onto the closed interval `[a, b]` is a continuous function, given that `a` is less than or equal to `b`. In mathematical terms, for any `α` in a topological space with an order topology, the function that maps any element to the closed interval `[a, b]` is continuous.

For any type `α` with a linear order and an order topology, the function mapping elements to their closed intervals `[a, b]` (with `a ≤ b`) is continuous.


theorem OfNat.ofNat_ne_zero :
    ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [inst_1 : CharZero R] (n : ℕ) [inst_2 : Nat.AtLeastTwo n],
      OfNat.ofNat n ≠ 0 :=
  by sorry

This theorem states that for any type `R` that is equipped with the structure of an additive monoid with a distinguished one element and has the property of "characteristic zero" (i.e., no natural number other than zero, when considered as an element of `R`, equals the zero element of `R`), and for any natural number `n` that is at least two, the natural number `n` as an element of `R` (obtained through the `OfNat.ofNat` function) is not equal to the zero element of `R`.

For any additive monoid `R` with unit and characteristic zero, the natural number `n` (as an element of `R`) is distinct from the zero element for any `n` greater than or equal to 2.


theorem or_left_comm : ∀ {a b c : Prop}, a ∨ b ∨ c ↔ b ∨ a ∨ c := by sorry

This theorem states that for any three propositions `a`, `b`, and `c`, the logical OR (`∨`) operation is left-commutative. This means that the order in which the propositions are OR'ed together does not affect the outcome: the proposition "a OR b OR c" is logically equivalent to "b OR a OR c". In other words, the result of the OR operation remains the same even when the first two propositions `a` and `b` are swapped.

For any propositions `a`, `b`, and `c`, the logical OR `(a ∨ b ∨ c)` is equivalent to `(b ∨ a ∨ c)`.


theorem Filter.Tendsto.atTop_mul_atTop :
    ∀ {α : Type u_3} {β : Type u_4} [inst : OrderedSemiring α] {l : Filter β} {f g : β → α},
      Filter.Tendsto f l Filter.atTop →
        Filter.Tendsto g l Filter.atTop → Filter.Tendsto (fun x => f x * g x) l Filter.atTop :=
  by sorry

The theorem `Filter.Tendsto.atTop_mul_atTop` states that for any ordered semiring `α`, any type `β`, any filter `l` of type `β`, and any two functions `f` and `g` from `β` to `α`, if `f` and `g` both tend towards infinity (represented by `Filter.atTop`) as the values from filter `l` go towards infinity, then the function that maps each value from filter `l` to the product of `f` and `g` at that value also tends towards infinity. In other words, the limit of the product of two functions, each tending to infinity, is also infinity.

If two functions from a filter over a type to an ordered semiring both tend towards infinity, then their product also tends towards infinity with respect to that filter.


theorem sInf_singleton : ∀ {α : Type u_1} [inst : CompleteSemilatticeInf α] {a : α}, sInf {a} = a := by sorry

This theorem, `sInf_singleton`, states that for any type `α`, which is a complete semilattice under the infimum operation, the infimum (greatest lower bound) of a singleton set containing an element 'a' of type `α` is 'a' itself. In other words, if we have a set with only one element 'a', the greatest lower bound or infimum of this set is simply the element 'a'.

For any complete semilattice `α` and its element `a`, the singleton set `{a}`, if non-empty, has `a` as its infimum.


theorem Real.exp_log_eq_abs : ∀ {x : ℝ}, x ≠ 0 → Real.exp (Real.log x) = |x| := by sorry

This theorem states that for any real number `x` that is not zero, applying the real exponential function to the real logarithm of `x` will yield the absolute value of `x`. In mathematical terms, for all real `x ≠ 0`, we have `exp(log(x)) = |x|`. This encapsulates the property that the exponential and logarithm functions are inverse of each other, with the absolute value involved due to the definition of real logarithm function in the negative real number territory.

For any non-zero real number `x`, the exponential function of the logarithm of `x` equals the absolute value of `x`: `exp(log(x)) = |x|`.


theorem Polynomial.revAt_invol : ∀ {N i : ℕ}, (Polynomial.revAt N) ((Polynomial.revAt N) i) = i := by sorry

This theorem states that for all natural numbers `N` and `i`, applying the `Polynomial.revAt N` function twice gives back the original natural number `i`. In other words, the `Polynomial.revAt N` function is an involution, meaning it is its own inverse.

For all natural numbers `N` and `i`, `Polynomial.revAt N (Polynomial.revAt N i) = i`.