theorem Mathlib.MeasureTheory.Measure.Regular._auxLemma.2 : ∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b) := by sorry

This theorem states that for any two propositions 'a' and 'b', the existence of 'b' assuming 'a', denoted as `∃ (_ : a), b`, is equivalent to the conjunction of 'a' and 'b', denoted as `a ∧ b`. In other words, 'a' being true and 'b' existing under that assumption is the same as 'a' and 'b' both being true simultaneously.

The theorem asserts that for any propositions 'a' and 'b' in Lean, `∃ (x : a), b` is logically equivalent to `a ∧ b`.


theorem Mathlib.CategoryTheory.Iso._auxLemma.14 :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {W X X' Y Z : C} (f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X')
      (g' : X' ⟶ Y) (h : Z ≅ Y),
      (CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h.inv) =
          CategoryTheory.CategoryStruct.comp f' (CategoryTheory.CategoryStruct.comp g' h.inv)) =
        (CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f' g') :=
  by sorry

The theorem, named `Mathlib.CategoryTheory.Iso._auxLemma.14`, states that for any category `C` of Type `u`, for any objects `W, X, X', Y, Z` in `C`, and for any morphisms `f: W ⟶ X`, `g: X ⟶ Y`, `f': W ⟶ X'`, `g': X' ⟶ Y`, and isomorphism `h: Z ≅ Y` in the category `C`, the composition of `f` and `g` followed by the inverse of `h` is equal to the composition of `f'` and `g'` followed by the inverse of `h` if and only if the composition of `f` and `g` is equal to the composition of `f'` and `g'`. In mathematical terms, it means `(f ⋅ (g ⋅ h.inv)) = (f' ⋅ (g' ⋅ h.inv))` if and only if `(f ⋅ g) = (f' ⋅ g')`.

For any objects and morphisms in a category, the composition of two morphisms followed by an isomorphism and its inverse is equal to the composition of two other morphisms and the same isomorphism and its inverse if and only if the original compositions are equal.


Poly.ext

/-- Extensionality for `Poly α`  -/
theorem Poly.ext : ∀ {α : Type u_1} {f g : Poly α}, (∀ (x : α → ℕ), f x = g x) → f = g := by sorry

The theorem `Poly.ext` states that for any type `α` and any two multivariate integer polynomials `f` and `g` of type `Poly α`, if `f` and `g` are equal for all functions from `α` to the natural numbers `ℕ`, then `f` is equal to `g`. This is an extensionality principle for the type of multivariate integer polynomials `Poly α`, asserting that two such polynomials are identical if they produce the same values for all possible inputs.

If two multivariate integer polynomials of type `Poly α` have the same value for all functions from `α` to the natural numbers `ℕ`, then they are equal.


theorem Function.Surjective.iInf_congr :
    ∀ {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [inst : InfSet α] {f : ι → α} {g : ι' → α} (h : ι → ι'),
      Function.Surjective h → (∀ (x : ι), g (h x) = f x) → ⨅ x, f x = ⨅ y, g y :=
  by sorry

The theorem `Function.Surjective.iInf_congr` states that for any types `α`, `ι`, and `ι'`, where `α` has an infimum defined, and any functions `f : ι → α` and `g : ι' → α`, if there exists a function `h : ι → ι'` which is surjective, and for each `x : ι`, `g (h x)` equals `f x`, then the infimum of the values of `f` over all `x` is equal to the infimum of the values of `g` over all `y`. In simpler terms, it says that if we can rearrange the inputs to the functions `f` and `g` such that they give the same outputs, then the infimum of `f`'s values is the same as the infimum of `g`'s values.

If `f : ι → α` and `g : ι' → α` have the same values for the images of a surjective function `h : ι → ι'`, then the infimum of `f` is equal to the infimum of `g`.


theorem Set.center_subset_centralizer : ∀ {M : Type u_1} [inst : Mul M] (S : Set M), Set.center M ⊆ Set.centralizer S :=
  by sorry

This theorem states that, for any type `M` equipped with a multiplication operation (represented by `Mul M`), and for any set `S` of elements of type `M`, the center of the magma `M` is a subset of the centralizer of the set `S`. In mathematical terms, this means that every element from the center of the magma (which is defined as those elements that commute with all other elements in the magma) also belongs to the centralizer of `S` (which is the set of all elements that commute with every element in `S`). Hence, every element in the center of the magma also commutes with every element in `S`.

The center of a magma is a subset of its centralizer for any set of elements.


theorem Mathlib.Data.Rat.Cast.CharZero._auxLemma.1 :
    ∀ {α : Type u_3} [inst : DivisionRing α] [inst_1 : CharZero α] {m n : ℚ}, (↑m = ↑n) = (m = n) := by sorry

This theorem states that for any type `α` which is a `DivisionRing` and has a characteristic of zero (denoted by `CharZero α`), and for any rational numbers `m` and `n`, the condition that the cast of `m` to `α` equals the cast of `n` to `α` is equivalent to `m` being equal to `n`. In other words, it says that if two rational numbers are equal, then their cast to any division ring with characteristic zero will also be equal, and vice versa.

For any division ring `α` of characteristic zero, the cast function from the rational numbers to `α` is a ring homomorphism preserving equalities.


theorem isPreconnected_connectedComponent :
    ∀ {α : Type u} [inst : TopologicalSpace α] {x : α}, IsPreconnected (connectedComponent x) := by sorry

This theorem states that, for any type `α` with a topological space structure and for any point `x` in `α`, the connected component of `x` is a preconnected set. In mathematical terms, the connected component of a point, which is the largest set containing the point where there is no non-trivial open partition, is always preconnected. This means that for any two open sets `u` and `v`, if the connected component of `x` is a subset of the union of `u` and `v` and has nonempty intersection with both `u` and `v`, then it also has nonempty intersection with the intersection of `u` and `v`.

For any topological space `(α, τ)` and point `x` in `α`, the connected component of `x` is a preconnected set, i.e., if it intersects both open sets `u` and `v` non-empty and is a subset of their union, then it also intersects their intersection non-empty.


Polynomial.SplittingField.splits

theorem Polynomial.SplittingField.splits :
    ∀ {K : Type v} [inst : Field K] (f : Polynomial K),
      Polynomial.Splits (algebraMap K (Polynomial.SplittingField f)) f :=
  by sorry

This theorem asserts that for any field `K` and for any polynomial `f` over `K`, the polynomial `f` splits over its splitting field. In other words, the polynomial `f` can be factored into linear polynomials in the splitting field of `f`. The splitting field of a polynomial is a field extension that contains all the roots of the polynomial. Specifically, the polynomial `f` splits means that it can be expressed as a product of irreducible factors in the splitting field, and since we are in a field, these irreducible factors are linear. The `algebraMap K (Polynomial.SplittingField f)` is a function mapping elements of `K` to their corresponding elements in the splitting field of `f`.

For any polynomial `f` over a field `K`, `f` splits in its splitting field, i.e., it can be expressed as a product of linear polynomials in the splitting field.


theorem Finset.disjoint_filter_filter_neg :
    ∀ {α : Type u_1} (s t : Finset α) (p : α → Prop) [inst : DecidablePred p] [inst_1 : (x : α) → Decidable ¬p x],
      Disjoint (Finset.filter p s) (Finset.filter (fun a => ¬p a) t) :=
  by sorry

The theorem `Finset.disjoint_filter_filter_neg` states that for any type `α`, any finite sets `s` and `t` of `α`, and any decidable predicate `p` on `α`, the subset of `s` where `p` holds is disjoint from the subset of `t` where `p` does not hold. In other words, there is no element in `s` that both satisfies `p` and is in `t` but does not satisfy `p`. This is equivalent to saying that the intersection of these two subsets (filtered sets) is an empty set.

For any type `α`, finite sets `s` and `t` of `α`, and decidable predicate `p` on `α`, the filters `{x : α | p x}` and `{x : α | ¬p x}` on `α` have empty intersection.


theorem Set.iInter_true : ∀ {α : Type u_1} {s : True → Set α}, Set.iInter s = s trivial := by sorry

This theorem states that for any type `α` and a function `s` from the truth value `True` to a set of elements of type `α`, the indexed intersection of this function (which represents a collection of sets) is equal to the set produced by applying the function `s` to the proof of `True`. In other words, for a set defined by a function from `True` (which is always true), the intersection of all such sets (indexed by the truth value) is simply the set associated with `True`.

For any function `s` from `True` to a type `α`, the indexed intersection of sets `{x | s (true) x}` is equal to the image of `s` applied to the proof of `True`.


theorem MeasureTheory.Measure.InnerRegular.map_of_continuous :
    ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : TopologicalSpace α]
      {μ : MeasureTheory.Measure α} [inst_2 : BorelSpace α] [inst_3 : MeasurableSpace β] [inst_4 : TopologicalSpace β]
      [inst_5 : BorelSpace β] [h : MeasureTheory.Measure.InnerRegular μ] {f : α → β},
      Continuous f → MeasureTheory.Measure.InnerRegular (MeasureTheory.Measure.map f μ) :=
  by sorry

The theorem `MeasureTheory.Measure.InnerRegular.map_of_continuous` states that for any types `α` and `β`, given that `α` is both a measurable space and a topological space, and `β` is a measurable space, a topological space and a Borel space, if `μ` is a measure on `α` that is inner regular and `f` is a continuous function from `α` to `β`, then the pushforward measure of `μ` by `f`, denoted as `MeasureTheory.Measure.map f μ`, is also inner regular. This means that the property of inner regularity of a measure is preserved under the action of continuous functions.

If `μ` is an inner regular measure on the measurable and topological space `α`, and `f` is a continuous function from `α` to the measurable and Borel space `β`, then `MeasureTheory.Measure.map f μ` is also an inner regular measure on `β`.


theorem Multiset.disjoint_comm : ∀ {α : Type u_1} {s t : Multiset α}, Multiset.Disjoint s t ↔ Multiset.Disjoint t s :=
  by sorry

The theorem `Multiset.disjoint_comm` states that for any type `α` and any two multisets `s` and `t` of that type, `s` and `t` are disjoint if and only if `t` and `s` are disjoint. In other words, the property of disjointness is commutative for multisets; the order in which we consider the two multisets does not affect whether or not they are disjoint. Disjointness here means that there is no element that belongs to both multisets.

For any type `α`, two multisets `s` and `t` of type `α` are disjoint if and only if `t` and `s` are disjoint. (That is, `s.disjoint t <--> t.disjoint s`.)


theorem nnnorm_one :
    ∀ {α : Type u_1} [inst : SeminormedAddCommGroup α] [inst_1 : One α] [inst_2 : NormOneClass α], ‖1‖₊ = 1 := by sorry

This theorem states that for any type `α` that is a semi-normed addition commutative group with a unit element and adheres to the `NormOneClass` (meaning the norm of the unit element equals one), the non-negative norm (notated as `‖1‖₊`) of the unit element (`1`) is `1`.

For any commutative semINormed additive group `α` with a unit element and `NormOneClass`, the norm of the unit element equals 1.


theorem Complex.sin_antiperiodic : Function.Antiperiodic Complex.sin ↑Real.pi := by sorry

The theorem `Complex.sin_antiperiodic` states that the complex sine function is antiperiodic with an antiperiod equal to the real number π. This means that for every complex number `x`, the value of `Complex.sin` at `x + π` is the negative of its value at `x`, i.e., `Complex.sin (x + π) = -Complex.sin x`.

The complex sine function is antiperiodic with a period of π, i.e., `Complex.sin (x + π) = -Complex.sin x` for all complex numbers `x`.


theorem Subgroup.subset_normalClosure :
    ∀ {G : Type u_1} [inst : Group G] {s : Set G}, s ⊆ ↑(Subgroup.normalClosure s) := by sorry

This theorem states that for any group `G` and any set `s` of elements from `G`, `s` is a subset of the normal closure of `s`. In other words, all elements of `s` are included in the smallest normal subgroup that contains all the conjugates of elements of `s`. This is a fundamental characteristic of the normal closure in group theory.

The normal closure of a set of elements in a group is the smallest normal subgroup containing all conjugates of those elements.


theorem ENNReal.inv_ne_zero : ∀ {a : ENNReal}, a⁻¹ ≠ 0 ↔ a ≠ ⊤ := by sorry

The theorem `ENNReal.inv_ne_zero` says that for all extended nonnegative real numbers `a`, the inverse of `a` is not equal to zero if and only if `a` is not equal to positive infinity. In other words, the only situation where the inverse of an extended nonnegative real number would be zero is if the original number was positive infinity.

The extended nonnegative real number `a` has a multiplicative inverse equal to zero if and only if `a` is not positive infinity.


theorem not_ne_iff : ∀ {α : Sort u_1} {a b : α}, ¬a ≠ b ↔ a = b := by sorry

This theorem, `not_ne_iff`, states that for any type `α` and any two elements `a` and `b` of this type, the negation of `a` being not equal to `b` is logically equivalent to `a` being equal to `b`. This essentially captures the intuitive idea that if it's not true that some `a` is not equal to some `b`, then `a` must be equal to `b`. This holds in general for any type `α`.

For all types `α` and elements `a` and `b` of type `α`, `a ≠ b` is logically equivalent to `a = b`.


theorem ack_succ_succ : ∀ (m n : ℕ), ack (m + 1) (n + 1) = ack m (ack (m + 1) n) := by sorry

The theorem `ack_succ_succ` states that for all natural numbers `m` and `n`, the value of the Ackermann function at `(m + 1, n + 1)` is equal to the Ackermann function at `(m, ack(m + 1, n))`. In other words, it is validating one of the recursive case definitions for the Ackermann function, specifically the case where both arguments are successors of some natural numbers.

The Ackermann function satisfies the recursive relation A(m + 1, n + 1) = A(m, A(m + 1, n)).


/-- Additive closure of an additive subgroup `K` equals `K` -/
theorem AddSubgroup.closure_eq : ∀ {G : Type u_1} [inst : AddGroup G] (K : AddSubgroup G), AddSubgroup.closure ↑K = K :=
  by sorry

This theorem states that for any additive group `G` and any additive subgroup `K` of `G`, the additive closure of `K` is equal to `K` itself. In Lean 4, the additive closure of a set is defined as the smallest additive subgroup that contains the set. Thus, this theorem asserts that the smallest additive subgroup containing an existing additive subgroup is the subgroup itself, which intuitively makes sense as the subgroup is already closed under addition. In mathematical terms, if `G` is an additive group and `K` is a subgroup of `G`, then the closure of `K` in `G` is `K`.

The additive closure of an additive subgroup of an additive group is equal to the subgroup itself.


theorem Matrix.det_diagonal :
    ∀ {n : Type u_2} [inst : DecidableEq n] [inst_1 : Fintype n] {R : Type v} [inst_2 : CommRing R] {d : n → R},
      Matrix.det (Matrix.diagonal d) = Finset.prod Finset.univ fun i => d i :=
  by sorry

The theorem `Matrix.det_diagonal` states that for any type `n` with decidable equality and finiteness, and any commutative ring `R`, given a function `d : n → R`, the determinant of a diagonal matrix created from `d` is equal to the product of all elements outputted by `d` when applied to each element of the universal finite set of `n`. In simpler terms, it's saying that the determinant of a diagonal matrix is the product of its diagonal elements.

For any commutative ring R and type n with decidable equality and finiteness, the determinant of a diagonal matrix with diagonal entries d : n → R equals the product of the images of d applied to all elements of the universal finite set of n.


theorem Function.iterate_id : ∀ {α : Type u} (n : ℕ), id^[n] = id := by sorry

This theorem, named `Function.iterate_id`, states that for any type `α` and natural number `n`, iterating the identity function (`id`) `n` times is equal to the identity function itself. In other words, no matter how many times you apply the identity function, the result is always the identity function.

For any type `α` and natural number `n`, the repeated application of the identity function `n` times equals the identity function itself. (function composition) `id^n = id`


theorem mul_self_pos :
    ∀ {α : Type u} [inst : LinearOrderedSemiring α] [inst_1 : ExistsAddOfLE α] {a : α}, 0 < a * a ↔ a ≠ 0 := by sorry

This theorem states that for any type `α` that is a linear ordered semiring and supports the existence of an addition operation for less than or equal to comparisons, the square of any instance `a` from this type is greater than zero if and only if `a` is not equal to zero. In mathematical terms, for any `a` in a linear ordered semiring, 0 < a² if and only if `a` ≠ 0.

In a linear ordered semiring, the square of any non-zero element is positive.


theorem inv_involutive : ∀ {G : Type u_3} [inst : InvolutiveInv G], Function.Involutive Inv.inv := by sorry

This theorem states that for any type `G` that has the structure of `InvolutiveInv`, the operation `Inv.inv`, which inverts elements of `G`, is an involutive function. In other words, applying `Inv.inv` twice to any element of `G` will give back the original element. In the language of mathematics, if `Inv.inv` is denoted as `f`, then for all `x` in `G`, `f(f(x)) = x`.

For any type `G` with an `InvolutiveInv` structure, the involution function `Inv.inv` is an involution, that is, `Inv.inv (Inv.inv x) = x` for all `x` in `G`.


theorem Polynomial.aroots_def :
    ∀ {T : Type w} [inst : CommRing T] (p : Polynomial T) (S : Type u_1) [inst_1 : CommRing S] [inst_2 : IsDomain S]
      [inst_3 : Algebra T S], Polynomial.aroots p S = Polynomial.roots (Polynomial.map (algebraMap T S) p) :=
  by sorry

The theorem `Polynomial.aroots_def` states that for any given polynomial `p` with coefficients in a commutative ring `T`, and any `T`-algebra `S`, the multiset of roots of `p` when regarded as a polynomial over `S` (as denoted by `Polynomial.aroots p S`) is equal to the multiset of roots of the polynomial obtained by mapping `p` via the algebraic structure from `T` to `S` (as denoted by `Polynomial.roots (Polynomial.map (algebraMap T S) p)`). In other words, mapping a polynomial to an algebra and then finding its roots is the same as finding the roots of the original polynomial in the context of that algebra.

For any polynomial `p` over a commutative ring `T` and `T`-algebra `S`, the multiset of roots of `p` over `S` equals the multiset of roots of the polynomial obtained by mapping `p` to `S`. (Equivalently, `Polynomial.aroots p S` = `Polynomial.roots (Polynomial.map (algebraMap T S) p)`)


theorem lt_mul_of_one_lt_right :
    ∀ {α : Type u_1} {a b : α} [inst : MulOneClass α] [inst_1 : Zero α] [inst_2 : Preorder α]
      [inst_3 : PosMulStrictMono α], 0 < a → 1 < b → a < a * b :=
  by sorry

The theorem `lt_mul_of_one_lt_right` states that for any type `α` that has a multiplication operation, a multiplicative identity (one), a zero, and a preorder relation (a binary relation that is reflexive and transitive), and also obeys the property of strict monotonicity of multiplication by positive elements on the left, if we have two elements `a` and `b` of this type such that `a` is positive and `b` is greater than the multiplicative identity, then `a` is less than the product of `a` and `b`. In simpler terms, if `a` is a positive number and `b` is greater than one, then `a` is less than `a * b`.

For any type with a multiplication operation, multiplicative identity, zero, preorder relation, and strict monotonicity of left multiplication by positives, if `a` is positive and `b` is greater than the identity, then `a` is strictly less than `a * b`.


theorem eq_natCast :
    ∀ {R : Type u_3} {F : Type u_5} [inst : NonAssocSemiring R] [inst_1 : FunLike F ℕ R] [inst_2 : RingHomClass F ℕ R]
      (f : F) (n : ℕ), f n = ↑n :=
  by sorry

The theorem `eq_natCast` states that for any type `R` that forms a non-associative semiring, for any type `F` that is function-like from natural numbers to `R` and for any ring homomorphism class from natural numbers to `R`, any function `f : F` when evaluated at a natural number `n` is equal to the cast of `n` to `R`. In other words, this theorem asserts that for these given conditions, the action of the function `f` on natural numbers is equivalent to just casting the natural numbers to the type `R`.

For any non-associative semiring type R, any function-like F from natural numbers to R, and any ring homomorphism f : F, the evaluation of f at a natural number n equals the cast of n to R.


/-- If the algebra map `R →+* A` is injective then `A` has the same characteristic as `R`.  -/
theorem charP_of_injective_algebraMap :
    ∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A],
      Function.Injective ⇑(algebraMap R A) → ∀ (p : ℕ) [inst : CharP R p], CharP A p :=
  by sorry

The theorem `charP_of_injective_algebraMap` states that for any types `R` and `A`, where `R` is a commutative semiring and `A` is a semiring with an algebra structure over `R`, if the algebra map from `R` to `A` (which is a map induced by the ring homomorphism defined by the Algebra structure) is injective, then whenever `R` has a certain characteristic `p`, `A` will also have the same characteristic `p`. In mathematical terms, the characteristic of a ring is the smallest number of times you need to add the multiplicative identity, 1, together to get the additive identity, 0. If no such number exists, we say the ring has characteristic 0. Therefore, this theorem is stating a property about how the characteristics of rings are preserved under injective ring homomorphisms.

If `R` is a commutative semiring with characteristic `p`, `A` is a semiring with an algebra structure over `R`, and the algebra map from `R` to `A` is injective, then `A` also has characteristic `p`.


theorem Filter.Tendsto.const_mul :
    ∀ {α : Type u_2} {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : Mul M] [inst_2 : ContinuousMul M] (b : M)
      {c : M} {f : α → M} {l : Filter α},
      Filter.Tendsto (fun k => f k) l (nhds c) → Filter.Tendsto (fun k => b * f k) l (nhds (b * c)) :=
  by sorry

The theorem `Filter.Tendsto.const_mul` states that for any type `α`, any type `M` that is a topological space and has multiplication and continuous multiplication defined on it, a constant `b` of type `M`, a function `f` from `α` to `M`, and a filter `l` on `α`, if the function `f` tends towards a neighborhood of a constant `c` (in the sense that the pre-image under `f` of any neighborhood of `c` is a neighborhood under `l`), then the function which maps `k` to the product of `b` and `f(k)` also tends towards the neighborhood of the product of `b` and `c`. In other words, if a function approaches a certain value in a topological space, then multiplying the function by a constant results in a new function that approaches the product of the constant and the original limit.

If a continuous function from a topological space to itself tends to a constant under a filter, then the constant multiplication of the function with another constant tends to the product of the constants.


theorem Mathlib.Data.Finset.Basic._auxLemma.95 : ∀ {a b c : Prop}, ((a ∧ b) ∧ c) = ((a ∧ c) ∧ b) := by sorry

This theorem states that for any three propositions `a`, `b`, and `c`, the logical conjunction of `a`, `b`, and `c` (in the order `a`, `b`, `c`) is equivalent to the logical conjunction of `a`, `c`, and `b` (in the order `a`, `c`, `b`). In other words, it states that 'and' is commutative when joining three items; the order of `b` and `c` does not affect the truth value of the entire expression. This is stated in mathematical logic as: $(a \land b) \land c = (a \land c) \land b$.

The logical conjunction of three propositions is commutative, that is, `(a ∧ b) ∧ c = a ∧ (b ∧ c)` for any propositions `a`, `b`, and `c`.


theorem Mathlib.Data.Finset.Basic._auxLemma.74 :
    ∀ {α : Type u_1} {s₁ s₂ : Finset α}, (s₁ ⊆ s₂) = ∀ ⦃x : α⦄, x ∈ s₁ → x ∈ s₂ := by sorry

This theorem, titled `Mathlib.Data.Finset.Basic._auxLemma.74`, states that for any two finite sets `s₁` and `s₂`, defined over an arbitrary type `α`, the condition that `s₁` is a subset of `s₂` is equivalent to the statement that every element `x` of `s₁` also belongs to `s₂`. In other words, if every element of `s₁` is also in `s₂`, then `s₁` is a subset of `s₂`, and vice versa.

For finite sets `s₁` and `s₂` over type `α`, `s₁ ⊆ s₂` if and only if `∀ x ∈ s₁, x ∈ s₂`.


theorem Real.hasDerivAt_sinh : ∀ (x : ℝ), HasDerivAt Real.sinh (Real.cosh x) x := by sorry

This theorem states that for all real numbers `x`, the function `Real.sinh` (the real hyperbolic sine function) has a derivative at the point `x`. The derivative of `Real.sinh` at the point `x` is given by `Real.cosh x` (the real hyperbolic cosine function evaluated at `x`). This means that `Real.cosh` is the rate of change of `Real.sinh` at any given point `x` in its domain.

The real hyperbolic sine function `Real.sinh` has a derivative equal to the real hyperbolic cosine function `Real.cosh` at every real number `x`.


theorem Mathlib.Data.Finset.Basic._auxLemma.45 : ∀ {a b c : Prop}, (a ∨ b → c) = ((a → c) ∧ (b → c)) := by sorry

This theorem, named `Mathlib.Data.Finset.Basic._auxLemma.45`, states that for any three propositional statements `a`, `b`, and `c`, the proposition that `a OR b` implies `c` is logically equivalent to the conjunction of two propositions, specifically that `a` implies `c` AND `b` implies `c`. In other words, if `c` follows from either `a` or `b`, it must follow from `a` and also from `b` separately.

The proposition `(a ∨ b) → c` is logically equivalent to `(a → c) ∧ (b → c)`.


theorem Mathlib.Data.Finset.Basic._auxLemma.51 :
    ∀ {α : Type u_1} {s t : Finset α}, Disjoint s t = ∀ ⦃a : α⦄, a ∈ s → a ∉ t := by sorry

This theorem states that for any type `α` and any two finite sets `s` and `t` of this type, `s` and `t` are disjoint if and only if for all elements `a` of type `α`, if `a` belongs to `s`, then `a` does not belong to `t`. In other words, no element can be in both `s` and `t` at the same time if the sets are disjoint. This is a generalization of the concept of disjoint sets in set theory.

For any type `α` and finite sets `s` and `t` of type `α`, `s` and `t` are disjoint if and only if no element of `α` belongs to both `s` and `t`.


theorem Mathlib.Data.Finset.Basic._auxLemma.23 : ∀ {α : Type u_1} {s : Finset α}, (s.val = 0) = (s = ∅) := by sorry

This theorem, titled `_auxLemma.23` from the module `Mathlib.Data.Finset.Basic`, states that for any type `α` and any finite set `s` of type `α`, the value of `s` being zero is equivalent to `s` being an empty set. In other words, in the context of finite sets, a set is empty if and only if its value is zero.

A finite set `s` of type `α` is empty if and only if its value is zero.


theorem Function.Injective.Prod_map :
    ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ} {g : β → δ},
      Function.Injective f → Function.Injective g → Function.Injective (Prod.map f g) :=
  by sorry

The theorem `Function.Injective.Prod_map` states that for any four types `α`, `β`, `γ`, `δ` and two functions `f : α → γ` and `g : β → δ`, if function `f` is injective and function `g` is also injective, then the function `Prod.map f g` is also injective. Here, `Prod.map f g` is a function that applies `f` to the first component and `g` to the second component of a pair from the Cartesian product of `α` and `β`. In mathematical terms, an injective function (or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. Hence, the theorem essentially states that the injectivity property is preserved under the operation of forming Cartesian products with functions.

If functions `f : α → γ` and `g : β → δ` are injective, then the function `Prod.map f g` that maps `(x, y)` to `(f x, g y)` is also injective.


theorem ENNReal.coe_nat : ∀ (n : ℕ), ↑↑n = ↑n := by sorry

This theorem states that for any natural number `n`, the process of first casting `n` to a non-negative real number (`ENNReal`), and then casting the result to a real number results in the same value as directly casting `n` to a real number. In other words, the two-step casting process doesn't change the value of `n` in any way when compared to direct casting. This is represented symbolically by the equality `↑↑n = ↑n`.

For all natural numbers `n`, the double casting from `Nat` to `Real` (`↑↑n`) is equal to the direct casting from `Nat` to `Real` (`↑n`).


theorem nsmul_le_nsmul_left :
    ∀ {M : Type u_3} [inst : AddMonoid M] [inst_1 : Preorder M]
      [inst_2 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} {n m : ℕ},
      0 ≤ a → n ≤ m → n • a ≤ m • a :=
  by sorry

This theorem, `nsmul_le_nsmul_left`, states that for any type `M` that is an additive monoid and has an order (`Preorder`) and covariance property, given any element `a` of `M` and two natural numbers `n` and `m`, if `a` is nonnegative (`0 ≤ a`) and `n` is less than or equal to `m` (`n ≤ m`), then multiplying `a` by `n` (denoted as `n • a`) is less than or equal to multiplying `a` by `m` (denoted as `m • a`). This is essentially a generalized version of the property that in any ordered monoid, multiplication respects the order of its operands.

For any additive monoid M with order and covariance property, if 0 ≤ a and n ≤ m in N, then n • a ≤ m • a.


theorem Primrec.option_bind :
    ∀ {α : Type u_1} {β : Type u_2} {σ : Type u_5} [inst : Primcodable α] [inst_1 : Primcodable β]
      [inst_2 : Primcodable σ] {f : α → Option β} {g : α → β → Option σ},
      Primrec f → Primrec₂ g → Primrec fun a => Option.bind (f a) (g a) :=
  by sorry

The `Primrec.option_bind` theorem states that for any types `α`, `β`, and `σ` that are `Primcodable` (meaning that there exists a way to encode objects of these types as natural numbers), if we have a primitive recursive function `f` from `α` to `Option β`, and a binary primitive recursive function `g` from `α` and `β` to `Option σ`, then the function that takes `a` in `α` and binds the result of `f a` to the second argument of `g a` using the `Option.bind` operation is also primitive recursive. This theorem essentially confirms that the `Option.bind` operation preserves the property of primitive recursion.

If `f` is a primitive recursive function from `α` to `Option β`, and `g` is a binary primitive recursive function from `α` and `β` to `Option σ`, then the function `x ↦ Option.bind (f x) g` is primitive recursive.


theorem Finsupp.toMultiset_single : ∀ {α : Type u_1} (a : α) (n : ℕ), (Finsupp.toMultiset fun₀ | a => n) = n • {a} := by
  sorry

The theorem `Finsupp.toMultiset_single` states that for any type `α`, any element `a` of that type, and any natural number `n`, the transformation `Finsupp.toMultiset` applied to the function that maps `a` to `n` (and anything else to zero) is equivalent to `n` copies of the multiset containing just `a`. In other words, creating a function that maps only one element to `n` and then converting this to a Multiset via `Finsupp.toMultiset` is the same as creating a multiset with `n` occurrences of that element.

For any type `α`, the function mapping an element `a` of `α` to natural number `n`, and everything else to zero, and its corresponding multiset obtained via `Finsupp.toMultiset`, both contain `n` occurrences of `a`.


theorem Hyperreal.isSt_refl_real : ∀ (r : ℝ), Hyperreal.IsSt (↑r) r := by sorry

This theorem states that for every real number `r`, `r` is the standard part of its hyperreal extension. In the context of hyperreal numbers, the standard part of a hyperreal number is the real number that is infinitesimally close to it. The `Hyperreal.IsSt` function captures this concept, and the theorem `Hyperreal.isSt_refl_real` asserts that the standard part of the hyperreal extension of a real number `r` is `r` itself. In other words, every real number is infinitesimally close to its hyperreal counterpart.

Every real number is equal to its standard part in its hyperreal extension.


theorem Matrix.mul_eq_one_comm :
    ∀ {n : Type u'} {α : Type v} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing α] {A B : Matrix n n α},
      A * B = 1 ↔ B * A = 1 :=
  by sorry

This theorem states that, for any type `n`, type `α`, where `n` is a finite type, `n` has decidable equality, and `α` is a commutative ring, if `A` and `B` are square matrices with entries in `α` and dimensions indexed by `n`, then the product of `A` and `B` equals the identity matrix if and only if the product of `B` and `A` equals the identity matrix. In other words, `A` and `B` are inverses with respect to matrix multiplication in either order.

For any finite type `n` and commutative ring `α`, if `n` has decidable equality and square matrices `A` and `B` of dimension `n` over `α` have equal products with the identity matrix in either order, then `A` and `B` are inverses of each other with respect to matrix multiplication.


theorem dist_ne_zero : ∀ {γ : Type w} [inst : MetricSpace γ] {x y : γ}, dist x y ≠ 0 ↔ x ≠ y := by sorry

This theorem states that for any type `γ` that is a metric space, the distance between two points `x` and `y` of type `γ` is not equal to zero if and only if `x` and `y` are not the same point. Essentially, in any metric space, two different points will always have a non-zero distance between them.

In any metric space, two distinct points have positive distance.


/-- The image under a monoid homomorphism of the subgroup generated by a set equals the subgroup
generated by the image of the set.  -/
theorem MonoidHom.map_closure :
    ∀ {G : Type u_1} [inst : Group G] {N : Type u_5} [inst_1 : Group N] (f : G →* N) (s : Set G),
      Subgroup.map f (Subgroup.closure s) = Subgroup.closure (⇑f '' s) :=
  by sorry

The theorem `MonoidHom.map_closure` states that for any two groups `G` and `N`, and a monoid homomorphism `f` from `G` to `N`, the image under the homomorphism `f` of the subgroup generated by a set `s` in `G` is equal to the subgroup generated by the image of the set `s` under `f` in `N`. In other words, if you have a subgroup generated by a set in one group and you map that subgroup through a group homomorphism, you get the same result as if you first mapped the original set through the homomorphism and then generated a subgroup in the target group. This theorem captures an important property of group homomorphisms and their interaction with the operation of taking the subgroup generated by a set.

For any group homomorphism between groups G and N, and any subset s of G, the image under the homomorphism of the subgroup generated by s in G is equal to the subgroup generated by the image of s in N.


AddGroupSeminorm.ext

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

This theorem states that for any type `E` that has a structure of an additive group, if we have two additive group semi-norms `p` and `q`, then if for every element `x` in `E` the semi-norm `p` of `x` is equal to the semi-norm `q` of `x`, we can conclude that `p` and `q` are the same semi-norm. In other words, two additive group semi-norms on a given additive group are equal if and only if they give the same value for every element of the group.

Two additive group semi-norms on an additive group are equal if and only if they agree on all group elements.


theorem CategoryTheory.Limits.KernelFork.condition_assoc :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
      {X Y : C} {f : X ⟶ Y} (s : CategoryTheory.Limits.KernelFork f) {Z : C} (h : Y ⟶ Z),
      CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Fork.ι s) (CategoryTheory.CategoryStruct.comp f h) =
        CategoryTheory.CategoryStruct.comp 0 h :=
  by sorry

The theorem `CategoryTheory.Limits.KernelFork.condition_assoc` states that for any category `C` that has zero morphisms, given objects `X`, `Y`, and `Z` in the category `C`, a morphism `f` from `X` to `Y`, and a kernel fork `s` of `f`, if we take another morphism `h` from `Y` to `Z`, the composition of the morphism from the fork `s` to `Y` with the composition of `f` and `h` is equal to the composition of the zero morphism and `h`. In mathematical terms, it states that $(\text{Fork}.ι \, s) \circ (f \circ h) = 0 \circ h$.

For any category with zero morphisms, given a morphism `f: X → Y`, a kernel fork `s` of `f`, and a morphism `h: Y → Z`, we have `(s ∘ f) ∘ h = 0 ∘ h`.


theorem List.rotate_rotate :
    ∀ {α : Type u} (l : List α) (n m : ℕ), List.rotate (List.rotate l n) m = List.rotate l (n + m) := by sorry

The theorem `List.rotate_rotate` states that for any list `l` of elements of some type `α`, and for any two natural numbers `n` and `m`, rotating `l` by `n` places and then by `m` places is equivalent to rotating `l` by `n + m` places. In other words, the sequence of rotations is commutative; the order of applying the rotations doesn't change the final result.

For any list `l` of length `n` and natural numbers `m` and `n`, rotating list `l` by `m` places and then by `n` places is equivalent to rotating it by `n + m` places.


theorem cauchy_nhds : ∀ {α : Type u} [uniformSpace : UniformSpace α] {a : α}, Cauchy (nhds a) := by sorry

This theorem states that for any type `α` that has a `UniformSpace` structure, and for any element `a` of type `α`, the neighborhood filter at `a` is a Cauchy filter. In terms of mathematics, it means that in any uniform space, every neighborhood of a point forms a Cauchy filter, which generalizes the concept of a Cauchy sequence to arbitrary uniform spaces.

In any uniform space, the neighborhood filter of any point is a Cauchy filter.


theorem Nat.pow_add : ∀ (a m n : ℕ), a ^ (m + n) = a ^ m * a ^ n := by sorry

This theorem, `Nat.pow_add`, states that for all natural numbers `a`, `m`, and `n`, the power of `a` raised to the sum of `m` and `n` is equal to the product of `a` raised to the power `m` and `a` raised to the power `n`. This is an important property in mathematics often referred to as the rule of exponents, specifically, the multiplication rule, which in this case, is applied to natural numbers. In mathematical notation, this theorem can be expressed as a^(m + n) = a^m * a^n.

For all natural numbers `a`, `m`, and `n`, `a^(m + n)` equals the product `a^m * a^n`.


theorem RingQuot.add_quot :
    ∀ {R : Type uR} [inst : Semiring R] (r : R → R → Prop) {a b : R},
      { toQuot := Quot.mk (RingQuot.Rel r) a } + { toQuot := Quot.mk (RingQuot.Rel r) b } =
        { toQuot := Quot.mk (RingQuot.Rel r) (a + b) } :=
  by sorry

This theorem states that for any semiring `R` and a binary relation `r` on `R`, the sum of two elements `a` and `b` in `R` after being mapped to the quotient ring is equivalent to the result obtained by first adding `a` and `b` in `R` and then mapping the sum to the quotient ring. In other words, the operation of addition commutes with the operation of taking quotients with respect to the ring quotient relation `RingQuot.Rel r`.

For any semiring `R` and relation `r` on `R`, the addition operation in the quotient ring of `R` modulo `r` commutes with the quotient operation, i.e., `(a + b) % r = a % r + b % r` for all `a, b` in `R`.


theorem map_seq :
    ∀ {α β γ : Type u} {F : Type u → Type v} [inst : Applicative F] [inst_1 : LawfulApplicative F] (f : β → γ)
      (x : F (α → β)) (y : F α), (f <$> Seq.seq x fun x => y) = Seq.seq ((fun x => f ∘ x) <$> x) fun x => y :=
  by sorry

The theorem `map_seq` is a statement about the interaction of function composition with sequential application in the context of an applicative functor. Specifically, for any types `α`, `β`, and `γ`, any applicative functor `F`, any lawful applicative `F`, any function `f` from `β` to `γ`, any value `x` of type `F (α → β)`, and any value `y` of type `F α`, the theorem states that mapping `f` over the result of applying `x` to `y` sequentially (using `Seq.seq`) is the same as sequentially applying the result of mapping the function `(fun x => f ∘ x)` over `x` to `y`. In other words, function composition can be "pulled out" of a sequential application, a fundamental property in functional programming and category theory.

For any applicative functor `F`, any lawful applicative `F`, any function `f` from `β` to `γ`, and any values `x` of type `F (α → β)` and `y` of type `F α`, `map (λ hx => f ∘ hx) x >>= fun z => f <$ y = map (λ x => f ∘ x) x >>= fun x => x >>= fun y => f <$ y`.


theorem Asymptotics.isBigO_norm_right :
    ∀ {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] fun x => ‖g' x‖) ↔ f =O[l] g' :=
  by sorry

This theorem, `Asymptotics.isBigO_norm_right`, states that for any types `α`, `E`, and `F'`, where `E` has a norm structure and `F'` has a seminormed additive commutative group structure, and given any functions `f : α → E` and `g' : α → F'`, and a filter `l` on `α`, the function `f` is big-O of the norm of `g'` at filter `l` if and only if `f` is big-O of `g'` at filter `l`. In other words, the big-O notation holds the same for a function and its normed version with respect to a specific filter.

For functions `f : α → E` and `g' : α → F'` between normed spaces `E` and seminormed additive commutative groups `F'`, and filter `l` on `α`, `f` is big-O of the norm of `g'` at filter `l` if and only if `f` is big-O of `g'` at filter `l`.


theorem contDiff_neg :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type uF} [inst_1 : NormedAddCommGroup F]
      [inst_2 : NormedSpace 𝕜 F] {n : ℕ∞}, ContDiff 𝕜 n fun p => -p :=
  by sorry

This theorem states that for any nontrivially normed field `𝕜` and any normed additive commutative group `F` that also forms a normed space over `𝕜`, the negation function (which maps each point `p` to its negation `-p`) is continuously differentiable of any order `n`, where `n` can be any natural number or infinity (`ℕ∞`). Thus, the negative of any point in this normed space is smooth, meaning it has derivatives of all orders.

For any nontrivially normed field `𝕜` and normed additive commutative group `F` over `𝕜` that is also a normed space, the negation function `- : F → F` is continuously differentiable of any order `n` (`n : ℕ∞`).


theorem Monotone.tendsto_leftLim :
    ∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : ConditionallyCompleteLinearOrder β]
      [inst_2 : TopologicalSpace β] [inst_3 : OrderTopology β] {f : α → β},
      Monotone f →
        ∀ [inst_4 : TopologicalSpace α] [inst_5 : OrderTopology α] (x : α),
          Filter.Tendsto f (nhdsWithin x (Set.Iio x)) (nhds (Function.leftLim f x)) :=
  by sorry

The theorem `Monotone.tendsto_leftLim` states that for a monotone function `f` from a linear order into a conditionally complete linear order that both have a topological structure compatible with the order (OrderTopology), the function `f` tends to its left limit as we approach any point `x` from the left. In other words, given any point `x` in the domain of `f`, if we consider the sequence of `f` values for points in the left-infinite right-open interval ending at `x` (the set of all points less than `x`), the limit of this sequence is the left limit of `f` at `x`.

Given a monotone function `f` from a linearly ordered and conditionally complete topological space into another such space, for any `x` in the domain of `f`, the sequence `{f(y) | y < x}` converges to the left limit of `f` at `x`.


theorem CategoryTheory.Limits.coprod.inr_desc :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {W X Y : C}
      [inst_1 : CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W),
      CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr (CategoryTheory.Limits.coprod.desc f g) = g :=
  by sorry

The theorem `CategoryTheory.Limits.coprod.inr_desc` states that in any category `C` which has binary coproducts, for any three objects `X`, `Y`, and `W` in `C`, and for any two morphisms `f` from `X` to `W` and `g` from `Y` to `W`, the composition of the inclusion map from the second component of the coproduct (`CategoryTheory.Limits.coprod.inr`) and the morphism induced by `f` and `g` (`CategoryTheory.Limits.coprod.desc f g`) is equal to `g`. In other words, when we map `Y` into the coproduct `X ⨿ Y` and then map to `W` using the morphism defined by `f` and `g`, we get the same result as if we had directly used `g` to map `Y` to `W`.

In any category with binary coproducts, for any objects X, Y, and W, and morphisms f : X -> W and g : Y -> W, the composition of the inclusion map of the second component of the coproduct (inr : Y -> X ⨿ Y) and the morphism induced by f and g (desc f g : Y -> W) equals g.


theorem Set.nonempty_Ico : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, Set.Nonempty (Set.Ico a b) ↔ a < b := by
  sorry

The theorem `Set.nonempty_Ico` states that for any type `α` that has a preorder relationship, and any two elements `a` and `b` of type `α`, the left-closed right-open interval `(Set.Ico a b)` is nonempty if and only if `a` is less than `b`. In other words, there exists an element in the interval from `a` to `b`, not including `b`, if and only if `a` is strictly less than `b`.

The set of elements between `a` and `b` (inclusive of `a` but exclusive of `b`) in a preordered type `α` is nonempty if and only if `a` is strictly less than `b`.


theorem Equiv.refl_apply : ∀ {α : Sort u} (x : α), (Equiv.refl α) x = x := by sorry

This theorem states that for any type `α` and any element `x` of type `α`, applying the identity equivalence on `α` to `x` results in `x` itself. In other words, an identity function when applied to any element of its domain will always return the element itself. This identity function is manifested here as an "equivalence relationship", denoted by `Equiv.refl α`, which captures the concept of reflexivity in the context of type theory.

For any type `α` and any of its elements `x`, `Equiv.refl α x` equals `x`.


theorem Metric.closedBall_subset_ball :
    ∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε₁ ε₂ : ℝ},
      ε₁ < ε₂ → Metric.closedBall x ε₁ ⊆ Metric.ball x ε₂ :=
  by sorry

The theorem `Metric.closedBall_subset_ball` states that for any type `α` that is a pseudo metric space, for any point `x` of type `α`, and for any two real numbers `ε₁` and `ε₂`, if `ε₁` is less than `ε₂`, then the closed ball centered at `x` with radius `ε₁` is a subset of the open ball centered at `x` with radius `ε₂`. In other words, all points `y` at a distance from `x` less than or equal to `ε₁` (forming a closed ball around `x`) are also at a distance from `x` less than `ε₂` (thus falling within an open ball around `x`).

For any pseudo metric space `α` and point `x` in `α`, the closed ball `Bcl(x, ε₁)` is a subset of the open ball `Bop(x, ε₂)` when `ε₁` is less than `ε₂`.


theorem Matrix.mulVec_diagonal :
    ∀ {m : Type u_2} {α : Type v} [inst : NonUnitalNonAssocSemiring α] [inst_1 : Fintype m] [inst_2 : DecidableEq m]
      (v w : m → α) (x : m), Matrix.mulVec (Matrix.diagonal v) w x = v x * w x :=
  by sorry

This theorem states that for any types `m` and `α`, where `α` is a non-unital non-associative semiring, `m` is a finite type, and `m` has decidable equality, if `v` and `w` are functions from `m` to `α` and `x` is an element of type `m`, then the result of multiplying the vector `w` by the diagonal matrix defined by `v` at index `x` is equal to the product of `v x` and `w x`.

In other words, in the context of matrix and vector multiplication, this theorem asserts that the multiplication of a vector `w` by a diagonal matrix (which only has non-zero elements along its main diagonal, as defined by function `v`) performs an element-wise multiplication of the vector `w` with the diagonal of the matrix.

For any non-unital non-associative semiring type `α`, finite type `m` with decidable equality, functions `v : m → α` and `w : m → α`, and element `x : m`, the diagonal matrix `v` multiplied by vector `w` equals the element-wise product of `v x` and `w x`.


theorem minpoly.algebraMap_eq :
    ∀ {A : Type u_1} {B' : Type u_3} [inst : CommRing A] [inst_1 : Ring B'] [inst_2 : Algebra A B'] {B : Type u_4}
      [inst_3 : CommRing B] [inst_4 : Algebra A B] [inst_5 : Algebra B B'] [inst_6 : IsScalarTower A B B'],
      Function.Injective ⇑(algebraMap B B') → ∀ (x : B), minpoly A ((algebraMap B B') x) = minpoly A x :=
  by sorry

This theorem, `minpoly.algebraMap_eq`, states that for any types `A`, `B'` and `B`, where `A` and `B` have a commutative ring structure and `B'` has a ring structure, and where `A`, `B` and `B'` have an algebra structure, with `B` and `B'` forming a scalar tower over `A`, if the algebra map from `B` to `B'` is injective, then for any element `x` of `B`, the minimum polynomial of `x` under the algebra map from `B` to `B'` is equal to the minimum polynomial of `x` in `A`.

In more simple terms, if we have two algebraic structures over a common base and a "nice" way (the injective algebra map) to go from one to the other, then the minimum polynomial of an element in the first structure is the same whether we consider it as an element of the first structure or map it to the second one.

If `A`, `B`, and `B'` are commutative rings with algebra structures over a common base, `B` and `B'` form a scalar tower over `A`, and the algebra map from `B` to `B'` is injective, then the minimum polynomial of any element in `B` under the algebra map is equal to its minimum polynomial in `A`.


theorem Polynomial.smeval_eq_sum :
    ∀ {R : Type u_1} [inst : Semiring R] (p : Polynomial R) {S : Type u_2} [inst_1 : AddCommMonoid S] [inst_2 : Pow S ℕ]
      [inst_3 : MulActionWithZero R S] (x : S), Polynomial.smeval p x = Polynomial.sum p (Polynomial.smul_pow x) :=
  by sorry

The theorem `Polynomial.smeval_eq_sum` states that for a given polynomial `p` from a semiring `R` and an element `x` from an additive commutative monoid `S` that is also equipped with a power operation and a scalar multiplication with zero, the operation of evaluating the polynomial at `x` is equivalent to summing the values of the function, which applies scalar multiplication and natural number power to `x`, on the coefficients of the polynomial `p`. In mathematical terms, if `p` is a polynomial with coefficients in `R` and `x` is in `S`, then the value of `p` at `x` is equal to the sum over all coefficients of `p` of the value at that coefficient times `x` to the power of the corresponding exponent.

For a polynomial $p$ with coefficients in a semiring $R$ and an element $x$ from an additive commutative monoid $S$ with power operation and scalar multiplication with zero, the value of $p$ at $x$ equals the sum of the products of coefficients and $x$ raised to the corresponding exponent.


theorem List.eq_of_perm_of_sorted :
    ∀ {α : Type uu} {r : α → α → Prop} [inst : IsAntisymm α r] {l₁ l₂ : List α},
      List.Perm l₁ l₂ → List.Sorted r l₁ → List.Sorted r l₂ → l₁ = l₂ :=
  by sorry

The theorem `List.eq_of_perm_of_sorted` states that for any type `α` and a binary relation `r` on `α` which is antisymmetric, given two lists `l₁` and `l₂` of elements of type `α`, if `l₁` is a permutation of `l₂` and both `l₁` and `l₂` are sorted with respect to the relation `r`, then `l₁` and `l₂` are equal. In other words, two sorted lists that are permutations of each other must be identical, given the sorting relation is antisymmetric. This theorem is important in scenarios where the order of elements matters, such as in sorting algorithms or computations related to order statistics.

Given two sorted lists `l₁` and `l₂` of elements from a type `α` with an antisymmetric relation `r`, if `l₁` is a permutation of `l₂` with respect to `r`, then `l₁ = l₂`.


theorem Part.bind_some_eq_map :
    ∀ {α : Type u_1} {β : Type u_2} (f : α → β) (x : Part α), Part.bind x (Part.some ∘ f) = Part.map f x := by sorry

This theorem, `Part.bind_some_eq_map`, states that for any types `α` and `β`, and any function `f` from `α` to `β`, and for any partial value `x` of type `α`, binding `x` to a function that applies `f` and then wraps the result in a `Part` with a `True` domain (done by `Part.some ∘ f`) is equivalent to just mapping `f` over `x` directly. In essence, the theorem establishes a link between the `bind` and `map` operations on partial values, where binding with a function that simply wraps the result is the same as mapping that function.

For any types `α` and `β`, and any function `f: α → β`, the application of `Part.bind` with the function `x => Part.some ∘ f` to a partial value `x: α` is equal to directly applying `f` to `x`.


theorem Set.Finite.image2 :
    ∀ {α : Type u} {β : Type v} {γ : Type x} {s : Set α} {t : Set β} (f : α → β → γ),
      Set.Finite s → Set.Finite t → Set.Finite (Set.image2 f s t) :=
  by sorry

This theorem states that for any types `α`, `β`, and `γ`, any sets `s` of `α` and `t` of `β`, and any binary function `f` from `α` and `β` to `γ`, if both `s` and `t` are finite, then the set formed by the image of the function `f` applied on all combinations of elements from sets `s` and `t` (mathematically considered as the image of the function `f` from `α × β` to `γ`) is also finite.

If `α`, `β`, and `γ` are types, `s` and `t` are finite sets of `α` and `β` respectively, and `f` is a binary function from `α × β` to `γ`, then the image of `f` on the cartesian product `s × t` is a finite set.


theorem List.reverse_involutive : ∀ {α : Type u}, Function.Involutive List.reverse := by sorry

The theorem `List.reverse_involutive` states that the `List.reverse` function is involutive for any type `α`. In other words, for any list of elements from type `α`, applying `List.reverse` twice returns the original list. The term "involutive" here refers to a function that is its own inverse, i.e., applying the function twice to any argument yields the original argument. This property is often written as `f(f(x)) = x` for all `x` in the domain of `f`. In this case, `f` is the `List.reverse` function.

For any list `l` of type `α`, `List.reverse (List.reverse l)` equals `l`.


Dilation.ext

theorem Dilation.ext :
    ∀ {α : Type u_1} {β : Type u_2} [inst : PseudoEMetricSpace α] [inst_1 : PseudoEMetricSpace β] {f g : α →ᵈ β},
      (∀ (x : α), f x = g x) → f = g :=
  by sorry

This theorem, `Dilation.ext`, states that for any two dilation functions `f` and `g` within a pseudo emetric space, if for all `x` in domain type `α`, `f(x)` equals `g(x)`, then `f` equals `g`. In other words, if two dilation functions produce the same output for every input in the domain, they are the same function. The pseudo emetric space ensures that the distances between points in the space adhere to certain conditions, like symmetry and triangle inequality.

In a pseudo-metric space, if two dilation functions are equal on all points in their domain, then they are identical functions.


theorem MvPolynomial.coeff_map :
    ∀ {R : Type u} {S₁ : Type v} {σ : Type u_1} [inst : CommSemiring R] [inst_1 : CommSemiring S₁] (f : R →+* S₁)
      (p : MvPolynomial σ R) (m : σ →₀ ℕ), MvPolynomial.coeff m ((MvPolynomial.map f) p) = f (MvPolynomial.coeff m p) :=
  by sorry

The theorem `MvPolynomial.coeff_map` states that for any commutative semiring `R` and `S₁`, for any index set `σ`, for any ring homomorphism `f` from `R` to `S₁`, for any multivariate polynomial `p` with coefficients in `R`, and for any monomial `m`, mapping the polynomial `p` across the ring homomorphism `f` and then taking the coefficient of the monomial `m` is the same as taking the coefficient of the monomial `m` in the original polynomial `p` and then applying the ring homomorphism `f` to it. In other words, the process of taking coefficients commutes with the process of mapping across a ring homomorphism.

For any commutative semiring homomorphism `f` and multivariate polynomial `p` over a commutative semiring `R`, the coefficient of any monomial `m` in `p` is equal to the image under `f` of the coefficient of `m` in `p`.


theorem Metric.cthickening_of_nonpos :
    ∀ {α : Type u} [inst : PseudoEMetricSpace α] {δ : ℝ}, δ ≤ 0 → ∀ (E : Set α), Metric.cthickening δ E = closure E :=
  by sorry

This theorem states that for any type `α` equipped with a `PseudoEMetricSpace` structure, and for a real number `δ` that is less than or equal to zero and any set `E` of this type `α`, the `δ`-thickening of `E` (denoted as `Metric.cthickening δ E`) is equal to the closure of `E`. In other words, if we consider all points that are at most `δ` away from `E` in terms of the pseudo emetric (where `δ` is nonpositive), we end up with the smallest closed set that contains `E`, which is the definition of the closure of `E`.

For any `PseudoEMetricSpace` `α` and `δ ≤ 0`, the `δ`-thickening of a set `E` in `α` equals the closure of `E`.


/-- **Alias** of `nndist_eq_nnnorm_sub`. -/
theorem nndist_eq_nnnorm : ∀ {E : Type u_6} [inst : SeminormedAddGroup E] (a b : E), nndist a b = ‖a - b‖₊ := by sorry

This theorem, called `nndist_eq_nnnorm`, states that for any type `E` which is a seminormed add group (a group with an additional structure called a seminorm), the non-negative distance (`nndist`) between any two elements `a` and `b` of `E` is equal to the non-negative norm (`nnnorm`) of the difference between `a` and `b`. In mathematical terms, for all `a` and `b` in `E`, `nndist a b = ‖a - b‖₊`.

For any seminormed add group `E`, the non-negative distance between two elements `a` and `b` is equal to the non-negative norm of their difference: `nndist a b = ‖a - b‖₊`.


theorem Finset.union_assoc :
    ∀ {α : Type u_1} [inst : DecidableEq α] (s₁ s₂ s₃ : Finset α), s₁ ∪ s₂ ∪ s₃ = s₁ ∪ (s₂ ∪ s₃) := by sorry

The theorem `Finset.union_assoc` states that for any type `α` that has decidable equality, and any three finite sets (`s₁`, `s₂`, `s₃`) of type `α`, the union operation (`∪`) is associative. In other words, taking the union of `s₁` and `s₂` first, and then taking the union with `s₃`, is the same as taking the union of `s₂` and `s₃` first, and then taking the union with `s₁`. In terms of set theory, this theorem states that $(s₁ ∪ s₂) ∪ s₃ = s₁ ∪ (s₂ ∪ s₃)$.

For any type `α` with decidable equality and any finite sets `s₁`, `s₂`, `s₃` of type `α`, `(s₁ ∪ s₂) ∪ s₃ = s₁ ∪ (s₂ ∪ s₃)`.


theorem Nat.leRecOn_succ :
    ∀ {C : ℕ → Sort u_1} {n m : ℕ} (h1 : n ≤ m) {h2 : n ≤ m + 1} {next : {k : ℕ} → C k → C (k + 1)} (x : C n),
      Nat.leRecOn h2 next x = next (Nat.leRecOn h1 (fun {k} => next) x) :=
  by sorry

The theorem `Nat.leRecOn_succ` is a property of recursively defined functions on natural numbers in Lean 4. Specifically, it defines how a function, denoted here as `next`, behaves when applied to a specific case of the recursive structure `Nat.leRecOn`. The theorem states that for any natural numbers `n` and `m` such that `n ≤ m`, a function `C` from natural numbers to a type, a function `next` modifying `C`, and an element `x : C n`, the result of applying `Nat.leRecOn` to `n ≤ m + 1` and `next` and `x`, is the same as applying `next` to the result of `Nat.leRecOn` applied to `n ≤ m` and the same `next` and `x`. This theorem ensures a consistent behavior of recursive functions across different values of natural numbers.

For any recursively defined function `next` on natural numbers, `Nat.leRecOn (n ≤ m + 1) (next) (x)` equals `next (Nat.leRecOn (n ≤ m) (next) (x))`, where `n ≤ m` holds and `x : C n`.


theorem LieSubmodule.mono_lie :
    ∀ {R : Type u} {L : Type v} {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
      [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] (N N' : LieSubmodule R L M)
      (I J : LieIdeal R L), I ≤ J → N ≤ N' → ⁅I, N⁆ ≤ ⁅J, N'⁆ :=
  by sorry

The theorem `LieSubmodule.mono_lie` states that for any commutative ring `R`, Lie ring `L`, additive commutative group `M`, and the related Lie algebra, module, and Lie ring module structures, given two Lie submodules `N` and `N'` of `M` and two Lie ideals `I` and `J` of `L`, if `I` is a subset of `J` and `N` is a subset of `N'`, then the Lie bracket of `I` and `N` is also a subset of the Lie bracket of `J` and `N'`. This theorem is a statement about the monotonicity of the Lie bracket in the context of Lie algebras and Lie submodules.

If `I` is a Lie ideal contained in `J` and `N` is a Lie submodule of `N'`, then the Lie bracket of `I` and `N` is contained in the Lie bracket of `J` and `N'`.


theorem AddMonoidHomClass.isometry_of_norm :
    ∀ {𝓕 : Type u_1} {E : Type u_6} {F : Type u_7} [inst : SeminormedAddGroup E] [inst_1 : SeminormedAddGroup F]
      [inst_2 : FunLike 𝓕 E F] [inst_3 : AddMonoidHomClass 𝓕 E F] (f : 𝓕), (∀ (x : E), ‖f x‖ = ‖x‖) → Isometry ⇑f :=
  by sorry

This theorem, named `AddMonoidHomClass.isometry_of_norm`, states that for any types `𝓕`, `E`, and `F`, where `E` and `F` are seminormed add groups, `𝓕` is a function-like object from `E` to `F`, and `𝓕` is an additive monoid homomorphism class, if a function `f` of type `𝓕` satisfies the property that the norm of `f(x)` is equal to the norm of `x` for all `x` in `E`, then this function `f` is an isometry. In other words, it is a map that preserves the distance (in this case, the edistance) between elements of the pseudometric spaces `E` and `F`.

If `𝓕` is an additive monoid homomorphism class from a seminormed add group `E` to another seminormed add group `F`, such that `𝓕` preserves the norms of elements in `E`, then `𝓕` is an isometry between the pseudometric spaces `(E, edist)` and `(F, edist')`, where `edist` and `edist'` denote the respective edistances.


theorem Filter.iInter_mem :
    ∀ {α : Type u} {f : Filter α} {β : Sort v} {s : β → Set α} [inst : Finite β], ⋂ i, s i ∈ f ↔ ∀ (i : β), s i ∈ f :=
  by sorry

This theorem states that for any type `α`, any filter `f` of type `Filter α`, and any function `s` mapping from type `β` to the set of `α`, given that `β` is finite, the intersection over all `i` of `s i` is in `f` if and only if for all `i` in `β`, `s i` is in `f`. In simpler terms, it's telling us that the condition of all sets `s i` being in a given filter `f` is equivalent to the intersection of all such sets being in filter `f`.

For any filter `f` on type `α`, any finite type `β`, and function `s` from `β` to `α`, if for all `i` in `β`, `s i` is in `f`, then the intersection of `s i` for all `i` is in `f`. Conversely, if the intersection of `s i` is in `f`, then each `s i` is in `f`.


theorem Metric.mem_uniformity_dist :
    ∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set (α × α)},
      s ∈ uniformity α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s :=
  by sorry

This theorem states that for any type `α` that has an instance of a PseudoMetricSpace and any set `s` of ordered pairs from `α`, the set `s` is an element of the uniformity of `α` if and only if there exists a positive real number `ε` such that for any two elements `a` and `b` of `α`, if the distance between `a` and `b` is less than `ε`, then the ordered pair `(a, b)` is an element of `s`. In other words, it describes the conditions under which a set of pairs of points in a pseudometric space can be considered uniformly close with respect to some positive real number.

A set of ordered pairs from a pseudometric space is an element of its uniformity if and only if it contains all ordered pairs of points with distance less than some positive real number.


theorem Nat.one_ne_zero : 1 ≠ 0 := by sorry

This theorem, `Nat.one_ne_zero`, asserts that one is not equal to zero in the natural numbers. It's a fundamental concept in arithmetic and number theory, stating that the numbers one and zero are distinct. This is a basic property of the natural number system, and it's used often in mathematical reasoning and proofs.

The natural number one is not equal to zero.


/-- The uniformity of `α →ᵤ β` admits the family `{(f, g) | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as
a filter basis, for any basis `𝓑` of `𝓤 β` (in the case `𝓑 = (𝓤 β).as_basis` this is true by
definition).  -/
theorem UniformFun.hasBasis_uniformity_of_basis :
    ∀ (α : Type u_1) (β : Type u_2) [inst : UniformSpace β] {ι : Sort u_5} {p : ι → Prop} {s : ι → Set (β × β)},
      Filter.HasBasis (uniformity β) p s → Filter.HasBasis (uniformity (UniformFun α β)) p (UniformFun.gen α β ∘ s) :=
  by sorry

This theorem asserts that the uniformity of the uniform space of functions from `α` to `β` (`α →ᵤ β`), has a filter basis `{(f, g) | ∀ x, (f x, g x) ∈ V}` for any `V` belonging to a basis `𝓑` of the uniformity `𝓤 β`. This filter basis is created through the operation `UniformFun.gen α β ∘ s` which maps every `s` in `𝓑` to a set of pairs of functions `(f, g)` such that for every `x` in `α`, the pair `(f(x), g(x))` belongs to `V`. This statement holds true especially when `𝓑` is the set of basis elements of the uniformity `𝓤 β`, i.e., `𝓑 = (𝓤 β).as_basis`.

The uniformity of functions from `α` to `β` has a filter basis consisting of pairs of functions `(f, g)` such that for all `x in α`, `(f x, g x)` belongs to any given basis element of the uniformity of `β`.


theorem Polynomial.toFinsupp_smul :
    ∀ {R : Type u} [inst : Semiring R] {S : Type u_1} [inst_1 : SMulZeroClass S R] (a : S) (b : Polynomial R),
      (a • b).toFinsupp = a • b.toFinsupp :=
  by sorry

This theorem states that for any type `R` with a semiring structure, and any `S` that forms a scalar multiplication zero class with `R`, given any scalar `a` of type `S` and any polynomial `b` with coefficients in `R`, the operation of converting the result of scalar multiplication of `a` and `b` to a `finsupp` representation (which maps each polynomial degree to its coefficient) is equivalent to first converting `b` to a `finsupp` representation and then performing the scalar multiplication. In essence, it says that scalar multiplication commutes with the conversion to the `finsupp` representation.

For any semiring `R`, scalar multiplication zero class `S`, scalar `a` of type `S`, and polynomial `b` with coefficients in `R`, the operation of converting `a * b` to `finsupp` is equivalent to first converting `b` to `finsupp` and then performing scalar multiplication.


theorem Action.hom_ext :
    ∀ {V : Type (u + 1)} [inst : CategoryTheory.LargeCategory V] {G : MonCat} {M N : Action V G} (φ₁ φ₂ : M ⟶ N),
      φ₁.hom = φ₂.hom → φ₁ = φ₂ :=
  by sorry

The theorem `Action.hom_ext` states that for any large category `V`, any monoid `G` in the category of monoids `MonCat`, and any two action objects `M` and `N` in the category of actions over `V` and `G`, if two morphisms `φ₁` and `φ₂` from `M` to `N` have the same underlying function (i.e., `φ₁.hom = φ₂.hom`), then `φ₁` and `φ₂` are the same morphism. This is a property of extensionality for morphisms in the category of actions.

In the category of actions over a large category and a monoid, morphisms with equal underlying functions are equal.


theorem List.prod_nil : ∀ {M : Type u_5} [inst : MulOneClass M], List.prod [] = 1 := by sorry

This theorem states that the product of an empty list, regardless of what type `M` of elements it might contain, is 1 - provided that `M` is a type for which the operations of multiplication and identity element (one) are defined. Specifically, the `List.prod []` function uses a left fold (`List.foldl`) operation to multiply all elements of the list, starting with the identity element (one). When the list is empty, this function simply returns the starting value, which is 1.

The product of an empty list of any type `M` with defined multiplication and identity element is equal to 1.


theorem ContDiffBump.rOut_pos : ∀ {E : Type u_1} {c : E} (f : ContDiffBump c), 0 < f.rOut := by sorry

This theorem, `ContDiffBump.rOut_pos`, states that for any type `E`, any element `c` of `E`, and any continuously differentiable 'bump' function `f` centered at `c`, the radius `rOut` (the outer radius of the support of `f`) is always positive. Essentially, it ensures that the bump function, which is nonzero only in a certain neighborhood of its center, actually 'extends' beyond the center point in the mathematical space `E`.

For any continuously differentiable bump function $f$ centered at an element $c$ in a type $E$, the outer radius of its support is positive.


Specializes.map

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

This theorem states that for any two types `X` and `Y`, given that `X` and `Y` are both topological spaces, and given a function `f` from `X` to `Y` and any two elements `x` and `y` of `X`, if `x` specializes to `y` and `f` is a continuous function, then `f(x)` also specializes to `f(y)`. This theorem captures the idea that the specialization relation, which is a form of ordering in topological spaces, preserves continuity under function mapping.

If `f` is a continuous function between topological spaces `X` and `Y`, and `x` specializes to `y` in `X`, then `f(x)` specializes to `f(y)` in `Y`.


theorem Ordinal.lift_card :
    ∀ (a : Ordinal.{v}), Cardinal.lift.{u, v} (Ordinal.card a) = Ordinal.card (Ordinal.lift.{u, v} a) := by sorry

The theorem `Ordinal.lift_card` states that for any ordinal `a`, lifting the cardinality of `a` (using the universe lift operation on cardinals) is equivalent to first lifting the ordinal `a` to a higher universe and then taking its cardinality. In mathematical terms, it means that the two operations - lifting the cardinality and lifting the ordinal - commute with each other.

For any ordinal `a`, the lifted cardinality of `a` is equal to the cardinality of the lifted ordinal `a`.


theorem HasSum.mul_left :
    ∀ {ι : Type u_1} {α : Type u_4} [inst : NonUnitalNonAssocSemiring α] [inst_1 : TopologicalSpace α]
      [inst_2 : TopologicalSemiring α] {f : ι → α} {a₁ : α} (a₂ : α),
      HasSum f a₁ → HasSum (fun i => a₂ * f i) (a₂ * a₁) :=
  by sorry

This theorem states that for any index type `ι`, any type `α` that is a non-unital non-associative semiring and a topological space, and any function `f` from `ι` to `α`, if the sum of the series defined by `f` converges to a value `a₁`, then the sum of the series defined by the function `λi, a₂ * f(i)` (i.e., the series obtained by multiplying every term in the original series by a constant `a₂`) converges to `a₂ * a₁`. This is in essence a formalization of the property of constant multiples in series summation in mathematics.

Given a non-unital non-associative semiring `α` and a topological space `ι`, if the series `∑ i : ι, f i` converges to `a₁` in `α`, then `∑ i : ι, a₂ * f i` converges to `a₂ * a₁` for any constant `a₂` in `α`.


theorem max_eq_left_of_lt : ∀ {α : Type u} [inst : LinearOrder α] {a b : α}, b < a → max a b = a := by sorry

This theorem, `max_eq_left_of_lt`, states that for any type `α` that has a linear order, given two objects `a` and `b` of this type, if `b` is less than `a`, then the maximum of `a` and `b` is equal to `a`. Essentially, it is proving the expected behavior of the `max` function: when applied to two arguments where the first is greater than the second, it will return the first argument.

For all types `α` with a linear order and all `a b` in `α`, if `b` is less than `a`, then `max a b = a`.


theorem Asymptotics.IsBigO.mul_isLittleO :
    ∀ {α : Type u_1} {R : Type u_13} {𝕜 : Type u_15} [inst : SeminormedRing R] [inst_1 : NormedDivisionRing 𝕜]
      {l : Filter α} {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜},
      f₁ =O[l] g₁ → f₂ =o[l] g₂ → (fun x => f₁ x * f₂ x) =o[l] fun x => g₁ x * g₂ x :=
  by sorry

This theorem states that for any types `α`, `R`, and `𝕜` (with `R` being a seminormed ring and `𝕜` a normed division ring), and for any filter `l`, if function `f₁` is big-O of `g₁` under filter `l` and `f₂` is little-o of `g₂` under the same filter, then the function that maps `x` to the product of `f₁(x)` and `f₂(x)` is also little-o of the function that maps `x` to the product of `g₁(x)` and `g₂(x)`. This theorem is a statement about asymptotic behavior of function multiplication under certain norms.

If $f\_1$ is big-O of $g\_1$ and $f\_2$ is little-o of $g\_2$ under filter $l$ in the seminormed ring $(R, ||\_.||)$ and normed division ring $(𝕜, ||\_.||)$, then $f\_1 \cdot f\_2$ is little-o of $g\_1 \cdot g\_2$ under filter $l$.


theorem Nat.bodd_bit : ∀ (b : Bool) (n : ℕ), Nat.bodd (Nat.bit b n) = b := by sorry

This theorem, `Nat.bodd_bit`, states that for any Boolean value `b` and any natural number `n`, the result of computing the binary oddness (`Nat.bodd`) of the number obtained by appending `b` as the least significant bit to the binary representation of `n` (`Nat.bit b n`) is equal to `b` itself. In other words, the least significant bit of a binary number determines its oddness: if the bit is `1` (true), the number is odd; if the bit is `0` (false), the number is even.

For any Boolean value $b$ and natural number $n$, the binary oddness of $n$ with its least significant bit set to $b$ equals $b$.


theorem Computation.think_mem : ∀ {α : Type u} {s : Computation α} {a : α}, a ∈ s → a ∈ Computation.think s := by sorry

The theorem `Computation.think_mem` states that for any type `α`, any computation `s` of type `α`, and any element `a` of type `α`, if `a` is a member of `s`, then `a` is also a member of the computation obtained by delaying `s` for one "tick" (i.e., `Computation.think s`). In other words, if a computation produces a certain result, then delaying that computation will also produce the same result.

For any computation `s` of type `α` and any element `a` of type `α` in `s`, `a` is also an element of `Computation.think s`.


theorem List.prod_hom :
    ∀ {M : Type u_5} {N : Type u_6} [inst : Monoid M] [inst_1 : Monoid N] (l : List M) {F : Type u_11}
      [inst_2 : FunLike F M N] [inst_3 : MonoidHomClass F M N] (f : F), List.prod (List.map (⇑f) l) = f (List.prod l) :=
  by sorry

The theorem `List.prod_hom` states that for any two types `M` and `N` that are monoids, a list `l` of elements of type `M`, an object `F` that is a function-like structure from `M` to `N` and also a monoid homomorphism, and a function `f` of this structure, the product of the list obtained by mapping `f` over `l` is equal to the result of applying `f` to the product of `l`. In other words, given a monoid homomorphism from `M` to `N`, the action of the monoid homomorphism on the product of a list is the same as the product of the list obtained by acting the monoid homomorphism on each element of the list. This is a formalization of the homomorphism property $f(\prod l) = \prod (f(l))$ in the context of lists and monoids.

Given a monoid homomorphism from type `M` to type `N`, the product of the image of a list `l` of elements from `M` under this homomorphism is equal to the image of the product of `l` under the same homomorphism. In other words, `f (prod l) = prod (map f l)` for a monoid homomorphism `f` from `M` to `N` and a list `l` of elements from `M`.


theorem Set.toFinset_setOf :
    ∀ {α : Type u_1} [inst : Fintype α] (p : α → Prop) [inst_1 : DecidablePred p] [inst_2 : Fintype ↑{x | p x}],
      Set.toFinset {x | p x} = Finset.filter p Finset.univ :=
  by sorry

The theorem `Set.toFinset_setOf` states that for any type `α` which has a `Fintype` instance, any predicate `p` over `α` for which `DecidablePred p` instance exists, and any subset of `α` defined by the set of elements that satisfy the predicate `p` for which a `Fintype` instance exists, the finset created from this subset using `Set.toFinset` is equal to the finset obtained by filtering the universal finset of `α` using the predicate `p`. In other words, it states the equivalence between the conversion of a subset defined by a predicate into a finset and the filtering of the universal set using the same predicate.

For any type `α` with a `Fintype` instance, any decidable predicate `p` over `α`, and any subset of `α` defined by `p`, the finset obtained from `p` using `Set.toFinset` is equal to the filtering of the universal finset of `α` through `p`.


theorem Nat.mul_lt_mul_of_le_of_lt : ∀ {a c b d : ℕ}, a ≤ c → b < d → 0 < c → a * b < c * d := by sorry

This theorem states that for any four natural numbers `a`, `b`, `c`, and `d`, if `a` is less than or equal to `c`, `b` is less than `d`, and `c` is greater than 0, then the product of `a` and `b` is less than the product of `c` and `d`. This theorem represents a basic inequality property of multiplication in the natural numbers.


theorem BoxIntegral.Box.coe_eq_pi :
    ∀ {ι : Type u_1} (I : BoxIntegral.Box ι), ↑I = Set.pi Set.univ fun i => Set.Ioc (I.lower i) (I.upper i) := by sorry

This theorem states that for any given box `I` in the space indexed by the type `ι`, the set corresponding to the box `I` equals the set of all dependent functions (indexed by `ι`) that map each index to a left-open right-closed interval. Specifically, the interval is from the lower limit of the box `I` at that index to the upper limit of the box `I` at that index. The theorem applies to all elements in the universal set, i.e., all elements of the type `ι`.

For any box `I` in the indexed space `ι`, the set of functions from `ι` to the real numbers with each function mapping an index `i` to the left-open right-closed interval `[I.lower i, I.upper i)` is equal to the set corresponding to the box `I`.


theorem Associates.mk_ne_zero : ∀ {α : Type u_1} [inst : MonoidWithZero α] {a : α}, Associates.mk a ≠ 0 ↔ a ≠ 0 := by
  sorry

This theorem states that for any type `α` that is a monoid with zero, a term `a` of type `α` is mapped to a non-zero element in the `Associates` of `α` if and only if `a` is not zero. In other words, the function `Associates.mk` preserves the non-zero property when mapping from a monoid to its associates.

For any monoid with zero `α` and non-zero element `a` of type `α`, `a` is mapped to a non-zero associate by `Associates.mk`.


theorem MeasureTheory.AEEqFun.induction_on :
    ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α}
      [inst_1 : TopologicalSpace β] (f : α →ₘ[μ] β) {p : (α →ₘ[μ] β) → Prop},
      (∀ (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ), p (MeasureTheory.AEEqFun.mk f hf)) → p f :=
  by sorry

This theorem, `MeasureTheory.AEEqFun.induction_on`, states that for any types `α` and `β`, where `α` has a measurable space structure and `β` has a topological space structure, for any measure `μ` on `α`, and for any equivalence class `f` of almost everywhere measurable functions from `α` to `β` with respect to `μ`, if a property `p` holds for all constructors `MeasureTheory.AEEqFun.mk f hf`, where `f` is a function from `α` to `β`, and `hf` is a proof that `f` is almost everywhere strongly measurable, then the property `p` holds for `f`. Essentially, it's a form of mathematical induction tailored for equivalence classes of almost everywhere measurable functions.

If `α` is a measurable space, `β` is a topological space, `μ` is a measure on `α`, `f` is an almost everywhere measurable function from `α` to `β`, and `p` is a property such that `p` holds for all constructors `MeasureTheory.AEEqFun.mk f hf`, then `p` holds for `f`.


theorem Set.Finite.of_finite_image :
    ∀ {α : Type u} {β : Type v} {s : Set α} {f : α → β}, Set.Finite (f '' s) → Set.InjOn f s → Set.Finite s := by sorry

This theorem states that for any types `α` and `β`, any set `s` of type `α`, and any function `f` from `α` to `β`, if the image of `s` under `f` (denoted by `f '' s`) is finite and if `f` is injective on `s` (i.e., no two distinct elements of `s` have the same image under `f`), then the original set `s` is also finite.

If `s` is a finite, invariant under `f` subset of type `α` such that the image of `s` under an injective function `f`: `α → β` is also finite, then `s` is finite.


theorem Equiv.hasSum_iff :
    ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {f : β → α}
      {a : α} (e : γ ≃ β), HasSum (f ∘ ⇑e) a ↔ HasSum f a :=
  by sorry

This theorem states that for any types `α`, `β`, and `γ`, where `α` is an additive commutative monoid and also has a topology, for any function `f` mapping `β` to `α` and any element `a` of `α`, given an equivalence `e` between `γ` and `β`, the function `f` has a sum equal to `a` when composed with the inverse of `e` if and only if the function `f` has a sum equal to `a`. This means that the sum total of the composed function does not change under the transformation of `e`.

For any additive commutative monoid `α` with a topology, function `f` from `β` to `α`, element `a` in `α`, and equivalence `e` between `γ` and `β`, if `f` has a sum equal to `a` in `α` then `f ∘ e^{-1}` also has a sum equal to `a`.


theorem Finset.prod_le_prod_of_subset_of_one_le' :
    ∀ {ι : Type u_1} {N : Type u_5} [inst : OrderedCommMonoid N] {f : ι → N} {s t : Finset ι},
      s ⊆ t → (∀ i ∈ t, i ∉ s → 1 ≤ f i) → (Finset.prod s fun i => f i) ≤ Finset.prod t fun i => f i :=
  by sorry

This theorem states that given any two finite sets `s` and `t` of elements from a type `ι`, with `s` being a subset of `t`, a function `f` from `ι` to a type `N` (where `N` forms an ordered commutative monoid), if for every element `i` in `t` but not in `s`, `f(i)` is greater than or equal to `1`, then the product of `f(i)` for all `i` in `s` is less than or equal to the product of `f(i)` for all `i` in `t`. In other words, if we have a function that maps elements of `t` not in `s` to values greater than or equal to `1`, then taking the product over all elements of the smaller set `s` will produce a value that is less than or equal to taking the product over all elements of the larger set `t`.

Given a finite subset `s` of a set `ι`, a function `f` from `ι` to an ordered commutative monoid `N` with `s ⊆ t` such that `f(i) ≥ 1` for all `i ∈ t \ s`, we have `∏(i ∈ s) f(i) ≤ ∏(i ∈ t) f(i)`.


theorem Submodule.span_eq_of_le :
    ∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
      (p : Submodule R M) {s : Set M}, s ⊆ ↑p → p ≤ Submodule.span R s → Submodule.span R s = p :=
  by sorry

This theorem states that for any given semiring `R`, additive commutative monoid `M` and a module `M` over `R`, if we have a submodule `p` of `M` and a set `s` of `M` such that `s` is a subset of `p` and `p` is less than or equal to the span of `s`, then the span of `s` is equal to `p`. In other words, if a submodule `p` contains a set `s` and is also contained within the span of `s`, then `p` is precisely the span of `s`. This defines an essential property of the span of a set in the context of module theory.

If `p` is a submodule of an `R`-module `M` containing a set `s` such that `p` is contained in the span of `s`, then `p` equals the span of `s`.


/-- The equalizer morphism in any limit cone is a monomorphism.  -/
theorem CategoryTheory.Limits.mono_of_isLimit_fork :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} {f g : X ⟶ Y}
      {c : CategoryTheory.Limits.Fork f g},
      CategoryTheory.Limits.IsLimit c → CategoryTheory.Mono (CategoryTheory.Limits.Fork.ι c) :=
  by sorry

This theorem states that in any category `C`, for any two morphisms `f` and `g` from object `X` to object `Y`, if `c` is a fork on `f` and `g` that is also a limit cone, then the fork's natural morphism, often termed the equalizer morphism, is a monomorphism. In category theory, a monomorphism is a morphism (or arrow) that is left-cancelable, meaning that if two morphisms are post-composed with the monomorphism, their equality implies that they were equal to start with.

In any category C, given morphisms f and g from object X to object Y, if c is a fork on f and g that is also a limit cone, then the equalizer morphism of c is a monomorphism.


theorem Mathlib.Combinatorics.SimpleGraph.Finite._auxLemma.3 :
    ∀ {V : Type u_1} (G : SimpleGraph V) (v : V) [inst : Fintype ↑(SimpleGraph.neighborSet G v)] (w : V),
      (w ∈ SimpleGraph.neighborFinset G v) = G.Adj v w :=
  by sorry

This theorem states that for any vertex 'v' of a simple graph 'G', where the set of vertices adjacent to 'v' (the neighbor set of 'v') is finite, a vertex 'w' belongs to the finite version (Finset) of this neighbor set if and only if 'v' is adjacent to 'w' in the graph 'G'. In other words, 'w' is in the finite neighbor set of 'v' precisely when 'w' and 'v' share an edge in 'G'. The adjacency relationship is defined by the function 'G.Adj'.

For any simple graph G and vertex v with finite neighbor set, vertex w is in v's neighbor set if and only if (v, w) is an edge in G.


/-- The closed thickening with radius zero is the closure of the set.  -/
theorem Metric.cthickening_zero :
    ∀ {α : Type u} [inst : PseudoEMetricSpace α] (E : Set α), Metric.cthickening 0 E = closure E := by sorry

This theorem states that, in any pseudo emetric space, the closed thickening of a set at radius zero is equivalent to the closure of the set. In other words, the set of all points that are at an infimum distance of zero from the set is the same as the smallest closed set containing the said set. This theorem holds for all sets in any pseudo emetric space.

In a pseudo metric space, the closed thickening of a set at radius 0 is equal to its closure.


theorem Set.iInter_of_empty : ∀ {α : Type u_1} {ι : Sort u_4} [inst : IsEmpty ι] (s : ι → Set α), ⋂ i, s i = Set.univ :=
  by sorry

The theorem `Set.iInter_of_empty` states that for any type `α` and an `ι` sort (an index set) such that `ι` is empty, any function `s` from `ι` to `Set α` (mapping indices to sets of `α`), the intersection of all sets `s i` (for `i` in `ι`) is the universal set `Set.univ`. In other words, if we have no sets to intersect (because `ι` is empty), then the intersection is the universal set, which includes all possible elements of type `α`.

For any empty index set `ι` and any function `s` from `ι` to sets of type `α`, the intersection of all sets `s i` is the universal set `Set.univ` of type `Set α`.


Matrix.IsHermitian.eq

theorem Matrix.IsHermitian.eq :
    ∀ {α : Type u_1} {n : Type u_4} [inst : Star α] {A : Matrix n n α},
      Matrix.IsHermitian A → Matrix.conjTranspose A = A :=
  by sorry

This theorem states that for any matrix `A` whose entries lie in a type `α` that is equipped with a `star` operation (conjugation), if `A` is Hermitian (that implies it is equal to its conjugate transpose), then the conjugate transpose of `A` is indeed `A` itself. In mathematical terms, this means that if a matrix is Hermitian (i.e., it is equal to its conjugate transpose), then the equality is true. The `star` operation in this context typically refers to complex conjugation for complex numbers, and this theorem captures a key property of Hermitian matrices over the complex numbers (or symmetric matrices over the reals).

If `A` is a matrixwhose entries lie in a type `α` with a `star` operation, and `A` is Hermitian (i.e., `A = A.transpose.star`), then `A` is equal to its conjugate transpose (`A = A.star.transpose`).


theorem MultilinearMap.map_coord_zero :
    ∀ {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [inst : Semiring R]
      [inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : AddCommMonoid M₂] [inst_3 : (i : ι) → Module R (M₁ i)]
      [inst_4 : Module R M₂] (f : MultilinearMap R M₁ M₂) {m : (i : ι) → M₁ i} (i : ι), m i = 0 → f m = 0 :=
  by sorry

This theorem states that for any given semiring `R`, type `ι`, and types `M₁` and `M₂` indexed by `ι`, if we have an additive commutative monoid for each `M₁ i` and `M₂`, and if each `M₁ i` and `M₂` are also `R`-modules. Then for any multilinear map `f` from `M₁` to `M₂`, if the `i-th` element of a collection `m` (indexed by `ι`) equals zero, then the multilinear map `f` applied to `m` will yield zero.

If `R` is a semiring, `ι` is a type, and `M₁ i` and `M₂` are additive commutative monoids and `R`-modules indexed by `ι`, then for any multilinear map `f` from `M₁` to `M₂`, if all `i-th` elements of `m` in `M₁` are zero, then `f` applied to `m` is zero.


Set.Subsingleton.image

/-- The image of a subsingleton is a subsingleton.  -/
theorem Set.Subsingleton.image :
    ∀ {α : Type u_1} {β : Type u_2} {s : Set α}, Set.Subsingleton s → ∀ (f : α → β), Set.Subsingleton (f '' s) := by
  sorry

The theorem `Set.Subsingleton.image` states that for any two types `α` and `β`, and a set `s` of elements of type `α`, if `s` is a subsingleton (i.e., it contains at most one element), then for any function `f` from `α` to `β`, the image of `s` under the function `f` (denoted as `f '' s`) is also a subsingleton. In other words, if a set has at most one element, then the image of that set under any function also has at most one element.

If `s` is a subsingleton set in type `α`, then the image `f '' s` of `s` under any function `f` from `α` to `β` is also a subsingleton.


theorem DFinsupp.coe_add :
    ∀ {ι : Type u} {β : ι → Type v} [inst : (i : ι) → AddZeroClass (β i)] (g₁ g₂ : Π₀ (i : ι), β i),
      ⇑(g₁ + g₂) = ⇑g₁ + ⇑g₂ :=
  by sorry

This theorem states that for any types `ι` and `β`, and function `β` from `ι` to `Type v`, where each `β i` forms an additive group with zero (`AddZeroClass`), the sum of two direct sum functions `g₁` and `g₂` (of type `Π₀ (i : ι), β i`) can be obtained by separately applying the functions `g₁` and `g₂` and then adding the results. In mathematical terms, this theorem ensures that the addition operation is compatible with the function application operation for direct sum functions.

For any types `ι` and `β`, and functions `g₁`, `g₂` : ι → β with each `β i` an additive group, the function `(g₁ + g₂) : ι → β` equals the application of `g₁` and `g₂` followed by the addition of their outputs: `(g₁ + g₂) i = g₁ i + g₂ i`.


theorem frontier_eq_closure_inter_closure :
    ∀ {X : Type u} {s : Set X} [inst : TopologicalSpace X], frontier s = closure s ∩ closure sᶜ := by sorry

This theorem states that for any set `s` within a topological space `X`, the frontier of `s` is equal to the intersection of the closure of `s` and the closure of the complement of `s`. In the context of topology, the frontier of a set is defined as the set of points that lie between the closure and the interior of the set. The closure of a set is the smallest closed set containing the set, and the complement of a set is a set of all points not in the original set. Hence, this theorem precisely defines the frontier in terms of these other fundamental operations in topology.

The frontier of a set in a topological space is equal to the intersection of its closure and the closure of its complement.


theorem Int.lt_floor_add_one :
    ∀ {α : Type u_2} [inst : LinearOrderedRing α] [inst_1 : FloorRing α] (a : α), a < ↑⌊a⌋ + 1 := by sorry

This theorem states that for any type `α` that is a linearly ordered ring and also a floor ring, any element `a` of type `α` is strictly less than the floor of `a` (denoted as ⌊a⌋) plus one. In other words, for any real number `a`, `a` is less than its floor value incremented by one.

For any linearly ordered ring and floor ring type `α`, and any `a` of type `α`, `a` is strictly less than the floor of `a` plus one.


Set.MapsTo.closure

/-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`.  -/
theorem Set.MapsTo.closure :
    ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y} {s : Set X}
      {t : Set Y}, Set.MapsTo f s t → Continuous f → Set.MapsTo f (closure s) (closure t) :=
  by sorry

The given theorem states that if there exists a continuous map `f` from a set `s` to a set `t`, then this map `f` also maps the closure of `s` to the closure of `t`. In other words, under a continuous map, the closure of a set is mapped to the closure of its image. Here, `Set.MapsTo f s t` means that the image of `s` under `f` is contained in `t`, `closure s` and `closure t` denote the smallest closed sets containing `s` and `t` respectively, and `Continuous f` means that the function `f` is continuous.

If `f` is a continuous function from a set `s` to a set `t` in Lean 4, then the closure of `s` is mapped to the closure of `t`, that is, `closure (Image f s) = Image (closure f) (closure s)`.


theorem Nat.dvd_add_iff_right : ∀ {k m n : ℕ}, k ∣ m → (k ∣ n ↔ k ∣ m + n) := by sorry

This theorem states that for any three natural numbers 'k', 'm', and 'n', if 'k' divides 'm', then 'k' divides 'n' if and only if 'k' divides the sum of 'm' and 'n'. In the language of number theory, this is essentially asserting a property of divisibility and is particularly important when dealing with concepts like greatest common divisors and factorization.

If k divides m and n, then k divides m + n. (This theorem is known as the Transitive Property of Divisibility or the Distributive Property of Divisibility.)


theorem Polynomial.monic_prod_of_monic :
    ∀ {R : Type u} {ι : Type y} [inst : CommSemiring R] (s : Finset ι) (f : ι → Polynomial R),
      (∀ i ∈ s, Polynomial.Monic (f i)) → Polynomial.Monic (Finset.prod s fun i => f i) :=
  by sorry

This theorem states that for any type `R` with a commutative semiring structure, and any finite set `s` of an arbitrary type `ι`, if you have a function `f` that assigns to each element of `s` a monic polynomial (i.e., a polynomial with leading coefficient 1), then the product of all these polynomials (where the product is taken over all elements of `s`) is also a monic polynomial. In other words, the product of monic polynomials, produced over a finite set, is always a monic polynomial.

Given a commutative semiring `R` and a finite set `s` of any type, if for each element `x` in `s`, we have a monic polynomial `p_x` in `R[x]`, then the product of `p_x` over all `x` in `s` is a monic polynomial in `R[x_1, ..., x_n]`.


theorem algebraMap.coe_zero :
    ∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A], ↑0 = 0 := by
  sorry

This theorem states that, for any commutative semiring `R`, any semiring `A`, and any algebra structure over `R` and `A`, the algebra map of `0` (from `R` to `A`) is equal to `0` in `A`. In other words, the image of `0` under any algebra mapping is `0`. This is a consequence of the definition of an algebra over a ring, where the algebra map must preserve addition and scalar multiplication, which in particular implies that it must send `0` to `0`.

For any commutative semiring `R`, any semiring `A`, and any algebra structure over `R` and `A`, the algebra map of `0` from `R` to `A` equals `0` in `A`.


theorem Mathlib.SetTheory.Ordinal.Arithmetic._auxLemma.10 :
    ∀ {α : Type u} [inst : LinearOrder α] {a b : α}, (¬a ≤ b) = (b < a) := by sorry

This theorem from the Set Theory Ordinal Arithmetic in Mathlib library states that for any type `α` that has a `LinearOrder` instance, and for any two elements `a` and `b` of that type, the statement that `a` is not less than or equal to `b` is equivalent to the statement that `b` is strictly less than `a`. This is a basic property of ordered sets.

In an ordered set, for all elements `a` and `b`, `a` is not less than or equal to `b` if and only if `b` is strictly less than `a`.


/-- A homomorphism `f` of seminormed groups is continuous, if there exists a constant `C`
such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖` -/
theorem AddMonoidHomClass.continuous_of_bound :
    ∀ {𝓕 : Type u_1} {E : Type u_6} {F : Type u_7} [inst : SeminormedAddGroup E] [inst_1 : SeminormedAddGroup F]
      [inst_2 : FunLike 𝓕 E F] [inst_3 : AddMonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ),
      (∀ (x : E), ‖f x‖ ≤ C * ‖x‖) → Continuous ⇑f :=
  by sorry

The theorem `AddMonoidHomClass.continuous_of_bound` states that for a given homomorphism `f` of seminormed groups, if there exists a real constant `C` such that the norm of `f x` is less than or equal to `C` times the norm of `x` for all `x`, then the function `f` is continuous. This theorem applies to any types `𝓕`, `E`, and `F` where `𝓕` is a function from `E` to `F`, both `E` and `F` are seminormed add groups, and `𝓕` demonstrates the properties of an additive monoid homomorphism.

If `f : E → F` is an additive homomorphism between seminormed add groups `E` and `F`, and there exists a constant `C` such that `‖f (x)‖ ≤ C * ‖x‖` for all `x` in `E`, then `f` is a continuous function.


theorem Nat.succ_injective : Function.Injective Nat.succ := by sorry

The theorem `Nat.succ_injective` states that the successor function `Nat.succ` on natural numbers is injective. In other words, for any two natural numbers `a` and `b`, if the successor of `a` (denoted as `Nat.succ a`) is equal to the successor of `b` (denoted as `Nat.succ b`), then `a` must be equal to `b`. This is a property of the "next number" or "successor" function in the set of natural numbers.

If for natural numbers `a` and `b`, `Nat.succ a = Nat.succ b`, then `a = b`.


theorem Cubic.of_b_eq_zero :
    ∀ {R : Type u_1} {P : Cubic R} [inst : Semiring R],
      P.a = 0 → P.b = 0 → Cubic.toPoly P = Polynomial.C P.c * Polynomial.X + Polynomial.C P.d :=
  by sorry

This theorem states that for any semiring `R`, and any cubic polynomial `P` over `R`, if the coefficients of `x^3` and `x^2` terms (`P.a` and `P.b`) in `P` are zero, then the polynomial obtained by converting the cubic polynomial `P` to a general polynomial is equivalent to `P.c * x + P.d`. In other words, if a cubic polynomial lacks `x^3` and `x^2` terms, it behaves exactly like a linear polynomial.

If `P` is a cubic polynomial over a semiring with zero coefficients for `x^3` and `x^2` terms, then `P` is equivalent to `P.c * x + P.d`.


theorem Mathlib.Algebra.Opposites._auxLemma.1 :
    ∀ {α : Type u_1} {x y : α}, (MulOpposite.op x = MulOpposite.op y) = (x = y) := by sorry

This theorem, named `_auxLemma.1` under the `Mathlib.Algebra.Opposites` namespace, states that for any type `α`, and any two elements `x` and `y` from this type, the operation `MulOpposite.op` applied to `x` being equal to the same operation applied to `y` is equivalent to `x` being equal to `y`. In other words, if the `MulOpposite.op` operation produces the same result for `x` and `y`, then `x` and `y` must be the same. This theorem plays a role in the manipulation of algebraic structures and their opposites in Lean 4, ensuring the integrity of operations on these structures.

For any type `α` and elements `x` and `y` from `α`, `MulOpposite.op x = MulOpposite.op y` implies `x = y`.