theorem intervalIntegral.integral_const :
    ∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : CompleteSpace E] [inst_2 : NormedSpace ℝ E] {a b : ℝ}
      (c : E), ∫ (x : ℝ) in a..b, c = (b - a) • c :=
  by sorry

This theorem states that for any type `E` that forms a normed additive commutative group, is a complete space, and also a normed space over the real numbers (`ℝ`), given any two real numbers `a` and `b` and any element `c` of type `E`, the definite integral from `a` to `b` of a constant function `c` is equal to `(b - a) • c`. In other words, the integral of a constant function over an interval is simply the width of the interval times the constant, scaled appropriately for the type `E`.

For any normed additive commutative group `E` complete over the real numbers `ℝ`, the definite integral of a constant function `c` over an interval `[a, b]` in `E` equals `(b - a) • c`.


theorem CategoryTheory.mono_comp :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} (f : X ⟶ Y) [inst_1 : CategoryTheory.Mono f]
      (g : Y ⟶ Z) [inst_2 : CategoryTheory.Mono g], CategoryTheory.Mono (CategoryTheory.CategoryStruct.comp f g) :=
  by sorry

This theorem, `CategoryTheory.mono_comp`, states that in any category `C`, given objects `X`, `Y`, and `Z`, if `f` is a morphism from `X` to `Y` and `g` is a morphism from `Y` to `Z` such that `f` and `g` are both monomorphisms (injective homomorphisms in the category), then the composition of `f` and `g` (denoted by `CategoryTheory.CategoryStruct.comp f g`, or simply `f ≫ g` in categorical terms) is also a monomorphism. In other words, it says that the composition of two monomorphisms is also a monomorphism in the category `C`.

In any category, the composition of two monomorphisms is a monomorphism.


Set.Finite.countable

theorem Set.Finite.countable : ∀ {α : Type u} {s : Set α}, Set.Finite s → Set.Countable s := by sorry

This theorem states that for any type `α` and any set `s` of type `α`, if the set `s` is finite, then it is also countable. In other words, if there exists a natural number `n` and an equivalence between the set `s` and the finite ordinal `Fin n`, there must also exist an injective map from `s` to the set of natural numbers `ℕ`. This theorem is a fundamental property in set theory and number theory.

If a type `α`'s set `s` is finite, then it has an injective map to the natural numbers `ℕ`.


theorem Monotone.map_min :
    ∀ {α : Type u} {β : Type v} [inst : LinearOrder α] [inst_1 : LinearOrder β] {f : α → β} {a b : α},
      Monotone f → f (min a b) = min (f a) (f b) :=
  by sorry

The theorem `Monotone.map_min` states that for any types `α` and `β` with a linear order, a function `f` from `α` to `β`, and any elements `a` and `b` of `α`, the following holds: if `f` is a monotone function (i.e., a function where `a ≤ b` implies `f(a) ≤ f(b)`), then the application of `f` to the minimum of `a` and `b` is equal to the minimum of `f(a)` and `f(b)`. In other words, a monotone function preserves the operation of taking the minimum.

If `f` is a monotonic function from a type `α` with a linear order to another type `β` also with a linear order, then `f(min(a, b)) = min(f(a), f(b))` for all `a, b ∈ α`.


theorem Filter.neBot_iff : ∀ {α : Type u} {f : Filter α}, Filter.NeBot f ↔ f ≠ ⊥ := by sorry

This theorem states that, for any given type `α` and a filter `f` on this type, the filter `f` is not minimal (or equivalently, is non-trivial or contains more than the empty set) if and only if `f` is not equal to the bottom element (denoted by ⊥). Here, the bottom element represents the smallest possible filter, which, in this context, is a filter that contains only the empty set. This theorem establishes an equivalence between the `Filter.NeBot` predicate and the inequality `f ≠ ⊥`, both of which are ways to express that a filter is non-trivial in Lean 4.

A filter `f` over type `α` is non-trivial if and only if it is not equal to the bottom filter (denoted by ⊥).


theorem CategoryTheory.Limits.isZero_zero :
    ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C],
      CategoryTheory.Limits.IsZero 0 :=
  by sorry

This theorem states that for any type `C` which is a Category, and given that `C` has a Zero object, the object `0` is a Zero object in the category `C`. In the context of category theory, a Zero object is an object that is both an initial object and a terminal object. This means that there's exactly one morphism going to (for terminal objects) and coming from (for initial objects) the Zero object to any other object in the category.

In any category with a Zero object, the Zero object is both initial and terminal.


theorem Filter.inf_principal :
    ∀ {α : Type u} {s t : Set α}, Filter.principal s ⊓ Filter.principal t = Filter.principal (s ∩ t) := by sorry

This theorem, named `Filter.inf_principal`, states that for any type `α` and any two sets `s` and `t` of type `α`, the intersection of the principal filters of `s` and `t` is equal to the principal filter of the intersection of `s` and `t`. In other words, the principal filter of the intersection of two sets is equivalent to the intersection of the principal filters of the individual sets.

For any type `α` and sets `s` and `t` of type `α`, the principal filters of `s` and `t` have equal elements if and only if the intersection of `s` and `t` has the same principal filter.


theorem Int.ofNat_zero_le : ∀ (n : ℕ), 0 ≤ ↑n := by sorry

This theorem, `Int.ofNat_zero_le`, states that for any natural number `n`, the value of `n` when converted (or 'lifted') to an integer is always greater than or equal to zero. This is a standard fact about natural numbers, as they are defined to start from zero and increase, so there are no negative values. The `↑n` notation is used in Lean to denote the act of 'lifting' or 'casting' a value from one type to another, in this case, from natural numbers (`ℕ`) to integers (`ℤ`).

For any natural number `n`, the integer obtained by lifting `n` is non-negative, i.e., `↑n ≥ 0`.


/-- The head of a vector obtained by prepending is the element prepended.  -/
theorem Vector.head_cons : ∀ {α : Type u} {n : ℕ} (a : α) (v : Vector α n), Vector.head (a ::ᵥ v) = a := by sorry

The theorem `Vector.head_cons` states that for any type `α`, any natural number `n`, any element `a` of type `α`, and any vector `v` of length `n` with elements of type `α`, if we prepend the element `a` to the vector `v`, the head of the resulting vector (i.e., the first element) is the element `a` that was prepended. In other words, this theorem is about the property of the `head` operation on vectors, asserting that the head of a vector obtained by prepending an element is indeed the element that was prepended.

For any type `α`, any natural number `n`, any `α` element `a`, and any vector `v` of length `n` in `α`, the head of the vector obtained by prepending `a` to `v` is equal to `a`.


theorem Nat.Prime.coprime_iff_not_dvd : ∀ {p n : ℕ}, Nat.Prime p → (Nat.Coprime p n ↔ ¬p ∣ n) := by sorry

The theorem `Nat.Prime.coprime_iff_not_dvd` states that for any two natural numbers `p` and `n`, if `p` is a prime number, then `p` and `n` are coprime (their greatest common divisor is `1`) if and only if `p` does not divide `n`. In other words, a prime number `p` and another number `n` share no common divisors other than `1` exactly when `p` is not a divisor of `n`.

A prime number is coprime with another natural number if and only if it does not divide that number.


theorem edist_dist : ∀ {α : Type u} [inst : PseudoMetricSpace α] (x y : α), edist x y = ENNReal.ofReal (dist x y) := by
  sorry

In a pseudometric space `α`, the theorem `edist_dist` states that for any two points `x` and `y` in `α`, the extended distance (or edistance) between `x` and `y` equals the application of the function `ENNReal.ofReal` to the ordinary distance between `x` and `y`. In plain words, the extended distance is the same as the ordinary distance, but converted to a non-negative extended real number with the help of `ENNReal.ofReal`. This function returns the original real number if it is non-negative, or zero otherwise.

In a pseudometric space, the extended distance between any two points equals the non-negative real number obtained by applying `ENNReal.ofReal` to the ordinary distance between them.


theorem iSup_mono :
    ∀ {α : Type u_1} {ι : Sort u_5} [inst : CompleteLattice α] {f g : ι → α},
      (∀ (i : ι), f i ≤ g i) → iSup f ≤ iSup g :=
  by sorry

The theorem `iSup_mono` states that for any type `α` that forms a complete lattice and for any index type `ι`, given two functions `f` and `g` from `ι` to `α`, if for every `i` in `ι`, `f(i)` is less than or equal to `g(i)`, then the supremum of the range of `f` is less than or equal to the supremum of the range of `g`. Here, `iSup` is the function that calculates the supremum of the set of values obtained by applying a function to an index set.

For any complete lattice `α` and index type `ι`, if `f` and `g` are functions from `ι` to `α` with `f(i) ≤ g(i)` for all `i`, then `sup(map f) ≤ sup(map g)`.


theorem sSup_eq_iSup : ∀ {α : Type u_1} [inst : CompleteLattice α] {s : Set α}, sSup s = ⨆ a ∈ s, a := by sorry

The theorem `sSup_eq_iSup` states that for any type `α`, which is a complete lattice, and for any set `s` of elements of type `α`, the supremum (also known as the least upper bound) of the set `s` is equal to the join operation (notated by `⨆`) over all elements `a` in the set `s`. In other words, the supremum of a set in a complete lattice can be computed by taking the join of all elements in that set.

For any complete lattice α and set s ⊆ α, the supremum of s equals the join of all elements in s. (Written mathematically as ⨆s.x ≡ ⨅{x ∈ s : x ≤ y ∀y ∈ s})


Nat.le.dest

theorem Nat.le.dest : ∀ {n m : ℕ}, n ≤ m → ∃ k, n + k = m := by sorry

This theorem, `Nat.le.dest`, states that for any two natural numbers 'n' and 'm', if 'n' is less than or equal to 'm', then there exists another natural number 'k' such that 'n' plus 'k' equals 'm'. This theorem essentially formalizes the concept that any natural number 'm' can be expressed as the sum of a smaller number 'n' and some natural number 'k'.

For all natural numbers n and m, if n ≤ m, then there exists a natural number k such that n + k = m.


theorem Mathlib.Data.Complex.Basic._auxLemma.1 : ∀ {z w : ℝ}, (↑z = ↑w) = (z = w) := by sorry

This theorem in the Lean 4 Mathlib library states that for any two real numbers `z` and `w`, their respective complex number representations (denoted by `↑z` and `↑w`) are equal if and only if the original real numbers `z` and `w` themselves are equal. This equivalence is a fundamental property reflecting the relationship between the real and complex number systems.

For any real numbers `z` and `w`, `↑z = ↑w` if and only if `z = w`.


theorem Complex.smul_re : ∀ {R : Type u_1} [inst : SMul R ℝ] (r : R) (z : ℂ), (r • z).re = r • z.re := by sorry

This theorem states that for any type `R` that can be multiplied with real numbers, and any instance `r` of this type and a complex number `z`, the real part of the result of scaling `z` by `r` is equal to scaling the real part of `z` by `r`. In other words, scaling distributes over the operation of taking the real part in complex numbers.

For any type `R` that supports real number multiplication, and `r` in `R` and complex number `z`, the real part of `r * z` equals the real part of `r * (re part z)`.


theorem Complex.smul_im : ∀ {R : Type u_1} [inst : SMul R ℝ] (r : R) (z : ℂ), (r • z).im = r • z.im := by sorry

This theorem states that for any real scalar `r` and any complex number `z`, the imaginary part of the scaled complex number (`r • z`) is equal to the scaled imaginary part of the complex number (`r • z.im`). In other words, scalar multiplication distributes over the imaginary part of a complex number.

For any real scalar r and complex number z, the imaginary parts of r * z and r * (z.im) are equal.


Continuous.snd'

/-- Precomposing `f` with `Prod.snd` is continuous  -/
theorem Continuous.snd' :
    ∀ {X : Type u} {Y : Type v} {Z : Type u_1} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
      [inst_2 : TopologicalSpace Z] {f : Y → Z}, Continuous f → Continuous fun x => f x.2 :=
  by sorry

The theorem `Continuous.snd'` states that for any three types `X`, `Y`, and `Z` that have a topological space structure and for any function `f` from `Y` to `Z` that is continuous, the function created by first applying the second part of a product (`Prod.snd`) and then applying `f` is also continuous. In mathematical terms, if we have a continuous function `f: Y -> Z`, then the function `g: X x Y -> Z` defined by `g(x, y) = f(y)` is also continuous.

If `f: Y �ERCHANT Z` is a continuous function and `g: X × Y �ERCHANT Z` is defined by `g(x, y) := f(y)`, then `g` is also continuous.


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

The theorem `Polynomial.map_C` states that for any semirings `R` and `S`, for any ring homomorphism `f` from `R` to `S`, and for any element `a` from `R`, the operation of mapping the constant polynomial `a` in `R` using `f` is the same as creating the constant polynomial in `S` using the image of `a` under `f`. In other words, if we map the constant polynomial `a` across the ring homomorphism `f`, we get the constant polynomial `f(a)` in the codomain.

For any semirings R and S, and any ring homomorphism f from R to S, the constant polynomial f(a) in S is the image of the constant polynomial a in R under the mapping f.


theorem Mathlib.Data.Set.Basic._auxLemma.9 : ∀ {α : Type u} (s : Set α), (∅ ⊆ s) = True := by sorry

This theorem states that for any set `s` of any type `α`, the empty set is a subset of `s`. In other words, this theorem confirms the general mathematical principle that the empty set is a subset of any given set. In mathematical notation, this would be represented as ∀s ∈ α, ∅ ⊆ s.

The empty set is a subset of any set. (Written mathematically as ∀α, ∅ ⊆ s, where α is the type of the set s and ∅ is the empty set.)


theorem CategoryTheory.Limits.prod.lift_map :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {V W X Y Z : C}
      [inst_1 : CategoryTheory.Limits.HasBinaryProduct W X] [inst_2 : CategoryTheory.Limits.HasBinaryProduct Y Z]
      (f : V ⟶ W) (g : V ⟶ X) (h : W ⟶ Y) (k : X ⟶ Z),
      CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.prod.lift f g) (CategoryTheory.Limits.prod.map h k) =
        CategoryTheory.Limits.prod.lift (CategoryTheory.CategoryStruct.comp f h)
          (CategoryTheory.CategoryStruct.comp g k) :=
  by sorry

This theorem in category theory states that for any objects `V`, `W`, `X`, `Y`, `Z` in a category `C` that has binary products of `W` and `X` and `Y` and `Z`, and given any morphisms `f : V ⟶ W`, `g : V ⟶ X`, `h : W ⟶ Y`, `k : X ⟶ Z`, the composition of the lift of the product of `f` and `g` with the map of the product of `h` and `k` is equal to the lift of the product of the composition of `f` and `h` with the composition of `g` and `k`. In mathematical terms, this can be written as `(f x g) ∘ (h x k) = (f ∘ h) x (g ∘ k)`, where `∘` denotes composition of morphisms, `x` denotes the product of two morphisms, and `f x g` and `h x k` are the lifts of the product of the morphisms.

For any objects `V`, `W`, `X`, `Y`, `Z` in a category with binary products, and morphisms `f : V ⟶ W`, `g : V ⟶ X`, `h : W ⟶ Y`, `k : X ⟶ Z`, the lifts of their products satisfy `(f x g) ∘ (h x k) = (f ∘ h) x (g ∘ k)`.


theorem Set.subset_sUnion_of_mem : ∀ {α : Type u_1} {S : Set (Set α)} {t : Set α}, t ∈ S → t ⊆ ⋃₀ S := by sorry

The theorem `Set.subset_sUnion_of_mem` states that for any type `α`, and for any set `S` of sets of type `α`, if another set `t` of type `α` is a member of `S`, then `t` is a subset of the union over all sets in `S`, denoted by `⋃₀ S`. In other words, any element of a collection of sets is itself a subset of the union of the entire collection.

For any type `α` and set `S` of subsets of `α`, if `t ∈ S`, then `t ⊆ ⋃₀ S`.


/-- See `inv_lt_inv_of_lt` for the implication from right-to-left with one fewer assumption.  -/
theorem inv_lt_inv :
    ∀ {α : Type u_2} [inst : LinearOrderedSemifield α] {a b : α}, 0 < a → 0 < b → (a⁻¹ < b⁻¹ ↔ b < a) := by sorry

The theorem `inv_lt_inv` states that for any linear ordered semifield `α` and any two elements `a` and `b` of `α` that are greater than zero, the inverse of `a` is less than the inverse of `b` if and only if `b` is less than `a`. This theorem captures a key property of inverses in ordered semifields: flipping the order of comparison and taking inverses are equivalent operations.

For any linear ordered semifield `α` and elements `a, b` in `α` with `0 < a, b`, `a ^-1 < b ^-1` if and only if `b < a`.


theorem vsub_ne_zero :
    ∀ {G : Type u_1} {P : Type u_2} [inst : AddGroup G] [T : AddTorsor G P] {p q : P}, p -ᵥ q ≠ 0 ↔ p ≠ q := by sorry

This theorem states that for any types `G` and `P`, where `G` has an addition group structure and `P` is an additive torsor over `G`, the difference between two points `p` and `q` of type `P` (represented as `p -ᵥ q`) is not zero if and only if `p` and `q` are not equal. In other words, in this context, two points are distinct if and only if their "vector difference" is non-zero.

For any additive torsors `P` over an additive group `G`, two points `p` and `q` in `P` differ, i.e., `p -ᵥ q ≠ 0`, if and only if `p` and `q` are not equal.


theorem csSup_le :
    ∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s : Set α} {a : α},
      Set.Nonempty s → (∀ b ∈ s, b ≤ a) → sSup s ≤ a :=
  by sorry

The theorem `csSup_le` states that for any type `α` that is a conditionally complete lattice, any non-empty set `s` of type `α`, and any element `a` of type `α`, if every element `b` in `s` is less than or equal to `a`, then the supremum (or least upper bound) of `s` is less than or equal to `a`. In other words, if `a` is an upper bound of a non-empty set in a conditionally complete lattice, then it is at least as large as the supremum of that set.

If `α` is a conditionally complete lattice and `s` is a non-empty subset of `α` with `a` as an upper bound, then `sup s ≤ a`.


theorem mul_le_mul_left :
    ∀ {α : Type u_1} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α] [inst_3 : PosMulMono α]
      [inst_4 : PosMulReflectLE α], 0 < a → (a * b ≤ a * c ↔ b ≤ c) :=
  by sorry

The theorem `mul_le_mul_left` states that for any type `α` equipped with multiplication, zero element, and a preorder structure, and satisfying both the properties of `PosMulMono` and `PosMulReflectLE`, whenever `a` is a positive element of `α`, the inequality `a * b ≤ a * c` holds if and only if `b ≤ c`. In other words, multiplication by a positive element on the left preserves the order of elements. This property is commonly known in the context of ordered semirings or ordered fields.

For any type `α` with multiplication, zero element, preorder structure, and satisfying `PosMulMono` and `PosMulReflectLE` properties, multiplication by a positive element on the left preserves the order relation: `a > 0 ∧ b ≤ c ↔ a * b ≤ a * c`.


theorem Filter.map_le_iff_le_comap :
    ∀ {α : Type u} {β : Type v} {f : Filter α} {g : Filter β} {m : α → β}, Filter.map m f ≤ g ↔ f ≤ Filter.comap m g :=
  by sorry

This theorem states that for any types `α` and `β`, any filters `f` on `α` and `g` on `β`, and any function `m` from `α` to `β`, the filter obtained by mapping `f` through `m` is a subfilter of `g` if and only if `f` is a subfilter of the filter obtained by co-mapping `g` through `m`. In more mathematical terms, `Filter.map m f ≤ g` if and only if `f ≤ Filter.comap m g`. This provides a relationship between the operations of filter mapping and co-mapping.

For filters $f$ on type $\alpha$ and $g$ on type $\beta$, and any function $m$ from $\alpha$ to $\beta$, the filter obtained by mapping $f$ through $m$ is a subfilter of $g$ if and only if $f$ is a subfilter of the filter obtained by co-mapping $g$ through $m$.


theorem Mathlib.Data.Finset.Basic._auxLemma.108 :
    ∀ {α : Type u_1} [inst : DecidableEq α] {s : Finset α} {a : α}, (Finset.erase s a = s) = (a ∉ s) := by sorry

This theorem states that for any type `α` with decidable equality, any finite set `s` of elements of type `α`, and any element `a` of type `α`, the statement that the result of "erasing" `a` from `s` is equal to `s` itself, is equivalent to the statement that `a` is not an element of `s`. In other words, if removing an element from a set results in the same set, then that element was not in the set to begin with.

For any type `α` with decidable equality, a finite set `s` of elements from `α`, and an element `a` not in `s`, the set `s` with `a` removed is equal to `s` if and only if `a` was not an element of `s` to begin with.


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

The theorem `isOpen_biUnion` states that for any type `X` with a topological space structure, a set `s` of another type `α`, and a function `f` from `α` to sets of `X`, if for all elements (`i`) in the set `s`, the set `f(i)` is open, then the union of all the sets `f(i)` (for `i` in `s`) is also open. In other words, it's saying that the union of an arbitrary collection of open sets is also an open set.

If every set in an arbitrary collection is open in a topological space, then their union is also open.


theorem Mathlib.Data.Set.Defs._auxLemma.12 :
    ∀ {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {f : (i : ι) → α i},
      (f ∈ Set.pi s t) = ∀ i ∈ s, f i ∈ t i :=
  by sorry

This theorem states that for any index set `ι`, a family of sets `t` indexed by `ι`, a set `s` of indices, and a dependent function `f`, the function `f` is in the set `Set.pi s t` (the set of dependent functions mapped from set `s` to `t`) if and only if for every index `i` in `s`, `f(i)` is in `t(i)`. Essentially, this theorem is clarifying the condition for a function to be an element of a set of functions, defined by `Set.pi`.

A function `f` is in the set `Set.pi s t` if and only if for all indices `i` in `s`, `f(i)` is in `t(i)`.


LawfulMonad.mk'

/-- An alternative constructor for `LawfulMonad` which has more
defaultable fields in the common case.
 -/
theorem LawfulMonad.mk' :
    ∀ (m : Type u → Type v) [inst : Monad m],
      (∀ {α : Type u} (x : m α), id <$> x = x) →
        (∀ {α β : Type u} (x : α) (f : α → m β), pure x >>= f = f x) →
          (∀ {α β γ : Type u} (x : m α) (f : α → m β) (g : β → m γ), x >>= f >>= g = x >>= fun x => f x >>= g) →
            autoParam (∀ {α β : Type u} (x : α) (y : m β), Functor.mapConst x y = Function.const β x <$> y) _auto✝ →
              autoParam
                  (∀ {α β : Type u} (x : m α) (y : m β),
                    (SeqLeft.seqLeft x fun x => y) = do
                      let a ← x
                      let _ ← y
                      pure a)
                  _auto✝¹ →
                autoParam
                    (∀ {α β : Type u} (x : m α) (y : m β),
                      (SeqRight.seqRight x fun x => y) = do
                        let _ ← x
                        y)
                    _auto✝² →
                  autoParam
                      (∀ {α β : Type u} (f : α → β) (x : m α),
                        (do
                            let y ← x
                            pure (f y)) =
                          f <$> x)
                      _auto✝³ →
                    autoParam
                        (∀ {α β : Type u} (f : m (α → β)) (x : m α),
                          (do
                              let x_1 ← f
                              x_1 <$> x) =
                            Seq.seq f fun x_1 => x)
                        _auto✝⁴ →
                      LawfulMonad m :=
  by sorry

The theorem `LawfulMonad.mk'` presents an alternative way to construct a lawful monad `m`. Given a type constructor `m` that forms a monad, it establishes certain laws:

1. The identity law: for any value `x` from the monad `m`, mapping the identity function over `x` should result in `x` itself.

2. The left identity law: for any value `x` and function `f` from `α` to `m β`, the operation `pure x >>= f` should be equivalent to `f x`.

3. The associativity law: for any value `x` from the monad `m`, and functions `f` and `g`, the operation `x >>= f >>= g` should be equivalent to `x >>= fun x => f x >>= g`.

4. The functor law: for any value `x` and `y` from the monad `m`, `Functor.mapConst x y` should be equivalent to `Function.const β x <$> y`.

5. The sequence left law: for any value `x` and `y` from the monad `m`, sequencing `x` and `y` and keeping the value of `x` should be equivalent to the do-notation `do let a ← x; let _ ← y; pure a`.

6. The sequence right law: for any value `x` and `y` from the monad `m`, sequencing `x` and `y` and keeping the value of `y` should be equivalent to the do-notation `do let _ ← x; y`.

7. The function application law: for any function `f` and value `x` from the monad `m`, the do-notation `do let y ← x; pure (f y)` should be equivalent to `f <$> x`.

8. The sequence law: for any function `f` and value `x` from the monad `m`, the do-notation `do let x_1 ← f; x_1 <$> x` should be equivalent to `Seq.seq f fun x_1 => x`.

These laws are expressed with `autoParam`, which allows Lean to fill in some arguments automatically, making it more convenient to use the theorem. If these laws hold, the monad `m` is a lawful monad `LawfulMonad m`.

The `LawfulMonad.mk` theorem constructs a lawful monad `m` if and only if the given monad satisfies the identity, left identity, associativity, functor, sequence left and right, function application, and sequence laws.


theorem neg_lt_zero :
    ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LT α]
      [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α}, -a < 0 ↔ 0 < a :=
  by sorry

This theorem, `neg_lt_zero`, states that for any type `α` that is an additive group and has a less than (`<`) operation, and where addition and less than operation satisfy the CovariantClass properties, the negative of any element `a` from `α` is less than zero if and only if zero is less than `a`. This establishes a relationship between the negative of a number being less than zero and zero being less than the number itself in this type of mathematical structure.

For any additive group type `α` with CovariantClass `<` operation, `-a < 0 <-> 0 < a`.


/-- The composition of a continuous function and an ae strongly measurable function is ae strongly
measurable.  -/
theorem Continuous.comp_aestronglyMeasurable :
    ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
      [inst : TopologicalSpace β] [inst_1 : TopologicalSpace γ] {g : β → γ} {f : α → β},
      Continuous g → MeasureTheory.AEStronglyMeasurable f μ → MeasureTheory.AEStronglyMeasurable (fun x => g (f x)) μ :=
  by sorry

The theorem states that if we have a continuous function `g` from a topological space `β` to another topological space `γ`, and another function `f` from a measurable space `α` to `β` that is `AEStronglyMeasurable` (almost everywhere strongly measurable) with respect to a measure `μ`, then the composition of `g` and `f` (given by `g(f(x))`) is also `AEStronglyMeasurable` with respect to the same measure `μ`. This is to say, almost everywhere, the composition of `g` and `f` matches the limit of a sequence of simple functions.

If `g: β -> γ` is continuous and `f: α -> β` is `AEStronglyMeasurable`, then the composition `g ∘ f: α -> γ` is also `AEStronglyMeasurable`.


theorem monotone_nat_of_le_succ :
    ∀ {α : Type u} [inst : Preorder α] {f : ℕ → α}, (∀ (n : ℕ), f n ≤ f (n + 1)) → Monotone f := by sorry

The theorem `monotone_nat_of_le_succ` states that for any type `α` that satisfies the `Preorder` condition (meaning it has defined operations for comparison and ordering), and a function `f` from natural numbers to `α`, if for every natural number `n`, the function `f` applied to `n` is less than or equal to `f` applied to `n + 1`, then the function `f` is monotone. In mathematical terms, this theorem asserts that a function `f : ℕ -> α` is monotone if `f(n) ≤ f(n+1)` for all `n` in natural numbers.

A function `f : ℕ -> α` is monotone if `f(n) ≤ f(n+1)` for all natural numbers `n`.


theorem Polynomial.natDegree_one : ∀ {R : Type u} [inst : Semiring R], Polynomial.natDegree 1 = 0 := by sorry

This theorem states that for any type `R` that is a semiring, the natural degree of the polynomial `1` is `0`. In mathematical terms, this means that the degree of the constant polynomial `1` (which is a polynomial that does not depend on the variable) in any semiring is `0`. This is consistent with the general rule that the degree of a nonzero constant polynomial is `0`.

For any semiring R, the constant polynomial 1 has degree 0.


theorem Finset.coe_preimage :
    ∀ {α : Type u} {β : Type v} {f : α → β} (s : Finset β) (hf : Set.InjOn f (f ⁻¹' ↑s)),
      ↑(Finset.preimage s f hf) = f ⁻¹' ↑s :=
  by sorry

The theorem `Finset.coe_preimage` states that for any types `α` and `β`, any function `f` from `α` to `β`, and any finite set `s` of type `β`, if `f` is injective on the preimage of `s` under `f`, then the preimage of `s` under `f` as a finite set, when coverted back to a set, is equal to the preimage of `s` under `f`. In simple terms, it asserts that the process of creating a finite set from the preimage doesn't change the underlying set, assuming `f` is injective on the preimage of `s`.

If `f` is an injective function from `α` to `β`, then the preimage of a finite set `s ∈ β` under `f` equals the set of elements in `α` mapping to `s` under `f`.


theorem CategoryTheory.Limits.limit.isoLimitCone_inv_π :
    ∀ {J : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [inst_1 : CategoryTheory.Category.{v, u} C]
      {F : CategoryTheory.Functor J C} [inst_2 : CategoryTheory.Limits.HasLimit F]
      (t : CategoryTheory.Limits.LimitCone F) (j : J),
      CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.isoLimitCone t).inv
          (CategoryTheory.Limits.limit.π F j) =
        t.cone.π.app j :=
  by sorry

The theorem `CategoryTheory.Limits.limit.isoLimitCone_inv_π` states that in a category `C` with a functor `F` from a category `J`, given `t` as a limit cone of `F` and `j` as an object from category `J`, the composition of the inverse of the isomorphism from limit to limit cone `t` and the projection from the limit to the functor value at `j` equals to the projection of the cone `t` at `j`. In simpler terms, it signifies the correspondence between the mappings from the limit object and the vertex of the limit cone in the context of category theory.

In a category with a functor, the inverse of the limit cone isomorphism to a limit and the projection from the limit to the functor value at an object are equal.


theorem CategoryTheory.Limits.Pi.hom_ext :
    ∀ {β : Type w} {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {f : β → C}
      [inst_1 : CategoryTheory.Limits.HasProduct f] {X : C} (g₁ g₂ : X ⟶ ∏ f),
      (∀ (b : β),
          CategoryTheory.CategoryStruct.comp g₁ (CategoryTheory.Limits.Pi.π f b) =
            CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.Pi.π f b)) →
        g₁ = g₂ :=
  by sorry

This theorem, `CategoryTheory.Limits.Pi.hom_ext`, states that for any category `C`, any indexing type `β`, and any function `f : β → C` that has a product in `C`, if we have two morphisms `g₁, g₂: X ⟶ ∏ f` from some object `X` to the product of `f`, then `g₁` is equal to `g₂` if and only if for all `b` in `β`, the composition of `g₁` and the `b`-th projection from the product of `f` is equal to the composition of `g₂` and the same projection. In essence, two morphisms to a product are the same if they make the same diagram commute with all projection maps from the product.

Given a category \(C\), a function \(f : \beta \rightarrow C\) with product \(\prod f\), and morphisms \(g\_1, g\_2 : X \rightarrow \prod f\), \(g\_1 = g\_2\) if and only if \(g\_1 \circ p\_b = g\_2 \circ p\_b\) for all \(b \in \beta\), where \(p\_b\) denotes the \(b\)-th projection from \(\prod f\).


theorem Real.log_pos : ∀ {x : ℝ}, 1 < x → 0 < Real.log x := by sorry

The theorem `Real.log_pos` states that for all real numbers `x`, if `x` is greater than 1, then the logarithm of `x` is greater than 0. This aligns with the properties of the logarithm function in mathematics, where log(x) increases from negative infinity to positive infinity as x increases from 0 to infinity, passing through 0 at x = 1.

For all real numbers `x` greater than 1, `log x` is positive.


theorem List.take_nil : ∀ {α : Type u_1} {i : ℕ}, List.take i [] = [] := by sorry

This theorem states that for any type `α` and any natural number `i`, when you try to take `i` elements from an empty list of type `α`, the result will always be an empty list. This hold regardless of what `i` is, because there are no elements to take from an empty list.

For any type `α` and natural number `i`, the `nth` element of an empty list of type `α` is an empty list.


theorem Mathlib.Order.Lattice._auxLemma.3 :
    ∀ {α : Type u} [inst : SemilatticeSup α] {a b c : α}, (a ⊔ b ≤ c) = (a ≤ c ∧ b ≤ c) := by sorry

This theorem is a proposition in the field of order theory, specifically concerning semilattices. It states that for any type `α` that is a semilattice sup (a structure with an associative, commutative, and idempotent binary operation called sup), and any elements `a`, `b`, and `c` of that type, `a` joined with `b` being less than or equal to `c` is equivalent to `a` being less than or equal to `c` and `b` being less than or equal to `c`. In terms of lattice theory, the join (`a ⊔ b`) of `a` and `b` is less than or equal to `c` if and only if both `a` and `b` are individually less than or equal to `c`.

In a semilattice `α`, for all elements `a`, `b`, and `c`, `a ⊔ b ≤ c` if and only if `a ≤ c` and `b ≤ c`.


theorem Set.univ_prod_univ : ∀ {α : Type u_1} {β : Type u_2}, Set.univ ×ˢ Set.univ = Set.univ := by sorry

This theorem states that the Cartesian product of the universal set on type `α` and the universal set on type `β` (denoted as `Set.univ ×ˢ Set.univ`) is also a universal set. In other words, the Cartesian product of all elements of two arbitrary types `α` and `β` includes every possible pair from `α` and `β`, which forms a new universal set. This theorem holds for any types `α` and `β`.

The Cartesian product of the universal sets on types `α` and `β` is itself a universal set.


/-- Everything in the image of `algebraMap` is a unit  -/
theorem IsLocalization.map_units :
    ∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} (S : Type u_2) [inst_1 : CommSemiring S]
      [inst_2 : Algebra R S] [inst_3 : IsLocalization M S] (y : ↥M), IsUnit ((algebraMap R S) ↑y) :=
  by sorry

This theorem states that for any commutative semiring `R`, a submonoid `M` of `R`, a commutative semiring `S` and a localization `S` of `M`, every element `y` of the submonoid `M` becomes a unit when mapped into `S` using the algebra mapping. In other words, every `y` in `M` has a multiplicative inverse in the image of `algebraMap` from `R` to `S`. This is an important property in the study of localizations in algebraic structures.

For any commutative semiring `R`, submonoid `M` of `R`, commutative semiring `S` with `M` as a local subsemigroup, the algebra map from `R` to `S` maps each element `y` in `M` to an element with a multiplicative inverse in `S`.


Associated.dvd

theorem Associated.dvd : ∀ {α : Type u_1} [inst : Monoid α] {a b : α}, Associated a b → a ∣ b := by sorry

This theorem states that, for any type `α` that is a Monoid, if two elements `a` and `b` of `α` are associated (i.e., `a` can be expressed as `b` multiplied by a unit on the right), then `a` divides `b`. In math notation, this would be "if a and b are associated (there exists a unit u such that a * u = b), then a divides b".

If `α` is a monoid and there exists a unit `u` such that `a = b * u`, then `a` divides `b`.


theorem Set.mem_compl_iff : ∀ {α : Type u} (s : Set α) (x : α), x ∈ sᶜ ↔ x ∉ s := by sorry

This theorem, `Set.mem_compl_iff`, states that for any type `α`, any set `s` of `α`, and any element `x` of `α`, the element `x` is a member of the complement of set `s` (denoted by `sᶜ`) if and only if `x` is not a member of set `s`. In other words, the theorem provides a property of set complementarity in the context of the membership (`∈`) of an element `x`.

For any type `α`, set `s` of `α`, and `x` in `α`, `x` belongs to the complement of `s` if and only if `x` is not an element of `s`.


/-- postcompose an equation between morphisms by another morphism  -/
theorem CategoryTheory.eq_whisker :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} {f g : X ⟶ Y},
      f = g → ∀ (h : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g h :=
  by sorry

The theorem `CategoryTheory.eq_whisker` states that in any category `C`, for any three objects `X`, `Y`, and `Z`, and for any two morphisms `f` and `g` from `X` to `Y`, if `f` is equal to `g`, then composing `f` (or equivalently `g` due to their equality) with any morphism `h` from `Y` to `Z` will result in the same morphism from `X` to `Z`. In other words, postcomposing an equality between morphisms with another morphism preserves the equality.

In any category, if two morphisms between objects have the same source and target and are equal, then their composition with any morphism from the target to another object is equal.


theorem Real.rpow_mul : ∀ {x : ℝ}, 0 ≤ x → ∀ (y z : ℝ), x ^ (y * z) = (x ^ y) ^ z := by sorry

This theorem states that for any real number `x` that is greater than or equal to zero, and for any two real numbers `y` and `z`, the power of `x` raised to the product of `y` and `z` is equal to `x` raised to the power `y` and then the result raised to the power `z`. In LaTeX, this is expressed as `x^{yz} = (x^y)^z` for all `x ≥ 0`, `y`, and `z` in the set of real numbers.

For any real number `x ≥ 0`, `x^(yz) = (x^y)^z`.


theorem Filter.Germ.coe_eq : ∀ {α : Type u_1} {β : Type u_2} {l : Filter α} {f g : α → β}, ↑f = ↑g ↔ f =ᶠ[l] g := by
  sorry

This theorem states that for any two functions 'f' and 'g' from a type 'α' to a type 'β', and any filter 'l' on 'α', the germs (elements in the ultrapower) of 'f' and 'g' are equal if and only if 'f' and 'g' are eventually equal almost everywhere with respect to the filter 'l'. In the context of this theorem, 'eventually equal almost everywhere' means that 'f' and 'g' are equal for all elements in the set, except possibly on a subset that the filter 'l' does not care about.

The theorem asserts that functions from type 'α' to type 'β' have equal germs in the ultrapower with respect to filter 'l' if and only if they are almost everywhere equal with respect to 'l'.


theorem div_self' : ∀ {G : Type u_3} [inst : Group G] (a : G), a / a = 1 := by sorry

This theorem states that for any given type `G` that has a group structure, the division of any element `a` of `G` by itself always equals the group's identity element `1`. This is a generalization of the usual mathematical rule that any number divided by itself equals 1, extended to any group structure.

For any group `G` and its element `a`, `a * inverse a = 1` holds.


theorem Mathlib.Data.Set.Intervals.Basic._auxLemma.1 :
    ∀ {α : Type u_1} [inst : Preorder α] {a b x : α}, (x ∈ Set.Ioo a b) = (a < x ∧ x < b) := by sorry

This theorem states that for any type `α` that has a preorder structure, and any three elements `a`, `b`, and `x` in `α`, `x` belongs to the left-open right-open interval between `a` and `b`, denoted as `Set.Ioo a b`, if and only if `a` is less than `x` and `x` is less than `b`. In other words, `x` being in the interval `(a, b)` is equivalent to `a < x` and `x < b`. This definition encapsulates the standard mathematical understanding of an open interval.

For any type `α` with a preorder structure, and any elements `a`, `b`, and `x` in `α`, `x` lies in the left-open right-open interval between `a` and `b` if and only if `a` is strictly less than `x` and `x` is strictly less than `b`.


theorem PartialEquiv.left_inv :
    ∀ {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β) {x : α}, x ∈ e.source → ↑(PartialEquiv.symm e) (↑e x) = x :=
  by sorry

The theorem `PartialEquiv.left_inv` states that: for any two types `α` and `β`, and given a partial equivalence `e` from `α` to `β`, if an element `x` from the source set of `e` is taken, then applying the original partial equivalence `e` to `x` and then applying the inverse of the partial equivalence (i.e., `PartialEquiv.symm e`) to the result will return the original element `x`. This is a statement of the left-inverse property for the defined partial equivalence.

For any partial equivalence \(e: \alpha \leftrightarrows \beta\), \(e \circ \text{PartialEquiv.symm } e = \text{id}_{\alpha}\).


theorem List.length_replicate : ∀ {α : Type u} (n : ℕ) (a : α), List.length (List.replicate n a) = n := by sorry

The theorem `List.length_replicate` states that for any type `α`, any natural number `n`, and any element `a` of type `α`, the length of the list obtained by replicating `a` `n` times is exactly `n`. This means that if you create a list by repeating a certain element a certain number of times, the length of that list will be equal to the number of times you repeated the element.

For any natural number `n` and element `a`, the length of the list obtained by repeating `a` `n` times is equal to `n`.


theorem Complex.exp_add : ∀ (x y : ℂ), Complex.exp (x + y) = Complex.exp x * Complex.exp y := by sorry

This theorem states that for any two complex numbers `x` and `y`, the exponential of the sum `x + y` is equal to the product of the exponential of `x` and the exponential of `y`. In mathematical terms, this corresponds to the well-known property of the exponential function, written as `exp(x + y) = exp(x) * exp(y)`, but extended to the realm of complex numbers.

For all complex numbers x and y, exp(x + y) = exp(x) * exp(y).


theorem List.mem_map :
    ∀ {α : Type u_1} {β : Type u_2} {b : β} {f : α → β} {l : List α}, b ∈ List.map f l ↔ ∃ a ∈ l, f a = b := by sorry

This theorem, `List.mem_map`, states that for any two types `α` and `β`, a function `f` from `α` to `β`, a list `l` of type `α`, and an element `b` of type `β`, `b` is an element of the list obtained by applying `f` to each element of `l` if and only if there exists an element `a` in the original list `l` such that `f a` equals `b`. In other words, `b` is in the mapped list if there is an element in the original list that is mapped to `b` by the function `f`.

For any function `f` from type `α` to type `β`, list `l` of type `α`, and element `b` of type `β`, `b` is in the image of `l` under `f` if and only if there exists an element `a` in `l` such that `f(a) = b`.


theorem Complex.ofReal_div : ∀ (r s : ℝ), ↑(r / s) = ↑r / ↑s := by sorry

This theorem states that for any two real numbers `r` and `s`, the complex representation (or 'embedding') of the division of `r` by `s` is equal to the division of the complex representations of `r` and `s`. In mathematical terms, it means for all real numbers `r` and `s`, we have `(r / s) = r / s` in the complex number space, where the left side is the real number division operation and the right side is the complex number division operation.

For all real numbers r and s, the complex representation of r divided by s equals the complex division of the complex representations of r and s.


theorem Equiv.Perm.mul_apply : ∀ {α : Type u} (f g : Equiv.Perm α) (x : α), (f * g) x = f (g x) := by sorry

The theorem `Equiv.Perm.mul_apply` states that for any type `α`, and for any two permutations `f` and `g` of `α`, the function application of the product `(f * g)` at any element `x` of `α` is equal to the function application of `f` at the result of the function application of `g` at `x`. In other words, this theorem is explaining the composition of two permutations: applying the permutation `(f * g)` to `x` is the same as first applying the permutation `g` to `x` and then applying the permutation `f` to the result.

For any type `α` and permutations `f` and `g` of `α`, `(f * g) x = f (g x)`.


theorem neg_le_neg : ∀ {α : Type u} [inst : OrderedAddCommGroup α] {a b : α}, a ≤ b → -b ≤ -a := by sorry

This theorem states that for any ordered additive commutative group `α`, and for any two elements `a` and `b` from `α`, if `a` is less than or equal to `b`, then the negation of `b` is less than or equal to the negation of `a`. In other words, it expresses that within this kind of group, inverting the order of elements by taking their negatives also inverts their inequality relation.

For any ordered additive commutative group `α`, if `a ≤ b`, then `-b ≤ -a`.


theorem isClosed_le :
    ∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : Preorder α] [t : OrderClosedTopology α]
      [inst_2 : TopologicalSpace β] {f g : β → α}, Continuous f → Continuous g → IsClosed {b | f b ≤ g b} :=
  by sorry

This theorem states that for any types `α` and `β`, given `α` is a topological space and a preorder with order-closed topology, and `β` is also a topological space, if `f` and `g` are two continuous functions from `β` to `α`, then the set of all elements `b` in `β` such that `f(b)` is less than or equal to `g(b)` is a closed set.

If `α` is a topological space and a preorder with order-closed topology, and `β` is a topological space, then the set of elements `b` in `β` such that `f(b) ≤ g(b)` for continuous functions `f` and `g` from `β` to `α` forms a closed set in `β`.


theorem Multiset.map_id' : ∀ {α : Type u_1} (s : Multiset α), Multiset.map (fun x => x) s = s := by sorry

The theorem `Multiset.map_id'` asserts that for any type `α` and any multiset `s` of `α`, applying the identity function to each element of `s` through the function `Multiset.map` returns the multiset `s` itself. In other words, `Multiset.map` with the identity function leaves the multiset unchanged. This is akin to the mathematical principle that applying the identity function to each element of a set yields the set itself.

For any type α and multiset s of α, the application of the identity function through Multiset.map leaves the multiset s unchanged.


List.Pairwise.imp

theorem List.Pairwise.imp :
    ∀ {α : Type u_1} {R S : α → α → Prop},
      (∀ {a b : α}, R a b → S a b) → ∀ {l : List α}, List.Pairwise R l → List.Pairwise S l :=
  by sorry

This theorem states that for any type `α` and any two relations `R` and `S` on `α`, if all pairs of elements in `α` that satisfy `R` also satisfy `S`, then for any list of `α`, if all consecutive pairs of elements in the list satisfy `R`, they also satisfy `S`. In other words, it describes the preservation of pairwise properties of lists under implication of relations.

If `R` implies `S` as relations on type `α`, then any list of `α` where consecutive pairs satisfy `R` also satisfies `S`.


theorem Int.sub_add_cancel : ∀ (a b : ℤ), a - b + b = a := by sorry

This theorem states that for any two integers `a` and `b`, if you subtract `b` from `a` and then add `b` back, you get `a` back. In mathematical terms, it states that for all integers `a` and `b`, the equation `a - b + b = a` holds. This essentially captures the property of subtraction and addition being inverse operations.

For all integers `a` and `b`, `a - b + b = a`.


theorem Complex.continuous_ofReal : Continuous Complex.ofReal' := by sorry

The theorem `Complex.continuous_ofReal` states that the function `Complex.ofReal'`, which maps real numbers to complex numbers, is continuous. In other words, the function that takes a real number and returns a complex number with that real number as the real part and zero as the imaginary part, is a continuous function with respect to the topology of real numbers and complex numbers.

The real-valued function `Complex.ofReal'` mapping reals to complex numbers with real part equal to the input and imaginary part equal to zero is continuous.


theorem CategoryTheory.Limits.coprod.map_desc :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {S T U V W : C}
      [inst_1 : CategoryTheory.Limits.HasBinaryCoproduct U W] [inst_2 : CategoryTheory.Limits.HasBinaryCoproduct T V]
      (f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W),
      CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map h k)
          (CategoryTheory.Limits.coprod.desc f g) =
        CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp h f)
          (CategoryTheory.CategoryStruct.comp k g) :=
  by sorry

The theorem `CategoryTheory.Limits.coprod.map_desc` states that in a category `C` with binary coproducts, for objects `S`, `T`, `U`, `V`, and `W`, given two morphisms `f: U ⟶ S` and `g: W ⟶ S`, and two other morphisms `h: T ⟶ U` and `k: V ⟶ W`, the composition of `CategoryTheory.Limits.coprod.map h k` and `CategoryTheory.Limits.coprod.desc f g` is equal to `CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp h f) (CategoryTheory.CategoryStruct.comp k g)`. In other words, this expresses a commutative property regarding the composition of morphisms involving the coproduct.

In a category with binary coproducts, the composition of morphisms `h` and `k` with `CategoryTheory.Limits.coprod.map` followed by `CategoryTheory.Limits.coprod.desc` is equal to `CategoryTheory.Limits.coprod.desc` of their composition.


theorem Real.sin_pi_div_two : Real.sin (Real.pi / 2) = 1 := by sorry

This theorem states that the sine of pi divided by 2, in real numbers, is equal to 1. In other words, it is expressing the well-known trigonometric fact that $\sin(\pi/2) = 1$ where $\pi$ is the mathematical constant Pi. The sine function here is defined as the real part of the complex sine.

The real part of the complex sine of π/2 is equal to 1.


theorem RingEquiv.apply_symm_apply :
    ∀ {R : Type u_4} {S : Type u_5} [inst : Mul R] [inst_1 : Mul S] [inst_2 : Add R] [inst_3 : Add S] (e : R ≃+* S)
      (x : S), e ((RingEquiv.symm e) x) = x :=
  by sorry

This theorem states that for any ring isomorphism `e` from a ring `R` to a ring `S`, and for any element `x` from `S`, applying the isomorphism `e` to the result of applying the inverse of `e` to `x` will yield `x` itself. In mathematical terms, for a ring isomorphism \(e: R \rightarrow S\), and any \(x \in S\), we have \(e(e^{-1}(x)) = x\). Essentially, this theorem is asserting the correctness of the inverse function of a ring isomorphism.

For any ring isomorphism \(e: R \rightarrow S\), \(e \circ e^{-1} = id\_S\), where \(id\_S\) is the identity function on \(S\).


theorem Set.Ioo_subset_Ico_self : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, Set.Ioo a b ⊆ Set.Ico a b := by sorry

The theorem `Set.Ioo_subset_Ico_self` states that for any type `α` with a preorder structure and any two elements `a` and `b` of that type, the left-open right-open interval `(a, b)`, denoted as `Set.Ioo a b`, is a subset of the left-closed right-open interval `[a, b)`, denoted as `Set.Ico a b`. In other words, all elements `x` satisfying `a < x < b` also satisfy `a ≤ x < b`.

For any type `α` with a preorder structure and any `a < b` in `α`, the left-open right-open interval `(a, b)` is included in the left-closed right-open interval `[a, b)`.


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

This theorem, `and_iff_right_of_imp`, states that for any two propositions `a` and `b`, if `b` implies `a`, then `a ∧ b` is equivalent to `b`. In other words, if `b` being true guarantees the truth of `a`, then the conjunction `a ∧ b` is true if and only if `b` is true.

If `b` implies `a`, then `a ∧ b` is equivalent to `b`.


/-- A subgroup contains the group's 1.  -/
theorem Subgroup.one_mem : ∀ {G : Type u_1} [inst : Group G] (H : Subgroup G), 1 ∈ H := by sorry

This theorem states that for any given subgroup `H` of a group `G`, the identity element (or the "1") of the group `G` is always an element of the subgroup `H`. This is a fundamental property of subgroups in group theory. In other words, no matter what specific subgroup you have, it always includes the identity element of the overall group.

For any subgroup H of a group G, the identity element of G belongs to H.


theorem Matrix.conjTranspose_conjTranspose :
    ∀ {m : Type u_2} {n : Type u_3} {α : Type v} [inst : InvolutiveStar α] (M : Matrix m n α),
      Matrix.conjTranspose (Matrix.conjTranspose M) = M :=
  by sorry

The theorem `Matrix.conjTranspose_conjTranspose` states that for any matrix `M` with indices `m` and `n` and entries in a type `α` that has an involutive star operation, the conjugate transpose of the conjugate transpose of `M` is equal to `M` itself. In mathematical terms, given a matrix `M` and denoting the conjugate transpose operation as `*`, the theorem asserts that `(M*)* = M`. This is analogous to the property of the conjugate transpose operation in complex matrix algebra, where taking the conjugate transpose twice returns the original matrix.

For any square matrix M with entries in a type having an involutive star operation, the conjugate transpose of its conjugate transpose equals the original matrix: (M*)* = M.


theorem mul_lt_mul_left :
    ∀ {α : Type u_1} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α] [inst_3 : PosMulStrictMono α]
      [inst_4 : PosMulReflectLT α], 0 < a → (a * b < a * c ↔ b < c) :=
  by sorry

This theorem, `mul_lt_mul_left`, states that for any type `α` equipped with multiplication, a zero element, and a preorder relation, and satisfying the properties of strict monotonicity of multiplication by positive elements on the left (`PosMulStrictMono α`) and strict reverse monotonicity of multiplication by nonnegative elements on the left (`PosMulReflectLT α`), if `0 < a`, then the product of `a` and `b` is less than the product of `a` and `c` if, and only if, `b` is less than `c`. This embeds the property of 'cancellation' in these algebraic structures, stating that multiplying a smaller number by a positive number yields a smaller product.

If `α` is a type with multiplication, a zero element, and preorder relation such that multiplication by positive elements is strictly monotonic on the left and multiplication by nonnegative elements is strictly reverse monotonic on the left, then `a > 0` implies `a * b < a * c` if and only if `b < c`.


theorem map_intCast :
    ∀ {F : Type u_1} {α : Type u_3} {β : Type u_4} [inst : NonAssocRing α] [inst_1 : NonAssocRing β]
      [inst_2 : FunLike F α β] [inst_3 : RingHomClass F α β] (f : F) (n : ℤ), f ↑n = ↑n :=
  by sorry

The theorem `map_intCast` states that for any types `F`, `α` and `β`, where `α` and `β` are non-associative rings, `F` is a function-like object from `α` to `β`, and `F` is a ring homomorphism from `α` to `β`, the function `f` of type `F` preserves the cast of an integer `n` to `α` and `β`. In other words, applying `f` to the cast of `n` to `α` is the same as the cast of `n` to `β`. This theorem expresses an important property of ring homomorphisms: they preserve the ring structure, in this case, the ability to cast integers to members of the ring.

For any ring homomorphism `F` from non-associative rings `α` and `β`, and any integer `n`, `F (int.cast α n) = int.cast β (n)`.


theorem Set.smul_mem_smul_set :
    ∀ {α : Type u_2} {β : Type u_3} [inst : SMul α β] {s : Set β} {a : α} {b : β}, b ∈ s → a • b ∈ a • s := by sorry

The theorem `Set.smul_mem_smul_set` states that for any types `α` and `β`, where `α` has a scalar multiplication operation with `β`, given a set `s` of type `β`, a scalar `a` of type `α`, and an element `b` of type `β`, if `b` is an element of `s`, then the result of scalar multiplication of `a` and `b` (`a • b`) is an element of the scalar multiplication of `a` and the set `s` (`a • s`). In other words, for every element in a set, its scalar multiplication is an element in the scalar multiplication of the set.

For any type `α` with scalar multiplication, set `β`, and `a ∈ α`, `b ∈ β`, if `b ∈ s`, then `a • b ∈ a • s` (scalar multiplication of an element by a set).


/-- The identity is `C^∞`.
 -/
theorem contDiff_id :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {n : ℕ∞}, ContDiff 𝕜 n id :=
  by sorry

This theorem states that the identity function is continuously differentiable an infinite number of times. More formally, for any non-trivial normed field 𝕜 and any normed additive commutative group E that is also a normed space over 𝕜, the identity function is continuously differentiable at every point, to any order n in ℕ∞ (the extended natural numbers, including positive infinity).

For any non-trivial normed field 𝕜 and normed additive commutative group E that is also a normed space over 𝕜, the identity function on E is continuously differentiable to any order in the extended natural numbers.


theorem le_tsub_add :
    ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : Add α] [inst_2 : Sub α] [inst_3 : OrderedSub α] {a b : α},
      b ≤ b - a + a :=
  by sorry

This theorem states that for any type `α` with a preorder relation (i.e., a binary relation that is reflexive and transitive), addition, and subtraction operations, and also satisfying the property of ordered subtraction, for any two elements `a` and `b` of type `α`, `b` is always less than or equal to the result of subtracting `a` from `b` and then adding `a` back. In mathematical terms, for all `a` and `b` in `α`, we have the inequality `b ≤ b - a + a`.

For any type `α` with a preorder relation, addition, subtraction, and the property of ordered subtraction, for all `a` and `b` in `α`, we have `b ≤ a + (b - a)`.


theorem Finset.mem_Ioc :
    ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : LocallyFiniteOrder α] {a b x : α},
      x ∈ Finset.Ioc a b ↔ a < x ∧ x ≤ b :=
  by sorry

The theorem `Finset.mem_Ioc` states that for any types `α` with a preorder and a locally finite order, and any elements `a`, `b`, and `x` of `α`, `x` is a member of the finite set `Finset.Ioc a b` if and only if `a` is less than `x` and `x` is less than or equal to `b`. In other words, `x` belongs to the interval `(a, b]` in `α`.

For any type `α` with a preorder and locally finite order, `x` belongs to the interval `(a, b]` if and only if `a` is strictly less than `x` and `x` is less than or equal to `b`.


theorem Mathlib.Analysis.Normed.Group.Basic._auxLemma.30 :
    ∀ {E : Type u_6} [inst : NormedAddGroup E] {a : E}, (‖a‖ = 0) = (a = 0) := by sorry

This theorem, named `Mathlib.Analysis.Normed.Group.Basic._auxLemma.30`, states that for any type `E` that is a normed additive group, the norm of an element `a` from `E` is zero if and only if `a` itself is the zero element of the group. In mathematical notation, this could be written as ‖a‖ = 0 if and only if a = 0.

For any normed additive group E and its element a, the norm of a is zero if and only if a is the additive identity of E.


theorem ENNReal.rpow_mul : ∀ (x : ENNReal) (y z : ℝ), x ^ (y * z) = (x ^ y) ^ z := by sorry

This theorem states that for any extended nonnegative real number `x` and any two real numbers `y` and `z`, the exponentiation of `x` by the product of `y` and `z` is equal to the exponentiation of `x` raised to the power of `y`, then raised to the power of `z`. In mathematical notation, this can be written as \(x^{(y * z)} = ((x^y)^z)\). This theorem is demonstrating the property of exponentiation in the context of extended nonnegative real numbers.

For any extended nonnegative real number x and real numbers y and z, x^(y * z) = (x^y)^z.


theorem MvPolynomial.algHom_ext :
    ∀ {R : Type u} {σ : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]
      {f g : MvPolynomial σ R →ₐ[R] A}, (∀ (i : σ), f (MvPolynomial.X i) = g (MvPolynomial.X i)) → f = g :=
  by sorry

This theorem states that for any type `R` with a commutative semiring structure, any type `σ`, any semiring `A` that also forms an algebra over `R`, and any two algebra homomorphisms `f` and `g` from the multivariate polynomial ring `MvPolynomial σ R` to `A`, if `f` and `g` agree on all degree `1` monomials `MvPolynomial.X i` (where `i` ranges over `σ`), then `f` and `g` are the same. In other words, an algebra homomorphism from a multivariate polynomial ring is uniquely determined by its values on the degree `1` monomials.

Given a commutative semiring `R`, a type `σ`, a semiring `A` that is an algebra over `R` for `σ`, and two algebra homomorphisms `f` and `g` from the multivariate polynomial ring `MvPolynomial σ R` to `A`, if `f(X i) = g(X i)` for all `i ∈ σ` and `X` being the indeterminate, then `f = g`.


theorem Asymptotics.isBigO_refl :
    ∀ {α : Type u_1} {E : Type u_3} [inst : Norm E] (f : α → E) (l : Filter α), f =O[l] f := by sorry

This theorem, `Asymptotics.isBigO_refl`, states that for any type `α`, any normed type `E`, any function `f` from `α` to `E`, and any filter `l` on `α`, the function `f` is asymptotically bounded by itself, i.e., `f` is Big O of `f` (`f =O[l] f`) at the filter `l`. In the language of theoretical computer science or mathematics, this means that the growth rate of the function `f` is at most as fast as the growth rate of the function `f`, which is always true.

For any type `α`, normed type `E`, function `f` from `α` to `E`, and filter `l` on `α`, `f` is in Big O notation with respect to `l`, i.e., `f = O(f)` at the filter `l`.


theorem Filter.HasBasis.to_hasBasis :
    ∀ {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop}
      {s' : ι' → Set α},
      Filter.HasBasis l p s →
        (∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i) →
          (∀ (i' : ι'), p' i' → ∃ i, p i ∧ s i ⊆ s' i') → Filter.HasBasis l p' s' :=
  by sorry

This theorem states that given a filter `l` on a type `α` that has a basis characterized by a predicate `p` on an indexing type `ι` and a mapping `s` from `ι` to sets of `α`, if for every element `i` of `ι` satisfying `p`, there exists an element `i'` satisfying a predicate `p'` on another indexing type `ι'` such that the set `s'(i')` is a subset of `s(i)`, and for every element `i'` of `ι'` satisfying `p'`, there exists an element `i` satisfying `p` such that the set `s(i)` is a subset of `s'(i')`, then `l` also has a basis characterized by the predicate `p'` and the mapping `s'`. This theorem essentially provides a necessary and sufficient condition for a filter to have a certain basis.

If a filter on a type has a basis characterized by a predicate and a mapping to sets, and there exists a dual mapping and predicate such that each element in the original indexing type satisfies the predicate if and only if there exists an element in the dual indexing type satisfying the dual predicate with a subset relation between their corresponding sets, then the filter has an equivalent basis characterized by the dual predicate and mapping.


theorem Nat.odd_iff_not_even : ∀ {n : ℕ}, Odd n ↔ ¬Even n := by sorry

This theorem states that for every natural number `n`, `n` being odd is equivalent to `n` not being even. In other words, a natural number is odd if and only if it is not even. Here, a number is considered odd if it can be expressed as `2*k + 1` for some `k`, and a number is considered even if it can be written as `r + r` for some `r`.

A natural number is odd if and only if it is not even, or equivalently, a natural number is odd if and only if it can be expressed as 2 times an integer plus one, while an even number can be expressed as the sum of two integers.


ULift.ext

theorem ULift.ext : ∀ {α : Type u} (x y : ULift.{u_1, u} α), x.down = y.down → x = y := by sorry

This theorem, `ULift.ext`, states that for any type `α`, given two objects `x` and `y` of the type `ULift α`, if the `down` property of `x` is equal to the `down` property of `y`, then `x` is equal to `y`. The `down` property of a `ULift` object is a value of type `α` that the `ULift` object encapsulates. Essentially, this theorem is asserting that `ULift` objects are uniquely determined by their encapsulated values.

If two objects `x` and `y` of type `ULift α` have equal `down` values, then `x` is equal to `y`.


theorem Int.emod_add_ediv : ∀ (a b : ℤ), a % b + b * (a / b) = a := by sorry

This theorem states that for all integer numbers `a` and `b`, the remainder of `a` divided by `b` added to `b` times the integer division of `a` by `b` equals `a`. In terms of mathematical notation, this would be: for all integers a and b, a % b + b * ⌊a / b⌋ = a. This is a key property of the integer division operation and its corresponding remainder operation.

For all integers `a` and `b`, the remainder of `a` divided by `b` is equal to `a` modulo `b` plus `b` times the quotient of `a` by `b`. (Or, more succinctly: `a % b = a // b * b + a % b`)


theorem ENNReal.rpow_zero : ∀ {x : ENNReal}, x ^ 0 = 1 := by sorry

This theorem states that for every extended nonnegative real number `x`, raising `x` to the power of `0` results in `1`. This is a reflection of the mathematical principle that any non-zero number raised to the power of zero is `1`. The extended nonnegative real numbers, denoted as `[0, ∞]`, include all nonnegative real numbers along with positive infinity.

For every extended nonnegative real number `x`, `x^0` equals 1.


theorem PartialHomeomorph.left_inv :
    ∀ {X : Type u_1} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
      (e : PartialHomeomorph X Y) {x : X}, x ∈ e.source → ↑(PartialHomeomorph.symm e) (↑e x) = x :=
  by sorry

This theorem states that for any partial homeomorphism `e` from a type `X` to a type `Y`, both endowed with a topological space structure, if an element `x` of type `X` lies within the source of `e`, then applying the homeomorphism `e` to `x` and then the inverse of the homeomorphism (denoted by `PartialHomeomorph.symm e`) to the result, will give back the original element `x`. Essentially, it is a statement about the left-inverse property of the partial homeomorphism `e`.

For any partial homeomorphism `e` between topological spaces `X` and `Y`, if `x ∈ X` lies in the domain of `e`, then `e (x) = e.symm (e x)`.


theorem Inducing.nhds_eq_comap :
    ∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
      Inducing f → ∀ (x : X), nhds x = Filter.comap f (nhds (f x)) :=
  by sorry

The theorem `Inducing.nhds_eq_comap` states that for any types `X` and `Y` with their corresponding topological spaces, and any function `f` from `X` to `Y` that induces the topology of `Y` from `X`, the neighborhood filter of any point `x` in `X` is equal to the pullback (also known as the inverse image under `f` or the `comap`) of the neighborhood filter of `f(x)` in `Y`. In simpler terms, the theorem is basically saying that the neighborhoods of point `x` in `X` correspond exactly to the pre-images under `f` of the neighborhoods of `f(x)` in `Y`, whenever `f` is a topology-inducing function.

For any topological spaces (X,τ\_X) and (Y,τ\_Y), and a topology-inducing function f : X -> Y, the neighborhood filter of x in X is equal to the pullback of the neighborhood filter of f(x) in Y.


theorem and_imp : ∀ {a b c : Prop}, a ∧ b → c ↔ a → b → c := by sorry

This theorem, `and_imp`, is a statement about logical implications in propositional logic. It asserts that for any propositions `a`, `b`, and `c`, the implication from the conjunction of `a` and `b` to `c` is logically equivalent to the double implication from `a` to `b` to `c`. In other words, if `a` and `b` together imply `c`, then it's the same as saying if `a` is true, then if `b` is true, `c` is also true.

The theorem `and_imp` asserts that for all propositions `a`, `b`, and `c`, `(a ∧ b) → c` is logically equivalent to `a → (b → c)`.


theorem Fin.cons_zero :
    ∀ {n : ℕ} {α : Fin (n + 1) → Type u} (x : α 0) (p : (i : Fin n) → α (Fin.succ i)), Fin.cons x p 0 = x := by sorry

The theorem `Fin.cons_zero` states that for any natural number `n`, and any indexed type `α` that depends on `Fin (n + 1)`, given an element `x` of type `α 0` and a function `p` that maps any value of type `Fin n` to `α` indexed by the successor of that value, if `Fin.cons` is applied to `x`, `p`, and `0`, the result will be `x`. In simpler terms, this theorem says that when we add an element at the beginning of an `n`-tuple using `Fin.cons`, the resulting tuple's first element (at index `0`) is the element we added.

For any natural number `n` and indexed type `α` over `Fin (n + 1)`, the application of `Fin.cons` to an element `x` of type `α 0` and a function `p` mapping `Fin n` to `α` yields a tuple with the first element equal to `x`.


antisymm

theorem antisymm : ∀ {α : Sort u} {r : α → α → Prop} [inst : IsAntisymm α r] {a b : α}, r a b → r b a → a = b := by
  sorry

The theorem `antisymm` states that for any type `α` and any relation `r` on `α`, if this relation `r` is antisymmetric (i.e., for all `a` and `b` in `α`, if `r a b` and `r b a` then `a` equals `b`), then for any `a` and `b` in `α`, if `r a b` and `r b a` are both true, it follows that `a` equals `b`. In simpler words, this theorem captures the property of antisymmetry: if two elements in a set relate to each other under a certain relation, they must be the same.

If a binary relation on a type is antisymmetric, then for all elements `a` and `b`, if `a` is related to `b` and `b` is related to `a`, then `a` equals `b`.


theorem Finset.card_product :
    ∀ {α : Type u_1} {β : Type u_2} (s : Finset α) (t : Finset β), (s ×ˢ t).card = s.card * t.card := by sorry

This theorem states that for any two finite sets `s` and `t` of any types `α` and `β` respectively, the cardinality (i.e., the number of elements) of the Cartesian product of `s` and `t` (denoted as `s ×ˢ t`) is equal to the product of the cardinalities of `s` and `t`. In other words, if you have `s.card` elements in set `s` and `t.card` elements in set `t`, then there will be `s.card * t.card` ordered pairs in the Cartesian product of `s` and `t`.

For any finite sets `s` and `t` of types `α` and `β` respectively, the cardinality of their Cartesian product `s ×ˢ t` equals the product of their cardinalities `s.card * t.card`.


MvPolynomial.ext

theorem MvPolynomial.ext :
    ∀ {R : Type u} {σ : Type u_1} [inst : CommSemiring R] (p q : MvPolynomial σ R),
      (∀ (m : σ →₀ ℕ), MvPolynomial.coeff m p = MvPolynomial.coeff m q) → p = q :=
  by sorry

This theorem states that for any commutative semiring `R` and any type `σ`, if we have two multivariate polynomials `p` and `q` over the variables indexed by `σ` and with coefficients in `R`, then `p` and `q` are equal if and only if their coefficients for each monomial `m` (represented as a function from `σ` to natural numbers) are equal. In other words, two multivariate polynomials are considered identical if they have the same coefficients for the same monomials.

For any commutative semiring R and type σ, two multivariate polynomials over R with variables indexed by σ are equal if and only if their corresponding coefficient functions agree on all monomials.


/-- An element is in the kernel if and only if it maps to zero. -/
theorem RingHom.mem_ker :
    ∀ {R : Type u} {S : Type v} {F : Type u_1} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S]
      [rcf : RingHomClass F R S] (f : F) {r : R}, r ∈ RingHom.ker f ↔ f r = 0 :=
  by sorry

This theorem states that for any semirings `R` and `S`, any function-like structure `F` from `R` to `S` that forms a ring homomorphism class, and any element `r` from `R`, the element `r` is in the kernel of the ring homomorphism `f` if and only if the application of `f` to `r` is zero. The kernel of a ring homomorphism is defined as the set of elements from the domain that map to zero in the codomain. Therefore, this theorem is showing the equivalence of membership in the kernel and mapping to zero.

For any ring homomorphism `f` from semiring `R` to semiring `S` and any element `r` in `R`, `r` is in the kernel of `f` if and only if `f(r)` is the additive identity in `S`.


theorem Nat.not_le_of_gt : ∀ {n m : ℕ}, n > m → ¬n ≤ m := by sorry

This theorem states that for any two natural numbers `n` and `m`, if `n` is greater than `m` then it is not possible for `n` to be less than or equal to `m`. This theorem embodies the intuitive concept that a larger number cannot also be smaller or equal to a smaller number.

For all natural numbers n and m, if n > m then it is false that n <= m.


theorem mul_lt_mul_iff_left :
    ∀ {α : Type u_1} [inst : Mul α] [inst_1 : LT α]
      [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1]
      [inst_3 : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b c : α},
      a * b < a * c ↔ b < c :=
  by sorry

This theorem, `mul_lt_mul_iff_left`, states that for any type `α` which supports multiplication (`Mul α`) and less than operation (`LT α`) and follows certain mathematical properties (`CovariantClass` and `ContravariantClass`), the product `a * b` is less than `a * c` if and only if `b` is less than `c`. This is assuming that `a`, `b`, and `c` are all of type `α`. In other words, it states that multiplying both sides of a strict inequality by the same positive number doesn't affect the inequality's direction.

For all types `α` with multiplication and less-than relations that are covariant and contravariant, `a * b` is less than `a * c` if and only if `b` is less than `c`.


theorem mul_le_mul_iff_left :
    ∀ {α : Type u_1} [inst : Mul α] [inst_1 : LE α]
      [inst_2 : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
      [inst_3 : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α) {b c : α},
      a * b ≤ a * c ↔ b ≤ c :=
  by sorry

This theorem, `mul_le_mul_iff_left`, states that for any type `α` that has multiplication and an order relation (denoted by `≤`), and that satisfies the `CovariantClass` and `ContravariantClass` properties with respect to multiplication and this order, for any elements `a`, `b`, and `c` of this type, the product of `a` and `b` is less than or equal to the product of `a` and `c` if and only if `b` is less than or equal to `c`. This can be interpreted as the order preservation property of multiplication when the multiplier is non-negative.

For any type equipped with multiplication and a total order such that multiplication is covariant and contravariant with respect to the order, the product of two elements is less than or equal to the product of another pair if and only if the first pair is less than or equal to the second in the order.


theorem Ideal.zero_mem : ∀ {α : Type u} [inst : Semiring α] (I : Ideal α), 0 ∈ I := by sorry

This theorem states that for any type `α` that is a semiring, the zero element is always a member of any ideal `I` of `α`. In mathematical terms, given any ideal in a semiring, the zero element (0) belongs to this ideal.

For any semiring `α` and ideal `I` of `α`, the zero element `0` belongs to `I`.


/-- The length of a list converted from a function is the size of the domain.  -/
theorem List.length_ofFn : ∀ {α : Type u} {n : ℕ} (f : Fin n → α), List.length (List.ofFn f) = n := by sorry

This theorem states that for any function `f` from the finite set `Fin n` to any type `α`, the length of the list created by applying `f` to each element in `Fin n` (using `List.ofFn`) is equal to `n`. In other words, when a function `f` is mapped over a finite domain `Fin n` to generate a list, the size (length) of the resulting list will be the same as the size of the domain `n`.

For any function from a finite set `Fin n` to type `α`, the length of the list obtained by applying `List.ofFn` is equal to `n`.


ProbabilityTheory.kernel.ext

theorem ProbabilityTheory.kernel.ext :
    ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
      {κ η : ↥(ProbabilityTheory.kernel α β)}, (∀ (a : α), κ a = η a) → κ = η :=
  by sorry

This theorem is a part of Probability Theory and states that for any two measurable spaces `α` and `β` (with their associated measurable spaces `mα` and `mβ`), if two probability kernels `κ` and `η` from `α` to `β` are such that for every element `a` in `α`, `κ a = η a`, then the two kernels `κ` and `η` are equal. In other words, if two probability kernels produce the same output for every input in the source space, those two kernels are identical.

If two probability kernels from one measurable space to another have identical output for every input in the source space, then they are equal.


theorem UniformContinuous.prod_mk :
    ∀ {α : Type ua} {β : Type ub} {γ : Type uc} [inst : UniformSpace α] [inst_1 : UniformSpace β]
      [inst_2 : UniformSpace γ] {f₁ : α → β} {f₂ : α → γ},
      UniformContinuous f₁ → UniformContinuous f₂ → UniformContinuous fun a => (f₁ a, f₂ a) :=
  by sorry

The theorem `UniformContinuous.prod_mk` states that for any types `α`, `β`, and `γ` equipped with uniform space structures, if we have two functions `f₁ : α → β` and `f₂: α → γ` that are both uniformly continuous, then the function that takes an element from `α` and maps it to a pair in `β × γ` where the first element of the pair is the image of the element under `f₁` and the second element of the pair is the image of the same element under `f₂` is also uniformly continuous. In other words, the uniform continuity of individual functions implies the uniform continuity of the function that produces pairs of their outputs.

If `f₁ : α → β` and `f₂ : α → γ` are uniformly continuous functions between uniform spaces `α`, `β`, and `γ`, then the function `x mapsto (f₁ x, f₂ x)` from `α` to `β × γ` is also uniformly continuous.


theorem MeasureTheory.StronglyMeasurable.tendsto_approx :
    ∀ {α : Type u_1} {β : Type u_2} {f : α → β} [inst : TopologicalSpace β] {x : MeasurableSpace α}
      (hf : MeasureTheory.StronglyMeasurable f) (x_1 : α),
      Filter.Tendsto (fun n => ↑(MeasureTheory.StronglyMeasurable.approx hf n) x_1) Filter.atTop (nhds (f x_1)) :=
  by sorry

The theorem `MeasureTheory.StronglyMeasurable.tendsto_approx` states that for all types `α` and `β`, and a function `f` from `α` to `β`, if the function is strongly measurable (i.e., it can be approximated by a sequence of simple functions), then for every point `x_1` of type `α`, the sequence of approximations at `x_1` tends to `f(x_1)` as `n` tends to infinity. Here, 'tends to' is defined in the sense of filter: for every neighborhood of `f(x_1)`, there exists some `n` such that for all `m ≥ n`, the `m`-th approximation at `x_1` is inside that neighborhood.

For any type `α` and `β`, and strongly measurable function `f : α → β`, for every `x_1 : α`, the sequence of approximations of `f` at `x_1` converges to `f(x_1)` in the sense of filters.


theorem Finmap.induction_on :
    ∀ {α : Type u} {β : α → Type v} {C : Finmap β → Prop} (s : Finmap β),
      (∀ (a : AList β), C (AList.toFinmap a)) → C s :=
  by sorry

The theorem `Finmap.induction_on` is a principle of induction for finite maps (`Finmap`). For every type `α`, a type `β` dependent on `α`, a property `C` about finite maps of `β`, and an instance `s` of a finite map of `β`, it states that if the property `C` holds for all finite maps that result from applying the function `AList.toFinmap` to any instance `a` of `AList β`, then the property `C` also holds for the finite map `s`.

If `C` is a property that holds for all finite maps obtained by applying `AList.toFinmap` to any `AList β`, then `C` holds for the finite map `s` of type `Finmap β`.


/-- This lemma has an abbreviated name because it is used frequently.  -/
theorem ENNReal.cancel_of_ne : ∀ {a : ENNReal}, a ≠ ⊤ → AddLECancellable a := by sorry

This theorem, named `ENNReal.cancel_of_ne`, states that for any element `a` in the extended nonnegative real numbers (usually denoted as [0, ∞]), if `a` is not equal to infinity, then `a` possesses the property of being `AddLECancellable`. In other words, the function `x ↦ a + x` is order-reflecting for such `a`. This implies that for all `b` and `c` in the same set, if `a + b` is less than or equal to `a + c`, then `b` itself is less than or equal to `c`.

For any finite element `a` in the extended nonnegative real numbers, the function `x mapsto a + x` is order-reflecting.


theorem add_zsmul : ∀ {G : Type w} [inst : AddGroup G] (a : G) (m n : ℤ), (m + n) • a = m • a + n • a := by sorry

This theorem, `add_zsmul`, in Lean 4 states that for any type `G` that has an addition group structure, and any element `a` of `G` and any two integers `m` and `n`, the operation of scalar multiplication by (`m + n`) on `a` is equal to the sum of the operations of scalar multiplication by `m` on `a` and scalar multiplication by `n` on `a`. In other words, the scalar multiplication distributes over integer addition in the context of addition groups.

For any additive group `G` and elements `a` in `G` and integers `m` and `n`, the scalar multiplication `(m + n) * a` equals the sum `(m * a) + (n * a)`.


theorem CategoryTheory.Sheaf.hom_ext :
    ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C}
      {A : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} A] {X Y : CategoryTheory.Sheaf J A} (x y : X ⟶ Y),
      x.val = y.val → x = y :=
  by sorry

The theorem `CategoryTheory.Sheaf.hom_ext` states that for any category `C` with a Grothendieck topology `J`, another category `A`, and two sheaves `X` and `Y` over `A` with respect to the topology `J`, if two morphisms `x` and `y` from `X` to `Y` are such that their underlying functions (obtained using `.val`) are equal, then the morphisms `x` and `y` themselves are also equal. This theorem ensures that in the context of category theory and sheaf theory, the equality of morphisms can be determined by the equality of their underlying functions.

If `C` is a category with a Grothendieck topology `J`, `A` is another category, and `X`, `Y` are sheaves over `A` with respect to `J`, then two morphisms `x : X -> Y` and `y : X -> Y` are equal if and only if their underlying functions `.val x` and `.val y` are equal.


theorem Functor.map_map :
    ∀ {α β γ : Type u} {f : Type u → Type v} [inst : Functor f] [inst_1 : LawfulFunctor f] (m : α → β) (g : β → γ)
      (x : f α), g <$> m <$> x = (g ∘ m) <$> x :=
  by sorry

The theorem `Functor.map_map` states that for any types `α`, `β`, and `γ`, and for a functor `f` of type `Type u → Type v` that obeys the laws of a `LawfulFunctor`, and given a function `m` from `α` to `β` and a function `g` from `β` to `γ`, and an instance `x` of functor `f` applied to `α`, the composition of applying function `m` then function `g` to `x` (denoted as `g <$> m <$> x`) is equivalent to applying the composed function `g ∘ m` to `x` (denoted as `(g ∘ m) <$> x`). In other words, the order of mapping functions over the functor does not change the final result, reflecting the law of composition in the context of functors.

For any type maps `m : α → β` and `g : β → γ`, functor `f : Type u → Type v` (as a lawful functor), and `x : f α`, the application of `g` followed by `m` to `x` is equal to the composition of `g` and `m` applied to `x` (`(g ∘ m) <$> x` is equal to `g <$> m <$> x`).


theorem abs_lt :
    ∀ {α : Type u_1} [inst : AddGroup α] [inst_1 : LinearOrder α]
      [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
      [inst_3 : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1],
      |a| < b ↔ -b < a ∧ a < b :=
  by sorry

The theorem `abs_lt` states that for any type `α` that is a linearly ordered additive group, and `a` and `b` being members of this type `α`, the absolute value of `a` is less than `b` if and only if `a` lies strictly between `-b` and `b`. Here, CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 denotes monotonicity of addition in α, and CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 represents that the function swap preserves the order.

For any linearly ordered additive group `α` and `a, b ∈ α`, `|a| < |b|` if and only if `-b < a < b`.


theorem CategoryTheory.Iso.eq_comp_inv :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X},
      g = CategoryTheory.CategoryStruct.comp f α.inv ↔ CategoryTheory.CategoryStruct.comp g α.hom = f :=
  by sorry

This theorem states that in any category `C`, for any objects `X`, `Y`, and `Z` in `C`, and any isomorphism `α` between `X` and `Y`, a morphism `g` from `Z` to `X` equals the composition of a morphism `f` from `Z` to `Y` and the inverse of `α` if and only if the composition of `g` and the isomorphism `α` equals `f`. In other words, this theorem presents a condition of equality for morphisms in terms of composition with an isomorphism and its inverse.

In any category, given objects X, Y, Z, an isomorphism α : X ↔ Y, and morphisms g : Z → X and f : Z → Y, the conditions g = f α and α . g = f hold if and only if g = f.


theorem Set.subset_empty_iff : ∀ {α : Type u} {s : Set α}, s ⊆ ∅ ↔ s = ∅ := by sorry

This theorem states that for any type `α` and any set `s` of type `α`, `s` is a subset of the empty set if and only if `s` is the empty set itself. Here, ⊆ denotes the subset relation, meaning every element of `s` is also an element of the empty set. The ⊆ ∅ condition can only be satisfied if `s` is the empty set (∅), as the empty set contains no elements.

A set `s` is a subset of the empty set if and only if `s` is the empty set.


theorem abs_one : ∀ {α : Type u_1} [inst : LinearOrderedRing α], |1| = 1 := by sorry

This theorem states that the absolute value of 1 is 1 for any type `α` that forms a linear ordered ring. In other words, no matter what specific number system we are working in, as long as it's a linear ordered ring, the absolute value of the number 1 will always be 1.

For all types `α` that are linear ordered rings, |1| = 1.


theorem Set.sUnion_eq_iUnion : ∀ {α : Type u_1} {s : Set (Set α)}, ⋃₀ s = ⋃ i, ↑i := by sorry

The theorem `Set.sUnion_eq_iUnion` states that for any type `α` and any set `s` of sets of type `α`, the set-theoretic union of all sets in `s` is equal to the union over all elements in the embedding of individual sets `i` in `s`. In mathematical terms, this means that for any collection of sets, the union of all those sets is the same as taking the union of each set one at a time.

For any type `α` and set `s` of sets of type `α`, the set-theoretic union of all sets in `s` equals the indexed union over `s`.


theorem Nat.mul_succ : ∀ (n m : ℕ), n * Nat.succ m = n * m + n := by sorry

This theorem states that for all natural numbers `n` and `m`, the product of `n` and the successor of `m` (`m` incremented by 1) is equal to the product of `n` and `m` plus `n`. In LaTeX, this would be written as `n * (m + 1) = n * m + n`. This is a property of multiplication over addition in natural numbers, reflecting the distributive law.

For all natural numbers `n` and `m`, the product of `n` and the successor of `m` (`m + 1`) equals the sum of `n` times `m` and `n`. In mathematical notation, `n * (m + 1) = n * m + n`.


/-- Makes a Galois connection from an order-preserving bijection.  -/
theorem OrderIso.to_galoisConnection :
    ∀ {α : Type u} {β : Type v} [inst : Preorder α] [inst_1 : Preorder β] (oi : α ≃o β),
      GaloisConnection ⇑oi ⇑(OrderIso.symm oi) :=
  by sorry

The theorem `OrderIso.to_galoisConnection` establishes that for any given order-preserving bijection `oi` between two types `α` and `β`, which are both preordered, there exists a Galois connection. Specifically, the Galois connection is established between the function `oi` and the inverse function of `oi`, represented by `OrderIso.symm oi`. This means that for any elements `a` from `α` and `b` from `β`, `oi a` is less than or equal to `b` if and only if `a` is less than or equal to the inverse of `oi` applied to `b`.

Given an order-preserving bijection between two preordered types `α` and `β`, there exists a Galois connection between the function and its inverse. Specifically, for all `a : α` and `b : β`, `a ≤ oi⁻¹(b)` if and only if `oi(a) ≤ b`.


Homeomorph.embedding

theorem Homeomorph.embedding :
    ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (h : X ≃ₜ Y),
      Embedding ⇑h :=
  by sorry

This theorem states that for every homeomorphism `h` between two types `X` and `Y`, where both `X` and `Y` are topological spaces, the function resulting from applying `h` is an embedding. In other words, homeomorphisms are always embeddings in the context of topological spaces.

Every homeomorphism between topological spaces is an embedding.


theorem Mon_.ext :
    ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
      {X Y : Mon_ C} {f g : X ⟶ Y}, f.hom = g.hom → f = g :=
  by sorry

This theorem states that for any given type `C` (which is a category and a monoidal category) and any two objects `X` and `Y` of the monoidal category `Mon_ C`, if you have two morphisms `f` and `g` from `X` to `Y` such that their corresponding 'hom' fields are equal, then `f` and `g` themselves must be equal. In other words, the 'hom' field uniquely determines the morphism in this context.

Given a monoidal category `C` and objects `X`, `Y` in `C`, if there exist morphisms `f` and `g` from `X` to `Y` with equal 'hom' fields, then `f` and `g` are equal.


theorem OrderIso.map_inf :
    ∀ {α : Type u_2} {β : Type u_3} [inst : SemilatticeInf α] [inst_1 : SemilatticeInf β] (f : α ≃o β) (x y : α),
      f (x ⊓ y) = f x ⊓ f y :=
  by sorry

This theorem states that for any two types `α` and `β`, given that they are both semilattices with respect to the infimum operation (a binary operation which for any two elements returns the greatest element that is less than or equal to both), an order isomorphism `f` from `α` to `β` preserves the infimum operation. That is, applying `f` to the infimum of `x` and `y` (denoted as `x ⊓ y`) results in the same output as the infimum of `f(x)` and `f(y)` (`f x ⊓ f y`). This is a statement about the structure-preserving property of order isomorphisms in the context of semilattices.

Given two semilattices `α` and `β` and an order isomorphism `f` from `α` to `β`, for all `x, y ∈ α`, we have `f (x ⊓ y) = f(x) ⊓ f(y)`.


theorem dvd_add :
    ∀ {α : Type u_1} [inst : Add α] [inst_1 : Semigroup α] [inst_2 : LeftDistribClass α] {a b c : α},
      a ∣ b → a ∣ c → a ∣ b + c :=
  by sorry

This theorem, `dvd_add`, states that for all types `α` with addition, semigroup, and left distributive properties, if a number `a` divides number `b` and also divides number `c`, then `a` also divides the sum of `b` and `c`. This theorem is a generalized form of the mathematical principle that if a number divides two numbers individually, it also divides their sum.

If `a` divides `b` and `c`, then `a` divides `b + c` for types `α` with addition, semigroup, and left distributive properties.


theorem Commute.mul_pow :
    ∀ {M : Type u} [inst : Monoid M] {a b : M}, Commute a b → ∀ (n : ℕ), (a * b) ^ n = a ^ n * b ^ n := by sorry

This theorem states that for any monoid `M`, given two elements `a` and `b` from `M` that commute (i.e., `a * b = b * a`), for any natural number `n`, the `n`th power of the product `a * b` is equal to the product of `a` to the power of `n` and `b` to the power of `n`. In mathematical notation, if `a` and `b` commute, then $(a * b) ^ n = a^n * b^n$ for all natural numbers `n`.

For any commuting elements `a` and `b` in a monoid and any natural number `n`, their `n`th powers satisfy the identity `(a * b)^n = a^n * b^n`.


theorem Finset.coe_add :
    ∀ {α : Type u_2} [inst : DecidableEq α] [inst_1 : Add α] (s t : Finset α), ↑(s + t) = ↑s + ↑t := by sorry

The theorem `Finset.coe_add` states that for any type `α` with decidable equality and an addition operation, and for any two finite sets `s` and `t` of type `α`, the set formed by adding each pair of elements, one from `s` and one from `t` (`s + t`), is equal to the set obtained by adding the sets `s` and `t` themselves (`↑s + ↑t`). In other words, the operation of summing the sets commutes with the operation of taking the sum of pairs. This is expressed in the context of "coercions" where `↑s` denotes the coercion (conversion) of the finite set `s` to a set.

For any type `α` with decidable equality and addition operation, the sum of finite sets `s` and `t` as sets (`↑s + ↑t`) is equal to the set of pairs `(x, y)` with `x ∈ s` and `y ∈ t` (`s + t`).


theorem Real.exp_neg : ∀ (x : ℝ), Real.exp (-x) = (Real.exp x)⁻¹ := by sorry

The theorem `Real.exp_neg` states that for all real numbers `x`, the exponential function evaluated at the negative of `x` is equal to the inverse of the exponential function evaluated at `x`. In mathematical notation, this can be written as: for all $x \in \mathbb{R}$, $\exp(-x) = \exp(x)^{-1}$. This is a fundamental property of the exponential function.

The exponential function's value with a negative real number argument is equal to the reciprocal of the exponential function's value with the same positive real number argument. (i.e., for all $x \in \mathbb{R}$, $\exp(-x) = \exp(x)^{-1}$)


theorem Mathlib.Algebra.Order.Sub.Defs._auxLemma.1 :
    ∀ {α : Type u_1} [inst : LE α] [inst_1 : Add α] [inst_2 : Sub α] [inst_3 : OrderedSub α] {a b c : α},
      (a - b ≤ c) = (a ≤ c + b) :=
  by sorry

This theorem, named `Mathlib.Algebra.Order.Sub.Defs._auxLemma.1`, states that for any type `α` that supports an order (`LE α`), addition (`Add α`), subtraction (`Sub α`), and ordered subtraction (`OrderedSub α`), and for any three elements `a`, `b`, and `c` of type `α`, the inequality `a - b ≤ c` holds true if and only if `a ≤ c + b`. This essentially states a property of ordered subtraction in the context of ordered algebraic structures.

For any type `α` with order, addition, subtraction, and ordered subtraction, the subtraction `a - b` is less than or equal to `c` if and only if `a` is less than or equal to `c + b`.


theorem Nat.factorization_zero : Nat.factorization 0 = 0 := by sorry

The theorem `Nat.factorization_zero` states that the factorization of zero, as defined by the `Nat.factorization` function, is equal to zero. In other words, this states that zero has no prime factors, since the factorization of a number is a function mapping each of its prime factor to its multiplicity, and for zero, this function is identically zero.

The theorem `Nat.factorization_zero` asserts that the factorization of zero, according to the `Nat.factorization` function, is the empty map.


Membership.mem.out

/-- If `h : a ∈ {x | p x}` then `h.out : p x`. These are definitionally equal, but this can
nevertheless be useful for various reasons, e.g. to apply further projection notation or in an
argument to `simp`.  -/
theorem Membership.mem.out : ∀ {α : Type u} {p : α → Prop} {a : α}, a ∈ {x | p x} → p a := by sorry

This theorem states that if `a` is an element of a set of elements `x` for which a condition `p` holds, then the condition `p` holds for `a`. In other words, if `a` belongs to the set `{x | p x}`, then `p a` is true. Despite these being definitionally equal, the theorem can be useful in various ways, for example, to apply further projection notation or in an argument to `simp`.

If a set contains an element x such that p(x) holds, then p(a) holds for any a in that set.


theorem Mathlib.Data.Set.Basic._auxLemma.12 : ∀ {α : Type u}, Set.Nonempty ∅ = False := by sorry

This theorem states that the empty set, denoted by ∅, is not nonempty for any type `α`. In other words, there does not exist an element `x` that belongs to the empty set ∅, regardless of the type of `x`. This is a fundamental property of the empty set in set theory.

The empty set is a subset of any type that has no elements.


theorem inner_neg_left :
    ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
      (x y : E), ⟪-x, y⟫_𝕜 = -⟪x, y⟫_𝕜 :=
  by sorry

This theorem states that for any inner product space `E` over a scalar field `𝕜` with a commutative group structure and a norm, if we have two elements `x` and `y` in `E`, then the inner product of `-x` and `y` is equal to the negation of the inner product of `x` and `y`. In mathematical notation, this would be expressed as `⟪-x, y⟫_𝕜 = -⟪x, y⟫_𝕜`. This property is one of the basic properties of inner product spaces.

The inner product of the negation of an element with another element in an inner product space is equal to the negation of their inner product. In other words, `⟪-x, y⟫_𝕜 = -⟪x, y⟫_𝕜`.


theorem associated_of_dvd_dvd :
    ∀ {α : Type u_1} [inst : CancelMonoidWithZero α] {a b : α}, a ∣ b → b ∣ a → Associated a b := by sorry

This theorem states that for any type `α` that forms a `CancelMonoidWithZero`, given any two elements `a` and `b` of this type, if `a` divides `b` and `b` divides `a`, then `a` and `b` are associated. In the context of monoids, two elements are considered associated if one can be obtained by multiplying the other by a unit on the right. In other words, `a` and `b` are associated if there exists a unit `u` such that `a * u = b`.

If `α` is a type forming a `CancelMonoidWithZero` and `a` and `b` are elements of `α` such that `a` divides `b` and `b` divides `a`, then there exists a unit `u` such that `a * u = b`.


Associated.refl

theorem Associated.refl : ∀ {α : Type u_1} [inst : Monoid α] (x : α), Associated x x := by sorry

This theorem states that for any given type `α` that forms a Monoid (a set equipped with an associative binary operation and an identity element), any element `x` of this type is associated with itself. In other words, there exists a unit such that when it is multiplied by `x` on the right, the result is `x` itself. This is a reflection property of the `Associated` relation in a Monoid.

For any monoid `α` and its element `x`, there exists an identity element `e` such that `x * e = x`.