theorem le_max_right : ∀ {α : Type u} [inst : LinearOrder α] (a b : α), b ≤ max a b := by sorry

This theorem, `le_max_right`, states that for all types `α` that have a linear ordering, for any two instances `a` and `b` of that type, `b` is less than or equal to the maximum of `a` and `b`. In other words, the theorem asserts that any element is always less than or equal to the maximum of itself and any other element of the same type, given that the type has a linear order structure.

For all types `α` with a linear order and all `a, b : α`, we have `b <= max a b`.


theorem iInf_congr_Prop :
    ∀ {α : Type u_1} [inst : InfSet α] {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q),
      (∀ (x : q), f₁ ⋯ = f₂ x) → iInf f₁ = iInf f₂ :=
  by sorry

This theorem states that for any type `α` that has an infimum and any propositions `p` and `q`, if there exist functions `f₁ : p → α` and `f₂ : q → α`, then the indexed infimum of `f₁` is equal to the indexed infimum of `f₂` if and only if two conditions are met: (1) `p` and `q` are logically equivalent (i.e., `p` if and only if `q`), and (2) for all `x` in the domain of `f₂`, the output of `f₁` applied to some value corresponding to `x` (under the logical equivalence) is equal to `f₂ x`. Essentially, this theorem guarantees that the indexed infimum is preserved under changes that maintain logical equivalence and function output equivalence.

For types `α` with an infimum and propositions `p` and `q`, if there exist functions `f₁ : p → α` and `f₂ : q → α` such that `p` is logically equivalent to `q` and `f₁(x) = f₂(y)` for all `x` in the domain of `f₂` and corresponding `y` under the logical equivalence, then the indexed infima of `f₁` and `f₂` are equal.


theorem div_zero : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] (a : G₀), a / 0 = 0 := by sorry

This theorem states that for any type `G₀`, which has an instance of the `GroupWithZero` structure, the result of dividing any element `a` of this type by zero is zero. In mathematical terms, for any group with zero (a mathematical structure that is a group, but also has a zero element and satisfies certain properties), the division of any element of the group by zero yields zero.

For any group with zero `G₀`, `0 * x = x * 0 => x = 0` for all `x` in `G₀`. (In other words, dividing any element of `G₀` by zero results in zero.)


theorem mul_le_mul_of_nonneg_left :
    ∀ {α : Type u_1} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α] [inst_3 : PosMulMono α],
      b ≤ c → 0 ≤ a → a * b ≤ a * c :=
  by sorry

This theorem states that for any type `α` that has multiplication and zero defined on it, and is preordered, and has the property of `PosMulMono`, for any three elements `a`, `b`, and `c` of `α`, if `b` is less than or equal to `c` and `a` is nonnegative (i.e., `a` is greater than or equal to zero), then the product of `a` and `b` is less than or equal to the product of `a` and `c`. This is the principle of monotonicity of multiplication by nonnegative elements on the left.

For any preordered type `α` with multiplication and zero, if `a` is nonnegative and `b` is less than or equal to `c`, then `a * b` is less than or equal to `a * c`.


theorem Eq.mpr_prop : ∀ {p q : Prop}, p = q → q → p := by sorry

This theorem, `Eq.mpr_prop`, states that for any two propositions `p` and `q`, if `p` is equal to `q` and `q` is true, then `p` is also true. It applies the principle of substitutability of identicals in a logical context, asserting the transferability of truth values between equivalent propositions.

If two propositions are equal and one is true, then the other is also true.


theorem continuous_apply :
    ∀ {ι : Type u_5} {π : ι → Type u_6} [T : (i : ι) → TopologicalSpace (π i)] (i : ι), Continuous fun p => p i := by
  sorry

This theorem states that for any index type `ι` and any type function `π` from `ι` to another type, given that each type `π i` is endowed with a topological space structure, the function that applies an element of `ι` to a point `p` (thus obtaining an element of type `π i`) is continuous. In other words, it's saying that projection maps in a product of topological spaces are continuous.

For any index type `ι` and type function `π : ι → Type`, if each `π i` is endowed with a topological space structure, then the projection map `π` is continuous.


/-- Additive group homomorphisms preserve negation. -/
theorem map_neg :
    ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_1 : AddGroup G]
      [inst_2 : SubtractionMonoid H] [inst_3 : AddMonoidHomClass F G H] (f : F) (a : G), f (-a) = -f a :=
  by sorry

This theorem states that homomorphisms of additive groups preserve negation. In more detail, for any types `G`, `H`, and `F`, if `F` is a function-like structure that maps elements from `G` to `H`, `G` is an additive group, `H` is a subtraction monoid, and `F` is an additive monoid homomorphism class from `G` to `H`, then for any element `a` of `G` and any function `f` of type `F`, applying `f` to the negation of `a` gives the same result as negating the result of applying `f` to `a`. Mathematically, this can be expressed as: for all `a` in `G`, `f(-a) = -f(a)`. This preservation of negation is a key property of group homomorphisms.

Given an additive group `G`, an additive monoid `H`, and an additive monoid homomorphism `F` from `G` to `H`, for all elements `a` in `G`, `F(-a) = -F(a)`.


/-- **Alias** of `le_of_eq_of_le`. -/
theorem Eq.trans_le : ∀ {α : Type u} {a b c : α} [inst : LE α], a = b → b ≤ c → a ≤ c := by sorry

This theorem, `Eq.trans_le`, states that for any type `α` that has a less than or equal to (`≤`) operation and for any elements `a`, `b`, and `c` of this type, if `a` is equal to `b` and `b` is less than or equal to `c`, then `a` is less than or equal to `c`. In other words, it asserts the transitivity of the `≤` relation over equality, allowing to replace `b` with `a` in the inequality `b ≤ c` when `a` and `b` are equal.

If `a` equals `b` and `b` is less than or equal to `c`, then `a` is less than or equal to `c` (transitivity of the `≤` relation over equality).


Measurable.aemeasurable

theorem Measurable.aemeasurable :
    ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β}
      {μ : MeasureTheory.Measure α}, Measurable f → AEMeasurable f μ :=
  by sorry

The `Measurable.aemeasurable` theorem states that for any two types `α` and `β`, given a measurable space over `α`, a measurable function `f` from `α` to `β`, and a measure `μ` over `α`, if `f` is measurable, then `f` is also almost everywhere measurable with respect to the measure `μ`. In other words, a measurable function coincides almost everywhere with itself, providing a bridge between the concepts of measurability and almost everywhere measurability.

A measurable function between measurable spaces is almost everywhere measurable with respect to the given measure.


theorem Mathlib.Data.Finset.Basic._auxLemma.28 : ∀ {α : Type u_1} {a b : α}, (b ∈ {a}) = (b = a) := by sorry

This theorem states that for any type `α` and any two elements `a` and `b` of type `α`, `b` is an element of the singleton set containing only `a` if and only if `b` is equal to `a`. In other words, in a set with only one element, the question of membership for any given element boils down to whether or not it is equal to the single member of the set.

For any type `α` and any elements `a, b` of type `α`, `b` is in the singleton set `{a}` if and only if `a = b`.


/-- Additive group homomorphisms preserve subtraction. -/
theorem map_sub :
    ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_1 : AddGroup G]
      [inst_2 : SubtractionMonoid H] [inst_3 : AddMonoidHomClass F G H] (f : F) (a b : G), f (a - b) = f a - f b :=
  by sorry

The theorem `map_sub` states that for any types `G`, `H`, and `F` where `F` is a function-like object from `G` to `H`, `G` is an additive group, `H` is a subtraction monoid, and `F` is a homomorphism class from an additive group `G` to a monoid `H`, then for any function `f` of type `F` and any elements `a` and `b` of `G`, the difference of `a` and `b` under `f` is equal to the difference of `f(a)` and `f(b)`. In other words, an additive group homomorphism preserves the operation of subtraction.

For any additive group homomorphism F from group G to monoid H and any elements a, b in G, we have F(a - b) = F(a) - F(b).


/-- Gives a `Set.Finite` for the `Finset` coerced to a `Set`.
This is a wrapper around `Set.toFinite`.  -/
theorem Finset.finite_toSet : ∀ {α : Type u} (s : Finset α), Set.Finite ↑s := by sorry

This theorem states that for any finset `s` of type `α`, when `s` is coerced to a set, it is a finite set. In other words, any finset can be considered a finite set. This is a wrapper around the `Set.toFinite` function, meaning it is providing a more specific or convenient interface for the existing function `Set.toFinite`.

For any finset `s` of type `α`, the coerced set `{x | x ∈ s}` is a finite set.


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

The theorem `iSup_le` states that for any type `α` that is a complete lattice, any function `f` from an arbitrary type `ι` to `α`, and any value `a` of type `α`, if every value `f(i)` for all `i` in `ι` is less than or equal to `a`, then the indexed supremum of the function `f`, denoted as `iSup f`, is also less than or equal to `a`. In other words, if all the values of a function are below a specific value, then their supremum (greatest value) is also below or equal to that specific value.

For any complete lattice α and function f from type ι to α, if for all i in ι, f(i) ≤ a, then iSup f ≤ a.


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

This theorem states that for any type `G` that forms a `DivInvOneMonoid` (a mathematical structure that combines aspects of a monoid, a group, and a ring with division and a multiplicative identity element 'one'), the division of any element `a` of type `G` by the identity element 'one' is equal to the element `a` itself. In essence, it's encapsulating the common mathematical truth that any number divided by one remains the same.

For any type `G` that is a `DivInvOneMonoid`, the operation of dividing an element `a` by the identity element 'one' results in `a` itself. In mathematical notation: `a / one = a` for all `a` in `G`.


theorem Set.range_comp :
    ∀ {α : Type u_1} {β : Type u_2} {ι : Sort u_4} (g : α → β) (f : ι → α), Set.range (g ∘ f) = g '' Set.range f := by
  sorry

The theorem `Set.range_comp` states that for any types `α`, `β`, and `ι`, and for any functions `g` from `α` to `β` and `f` from `ι` to `α`, the range of the composition of `g` and `f` is equal to the image of the range of `f` under the function `g`. In other words, if we first apply the function `f` and then apply the function `g`, it is the same as if we directly applied `g` to the set of all possible outputs of `f`. This theorem formalizes the idea of function composition in the context of set theory.

The range of the composition of functions `g : α → β` and `f : ι → α` is equal to the image of the range of `f` under `g`.


theorem zero_pow : ∀ {M₀ : Type u_2} [inst : MonoidWithZero M₀] {n : ℕ}, n ≠ 0 → 0 ^ n = 0 := by sorry

This theorem states that for any non-zero natural number `n` and for any type `M₀` that is a Monoid with zero (a mathematical structure with a certain set of operations and properties), the nth power of zero is zero. Essentially, it's saying that raising zero to the power of any non-zero natural number will always result in zero, given that our mathematical structure supports such operations.

For any monoid with zero `M₀` and natural number `n ≠ 0`, the power `(0 : M₀) ^ n` equals the zero element of `M₀`.


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

This theorem, `and_congr`, states that for any four propositional variables `a`, `b`, `c`, and `d`, if `a` is logically equivalent to `c` and `b` is logically equivalent to `d`, then the logical conjunction (AND operation) of `a` and `b` is also logically equivalent to the logical conjunction of `c` and `d`. In mathematical terms, if $a \Leftrightarrow c$ and $b \Leftrightarrow d$, then $(a \land b) \Leftrightarrow (c \land d)$.

If $a$ is logically equivalent to $c$ and $b$ is logically equivalent to $d$, then $(a \land b) \Leftrightarrow (c \land d)$.


theorem dite_cond_eq_false :
    ∀ {α : Sort u} {c : Prop} {x : Decidable c} {t : c → α} {e : ¬c → α} (h : c = False), dite c t e = e ⋯ := by sorry

This theorem, named "dite_cond_eq_false", states that for any type `α` and any proposition `c`, given a decidable instance of `c`, two functions `t` and `e` of types `c → α` and `¬c → α` respectively, if `c` is proven to be `False`, then the expression `dite c t e` (which stands for "if `c` then `t` else `e`") is equal to `e`. In other words, it says that if the condition of a conditional expression is false, the whole expression evaluates to the "else" part.

If `c` is a decidable proposition and `t : α → α` and `e : α` are functions, then `dite c t e = e` when `c` is false.


/-- Ring homomorphisms preserve multiplication.  -/
theorem RingHom.map_mul :
    ∀ {α : Type u_2} {β : Type u_3} {x : NonAssocSemiring α} {x_1 : NonAssocSemiring β} (f : α →+* β) (a b : α),
      f (a * b) = f a * f b :=
  by sorry

The theorem `RingHom.map_mul` states that for any two types `α` and `β` with their respective non-associative semirings `x` and `x_1`, and given a ring homomorphism `f` from `α` to `β` and two elements `a` and `b` of type `α`, the application of `f` on the product of `a` and `b` (i.e., `f (a * b)`) is equal to the product of `f a` and `f b`. This means that the multiplication operation is preserved under the ring homomorphism `f`.

Given ring homomorphisms `f : α → β` between semirings `x` and `x_1`, and elements `a` and `b` from `α`, we have `f (a * b) = f a * f b`.


theorem ne_of_lt : ∀ {α : Type u} [inst : Preorder α] {a b : α}, a < b → a ≠ b := by sorry

This theorem states that for any type `α` with a preorder (a set equipped with a binary relation that is reflexive and transitive), if `a` and `b` are elements of `α` and `a` is less than `b`, then `a` is not equal to `b`. In mathematical terms, this theorem asserts that for all `a` and `b` in a preordered set, `a < b` implies `a ≠ b`.

In a preordered set, if `a` is less than `b`, then `a` is strictly less than `b`, that is, `a < b` implies `a ≠ b`.


theorem Mathlib.Data.Finset.Basic._auxLemma.143 : ∀ {α : Sort u_1} {p : α → Prop}, (¬∀ (x : α), p x) = ∃ x, ¬p x := by
  sorry

This theorem states that for any type `α` and a property `p` that can be applied to elements of type `α`, the negation of the statement "for all elements `x` of type `α`, the property `p` holds" is equivalent to the statement "there exists an element `x` of type `α` for which the property `p` does not hold". In terms of mathematical logic, this is saying that ¬(∀x, P(x)) is equivalent to ∃x, ¬P(x).

The negation of "for all x in α, property p(x) holds" is equivalent to "there exists an x in α such that not p(x) holds".


theorem Mathlib.Order.SetNotation._auxLemma.3 :
    ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i := by sorry

This theorem states that for any type `α`, any sort `ι`, any element `x` of type `α`, and any function `s` that maps elements of sort `ι` to sets of type `α`, the statement "element `x` belongs to the union of sets `s i` for all `i`" is equivalent to the statement "there exists an `i` such that `x` belongs to the set `s i`". In mathematical terms, this is saying that $x \in \bigcup_{i} s_i$ is equivalent to $\exists i, x \in s_i$.

For any type `α`, sort `ι`, element `x` of type `α`, and function `s` mapping `ι` to sets of `α`, $x \in \bigcup\_{i} s\_i$ if and only if there exists an `i`, such that $x \in s\_i$.


theorem zero_sub : ∀ {G : Type u_3} [inst : SubNegMonoid G] (a : G), 0 - a = -a := by sorry

This theorem named `zero_sub` states that for all types `G` (which are instances of the `SubNegMonoid` class) and for every element `a` of type `G`, the result of the operation "0 minus a" is equal to the negative of `a`. In mathematical terms, this is similar to saying that in any subtractive negation monoid, subtracting any element from zero yields the negation of that element, i.e., `0 - a = -a`.

For all types `G` in the `SubNegMonoid` class and every element `a` in `G`, `0 - a = -a`.


theorem Function.comp_apply :
    ∀ {β : Sort u_1} {δ : Sort u_2} {α : Sort u_3} {f : β → δ} {g : α → β} {x : α}, (f ∘ g) x = f (g x) := by sorry

This theorem, named "Function.comp_apply", states that for every three types `β`, `δ`, and `α` and given any two functions `f : β → δ` and `g : α → β`, as well as any element `x` of type `α`, the composition of `f` and `g` applied to `x` (`(f ∘ g) x`) is equal to `f` applied to the result of `g` applied to `x` (`f (g x)`). This is essentially the definition of function composition in mathematics, often written as $(f \circ g)(x) = f(g(x))$.

The theorem asserts that for functions `f : β → δ` and `g : α → β`, `(f ∘ g) x = f (g x)` for all `x : α`.


/-- **Alias** of `eq_or_lt_of_le`. -/
theorem LE.le.eq_or_lt : ∀ {α : Type u} [inst : PartialOrder α] {a b : α}, a ≤ b → a = b ∨ a < b := by sorry

This theorem, named `LE.le.eq_or_lt`, states that for any given type `α` that has a partial order, and any two elements `a` and `b` of that type, if `a` is less than or equal to `b`, then either `a` is equal to `b` or `a` is strictly less than `b`. This is just an alias of the theorem `eq_or_lt_of_le`.

For any type with a partial order, if `a` is less than or equal to `b`, then `a` is equal to `b` or `a` is strictly less than `b`.


theorem Set.mem_iUnion : ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, x ∈ ⋃ i, s i ↔ ∃ i, x ∈ s i := by sorry

This theorem `Set.mem_iUnion` states that for all types `α` and a sort `ι`, and for any element `x` of type `α` and a function `s` mapping `ι` to a set of `α`, the element `x` is in the union of all the sets defined by `s i` for all `i` if and only if there exists some `i` such that `x` is in the set `s i`. In simple words, an element belongs to the union of a collection of sets if and only if it belongs to at least one set in that collection.

An element belongs to the union of a family of sets if and only if it is an element of at least one set in the family.


theorem add_smul :
    ∀ {R : Type u_2} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (r s : R)
      (x : M), (r + s) • x = r • x + s • x :=
  by sorry

This theorem, named `add_smul`, is a property of scalar multiplication in the context of modules over semirings. For any given types `R` and `M`, where `R` is a semiring and `M` is an additive commutative monoid and also a module over `R`, this theorem states that for any two elements `r` and `s` from `R` and an element `x` from `M`, the scalar multiplication of the sum of `r` and `s` with `x` is equal to the sum of the scalar multiplications of `r` and `s` with `x` individually. That is, `(r + s) • x = r • x + s • x`. This is a fundamental distributive law in the theory of modules.

For any semiring `R`, additive commutative monoid and `R`-module `M`, and elements `r`, `s` in `R` and `x` in `M`, the scalar multiplication distributes over addition: `(r + s) • x = r • x + s • x`.


/-- **Alias** of `Left.mul_nonneg`.

---

Assumes left covariance.  -/
theorem mul_nonneg :
    ∀ {α : Type u_1} {a b : α} [inst : MulZeroClass α] [inst_1 : Preorder α] [inst_2 : PosMulMono α],
      0 ≤ a → 0 ≤ b → 0 ≤ a * b :=
  by sorry

The theorem `mul_nonneg` is an alias of `Left.mul_nonneg`. It states that for any type `α` with a multiplication operation, a zero element, and a preorder relation, and that satisfies the property of monotonicity of multiplication by nonnegative elements on the left (i.e., is a `PosMulMono`), if `a` and `b` are elements of `α` such that `0 ≤ a` and `0 ≤ b`, then it must be the case that `0 ≤ a * b`. In other words, the product of two nonnegative elements in such a type is also nonnegative. This theorem assumes left covariance.

In a type with a monotonic multiplication by nonnegative elements on the left, the product of two nonnegative elements is nonnegative.


/-- **Alias** of `Set.forall_mem_range`. -/
theorem Set.forall_range_iff :
    ∀ {α : Type u_1} {ι : Sort u_4} {f : ι → α} {p : α → Prop}, (∀ a ∈ Set.range f, p a) ↔ ∀ (i : ι), p (f i) := by
  sorry

The theorem `Set.forall_range_iff` states that for any type `α`, any sort `ι`, any function `f: ι → α`, and any property `p` that holds for values of type `α`, it is equivalent to say that the property `p` holds for all elements in the range of the function `f`, and to say that the property `p` holds for all images of the function `f` when it is applied to any element of `ι`. In other words, checking a property on all elements in the range of a function is the same as checking the property on the image of the function for all possible inputs.

For any type `α`, sort `ι`, function `f: ι → α`, and property `p` on `α`, `∀ x ∈ ι, p(fx) ↔ ∀ x' ∈ range(f), p(x')`.


Set.Finite.subset

theorem Set.Finite.subset : ∀ {α : Type u} {s : Set α}, Set.Finite s → ∀ {t : Set α}, t ⊆ s → Set.Finite t := by sorry

This theorem states that for any type `α` and any set `s` of type `α` that is finite, if we have another set `t` of the same type `α` such that every element of `t` is also an element of `s` (i.e., `t` is a subset of `s`), then `t` is also finite. This is a generalization of the principle that a subset of a finite set is always finite.

If `s` is a finite set of type `α`, then any subset `t` of `s` also has finite size.


theorem Mathlib.Data.Finset.Basic._auxLemma.20 : ∀ {α : Type u_1} (a : α), (a ∈ ∅) = False := by sorry

This theorem, from the Mathlib library in Lean 4, states that for any type `α` and for any element `a` of type `α`, the element `a` does not belong to an empty set. In mathematical notation, this can be represented as `∀ α, a : α, a ∉ ∅`.

For any type `α` and element `a` of type `α`, `a` is not an element of the empty set. In mathematical notation, `∀ α a : α, a ∉ ∅`.


theorem eq_bot_iff : ∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderBot α] {a : α}, a = ⊥ ↔ a ≤ ⊥ := by sorry

This theorem states that for any type `α`, given it has a `PartialOrder` and `OrderBot` structures, an element `a` of type `α` is equal to the bottom element (⊥) if and only if `a` is less than or equal to this bottom element. This applies to any type that has a notion of partial order and a defined bottom element.

For any type `α` equipped with a `PartialOrder` and `OrderBot` structure, an element `a` is the bottom element if and only if `a ≤ ⊥`.


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

The theorem `le_iSup` asserts that for any type `α` which forms a complete lattice, for any function `f` from a type `ι` to `α`, and for any element `i` of `ι`, the value of the function `f` at `i` is less than or equal to the indexed supremum of the function `f`. In other words, it states that any individual element of the function `f` is not greater than the supremum of all elements in the range of `f`.

For any complete lattice `α` and function `f` from a type `ι` to `α`, the value `f i` is less than or equal to the supremum of `f` for all `i` in `ι`.


theorem Finset.mem_filter :
    ∀ {α : Type u_1} {p : α → Prop} [inst : DecidablePred p] {s : Finset α} {a : α},
      a ∈ Finset.filter p s ↔ a ∈ s ∧ p a :=
  by sorry

This theorem, `Finset.mem_filter`, states that for any type `α`, any decidable predicate `p` on `α`, any finite set `s` of `α`, and any element `a` of `α`, `a` is an element of the filtered set (`Finset.filter p s`) if and only if `a` is an element of the original set `s` and `a` satisfies the predicate `p`. In other words, filtering a finite set with a predicate in Lean only keeps the elements that satisfy the predicate.

For any type `α`, finite set `s` of `α`, decidable predicate `p` on `α`, and `a` in `α`, `a` is in `Finset.filter p s` if and only if `a` is in `s` and `p(a)` holds.


theorem CategoryTheory.comp_apply :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.ConcreteCategory C] {X Y Z : C}
      (f : X ⟶ Y) (g : Y ⟶ Z) (x : (CategoryTheory.forget C).obj X),
      (CategoryTheory.CategoryStruct.comp f g) x = g (f x) :=
  by sorry

This theorem, "CategoryTheory.comp_apply", states that for any category `C` that is also a concrete category, and for objects `X`, `Y`, and `Z` of this category, if we have morphisms `f` from `X` to `Y` and `g` from `Y` to `Z`, and an object `x` in the image of `X` under the forgetful functor from `C` to `Type w`, then applying the composition of `f` and `g` to `x` is the same as first applying `f` to `x` and then applying `g` to the result. In other words, the composition operation in the category matches function composition in the underlying type.

For any concrete category $C$, given objects $X$, $Y$, $Z$, morphisms $f : X \to Y$ and $g : Y \to Z$, and an object $x : X$ in the image of $X$ under the forgetful functor, we have $g \circ f(x) = g(f(x))$.


theorem pow_add : ∀ {M : Type u} [inst : Monoid M] (a : M) (m n : ℕ), a ^ (m + n) = a ^ m * a ^ n := by sorry

This theorem, named `pow_add`, states that for any type `M` that forms a monoid and any element `a` of type `M`, the power of `a` raised to the sum of two natural numbers `m` and `n` is equal to the product of `a` raised to the power `m` and `a` raised to the power `n`. In mathematical notation, this would be expressed as \(a^{m+n} = a^m \times a^n\). This theorem essentially mirrors the exponential law from elementary algebra.

For any monoid `M` and element `a` in `M`, `a^(m + n) = a^m * a^n`.


theorem map_pow :
    ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_1 : Monoid G] [inst_2 : Monoid H]
      [inst_3 : MonoidHomClass F G H] (f : F) (a : G) (n : ℕ), f (a ^ n) = f a ^ n :=
  by sorry

The theorem `map_pow` states that for any types `G`, `H`, and `F`, where `F` has a structure of `FunLike` (meaning terms of type `F` can be injectively cast into functions from `G` to `H`), `G` and `H` are both monoids and `F` is a monoid homomorphism from `G` to `H`, then for any homomorphism `f` of type `F`, any element `a` from `G` and any natural number `n`, the result of applying `f` to `a` raised to the power `n` is equal to the result of applying `f` to `a`, and then raising that result to the power `n`. In simple words, the homomorphism preserves the power operation.

For any monoid homomorphism `f` between monoids `G` and `H`, and for all `a` in `G` and natural number `n`, `f (a ^ n) = (f a) ^ n`.


theorem inf_comm : ∀ {α : Type u} [inst : SemilatticeInf α] (a b : α), a ⊓ b = b ⊓ a := by sorry

This theorem, named `inf_comm`, states that for any type `α` having a semilattice with infimum structure, the infimum operation, denoted by `⊓`, is commutative. In other words, for any two elements `a` and `b` of type `α`, the infimum of `a` and `b` is equal to the infimum of `b` and `a`. This is a fundamental property of semilattices that the order of elements does not affect the result of the infimum operation.

For any semilattice `α` with infimum operation `⊓`, the infimum of `a` and `b` equals the infimum of `b` and `a` for all `a, b ∈ α`.


theorem Finset.sum_singleton :
    ∀ {α : Type u_3} {β : Type u_4} [inst : AddCommMonoid β] (f : α → β) (a : α), (Finset.sum {a} fun x => f x) = f a :=
  by sorry

The theorem `Finset.sum_singleton` states that for any types `α` and `β`, given an addition-commutative monoid on `β`, a function `f` from `α` to `β` and a value `a` of type `α`, the sum over the singleton set `{a}` (i.e., a set containing only `a`) of the function `f` is equal to the function `f` applied to `a`. In mathematical terms, for a given function `f: α → β` and a single element `a` of `α`, this theorem states that `∑_{x in {a}} f(x) = f(a)`.

For any addition-commutative monoid β, function f : α → β, and element a of α, the sum of the function f over the singleton set {a} equals f(a).


theorem pow_pos : ∀ {α : Type u} [inst : StrictOrderedSemiring α] {a : α}, 0 < a → ∀ (n : ℕ), 0 < a ^ n := by sorry

This theorem states that for all types `α` that are strict ordered semi-rings, if `a` is an element of `α` and is greater than 0, then any natural number power of `a` (expressed as `a ^ n` where `n` is a natural number) is also greater than 0. In LaTeX, this would be expressed as: For all $a \in \alpha$, where $\alpha$ is a strict ordered semi-ring, if $a > 0$, then for all $n \in \mathbb{N}$, $a^n > 0$.

In a strict ordered semi-ring $\alpha$, if $0<a\in\alpha$, then $a^n>0$ for all natural numbers $n$.


Equiv.surjective

theorem Equiv.surjective : ∀ {α : Sort u} {β : Sort v} (e : α ≃ β), Function.Surjective ⇑e := by sorry

The theorem `Equiv.surjective` states that for any given types `α` and `β`, if there is an equivalence `e` between `α` and `β`, then the function corresponding to this equivalence, denoted by `⇑e`, is surjective. In other words, for every element `b` of type `β`, there exists an element `a` in type `α` such that `⇑e a = b`. This theorem therefore establishes the surjectivity of any function derived from an equivalence between two types.

If `e` is an equivalence between types `α` and `β`, then the function `⇑e` is surjective: for every `b` in `β`, there exists `a` in `α` such that `e a = b`.


theorem Linarith.lt_irrefl : ∀ {α : Type u} [inst : Preorder α] {a : α}, ¬a < a := by sorry

This theorem, `Linarith.lt_irrefl`, states that for any type `α` that is a preorder, for any element `a` of type `α`, it is not possible for `a` to be less than `a`. In other words, the "<" relation is irreflexive in a preorder: no element is less than itself.

In any preorder type `α`, for all `a` in `α`, it holds that `a` is not less than `a`.


Quotient.inductionOn'

/-- A version of `Quotient.inductionOn` taking `{s : Setoid α}` as an implicit argument instead
of an instance argument.  -/
theorem Quotient.inductionOn' :
    ∀ {α : Sort u_1} {s₁ : Setoid α} {p : Quotient s₁ → Prop} (q : Quotient s₁),
      (∀ (a : α), p (Quotient.mk'' a)) → p q :=
  by sorry

The theorem `Quotient.inductionOn'` states that for any sort `α` and setoid `s₁` on `α`, any property `p` on the quotient of `s₁`, and any element `q` of the quotient, if the property `p` holds for all elements `a` of `α` when passed through the `Quotient.mk''` function, then the property `p` also holds for `q`. In other words, this theorem provides a principle of induction on elements of the quotient sort, where the property extends from representatives of equivalence classes to their corresponding equivalence classes in the quotient.

If a property holds for all representatives of equivalence classes of a setoid on a sort, then it holds for any equivalence class in the quotient.


theorem iSup_congr_Prop :
    ∀ {α : Type u_1} [inst : SupSet α] {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q),
      (∀ (x : q), f₁ ⋯ = f₂ x) → iSup f₁ = iSup f₂ :=
  by sorry

The theorem `iSup_congr_Prop` states that for any type α with a supremum set structure, and for any propositions p and q along with functions f₁ from p to α and f₂ from q to α, if p is logically equivalent to q (denoted by `p ↔ q`), and for every instance x of q, the function f₁ applied to some value is equal to the result of function f₂ applied to x, then the indexed supremum (`iSup`) of f₁ is equal to the indexed supremum of f₂. Basically, it states that under these conditions, the supremums of the two functions with respect to their respective domains are equal.

If two propositions are logically equivalent and their corresponding functions to a type with a supremum set structure have equal values at every instance, then their indexed supremums are equal.


/-- If `t` is a measurable set, then the measure of `t` with respect to the restriction of
the measure to `s` equals the outer measure of `t ∩ s`. An alternate version requiring that `s`
be measurable instead of `t` exists as `Measure.restrict_apply'`.  -/
theorem MeasureTheory.Measure.restrict_apply :
    ∀ {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},
      MeasurableSet t → ↑↑(μ.restrict s) t = ↑↑μ (t ∩ s) :=
  by sorry

This theorem states that for any type `α`, equipped with a measurable space `m0` and a measure `μ`, and for any sets `s` and `t` of type `α`, if `t` is a measurable set, then the measure of `t` with respect to the restriction of the measure to `s` equals the outer measure of the intersection of `t` and `s`. There's a different version of this theorem where `s` is required to be measurable instead of `t`, known as `Measure.restrict_apply'`.

For any measurable set `t` of type `α` in a measurable space `(α, m0)` with measure `μ`, the measure of `t` under the restriction of `μ` to a set `s` equals the outer measure of `t ∩ s`.


theorem le_inf : ∀ {α : Type u} [inst : SemilatticeInf α] {a b c : α}, a ≤ b → a ≤ c → a ≤ b ⊓ c := by sorry

This theorem, `le_inf`, states that for any semilattice of infimum type `α` and for any elements `a`, `b`, `c` of type `α`, if `a` is less than or equal to both `b` and `c`, then `a` is also less than or equal to the infimum (greatest lower bound) of `b` and `c`. In terms of mathematical notation, given `a ≤ b` and `a ≤ c`, it follows that `a ≤ b ⊓ c`.

If `a` is an element of a semilattice of infimum type `α`, and `a ≤ b` and `a ≤ c`, then `a ≤ b ∧ c` implies `a ≤ b ⊓ c`.


theorem Mathlib.Data.Set.Basic._auxLemma.25 : ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) := by
  sorry

This theorem, named `Mathlib.Data.Set.Basic._auxLemma.25`, states that for any type `α`, and any element `x` of type `α`, and any two sets `a` and `b` of elements of type `α`, the membership of `x` in the intersection of `a` and `b` is equivalent to the conjunction of the membership of `x` in `a` and the membership of `x` in `b`. In other words, an element belongs to the intersection of two sets if and only if it belongs to both of these sets.

For any type `α` and sets `a` and `b` of elements of type `α`, an element `x` belongs to the intersection `a ∩ b` if and only if it belongs to both sets `a` and `b`: `x ∈ a ∧ x ∈ b ↔ x ∈ a ∧ b`.


theorem interior_subset : ∀ {X : Type u} {s : Set X} [inst : TopologicalSpace X], interior s ⊆ s := by sorry

This theorem states that for any given type `X` and a set `s` of `X` type, in the context of a topological space defined on `X`, the interior of the set `s` is always a subset of `s`. In other words, in any topological space, the interior of any set is contained within the set itself.

In any topological space, the interior of a set is a subset of the set itself.


theorem List.append_assoc : ∀ {α : Type u} (as bs cs : List α), as ++ bs ++ cs = as ++ (bs ++ cs) := by sorry

This theorem, `List.append_assoc`, declares that for all types `α` and any three lists `as`, `bs`, and `cs` of this type, the operation of appending the lists together is associative. In other words, appending `as` with the result of appending `bs` and `cs` (`as ++ (bs ++ cs)`) is the same as appending the result of appending `as` and `bs` with `cs` (`(as ++ bs) ++ cs`). This mirrors the associative property in algebra where `(a+b)+c = a+(b+c)` for any numbers a, b, and c.

For all types `α`, the operation of list concatenation `++` is associative, i.e., `(as ++ bs) ++ cs = as ++ (bs ++ cs)` for any lists `as`, `bs`, and `cs` of type `α`.


theorem continuous_iff_continuousAt :
    ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
      Continuous f ↔ ∀ (x : X), ContinuousAt f x :=
  by sorry

This theorem is stating that for any two types `X` and `Y` that have topological space structures, a function `f` between `X` and `Y` is continuous if and only if it is continuous at every point in `X`. In mathematical terms, a function `f` mapping `X` to `Y` is continuous if the preimage of every open set in `Y` is open in `X`. This is equivalent to saying that for any point `x` in `X`, the function `f` is continuous at `x`, meaning the function `f` applied to any sequence in `X` that converges to `x` also converges to `f(x)` in `Y`.

A function between two topological spaces is continuous if and only if it maps every open set to an open set.


Classical.byContradiction

theorem Classical.byContradiction : ∀ {p : Prop}, (¬p → False) → p := by sorry

This theorem, `Classical.byContradiction`, states that for any proposition `p`, if assuming the negation of `p` leads to a contradiction (expressed as `¬p → False`), then `p` must be true. This corresponds to the classical logic principle of proof by contradiction (also known as reductio ad absurdum), where you prove a statement to be true by showing that its negation would result in an unreasonable or impossible situation.

If assuming the negation of a proposition leads to a contradiction, then the proposition is true.


theorem CategoryTheory.types_ext : ∀ {α β : Type u} (f g : α ⟶ β), (∀ (a : α), f a = g a) → f = g := by sorry

This theorem states that in the category of types, for any two types `α` and `β`, if two morphisms (functions) `f` and `g` from `α` to `β` produce the same result for every element of `α`, then `f` and `g` are the same function. In other words, two functions are identical if they map every element of the domain to the same element in the codomain.

In the category of types and functions, two functions between types are equal if and only if they map every element of their domain to the same image in the codomain.


theorem sInf_le : ∀ {α : Type u_1} [inst : CompleteSemilatticeInf α] {s : Set α} {a : α}, a ∈ s → sInf s ≤ a := by sorry

The theorem `sInf_le` states that for any type `α` that forms a complete semilattice with respect to the infimum operation, any set `s` of elements of type `α`, and any element `a` of type `α`, if `a` is an element of the set `s`, then the infimum of the set `s` is less than or equal to `a`. In other words, in a complete semilattice, the infimum of a set is always less than or equal to any element of that set.

In a complete semilattice, the infimum of any set is less than or equal to every element in the set.


theorem Nat.mod_lt : ∀ (x : ℕ) {y : ℕ}, y > 0 → x % y < y := by sorry

This theorem states that for all natural numbers `x` and `y`, if `y` is greater than zero, then the modulus of `x` with respect to `y` (represented as `x % y`) is less than `y`. In mathematical terms, for any positive natural number `y`, the remainder of division of any natural number `x` by `y` is always less than `y`.

For all natural numbers `x` and `y` with `y > 0`, `x % y` is less than `y`.


theorem mul_le_mul_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] {b c : α},
      b ≤ c → ∀ (a : α), a * b ≤ a * c :=
  by sorry

This theorem states that for any type `α` endowed with a multiplication operation and an order relation, under the condition that the type `α` obeys the "covariant" property (namely, if `x` and `x_1` are elements of `α` and `x ≤ x_1`, then `x * z ≤ x_1 * z` for any `z` in `α`), the multiplication operation is monotonic. More specifically, if `b` and `c` are elements of `α` and `b ≤ c`, then for any `a` in `α`, `a * b` is less than or equal to `a * c`.

If `α` is a type with a covariant multiplication and order relation such that `x ≤ x_1` implies `x * z ≤ x_1 * z` for all `x, x_1, z` in `α`, then multiplication is monotonic, i.e., `b ≤ c` implies `a * b ≤ a * c` for all `a, b, c` in `α`.


Dvd.dvd.trans

/-- **Alias** of `dvd_trans`. -/
theorem Dvd.dvd.trans : ∀ {α : Type u_1} [inst : Semigroup α] {a b c : α}, a ∣ b → b ∣ c → a ∣ c := by sorry

This theorem, called `Dvd.dvd.trans`, is essentially an alias for the `dvd_trans` theorem. It states that for any type `α` that forms a semigroup, if `a` divides `b` and `b` divides `c`, then `a` also divides `c`. In mathematical notation, this is: \(a \,|\, b\), \(b \,|\, c\) ⇒ \(a \,|\, c\). This theorem is a fundamental property of divisibility in the realm of abstract algebra.

If `a` divides `b` and `b` divides `c`, then `a` divides `c` (transitivity of divisibility).


theorem Mathlib.Data.Set.Basic._auxLemma.32 : ∀ {α : Type u} {s t : Set α}, (t = s ∩ t) = (t ⊆ s) := by sorry

The theorem, named as `Mathlib.Data.Set.Basic._auxLemma.32`, states that for any type `α` and any two sets `s` and `t` of that type, the equality that `t` is the intersection of `s` and `t` is equivalent to saying that `t` is a subset of `s`. This captures the basic set theory idea that the intersection of a set with another set is a subset of the original set.

For any type α and sets s and t of that type, s intersect t = t if and only if t is a subset of s.


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

This theorem, `max_eq_right`, states that for any type `α` which has a linear order, given any two elements `a` and `b` of type `α`, if `a` is less than or equal to `b`, then the maximum of `a` and `b` is `b`. In other words, in a linearly ordered set, if one element is not greater than the other, the maximum of the two is the other element.

For any linearly ordered type `α` and elements `a` and `b` of `α`, if `a` is less than or equal to `b`, then the maximum of `a` and `b` equals `b`.


theorem sub_neg_eq_add : ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a b : α), a - -b = a + b := by sorry

This theorem states that for any type `α` that forms a subtraction monoid, for any two elements `a` and `b` of that type, subtracting the negation of `b` from `a` is equal to adding `b` to `a`. In mathematical terms, this theorem encapsulates the idea that `a - (-b) = a + b`.

For any type `α` that forms a subtraction monoid, `a - (-b) = a + b` for all `a, b` in `α`.


theorem List.append_nil : ∀ {α : Type u} (as : List α), as ++ [] = as := by sorry

This theorem, `List.append_nil`, states that for any type in Lean 4, denoted by `α`, and for any list of that type, `as`, appending an empty list to `as` will result in `as` itself. In other words, adding nothing to a list leaves the list unchanged. This theorem is a fundamental property of lists and holds for any type of list elements.

For any type `α` and list `as` of length zero or more elements `α`, `as ++ [] = as` holds.


theorem pow_succ' : ∀ {M : Type u_2} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a * a ^ n := by sorry

This theorem states that for any type `M` that is a monoid, and for any element `a` of that type and any natural number `n`, raising `a` to the power of `n+1` is equivalent to multiplying `a` by itself raised to the power of `n`. This is a property of exponentiation in a monoid structure.

For any monoid `M` and its element `a`, the exponentiation `a ^ (n + 1)` is equivalent to the repeated multiplication `a ^ n * a`.


/-- To prove a proposition about an arbitrary `Finset α`,
it suffices to prove it for the empty `Finset`,
and to show that if it holds for some `Finset α`,
then it holds for the `Finset` obtained by inserting a new element.
 -/
theorem Finset.induction_on :
    ∀ {α : Type u_4} {p : Finset α → Prop} [inst : DecidableEq α] (s : Finset α),
      p ∅ → (∀ ⦃a : α⦄ {s : Finset α}, a ∉ s → p s → p (insert a s)) → p s :=
  by sorry

The theorem `Finset.induction_on` states that to establish a property `p` about an arbitrary finite set `s` of type `α`, it is sufficient to demonstrate two things: (1) the property `p` holds for the empty set, and (2) for any element `a` of type `α` and any finite set `s` of type `α` where `a` is not in `s`, if the property `p` holds for `s`, then it also holds for the finite set obtained by inserting `a` into `s`. This theorem assumes that equality between elements of type `α` is decidable.

The theorem `Finset.induction_on` states that for any property `p` on finite sets `s` of type `α`, if `p` holds for the empty set and `p` holds for any `a` not in `s` and the set obtained by inserting `a` into `s`, then `p` holds for all finite sets `s` of type `α`. (Assumes decidable equality on `α`.)


MeasurableSet.univ

theorem MeasurableSet.univ : ∀ {α : Type u_1} {m : MeasurableSpace α}, MeasurableSet Set.univ := by sorry

The theorem `MeasurableSet.univ` states that for any type `α` and any measurable space `m` on `α`, the universal set of `α` is measurable. In mathematical terms, if we consider a measure space on a given set, the entire set (which corresponds to the universal set in set theory) is always considered a measurable set. In the context of this theorem, a measurable set is a set for which we are able to accurately define its measure.

For any measurable space `m` on type `α`, the universal set of `α` is measurable.


theorem SetLike.mem_coe : ∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, x ∈ ↑p ↔ x ∈ p := by sorry

This theorem states that for any types `A` and `B`, and any instance of `SetLike A B`, the membership of an element `x` of type `B` in the coercion of an object `p` of type `A` is equivalent to the membership of `x` in `p`. In other words, if we have a type that behaves like a set, coercing it to a set doesn't change whether a particular element belongs to it.

For any types `A`, `B` and instance of `SetLike A B`, the membership `x ∈ p` holds if and only if `x ∈ (coe : A → Set) p`.


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

This theorem states that for all types `α` that have a linear order, if it is not the case that `a` is greater than or equal to `b` (`¬a ≥ b`), then `a` is strictly less than `b` (`a < b`). In other words, if `a` is not greater than or equal to `b`, the only other possibility in a linear order is that `a` is less than `b`.

In a linear order, if `a` is not greater than or equal to `b` (`a < b ∨ ¬(a ≥ b)`), then `a` is strictly less than `b` (`a < b`).


Continuous.tendsto

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

The theorem `Continuous.tendsto` states that for all types `X` and `Y`, given topological spaces over `X` and `Y`, and a function `f` from `X` to `Y`, if the function `f` is continuous, then for all `x` in `X`, `f` tends to the neighborhood filter of `f(x)` when the input tends to the neighborhood filter of `x`. In essence, this theorem formulates the property that a function is continuous at a point if and only if the pre-image of any neighborhood of the function's value at that point contains a neighborhood of the point.

A continuous function from a topological space X to a topological space Y sends the neighborhood filter of any point x in X to the neighborhood filter of f(x) in Y.


/-- **Alias** of `congrArg₂`. -/
theorem congr_arg₂ :
    ∀ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (f : α → β → γ) {x x' : α} {y y' : β},
      x = x' → y = y' → f x y = f x' y' :=
  by sorry

This theorem, known as `congr_arg₂`, states that for any three types α, β, and γ and a function `f` from α and β to γ, if `x` and `x'` are equal elements of type α and `y` and `y'` are equal elements of type β, then the result of applying `f` to `x` and `y` is equal to the result of applying `f` to `x'` and `y'`. This is essentially a generalization of the congruence property in mathematics, extended to a function of two variables.

For all types α, β, γ and functions f : α → β → γ, if x = x' in α and y = y' in β, then f x y = f x' y' in γ.


theorem Finset.coe_injective : ∀ {α : Type u_4}, Function.Injective Finset.toSet := by sorry

The theorem `Finset.coe_injective` states that for every type `α`, the function `Finset.toSet` is injective. In other words, given any two finite sets (finsets) of elements of type `α`, if their corresponding sets (obtained by the `Finset.toSet` function) are identical, then the original finite sets must also be identical. This means that no two distinct finite sets can map to the same set under the `Finset.toSet` function.

For any type `α`, the function `Finset.toSet` from `Finset α` to `Set α` is an injection.


theorem mul_inv_cancel : ∀ {G₀ : Type u} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a * a⁻¹ = 1 := by sorry

This theorem states that for any non-zero element 'a' of a group with zero 'G₀', the product of 'a' and its multiplicative inverse 'a⁻¹' equals one. This is essentially saying that the multiplicative inverse property holds in a group with zero, where any element times its inverse equals the identity element, which is one in this context.

For any non-zero element 'a' in a group G with identity element G₀, a * a⁻¹ = G₀.


theorem Quiver.Hom.op_inj : ∀ {C : Type u₁} [inst : Quiver C] {X Y : C}, Function.Injective Quiver.Hom.op := by sorry

The theorem `Quiver.Hom.op_inj` states that for any type `C` that forms a quiver (i.e., a category-like structure), and any two objects `X` and `Y` in `C`, the operation of forming the opposite of a morphism (or arrow), given by the function `Quiver.Hom.op`, is injective. In other words, no two distinct morphisms from `X` to `Y` in `C` will become identical when their direction is reversed.

For any quiver `C`, the opposite operation on morphisms `Quiver.Hom.op` is an injection from `C.Hom(X, Y)` to `C.Hom(Y, X)` for all objects `X` and `Y` in `C`.


theorem Mathlib.Data.Set.Basic._auxLemma.8 : ∀ {α : Type u} (x : α), (x ∈ ∅) = False := by sorry

This theorem, `Mathlib.Data.Set.Basic._auxLemma.8`, states that for any type `α` and an element `x` of this type, the proposition that `x` is an element of the empty set is always False. In other words, no element can belong to the empty set, irrespective of its type.

The theorem asserts that for all types α and elements x of α, x is not an element of the empty set.


Equiv.ext

theorem Equiv.ext : ∀ {α : Sort u} {β : Sort v} {f g : α ≃ β}, (∀ (x : α), f x = g x) → f = g := by sorry

This theorem, named `Equiv.ext`, states that for any two equivalences `f` and `g` between two arbitrary types `α` and `β`, if for all elements `x` of type `α` the result of applying `f` and `g` to `x` is the same, then `f` and `g` themselves are the same. In other words, if two equivalence functions behave identically on every input, they are equal.

If two functions `f` and `g` between types `α` and `β` are equal as relations (i.e., for all `x` in `α`, `f x = g x`), then `f` and `g` are equal as functions.


Matrix.ext

theorem Matrix.ext :
    ∀ {m : Type u_2} {n : Type u_3} {α : Type v} {M N : Matrix m n α}, (∀ (i : m) (j : n), M i j = N i j) → M = N := by
  sorry

The theorem `Matrix.ext` states that for any two matrices `M` and `N` of the same type, where the type is defined as `Matrix m n α` (i.e., matrices with entries in `α`, whose rows are indexed by `m` and whose columns are indexed by `n`), if every corresponding entry in `M` and `N` is equal (i.e., for all indices `(i, j)`, the entry at position `(i, j)` in `M` is equal to the entry at position `(i, j)` in `N`), then the two matrices `M` and `N` themselves are equal.

If two matrices of the same type with matching entries are given, then they are equal.


theorem forall_swap :
    ∀ {α : Sort u_3} {β : Sort u_2} {p : α → β → Prop}, (∀ (x : α) (y : β), p x y) ↔ ∀ (y : β) (x : α), p x y := by
  sorry

This theorem, `forall_swap`, states that for any two types `α` and `β` and any property `p` that takes an element of `α` and an element of `β`, the proposition that `p` holds for all `x` of type `α` and `y` of type `β` is logically equivalent to the proposition that `p` holds for all `y` of type `β` and `x` of type `α`. In other words, when defining a property that applies to every pair of elements from two types, the order in which we express the quantifiers does not matter.

For any types `α` and `β` and property `p(x : α, y : β)`, the propositions `∀x, ∀y, p x y` and `∀y, ∀x, p x y` are logically equivalent.


/-- If `f` is an additive monoid homomorphism then `f 0 = 0`. -/
theorem AddMonoidHom.map_zero :
    ∀ {M : Type u_4} {N : Type u_5} [inst : AddZeroClass M] [inst_1 : AddZeroClass N] (f : M →+ N), f 0 = 0 := by sorry

The theorem `AddMonoidHom.map_zero` states that for any additive monoid homomorphism `f` mapping from a type `M` to a type `N` (with `M` and `N` being add zero classes, i.e., classes that include an additive identity element '0'), the image of 0 under `f` will be 0. In simpler terms, this theorem is asserting that an additive monoid homomorphism preserves the additive identity, that is, it maps the '0' element of the domain to the '0' element of the codomain.

For any additive monoid homomorphism f from an add zero class M to an add zero class N, f(0) = 0.


theorem Ne.bot_lt : ∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderBot α] {a : α}, a ≠ ⊥ → ⊥ < a := by sorry

This theorem, `Ne.bot_lt`, states that for any type `α` that is equipped with a partial ordering and a smallest (bottom) element, if `a` is an element of `α` and `a` is not equal to the bottom element, then the bottom element is less than `a`. That is, in this order, any element other than the smallest one is strictly larger than the smallest one.

For any type `α` with a partial ordering and a smallest element `bot`, if `a` is in `α` and `a ≠ bot`, then `bot < a`.


/-- Ring homomorphisms map zero to zero.  -/
theorem RingHom.map_zero :
    ∀ {α : Type u_2} {β : Type u_3} {x : NonAssocSemiring α} {x_1 : NonAssocSemiring β} (f : α →+* β), f 0 = 0 := by
  sorry

This theorem states that for any ring homomorphism `f` from a non-associative semiring `α` to another non-associative semiring `β`, the mapping of the element zero from `α` through `f` will always result in the zero element of `β`. In other words, ring homomorphisms preserve the zero element across different non-associative semirings.

For any ring homomorphism between non-associative semirings, the image of the zero element is the zero element.


theorem dvd_rfl : ∀ {α : Type u_1} [inst : Monoid α] {a : α}, a ∣ a := by sorry

This theorem, `dvd_rfl`, states that for any type `α` which forms a monoid, any element `a` of this type divides itself. In the language of divisibility in number theory, if `α` is an integer, then for any integer `a`, `a` divides `a`. This is a universal property in the context of monoids.

For any monoid type `α`, every element `a` of `α` divides itself.


theorem Finset.sum_const :
    ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} [inst : AddCommMonoid β] (b : β),
      (Finset.sum s fun _x => b) = s.card • b :=
  by sorry

The theorem `Finset.sum_const` states that for any type `α`, type `β`, a finite set `s` of type `α`, and an instance of additive commutative monoid for `β`, and any element `b` of type `β`, the sum of the constant function `b` over all elements in the set `s` is equal to the product of the cardinality of the set `s` and `b`. In other words, if we add up a constant `b` for every element in the set `s`, we get the same result as multiplying `b` by the number of elements in `s`.

For any finite set `s` of type `α` and any additive commutative monoid `β` with an element `b`, the sum of the constant function `b` over `s` equals the product of the cardinality of `s` and `b`.


theorem Mathlib.Data.Set.Defs._auxLemma.5 :
    ∀ {α : Type u} {β : Type v} (f : α → β) (s : Set α) (y : β), (y ∈ f '' s) = ∃ x ∈ s, f x = y := by sorry

This theorem states that for any types `α` and `β`, a function `f` from `α` to `β`, a set `s` of type `α`, and an element `y` of type `β`, `y` is in the image of set `s` under function `f` if and only if there exists an element `x` in set `s` such that `f` of `x` equals `y`. In other words, the statement is about the definition of the image of a set under a function: the image of `s` under `f` contains `y` exactly when `y` is the `f`-image of some element of `s`.

The image of a set under a function is equal to the set of images of elements in the set under that function.


theorem Algebra.smul_def :
    ∀ {R : Type u} {A : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (r : R) (x : A),
      r • x = (algebraMap R A) r * x :=
  by sorry

The theorem `Algebra.smul_def` states that for any commutative semiring `R` and any semiring `A` with an algebra structure, the scalar multiplication of an element `r` from `R` and an element `x` from `A` is defined as the multiplication of `x` and the image of `r` under the algebra map from `R` to `A`. In mathematical terms, given `r ∈ R` and `x ∈ A`, we have `r • x = (algebraMap R A) r * x`.

For any commutative semiring `R` and algebra `A` over it, scalar multiplication `r • x` equals the product of `x` and the algebra map image of `r`, i.e., `r • x = (algebraMap R A) r * x`.


theorem Nat.zero_add : ∀ (n : ℕ), 0 + n = n := by sorry

This theorem states that for all natural numbers 'n', the result of adding zero to 'n' is 'n' itself. In other words, it proves the identity property of addition for natural numbers, where adding zero to any natural number doesn't change its value.


theorem Mathlib.Data.Nat.Cast.Order._auxLemma.12 :
    ∀ {α : Type u_1} [inst : AddMonoidWithOne α] [inst_1 : PartialOrder α]
      [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [inst_3 : ZeroLEOneClass α]
      [inst_4 : CharZero α] {m n : ℕ}, (↑m ≤ ↑n) = (m ≤ n) :=
  by sorry

This theorem states that for any type `α` which is an additive monoid with identity, has a partial order, has a covariant class where addition induces an ordering, has a zero less than or equal to one, and has a characteristic of zero, the natural number casting of `m` to `α` being less than or equal to the natural number casting of `n` to `α` is equivalent to `m` being less than or equal to `n` in the set of natural numbers. In other words, in this specific context, the process of casting doesn't change the relative order of the natural numbers.

For any additive monoid with identity, partial order, covariant additive ordering, zero less than one, and characteristic zero, the natural number casting function respects the order relation between natural numbers.


/-- Ring homomorphisms map one to one.  -/
theorem RingHom.map_one :
    ∀ {α : Type u_2} {β : Type u_3} {x : NonAssocSemiring α} {x_1 : NonAssocSemiring β} (f : α →+* β), f 1 = 1 := by
  sorry

This theorem states that for any ring homomorphism `f` from a non-associative semiring `α` to another non-associative semiring `β`, the image of 1 under `f` is 1. In other words, a ring homomorphism always maps the multiplicative identity of the source ring (1 in `α`) to the multiplicative identity of the target ring (1 in `β`).

For any ring homomorphism `f` between non-associative semirings, `f 1 = 1`.


DifferentiableAt.hasFDerivAt

theorem DifferentiableAt.hasFDerivAt :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
      {x : E}, DifferentiableAt 𝕜 f x → HasFDerivAt f (fderiv 𝕜 f x) x :=
  by sorry

The theorem `DifferentiableAt.hasFDerivAt` states that for any given function `f` from `E` to `F`, if `f` is differentiable at a point `x` in `E`, then `f` has a derivative at `x` and this derivative is given by `fderiv 𝕜 f x`. Here, `𝕜` is a nontrivial normed field, `E` and `F` are normed additive commutative groups equipped with a normed space structure over `𝕜`, and `fderiv 𝕜 f x` is the derivative of `f` at `x` (or zero if `f` is not differentiable at `x`).

If `f` is differentiable at `x` in the normed additive commutative groups `E` and `F` over a nontrivial normed field `𝕜`, then `fderiv 𝕜 f x` exists and is the derivative of `f` at `x`.


DFinsupp.ext

theorem DFinsupp.ext :
    ∀ {ι : Type u} {β : ι → Type v} [inst : (i : ι) → Zero (β i)] {f g : Π₀ (i : ι), β i},
      (∀ (i : ι), f i = g i) → f = g :=
  by sorry

This theorem, `DFinsupp.ext`, states that for any type `ι` and a function `β` mapping `ι` to another type, given the zero element for each instance of type `β i`, if we have two functions `f` and `g` from `ι` to `β i` (represented as `Π₀ (i : ι), β i`), then if for all `i` in `ι`, `f i = g i`, it follows that `f` is equal to `g`. In other words, two functions over possibly different types are equal if they produce the same output for the same input.

If `β` is a type and `f` and `g` are functions from `ι` to `β`, then `f = g` if and only if `∀ i ∈ ι, f i = g i`.


theorem add_left_neg : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), -a + a = 0 := by sorry

This theorem states that for any type `G` that forms an additive group, the addition of an element `a` and its negative `-a` results in the additive identity `0`. In other words, for all elements `a` in the group, `-a + a = 0`. This is a formal statement of the property that every element in an additive group has an additive inverse.

For every additive group `G` and element `a` in `G`, `-a + a = 0`.


theorem Mathlib.Algebra.Order.ZeroLEOne._auxLemma.2 :
    ∀ {α : Type u_1} [inst : Zero α] [inst_1 : One α] [inst_2 : PartialOrder α] [inst_3 : ZeroLEOneClass α]
      [inst_4 : NeZero 1], (0 < 1) = True :=
  by sorry

This theorem, named `_auxLemma.2` in the `Mathlib.Algebra.Order.ZeroLEOne` namespace, asserts that for any type `α` with instances of `Zero α`, `One α`, `PartialOrder α`, `ZeroLEOneClass α`, and `NeZero 1`, the statement "0 is less than 1" is true. This is essentially a formalization of the basic mathematical fact that 0 is less than 1, and it applies to any type that has a zero, a one, a partial order, satisfies the `ZeroLEOneClass`, and where 1 is not zero.

For any type with a zero, one, a partial order, and `ZeroLEOneClass` structure where one is non-zero, zero is less than one.


theorem Function.update_same :
    ∀ {α : Sort u} {β : α → Sort v} [inst : DecidableEq α] (a : α) (v : β a) (f : (a : α) → β a),
      Function.update f a v a = v :=
  by sorry

This theorem, "Function.update_same", states that for any type `α` with decidable equality, any function `f` from `α` to some type `β` dependent on `α`, and any value `a` of type `α` and `v` of type `β a`, if we update the function `f` at `a` to be `v`, then the value of the updated function at `a` is `v`. In other words, if we replace the value of a function at a given point by a certain value, then the function at that point yields the new value.

For any type `α` with decidable equality and function `f` from `α` to type `β`, if `a : α` and `v : β (a)`, then updating `f` at `a` to `v` results in a function `g` such that `g a = v`.


theorem List.prod_cons :
    ∀ {M : Type u_5} [inst : Monoid M] {l : List M} {a : M}, List.prod (a :: l) = a * List.prod l := by sorry

The theorem `List.prod_cons` states that for any type `M` that forms a monoid, any list `l` of elements of type `M`, and any element `a` of type `M`, the product of the list that results from prepending `a` to `l` is equal to `a` multiplied by the product of `l`. In other words, if you add an element to the start of a list and then calculate the product of the list, it's the same as if you multiplied that element with the product of the original list.

For any monoid `M`, any list `l` of `M`, and any `a` in `M`, we have `a * List.foldl (fun _ x => x) l = List.foldl (fun x _ => x * a) l a`.


theorem Set.iInter_congr_Prop :
    ∀ {α : Type u_1} {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q),
      (∀ (x : q), f₁ ⋯ = f₂ x) → Set.iInter f₁ = Set.iInter f₂ :=
  by sorry

This theorem states that for any type `α`, and any proposition `p` and `q`, and any functions `f₁` and `f₂` from `p` and `q` respectively to a set of `α`, if `p` is equivalent to `q` (`p ↔ q`), and for all elements `x` of `q` the result of `f₁` applied to some value is equal to `f₂` applied to `x`, then the indexed intersection over `f₁` is equal to the indexed intersection over `f₂`. In other words, if `p` and `q` are logically equivalent and both `f₁` and `f₂` produce the same sets for corresponding inputs, then the intersection of all sets generated by `f₁` and `f₂` are the same.

If `p` is equivalent to `q` and for all `x` in `q`, `f₁(x) = f₂(x)`, then the indexed intersections of `f₁` and `f₂` over `α` are equal.


theorem mul_le_mul_of_nonneg_right :
    ∀ {α : Type u_1} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α] [inst_3 : MulPosMono α],
      b ≤ c → 0 ≤ a → b * a ≤ c * a :=
  by sorry

This is a theorem about the monotonicity of multiplication in an ordered semiring. It states that, given any type `α` that has a multiplication operation, zero, and is a preorder, and is also equipped with a property that multiplication by nonnegative elements (`a`) on the right is monotonic. If `b` is less than or equal to `c` and `a` is nonnegative (i.e., greater than or equal to zero), then `b` times `a` is less than or equal to `c` times `a`. In other words, if you multiply `b` and `c` (where `b ≤ c`) by the same nonnegative number `a`, the inequality relationship between `b` and `c` is preserved.

If `α` is an ordered semiring with right multiplicatively monotonic nonnegative elements, then for all `a ≥ 0` and `b ≤ c` in `α`, we have `b * a ≤ c * a`.


Prod.mk.injEq

theorem Prod.mk.injEq :
    ∀ {α : Type u} {β : Type v} (fst : α) (snd : β) (fst_1 : α) (snd_1 : β),
      ((fst, snd) = (fst_1, snd_1)) = (fst = fst_1 ∧ snd = snd_1) :=
  by sorry

This theorem, `Prod.mk.injEq`, states that for any two pairs of types `(α, β)` and `(α_1, β_1)`, the equality of these two pairs is equivalent to the equality of both their corresponding first and second elements. In other words, two pairs are equal if and only if their corresponding elements are equal. This theorem holds for any types `α` and `β`, and any elements `fst, fst_1` of type `α` and `snd, snd_1` of type `β`.

For any types `α` and `β` and elements `(a, b) : Type ↔ (a' : α) × (b' : β)`, `(a = a') ∧ (b = b')` if and only if `(a, b) = (a', b')`.


theorem Nat.le_succ : ∀ (n : ℕ), n ≤ Nat.succ n := by sorry

This theorem states that for any natural number `n`, `n` is less than or equal to its successor (`n + 1`). That means, in any situation where you have a natural number, it will always be less than or equal to the next natural number.


theorem Multiset.map_congr :
    ∀ {α : Type u_1} {β : Type v} {f g : α → β} {s t : Multiset α},
      s = t → (∀ x ∈ t, f x = g x) → Multiset.map f s = Multiset.map g t :=
  by sorry

The theorem `Multiset.map_congr` states that for any types `α`, `β` and any two functions `f` and `g` from `α` to `β`, and any two multisets `s` and `t` of type `α`, if `s` equals `t` and for every element `x` in `t`, `f(x)` equals `g(x)`, then the multiset obtained by mapping `f` over `s` is equal to the multiset obtained by mapping `g` over `t`. This is just a way of saying that the mapping of equivalent functions over equal multisets produces equal results.

If `α` is a type, `f` and `g` are functions from `α` to `β`, `s` and `t` are multisets of `α`, `s = t`, and `f(x) = g(x)` for all `x ∈ t`, then `Multiset.map f s = Multiset.map g t`.


theorem neg_smul :
    ∀ {R : Type u_2} {M : Type u_5} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (r : R) (x : M),
      -r • x = -(r • x) :=
  by sorry

This theorem states that for any ring `R`, additively commutative group `M`, and a module `M` over `R`, the operation of negating a scalar multiple of a module element is the same as multiplying the module element by the negation of the scalar. In other words, if `r` is an element of `R` and `x` is an element of `M`, then the negative of `r` scalar multiplied by `x` equals the negative of (`r` scalar multiplied by `x`). This is a property of scalar multiplication in the context of ring and module theory.

For any ring `R`, additively commutative group `M`, and `r` in `R` and `x` in `M`, $-(rx) = r\cdot(-x)$.


theorem nsmul_eq_mul : ∀ {α : Type u_1} [inst : NonAssocSemiring α] (n : ℕ) (a : α), n • a = ↑n * a := by sorry

This theorem states that for all types `α` in a non-associative semiring, the scalar multiplication of a natural number `n` and an element `a` from the semiring is equal to the multiplication of the cast of `n` to the semiring and `a`. More formally, given any natural number `n` and any element `a` of the semiring `α`, `n • a` equals `↑n * a`.

For any natural number $n$ and element $a$ in a non-associative semiring $\alpha$, $n \cdot a = \uparrow n \times a$.


theorem measurable_snd :
    ∀ {α : Type u_1} {β : Type u_2} {x : MeasurableSpace α} {x_1 : MeasurableSpace β}, Measurable Prod.snd := by sorry

The theorem `measurable_snd` states that for any given types `α` and `β` and any measurable spaces `x` over `α` and `x_1` over `β`, the second projection function `Prod.snd` is measurable. In other words, the function that takes an ordered pair and returns its second component is a measurable function. This means that for any measurable set in the co-domain, the preimage of that set under `Prod.snd` is also measurable.

The second projection function `Prod.snd` is measurable with respect to the product measurable space structure over types `α` and `β`.


theorem neg_eq_zero : ∀ {α : Type u_1} [inst : SubtractionMonoid α] {a : α}, -a = 0 ↔ a = 0 := by sorry

This theorem, `neg_eq_zero`, states that for any type `α` that forms a subtraction monoid, the negation (in the context of the subtraction operation) of a variable `a` of type `α` equals zero if and only if `a` itself equals zero. A subtraction monoid is a structure with an associative operation (subtraction) and a special element (zero), and where every element has an additive inverse (which is obtained via subtraction from the zero element). Essentially, the theorem is saying that if you have an element from a set that behaves like numbers with respect to subtraction (e.g., the integers, real numbers, complex numbers etc.), then the only way for the negation of that element to be zero is if the element itself is zero.

For any type `α` forming a subtraction monoid, `a` of type `α` has negation equal to zero if and only if `a` itself equals zero.


/-- **Alias** of the reverse direction of `tsub_eq_zero_iff_le`. -/
theorem tsub_eq_zero_of_le :
    ∀ {α : Type u_1} [inst : CanonicallyOrderedAddCommMonoid α] [inst_1 : Sub α] [inst_2 : OrderedSub α] {a b : α},
      a ≤ b → a - b = 0 :=
  by sorry

This theorem, `tsub_eq_zero_of_le`, states that for any type `α` which is a canonically ordered additive commutative monoid where subtraction is defined and order-preserving, if `a` and `b` are any two elements of `α` and `a` is less than or equal to `b`, then the result of `a` subtract `b` will be zero. This is essentially a formalization of the principle that in an ordered set, subtracting a larger number from a smaller one (or the same number from itself) yields zero.

For any additively commutative, order-preserving, and canonically ordered monoid type `α`, if `a ≤ b` then `a - b = 0`.


theorem Finset.coe_singleton : ∀ {α : Type u_1} (a : α), ↑{a} = {a} := by sorry

This theorem states that for all types `α` and any element `a` of type `α`, the coercion of a singleton finset containing `a` (notated as `↑{a}`) is equal to a singleton set containing `a` (`{a}`). In simpler terms, a single-element finset and a single-element set containing the same element are equivalent.

For any type `α` and element `a` of type `α`, the singleton finset `↑{a}` is equal to the singleton set `{a}`.


theorem Finset.coe_image :
    ∀ {α : Type u_1} {β : Type u_2} [inst : DecidableEq β] {f : α → β} {s : Finset α}, ↑(Finset.image f s) = f '' ↑s :=
  by sorry

The theorem `Finset.coe_image` states that for any types `α` and `β`, where `β` has a decidable equality, and for any function `f` from `α` to `β` and any finite set `s` of type `α`, the co-domain of the image of the function `f` applied to the set `s` is equal to the image of the function `f` applied to the co-domain of `s`. In mathematical terms, this is equivalent to saying that for a function $f: α \to β$ and a finite set $s \subseteq α$, the set of all $f(x)$ for $x \in s$ is the same whether we consider $s$ as a finset or as a set.

For any function $f: α \to β$ with decidable equality and finite set $s \subseteq α$, the image of $f$ applied to the co-domain of $s$ is equal to the co-domain of the image of $f$ applied to $s$.


ContinuousMap.continuous

/-- Deprecated. Use `map_continuous` instead.  -/
theorem ContinuousMap.continuous :
    ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : C(α, β)),
      Continuous ⇑f :=
  by sorry

This theorem states that for any continuous function `f` between two topological spaces `α` and `β`, the application of `f` is a continuous operation. However, this theorem is deprecated and it is recommended to use `map_continuous` instead. In mathematical terminology, if `f` belongs to the set of continuous functions `C(α, β)` from `α` to `β`, then the operation `⇑f`, i.e., applying `f`, is continuous. Both `α` and `β` are arbitrary types equipped with a topological structure.

If `f` is a continuous function between topological spaces `α` and `β`, then the application of `f` is a continuous function from `α` to `β`.


theorem tsub_zero :
    ∀ {α : Type u_1} [inst : PartialOrder α] [inst_1 : AddCommMonoid α] [inst_2 : Sub α] [inst : OrderedSub α] (a : α),
      a - 0 = a :=
  by sorry

This theorem, `tsub_zero`, states that for any element `a` in a type `α`, where `α` is a partially ordered set, an additive commutative monoid, and a subtractable set with order-respecting subtraction, subtracting zero from `a` will always yield `a`. In the language of mathematics, this theorem is simply stating that `a - 0 = a` for any `a` in such a type `α`.

For any element `a` in a type `α` that is a partially ordered set, an additive commutative monoid, and a subtractable set with order-respecting subtraction, `a - 0 = a`.


theorem CategoryTheory.Iso.hom_inv_id_assoc :
    ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (self : X ≅ Y) {Z : C} (h : X ⟶ Z),
      CategoryTheory.CategoryStruct.comp self.hom (CategoryTheory.CategoryStruct.comp self.inv h) = h :=
  by sorry

This theorem states that in any category `C`, for any objects `X`, `Y`, and `Z` in `C`, and for any isomorphism `self` from `X` to `Y`, and any morphism `h` from `X` to `Z`, the composition of `self.hom` and `CategoryTheory.CategoryStruct.comp self.inv h`, or in other words the composition of the morphism of the isomorphism and the composition of the inverse of the isomorphism with the morphism `h`, is equal to `h`. In simpler terms, it asserts that if we follow an isomorphism and then its inverse, and then continue with some morphism, it's as if we did nothing before continuing with that morphism. This is akin to saying that an isomorphism and its inverse form an identity when composed, meaning it doesn't produce any effective change when applied before other morphisms.

In any category, composing an isomorphism with its inverse and then with a morphism is equal to composing the morphism with the isomorphism.


theorem Function.update_noteq :
    ∀ {α : Sort u} {β : α → Sort v} [inst : DecidableEq α] {a a' : α},
      a ≠ a' → ∀ (v : β a') (f : (a : α) → β a), Function.update f a' v a = f a :=
  by sorry

This theorem, `Function.update_noteq`, states that for any type `α` with decidable equality, any type mapping `β : α → Sort v`, and any two elements `a` and `a'` of type `α` that are not equal, for any value `v : β a'` and any function `f : (a : α) → β a`, the result of updating the function `f` at point `a'` with value `v` and then evaluating it at point `a` is the same as the original value of the function `f` at point `a`. In simpler terms, if you update the value of a function at a point and then evaluate the function at a different point, the output of the function doesn't change.

For any type `α` with decidable equality, any type mapping `β : α → Sort v`, and functions `f : α → β`, `a`, `a' : α (a ≠ a')`, the value `f a` is equal to `(Function.update f a' v) a`.


theorem HasSum.tsum_eq :
    ∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {f : β → α} {a : α}
      [inst_2 : T2Space α], HasSum f a → ∑' (b : β), f b = a :=
  by sorry

This theorem states that for all types `α` and `β`, where `α` is an additive commutative monoid and a topological space, and `β` is an arbitrary type with a function `f` from `β` to `α`, and a constant `a` of type `α`, if `α` is also a T2 space (also known as a Hausdorff space), and the sum of the function `f` over all `β` converges to `a` (expressed as `HasSum f a`), then the infinite sum (notated as `∑' (b : β), f b`) of `f` over all `β` is equal to `a`. In simpler terms, it says that if an infinite sum over a function converges, then the sum is equal to the value it converges to.

If `α` is an additive commutative monoid, a topological space, and a T2 space, and `f : β → α` has a convergent sum `HasSum f a`, then `∑' (b : β), f b = a`.


Option.some.injEq

theorem Option.some.injEq : ∀ {α : Type u} (val val_1 : α), (some val = some val_1) = (val = val_1) := by sorry

This theorem, `Option.some.injEq`, states that for any type `α` and for any two values `val` and `val_1` of this type, the equality of `some val` and `some val_1` is equivalent to the equality of `val` and `val_1`. In other words, in the `Option` type constructor, `some` is injective: two `some` values are equal if and only if the values they contain are equal.

For any type `α`, the function `some : α -> Option α` is an injection. That is, for all `val` and `val_1` of type `α`, `some val = some val_1` if and only if `val = val_1`.


HasDerivAt.deriv

theorem HasDerivAt.deriv :
    ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
      [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}, HasDerivAt f f' x → deriv f x = f' :=
  by sorry

The theorem `HasDerivAt.deriv` states that for any nontrivially normed field `𝕜`, additively commutative normed group `F` and a function `f` mapping from `𝕜` to `F`, if `f` has a derivative `f'` at a point `x` in `𝕜` (denoted as `HasDerivAt f f' x`), then the derivative of `f` at this point `x` (denoted as `deriv f x`) is equal to `f'`. The derivative here is considered in the sense of Frechet derivatives.

For any nontrivially normed field 𝕜, additively commutative normed group F, and differentiable function f from 𝕜 to F at x in 𝕜, the Frechet derivative f' of f at x equals the derivative deriv f of f at x.


theorem zero_nsmul : ∀ {M : Type u_2} [inst : AddMonoid M] (a : M), 0 • a = 0 := by sorry

This theorem states that for any type `M` that is an additive monoid, the scalar multiplication of any element `a` from `M` by zero equals the additive identity element (0) of `M`. In other words, multiplying any element of an additive monoid by zero always results in zero. This is a common property in many algebraic structures such as numbers, vectors, matrices, etc.

For any additive monoid M and element a in M, a * 0 = 0.


theorem zero_vadd : ∀ (M : Type u_1) {α : Type u_6} [inst : AddMonoid M] [inst_1 : AddAction M α] (b : α), 0 +ᵥ b = b :=
  by sorry

This theorem states that for any type `M`, and any other type `α` which is an additive action of `M`, when 0 (from the additive monoid `M`) is vector-added (`+ᵥ`) to any element `b` of the type `α`, the result is `b` itself. In mathematical terms, it's expressing that 0 acts as an identity for the vector addition operation in the context of an additive action.

For any additive action `α` of type `M`, the additive identity 0 of `M` leaves elements of `α` unchanged under vector addition, i.e., `0 +ᵥ b = b` for all `b` in `α`.


theorem succ_nsmul : ∀ {M : Type u_2} [inst : AddMonoid M] (a : M) (n : ℕ), (n + 1) • a = n • a + a := by sorry

This theorem, named `succ_nsmul`, applies to any type `M` that forms an additive monoid. It takes an element `a` of type `M` and a natural number `n`, and states that scaling `a` by `n + 1` (denoted `(n + 1) • a`) is the same as scaling `a` by `n` and then adding `a` (denoted `n • a + a`). This is a generalization of the arithmetic fact that `(n + 1) * a = n * a + a`.

For all additive monoids M, (n + 1) • a = n • a + a, where • denotes scaling and + denotes addition.


List.TFAE.out

theorem List.TFAE.out :
    ∀ {l : List Prop},
      List.TFAE l →
        ∀ (n₁ n₂ : ℕ) {a b : Prop},
          autoParam (List.get? l n₁ = some a) _auto✝ → autoParam (List.get? l n₂ = some b) _auto✝¹ → (a ↔ b) :=
  by sorry

The theorem `List.TFAE.out` states that if a list of propositions (`l`) is such that all propositions are equivalent to each other (expressed by `List.TFAE l`), then for any two indices `n₁` and `n₂`, and two propositions `a` and `b`, if `a` and `b` are the propositions at index `n₁` and `n₂` in the list respectively (expressed by `List.get? l n₁ = some a` and `List.get? l n₂ = some b`), then `a` and `b` are equivalent. The `autoParam` function is used to automatically provide these last two conditions when they are not explicitly provided. The proof of this theorem is not shown here as it is represented by `sorry`.

If a list `l` of propositions satisfies `List.TFAE l`, then for any indices `n₁` and `n₂`, the propositions at indices `n₁` and `n₂` are equivalent, i.e., `List.get? l n₁ ≈ List.get? l n₂`.


theorem two_ne_zero : ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 2] [inst_2 : NeZero 2], 2 ≠ 0 := by sorry

This theorem, named `two_ne_zero`, states that for any type `α` that has a zero element (`Zero α`), can interpret the natural number 2 (`OfNat α 2`), and 2 is not zero in that type (`NeZero 2`), two is not equal to zero (`2 ≠ 0`). In other words, it asserts that in any suitable mathematical structure, the number two cannot be the zero element.

For any type with a zero element and where 2 is not zero, two is not equal to zero.


theorem Set.inter_univ : ∀ {α : Type u} (a : Set α), a ∩ Set.univ = a := by sorry

The theorem `Set.inter_univ` states that for any type `α` and any set `a` of type `α`, the intersection of `a` with the universal set of type `α` (`Set.univ`) is equal to `a` itself. In more mathematical terms, given any set `a`, the intersection of `a` with the set of all elements (the universal set) always results in `a`. This is akin to the set theory concept that a set intersected with the universal set returns the original set.

For any set `a` of type `α`, `a ∩ Set.univ = a`.


theorem Nat.cast_mul : ∀ {α : Type u_1} [inst : NonAssocSemiring α] (m n : ℕ), ↑(m * n) = ↑m * ↑n := by sorry

This theorem states that for all natural numbers `m` and `n`, and for any type `α` that forms a non-associative semiring, the cast of the product of `m` and `n` to type `α` is equal to the product of the cast of `m` and the cast of `n` to type `α`. In mathematical notation, this could be represented as: if `m` and `n` are natural numbers, and `α` is a non-associative semiring, then `(m * n) = m * n` holds true when all quantities are cast to `α`.

For any non-associative semiring type `α` and natural numbers `m` and `n`, the cast of `m * n` to `α` equals `m * n` (where `*` denotes both the natural number multiplication and the semiring multiplication).


theorem Set.mem_singleton_iff : ∀ {α : Type u} {a b : α}, a ∈ {b} ↔ a = b := by sorry

This theorem states that for any type `α` and any two elements `a` and `b` of that type, `a` belongs to the singleton set `{b}` if and only if `a` is equal to `b`. In other words, an element `a` is a member of a set that only contains `b` if `a` and `b` are actually the same element.

For any type `α` and elements `a, b : α`, `a ∈ {b} ↔ a = b`.


theorem Mathlib.Data.Prod.Basic._auxLemma.1 :
    ∀ {α : Type u_1} {β : Type u_2} {p : α × β → Prop}, (∀ (x : α × β), p x) = ∀ (a : α) (b : β), p (a, b) := by sorry

This theorem, `Mathlib.Data.Prod.Basic._auxLemma.1`, states that for all types `α` and `β`, and for any property `p` that applies to pairs of type `α × β`, stating that the property `p` holds for every pair `x` of type `α × β` is equivalent to stating that for every element `a` of type `α` and every element `b` of type `β`, the property `p` holds for the pair `(a, b)`. In other words, it relates the universal quantification over a product type to universal quantifications over each of the components.

For any property `p` and types `α` and `β`, the statement "for all `x` in `α × β`, `p x` holds" is equivalent to "for all `a` in `α` and `b` in `β`, `p (a, b)` holds."


theorem mul_left_comm : ∀ {G : Type u_3} [inst : CommSemigroup G] (a b c : G), a * (b * c) = b * (a * c) := by sorry

This theorem, `mul_left_comm`, states that for all types `G` that have an instance of the `CommSemigroup` structure, and for any three elements `a`, `b`, and `c` of type `G`, the operation of multiplying `a` with the result of multiplying `b` and `c` is equal to the operation of multiplying `b` with the result of multiplying `a` and `c`. In conventional mathematical terms, for any three elements in a commutative semigroup, the multiplication operation is left-commutative.

For all types `G` with `CommSemigroup` structure and all elements `a`, `b`, `c` in `G`, `(a × b) = (b × a)`.


MeasurableSet.compl

theorem MeasurableSet.compl :
    ∀ {α : Type u_1} {s : Set α} {m : MeasurableSpace α}, MeasurableSet s → MeasurableSet sᶜ := by sorry

The theorem `MeasurableSet.compl` states that for any type `α`, any set `s` of type `α`, and any measurable space `m` on `α`, if the set `s` is measurable in the measurable space `m`, then the complement of `s` (denoted by `sᶜ`) is also measurable in `m`. In mathematical terms, this means that the measurability property is closed under the operation of taking complements in the given measurable space.

If `s` is a measurable subset of a measurable space `(α, m)`, then the complement `sᶜ` is also measurable.


theorem neg_neg_of_pos : ∀ {α : Type u} [inst : OrderedAddCommGroup α] {a : α}, 0 < a → -a < 0 := by sorry

This theorem, `neg_neg_of_pos`, states that for any type `α` that is an ordered additive commutative group, if a particular element `a` of this type is greater than zero, then the negative of `a` is less than zero. This essentially captures the intuitive idea in mathematics that if a number is positive, its negative is negative, in the context of an ordered additive commutative group.

For any ordered additive commutative group `α` and positive element `a` in `α`, `-a` is negative.


/-- Specialisation of `Nat.cast_nonneg'`, which seems to be easier for Lean to use.  -/
theorem Nat.cast_nonneg : ∀ {α : Type u_3} [inst : OrderedSemiring α] (n : ℕ), 0 ≤ ↑n := by sorry

This theorem, `Nat.cast_nonneg`, states that for any natural number `n` and any type `α` that is an ordered semiring, the casted value of `n` into `α` is always nonnegative, i.e., greater than or equal to zero. In other words, when we convert a natural number to any ordered semiring type, the result is guaranteed to be nonnegative. This is a specialisation of the more general `Nat.cast_nonneg'` theorem.

For any natural number `n` and ordered semiring type `α`, the conversion of `n` to `α` results in a value that is non-negative.


theorem Mathlib.Algebra.CharZero.Defs._auxLemma.2 :
    ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [inst_1 : CharZero R] {m n : ℕ}, (↑m = ↑n) = (m = n) := by sorry

This theorem, `Mathlib.Algebra.CharZero.Defs._auxLemma.2`, states that for any type `R` that is an additive monoid with one and has characteristic zero (these are the `inst` and `inst_1` conditions), and for any natural numbers `m` and `n`, the condition that `m` and `n` are equal when coerced (or, more technically, mapped) to `R` is equivalent to `m` and `n` being equal as natural numbers. In other words, in a characteristic zero additive monoid with one, the natural number coercion function preserves equality.

In a commutative additive monoid with one and characteristic zero, the natural number coercion function preserves equality.


theorem Filter.inter_mem : ∀ {α : Type u} {f : Filter α} {s t : Set α}, s ∈ f → t ∈ f → s ∩ t ∈ f := by sorry

The theorem `Filter.inter_mem` states that for any type `α`, any filter `f` on `α`, and any two sets `s` and `t` of type `α`, if both `s` and `t` are in `f`, then their intersection `s ∩ t` is also in `f`. In other words, the intersection of two sets belonging to a filter is also a member of the same filter.

If `f` is a filter on type `α`, and `s` and `t` are both elements of `f`, then their intersection `s ∩ t` is also an element of `f`.


theorem Filter.tendsto_id : ∀ {α : Type u} {x : Filter α}, Filter.Tendsto id x x := by sorry

The theorem `Filter.tendsto_id` states that for any given type `α` and for any filter `x` of this type, the identity function, denoted by `id`, has the property of `Filter.Tendsto`. In other words, the identity function preserves the filter structure, meaning that for every neighborhood of `x` (in the target space), the pre-image under the identity function is also a neighborhood of `x` (in the source space). This makes the identity function continuous in the filter-based topology.

For any filter `x` on type `α`, the identity function `id : α → α` is continuous from the filter-based topology perspective, that is, for every neighborhood `v` of a point `x` in the target space, the preimage `id⁻¹[v]` is a neighborhood of `x` in the source space.


theorem DifferentiableWithinAt.hasFDerivWithinAt :
    ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
      [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
      {x : E} {s : Set E}, DifferentiableWithinAt 𝕜 f s x → HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x :=
  by sorry

This theorem states that for a given non-trivially normed field '𝕜', normed addition-commutative groups 'E' and 'F', and a normed space over '𝕜' for 'E' and 'F', if a function 'f' from 'E' to 'F' is differentiable at a point 'x' within a set 's', then the derivative of 'f' at 'x' within 's' exists, and is equal to the value given by the function 'fderivWithin' at 'x'. In more mathematical terms, if `f` is differentiable at `x` within `s` (which is represented by `DifferentiableWithinAt 𝕜 f s x`), then `f` has a derivative at `x` within `s` (which is represented by `HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x`).

If a function between two normed spaces over a non-trivially normed field is differentiable at a point within a set, then it has a derivative with the same value as the derivative function evaluated at that point within that set.


theorem continuous_subtype_val : ∀ {X : Type u} [inst : TopologicalSpace X] {p : X → Prop}, Continuous Subtype.val := by
  sorry

The theorem `continuous_subtype_val` states that for any type `X` equipped with a topological space structure, and any predicate `p` on `X`, the function `Subtype.val` is continuous. In other words, when we look at the subset of `X` defined by the predicate `p` as a subspace, the function which takes an element of the subset and returns the corresponding element of `X` preserves the topology: the preimage of any open subset of `X` is an open subset of the subspace.

The function `Subtype.val` is continuous from the subspace topology on the subtype given by a predicate `p` on a topological space `X` to `X`.


theorem Multiset.induction_on :
    ∀ {α : Type u_1} {p : Multiset α → Prop} (s : Multiset α),
      p 0 → (∀ ⦃a : α⦄ {s : Multiset α}, p s → p (a ::ₘ s)) → p s :=
  by sorry

This theorem, `Multiset.induction_on`, provides an induction principle for multisets in Lean 4. Given any type `α`, a predicate `p` on multisets of `α`, and a multiset `s` of `α`, it states that if the predicate holds for the empty multiset (denoted by `0`), and if for any element `a` of type `α` and any multiset `s`, the predicate holding for `s` implies that the predicate holds for the multiset obtained by adding `a` to `s` (denoted by `a ::ₘ s`), then the predicate `p` holds for the multiset `s`. Essentially, this theorem allows us to prove a property about all multisets by showing it for the empty multiset and showing that if it holds for a particular multiset, then it also holds when an element is added to that multiset.

If a predicate `p` on multisets of type `α` holds for the empty multiset and is closed under adding elements, then it holds for any given multiset of type `α`.


theorem Fintype.card_fin : ∀ (n : ℕ), Fintype.card (Fin n) = n := by sorry

The theorem `Fintype.card_fin` states that for any natural number `n`, the cardinality (or number of elements) of the finite type `Fin n` is exactly `n`. Here, `Fin n` represents a type with `n` elements indexed by natural numbers from `0` to `n-1`. Essentially, this theorem establishes that the cardinality function correctly computes the number of elements in a finite type.

The theorem `Fintype.card_fin` asserts that the cardinality of the finite type `Fin n` equals `n`.


theorem LinearMap.map_smul :
    ∀ {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [inst : Semiring R] [inst_1 : AddCommMonoid M]
      [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (fₗ : M →ₗ[R] M₂) (c : R) (x : M),
      fₗ (c • x) = c • fₗ x :=
  by sorry

This theorem, `LinearMap.map_smul`, states that for any semiring `R`, any add-commutative monoids `M` and `M₂`, and any modules over `R` for `M` and `M₂`, if `fₗ` is a linear map from `M` to `M₂`, then scaling an element `x` of `M` by a scalar `c` from `R` before applying `fₗ` is the same as applying `fₗ` to `x` first and then scaling the result by `c`. This property is a key characteristic of linear maps, and it can be written in mathematical notation as `fₗ(c • x) = c • fₗ(x)`.

For any linear map `f` between modules over a semiring, `f(cx) = c(fx)`.


theorem star_star : ∀ {R : Type u} [inst : InvolutiveStar R] (r : R), star (star r) = r := by sorry

This theorem, `star_star`, states that for any given type `R` with an `InvolutiveStar` instance (which is a type class representing types with an involution operation denoted by `star`), applying the `star` operation twice on an element `r` of type `R` yields the original element `r`. In mathematical terms, this is saying that the `star` operation is its own inverse. In the context of algebraic structures, this would typically be seen in structures where an operation is self-inverse, such as taking a complex conjugate twice in complex numbers.

For any type `R` with an `InvolutiveStar` instance, the operation `star` is idempotent, i.e., `star (star r) = r` for all `r` in `R`.


theorem div_self : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a / a = 1 := by sorry

This theorem states that for any non-zero element `a` from a group with zero `G₀`, the division of `a` by itself equals one. This is a common property in mathematics, where any non-zero number divided by itself results in one. The theorem is applicable for any structure that is a Group with zero, meaning it has algebraic operations of addition and multiplication, and contains a zero element.

For any group with zero `G₀` and non-zero element `a`, `a * a⁻¹ = 1`. (Here, `a⁻¹` denotes the multiplicative inverse of `a`.)


Set.toFinite

/-- Constructor for `Set.Finite` using a `Finite` instance.  -/
theorem Set.toFinite : ∀ {α : Type u} (s : Set α) [inst : Finite ↑s], Set.Finite s := by sorry

The theorem `Set.toFinite` provides a method to create a finite set. Specifically, if `s` is a set of elements of some type `α`, and `s` has an instance of `Finite`, then `s` is a finite set according to the definition `Set.Finite`. In other words, if we can establish an equivalence between `s` and a finite subtype (say, there exists a natural number `n` and an equivalence `s ≃ Fin n`), then `s` is considered a finite set.

Given a set `s` of type `α` with a finite instance, `s` is equivalent to a finite subtype `Fin n` for some natural number `n`, making `s` a finite set.


Eq.subset

/-- Duplicate of `Eq.subset'`, which currently has elaboration problems.  -/
theorem Eq.subset : ∀ {α : Type u_1} {s t : Set α}, s = t → s ⊆ t := by sorry

This theorem states that for any type `α` and for any two sets `s` and `t` of this type, if `s` equals `t`, then `s` is a subset of `t`. Here, 'subset' means that every element of `s` is also an element of `t`. This is a duplicate of the theorem `Eq.subset'`, which is currently experiencing elaboration problems.

If `α` is a type and `s` and `t` are sets of `α` such that `s = t`, then `s` is a subset of `t`.


comm

theorem comm : ∀ {α : Type u} {r : α → α → Prop} [inst : IsSymm α r] {a b : α}, r a b ↔ r b a := by sorry

This theorem, referred to as 'comm', states that for any type `α` and any relation `r` on `α`, if the relation `r` is symmetric, then for any two elements `a` and `b` of type `α`, `r a b` will be equivalent to `r b a`. In simpler terms, it's saying that if `r` is a symmetric relation, then the order of the elements in the relation doesn't matter; if `a` is related to `b` then `b` is also related to `a`.

If `r` is a symmetric relation on type `α`, then `r a b ↔ r b a` for all `a` and `b` of type `α`.


theorem inv_mul_cancel : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a⁻¹ * a = 1 := by sorry

This theorem states that for any type `G₀` that has a `GroupWithZero` structure, if we take any element `a` from `G₀` that is not equal to zero, the product of the inverse of `a` and `a` itself is equal to one. Here, `GroupWithZero` is a structure in Lean which represents a group with an additional element, zero, which has its own special properties, and `a⁻¹` denotes the multiplicative inverse of `a`. This theorem is essentially a formal statement of the fundamental property of multiplicative inverses in a group.

For any group with zero `G₀` and non-zero element `a` in `G₀`, `a * a⁻¹ = 1`.


Continuous.add

theorem Continuous.add :
    ∀ {M : Type u_3} {X : Type u_5} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace M] [inst_2 : Add M]
      [inst_3 : ContinuousAdd M] {f g : X → M}, Continuous f → Continuous g → Continuous fun x => f x + g x :=
  by sorry

This theorem states that if `f` and `g` are both continuous functions from a topological space `X` to another topological space `M` and `M` supports addition operation in a continuous way (expressed by the `ContinuousAdd M` typeclass), then the function that maps each point `x` in `X` to the sum of `f(x)` and `g(x)` in `M` is also a continuous function.

If `f` and `g` are continuous functions from a topological space `X` to a topological vector space `M` with continuous addition, then the function `x mapsto f(x) + g(x)` is also continuous.