fact stringlengths 10 19.8k | statement stringlengths 1 9.7k | proof stringlengths 0 19.6k | type stringclasses 14
values | symbolic_name stringlengths 0 110 | library stringclasses 165
values | filename stringclasses 857
values | imports listlengths 0 19 | deps listlengths 0 64 | docstring stringlengths 10 3.64k ⌀ | line_start int64 13 10.9k | line_end int64 15 10.9k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
recSubsingleton
{p : Prop} [h : Decidable p]
{h₁ : p → Sort u}
{h₂ : ¬p → Sort u}
[h₃ : ∀ (h : p), Subsingleton (h₁ h)]
[h₄ : ∀ (h : ¬p), Subsingleton (h₂ h)]
: Subsingleton (h.casesOn h₂ h₁) :=
match h with
| isTrue h => h₃ h
| isFalse h => h₄ h | recSubsingleton
{p : Prop} [h : Decidable p]
{h₁ : p → Sort u}
{h₂ : ¬p → Sort u}
[h₃ : ∀ (h : p), Subsingleton (h₁ h)]
[h₄ : ∀ (h : ¬p), Subsingleton (h₂ h)]
: Subsingleton (h.casesOn h₂ h₁) | match h with
| isTrue h => h₃ h
| isFalse h => h₄ h | theorem | recSubsingleton | Init | src/Init/Core.lean | [] | [
"Decidable",
"Subsingleton"
] | null | 1,290 | 1,299 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Equivalence {α : Sort u} (r : α → α → Prop) : Prop where
/-- An equivalence relation is reflexive: `r x x` -/
refl : ∀ x, r x x
/-- An equivalence relation is symmetric: `r x y` implies `r y x` -/
symm : ∀ {x y}, r x y → r y x
/-- An equivalence relation is transitive: `r x y` and `r y z` implies `r x z` -/... | Equivalence {α : Sort u} (r : α → α → Prop) : Prop where
/-- An equivalence relation is reflexive: `r x x` -/
refl : ∀ x, r x x
/-- An equivalence relation is symmetric: `r x y` implies `r y x` -/
symm : ∀ {x y}, r x y → r y x
/-- An equivalence relation is transitive: `r x y` and `r y z` implies `r x z` -/... | structure | Equivalence | Init | src/Init/Core.lean | [] | [] | An equivalence relation `r : α → α → Prop` is a relation that is
* reflexive: `r x x`,
* symmetric: `r x y` implies `r y x`, and
* transitive: `r x y` and `r y z` implies `r x z`.
Equality is an equivalence relation, and equivalence relations share many of the properties of
equality. | 1,311 | 1,317 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
emptyRelation {α : Sort u} (_ _ : α) : Prop :=
False | emptyRelation {α : Sort u} (_ _ : α) : Prop | False | def | emptyRelation | Init | src/Init/Core.lean | [] | [
"False"
] | The empty relation is the relation on `α` which is always `False`. | 1,320 | 1,321 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Subrelation {α : Sort u} (q r : α → α → Prop) :=
∀ {x y}, q x y → r x y | Subrelation {α : Sort u} (q r : α → α → Prop) | ∀ {x y}, q x y → r x y | def | Subrelation | Init | src/Init/Core.lean | [] | [] | `Subrelation q r` means that `q ⊆ r` or `∀ x y, q x y → r x y`.
It is the analogue of the subset relation on relations. | 1,327 | 1,328 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
InvImage {α : Sort u} {β : Sort v} (r : β → β → Prop) (f : α → β) : α → α → Prop :=
fun a₁ a₂ => r (f a₁) (f a₂) | InvImage {α : Sort u} {β : Sort v} (r : β → β → Prop) (f : α → β) : α → α → Prop | fun a₁ a₂ => r (f a₁) (f a₂) | def | InvImage | Init | src/Init/Core.lean | [] | [] | The inverse image of `r : β → β → Prop` by a function `α → β` is the relation
`s : α → α → Prop` defined by `s a b = r (f a) (f b)`. | 1,334 | 1,335 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Relation.TransGen {α : Sort u} (r : α → α → Prop) : α → α → Prop
/-- If `r a b`, then `TransGen r a b`. This is the base case of the transitive closure. -/
| single {a b : α} : r a b → TransGen r a b
/-- If `TransGen r a b` and `r b c`, then `TransGen r a c`.
This is the inductive case of the transitive closure... | Relation.TransGen {α : Sort u} (r : α → α → Prop) : α → α → Prop
/-- If `r a b`, then `TransGen r a b`. This is the base case of the transitive closure. -/
| single {a b : α} : r a b → TransGen r a b
/-- If `TransGen r a b` and `r b c`, then `TransGen r a c`.
This is the inductive case of the transitive closure... | inductive | Relation.TransGen | Init | src/Init/Core.lean | [] | [] | The transitive closure `TransGen r` of a relation `r` is the smallest relation which is
transitive and contains `r`. `TransGen r a z` if and only if there exists a sequence
`a r b r ... r z` of length at least 1 connecting `a` to `z`. | 1,342 | 1,347 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Relation.TransGen.trans {α : Sort u} {r : α → α → Prop} {a b c} :
TransGen r a b → TransGen r b c → TransGen r a c := by
intro hab hbc
induction hbc with
| single h => exact TransGen.tail hab h
| tail _ h ih => exact TransGen.tail ih h | Relation.TransGen.trans {α : Sort u} {r : α → α → Prop} {a b c} :
TransGen r a b → TransGen r b c → TransGen r a c | by
intro hab hbc
induction hbc with
| single h => exact TransGen.tail hab h
| tail _ h ih => exact TransGen.tail ih h | theorem | Relation.TransGen.trans | Init | src/Init/Core.lean | [] | [] | The transitive closure is transitive. | 1,350 | 1,355 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
exists_of_subtype {α : Type u} {p : α → Prop} : { x // p x } → Exists (fun x => p x)
| ⟨a, h⟩ => ⟨a, h⟩ | exists_of_subtype {α : Type u} {p : α → Prop} : { x // p x } → Exists (fun x => p x)
| ⟨a, h⟩ => ⟨a, h⟩ | theorem | Subtype.exists_of_subtype | Init | src/Init/Core.lean | [] | [
"Exists"
] | null | 1,361 | 1,362 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
ext : ∀ {a1 a2 : {x // p x}}, val a1 = val a2 → a1 = a2
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl | ext : ∀ {a1 a2 : {x // p x}}, val a1 = val a2 → a1 = a2
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl | theorem | Subtype.ext | Init | src/Init/Core.lean | [] | [
"rfl"
] | null | 1,366 | 1,367 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
eq : ∀ {a1 a2 : {x // p x}}, val a1 = val a2 → a1 = a2
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl | eq : ∀ {a1 a2 : {x // p x}}, val a1 = val a2 → a1 = a2
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl | theorem | Subtype.eq | Init | src/Init/Core.lean | [] | [
"rfl"
] | null | 1,369 | 1,371 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
eta (a : {x // p x}) (h : p (val a)) : mk (val a) h = a := by
cases a
exact rfl | eta (a : {x // p x}) (h : p (val a)) : mk (val a) h = a | by
cases a
exact rfl | theorem | Subtype.eta | Init | src/Init/Core.lean | [] | [
"rfl"
] | null | 1,373 | 1,375 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
{α : Type u} {p : α → Prop} [BEq α] : BEq {x : α // p x} :=
⟨fun x y => x.1 == y.1⟩ | {α : Type u} {p : α → Prop} [BEq α] : BEq {x : α // p x} | ⟨fun x y => x.1 == y.1⟩ | instance | Init | src/Init/Core.lean | [] | [
"BEq"
] | null | 1,377 | 1,378 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
{α : Type u} {p : α → Prop} [BEq α] [ReflBEq α] : ReflBEq {x : α // p x} where
rfl {x} := BEq.refl x.1 | {α : Type u} {p : α → Prop} [BEq α] [ReflBEq α] : ReflBEq {x : α // p x} where
rfl {x} := BEq.refl x.1 | instance | Init | src/Init/Core.lean | [] | [
"BEq",
"BEq.refl",
"ReflBEq",
"rfl"
] | null | 1,380 | 1,381 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
{α : Type u} {p : α → Prop} [BEq α] [LawfulBEq α] : LawfulBEq {x : α // p x} where
eq_of_beq h := Subtype.ext (eq_of_beq h) | {α : Type u} {p : α → Prop} [BEq α] [LawfulBEq α] : LawfulBEq {x : α // p x} where
eq_of_beq h := Subtype.ext (eq_of_beq h) | instance | Init | src/Init/Core.lean | [] | [
"BEq",
"LawfulBEq",
"Subtype.ext"
] | null | 1,383 | 1,384 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
{α : Sort u} {p : α → Prop} [DecidableEq α] : DecidableEq {x : α // p x} :=
fun ⟨a, h₁⟩ ⟨b, h₂⟩ =>
if h : a = b then isTrue (by subst h; exact rfl)
else isFalse (fun h' => Subtype.noConfusion rfl .rfl (heq_of_eq h') (fun h' => absurd (eq_of_heq h') h)) | {α : Sort u} {p : α → Prop} [DecidableEq α] : DecidableEq {x : α // p x} | fun ⟨a, h₁⟩ ⟨b, h₂⟩ =>
if h : a = b then isTrue (by subst h; exact rfl)
else isFalse (fun h' => Subtype.noConfusion rfl .rfl (heq_of_eq h') (fun h' => absurd (eq_of_heq h') h)) | instance | Init | src/Init/Core.lean | [] | [
"DecidableEq",
"absurd",
"eq_of_heq",
"heq_of_eq",
"rfl"
] | null | 1,386 | 1,389 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Sum.inhabitedLeft [Inhabited α] : Inhabited (Sum α β) where
default := Sum.inl default | Sum.inhabitedLeft [Inhabited α] : Inhabited (Sum α β) where
default := Sum.inl default | def | Sum.inhabitedLeft | Init | src/Init/Core.lean | [] | [
"Inhabited",
"Sum"
] | null | 1,398 | 1,400 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Sum.inhabitedRight [Inhabited β] : Inhabited (Sum α β) where
default := Sum.inr default | Sum.inhabitedRight [Inhabited β] : Inhabited (Sum α β) where
default := Sum.inr default | def | Sum.inhabitedRight | Init | src/Init/Core.lean | [] | [
"Inhabited",
"Sum"
] | null | 1,402 | 1,404 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Sum.nonemptyLeft [h : Nonempty α] : Nonempty (Sum α β) :=
Nonempty.elim h (fun a => ⟨Sum.inl a⟩) | Sum.nonemptyLeft [h : Nonempty α] : Nonempty (Sum α β) | Nonempty.elim h (fun a => ⟨Sum.inl a⟩) | instance | Sum.nonemptyLeft | Init | src/Init/Core.lean | [] | [
"Nonempty",
"Nonempty.elim",
"Sum"
] | null | 1,406 | 1,407 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Sum.nonemptyRight [h : Nonempty β] : Nonempty (Sum α β) :=
Nonempty.elim h (fun b => ⟨Sum.inr b⟩)
deriving instance DecidableEq for Sum | Sum.nonemptyRight [h : Nonempty β] : Nonempty (Sum α β) | Nonempty.elim h (fun b => ⟨Sum.inr b⟩)
deriving instance DecidableEq for Sum | instance | Sum.nonemptyRight | Init | src/Init/Core.lean | [] | [
"DecidableEq",
"Nonempty",
"Nonempty.elim",
"Sum"
] | null | 1,409 | 1,412 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
[h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (α × β) :=
Nonempty.elim h1 fun x =>
Nonempty.elim h2 fun y =>
⟨(x, y)⟩ | [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (α × β) | Nonempty.elim h1 fun x =>
Nonempty.elim h2 fun y =>
⟨(x, y)⟩ | instance | Init | src/Init/Core.lean | [] | [
"Nonempty",
"Nonempty.elim"
] | null | 1,418 | 1,421 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
[h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (MProd α β) :=
Nonempty.elim h1 fun x =>
Nonempty.elim h2 fun y =>
⟨⟨x, y⟩⟩ | [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (MProd α β) | Nonempty.elim h1 fun x =>
Nonempty.elim h2 fun y =>
⟨⟨x, y⟩⟩ | instance | Init | src/Init/Core.lean | [] | [
"MProd",
"Nonempty",
"Nonempty.elim"
] | null | 1,423 | 1,426 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
[h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (PProd α β) :=
Nonempty.elim h1 fun x =>
Nonempty.elim h2 fun y =>
⟨⟨x, y⟩⟩ | [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (PProd α β) | Nonempty.elim h1 fun x =>
Nonempty.elim h2 fun y =>
⟨⟨x, y⟩⟩ | instance | Init | src/Init/Core.lean | [] | [
"Nonempty",
"Nonempty.elim",
"PProd"
] | null | 1,428 | 1,431 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
[Inhabited α] [Inhabited β] : Inhabited (α × β) where
default := (default, default) | [Inhabited α] [Inhabited β] : Inhabited (α × β) where
default := (default, default) | instance | Init | src/Init/Core.lean | [] | [
"Inhabited"
] | null | 1,433 | 1,434 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
[Inhabited α] [Inhabited β] : Inhabited (MProd α β) where
default := ⟨default, default⟩ | [Inhabited α] [Inhabited β] : Inhabited (MProd α β) where
default := ⟨default, default⟩ | instance | Init | src/Init/Core.lean | [] | [
"Inhabited",
"MProd"
] | null | 1,436 | 1,437 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
[Inhabited α] [Inhabited β] : Inhabited (PProd α β) where
default := ⟨default, default⟩ | [Inhabited α] [Inhabited β] : Inhabited (PProd α β) where
default := ⟨default, default⟩ | instance | Init | src/Init/Core.lean | [] | [
"Inhabited",
"PProd"
] | null | 1,439 | 1,440 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
[DecidableEq α] [DecidableEq β] : DecidableEq (α × β) :=
fun (a, b) (a', b') =>
match decEq a a' with
| isTrue e₁ =>
match decEq b b' with
| isTrue e₂ => isTrue (e₁ ▸ e₂ ▸ rfl)
| isFalse n₂ => isFalse fun h => Prod.noConfusion rfl rfl (heq_of_eq h) fun _ e₂' => absurd (eq_of_heq e₂') n₂
... | [DecidableEq α] [DecidableEq β] : DecidableEq (α × β) | fun (a, b) (a', b') =>
match decEq a a' with
| isTrue e₁ =>
match decEq b b' with
| isTrue e₂ => isTrue (e₁ ▸ e₂ ▸ rfl)
| isFalse n₂ => isFalse fun h => Prod.noConfusion rfl rfl (heq_of_eq h) fun _ e₂' => absurd (eq_of_heq e₂') n₂
| isFalse n₁ => isFalse fun h => Prod.noConfusion rfl rf... | instance | Init | src/Init/Core.lean | [] | [
"DecidableEq",
"absurd",
"decEq",
"eq_of_heq",
"heq_of_eq",
"rfl"
] | null | 1,442 | 1,449 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
[BEq α] [BEq β] : BEq (α × β) where
beq := fun (a₁, b₁) (a₂, b₂) => a₁ == a₂ && b₁ == b₂ | [BEq α] [BEq β] : BEq (α × β) where
beq := fun (a₁, b₁) (a₂, b₂) => a₁ == a₂ && b₁ == b₂ | instance | Init | src/Init/Core.lean | [] | [
"BEq"
] | null | 1,451 | 1,452 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
Prod.lexLt [LT α] [LT β] (s : α × β) (t : α × β) : Prop :=
s.1 < t.1 ∨ (s.1 = t.1 ∧ s.2 < t.2) | Prod.lexLt [LT α] [LT β] (s : α × β) (t : α × β) : Prop | s.1 < t.1 ∨ (s.1 = t.1 ∧ s.2 < t.2) | def | Prod.lexLt | Init | src/Init/Core.lean | [] | [
"LT"
] | Lexicographical order for products.
Two pairs are lexicographically ordered if their first elements are ordered or if their first
elements are equal and their second elements are ordered. | 1,460 | 1,461 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Prod.lexLtDec
[LT α] [LT β] [DecidableEq α]
[(a b : α) → Decidable (a < b)] [(a b : β) → Decidable (a < b)]
: (s t : α × β) → Decidable (Prod.lexLt s t) :=
fun _ _ => inferInstanceAs (Decidable (_ ∨ _)) | Prod.lexLtDec
[LT α] [LT β] [DecidableEq α]
[(a b : α) → Decidable (a < b)] [(a b : β) → Decidable (a < b)]
: (s t : α × β) → Decidable (Prod.lexLt s t) | fun _ _ => inferInstanceAs (Decidable (_ ∨ _)) | instance | Prod.lexLtDec | Init | src/Init/Core.lean | [] | [
"Decidable",
"DecidableEq",
"LT",
"Prod.lexLt"
] | null | 1,463 | 1,467 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Prod.lexLt_def [LT α] [LT β] (s t : α × β) : (Prod.lexLt s t) = (s.1 < t.1 ∨ (s.1 = t.1 ∧ s.2 < t.2)) :=
rfl | Prod.lexLt_def [LT α] [LT β] (s t : α × β) : (Prod.lexLt s t) = (s.1 < t.1 ∨ (s.1 = t.1 ∧ s.2 < t.2)) | rfl | theorem | Prod.lexLt_def | Init | src/Init/Core.lean | [] | [
"LT",
"Prod.lexLt",
"rfl"
] | null | 1,469 | 1,470 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Prod.eta (p : α × β) : (p.1, p.2) = p := rfl | Prod.eta (p : α × β) : (p.1, p.2) = p | rfl | theorem | Prod.eta | Init | src/Init/Core.lean | [] | [
"rfl"
] | null | 1,472 | 1,472 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Prod.map {α₁ : Type u₁} {α₂ : Type u₂} {β₁ : Type v₁} {β₂ : Type v₂}
(f : α₁ → α₂) (g : β₁ → β₂) : α₁ × β₁ → α₂ × β₂
| (a, b) => (f a, g b) | Prod.map {α₁ : Type u₁} {α₂ : Type u₂} {β₁ : Type v₁} {β₂ : Type v₂}
(f : α₁ → α₂) (g : β₁ → β₂) : α₁ × β₁ → α₂ × β₂
| (a, b) => (f a, g b) | def | Prod.map | Init | src/Init/Core.lean | [] | [] | Transforms a pair by applying functions to both elements.
Examples:
* `(1, 2).map (· + 1) (· * 3) = (2, 6)`
* `(1, 2).map toString (· * 3) = ("1", 6)` | 1,481 | 1,483 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Prod.map_apply (f : α → β) (g : γ → δ) (x) (y) :
Prod.map f g (x, y) = (f x, g y) := rfl | Prod.map_apply (f : α → β) (g : γ → δ) (x) (y) :
Prod.map f g (x, y) = (f x, g y) | rfl | theorem | Prod.map_apply | Init | src/Init/Core.lean | [] | [
"Prod.map",
"rfl"
] | null | 1,485 | 1,486 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Prod.map_fst (f : α → β) (g : γ → δ) (x) : (Prod.map f g x).1 = f x.1 := rfl | Prod.map_fst (f : α → β) (g : γ → δ) (x) : (Prod.map f g x).1 = f x.1 | rfl | theorem | Prod.map_fst | Init | src/Init/Core.lean | [] | [
"Prod.map",
"rfl"
] | null | 1,489 | 1,489 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Prod.map_snd (f : α → β) (g : γ → δ) (x) : (Prod.map f g x).2 = g x.2 := rfl | Prod.map_snd (f : α → β) (g : γ → δ) (x) : (Prod.map f g x).2 = g x.2 | rfl | theorem | Prod.map_snd | Init | src/Init/Core.lean | [] | [
"Prod.map",
"rfl"
] | null | 1,490 | 1,490 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
{α : Type u} {β : α → Type v} [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] :
DecidableEq (Sigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFals... | {α : Type u} {β : α → Type v} [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] :
DecidableEq (Sigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFals... | instance | Init | src/Init/Core.lean | [] | [
"DecidableEq",
"Sigma",
"eq_of_heq",
"heq_of_eq",
"rfl"
] | null | 1,494 | 1,504 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
{α : Sort u} {β : α → Sort v} [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] : DecidableEq (PSigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFalse n... | {α : Sort u} {β : α → Sort v} [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] : DecidableEq (PSigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFalse n... | instance | Init | src/Init/Core.lean | [] | [
"DecidableEq",
"PSigma",
"eq_of_heq",
"heq_of_eq",
"rfl"
] | null | 1,506 | 1,515 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
Exists.of_psigma_prop {α : Sort u} {p : α → Prop} : (PSigma (fun x => p x)) → Exists (fun x => p x)
| ⟨x, hx⟩ => ⟨x, hx⟩ | Exists.of_psigma_prop {α : Sort u} {p : α → Prop} : (PSigma (fun x => p x)) → Exists (fun x => p x)
| ⟨x, hx⟩ => ⟨x, hx⟩ | theorem | Exists.of_psigma_prop | Init | src/Init/Core.lean | [] | [
"Exists",
"PSigma"
] | null | 1,517 | 1,518 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
PSigma.eta {α : Sort u} {β : α → Sort v} {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂}
(h₁ : a₁ = a₂) (h₂ : Eq.ndrec b₁ h₁ = b₂) : PSigma.mk a₁ b₁ = PSigma.mk a₂ b₂ := by
subst h₁
subst h₂
exact rfl | PSigma.eta {α : Sort u} {β : α → Sort v} {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂}
(h₁ : a₁ = a₂) (h₂ : Eq.ndrec b₁ h₁ = b₂) : PSigma.mk a₁ b₁ = PSigma.mk a₂ b₂ | by
subst h₁
subst h₂
exact rfl | theorem | PSigma.eta | Init | src/Init/Core.lean | [] | [
"rfl"
] | null | 1,520 | 1,524 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
PUnit.ext (a b : PUnit) : a = b := by
cases a; cases b; exact rfl | PUnit.ext (a b : PUnit) : a = b | by
cases a; cases b; exact rfl | theorem | PUnit.ext | Init | src/Init/Core.lean | [] | [
"PUnit",
"rfl"
] | null | 1,528 | 1,529 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
PUnit.subsingleton (a b : PUnit) : a = b := by
cases a; cases b; exact rfl | PUnit.subsingleton (a b : PUnit) : a = b | by
cases a; cases b; exact rfl | theorem | PUnit.subsingleton | Init | src/Init/Core.lean | [] | [
"PUnit",
"rfl"
] | null | 1,531 | 1,533 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
PUnit.eq_punit (a : PUnit) : a = ⟨⟩ :=
PUnit.ext a ⟨⟩ | PUnit.eq_punit (a : PUnit) : a = ⟨⟩ | PUnit.ext a ⟨⟩ | theorem | PUnit.eq_punit | Init | src/Init/Core.lean | [] | [
"PUnit",
"PUnit.ext"
] | null | 1,535 | 1,536 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
: Subsingleton PUnit :=
Subsingleton.intro PUnit.ext | : Subsingleton PUnit | Subsingleton.intro PUnit.ext | instance | Init | src/Init/Core.lean | [] | [
"PUnit",
"PUnit.ext",
"Subsingleton"
] | null | 1,538 | 1,539 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
: Inhabited PUnit where
default := ⟨⟩ | : Inhabited PUnit where
default := ⟨⟩ | instance | Init | src/Init/Core.lean | [] | [
"Inhabited",
"PUnit"
] | null | 1,541 | 1,542 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | ||
: DecidableEq PUnit :=
fun a b => isTrue (PUnit.ext a b) | : DecidableEq PUnit | fun a b => isTrue (PUnit.ext a b) | instance | Init | src/Init/Core.lean | [] | [
"DecidableEq",
"PUnit",
"PUnit.ext"
] | null | 1,544 | 1,545 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Setoid (α : Sort u) where
/-- `x ≈ y` is the distinguished equivalence relation of a setoid. -/
r : α → α → Prop
/-- The relation `x ≈ y` is an equivalence relation. -/
iseqv : Equivalence r | Setoid (α : Sort u) where
/-- `x ≈ y` is the distinguished equivalence relation of a setoid. -/
r : α → α → Prop
/-- The relation `x ≈ y` is an equivalence relation. -/
iseqv : Equivalence r | class | Setoid | Init | src/Init/Core.lean | [] | [
"Equivalence"
] | A setoid is a type with a distinguished equivalence relation, denoted `≈`.
The `Quotient` type constructor requires a `Setoid` instance. | 1,554 | 1,558 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
{α : Sort u} [Setoid α] : HasEquiv α :=
⟨Setoid.r⟩ | {α : Sort u} [Setoid α] : HasEquiv α | ⟨Setoid.r⟩ | instance | Init | src/Init/Core.lean | [] | [
"HasEquiv",
"Setoid"
] | null | 1,560 | 1,561 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
refl (a : α) : a ≈ a :=
iseqv.refl a | refl (a : α) : a ≈ a | iseqv.refl a | theorem | Setoid.refl | Init | src/Init/Core.lean | [] | [] | A setoid's equivalence relation is reflexive. | 1,568 | 1,569 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
symm {a b : α} (hab : a ≈ b) : b ≈ a :=
iseqv.symm hab | symm {a b : α} (hab : a ≈ b) : b ≈ a | iseqv.symm hab | theorem | Setoid.symm | Init | src/Init/Core.lean | [] | [] | A setoid's equivalence relation is symmetric. | 1,572 | 1,573 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
trans {a b c : α} (hab : a ≈ b) (hbc : b ≈ c) : a ≈ c :=
iseqv.trans hab hbc | trans {a b c : α} (hab : a ≈ b) (hbc : b ≈ c) : a ≈ c | iseqv.trans hab hbc | theorem | Setoid.trans | Init | src/Init/Core.lean | [] | [] | A setoid's equivalence relation is transitive. | 1,576 | 1,577 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
propext {a b : Prop} : (a ↔ b) → a = b | propext {a b : Prop} : (a ↔ b) → a = b | axiom | propext | Init | src/Init/Core.lean | [] | [] | The [axiom](lean-manual://section/axioms) of **propositional extensionality**. It asserts that if
propositions `a` and `b` are logically equivalent (that is, if `a` can be proved from `b` and vice
versa), then `a` and `b` are *equal*, meaning `a` can be replaced with `b` in all contexts.
The standard logical connectiv... | 1,593 | 1,593 | false | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Eq.propIntro {a b : Prop} (h₁ : a → b) (h₂ : b → a) : a = b :=
propext <| Iff.intro h₁ h₂ | Eq.propIntro {a b : Prop} (h₁ : a → b) (h₂ : b → a) : a = b | propext <| Iff.intro h₁ h₂ | theorem | Eq.propIntro | Init | src/Init/Core.lean | [] | [
"propext"
] | null | 1,595 | 1,596 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
{p q : Prop} [d : Decidable (p ↔ q)] : Decidable (p = q) :=
match d with
| isTrue h => isTrue (propext h)
| isFalse h => isFalse fun heq => h (heq ▸ Iff.rfl) | {p q : Prop} [d : Decidable (p ↔ q)] : Decidable (p = q) | match d with
| isTrue h => isTrue (propext h)
| isFalse h => isFalse fun heq => h (heq ▸ Iff.rfl) | instance | Init | src/Init/Core.lean | [] | [
"Decidable",
"Iff.rfl",
"propext"
] | null | 1,599 | 1,602 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 | |
Lean.injEq_helper {P Q R : Prop} :
(P → Q → R) → (P ∧ Q → R) := by intro h ⟨h₁,h₂⟩; exact h h₁ h₂
gen_injective_theorems% Array
gen_injective_theorems% BitVec
gen_injective_theorems% ByteArray
gen_injective_theorems% Char
gen_injective_theorems% DoResultBC
gen_injective_theorems% DoResultPR
gen_injective_theorems% D... | Lean.injEq_helper {P Q R : Prop} :
(P → Q → R) → (P ∧ Q → R) | by intro h ⟨h₁,h₂⟩; exact h h₁ h₂
gen_injective_theorems% Array
gen_injective_theorems% BitVec
gen_injective_theorems% ByteArray
gen_injective_theorems% Char
gen_injective_theorems% DoResultBC
gen_injective_theorems% DoResultPR
gen_injective_theorems% DoResultPRBC
gen_injective_theorems% DoResultSBC
gen_injective_theo... | theorem | Lean.injEq_helper | Init | src/Init/Core.lean | [] | [
"Array",
"BitVec",
"ByteArray",
"Char",
"DoResultBC",
"DoResultPR",
"DoResultPRBC",
"DoResultSBC",
"EStateM.Result",
"Except",
"Fin",
"ForInStep",
"Lean.Name",
"Lean.Syntax",
"List",
"MProd",
"NonScalar",
"Option",
"PLift",
"PNonScalar",
"PProd",
"PSigma",
"PSum",
"Prod... | Helper theorem for proving injectivity theorems | 1,605 | 1,646 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Nat.succ.inj {m n : Nat} : m.succ = n.succ → m = n :=
fun x => Nat.noConfusion x id | Nat.succ.inj {m n : Nat} : m.succ = n.succ → m = n | fun x => Nat.noConfusion x id | theorem | Nat.succ.inj | Init | src/Init/Core.lean | [] | [
"Nat",
"id"
] | null | 1,648 | 1,649 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Nat.succ.injEq (u v : Nat) : (u.succ = v.succ) = (u = v) :=
Eq.propIntro Nat.succ.inj (congrArg Nat.succ) | Nat.succ.injEq (u v : Nat) : (u.succ = v.succ) = (u = v) | Eq.propIntro Nat.succ.inj (congrArg Nat.succ) | theorem | Nat.succ.injEq | Init | src/Init/Core.lean | [] | [
"Eq.propIntro",
"Nat",
"Nat.succ.inj",
"congrArg"
] | null | 1,651 | 1,652 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
beq_iff_eq [BEq α] [LawfulBEq α] {a b : α} : a == b ↔ a = b :=
⟨eq_of_beq, beq_of_eq⟩ | beq_iff_eq [BEq α] [LawfulBEq α] {a b : α} : a == b ↔ a = b | ⟨eq_of_beq, beq_of_eq⟩ | theorem | beq_iff_eq | Init | src/Init/Core.lean | [] | [
"BEq",
"LawfulBEq"
] | null | 1,654 | 1,655 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Not.elim {α : Sort _} (H1 : ¬a) (H2 : a) : α := absurd H2 H1 | Not.elim {α : Sort _} (H1 : ¬a) (H2 : a) : α | absurd H2 H1 | def | Not.elim | Init | src/Init/Core.lean | [] | [
"absurd"
] | *Ex falso* for negation: from `¬a` and `a` anything follows. This is the same as `absurd` with
the arguments flipped, but it is in the `Not` namespace so that projection notation can be used. | 1,661 | 1,661 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
And.elim (f : a → b → α) (h : a ∧ b) : α := f h.left h.right | And.elim (f : a → b → α) (h : a ∧ b) : α | f h.left h.right | abbrev | And.elim | Init | src/Init/Core.lean | [] | [] | Non-dependent eliminator for `And`. | 1,664 | 1,664 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Iff.elim (f : (a → b) → (b → a) → α) (h : a ↔ b) : α := f h.mp h.mpr | Iff.elim (f : (a → b) → (b → a) → α) (h : a ↔ b) : α | f h.mp h.mpr | def | Iff.elim | Init | src/Init/Core.lean | [] | [] | Non-dependent eliminator for `Iff`. | 1,667 | 1,667 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Iff.subst {a b : Prop} {p : Prop → Prop} (h₁ : a ↔ b) (h₂ : p a) : p b :=
Eq.subst (propext h₁) h₂ | Iff.subst {a b : Prop} {p : Prop → Prop} (h₁ : a ↔ b) (h₂ : p a) : p b | Eq.subst (propext h₁) h₂ | theorem | Iff.subst | Init | src/Init/Core.lean | [] | [
"Eq.subst",
"propext"
] | Iff can now be used to do substitutions in a calculation | 1,670 | 1,671 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Not.intro {a : Prop} (h : a → False) : ¬a := h | Not.intro {a : Prop} (h : a → False) : ¬a | h | theorem | Not.intro | Init | src/Init/Core.lean | [] | [
"False"
] | null | 1,673 | 1,673 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Not.imp {a b : Prop} (H2 : ¬b) (H1 : a → b) : ¬a := mt H1 H2 | Not.imp {a b : Prop} (H2 : ¬b) (H1 : a → b) : ¬a | mt H1 H2 | theorem | Not.imp | Init | src/Init/Core.lean | [] | [
"mt"
] | null | 1,675 | 1,675 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_congr (h : a ↔ b) : ¬a ↔ ¬b := ⟨mt h.2, mt h.1⟩ | not_congr (h : a ↔ b) : ¬a ↔ ¬b | ⟨mt h.2, mt h.1⟩ | theorem | not_congr | Init | src/Init/Core.lean | [] | [
"mt"
] | null | 1,677 | 1,677 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_not_not : ¬¬¬a ↔ ¬a := ⟨mt not_not_intro, not_not_intro⟩ | not_not_not : ¬¬¬a ↔ ¬a | ⟨mt not_not_intro, not_not_intro⟩ | theorem | not_not_not | Init | src/Init/Core.lean | [] | [
"not_not_intro"
] | null | 1,679 | 1,679 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_of_true (ha : a) (hb : b) : a ↔ b := Iff.intro (fun _ => hb) (fun _ => ha) | iff_of_true (ha : a) (hb : b) : a ↔ b | Iff.intro (fun _ => hb) (fun _ => ha) | theorem | iff_of_true | Init | src/Init/Core.lean | [] | [] | null | 1,681 | 1,681 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_of_false (ha : ¬a) (hb : ¬b) : a ↔ b := Iff.intro ha.elim hb.elim | iff_of_false (ha : ¬a) (hb : ¬b) : a ↔ b | Iff.intro ha.elim hb.elim | theorem | iff_of_false | Init | src/Init/Core.lean | [] | [] | null | 1,682 | 1,682 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_true_left (ha : a) : (a ↔ b) ↔ b := Iff.intro (·.mp ha) (iff_of_true ha) | iff_true_left (ha : a) : (a ↔ b) ↔ b | Iff.intro (·.mp ha) (iff_of_true ha) | theorem | iff_true_left | Init | src/Init/Core.lean | [] | [
"iff_of_true"
] | null | 1,684 | 1,684 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_true_right (ha : a) : (b ↔ a) ↔ b := Iff.comm.trans (iff_true_left ha) | iff_true_right (ha : a) : (b ↔ a) ↔ b | Iff.comm.trans (iff_true_left ha) | theorem | iff_true_right | Init | src/Init/Core.lean | [] | [
"iff_true_left"
] | null | 1,685 | 1,685 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_false_left (ha : ¬a) : (a ↔ b) ↔ ¬b := Iff.intro (mt ·.mpr ha) (iff_of_false ha) | iff_false_left (ha : ¬a) : (a ↔ b) ↔ ¬b | Iff.intro (mt ·.mpr ha) (iff_of_false ha) | theorem | iff_false_left | Init | src/Init/Core.lean | [] | [
"iff_of_false",
"mt"
] | null | 1,687 | 1,687 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_false_right (ha : ¬a) : (b ↔ a) ↔ ¬b := Iff.comm.trans (iff_false_left ha) | iff_false_right (ha : ¬a) : (b ↔ a) ↔ ¬b | Iff.comm.trans (iff_false_left ha) | theorem | iff_false_right | Init | src/Init/Core.lean | [] | [
"iff_false_left"
] | null | 1,688 | 1,688 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
of_iff_true (h : a ↔ True) : a := h.mpr trivial | of_iff_true (h : a ↔ True) : a | h.mpr trivial | theorem | of_iff_true | Init | src/Init/Core.lean | [] | [
"True",
"trivial"
] | null | 1,690 | 1,690 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_true_intro (h : a) : a ↔ True := iff_of_true h trivial | iff_true_intro (h : a) : a ↔ True | iff_of_true h trivial | theorem | iff_true_intro | Init | src/Init/Core.lean | [] | [
"True",
"iff_of_true",
"trivial"
] | null | 1,691 | 1,691 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
eq_iff_true_of_subsingleton [Subsingleton α] (x y : α) : x = y ↔ True :=
iff_true_intro (Subsingleton.elim ..) | eq_iff_true_of_subsingleton [Subsingleton α] (x y : α) : x = y ↔ True | iff_true_intro (Subsingleton.elim ..) | theorem | eq_iff_true_of_subsingleton | Init | src/Init/Core.lean | [] | [
"Subsingleton",
"Subsingleton.elim",
"True",
"iff_true_intro"
] | null | 1,693 | 1,694 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_of_iff_false : (p ↔ False) → ¬p := Iff.mp | not_of_iff_false : (p ↔ False) → ¬p | Iff.mp | theorem | not_of_iff_false | Init | src/Init/Core.lean | [] | [
"False"
] | null | 1,696 | 1,696 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_false_intro (h : ¬a) : a ↔ False := iff_of_false h id | iff_false_intro (h : ¬a) : a ↔ False | iff_of_false h id | theorem | iff_false_intro | Init | src/Init/Core.lean | [] | [
"False",
"id",
"iff_of_false"
] | null | 1,697 | 1,697 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_iff_false_intro (h : a) : ¬a ↔ False := iff_false_intro (not_not_intro h) | not_iff_false_intro (h : a) : ¬a ↔ False | iff_false_intro (not_not_intro h) | theorem | not_iff_false_intro | Init | src/Init/Core.lean | [] | [
"False",
"iff_false_intro",
"not_not_intro"
] | null | 1,699 | 1,699 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_true : (¬True) ↔ False := iff_false_intro (not_not_intro trivial) | not_true : (¬True) ↔ False | iff_false_intro (not_not_intro trivial) | theorem | not_true | Init | src/Init/Core.lean | [] | [
"False",
"True",
"iff_false_intro",
"not_not_intro",
"trivial"
] | null | 1,700 | 1,700 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_false_iff : (¬False) ↔ True := iff_true_intro not_false | not_false_iff : (¬False) ↔ True | iff_true_intro not_false | theorem | not_false_iff | Init | src/Init/Core.lean | [] | [
"False",
"True",
"iff_true_intro",
"not_false"
] | null | 1,702 | 1,702 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
Eq.to_iff : a = b → (a ↔ b) := Iff.of_eq | Eq.to_iff : a = b → (a ↔ b) | Iff.of_eq | theorem | Eq.to_iff | Init | src/Init/Core.lean | [] | [
"Iff.of_eq"
] | null | 1,704 | 1,704 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_of_eq : a = b → (a ↔ b) := Iff.of_eq | iff_of_eq : a = b → (a ↔ b) | Iff.of_eq | theorem | iff_of_eq | Init | src/Init/Core.lean | [] | [
"Iff.of_eq"
] | null | 1,705 | 1,705 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
neq_of_not_iff : ¬(a ↔ b) → a ≠ b := mt Iff.of_eq | neq_of_not_iff : ¬(a ↔ b) → a ≠ b | mt Iff.of_eq | theorem | neq_of_not_iff | Init | src/Init/Core.lean | [] | [
"Iff.of_eq",
"mt"
] | null | 1,706 | 1,706 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_iff_eq : (a ↔ b) ↔ a = b := Iff.intro propext Iff.of_eq | iff_iff_eq : (a ↔ b) ↔ a = b | Iff.intro propext Iff.of_eq | theorem | iff_iff_eq | Init | src/Init/Core.lean | [] | [
"Iff.of_eq",
"propext"
] | null | 1,708 | 1,708 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
eq_iff_iff : (a = b) ↔ (a ↔ b) := iff_iff_eq.symm | eq_iff_iff : (a = b) ↔ (a ↔ b) | iff_iff_eq.symm | theorem | eq_iff_iff | Init | src/Init/Core.lean | [] | [] | null | 1,709 | 1,709 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
eq_self_iff_true (a : α) : a = a ↔ True := iff_true_intro rfl | eq_self_iff_true (a : α) : a = a ↔ True | iff_true_intro rfl | theorem | eq_self_iff_true | Init | src/Init/Core.lean | [] | [
"True",
"iff_true_intro",
"rfl"
] | null | 1,711 | 1,711 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
ne_self_iff_false (a : α) : a ≠ a ↔ False := not_iff_false_intro rfl | ne_self_iff_false (a : α) : a ≠ a ↔ False | not_iff_false_intro rfl | theorem | ne_self_iff_false | Init | src/Init/Core.lean | [] | [
"False",
"not_iff_false_intro",
"rfl"
] | null | 1,712 | 1,712 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
false_of_true_iff_false (h : True ↔ False) : False := h.mp trivial | false_of_true_iff_false (h : True ↔ False) : False | h.mp trivial | theorem | false_of_true_iff_false | Init | src/Init/Core.lean | [] | [
"False",
"True",
"trivial"
] | null | 1,714 | 1,714 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
false_of_true_eq_false (h : True = False) : False := false_of_true_iff_false (Iff.of_eq h) | false_of_true_eq_false (h : True = False) : False | false_of_true_iff_false (Iff.of_eq h) | theorem | false_of_true_eq_false | Init | src/Init/Core.lean | [] | [
"False",
"Iff.of_eq",
"True",
"false_of_true_iff_false"
] | null | 1,715 | 1,715 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
true_eq_false_of_false : False → (True = False) := False.elim | true_eq_false_of_false : False → (True = False) | False.elim | theorem | true_eq_false_of_false | Init | src/Init/Core.lean | [] | [
"False",
"False.elim",
"True"
] | null | 1,717 | 1,717 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_def : (a ↔ b) ↔ (a → b) ∧ (b → a) := iff_iff_implies_and_implies | iff_def : (a ↔ b) ↔ (a → b) ∧ (b → a) | iff_iff_implies_and_implies | theorem | iff_def | Init | src/Init/Core.lean | [] | [
"iff_iff_implies_and_implies"
] | null | 1,719 | 1,719 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_def' : (a ↔ b) ↔ (b → a) ∧ (a → b) := Iff.trans iff_def And.comm | iff_def' : (a ↔ b) ↔ (b → a) ∧ (a → b) | Iff.trans iff_def And.comm | theorem | iff_def' | Init | src/Init/Core.lean | [] | [
"And.comm",
"Iff.trans",
"iff_def"
] | null | 1,720 | 1,720 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
true_iff_false : (True ↔ False) ↔ False := iff_false_intro (·.mp True.intro) | true_iff_false : (True ↔ False) ↔ False | iff_false_intro (·.mp True.intro) | theorem | true_iff_false | Init | src/Init/Core.lean | [] | [
"False",
"True",
"iff_false_intro"
] | null | 1,722 | 1,722 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
false_iff_true : (False ↔ True) ↔ False := iff_false_intro (·.mpr True.intro) | false_iff_true : (False ↔ True) ↔ False | iff_false_intro (·.mpr True.intro) | theorem | false_iff_true | Init | src/Init/Core.lean | [] | [
"False",
"True",
"iff_false_intro"
] | null | 1,723 | 1,723 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
iff_not_self : ¬(a ↔ ¬a) | H => let f h := H.1 h h; f (H.2 f) | iff_not_self : ¬(a ↔ ¬a) | H => let f h | H.1 h h; f (H.2 f) | theorem | iff_not_self | Init | src/Init/Core.lean | [] | [] | null | 1,725 | 1,725 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
heq_self_iff_true (a : α) : a ≍ a ↔ True := iff_true_intro HEq.rfl | heq_self_iff_true (a : α) : a ≍ a ↔ True | iff_true_intro HEq.rfl | theorem | heq_self_iff_true | Init | src/Init/Core.lean | [] | [
"HEq.rfl",
"True",
"iff_true_intro"
] | null | 1,726 | 1,726 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_not_of_not_imp : ¬(a → b) → ¬¬a := mt Not.elim | not_not_of_not_imp : ¬(a → b) → ¬¬a | mt Not.elim | theorem | not_not_of_not_imp | Init | src/Init/Core.lean | [] | [
"Not.elim",
"mt"
] | null | 1,730 | 1,730 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
not_of_not_imp {a : Prop} : ¬(a → b) → ¬b := mt fun h _ => h | not_of_not_imp {a : Prop} : ¬(a → b) → ¬b | mt fun h _ => h | theorem | not_of_not_imp | Init | src/Init/Core.lean | [] | [
"mt"
] | null | 1,732 | 1,732 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
imp_not_self : (a → ¬a) ↔ ¬a := Iff.intro (fun h ha => h ha ha) (fun h _ => h) | imp_not_self : (a → ¬a) ↔ ¬a | Iff.intro (fun h ha => h ha ha) (fun h _ => h) | theorem | imp_not_self | Init | src/Init/Core.lean | [] | [] | null | 1,734 | 1,734 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
imp_intro {α β : Prop} (h : α) : β → α := fun _ => h | imp_intro {α β : Prop} (h : α) : β → α | fun _ => h | theorem | imp_intro | Init | src/Init/Core.lean | [] | [] | null | 1,736 | 1,736 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
imp_imp_imp {a b c d : Prop} (h₀ : c → a) (h₁ : b → d) : (a → b) → (c → d) := (h₁ ∘ · ∘ h₀) | imp_imp_imp {a b c d : Prop} (h₀ : c → a) (h₁ : b → d) : (a → b) → (c → d) | (h₁ ∘ · ∘ h₀) | theorem | imp_imp_imp | Init | src/Init/Core.lean | [] | [] | null | 1,738 | 1,738 | true | https://github.com/leanprover/lean4 | d265d1ca745e7741a7e7f7366c22ce9c9dda57b6 |
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