nl_statement stringlengths 6 6k | nl_proof stringlengths 55 8.29k ⌀ | lean_statement stringlengths 3 29.8k | lean_proof stringlengths 2 189k | labels listlengths 1 3 | source stringclasses 245
values |
|---|---|---|---|---|---|
For every signed 32-bit integer $x$, its integer value equals the signed integer value of its associated 32-bit vector. | null | @[grind hom] theorem Lean.Grind.Int32.toInt_eq_toBitVec_toInt (x : Int32) : x.toInt = x.toBitVec.toInt | := rfl | [
"theorem",
"core library"
] | lean4 |
For every signed 16-bit integer $x$, its integer value equals the signed integer value of its associated 16-bit vector. | null | @[grind hom] theorem Lean.Grind.Int16.toInt_eq_toBitVec_toInt (x : Int16) : x.toInt = x.toBitVec.toInt | := rfl | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, the bit vector associated with the unsigned 32-bit numeral $a$ equals the 32-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt32.toBitVec_OfNat_ofNat (a : Nat) :
(OfNat.ofNat a : UInt32).toBitVec = OfNat.ofNat a | :=
(UInt32.toBitVec_ofNat a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, converting $a$ to an unsigned 32-bit integer using `ofNat` and then to a bit vector gives the 32-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt32.toBitVec_ofNat (a : Nat) :
(UInt32.ofNat a).toBitVec = OfNat.ofNat a | :=
(UInt32.toBitVec_ofNat' a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every unsigned 32-bit integer $x$, its natural-number value equals the natural-number value of its associated bit vector. | null | @[grind hom] theorem Lean.Grind.UInt32.toNat_eq_toBitVec_toNat (x : UInt32) : x.toNat = x.toBitVec.toNat | := rfl | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, the bit vector associated with the unsigned 64-bit numeral $a$ equals the 64-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt64.toBitVec_OfNat_ofNat (a : Nat) :
(OfNat.ofNat a : UInt64).toBitVec = OfNat.ofNat a | :=
(UInt64.toBitVec_ofNat a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, converting $a$ to an unsigned 64-bit integer using `ofNat` and then to a bit vector gives the 64-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt64.toBitVec_ofNat (a : Nat) :
(UInt64.ofNat a).toBitVec = OfNat.ofNat a | :=
(UInt64.toBitVec_ofNat' a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every unsigned 64-bit integer $x$, its natural-number value equals the natural-number value of its associated bit vector. | null | @[grind hom] theorem Lean.Grind.UInt64.toNat_eq_toBitVec_toNat (x : UInt64) : x.toNat = x.toBitVec.toNat | := rfl | [
"theorem",
"core library"
] | lean4 |
For every signed machine-sized integer $x$ of type `ISize`, its integer value equals the signed integer value of its associated bit vector. | null | @[grind hom] theorem Lean.Grind.ISize.toInt_eq_toBitVec_toInt (x : ISize) : x.toInt = x.toBitVec.toInt | := rfl | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, the bit vector associated with the unsigned 16-bit numeral $a$ equals the 16-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt16.toBitVec_OfNat_ofNat (a : Nat) :
(OfNat.ofNat a : UInt16).toBitVec = OfNat.ofNat a | :=
(UInt16.toBitVec_ofNat a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, converting $a$ to an unsigned 16-bit integer using `ofNat` and then to a bit vector gives the 16-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt16.toBitVec_ofNat (a : Nat) :
(UInt16.ofNat a).toBitVec = OfNat.ofNat a | :=
(UInt16.toBitVec_ofNat' a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every unsigned 16-bit integer $x$, its natural-number value equals the natural-number value of its associated bit vector. | null | @[grind hom] theorem Lean.Grind.UInt16.toNat_eq_toBitVec_toNat (x : UInt16) : x.toNat = x.toBitVec.toNat | := rfl | [
"theorem",
"core library"
] | lean4 |
For every signed 64-bit integer $x$, its integer value equals the signed integer value of its associated 64-bit vector. | null | @[grind hom] theorem Lean.Grind.Int64.toInt_eq_toBitVec_toInt (x : Int64) : x.toInt = x.toBitVec.toInt | := rfl | [
"theorem",
"core library"
] | lean4 |
For all integers $m,n,k$, $((m\bmod n)k)\bmod n=(mk)\bmod n$, where $\bmod$ is the Euclidean remainder, with $a\bmod 0=a$. | null | @[grind hom] theorem Lean.Grind.Int.emod_mul_emod (m n k : Int) : m % n * k % n = m * k % n | := by
rw [Int.mul_emod, Int.emod_emod, ← Int.mul_emod] | [
"theorem",
"core library"
] | lean4 |
For all integers $m,n,k$, $(m(n\bmod k))\bmod k=(mn)\bmod k$, where $\bmod$ is the Euclidean remainder, with $a\bmod 0=a$. | null | @[grind hom] theorem Lean.Grind.Int.mul_emod_emod (m n k : Int) : m * (n % k) % k = m * n % k | := by
rw [Int.mul_emod, Int.emod_emod, ← Int.mul_emod] | [
"theorem",
"core library"
] | lean4 |
For all natural numbers $x,c,m,n$ satisfying $c=2^n-1$ and $m=2^n$, the bitwise AND satisfies $x\mathbin{\&}c=x\bmod m$. | null | theorem Lean.Grind.Nat.and_eq_mod (x c m n : Nat) (h₁ : c = 2^n - 1) (h₂ : m = 2^n) :
x &&& c = x % m | := by
subst h₁ h₂; exact Nat.and_two_pow_sub_one_eq_mod x n | [
"theorem",
"core library"
] | lean4 |
For all natural numbers $x,c,m,n$ satisfying $c=2^n-1$ and $m=2^n$, the bitwise AND satisfies $c\mathbin{\&}x=x\bmod m$. | null | theorem Lean.Grind.Nat.ones_and_eq_mod (x c m n : Nat) (h₁ : c = 2^n - 1) (h₂ : m = 2^n) :
c &&& x = x % m | := by
rw [Nat.and_comm]; exact and_eq_mod x c m n h₁ h₂ | [
"theorem",
"core library"
] | lean4 |
For all natural numbers $x,c,p,q,k,n$ satisfying $c=(2^n-1)2^k$, $p=2^k$, and $q=2^n$, the bitwise AND satisfies $x\mathbin{\&}c=(\lfloor x/p\rfloor\bmod q)p$. | null | theorem Lean.Grind.Nat.and_eq_div_mod_mul (x c p q k n : Nat)
(h₁ : c = (2^n - 1) * 2^k) (h₂ : p = 2^k) (h₃ : q = 2^n) :
x &&& c = x / p % q * p | := by
subst h₁ h₂ h₃
apply Nat.eq_of_testBit_eq
intro i
simp only [Nat.testBit_and, Nat.testBit_mul_two_pow, Nat.testBit_two_pow_sub_one,
Nat.testBit_mod_two_pow, Nat.testBit_div_two_pow]
cases Nat.lt_or_ge i k with
| inl h => simp [Nat.not_le_of_lt h]
| inr h => simp [h, Nat.sub_add_cancel h, Bool.an... | [
"theorem",
"core library"
] | lean4 |
For all natural numbers $x,c,p,q,k,n$ satisfying $c=(2^n-1)2^k$, $p=2^k$, and $q=2^n$, the bitwise AND satisfies $c\mathbin{\&}x=(\lfloor x/p\rfloor\bmod q)p$. | null | theorem Lean.Grind.Nat.ones_zeros_and_eq_div_mod_mul (x c p q k n : Nat)
(h₁ : c = (2^n - 1) * 2^k) (h₂ : p = 2^k) (h₃ : q = 2^n) :
c &&& x = x / p % q * p | := by
rw [Nat.and_comm]; exact and_eq_div_mod_mul x c p q k n h₁ h₂ h₃ | [
"theorem",
"core library"
] | lean4 |
For all natural numbers $x,c,c',m$ satisfying $c+c'=m$, $(cx)\bmod m=(m-((c'x)\bmod m))\bmod m$, where subtraction is truncated at zero and $a\bmod 0=a$. | null | theorem Lean.Grind.Nat.mul_mod_eq_sub_mul_mod (x c c' m : Nat) (h : c + c' = m) :
c * x % m = (m - c' * x % m) % m | := by
subst h
have h₁ : (c * x + c' * x) % (c + c') = 0 := by rw [← Nat.add_mul, Nat.mul_mod_right]
rw [Nat.add_mod] at h₁
cases Nat.eq_zero_or_pos (c + c') with
| inl h₀ =>
have ⟨hc, hc'⟩ := Nat.add_eq_zero_iff.mp h₀
subst hc hc'; simp
| inr hpos =>
have ha := Nat.mod_lt (c * x) hpos
have h... | [
"theorem",
"core library"
] | lean4 |
For all natural numbers $x,c,c',m$ satisfying $c+c'=m$, $(xc)\bmod m=(m-((c'x)\bmod m))\bmod m$, where subtraction is truncated at zero and $a\bmod 0=a$. | null | theorem Lean.Grind.Nat.mul_mod_eq_sub_mul_mod' (x c c' m : Nat) (h : c + c' = m) :
x * c % m = (m - c' * x % m) % m | := by
rw [Nat.mul_comm]; exact mul_mod_eq_sub_mul_mod x c c' m h | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, the bit vector associated with the unsigned machine-sized numeral $a$ of type `USize` equals the bit-vector numeral $a$ of the machine word width. | null | @[grind hom] theorem Lean.Grind.USize.toBitVec_OfNat_ofNat (a : Nat) :
(OfNat.ofNat a : USize).toBitVec = OfNat.ofNat a | :=
(USize.toBitVec_ofNat a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, converting $a$ to `USize` using `ofNat` and then to a bit vector gives the bit-vector numeral $a$ of the machine word width. | null | @[grind hom] theorem Lean.Grind.USize.toBitVec_ofNat (a : Nat) :
(USize.ofNat a).toBitVec = OfNat.ofNat a | :=
(USize.toBitVec_ofNat' a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every unsigned machine-sized integer $x$ of type `USize`, its natural-number value equals the natural-number value of its associated bit vector. | null | @[grind hom] theorem Lean.Grind.USize.toNat_eq_toBitVec_toNat (x : USize) : x.toNat = x.toBitVec.toNat | := rfl | [
"theorem",
"core library"
] | lean4 |
For every natural number $n$ and all $a,b\in\operatorname{Fin}(n)$, where $\operatorname{Fin}(n)$ consists of natural numbers less than $n$, $a=b$ if and only if their underlying natural-number values are equal. | null | @[grind hom] theorem Lean.Grind.Fin.eq_iff_val_eq {n : Nat} (a b : Fin n) : a = b ↔ a.val = b.val | :=
⟨Fin.val_eq_of_eq, Fin.eq_of_val_eq⟩ | [
"theorem",
"core library"
] | lean4 |
For every natural number $n$, decidable proposition $c$, and $x,y\in\operatorname{Fin}(n)$, the underlying natural-number value of $\text{if }c\text{ then }x\text{ else }y$ equals $\text{if }c\text{ then }\operatorname{val}(x)\text{ else }\operatorname{val}(y)$. | null | @[grind hom] theorem Lean.Grind.Fin.val_ite {n : Nat} (c : Prop) [Decidable c] (x y : Fin n) :
(if c then x else y).val = if c then x.val else y.val | := by
split <;> rfl | [
"theorem",
"core library"
] | lean4 |
For every nonzero natural number $n$ and natural number $a$, the underlying natural-number value of the numeral $a$ in $\operatorname{Fin}(n)$ is $a\bmod n$. | null | @[grind hom] theorem Lean.Grind.Fin.val_OfNat_ofNat (n : Nat) [NeZero n] (a : Nat) : (OfNat.ofNat a : Fin n).val = a % n | := by
dsimp [OfNat.ofNat] | [
"theorem",
"core library"
] | lean4 |
For every natural number $n$ and $a\in\operatorname{Fin}(n)$, $\operatorname{val}(\operatorname{log2}(a))=\operatorname{log2}(\operatorname{val}(a))$, using the natural-number `log2` on the right. | null | @[grind hom] theorem Lean.Grind.Fin.val_log2 {n : Nat} (a : Fin n) : a.log2.val = a.val.log2 | := by
simp [Fin.log2] | [
"theorem",
"core library"
] | lean4 |
For every nonzero natural number $n$ and natural number $a$, the underlying value of the natural-number cast of $a$ into $\operatorname{Fin}(n)$ is $a\bmod n$. | null | @[grind hom] theorem Lean.Grind.Fin.val_natCast (n : Nat) [NeZero n] (a : Nat) :
(NatCast.natCast a : Fin n).val = a % n | := rfl | [
"theorem",
"core library"
] | lean4 |
For every nonzero natural number $n$ and integer $i$, the underlying natural-number value of the integer cast of $i$ into $\operatorname{Fin}(n)$ is $\operatorname{toNat}(i\bmod n)$, where the remainder is Euclidean and $\operatorname{toNat}(z)=\max(z,0)$ viewed as a natural number. | null | @[grind hom] theorem Lean.Grind.Fin.val_intCast (n : Nat) [NeZero n] (i : Int) :
(IntCast.intCast i : Fin n).val = (i % n).toNat | := by
have hn : n ≠ 0 := NeZero.ne n
change (Fin.intCast i : Fin n).val = _
unfold Fin.intCast
split
next h => rw [Fin.val_ofNat, Int.emod_natAbs_of_nonneg h]
next h =>
have h : i < 0 := by omega
change (n - i.natAbs % n) % n = _
rw [Int.emod_natAbs_of_neg h hn]
have h₁ := Int.emod_nonneg i ... | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, the bit vector associated with the unsigned 8-bit numeral $a$ equals the 8-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt8.toBitVec_OfNat_ofNat (a : Nat) :
(OfNat.ofNat a : UInt8).toBitVec = OfNat.ofNat a | :=
(UInt8.toBitVec_ofNat a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every natural number $a$, converting $a$ to an unsigned 8-bit integer using `ofNat` and then to a bit vector gives the 8-bit vector numeral $a$. | null | @[grind hom] theorem Lean.Grind.UInt8.toBitVec_ofNat (a : Nat) :
(UInt8.ofNat a).toBitVec = OfNat.ofNat a | :=
(UInt8.toBitVec_ofNat' a).trans BitVec.ofNat_eq_ofNat.symm | [
"theorem",
"core library"
] | lean4 |
For every unsigned 8-bit integer $x$, its natural-number value equals the natural-number value of its associated bit vector. | null | @[grind hom] theorem Lean.Grind.UInt8.toNat_eq_toBitVec_toNat (x : UInt8) : x.toNat = x.toBitVec.toNat | := rfl | [
"theorem",
"core library"
] | lean4 |
For every signed 8-bit integer $x$, its integer value equals the signed integer value of its associated 8-bit vector. | null | @[grind hom] theorem Lean.Grind.Int8.toInt_eq_toBitVec_toInt (x : Int8) : x.toInt = x.toBitVec.toInt | := rfl | [
"theorem",
"core library"
] | lean4 |
The opaque function $\mathrm{getMajor}:\mathrm{Unit}\to\mathbb N$ supplies the major version number; $\mathrm{major}=\mathrm{getMajor}(())$. | null | @[extern "lean_version_get_major"]
private opaque version.getMajor (u : Unit) : Nat
def version.major : Nat | := version.getMajor () | [
"definition",
"core library"
] | lean4 |
The natural number $\mathrm{major}$ is $\mathrm{getMajor}(())$, the major version number. | null | def version.major : Nat | := version.getMajor () | [
"definition",
"core library"
] | lean4 |
The opaque function $\mathrm{getMinor}:\mathrm{Unit}\to\mathbb N$ supplies the minor version number; $\mathrm{minor}=\mathrm{getMinor}(())$. | null | @[extern "lean_version_get_minor"]
private opaque version.getMinor (u : Unit) : Nat
def version.minor : Nat | := version.getMinor () | [
"definition",
"core library"
] | lean4 |
The natural number $\mathrm{minor}$ is $\mathrm{getMinor}(())$, the minor version number. | null | def version.minor : Nat | := version.getMinor () | [
"definition",
"core library"
] | lean4 |
The opaque function $\mathrm{getPatch}:\mathrm{Unit}\to\mathbb N$ supplies the patch version number; $\mathrm{patch}=\mathrm{getPatch}(())$. | null | @[extern "lean_version_get_patch"]
private opaque version.getPatch (u : Unit) : Nat
def version.patch : Nat | := version.getPatch () | [
"definition",
"core library"
] | lean4 |
The natural number $\mathrm{patch}$ is $\mathrm{getPatch}(())$, the patch version number. | null | def version.patch : Nat | := version.getPatch () | [
"definition",
"core library"
] | lean4 |
The opaque function $\mathrm{getGithash}:\mathrm{Unit}\to\mathrm{String}$ supplies a Git hash string; $\mathrm{githash}=\mathrm{getGithash}(())$. | null | @[extern "lean_get_githash"]
opaque getGithash (u : Unit) : String
def githash : String | := getGithash () | [
"definition",
"core library"
] | lean4 |
The string $\mathrm{githash}$ is $\mathrm{getGithash}(())$. | null | def githash : String | := getGithash () | [
"definition",
"core library"
] | lean4 |
The opaque function $\mathrm{getIsRelease}:\mathrm{Unit}\to\mathrm{Bool}$ supplies a release flag; $\mathrm{isRelease}=\mathrm{getIsRelease}(())$. | null | @[extern "lean_version_get_is_release"]
opaque version.getIsRelease (u : Unit) : Bool
def version.isRelease : Bool | := version.getIsRelease () | [
"definition",
"core library"
] | lean4 |
The Boolean $\mathrm{isRelease}$ is $\mathrm{getIsRelease}(())$. | null | def version.isRelease : Bool | := version.getIsRelease () | [
"definition",
"core library"
] | lean4 |
Additional version description like "nightly-2018-03-11" | null | @[extern "lean_version_get_special_desc"]
opaque version.getSpecialDesc (u : Unit) : String
def version.specialDesc : String | := version.getSpecialDesc () | [
"definition",
"core library"
] | lean4 |
The string $\mathrm{specialDesc}$ is $\mathrm{getSpecialDesc}(())$, the special version description. | null | def version.specialDesc : String | := version.getSpecialDesc () | [
"definition",
"core library"
] | lean4 |
The string $\mathrm{versionStringCore}$ concatenates the decimal representations of $\mathrm{major}$, $\mathrm{minor}$, and $\mathrm{patch}$, separated by periods. | null | def versionStringCore | :=
String.Internal.append
(String.Internal.append
(String.Internal.append
(String.Internal.append (toString version.major) ".")
(toString version.minor))
".")
(toString version.patch) | [
"definition",
"core library"
] | lean4 |
The string $\mathrm{versionString}$ is $\mathrm{versionStringCore}$ followed by a hyphen and $\mathrm{specialDesc}$ when that description is nonempty; otherwise it is $\mathrm{versionStringCore}$ for a release, and $\mathrm{versionStringCore}$ followed by “, commit ” and $\mathrm{githash}$ otherwise. | null | def versionString | :=
if version.specialDesc ≠ "" then
String.Internal.append (String.Internal.append versionStringCore "-") version.specialDesc
else if version.isRelease then
versionStringCore
else
(String.Internal.append (String.Internal.append versionStringCore ", commit ") githash) | [
"definition",
"core library"
] | lean4 |
The string $\mathrm{origin}$ is “leanprover/lean4”. | null | def origin | :=
"leanprover/lean4" | [
"definition",
"core library"
] | lean4 |
The string $\mathrm{toolchain}$ is $\mathrm{origin}$ followed by “:v”, $\mathrm{versionStringCore}$, a hyphen, and $\mathrm{specialDesc}$ when the description is nonempty and the release flag is true; with a nonempty description and false release flag it is $\mathrm{origin}$ followed by a colon and the description. Wit... | null | def toolchain | :=
if version.specialDesc ≠ "" then
if version.isRelease then
String.Internal.append
(String.Internal.append
(String.Internal.append
(String.Internal.append origin ":v")
versionStringCore)
"-")
version.specialDesc
else
String.Internal.ap... | [
"definition",
"core library"
] | lean4 |
For a character $c$, the Boolean $\mathrm{isGreek}(c)$ is true exactly when its Unicode value lies in the inclusive hexadecimal interval $[391,3DD]$. | null | @[inline] def isGreek (c : Char) : Bool | :=
0x391 ≤ c.val && c.val ≤ 0x3dd | [
"definition",
"core library"
] | lean4 |
For a character $c$ with Unicode value $v$, $\mathrm{isLetterLike}(c)$ is true exactly when $v$ lies in one of the following inclusive hexadecimal ranges: $[3B1,3C9]\setminus\{3BB\}$, $[391,3A9]\setminus\{3A0,3A3\}$, $[3CA,3FB]$, $[1F00,1FFE]$, $[2100,214F]$, $[1D49C,1D59F]$, $[C0,FF]\setminus\{D7,F7\}$, or $[100,17F]$... | null | def isLetterLike (c : Char) : Bool | :=
(0x3b1 ≤ c.val && c.val ≤ 0x3c9 && c.val ≠ 0x3bb) ||
(0x391 ≤ c.val && c.val ≤ 0x3A9 && c.val ≠ 0x3A0 && c.val ≠ 0x3A3) ||
(0x3ca ≤ c.val && c.val ≤ 0x3fb) ||
(0x1f00 ≤ c.val && c.val ≤ 0x1ffe) ||
(0x2100 ≤ c.val && c.val ≤ 0x214f) ||
(0x1d49c ≤ c.val && c.val ≤ 0x1d59f) ||
(0x00c0 ≤ c.val && c.val ... | [
"definition",
"core library"
] | lean4 |
For a character $c$, $\mathrm{isNumericSubscript}(c)$ is true exactly when its Unicode value lies in the inclusive hexadecimal interval $[2080,2089]$. | null | @[inline] def isNumericSubscript (c : Char) : Bool | :=
0x2080 ≤ c.val && c.val ≤ 0x2089 | [
"definition",
"core library"
] | lean4 |
For a character $c$, $\mathrm{isSubScriptAlnum}(c)$ is true exactly when its Unicode value lies in one of the inclusive hexadecimal intervals $[2080,2089]$, $[2090,209C]$, or $[1D62,1D6A]$, or equals $2C7C$. | null | def isSubScriptAlnum (c : Char) : Bool | :=
isNumericSubscript c ||
(0x2090 ≤ c.val && c.val ≤ 0x209c) ||
(0x1d62 ≤ c.val && c.val ≤ 0x1d6a) ||
c.val == 0x2c7c | [
"definition",
"core library"
] | lean4 |
For a character $c$, $\mathrm{isIdFirst}(c)$ is true exactly when $c$ is alphabetic according to `isAlpha`, is an underscore, or satisfies `isLetterLike`. | null | @[inline] def isIdFirst (c : Char) : Bool | :=
c.isAlpha || c = '_' || isLetterLike c | [
"definition",
"core library"
] | lean4 |
For an unsigned byte $c$, $\mathrm{isAlphaAscii}(c)$ is true exactly when $c$ is the ASCII encoding of a letter from a through z or A through Z. | null | @[inline] private def isAlphaAscii (c : UInt8) : Bool | :=
'a'.toUInt8 ≤ c && c ≤ 'z'.toUInt8
|| 'A'.toUInt8 ≤ c && c ≤ 'Z'.toUInt8 | [
"definition",
"core library"
] | lean4 |
For an unsigned byte $c$, $\mathrm{isIdFirstAscii}(c)$ is true exactly when $c$ encodes an ASCII letter or an underscore. | null | @[inline] def isIdFirstAscii (c : UInt8) : Bool | :=
isAlphaAscii c || c = '_'.toUInt8 | [
"definition",
"core library"
] | lean4 |
For an unsigned byte $c$, $\mathrm{isAlphanumAscii}(c)$ is true exactly when $c$ encodes an ASCII letter or a digit from 0 through 9. | null | @[inline] private def isAlphanumAscii (c : UInt8) : Bool | :=
isAlphaAscii c || '0'.toUInt8 ≤ c && c ≤ '9'.toUInt8 | [
"definition",
"core library"
] | lean4 |
For a character $c$, $\mathrm{isIdRest}(c)$ is true exactly when $c$ satisfies `isAlphanum`, is one of underscore, apostrophe, exclamation mark, or question mark, or satisfies `isLetterLike` or `isSubScriptAlnum`. | null | @[inline] def isIdRest (c : Char) : Bool | :=
c.isAlphanum || c = '_' || c = '\'' || c == '!' || c == '?' || isLetterLike c || isSubScriptAlnum c | [
"definition",
"core library"
] | lean4 |
For an unsigned byte $c$, $\mathrm{isIdRestAscii}(c)$ is true exactly when $c$ encodes an ASCII letter, digit, underscore, apostrophe, exclamation mark, or question mark. | null | @[inline] def isIdRestAscii (c : UInt8) : Bool | :=
isAlphanumAscii c || c = '_'.toUInt8 || c = '\''.toUInt8 || c == '!'.toUInt8 || c == '?'.toUInt8 | [
"definition",
"core library"
] | lean4 |
The character $\mathrm{idBeginEscape}$ is «. | null | def idBeginEscape | := '«' | [
"definition",
"core library"
] | lean4 |
The character $\mathrm{idEndEscape}$ is ». | null | def idEndEscape | := '»' | [
"definition",
"core library"
] | lean4 |
For a character $c$, $\mathrm{isIdBeginEscape}(c)$ is true exactly when $c$ is «. | null | @[inline] def isIdBeginEscape (c : Char) : Bool | := c = idBeginEscape | [
"definition",
"core library"
] | lean4 |
For a character $c$, $\mathrm{isIdEndEscape}(c)$ is true exactly when $c$ is ». | null | @[inline] def isIdEndEscape (c : Char) : Bool | := c = idEndEscape | [
"definition",
"core library"
] | lean4 |
The function $\mathrm{getRoot}:\mathrm{Name}\to\mathrm{Name}$ sends the anonymous name to itself and every nonanonymous name to its first component, retaining whether that component is a string or a natural number. | null | def getRoot : Name → Name | | anonymous => anonymous
| n@(str anonymous _) => n
| n@(num anonymous _) => n
| str n _ => getRoot n
| num n _ => getRoot n | [
"definition",
"core library"
] | lean4 |
The Boolean function $\mathrm{isInaccessibleUserName}$ on names returns true when the last string component contains ✝ or equals “_inaccessible”; it skips trailing numeric components and returns false for the anonymous name. | null | def isInaccessibleUserName : Name → Bool | | Name.str _ s => (String.Internal.contains s '✝') || s == "_inaccessible"
| Name.num p _ => isInaccessibleUserName p
| _ => false | [
"definition",
"core library"
] | lean4 |
For a string $s$ and natural number $i$, $\mathrm{needsNoEscapeAsciiRest}(s,i)$ is true exactly when every UTF-8 byte of $s$ at an index at least $i$ satisfies `isIdRestAscii`; it is true when $i$ is at least the byte length. | null | private partial def needsNoEscapeAsciiRest (s : String) (i : Nat) : Bool | :=
if h : i < s.utf8ByteSize then
let c := String.Internal.getUTF8Byte s i h
isIdRestAscii c && needsNoEscapeAsciiRest s (i + 1)
else
true | [
"definition",
"core library"
] | lean4 |
For a string $s$ with positive UTF-8 byte length, $\mathrm{needsNoEscapeAscii}(s)$ is true exactly when its first byte satisfies `isIdFirstAscii` and all remaining bytes satisfy `isIdRestAscii`. | null | @[inline] private def needsNoEscapeAscii (s : String) (h : s.utf8ByteSize > 0) : Bool | :=
let c := String.Internal.getUTF8Byte s 0 h
isIdFirstAscii c && needsNoEscapeAsciiRest s 1 | [
"definition",
"core library"
] | lean4 |
For a string $s$ with positive UTF-8 byte length, $\mathrm{needsNoEscape}(s)$ is true when `needsNoEscapeAscii` holds, or when its first character satisfies `isIdFirst` and every remaining character satisfies `isIdRest`. | null | @[inline] private def needsNoEscape (s : String) (h : s.utf8ByteSize > 0) : Bool | :=
needsNoEscapeAscii s h || isIdFirst (String.Internal.get s 0) && Substring.Raw.Internal.all (Substring.Raw.Internal.drop s.toRawSubstring 1) isIdRest | [
"definition",
"core library"
] | lean4 |
For a string $s$, $\mathrm{escape}(s)$ is the concatenation of «, $s$, and ». | null | @[inline] private def escape (s : String) : String | :=
String.Internal.append (String.Internal.append idBeginEscape.toString s) idEndEscape.toString | [
"definition",
"core library"
] | lean4 |
For a string $s$ and Boolean $\mathrm{force}$, defaulting to false, $\mathrm{escapePart}(s,\mathrm{force})$ returns $\mathrm{some}(s)$ if $s$ is nonempty, forcing is disabled, and `needsNoEscape` holds. Otherwise it returns none if the nonempty string contains », and returns $\mathrm{some}(\text{«}s\text{»})$ in all ot... | null | @[inline]
private def Internal.Meta.escapePart (s : String) (force : Bool := false) : Option String | :=
if h : s.utf8ByteSize > 0 then
if !force && needsNoEscape s h then
some s
else if String.Internal.any s isIdEndEscape then
none
else
some <| escape s
else
some <| escape s | [
"definition",
"core library"
] | lean4 |
For a name $n$, separator string $\mathrm{sep}$, escape flag, and token test $\mathrm{isToken}$ defaulting to constantly false, `toStringWithSep` renders the anonymous name as “[anonymous]”, a root string component using `maybeEscape` with its token-test result, and a root numeric component in decimal. A further string... | null | private def Internal.Meta.toStringWithSep (n : Name) (isToken : String → Bool := fun _ => false) : String | :=
match n with
| anonymous => "[anonymous]"
| str anonymous s => maybeEscape s (isToken s)
| num anonymous v => toString v
| str n s =>
let r := toStringWithSep n isToken
let r' := String.Internal.append (String.Internal.append r sep) (maybeEscape s false)
if escape && isToken r' th... | [
"definition",
"core library"
] | lean4 |
For a name $n$, Boolean escape flag defaulting to true, and token test $\mathrm{isToken}:\mathrm{String}\to\mathrm{Bool}$, `toStringWithToken` applies `toStringWithSep` with period separators and the given token test, enabling escaping exactly when the flag is true, $n$ is neither an inaccessible user name nor a name w... | null | private def Internal.Meta.toStringWithToken (n : Name) (escape := true) (isToken : String → Bool) : String | :=
toStringWithSep "." (escape && !n.isInaccessibleUserName && !n.hasMacroScopes && !maybePseudoSyntax) n isToken | [
"definition",
"core library"
] | lean4 |
For a name $n$ and Boolean escape flag defaulting to true, $\mathrm{toString}(n)$ is `toStringWithToken` applied with that flag and a token test that is always false. | null | private def Internal.Meta.toString (n : Name) (escape := true) : String | :=
toStringWithToken n escape (fun _ => false) | [
"definition",
"core library"
] | lean4 |
Returns true if the name has any numeric components. | null | def hasNum : Name → Bool | | anonymous => false
| num .. => true
| str p .. => hasNum p | [
"definition",
"core library"
] | lean4 |
For a name $n$ and natural-number precedence $p$, $\mathrm{reprPrec}(n,p)$ produces a formatted representation: the anonymous name uses its constructor label; a numeric component uses an application of `mkNum` to the recursively represented prefix and the represented number; a string component whose prefix contains a n... | null | protected def reprPrec (n : Name) (prec : Nat) : Std.Format | :=
match n with
| anonymous => Std.Format.text "Lean.Name.anonymous"
| num p i => Repr.addAppParen ("Lean.Name.mkNum " ++ Name.reprPrec p max_prec ++ " " ++ repr i) prec
| str p s =>
if p.hasNum then
Repr.addAppParen ("Lean.Name.mkStr " ++ Name.reprPrec p max_prec ++ " " ++ repr s) prec
else
... | [
"definition",
"core library"
] | lean4 |
The function $\mathrm{capitalize}:\mathrm{Name}\to\mathrm{Name}$ applies string capitalization to the final component of a name when that component is a string, preserving its prefix; anonymous names and names ending in a numeric component remain unchanged. | null | def capitalize : Name → Name | | .str p s => .str p (String.Internal.capitalize s)
| n => n | [
"definition",
"core library"
] | lean4 |
For names $n,q,r$, $\mathrm{replacePrefix}(n,q,r)$ replaces a prefix equal to $q$ by $r$: it returns $r$ if $n=q$, otherwise recursively processes the parent of each string or numeric component and reattaches that component. At the anonymous name it returns $r$ if $q$ is anonymous, and the anonymous name otherwise. | null | def replacePrefix : Name → Name → Name → Name | | anonymous, anonymous, newP => newP
| anonymous, _, _ => anonymous
| n@(str p s), queryP, newP => if n == queryP then newP else Name.mkStr (p.replacePrefix queryP newP) s
| n@(num p s), queryP, newP => if n == queryP then newP else Name.mkNum (p.replacePrefix queryP newP) s | [
"definition",
"core library"
] | lean4 |
`eraseSuffix? n s` return `n'` if `n` is of the form `n == n' ++ s`. | null | def eraseSuffix? : Name → Name → Option Name | | n, anonymous => some n
| str p s, str p' s' => if s == s' then eraseSuffix? p p' else none
| num p s, num p' s' => if s == s' then eraseSuffix? p p' else none
| _, _ => none | [
"definition",
"core library"
] | lean4 |
Remove macros scopes, apply `f`, and put them back | null | @[inline] def modifyBase (n : Name) (f : Name → Name) : Name | :=
if n.hasMacroScopes then
let view := extractMacroScopes n
{ view with name := f view.name }.review
else
f n | [
"definition",
"core library"
] | lean4 |
For a name $n$ and string $s$, $\mathrm{appendAfter}(n,s)$ applies `modifyBase` to $n$, concatenating $s$ to a string base component and otherwise adding $s$ as a new string component after the base name. | null | @[export lean_name_append_after]
def appendAfter (n : Name) (suffix : String) : Name | :=
n.modifyBase fun
| str p s => Name.mkStr p (String.Internal.append s suffix)
| n => Name.mkStr n suffix | [
"definition",
"core library"
] | lean4 |
For a name $n$ and natural number $i$, $\mathrm{appendIndexAfter}(n,i)$ applies `modifyBase` to $n$, appending an underscore and the decimal representation of $i$ to a string base component, or adding that combined string as a new component otherwise. | null | @[export lean_name_append_index_after]
def appendIndexAfter (n : Name) (idx : Nat) : Name | :=
n.modifyBase fun
| str p s => Name.mkStr p (String.Internal.append (String.Internal.append s "_") (toString idx))
| n => Name.mkStr n (String.Internal.append "_" (toString idx)) | [
"definition",
"core library"
] | lean4 |
For a name $n$ and string $s$, $\mathrm{appendBefore}(n,s)$ applies `modifyBase` with the following transformation: an anonymous base becomes a root string component $s$; a string component has $s$ prepended; a numeric component retains its number and gains a string component $s$ immediately before it. | null | def appendBefore (n : Name) (pre : String) : Name | :=
n.modifyBase fun
| anonymous => Name.mkStr anonymous pre
| str p s => Name.mkStr p (String.Internal.append pre s)
| num p n => Name.mkNum (Name.mkStr p pre) n | [
"definition",
"core library"
] | lean4 |
For all $m,n$ of type `Name`, the Boolean equality test $m==n$ is true if and only if $m=n$. | null | protected theorem beq_iff_eq {m n : Name} : m == n ↔ m = n | := by
change m.beq n ↔ _
induction m generalizing n <;> cases n <;> simp_all [Name.beq, And.comm] | [
"theorem",
"core library"
] | lean4 |
For a name generator $g$, $\mathrm{curr}(g)$ is the name obtained by appending the numeric component $g.\mathrm{idx}$ to $g.\mathrm{namePrefix}$. | null | @[inline] def curr (g : NameGenerator) : Name | :=
Name.mkNum g.namePrefix g.idx | [
"definition",
"core library"
] | lean4 |
For a name generator $g$, $\mathrm{next}(g)$ preserves its other fields and replaces its index by $g.\mathrm{idx}+1$. | null | @[inline] def next (g : NameGenerator) : NameGenerator | :=
{ g with idx := g.idx + 1 } | [
"definition",
"core library"
] | lean4 |
For a name generator $g$, $\mathrm{mkChild}(g)$ returns a pair of generators: the first has prefix $\mathrm{curr}(g)$ and index $1$, and the second is $\mathrm{next}(g)$. | null | @[inline] def mkChild (g : NameGenerator) : NameGenerator × NameGenerator | :=
({ namePrefix := Name.mkNum g.namePrefix g.idx, idx := 1 },
{ g with idx := g.idx + 1 }) | [
"definition",
"core library"
] | lean4 |
For a type constructor $m:\mathrm{Type}\to\mathrm{Type}$, the class `MonadNameGenerator` provides operations $\mathrm{getNGen}:m\,\mathrm{NameGenerator}$ and $\mathrm{setNGen}:\mathrm{NameGenerator}\to m\,\mathrm{Unit}$, with no additional laws specified. | null | class MonadNameGenerator (m : Type → Type) | where
getNGen : m NameGenerator
setNGen : NameGenerator → m Unit | [
"definition",
"core library"
] | lean4 |
Creates a globally unique `Name`, without any semantic interpretation.
The names are not intended to be user-visible.
With the default name generator, names use `_uniq` as a base and have a numeric suffix.
This is used for example by `Lean.mkFreshFVarId`, `Lean.mkFreshMVarId`, and `Lean.mkFreshLMVarId`.
To create fres... | null | def mkFreshId {m : Type → Type} [Monad m] [MonadNameGenerator m] : m Name | := do
let ngen ← getNGen
let r := ngen.curr
setNGen ngen.next
pure r | [
"definition",
"core library"
] | lean4 |
For type constructors $m,n:\mathrm{Type}\to\mathrm{Type}$, a lift from $m$ to $n$ and a `MonadNameGenerator` structure on $m$ induce one on $n$ by lifting both the generator-reading action and each generator-setting action. | null | instance monadNameGeneratorLift (m n : Type → Type) [MonadLift m n] [MonadNameGenerator m] : MonadNameGenerator n | := {
getNGen := liftM (getNGen : m _),
setNGen := fun ngen => liftM (setNGen ngen : m _) | [
"definition",
"core library"
] | lean4 |
Syntax that represents a Lean term. | null | abbrev Term | := TSyntax `term | [
"definition",
"core library"
] | lean4 |
Syntax that represents a command. | null | abbrev Command | := TSyntax `command | [
"definition",
"core library"
] | lean4 |
Syntax that represents a universe level. | null | protected abbrev Level | := TSyntax `level | [
"definition",
"core library"
] | lean4 |
Syntax that represents a tactic. | null | protected abbrev Tactic | := TSyntax `tactic | [
"definition",
"core library"
] | lean4 |
Syntax that represents an element of a `do` sequence. | null | abbrev DoElem | := TSyntax `doElem | [
"definition",
"core library"
] | lean4 |
Syntax that represents a sequence of `do` elements. | null | abbrev DoSeq | := TSyntax `Lean.Parser.Term.doSeq | [
"definition",
"core library"
] | lean4 |
Syntax that represents a precedence (e.g. for an operator). | null | abbrev Prec | := TSyntax `prec | [
"definition",
"core library"
] | lean4 |
Syntax that represents a priority (e.g. for an instance declaration). | null | abbrev Prio | := TSyntax `prio | [
"definition",
"core library"
] | lean4 |
Syntax that represents an identifier. | null | abbrev Ident | := TSyntax identKind | [
"definition",
"core library"
] | lean4 |
Syntax that represents a string literal. | null | abbrev StrLit | := TSyntax strLitKind | [
"definition",
"core library"
] | lean4 |
End of preview. Expand in Data Studio
Lean 4 Proofs (public sources)
1,560,974 Lean 4 theorems and definitions from public Lean libraries and datasets. Each row is one declaration:
| column | content |
|---|---|
nl_statement |
the statement in natural language |
nl_proof |
a natural-language proof; null when there is none (42,408 rows have one) |
lean_statement |
the Lean 4 declaration header, e.g. theorem foo (x : ℕ) : x ≤ x + 1 |
lean_proof |
the Lean 4 proof (or definition body) that follows the header, e.g. := by omega |
labels |
tags: theorem or definition, then topic tags where known (e.g. number theory) |
source |
the library or dataset the row comes from (see the table below) |
Rows are theorems and lemmas with complete proofs, and the definitions (def, abbrev, structure,
class, inductive, instance, ...) they build on; for a definition, nl_statement describes what it defines.
lean_statement and lean_proof concatenate to the declaration. They are shown without comments, and
without the surrounding import, open, namespace and variable commands.
Every theorem/lemma with a complete (sorry-free) proof, and every named definition of the libraries, is included; exact duplicates are merged, and for machine-found proofs one proof is kept per statement. Natural-language statements come from the source (problem text, informal statement or docstring) where it has one, and were otherwise written by a language model from the Lean statement. Natural-language proofs are included only where the source provides them.
from datasets import load_dataset
ds = load_dataset("hmzprover/proofs-public", split="train")
Sources and licenses
| source | source value |
license | theorems | definitions |
|---|---|---|---|---|
| Lean 4 core library (Init, Std, Lean, Lake) | lean4 |
Apache-2.0 | 44,909 | 29,545 |
| Mathlib (incl. Archive, Counterexamples) | mathlib |
Apache-2.0 | 192,411 | 53,085 |
| Batteries | batteries |
Apache-2.0 | 2,104 | 1,295 |
| Aesop | aesop |
Apache-2.0 | 300 | 1,518 |
| Plausible | plausible |
Apache-2.0 | 4 | 229 |
| ProofWidgets4 | proofwidgets |
Apache-2.0 | 1 | 301 |
| Qq (quote4) | quote4 |
Apache-2.0 | 0 | 178 |
| Sphere eversion | sphere-eversion |
Apache-2.0 | 911 | 258 |
| CSLib | cslib |
Apache-2.0 | 3,040 | 1,532 |
| Fermat's Last Theorem project | flt |
Apache-2.0 | 2,204 | 891 |
| Polynomial Freiman–Ruzsa (PFR) | pfr |
Apache-2.0 | 916 | 104 |
| Carleson's theorem | carleson |
Apache-2.0 | 2,619 | 419 |
| Prime Number Theorem and more | pnt |
Apache-2.0 | 8,036 | 705 |
| PhysLean | physlean |
Apache-2.0 | 19,772 | 6,081 |
| FormalizedFormalLogic Foundation | foundation |
Apache-2.0 | 6,782 | 3,051 |
| Compfiles | compfiles |
Apache-2.0 | 5,784 | 1,761 |
| Formal Conjectures | formal-conjectures |
Apache-2.0 | 3,334 | 2,989 |
| ∞-cosmos | infinity-cosmos |
Apache-2.0 | 137 | 158 |
| LeanCamCombi | leancamcombi |
Apache-2.0 | 61 | 30 |
| Herald proofs | herald |
Apache-2.0 | 42,411 | 0 |
| NuminaMath-LEAN | numina-lean |
Apache-2.0 | 43,410 | 0 |
| Lean Workbook proofs (statements from Lean Workbook) | lean-workbook |
MIT; Lean Workbook: Apache-2.0 | 29,750 | 0 |
| DeepSeek-Prover-V1 | deepseek-prover-v1 |
DeepSeek License | 27,503 | 0 |
| Nemotron-Math-Proofs-v1 (Lean split) | nemotron |
CC-BY-SA-4.0 | 81,672 | 0 |
| STP (Self-play Theorem Prover) training data | stp |
MIT | 788,616 | 0 |
Lean Pool (third-party formalizations; upstream repository and license of each project in NOTICE) |
lean-pool/<project> |
Apache-2.0 or MIT per project | 109,161 | 40,996 |
The Lean code and texts remain under the licenses of their sources, listed above; the copyright belongs to their respective authors. Because part of the data is CC-BY-SA-4.0, this compilation is distributed under CC-BY-SA-4.0. Apache-2.0 sources: see https://www.apache.org/licenses/LICENSE-2.0 ; please also cite the original projects when you use their part of the data.
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