text stringlengths 0 3.34M |
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import topology.instances.real
open filter real
open_locale topological_space
theorem leibeck_23_3 (S : set β) (c : β) (hc : is_lub S c) :
β (f : β β β), (β n, f n β S) β§ tendsto f at_top (π c) :=
begin
sorry
end
|
informal statement Let $R$ be a ring, with $M$ an ideal of $R$. Suppose that every element of $R$ which is not in $M$ is a unit of $R$. Prove that $M$ is a maximal ideal and that moreover it is the only maximal ideal of $R$.formal statement theorem exercise_11_4_1b {F : Type*} [field F] [fintype F] (hF : card F = 2) :
... |
informal statement Suppose that $T \in \mathcal{L}(V)$ has $\operatorname{dim} V$ distinct eigenvalues and that $S \in \mathcal{L}(V)$ has the same eigenvectors as $T$ (not necessarily with the same eigenvalues). Prove that $S T=T S$.formal statement theorem exercise_6_2 {V : Type*} [add_comm_group V] [module β V]
[i... |
constants (p q : Prop)
theorem t_and_intro (Hp : p) (Hq : q) : p β§ q := and.intro Hp Hq
check t_and_intro -- t_and_intro : β p q, p β q β p β§ q
|
import LeanUtils
open Nat
theorem square_of_even_number_is_even (m : Nat) (hβ : even m) : (even (m ^ 2)) := by
have β¨n, hββ© : β (n : Nat), m = 2 * n := by
simp at *; assumption
have hβ : m^2 = 2*(2*n^2) := by
calc
m^2 = (2*n)^2 := by
repeat (first | ring | simp_all)
_ = 4*n^2 := by ... |
informal statement Show that the equation $7 x^{3}+2=y^{3}$ has no solution in integers.formal statement theorem exercise_3_5 : Β¬ β x y : β€, 7*x^3 + 2 = y^3 := |
informal statement Suppose (a) $f$ is continuous for $x \geq 0$, (b) $f^{\prime}(x)$ exists for $x>0$, (c) $f(0)=0$, (d) $f^{\prime}$ is monotonically increasing. Put $g(x)=\frac{f(x)}{x} \quad(x>0)$ and prove that $g$ is monotonically increasing.formal statement theorem exercise_5_6
{f : β β β}
(hf1 : continuous f... |
open Nat
theorem add_assoc (m n k : Nat) : m + n + k = m + (n + k) :=
Nat.recOn (motive := fun k => m + n + k = m + (n + k)) k rfl (fun k ih => by simp [Nat.add_succ, ih]; done)
|
informal statement Prove that the addition of residue classes $\mathbb{Z}/n\mathbb{Z}$ is associative.formal statement theorem exercise_1_1_3 (n : β€) :
β (a b c : β€), (a+b)+c β‘ a+(b+c) [ZMOD n] := |
informal statement Prove that if $H$ is a subgroup of $G$ then $H$ is generated by the set $H-\{1\}$.formal statement theorem exercise_2_4_4 {G : Type*} [group G] (H : subgroup G) :
subgroup.closure ((H : set G) \ {1}) = β€ := |
import advice
lemma conj (P Q : Prop)
(P β§ Q) : P :=
{! !}
|
-- Homework 5 due 9.26
theorem problem1 (A B : Prop) : (A β B) β (Β¬ B β Β¬ A) := sorry
theorem problem2 (A B : Prop) : Β¬ (A β¨ B) β Β¬ A β§ Β¬ B := sorry
theorem problem3 (A B C : Prop) (h : B β C) : (A β¨ B) β A β¨ C := sorry
theorem problem4 (A B : Prop) : (A β B) β Β¬ A β¨ B := sorry
theorem problem5 (A B : Prop) : Β¬ A... |
informal statement Prove that if $V$ is a complex inner-product space, then $\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}+\|u+i v\|^{2} i-\|u-i v\|^{2} i}{4}$ for all $u, v \in V$.formal statement theorem exercise_6_7 {V : Type*} [inner_product_space β V] (u v : V) :
βͺu, vβ«_β = (βu + vβ^2 - βu - vβ^2 + I*βu + Iβ’... |
informal statement Let $G$ be any group. Prove that the map from $G$ to itself defined by $g \mapsto g^{-1}$ is a homomorphism if and only if $G$ is abelian.formal statement theorem exercise_2_1_5 {G : Type*} [group G] [fintype G]
(hG : card G > 2) (H : subgroup G) [fintype H] :
card H β card G - 1 := |
informal statement Let $E$ be a bounded set in $R^{1}$. Prove that there exists a real function $f$ such that $f$ is uniformly continuous and is not bounded on $E$.formal statement theorem exercise_4_12
{Ξ± Ξ² Ξ³ : Type*} [uniform_space Ξ±] [uniform_space Ξ²] [uniform_space Ξ³]
{f : Ξ± β Ξ²} {g : Ξ² β Ξ³}
(hf : uniform_con... |
informal statement Prove that if $V$ is a complex inner-product space, then $\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}+\|u+i v\|^{2} i-\|u-i v\|^{2} i}{4}$ for all $u, v \in V$.formal statement theorem exercise_6_16 {K V : Type*} [is_R_or_C K] [inner_product_space K V]
{U : submodule K V} :
U.orthogonal = ... |
[GOAL]
β’ Β¬UnivLE.{u + 1, u}
[PROOFSTEP]
simp only [Small_iff, not_forall, not_exists, not_nonempty_iff]
[GOAL]
β’ β x, β (x_1 : Type u), IsEmpty (x β x_1)
[PROOFSTEP]
exact β¨Type u, fun Ξ± => β¨fun f => Function.not_surjective_Type.{u, u} f.symm f.symm.surjectiveβ©β©
|
import data.complex.exponential
import algebra.polynomial.big_operators
import analysis.special_functions.trigonometric.basic
theorem AMC_2021_A_22 (a b c :β)(P:β β β)
(hP: β (x:β), P x = x^3+a*x^2*b*x+c)
(hroots:(P (real.cos 2*real.pi/7) = 0)β§(P (real.cos 4*real.pi/7) = 0)β§(P (real.cos 6*real.pi/7) = 0)):
a*b*c = 1/3... |
informal statement Let $n$ be a positive integer, and let $f_{n}(z)=n+(n-1) z+$ $(n-2) z^{2}+\cdots+z^{n-1}$. Prove that $f_{n}$ has no roots in the closed unit disk $\{z \in \mathbb{C}:|z| \leq 1\}$.formal statement theorem exercise_2017_b3 (f : β β β) (c : β β β)
(hf : f = Ξ» x, (β' (i : β), (c i) * x^i))
(hc : β... |
informal statement Prove that there is no rational number whose square is $12$.formal statement theorem exercise_1_5 (A minus_A : set β) (hA : A.nonempty)
(hA_bdd_below : bdd_below A) (hminus_A : minus_A = {x | -x β A}) :
Inf A = Sup minus_A := |
informal statement Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p^{\prime}$ have no roots in common.formal statement theorem exercise_5_4 {F V : Type*} [add_comm_group V] [field F]
[module F V] (S T : V ββ[F] V) (hST : S β T = T β... |
informal statement Prove that the multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{C}-\{0\}$ are not isomorphic.formal statement theorem exercise_1_6_17 {G : Type*} [group G] (f : G β G)
(hf : f = Ξ» g, gβ»ΒΉ) :
β x y : G, f x * f y = f (x*y) β β x y : G, x*y = y*x := |
import smt2
lemma negation_of_conj :
forall (P Q : Prop),
not (P β§ Q) β not P β¨ not Q :=
by intros; z3 "d1.log"
lemma negation_of_disj :
forall (P Q : Prop),
Β¬ (P β¨ Q) β Β¬ P β§ Β¬ Q :=
begin
intros, z3
end
|
import data.real.basic
theorem IMO_Shortlist_A4_2001 (f : β β β) :
(β x y : β, f(x^2 - y^2) = x*f(x) - y * f(y)) β β l : β, β x : β, f(x) = l*x := sorry
|
theory Filter1
imports Main
begin
lemma "" |
informal statement Suppose $T \in \mathcal{L}(V)$ is self-adjoint, $\lambda \in \mathbf{F}$, and $\epsilon>0$. Prove that if there exists $v \in V$ such that $\|v\|=1$ and $\|T v-\lambda v\|<\epsilon,$ then $T$ has an eigenvalue $\lambda^{\prime}$ such that $\left|\lambda-\lambda^{\prime}\right|<\epsilon$.formal statem... |
informal statement Show that the rationals $\mathbb{Q}$ are not locally compact.formal statement theorem exercise_29_1 : Β¬ locally_compact_space β := |
informal statement Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple.formal statement theorem exercise_6_8_1 {G : Type*} [group G]
(a b : G) : closure ({a, b} : set G) = closure {b*a*b^2, b*a*b^3} := |
[GOAL]
a b : β€
β’ a = b β βa = βb
[PROOFSTEP]
simp only [Int.cast_inj]
[GOAL]
a b : β€
β’ a β b β βa β βb
[PROOFSTEP]
simp only [ne_eq, Int.cast_inj]
|
informal statement A map $f: X \rightarrow Y$ is said to be an open map if for every open set $U$ of $X$, the set $f(U)$ is open in $Y$. Show that $\pi_{1}: X \times Y \rightarrow X$ and $\pi_{2}: X \times Y \rightarrow Y$ are open maps.formal statement theorem exercise_17_4 {X : Type*} [topological_space X]
(U A : s... |
example : β x, x + 2 = 8 :=
begin
let a : β := 3 * 2,
existsi a,
reflexivity
end
|
[GOAL]
β’ AbsoluteValue.uniformSpace AbsoluteValue.abs = PseudoMetricSpace.toUniformSpace
[PROOFSTEP]
ext s
[GOAL]
case a.a
s : Set (β Γ β)
β’ s β uniformity β β s β uniformity β
[PROOFSTEP]
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
[GOAL]
case a.a
s : Set (β Γ β)
β’ (β i... |
universes u v
theorem eqLitOfSize0 {Ξ± : Type u} (a : Array Ξ±) (hsz : a.size = 0) : a = #[] :=
a.toArrayLitEq 0 hsz
theorem eqLitOfSize1 {Ξ± : Type u} (a : Array Ξ±) (hsz : a.size = 1) : a = #[a.getLit 0 hsz (ofDecideEqTrue rfl)] :=
a.toArrayLitEq 1 hsz
theorem eqLitOfSize2 {Ξ± : Type u} (a : Array Ξ±) (hsz : a.size = 2)... |
import LeanUtils
open Nat
theorem n_cube_plus_2_n (n : Nat) : divisible 3 (n^3 + 2*n) := by
match n with
| 0 => repeat (first | trivial | ring | simp_all)
| k+1 =>
have hβ : divisible 3 (k ^ 3 + 2 * k) := n_cube_plus_2_n _
have β¨z, hββ© : β z, k^3 + 2*k = 3*z := by
repeat (first | ring | simp_all)... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
! This file was ported from Lean 3 source module topology.category.Top.epi_mono
! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
! Please do not... |
lemma closure_empty [simp]: "closure {} = {}" |
informal statement Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname{dim} V=1$ and $T \in \mathcal{L}(V, V)$, then there exists $a \in \mathbf{F}$ such that $T v=a v$ for all $v \in V$.formal statement theorem exercise... |
informal statement Prove that characteristic subgroups are normal.formal statement theorem exercise_4_4_7 {G : Type*} [group G] {H : subgroup G} [fintype H]
(hH : β (K : subgroup G) (fK : fintype K), card H = @card K fK β H = K) :
H.characteristic := |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import analysis.calculus.conformal.normed_space
import analysis.inner_product_space.conformal_linear_map
/-!
# Conformal maps between inner product spaces
A function be... |
import Smt
theorem assoc (f : Bool β Bool β Bool) (p q r : Bool) :
f p (f q r) == f (f p q) r := by
smt
admit
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Data.Option.Basic
universe u v
theorem Option.eq_of_eq_some {Ξ± : Type u} : β {x y : Option Ξ±}, (βz, x = some z β y = some z) β x = y
... |
informal statement Show that the lower limit topology $\mathbb{R}_l$ and $K$-topology $\mathbb{R}_K$ are not comparable.formal statement theorem exercise_13_8b :
(topological_space.generate_from {S : set β | β a b : β, a < b β§ S = Ico a b}).is_open β
(lower_limit_topology β).is_open := |
def inc (x : Nat) := x + 1
@[simp] theorem inc_eq : inc x = x + 1 := rfl
theorem ex (a b : Fin (inc n)) (h : a = b) : b = a := by
simp only [inc_eq] at a
trace_state
exact h.symm
|
Formal statement is: lemma interior_hyperplane [simp]: assumes "a \<noteq> 0" shows "interior {x. a \<bullet> x = b} = {}" Informal statement is: If $a \neq 0$, then the interior of the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$ is empty. |
constants (p q : Prop) (H : p β§ q)
theorem t_proof_left : p := and.elim_left H
theorem t_proof_right : q := and.elim_right H
check t_proof_left -- t_proof_left : p
check t_proof_right -- t_proof_right : q
|
theorem Ex003_1 (a b: Prop) : Β¬a β a β b :=
assume A:Β¬a,
assume B:a,
have C:false, from A B,
show b, from false.elim C
theorem Ex003_2 (a b : Prop) : Β¬a β a β b :=
begin
intro,
intro,
contradiction
end
|
informal statement If $P$ is a $p$-Sylow subgroup of $G$ and $P \triangleleft G$, prove that $P$ is the only $p$-Sylow subgroup of $G$.formal statement theorem exercise_2_11_6 {G : Type*} [group G] {p : β} (hp : nat.prime p)
{P : sylow p G} (hP : P.normal) :
β (Q : sylow p G), P = Q := |
informal statement Let $\mathcal{T}_\alpha$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $\mathcal{T}_\alpha$.formal statement theorem exercise_13_5a {X : Type*}
[topological_space X] (A : set (set X)) (hA : is_topological_basis A) :
generate_... |
informal statement Prove that $\cos 1^{\circ}$ is algebraic over $\mathbb{Q}$.formal statement theorem exercise_5_5_2 : irreducible (X^3 - 3*X - 1 : polynomial β) := |
-- You can even write it as a function!
theorem contrapositive (P Q : Prop) :
(P β Q) β (Β¬ Q β Β¬ P) :=
Ξ» HPQ HnQ HP, HnQ (HPQ HP)
|
informal statement Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I \cap J$ in $R / I J$ is nilpotent.formal statement theorem exercise_10_7_10 {R : Type*} [ring R]
(M : ideal R) (hM : β (x : R), x β M β is_unit x) :
is_maximal M β§ β (N : ideal R), is_maximal N β N = M := |
informal statement Prove that if $R$ is an integral domain and $x^{2}=1$ for some $x \in R$ then $x=\pm 1$.formal statement theorem exercise_7_1_15 {R : Type*} [ring R] (hR : β a : R, a^2 = a) :
comm_ring R := |
informal statement A map $f: X \rightarrow Y$ is said to be an open map if for every open set $U$ of $X$, the set $f(U)$ is open in $Y$. Show that $\pi_{1}: X \times Y \rightarrow X$ and $\pi_{2}: X \times Y \rightarrow Y$ are open maps.formal statement theorem exercise_16_4 {X Y : Type*} [topological_space X] [topolog... |
Module first.
Polymorphic Record BAR (A:Type) :=
{ foo: A->Prop; bar: forall (x y: A), foo x -> foo y}.
Section A.
Context {A:Type}.
Set Printing Universes.
Hint Resolve bar.
Goal forall (P:BAR A) x y, foo _ P x -> foo _ P y.
intros.
eauto.
Qed.
End A.
End first.
Module firstbest.
Polymorphic Record BAR (A:Ty... |
import mynat.definition
import mynat.add
import mynat.mul
lemma example2 (x y : mynat) (h : y = x + 7) : 2 * y = 2 * (x + 7) :=
begin
rw h,
refl,
end
|
import data.real.basic
import data.real.nnreal
theorem USAMO_Problem_1_2011 (a b c : real) [a > 0] [b > 0] [c > 0]:
a^2 + b^2 + c^2 +(a+b+c)^4 β€ 4 β
((a*b + 1) / (a+b)^2) +(b*c+1) / (b+c)^2 + (c*a+1) / (c+a)^2 β₯ 3
:= sorry
|
informal statement Prove that $\lim_{n \rightarrow \infty} \sum_{i<n} a_i = \infty$, where $a_i = \sqrt{i + 1} -\sqrt{i}$.formal statement theorem exercise_3_6a
: tendsto (Ξ» (n : β), (β i in finset.range n, g i)) at_top at_top := |
def fact : Nat β Nat
| 0 => 1
| (n+1) => (n+1)*fact n
#check fact 6
#eval fact 10
-- set_option pp.all true
theorem tst1 : 100000000000 + 200000000000 = 300000000000 :=
rfl
theorem tst2 : 100000000000 * 200000000000 = 20000000000000000000000 :=
rfl
theorem tst3 : fact 7 = 5040 :=
rfl
theorem tst4 : fact 10 = ... |
def f (x : Nat) := 0
theorem ex1 (h : f x = 1) : False := by
simp [f] at h
def g (x : Nat) := [x]
theorem ex2 (h : g x = []) : 0 = 1 := by
simp [g] at h
theorem ex3 (x : Ξ±) (h : id x β x) : 0 = 1 := by
simp at h
|
informal statement Suppose $X$ is a nonempty complete metric space, and $\left\{G_{n}\right\}$ is a sequence of dense open sets of $X$. Prove Baire's theorem, namely, that $\bigcap_{1}^{\infty} G_{n}$ is not empty.formal statement theorem exercise_4_2a
{Ξ± : Type} [metric_space Ξ±]
{Ξ² : Type} [metric_space Ξ²]
(f : ... |
informal statement Suppose $u, v \in V$. Prove that $\langle u, v\rangle=0$ if and only if $\|u\| \leq\|u+a v\|$ for all $a \in \mathbf{F}$.formal statement theorem exercise_6_7 {V : Type*} [inner_product_space β V] (u v : V) :
βͺu, vβ«_β = (βu + vβ^2 - βu - vβ^2 + I*βu + Iβ’vβ^2 - I*βu-Iβ’vβ^2) / 4 := |
import algebra.big_operators.order
import tactic.ring
open_locale big_operators
theorem finset.mem_le_pos_sum {Ξ± : Type*} {Ξ² : Type*} [linear_ordered_add_comm_group Ξ²] (f : Ξ± β Ξ²) (s : finset Ξ±) (h1 : β x, x β s β 0 < f x)
: β y (H : y β s), f y β€ β x in s, f x :=
begin
apply finset.single_le_sum,
intros x h,... |
(* the only way to compute zero is for p to become canonical.
The only canonical member of A=A is eq_refl.
However, it is impossible to that p is propositionally equal to eq_refl.
In particular the univalence axiom allows for non-refl proofs. *)
Fixpoint zero (A : Set) (p : A = A) {struct p} : nat := 0.
(* although... |
import data.real.irrational
import topology.basic
import algebra.order.floor
--OUTPUT 1
theorem irrational_orbit_dense {Ξ± : β} (hΞ±_irrat : irrational Ξ±) : closure ((Ξ» m : β€, int.fract (Ξ± * βm)) '' (@set.univ β€)) = set.Icc 0 1 :=density_of_irrational_orbit (Ξ± : β) (hΞ± : Β¬ is_rat Ξ±) : β y β Icc 0 1, β x β Icc 0 1, x β ... |
informal statement Show that the collection $\{(a,b) \mid a < b, a \text{ and } b \text{ rational}\}$ is a basis that generates a topology different from the lower limit topology on $\mathbb{R}$.formal statement theorem exercise_16_4 {X Y : Type*} [topological_space X] [topological_space Y]
(Οβ : X Γ Y β X)
(Οβ : X... |
import logic.function
import data.fintype
def list.chain'' {Ξ±} (R : Ξ± β Ξ± β Prop) : (Ξ± β Prop) β list Ξ± β Ξ± β Prop
| P [] a := P a
| P (a::l) b := P a β§ list.chain'' (R a) l b
def flip_one {Ξ±} [decidable_eq Ξ±] (f : Ξ± β bool) (i : Ξ±) : Ξ± β bool :=
function.update f i (bnot (f i))
def admissible {Ξ±} [decidable_eq Ξ±] (... |
import AutograderTests.Util
in_exercise
inductive Example
| ctor (h : False)
theorem exercise : Example :=
sorry
in_solution
inductive Example
| ctor (h : True) -- clever change
theorem exercise : Example :=
β¨β¨β©β©
|
import tactic
lemma my_lemma: β n : β, n β₯ 0 :=
Ξ» n, nat.zero_le n
lemma my_lemma2: β n : β, n β₯ 0 :=
begin
intro n,
apply nat.zero_le,
end
#print my_lemma2
-- Comentario: Al colocar el cursor sobre print se obtiene
-- theorem my_lemma2 : β (n : β), n β₯ 0 :=
-- Ξ» (n : β), n.zero_le
lemma my_lemma3: β n :... |
informal statement If $f$ is a real continuous function defined on a closed set $E \subset \mathbb{R}$, prove that there exist continuous real functions $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$.formal statement theorem exercise_4_6
(f : β β β)
(E : set β)
(G : set (β Γ β))
(hβ : is_compact E)... |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
! This file was ported from Lean 3 source module analysis.inner_product_space.conformal_linear_map
! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd... |
informal statement Prove that if $f(x)$ and $g(x)$ are polynomials with rational coefficients whose product $f(x) g(x)$ has integer coefficients, then the product of any coefficient of $g(x)$ with any coefficient of $f(x)$ is an integer.formal statement theorem exercise_9_4_2b : irreducible
(X^6 + 30*X^5 - 15*X^3 + ... |
variable {Ξ± : Type*}
def is_prefix (lβ : list Ξ±) (lβ : list Ξ±) : Prop :=
β t, lβ ++ t = lβ
def list_has_le : has_le (list Ξ±) := β¨is_prefixβ©
section
local attribute [instance] list_has_le
theorem list.is_prefix_refl (l : list Ξ±) : l β€ l :=
β¨[], by simpβ©
end
-- error:
-- theorem bar (l : list Ξ±) : l β€ l :=... |
namespace Hidden
open Nat
theorem zero_add (n : Nat) : 0 + n = n :=
Nat.recOn (motive := fun x => 0 + x = x)
n
(show 0 + 0 = 0 from rfl)
(fun (n : Nat) (ih : 0 + n = n) =>
show 0 + succ n = succ n from
calc
0 + succ n = succ (0 + n) := rfl
_ = succ n := by rw [ih])
end... |
import Smt
theorem triv (p : Bool) : p β p := by
smt
simp_all
|
lemma maze (P Q R S T U: Prop)
(p : P)
(h : P β Q)
(i : Q β R)
(j : Q β T)
(k : S β T)
(l : T β U)
: U :=
begin
apply l,
apply j,
apply h,
exact p,
end
|
informal statement Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed connected subsets of $X$ that is simply ordered by proper inclusion. Then $Y=\bigcap_{A \in \mathcal{A}} A$ is connected.formal statement theorem exercise_26_11
{X : Type*} [topological_space X] [compact_space X] [t2_... |
(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
* SPDX-License-Identifier: BSD-2-Clause
*)
theory ptr_modifies
imports "Word_Lib.WordSetup" "CParser.CTranslation"
begin
external_file "ptr_modifies.c"
install_C_file "ptr_modifies.c"
context ptr_modifies
begin
thm foo_ptr_new_modifies
thm f_modifies
... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Data.Option.Basic
universe u v
theorem Option.eq_of_eq_some {Ξ± : Type u} : β {x y : Option Ξ±}, (βz, x = some z β y = some z) β x = y
... |
open import Relation.Binary.Core
module Heapsort.Impl1.Correctness.Order {A : Set}
(_β€_ : A β A β Set)
(totβ€ : Total _β€_)
(transβ€ : Transitive _β€_) where
open import Data.List
open import Function using (_β_)
open import Heapsort.Impl1 _β€_ totβ€ transβ€
open impor... |
informal statement Suppose $a, b \in R^k$. Find $c \in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$.formal statement theorem exercise_2_24 {X : Type*} [metric_space X]
(hX : β (A : set X), infinite A β β (x : X), x β closure A) :
separable_s... |
informal statement Prove that a group of order 312 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.formal statement theorem exercise_4_5_16 {p q r : β} {G : Type*} [group G]
[fintype G] (hpqr : p < q β§ q < r)
(hpqr1 : p.prime β§ q.prime β§ r.prime)(hG : card G = p*q*r) :
nonempty (sylow p ... |
import linear_algebra.finite_dimensional
lemma set.eq_empty_or_eq_insert {x : Type*} (s : set x) : s = β
β¨ β t (x β t), s = insert x t :=
begin
rcases s.eq_empty_or_nonempty with (rfl|β¨x, hxβ©),
{ exact or.inl rfl },
{ refine or.inr β¨s \ {x}, x, _, _β©; simp [set.insert_eq_of_mem hx] }
end
lemma csupr_subtype {Ξ± ... |
informal statement Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$.formal statement theorem exercise_1_1_18 {G : Type*} [group G]
(x y : G) : x * y = y * x β yβ»ΒΉ * x * y = x β xβ»ΒΉ * yβ»ΒΉ * x * y = 1 := |
lemma interior_empty [simp]: "interior {} = {}" |
informal statement Prove that two elements $a, b$ of a group generate the same subgroup as $b a b^2, b a b^3$.formal statement theorem exercise_6_8_1 {G : Type*} [group G]
(a b : G) : closure ({a, b} : set G) = closure {b*a*b^2, b*a*b^3} := |
/-
Practice with predicate logic in Lean
-/
variable {Ξ± : Type} (P Q : Ξ± β Prop)
theorem prob01 (aβ : Ξ±) (h : β a, P a) : β a, P a := sorry
theorem prob02 (h : β a, P a β§ Β¬ Q a) (h : β a, P a β Q a) : False := sorry
theorem prob03 (a a' : Ξ±) (h : a = a') (h' : P a) : P a' := sorry
|
[GOAL]
β’ 1 = 1
[PROOFSTEP]
sleep_heartbeats 1000
[GOAL]
β’ 1 = 1
[PROOFSTEP]
rfl
|
open tactic
lemma ex1 (a b c : nat) : a + 0 = 0 + a β§ 0 + b = b β§ c + b = b + c :=
begin
repeat {any_goals {constructor}},
show c + b = b + c, { apply add_comm }, -- third goal of three
show a + 0 = 0 + a, { simp }, -- first of two
show 0 + b = b, { rw [zero_add] }
end
/- Same example, but the local conte... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import algebra.algebra.subalgebra
import topology.algebra.polynomial
import topology.continuous_function.bounded
import analysis.special_functions.bernstein
/-!
# Th... |
import data.real.basic
open classical
attribute [instance] prop_decidable
/-
Rigorous definition of a limit
For a sequence x_n, we say that \lim_{n \to \infty} x_n = l if
β Ξ΅ > 0, β N, n β₯ N β |x_n - l| < Ξ΅
-/
def lim_to_inf (x : β β β) (l : β) :=
β Ξ΅ > 0, β N, β n β₯ N, abs (x n - l) < Ξ΅
/-
Bounded seque... |
informal statement Show that there are infinitely many primes congruent to $-1$ modulo 6 .formal statement theorem exercise_3_5 : Β¬ β x y : β€, 7*x^3 + 2 = y^3 := |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
! This file was ported from Lean 3 source module data.finite.set
! leanprover-community/mathlib commit 509de852e1de55e1efa8eacfa11df0823f26f226
! Please do not edit these lin... |
import data.real.basic
theorem challenge3 :
(2 : β) + 2 β 5 :=
begin
sorry
end
|
informal statement Show that 2 is divisible by $(1+i)^{2}$ in $\mathbb{Z}[i]$.formal statement theorem exercise_2_21 {l : β β β}
(hl : β p n : β, p.prime β l (p^n) = log p )
(hl1 : β m : β, Β¬ is_prime_pow m β l m = 0) :
l = Ξ» n, β d : divisors n, moebius (n/d) * log d := |
informal statement Define $\wedge(n)=\log p$ if $n$ is a power of $p$ and zero otherwise. Prove that $\sum_{A \mid n} \mu(n / d) \log d$ $=\wedge(n)$.formal statement theorem exercise_2_21 {l : β β β}
(hl : β p n : β, p.prime β l (p^n) = log p )
(hl1 : β m : β, Β¬ is_prime_pow m β l m = 0) :
l = Ξ» n, β d : diviso... |
informal statement For all odd $n$ show that $8 \mid n^{2}-1$.formal statement theorem exercise_1_31 : (β¨1, 1β© : gaussian_int) ^ 2 β£ 2 := |
import data.list.basic
open list
universe u
variables {Ξ± : Type} (x y z : Ξ±) (xs ys zs : list Ξ±)
def mk_symm (xs : list Ξ±) := xs ++ reverse xs
attribute [simp]
theorem reverse_mk_symm (xs : list Ξ±) :
reverse (mk_symm xs) = mk_symm xs :=
by simp [mk_symm]
example (xs ys : list β) :
reverse (xs ++ mk_symm ys) = ... |
import Meta.Boolean
import Meta.Resolution
theorem mpCvc5 (P Q : Prop) : Β¬ (P β (P β Q) β Q) β False :=
Ξ» lean_a0 =>
have lean_s0 := notImplies2 lean_a0
have lean_s1 := notImplies1 lean_s0
have lean_s2 := impliesElim lean_s1
have lean_s4 := notImplies1 lean_a0
have lean_s6 := ... |
example (x y : β) (h : x = y) : y = x :=
begin
revert x y,
intros,
symmetry,
assumption
end
|
informal statement Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}(x)\right)$ converges for each $x \in[0,1]$.formal statement theorem exercise_21_8
{X : Type*} [topological_space X] {Y : Type*} [metric_space Y]
{f : β β X β Y} {x : β β X}
(hf : β ... |
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