text stringlengths 0 3.34M |
|---|
lemma example2 (x y : mynat) (h : y = x + 7) : 2 * y = 2 * (x + 7) :=
begin
rw ← h,
refl,
end |
[GOAL]
⊢ DecidableRel fun x x_1 => x < x_1
[PROOFSTEP]
simp only [LT']
[GOAL]
⊢ DecidableRel fun s₁ s₂ => ltb (iter s₁) (iter s₂) = true
[PROOFSTEP]
infer_instance
-- short-circuit type class inference
[GOAL]
c : Char
cs₁ cs₂ : List Char
i₁ i₂ : Pos
⊢ ltb { s := { data := c :: cs₁ }, i := i₁ + c } { s := { data := c ... |
informal statement If $G$ is a group and $a, x \in G$, prove that $C\left(x^{-1} a x\right)=x^{-1} C(a) x$formal statement theorem exercise_2_4_36 {a n : ℕ} (h : a > 1) :
n ∣ (a ^ n - 1).totient := |
[GOAL]
a b : ℕ
⊢ a ≠ b ↔ ↑a ≠ ↑b
[PROOFSTEP]
simp only [ne_eq, Int.cast_eq_cast_iff_Nat]
|
universes v
/-
matcher for the following patterns
```
| "hello" => _
| "world" => _
| a => _
``` -/
def matchString (C : String → Sort v) (s : String)
(h₁ : Unit → C "hello")
(h₂ : Unit → C "world")
(h₃ : ∀ s, C s)
: C s :=
dite (s = "hello")
(fun h => @Eq.ndrec _ _ (fun x => C x) (h₁ ()) _ h.... |
import data.real.basic
theorem challenge3 :
(2 : ℝ) + 2 ≠ 5 :=
begin
norm_num
end
|
informal statement Let $X$ be a topological space and let $Y$ be a metric space. Let $f_{n}: X \rightarrow Y$ be a sequence of continuous functions. Let $x_{n}$ be a sequence of points of $X$ converging to $x$. Show that if the sequence $\left(f_{n}\right)$ converges uniformly to $f$, then $\left(f_{n}\left(x_{n}\right... |
import analysis.real tactic.norm_num algebra.group_power
theorem Q5a1 (S : set ℝ) : (∃ x : ℝ, x ∈ lower_bounds S)
↔ (∃ y : ℝ, y ∈ upper_bounds {t : ℝ | ∃ s ∈ S, t = -s }) := sorry
theorem Q5a2 (S : set ℝ) (x : ℝ) : is_glb S x ↔
is_lub {t : ℝ | ∃ s ∈ S, t = -s} (-x) := sorry
lemma Q5bhelper (S : set ℝ) (x₁ ... |
import data.real.basic
lemma mp (p q : Prop) :
p → (p → q) → q :=
λ hp hpq, hpq hp
|
informal statement Show that the subgroup of all rotations in a dihedral group is a maximal subgroup.formal statement theorem exercise_3_1_3a {A : Type*} [comm_group A] (B : subgroup A) :
∀ a b : A ⧸ B, a*b = b*a := |
informal statement Prove that characteristic subgroups are normal.formal statement theorem exercise_4_4_6a {G : Type*} [group G] (H : subgroup G)
[subgroup.characteristic H] : subgroup.normal H := |
import group_theory.subgroup
theorem cpge_groupe_9_a {G1 : Type*} [group G1] {G2 : Type*} [group G2]
(f : G1 →* G2) : ∀ (a : G1), ( (f a = 1) → (∀ (x : G1), f (x * a * x⁻¹) = 1)) := sorry
|
informal statement Prove that if an integer is the sum of two rational squares, then it is the sum of two integer squares.formal statement theorem exercise_8_3_6a {R : Type*} [ring R]
(hR : R = (gaussian_int ⧸ ideal.span ({⟨0, 1⟩} : set gaussian_int))) :
is_field R ∧ ∃ finR : fintype R, @card R finR = 2 := |
import data.real.basic tactic.ring tactic.tidy
/- Definitions -/
definition double (n : ℕ) : ℕ := n + n
#check double
#check double ∘ double
definition quadruple : ℕ → ℕ := double ∘ double
definition FLT : Prop :=
∀ n > 2, ∀ x y z, x^n + y^n = z^n → (x = 0 ∨ y = 0)
theorem Wiles : FLT :=
begin
unfold FLT,
in... |
import linear_algebra.basic
universes u v w x
variables {R : Type u} [ring R]
variables {M₁ : Type v} [add_comm_group M₁] [module R M₁]
variables {M₂ : Type w} [add_comm_group M₂] [module R M₂]
variables {M₃ : Type x} [add_comm_group M₃] [module R M₃]
open linear_map
open submodule
lemma linear_map.ker_le_range_iff... |
-- Andreas, 2017-01-14, issue #2405 reported by m0davis
-- Instance not found due to regression introduced by
-- parameter-refinement.
-- {-# OPTIONS --show-implicit #-}
-- {-# OPTIONS -v tc.instance:70 #-}
-- {-# OPTIONS -v tc.meta.assign:40 #-}
-- {-# OPTIONS -v tc.conv:40 #-}
-- {-# OPTIONS -v tc.sig.param:100 #-}
... |
/-
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... |
opaque f : Nat → Nat
opaque g : Nat → Nat
namespace Foo
@[scoped simp] axiom ax1 (x : Nat) : f (g x) = x
@[scoped simp] axiom ax2 (x : Nat) : g (g x) = g x
end Foo
theorem ex1 : f (g (g (g x))) = x := by
simp -- does not use ax1 and ax2
simp [Foo.ax1, Foo.ax2]
theorem ex2 : f (g (g (g x))) = x :=
have h₁ : f... |
theorem ex1 (p q r : Prop) (h1 : p ∨ q) (h2 : p → q) : q := by
theorem ex2 (p q r : Prop) (h1 : p ∨ q) (h2 : p → q) : q := by
cases h1
case inl =>
|
[STATEMENT]
lemma ensures_simple:
"\<lbrakk> \<turnstile> $P \<and> N \<longrightarrow> P` \<or> Q`;
\<turnstile> ($P \<and> N) \<and> A \<longrightarrow> Q`
\<rbrakk> \<Longrightarrow> \<turnstile> \<box>N \<and> \<box>\<diamond>A \<longrightarrow> (P \<leadsto> Q)"
[PROOF STATE]
proof (prove)
goal (1 subgo... |
import data.real.basic
import data.polynomial.basic
import data.polynomial.ring_division
import data.complex.basic
theorem USAMO_Problem_3_1977 (a b : ℂ):
(a ∈ polynomial.roots ((polynomial.monomial 4 (1:ℂ ) ) + (polynomial.monomial 3 (1:ℂ) ) - 1) ∧
b ∈ polynomial.roots ((polynomial.monomial 4 (1:ℂ) ) + (polynomial.... |
example : ∃ a : ℕ, 5 = a :=
begin
apply exists.intro,
reflexivity
end
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
! This file was ported from Lean 3 source module data.fin.tuple.bubble_sort_induction
! leanprover-community/mathlib commit 50832daea47b195a48b5b33b1c8b2162c48c3afc
! Ple... |
informal statement Let $H \leq K \leq G$. Prove that $|G: H|=|G: K| \cdot|K: H|$ (do not assume $G$ is finite).formal statement theorem exercise_3_2_11 {G : Type*} [group G] {H K : subgroup G}
(hHK : H ≤ K) :
H.index = K.index * H.relindex K := |
universe u
variable {α : Type u}
def split : List α → List α × List α
| [] => ([], [])
| [a] => ([a], [])
| a::b::as => (a :: (split as).1, b :: (split as).2)
theorem ex1 : split [1, 2, 3, 4, 5] = ([1, 3, 5], [2, 4]) :=
rfl
|
{-# OPTIONS --erased-cubical #-}
module Erased-cubical-Open-public.Erased (_ : Set₁) where
postulate
A : Set
|
theory Scratch
imports Main
begin
lemma "1=1" by simp
end |
universes u v
inductive Vec2 (α : Type u) (β : Type v) : Nat → Type (max u v)
| nil : Vec2 α β 0
| cons : α → β → forall {n}, Vec2 α β n → Vec2 α β (n+1)
inductive Fin2 : Nat → Type
| zero (n : Nat) : Fin2 (n+1)
| succ {n : Nat} (s : Fin2 n) : Fin2 (n+1)
theorem test1 {α β} {n} (a₁ a₂ : α) (b₁ b₂ : β) ... |
def f (a : Array Nat) (i : Nat) (v : Nat) (h : i < a.size) : Array Nat :=
a.set ⟨i, h⟩ (a.get ⟨i, h⟩ + v)
set_option pp.proofs true
theorem ex1 (h₃ : i = j) : f a i (0 + v) h₁ = f a j v h₂ := by
simp
trace_state
simp [h₃]
theorem ex2 (h₃ : i = j) : f a (0 + i) (0 + v) h₁ = f a j v h₂ := by
simp
trace_sta... |
@[simp] theorem liftOn_mk (a : α) (f : α → γ) (h : ∀ a₁ a₂, r a₁ a₂ → f a₁ = f a₂) :
Quot.liftOn (Quot.mk r a) f h = f a := rfl
theorem eq_iff_true_of_subsingleton [Subsingleton α] (x y : α) : x = y ↔ True :=
iff_true _ ▸ Subsingleton.elim ..
section attribute [simp] eq_iff_true_of_subsingleton end
@[simp] the... |
section {* \isaheader{Generic Compare Algorithms} *}
theory Gen_Comp
imports
"../Intf/Intf_Comp"
"../../../Automatic_Refinement/Automatic_Refinement"
begin
subsection {* Order for Product *}
(* TODO: Optimization? Or only go via prod_cmp? *)
lemma autoref_prod_cmp_dflt_id[autoref_rules_raw]:
"(dflt_cmp op \<l... |
--
/-
This example demonstratea that when we are using `nativeDecide`,
we are also trusting the correctness of `implementedBy` annotations,
foreign functions (i.e., `[extern]` annotations), etc.
-/
def g (b : Bool) := false
/-
The following `implementedBy` is telling the compiler
"trust me, `g` does implement `f`"
wh... |
[GOAL]
⊢ DivisionRing ℚ
[PROOFSTEP]
infer_instance
|
informal statement Show that there are infinitely many primes congruent to $-1$ modulo 6 .formal statement theorem exercise_3_1 : infinite {p : primes // p ≡ -1 [ZMOD 6]} := |
/-|
Hello World!
-/
#print "Hello World!"
/-|
A literate comment!
-/
theorem exampleTheorem (p q : Prop) (hp : p) (hq : q) : p ∧ q ∧ p := by
apply And.intro
. exact hp
. sorry |
class Preorder (α : Type u) extends LE α where
le_refl (a : α) : a ≤ a
le_trans {a b c : α} : a ≤ b → b ≤ c → a ≤ c
instance {α : Type u} {β : α → Type v} [(a : α) → Preorder (β a)] : Preorder ((a : α) → β a) where
le f g := ∀ a, f a ≤ g a
le_refl f := fun a => Preorder.le_refl (f a)
le_trans := fun h₁ h... |
From stdpp Require Import base tactics.
(** Test parsing of variants of [(≡)] notation. *)
Lemma test_equiv_annot_sections `{!Equiv A, !Equivalence (≡@{A})} (x : A) :
x ≡@{A} x ∧ (≡@{A}) x x ∧ (x ≡.) x ∧ (.≡ x) x ∧
((x ≡@{A} x)) ∧ ((≡@{A})) x x ∧ ((x ≡.)) x ∧ ((.≡ x)) x ∧
( x ≡@{A} x) ∧ ( x ≡.) x ∧
(x ≡@{A} x ... |
import LeanCodePrompts.FirstTacticFinder
import Mathlib.Tactic.Basic
import Mathlib.Tactic.Use
import Mathlib
-- example : ∀ n : ℕ, ∃ m : ℕ, n < 2 * m + 1 := by
-- repeat (aide_lookahead)
-- repeat (sorry)
set_option trace.Translate.info true
example : ∀ n : ℕ, ∃ m : ℕ, n < 2 * m + 1 := by
show_tactic_prompt
... |
informal statement Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint.formal statement theorem exercise_32_1 {X : Type*} [topological_space X]
(hX : normal_space X) (A : set X) (hA : is_closed A) :
normal_space {x // x ∈ A} := |
theorem P_implies_P (P : Prop) : P → P :=
begin
intro HP,
exact HP
end
|
import Mathlib.Data.Nat.Basic
theorem rangeDecompose (start mid stop : ℕ) (hs : start ≤ mid ∧ mid ≤ stop)
{f : ℕ → β → Id (ForInStep β)} :
STD.forIn (mkRange' start stop) init f =
STD.forIn (mkRange' mid stop) (Id.run (STD.forIn (mkRange' start mid) init f)) f := by
sorry |
import help data.nat.prime
open nat
theorem Euclid (n : ℕ) : ∃ p ≥ n, prime p :=
begin
let N := n.fact + 1,
let p := min_fac N,
use_this p,
have p_is_prime : prime p := min_fac_prime _,
split,
show p ≥ n,
by_contradiction,
have key_fact : p ∣ n.fact := dvd_fact _ _,
have oops : p ∣ 1,
all_... |
lemma lt_aux_one (a b : mynat) : a ≤ b ∧ ¬ (b ≤ a) → succ a ≤ b :=
begin
intro h,
cases h with h1 h2,
cases h1 with c h1,
cases c,
rw add_zero at h1,
rw h1 at h2,
exfalso,
exact h2 (le_refl a),
rwa [h1, add_succ],
apply succ_le_succ,
use c,
refl,
end
|
(* Title: HOL/ex/PresburgerEx.thy
Author: Amine Chaieb, TU Muenchen
*)
section {* Some examples for Presburger Arithmetic *}
theory PresburgerEx
imports Presburger
begin
lemma "\<And>m n ja ia. \<lbrakk>\<not> m \<le> j; \<not> (n::nat) \<le> i; (e::nat) \<noteq> 0; Suc j \<le> ja\<rbrakk> \<Longrighta... |
informal statement Show that any subgroup of order $p^{n-1}$ in a group $G$ of order $p^n$ is normal in $G$.formal statement theorem exercise_4_1_19 : infinite {x : quaternion ℝ | x^2 = -1} := |
import tactic
variable X : Type
theorem reflexive_of_symmetric_transitive_and_connected
(r : X → X → Prop)
(h_symm : ∀ x y : X, r x y → r y x)
(h_trans : ∀ x y z : X, r x y → r y z → r x z)
(h_connected : ∀ x, ∃ y, r x y)
: (∀ x : X, r x x) :=
begin
sorry,
end |
module Main
mutual
%inline
is_even : Int -> Int
is_even n =
if n == 0 then 1 else is_odd $ n-1
is_odd : Int -> Int
is_odd n =
if n == 0 then 0 else is_even $ n-1
main : JS_IO ()
main = do
putStr' $ show $ is_even 100001
|
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_3_1 : infinite {p : primes // p ≡ -1 [ZMOD 6]} := |
lemma contrapositive2 (P Q : Prop) : (¬ Q → ¬ P) → (P → Q) :=
begin
by_cases p : P; by_cases q : Q,
intros h p2,
exact q,
repeat { tauto! },
end |
informal statement If $C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0,$ where $C_{0}, \ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1.formal statement theorem exercise_5_6
{f : ℝ → ℝ}
(hf1 ... |
theorem tst0 (x : Nat) : x + 0 = x + 0 :=
by {
generalize x + 0 = y;
exact (Eq.refl y)
}
theorem tst1 (x : Nat) : x + 0 = x + 0 :=
by {
generalize h : x + 0 = y;
exact (Eq.refl y)
}
theorem tst2 (x y w : Nat) (h : y = w) : (x + x) + w = (x + x) + y :=
by {
generalize h' : x + x = z;
subst y;
exact Eq... |
open import Relation.Binary.Core
module TreeSort.Impl2.Correctness.Order {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_)
(trans≤ : Transitive _≤_) where
open import BBSTree _≤_
open import BBSTree.Properties _≤_ trans≤
open import Data.List
open import Functi... |
abbrev M := ExceptT String <| StateT Nat Id
def f (xs : List Nat) : M Unit := do
for x in xs do
if x == 0 then
throw "contains zero"
#eval f [1, 2, 3] |>.run' 0
#eval f [1, 0, 3] |>.run' 0
theorem ex1 : (f [1, 2, 3] |>.run' 0) = Except.ok () :=
rfl
theorem ex2 : (f [1, 0, 3] |>.run' 0) = Except.error "contain... |
lemma zero_le (a : mynat) : 0 ≤ a :=
begin
rw le_iff_exists_add,
use a,
rwa zero_add,
end
|
/-
# Advanced proposition world.
## Level 3: and_trans.
-/
/- Lemma
If $P$, $Q$ and $R$ are true/false statements, then $P\land Q$ and
$Q\land R$ together imply $P\land R$.
-/
lemma and_trans (P Q R : Prop) : P ∧ Q → Q ∧ R → P ∧ R :=
begin
intro hpq,
intro hqr,
cases hpq with p q,
cases hqr with q' r,
sp... |
informal statement Show that $\int_0^1 \log(\sin \pi x) dx = - \log 2$.formal statement theorem exercise_3_9 : ∫ x in 0..1, real.log (real.sin (real.pi * x)) = - real.log 2 := |
module ModusPonens where
modusPonens : ∀ {P Q : Set} → (P → Q) → P → Q
modusPonens x = x
|
[STATEMENT]
lemma HIm_nth [simp]:
"HIm x $ 1 = Im1 x" "HIm x $ 2 = Im2 x" "HIm x $ 3 = Im3 x" "HIm x $ 4 = Im4 x"
"HIm x $ 5 = Im5 x" "HIm x $ 6 = Im6 x" "HIm x $ 7 = Im7 x"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (HIm x $ 1 = Im1 x &&& HIm x $ 2 = Im2 x &&& HIm x $ 3 = Im3 x) &&& (HIm x $ 4 = Im4 x &&& ... |
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F
open_locale nat rat real big_operators topological_space
/--
Suppose that $f(x+3)=3x^2 + 7x + 4$ a... |
theorem tst1 (x y z : Nat) : y = z → x = x → x = y → x = z :=
by {
intros h1 h2 h3;
revert h2;
intro h2;
exact Eq.trans h3 h1
}
theorem tst2 (x y z : Nat) : y = z → x = x → x = y → x = z :=
by {
intros h1 h2 h3;
revert y;
intros y hb ha;
exact Eq.trans ha hb
}
theorem tst3 (x y z : Nat) : y = z → x ... |
universe u
theorem instForAll {A : Type u} {f : A → Prop} {a : A} :
(forall a' : A, f a') → f a := by
intro h
exact h a
theorem instEqual₁ {A : Type u} {P : A → Prop} {t : A} :
(forall x : A, x = t → P x) → P t := by
intro h
exact h t rfl
theorem instEqual₂ {A : Type u} {P : A → Prop} {t : A} :
P t → (... |
lemma ne_succ_self (n : ℕ) : n ≠ nat.succ n :=
begin
end |
import data.real.basic
import data.set.intervals.unordered_interval
theorem USAMO_Problem_3_1993 (f : ℝ -> ℝ) [f ≥ 0] [f(1) = 1]:
(∀ x y : ℝ, (x ∈ (set.interval (0 : ℝ) (1: ℝ)) ∧ y ∈ (set.interval (0 : ℝ) (1: ℝ))) →
x+y ∈ (set.interval (0 : ℝ) (1: ℝ)) → f(x) + f(y) ≤ f(x+y)) →
(∀ x : ℝ, x ∈ (set.interval (0 : ℝ) (... |
informal statement Prove that $\cos 1^{\circ}$ is algebraic over $\mathbb{Q}$.formal statement theorem exercise_5_3_10 : is_algebraic ℚ (cos (real.pi / 180)) := |
lemma convex_prod: assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}" shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}" |
section
variables (P Q : Prop)
theorem my_theorem : P ∧ Q → Q ∧ P :=
assume h : P ∧ Q,
have P, from and.left h,
have Q, from and.right h,
show Q ∧ P, from and.intro ‹Q› ‹P›
end
|
informal statement Let $F = \mathbb{Z}_p$ be the field of integers $\mod p$, where $p$ is a prime, and let $q(x) \in F[x]$ be irreducible of degree $n$. Show that $F[x]/(q(x))$ is a field having at exactly $p^n$ elements.formal statement theorem exercise_4_5_25 {p : ℕ} (hp : nat.prime p) :
irreducible (∑ i : finset.r... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import data.finset.basic
/-!
# Finsets of ordered types
-/
universes u v w
variables {α : Type u}
theorem directed.finset_le {r : α → α → Prop} [is_tra... |
lemma of_real_0 [simp]: "of_real 0 = 0" |
import PnP2023.Lec_01_25.Answer
namespace Waffle
theorem Answer.eq_of_le_le (a b : Answer) :
a ≤ b → b ≤ a → a = b := by sorry |
example : ∀ a b c : ℕ, a = b → a = c → c = b :=
begin
intros,
transitivity,
symmetry,
assumption,
assumption
end
|
lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0" |
lemma frontier_empty [simp]: "frontier {} = {}" |
/-
Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Jeremy Avigad
The square root function.
-/
import .ivt
open analysis real classical topology
noncomputable theory
private definition sqr_lb (x : ℝ) : ℝ := 0
privat... |
def g (x : Nat) (b : Bool) :=
if b then
x - 1
else
x + 1
theorem g_eq (x : Nat) (h : ¬ x = 0) : g x (x > 0) = x - 1 ∧ g x false = x + 1 := by
have : x > 0 := by match x with
| 0 => contradiction
| x+1 => apply Nat.zero_lt_succ
simp [g, this]
macro_rules
| `(tactic| decreasing_tactic) =>
`(tac... |
/-
lemma divides_p_times (r : ℕ) (n : ℕ) : p^r ∣ n ↔ p^(r+1) ∣ (p * n) :=
calc p^r ∣ n ↔ (p * p^r) ∣ (p * n) : (nat.mul_dvd_mul_iff_left (gt_zero hp)).symm
... = (p^r * p ∣ p * n) : by rw mul_comm
lemma exactly_divides_p_times (r : ℕ) (n : ℕ) : p^r ∣∣ n ↔ p^(r+1) ∣∣ (p * n) :=
have eq : (p * n) / p^(r+... |
section
variables (x y z : ℕ)
variables (h₁ : x = y) (h₂ : y = z)
include h₁ h₂
theorem foo : x = z :=
begin
rw [h₁, h₂]
end
omit h₁ h₂
theorem bar : x = z :=
eq.trans h₁ h₂
theorem baz : x = x := rfl
#check @foo
#check @bar
#check baz
end
|
/-
Copyright (c) Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
! This file was ported from Lean 3 source module analysis.complex.real_deriv
! leanprover-community/mathlib commit e9be2fa75faa22892937c275e27a91cd558cf8... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import algebra.char_p.basic
import algebra.ring_quot
/-!
# Characteristic of quotients rings
-/
universes u v
namespace char_p
theorem quotient (R : Type u) [... |
State Before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), IsCaratheodory m (s i)
hd : Pairwise (Disjoint on s)
t : Set α
⊢ ∑ i in Finset.range 0, ↑m (t ∩ s i) = ↑m (t ∩ ⋃ (i : ℕ) (_ : i < 0), s i) State After: no goals Tactic: simp [Nat.not_lt_zero, m.empty] State Before: α : Type u
m : ... |
From iris.proofmode Require Import tactics.
From aneris.aneris_lang Require Import network adequacy.
From aneris.aneris_lang.program_logic Require Export aneris_weakestpre.
Definition aneris_adequate (e :expr) (ip : ip_address) (σ : state)
(φ : val → Prop) :=
adequate NotStuck (mkExpr ip e) σ (λ v _, ∃ w... |
Class inhabited(A: Type): Type := mk_inhabited { default: A }.
Global Arguments mk_inhabited {_} _.
Global Hint Mode inhabited + : typeclass_instances.
Global Hint Extern 1 (inhabited _) =>
simple refine (mk_inhabited _); constructor
: typeclass_instances.
Module InhabitedTests.
Goal inhabited nat. typeclasses ... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import data.real.sqrt
import tactic.interval_cases
import ring_theory.algebraic
import data.rat.sqrt
... |
lemma coeffs_map_poly [code abstract]: "coeffs (map_poly f p) = strip_while ((=) 0) (map f (coeffs p))" |
informal statement Let $V$ be a vector space which is spanned by a countably infinite set. Prove that every linearly independent subset of $V$ is finite or countably infinite.formal statement theorem exercise_6_1_14 (G : Type*) [group G]
(hG : is_cyclic $ G ⧸ (center G)) :
center G = ⊤ := |
theorem very_easy : true :=
begin
sorry
end
|
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import tendsto_zero_of_sum_even_char
--import p_adic_L_function_def
-- import general_bernoulli_number.basic
-- import topology.algebra.nonarchimedean.bases
-- im... |
import .affine_space_type
import .affine_frame
import data.real.basic
def reals_as_affine_space := affine_space_type
ℝ
ℝ
ℝ
#check reals_as_affine_space
def reals_as_affine_space_1 : reals_as_affine_space := ⟨⟩
def reals_as_affine_space_2 : reals_as_affine_space := ⟨⟩
example : reals_as_affine_space_... |
[STATEMENT]
lemma min_enat_simps [simp]:
"min (enat m) (enat n) = enat (min m n)"
"min q 0 = 0"
"min 0 q = 0"
"min q (\<infinity>::enat) = q"
"min (\<infinity>::enat) q = q"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (min (enat m) (enat n) = enat (min m n) &&& min q 0 = 0) &&& min 0 q = 0 &&& min q \<i... |
/-
Copyright (c) 2021 Alain Verberkmoes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alain Verberkmoes
-/
import data.int.basic
/-!
# IMO 2011 Q5
Let `f` be a function from the set of integers to the set
of positive integers. Suppose that, for any two integers
`... |
import Mathlib.Data.Rat.Order
import Mathlib.Tactic.Ring
/- 4 points -/
theorem problem1 {x : ℚ} (hx : x = 2/3) : 3 * x ≠ 1 := by
apply ne_of_gt
calc 3 * x = 3 * (2 / 3) := by rw [hx]
_ > 1 := by rfl
/- 5 points -/
theorem problem2 {x y : ℚ} (h : x = 1 ∨ y = -1) :
x * y + x = y + 1 := by
cases' h with hx ... |
{-# OPTIONS --rewriting #-}
postulate
_↦_ : {A : Set} → A → A → Set
idr : {A : Set} {a : A} → a ↦ a
{-# BUILTIN REWRITE _↦_ #-}
record _×_ (A B : Set) : Set where
constructor _,_
field
fst : A
snd : B
postulate
A B C : Set
g : A → B → C
f : A × B → C
f (x , y) = g x y
postulate
D : Set
P ... |
/-
Description : Formalization of islanders problem for two people.
Copyright : (c) Daniel Selsam, 2018
License : GPL-3
-/
import .util .knows .problem
theorem islanders_n2 : islanders 2 :=
assume everyone_sees :
∀ (d₁ d₂ : day) (p₁ p₂ : person), p₂ ≠ p₁ → common_knowledge (is_marked p₂ → knows d₂ p₁ (is_... |
lemma of_real_1 [simp]: "of_real 1 = 1" |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import algebra.char_p.basic
import ring_theory.ideal.quotient
/-!
# Characteristic of quotients rings
-/
universes u v
namespace char_p
theorem quotient (R : ... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import data.real.sqrt
import data.rat.sqrt
import ring_theory.int.basic
import data.polynomial.eval
i... |
informal statement If $s_{1}=\sqrt{2}$, and $s_{n+1}=\sqrt{2+\sqrt{s_{n}}} \quad(n=1,2,3, \ldots),$ prove that $\left\{s_{n}\right\}$ converges, and that $s_{n}<2$ for $n=1,2,3, \ldots$.formal statement theorem exercise_3_6a
: tendsto (λ (n : ℕ), (∑ i in finset.range n, g i)) at_top at_top := |
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im] |
informal statement Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$.formal statement theorem exercise_4_4a
{α : Type} [metric_space α]
{β : Type} [metric_space β]
(f : α → β)
(s : set α)
(h₁ : conti... |
import tactic
-- the next two lines let us use the law of the excluded middle without trouble
noncomputable theory
open_locale classical
--BEGIN--
/--------------------------------------------------------------------------
Delete the ``sorry,`` below and replace them with a legitimate proof.
-----------------------... |
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