Zulip Chat Archive
Stream: lean4
Topic: I want to prove that b_i divides m. Why does the error 'ty
grandxuan pan (Aug 05 2025 at 14:18):
Kenny Lau (Aug 05 2025 at 14:19):
please copy your code (#backticks, #mwe)
grandxuan pan (Aug 05 2025 at 14:19):
import Mathlib.Algebra.Group.Defs -- 群论基础
import Mathlib.Data.Real.Basic -- 实数理论
import Mathlib.Analysis.Calculus.Deriv.Basic -- 微积分
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic -- 提供 Finset 和最小公倍数支持
import Mathlib.Data.Fintype.Basic -- 提供有限类型支持
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List -- 打开List命名空间
open Finset Nat
open Nat
namespace NumberTheory
-- 定义命名空间下全局变量和假设
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk0 : 0 < k := Nat.lt_of_lt_of_le (by decide) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ)
variable (ha_range : ∀ i, i ∈ Set.Icc (1 : ℕ) k)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
variable (ha_inj : Function.Injective a)
variable (ha_inji : Function.Injective b)
def a₀ : ℕ := a ⟨0, hk0⟩
def b₀ : ℕ := b ⟨0, hk0⟩
def a_i (i : Fin k) : ℕ := a i
def b_i (i : Fin k) : ℕ := b i
example (i : Fin k) : ℕ := a i -- aᵢ
example (i : Fin k) : ℕ := b i -- bᵢ
variable (h_coprime : ∀ (i : Fin k), gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values : Finset ℕ := Finset.image b (indices k)
def m (k : ℕ) (b : Fin k → ℕ): ℕ := Finset.fold lcm 1 b (indices k)
-- 步骤4: 证明每个bᵢ整除m
lemma b_dvd_m (i : Fin k) : b i ∣ m k b:= by
unfold m
have mem_b : b i ∈ b_values k b := by
{unfold b_values
unfold indices
rw [Finset.mem_image]
use i
constructor
· exact List.mem_finRange i -- i在finRange中
· rfl}
exact dvd_lcm
end NumberTheory
Kenny Lau (Aug 05 2025 at 14:21):
grandxuan pan (Aug 05 2025 at 14:37):
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List
open Finset Nat
open Nat
namespace NumberTheory
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk0 : 0 < k := Nat.lt_of_lt_of_le (by decide) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ)
variable (ha_range : ∀ i, i ∈ Set.Icc (1 : ℕ) k)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
variable (ha_inj : Function.Injective a)
variable (ha_inji : Function.Injective b)
def a₀ : ℕ := a ⟨0, hk0⟩
def b₀ : ℕ := b ⟨0, hk0⟩
def a_i (i : Fin k) : ℕ := a i
def b_i (i : Fin k) : ℕ := b i
example (i : Fin k) : ℕ := a i -- aᵢ
example (i : Fin k) : ℕ := b i -- bᵢ
variable (h_coprime : ∀ (i : Fin k), gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values : Finset ℕ := Finset.image b (indices k)
def m (k : ℕ) (b : Fin k → ℕ): ℕ := Finset.fold lcm 1 b (indices k)
'''
lemma b_dvd_m (i : Fin k) : b i ∣ m k b:= by
unfold m
have mem_b : b i ∈ b_values k b := by
{unfold b_values
unfold indices
rw [Finset.mem_image]
use i
constructor
· exact List.mem_finRange i -- i在finRange中
· rfl}
exact dvd_lcm
'''
end NumberTheory
Kenny Lau (Aug 05 2025 at 14:40):
Ruben Van de Velde (Aug 05 2025 at 14:45):
To clarify, that's a link. Follow it and follow the instructions
grandxuan pan (Aug 05 2025 at 14:49):
Apologies, this is my first time asking questions on Zulip.please bear with my newbie status.
grandxuan pan (Aug 05 2025 at 15:03):
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List
open Finset Nat
open Nat
namespace NumberTheory
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk₀ : 0 < k := Nat.lt_of_lt_of_le (Nat.zero_lt_one) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
variable (h_coprime : ∀ i, gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values (b : Fin k → ℕ) : Finset ℕ := Finset.image b (indices k)
def m (b : Fin k → ℕ) : ℕ := Finset.fold lcm 1 b (indices k)
lemma b_dvd_m (i : Fin k) : b i ∣ m b := by
have mem_b : b i ∈ b_values b := by
rw [Finset.mem_image]
use i
constructor
· exact List.mem_finRange i
· rfl
exact dvd_lcm -- Error occurs here
end NumberTheory
Aaron Liu (Aug 05 2025 at 15:04):
Can you post the code block without using no-break spaces?
grandxuan pan (Aug 05 2025 at 15:18):
'''
import Mathlib.Algebra.Group.Defs -- 群论基础
import Mathlib.Data.Real.Basic -- 实数理论
import Mathlib.Analysis.Calculus.Deriv.Basic -- 微积分
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic -- 提供 Finset 和最小公倍数支持
import Mathlib.Data.Fintype.Basic -- 提供有限类型支持
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List -- 打开List命名空间
open Finset Nat
open Nat
namespace NumberTheory
-- 定义命名空间下全局变量和假设
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk0 : 0 < k := Nat.lt_of_lt_of_le (by decide) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ) -- 修复箭头前的不间断空格
variable (ha_range : ∀ i, i ∈ Set.Icc (1 : ℕ) k)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
-- 添加必要的类型类声明
variable (N : Type) [CancelCommMonoidWithZero N] [NormalizedGCDMonoid N]
-- 添加这些实例声明 ↓
variable [inst1 : CancelCommMonoidWithZero N]
variable [inst2 : NormalizedGCDMonoid N]
variable (ha_inj : Function.Injective a)
variable (ha_inji : Function.Injective b)
def a₀ : ℕ := a ⟨0, hk0⟩
def b₀ : ℕ := b ⟨0, hk0⟩
def a_i (i : Fin k) : ℕ := a i
def b_i (i : Fin k) : ℕ := b i
example (i : Fin k) : ℕ := a i -- aᵢ
example (i : Fin k) : ℕ := b i -- bᵢ
variable (h_coprime : ∀ (i : Fin k), gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values : Finset ℕ := Finset.image b (indices k)
def m (k : ℕ) (b : Fin k → ℕ) : ℕ := Finset.fold lcm 1 b (indices k) -- 修复定义后的空格
-- 步骤4: 证明每个bᵢ整除m
lemma b_dvd_m (i : Fin k) : b i ∣ m k b := by -- 修复冒号后的空格
unfold m
have mem_b : b i ∈ b_values k b := by
{unfold b_values
unfold indices
rw [Finset.mem_image]
use i
constructor
· exact Finset.mem_univ i -- 修复：应该是mem_univ而不是range
· rfl}
exact dvd_lcm mem_b -- 修复dvd_lcm前的空格
end NumberTheory'''
grandxuan pan (Aug 05 2025 at 15:21):
'''import Mathlib.Algebra.Group.Defs -- 群论基础
import Mathlib.Data.Real.Basic -- 实数理论
import Mathlib.Analysis.Calculus.Deriv.Basic -- 微积分
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic -- 提供 Finset 和最小公倍数支持
import Mathlib.Data.Fintype.Basic -- 提供有限类型支持
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List -- 打开List命名空间
open Finset Nat
open Nat
namespace NumberTheory
-- 定义命名空间下全局变量和假设
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk0 : 0 < k := Nat.lt_of_lt_of_le (by decide) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ) -- 修复箭头前的不间断空格
variable (ha_range : ∀ i, i ∈ Set.Icc (1 : ℕ) k)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
-- 添加必要的类型类声明
variable (N : Type) [CancelCommMonoidWithZero N] [NormalizedGCDMonoid N]
-- 添加这些实例声明 ↓
variable [inst1 : CancelCommMonoidWithZero N]
variable [inst2 : NormalizedGCDMonoid N]
variable (ha_inj : Function.Injective a)
variable (ha_inji : Function.Injective b)
def a₀ : ℕ := a ⟨0, hk0⟩
def b₀ : ℕ := b ⟨0, hk0⟩
def a_i (i : Fin k) : ℕ := a i
def b_i (i : Fin k) : ℕ := b i
example (i : Fin k) : ℕ := a i -- aᵢ
example (i : Fin k) : ℕ := b i -- bᵢ
variable (h_coprime : ∀ (i : Fin k), gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values : Finset ℕ := Finset.image b (indices k)
def m (k : ℕ) (b : Fin k → ℕ) : ℕ := Finset.fold lcm 1 b (indices k) -- 修复定义后的空格
-- 步骤4: 证明每个bᵢ整除m
lemma b_dvd_m (i : Fin k) : b i ∣ m k b := by -- 修复冒号后的空格
unfold m
have mem_b : b i ∈ b_values k b := by
{unfold b_values
unfold indices
rw [Finset.mem_image]
use i
constructor
· exact Finset.mem_univ i -- 修复：应该是mem_univ而不是range
· rfl}
exact dvd_lcm mem_b -- 修复dvd_lcm前的空格
end NumberTheory'''
grandxuan pan (Aug 05 2025 at 15:23):
'''
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List -- 打开List命名空间
open Finset Nat
open Nat
namespace NumberTheory
-- 定义命名空间下全局变量和假设
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk0 : 0 < k := Nat.lt_of_lt_of_le (by decide) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ)
variable (ha_range : ∀ i, i ∈ Set.Icc (1 : ℕ) k)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
variable (N : Type) [CancelCommMonoidWithZero N] [NormalizedGCDMonoid N]
variable [inst1 : CancelCommMonoidWithZero N]
variable [inst2 : NormalizedGCDMonoid N]
variable (ha_inj : Function.Injective a)
variable (ha_inji : Function.Injective b)
def a₀ : ℕ := a ⟨0, hk0⟩
def b₀ : ℕ := b ⟨0, hk0⟩
def a_i (i : Fin k) : ℕ := a i
def b_i (i : Fin k) : ℕ := b i
example (i : Fin k) : ℕ := a i -- aᵢ
example (i : Fin k) : ℕ := b i -- bᵢ
variable (h_coprime : ∀ (i : Fin k), gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values : Finset ℕ := Finset.image b (indices k)
def m (k : ℕ) (b : Fin k → ℕ) : ℕ := Finset.fold lcm 1 b (indices k)
lemma b_dvd_m (i : Fin k) : b i ∣ m k b := by
unfold m
have mem_b : b i ∈ b_values k b := by
{unfold b_values
unfold indices
rw [Finset.mem_image]
use i
constructor
· exact Finset.mem_univ i
· rfl}
exact dvd_lcm mem_b
end NumberTheory'''
Henrik Böving (Aug 05 2025 at 15:30):
You want to use three backticks: ```, not the normal ticks ''', according to Wikipedia it should be on the top left key of most (but not all) chinese keyboards.
grandxuan pan (Aug 05 2025 at 15:36):
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List
open Finset Nat
open Nat
namespace NumberTheory
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk₀ : 0 < k := Nat.lt_of_lt_of_le (Nat.zero_lt_one) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
variable (h_coprime : ∀ i, gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values (b : Fin k → ℕ) : Finset ℕ := Finset.image b (indices k)
def m (b : Fin k → ℕ) : ℕ := Finset.fold lcm 1 b (indices k)
lemma b_dvd_m (i : Fin k) : b i ∣ m b := by
have mem_b : b i ∈ b_values b := by
rw [Finset.mem_image]
use i
constructor
· exact List.mem_finRange i
· rfl
exact dvd_lcm -- Error occurs here
end NumberTheory```
Aaron Liu (Aug 05 2025 at 15:38):
You should put a newline between the three backticks and your first import statement
grandxuan pan (Aug 05 2025 at 15:42):
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GCDMonoid.Finset
open List
open Finset Nat
open Nat
namespace NumberTheory
variable (k : ℕ) (hk : 1 ≤ k)
variable (hk₀ : 0 < k := Nat.lt_of_lt_of_le (Nat.zero_lt_one) hk)
variable (i : Fin k)
variable (a : Fin k → ℕ) (b : Fin k → ℕ)
variable (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0)
variable (h_coprime : ∀ i, gcd (a i) (b i) = 1)
def indices (k : ℕ) : Finset (Fin k) := Finset.univ
def b_values (b : Fin k → ℕ) : Finset ℕ := Finset.image b (indices k)
def m (b : Fin k → ℕ) : ℕ := Finset.fold lcm 1 b (indices k)
lemma b_dvd_m (i : Fin k) : b i ∣ m b := by
have mem_b : b i ∈ b_values b := by
rw [Finset.mem_image]
use i
constructor
· exact List.mem_finRange i
· rfl
exact dvd_lcm -- Error occurs here
end NumberTheory
Aaron Liu (Aug 05 2025 at 15:52):
When I click on the link to go the web editor the error I get is
Application type mismatch: In the application
m b
the argument
b
has type
Fin k → ℕ : Type
but is expected to have type
ℕ : Type
Last updated: Dec 20 2025 at 21:32 UTC