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pkuAI4M
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lean_github

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https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
apply dvd_trans _ (dvd_mul_left _ _)
n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ 4 * ∏ i in erase s 3, i
n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ ∏ i in erase s 3, i
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ 4 * ∏ i in erase s 3, i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
apply dvd_prod_of_mem
n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ ∏ i in erase s 3, i
case ha n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∈ erase s 3
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∣ ∏ i in erase s 3, i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
simp
case ha n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∈ erase s 3
case ha n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ Β¬p = 3 ∧ p ∈ s
Please generate a tactic in lean4 to solve the state. STATE: case ha n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ p ∈ erase s 3 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
constructor <;> assumption
case ha n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ Β¬p = 3 ∧ p ∈ s
no goals
Please generate a tactic in lean4 to solve the state. STATE: case ha n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 ⊒ Β¬p = 3 ∧ p ∈ s TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
convert Nat.dvd_sub' pdvd this
n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i in erase s 3, i ⊒ p ∣ 3
case h.e'_4 n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i in erase s 3, i ⊒ 3 = 4 * ∏ i in erase s 3, i + 3 - 4 * ∏ i in erase s 3, i
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i in erase s 3, i ⊒ p ∣ 3 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
simp
case h.e'_4 n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i in erase s 3, i ⊒ 3 = 4 * ∏ i in erase s 3, i + 3 - 4 * ∏ i in erase s 3, i
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_4 n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this : p ∣ 4 * ∏ i in erase s 3, i ⊒ 3 = 4 * ∏ i in erase s 3, i + 3 - 4 * ∏ i in erase s 3, i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean
C05S03.primes_mod_4_eq_3_infinite
[196, 1]
[235, 16]
apply pp.eq_of_dvd_of_prime Nat.prime_three this
n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝ : p ∣ 4 * ∏ i in erase s 3, i this : p ∣ 3 ⊒ p = 3
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : βˆ€ p > n, Nat.Prime p β†’ p % 4 β‰  3 s : Finset β„• hs : βˆ€ (p : β„•), Nat.Prime p ∧ p % 4 = 3 ↔ p ∈ s h₁ : (4 * ∏ i in erase s 3, i + 3) % 4 = 3 p : β„• pp : Nat.Prime p pdvd : p ∣ 4 * ∏ i in erase s 3, i + 3 p4eq : p % 4 = 3 ps : p ∈ s pne3 : p β‰  3 this✝ : p ∣ 4 * ∏ i in erase s 3, i this : p ∣ 3 ⊒ p = 3 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture9Before.lean
PointedFunction.comp
[84, 1]
[86, 24]
simp
X : PointedType Y : PointedType Z : PointedType g : Y β†’. Z f : X β†’. Y ⊒ (↑g ∘ ↑f) X.pt = Z.pt
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : PointedType Y : PointedType Z : PointedType g : Y β†’. Z f : X β†’. Y ⊒ (↑g ∘ ↑f) X.pt = Z.pt TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture9Before.lean
conjugate_one
[369, 1]
[369, 69]
sorry
G : Type u_1 H✝ : Type u_2 K : Type u_3 inst✝² : Group G inst✝¹ : Group H✝ inst✝ : Group K H : Subgroup G ⊒ conjugate 1 H = H
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 H✝ : Type u_2 K : Type u_3 inst✝² : Group G inst✝¹ : Group H✝ inst✝ : Group K H : Subgroup G ⊒ conjugate 1 H = H TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture9Before.lean
conjugate_mul
[371, 1]
[372, 66]
sorry
G : Type u_1 H✝ : Type u_2 K : Type u_3 inst✝² : Group G inst✝¹ : Group H✝ inst✝ : Group K x y : G H : Subgroup G ⊒ conjugate (x * y) H = conjugate x (conjugate y H)
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 H✝ : Type u_2 K : Type u_3 inst✝² : Group G inst✝¹ : Group H✝ inst✝ : Group K x y : G H : Subgroup G ⊒ conjugate (x * y) H = conjugate x (conjugate y H) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_inj
[98, 1]
[99, 8]
sorry
ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : ΞΉ β†’ Ideal R ⊒ Injective ↑(chineseMap I)
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : ΞΉ β†’ Ideal R ⊒ Injective ↑(chineseMap I) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := β¨… j ∈ s, J j calc 1 = I + K := sorry _ = I + K*(I + J i) := sorry _ = (1+K)*I + K*J i := sorry _ ≀ I + K βŠ“ J i := sorry
ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, IsCoprime I (J j) ⊒ IsCoprime I (β¨… j ∈ s, J j)
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, IsCoprime I (J j) ⊒ IsCoprime I (β¨… j ∈ s, J j) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
simp_rw [isCoprime_iff_add] at *
ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, IsCoprime I (J j) ⊒ IsCoprime I (β¨… j ∈ s, J j)
ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, I + J j = 1 ⊒ I + β¨… j ∈ s, J j = 1
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, IsCoprime I (J j) ⊒ IsCoprime I (β¨… j ∈ s, J j) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := β¨… j ∈ s, J j calc 1 = I + K := sorry _ = I + K*(I + J i) := sorry _ = (1+K)*I + K*J i := sorry _ ≀ I + K βŠ“ J i := sorry
ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, I + J j = 1 ⊒ I + β¨… j ∈ s, J j = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R s : Finset ΞΉ hf : βˆ€ j ∈ s, I + J j = 1 ⊒ I + β¨… j ∈ s, J j = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
simp
case empty ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R hf : βˆ€ j ∈ βˆ…, I + J j = 1 ⊒ I + β¨… j ∈ βˆ…, J j = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case empty ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R hf : βˆ€ j ∈ βˆ…, I + J j = 1 ⊒ I + β¨… j ∈ βˆ…, J j = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + β¨… j ∈ s, J j = 1 hf : βˆ€ j ∈ insert i s, I + J j = 1 ⊒ I + β¨… j ∈ insert i s, J j = 1
case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + β¨… j ∈ s, J j = 1 hf : βˆ€ j ∈ insert i s, I + J j = 1 ⊒ 1 ≀ I + (β¨… x ∈ s, J x) βŠ“ J i
Please generate a tactic in lean4 to solve the state. STATE: case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + β¨… j ∈ s, J j = 1 hf : βˆ€ j ∈ insert i s, I + J j = 1 ⊒ I + β¨… j ∈ insert i s, J j = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
set K := β¨… j ∈ s, J j
case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + β¨… j ∈ s, J j = 1 hf : βˆ€ j ∈ insert i s, I + J j = 1 ⊒ 1 ≀ I + (β¨… x ∈ s, J x) βŠ“ J i
case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hf : βˆ€ j ∈ insert i s, I + J j = 1 K : Ideal R := β¨… j ∈ s, J j hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + K = 1 ⊒ 1 ≀ I + K βŠ“ J i
Please generate a tactic in lean4 to solve the state. STATE: case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + β¨… j ∈ s, J j = 1 hf : βˆ€ j ∈ insert i s, I + J j = 1 ⊒ 1 ≀ I + (β¨… x ∈ s, J x) βŠ“ J i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[110, 1]
[124, 38]
calc 1 = I + K := sorry _ = I + K*(I + J i) := sorry _ = (1+K)*I + K*J i := sorry _ ≀ I + K βŠ“ J i := sorry
case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hf : βˆ€ j ∈ insert i s, I + J j = 1 K : Ideal R := β¨… j ∈ s, J j hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + K = 1 ⊒ 1 ≀ I + K βŠ“ J i
no goals
Please generate a tactic in lean4 to solve the state. STATE: case insert ΞΉ : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ΞΉ β†’ Ideal R i : ΞΉ s : Finset ΞΉ a✝ : i βˆ‰ s hf : βˆ€ j ∈ insert i s, I + J j = 1 K : Ideal R := β¨… j ∈ s, J j hs : (βˆ€ j ∈ s, I + J j = 1) β†’ I + K = 1 ⊒ 1 ≀ I + K βŠ“ J i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
classical intro g choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i) have key : βˆ€ i, βˆƒ e : R, mk (I i) e = 1 ∧ βˆ€ j, j β‰  i β†’ mk (I j) e = 0 := by intro i have hI' : βˆ€ j ∈ ({i} : Finset ΞΉ)ᢜ, IsCoprime (I i) (I j) := by sorry sorry choose e he using key use mk _ (βˆ‘ i, f i*e i) sorry
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) ⊒ Surjective ↑(chineseMap I)
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) ⊒ Surjective ↑(chineseMap I) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
intro g
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) ⊒ Surjective ↑(chineseMap I)
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i ⊒ βˆƒ a, ↑(chineseMap I) a = g
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) ⊒ Surjective ↑(chineseMap I) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i)
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i ⊒ βˆƒ a, ↑(chineseMap I) a = g
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i ⊒ βˆƒ a, ↑(chineseMap I) a = g
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i ⊒ βˆƒ a, ↑(chineseMap I) a = g TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
have key : βˆ€ i, βˆƒ e : R, mk (I i) e = 1 ∧ βˆ€ j, j β‰  i β†’ mk (I j) e = 0 := by intro i have hI' : βˆ€ j ∈ ({i} : Finset ΞΉ)ᢜ, IsCoprime (I i) (I j) := by sorry sorry
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i ⊒ βˆƒ a, ↑(chineseMap I) a = g
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i key : βˆ€ (i : ΞΉ), βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0 ⊒ βˆƒ a, ↑(chineseMap I) a = g
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i ⊒ βˆƒ a, ↑(chineseMap I) a = g TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
choose e he using key
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i key : βˆ€ (i : ΞΉ), βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0 ⊒ βˆƒ a, ↑(chineseMap I) a = g
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i e : ΞΉ β†’ R he : βˆ€ (i : ΞΉ), ↑(mk (I i)) (e i) = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) (e i) = 0 ⊒ βˆƒ a, ↑(chineseMap I) a = g
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i key : βˆ€ (i : ΞΉ), βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0 ⊒ βˆƒ a, ↑(chineseMap I) a = g TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
use mk _ (βˆ‘ i, f i*e i)
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i e : ΞΉ β†’ R he : βˆ€ (i : ΞΉ), ↑(mk (I i)) (e i) = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) (e i) = 0 ⊒ βˆƒ a, ↑(chineseMap I) a = g
case h ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i e : ΞΉ β†’ R he : βˆ€ (i : ΞΉ), ↑(mk (I i)) (e i) = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) (e i) = 0 ⊒ ↑(chineseMap I) (↑(mk (β¨… i, I i)) (βˆ‘ i : ΞΉ, f i * e i)) = g
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i e : ΞΉ β†’ R he : βˆ€ (i : ΞΉ), ↑(mk (I i)) (e i) = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) (e i) = 0 ⊒ βˆƒ a, ↑(chineseMap I) a = g TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
sorry
case h ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i e : ΞΉ β†’ R he : βˆ€ (i : ΞΉ), ↑(mk (I i)) (e i) = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) (e i) = 0 ⊒ ↑(chineseMap I) (↑(mk (β¨… i, I i)) (βˆ‘ i : ΞΉ, f i * e i)) = g
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i e : ΞΉ β†’ R he : βˆ€ (i : ΞΉ), ↑(mk (I i)) (e i) = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) (e i) = 0 ⊒ ↑(chineseMap I) (↑(mk (β¨… i, I i)) (βˆ‘ i : ΞΉ, f i * e i)) = g TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
intro i
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i ⊒ βˆ€ (i : ΞΉ), βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ ⊒ βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i ⊒ βˆ€ (i : ΞΉ), βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
have hI' : βˆ€ j ∈ ({i} : Finset ΞΉ)ᢜ, IsCoprime (I i) (I j) := by sorry
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ ⊒ βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ hI' : βˆ€ j ∈ {i}ᢜ, IsCoprime (I i) (I j) ⊒ βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ ⊒ βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
sorry
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ hI' : βˆ€ j ∈ {i}ᢜ, IsCoprime (I i) (I j) ⊒ βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ hI' : βˆ€ j ∈ {i}ᢜ, IsCoprime (I i) (I j) ⊒ βˆƒ e, ↑(mk (I i)) e = 1 ∧ βˆ€ (j : ΞΉ), j β‰  i β†’ ↑(mk (I j)) e = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[125, 1]
[137, 8]
sorry
ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ ⊒ βˆ€ j ∈ {i}ᢜ, IsCoprime (I i) (I j)
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ΞΉ I : ΞΉ β†’ Ideal R hI : βˆ€ (i j : ΞΉ), i β‰  j β†’ IsCoprime (I i) (I j) g : Ξ  (i : ΞΉ), R β§Έ I i f : ΞΉ β†’ R hf : βˆ€ (i : ΞΉ), ↑(mk (I i)) (f i) = g i i : ΞΉ ⊒ βˆ€ j ∈ {i}ᢜ, IsCoprime (I i) (I j) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean
C03S04.aux
[30, 1]
[32, 17]
linarith [pow_two_nonneg x, pow_two_nonneg y]
x y : ℝ h : x ^ 2 + y ^ 2 = 0 ⊒ x ^ 2 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : x ^ 2 + y ^ 2 = 0 ⊒ x ^ 2 = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[44, 1]
[47, 6]
rw [Monotone]
f : ℝ β†’ ℝ ⊒ Β¬Monotone f ↔ βˆƒ x y, x ≀ y ∧ f x > f y
f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
Please generate a tactic in lean4 to solve the state. STATE: f : ℝ β†’ ℝ ⊒ Β¬Monotone f ↔ βˆƒ x y, x ≀ y ∧ f x > f y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[44, 1]
[47, 6]
push_neg
f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
Please generate a tactic in lean4 to solve the state. STATE: f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[44, 1]
[47, 6]
rfl
f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C07_Hierarchies/solutions/Solutions_S02_Morphisms.lean
map_inv_of_inv
[75, 1]
[77, 61]
rw [← MonoidHomClassβ‚‚.map_mul, h, MonoidHomClassβ‚‚.map_one]
M N F : Type inst✝² : Monoid M inst✝¹ : Monoid N inst✝ : MonoidHomClassβ‚‚ F M N f : F m m' : M h : m * m' = 1 ⊒ ↑f m * ↑f m' = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: M N F : Type inst✝² : Monoid M inst✝¹ : Monoid N inst✝ : MonoidHomClassβ‚‚ F M N f : F m m' : M h : m * m' = 1 ⊒ ↑f m * ↑f m' = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture16.lean
my_equality
[38, 1]
[40, 11]
set c : β„• := a + b
a b : β„• ⊒ a + b + a ≀ 2 * (a + b)
a b : β„• c : β„• := a + b ⊒ c + a ≀ 2 * c
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ⊒ a + b + a ≀ 2 * (a + b) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture16.lean
my_equality
[38, 1]
[40, 11]
linarith
a b : β„• c : β„• := a + b ⊒ c + a ≀ 2 * c
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• c : β„• := a + b ⊒ c + a ≀ 2 * c TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture16.lean
my_lemma
[127, 1]
[129, 53]
rw [Nat.add_succ, my_lemma n]
n : β„• ⊒ 0 + Nat.succ n = Nat.succ n + 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• ⊒ 0 + Nat.succ n = Nat.succ n + 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.aux
[107, 1]
[109, 17]
sorry
x y : ℝ h : x ^ 2 + y ^ 2 = 0 ⊒ x ^ 2 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : x ^ 2 + y ^ 2 = 0 ⊒ x ^ 2 = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[127, 1]
[130, 6]
rw [Monotone]
f : ℝ β†’ ℝ ⊒ Β¬Monotone f ↔ βˆƒ x y, x ≀ y ∧ f x > f y
f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
Please generate a tactic in lean4 to solve the state. STATE: f : ℝ β†’ ℝ ⊒ Β¬Monotone f ↔ βˆƒ x y, x ≀ y ∧ f x > f y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[127, 1]
[130, 6]
push_neg
f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
Please generate a tactic in lean4 to solve the state. STATE: f : ℝ β†’ ℝ ⊒ (Β¬βˆ€ ⦃a b : ℝ⦄, a ≀ b β†’ f a ≀ f b) ↔ βˆƒ x y, x ≀ y ∧ f x > f y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C03_Logic/S04_Conjunction_and_Iff.lean
C03S04.not_monotone_iff
[127, 1]
[130, 6]
rfl
f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : ℝ β†’ ℝ ⊒ (βˆƒ a b, a ≀ b ∧ f b < f a) ↔ βˆƒ x y, x ≀ y ∧ f x > f y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.add_neg_cancel_right
[8, 1]
[9, 42]
rw [add_assoc, add_right_neg, add_zero]
R : Type u_1 inst✝ : Ring R a b : R ⊒ a + b + -b = a
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R ⊒ a + b + -b = a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.add_left_cancel
[11, 1]
[12, 57]
rw [← neg_add_cancel_left a b, h, neg_add_cancel_left]
R : Type u_1 inst✝ : Ring R a b c : R h : a + b = a + c ⊒ b = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b c : R h : a + b = a + c ⊒ b = c TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.add_right_cancel
[14, 1]
[15, 59]
rw [← add_neg_cancel_right a b, h, add_neg_cancel_right]
R : Type u_1 inst✝ : Ring R a b c : R h : a + b = c + b ⊒ a = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b c : R h : a + b = c + b ⊒ a = c TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.zero_mul
[17, 1]
[19, 25]
have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero]
R : Type u_1 inst✝ : Ring R a : R ⊒ 0 * a = 0
R : Type u_1 inst✝ : Ring R a : R h : 0 * a + 0 * a = 0 * a + 0 ⊒ 0 * a = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊒ 0 * a = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.zero_mul
[17, 1]
[19, 25]
rw [add_left_cancel h]
R : Type u_1 inst✝ : Ring R a : R h : 0 * a + 0 * a = 0 * a + 0 ⊒ 0 * a = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R h : 0 * a + 0 * a = 0 * a + 0 ⊒ 0 * a = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.zero_mul
[17, 1]
[19, 25]
rw [← add_mul, add_zero, add_zero]
R : Type u_1 inst✝ : Ring R a : R ⊒ 0 * a + 0 * a = 0 * a + 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊒ 0 * a + 0 * a = 0 * a + 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.neg_eq_of_add_eq_zero
[21, 1]
[22, 46]
rw [← neg_add_cancel_left a b, h, add_zero]
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ -a = b
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ -a = b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.eq_neg_of_add_eq_zero
[24, 1]
[27, 19]
symm
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ a = -b
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ -b = a
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ a = -b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.eq_neg_of_add_eq_zero
[24, 1]
[27, 19]
apply neg_eq_of_add_eq_zero
R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ -b = a
case h R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ b + a = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ -b = a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.eq_neg_of_add_eq_zero
[24, 1]
[27, 19]
rw [add_comm, h]
case h R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ b + a = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h R : Type u_1 inst✝ : Ring R a b : R h : a + b = 0 ⊒ b + a = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.neg_zero
[29, 1]
[31, 16]
apply neg_eq_of_add_eq_zero
R : Type u_1 inst✝ : Ring R ⊒ -0 = 0
case h R : Type u_1 inst✝ : Ring R ⊒ 0 + 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R ⊒ -0 = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.neg_zero
[29, 1]
[31, 16]
rw [add_zero]
case h R : Type u_1 inst✝ : Ring R ⊒ 0 + 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h R : Type u_1 inst✝ : Ring R ⊒ 0 + 0 = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.neg_neg
[33, 1]
[35, 20]
apply neg_eq_of_add_eq_zero
R : Type u_1 inst✝ : Ring R a : R ⊒ - -a = a
case h R : Type u_1 inst✝ : Ring R a : R ⊒ -a + a = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊒ - -a = a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.neg_neg
[33, 1]
[35, 20]
rw [add_left_neg]
case h R : Type u_1 inst✝ : Ring R a : R ⊒ -a + a = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h R : Type u_1 inst✝ : Ring R a : R ⊒ -a + a = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.self_sub
[42, 1]
[43, 37]
rw [sub_eq_add_neg, add_right_neg]
R : Type u_1 inst✝ : Ring R a : R ⊒ a - a = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊒ a - a = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.one_add_one_eq_two
[45, 1]
[46, 11]
norm_num
R : Type u_1 inst✝ : Ring R ⊒ 1 + 1 = 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R ⊒ 1 + 1 = 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyRing.two_mul
[48, 1]
[49, 46]
rw [← one_add_one_eq_two, add_mul, one_mul]
R : Type u_1 inst✝ : Ring R a : R ⊒ 2 * a = a + a
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : Ring R a : R ⊒ 2 * a = a + a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyGroup.mul_right_inv
[58, 1]
[61, 47]
have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv]
G : Type u_1 inst✝ : Group G a : G ⊒ a * a⁻¹ = 1
G : Type u_1 inst✝ : Group G a : G h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 ⊒ a * a⁻¹ = 1
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G a : G ⊒ a * a⁻¹ = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyGroup.mul_right_inv
[58, 1]
[61, 47]
rw [← h, ← mul_assoc, mul_left_inv, one_mul]
G : Type u_1 inst✝ : Group G a : G h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 ⊒ a * a⁻¹ = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G a : G h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 ⊒ a * a⁻¹ = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyGroup.mul_right_inv
[58, 1]
[61, 47]
rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv]
G : Type u_1 inst✝ : Group G a : G ⊒ (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G a : G ⊒ (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyGroup.mul_one
[63, 1]
[64, 61]
rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul]
G : Type u_1 inst✝ : Group G a : G ⊒ a * 1 = a
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G a : G ⊒ a * 1 = a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C02_Basics/solutions/Solutions_S02_Proving_Identities_in_Algebraic_Structures.lean
MyGroup.mul_inv_rev
[66, 1]
[68, 52]
rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹, mul_right_inv, one_mul, mul_right_inv, mul_one]
G : Type u_1 inst✝ : Group G a b : G ⊒ (a * b)⁻¹ = b⁻¹ * a⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G a b : G ⊒ (a * b)⁻¹ = b⁻¹ * a⁻¹ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
left_inv_eq_right_inv₁
[103, 1]
[104, 90]
rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b]
M : Type inst✝ : Monoid₁ M a b c : M hba : b β‹„ a = πŸ™ hac : a β‹„ c = πŸ™ ⊒ b = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type inst✝ : Monoid₁ M a b c : M hba : b β‹„ a = πŸ™ hac : a β‹„ c = πŸ™ ⊒ b = c TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
dia_inv
[119, 1]
[120, 51]
rw [← inv_dia a⁻¹, inv_eq_of_dia (inv_dia a)]
G : Type inst✝ : Group₁ G a : G ⊒ a β‹„ a⁻¹ = πŸ™
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝ : Group₁ G a : G ⊒ a β‹„ a⁻¹ = πŸ™ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
left_inv_eq_right_inv'
[146, 1]
[148, 54]
rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b]
M : Type inst✝ : Monoid₃ M a b c : M hba : b * a = 1 hac : a * c = 1 ⊒ b = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: M : Type inst✝ : Monoid₃ M a b c : M hba : b * a = 1 hac : a * c = 1 ⊒ b = c TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
Group₃.mul_inv
[181, 1]
[183, 48]
rw [← inv_mul a⁻¹, inv_eq_of_mul (inv_mul a)]
G : Type inst✝ : Group₃ G a : G ⊒ a * a⁻¹ = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝ : Group₃ G a : G ⊒ a * a⁻¹ = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture10.lean
units_ne_neg_self
[347, 1]
[347, 78]
sorry
R : Type u_1 M : Type u_2 inst✝¹ : Ring R inst✝ : CharZero R u : RΛ£ ⊒ u β‰  -u
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 inst✝¹ : Ring R inst✝ : CharZero R u : RΛ£ ⊒ u β‰  -u TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture10.lean
iterate_frobeniusMorphism
[383, 1]
[383, 96]
sorry
R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K n : β„• ⊒ (↑(frobeniusMorphism p K))^[n] x = x ^ p ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K n : β„• ⊒ (↑(frobeniusMorphism p K))^[n] x = x ^ p ^ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture10.lean
frobeniusMorphism_injective
[385, 1]
[387, 8]
have : βˆ€ x : K, x ^ p = 0 β†’ x = 0 := by exact?
R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K ⊒ Injective ↑(frobeniusMorphism p K)
R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K this : βˆ€ (x : K), x ^ p = 0 β†’ x = 0 ⊒ Injective ↑(frobeniusMorphism p K)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K ⊒ Injective ↑(frobeniusMorphism p K) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture10.lean
frobeniusMorphism_injective
[385, 1]
[387, 8]
sorry
R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K this : βˆ€ (x : K), x ^ p = 0 β†’ x = 0 ⊒ Injective ↑(frobeniusMorphism p K)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K this : βˆ€ (x : K), x ^ p = 0 β†’ x = 0 ⊒ Injective ↑(frobeniusMorphism p K) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture10.lean
frobeniusMorphism_injective
[385, 1]
[387, 8]
exact?
R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K ⊒ βˆ€ (x : K), x ^ p = 0 β†’ x = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝¹ : Field K inst✝ : CharP K p x : K ⊒ βˆ€ (x : K), x ^ p = 0 β†’ x = 0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Lectures/Lecture10.lean
frobeniusMorphism_bijective
[389, 1]
[390, 59]
sorry
R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝² : Field K inst✝¹ : CharP K p x : K inst✝ : Finite K ⊒ Bijective ↑(frobeniusMorphism p K)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 p : β„• hp : Fact (Nat.Prime p) K : Type u_3 inst✝² : Field K inst✝¹ : CharP K p x : K inst✝ : Finite K ⊒ Bijective ↑(frobeniusMorphism p K) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
by_contra h
p : Prop ⊒ p ∨ ¬p
p : Prop h : ¬(p ∨ ¬p) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: p : Prop ⊒ p ∨ ¬p TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
apply h
p : Prop h : ¬(p ∨ ¬p) ⊒ False
p : Prop h : ¬(p ∨ ¬p) ⊒ p ∨ ¬p
Please generate a tactic in lean4 to solve the state. STATE: p : Prop h : ¬(p ∨ ¬p) ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
right
p : Prop h : ¬(p ∨ ¬p) ⊒ p ∨ ¬p
case h p : Prop h : ¬(p ∨ ¬p) ⊒ ¬p
Please generate a tactic in lean4 to solve the state. STATE: p : Prop h : ¬(p ∨ ¬p) ⊒ p ∨ ¬p TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
intro h2
case h p : Prop h : ¬(p ∨ ¬p) ⊒ ¬p
case h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case h p : Prop h : ¬(p ∨ ¬p) ⊒ ¬p TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
apply h
case h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ False
case h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ p ∨ ¬p
Please generate a tactic in lean4 to solve the state. STATE: case h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
left
case h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ p ∨ ¬p
case h.h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ p
Please generate a tactic in lean4 to solve the state. STATE: case h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ p ∨ ¬p TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_1
[25, 1]
[32, 13]
assumption
case h.h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ p
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h p : Prop h : ¬(p ∨ ¬p) h2 : p ⊒ p TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_2
[44, 1]
[49, 31]
have h1 : β„€ Γ— β„• ≃ β„• := by exact Denumerable.eqv (β„€ Γ— β„•)
⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•)
h1 : β„€ Γ— β„• ≃ β„• ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•)
Please generate a tactic in lean4 to solve the state. STATE: ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_2
[44, 1]
[49, 31]
have h2 : β„€ Γ— β„€ ≃ β„• := by exact Denumerable.eqv (β„€ Γ— β„€)
h1 : β„€ Γ— β„• ≃ β„• ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•)
h1 : β„€ Γ— β„• ≃ β„• h2 : β„€ Γ— β„€ ≃ β„• ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•)
Please generate a tactic in lean4 to solve the state. STATE: h1 : β„€ Γ— β„• ≃ β„• ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_2
[44, 1]
[49, 31]
exact Equiv.arrowCongr h1 h2
h1 : β„€ Γ— β„• ≃ β„• h2 : β„€ Γ— β„€ ≃ β„• ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•)
no goals
Please generate a tactic in lean4 to solve the state. STATE: h1 : β„€ Γ— β„• ≃ β„• h2 : β„€ Γ— β„€ ≃ β„• ⊒ (β„€ Γ— β„• β†’ β„€ Γ— β„€) ≃ (β„• β†’ β„•) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_2
[44, 1]
[49, 31]
exact Denumerable.eqv (β„€ Γ— β„•)
⊒ β„€ Γ— β„• ≃ β„•
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊒ β„€ Γ— β„• ≃ β„• TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_2
[44, 1]
[49, 31]
exact Denumerable.eqv (β„€ Γ— β„€)
h1 : β„€ Γ— β„• ≃ β„• ⊒ β„€ Γ— β„€ ≃ β„•
no goals
Please generate a tactic in lean4 to solve the state. STATE: h1 : β„€ Γ— β„• ≃ β„• ⊒ β„€ Γ— β„€ ≃ β„• TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
intro Ξ΅ hΞ΅
s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m ⊒ SequentialLimit (s ∘ r) a
s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m ⊒ SequentialLimit (s ∘ r) a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
obtain ⟨N, hN⟩ := hs Ρ hΡ
s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅
case intro s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
obtain ⟨K, hK⟩ := hr N
case intro s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅
case intro.intro s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case intro s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
use K
case intro.intro s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅
case h s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N ⊒ βˆ€ n β‰₯ K, |(s ∘ r) n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N ⊒ βˆƒ N, βˆ€ n β‰₯ N, |(s ∘ r) n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
intro n hn
case h s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N ⊒ βˆ€ n β‰₯ K, |(s ∘ r) n - a| < Ξ΅
case h s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ |(s ∘ r) n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case h s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N ⊒ βˆ€ n β‰₯ K, |(s ∘ r) n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
apply hN
case h s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ |(s ∘ r) n - a| < Ξ΅
case h.a s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ r n β‰₯ N
Please generate a tactic in lean4 to solve the state. STATE: case h s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ |(s ∘ r) n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
apply hK
case h.a s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ r n β‰₯ N
case h.a.a s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ n β‰₯ K
Please generate a tactic in lean4 to solve the state. STATE: case h.a s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ r n β‰₯ N TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_3
[62, 1]
[72, 13]
assumption
case h.a.a s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ n β‰₯ K
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.a.a s : β„• β†’ ℝ r : β„• β†’ β„• a : ℝ hs : SequentialLimit s a hr : βˆ€ (m : β„•), βˆƒ N, βˆ€ n β‰₯ N, r n β‰₯ m Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s n - a| < Ξ΅ K : β„• hK : βˆ€ n β‰₯ K, r n β‰₯ N n : β„• hn : n β‰₯ K ⊒ n β‰₯ K TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
intro Ξ΅ hΞ΅
s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n ⊒ SequentialLimit sβ‚‚ a
s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n ⊒ SequentialLimit sβ‚‚ a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
obtain ⟨N, hN⟩ := hs₁ Ξ΅ hΞ΅
s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
case intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
obtain ⟨N', hN'⟩ := hs₃ Ξ΅ hΞ΅
case intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
case intro.intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
let N'' := max N N'
case intro.intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
case intro.intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
use N''
case intro.intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅
case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' ⊒ βˆ€ n β‰₯ N'', |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' ⊒ βˆƒ N, βˆ€ n β‰₯ N, |sβ‚‚ n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
intro n hn
case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' ⊒ βˆ€ n β‰₯ N'', |sβ‚‚ n - a| < Ξ΅
case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' n : β„• hn : n β‰₯ N'' ⊒ |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' ⊒ βˆ€ n β‰₯ N'', |sβ‚‚ n - a| < Ξ΅ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions3.lean
exercise3_4
[79, 1]
[98, 13]
specialize hN n (le_of_max_le_left hn)
case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' n : β„• hn : n β‰₯ N'' ⊒ |sβ‚‚ n - a| < Ξ΅
case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' n : β„• hn : n β‰₯ N'' hN : |s₁ n - a| < Ξ΅ ⊒ |sβ‚‚ n - a| < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case h s₁ sβ‚‚ s₃ : β„• β†’ ℝ a : ℝ hs₁ : SequentialLimit s₁ a hs₃ : SequentialLimit s₃ a hs₁sβ‚‚ : βˆ€ (n : β„•), s₁ n ≀ sβ‚‚ n hsβ‚‚s₃ : βˆ€ (n : β„•), sβ‚‚ n ≀ s₃ n Ξ΅ : ℝ hΞ΅ : Ξ΅ > 0 N : β„• hN : βˆ€ n β‰₯ N, |s₁ n - a| < Ξ΅ N' : β„• hN' : βˆ€ n β‰₯ N', |s₃ n - a| < Ξ΅ N'' : β„• := max N N' n : β„• hn : n β‰₯ N'' ⊒ |sβ‚‚ n - a| < Ξ΅ TACTIC: