url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet1.lean | Section3sheet1.injective_def | [52, 1] | [53, 6] | rfl | X Y Z : Type
f : X → Y
⊢ Injective f ↔ ∀ (a b : X), f a = f b → a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
f : X → Y
⊢ Injective f ↔ ∀ (a b : X), f a = f b → a = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet1.lean | Section3sheet1.surjective_def | [59, 1] | [60, 6] | rfl | X Y Z : Type
f : X → Y
⊢ Surjective f ↔ ∀ (b : Y), ∃ a, f a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
f : X → Y
⊢ Surjective f ↔ ∀ (b : Y), ∃ a, f a = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet1.lean | Section3sheet1.id_eval | [63, 1] | [64, 6] | rfl | X Y Z : Type
x : X
⊢ id x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
x : X
⊢ id x = x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet1.lean | Section3sheet1.comp_eval | [68, 1] | [69, 6] | rfl | X Y Z : Type
f : X → Y
g : Y → Z
x : X
⊢ (g ∘ f) x = g (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
f : X → Y
g : Y → Z
x : X
⊢ (g ∘ f) x = g (f x)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [one_mul] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ b = 1 * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ b = 1 * b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [inv_mul_self] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ 1 * b = a⁻¹ * a * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ 1 * b = a⁻¹ * a * b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [mul_assoc] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * b = a⁻¹ * (a * b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * b = a⁻¹ * (a * b)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [h] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * b) = a⁻¹ * (a * c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * b) = a⁻¹ * (a * c)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [mul_assoc] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * c) = a⁻¹ * a * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * c) = a⁻¹ * a * c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [inv_mul_self] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * c = 1 * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * c = 1 * c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_left_cancel | [56, 1] | [64, 29] | rw [one_mul] | G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ 1 * c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : a * b = a * c
⊢ 1 * c = c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_eq_of_eq_inv_mul | [66, 1] | [68, 43] | apply mul_left_cancel a⁻¹ | G : Type
inst✝ : WeakGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a * b = c | case h
G : Type
inst✝ : WeakGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a⁻¹ * (a * b) = a⁻¹ * c | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a * b = c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_eq_of_eq_inv_mul | [66, 1] | [68, 43] | rwa [← mul_assoc, inv_mul_self, one_mul] | case h
G : Type
inst✝ : WeakGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a⁻¹ * (a * b) = a⁻¹ * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : WeakGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a⁻¹ * (a * b) = a⁻¹ * c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_one | [70, 1] | [73, 20] | apply mul_eq_of_eq_inv_mul | G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a * 1 = a | case h
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ 1 = a⁻¹ * a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a * 1 = a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_one | [70, 1] | [73, 20] | rw [inv_mul_self] | case h
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ 1 = a⁻¹ * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ 1 = a⁻¹ * a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_inv_self | [75, 1] | [78, 15] | apply mul_eq_of_eq_inv_mul | G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a * a⁻¹ = 1 | case h
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a⁻¹ = a⁻¹ * 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section05groups/Sheet2.lean | Section5sheet2Solutions.WeakGroup.mul_inv_self | [75, 1] | [78, 15] | rw [mul_one] | case h
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a⁻¹ = a⁻¹ * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : WeakGroup G
a✝ b c a : G
⊢ a⁻¹ = a⁻¹ * 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate.one_mem | [109, 1] | [110, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ 1 ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate.inv_mem | [112, 1] | [114, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y⁻¹ ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate.mul_mem | [116, 1] | [119, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
hy : y ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
hz : z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
⊢ y * z ∈ {a | ∃ h ∈ H, a = x * h * x⁻¹}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.mem_conjugate_iff | [154, 1] | [156, 6] | rfl | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ a ∈ conjugate H x ↔ ∃ h ∈ H, a = x * h * x⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ a ∈ conjugate H x ↔ ∃ h ∈ H, a = x * h * x⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate_mono | [158, 1] | [159, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
⊢ conjugate H x ≤ conjugate K x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H✝ K✝ : Subgroup G
x y z : G
H K : Subgroup G
h : H ≤ K
⊢ conjugate H x ≤ conjugate K x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate_bot | [161, 1] | [162, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊥ x = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊥ x = ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate_top | [164, 1] | [165, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊤ x = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
⊢ conjugate ⊤ x = ⊤
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section07subgroupsAndHomomorphisms/Sheet1.lean | Section7sheet1.conjugate_eq_of_abelian | [167, 1] | [168, 8] | sorry | G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
⊢ conjugate H x = H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a b : G
H K : Subgroup G
x y z : G
habelian : ∀ (a b : G), a * b = b * a
⊢ conjugate H x = H
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet2.lean | lemma1 | [30, 1] | [36, 80] | constructor | K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a ↔ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) | case mp
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a → Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) → IsIntegral R a | Please generate a tactic in lean4 to solve the state.
STATE:
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a ↔ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet2.lean | lemma1 | [30, 1] | [36, 80] | exact IsIntegral.fg_adjoin_singleton | case mp
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a → Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a → Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet2.lean | lemma1 | [30, 1] | [36, 80] | intro h | case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) → IsIntegral R a | case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
h : Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
⊢ IsIntegral R a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) → IsIntegral R a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet2.lean | lemma1 | [30, 1] | [36, 80] | exact IsIntegral.of_mem_of_fg _ h _ (Algebra.self_mem_adjoin_singleton R a) | case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
h : Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
⊢ IsIntegral R a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
h : Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
⊢ IsIntegral R a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet2.lean | comp_id | [67, 1] | [70, 6] | ext x | G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp φ (MonoidHom.id G) = φ | case h
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
x : G
⊢ (MonoidHom.comp φ (MonoidHom.id G)) x = φ x | Please generate a tactic in lean4 to solve the state.
STATE:
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp φ (MonoidHom.id G) = φ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet2.lean | comp_id | [67, 1] | [70, 6] | rfl | case h
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
x : G
⊢ (MonoidHom.comp φ (MonoidHom.id G)) x = φ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
x : G
⊢ (MonoidHom.comp φ (MonoidHom.id G)) x = φ x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet2.lean | id_comp | [72, 1] | [75, 6] | ext x | G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp (MonoidHom.id H) φ = φ | case h
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
x : G
⊢ (MonoidHom.comp (MonoidHom.id H) φ) x = φ x | Please generate a tactic in lean4 to solve the state.
STATE:
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
⊢ MonoidHom.comp (MonoidHom.id H) φ = φ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet2.lean | id_comp | [72, 1] | [75, 6] | rfl | case h
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
x : G
⊢ (MonoidHom.comp (MonoidHom.id H) φ) x = φ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G H : Type
inst✝² : Group G
inst✝¹ : Group H
φ : G →* H
a : G
K : Type
inst✝ : Group K
ψ : H →* K
x : G
⊢ (MonoidHom.comp (MonoidHom.id H) φ) x = φ x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section07subgroupsAndHomomorphisms/Sheet2.lean | comp_assoc | [77, 1] | [78, 6] | rfl | G H : Type
inst✝³ : Group G
inst✝² : Group H
φ : G →* H
a : G
K : Type
inst✝¹ : Group K
ψ : H →* K
L : Type
inst✝ : Group L
ρ : K →* L
⊢ MonoidHom.comp (MonoidHom.comp ρ ψ) φ = MonoidHom.comp ρ (MonoidHom.comp ψ φ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G H : Type
inst✝³ : Group G
inst✝² : Group H
φ : G →* H
a : G
K : Type
inst✝¹ : Group K
ψ : H →* K
L : Type
inst✝ : Group L
ρ : K →* L
⊢ MonoidHom.comp (MonoidHom.comp ρ ψ) φ = MonoidHom.comp ρ (MonoidHom.comp ψ φ)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1.Real.isOpen_univ | [85, 1] | [86, 8] | sorry | X : Type
⊢ IsOpen Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
⊢ IsOpen Set.univ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1.Real.isOpen_inter | [88, 1] | [89, 8] | sorry | X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
⊢ IsOpen (s ∩ t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s t : Set ℝ
hs : IsOpen s
ht : IsOpen t
⊢ IsOpen (s ∩ t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section10TopologicalSpaces/Sheet1.lean | Section10sheet1.Real.isOpen_sUnion | [91, 1] | [92, 8] | sorry | X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
⊢ IsOpen (⋃₀ F) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
F : Set (Set ℝ)
hF : ∀ s ∈ F, IsOpen s
⊢ IsOpen (⋃₀ F)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | have h : x - 3 ∣ x ^ 3 - 27 | x : ℤ
⊢ x - 3 ∣ x ^ 3 - 3 ↔ x - 3 ∣ 24 | case h
x : ℤ
⊢ x - 3 ∣ x ^ 3 - 27
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ x ^ 3 - 3 ↔ x - 3 ∣ 24 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ x - 3 ∣ x ^ 3 - 3 ↔ x - 3 ∣ 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | constructor | x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ x ^ 3 - 3 ↔ x - 3 ∣ 24 | case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ x ^ 3 - 3 → x - 3 ∣ 24
case mpr
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ 24 → x - 3 ∣ x ^ 3 - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ x ^ 3 - 3 ↔ x - 3 ∣ 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | use x ^ 2 + 3 * x + 9 | case h
x : ℤ
⊢ x - 3 ∣ x ^ 3 - 27 | case h
x : ℤ
⊢ x ^ 3 - 27 = (x - 3) * (x ^ 2 + 3 * x + 9) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : ℤ
⊢ x - 3 ∣ x ^ 3 - 27
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | ring | case h
x : ℤ
⊢ x ^ 3 - 27 = (x - 3) * (x ^ 2 + 3 * x + 9) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : ℤ
⊢ x ^ 3 - 27 = (x - 3) * (x ^ 2 + 3 * x + 9)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | intro h1 | case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ x ^ 3 - 3 → x - 3 ∣ 24 | case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
⊢ x - 3 ∣ 24 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ x ^ 3 - 3 → x - 3 ∣ 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | have h2 := dvd_sub h1 h | case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
⊢ x - 3 ∣ 24 | case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
h2 : x - 3 ∣ x ^ 3 - 3 - (x ^ 3 - 27)
⊢ x - 3 ∣ 24 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
⊢ x - 3 ∣ 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | convert h2 | case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
h2 : x - 3 ∣ x ^ 3 - 3 - (x ^ 3 - 27)
⊢ x - 3 ∣ 24 | case h.e'_4
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
h2 : x - 3 ∣ x ^ 3 - 3 - (x ^ 3 - 27)
⊢ 24 = x ^ 3 - 3 - (x ^ 3 - 27) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
h2 : x - 3 ∣ x ^ 3 - 3 - (x ^ 3 - 27)
⊢ x - 3 ∣ 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | ring | case h.e'_4
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
h2 : x - 3 ∣ x ^ 3 - 3 - (x ^ 3 - 27)
⊢ 24 = x ^ 3 - 3 - (x ^ 3 - 27) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ x ^ 3 - 3
h2 : x - 3 ∣ x ^ 3 - 3 - (x ^ 3 - 27)
⊢ 24 = x ^ 3 - 3 - (x ^ 3 - 27)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | intro h1 | case mpr
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ 24 → x - 3 ∣ x ^ 3 - 3 | case mpr
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ 24
⊢ x - 3 ∣ x ^ 3 - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
⊢ x - 3 ∣ 24 → x - 3 ∣ x ^ 3 - 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | convert dvd_add h h1 using 1 | case mpr
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ 24
⊢ x - 3 ∣ x ^ 3 - 3 | case h.e'_4
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ 24
⊢ x ^ 3 - 3 = x ^ 3 - 27 + 24 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ 24
⊢ x - 3 ∣ x ^ 3 - 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma1 | [29, 1] | [40, 9] | ring | case h.e'_4
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ 24
⊢ x ^ 3 - 3 = x ^ 3 - 27 + 24 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
x : ℤ
h : x - 3 ∣ x ^ 3 - 27
h1 : x - 3 ∣ 24
⊢ x ^ 3 - 3 = x ^ 3 - 27 + 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.int_dvd_iff | [43, 1] | [44, 12] | simp [hn] | x n : ℤ
hn : n ≠ 0
⊢ x ∣ n ↔ Int.natAbs x ∈ Nat.divisors (Int.natAbs n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x n : ℤ
hn : n ≠ 0
⊢ x ∣ n ↔ Int.natAbs x ∈ Nat.divisors (Int.natAbs n)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma2 | [46, 1] | [55, 39] | suffices : x ∣ 24 ↔ x.natAbs ∈ ({1, 2, 3, 4, 6, 8, 12, 24} : Finset ℕ) | x : ℤ
⊢ x ∣ 24 ↔ x ∈ {-24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, 24} | x : ℤ
this : x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
⊢ x ∣ 24 ↔ x ∈ {-24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, 24}
case this
x : ℤ
⊢ x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24} | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ x ∣ 24 ↔ x ∈ {-24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, 24}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma2 | [46, 1] | [55, 39] | exact int_dvd_iff _ 24 (by norm_num) | case this
x : ℤ
⊢ x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
x : ℤ
⊢ x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma2 | [46, 1] | [55, 39] | simp only [Set.mem_insert_iff, Set.mem_singleton_iff] | x : ℤ
this : x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
⊢ x ∣ 24 ↔ x ∈ {-24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, 24} | x : ℤ
this : x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
⊢ x ∣ 24 ↔
x = -24 ∨
x = -12 ∨
x = -8 ∨
x = -6 ∨ x = -4 ∨ x = -3 ∨ x = -2 ∨ x = -1 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 6 ∨ x = 8 ∨ x = 12 ∨ x = 24 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
this : x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
⊢ x ∣ 24 ↔ x ∈ {-24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, 24}
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma2 | [46, 1] | [55, 39] | simp only [Finset.mem_insert, Int.natAbs_eq_iff, Nat.cast_one,
Nat.cast_ofNat, Finset.mem_singleton] at this | x : ℤ
this : x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
⊢ x ∣ 24 ↔
x = -24 ∨
x = -12 ∨
x = -8 ∨
x = -6 ∨ x = -4 ∨ x = -3 ∨ x = -2 ∨ x = -1 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 6 ∨ x = 8 ∨ x = 12 ∨ x = 24 | x : ℤ
this :
x ∣ 24 ↔
(x = 1 ∨ x = -1) ∨
(x = 2 ∨ x = -2) ∨
(x = 3 ∨ x = -3) ∨
(x = 4 ∨ x = -4) ∨ (x = 6 ∨ x = -6) ∨ (x = 8 ∨ x = -8) ∨ (x = 12 ∨ x = -12) ∨ x = 24 ∨ x = -24
⊢ x ∣ 24 ↔
x = -24 ∨
x = -12 ∨
x = -8 ∨
x = -6 ∨ x = -4 ∨ x = -3 ∨ x = -2 ∨ x = -1 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 6 ∨ x = 8 ∨ x = 12 ∨ x = 24 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
this : x ∣ 24 ↔ Int.natAbs x ∈ {1, 2, 3, 4, 6, 8, 12, 24}
⊢ x ∣ 24 ↔
x = -24 ∨
x = -12 ∨
x = -8 ∨
x = -6 ∨ x = -4 ∨ x = -3 ∨ x = -2 ∨ x = -1 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 6 ∨ x = 8 ∨ x = 12 ∨ x = 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma2 | [46, 1] | [55, 39] | tauto | x : ℤ
this :
x ∣ 24 ↔
(x = 1 ∨ x = -1) ∨
(x = 2 ∨ x = -2) ∨
(x = 3 ∨ x = -3) ∨
(x = 4 ∨ x = -4) ∨ (x = 6 ∨ x = -6) ∨ (x = 8 ∨ x = -8) ∨ (x = 12 ∨ x = -12) ∨ x = 24 ∨ x = -24
⊢ x ∣ 24 ↔
x = -24 ∨
x = -12 ∨
x = -8 ∨
x = -6 ∨ x = -4 ∨ x = -3 ∨ x = -2 ∨ x = -1 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 6 ∨ x = 8 ∨ x = 12 ∨ x = 24 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
this :
x ∣ 24 ↔
(x = 1 ∨ x = -1) ∨
(x = 2 ∨ x = -2) ∨
(x = 3 ∨ x = -3) ∨
(x = 4 ∨ x = -4) ∨ (x = 6 ∨ x = -6) ∨ (x = 8 ∨ x = -8) ∨ (x = 12 ∨ x = -12) ∨ x = 24 ∨ x = -24
⊢ x ∣ 24 ↔
x = -24 ∨
x = -12 ∨
x = -8 ∨
x = -6 ∨ x = -4 ∨ x = -3 ∨ x = -2 ∨ x = -1 ∨ x = 1 ∨ x = 2 ∨ x = 3 ∨ x = 4 ∨ x = 6 ∨ x = 8 ∨ x = 12 ∨ x = 24
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet2.lean | Section15sheet2Solutions.lemma2 | [46, 1] | [55, 39] | norm_num | x : ℤ
⊢ 24 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ 24 ≠ 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section04sets/Sheet4.lean | Section4sheet4Solutions.mem_def | [38, 1] | [40, 6] | rfl | X : Type
P : X → Prop
a : X
⊢ a ∈ {x | P x} ↔ P a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
P : X → Prop
a : X
⊢ a ∈ {x | P x} ↔ P a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section03functions/Sheet1.lean | Section3sheet1solutions.injective_def | [52, 1] | [53, 6] | rfl | X Y Z : Type
f : X → Y
⊢ Injective f ↔ ∀ (a b : X), f a = f b → a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
f : X → Y
⊢ Injective f ↔ ∀ (a b : X), f a = f b → a = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section03functions/Sheet1.lean | Section3sheet1solutions.surjective_def | [59, 1] | [60, 6] | rfl | X Y Z : Type
f : X → Y
⊢ Surjective f ↔ ∀ (b : Y), ∃ a, f a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
f : X → Y
⊢ Surjective f ↔ ∀ (b : Y), ∃ a, f a = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section03functions/Sheet1.lean | Section3sheet1solutions.id_eval | [63, 1] | [64, 6] | rfl | X Y Z : Type
x : X
⊢ id x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
x : X
⊢ id x = x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section03functions/Sheet1.lean | Section3sheet1solutions.comp_eval | [68, 1] | [69, 6] | rfl | X Y Z : Type
f : X → Y
g : Y → Z
x : X
⊢ (g ∘ f) x = g (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : Type
f : X → Y
g : Y → Z
x : X
⊢ (g ∘ f) x = g (f x)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section17curvesAndSurfaces/Sheet2.lean | Section17sheet2solutions.Complex.differentiableAt_coe | [22, 1] | [24, 53] | apply Complex.ofRealClm.differentiableAt.comp _ hf | f : ℝ → ℝ
x : ℝ
hf : DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ (fun y => ↑(f y)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
x : ℝ
hf : DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ (fun y => ↑(f y)) x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section18graphTheory/Sheet2.lean | e10 | [121, 1] | [121, 38] | decide | V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 1 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 1 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section18graphTheory/Sheet2.lean | e21 | [123, 1] | [123, 38] | decide | V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 2 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 2 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section18graphTheory/Sheet2.lean | e02 | [125, 1] | [125, 38] | decide | V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 0 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 0 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section18graphTheory/Sheet2.lean | e30 | [127, 1] | [127, 38] | decide | V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 3 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 3 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section18graphTheory/Sheet2.lean | e43 | [129, 1] | [129, 38] | decide | V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 4 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 4 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section18graphTheory/Sheet2.lean | e04 | [131, 1] | [131, 38] | decide | V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 0 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
G : SimpleGraph V
v w x : V
a : Walk G v w
⊢ Adj g5 0 4
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section17curvesAndSurfaces/Sheet2.lean | Section17sheet2.Complex.differentiableAt_coe | [22, 1] | [25, 53] | apply Complex.ofRealClm.differentiableAt.comp _ hf | f : ℝ → ℝ
x : ℝ
hf : DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ (fun y => ↑(f y)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
x : ℝ
hf : DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ (fun y => ↑(f y)) x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section04sets/Sheet1.lean | Section4sheet1Solutions.subset_def | [60, 1] | [62, 6] | rfl | X : Type
A B C D : Set X
x : X
⊢ A ⊆ B ↔ ∀ x ∈ A, x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
A B C D : Set X
x : X
⊢ A ⊆ B ↔ ∀ x ∈ A, x ∈ B
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section04sets/Sheet1.lean | Section4sheet1Solutions.mem_union_iff | [64, 1] | [65, 6] | rfl | X : Type
A B C D : Set X
x : X
⊢ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
A B C D : Set X
x : X
⊢ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet3.lean | Section3sheet1.Yb_ne_Yc | [50, 1] | [53, 10] | intro h | ⊢ Y.b ≠ Y.c | h : Y.b = Y.c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Y.b ≠ Y.c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet3.lean | Section3sheet1.Yb_ne_Yc | [50, 1] | [53, 10] | cases h | h : Y.b = Y.c
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
h : Y.b = Y.c
⊢ False
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet3.lean | Section3sheet1.gYb_eq_gYc | [56, 1] | [57, 8] | sorry | ⊢ g Y.b = g Y.c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ g Y.b = g Y.c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet3.lean | Section3sheet1.gf_injective | [61, 1] | [62, 8] | sorry | ⊢ Injective (g ∘ f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Injective (g ∘ f)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section03functions/Sheet3.lean | Section3sheet1.gf_surjective | [71, 1] | [72, 8] | sorry | ⊢ Surjective (g ∘ f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Surjective (g ∘ f)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section19algebraicNumberTheory/Sheet2.lean | Section19sheet2.lemma1 | [30, 1] | [36, 80] | constructor | K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a ↔ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) | case mp
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a → Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) → IsIntegral R a | Please generate a tactic in lean4 to solve the state.
STATE:
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a ↔ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section19algebraicNumberTheory/Sheet2.lean | Section19sheet2.lemma1 | [30, 1] | [36, 80] | exact IsIntegral.fg_adjoin_singleton | case mp
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a → Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ IsIntegral R a → Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section19algebraicNumberTheory/Sheet2.lean | Section19sheet2.lemma1 | [30, 1] | [36, 80] | intro h | case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) → IsIntegral R a | case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
h : Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
⊢ IsIntegral R a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
⊢ Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a})) → IsIntegral R a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section19algebraicNumberTheory/Sheet2.lean | Section19sheet2.lemma1 | [30, 1] | [36, 80] | exact IsIntegral.of_mem_of_fg _ h _ (Algebra.self_mem_adjoin_singleton R a) | case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
h : Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
⊢ IsIntegral R a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
K : Type
inst✝² : CommRing K
R : Type
inst✝¹ : CommRing R
inst✝ : Algebra R K
a : K
h : Submodule.FG (Subalgebra.toSubmodule (Algebra.adjoin R {a}))
⊢ IsIntegral R a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet1.lean | QSqrt2.ext | [237, 1] | [238, 29] | cases a | K : Type
inst✝¹ : Field K
inst✝ : NumberField K
a b : QSqrt2
h1 : a.R = b.R
h2 : a.i = b.i
⊢ a = b | case mk
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
b : QSqrt2
R✝ i✝ : ℚ
h1 : { R := R✝, i := i✝ }.R = b.R
h2 : { R := R✝, i := i✝ }.i = b.i
⊢ { R := R✝, i := i✝ } = b | Please generate a tactic in lean4 to solve the state.
STATE:
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
a b : QSqrt2
h1 : a.R = b.R
h2 : a.i = b.i
⊢ a = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet1.lean | QSqrt2.ext | [237, 1] | [238, 29] | cases b | case mk
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
b : QSqrt2
R✝ i✝ : ℚ
h1 : { R := R✝, i := i✝ }.R = b.R
h2 : { R := R✝, i := i✝ }.i = b.i
⊢ { R := R✝, i := i✝ } = b | case mk.mk
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
R✝¹ i✝¹ R✝ i✝ : ℚ
h1 : { R := R✝¹, i := i✝¹ }.R = { R := R✝, i := i✝ }.R
h2 : { R := R✝¹, i := i✝¹ }.i = { R := R✝, i := i✝ }.i
⊢ { R := R✝¹, i := i✝¹ } = { R := R✝, i := i✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
b : QSqrt2
R✝ i✝ : ℚ
h1 : { R := R✝, i := i✝ }.R = b.R
h2 : { R := R✝, i := i✝ }.i = b.i
⊢ { R := R✝, i := i✝ } = b
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section19algebraicNumberTheory/Sheet1.lean | QSqrt2.ext | [237, 1] | [238, 29] | simp_all | case mk.mk
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
R✝¹ i✝¹ R✝ i✝ : ℚ
h1 : { R := R✝¹, i := i✝¹ }.R = { R := R✝, i := i✝ }.R
h2 : { R := R✝¹, i := i✝¹ }.i = { R := R✝, i := i✝ }.i
⊢ { R := R✝¹, i := i✝¹ } = { R := R✝, i := i✝ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
K : Type
inst✝¹ : Field K
inst✝ : NumberField K
R✝¹ i✝¹ R✝ i✝ : ℚ
h1 : { R := R✝¹, i := i✝¹ }.R = { R := R✝, i := i✝ }.R
h2 : { R := R✝¹, i := i✝¹ }.i = { R := R✝, i := i✝ }.i
⊢ { R := R✝¹, i := i✝¹ } = { R := R✝, i := i✝ }
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_def | [60, 1] | [62, 6] | rfl | a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_thirtyseven | [77, 1] | [84, 11] | rw [tendsTo_def] | ⊢ TendsTo (fun n => 37) 37 | ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ TendsTo (fun n => 37) 37
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_thirtyseven | [77, 1] | [84, 11] | intro ε hε | ⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | ε : ℝ
hε : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_thirtyseven | [77, 1] | [84, 11] | use 100 | ε : ℝ
hε : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε | case h
ε : ℝ
hε : 0 < ε
⊢ ∀ (n : ℕ), 100 ≤ n → |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_thirtyseven | [77, 1] | [84, 11] | intro n hn | case h
ε : ℝ
hε : 0 < ε
⊢ ∀ (n : ℕ), 100 ≤ n → |37 - 37| < ε | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ |37 - 37| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε
⊢ ∀ (n : ℕ), 100 ≤ n → |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_thirtyseven | [77, 1] | [84, 11] | norm_num | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ |37 - 37| < ε | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ |37 - 37| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_thirtyseven | [77, 1] | [84, 11] | exact hε | case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε
n : ℕ
hn : 100 ≤ n
⊢ 0 < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | intro ε hε | c : ℝ
⊢ TendsTo (fun n => c) c | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c) n - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℝ
⊢ TendsTo (fun n => c) c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | dsimp only | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c) n - c| < ε | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |c - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => c) n - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | use 37 | c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |c - c| < ε | case h
c ε : ℝ
hε : ε > 0
⊢ ∀ (n : ℕ), 37 ≤ n → |c - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |c - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | intro n hn | case h
c ε : ℝ
hε : ε > 0
⊢ ∀ (n : ℕ), 37 ≤ n → |c - c| < ε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |c - c| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
⊢ ∀ (n : ℕ), 37 ≤ n → |c - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | ring_nf | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |c - c| < ε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |c - c| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | norm_num | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |0| < ε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ |0| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_const | [87, 1] | [95, 11] | exact hε | case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c ε : ℝ
hε : ε > 0
n : ℕ
hn : 37 ≤ n
⊢ 0 < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_add_const | [98, 1] | [103, 10] | rw [tendsTo_def] at h ⊢ | a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n + c) (t + c) | a : ℕ → ℝ
t c : ℝ
h : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + c - (t + c)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n + c) (t + c)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_add_const | [98, 1] | [103, 10] | ring_nf | a : ℕ → ℝ
t c : ℝ
h : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + c - (t + c)| < ε | a : ℕ → ℝ
t c : ℝ
h : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n + c - (t + c)| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section02reals/Sheet3.lean | Section2sheet3solutions.tendsTo_add_const | [98, 1] | [103, 10] | exact h | a : ℕ → ℝ
t c : ℝ
h : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section20representationTheory/Sheet2.lean | Section20sheet2.RepMap.comp_id | [92, 1] | [92, 65] | sorry | k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
⊢ comp φ (id ρ) = φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
⊢ comp φ (id ρ) = φ
TACTIC:
|
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