file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ico_subset_Ico_union_Ico | []
| [
1503,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1502,
1
]
|
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.Perm.zpow_aux | [
{
"state_after": "α : Type u\nβ : Type v\np : α → Prop\nf : Perm α\nhf : ∀ (x : α), p x ↔ p (↑f x)\nn : ℕ\n⊢ ∀ (x : α), p x ↔ p (↑(f ^ (n + 1))⁻¹ x)",
"state_before": "α : Type u\nβ : Type v\np : α → Prop\nf : Perm α\nhf : ∀ (x : α), p x ↔ p (↑f x)\nn : ℕ\n⊢ ∀ (x : α), p x ↔ p (↑(f ^ Int.negSucc n) x)",
"tactic": "rw [zpow_negSucc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\np : α → Prop\nf : Perm α\nhf : ∀ (x : α), p x ↔ p (↑f x)\nn : ℕ\n⊢ ∀ (x : α), p x ↔ p (↑(f ^ (n + 1))⁻¹ x)",
"tactic": "exact inv_aux.1 (pow_aux hf)"
}
]
| [
417,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
9
]
|
Mathlib/Algebra/Order/Hom/Basic.lean | le_map_add_map_div' | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.39551\nF : Type u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.39563\nδ : Type ?u.39566\ninst✝² : Group α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : GroupSeminormClass F α β\nf : F\nx y : α\n⊢ ↑f x ≤ ↑f y + ↑f (y / x)",
"tactic": "simpa only [add_comm, map_div_rev, div_mul_cancel'] using map_mul_le_add f (x / y) y"
}
]
| [
259,
87
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
]
|
Mathlib/Combinatorics/Additive/SalemSpencer.lean | mulSalemSpencer_insert_of_lt | [
{
"state_after": "F : Type ?u.90423\nα : Type u_1\nβ : Type ?u.90429\n𝕜 : Type ?u.90432\nE : Type ?u.90435\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\na : α\nhs : ∀ (i : α), i ∈ s → i < a\n⊢ (MulSalemSpencer s ∧\n (∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c) ↔\n MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b",
"state_before": "F : Type ?u.90423\nα : Type u_1\nβ : Type ?u.90429\n𝕜 : Type ?u.90432\nE : Type ?u.90435\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\na : α\nhs : ∀ (i : α), i ∈ s → i < a\n⊢ MulSalemSpencer (insert a s) ↔ MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b",
"tactic": "refine' mulSalemSpencer_insert.trans _"
},
{
"state_after": "F : Type ?u.90423\nα : Type u_1\nβ : Type ?u.90429\n𝕜 : Type ?u.90432\nE : Type ?u.90435\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\na : α\nhs : ∀ (i : α), i ∈ s → i < a\n⊢ ((MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧\n ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c) ↔\n MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b",
"state_before": "F : Type ?u.90423\nα : Type u_1\nβ : Type ?u.90429\n𝕜 : Type ?u.90432\nE : Type ?u.90435\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\na : α\nhs : ∀ (i : α), i ∈ s → i < a\n⊢ (MulSalemSpencer s ∧\n (∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c) ↔\n MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b",
"tactic": "rw [← and_assoc]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.90423\nα : Type u_1\nβ : Type ?u.90429\n𝕜 : Type ?u.90432\nE : Type ?u.90435\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\na : α\nhs : ∀ (i : α), i ∈ s → i < a\n⊢ ((MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧\n ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c) ↔\n MulSalemSpencer s ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b",
"tactic": "exact and_iff_left fun b c hb hc h => ((mul_lt_mul_of_lt_of_lt (hs _ hb) (hs _ hc)).ne h).elim"
}
]
| [
227,
97
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
]
|
Mathlib/Order/UpperLower/Basic.lean | UpperSet.mem_iSup_iff | [
{
"state_after": "α : Type u_1\nβ : Type ?u.59414\nγ : Type ?u.59417\nι : Sort u_2\nκ : ι → Sort ?u.59425\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : ι → UpperSet α\n⊢ (a ∈ ⋂ (i : ι), ↑(f i)) ↔ ∀ (i : ι), a ∈ f i",
"state_before": "α : Type u_1\nβ : Type ?u.59414\nγ : Type ?u.59417\nι : Sort u_2\nκ : ι → Sort ?u.59425\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : ι → UpperSet α\n⊢ (a ∈ ⨆ (i : ι), f i) ↔ ∀ (i : ι), a ∈ f i",
"tactic": "rw [← SetLike.mem_coe, coe_iSup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.59414\nγ : Type ?u.59417\nι : Sort u_2\nκ : ι → Sort ?u.59425\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\nf : ι → UpperSet α\n⊢ (a ∈ ⋂ (i : ι), ↑(f i)) ↔ ∀ (i : ι), a ∈ f i",
"tactic": "exact mem_iInter"
}
]
| [
603,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
601,
1
]
|
Mathlib/GroupTheory/Submonoid/Pointwise.lean | Submonoid.inv_iSup | []
| [
210,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
]
|
Mathlib/Data/Set/Basic.lean | SetCoe.forall' | []
| [
203,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
]
|
Mathlib/Data/Polynomial/UnitTrinomial.lean | Polynomial.IsUnitTrinomial.irreducible_aux1 | [
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * (↑C ↑u * X ^ (m + n) + ↑C ↑w * X ^ (n - m + k + n)) =\n { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) (p * mirror p).toFinsupp }",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\n⊢ ↑C ↑v * (↑C ↑u * X ^ (m + n) + ↑C ↑w * X ^ (n - m + k + n)) =\n { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) (p * mirror p).toFinsupp }",
"tactic": "have key : n - m + k < n := by rwa [← lt_tsub_iff_right, tsub_lt_tsub_iff_left_of_le hmn.le]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * (↑C ↑u * X ^ (m + n) + ↑C ↑w * X ^ (n - m + k + n)) =\n {\n toFinsupp :=\n Finsupp.filter (Set.Ioo (k + n) (n + n))\n (trinomial k m n ↑u ↑v ↑w * trinomial k (n - m + k) n ↑w ↑v ↑u).toFinsupp }",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * (↑C ↑u * X ^ (m + n) + ↑C ↑w * X ^ (n - m + k + n)) =\n { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) (p * mirror p).toFinsupp }",
"tactic": "rw [hp, trinomial_mirror hkm hmn u.ne_zero w.ne_zero]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n Finsupp.filter (Set.Ioo (k + n) (n + n))\n (Finsupp.single (k + k) (↑u * ↑w) + Finsupp.single (k + (n - m + k)) (↑u * ↑v) +\n Finsupp.single (k + n) (↑u * ↑u) +\n (Finsupp.single (m + k) (↑v * ↑w) + Finsupp.single (m + (n - m + k)) (↑v * ↑v) +\n Finsupp.single (m + n) (↑v * ↑u)) +\n (Finsupp.single (n + k) (↑w * ↑w) + Finsupp.single (n + (n - m + k)) (↑w * ↑v) +\n Finsupp.single (n + n) (↑w * ↑u))) }",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * (↑C ↑u * X ^ (m + n) + ↑C ↑w * X ^ (n - m + k + n)) =\n {\n toFinsupp :=\n Finsupp.filter (Set.Ioo (k + n) (n + n))\n (trinomial k m n ↑u ↑v ↑w * trinomial k (n - m + k) n ↑w ↑v ↑u).toFinsupp }",
"tactic": "simp_rw [trinomial_def, C_mul_X_pow_eq_monomial, add_mul, mul_add, monomial_mul_monomial,\n toFinsupp_add, toFinsupp_monomial, Finsupp.filter_add]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + k) (↑u * ↑w)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + (n - m + k)) (↑u * ↑v)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + n) (↑u * ↑u)) +\n (Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + k) (↑v * ↑w)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + (n - m + k)) (↑v * ↑v)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + n) (↑v * ↑u))) +\n (Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + k) (↑w * ↑w)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + (n - m + k)) (↑w * ↑v)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + n) (↑w * ↑u))) }",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n Finsupp.filter (Set.Ioo (k + n) (n + n))\n (Finsupp.single (k + k) (↑u * ↑w) + Finsupp.single (k + (n - m + k)) (↑u * ↑v) +\n Finsupp.single (k + n) (↑u * ↑u) +\n (Finsupp.single (m + k) (↑v * ↑w) + Finsupp.single (m + (n - m + k)) (↑v * ↑v) +\n Finsupp.single (m + n) (↑v * ↑u)) +\n (Finsupp.single (n + k) (↑w * ↑w) + Finsupp.single (n + (n - m + k)) (↑w * ↑v) +\n Finsupp.single (n + n) (↑w * ↑u))) }",
"tactic": "rw [Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add,\n Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n 0 + 0 + 0 + (0 + 0 + Finsupp.single (m + n) (↑v * ↑u)) + (0 + Finsupp.single (n + (n - m + k)) (↑w * ↑v) + 0) }\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (n + n)\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ Set.Ioo (k + n) (n + n) (n + (n - m + k))\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (n + k)\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ Set.Ioo (k + n) (n + n) (m + n)\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (m + (n - m + k))\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (m + k)\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (k + n)\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (k + (n - m + k))\n\ncase h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (k + k)",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + k) (↑u * ↑w)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + (n - m + k)) (↑u * ↑v)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + n) (↑u * ↑u)) +\n (Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + k) (↑v * ↑w)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + (n - m + k)) (↑v * ↑v)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + n) (↑v * ↑u))) +\n (Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + k) (↑w * ↑w)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + (n - m + k)) (↑w * ↑v)) +\n Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + n) (↑w * ↑u))) }",
"tactic": "rw [Finsupp.filter_single_of_neg, Finsupp.filter_single_of_neg, Finsupp.filter_single_of_neg,\n Finsupp.filter_single_of_neg, Finsupp.filter_single_of_neg, Finsupp.filter_single_of_pos,\n Finsupp.filter_single_of_neg, Finsupp.filter_single_of_pos, Finsupp.filter_single_of_neg]"
},
{
"state_after": "no goals",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\n⊢ n - m + k < n",
"tactic": "rwa [← lt_tsub_iff_right, tsub_lt_tsub_iff_left_of_le hmn.le]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n { toFinsupp := Finsupp.single (m + n) (↑v * ↑u) + Finsupp.single (n + (n - m + k)) (↑w * ↑v) }",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n 0 + 0 + 0 + (0 + 0 + Finsupp.single (m + n) (↑v * ↑u)) + (0 + Finsupp.single (n + (n - m + k)) (↑w * ↑v) + 0) }",
"tactic": "simp only [add_zero, zero_add, ofFinsupp_add, ofFinsupp_single]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n { toFinsupp := Finsupp.single (m + n) (↑v * ↑u) } + { toFinsupp := Finsupp.single (n + (n - m + k)) (↑w * ↑v) }",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n { toFinsupp := Finsupp.single (m + n) (↑v * ↑u) + Finsupp.single (n + (n - m + k)) (↑w * ↑v) }",
"tactic": "rw [ofFinsupp_add]"
},
{
"state_after": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n ↑(monomial (m + n)) (↑v * ↑u) + ↑(monomial (n + (n - m + k))) (↑w * ↑v)",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n { toFinsupp := Finsupp.single (m + n) (↑v * ↑u) } + { toFinsupp := Finsupp.single (n + (n - m + k)) (↑w * ↑v) }",
"tactic": "simp only [ofFinsupp_single]"
},
{
"state_after": "no goals",
"state_before": "p q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ↑C ↑v * ↑(monomial (m + n)) ↑u + ↑C ↑v * ↑(monomial (n - m + k + n)) ↑w =\n ↑(monomial (m + n)) (↑v * ↑u) + ↑(monomial (n + (n - m + k))) (↑w * ↑v)",
"tactic": "rw [C_mul_monomial, C_mul_monomial, mul_comm (v : ℤ) w, add_comm (n - m + k) n]"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (n + n)",
"tactic": "exact fun h => h.2.ne rfl"
},
{
"state_after": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ k + n < n + (n - m + k)",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ Set.Ioo (k + n) (n + n) (n + (n - m + k))",
"tactic": "refine' ⟨_, add_lt_add_left key n⟩"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ k + n < n + (n - m + k)",
"tactic": "rwa [add_comm, add_lt_add_iff_left, lt_add_iff_pos_left, tsub_pos_iff_lt]"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (n + k)",
"tactic": "exact fun h => h.1.ne (add_comm k n)"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ Set.Ioo (k + n) (n + n) (m + n)",
"tactic": "exact ⟨add_lt_add_right hkm n, add_lt_add_right hmn n⟩"
},
{
"state_after": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (n + k) (n + n) (n + k)",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (m + (n - m + k))",
"tactic": "rw [← add_assoc, add_tsub_cancel_of_le hmn.le, add_comm]"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (n + k) (n + n) (n + k)",
"tactic": "exact fun h => h.1.ne rfl"
},
{
"state_after": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\n⊢ False",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (m + k)",
"tactic": "intro h"
},
{
"state_after": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\nthis : k + n < m + k\n⊢ False",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\n⊢ False",
"tactic": "have := h.1"
},
{
"state_after": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\nthis : n < m\n⊢ False",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\nthis : k + n < m + k\n⊢ False",
"tactic": "rw [add_comm, add_lt_add_iff_right] at this"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\nthis : n < m\n⊢ False",
"tactic": "exact asymm this hmn"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (k + n)",
"tactic": "exact fun h => h.1.ne rfl"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (k + (n - m + k))",
"tactic": "exact fun h => asymm ((add_lt_add_iff_left k).mp h.1) key"
},
{
"state_after": "no goals",
"state_before": "case h\np q : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ ¬Set.Ioo (k + n) (n + n) (k + k)",
"tactic": "exact fun h => asymm ((add_lt_add_iff_left k).mp h.1) (hkm.trans hmn)"
}
]
| [
256,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
]
|
Mathlib/LinearAlgebra/FreeModule/PID.lean | Module.free_of_finite_type_torsion_free | [
{
"state_after": "case intro\nι : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nb : ι → M\ninst✝¹ : _root_.Finite ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : NoZeroSMulDivisors R M\nval✝ : Fintype ι\n⊢ Free R M",
"state_before": "ι : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nb : ι → M\ninst✝¹ : _root_.Finite ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : NoZeroSMulDivisors R M\n⊢ Free R M",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro.mk\nι : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nb✝ : ι → M\ninst✝¹ : _root_.Finite ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : NoZeroSMulDivisors R M\nval✝ : Fintype ι\nn : ℕ\nb : Basis (Fin n) R M\n⊢ Free R M",
"state_before": "case intro\nι : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nb : ι → M\ninst✝¹ : _root_.Finite ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : NoZeroSMulDivisors R M\nval✝ : Fintype ι\n⊢ Free R M",
"tactic": "obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nι : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nb✝ : ι → M\ninst✝¹ : _root_.Finite ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : NoZeroSMulDivisors R M\nval✝ : Fintype ι\nn : ℕ\nb : Basis (Fin n) R M\n⊢ Free R M",
"tactic": "exact Module.Free.of_basis b"
}
]
| [
410,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
406,
1
]
|
Mathlib/Analysis/Convex/Cone/Basic.lean | Submodule.coe_toConvexCone | []
| [
518,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
517,
1
]
|
Mathlib/Algebra/Divisibility/Basic.lean | MulHom.map_dvd | []
| [
103,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
]
|
Mathlib/Topology/Semicontinuous.lean | UpperSemicontinuousOn.upperSemicontinuousWithinAt | []
| [
687,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
685,
1
]
|
Mathlib/Topology/Order/Basic.lean | continuousWithinAt_Ioo_iff_Iio | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderClosedTopology α\na✝ b✝ : α\ninst✝ : TopologicalSpace γ\na b : α\nf : α → γ\nh : a < b\n⊢ ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b",
"tactic": "simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]"
}
]
| [
512,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
510,
1
]
|
Mathlib/Algebra/EuclideanDomain/Basic.lean | EuclideanDomain.gcd_one_left | []
| [
187,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | Set.Subsingleton.differentiableOn | []
| [
1132,
77
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1131,
1
]
|
Mathlib/Data/Fin/Basic.lean | Fin.castPred_zero | []
| [
2299,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2298,
1
]
|
Mathlib/CategoryTheory/StructuredArrow.lean | CategoryTheory.CostructuredArrow.eq_mk | [
{
"state_after": "case mk\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nleft✝ : C\nright✝ : Discrete PUnit\nhom✝ : S.obj left✝ ⟶ (Functor.fromPUnit T).obj right✝\n⊢ { left := left✝, right := right✝, hom := hom✝ } = mk { left := left✝, right := right✝, hom := hom✝ }.hom",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nf : CostructuredArrow S T\n⊢ f = mk f.hom",
"tactic": "cases f"
},
{
"state_after": "no goals",
"state_before": "case mk\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nleft✝ : C\nright✝ : Discrete PUnit\nhom✝ : S.obj left✝ ⟶ (Functor.fromPUnit T).obj right✝\n⊢ { left := left✝, right := right✝, hom := hom✝ } = mk { left := left✝, right := right✝, hom := hom✝ }.hom",
"tactic": "congr"
}
]
| [
401,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
399,
1
]
|
Mathlib/Data/Finset/NoncommProd.lean | Multiset.noncommProd_add | [
{
"state_after": "case mk\nF : Type ?u.83828\nι : Type ?u.83831\nα : Type u_1\nβ : Type ?u.83837\nγ : Type ?u.83840\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\ns t : Multiset α\na✝ : List α\ncomm : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝ + t} Commute\n⊢ noncommProd (Quot.mk Setoid.r a✝ + t) comm =\n noncommProd (Quot.mk Setoid.r a✝) (_ : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝} Commute) *\n noncommProd t (_ : Set.Pairwise {x | x ∈ t} Commute)",
"state_before": "F : Type ?u.83828\nι : Type ?u.83831\nα : Type u_1\nβ : Type ?u.83837\nγ : Type ?u.83840\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\ns t : Multiset α\ncomm : Set.Pairwise {x | x ∈ s + t} Commute\n⊢ noncommProd (s + t) comm =\n noncommProd s (_ : Set.Pairwise {x | x ∈ s} Commute) * noncommProd t (_ : Set.Pairwise {x | x ∈ t} Commute)",
"tactic": "rcases s with ⟨⟩"
},
{
"state_after": "case mk.mk\nF : Type ?u.83828\nι : Type ?u.83831\nα : Type u_1\nβ : Type ?u.83837\nγ : Type ?u.83840\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\ns t : Multiset α\na✝¹ a✝ : List α\ncomm : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝¹ + Quot.mk Setoid.r a✝} Commute\n⊢ noncommProd (Quot.mk Setoid.r a✝¹ + Quot.mk Setoid.r a✝) comm =\n noncommProd (Quot.mk Setoid.r a✝¹) (_ : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝¹} Commute) *\n noncommProd (Quot.mk Setoid.r a✝) (_ : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝} Commute)",
"state_before": "case mk\nF : Type ?u.83828\nι : Type ?u.83831\nα : Type u_1\nβ : Type ?u.83837\nγ : Type ?u.83840\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\ns t : Multiset α\na✝ : List α\ncomm : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝ + t} Commute\n⊢ noncommProd (Quot.mk Setoid.r a✝ + t) comm =\n noncommProd (Quot.mk Setoid.r a✝) (_ : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝} Commute) *\n noncommProd t (_ : Set.Pairwise {x | x ∈ t} Commute)",
"tactic": "rcases t with ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nF : Type ?u.83828\nι : Type ?u.83831\nα : Type u_1\nβ : Type ?u.83837\nγ : Type ?u.83840\nf : α → β → β\nop : α → α → α\ninst✝¹ : Monoid α\ninst✝ : Monoid β\ns t : Multiset α\na✝¹ a✝ : List α\ncomm : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝¹ + Quot.mk Setoid.r a✝} Commute\n⊢ noncommProd (Quot.mk Setoid.r a✝¹ + Quot.mk Setoid.r a✝) comm =\n noncommProd (Quot.mk Setoid.r a✝¹) (_ : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝¹} Commute) *\n noncommProd (Quot.mk Setoid.r a✝) (_ : Set.Pairwise {x | x ∈ Quot.mk Setoid.r a✝} Commute)",
"tactic": "simp"
}
]
| [
177,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
]
|
Mathlib/RingTheory/Congruence.lean | RingCon.coe_mul | []
| [
204,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
]
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | InnerProductGeometry.angle_smul_smul | [
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ : V\nc : ℝ\nhc : c ≠ 0\nx y : V\nthis : c * c ≠ 0\n⊢ angle (c • x) (c • y) = angle x y",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ : V\nc : ℝ\nhc : c ≠ 0\nx y : V\n⊢ angle (c • x) (c • y) = angle x y",
"tactic": "have : c * c ≠ 0 := mul_ne_zero hc hc"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ : V\nc : ℝ\nhc : c ≠ 0\nx y : V\nthis : c * c ≠ 0\n⊢ angle (c • x) (c • y) = angle x y",
"tactic": "rw [angle, angle, real_inner_smul_left, inner_smul_right, norm_smul, norm_smul, Real.norm_eq_abs,\n mul_mul_mul_comm _ ‖x‖, abs_mul_abs_self, ← mul_assoc c c, mul_div_mul_left _ _ this]"
}
]
| [
62,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
]
|
Mathlib/Analysis/NormedSpace/CompactOperator.lean | isCompactOperator_iff_exists_mem_nhds_image_subset_compact | []
| [
84,
59
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
]
|
Mathlib/Data/Real/Hyperreal.lean | Hyperreal.omega_ne_zero | []
| [
194,
16
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
]
|
Mathlib/RingTheory/Nilpotent.lean | isNilpotent_neg_iff | []
| [
58,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
]
|
Mathlib/Topology/MetricSpace/EMetricSpace.lean | ULift.edist_eq | []
| [
438,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
]
|
Mathlib/MeasureTheory/MeasurableSpaceDef.lean | Measurable.le | []
| [
571,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
570,
1
]
|
Mathlib/RingTheory/ChainOfDivisors.lean | DivisorChain.isPrimePow_of_has_chain | []
| [
221,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
]
|
Mathlib/RingTheory/DedekindDomain/Ideal.lean | FractionalIdeal.adjoinIntegral_eq_one_of_isUnit | [
{
"state_after": "R : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nI : FractionalIdeal A⁰ K := adjoinIntegral A⁰ x hx\nhI : IsUnit I\n⊢ I = 1",
"state_before": "R : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nhI : IsUnit (adjoinIntegral A⁰ x hx)\n⊢ adjoinIntegral A⁰ x hx = 1",
"tactic": "set I := adjoinIntegral A⁰ x hx"
},
{
"state_after": "R : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nI : FractionalIdeal A⁰ K := adjoinIntegral A⁰ x hx\nhI : IsUnit I\nmul_self : I * I = I\n⊢ I = 1",
"state_before": "R : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nI : FractionalIdeal A⁰ K := adjoinIntegral A⁰ x hx\nhI : IsUnit I\n⊢ I = 1",
"tactic": "have mul_self : I * I = I := by apply coeToSubmodule_injective; simp"
},
{
"state_after": "case a\nR : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nI : FractionalIdeal A⁰ K := adjoinIntegral A⁰ x hx\nhI : IsUnit I\n⊢ (fun I => ↑I) (I * I) = (fun I => ↑I) I",
"state_before": "R : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nI : FractionalIdeal A⁰ K := adjoinIntegral A⁰ x hx\nhI : IsUnit I\n⊢ I * I = I",
"tactic": "apply coeToSubmodule_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type ?u.114414\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : IsIntegral A x\nI : FractionalIdeal A⁰ K := adjoinIntegral A⁰ x hx\nhI : IsUnit I\n⊢ (fun I => ↑I) (I * I) = (fun I => ↑I) I",
"tactic": "simp"
}
]
| [
272,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
]
|
Std/Data/Array/Init/Lemmas.lean | Array.append_data | [
{
"state_after": "α : Type u_1\narr arr' : Array α\n⊢ (Array.append arr arr').data = arr.data ++ arr'.data",
"state_before": "α : Type u_1\narr arr' : Array α\n⊢ (arr ++ arr').data = arr.data ++ arr'.data",
"tactic": "rw [← append_eq_append]"
},
{
"state_after": "α : Type u_1\narr arr' : Array α\n⊢ (foldl (fun r v => push r v) arr arr' 0 (size arr')).data = arr.data ++ arr'.data",
"state_before": "α : Type u_1\narr arr' : Array α\n⊢ (Array.append arr arr').data = arr.data ++ arr'.data",
"tactic": "unfold Array.append"
},
{
"state_after": "α : Type u_1\narr arr' : Array α\n⊢ (List.foldl (fun r v => push r v) arr arr'.data).data = arr.data ++ arr'.data",
"state_before": "α : Type u_1\narr arr' : Array α\n⊢ (foldl (fun r v => push r v) arr arr' 0 (size arr')).data = arr.data ++ arr'.data",
"tactic": "rw [foldl_eq_foldl_data]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\narr arr' : Array α\n⊢ (List.foldl (fun r v => push r v) arr arr'.data).data = arr.data ++ arr'.data",
"tactic": "induction arr'.data generalizing arr <;> simp [*]"
}
]
| [
196,
52
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
192,
9
]
|
Mathlib/Data/IsROrC/Basic.lean | IsROrC.ofRealClm_coe | []
| [
1010,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1009,
1
]
|
Mathlib/Data/Nat/Order/Lemmas.lean | Nat.div_eq_iff_eq_of_dvd_dvd | [
{
"state_after": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\n⊢ n / x = n / y → x = y\n\ncase mpr\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\n⊢ x = y → n / x = n / y",
"state_before": "a b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\n⊢ n / x = n / y ↔ x = y",
"tactic": "constructor"
},
{
"state_after": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ x = y",
"state_before": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\n⊢ n / x = n / y → x = y",
"tactic": "intro h"
},
{
"state_after": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ n * x = n * y",
"state_before": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ x = y",
"tactic": "rw [← mul_right_inj' hn]"
},
{
"state_after": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ n * x / y = n",
"state_before": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ n * x = n * y",
"tactic": "apply Nat.eq_mul_of_div_eq_left (dvd_mul_of_dvd_left hy x)"
},
{
"state_after": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ n = x * (n / y)",
"state_before": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ n * x / y = n",
"tactic": "rw [eq_comm, mul_comm, Nat.mul_div_assoc _ hy]"
},
{
"state_after": "no goals",
"state_before": "case mp\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : n / x = n / y\n⊢ n = x * (n / y)",
"tactic": "exact Nat.eq_mul_of_div_eq_right hx h"
},
{
"state_after": "case mpr\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : x = y\n⊢ n / x = n / y",
"state_before": "case mpr\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\n⊢ x = y → n / x = n / y",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\na b m n✝ k n x y : ℕ\nhn : n ≠ 0\nhx : x ∣ n\nhy : y ∣ n\nh : x = y\n⊢ n / x = n / y",
"tactic": "rw [h]"
}
]
| [
80,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.range_sigma_eq_iUnion_range | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ✝ : Type ?u.148439\nι : Sort ?u.148442\nι' : Sort ?u.148445\nι₂ : Sort ?u.148448\nκ : ι → Sort ?u.148453\nκ₁ : ι → Sort ?u.148458\nκ₂ : ι → Sort ?u.148463\nκ' : ι' → Sort ?u.148468\nγ : α → Type u_1\nf : Sigma γ → β\n⊢ ∀ (x : β), x ∈ range f ↔ x ∈ ⋃ (a : α), range fun b => f { fst := a, snd := b }",
"tactic": "simp"
}
]
| [
1265,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1263,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.measure_biUnion_finset | []
| [
192,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
1
]
|
Mathlib/Data/Finset/Card.lean | Finset.card_union_le | []
| [
419,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
418,
1
]
|
Mathlib/RingTheory/Ideal/Basic.lean | Ideal.span_one | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\n⊢ span 1 = ⊤",
"tactic": "rw [← Set.singleton_one, span_singleton_one]"
}
]
| [
210,
91
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.zero_of_to_zero | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\nf : X ⟶ 0\n⊢ f = 0",
"tactic": "ext"
}
]
| [
378,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
]
|
Mathlib/CategoryTheory/Preadditive/Schur.lean | CategoryTheory.finrank_hom_simple_simple_le_one | [
{
"state_after": "case inl\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Subsingleton (X ⟶ Y)\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1\n\ncase inr\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"state_before": "C : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"tactic": "obtain (h|h) := subsingleton_or_nontrivial (X ⟶ Y)"
},
{
"state_after": "case inl\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Subsingleton (X ⟶ Y)\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"state_before": "case inl\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Subsingleton (X ⟶ Y)\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"tactic": "skip"
},
{
"state_after": "case inl\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Subsingleton (X ⟶ Y)\n⊢ 0 ≤ 1",
"state_before": "case inl\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Subsingleton (X ⟶ Y)\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"tactic": "rw [finrank_zero_of_subsingleton]"
},
{
"state_after": "no goals",
"state_before": "case inl\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Subsingleton (X ⟶ Y)\n⊢ 0 ≤ 1",
"tactic": "exact zero_le_one"
},
{
"state_after": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"state_before": "case inr\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"tactic": "obtain ⟨f, nz⟩ := (nontrivial_iff_exists_ne 0).mp h"
},
{
"state_after": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"state_before": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"tactic": "haveI fi := (isIso_iff_nonzero f).mpr nz"
},
{
"state_after": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\n⊢ ∀ (w : X ⟶ Y), ∃ c, c • f = w",
"state_before": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\n⊢ finrank 𝕜 (X ⟶ Y) ≤ 1",
"tactic": "refine' finrank_le_one f _"
},
{
"state_after": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\ng : X ⟶ Y\n⊢ ∃ c, c • f = g",
"state_before": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\n⊢ ∀ (w : X ⟶ Y), ∃ c, c • f = w",
"tactic": "intro g"
},
{
"state_after": "case inr.intro.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\ng : X ⟶ Y\nc : 𝕜\nw : c • 𝟙 X = g ≫ inv f\n⊢ ∃ c, c • f = g",
"state_before": "case inr.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\ng : X ⟶ Y\n⊢ ∃ c, c • f = g",
"tactic": "obtain ⟨c, w⟩ := endomorphism_simple_eq_smul_id 𝕜 (g ≫ inv f)"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nC : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\ng : X ⟶ Y\nc : 𝕜\nw : c • 𝟙 X = g ≫ inv f\n⊢ ∃ c, c • f = g",
"tactic": "exact ⟨c, by simpa using w =≫ f⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u_3\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_1\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIso f\ng : X ⟶ Y\nc : 𝕜\nw : c • 𝟙 X = g ≫ inv f\n⊢ c • f = g",
"tactic": "simpa using w =≫ f"
}
]
| [
183,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
172,
1
]
|
Mathlib/Data/Finset/PImage.lean | Finset.pimage_some | [
{
"state_after": "case a\nα : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\nf✝ g : α →. β\ninst✝² : (x : α) → Decidable (f✝ x).Dom\ninst✝¹ : (x : α) → Decidable (g x).Dom\ns✝ t : Finset α\nb : β\ns : Finset α\nf : α → β\ninst✝ : (x : α) → Decidable (Part.some (f x)).Dom\na✝ : β\n⊢ a✝ ∈ pimage (fun x => Part.some (f x)) s ↔ a✝ ∈ image f s",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\nf✝ g : α →. β\ninst✝² : (x : α) → Decidable (f✝ x).Dom\ninst✝¹ : (x : α) → Decidable (g x).Dom\ns✝ t : Finset α\nb : β\ns : Finset α\nf : α → β\ninst✝ : (x : α) → Decidable (Part.some (f x)).Dom\n⊢ pimage (fun x => Part.some (f x)) s = image f s",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\nf✝ g : α →. β\ninst✝² : (x : α) → Decidable (f✝ x).Dom\ninst✝¹ : (x : α) → Decidable (g x).Dom\ns✝ t : Finset α\nb : β\ns : Finset α\nf : α → β\ninst✝ : (x : α) → Decidable (Part.some (f x)).Dom\na✝ : β\n⊢ a✝ ∈ pimage (fun x => Part.some (f x)) s ↔ a✝ ∈ image f s",
"tactic": "simp [eq_comm]"
}
]
| [
82,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
]
|
Mathlib/Deprecated/Ring.lean | RingHom.coe_of | []
| [
158,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
]
|
Mathlib/Data/Set/Basic.lean | Set.mem_of_mem_diff | []
| [
1807,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1806,
1
]
|
Mathlib/Topology/DenseEmbedding.lean | denseEmbedding_id | []
| [
302,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
301,
1
]
|
Mathlib/Order/LiminfLimsup.lean | Filter.liminf_eq | []
| [
386,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
385,
1
]
|
Mathlib/Data/List/Basic.lean | List.map_comp_map | [
{
"state_after": "case h.a.a\nι : Type ?u.136607\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ng : β → γ\nf : α → β\nl : List α\nn✝ : ℕ\na✝ : γ\n⊢ a✝ ∈ get? ((map g ∘ map f) l) n✝ ↔ a✝ ∈ get? (map (g ∘ f) l) n✝",
"state_before": "ι : Type ?u.136607\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ng : β → γ\nf : α → β\n⊢ map g ∘ map f = map (g ∘ f)",
"tactic": "ext l"
},
{
"state_after": "no goals",
"state_before": "case h.a.a\nι : Type ?u.136607\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ng : β → γ\nf : α → β\nl : List α\nn✝ : ℕ\na✝ : γ\n⊢ a✝ ∈ get? ((map g ∘ map f) l) n✝ ↔ a✝ ∈ get? (map (g ∘ f) l) n✝",
"tactic": "rw [comp_map, Function.comp_apply]"
}
]
| [
1858,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1857,
1
]
|
Std/Data/Option/Lemmas.lean | Option.get_some | []
| [
35,
83
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
35,
9
]
|
Mathlib/Data/Matrix/Basic.lean | Matrix.scalar_inj | [
{
"state_after": "case mp\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\n⊢ ↑(scalar n) r = ↑(scalar n) s → r = s\n\ncase mpr\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\n⊢ r = s → ↑(scalar n) r = ↑(scalar n) s",
"state_before": "l : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\n⊢ ↑(scalar n) r = ↑(scalar n) s ↔ r = s",
"tactic": "constructor"
},
{
"state_after": "case mp\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\nh : ↑(scalar n) r = ↑(scalar n) s\n⊢ r = s",
"state_before": "case mp\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\n⊢ ↑(scalar n) r = ↑(scalar n) s → r = s",
"tactic": "intro h"
},
{
"state_after": "case mp\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\nh : ↑(scalar n) r = ↑(scalar n) s\ninhabited_h : Inhabited n\n⊢ r = s",
"state_before": "case mp\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\nh : ↑(scalar n) r = ↑(scalar n) s\n⊢ r = s",
"tactic": "inhabit n"
},
{
"state_after": "no goals",
"state_before": "case mp\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\nh : ↑(scalar n) r = ↑(scalar n) s\ninhabited_h : Inhabited n\n⊢ r = s",
"tactic": "rw [← scalar_apply_eq r (Inhabited.default (α := n)),\n ← scalar_apply_eq s (Inhabited.default (α := n)), h]"
},
{
"state_after": "case mpr\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr : α\n⊢ ↑(scalar n) r = ↑(scalar n) r",
"state_before": "case mpr\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr s : α\n⊢ r = s → ↑(scalar n) r = ↑(scalar n) s",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nl : Type ?u.458991\nm : Type ?u.458994\nn : Type u_1\no : Type ?u.459000\nm' : o → Type ?u.459005\nn' : o → Type ?u.459010\nR : Type ?u.459013\nS : Type ?u.459016\nα : Type v\nβ : Type w\nγ : Type ?u.459023\ninst✝³ : Semiring α\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Nonempty n\nr : α\n⊢ ↑(scalar n) r = ↑(scalar n) r",
"tactic": "rfl"
}
]
| [
1270,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1263,
1
]
|
Mathlib/Algebra/Order/Monoid/TypeTags.lean | Additive.toMul_le | []
| [
134,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
]
|
Mathlib/Algebra/FreeAlgebra.lean | FreeAlgebra.ι_ne_algebraMap | [
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\n⊢ False",
"tactic": "let f0 : FreeAlgebra R X →ₐ[R] R := lift R 0"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\n⊢ False",
"tactic": "let f1 : FreeAlgebra R X →ₐ[R] R := lift R 1"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : ↑f0 (ι R x) = 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\n⊢ False",
"tactic": "have hf0 : f0 (ι R x) = 0 := lift_ι_apply _ _"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : ↑f0 (ι R x) = 0\nhf1 : ↑f1 (ι R x) = 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : ↑f0 (ι R x) = 0\n⊢ False",
"tactic": "have hf1 : f1 (ι R x) = 1 := lift_ι_apply _ _"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : r = 0\nhf1 : ↑f1 (ι R x) = 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : ↑f0 (ι R x) = 0\nhf1 : ↑f1 (ι R x) = 1\n⊢ False",
"tactic": "rw [h, f0.commutes, Algebra.id.map_eq_self] at hf0"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : r = 0\nhf1 : r = 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : r = 0\nhf1 : ↑f1 (ι R x) = 1\n⊢ False",
"tactic": "rw [h, f1.commutes, Algebra.id.map_eq_self] at hf1"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.733161\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : X\nr : R\nh : ι R x = ↑(algebraMap R (FreeAlgebra R X)) r\nf0 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 0\nf1 : FreeAlgebra R X →ₐ[R] R := ↑(lift R) 1\nhf0 : r = 0\nhf1 : r = 1\n⊢ False",
"tactic": "exact zero_ne_one (hf0.symm.trans hf1)"
}
]
| [
461,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
]
|
Mathlib/Topology/Constructions.lean | continuous_quotient_mk' | []
| [
1162,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1161,
1
]
|
Mathlib/Order/UpperLower/Basic.lean | UpperSet.mem_sInf_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.58938\nγ : Type ?u.58941\nι : Sort ?u.58944\nκ : ι → Sort ?u.58949\ninst✝ : LE α\nS : Set (UpperSet α)\ns t : UpperSet α\na : α\n⊢ (∃ i j, a ∈ ↑i) ↔ ∃ s, s ∈ S ∧ a ∈ s",
"tactic": "simp only [exists_prop, SetLike.mem_coe]"
}
]
| [
597,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
596,
1
]
|
Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_le_of_le_one_of_le' | []
| [
859,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
857,
1
]
|
Mathlib/Algebra/Order/Field/Basic.lean | le_of_neg_of_one_div_le_one_div | []
| [
875,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
874,
1
]
|
Mathlib/Analysis/Calculus/ContDiff.lean | contDiffAt_const | []
| [
102,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
]
|
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.coe_update | []
| [
832,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
831,
1
]
|
Mathlib/Topology/LocalExtr.lean | IsLocalMax.comp_antitone | []
| [
238,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
8
]
|
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.ghostMap.bijective_of_invertible | []
| [
338,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
1
]
|
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_add_const_Ico | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x + a) ⁻¹' Ico b c = Ico (b - a) (c - a)",
"tactic": "simp [← Ici_inter_Iio]"
}
]
| [
114,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
113,
1
]
|
Mathlib/RingTheory/Nilpotent.lean | Commute.isNilpotent_sub | [
{
"state_after": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm✝ : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent y\n⊢ IsNilpotent (x - y)",
"state_before": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm : Commute x y\nhx : IsNilpotent x\nhy : IsNilpotent y\n⊢ IsNilpotent (x - y)",
"tactic": "rw [← neg_right_iff] at h_comm"
},
{
"state_after": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm✝ : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent (-y)\n⊢ IsNilpotent (x - y)",
"state_before": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm✝ : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent y\n⊢ IsNilpotent (x - y)",
"tactic": "rw [← isNilpotent_neg_iff] at hy"
},
{
"state_after": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm✝ : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent (-y)\n⊢ IsNilpotent (x + -y)",
"state_before": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm✝ : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent (-y)\n⊢ IsNilpotent (x - y)",
"tactic": "rw [sub_eq_add_neg]"
},
{
"state_after": "no goals",
"state_before": "R S : Type u\nx y : R\ninst✝ : Ring R\nh_comm✝ : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent (-y)\n⊢ IsNilpotent (x + -y)",
"tactic": "exact h_comm.isNilpotent_add hx hy"
}
]
| [
179,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.constantCoeff_zero | []
| [
508,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
507,
1
]
|
Mathlib/Algebra/Module/Bimodule.lean | Subbimodule.smul_mem' | [
{
"state_after": "R : Type u_4\nA : Type u_2\nB : Type u_1\nM : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScalarTower R B M\ninst✝ : SMulCommClass A B M\np : Submodule (A ⊗[R] B) M\nb : B\nm : M\nhm : m ∈ p\n⊢ b • m = 1 ⊗ₜ[R] b • m",
"state_before": "R : Type u_4\nA : Type u_2\nB : Type u_1\nM : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScalarTower R B M\ninst✝ : SMulCommClass A B M\np : Submodule (A ⊗[R] B) M\nb : B\nm : M\nhm : m ∈ p\n⊢ b • m ∈ p",
"tactic": "suffices b • m = (1 : A) ⊗ₜ[R] b • m by exact this.symm ▸ p.smul_mem _ hm"
},
{
"state_after": "no goals",
"state_before": "R : Type u_4\nA : Type u_2\nB : Type u_1\nM : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScalarTower R B M\ninst✝ : SMulCommClass A B M\np : Submodule (A ⊗[R] B) M\nb : B\nm : M\nhm : m ∈ p\n⊢ b • m = 1 ⊗ₜ[R] b • m",
"tactic": "simp [TensorProduct.Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_4\nA : Type u_2\nB : Type u_1\nM : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScalarTower R B M\ninst✝ : SMulCommClass A B M\np : Submodule (A ⊗[R] B) M\nb : B\nm : M\nhm : m ∈ p\nthis : b • m = 1 ⊗ₜ[R] b • m\n⊢ b • m ∈ p",
"tactic": "exact this.symm ▸ p.smul_mem _ hm"
}
]
| [
105,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
]
|
Mathlib/Data/Matrix/Basic.lean | Matrix.submatrix_mulVec_equiv | []
| [
2509,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2506,
1
]
|
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.add_eq_left_iff | [
{
"state_after": "a b : Cardinal\n⊢ a + b = a ↔ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0",
"state_before": "a b : Cardinal\n⊢ a + b = a ↔ max ℵ₀ b ≤ a ∨ b = 0",
"tactic": "rw [max_le_iff]"
},
{
"state_after": "case refine'_1\na b : Cardinal\nh : a + b = a\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0\n\ncase refine'_2\na b : Cardinal\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 → a + b = a",
"state_before": "a b : Cardinal\n⊢ a + b = a ↔ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0",
"tactic": "refine' ⟨fun h => _, _⟩"
},
{
"state_after": "case refine'_1.inl\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0\n\ncase refine'_1.inr\na b : Cardinal\nh : a + b = a\nha : a < ℵ₀\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0",
"state_before": "case refine'_1\na b : Cardinal\nh : a + b = a\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0",
"tactic": "cases' le_or_lt ℵ₀ a with ha ha"
},
{
"state_after": "case refine'_1.inr.h\na b : Cardinal\nh : a + b = a\nha : a < ℵ₀\n⊢ b = 0",
"state_before": "case refine'_1.inr\na b : Cardinal\nh : a + b = a\nha : a < ℵ₀\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0",
"tactic": "right"
},
{
"state_after": "case refine'_1.inr.h\na b : Cardinal\nh : a + b = a\nha : (∃ n, a = ↑n) ∧ ∃ n, b = ↑n\n⊢ b = 0",
"state_before": "case refine'_1.inr.h\na b : Cardinal\nh : a + b = a\nha : a < ℵ₀\n⊢ b = 0",
"tactic": "rw [← h, add_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha"
},
{
"state_after": "case refine'_1.inr.h.intro.intro.intro\nn m : ℕ\nh : ↑n + ↑m = ↑n\n⊢ ↑m = 0",
"state_before": "case refine'_1.inr.h\na b : Cardinal\nh : a + b = a\nha : (∃ n, a = ↑n) ∧ ∃ n, b = ↑n\n⊢ b = 0",
"tactic": "rcases ha with ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩"
},
{
"state_after": "case refine'_1.inr.h.intro.intro.intro\nn m : ℕ\nh : n + m = n\n⊢ m = 0",
"state_before": "case refine'_1.inr.h.intro.intro.intro\nn m : ℕ\nh : ↑n + ↑m = ↑n\n⊢ ↑m = 0",
"tactic": "norm_cast at h⊢"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr.h.intro.intro.intro\nn m : ℕ\nh : n + m = n\n⊢ m = 0",
"tactic": "rw [← add_right_inj, h, add_zero]"
},
{
"state_after": "case refine'_1.inl.h\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ ℵ₀ ≤ a ∧ b ≤ a",
"state_before": "case refine'_1.inl\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0",
"tactic": "left"
},
{
"state_after": "case refine'_1.inl.h\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ b ≤ a",
"state_before": "case refine'_1.inl.h\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ ℵ₀ ≤ a ∧ b ≤ a",
"tactic": "use ha"
},
{
"state_after": "case refine'_1.inl.h\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ ¬a < b",
"state_before": "case refine'_1.inl.h\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ b ≤ a",
"tactic": "rw [← not_lt]"
},
{
"state_after": "a b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ a < b → a < a + b",
"state_before": "case refine'_1.inl.h\na b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ ¬a < b",
"tactic": "apply fun hb => ne_of_gt _ h"
},
{
"state_after": "a b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\nhb : a < b\n⊢ a < a + b",
"state_before": "a b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\n⊢ a < b → a < a + b",
"tactic": "intro hb"
},
{
"state_after": "no goals",
"state_before": "a b : Cardinal\nh : a + b = a\nha : ℵ₀ ≤ a\nhb : a < b\n⊢ a < a + b",
"tactic": "exact hb.trans_le (self_le_add_left b a)"
},
{
"state_after": "case refine'_2.inl.intro\na b : Cardinal\nh1 : ℵ₀ ≤ a\nh2 : b ≤ a\n⊢ a + b = a\n\ncase refine'_2.inr\na b : Cardinal\nh3 : b = 0\n⊢ a + b = a",
"state_before": "case refine'_2\na b : Cardinal\n⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 → a + b = a",
"tactic": "rintro (⟨h1, h2⟩ | h3)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inl.intro\na b : Cardinal\nh1 : ℵ₀ ≤ a\nh2 : b ≤ a\n⊢ a + b = a",
"tactic": "rw [add_eq_max h1, max_eq_left h2]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inr\na b : Cardinal\nh3 : b = 0\n⊢ a + b = a",
"tactic": "rw [h3, add_zero]"
}
]
| [
804,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
787,
1
]
|
Mathlib/Algebra/Algebra/Equiv.lean | AlgEquiv.arrowCongr_refl | [
{
"state_after": "case H.H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nx✝¹ : A₁ →ₐ[R] A₂\nx✝ : A₁\n⊢ ↑(↑(arrowCongr refl refl) x✝¹) x✝ = ↑(↑(Equiv.refl (A₁ →ₐ[R] A₂)) x✝¹) x✝",
"state_before": "R : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\n⊢ arrowCongr refl refl = Equiv.refl (A₁ →ₐ[R] A₂)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nx✝¹ : A₁ →ₐ[R] A₂\nx✝ : A₁\n⊢ ↑(↑(arrowCongr refl refl) x✝¹) x✝ = ↑(↑(Equiv.refl (A₁ →ₐ[R] A₂)) x✝¹) x✝",
"tactic": "rfl"
}
]
| [
468,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
1
]
|
Mathlib/GroupTheory/Exponent.lean | card_dvd_exponent_pow_rank | [
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\n⊢ Nat.card G ∣ Monoid.exponent G ^ Group.rank G",
"state_before": "G : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\n⊢ Nat.card G ∣ Monoid.exponent G ^ Group.rank G",
"tactic": "obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\n⊢ Nat.card G ∣ Monoid.exponent G ^ Group.rank G",
"tactic": "rw [← hS1, ← Fintype.card_coe, ← Finset.card_univ, ← Finset.prod_const]"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"tactic": "let f : (∀ g : S, zpowers (g : G)) →* G := noncommPiCoprod fun s t _ x y _ _ => mul_comm x _"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf : Function.Surjective ↑f\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"tactic": "have hf : Function.Surjective f := by\n rw [← MonoidHom.range_top_iff_surjective, eq_top_iff, ← hS2, closure_le]\n exact fun g hg => ⟨Pi.mulSingle ⟨g, hg⟩ ⟨g, mem_zpowers g⟩, noncommPiCoprod_mulSingle _ _⟩"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf : Function.Surjective ↑f\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"tactic": "replace hf := nat_card_dvd_of_surjective f hf"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf✝ : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\nhf : Nat.card G ∣ ∏ a : { x // x ∈ S }, Nat.card { x // x ∈ zpowers ↑a }\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"tactic": "rw [Nat.card_pi] at hf"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf✝ : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\nhf : Nat.card G ∣ ∏ a : { x // x ∈ S }, Nat.card { x // x ∈ zpowers ↑a }\ng : { x // x ∈ S }\nx✝ : g ∈ Finset.univ\n⊢ Nat.card { x // x ∈ zpowers ↑g } ∣ Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf✝ : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\nhf : Nat.card G ∣ ∏ a : { x // x ∈ S }, Nat.card { x // x ∈ zpowers ↑a }\n⊢ Nat.card G ∣ ∏ _x : { x // x ∈ S }, Monoid.exponent G",
"tactic": "refine' hf.trans (Finset.prod_dvd_prod_of_dvd _ _ fun g _ => _)"
},
{
"state_after": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf✝ : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\nhf : Nat.card G ∣ ∏ a : { x // x ∈ S }, Nat.card { x // x ∈ zpowers ↑a }\ng : { x // x ∈ S }\nx✝ : g ∈ Finset.univ\n⊢ orderOf ↑g ∣ Monoid.exponent G",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf✝ : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\nhf : Nat.card G ∣ ∏ a : { x // x ∈ S }, Nat.card { x // x ∈ zpowers ↑a }\ng : { x // x ∈ S }\nx✝ : g ∈ Finset.univ\n⊢ Nat.card { x // x ∈ zpowers ↑g } ∣ Monoid.exponent G",
"tactic": "rw [← order_eq_card_zpowers']"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nG : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\nhf✝ : Nat.card G ∣ Nat.card ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g })\nhf : Nat.card G ∣ ∏ a : { x // x ∈ S }, Nat.card { x // x ∈ zpowers ↑a }\ng : { x // x ∈ S }\nx✝ : g ∈ Finset.univ\n⊢ orderOf ↑g ∣ Monoid.exponent G",
"tactic": "exact Monoid.order_dvd_exponent (g : G)"
},
{
"state_after": "G : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\n⊢ ↑S ⊆ ↑(MonoidHom.range f)",
"state_before": "G : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\n⊢ Function.Surjective ↑f",
"tactic": "rw [← MonoidHom.range_top_iff_surjective, eq_top_iff, ← hS2, closure_le]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝¹ : CommGroup G\ninst✝ : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure ↑S = ⊤\nf : ((g : { x // x ∈ S }) → { x // x ∈ zpowers ↑g }) →* G :=\n noncommPiCoprod (_ : ∀ (s t : { x // x ∈ S }), s ≠ t → ∀ (x y : G), x ∈ zpowers ↑s → y ∈ zpowers ↑t → x * y = y * x)\n⊢ ↑S ⊆ ↑(MonoidHom.range f)",
"tactic": "exact fun g hg => ⟨Pi.mulSingle ⟨g, hg⟩ ⟨g, mem_zpowers g⟩, noncommPiCoprod_mulSingle _ _⟩"
}
]
| [
379,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | Coplanar.finiteDimensional_vectorSpan | [
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.339734\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\nh : Coplanar k s\n⊢ 2 < Cardinal.aleph0",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.339734\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\nh : Coplanar k s\n⊢ FiniteDimensional k { x // x ∈ vectorSpan k s }",
"tactic": "refine' IsNoetherian.iff_fg.1 (IsNoetherian.iff_rank_lt_aleph0.2 (lt_of_le_of_lt h _))"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.339734\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\nh : Coplanar k s\n⊢ 2 < Cardinal.aleph0",
"tactic": "exact Cardinal.lt_aleph0.2 ⟨2, rfl⟩"
}
]
| [
614,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
611,
1
]
|
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.sub_snd | []
| [
285,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
]
|
Mathlib/Algebra/Group/WithOne/Defs.lean | WithOne.cases_on | []
| [
185,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
11
]
|
Mathlib/Data/Polynomial/EraseLead.lean | Polynomial.eraseLead_natDegree_lt | []
| [
194,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
]
|
Std/Logic.lean | iff_iff_eq | []
| [
51,
61
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
51,
1
]
|
Mathlib/Topology/Algebra/Group/Basic.lean | GroupTopology.continuous_inv' | [
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis : TopologicalSpace α := g.toTopologicalSpace\n⊢ Continuous Inv.inv",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\n⊢ Continuous Inv.inv",
"tactic": "letI := g.toTopologicalSpace"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis✝ : TopologicalSpace α := g.toTopologicalSpace\nthis : TopologicalGroup α\n⊢ Continuous Inv.inv",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis : TopologicalSpace α := g.toTopologicalSpace\n⊢ Continuous Inv.inv",
"tactic": "haveI := g.toTopologicalGroup"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis✝ : TopologicalSpace α := g.toTopologicalSpace\nthis : TopologicalGroup α\n⊢ Continuous Inv.inv",
"tactic": "exact continuous_inv"
}
]
| [
1868,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1863,
1
]
|
Mathlib/Algebra/AlgebraicCard.lean | Algebraic.infinite_of_charZero | []
| [
35,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
33,
1
]
|
Std/Data/Nat/Lemmas.lean | Nat.dvd_of_mul_dvd_mul_left | [
{
"state_after": "k m n : Nat\nkpos : 0 < k\nH✝ : k * m ∣ k * n\nl : Nat\nH : k * n = k * m * l\n⊢ m ∣ n",
"state_before": "k m n : Nat\nkpos : 0 < k\nH : k * m ∣ k * n\n⊢ m ∣ n",
"tactic": "let ⟨l, H⟩ := H"
},
{
"state_after": "k m n : Nat\nkpos : 0 < k\nH✝ : k * m ∣ k * n\nl : Nat\nH : k * n = k * (m * l)\n⊢ m ∣ n",
"state_before": "k m n : Nat\nkpos : 0 < k\nH✝ : k * m ∣ k * n\nl : Nat\nH : k * n = k * m * l\n⊢ m ∣ n",
"tactic": "rw [Nat.mul_assoc] at H"
},
{
"state_after": "no goals",
"state_before": "k m n : Nat\nkpos : 0 < k\nH✝ : k * m ∣ k * n\nl : Nat\nH : k * n = k * (m * l)\n⊢ m ∣ n",
"tactic": "exact ⟨_, Nat.eq_of_mul_eq_mul_left kpos H⟩"
}
]
| [
750,
46
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
746,
11
]
|
Mathlib/Data/Sum/Interval.lean | Sum.Icc_inl_inl | []
| [
142,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
]
|
Mathlib/Analysis/Convex/Topology.lean | isClosed_stdSimplex | []
| [
68,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.weightedVSubOfPoint_indicator_subset | [
{
"state_after": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.58761\ns₂✝ : Finset ι₂\nw : ι → k\np : ι → P\nb : P\ns₁ s₂ : Finset ι\nh : s₁ ⊆ s₂\n⊢ ∑ i in s₁, w i • (p i -ᵥ b) = ∑ i in s₂, Set.indicator (↑s₁) w i • (p i -ᵥ b)",
"state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.58761\ns₂✝ : Finset ι₂\nw : ι → k\np : ι → P\nb : P\ns₁ s₂ : Finset ι\nh : s₁ ⊆ s₂\n⊢ ↑(weightedVSubOfPoint s₁ p b) w = ↑(weightedVSubOfPoint s₂ p b) (Set.indicator (↑s₁) w)",
"tactic": "rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.58761\ns₂✝ : Finset ι₂\nw : ι → k\np : ι → P\nb : P\ns₁ s₂ : Finset ι\nh : s₁ ⊆ s₂\n⊢ ∑ i in s₁, w i • (p i -ᵥ b) = ∑ i in s₂, Set.indicator (↑s₁) w i • (p i -ᵥ b)",
"tactic": "exact\n Set.sum_indicator_subset_of_eq_zero w (fun i wi => wi • (p i -ᵥ b : V)) h fun i => zero_smul k _"
}
]
| [
174,
101
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
]
|
Mathlib/Algebra/Order/Monoid/WithTop.lean | WithBot.one_lt_coe | []
| [
521,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
520,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | Disjoint.edgeSet | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.125448\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nH₁ H₂ : Subgraph G\nh : Disjoint H₁ H₂\n⊢ SimpleGraph.Subgraph.edgeSet H₁ ⊓ SimpleGraph.Subgraph.edgeSet H₂ ≤ ⊥",
"tactic": "simpa using edgeSet_mono h.le_bot"
}
]
| [
639,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
1
]
|
Mathlib/Order/Max.lean | isTop_ofDual_iff | []
| [
258,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
]
|
Mathlib/Analysis/Calculus/Deriv/Comp.lean | HasStrictFDerivAt.comp_hasStrictDerivAt | []
| [
265,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
]
|
Mathlib/Order/Filter/Germ.lean | Filter.Germ.map_id | [
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39044\nδ : Type ?u.39047\nl : Filter α\nf✝ g h : α → β\nx✝ : Germ l β\nf : α → β\n⊢ map id (Quot.mk Setoid.r f) = id (Quot.mk Setoid.r f)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.39044\nδ : Type ?u.39047\nl : Filter α\nf g h : α → β\n⊢ map id = id",
"tactic": "ext ⟨f⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.39044\nδ : Type ?u.39047\nl : Filter α\nf✝ g h : α → β\nx✝ : Germ l β\nf : α → β\n⊢ map id (Quot.mk Setoid.r f) = id (Quot.mk Setoid.r f)",
"tactic": "rfl"
}
]
| [
199,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
]
|
Mathlib/Data/Complex/Basic.lean | Complex.sub_conj | [
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ (z - ↑(starRingEnd ℂ) z).re = (↑(2 * z.im) * I).re ∧ (z - ↑(starRingEnd ℂ) z).im = (↑(2 * z.im) * I).im",
"tactic": "simp [two_mul, sub_eq_add_neg, ofReal']"
}
]
| [
719,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
718,
1
]
|
Std/Data/List/Lemmas.lean | List.mem_diff_of_mem | [
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl₁ : List α\nb : α\nl₂ : List α\nh₁ : a ∈ l₁\nh₂ : ¬a ∈ b :: l₂\n⊢ a ∈ List.diff (List.erase l₁ b) l₂",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl₁ : List α\nb : α\nl₂ : List α\nh₁ : a ∈ l₁\nh₂ : ¬a ∈ b :: l₂\n⊢ a ∈ List.diff l₁ (b :: l₂)",
"tactic": "rw [diff_cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl₁ : List α\nb : α\nl₂ : List α\nh₁ : a ∈ l₁\nh₂ : ¬a ∈ b :: l₂\n⊢ a ∈ List.diff (List.erase l₁ b) l₂",
"tactic": "exact mem_diff_of_mem ((mem_erase_of_ne <| ne_of_not_mem_cons h₂).2 h₁) (mt (.tail _) h₂)"
}
]
| [
1539,
94
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1535,
1
]
|
Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.Compacts.coe_finset_sup | [
{
"state_after": "α : Type u_2\nβ : Type ?u.34252\nγ : Type ?u.34255\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\ns✝ : Finset ι\nf : ι → Compacts α\na : ι\ns : Finset ι\nx✝ : ¬a ∈ s\nh : ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i)\n⊢ ↑(Finset.sup (Finset.cons a s x✝) f) = Finset.sup (Finset.cons a s x✝) fun i => ↑(f i)",
"state_before": "α : Type u_2\nβ : Type ?u.34252\nγ : Type ?u.34255\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\ns : Finset ι\nf : ι → Compacts α\n⊢ ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i)",
"tactic": "refine Finset.cons_induction_on s rfl fun a s _ h => ?_"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.34252\nγ : Type ?u.34255\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\ns✝ : Finset ι\nf : ι → Compacts α\na : ι\ns : Finset ι\nx✝ : ¬a ∈ s\nh : ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i)\n⊢ ↑(f a) ∪ ↑(Finset.sup s f) = ↑(f a) ∪ Finset.sup s fun i => ↑(f i)",
"state_before": "α : Type u_2\nβ : Type ?u.34252\nγ : Type ?u.34255\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\ns✝ : Finset ι\nf : ι → Compacts α\na : ι\ns : Finset ι\nx✝ : ¬a ∈ s\nh : ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i)\n⊢ ↑(Finset.sup (Finset.cons a s x✝) f) = Finset.sup (Finset.cons a s x✝) fun i => ↑(f i)",
"tactic": "simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.34252\nγ : Type ?u.34255\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nι : Type u_1\ns✝ : Finset ι\nf : ι → Compacts α\na : ι\ns : Finset ι\nx✝ : ¬a ∈ s\nh : ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i)\n⊢ ↑(f a) ∪ ↑(Finset.sup s f) = ↑(f a) ∪ Finset.sup s fun i => ↑(f i)",
"tactic": "congr"
}
]
| [
131,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.inv_limsup | []
| [
525,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
523,
1
]
|
Mathlib/LinearAlgebra/Dual.lean | LinearMap.finrank_range_dualMap_eq_finrank_range | [
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K (V₂ ⧸ range f) + finrank K { x // x ∈ range f } = finrank K V₂\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"tactic": "have that := Submodule.finrank_quotient_add_finrank (LinearMap.range f)"
},
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K (V₂ ⧸ range f) + finrank K { x // x ∈ range f } = finrank K V₂\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K (V₂ ⧸ range f) + finrank K { x // x ∈ range f } = finrank K V₂\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"tactic": "let equiv := (Subspace.quotEquivAnnihilator <| LinearMap.range f)"
},
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K (V₂ ⧸ range f) + finrank K { x // x ∈ range f } = finrank K V₂\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K (V₂ ⧸ range f) + finrank K { x // x ∈ range f } = finrank K V₂\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"tactic": "have eq := LinearEquiv.finrank_eq (R := K) (M := (V₂ ⧸ range f))\n (M₂ := { x // x ∈ Submodule.dualAnnihilator (range f) }) equiv"
},
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K V₂\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K (V₂ ⧸ range f) + finrank K { x // x ∈ range f } = finrank K V₂\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"tactic": "rw [eq, ←ker_dualMap_eq_dualAnnihilator_range] at that"
},
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K (Dual K V₂)\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K V₂\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"tactic": "conv_rhs at that => rw [← Subspace.dual_finrank_eq]"
},
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K (Dual K V₂)\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ (fun x => x + finrank K { x // x ∈ ker (dualMap f) }) (finrank K { x // x ∈ range (dualMap f) }) =\n (fun x => x + finrank K { x // x ∈ ker (dualMap f) }) (finrank K { x // x ∈ range f })",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K (Dual K V₂)\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } = finrank K { x // x ∈ range f }",
"tactic": "refine' add_left_injective (finrank K <| LinearMap.ker f.dualMap) _"
},
{
"state_after": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K (Dual K V₂)\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } + finrank K { x // x ∈ ker (dualMap f) } =\n finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker (dualMap f) }",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K (Dual K V₂)\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ (fun x => x + finrank K { x // x ∈ ker (dualMap f) }) (finrank K { x // x ∈ range (dualMap f) }) =\n (fun x => x + finrank K { x // x ∈ ker (dualMap f) }) (finrank K { x // x ∈ range f })",
"tactic": "change _ + _ = _ + _"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\nthat : finrank K { x // x ∈ ker (dualMap f) } + finrank K { x // x ∈ range f } = finrank K (Dual K V₂)\nequiv : (V₂ ⧸ range f) ≃ₗ[K] { x // x ∈ Submodule.dualAnnihilator (range f) } := Subspace.quotEquivAnnihilator (range f)\neq : finrank K (V₂ ⧸ range f) = finrank K { x // x ∈ Submodule.dualAnnihilator (range f) }\n⊢ finrank K { x // x ∈ range (dualMap f) } + finrank K { x // x ∈ ker (dualMap f) } =\n finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker (dualMap f) }",
"tactic": "rw [finrank_range_add_finrank_ker f.dualMap, add_comm, that]"
}
]
| [
1494,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1481,
1
]
|
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Injective.summable_iff | []
| [
140,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
]
|
Mathlib/Algebra/Group/WithOne/Basic.lean | WithOne.lift_unique | []
| [
90,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
]
|
Mathlib/Topology/Order.lean | induced_compose | []
| [
474,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
468,
1
]
|
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.matrix_vecEmpty_coeff | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nR✝ : Type ?u.636245\nS : Type ?u.636248\nT : Type ?u.636251\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R✝\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.636266\nβ : Type ?u.636269\nx y : 𝕎 R✝\nR : Type u_1\ni : Fin 0\nj : ℕ\n⊢ coeff ![] j = ![]",
"tactic": "rcases i with ⟨_ | _ | _ | _ | i_val, ⟨⟩⟩"
}
]
| [
186,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
1
]
|
Mathlib/Data/Set/Pairwise/Basic.lean | Symmetric.pairwise_on | []
| [
55,
99
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
]
|
Mathlib/Data/Fin/Interval.lean | Fin.card_fintypeIci | [
{
"state_after": "case b\nn : ℕ\na b : Fin n\n⊢ Fin n",
"state_before": "n : ℕ\na b : Fin n\n⊢ Fintype.card ↑(Set.Ici a) = n - ↑a",
"tactic": "rw [Fintype.card_ofFinset, card_Ici]"
},
{
"state_after": "no goals",
"state_before": "case b\nn : ℕ\na b : Fin n\n⊢ Fin n",
"tactic": "assumption"
}
]
| [
203,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
]
|
Mathlib/RingTheory/TensorProduct.lean | Algebra.TensorProduct.lmul'_apply_tmul | []
| [
943,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
942,
1
]
|
Mathlib/Data/List/Basic.lean | List.nthLe_zero_scanl | []
| [
2631,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2630,
1
]
|
Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoDiv_sub | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a (b - p) = toIcoDiv hp a b - 1",
"tactic": "simpa only [one_zsmul] using toIcoDiv_sub_zsmul hp a b 1"
}
]
| [
340,
59
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
]
|
Mathlib/Analysis/Complex/Liouville.lean | Differentiable.exists_const_forall_eq_of_bounded | []
| [
129,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
]
|
Mathlib/Algebra/Order/Monoid/MinMax.lean | fn_min_mul_fn_max | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : CommSemigroup β\nf : α → β\nn m : α\n⊢ f (min n m) * f (max n m) = f n * f m",
"tactic": "cases' le_total n m with h h <;> simp [h, mul_comm]"
}
]
| [
29,
100
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
28,
1
]
|
Mathlib/Order/SymmDiff.lean | compl_bihimp | []
| [
743,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
742,
1
]
|
Mathlib/LinearAlgebra/Ray.lean | exists_nonneg_left_iff_sameRay | [
{
"state_after": "R : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\nh : ∃ r, 0 ≤ r ∧ r • x = y\n⊢ SameRay R x y",
"state_before": "R : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\n⊢ (∃ r, 0 ≤ r ∧ r • x = y) ↔ SameRay R x y",
"tactic": "refine' ⟨fun h => _, fun h => h.exists_nonneg_left hx⟩"
},
{
"state_after": "case intro.intro\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nhx : x ≠ 0\nr : R\nhr : 0 ≤ r\n⊢ SameRay R x (r • x)",
"state_before": "R : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\nh : ∃ r, 0 ≤ r ∧ r • x = y\n⊢ SameRay R x y",
"tactic": "rcases h with ⟨r, hr, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_2\ninst✝² : LinearOrderedField R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nhx : x ≠ 0\nr : R\nhr : 0 ≤ r\n⊢ SameRay R x (r • x)",
"tactic": "exact SameRay.sameRay_nonneg_smul_right x hr"
}
]
| [
731,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
727,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigOWith.exists_pos | []
| [
204,
88
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
]
|
Mathlib/Data/Matrix/Basic.lean | Matrix.diagonal_zero | [
{
"state_after": "case a.h\nl : Type ?u.47813\nm : Type ?u.47816\nn : Type u_1\no : Type ?u.47822\nm' : o → Type ?u.47827\nn' : o → Type ?u.47832\nR : Type ?u.47835\nS : Type ?u.47838\nα : Type v\nβ : Type w\nγ : Type ?u.47845\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\ni✝ x✝ : n\n⊢ diagonal (fun x => 0) i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"state_before": "l : Type ?u.47813\nm : Type ?u.47816\nn : Type u_1\no : Type ?u.47822\nm' : o → Type ?u.47827\nn' : o → Type ?u.47832\nR : Type ?u.47835\nS : Type ?u.47838\nα : Type v\nβ : Type w\nγ : Type ?u.47845\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\n⊢ (diagonal fun x => 0) = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.47813\nm : Type ?u.47816\nn : Type u_1\no : Type ?u.47822\nm' : o → Type ?u.47827\nn' : o → Type ?u.47832\nR : Type ?u.47835\nS : Type ?u.47838\nα : Type v\nβ : Type w\nγ : Type ?u.47845\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\ni✝ x✝ : n\n⊢ diagonal (fun x => 0) i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"tactic": "simp [diagonal]"
}
]
| [
461,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
]
|
Mathlib/Algebra/Order/Chebyshev.lean | Monovary.sum_smul_sum_le_card_smul_sum | []
| [
85,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
]
|
Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup | [
{
"state_after": "case H.h1.h\nX : Type u_1\nx y : X\n⊢ ↑(↑(AddMonoidHom.comp (AddMonoidHom.comp toFinsupp toFreeAbelianGroup) (singleAddHom x)) 1) y =\n ↑(↑(AddMonoidHom.comp (AddMonoidHom.id (X →₀ ℤ)) (singleAddHom x)) 1) y",
"state_before": "X : Type u_1\n⊢ AddMonoidHom.comp toFinsupp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ)",
"tactic": "ext (x y)"
},
{
"state_after": "case H.h1.h\nX : Type u_1\nx y : X\n⊢ ↑(↑(AddMonoidHom.comp (AddMonoidHom.comp toFinsupp toFreeAbelianGroup) (singleAddHom x)) 1) y =\n ↑(↑(singleAddHom x) 1) y",
"state_before": "case H.h1.h\nX : Type u_1\nx y : X\n⊢ ↑(↑(AddMonoidHom.comp (AddMonoidHom.comp toFinsupp toFreeAbelianGroup) (singleAddHom x)) 1) y =\n ↑(↑(AddMonoidHom.comp (AddMonoidHom.id (X →₀ ℤ)) (singleAddHom x)) 1) y",
"tactic": "simp only [AddMonoidHom.id_comp]"
},
{
"state_after": "case H.h1.h\nX : Type u_1\nx y : X\n⊢ ↑(↑(AddMonoidHom.comp toFinsupp (↑(AddMonoidHom.flip (smulAddHom ℤ (FreeAbelianGroup X))) (of x))) 1) y =\n ↑(↑(singleAddHom x) 1) y",
"state_before": "case H.h1.h\nX : Type u_1\nx y : X\n⊢ ↑(↑(AddMonoidHom.comp (AddMonoidHom.comp toFinsupp toFreeAbelianGroup) (singleAddHom x)) 1) y =\n ↑(↑(singleAddHom x) 1) y",
"tactic": "rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]"
},
{
"state_after": "no goals",
"state_before": "case H.h1.h\nX : Type u_1\nx y : X\n⊢ ↑(↑(AddMonoidHom.comp toFinsupp (↑(AddMonoidHom.flip (smulAddHom ℤ (FreeAbelianGroup X))) (of x))) 1) y =\n ↑(↑(singleAddHom x) 1) y",
"tactic": "simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,\n one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]"
}
]
| [
66,
89
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
]
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.