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/Polynomial/EraseLead.lean
|
Polynomial.eraseLead_add_C_mul_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf✝ f : R[X]\n⊢ eraseLead f + ↑C (leadingCoeff f) * X ^ natDegree f = f",
"tactic": "rw [C_mul_X_pow_eq_monomial, eraseLead_add_monomial_natDegree_leadingCoeff]"
}
] |
[
74,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Invertible.lean
|
invOf_eq_right_inv
|
[] |
[
151,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Quot.lean
|
Setoid.ext
|
[
{
"state_after": "α✝ : Sort ?u.8\nβ : Sort ?u.11\nα : Sort u_1\nr : α → α → Prop\niseqv✝¹ : Equivalence r\np : α → α → Prop\niseqv✝ : Equivalence p\nEq : ∀ (a b : α), Setoid.r a b ↔ Setoid.r a b\nthis : r = p\n⊢ { r := r, iseqv := iseqv✝¹ } = { r := p, iseqv := iseqv✝ }",
"state_before": "α✝ : Sort ?u.8\nβ : Sort ?u.11\nα : Sort u_1\nr : α → α → Prop\niseqv✝¹ : Equivalence r\np : α → α → Prop\niseqv✝ : Equivalence p\nEq : ∀ (a b : α), Setoid.r a b ↔ Setoid.r a b\n⊢ { r := r, iseqv := iseqv✝¹ } = { r := p, iseqv := iseqv✝ }",
"tactic": "have : r = p := funext fun a ↦ funext fun b ↦ propext <| Eq a b"
},
{
"state_after": "α✝ : Sort ?u.8\nβ : Sort ?u.11\nα : Sort u_1\nr : α → α → Prop\niseqv✝¹ iseqv✝ : Equivalence r\nEq : ∀ (a b : α), Setoid.r a b ↔ Setoid.r a b\n⊢ { r := r, iseqv := iseqv✝¹ } = { r := r, iseqv := iseqv✝ }",
"state_before": "α✝ : Sort ?u.8\nβ : Sort ?u.11\nα : Sort u_1\nr : α → α → Prop\niseqv✝¹ : Equivalence r\np : α → α → Prop\niseqv✝ : Equivalence p\nEq : ∀ (a b : α), Setoid.r a b ↔ Setoid.r a b\nthis : r = p\n⊢ { r := r, iseqv := iseqv✝¹ } = { r := p, iseqv := iseqv✝ }",
"tactic": "subst this"
},
{
"state_after": "no goals",
"state_before": "α✝ : Sort ?u.8\nβ : Sort ?u.11\nα : Sort u_1\nr : α → α → Prop\niseqv✝¹ iseqv✝ : Equivalence r\nEq : ∀ (a b : α), Setoid.r a b ↔ Setoid.r a b\n⊢ { r := r, iseqv := iseqv✝¹ } = { r := r, iseqv := iseqv✝ }",
"tactic": "rfl"
}
] |
[
33,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.getRight_swap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.23345\nδ : Type ?u.23348\nx : α ⊕ β\n⊢ getRight (swap x) = getLeft x",
"tactic": "cases x <;> rfl"
}
] |
[
381,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Fin.cycleRange_of_le
|
[
{
"state_after": "case zero\ni j : Fin (Nat.succ Nat.zero)\nh : j ≤ i\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1\n\ncase succ\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1",
"state_before": "n : ℕ\ni j : Fin (Nat.succ n)\nh : j ≤ i\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1",
"tactic": "cases n"
},
{
"state_after": "case succ\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1",
"state_before": "case succ\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1",
"tactic": "have : j = (Fin.castLE (Nat.succ_le_of_lt i.is_lt))\n ⟨j, lt_of_le_of_lt h (Nat.lt_succ_self i)⟩ :=\n by simp"
},
{
"state_after": "case succ.h\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ ↑(↑(cycleRange i) j) = ↑(if j = i then 0 else j + 1)",
"state_before": "case succ\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1",
"tactic": "ext"
},
{
"state_after": "case succ.h\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ (if { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } = last ↑i then 0\n else ↑{ val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } + 1) =\n ↑(if ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } = i then 0\n else ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } + 1)",
"state_before": "case succ.h\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ ↑(↑(cycleRange i) j) = ↑(if j = i then 0 else j + 1)",
"tactic": "erw [this, cycleRange, ofLeftInverse'_eq_ofInjective, ←\n Function.Embedding.toEquivRange_eq_ofInjective, ← viaFintypeEmbedding,\n viaFintypeEmbedding_apply_image, coe_castLE, coe_finRotate]"
},
{
"state_after": "case succ.h\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ (if ↑j = ↑i then 0 else ↑j + 1) = ↑(if ↑j = ↑i then 0 else j + 1)",
"state_before": "case succ.h\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ (if { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } = last ↑i then 0\n else ↑{ val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } + 1) =\n ↑(if ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } = i then 0\n else ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) } + 1)",
"tactic": "simp only [Fin.ext_iff, val_last, val_mk, val_zero, Fin.eta, castLE_mk]"
},
{
"state_after": "case succ.h.inl\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\nheq : ↑j = ↑i\n⊢ 0 = ↑0\n\ncase succ.h.inr\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\nheq : ¬↑j = ↑i\n⊢ ↑j + 1 = ↑(j + 1)",
"state_before": "case succ.h\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\n⊢ (if ↑j = ↑i then 0 else ↑j + 1) = ↑(if ↑j = ↑i then 0 else j + 1)",
"tactic": "split_ifs with heq"
},
{
"state_after": "no goals",
"state_before": "case zero\ni j : Fin (Nat.succ Nat.zero)\nh : j ≤ i\n⊢ ↑(cycleRange i) j = if j = i then 0 else j + 1",
"tactic": "exact Subsingleton.elim (α := Fin 1) _ _"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\n⊢ j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ.h.inl\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\nheq : ↑j = ↑i\n⊢ 0 = ↑0",
"tactic": "rfl"
},
{
"state_after": "case succ.h.inr\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\nheq : ¬↑j = ↑i\n⊢ j < last (n✝ + 1)",
"state_before": "case succ.h.inr\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\nheq : ¬↑j = ↑i\n⊢ ↑j + 1 = ↑(j + 1)",
"tactic": "rw [Fin.val_add_one_of_lt]"
},
{
"state_after": "no goals",
"state_before": "case succ.h.inr\nn✝ : ℕ\ni j : Fin (Nat.succ (Nat.succ n✝))\nh : j ≤ i\nthis : j = ↑(castLE (_ : Nat.succ ↑i ≤ Nat.succ (Nat.succ n✝))) { val := ↑j, isLt := (_ : ↑j < Nat.succ ↑i) }\nheq : ¬↑j = ↑i\n⊢ j < last (n✝ + 1)",
"tactic": "exact lt_of_lt_of_le (lt_of_le_of_ne h (mt (congr_arg _) heq)) (le_last i)"
}
] |
[
187,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasStrictDerivAt_exp_zero
|
[] |
[
212,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.ball_infDist_compl_subset
|
[] |
[
546,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.choose_spec
|
[] |
[
592,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
HasDerivAt.lhopital_zero_right_on_Ico
|
[
{
"state_after": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioi a] a) (𝓝 0)\n\ncase refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioi a] a) (𝓝 0)",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "refine' lhopital_zero_right_on_Ioo hab hff' hgg' hg' _ _ hdiv"
},
{
"state_after": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioo a b] a) (𝓝 (f a))",
"state_before": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioi a] a) (𝓝 0)",
"tactic": "rw [← hfa, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioo a b] a) (𝓝 (f a))",
"tactic": "exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto"
},
{
"state_after": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioo a b] a) (𝓝 (g a))",
"state_before": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioi a] a) (𝓝 0)",
"tactic": "rw [← hga, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioo a b] a) (𝓝 (g a))",
"tactic": "exact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto"
}
] |
[
108,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Combinatorics/SimpleGraph/Prod.lean
|
SimpleGraph.boxProd_neighborSet
|
[
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.9830\nG : SimpleGraph α\nH : SimpleGraph β\nx : α × β\na' : α\nb' : β\n⊢ (a', b') ∈ neighborSet (G □ H) x ↔ (a', b') ∈ neighborSet G x.fst ×ˢ {x.snd} ∪ {x.fst} ×ˢ neighborSet H x.snd",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9830\nG : SimpleGraph α\nH : SimpleGraph β\nx : α × β\n⊢ neighborSet (G □ H) x = neighborSet G x.fst ×ˢ {x.snd} ∪ {x.fst} ×ˢ neighborSet H x.snd",
"tactic": "ext ⟨a', b'⟩"
},
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.9830\nG : SimpleGraph α\nH : SimpleGraph β\nx : α × β\na' : α\nb' : β\n⊢ Adj G x.fst a' ∧ x.snd = b' ∨ Adj H x.snd b' ∧ x.fst = a' ↔ Adj G x.fst a' ∧ b' = x.snd ∨ a' = x.fst ∧ Adj H x.snd b'",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.9830\nG : SimpleGraph α\nH : SimpleGraph β\nx : α × β\na' : α\nb' : β\n⊢ (a', b') ∈ neighborSet (G □ H) x ↔ (a', b') ∈ neighborSet G x.fst ×ˢ {x.snd} ∪ {x.fst} ×ˢ neighborSet H x.snd",
"tactic": "simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.9830\nG : SimpleGraph α\nH : SimpleGraph β\nx : α × β\na' : α\nb' : β\n⊢ Adj G x.fst a' ∧ x.snd = b' ∨ Adj H x.snd b' ∧ x.fst = a' ↔ Adj G x.fst a' ∧ b' = x.snd ∨ a' = x.fst ∧ Adj H x.snd b'",
"tactic": "simp only [eq_comm, and_comm]"
}
] |
[
75,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.elim_nhds_subcover
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.14579\nπ : ι → Type ?u.14584\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nhs : IsCompact s\nU : α → Set α\nhU : ∀ (x : α), x ∈ s → U x ∈ 𝓝 x\nt : Finset ↑s\nht : s ⊆ ⋃ (x : ↑s) (_ : x ∈ t), U ↑x\n⊢ s ⊆ ⋃ (x : α) (_ : x ∈ Finset.image Subtype.val t), U x",
"tactic": "rwa [Finset.set_biUnion_finset_image]"
}
] |
[
213,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.transpose_eq_iff_eq_transpose
|
[
{
"state_after": "case mpr\nν : YoungDiagram\n⊢ transpose (transpose ν) = ν",
"state_before": "case mpr\nμ ν : YoungDiagram\n⊢ μ = transpose ν → transpose μ = ν",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nν : YoungDiagram\n⊢ transpose (transpose ν) = ν",
"tactic": "simp"
}
] |
[
231,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Algebra/RingQuot.lean
|
RingQuot.lift_unique
|
[
{
"state_after": "case w.a\nR : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\nT : Type u₄\ninst✝ : Semiring T\nf : R →+* T\nr : R → R → Prop\nw : ∀ ⦃x y : R⦄, r x y → ↑f x = ↑f y\ng : RingQuot r →+* T\nh : RingHom.comp g (mkRingHom r) = f\nx✝ : R\n⊢ ↑(RingHom.comp g (mkRingHom r)) x✝ = ↑(RingHom.comp (↑lift { val := f, property := w }) (mkRingHom r)) x✝",
"state_before": "R : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\nT : Type u₄\ninst✝ : Semiring T\nf : R →+* T\nr : R → R → Prop\nw : ∀ ⦃x y : R⦄, r x y → ↑f x = ↑f y\ng : RingQuot r →+* T\nh : RingHom.comp g (mkRingHom r) = f\n⊢ g = ↑lift { val := f, property := w }",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.a\nR : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\nT : Type u₄\ninst✝ : Semiring T\nf : R →+* T\nr : R → R → Prop\nw : ∀ ⦃x y : R⦄, r x y → ↑f x = ↑f y\ng : RingQuot r →+* T\nh : RingHom.comp g (mkRingHom r) = f\nx✝ : R\n⊢ ↑(RingHom.comp g (mkRingHom r)) x✝ = ↑(RingHom.comp (↑lift { val := f, property := w }) (mkRingHom r)) x✝",
"tactic": "simp [h]"
}
] |
[
470,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Data/Complex/Module.lean
|
Complex.reLm_coe
|
[] |
[
266,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
mem_vsub_const_affineSegment
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nV : Type u_1\nV' : Type ?u.55686\nP : Type u_2\nP' : Type ?u.55692\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z p : P\n⊢ z -ᵥ p ∈ affineSegment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affineSegment R x y",
"tactic": "rw [← affineSegment_vsub_const_image, (vsub_left_injective p).mem_set_image]"
}
] |
[
139,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.closure_eq_of_le
|
[] |
[
297,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.coeff_eq_c
|
[] |
[
116,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_def_of_nonpos
|
[
{
"state_after": "case inr\nx : ℝ\nhx : x ≤ 0\ny : ℝ\nh : ¬x = 0\n⊢ (↑x ^ ↑y).re = exp (log x * y) * cos (y * π)",
"state_before": "x : ℝ\nhx : x ≤ 0\ny : ℝ\n⊢ x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * π)",
"tactic": "split_ifs with h <;> simp [rpow_def, *]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nhx : x ≤ 0\ny : ℝ\nh : ¬x = 0\n⊢ (↑x ^ ↑y).re = exp (log x * y) * cos (y * π)",
"tactic": "exact rpow_def_of_neg (lt_of_le_of_ne hx h) _"
}
] |
[
92,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C
|
[] |
[
1125,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1121,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.top_mem
|
[] |
[
125,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.exists_intermediate_set
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA B : Finset α\ni : ℕ\nh₁ : i + card B ≤ card A\nh₂ : B ⊆ A\nk : ℕ\nh : i + card B + k = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA B : Finset α\ni : ℕ\nh₁ : i + card B ≤ card A\nh₂ : B ⊆ A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "rcases Nat.le.dest h₁ with ⟨k, h⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA B : Finset α\ni : ℕ\nh₂ : B ⊆ A\nk : ℕ\nh : i + card B + k = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA B : Finset α\ni : ℕ\nh₁ : i + card B ≤ card A\nh₂ : B ⊆ A\nk : ℕ\nh : i + card B + k = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "clear h₁"
},
{
"state_after": "case intro.zero\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk : ℕ\nh✝ : i + card B + k = card A✝\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + zero = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\n\ncase intro.succ\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA B : Finset α\ni : ℕ\nh₂ : B ⊆ A\nk : ℕ\nh : i + card B + k = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "induction' k with k ih generalizing A"
},
{
"state_after": "case intro.succ.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "case intro.succ\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "obtain ⟨a, ha⟩ : (A \\ B).Nonempty := by\n rw [← card_pos, card_sdiff h₂, ← h, Nat.add_right_comm, add_tsub_cancel_right, Nat.add_succ]\n apply Nat.succ_pos"
},
{
"state_after": "case intro.succ.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "case intro.succ.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "have z : i + card B + k = card (erase A a) := by\n rw [card_erase_of_mem (mem_sdiff.1 ha).1, ← h,\n Nat.add_sub_assoc (Nat.one_le_iff_ne_zero.mpr k.succ_ne_zero), ←pred_eq_sub_one,\n k.pred_succ]"
},
{
"state_after": "case intro.succ.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nthis : B ⊆ erase A a\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "case intro.succ.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "have : B ⊆ A.erase a := by\n rintro t th\n apply mem_erase_of_ne_of_mem _ (h₂ th)\n rintro rfl\n exact not_mem_sdiff_of_mem_right th ha"
},
{
"state_after": "case intro.succ.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nthis : B ⊆ erase A a\nB' : Finset α\nhB' : B ⊆ B'\nB'subA' : B' ⊆ erase A a\ncards : card B' = i + card B\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"state_before": "case intro.succ.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nthis : B ⊆ erase A a\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "rcases ih this z with ⟨B', hB', B'subA', cards⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.succ.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nthis : B ⊆ erase A a\nB' : Finset α\nhB' : B ⊆ B'\nB'subA' : B' ⊆ erase A a\ncards : card B' = i + card B\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "exact ⟨B', hB', B'subA'.trans (erase_subset _ _), cards⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.zero\nα : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk : ℕ\nh✝ : i + card B + k = card A✝\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + zero = card A\n⊢ ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B",
"tactic": "exact ⟨A, h₂, Subset.refl _, h.symm⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\n⊢ 0 < succ (i + k)",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\n⊢ Finset.Nonempty (A \\ B)",
"tactic": "rw [← card_pos, card_sdiff h₂, ← h, Nat.add_right_comm, add_tsub_cancel_right, Nat.add_succ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\n⊢ 0 < succ (i + k)",
"tactic": "apply Nat.succ_pos"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\n⊢ i + card B + k = card (erase A a)",
"tactic": "rw [card_erase_of_mem (mem_sdiff.1 ha).1, ← h,\n Nat.add_sub_assoc (Nat.one_le_iff_ne_zero.mpr k.succ_ne_zero), ←pred_eq_sub_one,\n k.pred_succ]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.43518\ns t✝ : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nt : α\nth : t ∈ B\n⊢ t ∈ erase A a",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\n⊢ B ⊆ erase A a",
"tactic": "rintro t th"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.43518\ns t✝ : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nt : α\nth : t ∈ B\n⊢ t ≠ a",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t✝ : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nt : α\nth : t ∈ B\n⊢ t ∈ erase A a",
"tactic": "apply mem_erase_of_ne_of_mem _ (h₂ th)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.43518\ns t✝ : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\nt : α\nth : t ∈ B\nha : t ∈ A \\ B\nz : i + card B + k = card (erase A t)\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t✝ : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\na : α\nha : a ∈ A \\ B\nz : i + card B + k = card (erase A a)\nt : α\nth : t ∈ B\n⊢ t ≠ a",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.43518\ns t✝ : Finset α\nf : α → β\nn : ℕ\nA✝ B : Finset α\ni : ℕ\nh₂✝ : B ⊆ A✝\nk✝ : ℕ\nh✝ : i + card B + k✝ = card A✝\nk : ℕ\nih : ∀ {A : Finset α}, B ⊆ A → i + card B + k = card A → ∃ C, B ⊆ C ∧ C ⊆ A ∧ card C = i + card B\nA : Finset α\nh₂ : B ⊆ A\nh : i + card B + succ k = card A\nt : α\nth : t ∈ B\nha : t ∈ A \\ B\nz : i + card B + k = card (erase A t)\n⊢ False",
"tactic": "exact not_mem_sdiff_of_mem_right th ha"
}
] |
[
486,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Additive.lean
|
BoxIntegral.BoxAdditiveMap.coe_injective
|
[] |
[
77,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.neg_fst
|
[] |
[
273,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Data/List/Basic.lean
|
List.splitOn_intercalate
|
[
{
"state_after": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls : ls ≠ []\n⊢ splitOn x (join (intersperse [x] ls)) = ls",
"state_before": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls : ls ≠ []\n⊢ splitOn x (intercalate [x] ls) = ls",
"tactic": "simp only [intercalate]"
},
{
"state_after": "case nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhx : ∀ (l : List α), l ∈ [] → ¬x ∈ l\nhls : [] ≠ []\n⊢ splitOn x (join (intersperse [x] [])) = []\n\ncase cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\ntl : List (List α)\nih : (∀ (l : List α), l ∈ tl → ¬x ∈ l) → tl ≠ [] → splitOn x (join (intersperse [x] tl)) = tl\nhx : ∀ (l : List α), l ∈ hd :: tl → ¬x ∈ l\nhls : hd :: tl ≠ []\n⊢ splitOn x (join (intersperse [x] (hd :: tl))) = hd :: tl",
"state_before": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls : ls ≠ []\n⊢ splitOn x (join (intersperse [x] ls)) = ls",
"tactic": "induction' ls with hd tl ih"
},
{
"state_after": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\n⊢ splitOn x (join (intersperse [x] [hd])) = [hd]\n\ncase cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ splitOn x (join (intersperse [x] (hd :: head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"state_before": "case cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\ntl : List (List α)\nih : (∀ (l : List α), l ∈ tl → ¬x ∈ l) → tl ≠ [] → splitOn x (join (intersperse [x] tl)) = tl\nhx : ∀ (l : List α), l ∈ hd :: tl → ¬x ∈ l\nhls : hd :: tl ≠ []\n⊢ splitOn x (join (intersperse [x] (hd :: tl))) = hd :: tl",
"tactic": "cases tl"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhx : ∀ (l : List α), l ∈ [] → ¬x ∈ l\nhls : [] ≠ []\n⊢ splitOn x (join (intersperse [x] [])) = []",
"tactic": "contradiction"
},
{
"state_after": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\n⊢ splitOn x hd = [hd]",
"state_before": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\n⊢ splitOn x (join (intersperse [x] [hd])) = [hd]",
"tactic": "suffices hd.splitOn x = [hd] by simpa [join]"
},
{
"state_after": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\n⊢ ∀ (x_1 : α), x_1 ∈ hd → ¬(x_1 == x) = true",
"state_before": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\n⊢ splitOn x hd = [hd]",
"tactic": "refine' splitOnP_eq_single _ _ _"
},
{
"state_after": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\ny : α\nhy : y ∈ hd\nH : (y == x) = true\n⊢ False",
"state_before": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\n⊢ ∀ (x_1 : α), x_1 ∈ hd → ¬(x_1 == x) = true",
"tactic": "intro y hy H"
},
{
"state_after": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\ny : α\nhy : x ∈ hd\nH : (y == x) = true\n⊢ False",
"state_before": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\ny : α\nhy : y ∈ hd\nH : (y == x) = true\n⊢ False",
"tactic": "rw [eq_of_beq H] at hy"
},
{
"state_after": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\ny : α\nhy : x ∈ hd\nH : (y == x) = true\n⊢ hd ∈ [hd]",
"state_before": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\ny : α\nhy : x ∈ hd\nH : (y == x) = true\n⊢ False",
"tactic": "refine' hx hd _ hy"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\ny : α\nhy : x ∈ hd\nH : (y == x) = true\n⊢ hd ∈ [hd]",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd : List α\nih : (∀ (l : List α), l ∈ [] → ¬x ∈ l) → [] ≠ [] → splitOn x (join (intersperse [x] [])) = []\nhx : ∀ (l : List α), l ∈ [hd] → ¬x ∈ l\nhls : [hd] ≠ []\nthis : splitOn x hd = [hd]\n⊢ splitOn x (join (intersperse [x] [hd])) = [hd]",
"tactic": "simpa [join]"
},
{
"state_after": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ splitOn x (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"state_before": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ splitOn x (join (intersperse [x] (hd :: head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"tactic": "simp only [intersperse_cons_cons, singleton_append, join]"
},
{
"state_after": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ ∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l\n\ncase cons.cons.specialize_2\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ head✝ :: tail✝ ≠ []\n\ncase cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\n⊢ splitOn x (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"state_before": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ splitOn x (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"tactic": "specialize ih _ _"
},
{
"state_after": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nthis : ∀ (as : List α), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as\n⊢ splitOn x (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"state_before": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nthis : ∀ (as : List α), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as\n⊢ splitOn x (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝\n\ncase h\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\n⊢ ∀ (x_1 : α), x_1 ∈ hd → ¬(fun x_2 => x_2 == x) x_1 = true",
"tactic": "case h =>\n intro y hy H\n rw [eq_of_beq H] at hy\n exact hx hd (.head _) hy"
},
{
"state_after": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOnP (fun x_1 => x_1 == x) (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nthis : ∀ (as : List α), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as\n⊢ splitOnP (fun x_1 => x_1 == x) (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"state_before": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nthis : ∀ (as : List α), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as\n⊢ splitOn x (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"tactic": "simp only [splitOn] at ih⊢"
},
{
"state_after": "no goals",
"state_before": "case cons.cons\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOnP (fun x_1 => x_1 == x) (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nthis : ∀ (as : List α), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as\n⊢ splitOnP (fun x_1 => x_1 == x) (hd ++ x :: join (intersperse [x] (head✝ :: tail✝))) = hd :: head✝ :: tail✝",
"tactic": "rw [this, ih]"
},
{
"state_after": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nl : List α\nhl : l ∈ head✝ :: tail✝\n⊢ ¬x ∈ l",
"state_before": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ ∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l",
"tactic": "intro l hl"
},
{
"state_after": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nl : List α\nhl : l ∈ head✝ :: tail✝\n⊢ l ∈ hd :: head✝ :: tail✝",
"state_before": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nl : List α\nhl : l ∈ head✝ :: tail✝\n⊢ ¬x ∈ l",
"tactic": "apply hx l"
},
{
"state_after": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nl : List α\nhl : l = head✝ ∨ l ∈ tail✝\n⊢ l = hd ∨ l = head✝ ∨ l ∈ tail✝",
"state_before": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nl : List α\nhl : l ∈ head✝ :: tail✝\n⊢ l ∈ hd :: head✝ :: tail✝",
"tactic": "simp at hl⊢"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.specialize_1\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nl : List α\nhl : l = head✝ ∨ l ∈ tail✝\n⊢ l = hd ∨ l = head✝ ∨ l ∈ tail✝",
"tactic": "exact Or.inr hl"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.specialize_2\nι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nih :\n (∀ (l : List α), l ∈ head✝ :: tail✝ → ¬x ∈ l) →\n head✝ :: tail✝ ≠ [] → splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\n⊢ head✝ :: tail✝ ≠ []",
"tactic": "exact List.noConfusion"
},
{
"state_after": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\ny : α\nhy : y ∈ hd\nH : (fun x_1 => x_1 == x) y = true\n⊢ False",
"state_before": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\n⊢ ∀ (x_1 : α), x_1 ∈ hd → ¬(fun x_2 => x_2 == x) x_1 = true",
"tactic": "intro y hy H"
},
{
"state_after": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\ny : α\nhy : x ∈ hd\nH : (fun x_1 => x_1 == x) y = true\n⊢ False",
"state_before": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\ny : α\nhy : y ∈ hd\nH : (fun x_1 => x_1 == x) y = true\n⊢ False",
"tactic": "rw [eq_of_beq H] at hy"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.310123\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\ninst✝ : DecidableEq α\nx : α\nhx✝ : ∀ (l : List α), l ∈ ls → ¬x ∈ l\nhls✝ : ls ≠ []\nhd head✝ : List α\ntail✝ : List (List α)\nhx : ∀ (l : List α), l ∈ hd :: head✝ :: tail✝ → ¬x ∈ l\nhls : hd :: head✝ :: tail✝ ≠ []\nih : splitOn x (join (intersperse [x] (head✝ :: tail✝))) = head✝ :: tail✝\ny : α\nhy : x ∈ hd\nH : (fun x_1 => x_1 == x) y = true\n⊢ False",
"tactic": "exact hx hd (.head _) hy"
}
] |
[
3024,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3000,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.empty_card
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\n⊢ card Language.empty = 0",
"tactic": "simp [card_eq_card_functions_add_card_relations]"
}
] |
[
240,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.toDeleteEdges_cons
|
[] |
[
1795,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1791,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.prod_dirac
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.4489185\nβ : Type u_2\nβ' : Type ?u.4489191\nγ : Type ?u.4489194\nE : Type ?u.4489197\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\ny : β\ns : Set α\nt : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑↑(map (fun x => (x, y)) μ) (s ×ˢ t) = ↑↑μ s * ↑↑(dirac y) t",
"state_before": "α : Type u_1\nα' : Type ?u.4489185\nβ : Type u_2\nβ' : Type ?u.4489191\nγ : Type ?u.4489194\nE : Type ?u.4489197\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\ny : β\n⊢ Measure.prod μ (dirac y) = map (fun x => (x, y)) μ",
"tactic": "refine' prod_eq fun s t hs ht => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.4489185\nβ : Type u_2\nβ' : Type ?u.4489191\nγ : Type ?u.4489194\nE : Type ?u.4489197\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\ny : β\ns : Set α\nt : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑↑(map (fun x => (x, y)) μ) (s ×ˢ t) = ↑↑μ s * ↑↑(dirac y) t",
"tactic": "simp_rw [map_apply measurable_prod_mk_right (hs.prod ht), mk_preimage_prod_left_eq_if, measure_if,\n dirac_apply' _ ht, ← indicator_mul_right _ fun _ => μ s, Pi.one_apply, mul_one]"
}
] |
[
558,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.const_lintegral
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\n⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ",
"state_before": "α : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\n⊢ lintegral (const α c) μ = c * ↑↑μ univ",
"tactic": "rw [lintegral]"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : IsEmpty α\n⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ\n\ncase inr\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : Nonempty α\n⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ",
"state_before": "α : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\n⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ",
"tactic": "cases isEmpty_or_nonempty α"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : IsEmpty α\n⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ",
"tactic": "simp [μ.eq_zero_of_isEmpty]"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : Nonempty α\n⊢ c * ↑↑μ (Function.const α c ⁻¹' {c}) = c * ↑↑μ univ",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : Nonempty α\n⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ",
"tactic": "simp"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : Nonempty α\n⊢ c * ↑↑μ ((fun x => c) ⁻¹' {c}) = c * ↑↑μ univ",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : Nonempty α\n⊢ c * ↑↑μ (Function.const α c ⁻¹' {c}) = c * ↑↑μ univ",
"tactic": "unfold Function.const"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.889363\nγ : Type ?u.889366\nδ : Type ?u.889369\nm : MeasurableSpace α\nμ ν : Measure α\nc : ℝ≥0∞\nh✝ : Nonempty α\n⊢ c * ↑↑μ ((fun x => c) ⁻¹' {c}) = c * ↑↑μ univ",
"tactic": "rw [preimage_const_of_mem (mem_singleton c)]"
}
] |
[
1094,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1090,
1
] |
Mathlib/GroupTheory/SchurZassenhaus.lean
|
Subgroup.SchurZassenhausInduction.step1
|
[
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\n⊢ ∃ H, IsComplement' N H",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nh3 : ∀ (H : Subgroup G), ¬IsComplement' N H\nK : Subgroup G\nhK : K ⊔ N = ⊤\n⊢ K = ⊤",
"tactic": "contrapose! h3"
},
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\n⊢ ∃ H, IsComplement' N H",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\n⊢ ∃ H, IsComplement' N H",
"tactic": "have h4 : (N.comap K.subtype).index = N.index := by\n rw [← N.relindex_top_right, ← hK]\n exact (relindex_sup_right K N).symm"
},
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\n⊢ ∃ H, IsComplement' N H",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\n⊢ ∃ H, IsComplement' N H",
"tactic": "have h5 : Fintype.card K < Fintype.card G := by\n rw [← K.index_mul_card]\n exact lt_mul_of_one_lt_left Fintype.card_pos (one_lt_index_of_ne_top h3)"
},
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\n⊢ ∃ H, IsComplement' N H",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\n⊢ ∃ H, IsComplement' N H",
"tactic": "have h6 : Nat.coprime (Fintype.card (N.comap K.subtype)) (N.comap K.subtype).index := by\n rw [h4]\n exact h1.coprime_dvd_left (card_comap_dvd_of_injective N K.subtype Subtype.coe_injective)"
},
{
"state_after": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : IsComplement' (comap (Subgroup.subtype K) N) H\n⊢ ∃ H, IsComplement' N H",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\n⊢ ∃ H, IsComplement' N H",
"tactic": "obtain ⟨H, hH⟩ := h2 K h5 h6"
},
{
"state_after": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\n⊢ ∃ H, IsComplement' N H",
"state_before": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : IsComplement' (comap (Subgroup.subtype K) N) H\n⊢ ∃ H, IsComplement' N H",
"tactic": "replace hH : Fintype.card (H.map K.subtype) = N.index := by\n rw [←relindex_bot_left_eq_card, ←relindex_comap, MonoidHom.comap_bot, Subgroup.ker_subtype,\n relindex_bot_left, ←IsComplement'.index_eq_card (IsComplement'.symm hH), index_comap,\n subtype_range, ←relindex_sup_right, hK, relindex_top_right]"
},
{
"state_after": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\nh7 : Fintype.card { x // x ∈ N } * Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = Fintype.card G\n⊢ ∃ H, IsComplement' N H",
"state_before": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\n⊢ ∃ H, IsComplement' N H",
"tactic": "have h7 : Fintype.card N * Fintype.card (H.map K.subtype) = Fintype.card G := by\n rw [hH, ← N.index_mul_card, mul_comm]"
},
{
"state_after": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\nh7 : Fintype.card { x // x ∈ N } * Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = Fintype.card G\nh8 : Nat.coprime (Fintype.card { x // x ∈ N }) (Fintype.card { x // x ∈ map (Subgroup.subtype K) H })\n⊢ ∃ H, IsComplement' N H",
"state_before": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\nh7 : Fintype.card { x // x ∈ N } * Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = Fintype.card G\n⊢ ∃ H, IsComplement' N H",
"tactic": "have h8 : (Fintype.card N).coprime (Fintype.card (H.map K.subtype)) := by\n rwa [hH]"
},
{
"state_after": "no goals",
"state_before": "case intro\nG : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\nh7 : Fintype.card { x // x ∈ N } * Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = Fintype.card G\nh8 : Nat.coprime (Fintype.card { x // x ∈ N }) (Fintype.card { x // x ∈ map (Subgroup.subtype K) H })\n⊢ ∃ H, IsComplement' N H",
"tactic": "exact ⟨H.map K.subtype, isComplement'_of_coprime h7 h8⟩"
},
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\n⊢ index (comap (Subgroup.subtype K) N) = relindex N (K ⊔ N)",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\n⊢ index (comap (Subgroup.subtype K) N) = index N",
"tactic": "rw [← N.relindex_top_right, ← hK]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\n⊢ index (comap (Subgroup.subtype K) N) = relindex N (K ⊔ N)",
"tactic": "exact (relindex_sup_right K N).symm"
},
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\n⊢ Fintype.card { x // x ∈ K } < index K * Fintype.card { x // x ∈ K }",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\n⊢ Fintype.card { x // x ∈ K } < Fintype.card G",
"tactic": "rw [← K.index_mul_card]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\n⊢ Fintype.card { x // x ∈ K } < index K * Fintype.card { x // x ∈ K }",
"tactic": "exact lt_mul_of_one_lt_left Fintype.card_pos (one_lt_index_of_ne_top h3)"
},
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\n⊢ Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index N)",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\n⊢ Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))",
"tactic": "rw [h4]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\n⊢ Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index N)",
"tactic": "exact h1.coprime_dvd_left (card_comap_dvd_of_injective N K.subtype Subtype.coe_injective)"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : IsComplement' (comap (Subgroup.subtype K) N) H\n⊢ Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N",
"tactic": "rw [←relindex_bot_left_eq_card, ←relindex_comap, MonoidHom.comap_bot, Subgroup.ker_subtype,\n relindex_bot_left, ←IsComplement'.index_eq_card (IsComplement'.symm hH), index_comap,\n subtype_range, ←relindex_sup_right, hK, relindex_top_right]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\n⊢ Fintype.card { x // x ∈ N } * Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = Fintype.card G",
"tactic": "rw [hH, ← N.index_mul_card, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nK : Subgroup G\nhK : K ⊔ N = ⊤\nh3 : K ≠ ⊤\nh4 : index (comap (Subgroup.subtype K) N) = index N\nh5 : Fintype.card { x // x ∈ K } < Fintype.card G\nh6 : Nat.coprime (Fintype.card { x // x ∈ comap (Subgroup.subtype K) N }) (index (comap (Subgroup.subtype K) N))\nH : Subgroup { x // x ∈ K }\nhH : Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = index N\nh7 : Fintype.card { x // x ∈ N } * Fintype.card { x // x ∈ map (Subgroup.subtype K) H } = Fintype.card G\n⊢ Nat.coprime (Fintype.card { x // x ∈ N }) (Fintype.card { x // x ∈ map (Subgroup.subtype K) H })",
"tactic": "rwa [hH]"
}
] |
[
196,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
9
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.as_sum_range
|
[] |
[
429,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.prodEquivOfIsCompl_symm_apply_right
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.90945\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.91461\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.92424\ninst✝² : Semiring S\nM : Type ?u.92430\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nh : IsCompl p q\nx : { x // x ∈ q }\n⊢ ↑x = ↑(prodEquivOfIsCompl p q h) (0, x)",
"tactic": "simp"
}
] |
[
130,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.div_mul_cancel
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.252386\nβ : Type ?u.252389\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh0 : a ≠ 0\nhI : a ≠ ⊤\n⊢ b / a * a = b",
"tactic": "rw [div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel h0 hI, mul_one]"
}
] |
[
1392,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1391,
11
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.closure_empty
|
[] |
[
422,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.symm_one
|
[
{
"state_after": "no goals",
"state_before": "⊢ ↑(σ 1) = ↑0",
"tactic": "simp [symm]"
}
] |
[
116,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
toLex_rat_cast
|
[] |
[
409,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.lsingle_apply
|
[] |
[
123,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.mapₗ_apply_of_measurable
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.176472\nδ : Type ?u.176475\nι : Type ?u.176478\nR : Type ?u.176481\nR' : Type ?u.176484\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf : α → β\nhf : Measurable f\nμ : Measure α\n⊢ ↑(mapₗ f) μ = ↑(mapₗ (AEMeasurable.mk f (_ : AEMeasurable f))) μ",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.176472\nδ : Type ?u.176475\nι : Type ?u.176478\nR : Type ?u.176481\nR' : Type ?u.176484\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf : α → β\nhf : Measurable f\nμ : Measure α\n⊢ ↑(mapₗ f) μ = map f μ",
"tactic": "simp only [← mapₗ_mk_apply_of_aemeasurable hf.aemeasurable]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.176472\nδ : Type ?u.176475\nι : Type ?u.176478\nR : Type ?u.176481\nR' : Type ?u.176484\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf : α → β\nhf : Measurable f\nμ : Measure α\n⊢ ↑(mapₗ f) μ = ↑(mapₗ (AEMeasurable.mk f (_ : AEMeasurable f))) μ",
"tactic": "exact mapₗ_congr hf hf.aemeasurable.measurable_mk hf.aemeasurable.ae_eq_mk"
}
] |
[
1167,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1164,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
continuousWithinAt_const_smul_iff₀
|
[] |
[
306,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
MulEquiv.map_finsupp_prod
|
[] |
[
232,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
11
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
WfDvdMonoid.iff_wellFounded_associates
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\n⊢ WfDvdMonoid α → WellFounded fun x x_1 => x < x_1",
"tactic": "apply WfDvdMonoid.wellFounded_associates"
}
] |
[
151,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/RingTheory/Localization/Module.lean
|
LinearIndependent.iff_fractionRing
|
[] |
[
159,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.subtype_mk
|
[] |
[
712,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.mem_const
|
[] |
[
239,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
|
CategoryTheory.Limits.WidePullback.π_arrow
|
[
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nj : J\n⊢ π arrows j ≫ arrows j = base arrows",
"tactic": "apply limit.w (WidePullbackShape.wideCospan _ _ _) (WidePullbackShape.Hom.term j)"
}
] |
[
326,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_const_mul_atTop_iff
|
[
{
"state_after": "case inl\nι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : NeBot l\nhr : r < 0\n⊢ Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot\n\ncase inr.inl\nι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\ninst✝ : NeBot l\n⊢ Tendsto (fun x => 0 * f x) l atTop ↔ 0 < 0 ∧ Tendsto f l atTop ∨ 0 < 0 ∧ Tendsto f l atBot\n\ncase inr.inr\nι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : NeBot l\nhr : 0 < r\n⊢ Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot",
"state_before": "ι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : NeBot l\n⊢ Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot",
"tactic": "rcases lt_trichotomy r 0 with (hr | rfl | hr)"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : NeBot l\nhr : r < 0\n⊢ Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot",
"tactic": "simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_neg]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\ninst✝ : NeBot l\n⊢ Tendsto (fun x => 0 * f x) l atTop ↔ 0 < 0 ∧ Tendsto f l atTop ∨ 0 < 0 ∧ Tendsto f l atBot",
"tactic": "simp [not_tendsto_const_atTop]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nι : Type ?u.233433\nι' : Type ?u.233436\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.233445\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : NeBot l\nhr : 0 < r\n⊢ Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot",
"tactic": "simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_pos]"
}
] |
[
1124,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1119,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.zero_inter
|
[] |
[
1759,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1758,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.quotientKerAlgEquivOfRightInverse.apply
|
[] |
[
320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Data/Polynomial/Expand.lean
|
Polynomial.expand_eq_sum
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np q : ℕ\nf : R[X]\n⊢ ↑(expand R p) f = sum f fun e a => ↑C a * (X ^ p) ^ e",
"tactic": "simp [expand, eval₂]"
}
] |
[
50,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.liftPred_const_iff
|
[] |
[
308,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.range_subtype
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.1533316\nR₂ : Type ?u.1533319\nR₃ : Type ?u.1533322\nR₄ : Type ?u.1533325\nS : Type ?u.1533328\nK : Type ?u.1533331\nK₂ : Type ?u.1533334\nM : Type u_1\nM' : Type ?u.1533340\nM₁ : Type ?u.1533343\nM₂ : Type ?u.1533346\nM₃ : Type ?u.1533349\nM₄ : Type ?u.1533352\nN : Type ?u.1533355\nN₂ : Type ?u.1533358\nι : Type ?u.1533361\nV : Type ?u.1533364\nV₂ : Type ?u.1533367\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\np p' : Submodule R M\nq : Submodule R₂ M₂\nτ₁₂ : R →+* R₂\nF : Type ?u.1533541\nsc : SemilinearMapClass F τ₁₂ M M₂\n⊢ range (Submodule.subtype p) = p",
"tactic": "simpa using map_comap_subtype p ⊤"
}
] |
[
1622,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1622,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.mem_keys_kunion
|
[
{
"state_after": "case nil\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion [] l₂) ↔ a ∈ keys [] ∨ a ∈ keys l₂\n\ncase cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ : ∀ {l₂ : List (Sigma β)}, a ∈ keys (kunion tail✝ l₂) ↔ a ∈ keys tail✝ ∨ a ∈ keys l₂\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion (head✝ :: tail✝) l₂) ↔ a ∈ keys (head✝ :: tail✝) ∨ a ∈ keys l₂",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List (Sigma β)\n⊢ a ∈ keys (kunion l₁ l₂) ↔ a ∈ keys l₁ ∨ a ∈ keys l₂",
"tactic": "induction l₁ generalizing l₂"
},
{
"state_after": "case cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ : ∀ {l₂ : List (Sigma β)}, a ∈ keys (kunion tail✝ l₂) ↔ a ∈ keys tail✝ ∨ a ∈ keys l₂\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion (head✝ :: tail✝) l₂) ↔ a ∈ keys (head✝ :: tail✝) ∨ a ∈ keys l₂",
"state_before": "case nil\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion [] l₂) ↔ a ∈ keys [] ∨ a ∈ keys l₂\n\ncase cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ : ∀ {l₂ : List (Sigma β)}, a ∈ keys (kunion tail✝ l₂) ↔ a ∈ keys tail✝ ∨ a ∈ keys l₂\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion (head✝ :: tail✝) l₂) ↔ a ∈ keys (head✝ :: tail✝) ∨ a ∈ keys l₂",
"tactic": "case nil => simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nhead✝ : Sigma β\ntail✝ : List (Sigma β)\ntail_ih✝ : ∀ {l₂ : List (Sigma β)}, a ∈ keys (kunion tail✝ l₂) ↔ a ∈ keys tail✝ ∨ a ∈ keys l₂\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion (head✝ :: tail✝) l₂) ↔ a ∈ keys (head✝ :: tail✝) ∨ a ∈ keys l₂",
"tactic": "case cons s l₁ ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion [] l₂) ↔ a ∈ keys [] ∨ a ∈ keys l₂",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\ns : Sigma β\nl₁ : List (Sigma β)\nih : ∀ {l₂ : List (Sigma β)}, a ∈ keys (kunion l₁ l₂) ↔ a ∈ keys l₁ ∨ a ∈ keys l₂\nl₂ : List (Sigma β)\n⊢ a ∈ keys (kunion (s :: l₁) l₂) ↔ a ∈ keys (s :: l₁) ∨ a ∈ keys l₂",
"tactic": "by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]"
}
] |
[
706,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_diff_Ici
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.61989\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioi a \\ Ici b = Ioo a b",
"tactic": "rw [diff_eq, compl_Ici, Ioi_inter_Iio]"
}
] |
[
1092,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1092,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformity_prod
|
[] |
[
1571,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1567,
1
] |
Mathlib/Analysis/LocallyConvex/StrongTopology.lean
|
ContinuousLinearMap.strongTopology.locallyConvexSpace
|
[
{
"state_after": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\n⊢ LocallyConvexSpace R (E →SL[σ] F)",
"state_before": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\n⊢ LocallyConvexSpace R (E →SL[σ] F)",
"tactic": "letI : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖"
},
{
"state_after": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\nthis : TopologicalAddGroup (E →SL[σ] F)\n⊢ LocallyConvexSpace R (E →SL[σ] F)",
"state_before": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\n⊢ LocallyConvexSpace R (E →SL[σ] F)",
"tactic": "haveI : TopologicalAddGroup (E →SL[σ] F) := strongTopology.topologicalAddGroup _ _ _"
},
{
"state_after": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\nthis : TopologicalAddGroup (E →SL[σ] F)\n⊢ ∀ (i : Set E × Set F),\n i.fst ∈ 𝔖 ∧ i.snd ∈ 𝓝 0 ∧ Convex R i.snd → Convex R {f | ∀ (x : E), x ∈ i.fst → ↑f x ∈ _root_.id i.snd}",
"state_before": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\nthis : TopologicalAddGroup (E →SL[σ] F)\n⊢ LocallyConvexSpace R (E →SL[σ] F)",
"tactic": "apply LocallyConvexSpace.ofBasisZero _ _ _ _\n (strongTopology.hasBasis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂\n (LocallyConvexSpace.convex_basis_zero R F)) _"
},
{
"state_after": "case mk.intro.intro\nR : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\nthis : TopologicalAddGroup (E →SL[σ] F)\nS : Set E\nV : Set F\nleft✝¹ : (S, V).fst ∈ 𝔖\nleft✝ : (S, V).snd ∈ 𝓝 0\nhVconvex : Convex R (S, V).snd\nf : E →SL[σ] F\nhf : f ∈ {f | ∀ (x : E), x ∈ (S, V).fst → ↑f x ∈ _root_.id (S, V).snd}\ng : E →SL[σ] F\nhg : g ∈ {f | ∀ (x : E), x ∈ (S, V).fst → ↑f x ∈ _root_.id (S, V).snd}\na b : R\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nx : E\nhx : x ∈ (S, V).fst\n⊢ ↑(a • f + b • g) x ∈ _root_.id (S, V).snd",
"state_before": "R : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\nthis : TopologicalAddGroup (E →SL[σ] F)\n⊢ ∀ (i : Set E × Set F),\n i.fst ∈ 𝔖 ∧ i.snd ∈ 𝓝 0 ∧ Convex R i.snd → Convex R {f | ∀ (x : E), x ∈ i.fst → ↑f x ∈ _root_.id i.snd}",
"tactic": "rintro ⟨S, V⟩ ⟨_, _, hVconvex⟩ f hf g hg a b ha hb hab x hx"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nR : Type u_2\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_1\nF : Type u_3\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : OrderedSemiring R\ninst✝⁷ : NormedField 𝕜₁\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : Module 𝕜₂ F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝³ : Module R F\ninst✝² : ContinuousConstSMul R F\ninst✝¹ : LocallyConvexSpace R F\ninst✝ : SMulCommClass 𝕜₂ R F\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nthis✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\nthis : TopologicalAddGroup (E →SL[σ] F)\nS : Set E\nV : Set F\nleft✝¹ : (S, V).fst ∈ 𝔖\nleft✝ : (S, V).snd ∈ 𝓝 0\nhVconvex : Convex R (S, V).snd\nf : E →SL[σ] F\nhf : f ∈ {f | ∀ (x : E), x ∈ (S, V).fst → ↑f x ∈ _root_.id (S, V).snd}\ng : E →SL[σ] F\nhg : g ∈ {f | ∀ (x : E), x ∈ (S, V).fst → ↑f x ∈ _root_.id (S, V).snd}\na b : R\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nx : E\nhx : x ∈ (S, V).fst\n⊢ ↑(a • f + b • g) x ∈ _root_.id (S, V).snd",
"tactic": "exact hVconvex (hf x hx) (hg x hx) ha hb hab"
}
] |
[
62,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
Filter.Tendsto.uniformity_symm
|
[] |
[
496,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.divInt_mul_left
|
[
{
"state_after": "no goals",
"state_before": "n d a : Int\na0 : a ≠ 0\n⊢ a * n /. (a * d) = n /. d",
"tactic": "if d0 : d = 0 then simp [d0] else\nsimp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm]"
},
{
"state_after": "no goals",
"state_before": "n d a : Int\na0 : a ≠ 0\nd0 : d = 0\n⊢ a * n /. (a * d) = n /. d",
"tactic": "simp [d0]"
},
{
"state_after": "no goals",
"state_before": "n d a : Int\na0 : a ≠ 0\nd0 : ¬d = 0\n⊢ a * n /. (a * d) = n /. d",
"tactic": "simp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm]"
}
] |
[
145,
84
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
143,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.iInf_mul
|
[
{
"state_after": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ (⨅ (i : ι), ↑(f i)) * ↑a = ⨅ (i : ι), ↑(f i * a)",
"state_before": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ iInf f * a = ⨅ (i : ι), f i * a",
"tactic": "rw [← NNReal.coe_eq, NNReal.coe_mul, coe_iInf, coe_iInf]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ (⨅ (i : ι), ↑(f i)) * ↑a = ⨅ (i : ι), ↑(f i * a)",
"tactic": "exact Real.iInf_mul_of_nonneg (NNReal.coe_nonneg _) _"
}
] |
[
955,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
953,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Normalized.lean
|
AlgebraicTopology.DoldKan.PInfty_comp_PInftyToNormalizedMooreComplex
|
[
{
"state_after": "no goals",
"state_before": "A : Type u_2\ninst✝¹ : Category A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\n⊢ PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X",
"tactic": "aesop_cat"
}
] |
[
93,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Data/List/Zip.lean
|
List.lt_length_right_of_zipWith
|
[
{
"state_after": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.19964\nε : Type ?u.19967\nf : α → β → γ\ni : ℕ\nl : List α\nl' : List β\nh : i < length l ∧ i < length l'\n⊢ i < length l'",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.19964\nε : Type ?u.19967\nf : α → β → γ\ni : ℕ\nl : List α\nl' : List β\nh : i < length (zipWith f l l')\n⊢ i < length l'",
"tactic": "rw [length_zipWith, lt_min_iff] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.19964\nε : Type ?u.19967\nf : α → β → γ\ni : ℕ\nl : List α\nl' : List β\nh : i < length l ∧ i < length l'\n⊢ i < length l'",
"tactic": "exact h.right"
}
] |
[
100,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.smul_apply
|
[] |
[
60,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/RingTheory/ChainOfDivisors.lean
|
DivisorChain.element_of_chain_not_isUnit_of_index_ne_zero
|
[] |
[
92,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Topology/Category/TopCat/EpiMono.lean
|
TopCat.epi_iff_surjective
|
[
{
"state_after": "X Y : TopCat\nf : X ⟶ Y\n⊢ Epi f ↔ Epi ((forget TopCat).map f)",
"state_before": "X Y : TopCat\nf : X ⟶ Y\n⊢ Epi f ↔ Function.Surjective ((forget TopCat).map f)",
"tactic": "suffices Epi f ↔ Epi ((forget TopCat).map f) by\n rw [this, CategoryTheory.epi_iff_surjective]"
},
{
"state_after": "case mp\nX Y : TopCat\nf : X ⟶ Y\n⊢ Epi f → Epi ((forget TopCat).map f)\n\ncase mpr\nX Y : TopCat\nf : X ⟶ Y\n⊢ Epi ((forget TopCat).map f) → Epi f",
"state_before": "X Y : TopCat\nf : X ⟶ Y\n⊢ Epi f ↔ Epi ((forget TopCat).map f)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "X Y : TopCat\nf : X ⟶ Y\nthis : Epi f ↔ Epi ((forget TopCat).map f)\n⊢ Epi f ↔ Function.Surjective ((forget TopCat).map f)",
"tactic": "rw [this, CategoryTheory.epi_iff_surjective]"
},
{
"state_after": "case mp\nX Y : TopCat\nf : X ⟶ Y\na✝ : Epi f\n⊢ Epi ((forget TopCat).map f)",
"state_before": "case mp\nX Y : TopCat\nf : X ⟶ Y\n⊢ Epi f → Epi ((forget TopCat).map f)",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case mp\nX Y : TopCat\nf : X ⟶ Y\na✝ : Epi f\n⊢ Epi ((forget TopCat).map f)",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "case mpr\nX Y : TopCat\nf : X ⟶ Y\n⊢ Epi ((forget TopCat).map f) → Epi f",
"tactic": "apply Functor.epi_of_epi_map"
}
] |
[
36,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsRat.neg_to_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nn d : ℕ\ninv✝ : Invertible ↑d\n⊢ ↑(Int.negOfNat n) * ⅟↑d = -(↑n / ↑d)",
"tactic": "simp [div_eq_mul_inv]"
}
] |
[
167,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
weightedVSub_mem_vectorSpan_pair
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ (∃ r,\n r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) =\n ↑(Finset.weightedVSub s p) w) ↔\n ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ ↑(Finset.weightedVSub s p) w ∈\n vectorSpan k {↑(Finset.affineCombination k s p) w₁, ↑(Finset.affineCombination k s p) w₂} ↔\n ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)",
"tactic": "rw [mem_vectorSpan_pair]"
},
{
"state_after": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh✝ : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh :\n ∃ r, r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w\n⊢ ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh✝ : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh : ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n⊢ ∃ r, r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ (∃ r,\n r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) =\n ↑(Finset.weightedVSub s p) w) ↔\n ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w\n⊢ ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)",
"state_before": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh✝ : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh :\n ∃ r, r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w\n⊢ ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)",
"tactic": "rcases h with ⟨r, hr⟩"
},
{
"state_after": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w\ni : ι\nhi : i ∈ s\n⊢ w i = r * (w₁ i - w₂ i)",
"state_before": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w\n⊢ ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)",
"tactic": "refine' ⟨r, fun i hi => _⟩"
},
{
"state_after": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\n⊢ w i = r * (w₁ i - w₂ i)",
"state_before": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w\ni : ι\nhi : i ∈ s\n⊢ w i = r * (w₁ i - w₂ i)",
"tactic": "rw [s.affineCombination_vsub, ← s.weightedVSub_const_smul, ← sub_eq_zero, ← map_sub] at hr"
},
{
"state_after": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\nhw' : ∑ j in s, (r • (w₁ - w₂) - w) j = 0\n⊢ w i = r * (w₁ i - w₂ i)",
"state_before": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\n⊢ w i = r * (w₁ i - w₂ i)",
"tactic": "have hw' : (∑ j in s, (r • (w₁ - w₂) - w) j) = 0 := by\n simp_rw [Pi.sub_apply, Pi.smul_apply, Pi.sub_apply, smul_sub, Finset.sum_sub_distrib, ←\n Finset.smul_sum, hw, hw₁, hw₂, sub_self]"
},
{
"state_after": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\nhw' : ∑ j in s, (r • (w₁ - w₂) - w) j = 0\nhr' : (r • (w₁ - w₂) - w) i = 0\n⊢ w i = r * (w₁ i - w₂ i)",
"state_before": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\nhw' : ∑ j in s, (r • (w₁ - w₂) - w) j = 0\n⊢ w i = r * (w₁ i - w₂ i)",
"tactic": "have hr' := h s _ hw' hr i hi"
},
{
"state_after": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\nhw' : ∑ j in s, (r • (w₁ - w₂) - w) j = 0\nhr' : (r • (w₁ - w₂) - w) i = 0\n⊢ r • (w₁ i - w₂ i) - w i = 0",
"state_before": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\nhw' : ∑ j in s, (r • (w₁ - w₂) - w) j = 0\nhr' : (r • (w₁ - w₂) - w) i = 0\n⊢ w i = r * (w₁ i - w₂ i)",
"tactic": "rw [eq_comm, ← sub_eq_zero, ← smul_eq_mul]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\nhw' : ∑ j in s, (r • (w₁ - w₂) - w) j = 0\nhr' : (r • (w₁ - w₂) - w) i = 0\n⊢ r • (w₁ i - w₂ i) - w i = 0",
"tactic": "exact hr'"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr✝ : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s p) w\nhr : ↑(Finset.weightedVSub s p) (r • (w₁ - w₂) - w) = 0\ni : ι\nhi : i ∈ s\n⊢ ∑ j in s, (r • (w₁ - w₂) - w) j = 0",
"tactic": "simp_rw [Pi.sub_apply, Pi.smul_apply, Pi.sub_apply, smul_sub, Finset.sum_sub_distrib, ←\n Finset.smul_sum, hw, hw₁, hw₂, sub_self]"
},
{
"state_after": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n⊢ ∃ r, r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"state_before": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh✝ : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh : ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n⊢ ∃ r, r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"tactic": "rcases h with ⟨r, hr⟩"
},
{
"state_after": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n⊢ r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"state_before": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n⊢ ∃ r, r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"tactic": "refine' ⟨r, _⟩"
},
{
"state_after": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\nw' : ι → k := fun i => r * (w₁ i - w₂ i)\n⊢ r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"state_before": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\n⊢ r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"tactic": "let w' i := r * (w₁ i - w₂ i)"
},
{
"state_after": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nw' : ι → k := fun i => r * (w₁ i - w₂ i)\nhr : ∀ (i : ι), i ∈ s → w i = w' i\n⊢ r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"state_before": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₁ i - w₂ i)\nw' : ι → k := fun i => r * (w₁ i - w₂ i)\n⊢ r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"tactic": "change ∀ i ∈ s, w i = w' i at hr"
},
{
"state_after": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nw' : ι → k := fun i => r * (w₁ i - w₂ i)\nhr : ∀ (i : ι), i ∈ s → w i = w' i\n⊢ ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s fun x => p x) fun i => w' i",
"state_before": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nw' : ι → k := fun i => r * (w₁ i - w₂ i)\nhr : ∀ (i : ι), i ∈ s → w i = w' i\n⊢ r • (↑(Finset.affineCombination k s p) w₁ -ᵥ ↑(Finset.affineCombination k s p) w₂) = ↑(Finset.weightedVSub s p) w",
"tactic": "rw [s.weightedVSub_congr hr fun _ _ => rfl, s.affineCombination_vsub, ←\n s.weightedVSub_const_smul]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 0\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nr : k\nw' : ι → k := fun i => r * (w₁ i - w₂ i)\nhr : ∀ (i : ι), i ∈ s → w i = w' i\n⊢ ↑(Finset.weightedVSub s p) (r • (w₁ - w₂)) = ↑(Finset.weightedVSub s fun x => p x) fun i => w' i",
"tactic": "congr"
}
] |
[
522,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.cons_ndinter_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns t : Multiset α\nh : a ∈ t\n⊢ ndinter (a ::ₘ s) t = a ::ₘ ndinter s t",
"tactic": "simp [ndinter, h]"
}
] |
[
234,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Init/Function.lean
|
Function.comp_const_right
|
[] |
[
60,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Topology/Inseparable.lean
|
Inseparable.mem_closed_iff
|
[] |
[
369,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_comm
|
[] |
[
823,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
11
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.Rel.mono
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\nh : ∀ (a : α), a ∈ 0 → ∀ (b : β), b ∈ 0 → r a b → p a b\n⊢ Rel p 0 0\n\ncase cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : (∀ (a : α), a ∈ as✝ → ∀ (b : β), b ∈ bs✝ → r a b → p a b) → Rel p as✝ bs✝\nh : ∀ (a : α), a ∈ a✝² ::ₘ as✝ → ∀ (b : β), b ∈ b✝ ::ₘ bs✝ → r a b → p a b\n⊢ Rel p (a✝² ::ₘ as✝) (b✝ ::ₘ bs✝)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\nhst : Rel r s t\nh : ∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → r a b → p a b\n⊢ Rel p s t",
"tactic": "induction hst"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : (∀ (a : α), a ∈ as✝ → ∀ (b : β), b ∈ bs✝ → r a b → p a b) → Rel p as✝ bs✝\nh : ∀ (a : α), a ∈ a✝² ::ₘ as✝ → ∀ (b : β), b ∈ b✝ ::ₘ bs✝ → r a b → p a b\n⊢ Rel p (a✝² ::ₘ as✝) (b✝ ::ₘ bs✝)",
"state_before": "case zero\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\nh : ∀ (a : α), a ∈ 0 → ∀ (b : β), b ∈ 0 → r a b → p a b\n⊢ Rel p 0 0\n\ncase cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : (∀ (a : α), a ∈ as✝ → ∀ (b : β), b ∈ bs✝ → r a b → p a b) → Rel p as✝ bs✝\nh : ∀ (a : α), a ∈ a✝² ::ₘ as✝ → ∀ (b : β), b ∈ b✝ ::ₘ bs✝ → r a b → p a b\n⊢ Rel p (a✝² ::ₘ as✝) (b✝ ::ₘ bs✝)",
"tactic": "case zero => exact Rel.zero"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\na✝² : α\nb✝ : β\nas✝ : Multiset α\nbs✝ : Multiset β\na✝¹ : r a✝² b✝\na✝ : Rel r as✝ bs✝\na_ih✝ : (∀ (a : α), a ∈ as✝ → ∀ (b : β), b ∈ bs✝ → r a b → p a b) → Rel p as✝ bs✝\nh : ∀ (a : α), a ∈ a✝² ::ₘ as✝ → ∀ (b : β), b ∈ b✝ ::ₘ bs✝ → r a b → p a b\n⊢ Rel p (a✝² ::ₘ as✝) (b✝ ::ₘ bs✝)",
"tactic": "case\n cons a b s t hab _hst ih =>\n apply Rel.cons (h a (mem_cons_self _ _) b (mem_cons_self _ _) hab)\n exact ih fun a' ha' b' hb' h' => h a' (mem_cons_of_mem ha') b' (mem_cons_of_mem hb') h'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns : Multiset α\nt : Multiset β\nh : ∀ (a : α), a ∈ 0 → ∀ (b : β), b ∈ 0 → r a b → p a b\n⊢ Rel p 0 0",
"tactic": "exact Rel.zero"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns✝ : Multiset α\nt✝ : Multiset β\na : α\nb : β\ns : Multiset α\nt : Multiset β\nhab : r a b\n_hst : Rel r s t\nih : (∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → r a b → p a b) → Rel p s t\nh : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → ∀ (b_1 : β), b_1 ∈ b ::ₘ t → r a_1 b_1 → p a_1 b_1\n⊢ Rel p s t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns✝ : Multiset α\nt✝ : Multiset β\na : α\nb : β\ns : Multiset α\nt : Multiset β\nhab : r a b\n_hst : Rel r s t\nih : (∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → r a b → p a b) → Rel p s t\nh : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → ∀ (b_1 : β), b_1 ∈ b ::ₘ t → r a_1 b_1 → p a_1 b_1\n⊢ Rel p (a ::ₘ s) (b ::ₘ t)",
"tactic": "apply Rel.cons (h a (mem_cons_self _ _) b (mem_cons_self _ _) hab)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.439427\nδ : Type ?u.439430\nr✝ : α → β → Prop\np✝ : γ → δ → Prop\nr p : α → β → Prop\ns✝ : Multiset α\nt✝ : Multiset β\na : α\nb : β\ns : Multiset α\nt : Multiset β\nhab : r a b\n_hst : Rel r s t\nih : (∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → r a b → p a b) → Rel p s t\nh : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → ∀ (b_1 : β), b_1 ∈ b ::ₘ t → r a_1 b_1 → p a_1 b_1\n⊢ Rel p s t",
"tactic": "exact ih fun a' ha' b' hb' h' => h a' (mem_cons_of_mem ha') b' (mem_cons_of_mem hb') h'"
}
] |
[
2699,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2692,
1
] |
Mathlib/Topology/Separation.lean
|
regularSpace_sInf
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\n⊢ RegularSpace X",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\n⊢ RegularSpace X",
"tactic": "let _ := sInf T"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\nthis :\n ∀ (a : X),\n HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i\n⊢ RegularSpace X",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\n⊢ RegularSpace X",
"tactic": "have : ∀ a, (𝓝 a).HasBasis\n (fun If : ΣI : Set T, I → Set X =>\n If.1.Finite ∧ ∀ i : If.1, If.2 i ∈ @nhds X i a ∧ @IsClosed X i (If.2 i))\n fun If => ⋂ i : If.1, If.snd i := by\n intro a\n rw [nhds_sInf, ← iInf_subtype'']\n exact hasBasis_iInf fun t : T => @closed_nhds_basis X t (h t t.2) a"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\nthis :\n ∀ (a : X),\n HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i\na : X\nIf : (I : Set ↑T) × (↑I → Set X)\nhIf : Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i)\ni : ↑If.fst\n⊢ IsClosed (Sigma.snd If i)",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\nthis :\n ∀ (a : X),\n HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i\n⊢ RegularSpace X",
"tactic": "refine' RegularSpace.ofBasis this fun a If hIf => isClosed_iInter fun i => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\nthis :\n ∀ (a : X),\n HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i\na : X\nIf : (I : Set ↑T) × (↑I → Set X)\nhIf : Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i)\ni : ↑If.fst\n⊢ IsClosed (Sigma.snd If i)",
"tactic": "exact (hIf.2 i).2.mono (sInf_le (i : T).2)"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\na : X\n⊢ HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\n⊢ ∀ (a : X),\n HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i",
"tactic": "intro a"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\na : X\n⊢ HasBasis (⨅ (i : ↑T), 𝓝 a)\n (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i)) fun If =>\n ⋂ (i : ↑If.fst), Sigma.snd If i",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\na : X\n⊢ HasBasis (𝓝 a) (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i))\n fun If => ⋂ (i : ↑If.fst), Sigma.snd If i",
"tactic": "rw [nhds_sInf, ← iInf_subtype'']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\nX : Type u_1\nT : Set (TopologicalSpace X)\nh : ∀ (t : TopologicalSpace X), t ∈ T → RegularSpace X\nx✝ : TopologicalSpace X := sInf T\na : X\n⊢ HasBasis (⨅ (i : ↑T), 𝓝 a)\n (fun If => Set.Finite If.fst ∧ ∀ (i : ↑If.fst), Sigma.snd If i ∈ 𝓝 a ∧ IsClosed (Sigma.snd If i)) fun If =>\n ⋂ (i : ↑If.fst), Sigma.snd If i",
"tactic": "exact hasBasis_iInf fun t : T => @closed_nhds_basis X t (h t t.2) a"
}
] |
[
1617,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1606,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
abs_add_eq_add_abs_le
|
[
{
"state_after": "case inl.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : 0 ≤ a\nb0 : 0 ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0\n\ncase inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : 0 ≤ a\nb0 : b < 0\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0\n\ncase inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0\n\ncase inr.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : b < 0\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "obtain a0 | a0 := le_or_lt 0 a <;> obtain b0 | b0 := le_or_lt 0 b"
},
{
"state_after": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"state_before": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0\n\ncase inr.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : b < 0\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "any_goals simp [a0.le, b0.le, abs_of_nonpos, add_nonpos, add_comm]"
},
{
"state_after": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"state_before": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "have : (|a + b| = -a + b ↔ b ≤ 0) ↔ (|a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) := by\n simp [a0, a0.le, a0.not_le, b0, abs_of_neg, abs_of_nonneg]"
},
{
"state_after": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\n⊢ b ≤ 0",
"state_before": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "refine' this.mp ⟨fun h => _, fun h => by simp only [le_antisymm h b0, abs_of_neg a0, add_zero]⟩"
},
{
"state_after": "case inr.inl.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : a + b ≤ 0\n⊢ b ≤ 0\n\ncase inr.inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : 0 < a + b\n⊢ b ≤ 0",
"state_before": "case inr.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\n⊢ b ≤ 0",
"tactic": "obtain ab | ab := le_or_lt (a + b) 0"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : 0 ≤ a\nb0 : 0 ≤ b\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "simp [a0, b0, abs_of_nonneg, add_nonneg a0 b0]"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : 0 ≤ a\nb0 : b < 0\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "exact (lt_irrefl (0 : α) <| a0.trans_lt <| hle.trans_lt b0).elim"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : b < 0\n⊢ abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"tactic": "simp [a0.le, b0.le, abs_of_nonpos, add_nonpos, add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\n⊢ (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)",
"tactic": "simp [a0, a0.le, a0.not_le, b0, abs_of_neg, abs_of_nonneg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : b ≤ 0\n⊢ abs (a + b) = -a + b",
"tactic": "simp only [le_antisymm h b0, abs_of_neg a0, add_zero]"
},
{
"state_after": "case inr.inl.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : a + b ≤ 0\n⊢ -b = b",
"state_before": "case inr.inl.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : a + b ≤ 0\n⊢ b ≤ 0",
"tactic": "refine' le_of_eq (eq_zero_of_neg_eq _)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inl\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : a + b ≤ 0\n⊢ -b = b",
"tactic": "rwa [abs_of_nonpos ab, neg_add_rev, add_comm, add_right_inj] at h"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : 0 < a + b\n⊢ 0 < 0",
"state_before": "case inr.inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : 0 < a + b\n⊢ b ≤ 0",
"tactic": "refine' (lt_irrefl (0 : α) _).elim"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : a = -a\nab : 0 < a + b\n⊢ 0 < 0",
"state_before": "case inr.inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : abs (a + b) = -a + b\nab : 0 < a + b\n⊢ 0 < 0",
"tactic": "rw [abs_of_pos ab, add_left_inj] at h"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inr\nα : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : LinearOrderedAddCommGroup α\na b : α\nhle : a ≤ b\na0 : a < 0\nb0 : 0 ≤ b\nthis : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) = abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0)\nh : a = -a\nab : 0 < a + b\n⊢ 0 < 0",
"tactic": "rwa [eq_zero_of_neg_eq h.symm] at a0"
}
] |
[
483,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.ext'
|
[] |
[
527,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.all_node4L
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nP : α → Prop\nl : Ordnode α\nx : α\nm : Ordnode α\ny : α\nr : Ordnode α\n⊢ All P (node4L l x m y r) ↔ All P l ∧ P x ∧ All P m ∧ P y ∧ All P r",
"tactic": "cases m <;> simp [node4L, all_node', All, all_node3L, and_assoc]"
}
] |
[
509,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.bodd_add_div2
|
[
{
"state_after": "n : ℕ\n⊢ (bif bodd n then 0 else 1) + (bif bodd n then succ (div2 n) else div2 n) * 2 = succ n",
"state_before": "n : ℕ\n⊢ (bif bodd (succ n) then 1 else 0) + 2 * div2 (succ n) = succ n",
"tactic": "simp"
},
{
"state_after": "n : ℕ\n⊢ (bif bodd n then 0 else 1) + (bif bodd n then succ (div2 n) else div2 n) * 2 =\n succ ((bif bodd n then 1 else 0) + 2 * div2 n)",
"state_before": "n : ℕ\n⊢ (bif bodd n then 0 else 1) + (bif bodd n then succ (div2 n) else div2 n) * 2 = succ n",
"tactic": "refine' Eq.trans _ (congr_arg succ (bodd_add_div2 n))"
},
{
"state_after": "case false\nn : ℕ\n⊢ 1 + div2 n * 2 = succ (div2 n * 2)\n\ncase true\nn : ℕ\n⊢ succ (div2 n) * 2 = succ (1 + div2 n * 2)",
"state_before": "n : ℕ\n⊢ (bif bodd n then 0 else 1) + (bif bodd n then succ (div2 n) else div2 n) * 2 =\n succ ((bif bodd n then 1 else 0) + 2 * div2 n)",
"tactic": "cases bodd n <;> simp [cond, not]"
},
{
"state_after": "no goals",
"state_before": "case false\nn : ℕ\n⊢ 1 + div2 n * 2 = succ (div2 n * 2)",
"tactic": "rw [Nat.add_comm, Nat.add_succ]"
},
{
"state_after": "no goals",
"state_before": "case true\nn : ℕ\n⊢ succ (div2 n) * 2 = succ (1 + div2 n * 2)",
"tactic": "rw [succ_mul, Nat.add_comm 1, Nat.add_succ]"
}
] |
[
137,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.out_nonempty_iff_ne_zero
|
[] |
[
285,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.range_val
|
[] |
[
508,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm'_add_le_of_le_one
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.2557321\nG : Type ?u.2557324\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhq0 : 0 ≤ q\nhq1 : q ≤ 1\n⊢ (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ≤ ∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2557321\nG : Type ?u.2557324\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhq0 : 0 ≤ q\nhq1 : q ≤ 1\n⊢ (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ^ (1 / q) ≤\n (∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ) ^ (1 / q)",
"tactic": "refine' ENNReal.rpow_le_rpow _ (by simp [hq0] : 0 ≤ 1 / q)"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.2557321\nG : Type ?u.2557324\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhq0 : 0 ≤ q\nhq1 : q ≤ 1\na : α\n⊢ ↑‖(f + g) a‖₊ ≤ ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2557321\nG : Type ?u.2557324\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhq0 : 0 ≤ q\nhq1 : q ≤ 1\n⊢ (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ≤ ∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ",
"tactic": "refine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ hq0"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2557321\nG : Type ?u.2557324\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhq0 : 0 ≤ q\nhq1 : q ≤ 1\na : α\n⊢ ↑‖(f + g) a‖₊ ≤ ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a",
"tactic": "simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.2557321\nG : Type ?u.2557324\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhq0 : 0 ≤ q\nhq1 : q ≤ 1\n⊢ 0 ≤ 1 / q",
"tactic": "simp [hq0]"
}
] |
[
766,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Order/Circular.lean
|
Set.compl_cIoo
|
[
{
"state_after": "case h\nα : Type u_1\ninst✝ : CircularOrder α\na b x✝ : α\n⊢ x✝ ∈ cIoo a bᶜ ↔ x✝ ∈ cIcc b a",
"state_before": "α : Type u_1\ninst✝ : CircularOrder α\na b : α\n⊢ cIoo a bᶜ = cIcc b a",
"tactic": "ext"
},
{
"state_after": "case h\nα : Type u_1\ninst✝ : CircularOrder α\na b x✝ : α\n⊢ x✝ ∈ cIoo a bᶜ ↔ ¬sbtw a x✝ b",
"state_before": "case h\nα : Type u_1\ninst✝ : CircularOrder α\na b x✝ : α\n⊢ x✝ ∈ cIoo a bᶜ ↔ x✝ ∈ cIcc b a",
"tactic": "rw [Set.mem_cIcc, btw_iff_not_sbtw]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : CircularOrder α\na b x✝ : α\n⊢ x✝ ∈ cIoo a bᶜ ↔ ¬sbtw a x✝ b",
"tactic": "rfl"
}
] |
[
381,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.FiniteAtFilter.filter_mono_ae
|
[] |
[
4125,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4124,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Module.algebraMap_end_apply
|
[] |
[
631,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
630,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.AbsolutelyContinuous.zero
|
[] |
[
1111,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1110,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.typevecCasesNil₂_appendFun
|
[] |
[
381,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinOn.on_preimage
|
[] |
[
416,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.continuousAt_iff'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.127935\nι : Type ?u.127938\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ninst✝ : TopologicalSpace β\nf : β → α\nb : β\n⊢ ContinuousAt f b ↔ ∀ (ε : ℝ), ε > 0 → ∀ᶠ (x : β) in 𝓝 b, dist (f x) (f b) < ε",
"tactic": "rw [ContinuousAt, tendsto_nhds]"
}
] |
[
1080,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/Order/BoundedOrder.lean
|
OrderBot.ext
|
[
{
"state_after": "case mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\nB : OrderBot α\ntoBot✝ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\n⊢ mk ha = B",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\nA B : OrderBot α\n⊢ A = B",
"tactic": "rcases A with ⟨ha⟩"
},
{
"state_after": "case mk.mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\ntoBot✝¹ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\ntoBot✝ : Bot α\nhb : ∀ (a : α), ⊥ ≤ a\n⊢ mk ha = mk hb",
"state_before": "case mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\nB : OrderBot α\ntoBot✝ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\n⊢ mk ha = B",
"tactic": "rcases B with ⟨hb⟩"
},
{
"state_after": "case mk.mk.e_toBot\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\ntoBot✝¹ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\ntoBot✝ : Bot α\nhb : ∀ (a : α), ⊥ ≤ a\n⊢ toBot✝¹ = toBot✝",
"state_before": "case mk.mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\ntoBot✝¹ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\ntoBot✝ : Bot α\nhb : ∀ (a : α), ⊥ ≤ a\n⊢ mk ha = mk hb",
"tactic": "congr"
},
{
"state_after": "case mk.mk.e_toBot.bot\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\ntoBot✝¹ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\ntoBot✝ : Bot α\nhb : ∀ (a : α), ⊥ ≤ a\n⊢ ⊥ = ⊥",
"state_before": "case mk.mk.e_toBot\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\ntoBot✝¹ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\ntoBot✝ : Bot α\nhb : ∀ (a : α), ⊥ ≤ a\n⊢ toBot✝¹ = toBot✝",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.e_toBot.bot\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25434\nδ : Type ?u.25437\nα : Type u_1\ninst✝ : PartialOrder α\ntoBot✝¹ : Bot α\nha : ∀ (a : α), ⊥ ≤ a\ntoBot✝ : Bot α\nhb : ∀ (a : α), ⊥ ≤ a\n⊢ ⊥ = ⊥",
"tactic": "exact le_antisymm (ha _) (hb _)"
}
] |
[
447,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.eventually_lt_of_limsup_lt
|
[] |
[
1134,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1130,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
Convex.norm_image_sub_le_of_norm_deriv_le
|
[] |
[
672,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean
|
WithTop.mul_lt_top'
|
[
{
"state_after": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Zero α\ninst✝¹ : Mul α\ninst✝ : LT α\na b : WithTop α\nha : a ≠ ⊤\nhb : b ≠ ⊤\n⊢ a * b ≠ ⊤",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Zero α\ninst✝¹ : Mul α\ninst✝ : LT α\na b : WithTop α\nha : a < ⊤\nhb : b < ⊤\n⊢ a * b < ⊤",
"tactic": "rw [WithTop.lt_top_iff_ne_top] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Zero α\ninst✝¹ : Mul α\ninst✝ : LT α\na b : WithTop α\nha : a ≠ ⊤\nhb : b ≠ ⊤\n⊢ a * b ≠ ⊤",
"tactic": "simp only [Ne.def, mul_eq_top_iff, *, and_false, false_and, false_or]"
}
] |
[
72,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.le_vanishingIdeal_zeroLocus
|
[] |
[
218,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Order/Filter/SmallSets.lean
|
Filter.HasBasis.smallSets
|
[] |
[
50,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Topology/Order/Basic.lean
|
countable_of_isolated_left'
|
[
{
"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 : OrderTopology α\ninst✝ : SecondCountableTopology α\n⊢ Set.Countable {x | ∃ y, y < x ∧ Ioo y x = ∅}",
"tactic": "simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left"
}
] |
[
1432,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1430,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure
|
[
{
"state_after": "case h₁\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ zeroLocus 𝒜 ↑(vanishingIdeal t) ⊆ closure t\n\ncase h₂\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ closure t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ zeroLocus 𝒜 ↑(vanishingIdeal t) = closure t",
"tactic": "apply Set.Subset.antisymm"
},
{
"state_after": "case h₁.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum 𝒜)\nht' : IsClosed t'\nht : t ⊆ t'\n⊢ x ∈ t'",
"state_before": "case h₁\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ zeroLocus 𝒜 ↑(vanishingIdeal t) ⊆ closure t",
"tactic": "rintro x hx t' ⟨ht', ht⟩"
},
{
"state_after": "case h₁.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : t ⊆ zeroLocus 𝒜 fs\n⊢ x ∈ zeroLocus 𝒜 fs",
"state_before": "case h₁.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum 𝒜)\nht' : IsClosed t'\nht : t ⊆ t'\n⊢ x ∈ t'",
"tactic": "obtain ⟨fs, rfl⟩ : ∃ s, t' = zeroLocus 𝒜 s := by rwa [isClosed_iff_zeroLocus] at ht'"
},
{
"state_after": "case h₁.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : fs ⊆ ↑(vanishingIdeal t)\n⊢ x ∈ zeroLocus 𝒜 fs",
"state_before": "case h₁.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : t ⊆ zeroLocus 𝒜 fs\n⊢ x ∈ zeroLocus 𝒜 fs",
"tactic": "rw [subset_zeroLocus_iff_subset_vanishingIdeal] at ht"
},
{
"state_after": "no goals",
"state_before": "case h₁.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : fs ⊆ ↑(vanishingIdeal t)\n⊢ x ∈ zeroLocus 𝒜 fs",
"tactic": "exact Set.Subset.trans ht hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum 𝒜)\nht' : IsClosed t'\nht : t ⊆ t'\n⊢ ∃ s, t' = zeroLocus 𝒜 s",
"tactic": "rwa [isClosed_iff_zeroLocus] at ht'"
},
{
"state_after": "case h₂\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)",
"state_before": "case h₂\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ closure t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)",
"tactic": "rw [(isClosed_zeroLocus _ _).closure_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case h₂\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nt : Set (ProjectiveSpectrum 𝒜)\n⊢ t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)",
"tactic": "exact subset_zeroLocus_vanishingIdeal 𝒜 t"
}
] |
[
371,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.HD_optimalGHDist_le
|
[] |
[
438,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
9
] |
Mathlib/Topology/Order/Basic.lean
|
exists_Ico_subset_of_mem_nhds
|
[] |
[
1245,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1241,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
tendsto_log_mul_rpow_nhds_zero
|
[
{
"state_after": "no goals",
"state_before": "r : ℝ\nhr : 0 < r\nx : ℝ\nhx : x ∈ Set.Ioi 0\n⊢ log x / x ^ (-r) = (fun x => log x * x ^ r) x",
"tactic": "rw [rpow_neg hx.out.le, div_inv_eq_mul]"
}
] |
[
336,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.Set.insert_symm_apply_inr
|
[] |
[
296,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
ContinuousMap.self_mem_compactConvNhd
|
[] |
[
106,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
TopologicalGroup.tendstoLocallyUniformlyOn_iff
|
[] |
[
650,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_comp_mul_add
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14872189\n𝕜 : Type ?u.14872192\nE : Type u_1\nF : Type ?u.14872198\nA : Type ?u.14872201\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c d✝ : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ (∫ (x : ℝ) in a..b, f (c * x + d)) = c⁻¹ • ∫ (x : ℝ) in c * a + d..c * b + d, f x",
"tactic": "rw [← integral_comp_add_right, ← integral_comp_mul_left _ hc]"
}
] |
[
755,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.Superset.trans
|
[] |
[
359,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
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