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/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.derivBFamily_fp
|
[
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) (derivBFamily o f a) =\n derivBFamily o f a",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ f i hi (derivBFamily o f a) = derivBFamily o f a",
"tactic": "rw [← familyOfBFamily_enum o f]"
},
{
"state_after": "case H\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ IsNormal (familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)))",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) (derivBFamily o f a) =\n derivBFamily o f a",
"tactic": "apply derivFamily_fp"
},
{
"state_after": "case H\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ IsNormal (f i ?H.hi)\n\ncase H.hi\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ i < o",
"state_before": "case H\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ IsNormal (familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)))",
"tactic": "rw [familyOfBFamily_enum]"
},
{
"state_after": "no goals",
"state_before": "case H\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ IsNormal (f i ?H.hi)\n\ncase H.hi\no : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\ni : Ordinal\nhi : i < o\nH : IsNormal (f i hi)\na : Ordinal\n⊢ i < o",
"tactic": "exact H"
}
] |
[
374,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_mul_le
|
[
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\n⊢ degree (∑ x in support p, sum q fun j a => ↑(monomial (x + j)) (coeff p x * a)) ≤\n sup (support p) fun i => degree (sum q fun j a => ↑(monomial (i + j)) (coeff p i * a))",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\n⊢ degree (p * q) ≤ sup (support p) fun i => degree (sum q fun j a => ↑C (coeff p i * a) * X ^ (i + j))",
"tactic": "simp only [← C_mul_X_pow_eq_monomial.symm, mul_eq_sum_sum (p := p) (q := q)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\n⊢ degree (∑ x in support p, sum q fun j a => ↑(monomial (x + j)) (coeff p x * a)) ≤\n sup (support p) fun i => degree (sum q fun j a => ↑(monomial (i + j)) (coeff p i * a))",
"tactic": "exact degree_sum_le _ _"
},
{
"state_after": "R : Type u\nS : Type v\na✝ b✝ c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\na : ℕ\nha : a ∈ support p\nb : ℕ\nhb : b ∈ support q\n⊢ ↑(a + b) ≤ degree p + degree q",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\n⊢ (sup (support p) fun i => sup (support q) fun j => degree (↑C (coeff p i * coeff q j) * X ^ (i + j))) ≤\n degree p + degree q",
"tactic": "refine'\n Finset.sup_le fun a ha => Finset.sup_le fun b hb => le_trans (degree_C_mul_X_pow_le _ _) _"
},
{
"state_after": "R : Type u\nS : Type v\na✝ b✝ c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\na : ℕ\nha : a ∈ support p\nb : ℕ\nhb : b ∈ support q\n⊢ ↑a + ↑b ≤ degree p + degree q",
"state_before": "R : Type u\nS : Type v\na✝ b✝ c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\na : ℕ\nha : a ∈ support p\nb : ℕ\nhb : b ∈ support q\n⊢ ↑(a + b) ≤ degree p + degree q",
"tactic": "rw [Nat.cast_withBot, WithBot.coe_add]"
},
{
"state_after": "R : Type u\nS : Type v\na✝ b✝ c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\na : ℕ\nha : coeff p a ≠ 0\nb : ℕ\nhb : coeff q b ≠ 0\n⊢ ↑a + ↑b ≤ degree p + degree q",
"state_before": "R : Type u\nS : Type v\na✝ b✝ c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\na : ℕ\nha : a ∈ support p\nb : ℕ\nhb : b ∈ support q\n⊢ ↑a + ↑b ≤ degree p + degree q",
"tactic": "rw [mem_support_iff] at ha hb"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b✝ c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.567630\np q : R[X]\na : ℕ\nha : coeff p a ≠ 0\nb : ℕ\nhb : coeff q b ≠ 0\n⊢ ↑a + ↑b ≤ degree p + degree q",
"tactic": "exact add_le_add (le_degree_of_ne_zero ha) (le_degree_of_ne_zero hb)"
}
] |
[
782,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
765,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
hasFDerivWithinAt_iff_hasDerivWithinAt
|
[] |
[
174,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.pred_eq_succ_iff
|
[
{
"state_after": "no goals",
"state_before": "m✝ n✝ k n m : ℕ\n⊢ pred n = succ m ↔ n = m + 2",
"tactic": "cases n <;> constructor <;> rintro ⟨⟩ <;> rfl"
}
] |
[
316,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContinuousLinearEquiv.comp_contDiffAt_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.297390\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ne : F ≃L[𝕜] G\n⊢ ContDiffAt 𝕜 n (↑e ∘ f) x ↔ ContDiffAt 𝕜 n f x",
"tactic": "simp only [← contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff]"
}
] |
[
338,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/GroupTheory/GroupAction/Basic.lean
|
MulAction.mem_orbit
|
[] |
[
64,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isLeast_univ_iff
|
[] |
[
826,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
825,
9
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
MultilinearMap.bound_of_shell
|
[
{
"state_after": "case inl.intro\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\ni : ι\nhi : m i = 0\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\n\ncase inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\nhm : ∀ (i : ι), m i ≠ 0\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"tactic": "rcases em (∃ i, m i = 0) with (⟨i, hi⟩ | hm) <;> [skip; push_neg at hm]"
},
{
"state_after": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\nhm : ∀ (i : ι), m i ≠ 0\nδ : ι → 𝕜\nhδ0 : ∀ (i : ι), δ i ≠ 0\nhδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i\nhle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖\n✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"state_before": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\nhm : ∀ (i : ι), m i ≠ 0\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"tactic": "choose δ hδ0 hδm_lt hle_δm _ using fun i => rescale_to_shell (hc i) (hε i) (hm i)"
},
{
"state_after": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\nhm : ∀ (i : ι), m i ≠ 0\nδ : ι → 𝕜\nhδ0✝ : ∀ (i : ι), δ i ≠ 0\nhδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i\nhle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖\n✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖\nhδ0 : 0 < ∏ i : ι, ‖δ i‖\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"state_before": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\nhm : ∀ (i : ι), m i ≠ 0\nδ : ι → 𝕜\nhδ0 : ∀ (i : ι), δ i ≠ 0\nhδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i\nhle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖\n✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"tactic": "have hδ0 : 0 < ∏ i, ‖δ i‖ := prod_pos fun i _ => norm_pos_iff.2 (hδ0 i)"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\nhm : ∀ (i : ι), m i ≠ 0\nδ : ι → 𝕜\nhδ0✝ : ∀ (i : ι), δ i ≠ 0\nhδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i\nhle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖\n✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖\nhδ0 : 0 < ∏ i : ι, ‖δ i‖\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"tactic": "simpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, mul_le_mul_left hδ0] using\n hf (fun i => δ i • m i) hle_δm hδm_lt"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : MultilinearMap 𝕜 E G\nε : ι → ℝ\nC : ℝ\nhε : ∀ (i : ι), 0 < ε i\nc : ι → 𝕜\nhc : ∀ (i : ι), 1 < ‖c i‖\nhf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖\nm : (i : ι) → E i\ni : ι\nhi : m i = 0\n⊢ ‖↑f m‖ ≤ C * ∏ i : ι, ‖m i‖",
"tactic": "simp [f.map_coord_zero i hi, prod_eq_zero (mem_univ i), hi]"
}
] |
[
118,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
continuousMultilinearCurryFin1_apply
|
[] |
[
1735,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1733,
1
] |
Mathlib/Data/PNat/Basic.lean
|
PNat.coe_coeMonoidHom
|
[] |
[
244,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Topology/Algebra/Constructions.lean
|
MulOpposite.comap_op_nhds
|
[] |
[
79,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.walkingParallelPairOp_left
|
[] |
[
151,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousWithinAt_union
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.322097\nδ : Type ?u.322100\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns t : Set α\nx : α\n⊢ ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x",
"tactic": "simp only [ContinuousWithinAt, nhdsWithin_union, tendsto_sup]"
}
] |
[
718,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
716,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsLUB.mem_closure
|
[] |
[
1985,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1984,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
exists_deriv_eq_zero'
|
[
{
"state_after": "f f' : ℝ → ℝ\na b l : ℝ\nhab : a < b\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 l)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 l)\nh : ∃ x, x ∈ Ioo a b ∧ ¬DifferentiableAt ℝ f x\n⊢ ∃ x, x ∈ Ioo a b ∧ ¬DifferentiableAt ℝ f x",
"state_before": "f f' : ℝ → ℝ\na b l : ℝ\nhab : a < b\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 l)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 l)\nh : ¬∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ f x\n⊢ ∃ x, x ∈ Ioo a b ∧ ¬DifferentiableAt ℝ f x",
"tactic": "push_neg at h"
},
{
"state_after": "no goals",
"state_before": "f f' : ℝ → ℝ\na b l : ℝ\nhab : a < b\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 l)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 l)\nh : ∃ x, x ∈ Ioo a b ∧ ¬DifferentiableAt ℝ f x\n⊢ ∃ x, x ∈ Ioo a b ∧ ¬DifferentiableAt ℝ f x",
"tactic": "exact h"
}
] |
[
345,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffOn.continuousLinearMap_comp
|
[] |
[
228,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Algebra/Group/Units.lean
|
divp_mul_divp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : CommMonoid α\nx y : α\nux uy : αˣ\n⊢ x /ₚ ux * (y /ₚ uy) = x * y /ₚ (ux * uy)",
"tactic": "rw [divp_mul_eq_mul_divp, divp_assoc', divp_divp_eq_divp_mul]"
}
] |
[
549,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.toSubgraph_rotate
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\ninst✝ : DecidableEq V\nc : Walk G v v\nh : u ∈ support c\n⊢ Walk.toSubgraph (rotate c h) = Walk.toSubgraph c",
"tactic": "rw [rotate, toSubgraph_append, sup_comm, ← toSubgraph_append, take_spec]"
}
] |
[
2271,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2269,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.not_isSuccLimit_succ
|
[] |
[
93,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.integrableOn_finset_iUnion
|
[] |
[
214,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/List/Zip.lean
|
List.get?_zip_eq_some
|
[
{
"state_after": "case mk\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ get? (zip l₁ l₂) i = some (fst✝, snd✝) ↔ get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\nz : α × β\ni : ℕ\n⊢ get? (zip l₁ l₂) i = some z ↔ get? l₁ i = some z.fst ∧ get? l₂ i = some z.snd",
"tactic": "cases z"
},
{
"state_after": "case mk\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ (∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)) ↔\n get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"state_before": "case mk\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ get? (zip l₁ l₂) i = some (fst✝, snd✝) ↔ get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"tactic": "rw [zip, get?_zip_with_eq_some]"
},
{
"state_after": "case mk.mp\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ (∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)) →\n get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd\n\ncase mk.mpr\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd →\n ∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)",
"state_before": "case mk\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ (∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)) ↔\n get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"tactic": "constructor"
},
{
"state_after": "case mk.mp.intro.intro.intro.intro\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\nx : α\ny : β\nh₀ : get? l₁ i = some x\nh₁ : get? l₂ i = some y\nh₂ : (x, y) = (fst✝, snd✝)\n⊢ get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"state_before": "case mk.mp\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ (∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)) →\n get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"tactic": "rintro ⟨x, y, h₀, h₁, h₂⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mp.intro.intro.intro.intro\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\nx : α\ny : β\nh₀ : get? l₁ i = some x\nh₁ : get? l₂ i = some y\nh₂ : (x, y) = (fst✝, snd✝)\n⊢ get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd",
"tactic": "simpa [h₀, h₁] using h₂"
},
{
"state_after": "case mk.mpr.intro\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\nh₀ : get? l₁ i = some (fst✝, snd✝).fst\nh₁ : get? l₂ i = some (fst✝, snd✝).snd\n⊢ ∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)",
"state_before": "case mk.mpr\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\n⊢ get? l₁ i = some (fst✝, snd✝).fst ∧ get? l₂ i = some (fst✝, snd✝).snd →\n ∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)",
"tactic": "rintro ⟨h₀, h₁⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mpr.intro\nα : Type u\nβ : Type u_1\nγ : Type ?u.155522\nδ : Type ?u.155525\nε : Type ?u.155528\nl₁ : List α\nl₂ : List β\ni : ℕ\nfst✝ : α\nsnd✝ : β\nh₀ : get? l₁ i = some (fst✝, snd✝).fst\nh₁ : get? l₂ i = some (fst✝, snd✝).snd\n⊢ ∃ x y, get? l₁ i = some x ∧ get? l₂ i = some y ∧ (x, y) = (fst✝, snd✝)",
"tactic": "exact ⟨_, _, h₀, h₁, rfl⟩"
}
] |
[
396,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.sub_eq
|
[] |
[
86,
52
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
86,
9
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.exists_mem_Ioc_zpow
|
[
{
"state_after": "case intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ny : ℝ≥0∞\nhy : 1 < y\nh'y : y ≠ ⊤\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ ∃ n, ↑x ∈ Ioc (y ^ n) (y ^ (n + 1))",
"state_before": "α : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y : ℝ≥0∞\nhx : x ≠ 0\nh'x : x ≠ ⊤\nhy : 1 < y\nh'y : y ≠ ⊤\n⊢ ∃ n, x ∈ Ioc (y ^ n) (y ^ (n + 1))",
"tactic": "lift x to ℝ≥0 using h'x"
},
{
"state_after": "case intro.intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ ∃ n, ↑x ∈ Ioc (↑y ^ n) (↑y ^ (n + 1))",
"state_before": "case intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ny : ℝ≥0∞\nhy : 1 < y\nh'y : y ≠ ⊤\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ ∃ n, ↑x ∈ Ioc (y ^ n) (y ^ (n + 1))",
"tactic": "lift y to ℝ≥0 using h'y"
},
{
"state_after": "case intro.intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ ∃ n, ↑x ∈ Ioc (↑y ^ n) (↑y ^ (n + 1))",
"state_before": "case intro.intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ ∃ n, ↑x ∈ Ioc (↑y ^ n) (↑y ^ (n + 1))",
"tactic": "have A : y ≠ 0 := by simpa only [Ne.def, coe_eq_zero] using (zero_lt_one.trans hy).ne'"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\nn : ℤ\nhn : y ^ n < x\nh'n : x ≤ y ^ (n + 1)\n⊢ ∃ n, ↑x ∈ Ioc (↑y ^ n) (↑y ^ (n + 1))",
"state_before": "case intro.intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ ∃ n, ↑x ∈ Ioc (↑y ^ n) (↑y ^ (n + 1))",
"tactic": "obtain ⟨n, hn, h'n⟩ : ∃ n : ℤ, y ^ n < x ∧ x ≤ y ^ (n + 1) := by\n refine' NNReal.exists_mem_Ioc_zpow _ (one_lt_coe_iff.1 hy)\n simpa only [Ne.def, coe_eq_zero] using hx"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\nn : ℤ\nhn : y ^ n < x\nh'n : x ≤ y ^ (n + 1)\n⊢ ↑y ^ n < ↑x\n\ncase intro.intro.intro.intro.refine'_2\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\nn : ℤ\nhn : y ^ n < x\nh'n : x ≤ y ^ (n + 1)\n⊢ ↑x ≤ ↑y ^ (n + 1)",
"state_before": "case intro.intro.intro.intro\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\nn : ℤ\nhn : y ^ n < x\nh'n : x ≤ y ^ (n + 1)\n⊢ ∃ n, ↑x ∈ Ioc (↑y ^ n) (↑y ^ (n + 1))",
"tactic": "refine' ⟨n, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ y ≠ 0",
"tactic": "simpa only [Ne.def, coe_eq_zero] using (zero_lt_one.trans hy).ne'"
},
{
"state_after": "α : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ x ≠ 0",
"state_before": "α : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ ∃ n, y ^ n < x ∧ x ≤ y ^ (n + 1)",
"tactic": "refine' NNReal.exists_mem_Ioc_zpow _ (one_lt_coe_iff.1 hy)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ x ≠ 0",
"tactic": "simpa only [Ne.def, coe_eq_zero] using hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\nn : ℤ\nhn : y ^ n < x\nh'n : x ≤ y ^ (n + 1)\n⊢ ↑y ^ n < ↑x",
"tactic": "rwa [← ENNReal.coe_zpow A, ENNReal.coe_lt_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2\nα : Type ?u.560921\nβ : Type ?u.560924\na b c d : ℝ≥0∞\nr p q x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\nn : ℤ\nhn : y ^ n < x\nh'n : x ≤ y ^ (n + 1)\n⊢ ↑x ≤ ↑y ^ (n + 1)",
"tactic": "rwa [← ENNReal.coe_zpow A, ENNReal.coe_le_coe]"
}
] |
[
1897,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1887,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
BiheytingHom.coe_id
|
[] |
[
553,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.cast_bit0
|
[
{
"state_after": "α : Type u_1\ninst✝ : Semiring α\nn : Num\n⊢ ↑n + ↑n = _root_.bit0 ↑n",
"state_before": "α : Type u_1\ninst✝ : Semiring α\nn : Num\n⊢ ↑(Num.bit0 n) = _root_.bit0 ↑n",
"tactic": "rw [← bit0_of_bit0, _root_.bit0, cast_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Semiring α\nn : Num\n⊢ ↑n + ↑n = _root_.bit0 ↑n",
"tactic": "rfl"
}
] |
[
736,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
algebraMap.coe_one
|
[] |
[
154,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_lt_mul_of_le_of_le'
|
[] |
[
228,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/MeasureTheory/Measure/Content.lean
|
MeasureTheory.Content.measure_apply
|
[] |
[
387,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.norm_integral_eq_norm_integral_Ioc
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.11586972\n𝕜 : Type ?u.11586975\nE : Type u_1\nF : Type ?u.11586981\nA : Type ?u.11586984\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf✝ g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nf : ℝ → E\n⊢ ‖∫ (x : ℝ) in a..b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖",
"tactic": "rw [← norm_integral_min_max, integral_of_le min_le_max, uIoc]"
}
] |
[
535,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
Set.iUnion_Ioc_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u\nα : Type v\nβ : Type w\ninst✝ : Preorder α\na✝ b c a : α\n⊢ (⋃ (b : α), Ioc a b) = Ioi a",
"tactic": "simp only [← Ioi_inter_Iic, ← inter_iUnion, iUnion_Iic, inter_univ]"
}
] |
[
83,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PNat.factorMultiset_mul
|
[
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"state_before": "n m : ℕ+\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"tactic": "let u := factorMultiset n"
},
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"tactic": "let v := factorMultiset m"
},
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"tactic": "have : n = u.prod := (prod_factorMultiset n).symm"
},
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (PrimeMultiset.prod u * m) = factorMultiset (PrimeMultiset.prod u) + factorMultiset m",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (n * m) = factorMultiset n + factorMultiset m",
"tactic": "rw [this]"
},
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ factorMultiset (PrimeMultiset.prod u * m) = factorMultiset (PrimeMultiset.prod u) + factorMultiset m",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (PrimeMultiset.prod u * m) = factorMultiset (PrimeMultiset.prod u) + factorMultiset m",
"tactic": "have : m = v.prod := (prod_factorMultiset m).symm"
},
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ factorMultiset (PrimeMultiset.prod u * PrimeMultiset.prod v) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ factorMultiset (PrimeMultiset.prod u * m) = factorMultiset (PrimeMultiset.prod u) + factorMultiset m",
"tactic": "rw [this]"
},
{
"state_after": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ factorMultiset (PrimeMultiset.prod (u + v)) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ factorMultiset (PrimeMultiset.prod u * PrimeMultiset.prod v) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)",
"tactic": "rw [← PrimeMultiset.prod_add]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ factorMultiset (PrimeMultiset.prod (u + v)) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)",
"tactic": "repeat' rw [PrimeMultiset.factorMultiset_prod]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis✝ : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n⊢ u + v = u + factorMultiset (PrimeMultiset.prod v)",
"tactic": "rw [PrimeMultiset.factorMultiset_prod]"
}
] |
[
306,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
|
MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.15886\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nn : ℕ\n⊢ seqTendstoAeSeq hfg (n + 1) = max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1)",
"tactic": "rw [seqTendstoAeSeq]"
}
] |
[
173,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.prod_span
|
[
{
"state_after": "ι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∏ i in s, span R (M i) = span R (∏ i in s, M i)",
"state_before": "ι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\n⊢ ∏ i in s, span R (M i) = span R (∏ i in s, M i)",
"tactic": "letI := Classical.decEq ι"
},
{
"state_after": "case refine'_1\nι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∏ i in ∅, span R (M i) = span R (∏ i in ∅, M i)\n\ncase refine'_2\nι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ∏ i in s, span R (M i) = span R (∏ i in s, M i) →\n ∏ i in insert a s, span R (M i) = span R (∏ i in insert a s, M i)",
"state_before": "ι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∏ i in s, span R (M i) = span R (∏ i in s, M i)",
"tactic": "refine' Finset.induction_on s _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∏ i in ∅, span R (M i) = span R (∏ i in ∅, M i)",
"tactic": "simp [one_eq_span, Set.singleton_one]"
},
{
"state_after": "case refine'_2\nι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\na✝ : ι\ns✝ : Finset ι\nH : ¬a✝ ∈ s✝\nih : ∏ i in s✝, span R (M i) = span R (∏ i in s✝, M i)\n⊢ ∏ i in insert a✝ s✝, span R (M i) = span R (∏ i in insert a✝ s✝, M i)",
"state_before": "case refine'_2\nι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ∏ i in s, span R (M i) = span R (∏ i in s, M i) →\n ∏ i in insert a s, span R (M i) = span R (∏ i in insert a s, M i)",
"tactic": "intro _ _ H ih"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM✝ N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nM : ι → Set A\nthis : DecidableEq ι := Classical.decEq ι\na✝ : ι\ns✝ : Finset ι\nH : ¬a✝ ∈ s✝\nih : ∏ i in s✝, span R (M i) = span R (∏ i in s✝, M i)\n⊢ ∏ i in insert a✝ s✝, span R (M i) = span R (∏ i in insert a✝ s✝, M i)",
"tactic": "rw [Finset.prod_insert H, Finset.prod_insert H, ih, span_mul_span]"
}
] |
[
614,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
MonoidHom.ker_eq_bot_of_cancel
|
[
{
"state_after": "no goals",
"state_before": "A : Type u\nB : Type v\ninst✝¹ : Group A\ninst✝ : Group B\nf : A →* B\nh : ∀ (u v : { x // x ∈ ker f } →* A), comp f u = comp f v → u = v\n⊢ ker f = ⊥",
"tactic": "simpa using _root_.congr_arg range (h f.ker.subtype 1 (by aesop_cat))"
},
{
"state_after": "no goals",
"state_before": "A : Type u\nB : Type v\ninst✝¹ : Group A\ninst✝ : Group B\nf : A →* B\nh : ∀ (u v : { x // x ∈ ker f } →* A), comp f u = comp f v → u = v\n⊢ comp f (Subgroup.subtype (ker f)) = comp f 1",
"tactic": "aesop_cat"
}
] |
[
37,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Dynamics/OmegaLimit.lean
|
omegaLimit_mono_left
|
[] |
[
85,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
|
AlgebraicTopology.DoldKan.PInfty_f_idem
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f PInfty n ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n",
"tactic": "simp only [PInfty_f, P_f_idem]"
}
] |
[
107,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toAdd_inv
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean
|
ContinuousAffineMap.smul_contLinear
|
[] |
[
153,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMaxFilter.neg
|
[] |
[
487,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/CategoryTheory/NatIso.lean
|
CategoryTheory.NatIso.cancel_natIso_inv_right_assoc
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nα : F ≅ G\nW X X' : D\nY : C\nf : W ⟶ X\ng : X ⟶ G.obj Y\nf' : W ⟶ X'\ng' : X' ⟶ G.obj Y\n⊢ f ≫ g ≫ α.inv.app Y = f' ≫ g' ≫ α.inv.app Y ↔ f ≫ g = f' ≫ g'",
"tactic": "simp only [← Category.assoc, cancel_mono, refl]"
}
] |
[
161,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
totallyBounded_iff_filter
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\n⊢ TotallyBounded s → ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c\n\ncase mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\n⊢ (∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c) → TotallyBounded s",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\n⊢ TotallyBounded s ↔ ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\n⊢ TotallyBounded s → ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c",
"tactic": "exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,\n (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nH : ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c\nd : Set (α × α)\nhd : d ∈ 𝓤 α\n⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\n⊢ (∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c) → TotallyBounded s",
"tactic": "intro H d hd"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\n⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ (c : Filter α), c ≤ f → ¬Cauchy c",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nH : ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c, c ≤ f ∧ Cauchy c\nd : Set (α × α)\nhd : d ∈ 𝓤 α\n⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}",
"tactic": "contrapose! H with hd_cover"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\n⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ (c : Filter α), c ≤ f → ¬Cauchy c",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\n⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ (c : Filter α), c ≤ f → ¬Cauchy c",
"tactic": "set f := ⨅ t : Finset α, 𝓟 (s \\ ⋃ y ∈ t, { x | (x, y) ∈ d })"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\n⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ (c : Filter α), c ≤ f → ¬Cauchy c",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis : NeBot f\n⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ (c : Filter α), c ≤ f → ¬Cauchy c",
"tactic": "have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\n⊢ False",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\n⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ (c : Filter α), c ≤ f → ¬Cauchy c",
"tactic": "refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩"
},
{
"state_after": "case mpr.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\n⊢ False",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\n⊢ False",
"tactic": "rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\n⊢ False",
"state_before": "case mpr.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\n⊢ False",
"tactic": "rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\n⊢ False",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\n⊢ False",
"tactic": "set ys := ⋃ y' ∈ ({y} : Finset α), { x | (x, y') ∈ d }"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\n⊢ False",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝ : NeBot f\nthis : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\n⊢ False",
"tactic": "have : c ≤ 𝓟 (s \\ ys) := hcf.trans (iInf_le_of_le {y} le_rfl)"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\n⊢ ∅ ∈ c",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\n⊢ False",
"tactic": "refine' hc.1.ne (empty_mem_iff_bot.mp _)"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\n⊢ ∀ (a : α), (a ∈ s \\ ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}) → a ∈ m → a ∈ ∅",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\n⊢ ∅ ∈ c",
"tactic": "filter_upwards [le_principal_iff.1 this, hm]"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\nx : α\nhx : x ∈ s \\ ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nhxm : x ∈ m\n⊢ x ∈ ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}",
"state_before": "case h\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\n⊢ ∀ (a : α), (a ∈ s \\ ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}) → a ∈ m → a ∈ ∅",
"tactic": "refine' fun x hx hxm => hx.2 _"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis✝¹ : NeBot f\nthis✝ : f ≤ 𝓟 s\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Set α\nhm : m ∈ c\nhmd : m ×ˢ m ⊆ d\ny : α\nhym : y ∈ m\nys : Set α := ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nthis : c ≤ 𝓟 (s \\ ys)\nx : α\nhx : x ∈ s \\ ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}\nhxm : x ∈ m\n⊢ x ∈ ⋃ (y' : α) (_ : y' ∈ {y}), {x | (x, y') ∈ d}",
"tactic": "simpa using hmd (mk_mem_prod hxm hym)"
},
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\n⊢ ∀ ⦃i j : Finset α⦄,\n i ≤ j → 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ i), {x | (x, y) ∈ d}) ≥ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ j), {x | (x, y) ∈ d})\n\ncase refine'_2\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\n⊢ ∀ (i : Finset α), NeBot (𝓟 (s \\ ⋃ (y : α) (_ : y ∈ i), {x | (x, y) ∈ d}))",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\n⊢ NeBot f",
"tactic": "refine' iInf_neBot_of_directed' (directed_of_sup _) _"
},
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nt₁ t₂ : Finset α\nh : t₁ ≤ t₂\n⊢ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t₁), {x | (x, y) ∈ d}) ≥ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t₂), {x | (x, y) ∈ d})",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\n⊢ ∀ ⦃i j : Finset α⦄,\n i ≤ j → 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ i), {x | (x, y) ∈ d}) ≥ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ j), {x | (x, y) ∈ d})",
"tactic": "intro t₁ t₂ h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nt₁ t₂ : Finset α\nh : t₁ ≤ t₂\n⊢ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t₁), {x | (x, y) ∈ d}) ≥ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t₂), {x | (x, y) ∈ d})",
"tactic": "exact principal_mono.2 (diff_subset_diff_right <| biUnion_subset_biUnion_left h)"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nt : Finset α\n⊢ NeBot (𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}))",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\n⊢ ∀ (i : Finset α), NeBot (𝓟 (s \\ ⋃ (y : α) (_ : y ∈ i), {x | (x, y) ∈ d}))",
"tactic": "intro t"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nt : Finset α\n⊢ NeBot (𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}))",
"tactic": "simpa [nonempty_diff] using hd_cover t t.finite_toSet"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\ns : Set α\nd : Set (α × α)\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d}\nf : Filter α := ⨅ (t : Finset α), 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ t), {x | (x, y) ∈ d})\nthis : NeBot f\n⊢ 𝓟 (s \\ ⋃ (y : α) (_ : y ∈ ∅), {x | (x, y) ∈ d}) ≤ 𝓟 s",
"tactic": "simp"
}
] |
[
565,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/MeasureTheory/Measure/OpenPos.lean
|
MeasureTheory.Measure.eqOn_Ioc_of_ae_eq
|
[] |
[
159,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.zero_imK
|
[] |
[
858,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
9
] |
Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
|
CategoryTheory.Functor.map_zsmul
|
[] |
[
98,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.mul_univ_of_one_mem
|
[] |
[
984,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
983,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
eq_iff_eq_of_div_eq_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.62821\nβ : Type ?u.62824\nG : Type u_1\ninst✝ : Group G\na b c d : G\nH : a / b = c / d\n⊢ a = b ↔ c = d",
"tactic": "rw [← div_eq_one, H, div_eq_one]"
}
] |
[
849,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/Analysis/NormedSpace/Dual.lean
|
NormedSpace.eq_zero_iff_forall_dual_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type v\ninst✝² : IsROrC 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\nhx : x = 0\n⊢ ∀ (g : Dual 𝕜 E), ↑g x = 0",
"tactic": "simp [hx]"
}
] |
[
155,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
Equiv.pointReflection_midpoint_left
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nV : Type u_2\nV' : Type ?u.17515\nP : Type u_1\nP' : Type ?u.17521\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\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 x y : P\n⊢ ↑(pointReflection (midpoint R x y)) x = y",
"tactic": "rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,\n mul_invOf_self, one_smul, vsub_vadd]"
}
] |
[
75,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.eqOn_union
|
[] |
[
226,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_lt_of_lt
|
[] |
[
361,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioo_subset_Ioo_right
|
[] |
[
207,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/CategoryTheory/Abelian/LeftDerived.lean
|
CategoryTheory.Abelian.Functor.preserves_exact_of_PreservesFiniteColimits_of_epi
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : PreservesFiniteColimits F\ninst✝ : Epi g\nex : Exact f g\n⊢ F.map f ≫ F.map g = 0",
"tactic": "simp [← Functor.map_comp, ex.w]"
}
] |
[
55,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean
|
CategoryTheory.Endofunctor.Algebra.Initial.left_inv'
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nF : C ⥤ C\nA A₀ A₁ A₂ : Algebra F\nf : A₀ ⟶ A₁\ng : A₁ ⟶ A₂\nh : Limits.IsInitial A\n⊢ F.map (strInv h ≫ A.str) ≫ A.str = A.str ≫ strInv h ≫ A.str",
"tactic": "rw [←Category.assoc, F.map_comp, strInv, ←Hom.h]"
}
] |
[
238,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
src/lean/Init/Control/StateCps.lean
|
StateCpsT.runK_pure
|
[] |
[
51,
135
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
51,
9
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
UniformCauchySeqOn.uniformCauchySeqOnFilter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nhF : UniformCauchySeqOn F p s\n⊢ UniformCauchySeqOnFilter F p (𝓟 s)",
"tactic": "rwa [← uniformCauchySeqOn_iff_uniformCauchySeqOnFilter]"
}
] |
[
418,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Topology/UrysohnsLemma.lean
|
Urysohns.CU.subset_right_C
|
[] |
[
131,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
SimpleGraph.le_card_nonuniformWitness
|
[
{
"state_after": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ ↑(card s) * ε ≤\n ↑(card (if WellOrderingRel s t then (nonuniformWitnesses G ε s t).fst else (nonuniformWitnesses G ε t s).snd))",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ ↑(card s) * ε ≤ ↑(card (nonuniformWitness G ε s t))",
"tactic": "unfold nonuniformWitness"
},
{
"state_after": "case inl\nα : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\nh✝ : WellOrderingRel s t\n⊢ ↑(card s) * ε ≤ ↑(card (nonuniformWitnesses G ε s t).fst)\n\ncase inr\nα : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\nh✝ : ¬WellOrderingRel s t\n⊢ ↑(card s) * ε ≤ ↑(card (nonuniformWitnesses G ε t s).snd)",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ ↑(card s) * ε ≤\n ↑(card (if WellOrderingRel s t then (nonuniformWitnesses G ε s t).fst else (nonuniformWitnesses G ε t s).snd))",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\nh✝ : WellOrderingRel s t\n⊢ ↑(card s) * ε ≤ ↑(card (nonuniformWitnesses G ε s t).fst)",
"tactic": "exact G.left_nonuniformWitnesses_card h"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\nh✝ : ¬WellOrderingRel s t\n⊢ ↑(card s) * ε ≤ ↑(card (nonuniformWitnesses G ε t s).snd)",
"tactic": "exact G.right_nonuniformWitnesses_card fun i => h i.symm"
}
] |
[
179,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Data/Part.lean
|
Part.some_toOption
|
[] |
[
254,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.iInf_ker_lapply_le_bot
|
[
{
"state_after": "α : Type u_1\nM : Type u_2\nN : Type ?u.32590\nP : Type ?u.32593\nR : Type u_3\nS : Type ?u.32599\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\n⊢ ∀ ⦃x : α →₀ M⦄, (∀ (i : α), ↑x i = 0) → x = 0",
"state_before": "α : Type u_1\nM : Type u_2\nN : Type ?u.32590\nP : Type ?u.32593\nR : Type u_3\nS : Type ?u.32599\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\n⊢ (⨅ (a : α), ker (lapply a)) ≤ ⊥",
"tactic": "simp only [SetLike.le_def, mem_iInf, mem_ker, mem_bot, lapply_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\nN : Type ?u.32590\nP : Type ?u.32593\nR : Type u_3\nS : Type ?u.32599\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\n⊢ ∀ ⦃x : α →₀ M⦄, (∀ (i : α), ↑x i = 0) → x = 0",
"tactic": "exact fun a h => Finsupp.ext h"
}
] |
[
148,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sInf_union
|
[] |
[
472,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_update
|
[
{
"state_after": "case h\nR : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn : ℕ\na : R\nx✝ : ℕ\n⊢ coeff (update p n a) x✝ = Function.update (coeff p) n a x✝",
"state_before": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn : ℕ\na : R\n⊢ coeff (update p n a) = Function.update (coeff p) n a",
"tactic": "ext"
},
{
"state_after": "case h.ofFinsupp\nR : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\na : R\nx✝ : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff (update { toFinsupp := toFinsupp✝ } n a) x✝ = Function.update (coeff { toFinsupp := toFinsupp✝ }) n a x✝",
"state_before": "case h\nR : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn : ℕ\na : R\nx✝ : ℕ\n⊢ coeff (update p n a) x✝ = Function.update (coeff p) n a x✝",
"tactic": "cases p"
},
{
"state_after": "no goals",
"state_before": "case h.ofFinsupp\nR : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\na : R\nx✝ : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff (update { toFinsupp := toFinsupp✝ } n a) x✝ = Function.update (coeff { toFinsupp := toFinsupp✝ }) n a x✝",
"tactic": "simp only [coeff, update, Function.update_apply, coe_update]"
}
] |
[
1088,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1084,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.prod_mono_right
|
[] |
[
1718,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1718,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.coe_rpow_def
|
[] |
[
369,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.erase_empty
|
[] |
[
1870,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1869,
1
] |
Std/Data/String/Lemmas.lean
|
String.takeWhile_eq
|
[] |
[
1102,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1101,
1
] |
Mathlib/FieldTheory/Normal.lean
|
AlgHom.fieldRange_of_normal
|
[
{
"state_after": "F : Type u_1\nK : Type u_2\ninst✝¹⁹ : Field F\ninst✝¹⁸ : Field K\ninst✝¹⁷ : Algebra F K\nK₁ : Type ?u.303481\nK₂ : Type ?u.303484\nK₃ : Type ?u.303487\ninst✝¹⁶ : Field F\ninst✝¹⁵ : Field K₁\ninst✝¹⁴ : Field K₂\ninst✝¹³ : Field K₃\ninst✝¹² : Algebra F K₁\ninst✝¹¹ : Algebra F K₂\ninst✝¹⁰ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\nω : K₂ ≃ₐ[F] K₃\nE✝ : Type ?u.303905\ninst✝⁹ : Field E✝\ninst✝⁸ : Algebra F E✝\ninst✝⁷ : Algebra E✝ K₁\ninst✝⁶ : Algebra E✝ K₂\ninst✝⁵ : Algebra E✝ K₃\ninst✝⁴ : IsScalarTower F E✝ K₁\ninst✝³ : IsScalarTower F E✝ K₂\ninst✝² : IsScalarTower F E✝ K₃\ninst✝¹ : Algebra F K\nE : IntermediateField F K\ninst✝ : Normal F { x // x ∈ E }\nf : { x // x ∈ E } →ₐ[F] K\nthis : Algebra { x // x ∈ E } { x // x ∈ E } := Algebra.id { x // x ∈ E }\n⊢ fieldRange f = E",
"state_before": "F : Type u_1\nK : Type u_2\ninst✝¹⁹ : Field F\ninst✝¹⁸ : Field K\ninst✝¹⁷ : Algebra F K\nK₁ : Type ?u.303481\nK₂ : Type ?u.303484\nK₃ : Type ?u.303487\ninst✝¹⁶ : Field F\ninst✝¹⁵ : Field K₁\ninst✝¹⁴ : Field K₂\ninst✝¹³ : Field K₃\ninst✝¹² : Algebra F K₁\ninst✝¹¹ : Algebra F K₂\ninst✝¹⁰ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\nω : K₂ ≃ₐ[F] K₃\nE✝ : Type ?u.303905\ninst✝⁹ : Field E✝\ninst✝⁸ : Algebra F E✝\ninst✝⁷ : Algebra E✝ K₁\ninst✝⁶ : Algebra E✝ K₂\ninst✝⁵ : Algebra E✝ K₃\ninst✝⁴ : IsScalarTower F E✝ K₁\ninst✝³ : IsScalarTower F E✝ K₂\ninst✝² : IsScalarTower F E✝ K₃\ninst✝¹ : Algebra F K\nE : IntermediateField F K\ninst✝ : Normal F { x // x ∈ E }\nf : { x // x ∈ E } →ₐ[F] K\n⊢ fieldRange f = E",
"tactic": "letI : Algebra E E := Algebra.id E"
},
{
"state_after": "F : Type u_1\nK : Type u_2\ninst✝¹⁹ : Field F\ninst✝¹⁸ : Field K\ninst✝¹⁷ : Algebra F K\nK₁ : Type ?u.303481\nK₂ : Type ?u.303484\nK₃ : Type ?u.303487\ninst✝¹⁶ : Field F\ninst✝¹⁵ : Field K₁\ninst✝¹⁴ : Field K₂\ninst✝¹³ : Field K₃\ninst✝¹² : Algebra F K₁\ninst✝¹¹ : Algebra F K₂\ninst✝¹⁰ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\nω : K₂ ≃ₐ[F] K₃\nE✝ : Type ?u.303905\ninst✝⁹ : Field E✝\ninst✝⁸ : Algebra F E✝\ninst✝⁷ : Algebra E✝ K₁\ninst✝⁶ : Algebra E✝ K₂\ninst✝⁵ : Algebra E✝ K₃\ninst✝⁴ : IsScalarTower F E✝ K₁\ninst✝³ : IsScalarTower F E✝ K₂\ninst✝² : IsScalarTower F E✝ K₃\ninst✝¹ : Algebra F K\nE : IntermediateField F K\ninst✝ : Normal F { x // x ∈ E }\nf : { x // x ∈ E } →ₐ[F] K\nthis : Algebra { x // x ∈ E } { x // x ∈ E } := Algebra.id { x // x ∈ E }\ng : { x // x ∈ E } ≃ₐ[F] { x // x ∈ E } := restrictNormal' f { x // x ∈ E }\n⊢ fieldRange f = E",
"state_before": "F : Type u_1\nK : Type u_2\ninst✝¹⁹ : Field F\ninst✝¹⁸ : Field K\ninst✝¹⁷ : Algebra F K\nK₁ : Type ?u.303481\nK₂ : Type ?u.303484\nK₃ : Type ?u.303487\ninst✝¹⁶ : Field F\ninst✝¹⁵ : Field K₁\ninst✝¹⁴ : Field K₂\ninst✝¹³ : Field K₃\ninst✝¹² : Algebra F K₁\ninst✝¹¹ : Algebra F K₂\ninst✝¹⁰ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\nω : K₂ ≃ₐ[F] K₃\nE✝ : Type ?u.303905\ninst✝⁹ : Field E✝\ninst✝⁸ : Algebra F E✝\ninst✝⁷ : Algebra E✝ K₁\ninst✝⁶ : Algebra E✝ K₂\ninst✝⁵ : Algebra E✝ K₃\ninst✝⁴ : IsScalarTower F E✝ K₁\ninst✝³ : IsScalarTower F E✝ K₂\ninst✝² : IsScalarTower F E✝ K₃\ninst✝¹ : Algebra F K\nE : IntermediateField F K\ninst✝ : Normal F { x // x ∈ E }\nf : { x // x ∈ E } →ₐ[F] K\nthis : Algebra { x // x ∈ E } { x // x ∈ E } := Algebra.id { x // x ∈ E }\n⊢ fieldRange f = E",
"tactic": "let g := f.restrictNormal' E"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type u_2\ninst✝¹⁹ : Field F\ninst✝¹⁸ : Field K\ninst✝¹⁷ : Algebra F K\nK₁ : Type ?u.303481\nK₂ : Type ?u.303484\nK₃ : Type ?u.303487\ninst✝¹⁶ : Field F\ninst✝¹⁵ : Field K₁\ninst✝¹⁴ : Field K₂\ninst✝¹³ : Field K₃\ninst✝¹² : Algebra F K₁\ninst✝¹¹ : Algebra F K₂\ninst✝¹⁰ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\nω : K₂ ≃ₐ[F] K₃\nE✝ : Type ?u.303905\ninst✝⁹ : Field E✝\ninst✝⁸ : Algebra F E✝\ninst✝⁷ : Algebra E✝ K₁\ninst✝⁶ : Algebra E✝ K₂\ninst✝⁵ : Algebra E✝ K₃\ninst✝⁴ : IsScalarTower F E✝ K₁\ninst✝³ : IsScalarTower F E✝ K₂\ninst✝² : IsScalarTower F E✝ K₃\ninst✝¹ : Algebra F K\nE : IntermediateField F K\ninst✝ : Normal F { x // x ∈ E }\nf : { x // x ∈ E } →ₐ[F] K\nthis : Algebra { x // x ∈ E } { x // x ∈ E } := Algebra.id { x // x ∈ E }\ng : { x // x ∈ E } ≃ₐ[F] { x // x ∈ E } := restrictNormal' f { x // x ∈ E }\n⊢ fieldRange f = E",
"tactic": "rw [← show E.val.comp ↑g = f from FunLike.ext_iff.mpr (f.restrictNormal_commutes E), ←\n IntermediateField.AlgHom.map_fieldRange, IntermediateField.AlgEquiv.fieldRange_eq_top g,\n ← IntermediateField.AlgHom.fieldRange_eq_map, IntermediateField.fieldRange_val]"
}
] |
[
351,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.linearMap_ext
|
[] |
[
326,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_add_const_Icc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x + a) ⁻¹' Icc b c = Icc (b - a) (c - a)",
"tactic": "simp [← Ici_inter_Iic]"
}
] |
[
109,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
LinearIsometryEquiv.hasFDerivAt
|
[] |
[
274,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
11
] |
Std/Data/List/Basic.lean
|
List.fillNones_eq_fillNonesTR
|
[
{
"state_after": "case h.h.h\nα : Type u_1\nas : List (Option α)\nas' : List α\n⊢ fillNones as as' = fillNonesTR as as'",
"state_before": "⊢ @fillNones = @fillNonesTR",
"tactic": "funext α as as'"
},
{
"state_after": "case h.h.h\nα : Type u_1\nas : List (Option α)\nas' : List α\n⊢ fillNones as as' = fillNonesTR.go as as' #[]",
"state_before": "case h.h.h\nα : Type u_1\nas : List (Option α)\nas' : List α\n⊢ fillNones as as' = fillNonesTR as as'",
"tactic": "simp [fillNonesTR]"
},
{
"state_after": "case h.h.h\nα : Type u_1\nas : List (Option α)\nas' : List α\n⊢ fillNones as as' = fillNonesTR.go as as' #[]",
"state_before": "case h.h.h\nα : Type u_1\nas : List (Option α)\nas' : List α\n⊢ fillNones as as' = fillNonesTR.go as as' #[]",
"tactic": "let rec go (acc) : ∀ as as', @fillNonesTR.go α as as' acc = acc.data ++ as.fillNones as'\n| [], _ => by simp [fillNonesTR.go]\n| some a :: as, as' => by simp [fillNonesTR.go, go _ as as']\n| none :: as, [] => by simp [fillNonesTR.go, reduceOption, filterMap_eq_filterMapTR.go]\n| none :: as, a :: as' => by simp [fillNonesTR.go, go _ as as']"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u_1\nas : List (Option α)\nas' : List α\n⊢ fillNones as as' = fillNonesTR.go as as' #[]",
"tactic": "simp [fillNonesTR, go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas : List (Option α)\nas' : List α\nacc : Array α\nx✝ : List α\n⊢ fillNonesTR.go [] x✝ acc = acc.data ++ fillNones [] x✝",
"tactic": "simp [fillNonesTR.go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas✝ : List (Option α)\nas'✝ : List α\nacc : Array α\na : α\nas : List (Option α)\nas' : List α\n⊢ fillNonesTR.go (some a :: as) as' acc = acc.data ++ fillNones (some a :: as) as'",
"tactic": "simp [fillNonesTR.go, go _ as as']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas✝ : List (Option α)\nas' : List α\nacc : Array α\nas : List (Option α)\n⊢ fillNonesTR.go (none :: as) [] acc = acc.data ++ fillNones (none :: as) []",
"tactic": "simp [fillNonesTR.go, reduceOption, filterMap_eq_filterMapTR.go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas✝ : List (Option α)\nas'✝ : List α\nacc : Array α\nas : List (Option α)\na : α\nas' : List α\n⊢ fillNonesTR.go (none :: as) (a :: as') acc = acc.data ++ fillNones (none :: as) (a :: as')",
"tactic": "simp [fillNonesTR.go, go _ as as']"
}
] |
[
1453,
25
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1446,
10
] |
Std/Data/List/Basic.lean
|
List.erase_eq_eraseTR
|
[
{
"state_after": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\n⊢ List.erase l a = eraseTR l a",
"state_before": "⊢ @List.erase = @eraseTR",
"tactic": "funext α _ l a"
},
{
"state_after": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\n⊢ List.erase l a = eraseTR.go l a l #[]",
"state_before": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\n⊢ List.erase l a = eraseTR l a",
"tactic": "simp [eraseTR]"
},
{
"state_after": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\n⊢ ∀ (xs : List α) (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a",
"state_before": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\n⊢ List.erase l a = eraseTR.go l a l #[]",
"tactic": "suffices ∀ xs acc, l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ xs.erase a from\n (this l #[] (by simp)).symm"
},
{
"state_after": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\nxs : List α\n⊢ ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a",
"state_before": "case h.h.h.h\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\n⊢ ∀ (xs : List α) (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a",
"tactic": "intro xs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx✝ : BEq α\nl : List α\na : α\nthis : ∀ (xs : List α) (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\n⊢ l = #[].data ++ l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.nil\nα : Type u_1\nx✝ : BEq α\nl : List α\na : α\nacc : Array α\nh : l = acc.data ++ []\n⊢ eraseTR.go l a [] acc = acc.data ++ List.erase [] a",
"tactic": "simp [List.erase, eraseTR.go, h]"
},
{
"state_after": "case h.h.h.h.cons\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ (bif x == a then acc.data ++ xs else eraseTR.go l a xs (Array.push acc x)) =\n acc.data ++\n match x == a with\n | true => xs\n | false => x :: List.erase xs a",
"state_before": "case h.h.h.h.cons\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ eraseTR.go l a (x :: xs) acc = acc.data ++ List.erase (x :: xs) a",
"tactic": "simp [List.erase, eraseTR.go]"
},
{
"state_after": "case h.h.h.h.cons.false\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ eraseTR.go l a xs (Array.push acc x) = acc.data ++ x :: List.erase xs a",
"state_before": "case h.h.h.h.cons\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ (bif x == a then acc.data ++ xs else eraseTR.go l a xs (Array.push acc x)) =\n acc.data ++\n match x == a with\n | true => xs\n | false => x :: List.erase xs a",
"tactic": "cases x == a <;> simp"
},
{
"state_after": "case h.h.h.h.cons.false\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ (Array.push acc x).data ++ List.erase xs a = acc.data ++ x :: List.erase xs a\n\ncase h.h.h.h.cons.false.a\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"state_before": "case h.h.h.h.cons.false\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ eraseTR.go l a xs (Array.push acc x) = acc.data ++ x :: List.erase xs a",
"tactic": "rw [IH]"
},
{
"state_after": "case h.h.h.h.cons.false.a\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"state_before": "case h.h.h.h.cons.false\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ (Array.push acc x).data ++ List.erase xs a = acc.data ++ x :: List.erase xs a\n\ncase h.h.h.h.cons.false.a\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"tactic": "simp"
},
{
"state_after": "case h.h.h.h.cons.false.a\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ l = acc.data ++ x :: xs",
"state_before": "case h.h.h.h.cons.false.a\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.cons.false.a\nα : Type u_1\nx✝ : BEq α\nl : List α\na x : α\nxs : List α\nIH : ∀ (acc : Array α), l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ List.erase xs a\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ l = acc.data ++ x :: xs",
"tactic": "exact h"
}
] |
[
50,
35
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
41,
10
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_sInf
|
[
{
"state_after": "case inl\nα β : Type u\n⊢ lift (sInf ∅) = sInf (lift '' ∅)\n\ncase inr\nα β : Type u\ns : Set Cardinal\nhs : Set.Nonempty s\n⊢ lift (sInf s) = sInf (lift '' s)",
"state_before": "α β : Type u\ns : Set Cardinal\n⊢ lift (sInf s) = sInf (lift '' s)",
"tactic": "rcases eq_empty_or_nonempty s with (rfl | hs)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα β : Type u\n⊢ lift (sInf ∅) = sInf (lift '' ∅)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nα β : Type u\ns : Set Cardinal\nhs : Set.Nonempty s\n⊢ lift (sInf s) = sInf (lift '' s)",
"tactic": "exact lift_monotone.map_csInf hs"
}
] |
[
1100,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1097,
1
] |
Mathlib/Topology/Basic.lean
|
interior_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ interior (sᶜ) = closure sᶜ",
"tactic": "simp [closure_eq_compl_interior_compl]"
}
] |
[
552,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.foldlM_append
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nβ : Type u_1\nα : Type u_3\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : β → α → m β\nb : β\nl l' : List α\n⊢ List.foldlM f b (l ++ l') = do\n let init ← List.foldlM f b l\n List.foldlM f init l'",
"tactic": "induction l generalizing b <;> simp [*]"
}
] |
[
164,
42
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
162,
9
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.det_nonsing_inv
|
[
{
"state_after": "case pos\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\n⊢ det A⁻¹ = Ring.inverse (det A)\n\ncase neg\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : ¬IsUnit (det A)\n⊢ det A⁻¹ = Ring.inverse (det A)",
"state_before": "l : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "by_cases h : IsUnit A.det"
},
{
"state_after": "case neg.inl\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : ¬IsUnit (det A)\nh✝ : IsEmpty n\n⊢ det A⁻¹ = Ring.inverse (det A)\n\ncase neg.inr\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : ¬IsUnit (det A)\nh✝ : Nonempty n\n⊢ det A⁻¹ = Ring.inverse (det A)",
"state_before": "case neg\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : ¬IsUnit (det A)\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "cases isEmpty_or_nonempty n"
},
{
"state_after": "case pos.intro\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\nval✝ : Invertible (det A)\n⊢ det A⁻¹ = Ring.inverse (det A)",
"state_before": "case pos\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "cases h.nonempty_invertible"
},
{
"state_after": "case pos.intro\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\nval✝ : Invertible (det A)\nthis : Invertible A := invertibleOfDetInvertible A\n⊢ det A⁻¹ = Ring.inverse (det A)",
"state_before": "case pos.intro\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\nval✝ : Invertible (det A)\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "letI := invertibleOfDetInvertible A"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : IsUnit (det A)\nval✝ : Invertible (det A)\nthis : Invertible A := invertibleOfDetInvertible A\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "rw [Ring.inverse_invertible, ← invOf_eq_nonsing_inv, det_invOf]"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : ¬IsUnit (det A)\nh✝ : IsEmpty n\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "rw [det_isEmpty, det_isEmpty, Ring.inverse_one]"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nl : Type ?u.242378\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nh : ¬IsUnit (det A)\nh✝ : Nonempty n\n⊢ det A⁻¹ = Ring.inverse (det A)",
"tactic": "rw [Ring.inverse_non_unit _ h, nonsing_inv_apply_not_isUnit _ h, det_zero ‹_›]"
}
] |
[
404,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.monotone_principal
|
[] |
[
658,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Data/Multiset/Functor.lean
|
Multiset.pure_def
|
[] |
[
73,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEFinStronglyMeasurable.aemeasurable
|
[] |
[
1866,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1863,
11
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Bounded.trans_right
|
[] |
[
964,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.finSuccEquiv_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.896628\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\n⊢ ↑(finSuccEquiv R n) p =\n ↑(eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (X k)) i) p",
"tactic": "rw [← finSuccEquiv_eq, RingHom.coe_coe]"
}
] |
[
339,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Data/Int/Parity.lean
|
Int.odd_sub
|
[
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ Odd (m - n) ↔ (Odd m ↔ Even n)",
"tactic": "rw [odd_iff_not_even, even_sub, not_iff, odd_iff_not_even]"
}
] |
[
189,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
PFunctor.comp_map
|
[] |
[
72,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
11
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_insert
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.1235297\nR : Type ?u.1235300\nK : Type u_1\nM : Type ?u.1235306\nM' : Type ?u.1235309\nM'' : Type ?u.1235312\nV : Type u\nV' : Type ?u.1235317\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nhxs : ¬x ∈ s\n⊢ (LinearIndependent K fun x => id ↑x) ∧ ¬id x ∈ span K (id '' s) ↔ (LinearIndependent K fun b => ↑b) ∧ ¬x ∈ span K s",
"tactic": "simp"
}
] |
[
1221,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1218,
1
] |
Mathlib/Algebra/Homology/Additive.lean
|
HomologicalComplex.nsmul_f_apply
|
[] |
[
89,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.Quotient.mk''_eq_mk
|
[] |
[
73,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean
|
ZMod.isQuadratic_χ₄
|
[
{
"state_after": "a : ZMod 4\n⊢ ↑χ₄ a = 0 ∨ ↑χ₄ a = 1 ∨ ↑χ₄ a = -1",
"state_before": "⊢ MulChar.IsQuadratic χ₄",
"tactic": "intro a"
},
{
"state_after": "case head\n\n⊢ ↑χ₄ { val := 0, isLt := (_ : 0 < 3 + 1) } = 0 ∨\n ↑χ₄ { val := 0, isLt := (_ : 0 < 3 + 1) } = 1 ∨ ↑χ₄ { val := 0, isLt := (_ : 0 < 3 + 1) } = -1\n\ncase tail.head\n\n⊢ ↑χ₄ { val := 1, isLt := (_ : (fun a => a < 3 + 1) 1) } = 0 ∨\n ↑χ₄ { val := 1, isLt := (_ : (fun a => a < 3 + 1) 1) } = 1 ∨\n ↑χ₄ { val := 1, isLt := (_ : (fun a => a < 3 + 1) 1) } = -1\n\ncase tail.tail.head\n\n⊢ ↑χ₄ { val := 2, isLt := (_ : (fun a => (fun a => a < 3 + 1) a) 2) } = 0 ∨\n ↑χ₄ { val := 2, isLt := (_ : (fun a => (fun a => a < 3 + 1) a) 2) } = 1 ∨\n ↑χ₄ { val := 2, isLt := (_ : (fun a => (fun a => a < 3 + 1) a) 2) } = -1\n\ncase tail.tail.tail.head\n\n⊢ ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = 0 ∨\n ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = 1 ∨\n ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = -1",
"state_before": "a : ZMod 4\n⊢ ↑χ₄ a = 0 ∨ ↑χ₄ a = 1 ∨ ↑χ₄ a = -1",
"tactic": "fin_cases a"
},
{
"state_after": "no goals",
"state_before": "case head\n\n⊢ ↑χ₄ { val := 0, isLt := (_ : 0 < 3 + 1) } = 0 ∨\n ↑χ₄ { val := 0, isLt := (_ : 0 < 3 + 1) } = 1 ∨ ↑χ₄ { val := 0, isLt := (_ : 0 < 3 + 1) } = -1\n\ncase tail.head\n\n⊢ ↑χ₄ { val := 1, isLt := (_ : (fun a => a < 3 + 1) 1) } = 0 ∨\n ↑χ₄ { val := 1, isLt := (_ : (fun a => a < 3 + 1) 1) } = 1 ∨\n ↑χ₄ { val := 1, isLt := (_ : (fun a => a < 3 + 1) 1) } = -1\n\ncase tail.tail.head\n\n⊢ ↑χ₄ { val := 2, isLt := (_ : (fun a => (fun a => a < 3 + 1) a) 2) } = 0 ∨\n ↑χ₄ { val := 2, isLt := (_ : (fun a => (fun a => a < 3 + 1) a) 2) } = 1 ∨\n ↑χ₄ { val := 2, isLt := (_ : (fun a => (fun a => a < 3 + 1) a) 2) } = -1\n\ncase tail.tail.tail.head\n\n⊢ ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = 0 ∨\n ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = 1 ∨\n ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = -1",
"tactic": "all_goals decide"
},
{
"state_after": "no goals",
"state_before": "case tail.tail.tail.head\n\n⊢ ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = 0 ∨\n ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = 1 ∨\n ↑χ₄ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 3 + 1) a) a) 3) } = -1",
"tactic": "decide"
}
] |
[
55,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.Disjoint.cycleType
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : Disjoint σ τ\n⊢ Multiset.Disjoint (cycleFactorsFinset σ).val (cycleFactorsFinset τ).val",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : Disjoint σ τ\n⊢ Perm.cycleType (σ * τ) = Perm.cycleType σ + Perm.cycleType τ",
"tactic": "rw [cycleType_def, cycleType_def, cycleType_def, h.cycleFactorsFinset_mul_eq_union, ←\n Multiset.map_add, Finset.union_val, Multiset.add_eq_union_iff_disjoint.mpr _]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : Disjoint σ τ\n⊢ Multiset.Disjoint (cycleFactorsFinset σ).val (cycleFactorsFinset τ).val",
"tactic": "exact Finset.disjoint_val.2 h.disjoint_cycleFactorsFinset"
}
] |
[
128,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.sin_add_pi
|
[] |
[
1145,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasDerivAt.unique
|
[] |
[
399,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.subset_infs
|
[] |
[
355,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infiniteNeg_neg
|
[] |
[
468,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
9
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
HasFTaylorSeriesUpToOn.hasStrictFDerivAt
|
[] |
[
1943,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1939,
1
] |
src/lean/Init/Prelude.lean
|
Or.elim
|
[] |
[
536,
24
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
533,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
isCoprime_mul_unit_right_right
|
[] |
[
269,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieSubmodule.lcs_succ
|
[] |
[
71,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
preimage_equivalence
|
[] |
[
221,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.wideCospan_map_term
|
[] |
[
586,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Topology/Separation.lean
|
Dense.diff_finite
|
[
{
"state_after": "case h.e'_3.h.e'_4\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : ∀ (x : α), NeBot (𝓝[{x}ᶜ] x)\ns : Set α\nhs : Dense s\nt : Set α\nht : Set.Finite t\n⊢ t = ↑(Finite.toFinset ht)",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : ∀ (x : α), NeBot (𝓝[{x}ᶜ] x)\ns : Set α\nhs : Dense s\nt : Set α\nht : Set.Finite t\n⊢ Dense (s \\ t)",
"tactic": "convert hs.diff_finset ht.toFinset"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_4\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : ∀ (x : α), NeBot (𝓝[{x}ᶜ] x)\ns : Set α\nhs : Dense s\nt : Set α\nht : Set.Finite t\n⊢ t = ↑(Finite.toFinset ht)",
"tactic": "exact (Finite.coe_toFinset _).symm"
}
] |
[
731,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/RingTheory/Adjoin/Basic.lean
|
Algebra.adjoin_univ
|
[] |
[
159,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
ContinuousAt.arsinh
|
[] |
[
220,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
8
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.