file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Finset/Image.lean
|
Finset.mem_map_equiv
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\nb : β\n⊢ (∃ a, a ∈ s ∧ ↑(Equiv.toEmbedding f) a = b) ↔ ↑f.symm b ∈ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\nb : β\n⊢ b ∈ map (Equiv.toEmbedding f) s ↔ ↑f.symm b ∈ s",
"tactic": "rw [mem_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\nb : β\n⊢ (∃ a, a ∈ s ∧ ↑(Equiv.toEmbedding f) a = b) ↔ ↑f.symm b ∈ s",
"tactic": "exact\n ⟨by\n rintro ⟨a, H, rfl⟩\n simpa, fun h => ⟨_, h, by simp⟩⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\na : α\nH : a ∈ s\n⊢ ↑f.symm (↑(Equiv.toEmbedding f) a) ∈ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\nb : β\n⊢ (∃ a, a ∈ s ∧ ↑(Equiv.toEmbedding f) a = b) → ↑f.symm b ∈ s",
"tactic": "rintro ⟨a, H, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\na : α\nH : a ∈ s\n⊢ ↑f.symm (↑(Equiv.toEmbedding f) a) ∈ s",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3584\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\nb : β\nh : ↑f.symm b ∈ s\n⊢ ↑(Equiv.toEmbedding f) (↑f.symm b) = b",
"tactic": "simp"
}
] |
[
80,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhdsWithin_Ioi_neBot
|
[] |
[
2388,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2387,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.disjoint_atTop_atBot
|
[] |
[
129,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/List/Lattice.lean
|
List.inter_subset_left
|
[] |
[
171,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.coe_mk
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_subtype
|
[] |
[
1198,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.toFinset_eq_univ
|
[] |
[
288,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
11
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.isOpen_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.209202\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\n⊢ IsOpen s ↔ ∀ (x : α), x ∈ s → ∃ ε, ε > 0 ∧ ball x ε ⊆ s",
"tactic": "simp [isOpen_iff_nhds, mem_nhds_iff]"
}
] |
[
687,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean
|
Polynomial.natDegree_bit1
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na✝ b : R\nm n : ℕ\ninst✝ : Semiring R\np q r a : R[X]\n⊢ max (natDegree (bit0 a)) (natDegree 1) ≤ natDegree a",
"tactic": "simp [natDegree_bit0]"
}
] |
[
263,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Analysis/SpecialFunctions/CompareExp.lean
|
Complex.IsExpCmpFilter.isLittleO_log_abs_re
|
[
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"tactic": "have h2 : 0 < Real.sqrt 2 := by simp"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"tactic": "have hz' : 1 ≤ abs z := hz.trans (re_le_abs z)"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"tactic": "have _ : 0 < abs z := one_pos.trans_le hz'"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"tactic": "have hm₀ : 0 < max z.re (|z.im|) := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ Real.log (↑abs z) ≤ ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ ‖Real.log (↑abs z)‖ ≤ 1 * ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"tactic": "rw [one_mul, Real.norm_eq_abs, _root_.abs_of_nonneg (Real.log_nonneg hz')]"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ Real.log (↑abs z) ≤ Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ Real.log (↑abs z) ≤ ‖Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))‖",
"tactic": "refine' le_trans _ (le_abs_self _)"
},
{
"state_after": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ ↑abs z ≤ Real.sqrt 2 * max (Abs.abs z.re) (Abs.abs z.im)\n\ncase h\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ 0 < ↑abs z\n\ncase h₁\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ 0 < Real.sqrt 2 * max z.re (Abs.abs z.im)\n\ncase hx\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ Real.sqrt 2 ≠ 0\n\ncase hy\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ max z.re (Abs.abs z.im) ≠ 0",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ Real.log (↑abs z) ≤ Real.log (Real.sqrt 2) + Real.log (max z.re (Abs.abs z.im))",
"tactic": "rw [← Real.log_mul, Real.log_le_log, ← _root_.abs_of_nonneg (le_trans zero_le_one hz)]"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ ↑abs z ≤ Real.sqrt 2 * max (Abs.abs z.re) (Abs.abs z.im)\n\ncase h\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ 0 < ↑abs z\n\ncase h₁\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ 0 < Real.sqrt 2 * max z.re (Abs.abs z.im)\n\ncase hx\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ Real.sqrt 2 ≠ 0\n\ncase hy\nl : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 ≤ ↑abs z\nx✝ : 0 < ↑abs z\nhm₀ : 0 < max z.re (Abs.abs z.im)\n⊢ max z.re (Abs.abs z.im) ≠ 0",
"tactic": "exacts [abs_le_sqrt_two_mul_max z, one_pos.trans_le hz', mul_pos h2 hm₀, h2.ne', hm₀.ne']"
},
{
"state_after": "no goals",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nz : ℂ\nhz : 1 ≤ z.re\n⊢ 0 < Real.sqrt 2",
"tactic": "simp"
},
{
"state_after": "case h\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"state_before": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\n⊢ ∀ᶠ (x : ℂ) in l, ↑n * ‖Real.log (max x.re (Abs.abs x.im))‖ ≤ ‖x.re‖",
"tactic": "filter_upwards [isLittleO_iff_nat_mul_le'.1 hl.isLittleO_log_re_re n,\n hl.abs_im_pow_eventuallyLE_exp_re n,\n hl.tendsto_re.eventually_gt_atTop 1] with z hre him h₁"
},
{
"state_after": "case h.inl\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : Abs.abs z.im ≤ z.re\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖\n\ncase h.inr\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : z.re ≤ Abs.abs z.im\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"state_before": "case h\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"tactic": "cases' le_total (|z.im|) z.re with hle hle"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : Abs.abs z.im ≤ z.re\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"tactic": "rwa [max_eq_left hle]"
},
{
"state_after": "case h.inr\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : z.re ≤ Abs.abs z.im\nH : 1 < Abs.abs z.im\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"state_before": "case h.inr\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : z.re ≤ Abs.abs z.im\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"tactic": "have H : 1 < |z.im| := h₁.trans_le hle"
},
{
"state_after": "case h.inr\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhle : z.re ≤ Abs.abs z.im\nhim : Abs.abs z.im ^ n ≤ Real.exp z.re\nh₁ : 1 < z.re\nH : 1 < Abs.abs z.im\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"state_before": "case h.inr\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : Abs.abs z.im ^ ↑n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : z.re ≤ Abs.abs z.im\nH : 1 < Abs.abs z.im\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"tactic": "norm_cast at *"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhle : z.re ≤ Abs.abs z.im\nhim : Abs.abs z.im ^ n ≤ Real.exp z.re\nh₁ : 1 < z.re\nH : 1 < Abs.abs z.im\n⊢ ↑n * ‖Real.log (max z.re (Abs.abs z.im))‖ ≤ ‖z.re‖",
"tactic": "rwa [max_eq_right hle, Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos (Real.log_pos H),\n ← Real.log_pow, Real.log_le_iff_le_exp (pow_pos (one_pos.trans H) _),\n abs_of_pos (one_pos.trans h₁)]"
}
] |
[
163,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Data/Fintype/Lattice.lean
|
Finite.exists_max
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x ≤ f x₀",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\n⊢ ∃ x₀, ∀ (x : α), f x ≤ f x₀",
"tactic": "cases nonempty_fintype α"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x ≤ f x₀",
"tactic": "simpa using exists_max_image univ f univ_nonempty"
}
] |
[
65,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.image.map_id
|
[
{
"state_after": "case e_self\nC : Type u\ninst✝⁴ : Category C\nf g : Arrow C\ninst✝³ : HasImage f.hom\ninst✝² : HasImage g.hom\nsq : f ⟶ g\ninst✝¹ : HasImageMap sq\ninst✝ : HasImageMap (𝟙 f)\n⊢ HasImageMap.imageMap (𝟙 f) = imageMapId f",
"state_before": "C : Type u\ninst✝⁴ : Category C\nf g : Arrow C\ninst✝³ : HasImage f.hom\ninst✝² : HasImage g.hom\nsq : f ⟶ g\ninst✝¹ : HasImageMap sq\ninst✝ : HasImageMap (𝟙 f)\n⊢ (HasImageMap.imageMap (𝟙 f)).map = (imageMapId f).map",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_self\nC : Type u\ninst✝⁴ : Category C\nf g : Arrow C\ninst✝³ : HasImage f.hom\ninst✝² : HasImage g.hom\nsq : f ⟶ g\ninst✝¹ : HasImageMap sq\ninst✝ : HasImageMap (𝟙 f)\n⊢ HasImageMap.imageMap (𝟙 f) = imageMapId f",
"tactic": "simp only [eq_iff_true_of_subsingleton]"
}
] |
[
843,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.tensorTensorTensorAssoc_symm_tmul
|
[] |
[
966,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
deriv_neg
|
[] |
[
263,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
tendsto_int_cast_atTop_iff
|
[
{
"state_after": "α : Type u_2\nR : Type u_1\ninst✝¹ : StrictOrderedRing R\ninst✝ : Archimedean R\nf : α → ℤ\nl : Filter α\n⊢ Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto (Int.cast ∘ f) l atTop",
"state_before": "α : Type u_2\nR : Type u_1\ninst✝¹ : StrictOrderedRing R\ninst✝ : Archimedean R\nf : α → ℤ\nl : Filter α\n⊢ Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto f l atTop",
"tactic": "rw [← @Int.comap_cast_atTop R, tendsto_comap_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\ninst✝¹ : StrictOrderedRing R\ninst✝ : Archimedean R\nf : α → ℤ\nl : Filter α\n⊢ Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto (Int.cast ∘ f) l atTop",
"tactic": "rfl"
}
] |
[
61,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.limZero_congr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : LimZero f\n⊢ LimZero g",
"tactic": "simpa using add_limZero (Setoid.symm h) l"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : LimZero g\n⊢ LimZero f",
"tactic": "simpa using add_limZero h l"
}
] |
[
495,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval_smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝³ : Semiring R\np✝ q r : R[X]\nx✝ : R\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S R\ninst✝ : IsScalarTower S R R\ns : S\np : R[X]\nx : R\n⊢ eval x (s • p) = s • eval x p",
"tactic": "rw [← smul_one_smul R s p, eval, eval₂_smul, RingHom.id_apply, smul_one_mul]"
}
] |
[
400,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Data/Matrix/Basis.lean
|
Matrix.StdBasisMatrix.diag_zero
|
[] |
[
153,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Std/Logic.lean
|
exists_congr
|
[] |
[
377,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
376,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_limsup
|
[] |
[
1341,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1339,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.bijOn_cos
|
[] |
[
646,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.symm_map_nhds_eq
|
[] |
[
1973,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1972,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.AddValuation.map_add
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\n⊢ min (addValuationDef x) (addValuationDef y) ≤ addValuationDef (x + y)",
"tactic": "simp only [addValuationDef]"
},
{
"state_after": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))\n\ncase neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "by_cases hxy : x + y = 0"
},
{
"state_after": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤ ⊤",
"state_before": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "rw [hxy, if_pos (Eq.refl _)]"
},
{
"state_after": "no goals",
"state_before": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤ ⊤",
"tactic": "exact le_top"
},
{
"state_after": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : x = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))\n\ncase neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"state_before": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "by_cases hx : x = 0"
},
{
"state_after": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : x = 0\n⊢ (if y = 0 then ⊤ else ↑(valuation y)) ≤ ⊤",
"state_before": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : x = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "rw [hx, if_pos (Eq.refl _), min_eq_right, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : x = 0\n⊢ (if y = 0 then ⊤ else ↑(valuation y)) ≤ ⊤",
"tactic": "exact le_top"
},
{
"state_after": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))\n\ncase neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"state_before": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "by_cases hy : y = 0"
},
{
"state_after": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : y = 0\n⊢ (if x = 0 then ⊤ else ↑(valuation x)) ≤ ⊤",
"state_before": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "rw [hy, if_pos (Eq.refl _), min_eq_left, add_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : y = 0\n⊢ (if x = 0 then ⊤ else ↑(valuation x)) ≤ ⊤",
"tactic": "exact le_top"
},
{
"state_after": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ min (valuation x) (valuation y) ≤ valuation (x + y)",
"state_before": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ min (if x = 0 then ⊤ else ↑(valuation x)) (if y = 0 then ⊤ else ↑(valuation y)) ≤\n if x + y = 0 then ⊤ else ↑(valuation (x + y))",
"tactic": "rw [if_neg hx, if_neg hy, if_neg hxy, ← WithTop.coe_min, WithTop.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ min (valuation x) (valuation y) ≤ valuation (x + y)",
"tactic": "exact valuation_map_add hxy"
}
] |
[
1144,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1131,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean
|
StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin
|
[] |
[
192,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.contDiffOn_log
|
[
{
"state_after": "x : ℝ\nn : ℕ∞\nthis : ContDiffOn ℝ ⊤ log ({0}ᶜ)\n⊢ ContDiffOn ℝ n log ({0}ᶜ)\n\ncase this\nx : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log ({0}ᶜ)",
"state_before": "x : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ n log ({0}ᶜ)",
"tactic": "suffices : ContDiffOn ℝ ⊤ log ({0}ᶜ)"
},
{
"state_after": "case this\nx : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log ({0}ᶜ)",
"state_before": "x : ℝ\nn : ℕ∞\nthis : ContDiffOn ℝ ⊤ log ({0}ᶜ)\n⊢ ContDiffOn ℝ n log ({0}ᶜ)\n\ncase this\nx : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log ({0}ᶜ)",
"tactic": "exact this.of_le le_top"
},
{
"state_after": "case this\nx : ℝ\nn : ℕ∞\n⊢ DifferentiableOn ℝ log ({0}ᶜ) ∧ ContDiffOn ℝ ⊤ (deriv log) ({0}ᶜ)",
"state_before": "case this\nx : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log ({0}ᶜ)",
"tactic": "refine' (contDiffOn_top_iff_deriv_of_open isOpen_compl_singleton).2 _"
},
{
"state_after": "no goals",
"state_before": "case this\nx : ℝ\nn : ℕ∞\n⊢ DifferentiableOn ℝ log ({0}ᶜ) ∧ ContDiffOn ℝ ⊤ (deriv log) ({0}ᶜ)",
"tactic": "simp [differentiableOn_log, contDiffOn_inv]"
}
] |
[
84,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.primPart_zero
|
[] |
[
264,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_Ioc
|
[] |
[
771,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Std/Data/Int/DivMod.lean
|
Int.zero_fmod
|
[
{
"state_after": "no goals",
"state_before": "b : Int\n⊢ fmod 0 b = 0",
"tactic": "cases b <;> rfl"
}
] |
[
250,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
250,
9
] |
Mathlib/Topology/LocalAtTarget.lean
|
isOpen_iff_coe_preimage_of_iSup_eq_top
|
[
{
"state_after": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\n⊢ (∀ (i : ι), IsOpen (s ∩ ↑(U i))) ↔ ∀ (i : ι), IsOpen (Subtype.val ⁻¹' s)",
"state_before": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\n⊢ IsOpen s ↔ ∀ (i : ι), IsOpen (Subtype.val ⁻¹' s)",
"tactic": "rw [isOpen_iff_inter_of_iSup_eq_top hU s]"
},
{
"state_after": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\ni : ι\n⊢ IsOpen (s ∩ ↑(U i)) ↔ IsOpen (Subtype.val ⁻¹' s)",
"state_before": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\n⊢ (∀ (i : ι), IsOpen (s ∩ ↑(U i))) ↔ ∀ (i : ι), IsOpen (Subtype.val ⁻¹' s)",
"tactic": "refine forall_congr' fun i => ?_"
},
{
"state_after": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\ni : ι\n⊢ IsOpen (s ∩ ↑(U i)) ↔ IsOpen (Subtype.val '' (Subtype.val ⁻¹' s))",
"state_before": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\ni : ι\n⊢ IsOpen (s ∩ ↑(U i)) ↔ IsOpen (Subtype.val ⁻¹' s)",
"tactic": "rw [(U _).2.openEmbedding_subtype_val.open_iff_image_open]"
},
{
"state_after": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\ni : ι\n⊢ IsOpen (s ∩ ↑(U i)) ↔ IsOpen (s ∩ range Subtype.val)",
"state_before": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\ni : ι\n⊢ IsOpen (s ∩ ↑(U i)) ↔ IsOpen (Subtype.val '' (Subtype.val ⁻¹' s))",
"tactic": "erw [Set.image_preimage_eq_inter_range]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.17624\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.17643\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\ni : ι\n⊢ IsOpen (s ∩ ↑(U i)) ↔ IsOpen (s ∩ range Subtype.val)",
"tactic": "rw [Subtype.range_coe, Opens.carrier_eq_coe]"
}
] |
[
100,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.map_fract
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : LinearOrderedRing β\ninst✝² : FloorRing α\ninst✝¹ : FloorRing β\ninst✝ : RingHomClass F α β\na✝ : α\nb : β\nf : F\nhf : StrictMono ↑f\na : α\n⊢ fract (↑f a) = ↑f (fract a)",
"tactic": "simp_rw [fract, map_sub, map_intCast, map_floor _ hf]"
}
] |
[
1543,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1542,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_sets
|
[] |
[
118,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
11
] |
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.haarContent_apply
|
[] |
[
560,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.inits_eq
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ ∀ (n : ℕ), nth (inits s) n = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) n",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ inits s = [head s] :: map (List.cons (head s)) (inits (tail s))",
"tactic": "apply Stream'.ext"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn : ℕ\n⊢ nth (inits s) n = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) n",
"state_before": "case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ ∀ (n : ℕ), nth (inits s) n = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) n",
"tactic": "intro n"
},
{
"state_after": "case a.zero\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ nth (inits s) zero = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) zero\n\ncase a.succ\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn✝ : ℕ\n⊢ nth (inits s) (succ n✝) = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) (succ n✝)",
"state_before": "case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn : ℕ\n⊢ nth (inits s) n = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case a.zero\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ nth (inits s) zero = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) zero",
"tactic": "rfl"
},
{
"state_after": "case a.succ\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn✝ : ℕ\n⊢ take (succ (succ n✝)) s = head s :: take (succ n✝) (tail s)",
"state_before": "case a.succ\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn✝ : ℕ\n⊢ nth (inits s) (succ n✝) = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) (succ n✝)",
"tactic": "rw [nth_inits, nth_succ, tail_cons, nth_map, nth_inits]"
},
{
"state_after": "no goals",
"state_before": "case a.succ\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn✝ : ℕ\n⊢ take (succ (succ n✝)) s = head s :: take (succ n✝) (tail s)",
"tactic": "rfl"
}
] |
[
728,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
722,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_range_eq_sUnion
|
[
{
"state_after": "case h\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\n⊢ (x ∈ ⋃ (y : β), range fun s => ↑(f s y)) ↔ x ∈ ⋃₀ C",
"state_before": "α✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\n⊢ (⋃ (y : β), range fun s => ↑(f s y)) = ⋃₀ C",
"tactic": "ext x"
},
{
"state_after": "case h.mp\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\n⊢ (x ∈ ⋃ (y : β), range fun s => ↑(f s y)) → x ∈ ⋃₀ C\n\ncase h.mpr\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\n⊢ x ∈ ⋃₀ C → x ∈ ⋃ (y : β), range fun s => ↑(f s y)",
"state_before": "case h\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\n⊢ (x ∈ ⋃ (y : β), range fun s => ↑(f s y)) ↔ x ∈ ⋃₀ C",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro.intro.intro.mk\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\ny : β\ns : Set α\nhs : s ∈ C\n⊢ (fun s => ↑(f s y)) { val := s, property := hs } ∈ ⋃₀ C",
"state_before": "case h.mp\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\n⊢ (x ∈ ⋃ (y : β), range fun s => ↑(f s y)) → x ∈ ⋃₀ C",
"tactic": "rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩"
},
{
"state_after": "case h.mp.intro.intro.intro.intro.mk\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\ny : β\ns : Set α\nhs : s ∈ C\n⊢ (fun s => ↑(f s y)) { val := s, property := hs } ∈ s",
"state_before": "case h.mp.intro.intro.intro.intro.mk\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\ny : β\ns : Set α\nhs : s ∈ C\n⊢ (fun s => ↑(f s y)) { val := s, property := hs } ∈ ⋃₀ C",
"tactic": "refine' ⟨_, hs, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.intro.mk\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\ny : β\ns : Set α\nhs : s ∈ C\n⊢ (fun s => ↑(f s y)) { val := s, property := hs } ∈ s",
"tactic": "exact (f ⟨s, hs⟩ y).2"
},
{
"state_after": "case h.mpr.intro.intro\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\ns : Set α\nhs : s ∈ C\nhx : x ∈ s\n⊢ x ∈ ⋃ (y : β), range fun s => ↑(f s y)",
"state_before": "case h.mpr\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\n⊢ x ∈ ⋃₀ C → x ∈ ⋃ (y : β), range fun s => ↑(f s y)",
"tactic": "rintro ⟨s, hs, hx⟩"
},
{
"state_after": "case h.mpr.intro.intro.intro\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\ns : Set α\nhs : s ∈ C\nhx : x ∈ s\ny : β\nhy : f { val := s, property := hs } y = { val := x, property := hx }\n⊢ x ∈ ⋃ (y : β), range fun s => ↑(f s y)",
"state_before": "case h.mpr.intro.intro\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\ns : Set α\nhs : s ∈ C\nhx : x ∈ s\n⊢ x ∈ ⋃ (y : β), range fun s => ↑(f s y)",
"tactic": "cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy"
},
{
"state_after": "case h.mpr.intro.intro.intro\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\ns : Set α\nhs : s ∈ C\nhx : x ∈ s\ny : β\nhy : f { val := s, property := hs } y = { val := x, property := hx }\n⊢ (fun s => ↑(f s y)) { val := s, property := hs } = x",
"state_before": "case h.mpr.intro.intro.intro\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\ns : Set α\nhs : s ∈ C\nhx : x ∈ s\ny : β\nhy : f { val := s, property := hs } y = { val := x, property := hx }\n⊢ x ∈ ⋃ (y : β), range fun s => ↑(f s y)",
"tactic": "refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.intro\nα✝ : Type ?u.180126\nβ✝ : Type ?u.180129\nγ : Type ?u.180132\nι : Sort ?u.180135\nι' : Sort ?u.180138\nι₂ : Sort ?u.180141\nκ : ι → Sort ?u.180146\nκ₁ : ι → Sort ?u.180151\nκ₂ : ι → Sort ?u.180156\nκ' : ι' → Sort ?u.180161\nα : Type u_1\nβ : Type u_2\nC : Set (Set α)\nf : (s : ↑C) → β → ↑↑s\nhf : ∀ (s : ↑C), Surjective (f s)\nx : α\ns : Set α\nhs : s ∈ C\nhx : x ∈ s\ny : β\nhy : f { val := s, property := hs } y = { val := x, property := hx }\n⊢ (fun s => ↑(f s y)) { val := s, property := hs } = x",
"tactic": "exact congr_arg Subtype.val hy"
}
] |
[
1384,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1375,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y = 0\n⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y = 0\n⊢ ‖x‖ / Real.cos (angle x (x + y)) = ‖x + y‖",
"tactic": "rw [cos_angle_add_of_inner_eq_zero h]"
},
{
"state_after": "case inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0\n⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖\n\ncase inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : y = 0\n⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y = 0\n⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖",
"tactic": "rcases h0 with (h0 | h0)"
},
{
"state_after": "no goals",
"state_before": "case inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0\n⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖",
"tactic": "rw [div_div_eq_mul_div, mul_comm, div_eq_mul_inv, mul_inv_cancel_right₀ (norm_ne_zero_iff.2 h0)]"
},
{
"state_after": "no goals",
"state_before": "case inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : y = 0\n⊢ ‖x‖ / (‖x‖ / ‖x + y‖) = ‖x + y‖",
"tactic": "simp [h0]"
}
] |
[
209,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Finset.coe_Ici
|
[] |
[
394,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Data/MvPolynomial/Supported.lean
|
MvPolynomial.supported_eq_range_rename
|
[
{
"state_after": "σ : Type u_1\nτ : Type ?u.560\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Set σ\n⊢ AlgHom.range (aeval fun x => X ↑x) = AlgHom.range (aeval (X ∘ Subtype.val))",
"state_before": "σ : Type u_1\nτ : Type ?u.560\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Set σ\n⊢ supported R s = AlgHom.range (rename Subtype.val)",
"tactic": "rw [supported, Set.image_eq_range, adjoin_range_eq_range_aeval, rename]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type ?u.560\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : Set σ\n⊢ AlgHom.range (aeval fun x => X ↑x) = AlgHom.range (aeval (X ∘ Subtype.val))",
"tactic": "congr"
}
] |
[
52,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.normSq_div
|
[] |
[
1397,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1396,
1
] |
Mathlib/Topology/Separation.lean
|
nhds_inter_eq_singleton_of_mem_discrete
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ns : Set α\ninst✝ : DiscreteTopology ↑s\nx : α\nhx : x ∈ s\n⊢ ∃ U, U ∈ 𝓝 x ∧ U ∩ s = {x}",
"tactic": "simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx"
}
] |
[
838,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
836,
1
] |
Mathlib/Logic/Equiv/Option.lean
|
Equiv.coe_optionSubtype_apply_apply
|
[] |
[
228,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sub_eq_of_eq_add_rev
|
[] |
[
1132,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1131,
11
] |
Mathlib/Order/Heyting/Boundary.lean
|
Coheyting.boundary_le_boundary_sup_sup_boundary_inf_right
|
[
{
"state_after": "α : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ∂ b ≤ ∂ (b ⊔ a) ⊔ ∂ (b ⊓ a)",
"state_before": "α : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ∂ b ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b)",
"tactic": "rw [@sup_comm _ _ a, inf_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ∂ b ≤ ∂ (b ⊔ a) ⊔ ∂ (b ⊓ a)",
"tactic": "exact boundary_le_boundary_sup_sup_boundary_inf_left"
}
] |
[
125,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.count_normalizedFactors_eq'
|
[
{
"state_after": "case inl\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nn : ℕ\nhnorm : ↑normalize 0 = 0\nhle : 0 ^ n ∣ x\nhlt : ¬0 ^ (n + 1) ∣ x\n⊢ count 0 (normalizedFactors x) = n\n\ncase inr\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\np x : R\nhnorm : ↑normalize p = p\nn : ℕ\nhle : p ^ n ∣ x\nhlt : ¬p ^ (n + 1) ∣ x\nhp : Irreducible p\n⊢ count p (normalizedFactors x) = n",
"state_before": "α : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\np x : R\nhp : p = 0 ∨ Irreducible p\nhnorm : ↑normalize p = p\nn : ℕ\nhle : p ^ n ∣ x\nhlt : ¬p ^ (n + 1) ∣ x\n⊢ count p (normalizedFactors x) = n",
"tactic": "rcases hp with (rfl | hp)"
},
{
"state_after": "case inl.zero\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nhnorm : ↑normalize 0 = 0\nhle : 0 ^ Nat.zero ∣ x\nhlt : ¬0 ^ (Nat.zero + 1) ∣ x\n⊢ count 0 (normalizedFactors x) = Nat.zero\n\ncase inl.succ\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nhnorm : ↑normalize 0 = 0\nn✝ : ℕ\nhle : 0 ^ Nat.succ n✝ ∣ x\nhlt : ¬0 ^ (Nat.succ n✝ + 1) ∣ x\n⊢ count 0 (normalizedFactors x) = Nat.succ n✝",
"state_before": "case inl\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nn : ℕ\nhnorm : ↑normalize 0 = 0\nhle : 0 ^ n ∣ x\nhlt : ¬0 ^ (n + 1) ∣ x\n⊢ count 0 (normalizedFactors x) = n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case inl.zero\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nhnorm : ↑normalize 0 = 0\nhle : 0 ^ Nat.zero ∣ x\nhlt : ¬0 ^ (Nat.zero + 1) ∣ x\n⊢ count 0 (normalizedFactors x) = Nat.zero",
"tactic": "exact count_eq_zero.2 (zero_not_mem_normalizedFactors _)"
},
{
"state_after": "case inl.succ\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nhnorm : ↑normalize 0 = 0\nn✝ : ℕ\nhle : 0 ∣ x\nhlt : ¬0 ∣ x\n⊢ count 0 (normalizedFactors x) = Nat.succ n✝",
"state_before": "case inl.succ\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nhnorm : ↑normalize 0 = 0\nn✝ : ℕ\nhle : 0 ^ Nat.succ n✝ ∣ x\nhlt : ¬0 ^ (Nat.succ n✝ + 1) ∣ x\n⊢ count 0 (normalizedFactors x) = Nat.succ n✝",
"tactic": "rw [zero_pow (Nat.succ_pos _)] at hle hlt"
},
{
"state_after": "no goals",
"state_before": "case inl.succ\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\nx : R\nhnorm : ↑normalize 0 = 0\nn✝ : ℕ\nhle : 0 ∣ x\nhlt : ¬0 ∣ x\n⊢ count 0 (normalizedFactors x) = Nat.succ n✝",
"tactic": "exact absurd hle hlt"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.1182347\nR : Type u_1\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : Nontrivial R\ninst✝¹ : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst✝ : DecidableEq R\np x : R\nhnorm : ↑normalize p = p\nn : ℕ\nhle : p ^ n ∣ x\nhlt : ¬p ^ (n + 1) ∣ x\nhp : Irreducible p\n⊢ count p (normalizedFactors x) = n",
"tactic": "exact count_normalizedFactors_eq hp hnorm hle hlt"
}
] |
[
1024,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1016,
1
] |
Mathlib/Algebra/Module/Opposites.lean
|
MulOpposite.opLinearEquiv_toAddEquiv
|
[] |
[
62,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean
|
SimpleGraph.dart_edge_fiber_card
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\n⊢ card (filter (fun d => Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (v, w)) univ) = 2",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh : e ∈ edgeSet G\n⊢ card (filter (fun d => Dart.edge d = e) univ) = 2",
"tactic": "refine' Sym2.ind (fun v w h => _) e h"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ card (filter (fun d => Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (v, w)) univ) = 2",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\n⊢ card (filter (fun d => Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (v, w)) univ) = 2",
"tactic": "let d : G.Dart := ⟨(v, w), h⟩"
},
{
"state_after": "case h.e'_3\nV : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ 2 = card {d, Dart.symm d}",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ card (filter (fun d => Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (v, w)) univ) = 2",
"tactic": "convert congr_arg card d.edge_fiber"
},
{
"state_after": "case h.e'_3\nV : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ ¬d ∈ {Dart.symm d}",
"state_before": "case h.e'_3\nV : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ 2 = card {d, Dart.symm d}",
"tactic": "rw [card_insert_of_not_mem, card_singleton]"
},
{
"state_after": "case h.e'_3\nV : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ ¬d = Dart.symm d",
"state_before": "case h.e'_3\nV : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ ¬d ∈ {Dart.symm d}",
"tactic": "rw [mem_singleton]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nV : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\ne : Sym2 V\nh✝ : e ∈ edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n⊢ ¬d = Dart.symm d",
"tactic": "exact d.symm_ne.symm"
}
] |
[
100,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.append_inj'
|
[
{
"state_after": "α✝ : Type u_1\ns₁ t₁ s₂ t₂ : List α✝\nh : s₁ ++ t₁ = s₂ ++ t₂\nhl : length t₁ = length t₂\nhap : length (s₁ ++ t₁) = length (s₂ ++ t₂) := congrArg length h\n⊢ length s₁ + length t₁ = length s₂ + length t₁",
"state_before": "α✝ : Type u_1\ns₁ t₁ s₂ t₂ : List α✝\nh : s₁ ++ t₁ = s₂ ++ t₂\nhl : length t₁ = length t₂\n⊢ length s₁ + length t₁ = length s₂ + length t₁",
"tactic": "let hap := congrArg length h"
},
{
"state_after": "α✝ : Type u_1\ns₁ t₁ s₂ t₂ : List α✝\nh : s₁ ++ t₁ = s₂ ++ t₂\nhl : length t₁ = length t₂\nhap : length s₁ + length t₁ = length s₂ + length t₁\n⊢ length s₁ + length t₁ = length s₂ + length t₁",
"state_before": "α✝ : Type u_1\ns₁ t₁ s₂ t₂ : List α✝\nh : s₁ ++ t₁ = s₂ ++ t₂\nhl : length t₁ = length t₂\nhap : length (s₁ ++ t₁) = length (s₂ ++ t₂) := congrArg length h\n⊢ length s₁ + length t₁ = length s₂ + length t₁",
"tactic": "simp only [length_append, ← hl] at hap"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\ns₁ t₁ s₂ t₂ : List α✝\nh : s₁ ++ t₁ = s₂ ++ t₂\nhl : length t₁ = length t₂\nhap : length s₁ + length t₁ = length s₂ + length t₁\n⊢ length s₁ + length t₁ = length s₂ + length t₁",
"tactic": "exact hap"
}
] |
[
70,
82
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
68,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lift_umax
|
[] |
[
694,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Multiset.toFinset_dedup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.453504\nγ : Type ?u.453507\ninst✝ : DecidableEq α\ns t m : Multiset α\n⊢ toFinset (dedup m) = toFinset m",
"tactic": "simp_rw [toFinset, dedup_idempotent]"
}
] |
[
3198,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3197,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.inductionOn₃'
|
[] |
[
689,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
11
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocDiv_add_zsmul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ b + m • p - (toIocDiv hp a b + m) • p ∈ Set.Ioc a (a + p)",
"tactic": "simpa only [add_smul, add_sub_add_right_eq_sub] using sub_toIocDiv_zsmul_mem_Ioc hp a b"
}
] |
[
251,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.min'_image
|
[
{
"state_after": "F : Type ?u.366451\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.366460\nι : Type ?u.366463\nκ : Type ?u.366466\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\ny : β\nhy : y ∈ image f s\n⊢ f (min' s (_ : Finset.Nonempty s)) ≤ y",
"state_before": "F : Type ?u.366451\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.366460\nι : Type ?u.366463\nκ : Type ?u.366466\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\n⊢ min' (image f s) h = f (min' s (_ : Finset.Nonempty s))",
"tactic": "refine'\n le_antisymm (min'_le _ _ (mem_image.mpr ⟨_, min'_mem _ _, rfl⟩)) (le_min' _ _ _ fun y hy => _)"
},
{
"state_after": "case intro.intro\nF : Type ?u.366451\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.366460\nι : Type ?u.366463\nκ : Type ?u.366466\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx✝ : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\nx : α\nhx : x ∈ s\nhy : f x ∈ image f s\n⊢ f (min' s (_ : Finset.Nonempty s)) ≤ f x",
"state_before": "F : Type ?u.366451\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.366460\nι : Type ?u.366463\nκ : Type ?u.366466\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\ny : β\nhy : y ∈ image f s\n⊢ f (min' s (_ : Finset.Nonempty s)) ≤ y",
"tactic": "obtain ⟨x, hx, rfl⟩ := mem_image.mp hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nF : Type ?u.366451\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.366460\nι : Type ?u.366463\nκ : Type ?u.366466\ninst✝¹ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx✝ : α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\ns : Finset α\nh : Finset.Nonempty (image f s)\nx : α\nhx : x ∈ s\nhy : f x ∈ image f s\n⊢ f (min' s (_ : Finset.Nonempty s)) ≤ f x",
"tactic": "exact hf (min'_le _ _ hx)"
}
] |
[
1512,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1507,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
balanced_iUnion
|
[] |
[
191,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.default_eq_empty
|
[] |
[
141,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.isGLB_sInf
|
[
{
"state_after": "case refine'_1\nC : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\n⊢ ∀ {x y : GrothendieckTopology C}, x.sieves ≤ y.sieves ↔ x ≤ y\n\ncase refine'_2\nC : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\n⊢ IsGLB (sieves '' s) (sInf s).sieves",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\n⊢ IsGLB s (sInf s)",
"tactic": "refine' @IsGLB.of_image _ _ _ _ sieves _ _ _ _"
},
{
"state_after": "case refine'_1\nC : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\nx✝ y✝ : GrothendieckTopology C\n⊢ x✝.sieves ≤ y✝.sieves ↔ x✝ ≤ y✝",
"state_before": "case refine'_1\nC : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\n⊢ ∀ {x y : GrothendieckTopology C}, x.sieves ≤ y.sieves ↔ x ≤ y",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nC : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\nx✝ y✝ : GrothendieckTopology C\n⊢ x✝.sieves ≤ y✝.sieves ↔ x✝ ≤ y✝",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\ns : Set (GrothendieckTopology C)\n⊢ IsGLB (sieves '' s) (sInf s).sieves",
"tactic": "exact _root_.isGLB_sInf _"
}
] |
[
293,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.update_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.148539\nγ : Type ?u.148542\nι : Type ?u.148545\nM : Type u_2\nM' : Type ?u.148551\nN : Type ?u.148554\nP : Type ?u.148557\nG : Type ?u.148560\nH : Type ?u.148563\nR : Type ?u.148566\nS : Type ?u.148569\ninst✝ : Zero M\nf : α →₀ M\na : α\nb : M\ni : α\n⊢ update f a (↑f a) = f",
"tactic": "classical\n ext\n simp"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.148539\nγ : Type ?u.148542\nι : Type ?u.148545\nM : Type u_2\nM' : Type ?u.148551\nN : Type ?u.148554\nP : Type ?u.148557\nG : Type ?u.148560\nH : Type ?u.148563\nR : Type ?u.148566\nS : Type ?u.148569\ninst✝ : Zero M\nf : α →₀ M\na : α\nb : M\ni a✝ : α\n⊢ ↑(update f a (↑f a)) a✝ = ↑f a✝",
"state_before": "α : Type u_1\nβ : Type ?u.148539\nγ : Type ?u.148542\nι : Type ?u.148545\nM : Type u_2\nM' : Type ?u.148551\nN : Type ?u.148554\nP : Type ?u.148557\nG : Type ?u.148560\nH : Type ?u.148563\nR : Type ?u.148566\nS : Type ?u.148569\ninst✝ : Zero M\nf : α →₀ M\na : α\nb : M\ni : α\n⊢ update f a (↑f a) = f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.148539\nγ : Type ?u.148542\nι : Type ?u.148545\nM : Type u_2\nM' : Type ?u.148551\nN : Type ?u.148554\nP : Type ?u.148557\nG : Type ?u.148560\nH : Type ?u.148563\nR : Type ?u.148566\nS : Type ?u.148569\ninst✝ : Zero M\nf : α →₀ M\na : α\nb : M\ni a✝ : α\n⊢ ↑(update f a (↑f a)) a✝ = ↑f a✝",
"tactic": "simp"
}
] |
[
573,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_finset_sum'
|
[] |
[
671,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.inter_eq_inter_iff_right
|
[] |
[
1815,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1814,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_pow
|
[] |
[
912,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
11
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousAt_update_same
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.328491\nδ : Type ?u.328494\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : DecidableEq α\nf : α → β\nx : α\ny : β\n⊢ ContinuousAt (update f x y) x ↔ Tendsto f (𝓝[{x}ᶜ] x) (𝓝 y)",
"tactic": "rw [← continuousWithinAt_univ, continuousWithinAt_update_same, compl_eq_univ_diff]"
}
] |
[
804,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
802,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_le_of_surjective
|
[] |
[
281,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.left_inv
|
[] |
[
158,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/RingTheory/Adjoin/Basic.lean
|
Algebra.adjoin_empty
|
[
{
"state_after": "case gc\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns t : Set A\n⊢ GaloisConnection (adjoin R) ?u\n\ncase u\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns t : Set A\n⊢ Subalgebra R A → Set A",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns t : Set A\n⊢ adjoin R ⊥ = ⊥",
"tactic": "apply GaloisConnection.l_bot"
},
{
"state_after": "no goals",
"state_before": "case gc\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns t : Set A\n⊢ GaloisConnection (adjoin R) ?u\n\ncase u\nR : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns t : Set A\n⊢ Subalgebra R A → Set A",
"tactic": "exact Algebra.gc"
}
] |
[
154,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.comap_iInf
|
[] |
[
349,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.count_inter
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.394964\nγ : Type ?u.394967\ninst✝ : DecidableEq α\na : α\ns t : Multiset α\n⊢ count a (s - t) + count a (s ∩ t) = count a (s - t) + min (count a s) (count a t)",
"state_before": "α : Type u_1\nβ : Type ?u.394964\nγ : Type ?u.394967\ninst✝ : DecidableEq α\na : α\ns t : Multiset α\n⊢ count a (s ∩ t) = min (count a s) (count a t)",
"tactic": "apply @Nat.add_left_cancel (count a (s - t))"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.394964\nγ : Type ?u.394967\ninst✝ : DecidableEq α\na : α\ns t : Multiset α\n⊢ count a (s - t) + count a (s ∩ t) = count a (s - t) + min (count a s) (count a t)",
"tactic": "rw [← count_add, sub_add_inter, count_sub, tsub_add_min]"
}
] |
[
2473,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2471,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.refl_restr_source
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.40873\nγ : Type ?u.40876\nδ : Type ?u.40879\ne : LocalEquiv α β\ne' : LocalEquiv β γ\ns : Set α\n⊢ (LocalEquiv.restr (LocalEquiv.refl α) s).source = s",
"tactic": "simp"
}
] |
[
629,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/Data/List/Basic.lean
|
List.reduceOption_cons_of_none
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.349293\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nl : List (Option α)\n⊢ reduceOption (none :: l) = reduceOption l",
"tactic": "simp only [reduceOption, filterMap, id.def]"
}
] |
[
3425,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3424,
1
] |
Mathlib/Data/Nat/Multiplicity.lean
|
Nat.Prime.pow_dvd_factorial_iff
|
[
{
"state_after": "no goals",
"state_before": "p n r b : ℕ\nhp : Prime p\nhbn : log p n < b\n⊢ p ^ r ∣ n ! ↔ r ≤ ∑ i in Ico 1 b, n / p ^ i",
"tactic": "rw [← PartENat.coe_le_coe, ← hp.multiplicity_factorial hbn, ← pow_dvd_iff_le_multiplicity]"
}
] |
[
166,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.add_le_add_left
|
[] |
[
388,
28
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
383,
11
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.coe_dual
|
[] |
[
287,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_singleton_le_iff_mem
|
[] |
[
176,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.sum_toPartitionOfUnity_eq
|
[] |
[
484,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.bliminf_antitone
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.132666\nι : Type ?u.132669\ninst✝ : CompleteLattice α\nf g : Filter β\np q : β → Prop\nu v : β → α\nh : ∀ (x : β), p x → q x\na : α\nha : a ∈ {a | ∀ᶠ (x : β) in f, q x → a ≤ u x}\n⊢ ∀ (x : β), (q x → a ≤ u x) → p x → a ≤ u x",
"tactic": "tauto"
}
] |
[
862,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
861,
1
] |
Mathlib/Algebra/Associated.lean
|
DvdNotUnit.not_associated
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np : α\na : αˣ\nh : DvdNotUnit p (p * ↑a)\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np q : α\nh : DvdNotUnit p q\n⊢ ¬p ~ᵤ q",
"tactic": "rintro ⟨a, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np : α\na : αˣ\nhp : p ≠ 0\nx : α\nhx : ¬IsUnit x\nhx' : p * ↑a = p * x\n⊢ False",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np : α\na : αˣ\nh : DvdNotUnit p (p * ↑a)\n⊢ False",
"tactic": "obtain ⟨hp, x, hx, hx'⟩ := h"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np : α\na : αˣ\nhp : p ≠ 0\nhx : ¬IsUnit ↑a\nhx' : p * ↑a = p * ↑a\n⊢ False",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np : α\na : αˣ\nhp : p ≠ 0\nx : α\nhx : ¬IsUnit x\nhx' : p * ↑a = p * x\n⊢ False",
"tactic": "rcases(mul_right_inj' hp).mp hx' with rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.354581\nγ : Type ?u.354584\nδ : Type ?u.354587\ninst✝ : CancelCommMonoidWithZero α\np : α\na : αˣ\nhp : p ≠ 0\nhx : ¬IsUnit ↑a\nhx' : p * ↑a = p * ↑a\n⊢ False",
"tactic": "exact hx a.isUnit"
}
] |
[
1197,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1192,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
IsSubmonoid.finset_prod_mem
|
[
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.56585\ninst✝² : Monoid M✝\ns✝ : Set M✝\nA✝ : Type ?u.56594\ninst✝¹ : AddMonoid A✝\nt : Set A✝\nM : Type u_1\nA : Type u_2\ninst✝ : CommMonoid M\ns : Set M\nhs : IsSubmonoid s\nf : A → M\nm : Multiset A\nhm : Multiset.Nodup m\nx✝ : ∀ (b : A), b ∈ { val := m, nodup := hm } → f b ∈ s\n⊢ ∀ (a : M), a ∈ Multiset.map (fun b => f b) { val := m, nodup := hm }.val → a ∈ s",
"tactic": "simpa"
}
] |
[
265,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/CategoryTheory/Limits/Pi.lean
|
CategoryTheory.pi.hasLimit_of_hasLimit_comp_eval
|
[] |
[
121,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean
|
CategoryTheory.MonoidalCategory.associator_conjugation
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\n𝒞 : Category C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z✝ X X' Y Y' Z Z' : C\nf : X ⟶ X'\ng : Y ⟶ Y'\nh : Z ⟶ Z'\n⊢ (f ⊗ g) ⊗ h = (α_ X Y Z).hom ≫ (f ⊗ g ⊗ h) ≫ (α_ X' Y' Z').inv",
"tactic": "rw [associator_inv_naturality, hom_inv_id_assoc]"
}
] |
[
330,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.image_Ioc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : α\n⊢ ↑e '' Ioc a b = Ioc (↑e a) (↑e b)",
"tactic": "rw [e.image_eq_preimage, e.symm.preimage_Ioc, e.symm_symm]"
}
] |
[
96,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Order/Interval.lean
|
Interval.coe_bot
|
[] |
[
498,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_biUnion
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.311298\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset γ\nt : γ → Finset α\nhs : Set.PairwiseDisjoint (↑s) t\n⊢ ∏ x in Finset.biUnion s t, f x = ∏ x in s, ∏ i in t x, f i",
"tactic": "rw [← disjiUnion_eq_biUnion _ _ hs, prod_disjiUnion]"
}
] |
[
519,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.dim_le_degree_of_root
|
[
{
"state_after": "R : Type ?u.124871\nS : Type u_2\nT : Type ?u.124877\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.125183\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.125605\ninst✝¹ : Field K\ninst✝ : Algebra A S\nh : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval h.gen) p = 0\n⊢ ↑h.dim ≤ ↑(natDegree p)",
"state_before": "R : Type ?u.124871\nS : Type u_2\nT : Type ?u.124877\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.125183\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.125605\ninst✝¹ : Field K\ninst✝ : Algebra A S\nh : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval h.gen) p = 0\n⊢ ↑h.dim ≤ degree p",
"tactic": "rw [degree_eq_natDegree ne_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.124871\nS : Type u_2\nT : Type ?u.124877\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.125183\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.125605\ninst✝¹ : Field K\ninst✝ : Algebra A S\nh : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval h.gen) p = 0\n⊢ ↑h.dim ≤ ↑(natDegree p)",
"tactic": "exact WithBot.coe_le_coe.2 (h.dim_le_natDegree_of_root ne_zero root)"
}
] |
[
188,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Int.prime_two
|
[] |
[
809,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
808,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.image_eq_preimage
|
[] |
[
44,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
11
] |
Mathlib/Algebra/Opposites.lean
|
AddOpposite.unop_eq_one_iff
|
[] |
[
388,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoDiv_sub'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp (a - p) b = toIcoDiv hp a b + 1",
"tactic": "simpa only [one_zsmul] using toIcoDiv_sub_zsmul' hp a b 1"
}
] |
[
345,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_eq_of_mul_eq_one_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.21054\nG : Type ?u.21057\ninst✝ : DivisionMonoid α\na b c : α\nh : a * b = 1\n⊢ b⁻¹ = a",
"tactic": "rw [← inv_eq_of_mul_eq_one_right h, inv_inv]"
}
] |
[
361,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.wOppSide_comm
|
[
{
"state_after": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n⊢ WOppSide s x y → WOppSide s y x\n\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n⊢ WOppSide s y x → WOppSide s x y",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n⊢ WOppSide s x y ↔ WOppSide s y x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ WOppSide s y x",
"state_before": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n⊢ WOppSide s x y → WOppSide s y x",
"tactic": "rintro ⟨p₁, hp₁, p₂, hp₂, h⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ SameRay R (y -ᵥ p₂) (p₁ -ᵥ x)",
"state_before": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ WOppSide s y x",
"tactic": "refine' ⟨p₂, hp₂, p₁, hp₁, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ SameRay R (y -ᵥ p₂) (p₁ -ᵥ x)",
"tactic": "rwa [SameRay.sameRay_comm, ← sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (y -ᵥ p₁) (p₂ -ᵥ x)\n⊢ WOppSide s x y",
"state_before": "case mpr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n⊢ WOppSide s y x → WOppSide s x y",
"tactic": "rintro ⟨p₁, hp₁, p₂, hp₂, h⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (y -ᵥ p₁) (p₂ -ᵥ x)\n⊢ SameRay R (x -ᵥ p₂) (p₁ -ᵥ y)",
"state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (y -ᵥ p₁) (p₂ -ᵥ x)\n⊢ WOppSide s x y",
"tactic": "refine' ⟨p₂, hp₂, p₁, hp₁, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.89985\nP : Type u_3\nP' : Type ?u.89991\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (y -ᵥ p₁) (p₂ -ᵥ x)\n⊢ SameRay R (x -ᵥ p₂) (p₁ -ᵥ y)",
"tactic": "rwa [SameRay.sameRay_comm, ← sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]"
}
] |
[
212,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.subset_coe_map
|
[] |
[
291,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
CochainComplex.cochainComplex_d_succ_succ_zero
|
[
{
"state_after": "case a\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ ¬ComplexShape.Rel (ComplexShape.up ℕ) 0 (i + 2)",
"state_before": "V : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ d C 0 (i + 2) = 0",
"tactic": "rw [C.shape]"
},
{
"state_after": "case a\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ ¬1 = i + 2",
"state_before": "case a\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ ¬ComplexShape.Rel (ComplexShape.up ℕ) 0 (i + 2)",
"tactic": "simp only [ComplexShape.up_Rel, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case a\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ ¬1 = i + 2",
"tactic": "exact (Nat.one_lt_succ_succ _).ne"
}
] |
[
327,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Topology/Separation.lean
|
t2_separation_nhds
|
[] |
[
937,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.C_mul
|
[] |
[
507,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/CategoryTheory/Sites/Closed.lean
|
CategoryTheory.GrothendieckTopology.close_eq_top_iff_mem
|
[
{
"state_after": "case mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\n⊢ close J₁ S = ⊤ → S ∈ sieves J₁ X\n\ncase mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\n⊢ S ∈ sieves J₁ X → close J₁ S = ⊤",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\n⊢ close J₁ S = ⊤ ↔ S ∈ sieves J₁ X",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\n⊢ S ∈ sieves J₁ X",
"state_before": "case mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\n⊢ close J₁ S = ⊤ → S ∈ sieves J₁ X",
"tactic": "intro h"
},
{
"state_after": "case mp.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, ⊤.arrows f → Sieve.pullback f S ∈ sieves J₁ Y",
"state_before": "case mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\n⊢ S ∈ sieves J₁ X",
"tactic": "apply J₁.transitive (J₁.top_mem X)"
},
{
"state_after": "case mp.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\nY : C\nf : Y ⟶ X\nhf : ⊤.arrows f\n⊢ Sieve.pullback f S ∈ sieves J₁ Y",
"state_before": "case mp.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, ⊤.arrows f → Sieve.pullback f S ∈ sieves J₁ Y",
"tactic": "intro Y f hf"
},
{
"state_after": "case mp.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\nY : C\nf : Y ⟶ X\nhf : ⊤.arrows f\n⊢ (close J₁ S).arrows f",
"state_before": "case mp.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\nY : C\nf : Y ⟶ X\nhf : ⊤.arrows f\n⊢ Sieve.pullback f S ∈ sieves J₁ Y",
"tactic": "change J₁.close S f"
},
{
"state_after": "no goals",
"state_before": "case mp.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nh : close J₁ S = ⊤\nY : C\nf : Y ⟶ X\nhf : ⊤.arrows f\n⊢ (close J₁ S).arrows f",
"tactic": "rwa [h]"
},
{
"state_after": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ sieves J₁ X\n⊢ close J₁ S = ⊤",
"state_before": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\n⊢ S ∈ sieves J₁ X → close J₁ S = ⊤",
"tactic": "intro hS"
},
{
"state_after": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ sieves J₁ X\n⊢ ⊤ ≤ close J₁ S",
"state_before": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ sieves J₁ X\n⊢ close J₁ S = ⊤",
"tactic": "rw [eq_top_iff]"
},
{
"state_after": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ sieves J₁ X\nY : C\nf : Y ⟶ X\na✝ : ⊤.arrows f\n⊢ (close J₁ S).arrows f",
"state_before": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ sieves J₁ X\n⊢ ⊤ ≤ close J₁ S",
"tactic": "intro Y f _"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ sieves J₁ X\nY : C\nf : Y ⟶ X\na✝ : ⊤.arrows f\n⊢ (close J₁ S).arrows f",
"tactic": "apply J₁.pullback_stable _ hS"
}
] |
[
165,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.two_lt_ncard_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.157357\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ 2 < ncard s ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c",
"tactic": "simp_rw [ncard_eq_toFinset_card _ hs, Finset.two_lt_card_iff, Finite.mem_toFinset]"
}
] |
[
710,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.cospan_one
|
[] |
[
222,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Order/Bounded.lean
|
Set.bounded_gt_inter_ge
|
[] |
[
440,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/Data/List/Func.lean
|
List.Func.get_nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nk : ℕ\n⊢ get k [] = default",
"tactic": "cases k <;> rfl"
}
] |
[
119,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
Filter.Realizer.top_σ
|
[] |
[
187,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.antitoneOn_singleton
|
[] |
[
2699,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2698,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atTop_pure
|
[] |
[
305,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMapClass.ker_eq_bot
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.1427240\nR₂ : Type u_3\nR₃ : Type ?u.1427246\nR₄ : Type ?u.1427249\nS : Type ?u.1427252\nK : Type ?u.1427255\nK₂ : Type ?u.1427258\nM : Type u_1\nM' : Type ?u.1427264\nM₁ : Type ?u.1427267\nM₂ : Type u_4\nM₃ : Type ?u.1427273\nM₄ : Type ?u.1427276\nN : Type ?u.1427279\nN₂ : Type ?u.1427282\nι : Type ?u.1427285\nV : Type ?u.1427288\nV₂ : Type ?u.1427291\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R₂\ninst✝⁷ : Ring R₃\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : AddCommGroup M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_5\nsc : SemilinearMapClass F τ₁₂ M M₂\nf : F\n⊢ ker f = ⊥ ↔ Injective ↑f",
"tactic": "simpa [disjoint_iff_inf_le] using @disjoint_ker' _ _ _ _ _ _ _ _ _ _ _ _ _ f ⊤"
}
] |
[
1506,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1505,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.dvd_to_nat
|
[
{
"state_after": "m n : Num\nx✝ : ↑m ∣ ↑n\nk : ℕ\ne : ↑n = ↑m * k\n⊢ ↑(↑m * k) = m * ↑k",
"state_before": "m n : Num\nx✝ : ↑m ∣ ↑n\nk : ℕ\ne : ↑n = ↑m * k\n⊢ n = m * ↑k",
"tactic": "rw [← of_to_nat n, e]"
},
{
"state_after": "no goals",
"state_before": "m n : Num\nx✝ : ↑m ∣ ↑n\nk : ℕ\ne : ↑n = ↑m * k\n⊢ ↑(↑m * k) = m * ↑k",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "m n : Num\nx✝ : m ∣ n\nk : Num\ne : n = m * k\n⊢ ↑n = ↑m * ↑k",
"tactic": "simp [e, mul_to_nat]"
}
] |
[
513,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.