Datasets:
Tags:
License:
full_name
stringlengths 2
100
| start
sequence | file_path
stringlengths 11
79
| end
sequence | url
stringclasses 4
values | traced_tactics
list | commit
stringclasses 4
values |
|---|---|---|---|---|---|---|
Pell.n_lt_xn | [
282,
1
] | Mathlib/NumberTheory/PellMatiyasevic.lean | [
283,
54
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
BoxIntegral.Box.withBotCoe_inj | [
307,
1
] | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | [
308,
76
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\nI J : WithBot (Box ι)\n⊢ ↑I = ↑J ↔ I = J",
"tactic": "simp only [Subset.antisymm_iff, ← le_antisymm_iff, withBotCoe_subset_iff]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
CategoryTheory.Comma.comp_left | [
130,
1
] | Mathlib/CategoryTheory/Comma.lean | [
131,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
VectorBundleCore.localTriv_symm_apply | [
723,
1
] | Mathlib/Topology/VectorBundle/Basic.lean | [
725,
40
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "R : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.410614\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i\nv : F\n⊢ Trivialization.symm (localTriv Z i) b v = ↑(coordChange Z i (indexAt Z b) b) v",
"tactic": "apply (Z.localTriv i).symm_apply hb v"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
WithTop.coe_nat | [
377,
1
] | Mathlib/Algebra/Order/Monoid/WithTop.lean | [
378,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.Integrable.toL1_add | [
1379,
1
] | Mathlib/MeasureTheory/Function/L1Space.lean | [
1381,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
lowerCentralSeries_succ_eq_bot | [
509,
1
] | Mathlib/GroupTheory/Nilpotent.lean | [
514,
36
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\n⊢ ∀ (y : G), y ∈ {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, q ∈ ⊤ ∧ p * q * p⁻¹ * q⁻¹ = x} → y = 1",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\n⊢ lowerCentralSeries G (n + 1) = ⊥",
"tactic": "rw [lowerCentralSeries_succ, closure_eq_bot_iff, Set.subset_singleton_iff]"
},
{
"state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ y * z * y⁻¹ * z⁻¹ = 1",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\n⊢ ∀ (y : G), y ∈ {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, q ∈ ⊤ ∧ p * q * p⁻¹ * q⁻¹ = x} → y = 1",
"tactic": "rintro x ⟨y, hy1, z, ⟨⟩, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ z * y = y * z",
"state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ y * z * y⁻¹ * z⁻¹ = 1",
"tactic": "rw [mul_assoc, ← mul_inv_rev, mul_inv_eq_one, eq_comm]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ z * y = y * z",
"tactic": "exact mem_center_iff.mp (h hy1) z"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff | [
1622,
1
] | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | [
1626,
52
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.393825\nγ : Type u_3\nι : Type ?u.393831\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β✝\ninst✝ : TopologicalSpace γ\nf✝ g✝ : α → β✝\nβ : Type u_1\nf : α → β\nmα : MeasurableSpace α\nμa : Measure α\nmβ : MeasurableSpace β\nμb : Measure β\nhf : MeasurePreserving f\nh₂ : MeasurableEmbedding f\ng : β → γ\n⊢ AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable g μb",
"tactic": "rw [← hf.map_eq, h₂.aestronglyMeasurable_map_iff]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
LipschitzOnWith.continuousOn | [
510,
11
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | [
511,
38
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
MulChar.ofUnitHom_coe | [
225,
1
] | Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean | [
225,
93
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommMonoid R\nR' : Type v\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\na : Rˣ\n⊢ ↑(ofUnitHom f) ↑a = ↑(↑f a)",
"tactic": "simp [ofUnitHom]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.finset_prod_subset_finset_prod | [
165,
1
] | Mathlib/Data/Set/Pointwise/BigOperators.lean | [
167,
46
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
LinearMap.ker_toAddSubmonoid | [
1337,
1
] | Mathlib/LinearAlgebra/Basic.lean | [
1338,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
iSup₂_mono | [
848,
1
] | Mathlib/Order/CompleteLattice.lean | [
850,
38
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Nat.pow_dvd_pow_iff_le_right' | [
215,
1
] | Mathlib/Data/Nat/Pow.lean | [
216,
55
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Nat.not_dvd_ord_compl | [
531,
1
] | Mathlib/Data/Nat/Factorization/Basic.lean | [
534,
26
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization (n / p ^ ↑(factorization n) p)) p",
"state_before": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬p ∣ n / p ^ ↑(factorization n) p",
"tactic": "rw [Nat.Prime.dvd_iff_one_le_factorization hp (ord_compl_pos p hn).ne']"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization n - factorization (p ^ ↑(factorization n) p)) p",
"state_before": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization (n / p ^ ↑(factorization n) p)) p",
"tactic": "rw [Nat.factorization_div (Nat.ord_proj_dvd n p)]"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization n - factorization (p ^ ↑(factorization n) p)) p",
"tactic": "simp [hp.factorization]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.Measure.InnerRegular.measurableSet_of_open | [
351,
1
] | Mathlib/MeasureTheory/Measure/Regular.lean | [
370,
48
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ r < ↑↑μ K",
"state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s : Set α\nε r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\n⊢ InnerRegular μ p fun s => MeasurableSet s ∧ ↑↑μ s ≠ ⊤",
"tactic": "rintro s ⟨hs, hμs⟩ r hr"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ r < ↑↑μ K",
"tactic": "obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by\n use (μ s - r) / 2\n simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"tactic": "rcases hs.exists_isOpen_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhsU' : U' ⊇ U \\ s\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"tactic": "rcases (U \\ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhsU' : U' ⊇ U \\ s\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"tactic": "replace hsU' := diff_subset_comm.1 hsU'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"tactic": "rcases H.exists_subset_lt_add h0 hUo hUt.ne hε with ⟨K, hKU, hKc, hKr⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ s < ↑↑μ (K \\ U') + (ε + ε)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K",
"tactic": "refine' ⟨K \\ U', fun x hx => hsU' ⟨hKU hx.1, hx.2⟩, hd hKc hU'o, ENNReal.sub_lt_of_lt_add hεs _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ s < ↑↑μ (K \\ U') + (ε + ε)",
"tactic": "calc\n μ s ≤ μ U := μ.mono hsU\n _ < μ K + ε := hKr\n _ ≤ μ (K \\ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)\n _ ≤ μ (K \\ U') + ε + ε := by\n apply add_le_add_right; apply add_le_add_left\n exact hμU'.le\n _ = μ (K \\ U') + (ε + ε) := add_assoc _ _ _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ x, (↑↑μ s - r) / 2 + (↑↑μ s - r) / 2 ≤ ↑↑μ s ∧ r = ↑↑μ s - ((↑↑μ s - r) / 2 + (↑↑μ s - r) / 2)",
"state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ ε x, ε + ε ≤ ↑↑μ s ∧ r = ↑↑μ s - (ε + ε)",
"tactic": "use (μ s - r) / 2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ x, (↑↑μ s - r) / 2 + (↑↑μ s - r) / 2 ≤ ↑↑μ s ∧ r = ↑↑μ s - ((↑↑μ s - r) / 2 + (↑↑μ s - r) / 2)",
"tactic": "simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]"
},
{
"state_after": "case bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ (K \\ U') + ↑↑μ U' ≤ ↑↑μ (K \\ U') + ε",
"state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ (K \\ U') + ↑↑μ U' + ε ≤ ↑↑μ (K \\ U') + ε + ε",
"tactic": "apply add_le_add_right"
},
{
"state_after": "case bc.bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ U' ≤ ε",
"state_before": "case bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ (K \\ U') + ↑↑μ U' ≤ ↑↑μ (K \\ U') + ε",
"tactic": "apply add_le_add_left"
},
{
"state_after": "no goals",
"state_before": "case bc.bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ U' ≤ ε",
"tactic": "exact hμU'.le"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.meas_ge_le_lintegral_div | [
848,
1
] | Mathlib/MeasureTheory/Integral/Lebesgue.lean | [
852,
41
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_1\nβ : Type ?u.966398\nγ : Type ?u.966401\nδ : Type ?u.966404\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε' : ε ≠ ⊤\n⊢ ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.966398\nγ : Type ?u.966401\nδ : Type ?u.966404\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε' : ε ≠ ⊤\n⊢ ↑↑μ {x | ε ≤ f x} * ε ≤ ∫⁻ (a : α), f a ∂μ",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.966398\nγ : Type ?u.966401\nδ : Type ?u.966404\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε' : ε ≠ ⊤\n⊢ ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ",
"tactic": "exact mul_meas_ge_le_lintegral₀ hf ε"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Complex.arg_cos_add_sin_mul_I_sub | [
454,
1
] | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | [
456,
82
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "θ : ℝ\n⊢ arg (cos ↑θ + sin ↑θ * I) - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋",
"tactic": "rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
LocalHomeomorph.eventually_right_inverse | [
246,
1
] | Mathlib/Topology/LocalHomeomorph.lean | [
248,
54
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.pairwise_top | [
78,
1
] | Mathlib/Data/Set/Pairwise/Basic.lean | [
79,
44
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
wittStructureInt_prop | [
306,
1
] | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | [
312,
48
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)",
"state_before": "p : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ ↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n) = ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ",
"tactic": "apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis :\n ↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ)\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)",
"state_before": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)",
"tactic": "have := wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n"
},
{
"state_after": "no goals",
"state_before": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis :\n ↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ)\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)",
"tactic": "simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,\n AlgHom.coe_toRingHom, map_wittStructureInt]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
SimpleGraph.Preconnected.ofBoxProdRight | [
189,
11
] | Mathlib/Combinatorics/SimpleGraph/Prod.lean | [
194,
27
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\n⊢ Preconnected H",
"tactic": "classical\nrintro b₁ b₂\nobtain ⟨w⟩ := h (Classical.arbitrary _, b₁) (Classical.arbitrary _, b₂)\nexact ⟨w.ofBoxProdRight⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\n⊢ Reachable H b₁ b₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\n⊢ Preconnected H",
"tactic": "rintro b₁ b₂"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\nw : Walk (G □ H) (Classical.arbitrary α, b₁) (Classical.arbitrary α, b₂)\n⊢ Reachable H b₁ b₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\n⊢ Reachable H b₁ b₂",
"tactic": "obtain ⟨w⟩ := h (Classical.arbitrary _, b₁) (Classical.arbitrary _, b₂)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\nw : Walk (G □ H) (Classical.arbitrary α, b₁) (Classical.arbitrary α, b₂)\n⊢ Reachable H b₁ b₂",
"tactic": "exact ⟨w.ofBoxProdRight⟩"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Function.invFunOn_pos | [
1213,
1
] | Mathlib/Data/Set/Function.lean | [
1215,
32
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64809\nι : Sort ?u.64812\nπ : α → Type ?u.64817\ninst✝ : Nonempty α\ns : Set α\nf : α → β\na : α\nb : β\nh : ∃ a, a ∈ s ∧ f a = b\n⊢ Classical.choose h ∈ s ∧ f (Classical.choose h) = b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64809\nι : Sort ?u.64812\nπ : α → Type ?u.64817\ninst✝ : Nonempty α\ns : Set α\nf : α → β\na : α\nb : β\nh : ∃ a, a ∈ s ∧ f a = b\n⊢ invFunOn f s b ∈ s ∧ f (invFunOn f s b) = b",
"tactic": "rw [invFunOn, dif_pos h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64809\nι : Sort ?u.64812\nπ : α → Type ?u.64817\ninst✝ : Nonempty α\ns : Set α\nf : α → β\na : α\nb : β\nh : ∃ a, a ∈ s ∧ f a = b\n⊢ Classical.choose h ∈ s ∧ f (Classical.choose h) = b",
"tactic": "exact Classical.choose_spec h"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.Measure.add_haar_eq_zero_of_disjoint_translates_aux | [
121,
1
] | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | [
134,
48
] | https://github.com/leanprover-community/mathlib4 | [
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}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
IsQuasiSeparated.image_of_embedding | [
69,
1
] | Mathlib/Topology/QuasiSeparated.lean | [
91,
48
] | https://github.com/leanprover-community/mathlib4 | [
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"state_after": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ f x ∈ f '' s",
"state_before": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ x ∈ s",
"tactic": "rw [← (h.inj.injOn _).mem_image_iff (Set.subset_univ _) trivial]"
},
{
"state_after": "no goals",
"state_before": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ f x ∈ f '' s",
"tactic": "exact hV hx"
},
{
"state_after": "case convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f '' (f ⁻¹' V))",
"state_before": "case convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f ⁻¹' V)",
"tactic": "rw [h.isCompact_iff_isCompact_image]"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' V) = V",
"state_before": "case convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f '' (f ⁻¹' V))",
"tactic": "convert hV''"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ V ⊆ Set.range f",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' V) = V",
"tactic": "rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ V ⊆ Set.range f",
"tactic": "exact hV.trans (Set.image_subset_range _ _)"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Order.Ici_succ | [
371,
1
] | Mathlib/Order/SuccPred/Basic.lean | [
372,
39
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
iSup_inf_le_sSup_inf | [
1936,
1
] | Mathlib/Order/CompleteLattice.lean | [
1937,
34
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.Iic.isCoatom_iff | [
193,
1
] | Mathlib/Order/Atoms.lean | [
196,
98
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ a ⋖ ⊤ ↔ ↑a ⋖ b",
"state_before": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ IsCoatom a ↔ ↑a ⋖ b",
"tactic": "rw [← covby_top_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ OrdConnected (range ↑(OrderEmbedding.subtype fun c => c ≤ b))",
"state_before": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ a ⋖ ⊤ ↔ ↑a ⋖ b",
"tactic": "refine' (Set.OrdConnected.apply_covby_apply_iff (OrderEmbedding.subtype fun c => c ≤ b) _).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ OrdConnected (range ↑(OrderEmbedding.subtype fun c => c ≤ b))",
"tactic": "simpa only [OrderEmbedding.subtype_apply, Subtype.range_coe_subtype] using Set.ordConnected_Iic"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
gauge_lt_eq | [
182,
1
] | Mathlib/Analysis/Convex/Gauge.lean | [
188,
48
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ (r : ℝ) (_ : r ∈ Ioo 0 a), r • s",
"state_before": "𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\n⊢ {x | gauge s x < a} = ⋃ (r : ℝ) (_ : r ∈ Ioo 0 a), r • s",
"tactic": "ext"
},
{
"state_after": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ gauge s x✝ < a ↔ ∃ i, 0 < i ∧ i < a ∧ x✝ ∈ i • s",
"state_before": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ (r : ℝ) (_ : r ∈ Ioo 0 a), r • s",
"tactic": "simp_rw [mem_setOf, mem_iUnion, exists_prop, mem_Ioo, and_assoc]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ gauge s x✝ < a ↔ ∃ i, 0 < i ∧ i < a ∧ x✝ ∈ i • s",
"tactic": "exact\n ⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>\n (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Filter.HasBasis.uniformEmbedding_iff' | [
143,
1
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [
148,
64
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nι : Sort u_1\nι' : Sort u_2\np : ι → Prop\np' : ι' → Prop\ns : ι → Set (α × α)\ns' : ι' → Set (β × β)\nh : HasBasis (𝓤 α) p s\nh' : HasBasis (𝓤 β) p' s'\nf : α → β\n⊢ UniformEmbedding f ↔\n Injective f ∧\n (∀ (i : ι'), p' i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧\n ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j",
"tactic": "rw [uniformEmbedding_iff, and_comm, h.uniformInducing_iff h']"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
compl_top | [
890,
1
] | Mathlib/Order/Heyting/Basic.lean | [
891,
93
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.161852\nα : Type u_1\nβ : Type ?u.161858\ninst✝ : HeytingAlgebra α\na✝ b c a : α\n⊢ a ≤ ⊤ᶜ ↔ a ≤ ⊥",
"tactic": "rw [le_compl_iff_disjoint_right, disjoint_top, le_bot_iff]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
SupBotHom.coe_comp | [
793,
1
] | Mathlib/Order/Hom/Lattice.lean | [
794,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
BinaryHeap.size_pos_of_max | [
128,
1
] | Mathlib/Data/BinaryHeap.lean | [
129,
95
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx : α\nlt : α → α → Bool\nself : BinaryHeap α lt\ne : max self = some x\nh : ¬0 < Array.size self.arr\n⊢ False",
"tactic": "simp [BinaryHeap.max, Array.get?, h] at e"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
SatisfiesM.map_pre | [
110,
1
] | Std/Classes/LawfulMonad.lean | [
112,
20
] | https://github.com/leanprover/std4 | [] | e68aa8f5fe47aad78987df45f99094afbcb5e936 |
count_le_of_ideal_ge | [
904,
1
] | Mathlib/RingTheory/DedekindDomain/Ideal.lean | [
908,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Algebra.TensorProduct.mulAux_apply | [
361,
1
] | Mathlib/RingTheory/TensorProduct.lean | [
363,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
isGδ_iInter | [
84,
1
] | Mathlib/Topology/GDelta.lean | [
88,
35
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\ns : ι → Set α\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), s i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), s i)",
"state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\ns : ι → Set α\nhs : ∀ (i : ι), IsGδ (s i)\n⊢ IsGδ (⋂ (i : ι), s i)",
"tactic": "choose T hTo hTc hTs using hs"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), (fun i => ⋂₀ T i) i)",
"state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\ns : ι → Set α\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), s i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), s i)",
"tactic": "obtain rfl : s = fun i => ⋂₀ T i := funext hTs"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ ∀ (t : Set α), (t ∈ ⋃ (i : ι), T i) → IsOpen t",
"state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), (fun i => ⋂₀ T i) i)",
"tactic": "refine' ⟨⋃ i, T i, _, countable_iUnion hTc, (sInter_iUnion _).symm⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ ∀ (t : Set α), (t ∈ ⋃ (i : ι), T i) → IsOpen t",
"tactic": "simpa [@forall_swap ι] using hTo"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Subtype.map_injective | [
219,
1
] | Mathlib/Data/Subtype.lean | [
221,
45
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Matrix.map_zero | [
347,
11
] | Mathlib/Data/Matrix/Basic.lean | [
350,
11
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "case a.h\nl : Type ?u.36987\nm : Type u_1\nn : Type u_2\no : Type ?u.36996\nm' : o → Type ?u.37001\nn' : o → Type ?u.37006\nR : Type ?u.37009\nS : Type ?u.37012\nα : Type v\nβ : Type w\nγ : Type ?u.37019\ninst✝¹ : Zero α\ninst✝ : Zero β\nf : α → β\nh : f 0 = 0\ni✝ : m\nx✝ : n\n⊢ map 0 f i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"state_before": "l : Type ?u.36987\nm : Type u_1\nn : Type u_2\no : Type ?u.36996\nm' : o → Type ?u.37001\nn' : o → Type ?u.37006\nR : Type ?u.37009\nS : Type ?u.37012\nα : Type v\nβ : Type w\nγ : Type ?u.37019\ninst✝¹ : Zero α\ninst✝ : Zero β\nf : α → β\nh : f 0 = 0\n⊢ map 0 f = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.36987\nm : Type u_1\nn : Type u_2\no : Type ?u.36996\nm' : o → Type ?u.37001\nn' : o → Type ?u.37006\nR : Type ?u.37009\nS : Type ?u.37012\nα : Type v\nβ : Type w\nγ : Type ?u.37019\ninst✝¹ : Zero α\ninst✝ : Zero β\nf : α → β\nh : f 0 = 0\ni✝ : m\nx✝ : n\n⊢ map 0 f i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"tactic": "simp [h]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
nhdsWithin_singleton | [
281,
1
] | Mathlib/Topology/ContinuousOn.lean | [
282,
72
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18761\nγ : Type ?u.18764\nδ : Type ?u.18767\ninst✝ : TopologicalSpace α\na : α\n⊢ 𝓝[{a}] a = pure a",
"tactic": "rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
finprod_eventually_eq_prod | [
800,
1
] | Mathlib/Topology/Algebra/Monoid.lean | [
804,
84
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
LipschitzOnWith.mono | [
81,
1
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | [
83,
46
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
not_le_of_gt | [
152,
1
] | Mathlib/Init/Algebra/Order.lean | [
153,
28
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Finsupp.prod_filter_mul_prod_filter_not | [
952,
1
] | Mathlib/Data/Finsupp/Basic.lean | [
955,
18
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.476654\nγ : Type ?u.476657\nι : Type ?u.476660\nM : Type u_3\nM' : Type ?u.476666\nN : Type u_1\nP : Type ?u.476672\nG : Type ?u.476675\nH : Type ?u.476678\nR : Type ?u.476681\nS : Type ?u.476684\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ prod (filter p f) g * prod (filter (fun a => ¬p a) f) g = prod f g",
"tactic": "classical simp_rw [prod_filter_index, support_filter, Finset.prod_filter_mul_prod_filter_not,\n Finsupp.prod]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.476654\nγ : Type ?u.476657\nι : Type ?u.476660\nM : Type u_3\nM' : Type ?u.476666\nN : Type u_1\nP : Type ?u.476672\nG : Type ?u.476675\nH : Type ?u.476678\nR : Type ?u.476681\nS : Type ?u.476684\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ prod (filter p f) g * prod (filter (fun a => ¬p a) f) g = prod f g",
"tactic": "simp_rw [prod_filter_index, support_filter, Finset.prod_filter_mul_prod_filter_not,\nFinsupp.prod]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Nat.bitwise'_zero_right | [
414,
1
] | Mathlib/Init/Data/Nat/Bitwise.lean | [
417,
29
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n bif f true false then m else 0",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ bitwise' f m 0 = bif f true false then m else 0",
"tactic": "unfold bitwise'"
},
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (b : Bool) (n : ℕ),\n binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b n) 0 =\n bif f true false then bit b n else 0",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n bif f true false then m else 0",
"tactic": "apply bitCasesOn m"
},
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b✝ n✝) 0 =\n bif f true false then bit b✝ n✝ else 0",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (b : Bool) (n : ℕ),\n binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b n) 0 =\n bif f true false then bit b n else 0",
"tactic": "intros"
},
{
"state_after": "case h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ (binaryRec (bif f true false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b✝ n✝) 0 =\n bif f true false then bit b✝ n✝ else 0",
"tactic": "rw [binaryRec_eq, binaryRec_zero]"
},
{
"state_after": "no goals",
"state_before": "case h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ (binaryRec (bif f true false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0",
"tactic": "exact bitwise'_bit_aux h"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Nat.succ_sub_succ_eq_sub | [
218,
1
] | src/lean/Init/Data/Nat/Basic.lean | [
221,
40
] | https://github.com/leanprover/lean4 | [
{
"state_after": "no goals",
"state_before": "n m : Nat\n⊢ succ n - succ m = n - m",
"tactic": "induction m with\n| zero => exact rfl\n| succ m ih => apply congrArg pred ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nn : Nat\n⊢ succ n - succ zero = n - zero",
"tactic": "exact rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nn m : Nat\nih : succ n - succ m = n - m\n⊢ succ n - succ (succ m) = n - succ m",
"tactic": "apply congrArg pred ih"
}
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 |
Part.Fix.approx_mono' | [
60,
1
] | Mathlib/Control/LawfulFix.lean | [
63,
56
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\ni : ℕ\n⊢ approx (↑f) i ≤ approx (↑f) (Nat.succ i)",
"tactic": "induction i with\n| zero => dsimp [approx]; apply @bot_le _ _ _ (f ⊥)\n| succ _ i_ih => intro ; apply f.monotone; apply i_ih"
},
{
"state_after": "case zero\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\n⊢ ⊥ ≤ ↑f ⊥",
"state_before": "case zero\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\n⊢ approx (↑f) Nat.zero ≤ approx (↑f) (Nat.succ Nat.zero)",
"tactic": "dsimp [approx]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\n⊢ ⊥ ≤ ↑f ⊥",
"tactic": "apply @bot_le _ _ _ (f ⊥)"
},
{
"state_after": "case succ\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) (Nat.succ n✝) i✝ ≤ approx (↑f) (Nat.succ (Nat.succ n✝)) i✝",
"state_before": "case succ\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\n⊢ approx (↑f) (Nat.succ n✝) ≤ approx (↑f) (Nat.succ (Nat.succ n✝))",
"tactic": "intro"
},
{
"state_after": "case succ.a\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)",
"state_before": "case succ\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) (Nat.succ n✝) i✝ ≤ approx (↑f) (Nat.succ (Nat.succ n✝)) i✝",
"tactic": "apply f.monotone"
},
{
"state_after": "no goals",
"state_before": "case succ.a\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)",
"tactic": "apply i_ih"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.einfsep_lt_iff | [
86,
1
] | Mathlib/Topology/MetricSpace/Infsep.lean | [
88,
33
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12733\ninst✝ : EDist α\nx y : α\ns t : Set α\nd : ℝ≥0∞\n⊢ einfsep s < d ↔ ∃ x x_1 y x_2 _h, edist x y < d",
"tactic": "simp_rw [einfsep, iInf_lt_iff]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Cardinal.powerlt_le | [
2287,
1
] | Mathlib/SetTheory/Cardinal/Basic.lean | [
2291,
47
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α β : Type u\na b c : Cardinal\n⊢ (∀ (i : ↑(Iio b)), a ^ ↑i ≤ c) ↔ ∀ (x : Cardinal), x < b → a ^ x ≤ c\n\ncase h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (range fun c => a ^ ↑c)",
"state_before": "α β : Type u\na b c : Cardinal\n⊢ a ^< b ≤ c ↔ ∀ (x : Cardinal), x < b → a ^ x ≤ c",
"tactic": "rw [powerlt, ciSup_le_iff']"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\na b c : Cardinal\n⊢ (∀ (i : ↑(Iio b)), a ^ ↑i ≤ c) ↔ ∀ (x : Cardinal), x < b → a ^ x ≤ c",
"tactic": "simp"
},
{
"state_after": "case h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (HPow.hPow a '' Iio b)",
"state_before": "case h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (range fun c => a ^ ↑c)",
"tactic": "rw [← image_eq_range]"
},
{
"state_after": "no goals",
"state_before": "case h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (HPow.hPow a '' Iio b)",
"tactic": "exact bddAbove_image.{u, u} _ bddAbove_Iio"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Stream'.drop_tail' | [
76,
9
] | Mathlib/Data/Stream/Init.lean | [
76,
83
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.Lp.meas_ge_le_mul_pow_norm | [
1804,
1
] | Mathlib/MeasureTheory/Function/LpSpace.lean | [
1807,
83
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
FirstOrder.Language.ElementarilyEquivalent.theory_model_iff | [
1118,
1
] | Mathlib/ModelTheory/Semantics.lean | [
1120,
25
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type ?u.909458\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nh : M ≅[L] N\n⊢ M ⊨ T ↔ N ⊨ T",
"tactic": "rw [Theory.model_iff_subset_completeTheory, Theory.model_iff_subset_completeTheory,\n h.completeTheory_eq]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Subsemiring.map_equiv_eq_comap_symm | [
836,
1
] | Mathlib/RingTheory/Subsemiring/Basic.lean | [
838,
56
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
add_abs_nonneg | [
162,
1
] | Mathlib/Algebra/Order/Group/Abs.lean | [
165,
26
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ a + -a ≤ a + abs a",
"state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ 0 ≤ a + abs a",
"tactic": "rw [← add_right_neg a]"
},
{
"state_after": "case bc\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ -a ≤ abs a",
"state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ a + -a ≤ a + abs a",
"tactic": "apply add_le_add_left"
},
{
"state_after": "no goals",
"state_before": "case bc\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ -a ≤ abs a",
"tactic": "exact neg_le_abs_self a"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Quotient.lift₂_mk | [
321,
1
] | Mathlib/Data/Quot.lean | [
326,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Complex.UnitDisc.coe_mul | [
76,
1
] | Mathlib/Analysis/Complex/UnitDisc/Basic.lean | [
77,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
LieHom.comp_apply | [
418,
1
] | Mathlib/Algebra/Lie/Basic.lean | [
419,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
CategoryTheory.Idempotents.idem_of_id_sub_idem | [
102,
1
] | Mathlib/CategoryTheory/Idempotents/Basic.lean | [
104,
75
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ (𝟙 X - p) ≫ (𝟙 X - p) = 𝟙 X - p",
"tactic": "simp only [comp_sub, sub_comp, id_comp, comp_id, hp, sub_self, sub_zero]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.sep_ext_iff | [
1420,
1
] | Mathlib/Data/Set/Basic.lean | [
1421,
54
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\np q : α → Prop\nx : α\n⊢ {x | x ∈ s ∧ p x} = {x | x ∈ s ∧ q x} ↔ ∀ (x : α), x ∈ s → (p x ↔ q x)",
"tactic": "simp_rw [ext_iff, mem_sep_iff, and_congr_right_iff]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Partrec.to₂ | [
468,
1
] | Mathlib/Computability/Partrec.lean | [
469,
29
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Finset.left_mem_uIcc | [
914,
1
] | Mathlib/Data/Finset/LocallyFinite.lean | [
915,
39
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
TopHom.coe_inf | [
336,
1
] | Mathlib/Order/Hom/Bounded.lean | [
337,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
convexOn_iff_convex_epigraph | [
262,
1
] | Mathlib/Analysis/Convex/Function.lean | [
264,
58
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
PythagoreanTriple.isClassified_of_normalize_isPrimitiveClassified | [
218,
1
] | Mathlib/NumberTheory/PythagoreanTriples.lean | [
225,
20
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "case h.e'_1\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ x\n\ncase h.e'_2\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ y\n\ncase h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ z",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ IsClassified h",
"tactic": "convert h.normalize.mul_isClassified (Int.gcd x y)\n (isClassified_of_isPrimitiveClassified h.normalize hc) <;>\n rw [Int.mul_ediv_cancel']"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ x",
"tactic": "exact Int.gcd_dvd_left x y"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ y",
"tactic": "exact Int.gcd_dvd_right x y"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ z",
"tactic": "exact h.gcd_dvd"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Wcovby.covby_of_ne | [
373,
1
] | Mathlib/Order/Cover.lean | [
374,
26
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
deriv_ccos | [
199,
1
] | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | [
201,
27
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Finset.empty_disjSum | [
43,
1
] | Mathlib/Data/Finset/Sum.lean | [
44,
39
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg' | [
543,
1
] | Mathlib/MeasureTheory/Integral/SetToL1.lean | [
554,
15
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i",
"state_before": "α : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\n⊢ 0 ≤ setToSimpleFunc T f",
"tactic": "refine' sum_nonneg fun i hi => _"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i\n\ncase neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : ¬i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i",
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"tactic": "by_cases h0 : i = 0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : ¬i = 0\n⊢ 0 ≤ i",
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"tactic": "refine'\n hT_nonneg _ (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable _ hfi h0) i _"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\n⊢ 0 ≤ i",
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"tactic": "rw [mem_range] at hi"
},
{
"state_after": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ i",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\n⊢ 0 ≤ i",
"tactic": "obtain ⟨y, hy⟩ := Set.mem_range.mp hi"
},
{
"state_after": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ ↑f y",
"state_before": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ i",
"tactic": "rw [← hy]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ ↑f y",
"tactic": "convert hf y"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i",
"tactic": "simp [h0]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Ideal.map_comap_of_equiv | [
1724,
1
] | Mathlib/RingTheory/Ideal/Operations.lean | [
1726,
78
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
CancelDenoms.cancel_factors_le | [
72,
1
] | Mathlib/Tactic/CancelDenoms.lean | [
77,
29
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < ad * bd\n\nα : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < 1 / gcd",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ (a ≤ b) = (1 / gcd * (bd * a') ≤ 1 / gcd * (ad * b'))",
"tactic": "rw [mul_le_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_le_mul_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < ad * bd",
"tactic": "exact mul_pos had hbd"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < 1 / gcd",
"tactic": "exact one_div_pos.2 hgcd"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Filter.principal_univ | [
668,
9
] | Mathlib/Order/Filter/Basic.lean | [
669,
75
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.78564\nι : Sort x\nf g : Filter α\ns t : Set α\n⊢ ⊤ ≤ 𝓟 univ",
"tactic": "simp only [le_principal_iff, mem_top, eq_self_iff_true]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
continuousWithinAt_Icc_iff_Ici | [
553,
1
] | Mathlib/Topology/Order/Basic.lean | [
555,
69
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderClosedTopology α\na✝ b✝ : α\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a",
"tactic": "simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
ConjClasses.carrier_eq_preimage_mk | [
329,
1
] | Mathlib/Algebra/Group/Conj.lean | [
330,
41
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
nonempty_ulift | [
100,
1
] | Mathlib/Logic/Nonempty.lean | [
101,
46
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Right.mul_le_one_of_le_of_le | [
852,
1
] | Mathlib/Algebra/Order/Ring/Lemmas.lean | [
854,
44
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.SimpleFunc.range_comp_subset_range | [
356,
1
] | Mathlib/MeasureTheory/Function/SimpleFunc.lean | [
358,
89
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.60622\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β →ₛ γ\ng : α → β\nhgm : Measurable g\n⊢ ↑(SimpleFunc.range (comp f g hgm)) ⊆ ↑(SimpleFunc.range f)",
"tactic": "simp only [coe_range, coe_comp, Set.range_comp_subset_range]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Pi.himp_apply | [
170,
1
] | Mathlib/Order/Heyting/Basic.lean | [
171,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.preimage_const_add_Ioo | [
78,
1
] | Mathlib/Data/Set/Pointwise/Interval.lean | [
79,
25
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a + x) ⁻¹' Ioo b c = Ioo (b - a) (c - a)",
"tactic": "simp [← Ioi_inter_Iio]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Basis.toDual_toDual | [
453,
1
] | Mathlib/LinearAlgebra/Dual.lean | [
456,
82
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ ↑(↑(LinearMap.comp (toDual (dualBasis b)) (toDual b)) (↑b i)) (↑(dualBasis b) j) =\n ↑(↑(Dual.eval R M) (↑b i)) (↑(dualBasis b) j)",
"state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\n⊢ LinearMap.comp (toDual (dualBasis b)) (toDual b) = Dual.eval R M",
"tactic": "refine' b.ext fun i => b.dualBasis.ext fun j => _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ ↑(↑(LinearMap.comp (toDual (dualBasis b)) (toDual b)) (↑b i)) (↑(dualBasis b) j) =\n ↑(↑(Dual.eval R M) (↑b i)) (↑(dualBasis b) j)",
"tactic": "rw [LinearMap.comp_apply, toDual_apply_left, coe_toDual_self, ← coe_dualBasis,\n Dual.eval_apply, Basis.repr_self, Finsupp.single_apply, dualBasis_apply_self]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
TopCat.pullbackIsoProdSubtype_hom_apply | [
138,
1
] | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | [
145,
71
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd",
"tactic": "simpa using ConcreteCategory.congr_hom pullback.condition x"
},
{
"state_after": "case a\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ ↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x) =\n ↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (forget TopCat).map (pullbackIsoProdSubtype f g).hom x =\n { val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }",
"tactic": "apply Subtype.ext"
},
{
"state_after": "case a.h₁\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).fst =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).fst\n\ncase a.h₂\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).snd =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).snd",
"state_before": "case a\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ ↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x) =\n ↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }",
"tactic": "apply Prod.ext"
},
{
"state_after": "no goals",
"state_before": "case a.h₁\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).fst =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).fst\n\ncase a.h₂\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).snd =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).snd",
"tactic": "exacts [ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_fst f g) x,\n ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_snd f g) x]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
CategoryTheory.Functor.preservesEpimorphsisms_of_adjunction | [
178,
1
] | Mathlib/CategoryTheory/Functor/EpiMono.lean | [
185,
41
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : F.map f ≫ g = F.map f ≫ h\n⊢ g = h",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\n⊢ ∀ {Z : D} (g h : F.obj Y ⟶ Z), F.map f ≫ g = F.map f ≫ h → g = h",
"tactic": "intro Z g h H"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ g) = ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ h)\n⊢ g = h",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : F.map f ≫ g = F.map f ≫ h\n⊢ g = h",
"tactic": "replace H := congr_arg (adj.homEquiv X Z) H"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ g) = ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ h)\n⊢ g = h",
"tactic": "rwa [adj.homEquiv_naturality_left, adj.homEquiv_naturality_left, cancel_epi,\n Equiv.apply_eq_iff_eq] at H"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
sSupHom.bot_apply | [
376,
1
] | Mathlib/Order/Hom/CompleteLattice.lean | [
377,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Subgroup.IsComplement.card_mul | [
518,
1
] | Mathlib/GroupTheory/Complement.lean | [
520,
67
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Finset.inter_subset_right | [
1589,
1
] | Mathlib/Data/Finset/Basic.lean | [
1589,
97
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict | [
802,
1
] | Mathlib/MeasureTheory/Function/SimpleFunc.lean | [
807,
52
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range (restrict f s)\nh0 : r ≠ 0\nhs : MeasurableSet s\n⊢ r ∈ ↑f '' s",
"tactic": "simpa [mem_restrict_range hs, h0, -mem_range] using hr"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range 0\nh0 : r ≠ 0\nhs : ¬MeasurableSet s\n⊢ r ∈ ↑f '' s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range (restrict f s)\nh0 : r ≠ 0\nhs : ¬MeasurableSet s\n⊢ r ∈ ↑f '' s",
"tactic": "rw [restrict_of_not_measurable hs] at hr"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range 0\nh0 : r ≠ 0\nhs : ¬MeasurableSet s\n⊢ r ∈ ↑f '' s",
"tactic": "exact (h0 <| eq_zero_of_mem_range_zero hr).elim"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
List.ofFn_nthLe | [
197,
1
] | Mathlib/Data/List/OfFn.lean | [
198,
11
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
gcd_eq_of_dvd_sub_right | [
981,
1
] | Mathlib/Algebra/GCDMonoid/Basic.lean | [
994,
95
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "case hab\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b ∣ c\n\ncase hba\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a c ∣ b",
"state_before": "α : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b = gcd a c",
"tactic": "apply dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _) <;>\n rw [dvd_gcd_iff] <;>\n refine' ⟨gcd_dvd_left _ _, _⟩"
},
{
"state_after": "case hab.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a b ∣ c",
"state_before": "case hab\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b ∣ c",
"tactic": "rcases h with ⟨d, hd⟩"
},
{
"state_after": "case hab.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\n⊢ gcd a b ∣ c",
"state_before": "case hab.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a b ∣ c",
"tactic": "rcases gcd_dvd_right a b with ⟨e, he⟩"
},
{
"state_after": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ gcd a b ∣ c",
"state_before": "case hab.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\n⊢ gcd a b ∣ c",
"tactic": "rcases gcd_dvd_left a b with ⟨f, hf⟩"
},
{
"state_after": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ c = gcd a b * (e - f * d)",
"state_before": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ gcd a b ∣ c",
"tactic": "use e - f * d"
},
{
"state_after": "no goals",
"state_before": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ c = gcd a b * (e - f * d)",
"tactic": "rw [mul_sub, ← he, ← mul_assoc, ← hf, ← hd, sub_sub_cancel]"
},
{
"state_after": "case hba.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a c ∣ b",
"state_before": "case hba\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a c ∣ b",
"tactic": "rcases h with ⟨d, hd⟩"
},
{
"state_after": "case hba.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\n⊢ gcd a c ∣ b",
"state_before": "case hba.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a c ∣ b",
"tactic": "rcases gcd_dvd_right a c with ⟨e, he⟩"
},
{
"state_after": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ gcd a c ∣ b",
"state_before": "case hba.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\n⊢ gcd a c ∣ b",
"tactic": "rcases gcd_dvd_left a c with ⟨f, hf⟩"
},
{
"state_after": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * (e + f * d)",
"state_before": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ gcd a c ∣ b",
"tactic": "use e + f * d"
},
{
"state_after": "no goals",
"state_before": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * (e + f * d)",
"tactic": "rw [mul_add, ← he, ← mul_assoc, ← hf, ← hd, ← add_sub_assoc, add_comm c b, add_sub_cancel]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
Set.iUnion_smul | [
243,
1
] | Mathlib/Data/Set/Pointwise/SMul.lean | [
244,
27
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
ContinuousLinearMap.integral_compLp | [
1065,
1
] | Mathlib/MeasureTheory/Integral/SetIntegral.lean | [
1067,
40
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
IsPrimitiveRoot.geom_sum_eq_zero | [
665,
1
] | Mathlib/RingTheory/RootsOfUnity/Basic.lean | [
668,
49
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "M : Type ?u.2777832\nN : Type ?u.2777835\nG : Type ?u.2777838\nR : Type u_1\nS : Type ?u.2777844\nF : Type ?u.2777847\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh : IsPrimitiveRoot ζ✝ k\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ (1 - ζ) * ∑ i in range k, ζ ^ i = 0",
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"tactic": "refine' eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (hζ.ne_one hk).symm) _"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.2777832\nN : Type ?u.2777835\nG : Type ?u.2777838\nR : Type u_1\nS : Type ?u.2777844\nF : Type ?u.2777847\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh : IsPrimitiveRoot ζ✝ k\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ (1 - ζ) * ∑ i in range k, ζ ^ i = 0",
"tactic": "rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
MeasureTheory.Measure.sub_le_of_le_add | [
45,
1
] | Mathlib/MeasureTheory/Measure/Sub.lean | [
46,
12
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
PiTensorProduct.isEmptyEquiv_apply_tprod | [
556,
1
] | Mathlib/LinearAlgebra/PiTensorProduct.lean | [
557,
15
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
List.append_inj_left' | [
75,
1
] | Std/Data/List/Init/Lemmas.lean | [
76,
26
] | https://github.com/leanprover/std4 | [] | e68aa8f5fe47aad78987df45f99094afbcb5e936 |
Subgroup.characteristic_iff_comap_eq | [
2000,
1
] | Mathlib/GroupTheory/Subgroup/Basic.lean | [
2001,
44
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
LinearMap.eq_adjoint_iff | [
425,
1
] | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | [
429,
76
] | https://github.com/leanprover-community/mathlib4 | [
{
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"tactic": "refine' ⟨fun h x y => by rw [h, adjoint_inner_left], fun h => _⟩"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1931672\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 F\ninst✝ : FiniteDimensional 𝕜 G\nA : E →ₗ[𝕜] F\nB : F →ₗ[𝕜] E\nh : ∀ (x : E) (y : (fun x => F) x), inner (↑A x) y = inner x (↑B y)\nx : E\n⊢ ↑A x = ↑(↑adjoint B) x",
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"tactic": "ext x"
},
{
"state_after": "no goals",
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"tactic": "exact ext_inner_right 𝕜 fun y => by simp only [adjoint_inner_left, h x y]"
},
{
"state_after": "no goals",
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"tactic": "rw [h, adjoint_inner_left]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1931672\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 F\ninst✝ : FiniteDimensional 𝕜 G\nA : E →ₗ[𝕜] F\nB : F →ₗ[𝕜] E\nh : ∀ (x : E) (y : (fun x => F) x), inner (↑A x) y = inner x (↑B y)\nx : E\ny : (fun x => F) x\n⊢ inner (↑A x) y = inner (↑(↑adjoint B) x) y",
"tactic": "simp only [adjoint_inner_left, h x y]"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
LieAlgebra.derivedSeriesOfIdeal_zero | [
63,
1
] | Mathlib/Algebra/Lie/Solvable.lean | [
64,
6
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
Convex.linear_preimage | [
200,
1
] | Mathlib/Analysis/Convex/Basic.lean | [
204,
27
] | https://github.com/leanprover-community/mathlib4 | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\nβ : Type ?u.40424\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ : Set E\nx✝ : E\ns : Set F\nhs : Convex 𝕜 s\nf : E →ₗ[𝕜] F\nx : E\nhx : x ∈ ↑f ⁻¹' s\ny : E\nhy : y ∈ ↑f ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • x + b • y ∈ ↑f ⁻¹' s",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\nβ : Type ?u.40424\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ : Set E\nx : E\ns : Set F\nhs : Convex 𝕜 s\nf : E →ₗ[𝕜] F\n⊢ Convex 𝕜 (↑f ⁻¹' s)",
"tactic": "intro x hx y hy a b ha hb hab"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\nβ : Type ?u.40424\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ : Set E\nx✝ : E\ns : Set F\nhs : Convex 𝕜 s\nf : E →ₗ[𝕜] F\nx : E\nhx : x ∈ ↑f ⁻¹' s\ny : E\nhy : y ∈ ↑f ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • ↑f x + b • ↑f y ∈ s",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\nβ : Type ?u.40424\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ : Set E\nx✝ : E\ns : Set F\nhs : Convex 𝕜 s\nf : E →ₗ[𝕜] F\nx : E\nhx : x ∈ ↑f ⁻¹' s\ny : E\nhy : y ∈ ↑f ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • x + b • y ∈ ↑f ⁻¹' s",
"tactic": "rw [mem_preimage, f.map_add, f.map_smul, f.map_smul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\nβ : Type ?u.40424\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns✝ : Set E\nx✝ : E\ns : Set F\nhs : Convex 𝕜 s\nf : E →ₗ[𝕜] F\nx : E\nhx : x ∈ ↑f ⁻¹' s\ny : E\nhy : y ∈ ↑f ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • ↑f x + b • ↑f y ∈ s",
"tactic": "exact hs hx hy ha hb hab"
}
] | 5a919533f110b7d76410134a237ee374f24eaaad |
RingHom.map_multiset_prod | [
251,
11
] | Mathlib/Algebra/BigOperators/Basic.lean | [
253,
24
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
IsROrC.ofReal_alg | [
95,
1
] | Mathlib/Data/IsROrC/Basic.lean | [
96,
35
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
ENNReal.toReal_mono | [
1990,
1
] | Mathlib/Data/Real/ENNReal.lean | [
1991,
55
] | https://github.com/leanprover-community/mathlib4 | [] | 5a919533f110b7d76410134a237ee374f24eaaad |
End of preview. Expand
in Dataset Viewer.
https://github.com/lean-dojo/LeanDojo
@article{yang2023leandojo,
title={{LeanDojo}: Theorem Proving with Retrieval-Augmented Language Models},
author={Yang, Kaiyu and Swope, Aidan and Gu, Alex and Chalamala, Rahul and Song, Peiyang and Yu, Shixing and Godil, Saad and Prenger, Ryan and Anandkumar, Anima},
journal={arXiv preprint arXiv:2306.15626},
year={2023}
}
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