file_path
stringlengths 11
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list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Algebra/Order/Hom/Ring.lean | OrderRingHom.coe_OrderAddMonoidHom_id | []
| [
293,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
]
|
Mathlib/Data/Multiset/Pi.lean | Multiset.Pi.cons_same | []
| [
44,
14
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
]
|
Mathlib/Data/Real/Sign.lean | Real.sign_zero | [
{
"state_after": "no goals",
"state_before": "⊢ sign 0 = 0",
"tactic": "rw [sign, if_neg (lt_irrefl _), if_neg (lt_irrefl _)]"
}
]
| [
46,
91
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
]
|
Mathlib/SetTheory/Cardinal/Cofinality.lean | Cardinal.derivFamily_lt_ord | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.166484\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\n⊢ lift (#ι) < c",
"tactic": "rwa [lift_id]"
}
]
| [
1185,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1182,
1
]
|
Mathlib/Data/List/Basic.lean | List.replicate_left_injective | []
| [
485,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
]
|
Mathlib/Order/Chain.lean | subset_succChain | [
{
"state_after": "α : Type u_1\nβ : Type ?u.9526\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\nh : ¬(IsChain r s ∧ ∃ x, SuperChain r s x)\n⊢ s ⊆ SuccChain r s",
"state_before": "α : Type u_1\nβ : Type ?u.9526\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\nh : ¬∃ t, IsChain r s ∧ SuperChain r s t\n⊢ s ⊆ SuccChain r s",
"tactic": "rw [exists_and_left] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9526\nr : α → α → Prop\nc c₁ c₂ c₃ s t : Set α\na b x y : α\nh : ¬(IsChain r s ∧ ∃ x, SuperChain r s x)\n⊢ s ⊆ SuccChain r s",
"tactic": "simp [SuccChain, dif_neg, h, Subset.rfl]"
}
]
| [
190,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
]
|
Mathlib/Data/Set/Function.lean | Set.eqOn_refl | []
| [
205,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
]
|
Mathlib/Data/Real/NNReal.lean | Real.nnabs_of_nonneg | [
{
"state_after": "case a\nx : ℝ\nh : 0 ≤ x\n⊢ ↑(↑nnabs x) = ↑(toNNReal x)",
"state_before": "x : ℝ\nh : 0 ≤ x\n⊢ ↑nnabs x = toNNReal x",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nx : ℝ\nh : 0 ≤ x\n⊢ ↑(↑nnabs x) = ↑(toNNReal x)",
"tactic": "rw [coe_toNNReal x h, coe_nnabs, abs_of_nonneg h]"
}
]
| [
1068,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1066,
1
]
|
Mathlib/Analysis/Normed/Field/Basic.lean | List.norm_prod_le' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.110463\nγ : Type ?u.110466\nι : Type ?u.110469\ninst✝ : SeminormedRing α\na : α\nx✝ : [a] ≠ []\n⊢ ‖prod [a]‖ ≤ prod (map norm [a])",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.110463\nγ : Type ?u.110466\nι : Type ?u.110469\ninst✝ : SeminormedRing α\na b : α\nl : List α\nx✝ : a :: b :: l ≠ []\n⊢ ‖a * prod (b :: l)‖ ≤ ‖a‖ * prod (map norm (b :: l))",
"state_before": "α : Type u_1\nβ : Type ?u.110463\nγ : Type ?u.110466\nι : Type ?u.110469\ninst✝ : SeminormedRing α\na b : α\nl : List α\nx✝ : a :: b :: l ≠ []\n⊢ ‖prod (a :: b :: l)‖ ≤ prod (map norm (a :: b :: l))",
"tactic": "rw [List.map_cons, List.prod_cons, @List.prod_cons _ _ _ ‖a‖]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.110463\nγ : Type ?u.110466\nι : Type ?u.110469\ninst✝ : SeminormedRing α\na b : α\nl : List α\nx✝ : a :: b :: l ≠ []\n⊢ ‖prod (b :: l)‖ ≤ prod (map norm (b :: l))",
"state_before": "α : Type u_1\nβ : Type ?u.110463\nγ : Type ?u.110466\nι : Type ?u.110469\ninst✝ : SeminormedRing α\na b : α\nl : List α\nx✝ : a :: b :: l ≠ []\n⊢ ‖a * prod (b :: l)‖ ≤ ‖a‖ * prod (map norm (b :: l))",
"tactic": "refine' le_trans (norm_mul_le _ _) (mul_le_mul_of_nonneg_left _ (norm_nonneg _))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.110463\nγ : Type ?u.110466\nι : Type ?u.110469\ninst✝ : SeminormedRing α\na b : α\nl : List α\nx✝ : a :: b :: l ≠ []\n⊢ ‖prod (b :: l)‖ ≤ prod (map norm (b :: l))",
"tactic": "exact List.norm_prod_le' (List.cons_ne_nil b l)"
}
]
| [
320,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_ite_one | [
{
"state_after": "case inl\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ∃ i, i ∈ s ∧ f i\n⊢ (∏ i in s, if f i then a else 1) = a\n\ncase inr\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ¬∃ i, i ∈ s ∧ f i\n⊢ (∏ i in s, if f i then a else 1) = 1",
"state_before": "ι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\n⊢ (∏ i in s, if f i then a else 1) = if ∃ i, i ∈ s ∧ f i then a else 1",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl.intro.intro\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\ni : α\nhi : i ∈ s\nhfi : f i\n⊢ (∏ i in s, if f i then a else 1) = a",
"state_before": "case inl\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ∃ i, i ∈ s ∧ f i\n⊢ (∏ i in s, if f i then a else 1) = a",
"tactic": "obtain ⟨i, hi, hfi⟩ := h"
},
{
"state_after": "case inl.intro.intro\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\ni : α\nhi : i ∈ s\nhfi : f i\n⊢ ∀ (b : α), b ∈ s → b ≠ i → (if f b then a else 1) = 1",
"state_before": "case inl.intro.intro\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\ni : α\nhi : i ∈ s\nhfi : f i\n⊢ (∏ i in s, if f i then a else 1) = a",
"tactic": "rw [prod_eq_single_of_mem _ hi, if_pos hfi]"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\ni : α\nhi : i ∈ s\nhfi : f i\n⊢ ∀ (b : α), b ∈ s → b ≠ i → (if f b then a else 1) = 1",
"tactic": "exact fun j hj h => if_neg fun hfj => (hf hj hi h).le_bot ⟨hfj, hfi⟩"
},
{
"state_after": "case inr\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ∀ (i : α), i ∈ s → ¬f i\n⊢ (∏ i in s, if f i then a else 1) = 1",
"state_before": "case inr\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ¬∃ i, i ∈ s ∧ f i\n⊢ (∏ i in s, if f i then a else 1) = 1",
"tactic": "push_neg at h"
},
{
"state_after": "case inr\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ∀ (i : α), i ∈ s → ¬f i\n⊢ ∀ (x : α), x ∈ s → (if f x then a else 1) = 1",
"state_before": "case inr\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ∀ (i : α), i ∈ s → ¬f i\n⊢ (∏ i in s, if f i then a else 1) = 1",
"tactic": "rw [prod_eq_one]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.798449\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → Prop\ninst✝ : DecidablePred f\nhf : Set.PairwiseDisjoint (↑s) f\na : β\nh : ∀ (i : α), i ∈ s → ¬f i\n⊢ ∀ (x : α), x ∈ s → (if f x then a else 1) = 1",
"tactic": "exact fun i hi => if_neg (h i hi)"
}
]
| [
1691,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1683,
1
]
|
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | MeasureTheory.Measure.add_haar_closedBall_mul | [
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1994959\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 ≤ r\ns : ℝ\nhs : 0 ≤ s\nthis : closedBall 0 (r * s) = r • closedBall 0 s\n⊢ ↑↑μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (closedBall 0 s)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1994959\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 ≤ r\ns : ℝ\nhs : 0 ≤ s\n⊢ ↑↑μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (closedBall 0 s)",
"tactic": "have : closedBall (0 : E) (r * s) = r • closedBall (0 : E) s := by\n simp [smul_closedBall r (0 : E) hs, abs_of_nonneg hr]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1994959\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 ≤ r\ns : ℝ\nhs : 0 ≤ s\nthis : closedBall 0 (r * s) = r • closedBall 0 s\n⊢ ↑↑μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (closedBall 0 s)",
"tactic": "simp only [this, add_haar_smul, abs_of_nonneg hr, add_haar_closedBall_center, abs_pow]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1994959\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 ≤ r\ns : ℝ\nhs : 0 ≤ s\n⊢ closedBall 0 (r * s) = r • closedBall 0 s",
"tactic": "simp [smul_closedBall r (0 : E) hs, abs_of_nonneg hr]"
}
]
| [
452,
89
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
448,
1
]
|
Mathlib/RingTheory/RootsOfUnity/Complex.lean | Complex.isPrimitiveRoot_exp_of_coprime | [
{
"state_after": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\n⊢ exp (2 * ↑π * I * (↑i / ↑n)) ^ n = 1 ∧ ∀ (l : ℕ), exp (2 * ↑π * I * (↑i / ↑n)) ^ l = 1 → n ∣ l",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\n⊢ IsPrimitiveRoot (exp (2 * ↑π * I * (↑i / ↑n))) n",
"tactic": "rw [IsPrimitiveRoot.iff_def]"
},
{
"state_after": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\n⊢ (∃ n_1, ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) ∧\n ∀ (l : ℕ), (∃ n_1, ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) → n ∣ l",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\n⊢ exp (2 * ↑π * I * (↑i / ↑n)) ^ n = 1 ∧ ∀ (l : ℕ), exp (2 * ↑π * I * (↑i / ↑n)) ^ l = 1 → n ∣ l",
"tactic": "simp only [← exp_nat_mul, exp_eq_one_iff]"
},
{
"state_after": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ (∃ n_1, ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) ∧\n ∀ (l : ℕ), (∃ n_1, ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) → n ∣ l",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\n⊢ (∃ n_1, ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) ∧\n ∀ (l : ℕ), (∃ n_1, ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) → n ∣ l",
"tactic": "have hn0 : (n : ℂ) ≠ 0 := by exact_mod_cast h0"
},
{
"state_after": "case left\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∃ n_1, ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)\n\ncase right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∀ (l : ℕ), (∃ n_1, ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) → n ∣ l",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ (∃ n_1, ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) ∧\n ∀ (l : ℕ), (∃ n_1, ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) → n ∣ l",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\n⊢ ↑n ≠ 0",
"tactic": "exact_mod_cast h0"
},
{
"state_after": "case left\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑↑i * (2 * ↑π * I)",
"state_before": "case left\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∃ n_1, ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)",
"tactic": "use i"
},
{
"state_after": "no goals",
"state_before": "case left\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ↑n * (2 * ↑π * I * (↑i / ↑n)) = ↑↑i * (2 * ↑π * I)",
"tactic": "field_simp [hn0, mul_comm (i : ℂ), mul_comm (n : ℂ)]"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∀ (l : ℕ) (x : ℤ), ↑i * ↑l * (2 * ↑π * I) = ↑x * (2 * ↑π * I) * ↑n → n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∀ (l : ℕ), (∃ n_1, ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑n_1 * (2 * ↑π * I)) → n ∣ l",
"tactic": "simp only [hn0, mul_right_comm _ _ ↑n, mul_left_inj' two_pi_I_ne_zero, Ne.def, not_false_iff,\n mul_comm _ (i : ℂ), ← mul_assoc _ (i : ℂ), exists_imp, field_simps]"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∀ (l : ℕ) (x : ℤ), ↑(i * l) * (↑(2 * π) * I) = ↑x * (↑(2 * π) * I) * ↑n → n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∀ (l : ℕ) (x : ℤ), ↑i * ↑l * (2 * ↑π * I) = ↑x * (2 * ↑π * I) * ↑n → n ∣ l",
"tactic": "norm_cast"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑k * (↑(2 * π) * I) * ↑n\n⊢ n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\n⊢ ∀ (l : ℕ) (x : ℤ), ↑(i * l) * (↑(2 * π) * I) = ↑x * (↑(2 * π) * I) * ↑n → n ∣ l",
"tactic": "rintro l k hk"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑n * ↑k * (↑(2 * π) * I)\n⊢ n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑k * (↑(2 * π) * I) * ↑n\n⊢ n ∣ l",
"tactic": "conv_rhs at hk => rw [mul_comm, ← mul_assoc]"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑n * ↑k * (↑(2 * π) * I)\nhz : 2 * ↑π * I ≠ 0\n⊢ n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑n * ↑k * (↑(2 * π) * I)\n⊢ n ∣ l",
"tactic": "have hz : 2 * ↑π * I ≠ 0 := by simp [pi_pos.ne.symm, I_ne_zero]"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑i * ↑l = ↑n * ↑k\n⊢ n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑n * ↑k * (↑(2 * π) * I)\nhz : 2 * ↑π * I ≠ 0\n⊢ n ∣ l",
"tactic": "field_simp [hz] at hk"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\n⊢ n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑i * ↑l = ↑n * ↑k\n⊢ n ∣ l",
"tactic": "norm_cast at hk"
},
{
"state_after": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\nthis : n ∣ i * l\n⊢ n ∣ l",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\n⊢ n ∣ l",
"tactic": "have : n ∣ i * l := by rw [← Int.coe_nat_dvd, hk, mul_comm]; apply dvd_mul_left"
},
{
"state_after": "no goals",
"state_before": "case right\ni n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\nthis : n ∣ i * l\n⊢ n ∣ l",
"tactic": "exact hi.symm.dvd_of_dvd_mul_left this"
},
{
"state_after": "no goals",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhk : ↑(i * l) * (↑(2 * π) * I) = ↑n * ↑k * (↑(2 * π) * I)\n⊢ 2 * ↑π * I ≠ 0",
"tactic": "simp [pi_pos.ne.symm, I_ne_zero]"
},
{
"state_after": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\n⊢ ↑n ∣ k * ↑n",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\n⊢ n ∣ i * l",
"tactic": "rw [← Int.coe_nat_dvd, hk, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "i n : ℕ\nh0 : n ≠ 0\nhi : Nat.coprime i n\nhn0 : ↑n ≠ 0\nl : ℕ\nk : ℤ\nhz : 2 * ↑π * I ≠ 0\nhk : ↑(i * l) = ↑n * k\n⊢ ↑n ∣ k * ↑n",
"tactic": "apply dvd_mul_left"
}
]
| [
52,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
35,
1
]
|
Mathlib/Topology/MetricSpace/Contracting.lean | ContractingWith.edist_le_of_fixedPoint | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : EMetricSpace α\ncs : CompleteSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nh : edist x y ≠ ⊤\nhy : IsFixedPt f y\n⊢ edist x y ≤ edist x (f x) / (1 - ↑K)",
"tactic": "simpa only [hy.eq, edist_self, add_zero] using hf.edist_inequality h"
}
]
| [
82,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
]
|
Mathlib/GroupTheory/Commutator.lean | Subgroup.commutator_def' | []
| [
147,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
]
|
Mathlib/ModelTheory/Syntax.lean | FirstOrder.Language.BoundedFormula.isPrenex_toPrenexImp | [
{
"state_after": "case of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α n✝\nhφ : IsQF φ✝\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp φ✝ ψ)\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nih1 : ∀ {ψ : BoundedFormula L α (n✝ + 1)}, IsPrenex ψ → IsPrenex (toPrenexImp φ✝ ψ)\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp (all φ✝) ψ)\n\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nih2 : ∀ {ψ : BoundedFormula L α (n✝ + 1)}, IsPrenex ψ → IsPrenex (toPrenexImp φ✝ ψ)\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp (BoundedFormula.ex φ✝) ψ)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝ ψ✝ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ : BoundedFormula L α n\nhφ : IsPrenex φ\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp φ ψ)",
"tactic": "induction' hφ with _ _ hφ _ _ _ ih1 _ _ _ ih2"
},
{
"state_after": "case of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α n✝\nhφ : IsQF φ✝\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImpRight φ✝ ψ)",
"state_before": "case of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α n✝\nhφ : IsQF φ✝\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp φ✝ ψ)",
"tactic": "rw [hφ.toPrenexImp]"
},
{
"state_after": "no goals",
"state_before": "case of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α n✝\nhφ : IsQF φ✝\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImpRight φ✝ ψ)",
"tactic": "exact isPrenex_toPrenexImpRight hφ hψ"
},
{
"state_after": "no goals",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nih1 : ∀ {ψ : BoundedFormula L α (n✝ + 1)}, IsPrenex ψ → IsPrenex (toPrenexImp φ✝ ψ)\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp (all φ✝) ψ)",
"tactic": "exact (ih1 hψ.liftAt).ex"
},
{
"state_after": "no goals",
"state_before": "case ex\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.138402\nP : Type ?u.138405\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.138433\nn l : ℕ\nφ✝¹ ψ✝¹ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ ψ✝ : BoundedFormula L α n\nhψ✝ : IsPrenex ψ✝\nn✝ : ℕ\nφ✝ : BoundedFormula L α (n✝ + 1)\nh✝ : IsPrenex φ✝\nih2 : ∀ {ψ : BoundedFormula L α (n✝ + 1)}, IsPrenex ψ → IsPrenex (toPrenexImp φ✝ ψ)\nψ : BoundedFormula L α n✝\nhψ : IsPrenex ψ\n⊢ IsPrenex (toPrenexImp (BoundedFormula.ex φ✝) ψ)",
"tactic": "exact (ih2 hψ.liftAt).all"
}
]
| [
850,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
844,
1
]
|
Std/Logic.lean | decide_eq_false_iff_not | []
| [
525,
40
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
524,
9
]
|
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.ofMulEquivOfDom_comp | []
| [
1534,
95
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1533,
1
]
|
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | MeasureTheory.SignedMeasure.someExistsOneDivLT_measurableSet | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬restrict s i ≤ restrict 0 i\n⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)\n\ncase neg\nα : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬¬restrict s i ≤ restrict 0 i\n⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)",
"state_before": "α : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\n⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)",
"tactic": "by_cases hi : ¬s ≤[i] 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬restrict s i ≤ restrict 0 i\n⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)",
"tactic": "exact\n let ⟨_, h, _⟩ := someExistsOneDivLT_spec hi\n h"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬¬restrict s i ≤ restrict 0 i\n⊢ MeasurableSet ∅",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬¬restrict s i ≤ restrict 0 i\n⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)",
"tactic": "rw [someExistsOneDivLT, dif_neg hi]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.7479\ninst✝³ : MeasurableSpace α\nM : Type ?u.7485\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬¬restrict s i ≤ restrict 0 i\n⊢ MeasurableSet ∅",
"tactic": "exact MeasurableSet.empty"
}
]
| [
153,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
9
]
|
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | CircleDeg1Lift.translationNumber_one | [
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\n⊢ Tendsto (fun n => (↑1^[n]) 0 / ↑n) atTop (𝓝 0)",
"tactic": "simp [tendsto_const_nhds]"
}
]
| [
713,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
712,
1
]
|
Mathlib/Data/Nat/Factors.lean | Nat.perm_factors_mul | [
{
"state_after": "case refine'_1\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ prod (factors a ++ factors b) = a * b\n\ncase refine'_2\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ∀ (p : ℕ), p ∈ factors a ++ factors b → Prime p",
"state_before": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ factors (a * b) ~ factors a ++ factors b",
"tactic": "refine' (factors_unique _ _).symm"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ prod (factors a ++ factors b) = a * b",
"tactic": "rw [List.prod_append, prod_factors ha, prod_factors hb]"
},
{
"state_after": "case refine'_2\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nhp : p ∈ factors a ++ factors b\n⊢ Prime p",
"state_before": "case refine'_2\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ∀ (p : ℕ), p ∈ factors a ++ factors b → Prime p",
"tactic": "intro p hp"
},
{
"state_after": "case refine'_2\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nhp : p ∈ factors a ∨ p ∈ factors b\n⊢ Prime p",
"state_before": "case refine'_2\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nhp : p ∈ factors a ++ factors b\n⊢ Prime p",
"tactic": "rw [List.mem_append] at hp"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\na b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\nhp : p ∈ factors a ∨ p ∈ factors b\n⊢ Prime p",
"tactic": "cases' hp with hp' hp' <;> exact prime_of_mem_factors hp'"
}
]
| [
207,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | hasFDerivAt_iff_tendsto | []
| [
318,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
316,
1
]
|
Mathlib/GroupTheory/FreeProduct.lean | FreeProduct.Word.prod_smul | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\nm : FreeProduct M\n⊢ ∀ (w : Word M), prod (m • w) = m * prod w",
"tactic": "induction m using FreeProduct.induction_on with\n| h_one =>\n intro\n rw [one_smul, one_mul]\n| h_of _ =>\n intros\n rw [of_smul_def, prod_rcons, of.map_mul, mul_assoc, ← prod_rcons, ← equivPair_symm,\n Equiv.symm_apply_apply]\n| h_mul x y hx hy =>\n intro w\n rw [mul_smul, hx, hy, mul_assoc]"
},
{
"state_after": "case h_one\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\nw✝ : Word M\n⊢ prod (1 • w✝) = 1 * prod w✝",
"state_before": "case h_one\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\n⊢ ∀ (w : Word M), prod (1 • w) = 1 * prod w",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case h_one\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\nw✝ : Word M\n⊢ prod (1 • w✝) = 1 * prod w✝",
"tactic": "rw [one_smul, one_mul]"
},
{
"state_after": "case h_of\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni✝ : ι\nm✝ : M i✝\nw✝ : Word M\n⊢ prod (↑of m✝ • w✝) = ↑of m✝ * prod w✝",
"state_before": "case h_of\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni✝ : ι\nm✝ : M i✝\n⊢ ∀ (w : Word M), prod (↑of m✝ • w) = ↑of m✝ * prod w",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case h_of\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\ni✝ : ι\nm✝ : M i✝\nw✝ : Word M\n⊢ prod (↑of m✝ • w✝) = ↑of m✝ * prod w✝",
"tactic": "rw [of_smul_def, prod_rcons, of.map_mul, mul_assoc, ← prod_rcons, ← equivPair_symm,\n Equiv.symm_apply_apply]"
},
{
"state_after": "case h_mul\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\nx y : FreeProduct M\nhx : ∀ (w : Word M), prod (x • w) = x * prod w\nhy : ∀ (w : Word M), prod (y • w) = y * prod w\nw : Word M\n⊢ prod ((x * y) • w) = x * y * prod w",
"state_before": "case h_mul\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\nx y : FreeProduct M\nhx : ∀ (w : Word M), prod (x • w) = x * prod w\nhy : ∀ (w : Word M), prod (y • w) = y * prod w\n⊢ ∀ (w : Word M), prod ((x * y) • w) = x * y * prod w",
"tactic": "intro w"
},
{
"state_after": "no goals",
"state_before": "case h_mul\nι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type ?u.466820\ninst✝² : Monoid N\ninst✝¹ : (i : ι) → DecidableEq (M i)\ninst✝ : DecidableEq ι\nx y : FreeProduct M\nhx : ∀ (w : Word M), prod (x • w) = x * prod w\nhy : ∀ (w : Word M), prod (y • w) = y * prod w\nw : Word M\n⊢ prod ((x * y) • w) = x * y * prod w",
"tactic": "rw [mul_smul, hx, hy, mul_assoc]"
}
]
| [
458,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
447,
1
]
|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | BoxIntegral.Prepartition.monotone_restrict | []
| [
506,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
505,
1
]
|
Mathlib/Order/Iterate.lean | Function.Commute.iterate_pos_le_iff_map_le | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\nf g : α → α\nh : Commute f g\nhf : Monotone f\nhg : StrictMono g\nx : α\nn : ℕ\nhn : 0 < n\n⊢ (f^[n]) x ≤ (g^[n]) x ↔ f x ≤ g x",
"tactic": "simpa only [not_lt] using not_congr (h.symm.iterate_pos_lt_iff_map_lt' hg hf hn)"
}
]
| [
211,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
]
|
Mathlib/Analysis/MeanInequalities.lean | Real.inner_le_Lp_mul_Lq | [
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\nthis :\n ↑(∑ i in s,\n (fun i => { val := abs (f i), property := (_ : 0 ≤ abs (f i)) }) i *\n (fun i => { val := abs (g i), property := (_ : 0 ≤ abs (g i)) }) i) ≤\n ↑((∑ i in s, (fun i => { val := abs (f i), property := (_ : 0 ≤ abs (f i)) }) i ^ p) ^ (1 / p) *\n (∑ i in s, (fun i => { val := abs (g i), property := (_ : 0 ≤ abs (g i)) }) i ^ q) ^ (1 / q))\n⊢ ∑ i in s, f i * g i ≤ (∑ i in s, abs (f i) ^ p) ^ (1 / p) * (∑ i in s, abs (g i) ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\n⊢ ∑ i in s, f i * g i ≤ (∑ i in s, abs (f i) ^ p) ^ (1 / p) * (∑ i in s, abs (g i) ^ q) ^ (1 / q)",
"tactic": "have :=\n NNReal.coe_le_coe.2\n (NNReal.inner_le_Lp_mul_Lq s (fun i => ⟨_, abs_nonneg (f i)⟩) (fun i => ⟨_, abs_nonneg (g i)⟩)\n hpq)"
},
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\nthis : ∑ x in s, abs (f x) * abs (g x) ≤ (∑ x in s, abs (f x) ^ p) ^ (1 / p) * (∑ x in s, abs (g x) ^ q) ^ (1 / q)\n⊢ ∑ i in s, f i * g i ≤ (∑ i in s, abs (f i) ^ p) ^ (1 / p) * (∑ i in s, abs (g i) ^ q) ^ (1 / q)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\nthis :\n ↑(∑ i in s,\n (fun i => { val := abs (f i), property := (_ : 0 ≤ abs (f i)) }) i *\n (fun i => { val := abs (g i), property := (_ : 0 ≤ abs (g i)) }) i) ≤\n ↑((∑ i in s, (fun i => { val := abs (f i), property := (_ : 0 ≤ abs (f i)) }) i ^ p) ^ (1 / p) *\n (∑ i in s, (fun i => { val := abs (g i), property := (_ : 0 ≤ abs (g i)) }) i ^ q) ^ (1 / q))\n⊢ ∑ i in s, f i * g i ≤ (∑ i in s, abs (f i) ^ p) ^ (1 / p) * (∑ i in s, abs (g i) ^ q) ^ (1 / q)",
"tactic": "push_cast at this"
},
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\nthis : ∑ x in s, abs (f x) * abs (g x) ≤ (∑ x in s, abs (f x) ^ p) ^ (1 / p) * (∑ x in s, abs (g x) ^ q) ^ (1 / q)\ni : ι\nx✝ : i ∈ s\n⊢ f i * g i ≤ abs (f i) * abs (g i)",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\nthis : ∑ x in s, abs (f x) * abs (g x) ≤ (∑ x in s, abs (f x) ^ p) ^ (1 / p) * (∑ x in s, abs (g x) ^ q) ^ (1 / q)\n⊢ ∑ i in s, f i * g i ≤ (∑ i in s, abs (f i) ^ p) ^ (1 / p) * (∑ i in s, abs (g i) ^ q) ^ (1 / q)",
"tactic": "refine' le_trans (sum_le_sum fun i _ => _) this"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : IsConjugateExponent p q\nthis : ∑ x in s, abs (f x) * abs (g x) ≤ (∑ x in s, abs (f x) ^ p) ^ (1 / p) * (∑ x in s, abs (g x) ^ q) ^ (1 / q)\ni : ι\nx✝ : i ∈ s\n⊢ f i * g i ≤ abs (f i) * abs (g i)",
"tactic": "simp only [← abs_mul, le_abs_self]"
}
]
| [
564,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
556,
1
]
|
Mathlib/Init/Logic.lean | heq_of_eq_rec_right | []
| [
76,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
]
|
Mathlib/RingTheory/OreLocalization/Basic.lean | OreLocalization.oreDiv_add_oreDiv | []
| [
607,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
1
]
|
Mathlib/Data/Fintype/Basic.lean | Finset.univ_eq_empty_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2002\nγ : Type ?u.2005\ninst✝ : Fintype α\ns t : Finset α\n⊢ univ = ∅ ↔ IsEmpty α",
"tactic": "rw [← not_nonempty_iff, ← univ_nonempty_iff, not_nonempty_iff_eq_empty]"
}
]
| [
113,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.sigmaFinite_iff | []
| [
3442,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3441,
1
]
|
Mathlib/Deprecated/Group.lean | Inv.isGroupHom | []
| [
402,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
401,
1
]
|
Mathlib/Topology/Algebra/Constructions.lean | Units.embedding_embedProduct | []
| [
111,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
]
|
Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.ext | []
| [
73,
5
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
]
|
Mathlib/Data/Fintype/Card.lean | Fintype.card_le_of_surjective | []
| [
486,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
1
]
|
Mathlib/Algebra/Order/Group/Defs.lean | inv_le_div_iff_le_mul' | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a⁻¹ ≤ b / c ↔ c ≤ a * b",
"tactic": "rw [inv_le_div_iff_le_mul, mul_comm]"
}
]
| [
834,
100
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
834,
1
]
|
Mathlib/Data/Finset/Basic.lean | List.toFinset.ext | []
| [
3289,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3288,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.hasSum_lt | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\nhsg : sg = ⊤\n⊢ sf < sg\n\ncase neg\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\nhsg : ¬sg = ⊤\n⊢ sf < sg",
"state_before": "α : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\n⊢ sf < sg",
"tactic": "by_cases hsg : sg = ⊤"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\nhsg : sg = ⊤\n⊢ sf < sg",
"tactic": "exact hsg.symm ▸ lt_of_le_of_ne le_top hsf"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\nhsg : ¬sg = ⊤\nhg' : ∀ (x : α), g x ≠ ⊤\n⊢ sf < sg",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\nhsg : ¬sg = ⊤\n⊢ sf < sg",
"tactic": "have hg' : ∀ x, g x ≠ ⊤ := ENNReal.ne_top_of_tsum_ne_top (hg.tsum_eq.symm ▸ hsg)"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\ng : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nhsf : sf ≠ ⊤\nhg : HasSum g sg\nhsg : ¬sg = ⊤\nhg' : ∀ (x : α), g x ≠ ⊤\nf : α → ℝ≥0\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ g a\nhi : (fun i => ↑(f i)) i < g i\nhf : HasSum (fun i => ↑(f i)) sf\n⊢ sf < sg",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nf g : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhsf : sf ≠ ⊤\nhf : HasSum f sf\nhg : HasSum g sg\nhsg : ¬sg = ⊤\nhg' : ∀ (x : α), g x ≠ ⊤\n⊢ sf < sg",
"tactic": "lift f to α → ℝ≥0 using fun x =>\n ne_of_lt (lt_of_le_of_lt (h x) <| lt_of_le_of_ne le_top (hg' x))"
},
{
"state_after": "case neg.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nsf sg : ℝ≥0∞\ni : α\nhsf : sf ≠ ⊤\nhsg : ¬sg = ⊤\nf : α → ℝ≥0\nhf : HasSum (fun i => ↑(f i)) sf\ng : α → ℝ≥0\nhg : HasSum (fun i => ↑(g i)) sg\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ (fun i => ↑(g i)) a\nhi : (fun i => ↑(f i)) i < (fun i => ↑(g i)) i\n⊢ sf < sg",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\ng : α → ℝ≥0∞\nsf sg : ℝ≥0∞\ni : α\nhsf : sf ≠ ⊤\nhg : HasSum g sg\nhsg : ¬sg = ⊤\nhg' : ∀ (x : α), g x ≠ ⊤\nf : α → ℝ≥0\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ g a\nhi : (fun i => ↑(f i)) i < g i\nhf : HasSum (fun i => ↑(f i)) sf\n⊢ sf < sg",
"tactic": "lift g to α → ℝ≥0 using hg'"
},
{
"state_after": "case neg.intro.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nsg : ℝ≥0∞\ni : α\nhsg : ¬sg = ⊤\nf g : α → ℝ≥0\nhg : HasSum (fun i => ↑(g i)) sg\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ (fun i => ↑(g i)) a\nhi : (fun i => ↑(f i)) i < (fun i => ↑(g i)) i\nsf : ℝ≥0\nhf : HasSum (fun i => ↑(f i)) ↑sf\n⊢ ↑sf < sg",
"state_before": "case neg.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nsf sg : ℝ≥0∞\ni : α\nhsf : sf ≠ ⊤\nhsg : ¬sg = ⊤\nf : α → ℝ≥0\nhf : HasSum (fun i => ↑(f i)) sf\ng : α → ℝ≥0\nhg : HasSum (fun i => ↑(g i)) sg\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ (fun i => ↑(g i)) a\nhi : (fun i => ↑(f i)) i < (fun i => ↑(g i)) i\n⊢ sf < sg",
"tactic": "lift sf to ℝ≥0 using hsf"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\ni : α\nf g : α → ℝ≥0\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ (fun i => ↑(g i)) a\nhi : (fun i => ↑(f i)) i < (fun i => ↑(g i)) i\nsf : ℝ≥0\nhf : HasSum (fun i => ↑(f i)) ↑sf\nsg : ℝ≥0\nhg : HasSum (fun i => ↑(g i)) ↑sg\n⊢ ↑sf < ↑sg",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\nsg : ℝ≥0∞\ni : α\nhsg : ¬sg = ⊤\nf g : α → ℝ≥0\nhg : HasSum (fun i => ↑(g i)) sg\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ (fun i => ↑(g i)) a\nhi : (fun i => ↑(f i)) i < (fun i => ↑(g i)) i\nsf : ℝ≥0\nhf : HasSum (fun i => ↑(f i)) ↑sf\n⊢ ↑sf < sg",
"tactic": "lift sg to ℝ≥0 using hsg"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\ni : α\nf g : α → ℝ≥0\nsf : ℝ≥0\nhf : HasSum (fun i => ↑(f i)) ↑sf\nsg : ℝ≥0\nhg : HasSum (fun i => ↑(g i)) ↑sg\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\n⊢ sf < sg",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\ni : α\nf g : α → ℝ≥0\nh : ∀ (a : α), (fun i => ↑(f i)) a ≤ (fun i => ↑(g i)) a\nhi : (fun i => ↑(f i)) i < (fun i => ↑(g i)) i\nsf : ℝ≥0\nhf : HasSum (fun i => ↑(f i)) ↑sf\nsg : ℝ≥0\nhg : HasSum (fun i => ↑(g i)) ↑sg\n⊢ ↑sf < ↑sg",
"tactic": "simp only [coe_le_coe, coe_lt_coe] at h hi⊢"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.336123\nγ : Type ?u.336126\ni : α\nf g : α → ℝ≥0\nsf : ℝ≥0\nhf : HasSum (fun i => ↑(f i)) ↑sf\nsg : ℝ≥0\nhg : HasSum (fun i => ↑(g i)) ↑sg\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\n⊢ sf < sg",
"tactic": "exact NNReal.hasSum_lt h hi (ENNReal.hasSum_coe.1 hf) (ENNReal.hasSum_coe.1 hg)"
}
]
| [
1276,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1265,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_sigma | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.312096\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nσ : α → Type u_1\ns : Finset α\nt : (a : α) → Finset (σ a)\nf : Sigma σ → β\n⊢ ∏ x in Finset.sigma s t, f x = ∏ a in s, ∏ s in t a, f { fst := a, snd := s }",
"tactic": "simp_rw [← disjiUnion_map_sigma_mk, prod_disjiUnion, prod_map, Function.Embedding.sigmaMk_apply]"
}
]
| [
529,
99
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
527,
1
]
|
Mathlib/Order/Filter/Extr.lean | IsMaxFilter.bicomp_mono | []
| [
380,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
377,
1
]
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | EulerSine.antideriv_cos_comp_const_mul | [
{
"state_after": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"state_before": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"tactic": "have a : HasDerivAt (fun y : ℂ => y * (2 * z)) _ x := hasDerivAt_mul_const _"
},
{
"state_after": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"state_before": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"tactic": "have b : HasDerivAt (fun y : ℂ => Complex.sin (y * (2 * z))) _ x :=\n HasDerivAt.comp (x : ℂ) (Complex.hasDerivAt_sin (x * (2 * z))) a"
},
{
"state_after": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\nc : HasDerivAt (fun x => Complex.sin (↑x * (2 * z)) / (2 * z)) (Complex.cos (↑x * (2 * z)) * (2 * z) / (2 * z)) x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"state_before": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"tactic": "have c := b.comp_ofReal.div_const (2 * z)"
},
{
"state_after": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\nc : HasDerivAt (fun x => Complex.sin (↑x * (2 * z)) / (2 * z)) (Complex.cos (↑x * (2 * z))) x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"state_before": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\nc : HasDerivAt (fun x => Complex.sin (↑x * (2 * z)) / (2 * z)) (Complex.cos (↑x * (2 * z)) * (2 * z) / (2 * z)) x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"tactic": "field_simp at c"
},
{
"state_after": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\nc : HasDerivAt (fun x => Complex.sin (2 * z * ↑x) / (2 * z)) (Complex.cos (2 * z * ↑x)) x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"state_before": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\nc : HasDerivAt (fun x => Complex.sin (↑x * (2 * z)) / (2 * z)) (Complex.cos (↑x * (2 * z))) x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"tactic": "simp only [fun y => mul_comm y (2 * z)] at c"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn : ℕ\nhz : z ≠ 0\nx : ℝ\na : HasDerivAt (fun y => y * (2 * z)) (2 * z) ↑x\nb : HasDerivAt (fun y => Complex.sin (y * (2 * z))) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x\nc : HasDerivAt (fun x => Complex.sin (2 * z * ↑x) / (2 * z)) (Complex.cos (2 * z * ↑x)) x\n⊢ HasDerivAt (fun y => Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x",
"tactic": "exact c"
}
]
| [
52,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
]
|
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.factors_pow | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\n⊢ Multiset.Rel Associated (factors (x ^ 0)) (0 • factors x)",
"tactic": "rw [zero_smul, pow_zero, factors_one, Multiset.rel_zero_right]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : x = 0\n⊢ Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) • factors x)\n\ncase neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) • factors x)",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\n⊢ Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) • factors x)",
"tactic": "by_cases h0 : x = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : x = 0\n⊢ Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) • factors x)",
"tactic": "simp [h0, zero_pow n.succ_pos, smul_zero]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors (x * x ^ n)) (factors x + n • factors x)",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) • factors x)",
"tactic": "rw [pow_succ, succ_nsmul]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors x + factors (x ^ n)) (factors x + n • factors x)",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors (x * x ^ n)) (factors x + n • factors x)",
"tactic": "refine' Multiset.Rel.trans _ (factors_mul h0 (pow_ne_zero n h0)) _"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors x) (factors x)",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors x + factors (x ^ n)) (factors x + n • factors x)",
"tactic": "refine' Multiset.Rel.add _ <| factors_pow n"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn✝ n : ℕ\nh0 : ¬x = 0\n⊢ Multiset.Rel Associated (factors x) (factors x)",
"tactic": "exact Multiset.rel_refl_of_refl_on fun y _ => Associated.refl _"
}
]
| [
530,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
520,
1
]
|
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.single_add | [
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\ni' : ι\nh : i = i'\n⊢ ↑(single i (b₁ + b₂)) i' = ↑(single i b₁ + single i b₂) i'\n\ncase neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\ni' : ι\nh : ¬i = i'\n⊢ ↑(single i (b₁ + b₂)) i' = ↑(single i b₁ + single i b₂) i'",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\ni' : ι\n⊢ ↑(single i (b₁ + b₂)) i' = ↑(single i b₁ + single i b₂) i'",
"tactic": "by_cases h : i = i'"
},
{
"state_after": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\n⊢ ↑(single i (b₁ + b₂)) i = ↑(single i b₁ + single i b₂) i",
"state_before": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\ni' : ι\nh : i = i'\n⊢ ↑(single i (b₁ + b₂)) i' = ↑(single i b₁ + single i b₂) i'",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\n⊢ ↑(single i (b₁ + b₂)) i = ↑(single i b₁ + single i b₂) i",
"tactic": "simp only [add_apply, single_eq_same]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nb₁ b₂ : β i\ni' : ι\nh : ¬i = i'\n⊢ ↑(single i (b₁ + b₂)) i' = ↑(single i b₁ + single i b₂) i'",
"tactic": "simp only [add_apply, single_eq_of_ne h, zero_add]"
}
]
| [
884,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
879,
1
]
|
Mathlib/ModelTheory/Types.lean | FirstOrder.Language.Theory.CompleteType.compl_setOf_mem | []
| [
114,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
]
|
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.nonneg_mul_lem | [
{
"state_after": "d x y : ℕ\na : ℤ√↑d\nha : Nonneg a\nthis : { re := ↑x, im := ↑y } * a = ↑x * a + sqrtd * (↑y * a)\n⊢ Nonneg ({ re := ↑x, im := ↑y } * a)",
"state_before": "d x y : ℕ\na : ℤ√↑d\nha : Nonneg a\n⊢ Nonneg ({ re := ↑x, im := ↑y } * a)",
"tactic": "have : (⟨x, y⟩ * a : ℤ√d) = (x : ℤ√d) * a + sqrtd * ((y : ℤ√d) * a) := by\n rw [decompose, right_distrib, mul_assoc, Int.cast_ofNat, Int.cast_ofNat]"
},
{
"state_after": "d x y : ℕ\na : ℤ√↑d\nha : Nonneg a\nthis : { re := ↑x, im := ↑y } * a = ↑x * a + sqrtd * (↑y * a)\n⊢ Nonneg (↑x * a + sqrtd * (↑y * a))",
"state_before": "d x y : ℕ\na : ℤ√↑d\nha : Nonneg a\nthis : { re := ↑x, im := ↑y } * a = ↑x * a + sqrtd * (↑y * a)\n⊢ Nonneg ({ re := ↑x, im := ↑y } * a)",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "d x y : ℕ\na : ℤ√↑d\nha : Nonneg a\nthis : { re := ↑x, im := ↑y } * a = ↑x * a + sqrtd * (↑y * a)\n⊢ Nonneg (↑x * a + sqrtd * (↑y * a))",
"tactic": "exact (nonneg_smul ha).add (nonneg_muld <| nonneg_smul ha)"
},
{
"state_after": "no goals",
"state_before": "d x y : ℕ\na : ℤ√↑d\nha : Nonneg a\n⊢ { re := ↑x, im := ↑y } * a = ↑x * a + sqrtd * (↑y * a)",
"tactic": "rw [decompose, right_distrib, mul_assoc, Int.cast_ofNat, Int.cast_ofNat]"
}
]
| [
825,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
821,
1
]
|
Mathlib/Topology/MetricSpace/EMetricSpace.lean | uniformity_basis_edist_inv_nat | []
| [
255,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
1
]
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.tendsto_lintegral_of_dominated_convergence' | [
{
"state_after": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))",
"tactic": "have : ∀ n, (∫⁻ a, F n a ∂μ) = ∫⁻ a, (hF_meas n).mk (F n) a ∂μ := fun n =>\n lintegral_congr_ae (hF_meas n).ae_eq_mk"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))",
"tactic": "simp_rw [this]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))\n\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) ≤ᵐ[μ] bound",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))",
"tactic": "apply\n tendsto_lintegral_of_dominated_convergence bound (fun n => (hF_meas n).measurable_mk) _ h_fin"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"tactic": "have : ∀ n, ∀ᵐ a ∂μ, (hF_meas n).mk (F n) a = F n a := fun n => (hF_meas n).ae_eq_mk.symm"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝¹ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"tactic": "have : ∀ᵐ a ∂μ, ∀ n, (hF_meas n).mk (F n) a = F n a := ae_all_iff.mpr this"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝¹ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\na : α\nH : ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nH' : Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝¹ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\n⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"tactic": "filter_upwards [this, h_lim] with a H H'"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝¹ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\na : α\nH : ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nH' : Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => F n a) atTop (𝓝 (f a))",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝¹ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\na : α\nH : ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nH' : Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a))",
"tactic": "simp_rw [H]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis✝¹ : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nthis✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\na : α\nH : ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nH' : Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => F n a) atTop (𝓝 (f a))",
"tactic": "exact H'"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nn : ℕ\n⊢ AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) ≤ᵐ[μ] bound",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\n⊢ ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) ≤ᵐ[μ] bound",
"tactic": "intro n"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nn : ℕ\na : α\nH : F n a ≤ bound a\nH' : F n a = AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a\n⊢ AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ≤ bound a",
"state_before": "α : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nn : ℕ\n⊢ AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) ≤ᵐ[μ] bound",
"tactic": "filter_upwards [h_bound n, (hF_meas n).ae_eq_mk] with a H H'"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1001538\nγ : Type ?u.1001541\nδ : Type ?u.1001544\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n)\nh_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound\nh_fin : (∫⁻ (a : α), bound a ∂μ) ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), (∫⁻ (a : α), F n a ∂μ) = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ\nn : ℕ\na : α\nH : F n a ≤ bound a\nH' : F n a = AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a\n⊢ AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ≤ bound a",
"tactic": "rwa [H'] at H"
}
]
| [
1077,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1061,
1
]
|
Mathlib/Data/Nat/Order/Basic.lean | Nat.mul_self_inj | []
| [
322,
95
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
320,
1
]
|
Mathlib/GroupTheory/Exponent.lean | Monoid.lcm_orderOf_dvd_exponent | [
{
"state_after": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Fintype G\n⊢ ∀ (b : G), b ∈ Finset.univ → orderOf b ∣ exponent G",
"state_before": "G : Type u\ninst✝¹ : Monoid G\ninst✝ : Fintype G\n⊢ Finset.lcm Finset.univ orderOf ∣ exponent G",
"tactic": "apply Finset.lcm_dvd"
},
{
"state_after": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Fintype G\ng : G\na✝ : g ∈ Finset.univ\n⊢ orderOf g ∣ exponent G",
"state_before": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Fintype G\n⊢ ∀ (b : G), b ∈ Finset.univ → orderOf b ∣ exponent G",
"tactic": "intro g _"
},
{
"state_after": "no goals",
"state_before": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Fintype G\ng : G\na✝ : g ∈ Finset.univ\n⊢ orderOf g ∣ exponent G",
"tactic": "exact order_dvd_exponent g"
}
]
| [
187,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
183,
1
]
|
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | IsCountablySpanning.prod | [
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.552\nβ : Type u_2\nβ' : Type ?u.558\nγ : Type ?u.561\nE : Type ?u.564\nC : Set (Set α)\nD : Set (Set β)\ns : ℕ → Set α\nh1s : ∀ (n : ℕ), s n ∈ C\nh2s : (⋃ (n : ℕ), s n) = univ\nt : ℕ → Set β\nh1t : ∀ (n : ℕ), t n ∈ D\nh2t : (⋃ (n : ℕ), t n) = univ\n⊢ IsCountablySpanning (image2 (fun x x_1 => x ×ˢ x_1) C D)",
"state_before": "α : Type u_1\nα' : Type ?u.552\nβ : Type u_2\nβ' : Type ?u.558\nγ : Type ?u.561\nE : Type ?u.564\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n⊢ IsCountablySpanning (image2 (fun x x_1 => x ×ˢ x_1) C D)",
"tactic": "rcases hC, hD with ⟨⟨s, h1s, h2s⟩, t, h1t, h2t⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.552\nβ : Type u_2\nβ' : Type ?u.558\nγ : Type ?u.561\nE : Type ?u.564\nC : Set (Set α)\nD : Set (Set β)\ns : ℕ → Set α\nh1s : ∀ (n : ℕ), s n ∈ C\nh2s : (⋃ (n : ℕ), s n) = univ\nt : ℕ → Set β\nh1t : ∀ (n : ℕ), t n ∈ D\nh2t : (⋃ (n : ℕ), t n) = univ\n⊢ (⋃ (n : ℕ), (fun n => s (Nat.unpair n).fst ×ˢ t (Nat.unpair n).snd) n) = univ",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.552\nβ : Type u_2\nβ' : Type ?u.558\nγ : Type ?u.561\nE : Type ?u.564\nC : Set (Set α)\nD : Set (Set β)\ns : ℕ → Set α\nh1s : ∀ (n : ℕ), s n ∈ C\nh2s : (⋃ (n : ℕ), s n) = univ\nt : ℕ → Set β\nh1t : ∀ (n : ℕ), t n ∈ D\nh2t : (⋃ (n : ℕ), t n) = univ\n⊢ IsCountablySpanning (image2 (fun x x_1 => x ×ˢ x_1) C D)",
"tactic": "refine' ⟨fun n => s n.unpair.1 ×ˢ t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.552\nβ : Type u_2\nβ' : Type ?u.558\nγ : Type ?u.561\nE : Type ?u.564\nC : Set (Set α)\nD : Set (Set β)\ns : ℕ → Set α\nh1s : ∀ (n : ℕ), s n ∈ C\nh2s : (⋃ (n : ℕ), s n) = univ\nt : ℕ → Set β\nh1t : ∀ (n : ℕ), t n ∈ D\nh2t : (⋃ (n : ℕ), t n) = univ\n⊢ (⋃ (n : ℕ), (fun n => s (Nat.unpair n).fst ×ˢ t (Nat.unpair n).snd) n) = univ",
"tactic": "rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]"
}
]
| [
87,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
]
|
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.mul_def | []
| [
172,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
]
|
Mathlib/RingTheory/Finiteness.lean | Submodule.fg_unit | [
{
"state_after": "R✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\n⊢ FG ↑I",
"state_before": "R✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\n⊢ FG ↑I",
"tactic": "have : (1 : A) ∈ (I * ↑I⁻¹ : Submodule R A) := by\n rw [I.mul_inv]\n exact one_le.mp le_rfl"
},
{
"state_after": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ FG ↑I",
"state_before": "R✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\n⊢ FG ↑I",
"tactic": "obtain ⟨T, T', hT, hT', one_mem⟩ := mem_span_mul_finite_of_mem_mul this"
},
{
"state_after": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑I ≤ span R ↑T",
"state_before": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ FG ↑I",
"tactic": "refine' ⟨T, span_eq_of_le _ hT _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑(1 * I) ≤ span R ↑T * 1",
"state_before": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑I ≤ span R ↑T",
"tactic": "rw [← one_mul I, ← mul_one (span R (T : Set A))]"
},
{
"state_after": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑(1 * I) ≤ span R ↑T * ↑I⁻¹ * ↑I",
"state_before": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑(1 * I) ≤ span R ↑T * 1",
"tactic": "conv_rhs => rw [← I.inv_mul, ← mul_assoc]"
},
{
"state_after": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑1 ≤ span R ↑T * span R ↑T'",
"state_before": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑(1 * I) ≤ span R ↑T * ↑I⁻¹ * ↑I",
"tactic": "refine' mul_le_mul_left (le_trans _ <| mul_le_mul_right <| span_le.mpr hT')"
},
{
"state_after": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ 1 ≤ span R (↑T * ↑T')",
"state_before": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ ↑1 ≤ span R ↑T * span R ↑T'",
"tactic": "simp only [Units.val_one, span_mul_span]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\nthis : 1 ∈ ↑I * ↑I⁻¹\nT T' : Finset A\nhT : ↑T ⊆ ↑↑I\nhT' : ↑T' ⊆ ↑↑I⁻¹\none_mem : 1 ∈ span R (↑T * ↑T')\n⊢ 1 ≤ span R (↑T * ↑T')",
"tactic": "rwa [one_le]"
},
{
"state_after": "R✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\n⊢ 1 ∈ 1",
"state_before": "R✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\n⊢ 1 ∈ ↑I * ↑I⁻¹",
"tactic": "rw [I.mul_inv]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.75765\nM : Type ?u.75768\ninst✝⁵ : Semiring R✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R✝ M\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nI : (Submodule R A)ˣ\n⊢ 1 ∈ 1",
"tactic": "exact one_le.mp le_rfl"
}
]
| [
166,
15
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.coeSubgraph_injective | []
| [
1014,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1012,
1
]
|
Mathlib/Data/List/AList.lean | AList.lookup_insert_ne | []
| [
311,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
]
|
Mathlib/Data/Fintype/Units.lean | Fintype.card_units | [
{
"state_after": "α : Type u_1\ninst✝² : GroupWithZero α\ninst✝¹ : Fintype α\ninst✝ : Fintype αˣ\n⊢ card α = card { a // a ≠ 0 } + Nat.succ 0",
"state_before": "α : Type u_1\ninst✝² : GroupWithZero α\ninst✝¹ : Fintype α\ninst✝ : Fintype αˣ\n⊢ card αˣ = card α - 1",
"tactic": "rw [eq_comm, Nat.sub_eq_iff_eq_add (Fintype.card_pos_iff.2 ⟨(0 : α)⟩),\n Fintype.card_congr (unitsEquivNeZero α)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : GroupWithZero α\ninst✝¹ : Fintype α\ninst✝ : Fintype αˣ\nthis : card α = card ({ a // a = 0 } ⊕ { a // ¬a = 0 })\n⊢ card α = card { a // a ≠ 0 } + Nat.succ 0",
"tactic": "rwa [Fintype.card_sum, add_comm, Fintype.card_subtype_eq] at this"
}
]
| [
45,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
]
|
Mathlib/Analysis/Calculus/LocalExtr.lean | IsLocalExtr.hasFDerivAt_eq_zero | []
| [
215,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
]
|
Mathlib/Algebra/Order/Ring/Lemmas.lean | zero_lt_mul_left | [
{
"state_after": "α : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : Preorder α\ninst✝¹ : PosMulStrictMono α\ninst✝ : PosMulReflectLT α\nh : 0 < c\n⊢ 0 < b ↔ c * 0 < b",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : Preorder α\ninst✝¹ : PosMulStrictMono α\ninst✝ : PosMulReflectLT α\nh : 0 < c\n⊢ 0 < c * b ↔ 0 < b",
"tactic": "rw [←mul_zero c, mul_lt_mul_left h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : Preorder α\ninst✝¹ : PosMulStrictMono α\ninst✝ : PosMulReflectLT α\nh : 0 < c\n⊢ 0 < b ↔ c * 0 < b",
"tactic": "simp"
}
]
| [
355,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
]
|
Mathlib/Topology/ContinuousFunction/Algebra.lean | ContinuousMap.tsum_apply | []
| [
442,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
1
]
|
Mathlib/Topology/Algebra/Constructions.lean | MulOpposite.map_op_nhds | []
| [
67,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
]
|
Mathlib/Data/Int/Cast/Field.lean | Int.cast_div | [
{
"state_after": "case intro\nα : Type u_1\ninst✝ : DivisionRing α\nn : ℤ\nn_nonzero : ↑n ≠ 0\nk : ℤ\n⊢ ↑(n * k / n) = ↑(n * k) / ↑n",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nm n : ℤ\nn_dvd : n ∣ m\nn_nonzero : ↑n ≠ 0\n⊢ ↑(m / n) = ↑m / ↑n",
"tactic": "rcases n_dvd with ⟨k, rfl⟩"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝ : DivisionRing α\nn : ℤ\nn_nonzero : ↑n ≠ 0\nk : ℤ\nthis : n ≠ 0\n⊢ ↑(n * k / n) = ↑(n * k) / ↑n",
"state_before": "case intro\nα : Type u_1\ninst✝ : DivisionRing α\nn : ℤ\nn_nonzero : ↑n ≠ 0\nk : ℤ\n⊢ ↑(n * k / n) = ↑(n * k) / ↑n",
"tactic": "have : n ≠ 0 := by\n rintro rfl\n simp at n_nonzero"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : DivisionRing α\nn : ℤ\nn_nonzero : ↑n ≠ 0\nk : ℤ\nthis : n ≠ 0\n⊢ ↑(n * k / n) = ↑(n * k) / ↑n",
"tactic": "rw [Int.mul_ediv_cancel_left _ this, mul_comm n k, Int.cast_mul, mul_div_cancel _ n_nonzero]"
},
{
"state_after": "α : Type u_1\ninst✝ : DivisionRing α\nk : ℤ\nn_nonzero : ↑0 ≠ 0\n⊢ False",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nn : ℤ\nn_nonzero : ↑n ≠ 0\nk : ℤ\n⊢ n ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nk : ℤ\nn_nonzero : ↑0 ≠ 0\n⊢ False",
"tactic": "simp at n_nonzero"
}
]
| [
48,
95
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
]
|
Mathlib/Data/Seq/Seq.lean | Stream'.Seq.ofList_append | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl l' : List α\n⊢ ↑(l ++ l') = append ↑l ↑l'",
"tactic": "induction l <;> simp [*]"
}
]
| [
820,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
819,
1
]
|
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.lift_mul_left | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_1\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nz : N\n⊢ ↑g ↑(sec f z).snd * ↑(lift f hg) z = ↑g (sec f z).fst",
"tactic": "rw [mul_comm, lift_mul_right]"
}
]
| [
1005,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1004,
1
]
|
Mathlib/Topology/Constructions.lean | exists_finset_piecewise_mem_of_mem_nhds | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.340703\nδ : Type ?u.340706\nε : Type ?u.340709\nζ : Type ?u.340712\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.340723\ninst✝² : TopologicalSpace α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → π a)\nx y : (a : ι) → π a\nhs : ∃ I t, (∀ (i : ι), t i ∈ 𝓝 (x i)) ∧ Set.pi (↑I) t ⊆ s\n⊢ ∃ I, Finset.piecewise I x y ∈ s",
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"tactic": "simp only [nhds_pi, Filter.mem_pi'] at hs"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.340703\nδ : Type ?u.340706\nε : Type ?u.340709\nζ : Type ?u.340712\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.340723\ninst✝² : TopologicalSpace α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → π a)\nx y : (a : ι) → π a\nI : Finset ι\nt : (i : ι) → Set (π i)\nhtx : ∀ (i : ι), t i ∈ 𝓝 (x i)\nhts : Set.pi (↑I) t ⊆ s\n⊢ ∃ I, Finset.piecewise I x y ∈ s",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.340703\nδ : Type ?u.340706\nε : Type ?u.340709\nζ : Type ?u.340712\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.340723\ninst✝² : TopologicalSpace α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → π a)\nx y : (a : ι) → π a\nhs : ∃ I t, (∀ (i : ι), t i ∈ 𝓝 (x i)) ∧ Set.pi (↑I) t ⊆ s\n⊢ ∃ I, Finset.piecewise I x y ∈ s",
"tactic": "rcases hs with ⟨I, t, htx, hts⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.340703\nδ : Type ?u.340706\nε : Type ?u.340709\nζ : Type ?u.340712\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.340723\ninst✝² : TopologicalSpace α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → π a)\nx y : (a : ι) → π a\nI : Finset ι\nt : (i : ι) → Set (π i)\nhtx : ∀ (i : ι), t i ∈ 𝓝 (x i)\nhts : Set.pi (↑I) t ⊆ s\ni : ι\nhi : i ∈ ↑I\n⊢ Finset.piecewise I x y i ∈ t i",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.340703\nδ : Type ?u.340706\nε : Type ?u.340709\nζ : Type ?u.340712\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.340723\ninst✝² : TopologicalSpace α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → π a)\nx y : (a : ι) → π a\nI : Finset ι\nt : (i : ι) → Set (π i)\nhtx : ∀ (i : ι), t i ∈ 𝓝 (x i)\nhts : Set.pi (↑I) t ⊆ s\n⊢ ∃ I, Finset.piecewise I x y ∈ s",
"tactic": "refine' ⟨I, hts fun i hi => _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.340703\nδ : Type ?u.340706\nε : Type ?u.340709\nζ : Type ?u.340712\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.340723\ninst✝² : TopologicalSpace α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → π a)\nx y : (a : ι) → π a\nI : Finset ι\nt : (i : ι) → Set (π i)\nhtx : ∀ (i : ι), t i ∈ 𝓝 (x i)\nhts : Set.pi (↑I) t ⊆ s\ni : ι\nhi : i ∈ ↑I\n⊢ Finset.piecewise I x y i ∈ t i",
"tactic": "simpa [Finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)"
}
]
| [
1365,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1360,
1
]
|
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | MeasureTheory.Measure.singularPart_zero | [
{
"state_after": "α : Type u_1\nβ : Type ?u.43416\nm : MeasurableSpace α\nμ ν✝ ν : Measure α\n⊢ 0 = 0 + withDensity ν 0",
"state_before": "α : Type u_1\nβ : Type ?u.43416\nm : MeasurableSpace α\nμ ν✝ ν : Measure α\n⊢ singularPart 0 ν = 0",
"tactic": "refine' (eq_singularPart measurable_zero MutuallySingular.zero_left _).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.43416\nm : MeasurableSpace α\nμ ν✝ ν : Measure α\n⊢ 0 = 0 + withDensity ν 0",
"tactic": "rw [zero_add, withDensity_zero]"
}
]
| [
268,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
]
|
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivAt.congr_of_eventuallyEq | []
| [
600,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
598,
1
]
|
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_eq_of_le_of_coeff_ne_zero | []
| [
194,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
]
|
Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_compl_mul_self_apply | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.58114\nι : Type ?u.58117\nM : Type u_2\nN : Type ?u.58123\ninst✝ : MulOneClass M\ns✝ t : Set α\nf✝ g : α → M\na✝ : α\ns : Set α\nf : α → M\na : α\nha : a ∈ s\n⊢ mulIndicator (sᶜ) f a * mulIndicator s f a = f a",
"tactic": "simp [ha]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.58114\nι : Type ?u.58117\nM : Type u_2\nN : Type ?u.58123\ninst✝ : MulOneClass M\ns✝ t : Set α\nf✝ g : α → M\na✝ : α\ns : Set α\nf : α → M\na : α\nha : ¬a ∈ s\n⊢ mulIndicator (sᶜ) f a * mulIndicator s f a = f a",
"tactic": "simp [ha]"
}
]
| [
419,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
417,
1
]
|
Mathlib/Order/Monotone/Monovary.lean | MonotoneOn.monovaryOn | []
| [
337,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
336,
11
]
|
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.of_mem_support_derivative | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : n ∈ support (↑derivative p)\nh1 : coeff p (n + 1) = 0\n⊢ coeff (↑derivative p) n = 0",
"tactic": "rw [coeff_derivative, h1, zero_mul]"
}
]
| [
202,
96
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
]
|
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.toAddEquiv_eq_coe | []
| [
193,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
]
|
Mathlib/Algebra/Lie/Classical.lean | LieAlgebra.matrix_trace_commutator_zero | []
| [
94,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
]
|
Mathlib/RingTheory/Ideal/QuotientOperations.lean | Ideal.Quotient.mk_comp_algebraMap | []
| [
219,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
]
|
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivAtFilter.mono | []
| [
362,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
360,
1
]
|
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.mem_center_iff | []
| [
732,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
731,
1
]
|
Mathlib/Topology/Sequences.lean | SeqContinuous.continuous | []
| [
225,
81
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
223,
11
]
|
Mathlib/Data/Set/Intervals/Basic.lean | Set.Icc_union_Ici' | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\n⊢ x ∈ Icc a b ∪ Ici c ↔ x ∈ Ici (min a c)",
"state_before": "α : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "ext1 x"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\n⊢ x ∈ Icc a b ∪ Ici c ↔ x ∈ Ici (min a c)",
"tactic": "simp_rw [mem_union, mem_Icc, mem_Ici, min_le_iff]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\nhc : c ≤ x\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x\n\ncase neg\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\nhc : ¬c ≤ x\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"tactic": "by_cases hc : c ≤ x"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\nhc : c ≤ x\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"tactic": "simp only [hc, or_true]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\nhc : ¬c ≤ x\nhxb : x ≤ b\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\nhc : ¬c ≤ x\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"tactic": "have hxb : x ≤ b := (le_of_not_ge hc).trans h₁"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.91582\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nx : α\nhc : ¬c ≤ x\nhxb : x ≤ b\n⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x",
"tactic": "simp only [hxb, and_true]"
}
]
| [
1348,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1342,
1
]
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | Measurable.norm | []
| [
2038,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2037,
1
]
|
src/lean/Init/Control/ExceptCps.lean | ExceptCpsT.runCatch_bind_lift | []
| [
69,
177
]
| d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
69,
9
]
|
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | intervalIntegral.deriv_integral_left | []
| [
809,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
806,
1
]
|
Mathlib/LinearAlgebra/Determinant.lean | Basis.isUnit_det | []
| [
571,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
570,
1
]
|
Mathlib/CategoryTheory/Quotient.lean | CategoryTheory.Quotient.lift_spec | [
{
"state_after": "case h_map\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ autoParam\n (∀ (X Y : C) (f : X ⟶ Y),\n (functor r ⋙ lift r F H).map f =\n eqToHom (_ : ?F.obj X = ?G.obj X) ≫ F.map f ≫ eqToHom (_ : F.obj Y = (functor r ⋙ lift r F H).obj Y))\n _auto✝\n\ncase h_obj\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ ∀ (X : C), (functor r ⋙ lift r F H).obj X = F.obj X",
"state_before": "C : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ functor r ⋙ lift r F H = F",
"tactic": "apply Functor.ext"
},
{
"state_after": "case h_obj\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ ∀ (X : C), (functor r ⋙ lift r F H).obj X = F.obj X\n\ncase h_map\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ autoParam\n (∀ (X Y : C) (f : X ⟶ Y),\n (functor r ⋙ lift r F H).map f =\n eqToHom (_ : ?F.obj X = ?G.obj X) ≫ F.map f ≫ eqToHom (_ : F.obj Y = (functor r ⋙ lift r F H).obj Y))\n _auto✝",
"state_before": "case h_map\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ autoParam\n (∀ (X Y : C) (f : X ⟶ Y),\n (functor r ⋙ lift r F H).map f =\n eqToHom (_ : ?F.obj X = ?G.obj X) ≫ F.map f ≫ eqToHom (_ : F.obj Y = (functor r ⋙ lift r F H).obj Y))\n _auto✝\n\ncase h_obj\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ ∀ (X : C), (functor r ⋙ lift r F H).obj X = F.obj X",
"tactic": "rotate_left"
},
{
"state_after": "case h_obj\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\nX : C\n⊢ (functor r ⋙ lift r F H).obj X = F.obj X",
"state_before": "case h_obj\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\n⊢ ∀ (X : C), (functor r ⋙ lift r F H).obj X = F.obj X",
"tactic": "rintro X"
},
{
"state_after": "no goals",
"state_before": "case h_obj\nC : Type u_1\ninst✝¹ : Category C\nr : HomRel C\nD : Type u_3\ninst✝ : Category D\nF : C ⥤ D\nH : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂\nX : C\n⊢ (functor r ⋙ lift r F H).obj X = F.obj X",
"tactic": "rfl"
},
{
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186,
9
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Mathlib/SetTheory/Game/PGame.lean | PGame.zero_lf_one | []
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1890,
23
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Mathlib/LinearAlgebra/Dimension.lean | LinearMap.rank_zero | [
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Mathlib/Data/MvPolynomial/Funext.lean | MvPolynomial.funext_iff | [
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"tactic": "simp only [forall_const, eq_self_iff_true]"
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67,
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Mathlib/Data/Complex/Exponential.lean | Complex.cos_two_mul | [
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1034,
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Mathlib/LinearAlgebra/LinearIndependent.lean | linearIndependent_iUnion_finite | [
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{
"state_after": "case pos\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ } = { fst := x₁, snd := y₂ }\n\ncase neg\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ { fst := x₁, snd := x₂ } = { fst := y₁, snd := y₂ }",
"state_before": "case pos\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : x₁ = y₁\n⊢ { fst := x₁, snd := x₂ } = { fst := y₁, snd := y₂ }\n\ncase neg\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ { fst := x₁, snd := x₂ } = { fst := y₁, snd := y₂ }",
"tactic": "subst h_cases"
},
{
"state_after": "case pos.a\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ Eq.recOn ?pos.h₁✝ { fst := x₁, snd := x₂ }.snd = { fst := x₁, snd := y₂ }.snd\n\ncase pos.h₁\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ }.fst = { fst := x₁, snd := y₂ }.fst",
"state_before": "case pos\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ } = { fst := x₁, snd := y₂ }",
"tactic": "apply Sigma.eq"
},
{
"state_after": "case pos.h₁\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ }.fst = { fst := x₁, snd := y₂ }.fst\n\ncase pos.h₁\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ }.fst = { fst := x₁, snd := y₂ }.fst",
"state_before": "case pos.a\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ Eq.recOn ?pos.h₁✝ { fst := x₁, snd := x₂ }.snd = { fst := x₁, snd := y₂ }.snd\n\ncase pos.h₁\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ }.fst = { fst := x₁, snd := y₂ }.fst",
"tactic": "rw [LinearIndependent.injective (hindep _) hxy]"
},
{
"state_after": "no goals",
"state_before": "case pos.h₁\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ }.fst = { fst := x₁, snd := y₂ }.fst\n\ncase pos.h₁\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ y₂ : ιs x₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := x₁, snd := y₂ }\n⊢ { fst := x₁, snd := x₂ }.fst = { fst := x₁, snd := y₂ }.fst",
"tactic": "rfl"
},
{
"state_after": "case neg\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\nh0 : f x₁ x₂ = 0\n⊢ { fst := x₁, snd := x₂ } = { fst := y₁, snd := y₂ }",
"state_before": "case neg\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ { fst := x₁, snd := x₂ } = { fst := y₁, snd := y₂ }",
"tactic": "have h0 : f x₁ x₂ = 0 := by\n apply\n disjoint_def.1 (hd x₁ {y₁} (finite_singleton y₁) fun h => h_cases (eq_of_mem_singleton h))\n (f x₁ x₂) (subset_span (mem_range_self _))\n rw [iSup_singleton]\n simp only at hxy\n rw [hxy]\n exact subset_span (mem_range_self y₂)"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\nh0 : f x₁ x₂ = 0\n⊢ { fst := x₁, snd := x₂ } = { fst := y₁, snd := y₂ }",
"tactic": "exact False.elim ((hindep x₁).ne_zero _ h0)"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ ∈ ⨆ (i : η) (_ : i ∈ {y₁}), span R (range (f i))",
"state_before": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ = 0",
"tactic": "apply\n disjoint_def.1 (hd x₁ {y₁} (finite_singleton y₁) fun h => h_cases (eq_of_mem_singleton h))\n (f x₁ x₂) (subset_span (mem_range_self _))"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ ∈ span R (range (f y₁))",
"state_before": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ ∈ ⨆ (i : η) (_ : i ∈ {y₁}), span R (range (f i))",
"tactic": "rw [iSup_singleton]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : f x₁ x₂ = f y₁ y₂\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ ∈ span R (range (f y₁))",
"state_before": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : (fun ji => f ji.fst ji.snd) { fst := x₁, snd := x₂ } = (fun ji => f ji.fst ji.snd) { fst := y₁, snd := y₂ }\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ ∈ span R (range (f y₁))",
"tactic": "simp only at hxy"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : f x₁ x₂ = f y₁ y₂\nh_cases : ¬x₁ = y₁\n⊢ f y₁ y₂ ∈ span R (range (f y₁))",
"state_before": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : f x₁ x₂ = f y₁ y₂\nh_cases : ¬x₁ = y₁\n⊢ f x₁ x₂ ∈ span R (range (f y₁))",
"tactic": "rw [hxy]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.387224\nR : Type u_3\nK : Type ?u.387230\nM : Type u_4\nM' : Type ?u.387236\nM'' : Type ?u.387239\nV : Type u\nV' : Type ?u.387244\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nη : Type u_1\nιs : η → Type u_2\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd :\n ∀ (i : η) (t : Set η),\n Set.Finite t → ¬i ∈ t → Disjoint (span R (range (f i))) (⨆ (i : η) (_ : i ∈ t), span R (range (f i)))\n✝ : Nontrivial R\nx₁ : η\nx₂ : ιs x₁\ny₁ : η\ny₂ : ιs y₁\nhxy : f x₁ x₂ = f y₁ y₂\nh_cases : ¬x₁ = y₁\n⊢ f y₁ y₂ ∈ span R (range (f y₁))",
"tactic": "exact subset_span (mem_range_self y₂)"
}
]
| [
736,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
713,
1
]
|
Mathlib/Analysis/Normed/Group/AddTorsor.lean | dist_vadd_cancel_right | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.8057\nV : Type u_2\nP : Type u_1\nW : Type ?u.8066\nQ : Type ?u.8069\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : NormedAddCommGroup W\ninst✝¹ : MetricSpace Q\ninst✝ : NormedAddTorsor W Q\nv₁ v₂ : V\nx : P\n⊢ dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂",
"tactic": "rw [dist_eq_norm_vsub V, dist_eq_norm, vadd_vsub_vadd_cancel_right]"
}
]
| [
108,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
]
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean | PadicSeq.norm_eq_of_equiv_aux | [
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\n⊢ False",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\n⊢ False",
"tactic": "have hpn : 0 < padicNorm p (f (stationaryPoint hf)) - padicNorm p (g (stationaryPoint hg)) :=\n sub_pos_of_lt hlt"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\n⊢ False",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\n⊢ False",
"tactic": "cases' hfg _ hpn with N hN"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\n⊢ False",
"tactic": "let i := max N (max (stationaryPoint hf) (stationaryPoint hg))"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\n⊢ False",
"tactic": "have hi : N ≤ i := le_max_left _ _"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\n⊢ False",
"tactic": "have hN' := hN _ hi"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑(f - g) i) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nh : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\n⊢ False",
"tactic": "rw [lift_index_left hf N (stationaryPoint hg), lift_index_right hg N (stationaryPoint hf)]\n at hN' h hlt"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑(f - g) i) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑(f - g) i) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\n⊢ False",
"tactic": "have hpne : padicNorm p (f i) ≠ padicNorm p (-g i) := by rwa [← padicNorm.neg (g i)] at h"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑(f - g) i) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ False",
"tactic": "rw [CauSeq.sub_apply, sub_eq_add_neg, add_eq_max_of_ne hpne, padicNorm.neg, max_eq_left_of_lt hlt]\n at hN'"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\nthis : padicNorm p (↑f i) < padicNorm p (↑f i)\n⊢ False",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ False",
"tactic": "have : padicNorm p (f i) < padicNorm p (f i) := by\n apply lt_of_lt_of_le hN'\n apply sub_le_self\n apply padicNorm.nonneg"
},
{
"state_after": "no goals",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\nthis : padicNorm p (↑f i) < padicNorm p (↑f i)\n⊢ False",
"tactic": "exact lt_irrefl _ this"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑(f - g) i) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\n⊢ padicNorm p (↑f i) ≠ padicNorm p (-↑g i)",
"tactic": "rwa [← padicNorm.neg (g i)] at h"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≤\n padicNorm p (↑f i)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ padicNorm p (↑f i) < padicNorm p (↑f i)",
"tactic": "apply lt_of_lt_of_le hN'"
},
{
"state_after": "case a\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ 0 ≤ padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≤\n padicNorm p (↑f i)",
"tactic": "apply sub_le_self"
},
{
"state_after": "no goals",
"state_before": "case a\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfg : f ≈ g\nhpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\nN : ℕ\nhlt :\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n ∀ (j : ℕ),\n j ≥ N → padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))\ni : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N ≤ i\nhN' :\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)\n⊢ 0 ≤ padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))",
"tactic": "apply padicNorm.nonneg"
}
]
| [
348,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
328,
9
]
|
Mathlib/CategoryTheory/Monoidal/Category.lean | CategoryTheory.MonoidalCategory.id_tensor_associator_naturality | [
{
"state_after": "no goals",
"state_before": "C✝ : Type u\n𝒞 : Category C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nU V W X✝ Y✝ Z✝ X Y Z Z' : C\nh : Z ⟶ Z'\n⊢ (𝟙 (X ⊗ Y) ⊗ h) ≫ (α_ X Y Z').hom = (α_ X Y Z).hom ≫ (𝟙 X ⊗ 𝟙 Y ⊗ h)",
"tactic": "rw [← tensor_id, associator_naturality]"
}
]
| [
344,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
]
|
Mathlib/Topology/ContinuousFunction/Compact.lean | BoundedContinuousFunction.dist_toContinuousMap | []
| [
123,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
]
|
Mathlib/LinearAlgebra/TensorProduct.lean | TensorProduct.lid_symm_apply | []
| [
638,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
1
]
|
Mathlib/LinearAlgebra/TensorProduct.lean | LinearMap.rTensor_id | []
| [
1106,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1105,
1
]
|
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.mk'_eq_iff_mk'_eq | []
| [
372,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
370,
1
]
|
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.inf_toSubalgebra | []
| [
135,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
]
|
Mathlib/Algebra/IndicatorFunction.lean | Set.indicator_eq_zero_iff_not_mem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.142655\nι : Type ?u.142658\nM : Type u_2\nN : Type ?u.142664\ninst✝¹ : MulZeroOneClass M\ninst✝ : Nontrivial M\nU : Set α\nx : α\n⊢ indicator U 1 x = 0 ↔ ¬x ∈ U",
"tactic": "classical simp [indicator_apply, imp_false]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.142655\nι : Type ?u.142658\nM : Type u_2\nN : Type ?u.142664\ninst✝¹ : MulZeroOneClass M\ninst✝ : Nontrivial M\nU : Set α\nx : α\n⊢ indicator U 1 x = 0 ↔ ¬x ∈ U",
"tactic": "simp [indicator_apply, imp_false]"
}
]
| [
734,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
733,
1
]
|
Mathlib/Topology/ContinuousFunction/Compact.lean | ContinuousMap.dist_apply_le_dist | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nE : Type ?u.70790\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : MetricSpace β\ninst✝ : NormedAddCommGroup E\nf g : C(α, β)\nC : ℝ\nx : α\n⊢ dist (↑f x) (↑g x) ≤ dist f g",
"tactic": "simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]"
}
]
| [
134,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
]
|
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | CircleDeg1Lift.tendsto_translation_number₀' | [
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\n⊢ dist (↑(f ^ (n + 1)) 0 / (↑n + 1)) (τ f) ≤ ((fun n => 1 / ↑n) ∘ fun a => a + 1) n",
"state_before": "f g : CircleDeg1Lift\n⊢ Tendsto (fun n => ↑(f ^ (n + 1)) 0 / (↑n + 1)) atTop (𝓝 (τ f))",
"tactic": "refine'\n tendsto_iff_dist_tendsto_zero.2 <|\n squeeze_zero (fun _ => dist_nonneg) (fun n => _)\n ((tendsto_const_div_atTop_nhds_0_nat 1).comp (tendsto_add_atTop_nat 1))"
},
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\n⊢ dist (↑(f ^ (n + 1)) 0 / (↑n + 1)) (τ f) ≤ 1 / ↑(n + 1)",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\n⊢ dist (↑(f ^ (n + 1)) 0 / (↑n + 1)) (τ f) ≤ ((fun n => 1 / ↑n) ∘ fun a => a + 1) n",
"tactic": "dsimp"
},
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < ↑n + 1\n⊢ dist (↑(f ^ (n + 1)) 0 / (↑n + 1)) (τ f) ≤ 1 / ↑(n + 1)",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\n⊢ dist (↑(f ^ (n + 1)) 0 / (↑n + 1)) (τ f) ≤ 1 / ↑(n + 1)",
"tactic": "have : (0 : ℝ) < n + 1 := n.cast_add_one_pos"
},
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < ↑n + 1\n⊢ dist (↑(f ^ (n + 1)) 0) (↑(n + 1) * τ f) ≤ 1",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < ↑n + 1\n⊢ dist (↑(f ^ (n + 1)) 0 / (↑n + 1)) (τ f) ≤ 1 / ↑(n + 1)",
"tactic": "rw [Real.dist_eq, div_sub' _ _ _ (ne_of_gt this), abs_div, ← Real.dist_eq, abs_of_pos this,\n Nat.cast_add_one, div_le_div_right this, ← Nat.cast_add_one]"
},
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < ↑n + 1\n⊢ dist (↑(f ^ (n + 1)) 0) (↑(n + 1) * τ f) ≤ 1",
"tactic": "apply dist_pow_map_zero_mul_translationNumber_le"
}
]
| [
785,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
775,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_sup | []
| [
654,
77
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
653,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigOWith.sum | [
{
"state_after": "case empty\nα : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns : Finset ι\nh✝ : ∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g\nh : ∀ (i : ι), i ∈ ∅ → IsBigOWith (C i) l (A i) g\n⊢ IsBigOWith (∑ i in ∅, C i) l (fun x => ∑ i in ∅, A i x) g\n\ncase insert\nα : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ s✝ → IsBigOWith (C i) l (A i) g\ni : ι\ns : Finset ι\nis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g) → IsBigOWith (∑ i in s, C i) l (fun x => ∑ i in s, A i x) g\nh : ∀ (i_1 : ι), i_1 ∈ Insert.insert i s → IsBigOWith (C i_1) l (A i_1) g\n⊢ IsBigOWith (∑ i in Insert.insert i s, C i) l (fun x => ∑ i in Insert.insert i s, A i x) g",
"state_before": "α : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns : Finset ι\nh : ∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g\n⊢ IsBigOWith (∑ i in s, C i) l (fun x => ∑ i in s, A i x) g",
"tactic": "induction' s using Finset.induction_on with i s is IH"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns : Finset ι\nh✝ : ∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g\nh : ∀ (i : ι), i ∈ ∅ → IsBigOWith (C i) l (A i) g\n⊢ IsBigOWith (∑ i in ∅, C i) l (fun x => ∑ i in ∅, A i x) g",
"tactic": "simp only [isBigOWith_zero', Finset.sum_empty, forall_true_iff]"
},
{
"state_after": "case insert\nα : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ s✝ → IsBigOWith (C i) l (A i) g\ni : ι\ns : Finset ι\nis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g) → IsBigOWith (∑ i in s, C i) l (fun x => ∑ i in s, A i x) g\nh : ∀ (i_1 : ι), i_1 ∈ Insert.insert i s → IsBigOWith (C i_1) l (A i_1) g\n⊢ IsBigOWith (C i + ∑ i in s, C i) l (fun x => A i x + ∑ i in s, A i x) g",
"state_before": "case insert\nα : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ s✝ → IsBigOWith (C i) l (A i) g\ni : ι\ns : Finset ι\nis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g) → IsBigOWith (∑ i in s, C i) l (fun x => ∑ i in s, A i x) g\nh : ∀ (i_1 : ι), i_1 ∈ Insert.insert i s → IsBigOWith (C i_1) l (A i_1) g\n⊢ IsBigOWith (∑ i in Insert.insert i s, C i) l (fun x => ∑ i in Insert.insert i s, A i x) g",
"tactic": "simp only [is, Finset.sum_insert, not_false_iff]"
},
{
"state_after": "no goals",
"state_before": "case insert\nα : Type u_2\nβ : Type ?u.590520\nE : Type ?u.590523\nF : Type u_4\nG : Type ?u.590529\nE' : Type u_3\nF' : Type ?u.590535\nG' : Type ?u.590538\nE'' : Type ?u.590541\nF'' : Type ?u.590544\nG'' : Type ?u.590547\nR : Type ?u.590550\nR' : Type ?u.590553\n𝕜 : Type ?u.590556\n𝕜' : Type ?u.590559\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nι : Type u_1\nA : ι → α → E'\nC : ι → ℝ\ns✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ s✝ → IsBigOWith (C i) l (A i) g\ni : ι\ns : Finset ι\nis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsBigOWith (C i) l (A i) g) → IsBigOWith (∑ i in s, C i) l (fun x => ∑ i in s, A i x) g\nh : ∀ (i_1 : ι), i_1 ∈ Insert.insert i s → IsBigOWith (C i_1) l (A i_1) g\n⊢ IsBigOWith (C i + ∑ i in s, C i) l (fun x => A i x + ∑ i in s, A i x) g",
"tactic": "exact (h _ (Finset.mem_insert_self i s)).add (IH fun j hj => h _ (Finset.mem_insert_of_mem hj))"
}
]
| [
1794,
100
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1789,
1
]
|
Mathlib/Analysis/InnerProductSpace/Calculus.lean | ContDiffAt.inner | []
| [
86,
16
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
8
]
|
Mathlib/Algebra/Lie/Nilpotent.lean | LieSubmodule.ucs_le_of_normalizer_eq_self | [
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : normalizer N₁ = N₁\nk : ℕ\n⊢ ucs k ⊥ ≤ ucs k N₁",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : normalizer N₁ = N₁\nk : ℕ\n⊢ ucs k ⊥ ≤ N₁",
"tactic": "rw [← ucs_eq_self_of_normalizer_eq_self h k]"
},
{
"state_after": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : normalizer N₁ = N₁\nk : ℕ\nh_symm : N₁ = normalizer N₁\n⊢ ⊥ ≤ N₁",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : normalizer N₁ = N₁\nk : ℕ\n⊢ ucs k ⊥ ≤ ucs k N₁",
"tactic": "mono"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : normalizer N₁ = N₁\nk : ℕ\nh_symm : N₁ = normalizer N₁\n⊢ ⊥ ≤ N₁",
"tactic": "simp"
}
]
| [
400,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
]
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | charmatrix_reindex | [
{
"state_after": "case a.h.a\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\n⊢ charmatrix (↑(reindex e e) M) = ↑(reindex e e) (charmatrix M)",
"tactic": "ext (i j x)"
},
{
"state_after": "case pos\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\nh : i = j\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x\n\ncase neg\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\nh : ¬i = j\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x",
"state_before": "case a.h.a\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x",
"tactic": "by_cases h : i = j"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\nh : i = j\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x\n\ncase neg\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\nh : ¬i = j\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x",
"tactic": "all_goals simp [h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\ninst✝⁴ : CommRing R\nn : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nm : Type v\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\ne : n ≃ m\nM : Matrix n n R\ni j : m\nx : ℕ\nh : ¬i = j\n⊢ coeff (charmatrix (↑(reindex e e) M) i j) x = coeff (↑(reindex e e) (charmatrix M) i j) x",
"tactic": "simp [h]"
}
]
| [
89,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
]
|
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