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Mathlib/Algebra/Algebra/Unitization.lean | Unitization.snd_inr | [] | [
128,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.eq_union_left | [] | [
1707,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1706,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.nat_cast_eq_nat_cast_iff | [
{
"state_after": "no goals",
"state_before": "a b c : ℕ\n⊢ ↑a = ↑b ↔ a ≡ b [MOD c]",
"tactic": "simpa [Int.coe_nat_modEq_iff] using ZMod.int_cast_eq_int_cast_iff a b c"
}
] | [
467,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.succ_succAbove_one | [
{
"state_after": "n✝ m n : ℕ\ninst✝ : NeZero n\ni : Fin (n + 1)\n⊢ ↑(succAbove (succ i)) (succ 0) = succ (↑(succAbove i) 0)",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\ni : Fin (n + 1)\n⊢ ↑(succAbove (succ i)) 1 = succ (↑(succAbove i) 0)",
"tactic": "rw [← succ_zero_eq_one]"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\ni : Fin (n + 1)\n⊢ ↑(succAbove (succ i)) (succ 0) = succ (↑(succAbove i) 0)",
"tactic": "exact succ_succAbove_succ i 0"
}
] | [
2240,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2237,
1
] |
Mathlib/Topology/Instances/EReal.lean | continuous_coe_ennreal_ereal | [] | [
123,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiffOn.continuousOn_deriv_of_open | [] | [
2144,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2142,
1
] |
Mathlib/Data/Int/Parity.lean | Int.add_one_ediv_two_mul_two_of_odd | [
{
"state_after": "case intro\nm c : ℤ\n⊢ 1 + (2 * c + 1) / 2 * 2 = 2 * c + 1",
"state_before": "m n : ℤ\n⊢ Odd n → 1 + n / 2 * 2 = n",
"tactic": "rintro ⟨c, rfl⟩"
},
{
"state_after": "case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1",
"state_before": "case intro\nm c : ℤ\n⊢ 1 + (2 * c + 1) / 2 * 2 = 2 * c + 1",
"tactic": "rw [add_comm]"
},
{
"state_after": "case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2",
"state_before": "case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1",
"tactic": "convert Int.ediv_add_emod' (2 * c + 1) 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2",
"tactic": "simp [Int.add_emod]"
}
] | [
277,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
] |
Mathlib/RingTheory/LaurentSeries.lean | PowerSeries.coe_one | [] | [
194,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.coe_nat | [] | [
679,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
679,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_split_single | [
{
"state_after": "case h.e'_3.h.e'_3\nα : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.134534\nγ : Type ?u.134537\nι : Sort ?u.134540\nι' : Sort ?u.134543\nκ : ι → Sort ?u.134548\nκ' : ι' → Sort ?u.134553\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : β → α\ni₀ : β\n⊢ f i₀ = ⨆ (i : β) (_ : i = i₀), f i",
"state_before": "α : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.134534\nγ : Type ?u.134537\nι : Sort ?u.134540\nι' : Sort ?u.134543\nκ : ι → Sort ?u.134548\nκ' : ι' → Sort ?u.134553\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : β → α\ni₀ : β\n⊢ (⨆ (i : β), f i) = f i₀ ⊔ ⨆ (i : β) (_ : i ≠ i₀), f i",
"tactic": "convert iSup_split f (fun i => i = i₀)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_3\nα : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.134534\nγ : Type ?u.134537\nι : Sort ?u.134540\nι' : Sort ?u.134543\nκ : ι → Sort ?u.134548\nκ' : ι' → Sort ?u.134553\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : β → α\ni₀ : β\n⊢ f i₀ = ⨆ (i : β) (_ : i = i₀), f i",
"tactic": "simp"
}
] | [
1434,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1432,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.add_sub_cancel_left | [] | [
1141,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1140,
11
] |
Mathlib/Algebra/Order/Group/Defs.lean | mul_inv_le_one_iff_le | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ a ≤ 1 * b ↔ a ≤ b",
"tactic": "rw [one_mul]"
}
] | [
259,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.mul_top | [] | [
753,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
752,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean | AlgHom.map_adjugate | [] | [
374,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
371,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfp_le | [] | [
458,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | dist_le_norm_mul_norm | [
{
"state_after": "𝓕 : Type ?u.82158\n𝕜 : Type ?u.82161\nα : Type ?u.82164\nι : Type ?u.82167\nκ : Type ?u.82170\nE : Type u_1\nF : Type ?u.82176\nG : Type ?u.82179\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ ‖a / b‖ ≤ ‖a‖ + ‖b‖",
"state_before": "𝓕 : Type ?u.82158\n𝕜 : Type ?u.82161\nα : Type ?u.82164\nι : Type ?u.82167\nκ : Type ?u.82170\nE : Type u_1\nF : Type ?u.82176\nG : Type ?u.82179\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ dist a b ≤ ‖a‖ + ‖b‖",
"tactic": "rw [dist_eq_norm_div]"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.82158\n𝕜 : Type ?u.82161\nα : Type ?u.82164\nι : Type ?u.82167\nκ : Type ?u.82170\nE : Type u_1\nF : Type ?u.82176\nG : Type ?u.82179\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ ‖a / b‖ ≤ ‖a‖ + ‖b‖",
"tactic": "apply norm_div_le"
}
] | [
554,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
552,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | MeasureTheory.aecover_Iio_of_Iio | [] | [
188,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic.isPrimitive | [] | [
329,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
328,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.natDegree_comp_le | [
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh0 : comp p q = 0\n⊢ 0 ≤ natDegree p * natDegree q",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh0 : comp p q = 0\n⊢ natDegree (comp p q) ≤ natDegree p * natDegree q",
"tactic": "rw [h0, natDegree_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh0 : comp p q = 0\n⊢ 0 ≤ natDegree p * natDegree q",
"tactic": "exact Nat.zero_le _"
},
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh0 : ¬comp p q = 0\nn : ℕ\nhn : n ∈ support p\n⊢ ↑n * ↑(natDegree q) = ↑(n * natDegree q)",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh0 : ¬comp p q = 0\nn : ℕ\nhn : n ∈ support p\n⊢ ↑(natDegree (↑C (coeff p n))) + n • ↑(natDegree q) = ↑(n * natDegree q)",
"tactic": "rw [natDegree_C, Nat.cast_zero, zero_add, nsmul_eq_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh0 : ¬comp p q = 0\nn : ℕ\nhn : n ∈ support p\n⊢ ↑n * ↑(natDegree q) = ↑(n * natDegree q)",
"tactic": "simp"
}
] | [
63,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
1
] |
Mathlib/Topology/MetricSpace/Closeds.lean | Metric.lipschitz_infDist_set | [
{
"state_after": "α : Type u\ninst✝ : MetricSpace α\nx : α\ns t : NonemptyCompacts α\n⊢ infDist x ↑s ≤ infDist x ↑t + dist t s",
"state_before": "α : Type u\ninst✝ : MetricSpace α\nx : α\ns t : NonemptyCompacts α\n⊢ infDist x ↑s ≤ infDist x ↑t + dist s t",
"tactic": "rw [dist_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : MetricSpace α\nx : α\ns t : NonemptyCompacts α\n⊢ infDist x ↑s ≤ infDist x ↑t + dist t s",
"tactic": "exact infDist_le_infDist_add_hausdorffDist (edist_ne_top t s)"
}
] | [
426,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/Data/Multiset/Dedup.lean | Multiset.dedup_nsmul | [
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.13180\ninst✝ : DecidableEq α\ns : Multiset α\nn : ℕ\nh0 : n ≠ 0\na : α\n⊢ count a (dedup (n • s)) = count a (dedup s)",
"state_before": "α : Type u_1\nβ : Type ?u.13180\ninst✝ : DecidableEq α\ns : Multiset α\nn : ℕ\nh0 : n ≠ 0\n⊢ dedup (n • s) = dedup s",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.13180\ninst✝ : DecidableEq α\ns : Multiset α\nn : ℕ\nh0 : n ≠ 0\na : α\n⊢ count a (dedup (n • s)) = count a (dedup s)",
"tactic": "by_cases h : a ∈ s <;> simp [h, h0]"
}
] | [
141,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean | LucasLehmer.sZMod_eq_s | [
{
"state_after": "case zero\np' : ℕ\n⊢ sZMod (p' + 2) zero = ↑(s zero)\n\ncase succ\np' i : ℕ\nih : sZMod (p' + 2) i = ↑(s i)\n⊢ sZMod (p' + 2) (succ i) = ↑(s (succ i))",
"state_before": "p' i : ℕ\n⊢ sZMod (p' + 2) i = ↑(s i)",
"tactic": "induction' i with i ih"
},
{
"state_after": "case zero\np' : ℕ\n⊢ 4 = ↑4",
"state_before": "case zero\np' : ℕ\n⊢ sZMod (p' + 2) zero = ↑(s zero)",
"tactic": "dsimp [s, sZMod]"
},
{
"state_after": "no goals",
"state_before": "case zero\np' : ℕ\n⊢ 4 = ↑4",
"tactic": "norm_num"
},
{
"state_after": "case succ\np' i : ℕ\nih : sZMod (p' + 2) i = ↑(s i)\n⊢ ↑(s i) ^ 2 - 2 = ↑(s i) ^ 2 - 2",
"state_before": "case succ\np' i : ℕ\nih : sZMod (p' + 2) i = ↑(s i)\n⊢ sZMod (p' + 2) (succ i) = ↑(s (succ i))",
"tactic": "push_cast [s, sZMod, ih]"
},
{
"state_after": "no goals",
"state_before": "case succ\np' i : ℕ\nih : sZMod (p' + 2) i = ↑(s i)\n⊢ ↑(s i) ^ 2 - 2 = ↑(s i) ^ 2 - 2",
"tactic": "rfl"
}
] | [
127,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean | IsometryEquiv.divRight_symm | [] | [
232,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/CategoryTheory/Limits/Fubini.lean | CategoryTheory.Limits.limitUncurryIsoLimitCompLim_inv_π | [
{
"state_after": "J K : Type v\ninst✝⁵ : SmallCategory J\ninst✝⁴ : SmallCategory K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ K ⥤ C\ninst✝² : HasLimitsOfShape K C\ninst✝¹ : HasLimit (uncurry.obj F)\ninst✝ : HasLimit (F ⋙ lim)\nj : J\nk : K\n⊢ (limitUncurryIsoLimitCompLim F).hom ≫ (limitUncurryIsoLimitCompLim F).inv ≫ limit.π (uncurry.obj F) (j, k) =\n (limitUncurryIsoLimitCompLim F).hom ≫ limit.π (F ⋙ lim) j ≫ limit.π (F.obj j) k",
"state_before": "J K : Type v\ninst✝⁵ : SmallCategory J\ninst✝⁴ : SmallCategory K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ K ⥤ C\ninst✝² : HasLimitsOfShape K C\ninst✝¹ : HasLimit (uncurry.obj F)\ninst✝ : HasLimit (F ⋙ lim)\nj : J\nk : K\n⊢ (limitUncurryIsoLimitCompLim F).inv ≫ limit.π (uncurry.obj F) (j, k) = limit.π (F ⋙ lim) j ≫ limit.π (F.obj j) k",
"tactic": "rw [← cancel_epi (limitUncurryIsoLimitCompLim F).hom]"
},
{
"state_after": "no goals",
"state_before": "J K : Type v\ninst✝⁵ : SmallCategory J\ninst✝⁴ : SmallCategory K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ K ⥤ C\ninst✝² : HasLimitsOfShape K C\ninst✝¹ : HasLimit (uncurry.obj F)\ninst✝ : HasLimit (F ⋙ lim)\nj : J\nk : K\n⊢ (limitUncurryIsoLimitCompLim F).hom ≫ (limitUncurryIsoLimitCompLim F).inv ≫ limit.π (uncurry.obj F) (j, k) =\n (limitUncurryIsoLimitCompLim F).hom ≫ limit.π (F ⋙ lim) j ≫ limit.π (F.obj j) k",
"tactic": "simp"
}
] | [
213,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.coe_smul | [] | [
1078,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1078,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.insert'.valid | [
{
"state_after": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : IsTotal α fun x x_1 => x ≤ x_1\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordnode α\nh : Valid t\n⊢ Valid (insertWith id x t)",
"state_before": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : IsTotal α fun x x_1 => x ≤ x_1\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordnode α\nh : Valid t\n⊢ Valid (insert' x t)",
"tactic": "rw [insert'_eq_insertWith]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : IsTotal α fun x x_1 => x ≤ x_1\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordnode α\nh : Valid t\n⊢ Valid (insertWith id x t)",
"tactic": "exact insertWith.valid _ _ (fun _ => id) h"
}
] | [
1561,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1559,
1
] |
Mathlib/Data/Set/Finite.lean | Set.finite_range | [] | [
858,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
857,
1
] |
Mathlib/Logic/Relation.lean | Relation.transitive_join | [] | [
636,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
632,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean | BilinForm.congr_congr | [] | [
698,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
696,
1
] |
Mathlib/CategoryTheory/Opposites.lean | Quiver.Hom.op_inj | [] | [
44,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/Analysis/Convex/Strict.lean | strictConvex_Ico | [] | [
197,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
1
] |
Mathlib/Topology/Homotopy/Basic.lean | ContinuousMap.HomotopyWith.apply_one | [] | [
460,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean | CategoryTheory.Bicategory.associator_inv_naturality_middle | [
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng g' : b ⟶ c\nη : g ⟶ g'\nh : c ⟶ d\n⊢ f ◁ η ▷ h ≫ (α_ f g' h).inv = (α_ f g h).inv ≫ (f ◁ η) ▷ h",
"tactic": "simp"
}
] | [
365,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
364,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.toFun_eq_coe | [] | [
130,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Data/PNat/Basic.lean | PNat.add_sub_of_lt | [
{
"state_after": "a b : ℕ+\nh : a < b\n⊢ ↑a + (↑b - ↑a) = ↑b",
"state_before": "a b : ℕ+\nh : a < b\n⊢ ↑(a + (b - a)) = ↑b",
"tactic": "rw [add_coe, sub_coe, if_pos h]"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ+\nh : a < b\n⊢ ↑a + (↑b - ↑a) = ↑b",
"tactic": "exact add_tsub_cancel_of_le h.le"
}
] | [
300,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | Real.cos_sq_arctan | [
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ cos (arctan x) ^ 2 = ↑1 / (↑1 + x ^ 2)",
"tactic": "rw_mod_cast [one_div, ← inv_one_add_tan_sq (cos_arctan_pos x).ne', tan_arctan]"
}
] | [
145,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Order/Concept.lean | Concept.fst_ssubset_fst_iff | [] | [
241,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.FamilyOfElements.Compatible.to_sieveCompatible | [] | [
242,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | PrimeSpectrum.vanishingIdeal_singleton | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nx : PrimeSpectrum R\n⊢ vanishingIdeal {x} = x.asIdeal",
"tactic": "simp [vanishingIdeal]"
}
] | [
178,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Std/Data/RBMap/WF.lean | Std.RBNode.All.ins | [
{
"state_after": "case node\nα : Type u_1\np : α → Prop\ncmp : α → α → Ordering\nx : α\nh₁ : p x\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nl_ih✝ : All p l✝ → All p (ins cmp x l✝)\nr_ih✝ : All p r✝ → All p (ins cmp x r✝)\nh₂ : All p (node c✝ l✝ v✝ r✝)\n⊢ All p\n (match node c✝ l✝ v✝ r✝ with\n | nil => node red nil x nil\n | node red a y b =>\n match cmp x y with\n | Ordering.lt => node red (ins cmp x a) y b\n | Ordering.gt => node red a y (ins cmp x b)\n | Ordering.eq => node red a x b\n | node black a y b =>\n match cmp x y with\n | Ordering.lt => balance1 (ins cmp x a) y b\n | Ordering.gt => balance2 a y (ins cmp x b)\n | Ordering.eq => node black a x b)",
"state_before": "α : Type u_1\np : α → Prop\ncmp : α → α → Ordering\nx : α\nt : RBNode α\nh₁ : p x\nh₂ : All p t\n⊢ All p (ins cmp x t)",
"tactic": "induction t <;> unfold ins <;> simp [*]"
},
{
"state_after": "no goals",
"state_before": "case node\nα : Type u_1\np : α → Prop\ncmp : α → α → Ordering\nx : α\nh₁ : p x\nc✝ : RBColor\nl✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nl_ih✝ : All p l✝ → All p (ins cmp x l✝)\nr_ih✝ : All p r✝ → All p (ins cmp x r✝)\nh₂ : All p (node c✝ l✝ v✝ r✝)\n⊢ All p\n (match node c✝ l✝ v✝ r✝ with\n | nil => node red nil x nil\n | node red a y b =>\n match cmp x y with\n | Ordering.lt => node red (ins cmp x a) y b\n | Ordering.gt => node red a y (ins cmp x b)\n | Ordering.eq => node red a x b\n | node black a y b =>\n match cmp x y with\n | Ordering.lt => balance1 (ins cmp x a) y b\n | Ordering.gt => balance2 a y (ins cmp x b)\n | Ordering.eq => node black a x b)",
"tactic": "split <;> cases ‹_=_› <;> split <;> simp at h₂ <;> simp [*]"
}
] | [
95,
62
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
92,
11
] |
Mathlib/Algebra/Group/Defs.lean | mul_ne_mul_right | [] | [
203,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
] |
Mathlib/Topology/MetricSpace/Completion.lean | UniformSpace.Completion.eq_of_dist_eq_zero | [
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\n⊢ x = y",
"state_before": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\n⊢ x = y",
"tactic": "have : SeparatedSpace (Completion α) := by infer_instance"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\ns : Set (Completion α × Completion α)\nhs : s ∈ 𝓤 (Completion α)\n⊢ (x, y) ∈ s",
"state_before": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\n⊢ x = y",
"tactic": "refine' separated_def.1 this x y fun s hs ↦ _"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\ns : Set (Completion α × Completion α)\nhs : s ∈ 𝓤 (Completion α)\nε : ℝ\nεpos : ε > 0\nhε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s\n⊢ (x, y) ∈ s",
"state_before": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\ns : Set (Completion α × Completion α)\nhs : s ∈ 𝓤 (Completion α)\n⊢ (x, y) ∈ s",
"tactic": "rcases (Completion.mem_uniformity_dist s).1 hs with ⟨ε, εpos, hε⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\ns : Set (Completion α × Completion α)\nhs : s ∈ 𝓤 (Completion α)\nε : ℝ\nεpos : ε > dist x y\nhε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s\n⊢ (x, y) ∈ s",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\ns : Set (Completion α × Completion α)\nhs : s ∈ 𝓤 (Completion α)\nε : ℝ\nεpos : ε > 0\nhε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s\n⊢ (x, y) ∈ s",
"tactic": "rw [← h] at εpos"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\nthis : SeparatedSpace (Completion α)\ns : Set (Completion α × Completion α)\nhs : s ∈ 𝓤 (Completion α)\nε : ℝ\nεpos : ε > dist x y\nhε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s\n⊢ (x, y) ∈ s",
"tactic": "exact hε εpos"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\nh : dist x y = 0\n⊢ SeparatedSpace (Completion α)",
"tactic": "infer_instance"
}
] | [
153,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
11
] |
Mathlib/Topology/LocallyConstant/Algebra.lean | LocallyConstant.coe_smul | [] | [
162,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Order/WithBot.lean | WithTop.monotone_map_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.94162\nδ : Type ?u.94165\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\n⊢ ((Monotone fun a => map f ↑a) ∧ ∀ (x : α), map f ↑x ≤ map f ⊤) ↔ Monotone f",
"tactic": "simp [Monotone]"
}
] | [
1130,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1128,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean | MeasureTheory.IsFundamentalDomain.mk'' | [] | [
102,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean | tsum_le_of_sum_le' | [
{
"state_after": "case pos\nι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\nhf : Summable f\n⊢ (∑' (i : ι), f i) ≤ a₂\n\ncase neg\nι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\nhf : ¬Summable f\n⊢ (∑' (i : ι), f i) ≤ a₂",
"state_before": "ι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\n⊢ (∑' (i : ι), f i) ≤ a₂",
"tactic": "by_cases hf : Summable f"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\nhf : Summable f\n⊢ (∑' (i : ι), f i) ≤ a₂",
"tactic": "exact tsum_le_of_sum_le hf h"
},
{
"state_after": "case neg\nι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\nhf : ¬Summable f\n⊢ 0 ≤ a₂",
"state_before": "case neg\nι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\nhf : ¬Summable f\n⊢ (∑' (i : ι), f i) ≤ a₂",
"tactic": "rw [tsum_eq_zero_of_not_summable hf]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u_2\nκ : Type ?u.25943\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nha₂ : 0 ≤ a₂\nh : ∀ (s : Finset ι), ∑ i in s, f i ≤ a₂\nhf : ¬Summable f\n⊢ 0 ≤ a₂",
"tactic": "exact ha₂"
}
] | [
125,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/Deprecated/Group.lean | IsAddMonoidHom.isAddMonoidHom_mul_left | [] | [
232,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
229,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | affineIndependent_iff_not_collinear_of_ne | [
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.280484\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nhu : Finset.univ = {i₁, i₂, i₃}\n⊢ AffineIndependent k p ↔ ¬Collinear k {p i₁, p i₂, p i₃}",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.280484\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ AffineIndependent k p ↔ ¬Collinear k {p i₁, p i₂, p i₃}",
"tactic": "have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by\n fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp only at h₁₂ h₁₃ h₂₃ ⊢"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.280484\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nhu : Finset.univ = {i₁, i₂, i₃}\n⊢ AffineIndependent k p ↔ ¬Collinear k {p i₁, p i₂, p i₃}",
"tactic": "rw [affineIndependent_iff_not_collinear, ← Set.image_univ, ← Finset.coe_univ, hu,\n Finset.coe_insert, Finset.coe_insert, Finset.coe_singleton, Set.image_insert_eq, Set.image_pair]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.280484\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ Finset.univ = {i₁, i₂, i₃}",
"tactic": "fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp only at h₁₂ h₁₃ h₂₃ ⊢"
}
] | [
481,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
474,
1
] |
Std/Data/List/Init/Lemmas.lean | List.foldr_append | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\nb : β\nl l' : List α\n⊢ foldr f b (l ++ l') = foldr f (foldr f b l') l",
"tactic": "simp [foldr_eq_foldrM]"
}
] | [
200,
80
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
199,
9
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic_coeff_zero | [
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\n⊢ coeff (cyclotomic n R) 0 = 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhn : 1 < n\n⊢ coeff (cyclotomic n R) 0 = 1",
"tactic": "induction' n using Nat.strong_induction_on with n hi"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\n⊢ coeff (cyclotomic n R) 0 = 1",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\n⊢ coeff (cyclotomic n R) 0 = 1",
"tactic": "have hprod : (∏ i in Nat.properDivisors n, (Polynomial.cyclotomic i R).coeff 0) = -1 := by\n rw [← Finset.insert_erase (Nat.one_mem_properDivisors_iff_one_lt.2\n (lt_of_lt_of_le one_lt_two hn)), Finset.prod_insert (Finset.not_mem_erase 1 _),\n cyclotomic_one R]\n have hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j := by\n intro j hj\n apply Nat.succ_le_of_lt\n exact (Ne.le_iff_lt (Finset.mem_erase.1 hj).1.symm).mp\n (Nat.succ_le_of_lt (Nat.pos_of_mem_properDivisors (Finset.mem_erase.1 hj).2))\n have hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1 := by\n intro j hj\n exact hi j (Nat.mem_properDivisors.1 (Finset.mem_erase.1 hj).2).2 (hleq j hj)\n have hrw : (∏ x : ℕ in n.properDivisors.erase 1, (cyclotomic x R).coeff 0) = 1 := by\n rw [Finset.prod_congr (refl (n.properDivisors.erase 1)) hcongr]\n simp only [Finset.prod_const_one]\n simp only [hrw, mul_one, zero_sub, coeff_one_zero, coeff_X_zero, coeff_sub]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\n⊢ coeff (cyclotomic n R) 0 = 1",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\n⊢ coeff (cyclotomic n R) 0 = 1",
"tactic": "have heq : (X ^ n - 1 : R[X]).coeff 0 = -(cyclotomic n R).coeff 0 := by\n rw [← prod_cyclotomic_eq_X_pow_sub_one (zero_le_one.trans_lt hn), ←\n Nat.cons_self_properDivisors hn.ne_bot, Finset.prod_cons, mul_coeff_zero, coeff_zero_prod,\n hprod, mul_neg, mul_one]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\nhzero : coeff (X ^ n - 1) 0 = -1\n⊢ coeff (cyclotomic n R) 0 = 1",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\n⊢ coeff (cyclotomic n R) 0 = 1",
"tactic": "have hzero : (X ^ n - 1 : R[X]).coeff 0 = (-1 : R) := by\n rw [coeff_zero_eq_eval_zero _]\n simp only [zero_pow (lt_of_lt_of_le zero_lt_two hn), eval_X, eval_one, zero_sub, eval_pow,\n eval_sub]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : -1 = -coeff (cyclotomic n R) 0\nhzero : coeff (X ^ n - 1) 0 = -1\n⊢ coeff (cyclotomic n R) 0 = 1",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\nhzero : coeff (X ^ n - 1) 0 = -1\n⊢ coeff (cyclotomic n R) 0 = 1",
"tactic": "rw [hzero] at heq"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : -1 = -coeff (cyclotomic n R) 0\nhzero : coeff (X ^ n - 1) 0 = -1\n⊢ coeff (cyclotomic n R) 0 = 1",
"tactic": "exact neg_inj.mp (Eq.symm heq)"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\n⊢ ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1",
"tactic": "rw [← Finset.insert_erase (Nat.one_mem_properDivisors_iff_one_lt.2\n (lt_of_lt_of_le one_lt_two hn)), Finset.prod_insert (Finset.not_mem_erase 1 _),\n cyclotomic_one R]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"tactic": "have hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j := by\n intro j hj\n apply Nat.succ_le_of_lt\n exact (Ne.le_iff_lt (Finset.mem_erase.1 hj).1.symm).mp\n (Nat.succ_le_of_lt (Nat.pos_of_mem_properDivisors (Finset.mem_erase.1 hj).2))"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"tactic": "have hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1 := by\n intro j hj\n exact hi j (Nat.mem_properDivisors.1 (Finset.mem_erase.1 hj).2).2 (hleq j hj)"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\nhrw : ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = 1\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"tactic": "have hrw : (∏ x : ℕ in n.properDivisors.erase 1, (cyclotomic x R).coeff 0) = 1 := by\n rw [Finset.prod_congr (refl (n.properDivisors.erase 1)) hcongr]\n simp only [Finset.prod_const_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\nhrw : ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = 1\n⊢ coeff (X - 1) 0 * ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = -1",
"tactic": "simp only [hrw, mul_one, zero_sub, coeff_one_zero, coeff_X_zero, coeff_sub]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nj : ℕ\nhj : j ∈ Finset.erase (Nat.properDivisors n) 1\n⊢ 2 ≤ j",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\n⊢ ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j",
"tactic": "intro j hj"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nj : ℕ\nhj : j ∈ Finset.erase (Nat.properDivisors n) 1\n⊢ 1 < j",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nj : ℕ\nhj : j ∈ Finset.erase (Nat.properDivisors n) 1\n⊢ 2 ≤ j",
"tactic": "apply Nat.succ_le_of_lt"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nj : ℕ\nhj : j ∈ Finset.erase (Nat.properDivisors n) 1\n⊢ 1 < j",
"tactic": "exact (Ne.le_iff_lt (Finset.mem_erase.1 hj).1.symm).mp\n (Nat.succ_le_of_lt (Nat.pos_of_mem_properDivisors (Finset.mem_erase.1 hj).2))"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nj : ℕ\nhj : j ∈ Finset.erase (Nat.properDivisors n) 1\n⊢ coeff (cyclotomic j R) 0 = 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\n⊢ ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1",
"tactic": "intro j hj"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nj : ℕ\nhj : j ∈ Finset.erase (Nat.properDivisors n) 1\n⊢ coeff (cyclotomic j R) 0 = 1",
"tactic": "exact hi j (Nat.mem_properDivisors.1 (Finset.mem_erase.1 hj).2).2 (hleq j hj)"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\n⊢ ∏ x in Finset.erase (Nat.properDivisors n) 1, 1 = 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\n⊢ ∏ x in Finset.erase (Nat.properDivisors n) 1, coeff (cyclotomic x R) 0 = 1",
"tactic": "rw [Finset.prod_congr (refl (n.properDivisors.erase 1)) hcongr]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhleq : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → 2 ≤ j\nhcongr : ∀ (j : ℕ), j ∈ Finset.erase (Nat.properDivisors n) 1 → coeff (cyclotomic j R) 0 = 1\n⊢ ∏ x in Finset.erase (Nat.properDivisors n) 1, 1 = 1",
"tactic": "simp only [Finset.prod_const_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\n⊢ coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0",
"tactic": "rw [← prod_cyclotomic_eq_X_pow_sub_one (zero_le_one.trans_lt hn), ←\n Nat.cons_self_properDivisors hn.ne_bot, Finset.prod_cons, mul_coeff_zero, coeff_zero_prod,\n hprod, mul_neg, mul_one]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\n⊢ eval 0 (X ^ n - 1) = -1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\n⊢ coeff (X ^ n - 1) 0 = -1",
"tactic": "rw [coeff_zero_eq_eval_zero _]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ : ℕ\nhn✝ : 1 < n✝\nn : ℕ\nhi : ∀ (m : ℕ), m < n → 1 < m → coeff (cyclotomic m R) 0 = 1\nhn : 1 < n\nhprod : ∏ i in Nat.properDivisors n, coeff (cyclotomic i R) 0 = -1\nheq : coeff (X ^ n - 1) 0 = -coeff (cyclotomic n R) 0\n⊢ eval 0 (X ^ n - 1) = -1",
"tactic": "simp only [zero_pow (lt_of_lt_of_le zero_lt_two hn), eval_X, eval_one, zero_sub, eval_pow,\n eval_sub]"
}
] | [
615,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
587,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.star_leftMoves | [] | [
1839,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1838,
1
] |
Mathlib/Topology/Order/Basic.lean | nhdsWithin_Ioi_self_neBot | [] | [
2397,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2396,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean | norm_pow | [] | [
548,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
547,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean | Submodule.topologicalClosure_eq_top_iff | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ Kᗮᗮ = ⊤ ↔ Kᗮ = ⊥",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ topologicalClosure K = ⊤ ↔ Kᗮ = ⊥",
"tactic": "rw [← Submodule.orthogonal_orthogonal_eq_closure]"
},
{
"state_after": "case mp\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\nh : Kᗮᗮ = ⊤\n⊢ Kᗮ = ⊥\n\ncase mpr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\nh : Kᗮ = ⊥\n⊢ Kᗮᗮ = ⊤",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ Kᗮᗮ = ⊤ ↔ Kᗮ = ⊥",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\nh : Kᗮᗮ = ⊤\n⊢ Kᗮ = ⊥",
"tactic": "rw [← Submodule.triorthogonal_eq_orthogonal, h, Submodule.top_orthogonal_eq_bot]"
},
{
"state_after": "no goals",
"state_before": "case mpr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.886369\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\nh : Kᗮ = ⊥\n⊢ Kᗮᗮ = ⊤",
"tactic": "rw [h, Submodule.bot_orthogonal_eq_top]"
}
] | [
936,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
931,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean | soln_deriv_norm | [] | [
396,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
395,
9
] |
Mathlib/Analysis/Convex/Gauge.lean | Convex.gauge_le | [
{
"state_after": "case pos\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : 0 ≤ a\n⊢ Convex ℝ {x | gauge s x ≤ a}\n\ncase neg\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : ¬0 ≤ a\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"state_before": "𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"tactic": "by_cases ha : 0 ≤ a"
},
{
"state_after": "case pos\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : 0 ≤ a\n⊢ Convex ℝ (⋂ (r : ℝ) (_ : a < r), r • s)",
"state_before": "case pos\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : 0 ≤ a\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"tactic": "rw [gauge_le_eq hs h₀ absorbs ha]"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : 0 ≤ a\n⊢ Convex ℝ (⋂ (r : ℝ) (_ : a < r), r • s)",
"tactic": "exact convex_iInter fun i => convex_iInter fun _ => hs.smul _"
},
{
"state_after": "case h.e'_6\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : ¬0 ≤ a\n⊢ {x | gauge s x ≤ a} = ∅",
"state_before": "case neg\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : ¬0 ≤ a\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"tactic": "convert convex_empty (𝕜 := ℝ) (E := E)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\n𝕜 : Type ?u.77419\nE : Type u_1\nF : Type ?u.77425\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : ¬0 ≤ a\n⊢ {x | gauge s x ≤ a} = ∅",
"tactic": "exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx"
}
] | [
213,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
1
] |
Mathlib/Order/Monotone/Basic.lean | antitoneOn_const | [] | [
538,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
536,
1
] |
Mathlib/Topology/PathConnected.lean | Joined.refl | [] | [
774,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
773,
1
] |
Mathlib/ModelTheory/FinitelyGenerated.lean | FirstOrder.Language.Substructure.cg_iff_structure_cg | [
{
"state_after": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\n⊢ CG S ↔ CG ⊤",
"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\n⊢ CG S ↔ Structure.CG L { x // x ∈ S }",
"tactic": "rw [Structure.cg_def]"
},
{
"state_after": "case refine'_1\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh : CG S\n⊢ CG (map (Embedding.toHom (subtype S)) ⊤)\n\ncase refine'_2\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh : CG ⊤\n⊢ CG S",
"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\n⊢ CG S ↔ CG ⊤",
"tactic": "refine' ⟨fun h => CG.of_map_embedding S.subtype _, fun h => _⟩"
},
{
"state_after": "case refine'_1\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh : CG S\n⊢ CG S",
"state_before": "case refine'_1\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh : CG S\n⊢ CG (map (Embedding.toHom (subtype S)) ⊤)",
"tactic": "rw [← Hom.range_eq_map, range_subtype]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh : CG S\n⊢ CG S",
"tactic": "exact h"
},
{
"state_after": "case refine'_2\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh✝ : CG ⊤\nh : CG (map (Embedding.toHom (subtype S)) ⊤)\n⊢ CG S",
"state_before": "case refine'_2\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh : CG ⊤\n⊢ CG S",
"tactic": "have h := h.map S.subtype.toHom"
},
{
"state_after": "case refine'_2\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh✝ : CG ⊤\nh : CG S\n⊢ CG S",
"state_before": "case refine'_2\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh✝ : CG ⊤\nh : CG (map (Embedding.toHom (subtype S)) ⊤)\n⊢ CG S",
"tactic": "rw [← Hom.range_eq_map, range_subtype] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nS : Substructure L M\nh✝ : CG ⊤\nh : CG S\n⊢ CG S",
"tactic": "exact h"
}
] | [
294,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.mk_le_mk | [] | [
199,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearMap.coe_add | [] | [
715,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
714,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.range_fst | [] | [
1280,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1279,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean | Algebra.leftMulMatrix_mulVec_repr | [] | [
900,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
898,
1
] |
Mathlib/Order/Disjoint.lean | Codisjoint.mono_left | [] | [
257,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Data/PNat/Prime.lean | PNat.Coprime.symm | [
{
"state_after": "m n : ℕ+\n⊢ gcd m n = 1 → gcd n m = 1",
"state_before": "m n : ℕ+\n⊢ Coprime m n → Coprime n m",
"tactic": "unfold Coprime"
},
{
"state_after": "m n : ℕ+\n⊢ gcd n m = 1 → gcd n m = 1",
"state_before": "m n : ℕ+\n⊢ gcd m n = 1 → gcd n m = 1",
"tactic": "rw [gcd_comm]"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ+\n⊢ gcd n m = 1 → gcd n m = 1",
"tactic": "simp"
}
] | [
245,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivWithinAt.congr_of_eventuallyEq_of_mem | [] | [
595,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
593,
1
] |
Mathlib/Data/Polynomial/Coeff.lean | Polynomial.coeff_X_pow_mul | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ coeff (X ^ n * p) (d + n) = coeff p d",
"tactic": "rw [(commute_X_pow p n).eq, coeff_mul_X_pow]"
}
] | [
246,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
245,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.zero_linear | [] | [
290,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean | Polynomial.card_rootSet_le_derivative | [
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : CommRing F\ninst✝ : Algebra F ℝ\np : F[X]\n⊢ Fintype.card ↑(rootSet p ℝ) ≤ Fintype.card ↑(rootSet (↑derivative p) ℝ) + 1",
"tactic": "simpa only [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe, derivative_map] using\n card_roots_toFinset_le_derivative (p.map (algebraMap F ℝ))"
}
] | [
415,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
412,
1
] |
Mathlib/GroupTheory/PGroup.lean | IsPGroup.coprime_card_of_ne | [
{
"state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ : ℕ\nheq₁ : card { x // x ∈ H₁ } = p₁ ^ n₁\n⊢ Nat.coprime (card { x // x ∈ H₁ }) (card { x // x ∈ H₂ })",
"state_before": "p : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\n⊢ Nat.coprime (card { x // x ∈ H₁ }) (card { x // x ∈ H₂ })",
"tactic": "obtain ⟨n₁, heq₁⟩ := iff_card.mp hH₁"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ : ℕ\nheq₁ : card { x // x ∈ H₁ } = p₁ ^ n₁\n⊢ Nat.coprime (p₁ ^ n₁) (card { x // x ∈ H₂ })",
"state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ : ℕ\nheq₁ : card { x // x ∈ H₁ } = p₁ ^ n₁\n⊢ Nat.coprime (card { x // x ∈ H₁ }) (card { x // x ∈ H₂ })",
"tactic": "rw [heq₁]"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ : ℕ\n⊢ Nat.coprime (p₁ ^ n₁) (card { x // x ∈ H₂ })",
"state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ : ℕ\nheq₁ : card { x // x ∈ H₁ } = p₁ ^ n₁\n⊢ Nat.coprime (p₁ ^ n₁) (card { x // x ∈ H₂ })",
"tactic": "clear heq₁"
},
{
"state_after": "case intro.intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ n₂ : ℕ\nheq₂ : card { x // x ∈ H₂ } = p₂ ^ n₂\n⊢ Nat.coprime (p₁ ^ n₁) (card { x // x ∈ H₂ })",
"state_before": "case intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ : ℕ\n⊢ Nat.coprime (p₁ ^ n₁) (card { x // x ∈ H₂ })",
"tactic": "obtain ⟨n₂, heq₂⟩ := iff_card.mp hH₂"
},
{
"state_after": "case intro.intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ n₂ : ℕ\nheq₂ : card { x // x ∈ H₂ } = p₂ ^ n₂\n⊢ Nat.coprime (p₁ ^ n₁) (p₂ ^ n₂)",
"state_before": "case intro.intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ n₂ : ℕ\nheq₂ : card { x // x ∈ H₂ } = p₂ ^ n₂\n⊢ Nat.coprime (p₁ ^ n₁) (card { x // x ∈ H₂ })",
"tactic": "rw [heq₂]"
},
{
"state_after": "case intro.intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ n₂ : ℕ\n⊢ Nat.coprime (p₁ ^ n₁) (p₂ ^ n₂)",
"state_before": "case intro.intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ n₂ : ℕ\nheq₂ : card { x // x ∈ H₂ } = p₂ ^ n₂\n⊢ Nat.coprime (p₁ ^ n₁) (p₂ ^ n₂)",
"tactic": "clear heq₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\np : ℕ\nG : Type u_2\ninst✝³ : Group G\nG₂ : Type u_1\ninst✝² : Group G₂\np₁ p₂ : ℕ\nhp₁ : Fact (Nat.Prime p₁)\nhp₂ : Fact (Nat.Prime p₂)\nhne : p₁ ≠ p₂\nH₁ : Subgroup G\nH₂ : Subgroup G₂\ninst✝¹ : Fintype { x // x ∈ H₁ }\ninst✝ : Fintype { x // x ∈ H₂ }\nhH₁ : IsPGroup p₁ { x // x ∈ H₁ }\nhH₂ : IsPGroup p₂ { x // x ∈ H₂ }\nn₁ n₂ : ℕ\n⊢ Nat.coprime (p₁ ^ n₁) (p₂ ^ n₂)",
"tactic": "exact Nat.coprime_pow_primes _ _ hp₁.elim hp₂.elim hne"
}
] | [
350,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
344,
1
] |
Mathlib/GroupTheory/Index.lean | Subgroup.relindex_inf_mul_relindex | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\n⊢ relindex H (K ⊓ L) * relindex K L = relindex (H ⊓ K) L",
"tactic": "rw [← inf_relindex_right H (K ⊓ L), ← inf_relindex_right K L, ← inf_relindex_right (H ⊓ K) L,\n inf_assoc, relindex_mul_relindex (H ⊓ (K ⊓ L)) (K ⊓ L) L inf_le_right inf_le_right]"
}
] | [
150,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Algebra/Lie/Normalizer.lean | LieSubalgebra.le_normalizer | [] | [
131,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean | padicValRat.of_int_multiplicity | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ padicValRat p ↑z = ↑(Part.get (multiplicity (↑p) z) (_ : multiplicity.Finite (↑p) z))",
"tactic": "rw [of_int, padicValInt.of_ne_one_ne_zero hp hz]"
}
] | [
184,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.bounded_iff_mem_bounded | [] | [
2308,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2306,
1
] |
Mathlib/RingTheory/LaurentSeries.lean | PowerSeries.coe_C | [] | [
231,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
] |
Mathlib/Data/Nat/Lattice.lean | Nat.sInf_le | [
{
"state_after": "s : Set ℕ\nm : ℕ\nhm : m ∈ s\n⊢ Nat.find (_ : ∃ x, x ∈ s) ≤ m",
"state_before": "s : Set ℕ\nm : ℕ\nhm : m ∈ s\n⊢ sInf s ≤ m",
"tactic": "rw [Nat.sInf_def ⟨m, hm⟩]"
},
{
"state_after": "no goals",
"state_before": "s : Set ℕ\nm : ℕ\nhm : m ∈ s\n⊢ Nat.find (_ : ∃ x, x ∈ s) ≤ m",
"tactic": "exact Nat.find_min' ⟨m, hm⟩ hm"
}
] | [
85,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
11
] |
Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | SimplicialObject.Splitting.IndexSet.ext' | [] | [
77,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Order/Filter/Archimedean.lean | tendsto_int_cast_atTop_atTop | [] | [
71,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
] |
Mathlib/Computability/TMToPartrec.lean | Turing.ToPartrec.stepNormal_then | [
{
"state_after": "case cons\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝¹ (Cont.cons₁ a✝ v (Cont.then k k')) v = Cfg.then (stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k'\n\ncase comp\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k'\n\ncase case\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k')\n (List.headI v) =\n Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k'\n\ncase fix\nk' : Cont\na✝ : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k'",
"state_before": "c : Code\nk k' : Cont\nv : List ℕ\n⊢ stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k'",
"tactic": "induction c generalizing k v <;> simp only [Cont.then, stepNormal, *] <;>\n try { simp only [Cfg.then]; done }"
},
{
"state_after": "case comp\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k'\n\ncase case\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k')\n (List.headI v) =\n Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k'\n\ncase fix\nk' : Cont\na✝ : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k'",
"state_before": "case cons\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝¹ (Cont.cons₁ a✝ v (Cont.then k k')) v = Cfg.then (stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k'\n\ncase comp\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k'\n\ncase case\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k')\n (List.headI v) =\n Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k'\n\ncase fix\nk' : Cont\na✝ : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k'",
"tactic": "case cons c c' ih _ => rw [← ih, Cont.then]"
},
{
"state_after": "case case\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k')\n (List.headI v) =\n Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k'\n\ncase fix\nk' : Cont\na✝ : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k'",
"state_before": "case comp\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k'\n\ncase case\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k')\n (List.headI v) =\n Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k'\n\ncase fix\nk' : Cont\na✝ : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k'",
"tactic": "case comp c c' _ ih' => rw [← ih', Cont.then]"
},
{
"state_after": "no goals",
"state_before": "case fix\nk' : Cont\na✝ : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k'",
"tactic": "case fix c ih => rw [← ih, Cont.then]"
},
{
"state_after": "no goals",
"state_before": "k' : Cont\nc c' : Code\nih : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal c' (Cont.then k k') v = Cfg.then (stepNormal c' k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal c (Cont.cons₁ c' v (Cont.then k k')) v = Cfg.then (stepNormal c (Cont.cons₁ c' v k) v) k'",
"tactic": "rw [← ih, Cont.then]"
},
{
"state_after": "no goals",
"state_before": "k' : Cont\nc c' : Code\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k'\nih' : ∀ (k : Cont) (v : List ℕ), stepNormal c' (Cont.then k k') v = Cfg.then (stepNormal c' k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal c' (Cont.comp c (Cont.then k k')) v = Cfg.then (stepNormal c' (Cont.comp c k) v) k'",
"tactic": "rw [← ih', Cont.then]"
},
{
"state_after": "no goals",
"state_before": "case case\nk' : Cont\na✝¹ a✝ : Code\na_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k'\na_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k'\nk : Cont\nv : List ℕ\n⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k')\n (List.headI v) =\n Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v))\n k'",
"tactic": "cases v.headI <;> simp only [Nat.rec]"
},
{
"state_after": "no goals",
"state_before": "k' : Cont\nc : Code\nih : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k'\nk : Cont\nv : List ℕ\n⊢ stepNormal c (Cont.fix c (Cont.then k k')) v = Cfg.then (stepNormal c (Cont.fix c k) v) k'",
"tactic": "rw [← ih, Cont.then]"
}
] | [
577,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
570,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean | AddSubmonoid.mul_le_mul_right | [] | [
595,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
594,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.one_trans | [] | [
914,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
913,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.coe_prodAssoc_symm | [] | [
1156,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1154,
1
] |
Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | AffineMap.deriv | [] | [
47,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
19
] |
Mathlib/MeasureTheory/Constructions/Pi.lean | MeasureTheory.volume_preserving_funUnique | [] | [
789,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
787,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean | Homeomorph.coe_refl | [] | [
481,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
480,
1
] |
Mathlib/Topology/CompactOpen.lean | ContinuousMap.image_coev | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40917\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ny : β\ns : Set α\n⊢ ↑(coev α β y) '' s = {y} ×ˢ s",
"tactic": "aesop"
}
] | [
325,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
324,
1
] |
Mathlib/Order/CompleteLattice.lean | iInf_apply | [] | [
1778,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1776,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_ofAdd | [] | [
1456,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1455,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean | abs_le_abs_of_nonpos | [
{
"state_after": "α : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nha : a ≤ 0\nhab : b ≤ a\n⊢ -a ≤ -b",
"state_before": "α : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nha : a ≤ 0\nhab : b ≤ a\n⊢ abs a ≤ abs b",
"tactic": "rw [abs_of_nonpos ha, abs_of_nonpos (hab.trans ha)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : AddGroup α\ninst✝² : LinearOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\ninst✝ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nha : a ≤ 0\nhab : b ≤ a\n⊢ -a ≤ -b",
"tactic": "exact neg_le_neg_iff.mpr hab"
}
] | [
196,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean | quasiconcaveOn_iff_min_le | [] | [
136,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | Right.one_le_mul_of_le_of_le | [] | [
936,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
934,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.PushoutCocone.inr_colimit_cocone | [] | [
1166,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1165,
1
] |
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | MeasureTheory.SignedMeasure.findExistsOneDivLT_min | [
{
"state_after": "α : Type u_1\nβ : Type ?u.3427\ninst✝³ : MeasurableSpace α\nM : Type ?u.3433\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\nm : ℕ\nhm : m < Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n)\n⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m",
"state_before": "α : Type u_1\nβ : Type ?u.3427\ninst✝³ : MeasurableSpace α\nM : Type ?u.3433\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\nm : ℕ\nhm : m < MeasureTheory.SignedMeasure.findExistsOneDivLT s i\n⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m",
"tactic": "rw [findExistsOneDivLT, dif_pos hi] at hm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3427\ninst✝³ : MeasurableSpace α\nM : Type ?u.3433\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\nm : ℕ\nhm : m < Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n)\n⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m",
"tactic": "exact Nat.find_min _ hm"
}
] | [
121,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.mk_uniformity_basis_le | [
{
"state_after": "α : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\n⊢ (∃ i, 0 < i ∧ {p | dist p.fst p.snd < i} ⊆ s) ↔ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"state_before": "α : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\n⊢ HasBasis (𝓤 α) p fun x => {p | dist p.fst p.snd ≤ f x}",
"tactic": "refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩"
},
{
"state_after": "case mp\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\n⊢ (∃ i, 0 < i ∧ {p | dist p.fst p.snd < i} ⊆ s) → ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s\n\ncase mpr\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\n⊢ (∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s) → ∃ i, 0 < i ∧ {p | dist p.fst p.snd < i} ⊆ s",
"state_before": "α : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\n⊢ (∃ i, 0 < i ∧ {p | dist p.fst p.snd < i} ⊆ s) ↔ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\nε : ℝ\nε₀ : 0 < ε\nhε : {p | dist p.fst p.snd < ε} ⊆ s\n⊢ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"state_before": "case mp\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\n⊢ (∃ i, 0 < i ∧ {p | dist p.fst p.snd < i} ⊆ s) → ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"tactic": "rintro ⟨ε, ε₀, hε⟩"
},
{
"state_after": "case mp.intro.intro.intro\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\nε : ℝ\nε₀ : 0 < ε\nhε : {p | dist p.fst p.snd < ε} ⊆ s\nε' : ℝ\nhε' : 0 < ε' ∧ ε' < ε\n⊢ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"state_before": "case mp.intro.intro\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\nε : ℝ\nε₀ : 0 < ε\nhε : {p | dist p.fst p.snd < ε} ⊆ s\n⊢ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"tactic": "rcases exists_between ε₀ with ⟨ε', hε'⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\nε : ℝ\nε₀ : 0 < ε\nhε : {p | dist p.fst p.snd < ε} ⊆ s\nε' : ℝ\nhε' : 0 < ε' ∧ ε' < ε\ni : β\nhi : p i\nH : f i ≤ ε'\n⊢ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"state_before": "case mp.intro.intro.intro\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\nε : ℝ\nε₀ : 0 < ε\nhε : {p | dist p.fst p.snd < ε} ⊆ s\nε' : ℝ\nhε' : 0 < ε' ∧ ε' < ε\n⊢ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"tactic": "rcases hf ε' hε'.1 with ⟨i, hi, H⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\nε : ℝ\nε₀ : 0 < ε\nhε : {p | dist p.fst p.snd < ε} ⊆ s\nε' : ℝ\nhε' : 0 < ε' ∧ ε' < ε\ni : β\nhi : p i\nH : f i ≤ ε'\n⊢ ∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s",
"tactic": "exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <| lt_of_le_of_lt (le_trans hx H) hε'.2⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nβ✝ : Type v\nX : Type ?u.55514\nι : Type ?u.55517\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\nβ : Type u_1\np : β → Prop\nf : β → ℝ\nhf₀ : ∀ (x : β), p x → 0 < f x\nhf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε\ns : Set (α × α)\n⊢ (∃ i, p i ∧ {p | dist p.fst p.snd ≤ f i} ⊆ s) → ∃ i, 0 < i ∧ {p | dist p.fst p.snd < i} ⊆ s",
"tactic": "exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)⟩"
}
] | [
792,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
783,
11
] |
Mathlib/Analysis/Normed/Group/Basic.lean | dist_inv | [
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.635485\n𝕜 : Type ?u.635488\nα : Type ?u.635491\nι : Type ?u.635494\nκ : Type ?u.635497\nE : Type u_1\nF : Type ?u.635503\nG : Type ?u.635506\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx y : E\n⊢ dist x⁻¹ y = dist x y⁻¹",
"tactic": "simp_rw [dist_eq_norm_div, ← norm_inv' (x⁻¹ / y), inv_div, div_inv_eq_mul, mul_comm]"
}
] | [
1397,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1396,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.lt_of_equiv_of_lt | [] | [
846,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
845,
1
] |
src/lean/Init/Core.lean | if_pos | [] | [
793,
33
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
790,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean | PadicInt.norm_int_cast_eq_padic_norm | [
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nz : ℤ\n⊢ ‖↑z‖ = ‖↑z‖",
"tactic": "simp [norm_def]"
}
] | [
304,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | HasDerivAt.clog | [
{
"state_after": "α : Type ?u.36688\ninst✝² : TopologicalSpace α\nE : Type ?u.36694\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\nf' x : ℂ\nh₁ : HasDerivAt f f' x\nh₂ : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivAt (fun t => log (f t)) ((f x)⁻¹ * f') x",
"state_before": "α : Type ?u.36688\ninst✝² : TopologicalSpace α\nE : Type ?u.36694\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\nf' x : ℂ\nh₁ : HasDerivAt f f' x\nh₂ : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivAt (fun t => log (f t)) (f' / f x) x",
"tactic": "rw [div_eq_inv_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.36688\ninst✝² : TopologicalSpace α\nE : Type ?u.36694\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\nf' x : ℂ\nh₁ : HasDerivAt f f' x\nh₂ : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivAt (fun t => log (f t)) ((f x)⁻¹ * f') x",
"tactic": "exact (hasStrictDerivAt_log h₂).hasDerivAt.comp x h₁"
}
] | [
104,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | Antitone.mul | [] | [
452,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
451,
1
] |
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