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stringlengths 11
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|---|---|---|---|---|---|---|
Mathlib/MeasureTheory/Function/SpecialFunctions/IsROrC.lean | AEMeasurable.im | []
| [
61,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
]
|
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.Perm.mul_apply | []
| [
71,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
]
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.sup_eq_sSup | []
| [
1395,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1391,
1
]
|
Mathlib/Data/Set/Pointwise/Basic.lean | Set.mul_inter_subset | []
| [
460,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
]
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.sub_eq_of_add_eq | []
| [
558,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
557,
1
]
|
Mathlib/Data/Real/Basic.lean | Real.sSup_empty | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ¬(Set.Nonempty ∅ ∧ BddAbove ∅)",
"tactic": "simp"
}
]
| [
799,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
798,
1
]
|
Mathlib/LinearAlgebra/Matrix/Basis.lean | Basis.toMatrix_map | [
{
"state_after": "case a.h\nι : Type u_1\nι' : Type ?u.641315\nκ : Type ?u.641318\nκ' : Type ?u.641321\nR : Type u_2\nM : Type u_3\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nR₂ : Type ?u.641514\nM₂ : Type ?u.641517\ninst✝⁴ : CommRing R₂\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R₂ M₂\ne : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\nN : Type u_4\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf✝ : M →ₗ[R] N\nb : Basis ι R M\nf : M ≃ₗ[R] N\nv : ι → N\ni✝ x✝ : ι\n⊢ toMatrix (Basis.map b f) v i✝ x✝ = toMatrix b (↑(LinearEquiv.symm f) ∘ v) i✝ x✝",
"state_before": "ι : Type u_1\nι' : Type ?u.641315\nκ : Type ?u.641318\nκ' : Type ?u.641321\nR : Type u_2\nM : Type u_3\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nR₂ : Type ?u.641514\nM₂ : Type ?u.641517\ninst✝⁴ : CommRing R₂\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R₂ M₂\ne : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\nN : Type u_4\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf✝ : M →ₗ[R] N\nb : Basis ι R M\nf : M ≃ₗ[R] N\nv : ι → N\n⊢ toMatrix (Basis.map b f) v = toMatrix b (↑(LinearEquiv.symm f) ∘ v)",
"tactic": "ext"
}
]
| [
278,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
275,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.continuous_sub_right | [
{
"state_after": "case pos\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : a = ⊤\n⊢ Continuous fun x => x - a\n\ncase neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ Continuous fun x => x - a",
"state_before": "α : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ Continuous fun x => x - a",
"tactic": "by_cases a_infty : a = ∞"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : a = ⊤\n⊢ Continuous fun x => x - a",
"tactic": "simp [a_infty, continuous_const]"
},
{
"state_after": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ Continuous ((fun p => p.fst - p.snd) ∘ fun x => (x, a))",
"state_before": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ Continuous fun x => x - a",
"tactic": "rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]"
},
{
"state_after": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ ∀ (x : ℝ≥0∞), (x, a) ∈ {p | p ≠ (⊤, ⊤)}",
"state_before": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ Continuous ((fun p => p.fst - p.snd) ∘ fun x => (x, a))",
"tactic": "apply ContinuousOn.comp_continuous continuousOn_sub (continuous_id'.prod_mk continuous_const)"
},
{
"state_after": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\nx : ℝ≥0∞\n⊢ (x, a) ∈ {p | p ≠ (⊤, ⊤)}",
"state_before": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ ∀ (x : ℝ≥0∞), (x, a) ∈ {p | p ≠ (⊤, ⊤)}",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx✝ y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\nx : ℝ≥0∞\n⊢ (x, a) ∈ {p | p ≠ (⊤, ⊤)}",
"tactic": "simp only [a_infty, Ne.def, mem_setOf_eq, Prod.mk.inj_iff, and_false_iff, not_false_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.110881\nβ : Type ?u.110884\nγ : Type ?u.110887\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\na_infty : ¬a = ⊤\n⊢ (fun x => x - a) = (fun p => p.fst - p.snd) ∘ fun x => (x, a)",
"tactic": "rfl"
}
]
| [
471,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
465,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.mem_of_le | []
| [
525,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
524,
1
]
|
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.interpolate_eq_nodalWeight_mul_nodal_div_X_sub_C | [
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\ninst✝ : DecidableEq ι\nj : ι\nhj : j ∈ s\n⊢ ↑C (r j) * Lagrange.basis s v j = ↑C (nodalWeight s v j) * (nodal s v / (X - ↑C (v j))) * ↑C (r j)",
"tactic": "rw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hj]"
}
]
| [
595,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
593,
1
]
|
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.comap_of_equiv | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nF : Type ?u.1390279\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf✝ : F\nI✝ I : Ideal R\nf : R ≃+* S\n⊢ comap (↑f) (comap (↑(RingEquiv.symm f)) I) = I",
"tactic": "simp [← RingEquiv.toRingHom_eq_coe, comap_comap]"
}
]
| [
1720,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1718,
1
]
|
Mathlib/Data/Set/Image.lean | Subtype.coe_preimage_self | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Set α\n⊢ val ⁻¹' s = univ",
"tactic": "rw [← preimage_range, range_coe]"
}
]
| [
1413,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1412,
1
]
|
Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.one_apply_ne | []
| [
137,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_sigma' | []
| [
536,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
534,
1
]
|
Mathlib/Analysis/SpecificLimits/Normed.lean | hasSum_geometric_of_abs_lt_1 | []
| [
306,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
]
|
Mathlib/NumberTheory/LucasPrimality.lean | lucas_primality | [
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\n⊢ Nat.Prime p",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\n⊢ Nat.Prime p",
"tactic": "have h0 : p ≠ 0 := by\n rintro ⟨⟩\n exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\n⊢ Nat.Prime p",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\n⊢ Nat.Prime p",
"tactic": "have h1 : p ≠ 1 := by\n rintro ⟨⟩\n exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\n⊢ Nat.Prime p",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\n⊢ Nat.Prime p",
"tactic": "have hp1 : 1 < p := lt_of_le_of_ne h0.bot_lt h1.symm"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\n⊢ Nat.Prime p",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\n⊢ Nat.Prime p",
"tactic": "have order_of_a : orderOf a = p - 1 := by\n apply orderOf_eq_of_pow_and_pow_div_prime _ ha hd\n exact tsub_pos_of_lt hp1"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n⊢ Nat.Prime p",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\n⊢ Nat.Prime p",
"tactic": "haveI : NeZero p := ⟨h0⟩"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n⊢ Fintype.card (ZMod p)ˣ = p - 1",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n⊢ Nat.Prime p",
"tactic": "rw [Nat.prime_iff_card_units]"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n⊢ p - 1 ≤ Fintype.card (ZMod p)ˣ",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n⊢ Fintype.card (ZMod p)ˣ = p - 1",
"tactic": "refine' le_antisymm (Nat.card_units_zmod_lt_sub_one hp1) _"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\n⊢ p - 1 ≤ Fintype.card (ZMod p)ˣ",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n⊢ p - 1 ≤ Fintype.card (ZMod p)ˣ",
"tactic": "have hp' : p - 2 + 1 = p - 1 := tsub_add_eq_add_tsub hp1"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\na' : (ZMod p)ˣ := Units.mkOfMulEqOne a (a ^ (p - 2)) (_ : a * a ^ (p - 2) = 1)\n⊢ p - 1 ≤ Fintype.card (ZMod p)ˣ",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\n⊢ p - 1 ≤ Fintype.card (ZMod p)ˣ",
"tactic": "let a' : (ZMod p)ˣ := Units.mkOfMulEqOne a (a ^ (p - 2)) (by rw [← pow_succ, hp', ha])"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\na' : (ZMod p)ˣ := Units.mkOfMulEqOne a (a ^ (p - 2)) (_ : a * a ^ (p - 2) = 1)\n⊢ p - 1 ≤ Fintype.card (ZMod p)ˣ",
"tactic": "calc\n p - 1 = orderOf a := order_of_a.symm\n _ = orderOf a' := (orderOf_injective (Units.coeHom (ZMod p)) Units.ext a')\n _ ≤ Fintype.card (ZMod p)ˣ := orderOf_le_card_univ"
},
{
"state_after": "case refl\na : ZMod 0\nha : a ^ (0 - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ 0 - 1 → a ^ ((0 - 1) / q) ≠ 1\n⊢ False",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\n⊢ p ≠ 0",
"tactic": "rintro ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case refl\na : ZMod 0\nha : a ^ (0 - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ 0 - 1 → a ^ ((0 - 1) / q) ≠ 1\n⊢ False",
"tactic": "exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)"
},
{
"state_after": "case refl\na : ZMod 1\nha : a ^ (1 - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ 1 - 1 → a ^ ((1 - 1) / q) ≠ 1\nh0 : 1 ≠ 0\n⊢ False",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\n⊢ p ≠ 1",
"tactic": "rintro ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case refl\na : ZMod 1\nha : a ^ (1 - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ 1 - 1 → a ^ ((1 - 1) / q) ≠ 1\nh0 : 1 ≠ 0\n⊢ False",
"tactic": "exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)"
},
{
"state_after": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\n⊢ 0 < p - 1",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\n⊢ orderOf a = p - 1",
"tactic": "apply orderOf_eq_of_pow_and_pow_div_prime _ ha hd"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\n⊢ 0 < p - 1",
"tactic": "exact tsub_pos_of_lt hp1"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1\nh0 : p ≠ 0\nh1 : p ≠ 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\n⊢ a * a ^ (p - 2) = 1",
"tactic": "rw [← pow_succ, hp', ha]"
}
]
| [
66,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
]
|
Mathlib/Data/Multiset/Fold.lean | Multiset.fold_cons'_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6065\nop : α → α → α\nhc : IsCommutative α op\nha : IsAssociative α op\nb a : α\ns : Multiset α\n⊢ fold op b (a ::ₘ s) = fold op (op b a) s",
"tactic": "rw [fold_eq_foldl, foldl_cons, ← fold_eq_foldl]"
}
]
| [
72,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
]
|
Mathlib/Order/Filter/Germ.lean | Filter.Germ.coe_le | []
| [
662,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
661,
1
]
|
Mathlib/Algebra/Module/LocalizedModule.lean | LocalizedModule.zero_add' | [
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : LocalizedModule S M\nm : M\ns : { x // x ∈ S }\n⊢ ∃ u, u • s • s • m = u • (s * s) • m",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : LocalizedModule S M\nm : M\ns : { x // x ∈ S }\n⊢ 0 + mk m s = mk m s",
"tactic": "rw [← zero_mk s, mk_add_mk, smul_zero, zero_add, mk_eq]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : LocalizedModule S M\nm : M\ns : { x // x ∈ S }\n⊢ ∃ u, u • s • s • m = u • (s * s) • m",
"tactic": "exact ⟨1, by rw [one_smul, mul_smul, one_smul]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : LocalizedModule S M\nm : M\ns : { x // x ∈ S }\n⊢ 1 • s • s • m = 1 • (s * s) • m",
"tactic": "rw [one_smul, mul_smul, one_smul]"
}
]
| [
190,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
9
]
|
Mathlib/Control/Traversable/Equiv.lean | Equiv.comp_map | [
{
"state_after": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝¹ : Functor t\ninst✝ : LawfulFunctor t\nα β γ : Type u\ng : α → β\nh : β → γ\nx : t' α\n⊢ (h ∘ g) <$> ↑(eqv α).symm x = h <$> g <$> ↑(eqv α).symm x",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝¹ : Functor t\ninst✝ : LawfulFunctor t\nα β γ : Type u\ng : α → β\nh : β → γ\nx : t' α\n⊢ Equiv.map eqv (h ∘ g) x = Equiv.map eqv h (Equiv.map eqv g x)",
"tactic": "simp [Equiv.map]"
},
{
"state_after": "no goals",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝¹ : Functor t\ninst✝ : LawfulFunctor t\nα β γ : Type u\ng : α → β\nh : β → γ\nx : t' α\n⊢ (h ∘ g) <$> ↑(eqv α).symm x = h <$> g <$> ↑(eqv α).symm x",
"tactic": "apply comp_map"
}
]
| [
66,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
64,
11
]
|
Mathlib/RingTheory/HahnSeries.lean | HahnSeries.mul_coeff_left' | [
{
"state_after": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\n⊢ ∑ ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd =\n ∑ ij in addAntidiagonal hs (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd",
"state_before": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\n⊢ coeff (x * y) a = ∑ ij in addAntidiagonal hs (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd",
"tactic": "rw [mul_coeff]"
},
{
"state_after": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\n⊢ ∀ (x_1 : Γ × Γ),\n x_1 ∈\n addAntidiagonal hs (_ : Set.IsPwo (support y)) a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a →\n coeff x x_1.fst * coeff y x_1.snd = 0",
"state_before": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\n⊢ ∑ ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd =\n ∑ ij in addAntidiagonal hs (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd",
"tactic": "apply sum_subset_zero_on_sdiff (addAntidiagonal_mono_left hxs) _ fun _ _ => rfl"
},
{
"state_after": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\nb : Γ × Γ\nhb :\n b ∈\n addAntidiagonal hs (_ : Set.IsPwo (support y)) a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a\n⊢ coeff x b.fst * coeff y b.snd = 0",
"state_before": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\n⊢ ∀ (x_1 : Γ × Γ),\n x_1 ∈\n addAntidiagonal hs (_ : Set.IsPwo (support y)) a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a →\n coeff x x_1.fst * coeff y x_1.snd = 0",
"tactic": "intro b hb"
},
{
"state_after": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\nb : Γ × Γ\nhb : (b.fst ∈ s ∧ coeff y b.snd ≠ 0 ∧ b.fst + b.snd = a) ∧ (coeff y b.snd ≠ 0 ∧ b.fst + b.snd = a → coeff x b.fst = 0)\n⊢ coeff x b.fst * coeff y b.snd = 0",
"state_before": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\nb : Γ × Γ\nhb :\n b ∈\n addAntidiagonal hs (_ : Set.IsPwo (support y)) a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a\n⊢ coeff x b.fst * coeff y b.snd = 0",
"tactic": "simp only [not_and', mem_sdiff, mem_addAntidiagonal, mem_support, not_ne_iff] at hb"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : HahnSeries Γ R\na : Γ\ns : Set Γ\nhs : Set.IsPwo s\nhxs : support x ⊆ s\nb : Γ × Γ\nhb : (b.fst ∈ s ∧ coeff y b.snd ≠ 0 ∧ b.fst + b.snd = a) ∧ (coeff y b.snd ≠ 0 ∧ b.fst + b.snd = a → coeff x b.fst = 0)\n⊢ coeff x b.fst * coeff y b.snd = 0",
"tactic": "rw [hb.2 ⟨hb.1.2.1, hb.1.2.2⟩, MulZeroClass.zero_mul]"
}
]
| [
661,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
653,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.coe_symmDiff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.286326\nγ : Type ?u.286329\ninst✝ : DecidableEq α\ns t : Finset α\na b x : α\n⊢ x ∈ ↑(s ∆ t) ↔ x ∈ ↑s ∆ ↑t",
"tactic": "simp [mem_symmDiff, Set.mem_symmDiff]"
}
]
| [
2382,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2381,
1
]
|
Mathlib/Analysis/NormedSpace/lpSpace.lean | lp.eq_zero_iff_coeFn_eq_zero | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E p }\n⊢ f = 0 ↔ ↑f = 0",
"tactic": "rw [lp.ext_iff, coeFn_zero]"
}
]
| [
481,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
480,
1
]
|
Mathlib/Computability/Partrec.lean | Decidable.Partrec.const' | []
| [
449,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
448,
1
]
|
Mathlib/GroupTheory/FreeProduct.lean | FreeProduct.NeWord.append_last | []
| [
619,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
617,
1
]
|
Mathlib/Algebra/Ring/Prod.lean | RingEquiv.coe_prod_comm | []
| [
286,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
]
|
Mathlib/Logic/Nontrivial.lean | Function.Injective.nontrivial | []
| [
148,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
11
]
|
Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_apply_le | []
| [
868,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
866,
1
]
|
Mathlib/Algebra/Associated.lean | Associates.quotient_mk_eq_mk | []
| [
754,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
753,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.disjoint_singleton | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.484138\nγ : Type ?u.484141\nl : Multiset α\na : α\n⊢ Disjoint l {a} ↔ ¬a ∈ l",
"tactic": "rw [disjoint_comm, singleton_disjoint]"
}
]
| [
2944,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2943,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.toNNReal_apply_of_tsum_ne_top | []
| [
940,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
938,
1
]
|
Mathlib/CategoryTheory/Sites/Subsheaf.lean | CategoryTheory.GrothendieckTopology.top_subpresheaf_obj | []
| [
354,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
]
|
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | Matrix.SpecialLinearGroup.det_coe | []
| [
156,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
]
|
Mathlib/Algebra/Group/Basic.lean | mul_eq_of_eq_div | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.59709\nβ : Type ?u.59712\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a = c / b\n⊢ a * b = c",
"tactic": "simp [h]"
}
]
| [
785,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
785,
1
]
|
Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean | CategoryTheory.Limits.has_limits_of_hasEqualizers_and_products | []
| [
143,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
]
|
Std/Data/Int/Lemmas.lean | Int.neg_add_cancel_left | [
{
"state_after": "no goals",
"state_before": "a b : Int\n⊢ -a + (a + b) = b",
"tactic": "rw [← Int.add_assoc, Int.add_left_neg, Int.zero_add]"
}
]
| [
318,
55
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
317,
11
]
|
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.zero_nmem_multiset_map_X_sub_C | []
| [
1497,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1494,
1
]
|
Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | MeasureTheory.Measure.withDensity_rnDeriv_toReal_eq | [
{
"state_after": "case hfm\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ AEMeasurable fun x => rnDeriv μ ν x\n\ncase hf\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ ∀ᵐ (x : α) ∂restrict ν i, rnDeriv μ ν x < ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, ENNReal.toReal (rnDeriv μ ν x) ∂ν) = ENNReal.toReal (↑↑μ i)",
"tactic": "rw [integral_toReal, ← withDensity_apply _ hi, withDensity_rnDeriv_eq μ ν h]"
},
{
"state_after": "no goals",
"state_before": "case hfm\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ AEMeasurable fun x => rnDeriv μ ν x",
"tactic": "measurability"
},
{
"state_after": "case hf\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ (∫⁻ (x : α) in Set.univ, rnDeriv μ ν x ∂ν) < ⊤",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ ∀ᵐ (x : α) ∂restrict ν i, rnDeriv μ ν x < ⊤",
"tactic": "refine' ae_lt_top (μ.measurable_rnDeriv ν)\n (lt_of_le_of_lt (lintegral_mono_set i.subset_univ) _).ne"
},
{
"state_after": "case hf\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ ↑↑μ Set.univ < ⊤",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ (∫⁻ (x : α) in Set.univ, rnDeriv μ ν x ∂ν) < ⊤",
"tactic": "rw [← withDensity_apply _ MeasurableSet.univ, withDensity_rnDeriv_eq μ ν h]"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.2764\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : HaveLebesgueDecomposition μ ν\nh : μ ≪ ν\ni : Set α\nhi : MeasurableSet i\n⊢ ↑↑μ Set.univ < ⊤",
"tactic": "exact measure_lt_top _ _"
}
]
| [
81,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
]
|
Mathlib/Algebra/Star/Subalgebra.lean | StarSubalgebra.adjoin_le_iff | []
| [
482,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
481,
1
]
|
Mathlib/Data/Quot.lean | Quot.out_eq | []
| [
365,
44
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
364,
1
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|
Mathlib/Analysis/Calculus/ContDiff.lean | hasFTaylorSeriesUpToOn_pi' | [
{
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"tactic": "convert hasFTaylorSeriesUpToOn_pi (𝕜 := 𝕜) (φ := fun i x ↦ Φ x i)"
},
{
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{
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"tactic": "rfl"
}
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1143,
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1138,
1
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|
Mathlib/RingTheory/NonZeroDivisors.lean | mul_mem_nonZeroDivisors | [
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"tactic": "constructor"
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{
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"tactic": "intro h"
},
{
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"tactic": "rw [← mul_assoc, h', zero_mul]"
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{
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{
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"tactic": "apply hb"
},
{
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"state_before": "case mpr.intro.a.a\nM : Type ?u.20654\nM' : Type ?u.20657\nM₁ : Type u_1\nR : Type ?u.20663\nR' : Type ?u.20666\nF : Type ?u.20669\ninst✝⁴ : MonoidWithZero M\ninst✝³ : MonoidWithZero M'\ninst✝² : CommMonoidWithZero M₁\ninst✝¹ : Ring R\ninst✝ : CommRing R'\na b : M₁\nha : a ∈ M₁⁰\nhb : b ∈ M₁⁰\nx : M₁\nhx : x * (a * b) = 0\n⊢ x * a * b = 0",
"tactic": "rw [mul_assoc, hx]"
}
]
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103,
23
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
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|
Mathlib/Data/Setoid/Basic.lean | Setoid.injective_iff_ker_bot | []
| [
285,
44
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
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|
Mathlib/Deprecated/Subgroup.lean | Group.subset_normalClosure | []
| [
708,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
707,
1
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|
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ioo_subset_Ioi_self | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.49166\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : LocallyFiniteOrderTop α\n⊢ Ioo a b ⊆ Ioi a",
"tactic": "simpa [← coe_subset] using Set.Ioo_subset_Ioi_self"
}
]
| [
409,
53
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Topology/Sets/Opens.lean | TopologicalSpace.Opens.iSup_mk | []
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242,
13
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
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|
Mathlib/Algebra/Ring/Defs.lean | ite_and_mul_zero | [
{
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"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nR : Type x\nα : Type u_1\ninst✝² : MulZeroClass α\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b : α\n⊢ (if P ∧ Q then a * b else 0) = (if P then a else 0) * if Q then b else 0",
"tactic": "simp only [← ite_and, ite_mul, mul_ite, mul_zero, zero_mul, and_comm]"
}
]
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239,
72
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/Multiset/FinsetOps.lean | Multiset.subset_ndunion_left | []
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187,
28
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
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|
Mathlib/Algebra/Algebra/Spectrum.lean | spectrum.subset_starSubalgebra | []
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291,
53
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
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|
Mathlib/Data/Polynomial/Basic.lean | Polynomial.support_X_empty | [
{
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"tactic": "rw [X, H, monomial_zero_right, support_zero]"
}
]
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908,
47
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
907,
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|
Mathlib/Analysis/PSeries.lean | Real.summable_abs_int_rpow | [
{
"state_after": "case refine'_1\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)\n\ncase refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑(-↑n) ^ (-b)",
"state_before": "b : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)",
"tactic": "refine'\n summable_int_of_summable_nat (_ : Summable fun n : ℕ => |(n : ℝ)| ^ _)\n (_ : Summable fun n : ℕ => |((-n : ℤ) : ℝ)| ^ _)"
},
{
"state_after": "case refine'_1\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)\n\ncase refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)",
"state_before": "case refine'_1\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)\n\ncase refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑(-↑n) ^ (-b)",
"tactic": "on_goal 2 => simp_rw [Int.cast_neg, Int.cast_ofNat, abs_neg]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)\n\ncase refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)",
"tactic": "all_goals\n simp_rw [fun n : ℕ => abs_of_nonneg (n.cast_nonneg : 0 ≤ (n : ℝ))]\n rwa [Real.summable_nat_rpow, neg_lt_neg_iff]"
},
{
"state_after": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)",
"state_before": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑(-↑n) ^ (-b)",
"tactic": "simp_rw [Int.cast_neg, Int.cast_ofNat, abs_neg]"
},
{
"state_after": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)",
"state_before": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑(-↑n) ^ (-b)",
"tactic": "simp_rw [Int.cast_neg, Int.cast_ofNat, abs_neg]"
},
{
"state_after": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => ↑n ^ (-b)",
"state_before": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => abs ↑n ^ (-b)",
"tactic": "simp_rw [fun n : ℕ => abs_of_nonneg (n.cast_nonneg : 0 ≤ (n : ℝ))]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nb : ℝ\nhb : 1 < b\n⊢ Summable fun n => ↑n ^ (-b)",
"tactic": "rwa [Real.summable_nat_rpow, neg_lt_neg_iff]"
}
]
| [
243,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
]
|
Mathlib/Algebra/Regular/SMul.lean | IsLeftRegular.isSMulRegular | []
| [
43,
4
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
]
|
Mathlib/Algebra/Order/Floor.lean | Nat.floor_lt' | []
| [
210,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
]
|
Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | Matrix.Represents.one | [
{
"state_after": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\n⊢ ↑(LinearMap.comp (↑(LinearMap.llcomp R (ι → R) (ι → R) M) (↑(Fintype.total R R) b))\n (AlgEquiv.toLinearMap (AlgEquiv.symm algEquivMatrix')))\n 1 =\n ↑(PiToModule.fromEnd R b) 1",
"state_before": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\n⊢ Represents b 1 1",
"tactic": "delta Matrix.Represents PiToModule.fromMatrix"
},
{
"state_after": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\n⊢ ↑(↑(LinearMap.llcomp R (ι → R) (ι → R) M) (↑(Fintype.total R R) b)) 1 = ↑(PiToModule.fromEnd R b) 1",
"state_before": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\n⊢ ↑(LinearMap.comp (↑(LinearMap.llcomp R (ι → R) (ι → R) M) (↑(Fintype.total R R) b))\n (AlgEquiv.toLinearMap (AlgEquiv.symm algEquivMatrix')))\n 1 =\n ↑(PiToModule.fromEnd R b) 1",
"tactic": "rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_one]"
},
{
"state_after": "case h.h\nι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\ni✝ : ι\n⊢ ↑(LinearMap.comp (↑(↑(LinearMap.llcomp R (ι → R) (ι → R) M) (↑(Fintype.total R R) b)) 1) (LinearMap.single i✝)) 1 =\n ↑(LinearMap.comp (↑(PiToModule.fromEnd R b) 1) (LinearMap.single i✝)) 1",
"state_before": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\n⊢ ↑(↑(LinearMap.llcomp R (ι → R) (ι → R) M) (↑(Fintype.total R R) b)) 1 = ↑(PiToModule.fromEnd R b) 1",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\ni✝ : ι\n⊢ ↑(LinearMap.comp (↑(↑(LinearMap.llcomp R (ι → R) (ι → R) M) (↑(Fintype.total R R) b)) 1) (LinearMap.single i✝)) 1 =\n ↑(LinearMap.comp (↑(PiToModule.fromEnd R b) 1) (LinearMap.single i✝)) 1",
"tactic": "rfl"
}
]
| [
133,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
]
|
Mathlib/Data/ZMod/Basic.lean | ZMod.valMinAbs_natCast_of_half_lt | [
{
"state_after": "case zero\na : ℕ\nha : Nat.zero / 2 < a\nha' : a < Nat.zero\n⊢ valMinAbs ↑a = ↑a - ↑Nat.zero\n\ncase succ\na n✝ : ℕ\nha : Nat.succ n✝ / 2 < a\nha' : a < Nat.succ n✝\n⊢ valMinAbs ↑a = ↑a - ↑(Nat.succ n✝)",
"state_before": "n a : ℕ\nha : n / 2 < a\nha' : a < n\n⊢ valMinAbs ↑a = ↑a - ↑n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\na : ℕ\nha : Nat.zero / 2 < a\nha' : a < Nat.zero\n⊢ valMinAbs ↑a = ↑a - ↑Nat.zero",
"tactic": "cases not_lt_bot ha'"
},
{
"state_after": "no goals",
"state_before": "case succ\na n✝ : ℕ\nha : Nat.succ n✝ / 2 < a\nha' : a < Nat.succ n✝\n⊢ valMinAbs ↑a = ↑a - ↑(Nat.succ n✝)",
"tactic": "simp [valMinAbs_def_pos, val_nat_cast, Nat.mod_eq_of_lt ha', ha.not_le]"
}
]
| [
1079,
76
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1075,
1
]
|
Std/Data/String/Lemmas.lean | String.Pos.Valid.mk | []
| [
155,
94
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
155,
1
]
|
Mathlib/Combinatorics/Quiver/Symmetric.lean | Prefunctor.map_reverse | []
| [
98,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
]
|
Mathlib/LinearAlgebra/Finsupp.lean | Finsupp.mapRange.linearEquiv_refl | []
| [
864,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
862,
1
]
|
Mathlib/GroupTheory/GroupAction/Units.lean | Units.val_smul | []
| [
113,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
]
|
Mathlib/MeasureTheory/Measure/AEDisjoint.lean | MeasureTheory.aedisjoint_compl_left | []
| [
161,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
]
|
Mathlib/Data/Fin/Basic.lean | Fin.predAbove_last_apply | [
{
"state_after": "no goals",
"state_before": "n m : ℕ\ni : Fin n\n⊢ predAbove (last n) ↑↑i = castPred ↑↑i",
"tactic": "rw [predAbove_last]"
}
]
| [
2353,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2352,
1
]
|
Mathlib/Topology/Connected.lean | PreconnectedSpace.constant | []
| [
1596,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1594,
1
]
|
Mathlib/Data/PFun.lean | PFun.prodMap_apply | []
| [
681,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
679,
1
]
|
Mathlib/Algebra/Order/WithZero.lean | mul_le_one₀ | []
| [
128,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
]
|
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | stronglyMeasurable_const_smul_iff₀ | []
| [
506,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
504,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.cons_isCycle_iff | [
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G v u\nh : Adj G u v\n⊢ (¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔\n List.Nodup (support p) ∧ ¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G v u\nh : Adj G u v\n⊢ IsCycle (cons h p) ↔ IsPath p ∧ ¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p",
"tactic": "simp only [Walk.isCycle_def, Walk.isPath_def, Walk.isTrail_def, edges_cons, List.nodup_cons,\n support_cons, List.tail_cons]"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G v u\nh : Adj G u v\nthis : List.Nodup (support p) → List.Nodup (edges p)\n⊢ (¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔\n List.Nodup (support p) ∧ ¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G v u\nh : Adj G u v\n⊢ (¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔\n List.Nodup (support p) ∧ ¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p",
"tactic": "have : p.support.Nodup → p.edges.Nodup := edges_nodup_of_support_nodup"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G v u\nh : Adj G u v\nthis : List.Nodup (support p) → List.Nodup (edges p)\n⊢ (¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔\n List.Nodup (support p) ∧ ¬Quotient.mk (Sym2.Rel.setoid V) (u, v) ∈ edges p",
"tactic": "tauto"
}
]
| [
1014,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1009,
1
]
|
Mathlib/Algebra/Algebra/Tower.lean | Submodule.span_smul_of_span_eq_top | []
| [
319,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
]
|
Mathlib/Computability/Partrec.lean | Computable.unpair | []
| [
318,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
]
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | Real.lt_rpow_inv_iff_of_neg | [
{
"state_after": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"tactic": "have hz' : 0 < -z := by rwa [lt_neg, neg_zero]"
},
{
"state_after": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"tactic": "have hxz : 0 < x ^ (-z) := Real.rpow_pos_of_pos hx _"
},
{
"state_after": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"tactic": "have hyz : 0 < y ^ z⁻¹ := Real.rpow_pos_of_pos hy _"
},
{
"state_after": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ x ^ (-z) < y ^ (z⁻¹ * -z) ↔ y < x ^ z",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ x < y ^ z⁻¹ ↔ y < x ^ z",
"tactic": "rw [← Real.rpow_lt_rpow_iff hx.le hyz.le hz', ← Real.rpow_mul hy.le]"
},
{
"state_after": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ x ^ (-z) < y⁻¹ ↔ y < x ^ z",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ x ^ (-z) < y ^ (z⁻¹ * -z) ↔ y < x ^ z",
"tactic": "simp only [ne_of_lt hz, Real.rpow_neg_one, mul_neg, inv_mul_cancel, Ne.def, not_false_iff]"
},
{
"state_after": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ y < x ^ (-z * -1) ↔ y < x ^ z",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ x ^ (-z) < y⁻¹ ↔ y < x ^ z",
"tactic": "rw [lt_inv hxz hy, ← Real.rpow_neg_one, ← Real.rpow_mul hx.le]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z⁻¹\n⊢ y < x ^ (-z * -1) ↔ y < x ^ z",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n⊢ 0 < -z",
"tactic": "rwa [lt_neg, neg_zero]"
}
]
| [
455,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
447,
1
]
|
Mathlib/CategoryTheory/EqToHom.lean | CategoryTheory.eqToHom_refl | []
| [
52,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
]
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean | measure_le_lintegral_thickenedIndicator | [
{
"state_after": "case h.e'_4.h.e'_4.h\nα : Type u_1\nβ : Type ?u.5010653\nE✝ : Type ?u.5010656\nF : Type ?u.5010659\ninst✝² : MeasurableSpace α\nι : Type ?u.5010665\ninst✝¹ : NormedAddCommGroup E✝\ninst✝ : PseudoEMetricSpace α\nμ : MeasureTheory.Measure α\nE : Set α\nE_mble : MeasurableSet E\nδ : ℝ\nδ_pos : 0 < δ\nx✝ : α\n⊢ ↑(↑(thickenedIndicator δ_pos E) x✝) = thickenedIndicatorAux δ E x✝",
"state_before": "α : Type u_1\nβ : Type ?u.5010653\nE✝ : Type ?u.5010656\nF : Type ?u.5010659\ninst✝² : MeasurableSpace α\nι : Type ?u.5010665\ninst✝¹ : NormedAddCommGroup E✝\ninst✝ : PseudoEMetricSpace α\nμ : MeasureTheory.Measure α\nE : Set α\nE_mble : MeasurableSet E\nδ : ℝ\nδ_pos : 0 < δ\n⊢ ↑↑μ E ≤ ∫⁻ (a : α), ↑(↑(thickenedIndicator δ_pos E) a) ∂μ",
"tactic": "convert measure_le_lintegral_thickenedIndicatorAux μ E_mble δ"
},
{
"state_after": "case h.e'_4.h.e'_4.h\nα : Type u_1\nβ : Type ?u.5010653\nE✝ : Type ?u.5010656\nF : Type ?u.5010659\ninst✝² : MeasurableSpace α\nι : Type ?u.5010665\ninst✝¹ : NormedAddCommGroup E✝\ninst✝ : PseudoEMetricSpace α\nμ : MeasureTheory.Measure α\nE : Set α\nE_mble : MeasurableSet E\nδ : ℝ\nδ_pos : 0 < δ\nx✝ : α\n⊢ ↑(ENNReal.toNNReal (thickenedIndicatorAux δ E x✝)) = thickenedIndicatorAux δ E x✝",
"state_before": "case h.e'_4.h.e'_4.h\nα : Type u_1\nβ : Type ?u.5010653\nE✝ : Type ?u.5010656\nF : Type ?u.5010659\ninst✝² : MeasurableSpace α\nι : Type ?u.5010665\ninst✝¹ : NormedAddCommGroup E✝\ninst✝ : PseudoEMetricSpace α\nμ : MeasureTheory.Measure α\nE : Set α\nE_mble : MeasurableSet E\nδ : ℝ\nδ_pos : 0 < δ\nx✝ : α\n⊢ ↑(↑(thickenedIndicator δ_pos E) x✝) = thickenedIndicatorAux δ E x✝",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_4.h\nα : Type u_1\nβ : Type ?u.5010653\nE✝ : Type ?u.5010656\nF : Type ?u.5010659\ninst✝² : MeasurableSpace α\nι : Type ?u.5010665\ninst✝¹ : NormedAddCommGroup E✝\ninst✝ : PseudoEMetricSpace α\nμ : MeasureTheory.Measure α\nE : Set α\nE_mble : MeasurableSet E\nδ : ℝ\nδ_pos : 0 < δ\nx✝ : α\n⊢ ↑(ENNReal.toNNReal (thickenedIndicatorAux δ E x✝)) = thickenedIndicatorAux δ E x✝",
"tactic": "simp only [thickenedIndicatorAux_lt_top.ne, ENNReal.coe_toNNReal, Ne.def, not_false_iff]"
}
]
| [
1312,
91
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1307,
1
]
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_biUnion_finset | []
| [
1218,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1215,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigOWith.exists_eq_mul | []
| [
1952,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1950,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.map_monomial | [
{
"state_after": "case h\nσ : Type u_1\nR : Type u_3\nS : Type u_2\nT : Type ?u.869997\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\n⊢ ↑(coeff S m) (↑(map σ f) (↑(monomial R n) a)) = ↑(coeff S m) (↑(monomial S n) (↑f a))",
"state_before": "σ : Type u_1\nR : Type u_3\nS : Type u_2\nT : Type ?u.869997\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn : σ →₀ ℕ\na : R\n⊢ ↑(map σ f) (↑(monomial R n) a) = ↑(monomial S n) (↑f a)",
"tactic": "ext m"
},
{
"state_after": "no goals",
"state_before": "case h\nσ : Type u_1\nR : Type u_3\nS : Type u_2\nT : Type ?u.869997\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\n⊢ ↑(coeff S m) (↑(map σ f) (↑(monomial R n) a)) = ↑(coeff S m) (↑(monomial S n) (↑f a))",
"tactic": "simp [coeff_monomial, apply_ite f]"
}
]
| [
614,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
1
]
|
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Icc_diff_Ico_self | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.95633\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b c : α\ninst✝ : DecidableEq α\nh : a ≤ b\n⊢ Icc a b \\ Ico a b = {b}",
"tactic": "simp [← coe_inj, h]"
}
]
| [
585,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
585,
1
]
|
Mathlib/Order/Monotone/Monovary.lean | AntivaryOn.dual | []
| [
228,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.erase_zero | []
| [
1009,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1008,
1
]
|
Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean | CategoryTheory.OplaxNatTrans.Modification.whiskerLeft_naturality | [
{
"state_after": "no goals",
"state_before": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G : OplaxFunctor B C\nη θ ι : F ⟶ G\nΓ : Modification η θ\na b c : B\na' : C\nf : a' ⟶ (↑F.toPrelaxFunctor).obj b\ng : b ⟶ c\n⊢ f ◁ (↑F.toPrelaxFunctor).map g ◁ app Γ c ≫ f ◁ OplaxNatTrans.naturality θ g =\n f ◁ OplaxNatTrans.naturality η g ≫ f ◁ app Γ b ▷ (↑G.toPrelaxFunctor).map g",
"tactic": "simp_rw [← whiskerLeft_comp, naturality]"
}
]
| [
268,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
]
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | OrderIso.map_ciInf_set | []
| [
1361,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1359,
1
]
|
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.coe_mulₗᵢ | []
| [
1178,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1177,
1
]
|
Mathlib/Order/Basic.lean | le_iff_le_iff_lt_iff_lt | []
| [
521,
87
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
519,
1
]
|
Mathlib/Topology/ContinuousFunction/Compact.lean | ContinuousMap.uniformEmbedding_equivBoundedOfCompact | []
| [
75,
98
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
]
|
Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | CategoryTheory.Limits.pullbackZeroZeroIso_hom_snd | [
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝³ : Category C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryProduct X Y\n⊢ (pullbackZeroZeroIso X Y).hom ≫ prod.snd = pullback.snd",
"tactic": "simp [← Iso.eq_inv_comp]"
}
]
| [
193,
91
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
]
|
Mathlib/Algebra/Star/Module.lean | star_inv_int_cast_smul | []
| [
61,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
]
|
Mathlib/Topology/Constructions.lean | nhds_subtype_eq_comap | []
| [
1076,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1075,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.count_union | []
| [
2467,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2466,
1
]
|
Mathlib/Order/RelClasses.lean | IsAsymm.isIrrefl | []
| [
124,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
11
]
|
Mathlib/Data/Set/Lattice.lean | Set.preimage_sInter | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.226389\nι : Sort ?u.226392\nι' : Sort ?u.226395\nι₂ : Sort ?u.226398\nκ : ι → Sort ?u.226403\nκ₁ : ι → Sort ?u.226408\nκ₂ : ι → Sort ?u.226413\nκ' : ι' → Sort ?u.226418\nf : α → β\ns : Set (Set β)\n⊢ f ⁻¹' ⋂₀ s = ⋂ (t : Set β) (_ : t ∈ s), f ⁻¹' t",
"tactic": "rw [sInter_eq_biInter, preimage_iInter₂]"
}
]
| [
1742,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1741,
1
]
|
Mathlib/Order/WellFounded.lean | Function.argminOn_mem | []
| [
206,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
]
|
Mathlib/Algebra/Order/CompleteField.lean | LinearOrderedField.mem_cutMap_iff | []
| [
116,
88
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
]
|
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | Complex.map_exp_comap_re_atTop | [
{
"state_after": "⊢ {0}ᶜ ∈ comap (↑abs) atTop",
"state_before": "⊢ map exp (comap re atTop) = comap (↑abs) atTop",
"tactic": "rw [← comap_exp_comap_abs_atTop, map_comap, range_exp, inf_eq_left, le_principal_iff]"
},
{
"state_after": "no goals",
"state_before": "⊢ {0}ᶜ ∈ comap (↑abs) atTop",
"tactic": "exact eventually_ne_of_tendsto_norm_atTop tendsto_comap 0"
}
]
| [
224,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
]
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.cos_nat_mul_two_pi | []
| [
1232,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1231,
1
]
|
Mathlib/Data/List/Perm.lean | List.Perm.union_right | [
{
"state_after": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ : List α\na : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nih : List.union l₁✝ t₁ ~ List.union l₂✝ t₁\n⊢ List.insert a (List.union l₁✝ t₁) ~ List.insert a (List.union l₂✝ t₁)\n\ncase swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ : List α\nx✝ y✝ : α\nl✝ : List α\n⊢ List.insert y✝ (List.insert x✝ (List.union l✝ t₁)) ~ List.insert x✝ (List.insert y✝ (List.union l✝ t₁))\n\ncase trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\nih_1 : List.union l₁✝ t₁ ~ List.union l₂✝ t₁\nih_2 : List.union l₂✝ t₁ ~ List.union l₃✝ t₁\n⊢ List.union l₁✝ t₁ ~ List.union l₃✝ t₁",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ : List α\nh : l₁ ~ l₂\n⊢ List.union l₁ t₁ ~ List.union l₂ t₁",
"tactic": "induction' h with a _ _ _ ih _ _ _ _ _ _ _ _ ih_1 ih_2 <;> try simp"
},
{
"state_after": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\nih_1 : List.union l₁✝ t₁ ~ List.union l₂✝ t₁\nih_2 : List.union l₂✝ t₁ ~ List.union l₃✝ t₁\n⊢ List.union l₁✝ t₁ ~ List.union l₃✝ t₁",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\nih_1 : List.union l₁✝ t₁ ~ List.union l₂✝ t₁\nih_2 : List.union l₂✝ t₁ ~ List.union l₃✝ t₁\n⊢ List.union l₁✝ t₁ ~ List.union l₃✝ t₁",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ : List α\na : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nih : List.union l₁✝ t₁ ~ List.union l₂✝ t₁\n⊢ List.insert a (List.union l₁✝ t₁) ~ List.insert a (List.union l₂✝ t₁)",
"tactic": "exact ih.insert a"
},
{
"state_after": "no goals",
"state_before": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ : List α\nx✝ y✝ : α\nl✝ : List α\n⊢ List.insert y✝ (List.insert x✝ (List.union l✝ t₁)) ~ List.insert x✝ (List.insert y✝ (List.union l✝ t₁))",
"tactic": "apply perm_insert_swap"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\ninst✝ : DecidableEq α\nl₁ l₂ t₁ l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\nih_1 : List.union l₁✝ t₁ ~ List.union l₂✝ t₁\nih_2 : List.union l₂✝ t₁ ~ List.union l₃✝ t₁\n⊢ List.union l₁✝ t₁ ~ List.union l₃✝ t₁",
"tactic": "exact ih_1.trans ih_2"
}
]
| [
995,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
990,
1
]
|
Mathlib/Analysis/NormedSpace/ENorm.lean | ENorm.ext | []
| [
72,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
]
|
Mathlib/Order/Filter/AtTopBot.lean | Filter.inf_map_atBot_neBot_iff | []
| [
456,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
]
|
Mathlib/LinearAlgebra/Span.lean | Submodule.submodule_eq_sSup_le_nonzero_spans | [
{
"state_after": "R : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ p = sSup {T | ∃ m x x, T = span R {m}}",
"state_before": "R : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\n⊢ p = sSup {T | ∃ m x x, T = span R {m}}",
"tactic": "let S := { T : Submodule R M | ∃ (m : M) (_ : m ∈ p) (_ : m ≠ 0), T = span R {m} }"
},
{
"state_after": "case a\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ p ≤ sSup {T | ∃ m x x, T = span R {m}}\n\ncase a\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ sSup {T | ∃ m x x, T = span R {m}} ≤ p",
"state_before": "R : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ p = sSup {T | ∃ m x x, T = span R {m}}",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\nm : M\nhm : m ∈ p\n⊢ m ∈ sSup {T | ∃ m x x, T = span R {m}}",
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"tactic": "by_cases h : m = 0"
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{
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"state_before": "case pos\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\nm : M\nhm : m ∈ p\nh : m = 0\n⊢ m ∈ sSup {T | ∃ m x x, T = span R {m}}",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\nm : M\nhm : m ∈ p\nh : m = 0\n⊢ 0 ∈ sSup {T | ∃ m x x, T = span R {m}}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\nm : M\nhm : m ∈ p\nh : ¬m = 0\n⊢ m ∈ sSup {T | ∃ m x x, T = span R {m}}",
"tactic": "exact @le_sSup _ _ S _ ⟨m, ⟨hm, ⟨h, rfl⟩⟩⟩ m (mem_span_singleton_self m)"
},
{
"state_after": "case a\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ ∀ (b : Submodule R M), b ∈ {T | ∃ m x x, T = span R {m}} → b ≤ p",
"state_before": "case a\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ sSup {T | ∃ m x x, T = span R {m}} ≤ p",
"tactic": "rw [sSup_le_iff]"
},
{
"state_after": "case a.intro.intro.intro\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\nw✝² : M\nw✝¹ : w✝² ∈ p\nw✝ : w✝² ≠ 0\n⊢ span R {w✝²} ≤ p",
"state_before": "case a\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\n⊢ ∀ (b : Submodule R M), b ∈ {T | ∃ m x x, T = span R {m}} → b ≤ p",
"tactic": "rintro S ⟨_, ⟨_, ⟨_, rfl⟩⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro.intro\nR : Type u_1\nR₂ : Type ?u.227879\nK : Type ?u.227882\nM : Type u_2\nM₂ : Type ?u.227888\nV : Type ?u.227891\nS✝ : Type ?u.227894\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m x x, T = span R {m}}\nw✝² : M\nw✝¹ : w✝² ∈ p\nw✝ : w✝² ≠ 0\n⊢ span R {w✝²} ≤ p",
"tactic": "rwa [span_singleton_le_iff_mem]"
}
]
| [
702,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
691,
1
]
|
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.nhdsWithin_Ioi_ofNat_nebot | []
| [
227,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
226,
1
]
|
Std/Data/String/Lemmas.lean | String.get_cons_addChar | [
{
"state_after": "no goals",
"state_before": "c : Char\ncs : List Char\ni : Pos\n⊢ get { data := c :: cs } (i + c) = get { data := cs } i",
"tactic": "simp [get, utf8GetAux, Pos.zero_ne_addChar, utf8GetAux_addChar_right_cancel]"
}
]
| [
215,
79
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
213,
1
]
|
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | CircleDeg1Lift.tendsto_atTop | [
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\n⊢ Tendsto (fun x => x - 1) atTop atTop",
"tactic": "simpa [sub_eq_add_neg] using tendsto_atTop_add_const_right _ _ tendsto_id"
}
]
| [
553,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
550,
11
]
|
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | differentiableWithinAt_id | []
| [
1017,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1016,
1
]
|
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