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Mathlib/CategoryTheory/CommSq.lean | CategoryTheory.CommSq.of_arrow | []
| [
63,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
]
|
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | Ideal.adic_basis | [
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
"tactic": "suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by\n simpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\ni j : ℕ\n⊢ ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
"tactic": "intro i j"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\ni j : ℕ\n⊢ ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
"tactic": "exact ⟨max i j, pow_le_pow (le_max_left i j), pow_le_pow (le_max_right i j)⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
"tactic": "simpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j ≤ I ^ i",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j • ⊤ ≤ I ^ i • ⊤",
"tactic": "suffices ∀ (a : R) (i : ℕ), ∃ j : ℕ, a • I ^ j ≤ I ^ i by\n simpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\n⊢ ∃ j, r • I ^ j ≤ I ^ n",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j ≤ I ^ i",
"tactic": "intro r n"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\n⊢ r • I ^ n ≤ I ^ n",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\n⊢ ∃ j, r • I ^ j ≤ I ^ n",
"tactic": "use n"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\nx : R\nhx : x ∈ ↑(I ^ n)\n⊢ ↑(DistribMulAction.toLinearMap R R r) x ∈ I ^ n",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\n⊢ r • I ^ n ≤ I ^ n",
"tactic": "rintro a ⟨x, hx, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\nx : R\nhx : x ∈ ↑(I ^ n)\n⊢ ↑(DistribMulAction.toLinearMap R R r) x ∈ I ^ n",
"tactic": "exact (I ^ n).smul_mem r hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j ≤ I ^ i\n⊢ ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j • ⊤ ≤ I ^ i • ⊤",
"tactic": "simpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i : ℕ), ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ i)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i : ℕ), ∃ j, ↑(I ^ j • ⊤) * ↑(I ^ j • ⊤) ⊆ ↑(I ^ i • ⊤)",
"tactic": "suffices ∀ i : ℕ, ∃ j : ℕ, (I ^ j: Set R) * (I ^ j : Set R) ⊆ (I ^ i : Set R) by\n simpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\n⊢ ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ n)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (i : ℕ), ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ i)",
"tactic": "intro n"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\n⊢ ↑(I ^ n) * ↑(I ^ n) ⊆ ↑(I ^ n)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\n⊢ ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ n)",
"tactic": "use n"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nx b : R\n_hx : x ∈ ↑(I ^ n)\nhb : b ∈ ↑(I ^ n)\n⊢ (fun x x_1 => x * x_1) x b ∈ ↑(I ^ n)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\n⊢ ↑(I ^ n) * ↑(I ^ n) ⊆ ↑(I ^ n)",
"tactic": "rintro a ⟨x, b, _hx, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nx b : R\n_hx : x ∈ ↑(I ^ n)\nhb : b ∈ ↑(I ^ n)\n⊢ (fun x x_1 => x * x_1) x b ∈ ↑(I ^ n)",
"tactic": "exact (I ^ n).smul_mem x hb"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (i : ℕ), ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ i)\n⊢ ∀ (i : ℕ), ∃ j, ↑(I ^ j • ⊤) * ↑(I ^ j • ⊤) ⊆ ↑(I ^ i • ⊤)",
"tactic": "simpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this"
}
]
| [
77,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.sum_smul_vsub_const_eq_weightedVSub | [
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_3\nP : Type u_4\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.202333\ns₂ : Finset ι₂\nw : ι → k\np₁ : ι → P\np₂ : P\nh : ∑ i in s, w i = 0\n⊢ ∑ i in s, w i • (p₁ i -ᵥ p₂) = ↑(weightedVSub s p₁) w",
"tactic": "rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, sub_zero]"
}
]
| [
324,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
]
|
Mathlib/Data/Polynomial/Mirror.lean | Polynomial.mirror_eq_iff | []
| [
138,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | differentiable_id' | []
| [
1025,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1025,
1
]
|
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | Submodule.orthogonal_le | []
| [
159,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
]
|
Mathlib/RingTheory/Ideal/Cotangent.lean | Ideal.toCotangent_eq | [
{
"state_after": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx y : { x // x ∈ I }\n⊢ ↑(toCotangent I) x - ↑(toCotangent I) y = 0 ↔ ↑x - ↑y ∈ I ^ 2",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx y : { x // x ∈ I }\n⊢ ↑(toCotangent I) x = ↑(toCotangent I) y ↔ ↑x - ↑y ∈ I ^ 2",
"tactic": "rw [← sub_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx y : { x // x ∈ I }\n⊢ ↑(toCotangent I) x - ↑(toCotangent I) y = 0 ↔ ↑x - ↑y ∈ I ^ 2",
"tactic": "exact I.mem_toCotangent_ker"
}
]
| [
81,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
]
|
Mathlib/Topology/Algebra/Monoid.lean | LocallyFinite.exists_finset_mulSupport | [
{
"state_after": "case intro.intro\nι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\nU : Set X\nhxU : U ∈ 𝓝 x₀\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}\n⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ ↑I",
"state_before": "ι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\n⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ ↑I",
"tactic": "rcases hf x₀ with ⟨U, hxU, hUf⟩"
},
{
"state_after": "case intro.intro\nι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\nU : Set X\nhxU : U ∈ 𝓝 x₀\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}\ny : X\nhy : y ∈ U\ni : ι\nhi : i ∈ mulSupport fun i => f i y\n⊢ i ∈ ↑(Finite.toFinset hUf)",
"state_before": "case intro.intro\nι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\nU : Set X\nhxU : U ∈ 𝓝 x₀\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}\n⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ ↑I",
"tactic": "refine' ⟨hUf.toFinset, mem_of_superset hxU fun y hy i hi => _⟩"
},
{
"state_after": "case intro.intro\nι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\nU : Set X\nhxU : U ∈ 𝓝 x₀\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}\ny : X\nhy : y ∈ U\ni : ι\nhi : i ∈ mulSupport fun i => f i y\n⊢ i ∈ {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}",
"state_before": "case intro.intro\nι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\nU : Set X\nhxU : U ∈ 𝓝 x₀\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}\ny : X\nhy : y ∈ U\ni : ι\nhi : i ∈ mulSupport fun i => f i y\n⊢ i ∈ ↑(Finite.toFinset hUf)",
"tactic": "rw [hUf.coe_toFinset]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_2\nα : Type ?u.448737\nX : Type u_3\nM✝ : Type ?u.448743\nN : Type ?u.448746\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace M✝\ninst✝² : CommMonoid M✝\ninst✝¹ : ContinuousMul M✝\nM : Type u_1\ninst✝ : CommMonoid M\nf : ι → X → M\nhf : LocallyFinite fun i => mulSupport (f i)\nx₀ : X\nU : Set X\nhxU : U ∈ 𝓝 x₀\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}\ny : X\nhy : y ∈ U\ni : ι\nhi : i ∈ mulSupport fun i => f i y\n⊢ i ∈ {i | Set.Nonempty ((fun i => mulSupport (f i)) i ∩ U)}",
"tactic": "exact ⟨y, hi, hy⟩"
}
]
| [
795,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
789,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.not_mem_mono | []
| [
529,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
528,
1
]
|
Mathlib/LinearAlgebra/Matrix/Determinant.lean | Matrix.det_unique | [
{
"state_after": "no goals",
"state_before": "m : Type ?u.179266\nn✝ : Type ?u.179269\ninst✝⁷ : DecidableEq n✝\ninst✝⁶ : Fintype n✝\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\nR : Type v\ninst✝³ : CommRing R\nn : Type u_1\ninst✝² : Unique n\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nA : Matrix n n R\n⊢ det A = A default default",
"tactic": "simp [det_apply, univ_unique]"
}
]
| [
122,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
]
|
Mathlib/Computability/Primrec.lean | PrimrecPred.comp | []
| [
494,
15
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
492,
1
]
|
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean | StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin | []
| [
113,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
]
|
Mathlib/Computability/TuringMachine.lean | Turing.Tape.write_move_right_n | [
{
"state_after": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL✝ R✝ L R : ListBlank Γ\n⊢ write (f (ListBlank.nth R Nat.zero)) ((move Dir.right^[Nat.zero]) (mk' L R)) =\n (move Dir.right^[Nat.zero]) (mk' L (ListBlank.modifyNth f Nat.zero R))\n\ncase succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL✝ R✝ : ListBlank Γ\nn : ℕ\nIH :\n ∀ (L R : ListBlank Γ),\n write (f (ListBlank.nth R n)) ((move Dir.right^[n]) (mk' L R)) =\n (move Dir.right^[n]) (mk' L (ListBlank.modifyNth f n R))\nL R : ListBlank Γ\n⊢ write (f (ListBlank.nth R (Nat.succ n))) ((move Dir.right^[Nat.succ n]) (mk' L R)) =\n (move Dir.right^[Nat.succ n]) (mk' L (ListBlank.modifyNth f (Nat.succ n) R))",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL R : ListBlank Γ\nn : ℕ\n⊢ write (f (ListBlank.nth R n)) ((move Dir.right^[n]) (mk' L R)) =\n (move Dir.right^[n]) (mk' L (ListBlank.modifyNth f n R))",
"tactic": "induction' n with n IH generalizing L R"
},
{
"state_after": "no goals",
"state_before": "case succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL✝ R✝ : ListBlank Γ\nn : ℕ\nIH :\n ∀ (L R : ListBlank Γ),\n write (f (ListBlank.nth R n)) ((move Dir.right^[n]) (mk' L R)) =\n (move Dir.right^[n]) (mk' L (ListBlank.modifyNth f n R))\nL R : ListBlank Γ\n⊢ write (f (ListBlank.nth R (Nat.succ n))) ((move Dir.right^[Nat.succ n]) (mk' L R)) =\n (move Dir.right^[Nat.succ n]) (mk' L (ListBlank.modifyNth f (Nat.succ n) R))",
"tactic": "simp only [ListBlank.head_cons, ListBlank.nth_succ, ListBlank.modifyNth, Tape.move_right_mk',\n ListBlank.tail_cons, iterate_succ_apply, IH]"
},
{
"state_after": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL✝ R✝ L R : ListBlank Γ\n⊢ write (f (ListBlank.head R)) (mk' L R) = mk' L (ListBlank.cons (f (ListBlank.head R)) (ListBlank.tail R))",
"state_before": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL✝ R✝ L R : ListBlank Γ\n⊢ write (f (ListBlank.nth R Nat.zero)) ((move Dir.right^[Nat.zero]) (mk' L R)) =\n (move Dir.right^[Nat.zero]) (mk' L (ListBlank.modifyNth f Nat.zero R))",
"tactic": "simp only [ListBlank.nth_zero, ListBlank.modifyNth, iterate_zero_apply, Nat.zero_eq]"
},
{
"state_after": "no goals",
"state_before": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nL✝ R✝ L R : ListBlank Γ\n⊢ write (f (ListBlank.head R)) (mk' L R) = mk' L (ListBlank.cons (f (ListBlank.head R)) (ListBlank.tail R))",
"tactic": "rw [← Tape.write_mk', ListBlank.cons_head_tail]"
}
]
| [
711,
49
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
704,
1
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|
Mathlib/FieldTheory/IntermediateField.lean | toSubalgebra_toIntermediateField | [
{
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"tactic": "ext"
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{
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"state_before": "case h\nK : Type u_1\nL : Type u_2\nL' : Type ?u.77095\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS✝ : IntermediateField K L\nS : Subalgebra K L\ninv_mem : ∀ (x : L), x ∈ S → x⁻¹ ∈ S\nx✝ : L\n⊢ x✝ ∈ (Subalgebra.toIntermediateField S inv_mem).toSubalgebra ↔ x✝ ∈ S",
"tactic": "rfl"
}
]
| [
298,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
1
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|
Mathlib/Logic/IsEmpty.lean | Subtype.isEmpty_of_false | []
| [
74,
21
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
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|
Mathlib/Data/Holor.lean | Holor.slice_unitVec_mul | [
{
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},
{
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{
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"tactic": "rfl"
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]
| [
269,
81
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
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|
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.cof_blsub_le | [
{
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"state_before": "α : Type ?u.39412\nr : α → α → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ cof (blsub o f) ≤ card o",
"tactic": "rw [← o.card.lift_id]"
},
{
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"tactic": "exact cof_blsub_le_lift f"
}
]
| [
440,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
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|
Mathlib/Data/Finset/Basic.lean | Finset.insert_sdiff_of_mem | [
{
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"tactic": "rw [← coe_inj, coe_sdiff, coe_sdiff, coe_insert]"
},
{
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"state_before": "α : Type u_1\nβ : Type ?u.241033\nγ : Type ?u.241036\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na b : α\ns : Finset α\nx : α\nh : x ∈ t\n⊢ insert x ↑s \\ ↑t = ↑s \\ ↑t",
"tactic": "exact Set.insert_diff_of_mem _ h"
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| [
2183,
35
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2181,
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|
Mathlib/LinearAlgebra/LinearIndependent.lean | eq_of_linearIndependent_of_span_subtype | [
{
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"tactic": "let f : t ↪ s :=\n ⟨fun x => ⟨x.1, h x.2⟩, fun a b hab => Subtype.coe_injective (Subtype.mk.inj hab)⟩"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\n⊢ s = t",
"state_before": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\n⊢ s = t",
"tactic": "have h_surj : Surjective f := by\n apply surjective_of_linearIndependent_of_span hs f _\n convert hst <;> simp [comp]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\n⊢ s = t",
"state_before": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\n⊢ s = t",
"tactic": "show s = t"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\n⊢ (range fun x => ↑x) ⊆ ↑(span R (range ((fun x => ↑x) ∘ ↑f)))",
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"tactic": "apply surjective_of_linearIndependent_of_span hs f _"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\n⊢ (range fun x => ↑x) ⊆ ↑(span R (range ((fun x => ↑x) ∘ ↑f)))",
"tactic": "convert hst <;> simp [comp]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\n⊢ s ⊆ t",
"state_before": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\n⊢ s = t",
"tactic": "apply Subset.antisymm _ h"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\nx : M\nhx : x ∈ s\n⊢ x ∈ t",
"state_before": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\n⊢ s ⊆ t",
"tactic": "intro x hx"
},
{
"state_after": "case intro\nι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y✝ : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\nx : M\nhx : x ∈ s\ny : ↑t\nhy : ↑f y = { val := x, property := hx }\n⊢ x ∈ t",
"state_before": "ι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\nx : M\nhx : x ∈ s\n⊢ x ∈ t",
"tactic": "rcases h_surj ⟨x, hx⟩ with ⟨y, hy⟩"
},
{
"state_after": "case h.e'_4\nι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y✝ : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\nx : M\nhx : x ∈ s\ny : ↑t\nhy : ↑f y = { val := x, property := hx }\n⊢ x = ↑y",
"state_before": "case intro\nι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y✝ : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\nx : M\nhx : x ∈ s\ny : ↑t\nhy : ↑f y = { val := x, property := hx }\n⊢ x ∈ t",
"tactic": "convert y.mem"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type u'\nι' : Type ?u.605779\nR : Type u_1\nK : Type ?u.605785\nM : Type u_2\nM' : Type ?u.605791\nM'' : Type ?u.605794\nV : Type u\nV' : Type ?u.605799\nv : ι → M\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\na b : R\nx✝ y✝ : M\ninst✝ : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => ↑x\nh : t ⊆ s\nhst : s ⊆ ↑(span R t)\nf : ↑t ↪ ↑s :=\n { toFun := fun x => { val := ↑x, property := (_ : ↑x ∈ s) },\n inj' :=\n (_ :\n ∀ (a b : ↑t),\n (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) a = (fun x => { val := ↑x, property := (_ : ↑x ∈ s) }) b →\n a = b) }\nh_surj : Surjective ↑f\nx : M\nhx : x ∈ s\ny : ↑t\nhy : ↑f y = { val := x, property := hx }\n⊢ x = ↑y",
"tactic": "rw [← Subtype.mk.inj hy]"
}
]
| [
967,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
955,
1
]
|
Mathlib/Analysis/LocallyConvex/Polar.lean | LinearMap.polar_union | []
| [
97,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
]
|
Mathlib/Data/Set/Basic.lean | Set.union_compl_self | []
| [
1718,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1717,
1
]
|
Mathlib/Topology/Inseparable.lean | inseparable_iff_specializes_and | []
| [
273,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
272,
1
]
|
Mathlib/Algebra/Star/StarAlgHom.lean | StarAlgEquiv.toRingEquiv_eq_coe | []
| [
772,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
771,
1
]
|
Mathlib/LinearAlgebra/Basis.lean | Basis.smul_eq_zero | []
| [
789,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
787,
11
]
|
Mathlib/Algebra/Hom/Equiv/Basic.lean | MulEquiv.eq_symm_comp | []
| [
439,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
437,
1
]
|
Mathlib/Algebra/Module/LinearMap.lean | LinearMap.id_apply | []
| [
274,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
]
|
Mathlib/Data/ZMod/Basic.lean | ZMod.ringHom_map_cast | [
{
"state_after": "case zero\nR : Type u_1\ninst✝ : Ring R\nf : R →+* ZMod Nat.zero\nk : ZMod Nat.zero\n⊢ ↑f ↑k = k\n\ncase succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* ZMod (Nat.succ n✝)\nk : ZMod (Nat.succ n✝)\n⊢ ↑f ↑k = k",
"state_before": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nf : R →+* ZMod n\nk : ZMod n\n⊢ ↑f ↑k = k",
"tactic": "cases n"
},
{
"state_after": "case succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* ZMod (Nat.succ n✝)\nk : ZMod (Nat.succ n✝)\n⊢ ↑f ↑k = k",
"state_before": "case zero\nR : Type u_1\ninst✝ : Ring R\nf : R →+* ZMod Nat.zero\nk : ZMod Nat.zero\n⊢ ↑f ↑k = k\n\ncase succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* ZMod (Nat.succ n✝)\nk : ZMod (Nat.succ n✝)\n⊢ ↑f ↑k = k",
"tactic": ". dsimp [ZMod, ZMod.cast] at f k ⊢; simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* ZMod (Nat.succ n✝)\nk : ZMod (Nat.succ n✝)\n⊢ ↑f ↑k = k",
"tactic": ". dsimp [ZMod, ZMod.cast] at f k ⊢\n erw [map_natCast, Fin.cast_val_eq_self]"
},
{
"state_after": "case zero\nR : Type u_1\ninst✝ : Ring R\nf : R →+* ℤ\nk : ℤ\n⊢ ↑f ↑k = k",
"state_before": "case zero\nR : Type u_1\ninst✝ : Ring R\nf : R →+* ZMod Nat.zero\nk : ZMod Nat.zero\n⊢ ↑f ↑k = k",
"tactic": "dsimp [ZMod, ZMod.cast] at f k ⊢"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝ : Ring R\nf : R →+* ℤ\nk : ℤ\n⊢ ↑f ↑k = k",
"tactic": "simp"
},
{
"state_after": "case succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* Fin (n✝ + 1)\nk : Fin (n✝ + 1)\n⊢ ↑f ↑(val k) = k",
"state_before": "case succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* ZMod (Nat.succ n✝)\nk : ZMod (Nat.succ n✝)\n⊢ ↑f ↑k = k",
"tactic": "dsimp [ZMod, ZMod.cast] at f k ⊢"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nf : R →+* Fin (n✝ + 1)\nk : Fin (n✝ + 1)\n⊢ ↑f ↑(val k) = k",
"tactic": "erw [map_natCast, Fin.cast_val_eq_self]"
}
]
| [
1167,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1163,
1
]
|
Mathlib/Data/Polynomial/FieldDivision.lean | Polynomial.X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic | [
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\nkey : f /ₘ (X - ↑C a) + (X - ↑C a) * ↑derivative (f /ₘ (X - ↑C a)) = ↑derivative f\n⊢ X - ↑C a ∣ ↑derivative f",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\n⊢ X - ↑C a ∣ ↑derivative f",
"tactic": "have key := divByMonic_add_X_sub_C_mul_derivate_divByMonic_eq_derivative f a"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\nkey : f /ₘ (X - ↑C a) + (X - ↑C a) * ↑derivative (f /ₘ (X - ↑C a)) = ↑derivative f\nu : K[X]\nhu : f /ₘ (X - ↑C a) = (X - ↑C a) * u\n⊢ X - ↑C a ∣ ↑derivative f",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\nkey : f /ₘ (X - ↑C a) + (X - ↑C a) * ↑derivative (f /ₘ (X - ↑C a)) = ↑derivative f\n⊢ X - ↑C a ∣ ↑derivative f",
"tactic": "have ⟨u,hu⟩ := hf"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\nkey : f /ₘ (X - ↑C a) + (X - ↑C a) * ↑derivative (f /ₘ (X - ↑C a)) = ↑derivative f\nu : K[X]\nhu : f /ₘ (X - ↑C a) = (X - ↑C a) * u\n⊢ X - ↑C a ∣ (X - ↑C a) * (u + ↑derivative ((X - ↑C a) * u))",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\nkey : f /ₘ (X - ↑C a) + (X - ↑C a) * ↑derivative (f /ₘ (X - ↑C a)) = ↑derivative f\nu : K[X]\nhu : f /ₘ (X - ↑C a) = (X - ↑C a) * u\n⊢ X - ↑C a ∣ ↑derivative f",
"tactic": "rw [←key, hu, ←mul_add (X - C a) u _]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn : ℕ\ninst✝¹ : Field R\np q : R[X]\nK : Type u_1\ninst✝ : Field K\nf : K[X]\na : K\nhf : X - ↑C a ∣ f /ₘ (X - ↑C a)\nkey : f /ₘ (X - ↑C a) + (X - ↑C a) * ↑derivative (f /ₘ (X - ↑C a)) = ↑derivative f\nu : K[X]\nhu : f /ₘ (X - ↑C a) = (X - ↑C a) * u\n⊢ X - ↑C a ∣ (X - ↑C a) * (u + ↑derivative ((X - ↑C a) * u))",
"tactic": "use (u + derivative ((X - C a) * u))"
}
]
| [
527,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
522,
1
]
|
Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | CategoryTheory.MonoidalOfChosenFiniteProducts.hexagon_reverse | [
{
"state_after": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ (BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv =\n IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ (BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (BinaryFan.braiding (ℬ (tensorObj ℬ X Y) Z).isLimit (ℬ Z (tensorObj ℬ X Y)).isLimit).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv =\n tensorHom ℬ (𝟙 X) (BinaryFan.braiding (ℬ Y Z).isLimit (ℬ Z Y).isLimit).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n tensorHom ℬ (BinaryFan.braiding (ℬ X Z).isLimit (ℬ Z X).isLimit).hom (𝟙 Y)",
"tactic": "dsimp [tensorHom, Limits.BinaryFan.braiding]"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ∀ (j : Discrete WalkingPair),\n ((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app j =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app j",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ (BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv =\n IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))",
"tactic": "apply (ℬ _ _).isLimit.hom_ext"
},
{
"state_after": "case mk.left\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left } =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left }\n\ncase mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.right } =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.right }",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ∀ (j : Discrete WalkingPair),\n ((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app j =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app j",
"tactic": "rintro ⟨⟨⟩⟩"
},
{
"state_after": "case mk.left\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ∀ (j : Discrete WalkingPair),\n (((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left }) ≫\n (ℬ Z X).cone.π.app j =\n ((IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left }) ≫\n (ℬ Z X).cone.π.app j",
"state_before": "case mk.left\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left } =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left }",
"tactic": "apply (ℬ _ _).isLimit.hom_ext"
},
{
"state_after": "case mk.left.mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ((IsLimit.lift (ℬ (ℬ X Y).cone.pt Z).isLimit (BinaryFan.assocInv (ℬ X Y).isLimit (ℬ X (ℬ Y Z).cone.pt).cone) ≫\n IsLimit.lift (ℬ Z (tensorObj ℬ X Y)).isLimit (BinaryFan.swap (ℬ (tensorObj ℬ X Y) Z).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.assocInv (ℬ Z X).isLimit (ℬ Z (ℬ X Y).cone.pt).cone)) ≫\n BinaryFan.fst (ℬ (ℬ Z X).cone.pt Y).cone) ≫\n BinaryFan.snd (ℬ Z X).cone =\n ((IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫ IsLimit.lift (ℬ Z Y).isLimit (BinaryFan.swap (ℬ Y Z).cone))) ≫\n IsLimit.lift (ℬ (ℬ X Z).cone.pt Y).isLimit (BinaryFan.assocInv (ℬ X Z).isLimit (ℬ X (ℬ Z Y).cone.pt).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫ IsLimit.lift (ℬ Z X).isLimit (BinaryFan.swap (ℬ X Z).cone))\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n BinaryFan.fst (ℬ (ℬ Z X).cone.pt Y).cone) ≫\n BinaryFan.snd (ℬ Z X).cone",
"state_before": "case mk.left.mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ (((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left }) ≫\n (ℬ Z X).cone.π.app { as := WalkingPair.right } =\n ((IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.left }) ≫\n (ℬ Z X).cone.π.app { as := WalkingPair.right }",
"tactic": "dsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator,\n Limits.IsLimit.conePointUniqueUpToIso]"
},
{
"state_after": "no goals",
"state_before": "case mk.left.mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ((IsLimit.lift (ℬ (ℬ X Y).cone.pt Z).isLimit (BinaryFan.assocInv (ℬ X Y).isLimit (ℬ X (ℬ Y Z).cone.pt).cone) ≫\n IsLimit.lift (ℬ Z (tensorObj ℬ X Y)).isLimit (BinaryFan.swap (ℬ (tensorObj ℬ X Y) Z).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.assocInv (ℬ Z X).isLimit (ℬ Z (ℬ X Y).cone.pt).cone)) ≫\n BinaryFan.fst (ℬ (ℬ Z X).cone.pt Y).cone) ≫\n BinaryFan.snd (ℬ Z X).cone =\n ((IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫ IsLimit.lift (ℬ Z Y).isLimit (BinaryFan.swap (ℬ Y Z).cone))) ≫\n IsLimit.lift (ℬ (ℬ X Z).cone.pt Y).isLimit (BinaryFan.assocInv (ℬ X Z).isLimit (ℬ X (ℬ Z Y).cone.pt).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫ IsLimit.lift (ℬ Z X).isLimit (BinaryFan.swap (ℬ X Z).cone))\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n BinaryFan.fst (ℬ (ℬ Z X).cone.pt Y).cone) ≫\n BinaryFan.snd (ℬ Z X).cone",
"tactic": "simp"
},
{
"state_after": "case mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ (IsLimit.lift (ℬ (ℬ X Y).cone.pt Z).isLimit (BinaryFan.assocInv (ℬ X Y).isLimit (ℬ X (ℬ Y Z).cone.pt).cone) ≫\n IsLimit.lift (ℬ Z (tensorObj ℬ X Y)).isLimit (BinaryFan.swap (ℬ (tensorObj ℬ X Y) Z).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit (BinaryFan.assocInv (ℬ Z X).isLimit (ℬ Z (ℬ X Y).cone.pt).cone)) ≫\n BinaryFan.snd (ℬ (ℬ Z X).cone.pt Y).cone =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫ IsLimit.lift (ℬ Z Y).isLimit (BinaryFan.swap (ℬ Y Z).cone))) ≫\n IsLimit.lift (ℬ (ℬ X Z).cone.pt Y).isLimit (BinaryFan.assocInv (ℬ X Z).isLimit (ℬ X (ℬ Z Y).cone.pt).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫ IsLimit.lift (ℬ Z X).isLimit (BinaryFan.swap (ℬ X Z).cone))\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n BinaryFan.snd (ℬ (ℬ Z X).cone.pt Y).cone",
"state_before": "case mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ ((BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫\n (IsLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ X Y) Z).isLimit\n (IsLimit.swapBinaryFan (ℬ Z (tensorObj ℬ X Y)).isLimit)).hom ≫\n (BinaryFan.associatorOfLimitCone ℬ Z X Y).inv) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.right } =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ Y Z).isLimit (IsLimit.swapBinaryFan (ℬ Z Y).isLimit)).hom)) ≫\n (BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫\n (IsLimit.conePointUniqueUpToIso (ℬ X Z).isLimit (IsLimit.swapBinaryFan (ℬ Z X).isLimit)).hom)\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n (ℬ (ℬ Z X).cone.pt Y).cone.π.app { as := WalkingPair.right }",
"tactic": "dsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator,\n Limits.IsLimit.conePointUniqueUpToIso]"
},
{
"state_after": "no goals",
"state_before": "case mk.right\nC : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y Z : C\n⊢ (IsLimit.lift (ℬ (ℬ X Y).cone.pt Z).isLimit (BinaryFan.assocInv (ℬ X Y).isLimit (ℬ X (ℬ Y Z).cone.pt).cone) ≫\n IsLimit.lift (ℬ Z (tensorObj ℬ X Y)).isLimit (BinaryFan.swap (ℬ (tensorObj ℬ X Y) Z).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit (BinaryFan.assocInv (ℬ Z X).isLimit (ℬ Z (ℬ X Y).cone.pt).cone)) ≫\n BinaryFan.snd (ℬ (ℬ Z X).cone.pt Y).cone =\n (IsLimit.lift (ℬ X (ℬ Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ Y Z).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ Y Z).cone.pt).cone ≫ IsLimit.lift (ℬ Z Y).isLimit (BinaryFan.swap (ℬ Y Z).cone))) ≫\n IsLimit.lift (ℬ (ℬ X Z).cone.pt Y).isLimit (BinaryFan.assocInv (ℬ X Z).isLimit (ℬ X (ℬ Z Y).cone.pt).cone) ≫\n IsLimit.lift (ℬ (ℬ Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (ℬ (ℬ X Z).cone.pt Y).cone ≫ IsLimit.lift (ℬ Z X).isLimit (BinaryFan.swap (ℬ X Z).cone))\n (BinaryFan.snd (ℬ (ℬ X Z).cone.pt Y).cone ≫ 𝟙 Y))) ≫\n BinaryFan.snd (ℬ (ℬ Z X).cone.pt Y).cone",
"tactic": "simp"
}
]
| [
78,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
]
|
Mathlib/Data/Set/Pointwise/Basic.lean | Set.mem_mul | []
| [
338,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
1
]
|
Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | MeasureTheory.WithDensityᵥEq.congr_ae | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\n⊢ withDensityᵥ μ f = withDensityᵥ μ g\n\ncase neg\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\n⊢ withDensityᵥ μ f = withDensityᵥ μ g",
"state_before": "α : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\n⊢ withDensityᵥ μ f = withDensityᵥ μ g",
"tactic": "by_cases hf : Integrable f μ"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\ni : Set α\n⊢ MeasurableSet i → ↑(withDensityᵥ μ f) i = ↑(withDensityᵥ μ g) i",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\n⊢ withDensityᵥ μ f = withDensityᵥ μ g",
"tactic": "ext i"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(withDensityᵥ μ f) i = ↑(withDensityᵥ μ g) i",
"state_before": "case pos.h\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\ni : Set α\n⊢ MeasurableSet i → ↑(withDensityᵥ μ f) i = ↑(withDensityᵥ μ g) i",
"tactic": "intro hi"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, f x ∂μ) = ∫ (x : α) in i, g x ∂μ",
"state_before": "case pos.h\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(withDensityᵥ μ f) i = ↑(withDensityᵥ μ g) i",
"tactic": "rw [withDensityᵥ_apply hf hi, withDensityᵥ_apply (hf.congr h) hi]"
},
{
"state_after": "no goals",
"state_before": "case pos.h\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : Integrable f\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, f x ∂μ) = ∫ (x : α) in i, g x ∂μ",
"tactic": "exact integral_congr_ae (ae_restrict_of_ae h)"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\nhg : ¬Integrable g\n⊢ withDensityᵥ μ f = withDensityᵥ μ g",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\n⊢ withDensityᵥ μ f = withDensityᵥ μ g",
"tactic": "have hg : ¬Integrable g μ := by intro hg; exact hf (hg.congr h.symm)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\nhg : ¬Integrable g\n⊢ withDensityᵥ μ f = withDensityᵥ μ g",
"tactic": "rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg hg]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\nhg : Integrable g\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\n⊢ ¬Integrable g",
"tactic": "intro hg"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.63324\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g✝ f g : α → E\nh : f =ᶠ[ae μ] g\nhf : ¬Integrable f\nhg : Integrable g\n⊢ False",
"tactic": "exact hf (hg.congr h.symm)"
}
]
| [
159,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
]
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | Matrix.toLinearMap₂_compl₁₂ | [
{
"state_after": "no goals",
"state_before": "R : Type u_3\nR₁ : Type ?u.1948955\nR₂ : Type ?u.1948958\nM✝ : Type ?u.1948961\nM₁ : Type u_8\nM₂ : Type u_9\nM₁' : Type u_6\nM₂' : Type u_7\nn : Type u_1\nm : Type u_2\nn' : Type u_4\nm' : Type u_5\nι : Type ?u.1948988\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommMonoid M₁\ninst✝¹⁴ : Module R M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R M₂\ninst✝¹¹ : DecidableEq n\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq m\ninst✝⁸ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\ninst✝⁷ : AddCommMonoid M₁'\ninst✝⁶ : Module R M₁'\ninst✝⁵ : AddCommMonoid M₂'\ninst✝⁴ : Module R M₂'\nb₁' : Basis n' R M₁'\nb₂' : Basis m' R M₂'\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nM : Matrix n m R\nP : Matrix n n' R\nQ : Matrix m m' R\n⊢ ↑(toMatrix₂ b₁' b₂') (compl₁₂ (↑(toLinearMap₂ b₁ b₂) M) (↑(toLin b₁' b₁) P) (↑(toLin b₂' b₂) Q)) =\n ↑(toMatrix₂ b₁' b₂') (↑(toLinearMap₂ b₁' b₂') (Pᵀ ⬝ M ⬝ Q))",
"tactic": "simp only [LinearMap.toMatrix₂_compl₁₂ b₁ b₂, LinearMap.toMatrix₂_toLinearMap₂,\n toMatrix_toLin]"
}
]
| [
512,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
506,
1
]
|
Mathlib/Algebra/Hom/Group.lean | map_one | []
| [
226,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
]
|
Mathlib/Data/Set/Basic.lean | Set.inclusion_injective | []
| [
2809,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2808,
1
]
|
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.spanSingleton_le_iff_mem | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1304874\ninst✝³ : CommRing R₁\nK : Type ?u.1304880\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : P\nI : FractionalIdeal S P\n⊢ spanSingleton S x ≤ I ↔ x ∈ I",
"tactic": "rw [← coe_le_coe, coe_spanSingleton, Submodule.span_singleton_le_iff_mem, mem_coe]"
}
]
| [
1320,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1318,
1
]
|
Mathlib/Algebra/DirectLimit.lean | Module.DirectLimit.induction_on | []
| [
142,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
11
]
|
Mathlib/Analysis/Calculus/FDeriv/Add.lean | fderiv_neg | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.436745\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.436840\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ fderiv 𝕜 (fun y => -f y) x = -fderiv 𝕜 f x",
"tactic": "simp only [← fderivWithin_univ, fderivWithin_neg uniqueDiffWithinAt_univ]"
}
]
| [
466,
76
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
465,
1
]
|
Mathlib/Algebra/Order/Nonneg/Ring.lean | Nonneg.coe_mul | []
| [
217,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
11
]
|
Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.convex_basis_zero | []
| [
71,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
]
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.IsCycle.cycleOf_eq | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2371171\nα : Type u_1\nβ : Type ?u.2371177\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nx y✝ : α\nhf : IsCycle f\nhx : ↑f x ≠ x\ny : α\nh : SameCycle f x y\n⊢ ↑(cycleOf f x) y = ↑f y",
"tactic": "rw [h.cycleOf_apply]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2371171\nα : Type u_1\nβ : Type ?u.2371177\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nx y✝ : α\nhf : IsCycle f\nhx : ↑f x ≠ x\ny : α\nh : ¬SameCycle f x y\n⊢ ↑(cycleOf f x) y = ↑f y",
"tactic": "rw [cycleOf_apply_of_not_sameCycle h,\n Classical.not_not.1 (mt ((isCycle_iff_sameCycle hx).1 hf).2 h)]"
}
]
| [
1045,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1040,
1
]
|
Mathlib/Data/Real/NNReal.lean | NNReal.ne_iff | []
| [
107,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
]
|
Mathlib/Data/Set/Finite.lean | Set.Finite.infinite_compl | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝ : Infinite α\ns : Set α\nhs : Set.Finite s\nh : Set.Finite (sᶜ)\n⊢ Set.Finite univ",
"tactic": "simpa using hs.union h"
}
]
| [
1304,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1303,
1
]
|
Mathlib/Init/Data/Bool/Lemmas.lean | Bool.coe_false | [
{
"state_after": "no goals",
"state_before": "⊢ (false = true) = False",
"tactic": "simp"
}
]
| [
112,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
]
|
Mathlib/Topology/Algebra/GroupWithZero.lean | continuousOn_zpow₀ | []
| [
326,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
]
|
Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousWithinAt.inv₀ | []
| [
126,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
8
]
|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | WithSeminorms.continuous_seminorm | [
{
"state_after": "𝕜 : Type ?u.440749\n𝕜₂ : Type ?u.440752\n𝕝 : Type u_1\n𝕝₂ : Type ?u.440758\nE : Type u_2\nF : Type ?u.440764\nG : Type ?u.440767\nι : Type u_3\nι' : Type ?u.440773\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Nonempty ι\nt : TopologicalSpace E\ninst✝³ : TopologicalAddGroup E\ninst✝² : NontriviallyNormedField 𝕝\ninst✝¹ : Module 𝕝 E\ninst✝ : ContinuousConstSMul 𝕝 E\np : SeminormFamily 𝕝 E ι\nhp : WithSeminorms p\ni : ι\n⊢ ball (p i) 0 1 ∈ 𝓝 0",
"state_before": "𝕜 : Type ?u.440749\n𝕜₂ : Type ?u.440752\n𝕝 : Type u_1\n𝕝₂ : Type ?u.440758\nE : Type u_2\nF : Type ?u.440764\nG : Type ?u.440767\nι : Type u_3\nι' : Type ?u.440773\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Nonempty ι\nt : TopologicalSpace E\ninst✝³ : TopologicalAddGroup E\ninst✝² : NontriviallyNormedField 𝕝\ninst✝¹ : Module 𝕝 E\ninst✝ : ContinuousConstSMul 𝕝 E\np : SeminormFamily 𝕝 E ι\nhp : WithSeminorms p\ni : ι\n⊢ Continuous ↑(p i)",
"tactic": "refine' Seminorm.continuous one_pos _"
},
{
"state_after": "𝕜 : Type ?u.440749\n𝕜₂ : Type ?u.440752\n𝕝 : Type u_1\n𝕝₂ : Type ?u.440758\nE : Type u_2\nF : Type ?u.440764\nG : Type ?u.440767\nι : Type u_3\nι' : Type ?u.440773\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Nonempty ι\nt : TopologicalSpace E\ninst✝³ : TopologicalAddGroup E\ninst✝² : NontriviallyNormedField 𝕝\ninst✝¹ : Module 𝕝 E\ninst✝ : ContinuousConstSMul 𝕝 E\np : SeminormFamily 𝕝 E ι\nhp : WithSeminorms p\ni : ι\n⊢ ↑(p i) ⁻¹' Metric.ball 0 1 ∈ ⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0)",
"state_before": "𝕜 : Type ?u.440749\n𝕜₂ : Type ?u.440752\n𝕝 : Type u_1\n𝕝₂ : Type ?u.440758\nE : Type u_2\nF : Type ?u.440764\nG : Type ?u.440767\nι : Type u_3\nι' : Type ?u.440773\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Nonempty ι\nt : TopologicalSpace E\ninst✝³ : TopologicalAddGroup E\ninst✝² : NontriviallyNormedField 𝕝\ninst✝¹ : Module 𝕝 E\ninst✝ : ContinuousConstSMul 𝕝 E\np : SeminormFamily 𝕝 E ι\nhp : WithSeminorms p\ni : ι\n⊢ ball (p i) 0 1 ∈ 𝓝 0",
"tactic": "rw [p.withSeminorms_iff_nhds_eq_iInf.mp hp, ball_zero_eq_preimage_ball]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.440749\n𝕜₂ : Type ?u.440752\n𝕝 : Type u_1\n𝕝₂ : Type ?u.440758\nE : Type u_2\nF : Type ?u.440764\nG : Type ?u.440767\nι : Type u_3\nι' : Type ?u.440773\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Nonempty ι\nt : TopologicalSpace E\ninst✝³ : TopologicalAddGroup E\ninst✝² : NontriviallyNormedField 𝕝\ninst✝¹ : Module 𝕝 E\ninst✝ : ContinuousConstSMul 𝕝 E\np : SeminormFamily 𝕝 E ι\nhp : WithSeminorms p\ni : ι\n⊢ ↑(p i) ⁻¹' Metric.ball 0 1 ∈ ⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0)",
"tactic": "exact Filter.mem_iInf_of_mem i (Filter.preimage_mem_comap <| Metric.ball_mem_nhds _ one_pos)"
}
]
| [
447,
95
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
1
]
|
Mathlib/Logic/Function/Basic.lean | Function.Injective.of_comp | []
| [
129,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
]
|
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.lift_mk_eq | []
| [
321,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
]
|
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.neg_apply | []
| [
304,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
]
|
Mathlib/RingTheory/RootsOfUnity/Basic.lean | IsPrimitiveRoot.coe_units_iff | [
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.1881352\nG : Type ?u.1881355\nR : Type ?u.1881358\nS : Type ?u.1881361\nF : Type ?u.1881364\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ✝ : M\nf : F\nh : IsPrimitiveRoot ζ✝ k\nζ : Mˣ\n⊢ IsPrimitiveRoot (↑ζ) k ↔ IsPrimitiveRoot ζ k",
"tactic": "simp only [iff_def, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one]"
}
]
| [
414,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
]
|
Mathlib/Data/Polynomial/CancelLeads.lean | Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm | [
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : p = 0\n⊢ natDegree (cancelLeads p q) < natDegree q\n\ncase neg\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\n⊢ natDegree (cancelLeads p q) < natDegree q",
"state_before": "R : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\n⊢ natDegree (cancelLeads p q) < natDegree q",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q",
"state_before": "case neg\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\n⊢ natDegree (cancelLeads p q) < natDegree q",
"tactic": "rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q\n\ncase neg\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q",
"state_before": "case neg\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q",
"tactic": "by_cases h0 :\n C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0"
},
{
"state_after": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≤ natDegree q\n\ncase neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≠ natDegree q",
"state_before": "case neg\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q",
"tactic": "apply lt_of_le_of_ne"
},
{
"state_after": "case h.e'_3\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : p = 0\n⊢ natDegree (cancelLeads p q) = 0",
"state_before": "case pos\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : p = 0\n⊢ natDegree (cancelLeads p q) < natDegree q",
"tactic": "convert hq"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : p = 0\n⊢ natDegree (cancelLeads p q) = 0",
"tactic": "simp [hp, cancelLeads]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q",
"tactic": "exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = 0",
"tactic": "simp only [h0, natDegree_zero]"
},
{
"state_after": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q) ≤ natDegree q\n\ncase neg.a.qn\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (-(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≤ natDegree q",
"state_before": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≤ natDegree q",
"tactic": "rw [natDegree_add_le_iff_left]"
},
{
"state_after": "case neg.a.qn\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) ≤ natDegree q",
"state_before": "case neg.a.qn\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (-(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≤ natDegree q",
"tactic": "refine (natDegree_neg (C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p)).le.trans ?_"
},
{
"state_after": "no goals",
"state_before": "case neg.a.qn\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) ≤ natDegree q",
"tactic": "exact natDegree_mul_le.trans <| Nat.add_le_of_le_sub h <| natDegree_C_mul_X_pow_le _ _"
},
{
"state_after": "no goals",
"state_before": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q) ≤ natDegree q",
"tactic": "apply natDegree_C_mul_le"
},
{
"state_after": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n⊢ ↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0",
"state_before": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : ¬↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n⊢ natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≠ natDegree q",
"tactic": "contrapose! h0"
},
{
"state_after": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n⊢ coeff\n (↑C (leadingCoeff p) * q +\n -(↑C (leadingCoeff q) * (p * X ^ (natDegree p + (natDegree q - natDegree p) - natDegree p))))\n (natDegree p + (natDegree q - natDegree p)) =\n 0",
"state_before": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n⊢ ↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0",
"tactic": "rw [← leadingCoeff_eq_zero, leadingCoeff, h0, mul_assoc, X_pow_mul, ← tsub_add_cancel_of_le h,\n add_comm _ p.natDegree]"
},
{
"state_after": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n⊢ leadingCoeff p * coeff q (natDegree p + (natDegree q - natDegree p)) + -(leadingCoeff q * coeff p (natDegree p)) = 0",
"state_before": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n⊢ coeff\n (↑C (leadingCoeff p) * q +\n -(↑C (leadingCoeff q) * (p * X ^ (natDegree p + (natDegree q - natDegree p) - natDegree p))))\n (natDegree p + (natDegree q - natDegree p)) =\n 0",
"tactic": "simp only [coeff_mul_X_pow, coeff_neg, coeff_C_mul, add_tsub_cancel_left, coeff_add]"
},
{
"state_after": "no goals",
"state_before": "case neg.a\nR : Type u_1\ninst✝ : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p ≤ natDegree q\nhq : 0 < natDegree q\nhp : ¬p = 0\nh0 : natDegree (↑C (leadingCoeff p) * q + -(↑C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n⊢ leadingCoeff p * coeff q (natDegree p + (natDegree q - natDegree p)) + -(leadingCoeff q * coeff p (natDegree p)) = 0",
"tactic": "rw [add_comm p.natDegree, tsub_add_cancel_of_le h, ← leadingCoeff, ← leadingCoeff, comm,\n add_right_neg]"
}
]
| [
76,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
]
|
Mathlib/GroupTheory/Solvable.lean | map_derivedSeries_eq | []
| [
97,
97
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
]
|
Mathlib/Algebra/DirectSum/Module.lean | DirectSum.IsInternal.addSubmonoid_independent | []
| [
429,
76
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
]
|
Mathlib/Data/Nat/Basic.lean | Nat.pred_eq_self_iff | [
{
"state_after": "no goals",
"state_before": "m n✝ k n : ℕ\n⊢ pred n = n ↔ n = 0",
"tactic": "cases n <;> simp [(Nat.succ_ne_self _).symm]"
}
]
| [
809,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
808,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.mem_support_iff_exists_append | [
{
"state_after": "case mp\nV✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\np : Walk G u v\n⊢ w ∈ support p → ∃ q r, p = append q r\n\ncase mpr\nV✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\np : Walk G u v\n⊢ (∃ q r, p = append q r) → w ∈ support p",
"state_before": "V✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\np : Walk G u v\n⊢ w ∈ support p ↔ ∃ q r, p = append q r",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp\nV✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\np : Walk G u v\n⊢ w ∈ support p → ∃ q r, p = append q r",
"tactic": "exact fun h => ⟨_, _, (p.take_spec h).symm⟩"
},
{
"state_after": "case mpr.intro.intro\nV✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\nq : Walk G u w\nr : Walk G w v\n⊢ w ∈ support (append q r)",
"state_before": "case mpr\nV✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\np : Walk G u v\n⊢ (∃ q r, p = append q r) → w ∈ support p",
"tactic": "rintro ⟨q, r, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nV✝ : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V✝\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V✝\nV : Type u\nG : SimpleGraph V\nu v w : V\nq : Walk G u w\nr : Walk G w v\n⊢ w ∈ support (append q r)",
"tactic": "simp only [mem_support_append_iff, end_mem_support, start_mem_support, or_self_iff]"
}
]
| [
1080,
88
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1074,
1
]
|
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.lsub_lt_ord_lift | [
{
"state_after": "α : Type ?u.28987\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal\nhι : Cardinal.lift (#ι) < cof (lsub f)\nhf : ∀ (i : ι), f i < lsub f\n⊢ False",
"state_before": "α : Type ?u.28987\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal\nc : Ordinal\nhι : Cardinal.lift (#ι) < cof c\nhf : ∀ (i : ι), f i < c\nh : lsub f = c\n⊢ False",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.28987\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal\nhι : Cardinal.lift (#ι) < cof (lsub f)\nhf : ∀ (i : ι), f i < lsub f\n⊢ False",
"tactic": "exact (cof_lsub_le_lift.{u, v} f).not_lt hι"
}
]
| [
335,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.union_eq_union_iff_left | []
| [
1494,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1493,
1
]
|
Mathlib/Algebra/Order/Group/Defs.lean | Left.inv_le_self | []
| [
444,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
443,
1
]
|
Mathlib/Computability/TMToPartrec.lean | Turing.PartrecToTM2.K'.elim_update_main | [
{
"state_after": "case h\na b c d a' : List Γ'\nx : K'\n⊢ update (elim a b c d) main a' x = elim a' b c d x",
"state_before": "a b c d a' : List Γ'\n⊢ update (elim a b c d) main a' = elim a' b c d",
"tactic": "funext x"
},
{
"state_after": "no goals",
"state_before": "case h\na b c d a' : List Γ'\nx : K'\n⊢ update (elim a b c d) main a' x = elim a' b c d x",
"tactic": "cases x <;> rfl"
}
]
| [
1283,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1282,
1
]
|
Mathlib/Analysis/InnerProductSpace/Basic.lean | InnerProductSpace.Core.inner_self_ne_zero | []
| [
271,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
270,
1
]
|
Mathlib/Topology/MetricSpace/Basic.lean | Metric.tendsto_nhdsWithin_nhds | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.99263\nι : Type ?u.99266\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ninst✝ : PseudoMetricSpace β\nf : α → β\na : α\nb : β\n⊢ (∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ univ ∧ dist (f x) b < ε) ↔\n ∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.99263\nι : Type ?u.99266\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ninst✝ : PseudoMetricSpace β\nf : α → β\na : α\nb : β\n⊢ Tendsto f (𝓝[s] a) (𝓝 b) ↔ ∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε",
"tactic": "rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.99263\nι : Type ?u.99266\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ninst✝ : PseudoMetricSpace β\nf : α → β\na : α\nb : β\n⊢ (∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ univ ∧ dist (f x) b < ε) ↔\n ∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε",
"tactic": "simp only [mem_univ, true_and_iff]"
}
]
| [
1044,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1040,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | affineSpan_eq_bot | [
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.336276\ns : Set P\n⊢ affineSpan k s = ⊥ ↔ s = ∅",
"tactic": "rw [← not_iff_not, ← Ne.def, ← Ne.def, ← nonempty_iff_ne_bot, affineSpan_nonempty,\n nonempty_iff_ne_empty]"
}
]
| [
1166,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1164,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_union' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.297875\nγ : Type ?u.297878\nδ : Type ?u.297881\nι : Type ?u.297884\nR : Type ?u.297887\nR' : Type ?u.297890\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : Disjoint s t\nhs : MeasurableSet s\n⊢ restrict μ (s ∪ t) = restrict μ s + restrict μ t",
"tactic": "rw [union_comm, restrict_union h.symm hs, add_comm]"
}
]
| [
1734,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1732,
1
]
|
Mathlib/Analysis/Convex/Gauge.lean | gauge_smul | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.169480\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns t : Set E\na : ℝ\ninst✝² : IsROrC 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsScalarTower ℝ 𝕜 E\nhs : Balanced 𝕜 s\nr : 𝕜\nx : E\n⊢ gauge s (r • x) = ‖r‖ * gauge s x",
"tactic": "rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]"
}
]
| [
324,
81
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
]
|
Mathlib/Algebra/Order/Group/MinMax.lean | max_inv_inv' | []
| [
50,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
]
|
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiff.clm_comp | []
| [
886,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
884,
1
]
|
Mathlib/GroupTheory/Index.lean | Subgroup.relindex_subgroupOf | []
| [
121,
98
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
119,
1
]
|
Mathlib/LinearAlgebra/SesquilinearForm.lean | LinearMap.IsAdjointPair.sub | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.337838\nR₂ : Type ?u.337841\nR₃ : Type ?u.337844\nM : Type u_2\nM₁ : Type u_3\nM₂ : Type ?u.337853\nMₗ₁ : Type ?u.337856\nMₗ₁' : Type ?u.337859\nMₗ₂ : Type ?u.337862\nMₗ₂' : Type ?u.337865\nK : Type ?u.337868\nK₁ : Type ?u.337871\nK₂ : Type ?u.337874\nV : Type ?u.337877\nV₁ : Type ?u.337880\nV₂ : Type ?u.337883\nn : Type ?u.337886\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\nB' : M₁ →ₗ[R] M₁ →ₗ[R] R\nf f' : M →ₗ[R] M₁\ng g' : M₁ →ₗ[R] M\nh : IsAdjointPair B B' f g\nh' : IsAdjointPair B B' f' g'\nx : M\nx✝ : M₁\n⊢ ↑(↑B' (↑(f - f') x)) x✝ = ↑(↑B x) (↑(g - g') x✝)",
"tactic": "rw [f.sub_apply, g.sub_apply, B'.map_sub₂, (B x).map_sub, h, h']"
}
]
| [
484,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
482,
1
]
|
Mathlib/Order/CompleteLattice.lean | OrderIso.map_iSup | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nβ₂ : Type ?u.79767\nγ : Type ?u.79770\nι : Sort u_3\nι' : Sort ?u.79776\nκ : ι → Sort ?u.79781\nκ' : ι' → Sort ?u.79786\ninst✝¹ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\ninst✝ : CompleteLattice β\nf : α ≃o β\nx✝ : ι → α\nx : α\n⊢ ↑f (⨆ (i : ι), x✝ i) ≤ ↑f x ↔ (⨆ (i : ι), ↑f (x✝ i)) ≤ ↑f x",
"tactic": "simp only [f.le_iff_le, iSup_le_iff]"
}
]
| [
968,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
965,
1
]
|
Mathlib/Analysis/Asymptotics/Theta.lean | Asymptotics.IsTheta.isLittleO_congr_right | []
| [
179,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
]
|
Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.toStalk_comp_stalkToFiberRingHom | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nx : ↑(PrimeSpectrum.Top R)\n⊢ toOpen R ⊤ ≫\n openToLocalization R ⊤ ↑{ val := x, property := True.intro } (_ : ↑{ val := x, property := True.intro } ∈ ⊤) =\n algebraMap R (Localization.AtPrime x.asIdeal)",
"state_before": "R : Type u\ninst✝ : CommRing R\nx : ↑(PrimeSpectrum.Top R)\n⊢ toStalk R x ≫ stalkToFiberRingHom R x = algebraMap R (Localization.AtPrime x.asIdeal)",
"tactic": "erw [toStalk, Category.assoc, germ_comp_stalkToFiberRingHom]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nx : ↑(PrimeSpectrum.Top R)\n⊢ toOpen R ⊤ ≫\n openToLocalization R ⊤ ↑{ val := x, property := True.intro } (_ : ↑{ val := x, property := True.intro } ∈ ⊤) =\n algebraMap R (Localization.AtPrime x.asIdeal)",
"tactic": "rfl"
}
]
| [
562,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
559,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.iInter_eq_univ | []
| [
716,
14
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
715,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.coe_finsetWalkLength_eq | [
{
"state_after": "case zero\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ u v : V\n⊢ ↑(finsetWalkLength G Nat.zero u v) = {p | Walk.length p = Nat.zero}\n\ncase succ\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\n⊢ ↑(finsetWalkLength G (Nat.succ n) u v) = {p | Walk.length p = Nat.succ n}",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nn : ℕ\nu v : V\n⊢ ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}",
"tactic": "induction' n with n ih generalizing u v"
},
{
"state_after": "no goals",
"state_before": "case zero\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ u v : V\n⊢ ↑(finsetWalkLength G Nat.zero u v) = {p | Walk.length p = Nat.zero}",
"tactic": "obtain rfl | huv := eq_or_ne u v <;> simp [finsetWalkLength, set_walk_length_zero_eq_of_ne, *]"
},
{
"state_after": "case succ\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\n⊢ (⋃ (x : ↑(neighborSet G u)),\n ↑(Finset.map\n { toFun := fun p => Walk.cons (_ : ↑x ∈ neighborSet G u) p,\n inj' :=\n (_ :\n ∀ (x_1 x_2 : Walk G (↑x) v),\n Walk.cons (_ : ↑x ∈ neighborSet G u) x_1 = Walk.cons (_ : ↑x ∈ neighborSet G u) x_2 → x_1 = x_2) }\n (finsetWalkLength G n (↑x) v))) =\n ⋃ (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}",
"state_before": "case succ\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\n⊢ ↑(finsetWalkLength G (Nat.succ n) u v) = {p | Walk.length p = Nat.succ n}",
"tactic": "simp only [finsetWalkLength, set_walk_length_succ_eq, Finset.coe_biUnion, Finset.mem_coe,\n Finset.mem_univ, Set.iUnion_true]"
},
{
"state_after": "case succ.h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\n⊢ (p ∈\n ⋃ (x : ↑(neighborSet G u)),\n ↑(Finset.map\n { toFun := fun p => Walk.cons (_ : ↑x ∈ neighborSet G u) p,\n inj' :=\n (_ :\n ∀ (x_1 x_2 : Walk G (↑x) v),\n Walk.cons (_ : ↑x ∈ neighborSet G u) x_1 = Walk.cons (_ : ↑x ∈ neighborSet G u) x_2 → x_1 = x_2) }\n (finsetWalkLength G n (↑x) v))) ↔\n p ∈ ⋃ (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}",
"state_before": "case succ\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\n⊢ (⋃ (x : ↑(neighborSet G u)),\n ↑(Finset.map\n { toFun := fun p => Walk.cons (_ : ↑x ∈ neighborSet G u) p,\n inj' :=\n (_ :\n ∀ (x_1 x_2 : Walk G (↑x) v),\n Walk.cons (_ : ↑x ∈ neighborSet G u) x_1 = Walk.cons (_ : ↑x ∈ neighborSet G u) x_2 → x_1 = x_2) }\n (finsetWalkLength G n (↑x) v))) =\n ⋃ (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}",
"tactic": "ext p"
},
{
"state_after": "case succ.h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\n⊢ (∃ i h x,\n x ∈ finsetWalkLength G n i v ∧\n Walk.cons (_ : ↑{ val := i, property := (_ : i ∈ neighborSet G u) } ∈ neighborSet G u) x = p) ↔\n ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : Adj G u i) x = p",
"state_before": "case succ.h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\n⊢ (p ∈\n ⋃ (x : ↑(neighborSet G u)),\n ↑(Finset.map\n { toFun := fun p => Walk.cons (_ : ↑x ∈ neighborSet G u) p,\n inj' :=\n (_ :\n ∀ (x_1 x_2 : Walk G (↑x) v),\n Walk.cons (_ : ↑x ∈ neighborSet G u) x_1 = Walk.cons (_ : ↑x ∈ neighborSet G u) x_2 → x_1 = x_2) }\n (finsetWalkLength G n (↑x) v))) ↔\n p ∈ ⋃ (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}",
"tactic": "simp only [mem_neighborSet, Finset.coe_map, Embedding.coeFn_mk, Set.iUnion_coe_set,\n Set.mem_iUnion, Set.mem_image, Finset.mem_coe, Set.mem_setOf_eq]"
},
{
"state_after": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nx✝² : V\nx✝¹ : Adj G u x✝²\nx✝ : Walk G x✝² v\n⊢ x✝ ∈ finsetWalkLength G n x✝² v ↔ Walk.length x✝ = n",
"state_before": "case succ.h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\n⊢ (∃ i h x,\n x ∈ finsetWalkLength G n i v ∧\n Walk.cons (_ : ↑{ val := i, property := (_ : i ∈ neighborSet G u) } ∈ neighborSet G u) x = p) ↔\n ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : Adj G u i) x = p",
"tactic": "congr!"
},
{
"state_after": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nw : V\nx✝ : Adj G u w\nq : Walk G w v\n⊢ q ∈ finsetWalkLength G n w v ↔ Walk.length q = n",
"state_before": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nx✝² : V\nx✝¹ : Adj G u x✝²\nx✝ : Walk G x✝² v\n⊢ x✝ ∈ finsetWalkLength G n x✝² v ↔ Walk.length x✝ = n",
"tactic": "rename_i w _ q"
},
{
"state_after": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nw : V\nx✝ : Adj G u w\nq : Walk G w v\nthis : q ∈ ↑(finsetWalkLength G n w v) ↔ q ∈ {p | Walk.length p = n}\n⊢ q ∈ finsetWalkLength G n w v ↔ Walk.length q = n",
"state_before": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nw : V\nx✝ : Adj G u w\nq : Walk G w v\n⊢ q ∈ finsetWalkLength G n w v ↔ Walk.length q = n",
"tactic": "have := Set.ext_iff.mp (ih w v) q"
},
{
"state_after": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nw : V\nx✝ : Adj G u w\nq : Walk G w v\nthis : q ∈ finsetWalkLength G n w v ↔ Walk.length q = n\n⊢ q ∈ finsetWalkLength G n w v ↔ Walk.length q = n",
"state_before": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nw : V\nx✝ : Adj G u w\nq : Walk G w v\nthis : q ∈ ↑(finsetWalkLength G n w v) ↔ q ∈ {p | Walk.length p = n}\n⊢ q ∈ finsetWalkLength G n w v ↔ Walk.length q = n",
"tactic": "simp only [Finset.mem_coe, Set.mem_setOf_eq] at this"
},
{
"state_after": "no goals",
"state_before": "case succ.h.a.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_1.a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nu✝ v✝ : V\nn : ℕ\nih : ∀ (u v : V), ↑(finsetWalkLength G n u v) = {p | Walk.length p = n}\nu v : V\np : Walk G u v\nw : V\nx✝ : Adj G u w\nq : Walk G w v\nthis : q ∈ finsetWalkLength G n w v ↔ Walk.length q = n\n⊢ q ∈ finsetWalkLength G n w v ↔ Walk.length q = n",
"tactic": "rw [← this]"
}
]
| [
2363,
16
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2350,
1
]
|
src/lean/Init/Prelude.lean | ne_false_of_eq_true | []
| [
653,
35
]
| d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
651,
1
]
|
Std/Data/String/Lemmas.lean | Substring.prevn_zero | [
{
"state_after": "no goals",
"state_before": "s : Substring\nn : Nat\n⊢ prevn s (n + 1) 0 = 0",
"tactic": "simp [prevn, prevn_zero s n]"
}
]
| [
769,
43
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
767,
9
]
|
Mathlib/FieldTheory/RatFunc.lean | RatFunc.eval_eq_zero_of_eval₂_denom_eq_zero | [
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : Field K\nL : Type u_1\ninst✝ : Field L\nf : K →+* L\na : L\nx : RatFunc K\nh : Polynomial.eval₂ f a (denom x) = 0\n⊢ eval f a x = 0",
"tactic": "rw [eval, h, div_zero]"
}
]
| [
1478,
91
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1477,
1
]
|
Mathlib/Data/Sym/Sym2.lean | Sym2.out_fst_mem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.34339\nγ : Type ?u.34342\ne : Sym2 α\n⊢ e = Quotient.mk (Rel.setoid α) ((Quotient.out e).fst, (Quotient.out e).snd)",
"tactic": "rw [Prod.mk.eta, e.out_eq]"
}
]
| [
343,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
]
|
Mathlib/Algebra/Order/Archimedean.lean | exists_floor | [
{
"state_after": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis : (a : Prop) → Decidable a\n⊢ ∃ fl, ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\n⊢ ∃ fl, ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x",
"tactic": "haveI := Classical.propDecidable"
},
{
"state_after": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis✝ : (a : Prop) → Decidable a\nthis : ∃ ub, ↑ub ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ ub\n⊢ ∃ fl, ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis : (a : Prop) → Decidable a\n⊢ ∃ fl, ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x",
"tactic": "have : ∃ ub : ℤ, (ub : α) ≤ x ∧ ∀ z : ℤ, (z : α) ≤ x → z ≤ ub :=\n Int.exists_greatest_of_bdd\n (let ⟨n, hn⟩ := exists_int_gt x\n ⟨n, fun z h' => Int.cast_le.1 <| le_trans h' <| le_of_lt hn⟩)\n (let ⟨n, hn⟩ := exists_int_lt x\n ⟨n, le_of_lt hn⟩)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis✝ : (a : Prop) → Decidable a\nthis : ∃ ub, ↑ub ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ ub\nfl : ℤ\nh : ↑fl ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ fl\nz : ℤ\n⊢ z ≤ fl ↔ ↑z ≤ x",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis✝ : (a : Prop) → Decidable a\nthis : ∃ ub, ↑ub ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ ub\n⊢ ∃ fl, ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x",
"tactic": "refine' this.imp fun fl h z => _"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis✝ : (a : Prop) → Decidable a\nthis : ∃ ub, ↑ub ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ ub\nfl z : ℤ\nh₁ : ↑fl ≤ x\nh₂ : ∀ (z : ℤ), ↑z ≤ x → z ≤ fl\n⊢ z ≤ fl ↔ ↑z ≤ x",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis✝ : (a : Prop) → Decidable a\nthis : ∃ ub, ↑ub ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ ub\nfl : ℤ\nh : ↑fl ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ fl\nz : ℤ\n⊢ z ≤ fl ↔ ↑z ≤ x",
"tactic": "cases' h with h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nthis✝ : (a : Prop) → Decidable a\nthis : ∃ ub, ↑ub ≤ x ∧ ∀ (z : ℤ), ↑z ≤ x → z ≤ ub\nfl z : ℤ\nh₁ : ↑fl ≤ x\nh₂ : ∀ (z : ℤ), ↑z ≤ x → z ≤ fl\n⊢ z ≤ fl ↔ ↑z ≤ x",
"tactic": "exact ⟨fun h => le_trans (Int.cast_le.2 h) h₁, h₂ z⟩"
}
]
| [
169,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
]
|
Mathlib/Data/Set/Intervals/Group.lean | Set.add_mem_Ioo_iff_right | []
| [
97,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
]
|
Std/Data/Int/Lemmas.lean | Int.mul_nonneg | [
{
"state_after": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\nn : Nat\nhn : a = ↑n\n⊢ 0 ≤ a * b",
"state_before": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ 0 ≤ a * b",
"tactic": "let ⟨n, hn⟩ := eq_ofNat_of_zero_le ha"
},
{
"state_after": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\nn : Nat\nhn : a = ↑n\nm : Nat\nhm : b = ↑m\n⊢ 0 ≤ a * b",
"state_before": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\nn : Nat\nhn : a = ↑n\n⊢ 0 ≤ a * b",
"tactic": "let ⟨m, hm⟩ := eq_ofNat_of_zero_le hb"
},
{
"state_after": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\nn : Nat\nhn : a = ↑n\nm : Nat\nhm : b = ↑m\n⊢ 0 ≤ ↑(n * m)",
"state_before": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\nn : Nat\nhn : a = ↑n\nm : Nat\nhm : b = ↑m\n⊢ 0 ≤ a * b",
"tactic": "rw [hn, hm, ← ofNat_mul]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nha : 0 ≤ a\nhb : 0 ≤ b\nn : Nat\nhn : a = ↑n\nm : Nat\nhm : b = ↑m\n⊢ 0 ≤ ↑(n * m)",
"tactic": "apply ofNat_nonneg"
}
]
| [
647,
47
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
644,
11
]
|
Mathlib/Data/Finset/Lattice.lean | Finset.le_sup_iff | [
{
"state_after": "case mp\nF : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\n⊢ a ≤ sup s f → ∃ b, b ∈ s ∧ a ≤ f b\n\ncase mpr\nF : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\n⊢ (∃ b, b ∈ s ∧ a ≤ f b) → a ≤ sup s f",
"state_before": "F : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\n⊢ a ≤ sup s f ↔ ∃ b, b ∈ s ∧ a ≤ f b",
"tactic": "apply Iff.intro"
},
{
"state_after": "case mp.cons\nF : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\nc : ι\nt : Finset ι\nhc : ¬c ∈ t\nih : a ≤ sup t f → ∃ b, b ∈ t ∧ a ≤ f b\n⊢ a ≤ f c ∨ a ≤ sup t f → ∃ b, b ∈ cons c t hc ∧ a ≤ f b",
"state_before": "case mp.cons\nF : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\nc : ι\nt : Finset ι\nhc : ¬c ∈ t\nih : a ≤ sup t f → ∃ b, b ∈ t ∧ a ≤ f b\n⊢ a ≤ sup (cons c t hc) f → ∃ b, b ∈ cons c t hc ∧ a ≤ f b",
"tactic": "rw [sup_cons, le_sup_iff]"
},
{
"state_after": "no goals",
"state_before": "case mp.cons\nF : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\nc : ι\nt : Finset ι\nhc : ¬c ∈ t\nih : a ≤ sup t f → ∃ b, b ∈ t ∧ a ≤ f b\n⊢ a ≤ f c ∨ a ≤ sup t f → ∃ b, b ∈ cons c t hc ∧ a ≤ f b",
"tactic": "exact fun\n| Or.inl h => ⟨c, mem_cons.2 (Or.inl rfl), h⟩\n| Or.inr h => let ⟨b, hb, hle⟩ := ih h; ⟨b, mem_cons.2 (Or.inr hb), hle⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr\nF : Type ?u.208752\nα : Type u_1\nβ : Type ?u.208758\nγ : Type ?u.208761\nι : Type u_2\nκ : Type ?u.208767\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nha : ⊥ < a\n⊢ (∃ b, b ∈ s ∧ a ≤ f b) → a ≤ sup s f",
"tactic": "exact fun ⟨b, hb, hle⟩ => le_trans hle (le_sup hb)"
}
]
| [
688,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
679,
11
]
|
Mathlib/Data/List/Basic.lean | List.foldlRecOn_nil | []
| [
2591,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2589,
1
]
|
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | convexOn_pow | [
{
"state_after": "case zero\n\n⊢ ConvexOn ℝ (Ici 0) fun x => x ^ Nat.zero\n\ncase succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\n⊢ ConvexOn ℝ (Ici 0) fun x => x ^ Nat.succ k",
"state_before": "n : ℕ\n⊢ ConvexOn ℝ (Ici 0) fun x => x ^ n",
"tactic": "induction' n with k IH"
},
{
"state_after": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\n⊢ ∀ ⦃x : ℝ⦄,\n x ∈ Ici 0 →\n ∀ ⦃y : ℝ⦄,\n y ∈ Ici 0 →\n ∀ ⦃a b : ℝ⦄,\n 0 ≤ a →\n 0 ≤ b →\n a + b = 1 →\n (fun x => x ^ Nat.succ k) (a • x + b • y) ≤\n a • (fun x => x ^ Nat.succ k) x + b • (fun x => x ^ Nat.succ k) y",
"state_before": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\n⊢ ConvexOn ℝ (Ici 0) fun x => x ^ Nat.succ k",
"tactic": "refine' ⟨convex_Ici _, _⟩"
},
{
"state_after": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (fun x => x ^ Nat.succ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ Nat.succ k) a + ν • (fun x => x ^ Nat.succ k) b",
"state_before": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\n⊢ ∀ ⦃x : ℝ⦄,\n x ∈ Ici 0 →\n ∀ ⦃y : ℝ⦄,\n y ∈ Ici 0 →\n ∀ ⦃a b : ℝ⦄,\n 0 ≤ a →\n 0 ≤ b →\n a + b = 1 →\n (fun x => x ^ Nat.succ k) (a • x + b • y) ≤\n a • (fun x => x ^ Nat.succ k) x + b • (fun x => x ^ Nat.succ k) y",
"tactic": "rintro a (ha : 0 ≤ a) b (hb : 0 ≤ b) μ ν hμ hν h"
},
{
"state_after": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\n⊢ (fun x => x ^ Nat.succ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ Nat.succ k) a + ν • (fun x => x ^ Nat.succ k) b",
"state_before": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (fun x => x ^ Nat.succ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ Nat.succ k) a + ν • (fun x => x ^ Nat.succ k) b",
"tactic": "have H := IH.2 ha hb hμ hν h"
},
{
"state_after": "no goals",
"state_before": "case succ\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (fun x => x ^ Nat.succ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ Nat.succ k) a + ν • (fun x => x ^ Nat.succ k) b",
"tactic": "calc\n (μ * a + ν * b) ^ k.succ = (μ * a + ν * b) * (μ * a + ν * b) ^ k := pow_succ _ _\n _ ≤ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) := by gcongr; exact H\n _ ≤ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) + (b ^ k - a ^ k) * (b - a) * μ * ν := by linarith\n _ = (μ + ν) * (μ * a ^ k.succ + ν * b ^ k.succ) := by rw [Nat.succ_eq_add_one]; ring\n _ = μ * a ^ k.succ + ν * b ^ k.succ := by rw [h]; ring"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ ConvexOn ℝ (Ici 0) fun x => x ^ Nat.zero",
"tactic": "exact convexOn_const (1 : ℝ) (convex_Ici _)"
},
{
"state_after": "case inl\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n\ncase inr\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "cases' le_or_lt a b with hab hab"
},
{
"state_after": "case inl\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\nthis : a ^ k ≤ b ^ k\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"state_before": "case inl\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "have : a ^ k ≤ b ^ k := by gcongr"
},
{
"state_after": "case inl\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\nthis✝ : a ^ k ≤ b ^ k\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a)\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"state_before": "case inl\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\nthis : a ^ k ≤ b ^ k\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "have : 0 ≤ (b ^ k - a ^ k) * (b - a) := by nlinarith"
},
{
"state_after": "no goals",
"state_before": "case inl\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\nthis✝ : a ^ k ≤ b ^ k\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a)\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\n⊢ a ^ k ≤ b ^ k",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : a ≤ b\nthis : a ^ k ≤ b ^ k\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a)",
"tactic": "nlinarith"
},
{
"state_after": "case inr\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\nthis : b ^ k ≤ a ^ k\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"state_before": "case inr\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "have : b ^ k ≤ a ^ k := by gcongr"
},
{
"state_after": "case inr\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\nthis✝ : b ^ k ≤ a ^ k\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a)\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"state_before": "case inr\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\nthis : b ^ k ≤ a ^ k\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "have : 0 ≤ (b ^ k - a ^ k) * (b - a) := by nlinarith"
},
{
"state_after": "no goals",
"state_before": "case inr\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\nthis✝ : b ^ k ≤ a ^ k\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a)\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\n⊢ b ^ k ≤ a ^ k",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nhab : b < a\nthis : b ^ k ≤ a ^ k\n⊢ 0 ≤ (b ^ k - a ^ k) * (b - a)",
"tactic": "nlinarith"
},
{
"state_after": "case h\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) ^ k ≤ μ * a ^ k + ν * b ^ k",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) * (μ * a + ν * b) ^ k ≤ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\nk : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) ^ k ≤ μ * a ^ k + ν * b ^ k",
"tactic": "exact H"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) ≤\n (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) + (b ^ k - a ^ k) * (b - a) * μ * ν",
"tactic": "linarith"
},
{
"state_after": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) + (b ^ k - a ^ k) * (b - a) * μ * ν =\n (μ + ν) * (μ * a ^ (k + 1) + ν * b ^ (k + 1))",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) + (b ^ k - a ^ k) * (b - a) * μ * ν =\n (μ + ν) * (μ * a ^ Nat.succ k + ν * b ^ Nat.succ k)",
"tactic": "rw [Nat.succ_eq_add_one]"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ * a + ν * b) * (μ * a ^ k + ν * b ^ k) + (b ^ k - a ^ k) * (b - a) * μ * ν =\n (μ + ν) * (μ * a ^ (k + 1) + ν * b ^ (k + 1))",
"tactic": "ring"
},
{
"state_after": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ 1 * (μ * a ^ Nat.succ k + ν * b ^ Nat.succ k) = μ * a ^ Nat.succ k + ν * b ^ Nat.succ k",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ (μ + ν) * (μ * a ^ Nat.succ k + ν * b ^ Nat.succ k) = μ * a ^ Nat.succ k + ν * b ^ Nat.succ k",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nIH : ConvexOn ℝ (Ici 0) fun x => x ^ k\na : ℝ\nha : 0 ≤ a\nb : ℝ\nhb : 0 ≤ b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nH : (fun x => x ^ k) (μ • a + ν • b) ≤ μ • (fun x => x ^ k) a + ν • (fun x => x ^ k) b\nthis : 0 ≤ (b ^ k - a ^ k) * (b - a) * μ * ν\n⊢ 1 * (μ * a ^ Nat.succ k + ν * b ^ Nat.succ k) = μ * a ^ Nat.succ k + ν * b ^ Nat.succ k",
"tactic": "ring"
}
]
| [
99,
59
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
]
|
Mathlib/Data/Fintype/Units.lean | Fintype.card_units_int | []
| [
31,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
31,
1
]
|
Mathlib/Geometry/Euclidean/Basic.lean | EuclideanGeometry.inter_eq_singleton_orthogonalProjectionFn | [
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ IsCompl (direction s) (direction s)ᗮ",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ IsCompl (direction s) (direction (mk' p (direction s)ᗮ))",
"tactic": "rw [direction_mk' p s.directionᗮ]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ IsCompl (direction s) (direction s)ᗮ",
"tactic": "exact Submodule.isCompl_orthogonal_of_completeSpace"
}
]
| [
258,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
]
|
Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean | Projectivization.map_injective | [
{
"state_after": "case h\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v : ℙ K V\nh✝ : map f hf u✝ = map f hf v\nu : V\nhu : u ≠ 0\nh : map f hf (mk K u hu) = map f hf v\n⊢ mk K u hu = v",
"state_before": "K : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu v : ℙ K V\nh : map f hf u = map f hf v\n⊢ u = v",
"tactic": "induction' u using ind with u hu"
},
{
"state_after": "case h.h\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝² : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝¹ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh✝ : map f hf u✝ = map f hf (mk K v hv)\nh : map f hf (mk K u hu) = map f hf (mk K v hv)\n⊢ mk K u hu = mk K v hv",
"state_before": "case h\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v : ℙ K V\nh✝ : map f hf u✝ = map f hf v\nu : V\nhu : u ≠ 0\nh : map f hf (mk K u hu) = map f hf v\n⊢ mk K u hu = v",
"tactic": "induction' v using ind with v hv"
},
{
"state_after": "case h.h\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝² : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝¹ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh✝ : map f hf u✝ = map f hf (mk K v hv)\nh : ∃ a, a • ↑f v = ↑f u\n⊢ ∃ a, a • v = u",
"state_before": "case h.h\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝² : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝¹ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh✝ : map f hf u✝ = map f hf (mk K v hv)\nh : map f hf (mk K u hu) = map f hf (mk K v hv)\n⊢ mk K u hu = mk K v hv",
"tactic": "simp only [map_mk, mk_eq_mk_iff'] at h ⊢"
},
{
"state_after": "case h.h.intro\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝¹ : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh : map f hf u✝ = map f hf (mk K v hv)\na : L\nha : a • ↑f v = ↑f u\n⊢ ∃ a, a • v = u",
"state_before": "case h.h\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝² : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝¹ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh✝ : map f hf u✝ = map f hf (mk K v hv)\nh : ∃ a, a • ↑f v = ↑f u\n⊢ ∃ a, a • v = u",
"tactic": "rcases h with ⟨a, ha⟩"
},
{
"state_after": "case h.h.intro\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝¹ : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh : map f hf u✝ = map f hf (mk K v hv)\na : L\nha : a • ↑f v = ↑f u\n⊢ ↑f (↑τ a • v) = ↑f u",
"state_before": "case h.h.intro\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝¹ : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh : map f hf u✝ = map f hf (mk K v hv)\na : L\nha : a • ↑f v = ↑f u\n⊢ ∃ a, a • v = u",
"tactic": "refine ⟨τ a, hf ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case h.h.intro\nK : Type u_1\nV : Type u_3\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nL : Type u_2\nW : Type u_4\ninst✝³ : DivisionRing L\ninst✝² : AddCommGroup W\ninst✝¹ : Module L W\nσ : K →+* L\nτ : L →+* K\ninst✝ : RingHomInvPair σ τ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\nu✝ v✝ : ℙ K V\nh✝¹ : map f hf u✝ = map f hf v✝\nu : V\nhu : u ≠ 0\nh✝ : map f hf (mk K u hu) = map f hf v✝\nv : V\nhv : v ≠ 0\nh : map f hf u✝ = map f hf (mk K v hv)\na : L\nha : a • ↑f v = ↑f u\n⊢ ↑f (↑τ a • v) = ↑f u",
"tactic": "rwa [f.map_smulₛₗ, RingHomInvPair.comp_apply_eq₂]"
}
]
| [
217,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
]
|
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.support_sum | []
| [
568,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
566,
1
]
|
Mathlib/RingTheory/MvPolynomial/Symmetric.lean | MvPolynomial.esymm_eq_sum_monomial | [
{
"state_after": "σ : Type u_1\nR : Type u_2\nτ : Type ?u.91991\nS : Type ?u.91994\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\n⊢ esymm σ R n = ∑ x in powersetLen n univ, ∏ x in x, ↑(monomial (Finsupp.single x 1)) 1",
"state_before": "σ : Type u_1\nR : Type u_2\nτ : Type ?u.91991\nS : Type ?u.91994\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\n⊢ esymm σ R n = ∑ t in powersetLen n univ, ↑(monomial (∑ i in t, Finsupp.single i 1)) 1",
"tactic": "simp_rw [monomial_sum_one]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\nτ : Type ?u.91991\nS : Type ?u.91994\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\n⊢ esymm σ R n = ∑ x in powersetLen n univ, ∏ x in x, ↑(monomial (Finsupp.single x 1)) 1",
"tactic": "rfl"
}
]
| [
190,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.le_def | []
| [
625,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
624,
1
]
|
Mathlib/Data/Set/Pointwise/Basic.lean | Set.isUnit_iff | [
{
"state_after": "case mp\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ IsUnit s → ∃ a, s = {a} ∧ IsUnit a\n\ncase mpr\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ (∃ a, s = {a} ∧ IsUnit a) → IsUnit s",
"state_before": "F : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\n⊢ ∃ a, ↑u = {a} ∧ IsUnit a",
"state_before": "case mp\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ IsUnit s → ∃ a, s = {a} ∧ IsUnit a",
"tactic": "rintro ⟨u, rfl⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\na b : α\nha : ↑u = {a}\nhb : ↑u⁻¹ = {b}\nh : a * b = 1\n⊢ ∃ a, ↑u = {a} ∧ IsUnit a",
"state_before": "case mp.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\n⊢ ∃ a, ↑u = {a} ∧ IsUnit a",
"tactic": "obtain ⟨a, b, ha, hb, h⟩ := Set.mul_eq_one_iff.1 u.mul_inv"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\na b : α\nha : ↑u = {a}\nhb : ↑u⁻¹ = {b}\nh : a * b = 1\n⊢ {b * a} = {1}",
"state_before": "case mp.intro.intro.intro.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\na b : α\nha : ↑u = {a}\nhb : ↑u⁻¹ = {b}\nh : a * b = 1\n⊢ ∃ a, ↑u = {a} ∧ IsUnit a",
"tactic": "refine' ⟨a, ha, ⟨a, b, h, singleton_injective _⟩, rfl⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\na b : α\nha : ↑u = {a}\nhb : ↑u⁻¹ = {b}\nh : a * b = 1\n⊢ ↑u⁻¹ * ↑u = {1}",
"state_before": "case mp.intro.intro.intro.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\na b : α\nha : ↑u = {a}\nhb : ↑u⁻¹ = {b}\nh : a * b = 1\n⊢ {b * a} = {1}",
"tactic": "rw [← singleton_mul_singleton, ← ha, ← hb]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\nu : (Set α)ˣ\na b : α\nha : ↑u = {a}\nhb : ↑u⁻¹ = {b}\nh : a * b = 1\n⊢ ↑u⁻¹ * ↑u = {1}",
"tactic": "exact u.inv_mul"
},
{
"state_after": "case mpr.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\na : α\nha : IsUnit a\n⊢ IsUnit {a}",
"state_before": "case mpr\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ (∃ a, s = {a} ∧ IsUnit a) → IsUnit s",
"tactic": "rintro ⟨a, rfl, ha⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nF : Type ?u.88413\nα : Type u_1\nβ : Type ?u.88419\nγ : Type ?u.88422\ninst✝ : DivisionMonoid α\nt : Set α\na : α\nha : IsUnit a\n⊢ IsUnit {a}",
"tactic": "exact ha.set"
}
]
| [
1080,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1072,
1
]
|
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.power_def | []
| [
494,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
493,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsLittleO.congr_left | []
| [
363,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
]
|
Mathlib/RingTheory/HahnSeries.lean | HahnSeries.mul_coeff | []
| [
639,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
636,
1
]
|
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_X_sub_C_le | []
| [
1324,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1323,
1
]
|
Mathlib/Data/Sym/Card.lean | Finset.card_sym2 | [
{
"state_after": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Disjoint (image Quotient.mk' (Finset.diag s)) (image Quotient.mk' (offDiag s))",
"state_before": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Finset.card (Finset.sym2 s) = Finset.card s * (Finset.card s + 1) / 2",
"tactic": "rw [← image_diag_union_image_offDiag, card_union_eq, Sym2.card_image_diag,\n Sym2.card_image_offDiag, Nat.choose_two_right, add_comm, ← Nat.triangle_succ, Nat.succ_sub_one,\n mul_comm]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\n⊢ ∀ ⦃a : Quotient (Rel.setoid α)⦄, a ∈ image Quotient.mk' (Finset.diag s) → ¬a ∈ image Quotient.mk' (offDiag s)",
"state_before": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Disjoint (image Quotient.mk' (Finset.diag s)) (image Quotient.mk' (offDiag s))",
"tactic": "rw [disjoint_left]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\nm : Quotient (Rel.setoid α)\nha : m ∈ image Quotient.mk' (Finset.diag s)\nhb : m ∈ image Quotient.mk' (offDiag s)\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\n⊢ ∀ ⦃a : Quotient (Rel.setoid α)⦄, a ∈ image Quotient.mk' (Finset.diag s) → ¬a ∈ image Quotient.mk' (offDiag s)",
"tactic": "rintro m ha hb"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\nm : Quotient (Rel.setoid α)\nha : ∃ a, a ∈ Finset.diag s ∧ Quotient.mk' a = m\nhb : ∃ a, a ∈ offDiag s ∧ Quotient.mk' a = m\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\nm : Quotient (Rel.setoid α)\nha : m ∈ image Quotient.mk' (Finset.diag s)\nhb : m ∈ image Quotient.mk' (offDiag s)\n⊢ False",
"tactic": "rw [mem_image] at ha hb"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk' b = Quotient.mk' a\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\nm : Quotient (Rel.setoid α)\nha : ∃ a, a ∈ Finset.diag s ∧ Quotient.mk' a = m\nhb : ∃ a, a ∈ offDiag s ∧ Quotient.mk' a = m\n⊢ False",
"tactic": "obtain ⟨⟨a, ha, rfl⟩, ⟨b, hb, hab⟩⟩ := ha, hb"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk' b = Quotient.mk' a\n⊢ IsDiag (Quotient.mk (Rel.setoid α) b)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk' b = Quotient.mk' a\n⊢ False",
"tactic": "refine' not_isDiag_mk'_of_mem_offDiag hb _"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk (Rel.setoid α) b = Quotient.mk (Rel.setoid α) a\n⊢ IsDiag (Quotient.mk (Rel.setoid α) b)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk' b = Quotient.mk' a\n⊢ IsDiag (Quotient.mk (Rel.setoid α) b)",
"tactic": "dsimp [Quotient.mk'] at hab"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk (Rel.setoid α) b = Quotient.mk (Rel.setoid α) a\n⊢ IsDiag (Quotient.mk (Rel.setoid α) a)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk (Rel.setoid α) b = Quotient.mk (Rel.setoid α) a\n⊢ IsDiag (Quotient.mk (Rel.setoid α) b)",
"tactic": "rw [hab]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.120254\ninst✝ : DecidableEq α\ns : Finset α\na : α × α\nha : a ∈ Finset.diag s\nb : α × α\nhb : b ∈ offDiag s\nhab : Quotient.mk (Rel.setoid α) b = Quotient.mk (Rel.setoid α) a\n⊢ IsDiag (Quotient.mk (Rel.setoid α) a)",
"tactic": "exact isDiag_mk'_of_mem_diag ha"
}
]
| [
208,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
]
|
Mathlib/Topology/ContinuousOn.lean | ContinuousWithinAt.prod_map | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → γ\ng : β → δ\ns : Set α\nt : Set β\nx : α\ny : β\nhf : Tendsto f (𝓝[s] x) (𝓝 (f x))\nhg : Tendsto g (𝓝[t] y) (𝓝 (g y))\n⊢ Tendsto (Prod.map f g) (𝓝[s ×ˢ t] (x, y)) (𝓝 (Prod.map f g (x, y)))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → γ\ng : β → δ\ns : Set α\nt : Set β\nx : α\ny : β\nhf : ContinuousWithinAt f s x\nhg : ContinuousWithinAt g t y\n⊢ ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y)",
"tactic": "unfold ContinuousWithinAt at *"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → γ\ng : β → δ\ns : Set α\nt : Set β\nx : α\ny : β\nhf : Tendsto f (𝓝[s] x) (𝓝 (f x))\nhg : Tendsto g (𝓝[t] y) (𝓝 (g y))\n⊢ Tendsto (Prod.map f g) (𝓝[s] x ×ˢ 𝓝[t] y) (𝓝 (f x) ×ˢ 𝓝 (g y))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → γ\ng : β → δ\ns : Set α\nt : Set β\nx : α\ny : β\nhf : Tendsto f (𝓝[s] x) (𝓝 (f x))\nhg : Tendsto g (𝓝[t] y) (𝓝 (g y))\n⊢ Tendsto (Prod.map f g) (𝓝[s ×ˢ t] (x, y)) (𝓝 (Prod.map f g (x, y)))",
"tactic": "rw [nhdsWithin_prod_eq, Prod.map, nhds_prod_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → γ\ng : β → δ\ns : Set α\nt : Set β\nx : α\ny : β\nhf : Tendsto f (𝓝[s] x) (𝓝 (f x))\nhg : Tendsto g (𝓝[t] y) (𝓝 (g y))\n⊢ Tendsto (Prod.map f g) (𝓝[s] x ×ˢ 𝓝[t] y) (𝓝 (f x) ×ˢ 𝓝 (g y))",
"tactic": "exact hf.prod_map hg"
}
]
| [
567,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
562,
1
]
|
Mathlib/Data/PNat/Prime.lean | PNat.not_prime_one | []
| [
145,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
]
|
Mathlib/Analysis/Calculus/Deriv/Add.lean | deriv_const_sub | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nc : F\n⊢ deriv (fun y => c - f y) x = -deriv f x",
"tactic": "simp only [← derivWithin_univ,\n derivWithin_const_sub (uniqueDiffWithinAt_univ : UniqueDiffWithinAt 𝕜 _ _)]"
}
]
| [
377,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
1
]
|
Mathlib/Algebra/Order/Floor.lean | Nat.floor_div_nat | [
{
"state_after": "case inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : a ≤ 0\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n\n\ncase inr\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n",
"state_before": "F : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n",
"tactic": "cases' le_total a 0 with ha ha"
},
{
"state_after": "case inr.inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nha : 0 ≤ a\n⊢ ⌊a / ↑0⌋₊ = ⌊a⌋₊ / 0\n\ncase inr.inr\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n",
"state_before": "case inr\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n",
"tactic": "obtain rfl | hn := n.eq_zero_or_pos"
},
{
"state_after": "case inr.inr.refine'_1\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ 0 ≤ a / ↑n\n\ncase inr.inr.refine'_2\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ↑(⌊a⌋₊ / n) ≤ a / ↑n ∧ a / ↑n < ↑(⌊a⌋₊ / n) + 1",
"state_before": "case inr.inr\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n",
"tactic": "refine' (floor_eq_iff _).2 _"
},
{
"state_after": "case inr.inr.refine'_2.left\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ↑(⌊a⌋₊ / n) ≤ a / ↑n\n\ncase inr.inr.refine'_2.right\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ a / ↑n < ↑(⌊a⌋₊ / n) + 1",
"state_before": "case inr.inr.refine'_2\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ↑(⌊a⌋₊ / n) ≤ a / ↑n ∧ a / ↑n < ↑(⌊a⌋₊ / n) + 1",
"tactic": "constructor"
},
{
"state_after": "case inr.inr.refine'_2.right\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ⌊a⌋₊ < ⌊a⌋₊ / n * n + n\n\ncase inr.inr.refine'_2.right\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ 0 < ↑n",
"state_before": "case inr.inr.refine'_2.right\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ a / ↑n < ↑(⌊a⌋₊ / n) + 1",
"tactic": "rw [div_lt_iff, add_mul, one_mul, ← cast_mul, ← cast_add, ← floor_lt ha]"
},
{
"state_after": "case inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : a ≤ 0\n⊢ 0 = 0 / n\n\ncase inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : a ≤ 0\n⊢ a / ↑n ≤ 0",
"state_before": "case inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : a ≤ 0\n⊢ ⌊a / ↑n⌋₊ = ⌊a⌋₊ / n",
"tactic": "rw [floor_of_nonpos, floor_of_nonpos ha]"
},
{
"state_after": "no goals",
"state_before": "case inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : a ≤ 0\n⊢ a / ↑n ≤ 0",
"tactic": "apply div_nonpos_of_nonpos_of_nonneg ha n.cast_nonneg"
},
{
"state_after": "no goals",
"state_before": "case inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : a ≤ 0\n⊢ 0 = 0 / n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nha : 0 ≤ a\n⊢ ⌊a / ↑0⌋₊ = ⌊a⌋₊ / 0",
"tactic": "rw [cast_zero, div_zero, Nat.div_zero, floor_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.refine'_1\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ 0 ≤ a / ↑n",
"tactic": "exact div_nonneg ha n.cast_nonneg"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.refine'_2.left\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ↑(⌊a⌋₊ / n) ≤ a / ↑n",
"tactic": "exact cast_div_le.trans (div_le_div_of_le_of_nonneg (floor_le ha) n.cast_nonneg)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.refine'_2.right\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ ⌊a⌋₊ < ⌊a⌋₊ / n * n + n",
"tactic": "exact lt_div_mul_add hn"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.refine'_2.right\nF : Type ?u.94914\nα : Type u_1\nβ : Type ?u.94920\ninst✝¹ : LinearOrderedSemifield α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nhn : n > 0\n⊢ 0 < ↑n",
"tactic": "exact cast_pos.2 hn"
}
]
| [
520,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
507,
1
]
|
Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean | Set.nonempty_ordConnectedComponent | []
| [
61,
89
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
]
|
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