file_path
stringlengths 11
79
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stringlengths 2
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list | end
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stringclasses 4
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stringclasses 4
values | start
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Mathlib/Analysis/Convex/Slope.lean | StrictConvexOn.secant_strict_mono_aux3 | [
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)",
"tactic": "have hyz' : 0 < z - y := by linarith"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)",
"tactic": "have hxz' : 0 < z - x := by linarith"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (f z - f x) * (z - y) < (f z - f y) * (z - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)",
"tactic": "rw [div_lt_div_iff hxz' hyz']"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (f z - f x) * (z - y) < (f z - f y) * (z - x)",
"tactic": "linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ 0 < z - y",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n⊢ 0 < z - x",
"tactic": "linarith"
}
] | [
324,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
] |
Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | SimpleGraph.IsSRGWith.card_commonNeighbors_eq_of_adj_compl | [
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\n⊢ Finset.card (neighborFinset G vᶜ \\ {v} ∩ (neighborFinset G wᶜ \\ {w})) = n - (2 * k - μ) - 2",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\n⊢ Fintype.card ↑(commonNeighbors (Gᶜ) v w) = n - (2 * k - μ) - 2",
"tactic": "simp only [← Set.toFinset_card, commonNeighbors, Set.toFinset_inter, neighborSet_compl,\n Set.toFinset_diff, Set.toFinset_singleton, Set.toFinset_compl, ← neighborFinset_def]"
},
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\n⊢ Finset.card ((neighborFinset G vᶜ ∩ neighborFinset G wᶜ) \\ ({w} ∪ {v})) = n - (2 * k - μ) - 2",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\n⊢ Finset.card (neighborFinset G vᶜ \\ {v} ∩ (neighborFinset G wᶜ \\ {w})) = n - (2 * k - μ) - 2",
"tactic": "simp_rw [compl_neighborFinset_sdiff_inter_eq]"
},
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\nhne : v ≠ w\n⊢ Finset.card ((neighborFinset G vᶜ ∩ neighborFinset G wᶜ) \\ ({w} ∪ {v})) = n - (2 * k - μ) - 2",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\n⊢ Finset.card ((neighborFinset G vᶜ ∩ neighborFinset G wᶜ) \\ ({w} ∪ {v})) = n - (2 * k - μ) - 2",
"tactic": "have hne : v ≠ w := ne_of_adj _ ha"
},
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ Finset.card ((neighborFinset G vᶜ ∩ neighborFinset G wᶜ) \\ ({w} ∪ {v})) = n - (2 * k - μ) - 2",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : Adj (Gᶜ) v w\nhne : v ≠ w\n⊢ Finset.card ((neighborFinset G vᶜ ∩ neighborFinset G wᶜ) \\ ({w} ∪ {v})) = n - (2 * k - μ) - 2",
"tactic": "rw [compl_adj] at ha"
},
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ Finset.card ((neighborFinset G v ∪ neighborFinset G w)ᶜ) - (1 + 1) = n - (2 * k - μ) - 2\n\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ ¬w ∈ {v}\n\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ {w} ∪ {v} ⊆ neighborFinset G vᶜ ∩ neighborFinset G wᶜ",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ Finset.card ((neighborFinset G vᶜ ∩ neighborFinset G wᶜ) \\ ({w} ∪ {v})) = n - (2 * k - μ) - 2",
"tactic": "rw [card_sdiff, ← insert_eq, card_insert_of_not_mem, card_singleton, ← Finset.compl_union]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ Finset.card ((neighborFinset G v ∪ neighborFinset G w)ᶜ) - (1 + 1) = n - (2 * k - μ) - 2",
"tactic": "rw [card_compl, h.card_neighborFinset_union_of_not_adj hne ha.2, ← h.card]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ ¬w ∈ {v}",
"tactic": "simp only [hne.symm, not_false_iff, mem_singleton]"
},
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\nu : V\n⊢ u ∈ {w} ∪ {v} → u ∈ neighborFinset G vᶜ ∩ neighborFinset G wᶜ",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\n⊢ {w} ∪ {v} ⊆ neighborFinset G vᶜ ∩ neighborFinset G wᶜ",
"tactic": "intro u"
},
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\nu : V\n⊢ u = w ∨ u = v → ¬Adj G v u ∧ ¬Adj G w u",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\nu : V\n⊢ u ∈ {w} ∪ {v} → u ∈ neighborFinset G vᶜ ∩ neighborFinset G wᶜ",
"tactic": "simp only [mem_union, mem_compl, mem_neighborFinset, mem_inter, mem_singleton]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nh : IsSRGWith G n k ℓ μ\nv w : V\nha : v ≠ w ∧ ¬Adj G v w\nhne : v ≠ w\nu : V\n⊢ u = w ∨ u = v → ¬Adj G v u ∧ ¬Adj G w u",
"tactic": "rintro (rfl | rfl) <;> simpa [adj_comm] using ha.2"
}
] | [
156,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | NNReal.rpow_le_rpow | [] | [
145,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean | linearIndependent_fin_succ | [
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.1249118\nR : Type ?u.1249121\nK : Type u_1\nM : Type ?u.1249127\nM' : Type ?u.1249130\nM'' : Type ?u.1249133\nV : Type u\nV' : Type ?u.1249138\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nn : ℕ\nv : Fin (n + 1) → V\n⊢ LinearIndependent K v ↔ LinearIndependent K (Fin.tail v) ∧ ¬v 0 ∈ span K (range (Fin.tail v))",
"tactic": "rw [← linearIndependent_fin_cons, Fin.cons_self_tail]"
}
] | [
1255,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1252,
1
] |
Mathlib/Data/List/Count.lean | List.countp_pos | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\np q : α → Bool\n⊢ 0 < countp p l ↔ ∃ a, a ∈ l ∧ p a = true",
"tactic": "simp only [countp_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]"
}
] | [
102,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/Data/Nat/Choose/Multinomial.lean | Nat.binomial_eq_choose | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ multinomial {a, b} f = choose (f a + f b) (f a)",
"tactic": "simp [binomial_eq _ h, choose_eq_factorial_div_factorial (Nat.le_add_right _ _)]"
}
] | [
113,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Data/Bool/Count.lean | List.Chain'.two_mul_count_bool_of_even | [
{
"state_after": "no goals",
"state_before": "l : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nh2 : Even (length l)\nb : Bool\n⊢ 2 * count b l = length l",
"tactic": "rw [← count_not_add_count l b, hl.count_not_eq_count h2, two_mul]"
}
] | [
105,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.image_ι_eq | [
{
"state_after": "Δ Δ'' : SimplexCategory\nφ : Δ ⟶ Δ''\ne : Δ ⟶ image φ\ninst✝¹ : Epi e\ni : image φ ⟶ Δ''\ninst✝ : Mono i\nfac : e ≫ i = φ\nthis : StrongEpi e\n⊢ image.ι φ = i",
"state_before": "Δ Δ'' : SimplexCategory\nφ : Δ ⟶ Δ''\ne : Δ ⟶ image φ\ninst✝¹ : Epi e\ni : image φ ⟶ Δ''\ninst✝ : Mono i\nfac : e ≫ i = φ\n⊢ image.ι φ = i",
"tactic": "haveI := strongEpi_of_epi e"
},
{
"state_after": "no goals",
"state_before": "Δ Δ'' : SimplexCategory\nφ : Δ ⟶ Δ''\ne : Δ ⟶ image φ\ninst✝¹ : Epi e\ni : image φ ⟶ Δ''\ninst✝ : Mono i\nfac : e ≫ i = φ\nthis : StrongEpi e\n⊢ image.ι φ = i",
"tactic": "rw [← image.isoStrongEpiMono_hom_comp_ι e i fac,\n SimplexCategory.eq_id_of_isIso (image.isoStrongEpiMono e i fac).hom, Category.id_comp]"
}
] | [
793,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
789,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousInv.of_nhds_one | [
{
"state_after": "α : Type u\nβ : Type v\nG✝ : Type w\nH : Type x\ninst✝⁵ : TopologicalSpace G✝\ninst✝⁴ : Group G✝\ninst✝³ : TopologicalGroup G✝\ninst✝² : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : TopologicalSpace G\nhinv : Tendsto (fun x => x⁻¹) (𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : G), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\nhconj : ∀ (x₀ : G), Tendsto (fun x => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)\nx₀ : G\n⊢ ContinuousAt (fun a => a⁻¹) x₀",
"state_before": "α : Type u\nβ : Type v\nG✝ : Type w\nH : Type x\ninst✝⁵ : TopologicalSpace G✝\ninst✝⁴ : Group G✝\ninst✝³ : TopologicalGroup G✝\ninst✝² : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : TopologicalSpace G\nhinv : Tendsto (fun x => x⁻¹) (𝓝 1) (𝓝 1)\nhleft : ∀ (x₀ : G), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)\nhconj : ∀ (x₀ : G), Tendsto (fun x => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)\n⊢ ContinuousInv G",
"tactic": "refine' ⟨continuous_iff_continuousAt.2 fun x₀ => _⟩"
}
] | [
915,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
907,
1
] |
Mathlib/Algebra/Lie/Quotient.lean | LieSubmodule.Quotient.mk_eq_zero' | [] | [
199,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/Data/Real/NNReal.lean | NNReal.coe_nonneg | [] | [
137,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/FieldTheory/RatFunc.lean | RatFunc.ofFractionRing_injective | [] | [
131,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Std/Data/List/Lemmas.lean | List.Pairwise.imp | [] | [
1298,
50
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1295,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean | Equiv.Perm.sign_refl | [] | [
565,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
564,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean | LinearMap.nondegenerateRestrictOfDisjointOrthogonal | [
{
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Mathlib/GroupTheory/Submonoid/Membership.lean | Submonoid.closure_singleton_one | [
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Std/Data/RBMap/Lemmas.lean | Std.RBNode.Stream.next?_toList | [
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Mathlib/Order/Filter/Extr.lean | isExtrFilter_dual_iff | [] | [
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Mathlib/CategoryTheory/Iso.lean | CategoryTheory.Iso.inv_ext' | [] | [
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Mathlib/Analysis/Convex/Star.lean | starConvex_iff_div | [
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"tactic": "exact div_self hab.ne'"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.140518\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nh : ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh' : 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s\n⊢ a • x + b • y ∈ s",
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"tactic": "have h' := h hy ha hb"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.140518\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nh : ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh' : 0 < 1 → a • x + b • y ∈ s\n⊢ a • x + b • y ∈ s",
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"tactic": "rw [hab, div_one, div_one] at h'"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.140518\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nh : ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh' : 0 < 1 → a • x + b • y ∈ s\n⊢ a • x + b • y ∈ s",
"tactic": "exact h' zero_lt_one"
}
] | [
405,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
394,
1
] |
Mathlib/Topology/Separation.lean | closure_singleton | [] | [
627,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
626,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.coe_mk' | [] | [
279,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Std/Logic.lean | imp_intro | [] | [
109,
61
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
109,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean | gcd_isUnit_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : GCDMonoid R\nx y : R\n⊢ IsUnit (gcd x y) ↔ IsCoprime x y",
"tactic": "rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]"
}
] | [
388,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
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] |
Mathlib/RingTheory/AlgebraTower.lean | Basis.coe_algebraMapCoeffs | [] | [
100,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/GroupTheory/Abelianization.lean | commutator_eq_normalClosure | [
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Group G\n⊢ commutator G = Subgroup.normalClosure (commutatorSet G)",
"tactic": "simp [commutator, Subgroup.commutator_def', commutatorSet]"
}
] | [
55,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
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Mathlib/Algebra/AddTorsor.lean | vadd_vsub | [] | [
96,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.norm_toSpanSingleton | [
{
"state_after": "case refine'_1\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx✝ : E\nx : 𝕜\n⊢ ‖↑(toSpanSingleton 𝕜 x✝) x‖ ≤ ‖x✝‖ * ‖x‖\n\ncase refine'_2\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ∀ (x_1 : 𝕜), ‖↑(toSpanSingleton 𝕜 x) x_1‖ ≤ N * ‖x_1‖\n⊢ ‖x‖ ≤ N",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\n⊢ ‖toSpanSingleton 𝕜 x‖ = ‖x‖",
"tactic": "refine' op_norm_eq_of_bounds (norm_nonneg _) (fun x => _) fun N _ h => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx✝ : E\nx : 𝕜\n⊢ ‖↑(toSpanSingleton 𝕜 x✝) x‖ ≤ ‖x✝‖ * ‖x‖",
"tactic": "rw [toSpanSingleton_apply, norm_smul, mul_comm]"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ‖↑(toSpanSingleton 𝕜 x) 1‖ ≤ N * ‖1‖\n⊢ ‖x‖ ≤ N",
"state_before": "case refine'_2\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ∀ (x_1 : 𝕜), ‖↑(toSpanSingleton 𝕜 x) x_1‖ ≤ N * ‖x_1‖\n⊢ ‖x‖ ≤ N",
"tactic": "specialize h 1"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ‖x‖ * ‖1‖ ≤ N * ‖1‖\n⊢ ‖x‖ ≤ N",
"state_before": "case refine'_2\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ‖↑(toSpanSingleton 𝕜 x) 1‖ ≤ N * ‖1‖\n⊢ ‖x‖ ≤ N",
"tactic": "rw [toSpanSingleton_apply, norm_smul, mul_comm] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\n𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ‖x‖ * ‖1‖ ≤ N * ‖1‖\n⊢ ‖x‖ ≤ N",
"tactic": "exact (mul_le_mul_right (by simp)).mp h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.2121574\n𝕜₃ : Type ?u.2121577\nE : Type u_2\nEₗ : Type ?u.2121583\nF : Type ?u.2121586\nFₗ : Type ?u.2121589\nG : Type ?u.2121592\nGₗ : Type ?u.2121595\n𝓕 : Type ?u.2121598\ninst✝²² : SeminormedAddCommGroup E\ninst✝²¹ : SeminormedAddCommGroup Eₗ\ninst✝²⁰ : SeminormedAddCommGroup F\ninst✝¹⁹ : SeminormedAddCommGroup Fₗ\ninst✝¹⁸ : SeminormedAddCommGroup G\ninst✝¹⁷ : SeminormedAddCommGroup Gₗ\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : NontriviallyNormedField 𝕜₂\ninst✝¹⁴ : NontriviallyNormedField 𝕜₃\ninst✝¹³ : NormedSpace 𝕜 E\ninst✝¹² : NormedSpace 𝕜 Eₗ\ninst✝¹¹ : NormedSpace 𝕜₂ F\ninst✝¹⁰ : NormedSpace 𝕜 Fₗ\ninst✝⁹ : NormedSpace 𝕜₃ G\ninst✝⁸ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁶ : RingHomIsometric σ₂₃\ninst✝⁵ : RingHomIsometric σ₁₃\ninst✝⁴ : RingHomIsometric σ₁₂\n𝕜' : Type ?u.2124098\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' E\ninst✝ : IsScalarTower 𝕜 𝕜' E\nx : E\nN : ℝ\nx✝ : N ≥ 0\nh : ‖x‖ * ‖1‖ ≤ N * ‖1‖\n⊢ 0 < ‖1‖",
"tactic": "simp"
}
] | [
1214,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1209,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean | CategoryTheory.Presieve.singleton_eq_iff_domain | [
{
"state_after": "case mp\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f g : Y ⟶ X\n⊢ singleton f g → f = g\n\ncase mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f g : Y ⟶ X\n⊢ f = g → singleton f g",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f g : Y ⟶ X\n⊢ singleton f g ↔ f = g",
"tactic": "constructor"
},
{
"state_after": "case mp.mk\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y✝ Z : C\nf✝ f : Y✝ ⟶ X\nY : C\n⊢ f = f",
"state_before": "case mp\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f g : Y ⟶ X\n⊢ singleton f g → f = g",
"tactic": "rintro ⟨a, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.mk\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y✝ Z : C\nf✝ f : Y✝ ⟶ X\nY : C\n⊢ f = f",
"tactic": "rfl"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f : Y ⟶ X\n⊢ singleton f f",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f g : Y ⟶ X\n⊢ f = g → singleton f g",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ f : Y ⟶ X\n⊢ singleton f f",
"tactic": "apply singleton.mk"
}
] | [
104,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean | WittVector.constantCoeff_wittAdd | [
{
"state_after": "p : ℕ\nR : Type ?u.75316\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\n⊢ ↑constantCoeff (X 0 + X 1) = 0",
"state_before": "p : ℕ\nR : Type ?u.75316\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\n⊢ ↑constantCoeff (wittAdd p n) = 0",
"tactic": "apply constantCoeff_wittStructureInt p _ _ n"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type ?u.75316\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\n⊢ ↑constantCoeff (X 0 + X 1) = 0",
"tactic": "simp only [add_zero, RingHom.map_add, constantCoeff_X]"
}
] | [
296,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/Order/Basic.lean | Prod.mk_lt_mk_iff_right | [] | [
1233,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1232,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | ContinuousAt.log | [] | [
394,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
392,
8
] |
Mathlib/Algebra/Group/Units.lean | Units.val_mk | [] | [
130,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | nhds_eq_uniformity_prod | [
{
"state_after": "case hg₁\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\n⊢ ∀ (s : Set (α × α)), Monotone fun t => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}\n\ncase hg₂\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\n⊢ ∀ (t : Set (α × α)), Monotone fun s => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\n⊢ 𝓝 (a, b) = Filter.lift' (𝓤 α) fun s => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ s}",
"tactic": "rw [nhds_prod_eq, nhds_nhds_eq_uniformity_uniformity_prod, lift_lift'_same_eq_lift']"
},
{
"state_after": "no goals",
"state_before": "case hg₁\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\n⊢ ∀ (s : Set (α × α)), Monotone fun t => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}",
"tactic": "exact fun s => monotone_const.set_prod monotone_preimage"
},
{
"state_after": "case hg₂\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\nt : Set (α × α)\n⊢ Monotone fun s => {y | (y, a) ∈ s}",
"state_before": "case hg₂\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\n⊢ ∀ (t : Set (α × α)), Monotone fun s => {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}",
"tactic": "refine fun t => Monotone.set_prod ?_ monotone_const"
},
{
"state_after": "no goals",
"state_before": "case hg₂\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.78723\ninst✝ : UniformSpace α\na b : α\nt : Set (α × α)\n⊢ Monotone fun s => {y | (y, a) ∈ s}",
"tactic": "exact monotone_preimage (f := fun y => (y, a))"
}
] | [
892,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
886,
1
] |
Mathlib/Topology/Constructions.lean | continuous_apply | [] | [
1199,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1198,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean | smul_one_mul | [
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.27123\nN✝ : Type ?u.27126\nG : Type ?u.27129\nA : Type ?u.27132\nB : Type ?u.27135\nα : Type ?u.27138\nβ : Type ?u.27141\nγ : Type ?u.27144\nδ : Type ?u.27147\nM : Type u_1\nN : Type u_2\ninst✝² : MulOneClass N\ninst✝¹ : SMul M N\ninst✝ : IsScalarTower M N N\nx : M\ny : N\n⊢ x • 1 * y = x • y",
"tactic": "rw [smul_mul_assoc, one_mul]"
}
] | [
649,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
648,
1
] |
Mathlib/RingTheory/MvPolynomial/Basic.lean | Polynomial.coe_basisMonomials | [] | [
142,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean | WeakDual.eval_continuous | [] | [
284,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
283,
1
] |
Mathlib/FieldTheory/AxGrothendieck.lean | ax_grothendieck_of_locally_finite | [
{
"state_after": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\n⊢ Surjective fun v i => ↑(MvPolynomial.eval v) (ps i)",
"state_before": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\n⊢ Surjective fun v i => ↑(MvPolynomial.eval v) (ps i)",
"tactic": "have is_int : ∀ x : R, IsIntegral K x := fun x => isAlgebraic_iff_isIntegral.1 (alg x)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\n⊢ Surjective fun v i => ↑(MvPolynomial.eval v) (ps i)",
"tactic": "classical\n intro v\n cases nonempty_fintype ι\n \n let s : Finset R :=\n (Finset.biUnion (univ : Finset ι) fun i => (ps i).support.image fun x => coeff x (ps i)) ∪\n (univ : Finset ι).image v\n have hv : ∀ i, v i ∈ Algebra.adjoin K (s : Set R) := fun j =>\n Algebra.subset_adjoin (mem_union_right _ (mem_image.2 ⟨j, mem_univ _, rfl⟩))\n have hs₁ : ∀ (i : ι) (k : ι →₀ ℕ),\n k ∈ (ps i).support → coeff k (ps i) ∈ Algebra.adjoin K (s : Set R) :=\n fun i k hk => Algebra.subset_adjoin\n (mem_union_left _ (mem_biUnion.2 ⟨i, mem_univ _, mem_image_of_mem _ hk⟩))\n letI := isNoetherian_adjoin_finset s fun x _ => is_int x\n letI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))\n letI : Finite (Algebra.adjoin K (s : Set R)) :=\n FiniteDimensional.finite_of_finite K (Algebra.adjoin K (s : Set R))\n let res : (ι → Algebra.adjoin K (s : Set R)) → ι → Algebra.adjoin K (s : Set R) := fun x i =>\n ⟨eval (fun j : ι => (x j : R)) (ps i), eval_mem (hs₁ _) fun i => (x i).2⟩\n have hres_inj : Injective res := by\n intro x y hxy\n ext i\n simp only [Subtype.ext_iff, funext_iff] at hxy\n exact congr_fun (hinj (funext hxy)) i\n have hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj\n cases' hres_surj fun i => ⟨v i, hv i⟩ with w hw\n use fun i => w i\n simpa only [Subtype.ext_iff, funext_iff] using hw"
},
{
"state_after": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\n⊢ Surjective fun v i => ↑(MvPolynomial.eval v) (ps i)",
"tactic": "intro v"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "let s : Finset R :=\n (Finset.biUnion (univ : Finset ι) fun i => (ps i).support.image fun x => coeff x (ps i)) ∪\n (univ : Finset ι).image v"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "have hv : ∀ i, v i ∈ Algebra.adjoin K (s : Set R) := fun j =>\n Algebra.subset_adjoin (mem_union_right _ (mem_image.2 ⟨j, mem_univ _, rfl⟩))"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "have hs₁ : ∀ (i : ι) (k : ι →₀ ℕ),\n k ∈ (ps i).support → coeff k (ps i) ∈ Algebra.adjoin K (s : Set R) :=\n fun i k hk => Algebra.subset_adjoin\n (mem_union_left _ (mem_biUnion.2 ⟨i, mem_univ _, mem_image_of_mem _ hk⟩))"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "letI := isNoetherian_adjoin_finset s fun x _ => is_int x"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } := Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "letI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } := Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "letI : Finite (Algebra.adjoin K (s : Set R)) :=\n FiniteDimensional.finite_of_finite K (Algebra.adjoin K (s : Set R))"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "let res : (ι → Algebra.adjoin K (s : Set R)) → ι → Algebra.adjoin K (s : Set R) := fun x i =>\n ⟨eval (fun j : ι => (x j : R)) (ps i), eval_mem (hs₁ _) fun i => (x i).2⟩"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "have hres_inj : Injective res := by\n intro x y hxy\n ext i\n simp only [Subtype.ext_iff, funext_iff] at hxy\n exact congr_fun (hinj (funext hxy)) i"
},
{
"state_after": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\nhres_surj : Surjective res\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "have hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj"
},
{
"state_after": "case intro.intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\nhres_surj : Surjective res\nw : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhw : res w = fun i => { val := v i, property := (_ : v i ∈ Algebra.adjoin K ↑s) }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"state_before": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\nhres_surj : Surjective res\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "cases' hres_surj fun i => ⟨v i, hv i⟩ with w hw"
},
{
"state_after": "case intro.intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\nhres_surj : Surjective res\nw : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhw : res w = fun i => { val := v i, property := (_ : v i ∈ Algebra.adjoin K ↑s) }\n⊢ ((fun v i => ↑(MvPolynomial.eval v) (ps i)) fun i => ↑(w i)) = v",
"state_before": "case intro.intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\nhres_surj : Surjective res\nw : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhw : res w = fun i => { val := v i, property := (_ : v i ∈ Algebra.adjoin K ↑s) }\n⊢ ∃ a, (fun v i => ↑(MvPolynomial.eval v) (ps i)) a = v",
"tactic": "use fun i => w i"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nhres_inj : Injective res\nhres_surj : Surjective res\nw : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhw : res w = fun i => { val := v i, property := (_ : v i ∈ Algebra.adjoin K ↑s) }\n⊢ ((fun v i => ↑(MvPolynomial.eval v) (ps i)) fun i => ↑(w i)) = v",
"tactic": "simpa only [Subtype.ext_iff, funext_iff] using hw"
},
{
"state_after": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nx y : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhxy : res x = res y\n⊢ x = y",
"state_before": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\n⊢ Injective res",
"tactic": "intro x y hxy"
},
{
"state_after": "case h.a\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nx y : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhxy : res x = res y\ni : ι\n⊢ ↑(x i) = ↑(y i)",
"state_before": "ι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nx y : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhxy : res x = res y\n⊢ x = y",
"tactic": "ext i"
},
{
"state_after": "case h.a\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nx y : ι → { x // x ∈ Algebra.adjoin K ↑s }\ni : ι\nhxy : ∀ (a : ι), ↑(MvPolynomial.eval fun j => ↑(x j)) (ps a) = ↑(MvPolynomial.eval fun j => ↑(y j)) (ps a)\n⊢ ↑(x i) = ↑(y i)",
"state_before": "case h.a\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nx y : ι → { x // x ∈ Algebra.adjoin K ↑s }\nhxy : res x = res y\ni : ι\n⊢ ↑(x i) = ↑(y i)",
"tactic": "simp only [Subtype.ext_iff, funext_iff] at hxy"
},
{
"state_after": "no goals",
"state_before": "case h.a\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nhinj : Injective fun v i => ↑(MvPolynomial.eval v) (ps i)\nis_int : ∀ (x : R), IsIntegral K x\nv : (i : ι) → (fun x => R) (ps i)\nval✝ : Fintype ι\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ\nhv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s\nhs₁ : ∀ (i : ι) (k : ι →₀ ℕ), k ∈ MvPolynomial.support (ps i) → coeff k (ps i) ∈ Algebra.adjoin K ↑s\nthis✝¹ : IsNoetherian K { x // x ∈ Algebra.adjoin K ↑s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis✝ : Module.Finite K { x // x ∈ Algebra.adjoin K ↑s } :=\n Module.IsNoetherian.finite K { x // x ∈ Algebra.adjoin K ↑s }\nthis : Finite { x // x ∈ Algebra.adjoin K ↑s } := FiniteDimensional.finite_of_finite K { x // x ∈ Algebra.adjoin K ↑s }\nres : (ι → { x // x ∈ Algebra.adjoin K ↑s }) → ι → { x // x ∈ Algebra.adjoin K ↑s } :=\n fun x i =>\n { val := ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i),\n property := (_ : ↑(MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }\nx y : ι → { x // x ∈ Algebra.adjoin K ↑s }\ni : ι\nhxy : ∀ (a : ι), ↑(MvPolynomial.eval fun j => ↑(x j)) (ps a) = ↑(MvPolynomial.eval fun j => ↑(y j)) (ps a)\n⊢ ↑(x i) = ↑(y i)",
"tactic": "exact congr_fun (hinj (funext hxy)) i"
}
] | [
70,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
] |
Mathlib/LinearAlgebra/Finrank.lean | finrank_range_le_card | [
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ Finset.card (Finset.image b Finset.univ) ≤ Fintype.card ι",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ Finset.card (Set.toFinset (Set.range b)) ≤ Fintype.card ι",
"tactic": "rw [Set.toFinset_range]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ Finset.card (Finset.image b Finset.univ) ≤ Fintype.card ι",
"tactic": "exact Finset.card_image_le"
}
] | [
328,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
324,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_iUnion_le | [
{
"state_after": "α : Type u_2\nβ : Type ?u.360947\nγ : Type ?u.360950\nδ : Type ?u.360953\nι : Type u_1\nR : Type ?u.360959\nR' : Type ?u.360962\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(restrict μ (⋃ (i : ι), s i)) t ≤ ↑↑(sum fun i => restrict μ (s i)) t",
"state_before": "α : Type u_2\nβ : Type ?u.360947\nγ : Type ?u.360950\nδ : Type ?u.360953\nι : Type u_1\nR : Type ?u.360959\nR' : Type ?u.360962\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : Countable ι\ns : ι → Set α\n⊢ restrict μ (⋃ (i : ι), s i) ≤ sum fun i => restrict μ (s i)",
"tactic": "intro t ht"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.360947\nγ : Type ?u.360950\nδ : Type ?u.360953\nι : Type u_1\nR : Type ?u.360959\nR' : Type ?u.360962\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑μ (⋃ (i : ι), t ∩ s i) ≤ ∑' (i : ι), ↑↑μ (t ∩ s i)",
"state_before": "α : Type u_2\nβ : Type ?u.360947\nγ : Type ?u.360950\nδ : Type ?u.360953\nι : Type u_1\nR : Type ?u.360959\nR' : Type ?u.360962\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(restrict μ (⋃ (i : ι), s i)) t ≤ ↑↑(sum fun i => restrict μ (s i)) t",
"tactic": "suffices μ (⋃ i, t ∩ s i) ≤ ∑' i, μ (t ∩ s i) by simpa [ht, inter_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.360947\nγ : Type ?u.360950\nδ : Type ?u.360953\nι : Type u_1\nR : Type ?u.360959\nR' : Type ?u.360962\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑μ (⋃ (i : ι), t ∩ s i) ≤ ∑' (i : ι), ↑↑μ (t ∩ s i)",
"tactic": "apply measure_iUnion_le"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.360947\nγ : Type ?u.360950\nδ : Type ?u.360953\nι : Type u_1\nR : Type ?u.360959\nR' : Type ?u.360962\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nt : Set α\nht : MeasurableSet t\nthis : ↑↑μ (⋃ (i : ι), t ∩ s i) ≤ ∑' (i : ι), ↑↑μ (t ∩ s i)\n⊢ ↑↑(restrict μ (⋃ (i : ι), s i)) t ≤ ↑↑(sum fun i => restrict μ (s i)) t",
"tactic": "simpa [ht, inter_iUnion]"
}
] | [
2176,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2172,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean | expNegInvGlue.tendsto_polynomial_inv_mul_zero | [
{
"state_after": "p : ℝ[X]\n⊢ Tendsto (fun x => if x ≤ 0 then 0 else Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0) (𝓝 0)",
"state_before": "p : ℝ[X]\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) (𝓝 0) (𝓝 0)",
"tactic": "simp only [expNegInvGlue, mul_ite, mul_zero]"
},
{
"state_after": "p : ℝ[X]\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | ¬x ≤ 0}) (𝓝 0)",
"state_before": "p : ℝ[X]\n⊢ Tendsto (fun x => if x ≤ 0 then 0 else Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0) (𝓝 0)",
"tactic": "refine tendsto_const_nhds.if ?_"
},
{
"state_after": "p : ℝ[X]\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)",
"state_before": "p : ℝ[X]\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | ¬x ≤ 0}) (𝓝 0)",
"tactic": "simp only [not_le]"
},
{
"state_after": "p : ℝ[X]\nthis : Tendsto (fun x => Polynomial.eval x⁻¹ p / exp x⁻¹) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)",
"state_before": "p : ℝ[X]\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)",
"tactic": "have : Tendsto (fun x ↦ p.eval x⁻¹ / exp x⁻¹) (𝓝[>] 0) (𝓝 0) :=\n p.tendsto_div_exp_atTop.comp tendsto_inv_zero_atTop"
},
{
"state_after": "p : ℝ[X]\nthis : Tendsto (fun x => Polynomial.eval x⁻¹ p / exp x⁻¹) (𝓝[Ioi 0] 0) (𝓝 0)\nx : ℝ\nhx : x ∈ {x | 0 < x}\n⊢ x ∈ {x | (fun x => Polynomial.eval x⁻¹ p / exp x⁻¹ = (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) x) x}",
"state_before": "p : ℝ[X]\nthis : Tendsto (fun x => Polynomial.eval x⁻¹ p / exp x⁻¹) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)",
"tactic": "refine this.congr' <| mem_of_superset self_mem_nhdsWithin fun x hx ↦ ?_"
},
{
"state_after": "no goals",
"state_before": "p : ℝ[X]\nthis : Tendsto (fun x => Polynomial.eval x⁻¹ p / exp x⁻¹) (𝓝[Ioi 0] 0) (𝓝 0)\nx : ℝ\nhx : x ∈ {x | 0 < x}\n⊢ x ∈ {x | (fun x => Polynomial.eval x⁻¹ p / exp x⁻¹ = (fun x => Polynomial.eval x⁻¹ p * exp (-x⁻¹)) x) x}",
"tactic": "simp [expNegInvGlue, hx.out.not_le, exp_neg, div_eq_mul_inv]"
}
] | [
120,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | Real.ediam_Ioc | [] | [
1552,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1550,
1
] |
Mathlib/MeasureTheory/Function/LpOrder.lean | MeasureTheory.Lp.coeFn_le | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\ninst✝ : NormedLatticeAddCommGroup E\nf g : { x // x ∈ Lp E p }\n⊢ ↑↑f ≤ᵐ[μ] ↑↑g ↔ f ≤ g",
"tactic": "rw [← Subtype.coe_le_coe, ← AEEqFun.coeFn_le]"
}
] | [
46,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/Data/Finset/Fold.lean | Finset.fold_min_lt | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc : β\n⊢ c > fold min b f s ↔ b < c ∨ ∃ x, x ∈ s ∧ f x < c",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc : β\n⊢ fold min b f s < c ↔ b < c ∨ ∃ x, x ∈ s ∧ f x < c",
"tactic": "show _ > _ ↔ _"
},
{
"state_after": "case hr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc : β\n⊢ ∀ {x y z : β}, x > min y z ↔ x > y ∨ x > z",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc : β\n⊢ c > fold min b f s ↔ b < c ∨ ∃ x, x ∈ s ∧ f x < c",
"tactic": "apply fold_op_rel_iff_or"
},
{
"state_after": "case hr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc x y z : β\n⊢ x > min y z ↔ x > y ∨ x > z",
"state_before": "case hr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc : β\n⊢ ∀ {x y z : β}, x > min y z ↔ x > y ∨ x > z",
"tactic": "intro x y z"
},
{
"state_after": "case hr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc x y z : β\n⊢ min y z < x ↔ x > y ∨ x > z",
"state_before": "case hr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc x y z : β\n⊢ x > min y z ↔ x > y ∨ x > z",
"tactic": "show _ < _ ↔ _"
},
{
"state_after": "no goals",
"state_before": "case hr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67307\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : LinearOrder β\nc x y z : β\n⊢ min y z < x ↔ x > y ∨ x > z",
"tactic": "exact min_lt_iff"
}
] | [
239,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
234,
1
] |
Mathlib/Data/Opposite.lean | Opposite.equivToOpposite_coe | [] | [
97,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean | FDerivMeasurableAux.isOpen_a | [
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\n⊢ ∀ (x : E), x ∈ A f L r ε → ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\n⊢ IsOpen (A f L r ε)",
"tactic": "rw [Metric.isOpen_iff]"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\n⊢ ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\n⊢ ∀ (x : E), x ∈ A f L r ε → ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"tactic": "rintro x ⟨r', r'_mem, hr'⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\n⊢ ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"state_before": "case intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\n⊢ ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"tactic": "obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between r'_mem.1"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\n⊢ ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\n⊢ ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"tactic": "have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : E\nhx' : x' ∈ ball x (r' - s)\n⊢ ∀ (y : E), y ∈ ball x' s → ∀ (z : E), z ∈ ball x' s → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\n⊢ ∃ ε_1, ε_1 > 0 ∧ ball x ε_1 ⊆ A f L r ε",
"tactic": "refine' ⟨r' - s, by linarith, fun x' hx' => ⟨s, this, _⟩⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : E\nhx' : x' ∈ ball x (r' - s)\nB : ball x' s ⊆ ball x r'\n⊢ ∀ (y : E), y ∈ ball x' s → ∀ (z : E), z ∈ ball x' s → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : E\nhx' : x' ∈ ball x (r' - s)\n⊢ ∀ (y : E), y ∈ ball x' s → ∀ (z : E), z ∈ ball x' s → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r",
"tactic": "have B : ball x' s ⊆ ball x r' := ball_subset (le_of_lt hx')"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : E\nhx' : x' ∈ ball x (r' - s)\nB : ball x' s ⊆ ball x r'\ny : E\nhy : y ∈ ball x' s\nz : E\nhz : z ∈ ball x' s\n⊢ ‖f z - f y - ↑L (z - y)‖ ≤ ε * r",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : E\nhx' : x' ∈ ball x (r' - s)\nB : ball x' s ⊆ ball x r'\n⊢ ∀ (y : E), y ∈ ball x' s → ∀ (z : E), z ∈ ball x' s → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r",
"tactic": "intro y hy z hz"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : E\nhx' : x' ∈ ball x (r' - s)\nB : ball x' s ⊆ ball x r'\ny : E\nhy : y ∈ ball x' s\nz : E\nhz : z ∈ ball x' s\n⊢ ‖f z - f y - ↑L (z - y)‖ ≤ ε * r",
"tactic": "exact hr' y (B hy) z (B hz)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε : ℝ\nx : E\nr' : ℝ\nr'_mem : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\n⊢ r' - s > 0",
"tactic": "linarith"
}
] | [
143,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean | LieIdeal.lowerCentralSeries_map_eq | [
{
"state_after": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\n⊢ map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\n⊢ map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k",
"tactic": "have h' : (⊤ : LieIdeal R L).map f = ⊤ := by\n rw [← f.idealRange_eq_map]\n exact f.idealRange_eq_top_of_surjective h"
},
{
"state_after": "case zero\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\n⊢ map f (lowerCentralSeries R L L Nat.zero) = lowerCentralSeries R L' L' Nat.zero\n\ncase succ\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\nk : ℕ\nih : map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k\n⊢ map f (lowerCentralSeries R L L (Nat.succ k)) = lowerCentralSeries R L' L' (Nat.succ k)",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\n⊢ map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k",
"tactic": "induction' k with k ih"
},
{
"state_after": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\n⊢ LieHom.idealRange f = ⊤",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\n⊢ map f ⊤ = ⊤",
"tactic": "rw [← f.idealRange_eq_map]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\n⊢ LieHom.idealRange f = ⊤",
"tactic": "exact f.idealRange_eq_top_of_surjective h"
},
{
"state_after": "case zero\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\n⊢ map f (lowerCentralSeries R L L Nat.zero) = lowerCentralSeries R L' L' Nat.zero",
"state_before": "case zero\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\n⊢ map f (lowerCentralSeries R L L Nat.zero) = lowerCentralSeries R L' L' Nat.zero",
"tactic": "simp only [LieModule.lowerCentralSeries_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\n⊢ map f (lowerCentralSeries R L L Nat.zero) = lowerCentralSeries R L' L' Nat.zero",
"tactic": "exact h'"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ↑f\nh' : map f ⊤ = ⊤\nk : ℕ\nih : map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k\n⊢ map f (lowerCentralSeries R L L (Nat.succ k)) = lowerCentralSeries R L' L' (Nat.succ k)",
"tactic": "simp only [LieModule.lowerCentralSeries_succ, LieIdeal.map_bracket_eq f h, ih, h']"
}
] | [
618,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
611,
1
] |
Mathlib/Data/List/Palindrome.lean | List.Palindrome.append_reverse | [
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.1074\nl✝ l : List α\n⊢ reverse (l ++ reverse l) = l ++ reverse l",
"state_before": "α : Type u_1\nβ : Type ?u.1074\nl✝ l : List α\n⊢ Palindrome (l ++ reverse l)",
"tactic": "apply of_reverse_eq"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.1074\nl✝ l : List α\n⊢ reverse (l ++ reverse l) = l ++ reverse l",
"tactic": "rw [reverse_append, reverse_reverse]"
}
] | [
73,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | ContDiff.contDiffAt | [] | [
1422,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1421,
1
] |
Mathlib/Algebra/GradedMonoid.lean | GradedMonoid.list_prod_ofFn_eq_dProd | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type ?u.51491\nA : ι → Type u_2\ninst✝¹ : AddMonoid ι\ninst✝ : GMonoid A\nn : ℕ\nf : Fin n → GradedMonoid A\n⊢ List.prod (List.ofFn f) =\n mk (List.dProdIndex (List.finRange n) fun i => (f i).fst)\n (List.dProd (List.finRange n) (fun i => (f i).fst) fun i => (f i).snd)",
"tactic": "rw [List.ofFn_eq_map, GradedMonoid.list_prod_map_eq_dProd]"
}
] | [
424,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
421,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.setToFun_neg | [
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T hT (-f) = -setToFun μ T hT f\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T hT (-f) = -setToFun μ T hT f",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\n⊢ setToFun μ T hT (-f) = -setToFun μ T hT f",
"tactic": "by_cases hf : Integrable f μ"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T hT (-f) = -setToFun μ T hT f",
"tactic": "rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,\n (L1.setToL1 hT).map_neg]"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhf : ¬Integrable f\n⊢ ¬Integrable (-f)",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T hT (-f) = -setToFun μ T hT f",
"tactic": "rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1458556\nG : Type ?u.1458559\n𝕜 : Type ?u.1458562\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhf : ¬Integrable f\n⊢ ¬Integrable (-f)",
"tactic": "rwa [← integrable_neg_iff] at hf"
}
] | [
1404,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1398,
1
] |
Mathlib/Logic/Basic.lean | forall_prop_congr | [] | [
915,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
913,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean | TensorProduct.map_smul_right | [
{
"state_after": "case H.h.h\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.1007951\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.1007957\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type ?u.1007975\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.1009102\nQ' : Type ?u.1009105\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nr : R\nf : M →ₗ[R] P\ng : N →ₗ[R] Q\nx✝¹ : M\nx✝ : N\n⊢ ↑(↑(compr₂ (mk R M N) (map f (r • g))) x✝¹) x✝ = ↑(↑(compr₂ (mk R M N) (r • map f g)) x✝¹) x✝",
"state_before": "R : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.1007951\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.1007957\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type ?u.1007975\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.1009102\nQ' : Type ?u.1009105\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nr : R\nf : M →ₗ[R] P\ng : N →ₗ[R] Q\n⊢ map f (r • g) = r • map f g",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.h.h\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.1007951\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.1007957\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type ?u.1007975\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.1009102\nQ' : Type ?u.1009105\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nr : R\nf : M →ₗ[R] P\ng : N →ₗ[R] Q\nx✝¹ : M\nx✝ : N\n⊢ ↑(↑(compr₂ (mk R M N) (map f (r • g))) x✝¹) x✝ = ↑(↑(compr₂ (mk R M N) (r • map f g)) x✝¹) x✝",
"tactic": "simp only [smul_tmul, compr₂_apply, mk_apply, map_tmul, smul_apply, tmul_smul]"
}
] | [
810,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
808,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | strictConcaveOn_log_Iio | [
{
"state_after": "⊢ ∀ ⦃x : ℝ⦄,\n x ∈ Iio 0 →\n ∀ ⦃y : ℝ⦄,\n y ∈ Iio 0 → x ≠ y → ∀ ⦃a b : ℝ⦄, 0 < a → 0 < b → a + b = 1 → a • log x + b • log y < log (a • x + b • y)",
"state_before": "⊢ StrictConcaveOn ℝ (Iio 0) log",
"tactic": "refine' ⟨convex_Iio _, _⟩"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • log x + b • log y < log (a • x + b • y)",
"state_before": "⊢ ∀ ⦃x : ℝ⦄,\n x ∈ Iio 0 →\n ∀ ⦃y : ℝ⦄,\n y ∈ Iio 0 → x ≠ y → ∀ ⦃a b : ℝ⦄, 0 < a → 0 < b → a + b = 1 → a • log x + b • log y < log (a • x + b • y)",
"tactic": "rintro x (hx : x < 0) y (hy : y < 0) hxy a b ha hb hab"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\n⊢ a • log x + b • log y < log (a • x + b • y)",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • log x + b • log y < log (a • x + b • y)",
"tactic": "have hx' : 0 < -x := by linarith"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\n⊢ a • log x + b • log y < log (a • x + b • y)",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\n⊢ a • log x + b • log y < log (a • x + b • y)",
"tactic": "have hy' : 0 < -y := by linarith"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a • log x + b • log y < log (a • x + b • y)",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\n⊢ a • log x + b • log y < log (a • x + b • y)",
"tactic": "have hxy' : -x ≠ -y := by contrapose! hxy; linarith"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a • log x + b • log y < log (a • x + b • y)",
"tactic": "calc\n a • log x + b • log y = a • log (-x) + b • log (-y) := by simp_rw [log_neg_eq_log]\n _ < log (a • -x + b • -y) := (strictConcaveOn_log_Ioi.2 hx' hy' hxy' ha hb hab)\n _ = log (-(a • x + b • y)) := by congr 1; simp only [Algebra.id.smul_eq_mul]; ring\n _ = _ := by rw [log_neg_eq_log]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ 0 < -x",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\n⊢ 0 < -y",
"tactic": "linarith"
},
{
"state_after": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy : -x = -y\n⊢ x = y",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\n⊢ -x ≠ -y",
"tactic": "contrapose! hxy"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy : -x = -y\n⊢ x = y",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a • log x + b • log y = a • log (-x) + b • log (-y)",
"tactic": "simp_rw [log_neg_eq_log]"
},
{
"state_after": "case e_x\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a • -x + b • -y = -(a • x + b • y)",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ log (a • -x + b • -y) = log (-(a • x + b • y))",
"tactic": "congr 1"
},
{
"state_after": "case e_x\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a * -x + b * -y = -(a * x + b * y)",
"state_before": "case e_x\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a • -x + b • -y = -(a • x + b • y)",
"tactic": "simp only [Algebra.id.smul_eq_mul]"
},
{
"state_after": "no goals",
"state_before": "case e_x\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ a * -x + b * -y = -(a * x + b * y)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x ≠ -y\n⊢ log (-(a • x + b • y)) = log (a • x + b • y)",
"tactic": "rw [log_neg_eq_log]"
}
] | [
295,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/Algebra/Algebra/Operations.lean | Submodule.mem_span_mul_finite_of_mem_span_mul | [
{
"state_after": "case intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\n⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (↑T * ↑T')",
"state_before": "ι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\n⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (↑T * ↑T')",
"tactic": "obtain ⟨U, h, hU⟩ := mem_span_finite_of_mem_span hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\n⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (↑T * ↑T')",
"state_before": "case intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\n⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (↑T * ↑T')",
"tactic": "obtain ⟨T, T', hS, hS', h⟩ := Finset.subset_mul h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\n⊢ x ∈ span R (↑T * ↑T')",
"state_before": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\n⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (↑T * ↑T')",
"tactic": "use T, T', hS, hS'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\nh' : ↑U ⊆ ↑T * ↑T'\n⊢ x ∈ span R (↑T * ↑T')",
"state_before": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\n⊢ x ∈ span R (↑T * ↑T')",
"tactic": "have h' : (U : Set A) ⊆ T * T' := by assumption_mod_cast"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\nh' : ↑U ⊆ ↑T * ↑T'\nh'' : x ∈ span R (↑T * ↑T')\n⊢ x ∈ span R (↑T * ↑T')",
"state_before": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\nh' : ↑U ⊆ ↑T * ↑T'\n⊢ x ∈ span R (↑T * ↑T')",
"tactic": "have h'' := span_mono h' hU"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\nh' : ↑U ⊆ ↑T * ↑T'\nh'' : x ∈ span R (↑T * ↑T')\n⊢ x ∈ span R (↑T * ↑T')",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR✝ : Type u\ninst✝⁶ : CommSemiring R✝\nA✝ : Type v\ninst✝⁵ : Semiring A✝\ninst✝⁴ : Algebra R✝ A✝\nS✝ T✝ : Set A✝\nM N P Q : Submodule R✝ A✝\nm n : A✝\nR : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Mul A\ninst✝ : Module R A\nS S' : Set A\nx : A\nhx : x ∈ span R (S * S')\nU : Finset A\nh✝ : ↑U ⊆ S * S'\nhU : x ∈ span R ↑U\nT T' : Finset A\nhS : ↑T ⊆ S\nhS' : ↑T' ⊆ S'\nh : U ⊆ T * T'\n⊢ ↑U ⊆ ↑T * ↑T'",
"tactic": "assumption_mod_cast"
}
] | [
348,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/LinearAlgebra/Determinant.lean | LinearMap.associated_det_comp_equiv | [
{
"state_after": "R : Type u_2\ninst✝⁸ : CommRing R\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type ?u.2243677\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type ?u.2244219\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne✝ : Basis ι R M\nN : Type u_1\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\ne e' : M ≃ₗ[R] N\n⊢ ∀ (x : M), ↑(comp f ↑e) x = ↑(comp f ↑e') (↑(LinearEquiv.trans e (LinearEquiv.symm e')) x)",
"state_before": "R : Type u_2\ninst✝⁸ : CommRing R\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type ?u.2243677\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type ?u.2244219\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne✝ : Basis ι R M\nN : Type u_1\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\ne e' : M ≃ₗ[R] N\n⊢ Associated (↑LinearMap.det (comp f ↑e)) (↑LinearMap.det (comp f ↑e'))",
"tactic": "refine' LinearMap.associated_det_of_eq_comp (e.trans e'.symm) _ _ _"
},
{
"state_after": "R : Type u_2\ninst✝⁸ : CommRing R\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type ?u.2243677\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type ?u.2244219\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne✝ : Basis ι R M\nN : Type u_1\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\ne e' : M ≃ₗ[R] N\nx : M\n⊢ ↑(comp f ↑e) x = ↑(comp f ↑e') (↑(LinearEquiv.trans e (LinearEquiv.symm e')) x)",
"state_before": "R : Type u_2\ninst✝⁸ : CommRing R\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type ?u.2243677\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type ?u.2244219\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne✝ : Basis ι R M\nN : Type u_1\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\ne e' : M ≃ₗ[R] N\n⊢ ∀ (x : M), ↑(comp f ↑e) x = ↑(comp f ↑e') (↑(LinearEquiv.trans e (LinearEquiv.symm e')) x)",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁸ : CommRing R\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type ?u.2243677\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι : Type ?u.2244219\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\ne✝ : Basis ι R M\nN : Type u_1\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\ne e' : M ≃ₗ[R] N\nx : M\n⊢ ↑(comp f ↑e) x = ↑(comp f ↑e') (↑(LinearEquiv.trans e (LinearEquiv.symm e')) x)",
"tactic": "simp only [LinearMap.comp_apply, LinearEquiv.coe_coe, LinearEquiv.trans_apply,\n LinearEquiv.apply_symm_apply]"
}
] | [
505,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
499,
1
] |
Mathlib/Topology/Order/Basic.lean | Ioc_mem_nhdsWithin_Ioi | [] | [
409,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
] |
Mathlib/CategoryTheory/Abelian/Homology.lean | homology.map_eq_lift_desc'_left | [
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ f ≫ α.right ≫ cokernel.π f' = 0",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0",
"tactic": "simp only [kernel.lift_ι_assoc, ← h]"
},
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ f ≫ α.right ≫ cokernel.π f' = 0",
"tactic": "erw [← reassoc_of% α.w]"
},
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0",
"tactic": "simp"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ π' f g w ≫\n desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n π' f g w ≫ 0",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f') (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n 0",
"tactic": "apply homology.hom_from_ext"
},
{
"state_after": "no goals",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ π' f g w ≫\n desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n π' f g w ≫ 0",
"tactic": "simp"
},
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0) =\n lift f' g' w'\n (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))\n (_ :\n desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n 0)",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ map w w' α β h =\n lift f' g' w'\n (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))\n (_ :\n desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n 0)",
"tactic": "rw [map_eq_desc'_lift_left]"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0) ≫\n ι f' g' w' =\n lift f' g' w'\n (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))\n (_ :\n desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n 0) ≫\n ι f' g' w'",
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},
{
"state_after": "case h.h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0) ≫\n ι f' g' w' =\n π' f g w ≫\n lift f' g' w'\n (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))\n (_ :\n desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫\n cokernel.desc f' g' w' =\n 0) ≫\n ι f' g' w'",
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"tactic": "apply homology.hom_from_ext"
},
{
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}
] | [
276,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
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Mathlib/Algebra/ContinuedFractions/Translations.lean | GeneralizedContinuedFraction.zeroth_continuant_aux_eq_one_zero | [] | [
123,
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122,
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Mathlib/CategoryTheory/Monoidal/Opposite.lean | CategoryTheory.mop_tensorUnit | [] | [
215,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
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Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.Cofork.app_zero_eq_comp_π_right | [
{
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] | [
356,
61
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Mathlib/Data/Set/UnionLift.lean | Set.preimage_iUnionLift | [
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{
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},
{
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"tactic": "constructor"
},
{
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"tactic": "rcases mem_iUnion.1 (hT x.prop) with ⟨i, hi⟩"
},
{
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"tactic": "refine fun h => ⟨i, ⟨x, hi⟩, ?_, rfl⟩"
},
{
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"tactic": "rwa [iUnionLift_of_mem x hi] at h"
},
{
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"tactic": "rintro ⟨i, ⟨y, hi⟩, h, hxy⟩"
},
{
"state_after": "case h.mpr.intro.intro.mk.intro\nα : Type u_2\nι : Sort u_3\nβ : Type u_1\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nt : Set β\nx : ↑T\ni : ι\nhi : ↑x ∈ S i\nh : f i { val := ↑x, property := hi } ∈ t\nhxy : inclusion (_ : S i ⊆ ⋃ (i : ι), S i) { val := ↑x, property := hi } = inclusion hT x\n⊢ iUnionLift S f hf T hT x ∈ t",
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"tactic": "obtain rfl : y = x := congr_arg Subtype.val hxy"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.mk.intro\nα : Type u_2\nι : Sort u_3\nβ : Type u_1\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nt : Set β\nx : ↑T\ni : ι\nhi : ↑x ∈ S i\nh : f i { val := ↑x, property := hi } ∈ t\nhxy : inclusion (_ : S i ⊆ ⋃ (i : ι), S i) { val := ↑x, property := hi } = inclusion hT x\n⊢ iUnionLift S f hf T hT x ∈ t",
"tactic": "rwa [iUnionLift_of_mem x hi]"
}
] | [
93,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.coe_iSup | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.69615\nγ : Type ?u.69618\nι : Sort u_2\nκ : ι → Sort ?u.69626\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\nf : ι → LowerSet α\n⊢ ↑(⨆ (i : ι), f i) = ⋃ (i : ι), ↑(f i)",
"tactic": "simp_rw [iSup, coe_sSup, mem_range, iUnion_exists, iUnion_iUnion_eq']"
}
] | [
703,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
702,
1
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.symm_symm | [] | [
269,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.cast_addNat_left | [
{
"state_after": "case h\nn✝ m✝ n n' m : ℕ\ni : Fin n'\nh : n' + m = n + m\n⊢ ↑(↑(cast h) (↑(addNat m) i)) = ↑(↑(addNat m) (↑(cast (_ : n' = n)) i))",
"state_before": "n✝ m✝ n n' m : ℕ\ni : Fin n'\nh : n' + m = n + m\n⊢ ↑(cast h) (↑(addNat m) i) = ↑(addNat m) (↑(cast (_ : n' = n)) i)",
"tactic": "ext"
},
{
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"state_before": "case h\nn✝ m✝ n n' m : ℕ\ni : Fin n'\nh : n' + m = n + m\n⊢ ↑(↑(cast h) (↑(addNat m) i)) = ↑(↑(addNat m) (↑(cast (_ : n' = n)) i))",
"tactic": "simp"
}
] | [
1381,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1378,
1
] |
Mathlib/RingTheory/IntegralClosure.lean | RingHom.is_integral_mul | [] | [
524,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.eventually_le_atBot | [] | [
190,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | ENNReal.rpow_neg | [
{
"state_after": "case none\ny : ℝ\n⊢ none ^ (-y) = (none ^ y)⁻¹\n\ncase some\ny : ℝ\nx : ℝ≥0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"state_before": "x : ℝ≥0∞\ny : ℝ\n⊢ x ^ (-y) = (x ^ y)⁻¹",
"tactic": "cases' x with x"
},
{
"state_after": "no goals",
"state_before": "case none\ny : ℝ\n⊢ none ^ (-y) = (none ^ y)⁻¹",
"tactic": "rcases lt_trichotomy y 0 with (H | H | H) <;>\n simp [top_rpow_of_pos, top_rpow_of_neg, H, neg_pos.mpr]"
},
{
"state_after": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹\n\ncase neg\ny : ℝ\nx : ℝ≥0\nh : ¬x = 0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"state_before": "case some\ny : ℝ\nx : ℝ≥0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"tactic": "by_cases h : x = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"tactic": "rcases lt_trichotomy y 0 with (H | H | H) <;>\n simp [h, zero_rpow_of_pos, zero_rpow_of_neg, H, neg_pos.mpr]"
},
{
"state_after": "case neg\ny : ℝ\nx : ℝ≥0\nh : ¬x = 0\nA : x ^ y ≠ 0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"state_before": "case neg\ny : ℝ\nx : ℝ≥0\nh : ¬x = 0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"tactic": "have A : x ^ y ≠ 0 := by simp [h]"
},
{
"state_after": "no goals",
"state_before": "case neg\ny : ℝ\nx : ℝ≥0\nh : ¬x = 0\nA : x ^ y ≠ 0\n⊢ Option.some x ^ (-y) = (Option.some x ^ y)⁻¹",
"tactic": "simp [coe_rpow_of_ne_zero h, ← coe_inv A, NNReal.rpow_neg]"
},
{
"state_after": "no goals",
"state_before": "y : ℝ\nx : ℝ≥0\nh : ¬x = 0\n⊢ x ^ y ≠ 0",
"tactic": "simp [h]"
}
] | [
445,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
437,
1
] |
Mathlib/Order/Basic.lean | min_rec' | [] | [
949,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
948,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean | HasDerivAtFilter.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\nhf : HasDerivAtFilter f f' x L\nhg : HasDerivAtFilter g g' x L\n⊢ HasDerivAtFilter (fun x => f x - g x) (f' - g') x L",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] | [
296,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | QuadraticForm.zero_apply | [] | [
408,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
] |
Mathlib/Logic/Equiv/Fin.lean | finCongr_symm_apply_coe | [] | [
126,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/LinearAlgebra/Basis.lean | Basis.constr_apply | [
{
"state_after": "ι : Type u_4\nι' : Type ?u.502339\nR : Type u_3\nR₂ : Type ?u.502345\nK : Type ?u.502348\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.502357\nV : Type u\nV' : Type ?u.502362\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nS : Type u_5\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : ι → M'\nx : M\n⊢ (Finsupp.sum (Finsupp.mapDomain f (↑↑b.repr x)) fun i a => a • id i) = Finsupp.sum (↑b.repr x) fun b a => a • f b",
"state_before": "ι : Type u_4\nι' : Type ?u.502339\nR : Type u_3\nR₂ : Type ?u.502345\nK : Type ?u.502348\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.502357\nV : Type u\nV' : Type ?u.502362\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nS : Type u_5\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : ι → M'\nx : M\n⊢ ↑(↑(constr b S) f) x = Finsupp.sum (↑b.repr x) fun b a => a • f b",
"tactic": "simp only [constr_def, LinearMap.comp_apply, Finsupp.lmapDomain_apply, Finsupp.total_apply]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_4\nι' : Type ?u.502339\nR : Type u_3\nR₂ : Type ?u.502345\nK : Type ?u.502348\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.502357\nV : Type u\nV' : Type ?u.502362\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nS : Type u_5\ninst✝² : Semiring S\ninst✝¹ : Module S M'\ninst✝ : SMulCommClass R S M'\nf : ι → M'\nx : M\n⊢ (Finsupp.sum (Finsupp.mapDomain f (↑↑b.repr x)) fun i a => a • id i) = Finsupp.sum (↑b.repr x) fun b a => a • f b",
"tactic": "rw [Finsupp.sum_mapDomain_index] <;> simp [add_smul]"
}
] | [
632,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
629,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean | exp_sum_of_commute | [
{
"state_after": "case empty\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s fun i j => Commute (f i) (f j)\nh : Set.Pairwise ↑∅ fun i j => Commute (f i) (f j)\n⊢ exp 𝕂 (∑ i in ∅, f i) =\n Finset.noncommProd ∅ (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑∅ → ∀ (j : ι), j ∈ ↑∅ → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\n\ncase insert\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns✝ : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s✝ fun i j => Commute (f i) (f j)\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n ∀ (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)),\n exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\nh : Set.Pairwise ↑(insert a s) fun i j => Commute (f i) (f j)\n⊢ exp 𝕂 (∑ i in insert a s, f i) =\n Finset.noncommProd (insert a s) (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑(insert a s) → ∀ (j : ι), j ∈ ↑(insert a s) → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))",
"state_before": "𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns : Finset ι\nf : ι → 𝔸\nh : Set.Pairwise ↑s fun i j => Commute (f i) (f j)\n⊢ exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))",
"tactic": "induction' s using Finset.induction_on with a s ha ih"
},
{
"state_after": "case insert\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns✝ : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s✝ fun i j => Commute (f i) (f j)\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n ∀ (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)),\n exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\nh : Set.Pairwise ↑(insert a s) fun i j => Commute (f i) (f j)\n⊢ Commute (f a) (∑ x in s, f x)",
"state_before": "case insert\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns✝ : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s✝ fun i j => Commute (f i) (f j)\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n ∀ (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)),\n exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\nh : Set.Pairwise ↑(insert a s) fun i j => Commute (f i) (f j)\n⊢ exp 𝕂 (∑ i in insert a s, f i) =\n Finset.noncommProd (insert a s) (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑(insert a s) → ∀ (j : ι), j ∈ ↑(insert a s) → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))",
"tactic": "rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,\n ih (h.mono <| Finset.subset_insert _ _)]"
},
{
"state_after": "case insert\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns✝ : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s✝ fun i j => Commute (f i) (f j)\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n ∀ (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)),\n exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\nh : Set.Pairwise ↑(insert a s) fun i j => Commute (f i) (f j)\ni : ι\nhi : i ∈ s\n⊢ Commute (f a) (f i)",
"state_before": "case insert\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns✝ : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s✝ fun i j => Commute (f i) (f j)\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n ∀ (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)),\n exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\nh : Set.Pairwise ↑(insert a s) fun i j => Commute (f i) (f j)\n⊢ Commute (f a) (∑ x in s, f x)",
"tactic": "refine' Commute.sum_right _ _ _ fun i hi => _"
},
{
"state_after": "no goals",
"state_before": "case insert\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns✝ : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s✝ fun i j => Commute (f i) (f j)\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n ∀ (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)),\n exp 𝕂 (∑ i in s, f i) =\n Finset.noncommProd s (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))\nh : Set.Pairwise ↑(insert a s) fun i j => Commute (f i) (f j)\ni : ι\nhi : i ∈ s\n⊢ Commute (f a) (f i)",
"tactic": "exact h.of_refl (Finset.mem_insert_self _ _) (Finset.mem_insert_of_mem hi)"
},
{
"state_after": "no goals",
"state_before": "case empty\n𝕂 : Type u_3\n𝔸 : Type u_2\n𝔹 : Type ?u.372921\ninst✝⁵ : IsROrC 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedRing 𝔹\ninst✝¹ : NormedAlgebra 𝕂 𝔹\ninst✝ : CompleteSpace 𝔸\nι : Type u_1\ns : Finset ι\nf : ι → 𝔸\nh✝ : Set.Pairwise ↑s fun i j => Commute (f i) (f j)\nh : Set.Pairwise ↑∅ fun i j => Commute (f i) (f j)\n⊢ exp 𝕂 (∑ i in ∅, f i) =\n Finset.noncommProd ∅ (fun i => exp 𝕂 (f i))\n (_ : ∀ (i : ι), i ∈ ↑∅ → ∀ (j : ι), j ∈ ↑∅ → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))",
"tactic": "simp"
}
] | [
509,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
499,
1
] |
Mathlib/RingTheory/FreeRing.lean | FreeRing.lift_comp_of | [] | [
93,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | QuadraticForm.sub_apply | [] | [
522,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | Subgroup.norm_coe | [] | [
2674,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2673,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.prod_filter_index | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.475516\nγ : Type ?u.475519\nι : Type ?u.475522\nM : Type u_3\nM' : Type ?u.475528\nN : Type u_1\nP : Type ?u.475534\nG : Type ?u.475537\nH : Type ?u.475540\nR : Type ?u.475543\nS : Type ?u.475546\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ prod (filter p f) g = ∏ x in (filter p f).support, g x (↑f x)",
"tactic": "classical\n refine' Finset.prod_congr rfl fun x hx => _\n rw [support_filter, Finset.mem_filter] at hx\n rw [filter_apply_pos _ _ hx.2]"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.475516\nγ : Type ?u.475519\nι : Type ?u.475522\nM : Type u_3\nM' : Type ?u.475528\nN : Type u_1\nP : Type ?u.475534\nG : Type ?u.475537\nH : Type ?u.475540\nR : Type ?u.475543\nS : Type ?u.475546\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\nx : α\nhx : x ∈ (filter p f).support\n⊢ g x (↑(filter p f) x) = g x (↑f x)",
"state_before": "α : Type u_2\nβ : Type ?u.475516\nγ : Type ?u.475519\nι : Type ?u.475522\nM : Type u_3\nM' : Type ?u.475528\nN : Type u_1\nP : Type ?u.475534\nG : Type ?u.475537\nH : Type ?u.475540\nR : Type ?u.475543\nS : Type ?u.475546\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ prod (filter p f) g = ∏ x in (filter p f).support, g x (↑f x)",
"tactic": "refine' Finset.prod_congr rfl fun x hx => _"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.475516\nγ : Type ?u.475519\nι : Type ?u.475522\nM : Type u_3\nM' : Type ?u.475528\nN : Type u_1\nP : Type ?u.475534\nG : Type ?u.475537\nH : Type ?u.475540\nR : Type ?u.475543\nS : Type ?u.475546\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\nx : α\nhx : x ∈ f.support ∧ p x\n⊢ g x (↑(filter p f) x) = g x (↑f x)",
"state_before": "α : Type u_2\nβ : Type ?u.475516\nγ : Type ?u.475519\nι : Type ?u.475522\nM : Type u_3\nM' : Type ?u.475528\nN : Type u_1\nP : Type ?u.475534\nG : Type ?u.475537\nH : Type ?u.475540\nR : Type ?u.475543\nS : Type ?u.475546\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\nx : α\nhx : x ∈ (filter p f).support\n⊢ g x (↑(filter p f) x) = g x (↑f x)",
"tactic": "rw [support_filter, Finset.mem_filter] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.475516\nγ : Type ?u.475519\nι : Type ?u.475522\nM : Type u_3\nM' : Type ?u.475528\nN : Type u_1\nP : Type ?u.475534\nG : Type ?u.475537\nH : Type ?u.475540\nR : Type ?u.475543\nS : Type ?u.475546\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\nx : α\nhx : x ∈ f.support ∧ p x\n⊢ g x (↑(filter p f) x) = g x (↑f x)",
"tactic": "rw [filter_apply_pos _ _ hx.2]"
}
] | [
947,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
942,
1
] |
Mathlib/CategoryTheory/Monoidal/Functor.lean | CategoryTheory.MonoidalFunctor.map_leftUnitor | [
{
"state_after": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n⊢ F.map (λ_ X).hom =\n inv (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫\n (inv F.ε ⊗ 𝟙 (F.obj X)) ≫\n (F.ε ⊗ 𝟙 (F.obj X)) ≫ LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X ≫ F.map (λ_ X).hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n⊢ F.map (λ_ X).hom =\n inv (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫ (inv F.ε ⊗ 𝟙 (F.obj X)) ≫ (λ_ (F.obj X)).hom",
"tactic": "simp only [LaxMonoidalFunctor.left_unitality]"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n⊢ F.map (λ_ X).hom =\n inv (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫\n (𝟙 (F.obj (𝟙_ C) ⊗ F.obj X) ≫ LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫ F.map (λ_ X).hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n⊢ F.map (λ_ X).hom =\n inv (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫\n (inv F.ε ⊗ 𝟙 (F.obj X)) ≫\n (F.ε ⊗ 𝟙 (F.obj X)) ≫ LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X ≫ F.map (λ_ X).hom",
"tactic": "slice_rhs 2 3 =>\n rw [← comp_tensor_id]\n simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n⊢ F.map (λ_ X).hom =\n inv (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫\n (𝟙 (F.obj (𝟙_ C) ⊗ F.obj X) ≫ LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ C) X) ≫ F.map (λ_ X).hom",
"tactic": "simp"
}
] | [
214,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Data/Finset/Slice.lean | Set.Sized.univ_mem_iff | [] | [
94,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.smul_eq_bot_iff | [] | [
989,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
988,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.coe_zsmul | [] | [
336,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
335,
1
] |
Mathlib/Computability/Primrec.lean | Primrec.fin_succ | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.360281\nβ : Type ?u.360284\nγ : Type ?u.360287\nσ : Type ?u.360290\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\n⊢ Primrec fun a => ↑(Fin.succ a)",
"tactic": "simp [succ.comp fin_val]"
}
] | [
1281,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1280,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean | Real.rpow_arith_mean_le_arith_mean_rpow | [] | [
102,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/Analysis/Calculus/Deriv/Linear.lean | ContinuousLinearMap.deriv | [] | [
70,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
11
] |
Mathlib/Data/Real/CauSeq.lean | CauSeq.cauchy | [] | [
160,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfpFamily_monotone | [] | [
87,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean | Pretrivialization.linearMapAt_eq_zero | [] | [
151,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | uniformSpace_eq | [
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.27706\nt₁ : TopologicalSpace α\nu₁ : UniformSpace.Core α\no₁ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nt₂ : TopologicalSpace α\no₂ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nh : uniformity α = uniformity α\n⊢ UniformSpace.mk' t₁ u₁ o₁ = UniformSpace.mk' t₂ u₁ o₂",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.27706\nt₁ : TopologicalSpace α\nu₁ : UniformSpace.Core α\no₁ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nt₂ : TopologicalSpace α\nu₂ : UniformSpace.Core α\no₂ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₂.uniformity\nh : uniformity α = uniformity α\n⊢ UniformSpace.mk' t₁ u₁ o₁ = UniformSpace.mk' t₂ u₂ o₂",
"tactic": "obtain rfl : u₁ = u₂ := UniformSpace.core_eq h"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.27706\nt₁ : TopologicalSpace α\nu₁ : UniformSpace.Core α\no₁ o₂ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nh : uniformity α = uniformity α\n⊢ UniformSpace.mk' t₁ u₁ o₁ = UniformSpace.mk' t₁ u₁ o₂",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.27706\nt₁ : TopologicalSpace α\nu₁ : UniformSpace.Core α\no₁ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nt₂ : TopologicalSpace α\no₂ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nh : uniformity α = uniformity α\n⊢ UniformSpace.mk' t₁ u₁ o₁ = UniformSpace.mk' t₂ u₁ o₂",
"tactic": "obtain rfl : t₁ = t₂ := topologicalSpace_eq <| funext fun s => by rw [o₁, o₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.27706\nt₁ : TopologicalSpace α\nu₁ : UniformSpace.Core α\no₁ o₂ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nh : uniformity α = uniformity α\n⊢ UniformSpace.mk' t₁ u₁ o₁ = UniformSpace.mk' t₁ u₁ o₂",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.27706\nt₁ : TopologicalSpace α\nu₁ : UniformSpace.Core α\no₁ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nt₂ : TopologicalSpace α\no₂ : ∀ (s : Set α), IsOpen s ↔ ∀ (x : α), x ∈ s → {p | p.fst = x → p.snd ∈ s} ∈ u₁.uniformity\nh : uniformity α = uniformity α\ns : Set α\n⊢ IsOpen s = IsOpen s",
"tactic": "rw [o₁, o₂]"
}
] | [
369,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
] |
Mathlib/Topology/PathConnected.lean | JoinedIn.joined | [] | [
859,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
858,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoMod_apply_right | [
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a < a + p ∧ ∃ z, a + p = a + z • p",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ toIcoMod hp a (a + p) = a",
"tactic": "rw [toIcoMod_eq_iff hp, Set.left_mem_Ico]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a < a + p ∧ ∃ z, a + p = a + z • p",
"tactic": "exact ⟨lt_add_of_pos_right _ hp, 1, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a + p = a + 1 • p",
"tactic": "simp"
}
] | [
226,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean | quotient_nhd_basis | [
{
"state_after": "case this\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\n⊢ ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε}\n\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nthis : ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε}\n⊢ Filter.HasBasis (𝓝 0) (fun ε => 0 < ε) fun ε => {x | ‖x‖ < ε}",
"state_before": "M : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\n⊢ Filter.HasBasis (𝓝 0) (fun ε => 0 < ε) fun ε => {x | ‖x‖ < ε}",
"tactic": "have : ∀ ε : ℝ, mk '' ball (0 : M) ε = { x : M ⧸ S | ‖x‖ < ε }"
},
{
"state_after": "M : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nthis : ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε}\n⊢ Filter.HasBasis (Filter.map mk (𝓝 0)) (fun ε => 0 < ε) fun x => mk '' ball 0 x",
"state_before": "M : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nthis : ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε}\n⊢ Filter.HasBasis (𝓝 0) (fun ε => 0 < ε) fun ε => {x | ‖x‖ < ε}",
"tactic": "rw [← mk_zero, nhds_eq, ← funext this]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nthis : ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε}\n⊢ Filter.HasBasis (Filter.map mk (𝓝 0)) (fun ε => 0 < ε) fun x => mk '' ball 0 x",
"tactic": "exact .map _ Metric.nhds_basis_ball"
},
{
"state_after": "case this\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nε : ℝ\nx : M\n⊢ ↑x ∈ mk '' ball 0 ε ↔ ↑x ∈ {x | ‖x‖ < ε}",
"state_before": "case this\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\n⊢ ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε}",
"tactic": "refine fun ε ↦ Set.ext <| forall_mk.2 fun x ↦ ?_"
},
{
"state_after": "case this\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nε : ℝ\nx : M\n⊢ (∃ x_1, x_1 ∈ {x | ‖x‖ < ε} ∧ ↑x_1 = ↑x) ↔ ∃ m, ↑m = ↑x ∧ ‖m‖ < ε",
"state_before": "case this\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nε : ℝ\nx : M\n⊢ ↑x ∈ mk '' ball 0 ε ↔ ↑x ∈ {x | ‖x‖ < ε}",
"tactic": "rw [ball_zero_eq, mem_setOf_eq, norm_lt_iff, mem_image]"
},
{
"state_after": "no goals",
"state_before": "case this\nM : Type u_1\nN : Type ?u.386793\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nε : ℝ\nx : M\n⊢ (∃ x_1, x_1 ∈ {x | ‖x‖ < ε} ∧ ↑x_1 = ↑x) ↔ ∃ m, ↑m = ↑x ∧ ‖m‖ < ε",
"tactic": "exact exists_congr fun _ ↦ and_comm"
}
] | [
239,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
1
] |
Mathlib/Data/Nat/Order/Basic.lean | Nat.mul_self_le_mul_self_iff | [] | [
294,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Data/Real/NNReal.lean | Real.toNNReal_lt_toNNReal_iff | [] | [
646,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
644,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.continuousOn | [] | [
102,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
11
] |
Mathlib/GroupTheory/Perm/Sign.lean | Equiv.Perm.Disjoint.orderOf | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nσ τ : Perm α\nhστ : Disjoint σ τ\nn : ℕ\n⊢ (σ * τ) ^ n = 1 ↔ σ ^ n = 1 ∧ τ ^ n = 1",
"tactic": "rw [hστ.commute.mul_pow, Disjoint.mul_eq_one_iff (hστ.pow_disjoint_pow n n)]"
}
] | [
180,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
8
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.card_roots_of_disc_ne_zero | [
{
"state_after": "R : Type ?u.1301913\nS : Type ?u.1301916\nF : Type u_2\nK : Type u_1\nP : Cubic F\ninst✝² : Field F\ninst✝¹ : Field K\nφ : F →+* K\nx y z : K\ninst✝ : DecidableEq K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\nhd : disc P ≠ 0\n⊢ ∃ x y z, roots (map φ P) = {x, y, z}",
"state_before": "R : Type ?u.1301913\nS : Type ?u.1301916\nF : Type u_2\nK : Type u_1\nP : Cubic F\ninst✝² : Field F\ninst✝¹ : Field K\nφ : F →+* K\nx y z : K\ninst✝ : DecidableEq K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\nhd : disc P ≠ 0\n⊢ Finset.card (toFinset (roots (map φ P))) = 3",
"tactic": "rw [toFinset_card_of_nodup <| (disc_ne_zero_iff_roots_nodup ha h3).mp hd,\n ← splits_iff_card_roots ha, splits_iff_roots_eq_three ha]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1301913\nS : Type ?u.1301916\nF : Type u_2\nK : Type u_1\nP : Cubic F\ninst✝² : Field F\ninst✝¹ : Field K\nφ : F →+* K\nx y z : K\ninst✝ : DecidableEq K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\nhd : disc P ≠ 0\n⊢ ∃ x y z, roots (map φ P) = {x, y, z}",
"tactic": "exact ⟨x, ⟨y, ⟨z, h3⟩⟩⟩"
}
] | [
601,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
597,
1
] |
Mathlib/Data/Set/Basic.lean | Set.union_eq_left_iff_subset | [] | [
800,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
799,
1
] |
Mathlib/Data/Polynomial/Basic.lean | Polynomial.erase_ne | [
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn i : ℕ\nh : i ≠ n\n⊢ coeff (erase n p) i = coeff p i",
"tactic": "simp [coeff_erase, h]"
}
] | [
1072,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1071,
1
] |
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