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Mathlib/Order/Hom/Basic.lean | OrderIso.withBotCongr_symm | [] | [
1353,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1352,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean | StrictAntiOn.inv | [] | [
1336,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1335,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean | CategoryTheory.Preadditive.sum_comp | [] | [
194,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.Theory.isSatisfiable_directed_union_iff | [
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\n⊢ IsSatisfiable (⋃ (i : ι), T i)",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\n⊢ IsSatisfiable (⋃ (i : ι), T i) ↔ ∀ (i : ι), IsSatisfiable (T i)",
"tactic": "refine' ⟨fun h' i => h'.mono (Set.subset_iUnion _ _), fun h' => _⟩"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\n⊢ ∀ (T0 : Finset (Sentence L)), (↑T0 ⊆ ⋃ (i : ι), T i) → IsSatisfiable ↑T0",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\n⊢ IsSatisfiable (⋃ (i : ι), T i)",
"tactic": "rw [isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\nT0 : Finset (Sentence L)\nhT0 : ↑T0 ⊆ ⋃ (i : ι), T i\n⊢ IsSatisfiable ↑T0",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\n⊢ ∀ (T0 : Finset (Sentence L)), (↑T0 ⊆ ⋃ (i : ι), T i) → IsSatisfiable ↑T0",
"tactic": "intro T0 hT0"
},
{
"state_after": "case intro\nL : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\nT0 : Finset (Sentence L)\nhT0 : ↑T0 ⊆ ⋃ (i : ι), T i\ni : ι\nhi : ↑T0 ⊆ T i\n⊢ IsSatisfiable ↑T0",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\nT0 : Finset (Sentence L)\nhT0 : ↑T0 ⊆ ⋃ (i : ι), T i\n⊢ IsSatisfiable ↑T0",
"tactic": "obtain ⟨i, hi⟩ := h.exists_mem_subset_of_finset_subset_biUnion hT0"
},
{
"state_after": "no goals",
"state_before": "case intro\nL : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' : Theory L\nι : Type u_1\ninst✝ : Nonempty ι\nT : ι → Theory L\nh : Directed (fun x x_1 => x ⊆ x_1) T\nh' : ∀ (i : ι), IsSatisfiable (T i)\nT0 : Finset (Sentence L)\nhT0 : ↑T0 ⊆ ⋃ (i : ι), T i\ni : ι\nhi : ↑T0 ⊆ T i\n⊢ IsSatisfiable ↑T0",
"tactic": "exact (h' i).mono hi"
}
] | [
138,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
132,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean | LocalRing.ResidueField.map_id_apply | [] | [
440,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
1
] |
Mathlib/CategoryTheory/Opposites.lean | CategoryTheory.op_id | [] | [
86,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.coe_castPred_le_self | [
{
"state_after": "case inl\nn m : ℕ\n⊢ ↑(castPred (last (n + 1))) ≤ ↑(last (n + 1))\n\ncase inr\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑(castPred i) ≤ ↑i",
"state_before": "n m : ℕ\ni : Fin (n + 2)\n⊢ ↑(castPred i) ≤ ↑i",
"tactic": "rcases i.le_last.eq_or_lt with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nn m : ℕ\n⊢ ↑(castPred (last (n + 1))) ≤ ↑(last (n + 1))",
"tactic": "simp"
},
{
"state_after": "case inr\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑(castLT i (_ : ↑i < n + 1)) ≤ ↑i\n\ncase inr.hnc\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ¬↑castSucc (last n) < i",
"state_before": "case inr\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑(castPred i) ≤ ↑i",
"tactic": "rw [castPred, predAbove, dif_neg]"
},
{
"state_after": "no goals",
"state_before": "case inr\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ↑(castLT i (_ : ↑i < n + 1)) ≤ ↑i",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.hnc\nn m : ℕ\ni : Fin (n + 2)\nh : i < last (n + 1)\n⊢ ¬↑castSucc (last n) < i",
"tactic": "simpa [lt_iff_val_lt_val, le_iff_val_le_val, lt_succ_iff] using h"
}
] | [
2467,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2462,
1
] |
Mathlib/Algebra/Quaternion.lean | QuaternionAlgebra.coe_mul_eq_smul | [] | [
599,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
598,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | inner_self_eq_zero | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1685745\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : E\n⊢ inner x x = 0 ↔ x = 0",
"tactic": "rw [inner_self_eq_norm_sq_to_K, sq_eq_zero_iff, ofReal_eq_zero, norm_eq_zero]"
}
] | [
614,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean | Localization.mk_eq_monoidOf_mk' | [] | [
1655,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1654,
1
] |
Mathlib/CategoryTheory/Preadditive/Mat.lean | CategoryTheory.Mat_.hom_ext | [] | [
124,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Data/Nat/Totient.lean | Nat.totient_mul_of_prime_of_not_dvd | [
{
"state_after": "p n : ℕ\nhp : Prime p\nh : ¬p ∣ n\n⊢ coprime p n",
"state_before": "p n : ℕ\nhp : Prime p\nh : ¬p ∣ n\n⊢ φ (p * n) = (p - 1) * φ n",
"tactic": "rw [totient_mul _, totient_prime hp]"
},
{
"state_after": "no goals",
"state_before": "p n : ℕ\nhp : Prime p\nh : ¬p ∣ n\n⊢ coprime p n",
"tactic": "simpa [h] using coprime_or_dvd_of_prime hp n"
}
] | [
386,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
383,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.Sized.pos | [
{
"state_after": "α : Type u_1\ns : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\nh : Sized (node s l x r)\n⊢ 0 < size l + size r + 1",
"state_before": "α : Type u_1\ns : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\nh : Sized (node s l x r)\n⊢ 0 < s",
"tactic": "rw [h.1]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\nh : Sized (node s l x r)\n⊢ 0 < size l + size r + 1",
"tactic": "apply Nat.le_add_left"
}
] | [
147,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.map_singletonSubgraph | [
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.208301\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv : V\nx✝¹ x✝ : W\n⊢ ⊥ → ∃ x, ↑f x = x✝¹ ∧ ∃ x, ↑f x = x✝",
"state_before": "ι : Sort ?u.208301\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv : V\n⊢ Subgraph.map f (SimpleGraph.singletonSubgraph G v) = SimpleGraph.singletonSubgraph G' (↑f v)",
"tactic": "ext <;> simp only [Relation.Map, Subgraph.map_Adj, singletonSubgraph_Adj, Pi.bot_apply,\n exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts,\n singletonSubgraph_verts, Set.image_singleton]"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nι : Sort ?u.208301\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv : V\nx✝¹ x✝ : W\n⊢ ⊥ → ∃ x, ↑f x = x✝¹ ∧ ∃ x, ↑f x = x✝",
"tactic": "exact False.elim"
}
] | [
867,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
862,
1
] |
Mathlib/Topology/Instances/Rat.lean | Int.closedEmbedding_coe_rat | [
{
"state_after": "no goals",
"state_before": "⊢ Pairwise fun x y => 1 ≤ dist ↑x ↑y",
"tactic": "simpa using Int.pairwise_one_le_dist"
}
] | [
84,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Data/Fin/Interval.lean | Fin.card_fintypeIoc | [
{
"state_after": "no goals",
"state_before": "n : ℕ\na b : Fin n\n⊢ Fintype.card ↑(Set.Ioc a b) = ↑b - ↑a",
"tactic": "rw [← card_Ioc, Fintype.card_ofFinset]"
}
] | [
113,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.aleph_le | [] | [
257,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.Integrable.integrableAtFilter | [] | [
398,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
] |
Mathlib/Data/List/NatAntidiagonal.lean | List.Nat.antidiagonal_succ | [
{
"state_after": "n : ℕ\n⊢ (0, n + 1) :: (succ 0, n + 1 - succ 0) :: map ((fun i => (i, n + 1 - i)) ∘ succ ∘ succ) (range n) =\n (0, n + 1) :: (succ 0, n) :: map (Prod.map succ id ∘ (fun i => (i, n - i)) ∘ succ) (range n)",
"state_before": "n : ℕ\n⊢ antidiagonal (n + 1) = (0, n + 1) :: map (Prod.map succ id) (antidiagonal n)",
"tactic": "simp only [antidiagonal, range_succ_eq_map, map_cons, true_and_iff, Nat.add_succ_sub_one,\n add_zero, id.def, eq_self_iff_true, tsub_zero, map_map, Prod.map_mk]"
},
{
"state_after": "n : ℕ\n⊢ map ((fun i => (i, n + 1 - i)) ∘ succ ∘ succ) (range n) =\n map (Prod.map succ id ∘ (fun i => (i, n - i)) ∘ succ) (range n)",
"state_before": "n : ℕ\n⊢ (0, n + 1) :: (succ 0, n + 1 - succ 0) :: map ((fun i => (i, n + 1 - i)) ∘ succ ∘ succ) (range n) =\n (0, n + 1) :: (succ 0, n) :: map (Prod.map succ id ∘ (fun i => (i, n - i)) ∘ succ) (range n)",
"tactic": "apply congr rfl (congr rfl _)"
},
{
"state_after": "case a.a\nn n✝ : ℕ\na✝ : ℕ × ℕ\n⊢ a✝ ∈ get? (map ((fun i => (i, n + 1 - i)) ∘ succ ∘ succ) (range n)) n✝ ↔\n a✝ ∈ get? (map (Prod.map succ id ∘ (fun i => (i, n - i)) ∘ succ) (range n)) n✝",
"state_before": "n : ℕ\n⊢ map ((fun i => (i, n + 1 - i)) ∘ succ ∘ succ) (range n) =\n map (Prod.map succ id ∘ (fun i => (i, n - i)) ∘ succ) (range n)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.a\nn n✝ : ℕ\na✝ : ℕ × ℕ\n⊢ a✝ ∈ get? (map ((fun i => (i, n + 1 - i)) ∘ succ ∘ succ) (range n)) n✝ ↔\n a✝ ∈ get? (map (Prod.map succ id ∘ (fun i => (i, n - i)) ∘ succ) (range n)) n✝",
"tactic": "simp"
}
] | [
76,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.image_sub_const_Ici | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) '' Ici b = Ici (b - a)",
"tactic": "simp [sub_eq_neg_add]"
}
] | [
374,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
374,
1
] |
Mathlib/Data/List/Func.lean | List.Func.forall_val_of_forall_mem | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\n⊢ p (get n as)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\n⊢ p default → (∀ (x : α), x ∈ as → p x) → ∀ (n : ℕ), p (get n as)",
"tactic": "intro h1 h2 n"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : n < length as\n⊢ p (get n as)\n\ncase neg\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : ¬n < length as\n⊢ p (get n as)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\n⊢ p (get n as)",
"tactic": "by_cases h3 : n < as.length"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : n < length as\n⊢ p (get n as)",
"tactic": "apply h2 _ (mem_get_of_le h3)"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : length as ≤ n\n⊢ p (get n as)",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : ¬n < length as\n⊢ p (get n as)",
"tactic": "rw [not_lt] at h3"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : length as ≤ n\n⊢ p default",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : length as ≤ n\n⊢ p (get n as)",
"tactic": "rw [get_eq_default_of_le _ h3]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nas : List α\np : α → Prop\nh1 : p default\nh2 : ∀ (x : α), x ∈ as → p x\nn : ℕ\nh3 : length as ≤ n\n⊢ p default",
"tactic": "apply h1"
}
] | [
225,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.sin_nat_mul_two_pi_sub | [] | [
1193,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1192,
1
] |
Mathlib/Topology/LocalExtr.lean | IsLocalExtr.congr | [] | [
625,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
623,
8
] |
Mathlib/Algebra/Lie/Basic.lean | LieEquiv.symm_trans | [] | [
658,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
656,
1
] |
Mathlib/Algebra/Parity.lean | Even.isSquare_pow | [
{
"state_after": "case intro\nF : Type ?u.14072\nα : Type u_1\nβ : Type ?u.14078\nR : Type ?u.14081\ninst✝ : Monoid α\na✝ : α\nn : ℕ\na : α\n⊢ IsSquare (a ^ (n + n))",
"state_before": "F : Type ?u.14072\nα : Type u_1\nβ : Type ?u.14078\nR : Type ?u.14081\ninst✝ : Monoid α\nn : ℕ\na : α\n⊢ Even n → ∀ (a : α), IsSquare (a ^ n)",
"tactic": "rintro ⟨n, rfl⟩ a"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type ?u.14072\nα : Type u_1\nβ : Type ?u.14078\nR : Type ?u.14081\ninst✝ : Monoid α\na✝ : α\nn : ℕ\na : α\n⊢ IsSquare (a ^ (n + n))",
"tactic": "exact ⟨a ^ n, pow_add _ _ _⟩"
}
] | [
119,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | edist_triangle_left | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.4281\ninst✝ : PseudoEMetricSpace α\nx y z : α\n⊢ edist x y ≤ edist x z + edist z y",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.4281\ninst✝ : PseudoEMetricSpace α\nx y z : α\n⊢ edist x y ≤ edist z x + edist z y",
"tactic": "rw [edist_comm z]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.4281\ninst✝ : PseudoEMetricSpace α\nx y z : α\n⊢ edist x y ≤ edist x z + edist z y",
"tactic": "apply edist_triangle"
}
] | [
101,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/RingTheory/Localization/Integral.lean | IsFractionRing.isAlgebraic_iff | [
{
"state_after": "case mp.intro.intro\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\n⊢ IsAlgebraic K x\n\ncase mpr.intro.intro\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : K[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\n⊢ IsAlgebraic A x",
"state_before": "R : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\n⊢ IsAlgebraic A x ↔ IsAlgebraic K x",
"tactic": "constructor <;> rintro ⟨p, hp, px⟩"
},
{
"state_after": "case mp.intro.intro.refine'_1\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : ℕ\n⊢ coeff p i = coeff 0 i\n\ncase mp.intro.intro.refine'_2\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\n⊢ ↑(aeval x) (Polynomial.map (algebraMap A K) p) = 0",
"state_before": "case mp.intro.intro\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\n⊢ IsAlgebraic K x",
"tactic": "refine' ⟨p.map (algebraMap A K), fun h => hp (Polynomial.ext fun i => _), _⟩"
},
{
"state_after": "case mp.intro.intro.refine'_1\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : ℕ\nthis : ↑(algebraMap A K) (coeff p i) = 0\n⊢ coeff p i = coeff 0 i",
"state_before": "case mp.intro.intro.refine'_1\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : ℕ\n⊢ coeff p i = coeff 0 i",
"tactic": "have : algebraMap A K (p.coeff i) = 0 :=\n _root_.trans (Polynomial.coeff_map _ _).symm (by simp [h])"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refine'_1\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : ℕ\nthis : ↑(algebraMap A K) (coeff p i) = 0\n⊢ coeff p i = coeff 0 i",
"tactic": "exact to_map_eq_zero_iff.mp this"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : ℕ\n⊢ coeff (Polynomial.map (algebraMap A K) p) i = 0",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refine'_2\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : A[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\n⊢ ↑(aeval x) (Polynomial.map (algebraMap A K) p) = 0",
"tactic": "exact (Polynomial.aeval_map_algebraMap K _ _).trans px"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nR : Type ?u.95350\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type ?u.95557\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.95813\ninst✝⁹ : CommRing P\nA : Type u_1\nK : Type u_3\nC : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsFractionRing A K\ninst✝³ : CommRing C\ninst✝² : Algebra A C\ninst✝¹ : Algebra K C\ninst✝ : IsScalarTower A K C\nx : C\np : K[X]\nhp : p ≠ 0\npx : ↑(aeval x) p = 0\n⊢ IsAlgebraic A x",
"tactic": "exact\n ⟨integerNormalization _ p, mt integerNormalization_eq_zero_iff.mp hp,\n integerNormalization_aeval_eq_zero _ p px⟩"
}
] | [
165,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Data/Set/Basic.lean | Set.subset_union_compl_iff_inter_subset | [] | [
1767,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1766,
1
] |
Mathlib/Data/Finset/Sort.lean | Finset.sort_perm_toList | [
{
"state_after": "α : Type u_1\nβ : Type ?u.3947\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : IsAntisymm α r\ninst✝ : IsTotal α r\ns : Finset α\n⊢ ↑(sort r s) = ↑(toList s)",
"state_before": "α : Type u_1\nβ : Type ?u.3947\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : IsAntisymm α r\ninst✝ : IsTotal α r\ns : Finset α\n⊢ sort r s ~ toList s",
"tactic": "rw [← Multiset.coe_eq_coe]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3947\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : IsAntisymm α r\ninst✝ : IsTotal α r\ns : Finset α\n⊢ ↑(sort r s) = ↑(toList s)",
"tactic": "simp only [coe_toList, sort_eq]"
}
] | [
82,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/Analysis/Complex/Basic.lean | Complex.dist_conj_comm | [
{
"state_after": "no goals",
"state_before": "E : Type ?u.548925\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz w : ℂ\n⊢ dist (↑(starRingEnd ℂ) z) w = dist z (↑(starRingEnd ℂ) w)",
"tactic": "rw [← dist_conj_conj, conj_conj]"
}
] | [
340,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | fderiv_const_add | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.289578\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.289673\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\n⊢ fderiv 𝕜 (fun y => c + f y) x = fderiv 𝕜 f x",
"tactic": "simp only [add_comm c, fderiv_add_const]"
}
] | [
320,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
] |
Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.zsmul_mem | [] | [
244,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
243,
11
] |
Mathlib/GroupTheory/Submonoid/Membership.lean | Submonoid.mem_sup_right | [
{
"state_after": "M : Type u_1\nA : Type ?u.44116\nB : Type ?u.44119\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ T ≤ S ⊔ T",
"state_before": "M : Type u_1\nA : Type ?u.44116\nB : Type ?u.44119\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"tactic": "rw [←SetLike.le_def]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nA : Type ?u.44116\nB : Type ?u.44119\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ T ≤ S ⊔ T",
"tactic": "exact le_sup_right"
}
] | [
246,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
244,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean | Subgroup.Normal.conjAct | [] | [
390,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
386,
1
] |
Mathlib/FieldTheory/Separable.lean | Polynomial.separable_of_subsingleton | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Subsingleton R\nf : R[X]\n⊢ Separable f",
"tactic": "simp [Separable, IsCoprime]"
}
] | [
65,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
64,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | EMetric.countable_closure_of_compact | [
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.313712\ninst✝¹ : PseudoEMetricSpace α\nγ : Type w\ninst✝ : EMetricSpace γ\ns : Set γ\nhs : IsCompact s\nt : Set γ\nhts : t ⊆ s\nhtc : Set.Countable t\nhsub : s ⊆ closure t\n⊢ ∃ t, t ⊆ s ∧ Set.Countable t ∧ s = closure t",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.313712\ninst✝¹ : PseudoEMetricSpace α\nγ : Type w\ninst✝ : EMetricSpace γ\ns : Set γ\nhs : IsCompact s\n⊢ ∃ t, t ⊆ s ∧ Set.Countable t ∧ s = closure t",
"tactic": "rcases subset_countable_closure_of_compact hs with ⟨t, hts, htc, hsub⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.313712\ninst✝¹ : PseudoEMetricSpace α\nγ : Type w\ninst✝ : EMetricSpace γ\ns : Set γ\nhs : IsCompact s\nt : Set γ\nhts : t ⊆ s\nhtc : Set.Countable t\nhsub : s ⊆ closure t\n⊢ ∃ t, t ⊆ s ∧ Set.Countable t ∧ s = closure t",
"tactic": "exact ⟨t, hts, htc, hsub.antisymm (closure_minimal hts hs.isClosed)⟩"
}
] | [
1122,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1119,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiffWithinAt.fderivWithin'' | [
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n⊢ ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\n⊢ ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "have : ∀ k : ℕ, (k : ℕ∞) ≤ m →\n ContDiffWithinAt 𝕜 k (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := fun k hkm ↦ by\n obtain ⟨v, hv, -, f', hvf', hf'⟩ :=\n (hf.of_le <| (add_le_add_right hkm 1).trans hmn).hasFDerivWithinAt_nhds (hg.of_le hkm) hgt\n refine hf'.congr_of_eventuallyEq_insert ?_\n filter_upwards [hv, ht]\n exact fun y hy h2y => (hvf' y hy).fderivWithin h2y"
},
{
"state_after": "case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg✝ : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn✝ : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhg : ContDiffWithinAt 𝕜 ⊤ g s x₀\nhmn : ⊤ + 1 ≤ n\nthis : ∀ (k : ℕ), ↑k ≤ ⊤ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n⊢ ContDiffWithinAt 𝕜 ⊤ (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n\ncase coe\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg✝ : ContDiffWithinAt 𝕜 m✝ g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn✝ : m✝ + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m✝ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nm : ℕ\nhg : ContDiffWithinAt 𝕜 (↑m) g s x₀\nhmn : ↑m + 1 ≤ n\nthis : ∀ (k : ℕ), ↑k ≤ ↑m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n⊢ ContDiffWithinAt 𝕜 (↑m) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n⊢ ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "induction' m using WithTop.recTopCoe with m"
},
{
"state_after": "no goals",
"state_before": "case coe\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg✝ : ContDiffWithinAt 𝕜 m✝ g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn✝ : m✝ + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m✝ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nm : ℕ\nhg : ContDiffWithinAt 𝕜 (↑m) g s x₀\nhmn : ↑m + 1 ≤ n\nthis : ∀ (k : ℕ), ↑k ≤ ↑m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n⊢ ContDiffWithinAt 𝕜 (↑m) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "exact this _ le_rfl"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\nv : Set E\nhv : v ∈ 𝓝[insert x₀ s] x₀\nf' : E → F →L[𝕜] G\nhvf' : ∀ (x : E), x ∈ v → HasFDerivWithinAt (f x) (f' x) t (g x)\nhf' : ContDiffWithinAt 𝕜 (↑k) (fun x => f' x) s x₀\n⊢ ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\n⊢ ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "obtain ⟨v, hv, -, f', hvf', hf'⟩ :=\n (hf.of_le <| (add_le_add_right hkm 1).trans hmn).hasFDerivWithinAt_nhds (hg.of_le hkm) hgt"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\nv : Set E\nhv : v ∈ 𝓝[insert x₀ s] x₀\nf' : E → F →L[𝕜] G\nhvf' : ∀ (x : E), x ∈ v → HasFDerivWithinAt (f x) (f' x) t (g x)\nhf' : ContDiffWithinAt 𝕜 (↑k) (fun x => f' x) s x₀\n⊢ (fun x => fderivWithin 𝕜 (f x) t (g x)) =ᶠ[𝓝[insert x₀ s] x₀] fun x => f' x",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\nv : Set E\nhv : v ∈ 𝓝[insert x₀ s] x₀\nf' : E → F →L[𝕜] G\nhvf' : ∀ (x : E), x ∈ v → HasFDerivWithinAt (f x) (f' x) t (g x)\nhf' : ContDiffWithinAt 𝕜 (↑k) (fun x => f' x) s x₀\n⊢ ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "refine hf'.congr_of_eventuallyEq_insert ?_"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\nv : Set E\nhv : v ∈ 𝓝[insert x₀ s] x₀\nf' : E → F →L[𝕜] G\nhvf' : ∀ (x : E), x ∈ v → HasFDerivWithinAt (f x) (f' x) t (g x)\nhf' : ContDiffWithinAt 𝕜 (↑k) (fun x => f' x) s x₀\n⊢ ∀ (a : E), a ∈ v → UniqueDiffWithinAt 𝕜 t (g a) → fderivWithin 𝕜 (f a) t (g a) = f' a",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\nv : Set E\nhv : v ∈ 𝓝[insert x₀ s] x₀\nf' : E → F →L[𝕜] G\nhvf' : ∀ (x : E), x ∈ v → HasFDerivWithinAt (f x) (f' x) t (g x)\nhf' : ContDiffWithinAt 𝕜 (↑k) (fun x => f' x) s x₀\n⊢ (fun x => fderivWithin 𝕜 (f x) t (g x)) =ᶠ[𝓝[insert x₀ s] x₀] fun x => f' x",
"tactic": "filter_upwards [hv, ht]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nk : ℕ\nhkm : ↑k ≤ m\nv : Set E\nhv : v ∈ 𝓝[insert x₀ s] x₀\nf' : E → F →L[𝕜] G\nhvf' : ∀ (x : E), x ∈ v → HasFDerivWithinAt (f x) (f' x) t (g x)\nhf' : ContDiffWithinAt 𝕜 (↑k) (fun x => f' x) s x₀\n⊢ ∀ (a : E), a ∈ v → UniqueDiffWithinAt 𝕜 t (g a) → fderivWithin 𝕜 (f a) t (g a) = f' a",
"tactic": "exact fun y hy h2y => (hvf' y hy).fderivWithin h2y"
},
{
"state_after": "case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nhg✝ : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhg : ContDiffWithinAt 𝕜 ⊤ g s x₀\nthis : ∀ (k : ℕ), ↑k ≤ ⊤ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhf : ContDiffWithinAt 𝕜 ⊤ (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhmn✝ : m + 1 ≤ ⊤\nhmn : ⊤ + 1 ≤ ⊤\n⊢ ContDiffWithinAt 𝕜 ⊤ (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"state_before": "case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nn : ℕ∞\nhf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhg✝ : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhmn✝ : m + 1 ≤ n\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhg : ContDiffWithinAt 𝕜 ⊤ g s x₀\nhmn : ⊤ + 1 ≤ n\nthis : ∀ (k : ℕ), ↑k ≤ ⊤ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\n⊢ ContDiffWithinAt 𝕜 ⊤ (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "obtain rfl := eq_top_iff.mpr hmn"
},
{
"state_after": "case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nhg✝ : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhg : ContDiffWithinAt 𝕜 ⊤ g s x₀\nthis : ∀ (k : ℕ), ↑k ≤ ⊤ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhf : ContDiffWithinAt 𝕜 ⊤ (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhmn✝ : m + 1 ≤ ⊤\nhmn : ⊤ + 1 ≤ ⊤\n⊢ ∀ (n : ℕ), ContDiffWithinAt 𝕜 (↑n) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"state_before": "case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nhg✝ : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhg : ContDiffWithinAt 𝕜 ⊤ g s x₀\nthis : ∀ (k : ℕ), ↑k ≤ ⊤ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhf : ContDiffWithinAt 𝕜 ⊤ (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhmn✝ : m + 1 ≤ ⊤\nhmn : ⊤ + 1 ≤ ⊤\n⊢ ContDiffWithinAt 𝕜 ⊤ (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "rw [contDiffWithinAt_top]"
},
{
"state_after": "no goals",
"state_before": "case top\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.1044172\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F → G\ng : E → F\nt : Set F\nhg✝ : ContDiffWithinAt 𝕜 m g s x₀\nht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)\nhgt : t ∈ 𝓝[g '' s] g x₀\nthis✝ : ∀ (k : ℕ), ↑k ≤ m → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhg : ContDiffWithinAt 𝕜 ⊤ g s x₀\nthis : ∀ (k : ℕ), ↑k ≤ ⊤ → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀\nhf : ContDiffWithinAt 𝕜 ⊤ (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)\nhmn✝ : m + 1 ≤ ⊤\nhmn : ⊤ + 1 ≤ ⊤\n⊢ ∀ (n : ℕ), ContDiffWithinAt 𝕜 (↑n) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀",
"tactic": "exact fun m => this m le_top"
}
] | [
997,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
980,
1
] |
Mathlib/Order/WithBot.lean | WithBot.toDual_map | [] | [
964,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
962,
1
] |
Mathlib/Algebra/Group/Basic.lean | inv_mul_eq_div | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.31497\nG : Type ?u.31500\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a⁻¹ * b = b / a",
"tactic": "simp"
}
] | [
510,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
510,
1
] |
Mathlib/RingTheory/Localization/FractionRing.lean | IsFractionRing.div_surjective | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.139727\ninst✝¹¹ : CommRing R\nM : Submonoid R\nS : Type ?u.139919\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nP : Type ?u.140156\ninst✝⁸ : CommRing P\nA : Type u_1\ninst✝⁷ : CommRing A\ninst✝⁶ : IsDomain A\nK : Type u_2\nB : Type ?u.140339\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDomain B\ninst✝³ : Field K\nL : Type ?u.140516\ninst✝² : Field L\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\ng : A →+* L\nz : K\nx y : A\nhy : y ∈ nonZeroDivisors A\nh : mk' K x { val := y, property := hy } = z\n⊢ ↑(algebraMap A K) x / ↑(algebraMap A K) y = z",
"tactic": "rwa [mk'_eq_div] at h"
}
] | [
171,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean | Unitization.snd_neg | [] | [
239,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.val_bit0 | [
{
"state_after": "case mk\nn✝ m n val✝ : ℕ\nisLt✝ : val✝ < n\n⊢ ↑(bit0 { val := val✝, isLt := isLt✝ }) = bit0 ↑{ val := val✝, isLt := isLt✝ } % n",
"state_before": "n✝ m n : ℕ\nk : Fin n\n⊢ ↑(bit0 k) = bit0 ↑k % n",
"tactic": "cases k"
},
{
"state_after": "no goals",
"state_before": "case mk\nn✝ m n val✝ : ℕ\nisLt✝ : val✝ < n\n⊢ ↑(bit0 { val := val✝, isLt := isLt✝ }) = bit0 ↑{ val := val✝, isLt := isLt✝ } % n",
"tactic": "rfl"
}
] | [
721,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
719,
1
] |
Mathlib/Topology/Bases.lean | TopologicalSpace.IsSeparable.union | [
{
"state_after": "case intro.intro\nα : Type u\nt : TopologicalSpace α\ns u : Set α\nhu : IsSeparable u\ncs : Set α\ncs_count : Set.Countable cs\nhcs : s ⊆ closure cs\n⊢ IsSeparable (s ∪ u)",
"state_before": "α : Type u\nt : TopologicalSpace α\ns u : Set α\nhs : IsSeparable s\nhu : IsSeparable u\n⊢ IsSeparable (s ∪ u)",
"tactic": "rcases hs with ⟨cs, cs_count, hcs⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nt : TopologicalSpace α\ns u cs : Set α\ncs_count : Set.Countable cs\nhcs : s ⊆ closure cs\ncu : Set α\ncu_count : Set.Countable cu\nhcu : u ⊆ closure cu\n⊢ IsSeparable (s ∪ u)",
"state_before": "case intro.intro\nα : Type u\nt : TopologicalSpace α\ns u : Set α\nhu : IsSeparable u\ncs : Set α\ncs_count : Set.Countable cs\nhcs : s ⊆ closure cs\n⊢ IsSeparable (s ∪ u)",
"tactic": "rcases hu with ⟨cu, cu_count, hcu⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nt : TopologicalSpace α\ns u cs : Set α\ncs_count : Set.Countable cs\nhcs : s ⊆ closure cs\ncu : Set α\ncu_count : Set.Countable cu\nhcu : u ⊆ closure cu\n⊢ s ∪ u ⊆ closure (cs ∪ cu)",
"state_before": "case intro.intro.intro.intro\nα : Type u\nt : TopologicalSpace α\ns u cs : Set α\ncs_count : Set.Countable cs\nhcs : s ⊆ closure cs\ncu : Set α\ncu_count : Set.Countable cu\nhcu : u ⊆ closure cu\n⊢ IsSeparable (s ∪ u)",
"tactic": "refine' ⟨cs ∪ cu, cs_count.union cu_count, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u\nt : TopologicalSpace α\ns u cs : Set α\ncs_count : Set.Countable cs\nhcs : s ⊆ closure cs\ncu : Set α\ncu_count : Set.Countable cu\nhcu : u ⊆ closure cu\n⊢ s ∪ u ⊆ closure (cs ∪ cu)",
"tactic": "exact\n union_subset (hcs.trans (closure_mono (subset_union_left _ _)))\n (hcu.trans (closure_mono (subset_union_right _ _)))"
}
] | [
397,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
390,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Real.pi_le_four | [
{
"state_after": "no goals",
"state_before": "⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "⊢ 2 = 4 / 2",
"tactic": "norm_num"
}
] | [
162,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | MvPolynomial.isHomogeneous_zero | [] | [
147,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Data/Nat/PartENat.lean | PartENat.withTopEquiv_natCast | [] | [
696,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
695,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | exists_between_of_forall_le | [] | [
646,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
644,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | gramSchmidtNormed_unit_length' | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nn : ι\nhn : (↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n ≠ 0\n⊢ ‖(↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n‖ = 1",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nn : ι\nhn : gramSchmidtNormed 𝕜 f n ≠ 0\n⊢ ‖gramSchmidtNormed 𝕜 f n‖ = 1",
"tactic": "rw [gramSchmidtNormed] at *"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nn : ι\nhn : (↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n ≠ 0\n⊢ gramSchmidt 𝕜 f n ≠ 0",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nn : ι\nhn : (↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n ≠ 0\n⊢ ‖(↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n‖ = 1",
"tactic": "rw [norm_smul_inv_norm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nn : ι\nhn : (↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n ≠ 0\n⊢ gramSchmidt 𝕜 f n ≠ 0",
"tactic": "simpa using hn"
}
] | [
290,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
] |
Mathlib/Topology/Constructions.lean | isClosed_sigma_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.514368\nδ : Type ?u.514371\nε : Type ?u.514374\nζ : Type ?u.514377\nι : Type u_2\nκ : Type ?u.514383\nσ : ι → Type u_1\nτ : κ → Type ?u.514393\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\ns : Set (Sigma σ)\n⊢ IsClosed s ↔ ∀ (i : ι), IsClosed (Sigma.mk i ⁻¹' s)",
"tactic": "simp only [← isOpen_compl_iff, isOpen_sigma_iff, preimage_compl]"
}
] | [
1456,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1455,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.measure_setOf_frequently_eq_zero | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.590790\nγ : Type ?u.590793\nδ : Type ?u.590796\nι : Type ?u.590799\nR : Type ?u.590802\nR' : Type ?u.590805\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : ℕ → α → Prop\nhp : (∑' (i : ℕ), ↑↑μ {x | p i x}) ≠ ⊤\n⊢ ↑↑μ {x | ∃ᶠ (n : ℕ) in atTop, p n x} = 0",
"tactic": "simpa only [limsup_eq_iInf_iSup_of_nat, frequently_atTop, ← bex_def, setOf_forall,\n setOf_exists] using measure_limsup_eq_zero hp"
}
] | [
2943,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2940,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.trim_le_trim_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm m₁ m₂ : OuterMeasure α\ns : Set α\nhs : MeasurableSet s\n⊢ ↑(trim m₁) s ≤ ↑m₂ s ↔ ↑m₁ s ≤ ↑m₂ s",
"tactic": "rw [trim_eq _ hs]"
}
] | [
1647,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1645,
1
] |
Mathlib/Data/Vector/Basic.lean | Vector.removeNth_insertNth | [] | [
539,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
537,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean | add_tsub_le_left | [] | [
105,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/Data/Set/Basic.lean | Set.compl_setOf | [] | [
1632,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1631,
1
] |
Mathlib/CategoryTheory/CofilteredSystem.lean | CategoryTheory.Functor.eval_section_surjective_of_surjective | [
{
"state_after": "J : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\n⊢ ∃ a, (fun s => ↑s i) a = x",
"state_before": "J : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\n⊢ ∃ a, (fun s => ↑s i) a = x",
"tactic": "let s : Set (F.obj i) := {x}"
},
{
"state_after": "J : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\n⊢ ∃ a, (fun s => ↑s i) a = x",
"state_before": "J : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\n⊢ ∃ a, (fun s => ↑s i) a = x",
"tactic": "haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)"
},
{
"state_after": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\n⊢ ∃ a, (fun s => ↑s i) a = x",
"state_before": "J : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\n⊢ ∃ a, (fun s => ↑s i) a = x",
"tactic": "obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)"
},
{
"state_after": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"state_before": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\n⊢ ∃ a, (fun s => ↑s i) a = x",
"tactic": "refine' ⟨⟨fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nj✝ j'✝ : J\njk : j✝ ⟶ j'✝\n⊢ F.map jk ((fun j => ↑(sec j)) j✝) = (fun j => ↑(sec j)) j'✝",
"tactic": "simpa [Subtype.ext_iff] using h jk"
},
{
"state_after": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis✝ : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nthis : ↑(sec i) ∈ ⋂ (f : i ⟶ i), F.map f ⁻¹' s\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"state_before": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"tactic": "have := (sec i).prop"
},
{
"state_after": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis✝ : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nthis : ∀ (i_1 : i ⟶ i), F.map i_1 ↑(sec i) = x\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"state_before": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis✝ : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nthis : ↑(sec i) ∈ ⋂ (f : i ⟶ i), F.map f ⁻¹' s\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"tactic": "simp only [mem_iInter, mem_preimage, mem_singleton_iff] at this"
},
{
"state_after": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis✝¹ : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nthis✝ : ∀ (i_1 : i ⟶ i), F.map i_1 ↑(sec i) = x\nthis : F.map (𝟙 i) ↑(sec i) = x\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"state_before": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis✝ : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nthis : ∀ (i_1 : i ⟶ i), F.map i_1 ↑(sec i) = x\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"tactic": "have := this (𝟙 i)"
},
{
"state_after": "no goals",
"state_before": "case intro\nJ : Type u\ninst✝³ : Category J\nF : J ⥤ Type v\ni✝ j k : J\ns✝ : Set (F.obj i✝)\ninst✝² : IsCofilteredOrEmpty J\ninst✝¹ : ∀ (j : J), _root_.Nonempty (F.obj j)\ninst✝ : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis✝¹ : ∀ (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) → (toPreimages F s).obj j\nh : sec ∈ sections (toPreimages F s)\nthis✝ : ∀ (i_1 : i ⟶ i), F.map i_1 ↑(sec i) = x\nthis : F.map (𝟙 i) ↑(sec i) = x\n⊢ (fun s => ↑s i)\n { val := fun j => ↑(sec j),\n property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =\n x",
"tactic": "rwa [map_id_apply] at this"
}
] | [
362,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/Data/Matrix/Block.lean | Matrix.blockDiag_map | [] | [
514,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
512,
1
] |
Mathlib/Data/Set/Prod.lean | Set.diag_image | [
{
"state_after": "case h\nα : Type u_1\ns✝ t s : Set α\nx : α × α\n⊢ x ∈ (fun x => (x, x)) '' s ↔ x ∈ diagonal α ∩ s ×ˢ s",
"state_before": "α : Type u_1\ns✝ t s : Set α\n⊢ (fun x => (x, x)) '' s = diagonal α ∩ s ×ˢ s",
"tactic": "ext x"
},
{
"state_after": "case h.mp\nα : Type u_1\ns✝ t s : Set α\nx : α × α\n⊢ x ∈ (fun x => (x, x)) '' s → x ∈ diagonal α ∩ s ×ˢ s\n\ncase h.mpr\nα : Type u_1\ns✝ t s : Set α\nx : α × α\n⊢ x ∈ diagonal α ∩ s ×ˢ s → x ∈ (fun x => (x, x)) '' s",
"state_before": "case h\nα : Type u_1\ns✝ t s : Set α\nx : α × α\n⊢ x ∈ (fun x => (x, x)) '' s ↔ x ∈ diagonal α ∩ s ×ˢ s",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro\nα : Type u_1\ns✝ t s : Set α\nx : α\nhx : x ∈ s\n⊢ (fun x => (x, x)) x ∈ diagonal α ∩ s ×ˢ s",
"state_before": "case h.mp\nα : Type u_1\ns✝ t s : Set α\nx : α × α\n⊢ x ∈ (fun x => (x, x)) '' s → x ∈ diagonal α ∩ s ×ˢ s",
"tactic": "rintro ⟨x, hx, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro\nα : Type u_1\ns✝ t s : Set α\nx : α\nhx : x ∈ s\n⊢ (fun x => (x, x)) x ∈ diagonal α ∩ s ×ˢ s",
"tactic": "exact ⟨rfl, hx, hx⟩"
},
{
"state_after": "case h.mpr.mk\nα : Type u_1\ns✝ t s : Set α\nx y : α\n⊢ (x, y) ∈ diagonal α ∩ s ×ˢ s → (x, y) ∈ (fun x => (x, x)) '' s",
"state_before": "case h.mpr\nα : Type u_1\ns✝ t s : Set α\nx : α × α\n⊢ x ∈ diagonal α ∩ s ×ˢ s → x ∈ (fun x => (x, x)) '' s",
"tactic": "obtain ⟨x, y⟩ := x"
},
{
"state_after": "case h.mpr.mk.intro\nα : Type u_1\ns✝ t s : Set α\nx : α\nh2x : (x, x) ∈ s ×ˢ s\n⊢ (x, x) ∈ (fun x => (x, x)) '' s",
"state_before": "case h.mpr.mk\nα : Type u_1\ns✝ t s : Set α\nx y : α\n⊢ (x, y) ∈ diagonal α ∩ s ×ˢ s → (x, y) ∈ (fun x => (x, x)) '' s",
"tactic": "rintro ⟨rfl : x = y, h2x⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.mk.intro\nα : Type u_1\ns✝ t s : Set α\nx : α\nh2x : (x, x) ∈ s ×ˢ s\n⊢ (x, x) ∈ (fun x => (x, x)) '' s",
"tactic": "exact mem_image_of_mem _ h2x.1"
}
] | [
525,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
518,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean | PiNat.cylinder_zero | [
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\nx : (n : ℕ) → E n\n⊢ cylinder x 0 = univ",
"tactic": "simp [cylinder_eq_pi]"
}
] | [
121,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.basis_pair_right | [
{
"state_after": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\n⊢ Lagrange.basis {j, i} v j = basisDivisor (v j) (v i)",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\n⊢ Lagrange.basis {i, j} v j = basisDivisor (v j) (v i)",
"tactic": "rw [pair_comm]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\n⊢ Lagrange.basis {j, i} v j = basisDivisor (v j) (v i)",
"tactic": "exact basis_pair_left hij.symm"
}
] | [
213,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean | finprod_mem_mul_diff | [] | [
1046,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1044,
1
] |
Mathlib/Order/Filter/Lift.lean | Filter.lift_map_le | [] | [
129,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean | NatOrdinal.toOrdinal_toNatOrdinal | [] | [
90,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | SeminormFamily.basisSets_nonempty | [
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type ?u.18792\n𝕝 : Type ?u.18795\n𝕝₂ : Type ?u.18798\nE : Type u_2\nF : Type ?u.18804\nG : Type ?u.18807\nι : Type u_1\nι' : Type ?u.18813\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : Nonempty ι\ni : ι := Classical.arbitrary ι\n⊢ Set.Nonempty (basisSets p)",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type ?u.18792\n𝕝 : Type ?u.18795\n𝕝₂ : Type ?u.18798\nE : Type u_2\nF : Type ?u.18804\nG : Type ?u.18807\nι : Type u_1\nι' : Type ?u.18813\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : Nonempty ι\n⊢ Set.Nonempty (basisSets p)",
"tactic": "let i := Classical.arbitrary ι"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type ?u.18792\n𝕝 : Type ?u.18795\n𝕝₂ : Type ?u.18798\nE : Type u_2\nF : Type ?u.18804\nG : Type ?u.18807\nι : Type u_1\nι' : Type ?u.18813\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : Nonempty ι\ni : ι := Classical.arbitrary ι\n⊢ ball (p i) 0 1 ∈ basisSets p",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type ?u.18792\n𝕝 : Type ?u.18795\n𝕝₂ : Type ?u.18798\nE : Type u_2\nF : Type ?u.18804\nG : Type ?u.18807\nι : Type u_1\nι' : Type ?u.18813\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : Nonempty ι\ni : ι := Classical.arbitrary ι\n⊢ Set.Nonempty (basisSets p)",
"tactic": "refine' nonempty_def.mpr ⟨(p i).ball 0 1, _⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type ?u.18792\n𝕝 : Type ?u.18795\n𝕝₂ : Type ?u.18798\nE : Type u_2\nF : Type ?u.18804\nG : Type ?u.18807\nι : Type u_1\nι' : Type ?u.18813\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : Nonempty ι\ni : ι := Classical.arbitrary ι\n⊢ ball (p i) 0 1 ∈ basisSets p",
"tactic": "exact p.basisSets_singleton_mem i zero_lt_one"
}
] | [
98,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | Padic.valuation_map_mul | [
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\n⊢ valuation (x * y) = valuation x + valuation y",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ valuation (x * y) = valuation x + valuation y",
"tactic": "have h_norm : ‖x * y‖ = ‖x‖ * ‖y‖ := norm_mul x y"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\n⊢ valuation (x * y) = valuation x + valuation y",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\n⊢ valuation (x * y) = valuation x + valuation y",
"tactic": "have hp_ne_one : (p : ℝ) ≠ 1 := by\n rw [← Nat.cast_one, Ne.def, Nat.cast_inj]\n exact Nat.Prime.ne_one hp.elim"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\nhp_pos : 0 < ↑p\n⊢ valuation (x * y) = valuation x + valuation y",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\n⊢ valuation (x * y) = valuation x + valuation y",
"tactic": "have hp_pos : (0 : ℝ) < p := by\n rw [← Nat.cast_zero, Nat.cast_lt]\n exact Nat.Prime.pos hp.elim"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : valuation (x * y) = valuation x + valuation y\nhp_ne_one : ↑p ≠ 1\nhp_pos : 0 < ↑p\n⊢ valuation (x * y) = valuation x + valuation y",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\nhp_pos : 0 < ↑p\n⊢ valuation (x * y) = valuation x + valuation y",
"tactic": "rw [norm_eq_pow_val hx, norm_eq_pow_val hy, norm_eq_pow_val (mul_ne_zero hx hy), ←\n zpow_add₀ (ne_of_gt hp_pos), zpow_inj hp_pos hp_ne_one, ← neg_add, neg_inj] at h_norm"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : valuation (x * y) = valuation x + valuation y\nhp_ne_one : ↑p ≠ 1\nhp_pos : 0 < ↑p\n⊢ valuation (x * y) = valuation x + valuation y",
"tactic": "exact h_norm"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\n⊢ ¬p = 1",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\n⊢ ↑p ≠ 1",
"tactic": "rw [← Nat.cast_one, Ne.def, Nat.cast_inj]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\n⊢ ¬p = 1",
"tactic": "exact Nat.Prime.ne_one hp.elim"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\n⊢ 0 < p",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\n⊢ 0 < ↑p",
"tactic": "rw [← Nat.cast_zero, Nat.cast_lt]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\nh_norm : ‖x * y‖ = ‖x‖ * ‖y‖\nhp_ne_one : ↑p ≠ 1\n⊢ 0 < p",
"tactic": "exact Nat.Prime.pos hp.elim"
}
] | [
1102,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1091,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean | LinearMap.IsOrthoᵢ.nondegenerate_of_not_isOrtho_basis_self | [] | [
836,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
833,
1
] |
Mathlib/Algebra/Bounds.lean | subset_upperBounds_mul | [] | [
111,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Topology/Connected.lean | Embedding.isTotallyDisconnected | [] | [
1347,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1345,
1
] |
Mathlib/Analysis/Convex/Function.lean | LinearOrder.strictConvexOn_of_lt | [
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.242524\nα : Type ?u.242527\nβ : Type u_3\nι : Type ?u.242533\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • x + b • y) < a • f x + b • f y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.242524\nα : Type ?u.242527\nβ : Type u_3\nι : Type ?u.242533\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\n⊢ StrictConvexOn 𝕜 s f",
"tactic": "refine' ⟨hs, fun x hx y hy hxy a b ha hb hab => _⟩"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • x + b • y) < a • f x + b • f y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.242524\nα : Type ?u.242527\nβ : Type u_3\nι : Type ?u.242533\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • x + b • y) < a • f x + b • f y",
"tactic": "clear! α F ι"
},
{
"state_after": "case inr\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis :\n ∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} [inst : OrderedSemiring 𝕜] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid β] [inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 β] [inst_5 : LinearOrder E] {s : Set E}\n {f : E → β},\n Convex 𝕜 s →\n (∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y) →\n ∀ (x : E),\n x ∈ s →\n ∀ (y : E),\n y ∈ s → x ≠ y → ∀ (a b : 𝕜), 0 < a → 0 < b → a + b = 1 → x < y → f (a • x + b • y) < a • f x + b • f y\nh : ¬x < y\n⊢ f (a • x + b • y) < a • f x + b • f y\n\n𝕜✝ : Type u_1\nE✝ : Type u_2\nβ✝ : Type u_3\ninst✝¹¹ : OrderedSemiring 𝕜✝\ninst✝¹⁰ : AddCommMonoid E✝\ninst✝⁹ : OrderedAddCommMonoid β✝\ninst✝⁸ : Module 𝕜✝ E✝\ninst✝⁷ : Module 𝕜✝ β✝\ninst✝⁶ : LinearOrder E✝\ns✝ : Set E✝\nf✝ : E✝ → β✝\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nh : x < y\n⊢ f (a • x + b • y) < a • f x + b • f y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • x + b • y) < a • f x + b • f y",
"tactic": "wlog h : x < y"
},
{
"state_after": "no goals",
"state_before": "𝕜✝ : Type u_1\nE✝ : Type u_2\nβ✝ : Type u_3\ninst✝¹¹ : OrderedSemiring 𝕜✝\ninst✝¹⁰ : AddCommMonoid E✝\ninst✝⁹ : OrderedAddCommMonoid β✝\ninst✝⁸ : Module 𝕜✝ E✝\ninst✝⁷ : Module 𝕜✝ β✝\ninst✝⁶ : LinearOrder E✝\ns✝ : Set E✝\nf✝ : E✝ → β✝\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nh : x < y\n⊢ f (a • x + b • y) < a • f x + b • f y",
"tactic": "exact hf hx hy h ha hb hab"
},
{
"state_after": "case inr\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis :\n ∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} [inst : OrderedSemiring 𝕜] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid β] [inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 β] [inst_5 : LinearOrder E] {s : Set E}\n {f : E → β},\n Convex 𝕜 s →\n (∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y) →\n ∀ (x : E),\n x ∈ s →\n ∀ (y : E),\n y ∈ s → x ≠ y → ∀ (a b : 𝕜), 0 < a → 0 < b → a + b = 1 → x < y → f (a • x + b • y) < a • f x + b • f y\nh : ¬x < y\n⊢ f (b • y + a • x) < b • f y + a • f x",
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Mathlib/Data/Multiset/Basic.lean | Multiset.empty_eq_zero | [] | [
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Mathlib/Data/Set/Image.lean | Set.nonempty_of_nonempty_preimage | [] | [
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Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.map_sup | [] | [
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Mathlib/Data/Seq/Computation.lean | Computation.mem_map | [
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Mathlib/RingTheory/Ideal/LocalRing.lean | LocalRing.ResidueField.map_id | [] | [
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Mathlib/Data/QPF/Univariate/Basic.lean | Qpf.supp_eq_of_isUniform | [
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Mathlib/Topology/MetricSpace/Basic.lean | CauchySeq.bounded_range | [
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Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | Real.differentiable_sin | [] | [
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Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | csSup_image2_eq_csSup_csInf | [] | [
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Mathlib/Order/Monotone/Basic.lean | StrictAnti.le_iff_le | [] | [
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Mathlib/Topology/Bases.lean | TopologicalSpace.isSeparable_of_separableSpace_subtype | [
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Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_const_add_Ici | [] | [
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Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | CategoryTheory.Limits.terminal.comp_from | [
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Mathlib/GroupTheory/Perm/Sign.lean | Equiv.Perm.sign_trans_trans_symm | [] | [
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Mathlib/Data/Polynomial/Basic.lean | Polynomial.C_bit1 | [
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}
] | [
527,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
527,
1
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Mathlib/Order/Hom/Lattice.lean | SupHom.comp_assoc | [] | [
418,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
416,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean | Polynomial.trailingDegree_X | [] | [
444,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
443,
1
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Mathlib/RingTheory/Finiteness.lean | Submodule.fg_of_isUnit | [] | [
171,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
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Mathlib/Data/Set/Basic.lean | Set.mem_inter | [] | [
899,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
898,
1
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Mathlib/Analysis/NormedSpace/ENorm.lean | ENorm.map_sub_le | [
{
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"tactic": "rw [sub_eq_add_neg]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nV : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne : ENorm 𝕜 V\nx y : V\n⊢ ↑e x + ↑e (-y) = ↑e x + ↑e y",
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}
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127,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
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Mathlib/Algebra/Order/Floor.lean | Int.fract_lt_one | [] | [
883,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
882,
1
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Mathlib/Data/MvPolynomial/Variables.lean | MvPolynomial.degrees_monomial | [
{
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},
{
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},
{
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"tactic": "have := Finsupp.support_single_subset h"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.3088\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns : σ →₀ ℕ\na : R\nt : σ →₀ ℕ\nh : t ∈ support (↑(monomial s) a)\nthis : t = s\n⊢ ↑toMultiset t ≤ ↑toMultiset s",
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},
{
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103,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/Finset/Pointwise.lean | Finset.biUnion_op_smul_finset | [] | [
1844,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1843,
1
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Mathlib/Algebra/Support.lean | Function.mulSupport_iSup | [
{
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"tactic": "rw [mulSupport_subset_iff']"
},
{
"state_after": "α : Type u_3\nβ : Type ?u.13009\nA : Type ?u.13012\nB : Type ?u.13015\nM : Type u_1\nN : Type ?u.13021\nP : Type ?u.13024\nR : Type ?u.13027\nS : Type ?u.13030\nG : Type ?u.13033\nM₀ : Type ?u.13036\nG₀ : Type ?u.13039\nι : Sort u_2\ninst✝⁴ : One M\ninst✝³ : One N\ninst✝² : One P\ninst✝¹ : ConditionallyCompleteLattice M\ninst✝ : Nonempty ι\nf : ι → α → M\n⊢ ∀ (x : α), (∀ (x_1 : ι), f x_1 x = 1) → (⨆ (i : ι), f i x) = 1",
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},
{
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"tactic": "intro x hx"
},
{
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"tactic": "simp only [hx, ciSup_const]"
}
] | [
190,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
] |
Std/Data/Int/DivMod.lean | Int.ofNat_dvd_left | [
{
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"tactic": "rw [← natAbs_dvd_natAbs, natAbs_ofNat]"
}
] | [
662,
41
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
661,
1
] |
Mathlib/NumberTheory/Padics/PadicNorm.lean | padicNorm.nonarchimedean | [
{
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},
{
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"tactic": "exact nonarchimedean_aux hle"
},
{
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"tactic": "rw [add_comm, max_comm]"
},
{
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"tactic": "exact this (le_of_not_le hle)"
}
] | [
211,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
11
] |
Mathlib/Data/Polynomial/Monic.lean | Polynomial.monic_X_add_C | [] | [
111,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Data/Semiquot.lean | Semiquot.ext | [] | [
57,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean | LinearMap.IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft | [
{
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"tactic": "intro ho"
},
{
"state_after": "R : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nm : M\n⊢ ↑(↑B (↑v i)) m = 0",
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"tactic": "refine' v.ne_zero i (hB (v i) fun m ↦ _)"
},
{
"state_after": "case intro\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\n⊢ ↑(↑B (↑v i)) (↑(LinearEquiv.symm v.repr) vi) = 0",
"state_before": "R : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nm : M\n⊢ ↑(↑B (↑v i)) m = 0",
"tactic": "obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m"
},
{
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"tactic": "rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, map_sum]"
},
{
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"tactic": "apply Finset.sum_eq_zero"
},
{
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"tactic": "rintro j -"
},
{
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"tactic": "rw [map_smulₛₗ]"
},
{
"state_after": "case intro.h\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\nj : n\nthis : ↑(↑B (↑v i)) (↑v j) = 0\n⊢ ↑I' (↑vi j) • ↑(↑B (↑v i)) (↑v j) = 0\n\ncase this\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\nj : n\n⊢ ↑(↑B (↑v i)) (↑v j) = 0",
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"tactic": "suffices : B (v i) (v j) = 0"
},
{
"state_after": "case this.inl\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\n⊢ ↑(↑B (↑v i)) (↑v i) = 0\n\ncase this.inr\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\nj : n\nhij : i ≠ j\n⊢ ↑(↑B (↑v i)) (↑v j) = 0",
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"tactic": "obtain rfl | hij := eq_or_ne i j"
},
{
"state_after": "no goals",
"state_before": "case intro.h\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\nj : n\nthis : ↑(↑B (↑v i)) (↑v j) = 0\n⊢ ↑I' (↑vi j) • ↑(↑B (↑v i)) (↑v j) = 0",
"tactic": "rw [this, smul_eq_mul, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case this.inl\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\n⊢ ↑(↑B (↑v i)) (↑v i) = 0",
"tactic": "exact ho"
},
{
"state_after": "no goals",
"state_before": "case this.inr\nR : Type u_1\nR₁ : Type ?u.588000\nR₂ : Type ?u.588003\nR₃ : Type ?u.588006\nM : Type u_2\nM₁ : Type ?u.588012\nM₂ : Type ?u.588015\nMₗ₁ : Type ?u.588018\nMₗ₁' : Type ?u.588021\nMₗ₂ : Type ?u.588024\nMₗ₂' : Type ?u.588027\nK : Type ?u.588030\nK₁ : Type ?u.588033\nK₂ : Type ?u.588036\nV : Type ?u.588039\nV₁ : Type ?u.588042\nV₂ : Type ?u.588045\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI I' : R →+* R\ninst✝ : Nontrivial R\nB : M →ₛₗ[I] M →ₛₗ[I'] R\nv : Basis n R M\nh : IsOrthoᵢ B ↑v\nhB : SeparatingLeft B\ni : n\nho : IsOrtho B (↑v i) (↑v i)\nvi : n →₀ R\nj : n\nhij : i ≠ j\n⊢ ↑(↑B (↑v i)) (↑v j) = 0",
"tactic": "exact h hij"
}
] | [
780,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
766,
1
] |
Mathlib/Logic/Basic.lean | Exists₂.imp | [] | [
639,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | LinearMap.mul_toMatrix' | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1026117\nR₂ : Type ?u.1026120\nM✝ : Type ?u.1026123\nM₁ : Type ?u.1026126\nM₂ : Type ?u.1026129\nM₁' : Type ?u.1026132\nM₂' : Type ?u.1026135\nn : Type u_2\nm : Type u_3\nn' : Type u_4\nm' : Type ?u.1026147\nι : Type ?u.1026150\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : Fintype n\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq n\ninst✝⁴ : DecidableEq m\nσ₁ : R₁ →+* R\nσ₂ : R₂ →+* R\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nB : (n → R) →ₗ[R] (m → R) →ₗ[R] R\nM : Matrix n' n R\n⊢ M ⬝ ↑toMatrix₂' B = ↑toMatrix₂' (comp B (↑toLin' Mᵀ))",
"tactic": "simp only [B.toMatrix₂'_comp, transpose_transpose, toMatrix'_toLin']"
}
] | [
327,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
] |
Mathlib/GroupTheory/Subsemigroup/Centralizer.lean | Set.div_mem_centralizer₀ | [
{
"state_after": "M : Type u_1\nS T : Set M\na b : M\ninst✝ : GroupWithZero M\nha : a ∈ centralizer S\nhb : b ∈ centralizer S\n⊢ a * b⁻¹ ∈ centralizer S",
"state_before": "M : Type u_1\nS T : Set M\na b : M\ninst✝ : GroupWithZero M\nha : a ∈ centralizer S\nhb : b ∈ centralizer S\n⊢ a / b ∈ centralizer S",
"tactic": "rw [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nS T : Set M\na b : M\ninst✝ : GroupWithZero M\nha : a ∈ centralizer S\nhb : b ∈ centralizer S\n⊢ a * b⁻¹ ∈ centralizer S",
"tactic": "exact mul_mem_centralizer ha (inv_mem_centralizer₀ hb)"
}
] | [
118,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
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