file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.spanExt_inv_app_right
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nX Y Z X' Y' Z' : C\niX : X ≅ X'\niY : Y ≅ Y'\niZ : Z ≅ Z'\nf : X ⟶ Y\ng : X ⟶ Z\nf' : X' ⟶ Y'\ng' : X' ⟶ Z'\nwf : iX.hom ≫ f' = f ≫ iY.hom\nwg : iX.hom ≫ g' = g ≫ iZ.hom\n⊢ (spanExt iX iY iZ wf wg).inv.app WalkingSpan.right = iZ.inv",
"tactic": "dsimp [spanExt]"
}
] |
[
496,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Topology/Algebra/MulAction.lean
|
ContinuousWithinAt.smul
|
[] |
[
105,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
Subgroup.nilpotencyClass_le
|
[
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ Nat.find (_ : ∃ n, lowerCentralSeries { x // x ∈ H } n = ⊥) ≤ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥)",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ nilpotencyClass { x // x ∈ H } ≤ nilpotencyClass G",
"tactic": "repeat rw [← lowerCentralSeries_length_eq_nilpotencyClass]"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ ∀ (n : ℕ), lowerCentralSeries G n = ⊥ → lowerCentralSeries { x // x ∈ H } n = ⊥",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ Nat.find (_ : ∃ n, lowerCentralSeries { x // x ∈ H } n = ⊥) ≤ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥)",
"tactic": "refine @Nat.find_mono _ _ (Classical.decPred _) (Classical.decPred _) ?_ _ _"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG✝ : Group.IsNilpotent G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\n⊢ lowerCentralSeries { x // x ∈ H } n = ⊥",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ ∀ (n : ℕ), lowerCentralSeries G n = ⊥ → lowerCentralSeries { x // x ∈ H } n = ⊥",
"tactic": "intro n hG"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG✝ : Group.IsNilpotent G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\nthis : map (Subgroup.subtype H) (lowerCentralSeries { x // x ∈ H } n) ≤ lowerCentralSeries G n\n⊢ lowerCentralSeries { x // x ∈ H } n = ⊥",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG✝ : Group.IsNilpotent G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\n⊢ lowerCentralSeries { x // x ∈ H } n = ⊥",
"tactic": "have := lowerCentralSeries_map_subtype_le H n"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG✝ : Group.IsNilpotent G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\nthis :\n ∀ ⦃x : G⦄ (x_1 : { x // x ∈ H }), x_1 ∈ lowerCentralSeries { x // x ∈ H } n ∧ ↑(Subgroup.subtype H) x_1 = x → x ∈ ⊥\n⊢ lowerCentralSeries { x // x ∈ H } n = ⊥",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG✝ : Group.IsNilpotent G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\nthis : map (Subgroup.subtype H) (lowerCentralSeries { x // x ∈ H } n) ≤ lowerCentralSeries G n\n⊢ lowerCentralSeries { x // x ∈ H } n = ⊥",
"tactic": "simp only [hG, SetLike.le_def, mem_map, forall_apply_eq_imp_iff₂, exists_imp] at this"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG✝ : Group.IsNilpotent G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\nthis :\n ∀ ⦃x : G⦄ (x_1 : { x // x ∈ H }), x_1 ∈ lowerCentralSeries { x // x ∈ H } n ∧ ↑(Subgroup.subtype H) x_1 = x → x ∈ ⊥\n⊢ lowerCentralSeries { x // x ∈ H } n = ⊥",
"tactic": "exact eq_bot_iff.mpr fun x hx => Subtype.ext (this x ⟨hx, rfl⟩)"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ Nat.find (_ : ∃ n, lowerCentralSeries { x // x ∈ H } n = ⊥) ≤ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥)",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : Normal H✝\nH : Subgroup G\nhG : Group.IsNilpotent G\n⊢ Nat.find (_ : ∃ n, lowerCentralSeries { x // x ∈ H } n = ⊥) ≤ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥)",
"tactic": "rw [← lowerCentralSeries_length_eq_nilpotencyClass]"
}
] |
[
480,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.Measure.toSignedMeasure_congr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.135192\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : μ = ν\n⊢ toSignedMeasure μ = toSignedMeasure ν",
"tactic": "congr"
}
] |
[
447,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivAtFilter.neg
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nh : HasDerivAtFilter f f' x L\n⊢ HasDerivAtFilter (fun x => -f x) (-f') x L",
"tactic": "simpa using h.neg.hasDerivAtFilter"
}
] |
[
204,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
8
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.bot_coe_submodule
|
[] |
[
442,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.ne_zero_of_degree_ge_degree
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.466413\nhpq : degree p ≤ degree q\nhp : p ≠ 0\n⊢ degree p ≠ ⊥",
"tactic": "rwa [Ne.def, Polynomial.degree_eq_bot]"
}
] |
[
592,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
589,
1
] |
Mathlib/NumberTheory/Liouville/Residual.lean
|
eventually_residual_liouville
|
[
{
"state_after": "⊢ ({x | Irrational x} ∩ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)) ∈ residual ℝ",
"state_before": "⊢ ∀ᶠ (x : ℝ) in residual ℝ, Liouville x",
"tactic": "rw [Filter.Eventually, setOf_liouville_eq_irrational_inter_iInter_iUnion]"
},
{
"state_after": "⊢ ∀ᶠ (x : ℝ) in residual ℝ, x ∈ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)",
"state_before": "⊢ ({x | Irrational x} ∩ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)) ∈ residual ℝ",
"tactic": "refine eventually_residual_irrational.and ?_"
},
{
"state_after": "case refine_1\n\n⊢ IsGδ (⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n))\n\ncase refine_2\n\n⊢ range Rat.cast ⊆ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)",
"state_before": "⊢ ∀ᶠ (x : ℝ) in residual ℝ, x ∈ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)",
"tactic": "refine eventually_residual.2 ⟨_, ?_, Rat.denseEmbedding_coe_real.dense.mono ?_, Subset.rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_1\n\n⊢ IsGδ (⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n))",
"tactic": "exact isGδ_iInter fun n => IsOpen.isGδ <|\n isOpen_iUnion fun a => isOpen_iUnion fun b => isOpen_iUnion fun _hb => isOpen_ball"
},
{
"state_after": "case refine_2.intro\nr : ℚ\n⊢ ↑r ∈ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)",
"state_before": "case refine_2\n\n⊢ range Rat.cast ⊆ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)",
"tactic": "rintro _ ⟨r, rfl⟩"
},
{
"state_after": "case refine_2.intro\nr : ℚ\n⊢ ∀ (i : ℕ), ∃ i_1 i_2 i_3, ↑r ∈ ball (↑i_1 / ↑i_2) (1 / ↑i_2 ^ i)",
"state_before": "case refine_2.intro\nr : ℚ\n⊢ ↑r ∈ ⋂ (n : ℕ), ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)",
"tactic": "simp only [mem_iInter, mem_iUnion]"
},
{
"state_after": "case refine_2.intro.refine_1\nr : ℚ\nn : ℕ\n⊢ 1 < ↑r.den * 2\n\ncase refine_2.intro.refine_2\nr : ℚ\nn : ℕ\n⊢ ↑r ∈ ball (↑(r.num * 2) / ↑(↑r.den * 2)) (1 / ↑(↑r.den * 2) ^ n)",
"state_before": "case refine_2.intro\nr : ℚ\n⊢ ∀ (i : ℕ), ∃ i_1 i_2 i_3, ↑r ∈ ball (↑i_1 / ↑i_2) (1 / ↑i_2 ^ i)",
"tactic": "refine fun n => ⟨r.num * 2, r.den * 2, ?_, ?_⟩"
},
{
"state_after": "case refine_2.intro.refine_1\nr : ℚ\nn : ℕ\nthis : ↑(Nat.succ 0) ≤ ↑r.den\n⊢ 1 < ↑r.den * 2",
"state_before": "case refine_2.intro.refine_1\nr : ℚ\nn : ℕ\n⊢ 1 < ↑r.den * 2",
"tactic": "have := Int.ofNat_le.2 r.pos"
},
{
"state_after": "case refine_2.intro.refine_1\nr : ℚ\nn : ℕ\nthis : 1 ≤ ↑r.den\n⊢ 1 < ↑r.den * 2",
"state_before": "case refine_2.intro.refine_1\nr : ℚ\nn : ℕ\nthis : ↑(Nat.succ 0) ≤ ↑r.den\n⊢ 1 < ↑r.den * 2",
"tactic": "rw [Int.ofNat_one] at this"
},
{
"state_after": "no goals",
"state_before": "case refine_2.intro.refine_1\nr : ℚ\nn : ℕ\nthis : 1 ≤ ↑r.den\n⊢ 1 < ↑r.den * 2",
"tactic": "linarith"
},
{
"state_after": "case h.e'_5.h.e'_3\nr : ℚ\nn : ℕ\n⊢ ↑(r.num * 2) / ↑(↑r.den * 2) = ↑r\n\ncase refine_2.intro.refine_2.convert_2\nr : ℚ\nn : ℕ\n⊢ 0 < 1 / ↑(↑r.den * 2) ^ n",
"state_before": "case refine_2.intro.refine_2\nr : ℚ\nn : ℕ\n⊢ ↑r ∈ ball (↑(r.num * 2) / ↑(↑r.den * 2)) (1 / ↑(↑r.den * 2) ^ n)",
"tactic": "convert @mem_ball_self ℝ _ (r : ℝ) _ _"
},
{
"state_after": "case h.e'_5.h.e'_3\nr : ℚ\nn : ℕ\n⊢ ↑r.num * 2 / (↑r.den * 2) = ↑r",
"state_before": "case h.e'_5.h.e'_3\nr : ℚ\nn : ℕ\n⊢ ↑(r.num * 2) / ↑(↑r.den * 2) = ↑r",
"tactic": "push_cast"
},
{
"state_after": "case h.e'_5.h.e'_3\nr : ℚ\nn : ℕ\n⊢ Rat.divInt (r.num * 2) ↑(r.den * 2) = r",
"state_before": "case h.e'_5.h.e'_3\nr : ℚ\nn : ℕ\n⊢ ↑r.num * 2 / (↑r.den * 2) = ↑r",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_3\nr : ℚ\nn : ℕ\n⊢ Rat.divInt (r.num * 2) ↑(r.den * 2) = r",
"tactic": "simp [Rat.divInt_mul_right (two_ne_zero), Rat.mkRat_self]"
},
{
"state_after": "case refine_2.intro.refine_2.convert_2\nr : ℚ\nn : ℕ\n⊢ 0 < ↑r.den * 2",
"state_before": "case refine_2.intro.refine_2.convert_2\nr : ℚ\nn : ℕ\n⊢ 0 < 1 / ↑(↑r.den * 2) ^ n",
"tactic": "refine' one_div_pos.2 (pow_pos (Int.cast_pos.2 _) _)"
},
{
"state_after": "no goals",
"state_before": "case refine_2.intro.refine_2.convert_2\nr : ℚ\nn : ℕ\n⊢ 0 < ↑r.den * 2",
"tactic": "exact mul_pos (Int.coe_nat_pos.2 r.pos) zero_lt_two"
}
] |
[
72,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_inf_distrib_left
|
[
{
"state_after": "case empty\nF : Type ?u.138129\nα : Type u_2\nβ : Type ?u.138135\nγ : Type ?u.138138\nι : Type u_1\nκ : Type ?u.138144\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\nf : ι → α\na : α\n⊢ a ⊓ sup ∅ f = sup ∅ fun i => a ⊓ f i\n\ncase cons\nF : Type ?u.138129\nα : Type u_2\nβ : Type ?u.138135\nγ : Type ?u.138138\nι : Type u_1\nκ : Type ?u.138144\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\nf : ι → α\na : α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nh : a ⊓ sup s f = sup s fun i => a ⊓ f i\n⊢ a ⊓ sup (cons i s hi) f = sup (cons i s hi) fun i => a ⊓ f i",
"state_before": "F : Type ?u.138129\nα : Type u_2\nβ : Type ?u.138135\nγ : Type ?u.138138\nι : Type u_1\nκ : Type ?u.138144\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\ns : Finset ι\nf : ι → α\na : α\n⊢ a ⊓ sup s f = sup s fun i => a ⊓ f i",
"tactic": "induction' s using Finset.cons_induction with i s hi h"
},
{
"state_after": "no goals",
"state_before": "case empty\nF : Type ?u.138129\nα : Type u_2\nβ : Type ?u.138135\nγ : Type ?u.138138\nι : Type u_1\nκ : Type ?u.138144\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\nf : ι → α\na : α\n⊢ a ⊓ sup ∅ f = sup ∅ fun i => a ⊓ f i",
"tactic": "simp_rw [Finset.sup_empty, inf_bot_eq]"
},
{
"state_after": "no goals",
"state_before": "case cons\nF : Type ?u.138129\nα : Type u_2\nβ : Type ?u.138135\nγ : Type ?u.138138\nι : Type u_1\nκ : Type ?u.138144\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\nf : ι → α\na : α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nh : a ⊓ sup s f = sup s fun i => a ⊓ f i\n⊢ a ⊓ sup (cons i s hi) f = sup (cons i s hi) fun i => a ⊓ f i",
"tactic": "rw [sup_cons, sup_cons, inf_sup_left, h]"
}
] |
[
518,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_plift_down
|
[] |
[
242,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.filter_apply_neg
|
[] |
[
918,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
918,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_append_left
|
[] |
[
1079,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.surjOn_log
|
[] |
[
85,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.coeFn_sub
|
[] |
[
240,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.sublists'_eq_sublists'Aux
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ Array.toList\n (foldr (fun a arr => Array.foldl (fun r l => Array.push r (a :: l)) arr arr 0 (Array.size arr)) #[[]] l) =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ sublists' l = foldr (fun a r => sublists'Aux a r r) [[]] l",
"tactic": "simp only [sublists', sublists'Aux_eq_array_foldl]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ Array.toList\n (foldr (fun a arr => Array.foldl (fun r l => Array.push r (a :: l)) arr arr 0 (Array.size arr)) #[[]] l) =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ Array.toList\n (foldr (fun a arr => Array.foldl (fun r l => Array.push r (a :: l)) arr arr 0 (Array.size arr)) #[[]] l) =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l",
"tactic": "dsimp only"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ foldr ?g₂ (Array.toList #[[]]) l =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l\n\ncase g₂\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ α → List (List α) → List (List α)\n\ncase H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n ?g₂ x (Array.toList y) = Array.toList (Array.foldl (fun r l => Array.push r (x :: l)) y y 0 (Array.size y))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ Array.toList\n (foldr (fun a arr => Array.foldl (fun r l => Array.push r (a :: l)) arr arr 0 (Array.size arr)) #[[]] l) =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l",
"tactic": "rw [← List.foldr_hom Array.toList]"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n Array.toList\n (Array.foldl (fun r l => Array.push r (x :: l)) (toArray (Array.toList y)) (toArray (Array.toList y)) 0\n (Array.size (toArray (Array.toList y)))) =\n Array.toList (Array.foldl (fun r l => Array.push r (x :: l)) y y 0 (Array.size y))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ foldr ?g₂ (Array.toList #[[]]) l =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l\n\ncase g₂\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ α → List (List α) → List (List α)\n\ncase H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n ?g₂ x (Array.toList y) = Array.toList (Array.foldl (fun r l => Array.push r (x :: l)) y y 0 (Array.size y))",
"tactic": ". rfl"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n Array.toList\n (Array.foldl (fun r l => Array.push r (x :: l)) (toArray (Array.toList y)) (toArray (Array.toList y)) 0\n (Array.size (toArray (Array.toList y)))) =\n Array.toList (Array.foldl (fun r l => Array.push r (x :: l)) y y 0 (Array.size y))",
"tactic": ". intros _ _; congr <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ foldr ?g₂ (Array.toList #[[]]) l =\n foldr\n (fun a r =>\n Array.toList\n (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n [[]] l",
"tactic": "rfl"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\nx✝ : α\ny✝ : Array (List α)\n⊢ Array.toList\n (Array.foldl (fun r l => Array.push r (x✝ :: l)) (toArray (Array.toList y✝)) (toArray (Array.toList y✝)) 0\n (Array.size (toArray (Array.toList y✝)))) =\n Array.toList (Array.foldl (fun r l => Array.push r (x✝ :: l)) y✝ y✝ 0 (Array.size y✝))",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n Array.toList\n (Array.foldl (fun r l => Array.push r (x :: l)) (toArray (Array.toList y)) (toArray (Array.toList y)) 0\n (Array.size (toArray (Array.toList y)))) =\n Array.toList (Array.foldl (fun r l => Array.push r (x :: l)) y y 0 (Array.size y))",
"tactic": "intros _ _"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\nl : List α\nx✝ : α\ny✝ : Array (List α)\n⊢ Array.toList\n (Array.foldl (fun r l => Array.push r (x✝ :: l)) (toArray (Array.toList y✝)) (toArray (Array.toList y✝)) 0\n (Array.size (toArray (Array.toList y✝)))) =\n Array.toList (Array.foldl (fun r l => Array.push r (x✝ :: l)) y✝ y✝ 0 (Array.size y✝))",
"tactic": "congr <;> simp"
}
] |
[
69,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.sum_mul
|
[] |
[
1078,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1076,
11
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
mul_le_of_le_one_right'
|
[] |
[
346,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.zero_im
|
[] |
[
861,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
861,
9
] |
Mathlib/GroupTheory/GroupAction/Pi.lean
|
Pi.single_smul₀
|
[] |
[
199,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.lt_div_mul_add
|
[
{
"state_after": "m n k a b : ℕ\nhb : 0 < b\n⊢ a / b < succ (a / b)",
"state_before": "m n k a b : ℕ\nhb : 0 < b\n⊢ a < a / b * b + b",
"tactic": "rw [← Nat.succ_mul, ← Nat.div_lt_iff_lt_mul hb]"
},
{
"state_after": "no goals",
"state_before": "m n k a b : ℕ\nhb : 0 < b\n⊢ a / b < succ (a / b)",
"tactic": "exact Nat.lt_succ_self _"
}
] |
[
681,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean
|
Bool.eq_true_eq_not_eq_false
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (¬b = false) = (b = true)",
"tactic": "simp"
}
] |
[
75,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.map_inl
|
[] |
[
218,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Finsupp/BigOperators.lean
|
Multiset.support_sum_eq
|
[
{
"state_after": "case h\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs✝ : Pairwise (_root_.Disjoint on Finsupp.support) s\na : List (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) (Quot.mk Setoid.r a)\n⊢ (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))",
"state_before": "ι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\n⊢ (sum s).support = sup (map Finsupp.support s)",
"tactic": "induction' s using Quot.inductionOn with a"
},
{
"state_after": "case h.intro.intro\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\n⊢ (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))",
"state_before": "case h\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs✝ : Pairwise (_root_.Disjoint on Finsupp.support) s\na : List (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) (Quot.mk Setoid.r a)\n⊢ (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))",
"tactic": "obtain ⟨l, hl, hd⟩ := hs"
},
{
"state_after": "case h.intro.intro\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))\n\ncase this\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\n⊢ List.Pairwise (_root_.Disjoint on Finsupp.support) a",
"state_before": "case h.intro.intro\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\n⊢ (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))",
"tactic": "suffices : a.Pairwise (_root_.Disjoint on Finsupp.support)"
},
{
"state_after": "case h.e'_2.h.e'_4\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ sum (Quot.mk Setoid.r a) = List.sum a\n\ncase h.e'_3\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr ((fun x x_1 => x ⊔ x_1) ∘ Finsupp.support) ∅ a",
"state_before": "case h.intro.intro\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))",
"tactic": "convert List.support_sum_eq a this"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_4\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ sum (Quot.mk Setoid.r a) = List.sum a",
"tactic": "simp only [Multiset.quot_mk_to_coe'', Multiset.coe_sum]"
},
{
"state_after": "case h.e'_3\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr (fun x x_1 => x.support ⊔ x_1) ∅ a",
"state_before": "case h.e'_3\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr ((fun x x_1 => x ⊔ x_1) ∘ Finsupp.support) ∅ a",
"tactic": "dsimp only [Function.comp]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n⊢ sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr (fun x x_1 => x.support ⊔ x_1) ∅ a",
"tactic": "simp only [quot_mk_to_coe'', coe_map, sup_coe, ge_iff_le, Finset.le_eq_subset,\n Finset.sup_eq_union, Finset.bot_eq_empty, List.foldr_map]"
},
{
"state_after": "case this\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nhl : a ~ l\n⊢ List.Pairwise (_root_.Disjoint on Finsupp.support) a",
"state_before": "case this\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhl : Quot.mk Setoid.r a = ↑l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\n⊢ List.Pairwise (_root_.Disjoint on Finsupp.support) a",
"tactic": "simp only [Multiset.quot_mk_to_coe'', Multiset.coe_map, Multiset.coe_eq_coe] at hl"
},
{
"state_after": "no goals",
"state_before": "case this\nι : Type u_2\nM : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommMonoid M\ns : Multiset (ι →₀ M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (ι →₀ M)\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nhl : a ~ l\n⊢ List.Pairwise (_root_.Disjoint on Finsupp.support) a",
"tactic": "exact hl.symm.pairwise hd fun _ _ h ↦ _root_.Disjoint.symm h"
}
] |
[
113,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.680056\nF : Type u_2\nG : Type ?u.680062\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nhq0_lt : 0 < q\nhfq : snorm' f q μ < ⊤\n⊢ snorm' (fun a => f a) q μ ^ q < ⊤",
"state_before": "α : Type u_1\nE : Type ?u.680056\nF : Type u_2\nG : Type ?u.680062\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nhq0_lt : 0 < q\nhfq : snorm' f q μ < ⊤\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) < ⊤",
"tactic": "rw [lintegral_rpow_nnnorm_eq_rpow_snorm' hq0_lt]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.680056\nF : Type u_2\nG : Type ?u.680062\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nhq0_lt : 0 < q\nhfq : snorm' f q μ < ⊤\n⊢ snorm' (fun a => f a) q μ ^ q < ⊤",
"tactic": "exact ENNReal.rpow_lt_top_of_nonneg (le_of_lt hq0_lt) (ne_of_lt hfq)"
}
] |
[
149,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.pow_eq_closure_pow_set
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.290494\nG : Type ?u.290497\nM : Type ?u.290500\nR : Type u_1\nA : Type ?u.290506\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : Semiring R\ns : AddSubmonoid R\nn : ℕ\n⊢ s ^ n = closure (↑s ^ n)",
"tactic": "rw [← closure_pow, closure_eq]"
}
] |
[
688,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
1
] |
src/lean/Init/Data/Array/Basic.lean
|
Array.size_swap
|
[
{
"state_after": "α : Type u_1\na : Array α\ni j : Fin (size a)\n⊢ size (set (set a i (get a j)) (Eq.rec j (_ : size a = size (set a i (get a j)))) (get a i)) = size a",
"state_before": "α : Type u_1\na : Array α\ni j : Fin (size a)\n⊢ size (swap a i j) = size a",
"tactic": "show ((a.set i (a.get j)).set (size_set a i _ ▸ j) (a.get i)).size = a.size"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\ni j : Fin (size a)\n⊢ size (set (set a i (get a j)) (Eq.rec j (_ : size a = size (set a i (get a j)))) (get a i)) = size a",
"tactic": "rw [size_set, size_set]"
}
] |
[
625,
26
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
623,
9
] |
Mathlib/Topology/Maps.lean
|
OpenEmbedding.comp
|
[] |
[
615,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
11
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.symm_mk
|
[] |
[
544,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.smul_imI
|
[] |
[
1038,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1038,
9
] |
Mathlib/LinearAlgebra/Dual.lean
|
Module.DualBases.coe_basis
|
[
{
"state_after": "case h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni : ι\n⊢ ↑(basis h) i = e i",
"state_before": "R : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\n⊢ ↑(basis h) = e",
"tactic": "ext i"
},
{
"state_after": "case h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni : ι\n⊢ ↑(basis h).repr (e i) = Finsupp.single i 1",
"state_before": "case h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni : ι\n⊢ ↑(basis h) i = e i",
"tactic": "rw [Basis.apply_eq_iff]"
},
{
"state_after": "case h.h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni j : ι\n⊢ ↑(↑(basis h).repr (e i)) j = ↑(Finsupp.single i 1) j",
"state_before": "case h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni : ι\n⊢ ↑(basis h).repr (e i) = Finsupp.single i 1",
"tactic": "ext j"
},
{
"state_after": "case h.h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni j : ι\n⊢ (if j = i then 1 else 0) = if i = j then 1 else 0",
"state_before": "case h.h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni j : ι\n⊢ ↑(↑(basis h).repr (e i)) j = ↑(Finsupp.single i 1) j",
"tactic": "rw [h.basis_repr_apply, coeffs_apply, h.eval, Finsupp.single_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u_3\nM : Type u_1\nι : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ne : ι → M\nε : ι → Dual R M\ninst✝ : DecidableEq ι\nh : DualBases e ε\ni j : ι\n⊢ (if j = i then 1 else 0) = if i = j then 1 else 0",
"tactic": "convert if_congr (eq_comm (a := j) (b := i)) rfl rfl"
}
] |
[
751,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.rel_filterMap
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nf : α → Option γ\ng : β → Option δ\nhfg : (R ⇒ Option.Rel P) f g\na : α\nas : List α\nb : β\nbs : List β\nh₁ : R a b\nh₂ : Forall₂ R as bs\n⊢ Forall₂ P\n (match f a with\n | none => filterMap f as\n | some b => b :: filterMap f as)\n (match g b with\n | none => filterMap g bs\n | some b => b :: filterMap g bs)",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nf : α → Option γ\ng : β → Option δ\nhfg : (R ⇒ Option.Rel P) f g\na : α\nas : List α\nb : β\nbs : List β\nh₁ : R a b\nh₂ : Forall₂ R as bs\n⊢ Forall₂ P (filterMap f (a :: as)) (filterMap g (b :: bs))",
"tactic": "rw [filterMap_cons, filterMap_cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type u_4\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nf : α → Option γ\ng : β → Option δ\nhfg : (R ⇒ Option.Rel P) f g\na : α\nas : List α\nb : β\nbs : List β\nh₁ : R a b\nh₂ : Forall₂ R as bs\n⊢ Forall₂ P\n (match f a with\n | none => filterMap f as\n | some b => b :: filterMap f as)\n (match g b with\n | none => filterMap g bs\n | some b => b :: filterMap g bs)",
"tactic": "exact\n match f a, g b, hfg h₁ with\n | _, _, Option.Rel.none => rel_filterMap (@hfg) h₂\n | _, _, Option.Rel.some h => Forall₂.cons h (rel_filterMap (@hfg) h₂)"
}
] |
[
312,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_le_add
|
[] |
[
2030,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2028,
1
] |
Mathlib/Data/Dfinsupp/Lex.lean
|
Dfinsupp.lex_lt_of_lt_of_preorder
|
[
{
"state_after": "case intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt : x < y\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"tactic": "obtain ⟨hle, j, hlt⟩ := Pi.lt_def.1 hlt"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"tactic": "classical\nhave : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn\nobtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩\nrefine' ⟨i, fun k hk ↦ ⟨hle k, _⟩, hi⟩\nexact of_not_not fun h ↦ hl ⟨k, mem_neLocus.2 (ne_of_not_le h).symm⟩ ((hle k).lt_of_not_le h) hk"
},
{
"state_after": "case intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\nthis : Set.WellFoundedOn (↑(neLocus x y)) r\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"state_before": "case intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"tactic": "have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\nthis : Set.WellFoundedOn (↑(neLocus x y)) r\ni : ↑↑(neLocus x y)\nhi : i ∈ {i | ↑x ↑i < ↑y ↑i}\nhl : ∀ (x_1 : ↑↑(neLocus x y)), x_1 ∈ {i | ↑x ↑i < ↑y ↑i} → ¬r ↑x_1 ↑i\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"state_before": "case intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\nthis : Set.WellFoundedOn (↑(neLocus x y)) r\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"tactic": "obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\nthis : Set.WellFoundedOn (↑(neLocus x y)) r\ni : ↑↑(neLocus x y)\nhi : i ∈ {i | ↑x ↑i < ↑y ↑i}\nhl : ∀ (x_1 : ↑↑(neLocus x y)), x_1 ∈ {i | ↑x ↑i < ↑y ↑i} → ¬r ↑x_1 ↑i\nk : ι\nhk : r k ↑i\n⊢ ↑y k ≤ ↑x k",
"state_before": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\nthis : Set.WellFoundedOn (↑(neLocus x y)) r\ni : ↑↑(neLocus x y)\nhi : i ∈ {i | ↑x ↑i < ↑y ↑i}\nhl : ∀ (x_1 : ↑↑(neLocus x y)), x_1 ∈ {i | ↑x ↑i < ↑y ↑i} → ¬r ↑x_1 ↑i\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"tactic": "refine' ⟨i, fun k hk ↦ ⟨hle k, _⟩, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → Preorder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt✝ : x < y\nhle : (fun i => ↑x i) ≤ fun i => ↑y i\nj : ι\nhlt : ↑x j < ↑y j\nthis : Set.WellFoundedOn (↑(neLocus x y)) r\ni : ↑↑(neLocus x y)\nhi : i ∈ {i | ↑x ↑i < ↑y ↑i}\nhl : ∀ (x_1 : ↑↑(neLocus x y)), x_1 ∈ {i | ↑x ↑i < ↑y ↑i} → ¬r ↑x_1 ↑i\nk : ι\nhk : r k ↑i\n⊢ ↑y k ≤ ↑x k",
"tactic": "exact of_not_not fun h ↦ hl ⟨k, mem_neLocus.2 (ne_of_not_le h).symm⟩ ((hle k).lt_of_not_le h) hk"
}
] |
[
60,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean
|
spectrum.unit_smul_eq_smul
|
[
{
"state_after": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\n⊢ x ∈ σ (r • a) ↔ x ∈ r • σ a",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\n⊢ σ (r • a) = r • σ a",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ x ∈ σ (r • a) ↔ x ∈ r • σ a",
"state_before": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\n⊢ x ∈ σ (r • a) ↔ x ∈ r • σ a",
"tactic": "have x_eq : x = r • r⁻¹ • x := by simp"
},
{
"state_after": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ r • r⁻¹ • x ∈ σ (r • a) ↔ x ∈ r • σ a",
"state_before": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ x ∈ σ (r • a) ↔ x ∈ r • σ a",
"tactic": "nth_rw 1 [x_eq]"
},
{
"state_after": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ r⁻¹ • x ∈ σ a ↔ x ∈ r • σ a",
"state_before": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ r • r⁻¹ • x ∈ σ (r • a) ↔ x ∈ r • σ a",
"tactic": "rw [smul_mem_smul_iff]"
},
{
"state_after": "case h.mp\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ r⁻¹ • x ∈ σ a → x ∈ r • σ a\n\ncase h.mpr\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ x ∈ r • σ a → r⁻¹ • x ∈ σ a",
"state_before": "case h\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ r⁻¹ • x ∈ σ a ↔ x ∈ r • σ a",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\n⊢ x = r • r⁻¹ • x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.mp\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ r⁻¹ • x ∈ σ a → x ∈ r • σ a",
"tactic": "exact fun h => ⟨r⁻¹ • x, ⟨h, show r • r⁻¹ • x = x by simp⟩⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\nh : r⁻¹ • x ∈ σ a\n⊢ r • r⁻¹ • x = x",
"tactic": "simp"
},
{
"state_after": "case h.mpr.intro.intro\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\nw : R\nleft✝ : w ∈ σ a\nx'_eq : r • w = x\n⊢ r⁻¹ • x ∈ σ a",
"state_before": "case h.mpr\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\n⊢ x ∈ r • σ a → r⁻¹ • x ∈ σ a",
"tactic": "rintro ⟨w, _, (x'_eq : r • w = x)⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : Rˣ\nx : R\nx_eq : x = r • r⁻¹ • x\nw : R\nleft✝ : w ∈ σ a\nx'_eq : r • w = x\n⊢ r⁻¹ • x ∈ σ a",
"tactic": "simpa [← x'_eq ]"
}
] |
[
225,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.isPreconnected_setOf_sSameSide
|
[
{
"state_after": "case inl\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : ↑s = ∅\n⊢ IsPreconnected {y | SSameSide s x y}\n\ncase inr\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\n⊢ IsPreconnected {y | SSameSide s x y}",
"state_before": "R : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\n⊢ IsPreconnected {y | SSameSide s x y}",
"tactic": "rcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)"
},
{
"state_after": "case inl\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : s = ⊥\n⊢ IsPreconnected {y | SSameSide s x y}",
"state_before": "case inl\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : ↑s = ∅\n⊢ IsPreconnected {y | SSameSide s x y}",
"tactic": "rw [coe_eq_bot_iff] at h"
},
{
"state_after": "case inl\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : s = ⊥\n⊢ IsPreconnected {y | False}",
"state_before": "case inl\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : s = ⊥\n⊢ IsPreconnected {y | SSameSide s x y}",
"tactic": "simp only [h, not_sSameSide_bot]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : s = ⊥\n⊢ IsPreconnected {y | False}",
"tactic": "exact isPreconnected_empty"
},
{
"state_after": "case pos\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\nhx : x ∈ s\n⊢ IsPreconnected {y | SSameSide s x y}\n\ncase neg\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\nhx : ¬x ∈ s\n⊢ IsPreconnected {y | SSameSide s x y}",
"state_before": "case inr\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\n⊢ IsPreconnected {y | SSameSide s x y}",
"tactic": "by_cases hx : x ∈ s"
},
{
"state_after": "case pos\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\nhx : x ∈ s\n⊢ IsPreconnected {y | False}",
"state_before": "case pos\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\nhx : x ∈ s\n⊢ IsPreconnected {y | SSameSide s x y}",
"tactic": "simp only [hx, SSameSide, not_true, false_and_iff, and_false_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\nhx : x ∈ s\n⊢ IsPreconnected {y | False}",
"tactic": "exact isPreconnected_empty"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type ?u.485293\nV : Type u_1\nV' : Type ?u.485299\nP : Type u_2\nP' : Type ?u.485305\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : Set.Nonempty ↑s\nhx : ¬x ∈ s\n⊢ IsPreconnected {y | SSameSide s x y}",
"tactic": "exact (isConnected_setOf_sSameSide hx h).isPreconnected"
}
] |
[
903,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Sym/Basic.lean
|
SymOptionSuccEquiv.decode_inr
|
[] |
[
642,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.coeff_toMvPowerSeries
|
[] |
[
1270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1268,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
smoothManifoldWithCorners_of_contDiffOn
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nh :\n ∀ (e e' : LocalHomeomorph M H),\n e ∈ atlas H M →\n e' ∈ atlas H M →\n ContDiffOn 𝕜 ⊤ (↑I ∘ ↑(LocalHomeomorph.symm e ≫ₕ e') ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ range ↑I)\nthis : HasGroupoid M (contDiffGroupoid ⊤ I)\n⊢ ∀ {e e' : LocalHomeomorph M H}, e ∈ atlas H M → e' ∈ atlas H M → LocalHomeomorph.symm e ≫ₕ e' ∈ contDiffGroupoid ⊤ I",
"state_before": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nh :\n ∀ (e e' : LocalHomeomorph M H),\n e ∈ atlas H M →\n e' ∈ atlas H M →\n ContDiffOn 𝕜 ⊤ (↑I ∘ ↑(LocalHomeomorph.symm e ≫ₕ e') ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ range ↑I)\n⊢ ∀ {e e' : LocalHomeomorph M H}, e ∈ atlas H M → e' ∈ atlas H M → LocalHomeomorph.symm e ≫ₕ e' ∈ contDiffGroupoid ⊤ I",
"tactic": "haveI : HasGroupoid M (contDiffGroupoid ∞ I) := hasGroupoid_of_pregroupoid _ (h _ _)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nh :\n ∀ (e e' : LocalHomeomorph M H),\n e ∈ atlas H M →\n e' ∈ atlas H M →\n ContDiffOn 𝕜 ⊤ (↑I ∘ ↑(LocalHomeomorph.symm e ≫ₕ e') ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source ∩ range ↑I)\nthis : HasGroupoid M (contDiffGroupoid ⊤ I)\n⊢ ∀ {e e' : LocalHomeomorph M H}, e ∈ atlas H M → e' ∈ atlas H M → LocalHomeomorph.symm e ≫ₕ e' ∈ contDiffGroupoid ⊤ I",
"tactic": "apply StructureGroupoid.compatible"
}
] |
[
666,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.exists_ne_forall_mem_nhds_pos_measure_preimage
|
[
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nh : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nh : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "set m : OuterMeasure β := OuterMeasure.map f μ.toOuterMeasure"
},
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nh : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "replace h : ∀ b : β, m ({b}ᶜ) ≠ 0 := fun b => not_eventually.mpr (h b)"
},
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "inhabit β"
},
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "have : m univ ≠ 0 := ne_bot_of_le_ne_bot (h default) (m.mono' <| subset_univ _)"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝[univ] b → 0 < ↑m t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "α : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "rcases m.exists_mem_forall_mem_nhds_within_pos this with ⟨b, -, hb⟩"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "case intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝[univ] b → 0 < ↑m t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "simp only [nhdsWithin_univ] at hb"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\na : β\nhab : a ≠ b\nha : ∀ (t : Set β), t ∈ 𝓝[{b}ᶜ] a → 0 < ↑m t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "case intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "rcases m.exists_mem_forall_mem_nhds_within_pos (h b) with ⟨a, hab : a ≠ b, ha⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\na : β\nhab : a ≠ b\nha : ∀ (t : Set β), t ∈ 𝓝 a → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\na : β\nhab : a ≠ b\nha : ∀ (t : Set β), t ∈ 𝓝[{b}ᶜ] a → 0 < ↑m t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "simp only [isOpen_compl_singleton.nhdsWithin_eq hab] at ha"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ✝ : Type ?u.839313\nγ : Type ?u.839316\nδ : Type ?u.839319\nι : Type ?u.839322\nR : Type ?u.839325\nR' : Type ?u.839328\nm0 : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β✝\ninst✝⁴ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := ↑(OuterMeasure.map f) ↑μ\nh : ∀ (b : β), ↑m ({b}ᶜ) ≠ 0\ninhabited_h : Inhabited β\nthis : ↑m univ ≠ 0\nb : β\nhb : ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\na : β\nhab : a ≠ b\nha : ∀ (t : Set β), t ∈ 𝓝 a → 0 < ↑(↑(OuterMeasure.map f) ↑μ) t\n⊢ ∃ a b, a ≠ b ∧ (∀ (s : Set β), s ∈ 𝓝 a → 0 < ↑↑μ (f ⁻¹' s)) ∧ ∀ (t : Set β), t ∈ 𝓝 b → 0 < ↑↑μ (f ⁻¹' t)",
"tactic": "exact ⟨a, b, hab, ha, hb⟩"
}
] |
[
4035,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4023,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.nhds_top
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.8600\nα : Type u_1\nβ : Type ?u.8606\nX : Type ?u.8609\nY : Type ?u.8612\n⊢ 𝓝 ⊤ = ⊤",
"tactic": "simp [nhds_eq]"
}
] |
[
122,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Order/Disjoint.lean
|
IsCompl.inf_sup
|
[] |
[
575,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Covby.nhdsWithin_Iio
|
[] |
[
468,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.IsPrime.radical
|
[] |
[
916,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_top
|
[] |
[
870,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
dist_eq_norm_vsub'
|
[] |
[
89,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.unique_single_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.132948\nγ : Type ?u.132951\nι : Type ?u.132954\nM : Type u_2\nM' : Type ?u.132960\nN : Type ?u.132963\nP : Type ?u.132966\nG : Type ?u.132969\nH : Type ?u.132972\nR : Type ?u.132975\nS : Type ?u.132978\ninst✝¹ : Zero M\na a' : α\nb : M\ninst✝ : Unique α\nb' : M\n⊢ single a b = single a' b' ↔ b = b'",
"tactic": "rw [unique_ext_iff, Unique.eq_default a, Unique.eq_default a', single_eq_same, single_eq_same]"
}
] |
[
461,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
1
] |
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
Complex.integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real
|
[] |
[
225,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
DirectSum.coe_mul_of_apply_add
|
[] |
[
219,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffOn_id
|
[] |
[
186,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.mk'_linear
|
[] |
[
224,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.of_smul
|
[] |
[
90,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.subgroupOf_map_subtype
|
[] |
[
1633,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1632,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.sups_sups_sups_comm
|
[] |
[
216,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.apply_lt_nfpFamily
|
[] |
[
93,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.linearMap_ext
|
[] |
[
142,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Computability/RegularExpressions.lean
|
RegularExpression.deriv_char_self
|
[] |
[
193,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.BlankRel.symm
|
[] |
[
131,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.le_seq
|
[] |
[
2635,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2633,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
IsPreconnected.mem_intervals
|
[
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\nhs : IsPreconnected ∅\n⊢ ∅ ∈\n {Icc (sInf ∅) (sSup ∅), Ico (sInf ∅) (sSup ∅), Ioc (sInf ∅) (sSup ∅), Ioo (sInf ∅) (sSup ∅), Ici (sInf ∅),\n Ioi (sInf ∅), Iic (sSup ∅), Iio (sSup ∅), univ, ∅}\n\ncase inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "rcases s.eq_empty_or_nonempty with (rfl | hne)"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "have hs' : IsConnected s := ⟨hne, hs⟩"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase pos\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : ¬BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "by_cases hb : BddBelow s <;> by_cases ha : BddAbove s"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\nhs : IsPreconnected ∅\n⊢ ∅ ∈\n {Icc (sInf ∅) (sSup ∅), Ico (sInf ∅) (sSup ∅), Ioc (sInf ∅) (sSup ∅), Ioo (sInf ∅) (sSup ∅), Ici (sInf ∅),\n Ioi (sInf ∅), Iic (sSup ∅), Iio (sSup ∅), univ, ∅}",
"tactic": "apply_rules [Or.inr, mem_singleton]"
},
{
"state_after": "case pos.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s = Icc (sInf s) (sSup s)\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase pos.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s = Ico (sInf s) (sSup s)\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase pos.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s = Ioc (sInf s) (sSup s)\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase pos.inr.inr.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s ∈ {Ioo (sInf s) (sSup s)}\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
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"tactic": "rcases mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset (hs'.Ioo_csInf_csSup_subset hb ha)\n (subset_Icc_csInf_csSup hb ha) with (hs | hs | hs | hs)"
},
{
"state_after": "no goals",
"state_before": "case pos.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s = Icc (sInf s) (sSup s)\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case pos.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s = Ico (sInf s) (sSup s)\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inr <| Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case pos.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s = Ioc (sInf s) (sSup s)\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inr <| Or.inr <| Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case pos.inr.inr.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\nhs : s ∈ {Ioo (sInf s) (sSup s)}\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inr <| Or.inr <| Or.inr <| Or.inl hs"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\n⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "refine' Or.inr <| Or.inr <| Or.inr <| Or.inr _"
},
{
"state_after": "case neg.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\nhs : s = Ici (sInf s)\n⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase neg.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\nhs : s ∈ {Ioi (sInf s)}\n⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\n⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "cases'\n mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_csInf_subset hb ha) fun x hx => csInf_le hb hx with\n hs hs"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\nhs : s = Ici (sInf s)\n⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\nhs : s ∈ {Ioi (sInf s)}\n⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inr (Or.inl hs)"
},
{
"state_after": "case pos.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case pos\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "iterate 6 apply Or.inr"
},
{
"state_after": "case pos.h.h.h.h.h.h.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\nhs : s = Iic (sSup s)\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}\n\ncase pos.h.h.h.h.h.h.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\nhs : s ∈ {Iio (sSup s)}\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case pos.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "cases' mem_Iic_Iio_of_subset_of_subset (hs.Iio_csSup_subset hb ha) fun x hx => le_csSup ha hx\n with hs hs"
},
{
"state_after": "case pos.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}",
"state_before": "case pos.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈ {Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "apply Or.inr"
},
{
"state_after": "no goals",
"state_before": "case pos.h.h.h.h.h.h.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\nhs : s = Iic (sSup s)\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case pos.h.h.h.h.h.h.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\nhs : s ∈ {Iio (sSup s)}\n⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "exact Or.inr (Or.inl hs)"
},
{
"state_after": "case neg.h.h.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : ¬BddAbove s\n⊢ s ∈ {univ, ∅}",
"state_before": "case neg\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : ¬BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),\n Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}",
"tactic": "iterate 8 apply Or.inr"
},
{
"state_after": "no goals",
"state_before": "case neg.h.h.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : ¬BddAbove s\n⊢ s ∈ {univ, ∅}",
"tactic": "exact Or.inl (hs.eq_univ_of_unbounded hb ha)"
},
{
"state_after": "case neg.h.h.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : ¬BddAbove s\n⊢ s ∈ {univ, ∅}",
"state_before": "case neg.h.h.h.h.h.h.h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : IsPreconnected s\nhne : Set.Nonempty s\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : ¬BddAbove s\n⊢ s ∈ {Iio (sSup s), univ, ∅}",
"tactic": "apply Or.inr"
}
] |
[
307,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
Monotone.pow_right
|
[
{
"state_after": "no goals",
"state_before": "β : Type u_1\nA : Type ?u.79462\nG : Type ?u.79465\nM : Type u_2\nR : Type ?u.79471\ninst✝⁴ : Monoid M\ninst✝³ : Preorder M\ninst✝² : Preorder β\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : β → M\nhf : Monotone f\n⊢ Monotone fun a => f a ^ 0",
"tactic": "simpa using monotone_const"
},
{
"state_after": "β : Type u_1\nA : Type ?u.79462\nG : Type ?u.79465\nM : Type u_2\nR : Type ?u.79471\ninst✝⁴ : Monoid M\ninst✝³ : Preorder M\ninst✝² : Preorder β\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : β → M\nhf : Monotone f\nn : ℕ\n⊢ Monotone fun a => f a * f a ^ n",
"state_before": "β : Type u_1\nA : Type ?u.79462\nG : Type ?u.79465\nM : Type u_2\nR : Type ?u.79471\ninst✝⁴ : Monoid M\ninst✝³ : Preorder M\ninst✝² : Preorder β\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : β → M\nhf : Monotone f\nn : ℕ\n⊢ Monotone fun a => f a ^ (n + 1)",
"tactic": "simp_rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "β : Type u_1\nA : Type ?u.79462\nG : Type ?u.79465\nM : Type u_2\nR : Type ?u.79471\ninst✝⁴ : Monoid M\ninst✝³ : Preorder M\ninst✝² : Preorder β\ninst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : β → M\nhf : Monotone f\nn : ℕ\n⊢ Monotone fun a => f a * f a ^ n",
"tactic": "exact hf.mul' (Monotone.pow_right hf _)"
}
] |
[
192,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_invOfUnit
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\nu : Rˣ\n⊢ ↑(coeff R n) (invOfUnit φ u) =\n if n = 0 then ↑u⁻¹\n else -↑u⁻¹ * ∑ x in antidiagonal n, if x.snd < n then ↑(coeff R x.fst) φ * ↑(coeff R x.snd) (invOfUnit φ u) else 0",
"tactic": "convert coeff_inv_aux n (↑u⁻¹) φ"
}
] |
[
850,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.IsCycle.toDeleteEdges
|
[] |
[
1821,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1819,
11
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.isDiag_diagonal
|
[] |
[
45,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/RingTheory/SimpleModule.lean
|
LinearMap.injective_of_ne_zero
|
[] |
[
140,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Topology/Basic.lean
|
ContinuousAt.comp
|
[] |
[
1652,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1650,
8
] |
Mathlib/Data/Finset/Lattice.lean
|
Set.iInter_eq_iInter_finset
|
[] |
[
1870,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1869,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
CategoryTheory.Limits.Types.binaryCofan_isColimit_iff
|
[
{
"state_after": "case mp\nX Y : Type u\nc : BinaryCofan X Y\n⊢ Nonempty (IsColimit c) →\n Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n\ncase mpr\nX Y : Type u\nc : BinaryCofan X Y\n⊢ Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) →\n Nonempty (IsColimit c)",
"state_before": "X Y : Type u\nc : BinaryCofan X Y\n⊢ Nonempty (IsColimit c) ↔\n Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))",
"state_before": "case mp\nX Y : Type u\nc : BinaryCofan X Y\n⊢ Nonempty (IsColimit c) →\n Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))",
"tactic": "rintro ⟨h⟩"
},
{
"state_after": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ Injective\n ((binaryCoproductCocone X Y).ι.app { as := left } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n Injective\n ((binaryCoproductCocone X Y).ι.app { as := right } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n IsCompl\n (Set.range\n ((binaryCoproductCocone X Y).ι.app { as := left } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range\n ((binaryCoproductCocone X Y).ι.app { as := right } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))",
"state_before": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))",
"tactic": "rw [← show _ = c.inl from\n h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.left⟩,\n ← show _ = c.inr from\n h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.right⟩]"
},
{
"state_after": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ Injective (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n Injective (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n IsCompl (Set.range (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))",
"state_before": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ Injective\n ((binaryCoproductCocone X Y).ι.app { as := left } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n Injective\n ((binaryCoproductCocone X Y).ι.app { as := right } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n IsCompl\n (Set.range\n ((binaryCoproductCocone X Y).ι.app { as := left } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range\n ((binaryCoproductCocone X Y).ι.app { as := right } ≫\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))",
"tactic": "dsimp [binaryCoproductCocone]"
},
{
"state_after": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ IsCompl (Set.range (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))",
"state_before": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ Injective (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n Injective (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧\n IsCompl (Set.range (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))",
"tactic": "refine'\n ⟨(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp\n Sum.inl_injective,\n (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp\n Sum.inr_injective, _⟩"
},
{
"state_after": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv '' Set.range Sum.inl =\n ↑(IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).symm.toEquiv '' Set.range Sum.inrᶜ",
"state_before": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ IsCompl (Set.range (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))",
"tactic": "erw [Set.range_comp, ← eq_compl_iff_isCompl, Set.range_comp _ Sum.inr, ←\n Set.image_compl_eq\n (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.bijective]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv '' Set.range Sum.inl =\n ↑(IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).symm.toEquiv '' Set.range Sum.inrᶜ",
"tactic": "simp"
},
{
"state_after": "case mpr.intro.intro\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n⊢ Nonempty (IsColimit c)",
"state_before": "case mpr\nX Y : Type u\nc : BinaryCofan X Y\n⊢ Injective (BinaryCofan.inl c) ∧\n Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) →\n Nonempty (IsColimit c)",
"tactic": "rintro ⟨h₁, h₂, h₃⟩"
},
{
"state_after": "case mpr.intro.intro\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ Nonempty (IsColimit c)",
"state_before": "case mpr.intro.intro\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n⊢ Nonempty (IsColimit c)",
"tactic": "have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by\n rw [eq_compl_iff_isCompl.mpr h₃.symm]\n exact fun _ => or_not"
},
{
"state_after": "case mpr.intro.intro.refine'_1\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ {T : Type u} → (X ⟶ T) → (Y ⟶ T) → (c.pt ⟶ T)\n\ncase mpr.intro.intro.refine'_2\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ ∀ {T : Type u} (f : X ⟶ T) (g : Y ⟶ T), BinaryCofan.inl c ≫ ?mpr.intro.intro.refine'_1 f g = f\n\ncase mpr.intro.intro.refine'_3\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ ∀ {T : Type u} (f : X ⟶ T) (g : Y ⟶ T), BinaryCofan.inr c ≫ ?mpr.intro.intro.refine'_1 f g = g\n\ncase mpr.intro.intro.refine'_4\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ ∀ {T : Type u} (f : X ⟶ T) (g : Y ⟶ T) (m : c.pt ⟶ T),\n BinaryCofan.inl c ≫ m = f → BinaryCofan.inr c ≫ m = g → m = ?mpr.intro.intro.refine'_1 f g",
"state_before": "case mpr.intro.intro\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ Nonempty (IsColimit c)",
"tactic": "refine' ⟨BinaryCofan.IsColimit.mk _ _ _ _ _⟩"
},
{
"state_after": "X Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n⊢ ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inl c)ᶜ",
"state_before": "X Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n⊢ ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)",
"tactic": "rw [eq_compl_iff_isCompl.mpr h₃.symm]"
},
{
"state_after": "no goals",
"state_before": "X Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n⊢ ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inl c)ᶜ",
"tactic": "exact fun _ => or_not"
},
{
"state_after": "case mpr.intro.intro.refine'_1\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : c.pt\n⊢ T",
"state_before": "case mpr.intro.intro.refine'_1\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ {T : Type u} → (X ⟶ T) → (Y ⟶ T) → (c.pt ⟶ T)",
"tactic": "intro T f g x"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.refine'_1\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : c.pt\n⊢ T",
"tactic": "exact\n if h : x ∈ Set.range c.inl then f ((Equiv.ofInjective _ h₁).symm ⟨x, h⟩)\n else g ((Equiv.ofInjective _ h₂).symm ⟨x, (this x).resolve_left h⟩)"
},
{
"state_after": "case mpr.intro.intro.refine'_2\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\n⊢ (BinaryCofan.inl c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })) =\n f",
"state_before": "case mpr.intro.intro.refine'_2\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ ∀ {T : Type u} (f : X ⟶ T) (g : Y ⟶ T),\n (BinaryCofan.inl c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })) =\n f",
"tactic": "intro T f g"
},
{
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"tactic": "funext x"
},
{
"state_after": "case mpr.intro.intro.refine'_2.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := left }\n⊢ (if h : BinaryCofan.inl c x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := BinaryCofan.inl c x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := BinaryCofan.inl c x, property := (_ : BinaryCofan.inl c x ∈ Set.range (BinaryCofan.inr c)) })) =\n f x",
"state_before": "case mpr.intro.intro.refine'_2.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := left }\n⊢ (BinaryCofan.inl c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) }))\n x =\n f x",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.refine'_2.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := left }\n⊢ (if h : BinaryCofan.inl c x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := BinaryCofan.inl c x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := BinaryCofan.inl c x, property := (_ : BinaryCofan.inl c x ∈ Set.range (BinaryCofan.inr c)) })) =\n f x",
"tactic": "simp [h₁.eq_iff]"
},
{
"state_after": "case mpr.intro.intro.refine'_3\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\n⊢ (BinaryCofan.inr c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })) =\n g",
"state_before": "case mpr.intro.intro.refine'_3\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ ∀ {T : Type u} (f : X ⟶ T) (g : Y ⟶ T),\n (BinaryCofan.inr c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })) =\n g",
"tactic": "intro T f g"
},
{
"state_after": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\n⊢ (BinaryCofan.inr c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) }))\n x =\n g x",
"state_before": "case mpr.intro.intro.refine'_3\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\n⊢ (BinaryCofan.inr c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })) =\n g",
"tactic": "funext x"
},
{
"state_after": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\n⊢ (if h : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := BinaryCofan.inr c x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inr c)) })) =\n g x",
"state_before": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\n⊢ (BinaryCofan.inr c ≫ fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) }))\n x =\n g x",
"tactic": "dsimp"
},
{
"state_after": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\n⊢ ∀ (x_1 : X) (h : BinaryCofan.inl c x_1 = BinaryCofan.inr c x),\n f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"state_before": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\n⊢ (if h : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := BinaryCofan.inr c x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inr c)) })) =\n g x",
"tactic": "simp only [Set.mem_range, Equiv.ofInjective_symm_apply,\n dite_eq_right_iff, forall_exists_index]"
},
{
"state_after": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\n⊢ f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"state_before": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\n⊢ ∀ (x_1 : X) (h : BinaryCofan.inl c x_1 = BinaryCofan.inr c x),\n f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"tactic": "intro y e"
},
{
"state_after": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis✝ :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\nthis : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c) ⊓ Set.range (BinaryCofan.inr c)\n⊢ f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"state_before": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\n⊢ f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"tactic": "have : c.inr x ∈ Set.range c.inl ⊓ Set.range c.inr := ⟨⟨_, e⟩, ⟨_, rfl⟩⟩"
},
{
"state_after": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis✝ :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\nthis : BinaryCofan.inr c x ∈ ⊥\n⊢ f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"state_before": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis✝ :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\nthis : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c) ⊓ Set.range (BinaryCofan.inr c)\n⊢ f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"tactic": "rw [disjoint_iff.mp h₃.1] at this"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis✝ :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nf : X ⟶ T\ng : Y ⟶ T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\nthis : BinaryCofan.inr c x ∈ ⊥\n⊢ f\n (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x ∈ Set.range (BinaryCofan.inl c)) }) =\n g x",
"tactic": "exact this.elim"
},
{
"state_after": "case mpr.intro.intro.refine'_4\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt ⟶ T\n⊢ m = fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n (BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n (BinaryCofan.inr c ≫ m)\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })",
"state_before": "case mpr.intro.intro.refine'_4\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\n⊢ ∀ {T : Type u} (f : X ⟶ T) (g : Y ⟶ T) (m : c.pt ⟶ T),\n BinaryCofan.inl c ≫ m = f →\n BinaryCofan.inr c ≫ m = g →\n m = fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n f (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n g\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })",
"tactic": "rintro T _ _ m rfl rfl"
},
{
"state_after": "case mpr.intro.intro.refine'_4.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt ⟶ T\nx : c.pt\n⊢ m x =\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n (BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n (BinaryCofan.inr c ≫ m)\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })",
"state_before": "case mpr.intro.intro.refine'_4\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt ⟶ T\n⊢ m = fun x =>\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n (BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n (BinaryCofan.inr c ≫ m)\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })",
"tactic": "funext x"
},
{
"state_after": "case mpr.intro.intro.refine'_4.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt ⟶ T\nx : c.pt\n⊢ m x =\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n m (BinaryCofan.inl c (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h }))\n else\n m\n (BinaryCofan.inr c\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) }))",
"state_before": "case mpr.intro.intro.refine'_4.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt ⟶ T\nx : c.pt\n⊢ m x =\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n (BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h })\n else\n (BinaryCofan.inr c ≫ m)\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) })",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.refine'_4.h\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective (BinaryCofan.inl c)\nh₂ : Injective (BinaryCofan.inr c)\nh₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n ∀ (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x ∈ Set.range (BinaryCofan.inl c) ∨ x ∈ Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt ⟶ T\nx : c.pt\n⊢ m x =\n if h : x ∈ Set.range (BinaryCofan.inl c) then\n m (BinaryCofan.inl c (↑(Equiv.ofInjective (BinaryCofan.inl c) h₁).symm { val := x, property := h }))\n else\n m\n (BinaryCofan.inr c\n (↑(Equiv.ofInjective (BinaryCofan.inr c) h₂).symm\n { val := x, property := (_ : x ∈ Set.range (BinaryCofan.inr c)) }))",
"tactic": "split_ifs <;> exact congr_arg _ (Equiv.apply_ofInjective_symm _ ⟨_, _⟩).symm"
}
] |
[
304,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.finrank_adjoin_simple_eq_one_iff
|
[
{
"state_after": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ Insert.insert ∅ α ⊆ ↑⊥ ↔ α ∈ ⊥",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ finrank F { x // x ∈ F⟮α⟯ } = 1 ↔ α ∈ ⊥",
"tactic": "rw [finrank_adjoin_eq_one_iff]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ Insert.insert ∅ α ⊆ ↑⊥ ↔ α ∈ ⊥",
"tactic": "exact Set.singleton_subset_iff"
}
] |
[
745,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.of_iUnion_nonpos
|
[] |
[
233,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
fderivWithin_add_const
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.226279\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.226374\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\nhxs : UniqueDiffWithinAt 𝕜 s x\nc : F\nhf : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ ¬DifferentiableWithinAt 𝕜 (fun y => f y + c) s x",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.226279\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.226374\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\nhxs : UniqueDiffWithinAt 𝕜 s x\nc : F\nhf : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ fderivWithin 𝕜 (fun y => f y + c) s x = fderivWithin 𝕜 f s x",
"tactic": "rw [fderivWithin_zero_of_not_differentiableWithinAt hf,\n fderivWithin_zero_of_not_differentiableWithinAt]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.226279\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.226374\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\nhxs : UniqueDiffWithinAt 𝕜 s x\nc : F\nhf : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ ¬DifferentiableWithinAt 𝕜 (fun y => f y + c) s x",
"tactic": "simpa"
}
] |
[
245,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
Real.contDiff_arsinh
|
[] |
[
197,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.zero_val
|
[] |
[
403,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_preimage_of_injective_lift
|
[
{
"state_after": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ Nonempty (↑(f ⁻¹' s) ↪ ↑s)",
"state_before": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ lift (#↑(f ⁻¹' s)) ≤ lift (#↑s)",
"tactic": "rw [lift_mk_le.{u, v, 0}]"
},
{
"state_after": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ Injective (Subtype.coind (fun x => f ↑x) (_ : ∀ (x : ↑(f ⁻¹' s)), f ↑x ∈ s))",
"state_before": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ Nonempty (↑(f ⁻¹' s) ↪ ↑s)",
"tactic": "use Subtype.coind (fun x => f x.1) fun x => mem_preimage.mp x.2"
},
{
"state_after": "case hf\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ Injective fun x => f ↑x",
"state_before": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ Injective (Subtype.coind (fun x => f ↑x) (_ : ∀ (x : ↑(f ⁻¹' s)), f ↑x ∈ s))",
"tactic": "apply Subtype.coind_injective"
},
{
"state_after": "no goals",
"state_before": "case hf\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : Injective f\n⊢ Injective fun x => f ↑x",
"tactic": "exact h.comp Subtype.val_injective"
}
] |
[
2169,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2164,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelEmbedding.ext
|
[] |
[
291,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
OrthogonalFamily.norm_sq_diff_sum
|
[
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\n⊢ ‖∑ i in s₁, ↑(V i) (f i) - ∑ i in s₂, ↑(V i) (f i)‖ ^ 2 = ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "rw [← Finset.sum_sdiff_sub_sum_sdiff, sub_eq_add_neg, ← Finset.sum_neg_distrib]"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "let F : ∀ i, G i := fun i => if i ∈ s₁ then f i else -f i"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "have hF₁ : ∀ i ∈ s₁ \\ s₂, F i = f i := fun i hi => if_pos (Finset.sdiff_subset _ _ hi)"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "have hF₂ : ∀ i ∈ s₂ \\ s₁, F i = -f i := fun i hi => if_neg (Finset.mem_sdiff.mp hi).2"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "have hF : ∀ i, ‖F i‖ = ‖f i‖ := by\n intro i\n dsimp only\n split_ifs <;> simp only [eq_self_iff_true, norm_neg]"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "have :\n ‖(∑ i in s₁ \\ s₂, V i (F i)) + ∑ i in s₂ \\ s₁, V i (F i)‖ ^ 2 =\n (∑ i in s₁ \\ s₂, ‖F i‖ ^ 2) + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2 := by\n have hs : Disjoint (s₁ \\ s₂) (s₂ \\ s₁) := disjoint_sdiff_sdiff\n simpa only [Finset.sum_union hs] using hV.norm_sum F (s₁ \\ s₂ ∪ s₂ \\ s₁)"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_3.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\n⊢ ∑ x in s₁ \\ s₂, ↑(V x) (f x) = ∑ i in s₁ \\ s₂, ↑(V i) (F i)\n\ncase h.e'_2.h.e'_5.h.e'_3.h.e'_6\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\n⊢ ∑ x in s₂ \\ s₁, -↑(V x) (f x) = ∑ i in s₂ \\ s₁, ↑(V i) (F i)\n\ncase h.e'_3.h.e'_5.a.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\nx✝ : ι\na✝ : x✝ ∈ s₁ \\ s₂\n⊢ ‖f x✝‖ = ‖F x✝‖\n\ncase h.e'_3.h.e'_6.a.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\nx✝ : ι\na✝ : x✝ ∈ s₂ \\ s₁\n⊢ ‖f x✝‖ = ‖F x✝‖",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\n⊢ ‖∑ x in s₁ \\ s₂, ↑(V x) (f x) + ∑ x in s₂ \\ s₁, -↑(V x) (f x)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖f i‖ ^ 2",
"tactic": "convert this using 4"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\ni : ι\n⊢ ‖F i‖ = ‖f i‖",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\n⊢ ∀ (i : ι), ‖F i‖ = ‖f i‖",
"tactic": "intro i"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\ni : ι\n⊢ ‖if i ∈ s₁ then f i else -f i‖ = ‖f i‖",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\ni : ι\n⊢ ‖F i‖ = ‖f i‖",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\ni : ι\n⊢ ‖if i ∈ s₁ then f i else -f i‖ = ‖f i‖",
"tactic": "split_ifs <;> simp only [eq_self_iff_true, norm_neg]"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nhs : Disjoint (s₁ \\ s₂) (s₂ \\ s₁)\n⊢ ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\n⊢ ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2",
"tactic": "have hs : Disjoint (s₁ \\ s₂) (s₂ \\ s₁) := disjoint_sdiff_sdiff"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nhs : Disjoint (s₁ \\ s₂) (s₂ \\ s₁)\n⊢ ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2",
"tactic": "simpa only [Finset.sum_union hs] using hV.norm_sum F (s₁ \\ s₂ ∪ s₂ \\ s₁)"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_3.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\ni : ι\nhi : i ∈ s₁ \\ s₂\n⊢ ↑(V i) (f i) = ↑(V i) (F i)",
"state_before": "case h.e'_2.h.e'_5.h.e'_3.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\n⊢ ∑ x in s₁ \\ s₂, ↑(V x) (f x) = ∑ i in s₁ \\ s₂, ↑(V i) (F i)",
"tactic": "refine' Finset.sum_congr rfl fun i hi => _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.h.e'_3.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\ni : ι\nhi : i ∈ s₁ \\ s₂\n⊢ ↑(V i) (f i) = ↑(V i) (F i)",
"tactic": "simp only [hF₁ i hi]"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_3.h.e'_6\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\ni : ι\nhi : i ∈ s₂ \\ s₁\n⊢ -↑(V i) (f i) = ↑(V i) (F i)",
"state_before": "case h.e'_2.h.e'_5.h.e'_3.h.e'_6\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\n⊢ ∑ x in s₂ \\ s₁, -↑(V x) (f x) = ∑ i in s₂ \\ s₁, ↑(V i) (F i)",
"tactic": "refine' Finset.sum_congr rfl fun i hi => _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.h.e'_3.h.e'_6\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\ni : ι\nhi : i ∈ s₂ \\ s₁\n⊢ -↑(V i) (f i) = ↑(V i) (F i)",
"tactic": "simp only [hF₂ i hi, LinearIsometry.map_neg]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_5.a.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\nx✝ : ι\na✝ : x✝ ∈ s₁ \\ s₂\n⊢ ‖f x✝‖ = ‖F x✝‖",
"tactic": "simp only [hF]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.a.h.e'_5\n𝕜 : Type u_4\nE : Type u_2\nF✝ : Type ?u.3696068\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : InnerProductSpace ℝ F✝\ndec_E : DecidableEq E\nι : Type u_1\ndec_ι : DecidableEq ι\nG : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nf : (i : ι) → G i\ns₁ s₂ : Finset ι\nF : (i : ι) → G i := fun i => if i ∈ s₁ then f i else -f i\nhF₁ : ∀ (i : ι), i ∈ s₁ \\ s₂ → F i = f i\nhF₂ : ∀ (i : ι), i ∈ s₂ \\ s₁ → F i = -f i\nhF : ∀ (i : ι), ‖F i‖ = ‖f i‖\nthis :\n ‖∑ i in s₁ \\ s₂, ↑(V i) (F i) + ∑ i in s₂ \\ s₁, ↑(V i) (F i)‖ ^ 2 =\n ∑ i in s₁ \\ s₂, ‖F i‖ ^ 2 + ∑ i in s₂ \\ s₁, ‖F i‖ ^ 2\nx✝ : ι\na✝ : x✝ ∈ s₂ \\ s₁\n⊢ ‖f x✝‖ = ‖F x✝‖",
"tactic": "simp only [hF]"
}
] |
[
2106,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2084,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.smul_apply
|
[] |
[
359,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
LinearIndependent.restrict_scalars_algebras
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\nι : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Algebra R S\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nhinj : Function.Injective ↑(algebraMap R S)\nv : ι → M\nli : LinearIndependent S v\n⊢ Function.Injective fun r => r • 1",
"tactic": "rwa [Algebra.algebraMap_eq_smul_one'] at hinj"
}
] |
[
641,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Injective2.uncurry
|
[] |
[
940,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
11
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.const_smul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.538090\nE : Type ?u.538093\nF : Type u_3\nG : Type ?u.538099\nE' : Type u_2\nF' : Type ?u.538105\nG' : Type ?u.538108\nE'' : Type ?u.538111\nF'' : Type ?u.538114\nG'' : Type ?u.538117\nR : Type ?u.538120\nR' : Type ?u.538123\n𝕜 : Type u_4\n𝕜' : Type ?u.538129\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝ : NormedSpace 𝕜 E'\nh : IsBigOWith c l f' g\nc' : 𝕜\n⊢ IsBigOWith (‖c'‖ * c) l (fun x => ‖c' • f' x‖) g",
"tactic": "simpa only [norm_smul, _root_.norm_norm] using h.norm_left.const_mul_left ‖c'‖"
}
] |
[
1699,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1696,
1
] |
Mathlib/Topology/ContinuousFunction/Algebra.lean
|
ContinuousMap.inv_apply
|
[] |
[
186,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.IsImage.symm_mapsTo
|
[] |
[
499,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_op_norm
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.392348\nF : Type u_3\nF' : Type u_2\nG : Type ?u.392357\n𝕜 : Type ?u.392360\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT : Set α → F' →L[ℝ] F\nf : α →ₛ F'\nb : F'\nx✝ : b ∈ SimpleFunc.range f\n⊢ ‖↑(T (↑f ⁻¹' {b})) b‖ ≤ ‖T (↑f ⁻¹' {b})‖ * ‖b‖",
"state_before": "α : Type u_1\nE : Type ?u.392348\nF : Type u_3\nF' : Type u_2\nG : Type ?u.392357\n𝕜 : Type ?u.392360\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT : Set α → F' →L[ℝ] F\nf : α →ₛ F'\n⊢ ∑ x in SimpleFunc.range f, ‖↑(T (↑f ⁻¹' {x})) x‖ ≤ ∑ x in SimpleFunc.range f, ‖T (↑f ⁻¹' {x})‖ * ‖x‖",
"tactic": "refine' Finset.sum_le_sum fun b _ => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.392348\nF : Type u_3\nF' : Type u_2\nG : Type ?u.392357\n𝕜 : Type ?u.392360\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT : Set α → F' →L[ℝ] F\nf : α →ₛ F'\nb : F'\nx✝ : b ∈ SimpleFunc.range f\n⊢ ‖↑(T (↑f ⁻¹' {b})) b‖ ≤ ‖T (↑f ⁻¹' {b})‖ * ‖b‖",
"tactic": "simp_rw [ContinuousLinearMap.le_op_norm]"
}
] |
[
575,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.cos_angle_add_of_inner_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ ‖x‖ ≤ ‖x + y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ Real.cos (angle x (x + y)) = ‖x‖ / ‖x + y‖",
"tactic": "rw [angle_add_eq_arccos_of_inner_eq_zero h,\n Real.cos_arccos (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))\n (div_le_one_of_le _ (norm_nonneg _))]"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ ‖x‖ * ‖x‖ ≤ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ ‖x‖ ≤ ‖x + y‖",
"tactic": "rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),\n norm_add_sq_eq_norm_sq_add_norm_sq_real h]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ ‖x‖ * ‖x‖ ≤ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"tactic": "exact le_add_of_nonneg_right (mul_self_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ -1 ≤ 0",
"tactic": "norm_num"
}
] |
[
146,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Algebra/Bounds.lean
|
BddBelow.inv
|
[] |
[
54,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.exists_mem_ne_one_of_prod_ne_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.145057\nα : Type ?u.145060\nM : Type u_1\nN : Type ?u.145066\nP : Type ?u.145069\nM₀ : Type ?u.145072\nG : Type ?u.145075\nR : Type ?u.145078\ninst✝ : Monoid M\nl : List M\nh : prod l ≠ 1\n⊢ ∃ x, x ∈ l ∧ x ≠ 1",
"tactic": "simpa only [not_forall, exists_prop] using mt prod_eq_one h"
}
] |
[
538,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Init/Algebra/Order.lean
|
lt_or_ge
|
[] |
[
349,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/CategoryTheory/Subobject/MonoOver.lean
|
CategoryTheory.MonoOver.forget_obj_hom
|
[] |
[
102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.any_cons
|
[] |
[
35,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
35,
9
] |
Mathlib/Topology/Algebra/Module/LinearPMap.lean
|
LinearPMap.IsClosed.isClosable
|
[] |
[
80,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
MulMemClass.coe_mul
|
[] |
[
534,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.bounded_image
|
[] |
[
396,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.mem_map
|
[] |
[
238,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.cast_le
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : Num\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : Num\n⊢ ↑m ≤ ↑n ↔ m ≤ n",
"tactic": "rw [← not_lt]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : Num\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"tactic": "exact not_congr cast_lt"
}
] |
[
872,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
IsROrC.summable_ofReal
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\n𝕜 : Type u_1\ninst✝ : IsROrC 𝕜\nf : α → ℝ\nh : Summable fun x => ↑(f x)\n⊢ Summable f",
"tactic": "simpa only [IsROrC.reClm_apply, IsROrC.ofReal_re] using reClm.summable h"
}
] |
[
515,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
ContinuousOn.mul
|
[] |
[
112,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.vsub_inter_subset
|
[] |
[
1551,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1550,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_pi_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.101571\nγ : Type ?u.101574\nδ : Type u_2\nδ' : Type ?u.101580\nι : Sort uι\ns t u : Set α\nπ : δ → Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : (a : δ) → MeasurableSpace (π a)\ninst✝ : MeasurableSpace γ\ng : α → (a : δ) → π a\n⊢ Measurable g ↔ ∀ (a : δ), Measurable fun x => g x a",
"tactic": "simp_rw [measurable_iff_comap_le, MeasurableSpace.pi, MeasurableSpace.comap_iSup,\n MeasurableSpace.comap_comp, Function.comp, iSup_le_iff]"
}
] |
[
842,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
840,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.coe_eq_zero
|
[] |
[
114,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
antilipschitzWith_lineMap
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.81137\nV : Type ?u.81140\nP : Type ?u.81143\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\np₁ p₂ : Q\nh : p₁ ≠ p₂\nc₁ c₂ : 𝕜\n⊢ dist c₁ c₂ ≤ ↑(nndist p₁ p₂)⁻¹ * dist (↑(lineMap p₁ p₂) c₁) (↑(lineMap p₁ p₂) c₂)",
"tactic": "rw [dist_lineMap_lineMap, NNReal.coe_inv, ← dist_nndist, mul_left_comm,\n inv_mul_cancel (dist_ne_zero.2 h), mul_one]"
}
] |
[
234,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.SimpleFunc.integral_eq_sum
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.1250929\n𝕜 : Type ?u.1250932\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1253623\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α →ₛ E\nhfi : Integrable ↑f\n⊢ setToSimpleFunc (weightedSMul μ) f = integral μ f",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.1250929\n𝕜 : Type ?u.1250932\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1253623\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α →ₛ E\nhfi : Integrable ↑f\n⊢ (∫ (x : α), ↑f x ∂μ) = ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x",
"tactic": "rw [← f.integral_eq_integral hfi, SimpleFunc.integral, ← SimpleFunc.integral_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.1250929\n𝕜 : Type ?u.1250932\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1253623\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α →ₛ E\nhfi : Integrable ↑f\n⊢ setToSimpleFunc (weightedSMul μ) f = integral μ f",
"tactic": "rfl"
}
] |
[
1357,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1355,
1
] |
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