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/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.closure_preimage_le
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.550439\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.550448\ninst✝² : AddGroup A\nN : Type u_2\nP : Type ?u.550457\ninst✝¹ : Group N\ninst✝ : Group P\nK : Subgroup G\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ↑f ⁻¹' s\n⊢ ↑f x ∈ closure s",
"state_before": "G : Type u_1\nG' : Type ?u.550439\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.550448\ninst✝² : AddGroup A\nN : Type u_2\nP : Type ?u.550457\ninst✝¹ : Group N\ninst✝ : Group P\nK : Subgroup G\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ↑f ⁻¹' s\n⊢ x ∈ ↑(comap f (closure s))",
"tactic": "rw [SetLike.mem_coe, mem_comap]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.550439\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.550448\ninst✝² : AddGroup A\nN : Type u_2\nP : Type ?u.550457\ninst✝¹ : Group N\ninst✝ : Group P\nK : Subgroup G\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ↑f ⁻¹' s\n⊢ ↑f x ∈ closure s",
"tactic": "exact subset_closure hx"
}
] |
[
2926,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2925,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.coe_ne_empty
|
[] |
[
142,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.compl_erase
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.11642\nγ : Type ?u.11645\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na a✝ : α\n⊢ a✝ ∈ erase s aᶜ ↔ a✝ ∈ insert a (sᶜ)",
"state_before": "α : Type u_1\nβ : Type ?u.11642\nγ : Type ?u.11645\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na : α\n⊢ erase s aᶜ = insert a (sᶜ)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.11642\nγ : Type ?u.11645\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na a✝ : α\n⊢ a✝ ∈ erase s aᶜ ↔ a✝ ∈ insert a (sᶜ)",
"tactic": "simp only [or_iff_not_imp_left, mem_insert, not_and, mem_compl, mem_erase]"
}
] |
[
224,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.Set.union_apply_right
|
[] |
[
247,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.extend_eq_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\ns : α\nh : ¬P s\n⊢ extend m s = ⊤",
"tactic": "simp [extend, h]"
}
] |
[
1325,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1325,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
fderiv_clm_comp
|
[] |
[
113,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_add_le_of_le_add
|
[
{
"state_after": "a b c : Int\nh✝ : a ≤ b + c\nh : -b + a ≤ -b + (b + c)\n⊢ -b + a ≤ c",
"state_before": "a b c : Int\nh : a ≤ b + c\n⊢ -b + a ≤ c",
"tactic": "have h := Int.add_le_add_left h (-b)"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh✝ : a ≤ b + c\nh : -b + a ≤ -b + (b + c)\n⊢ -b + a ≤ c",
"tactic": "rwa [Int.neg_add_cancel_left] at h"
}
] |
[
980,
37
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
978,
11
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
IsAbsoluteValue.abv_one
|
[] |
[
380,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toAdd_div
|
[] |
[
337,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.toDual_apply_bot
|
[] |
[
934,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
933,
1
] |
Mathlib/Topology/Category/Profinite/Basic.lean
|
Profinite.epi_iff_surjective
|
[
{
"state_after": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective ((forget Profinite).map f)\n\ncase mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Function.Surjective ((forget Profinite).map f) → Epi f",
"state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f ↔ Function.Surjective ((forget Profinite).map f)",
"tactic": "constructor"
},
{
"state_after": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → ∀ (b : (forget Profinite).obj Y), ∃ a, (forget Profinite).map f a = b",
"state_before": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective ((forget Profinite).map f)",
"tactic": "dsimp [Function.Surjective]"
},
{
"state_after": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ (∃ b, ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ b) → ¬Epi f",
"state_before": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi f → ∀ (b : (forget Profinite).obj Y), ∃ a, (forget Profinite).map f a = b",
"tactic": "contrapose!"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False",
"state_before": "case mp\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ (∃ b, ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ b) → ¬Epi f",
"tactic": "rintro ⟨y, hy⟩ hf"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False",
"tactic": "skip"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\n⊢ False",
"tactic": "let C := Set.range f"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\n⊢ False",
"tactic": "have hC : IsClosed C := (isCompact_range f.continuous).isClosed"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\n⊢ False",
"tactic": "let U := Cᶜ"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ False",
"tactic": "have hyU : y ∈ U := by\n refine' Set.mem_compl _\n rintro ⟨y', hy'⟩\n exact hy y' hy'"
},
{
"state_after": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\n⊢ False",
"tactic": "have hUy : U ∈ 𝓝 y := hC.compl_mem_nhds hyU"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\n⊢ False",
"state_before": "case mp.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\n⊢ False",
"tactic": "obtain ⟨V, hV, hyV, hVU⟩ := isTopologicalBasis_clopen.mem_nhds_iff.mp hUy"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\n⊢ False",
"tactic": "classical\n let Z := of (ULift.{u} <| Fin 2)\n let g : Y ⟶ Z := ⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩\n let h : Y ⟶ Z := ⟨fun _ => ⟨1⟩, continuous_const⟩\n have H : h = g := by\n rw [← cancel_epi f]\n ext x\n apply ULift.ext\n dsimp [LocallyConstant.ofClopen]\n simp only [FunctorToTypes.map_comp_apply, forget_ContinuousMap_mk, Function.comp_apply]\n rw [if_neg]\n refine' mt (fun α => hVU α) _\n simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]\n apply_fun fun e => (e y).down at H\n dsimp [LocallyConstant.ofClopen] at H\n rw [if_pos hyV] at H\n exact top_ne_bot H"
},
{
"state_after": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ ¬y ∈ C",
"state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ y ∈ U",
"tactic": "refine' Set.mem_compl _"
},
{
"state_after": "case intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\ny' : (forget Profinite).obj X\nhy' : (forget Profinite).map f y' = y\n⊢ False",
"state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\n⊢ ¬y ∈ C",
"tactic": "rintro ⟨y', hy'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\ny' : (forget Profinite).obj X\nhy' : (forget Profinite).map f y' = y\n⊢ False",
"tactic": "exact hy y' hy'"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\n⊢ False",
"tactic": "let Z := of (ULift.{u} <| Fin 2)"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\n⊢ False",
"tactic": "let g : Y ⟶ Z := ⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\n⊢ False",
"tactic": "let h : Y ⟶ Z := ⟨fun _ => ⟨1⟩, continuous_const⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : h = g\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ False",
"tactic": "have H : h = g := by\n rw [← cancel_epi f]\n ext x\n apply ULift.ext\n dsimp [LocallyConstant.ofClopen]\n simp only [FunctorToTypes.map_comp_apply, forget_ContinuousMap_mk, Function.comp_apply]\n rw [if_neg]\n refine' mt (fun α => hVU α) _\n simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : ((forget Profinite).map h y).down = ((forget Profinite).map g y).down\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : h = g\n⊢ False",
"tactic": "apply_fun fun e => (e y).down at H"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = if y ∈ V then 0 else 1\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : ((forget Profinite).map h y).down = ((forget Profinite).map g y).down\n⊢ False",
"tactic": "dsimp [LocallyConstant.ofClopen] at H"
},
{
"state_after": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = 0\n⊢ False",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = if y ∈ V then 0 else 1\n⊢ False",
"tactic": "rw [if_pos hyV] at H"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nH : 1 = 0\n⊢ False",
"tactic": "exact top_ne_bot H"
},
{
"state_after": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ f ≫ h = f ≫ g",
"state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ h = g",
"tactic": "rw [← cancel_epi f]"
},
{
"state_after": "case w\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ (forget Profinite).map (f ≫ h) x = (forget Profinite).map (f ≫ g) x",
"state_before": "X✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\n⊢ f ≫ h = f ≫ g",
"tactic": "ext x"
},
{
"state_after": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ h) x).down = ((forget Profinite).map (f ≫ g) x).down",
"state_before": "case w\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ (forget Profinite).map (f ≫ h) x = (forget Profinite).map (f ≫ g) x",
"tactic": "apply ULift.ext"
},
{
"state_after": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ ContinuousMap.mk fun x => { down := 1 }) x).down =\n ((forget Profinite).map (f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down",
"state_before": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ h) x).down = ((forget Profinite).map (f ≫ g) x).down",
"tactic": "dsimp [LocallyConstant.ofClopen]"
},
{
"state_after": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ 1 = if (forget Profinite).map f x ∈ V then 0 else 1",
"state_before": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ((forget Profinite).map (f ≫ ContinuousMap.mk fun x => { down := 1 }) x).down =\n ((forget Profinite).map (f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down",
"tactic": "simp only [FunctorToTypes.map_comp_apply, forget_ContinuousMap_mk, Function.comp_apply]"
},
{
"state_after": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ V",
"state_before": "case w.h\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ 1 = if (forget Profinite).map f x ∈ V then 0 else 1",
"tactic": "rw [if_neg]"
},
{
"state_after": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ U",
"state_before": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ V",
"tactic": "refine' mt (fun α => hVU α) _"
},
{
"state_after": "no goals",
"state_before": "case w.h.hnc\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\ny : (forget Profinite).obj Y\nhy : ∀ (a : (forget Profinite).obj X), (forget Profinite).map f a ≠ y\nhf : Epi f\nC : Set ((forget Profinite).obj Y) := Set.range ((forget Profinite).map f)\nhC : IsClosed C\nU : Set ((forget Profinite).obj Y) := Cᶜ\nhyU : y ∈ U\nhUy : U ∈ 𝓝 y\nV : Set ((forget Profinite).obj Y)\nhV : V ∈ {s | IsClopen s}\nhyV : y ∈ V\nhVU : V ⊆ U\nZ : Profinite := of (ULift (Fin 2))\ng : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV))\nh : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 }\nx : (forget Profinite).obj X\n⊢ ¬(forget Profinite).map f x ∈ U",
"tactic": "simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]"
},
{
"state_after": "case mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi ((forget Profinite).map f) → Epi f",
"state_before": "case mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Function.Surjective ((forget Profinite).map f) → Epi f",
"tactic": "rw [← CategoryTheory.epi_iff_surjective]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nX✝ Y✝ : Profinite\nf✝ : X✝ ⟶ Y✝\nX Y : Profinite\nf : X ⟶ Y\n⊢ Epi ((forget Profinite).map f) → Epi f",
"tactic": "apply (forget Profinite).epi_of_epi_map"
}
] |
[
413,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Std/Data/HashMap/WF.lean
|
Std.HashMap.Imp.modify_size
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))\n (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val))",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ (modify m k f).size = Bucket.size (modify m k f).buckets",
"tactic": "dsimp [modify, cond]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))\n (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n AssocList.nil\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val))",
"tactic": "rw [Bucket.update_update]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data)",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ m.size =\n Bucket.size\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val))",
"tactic": "simp [h, Bucket.size]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝))",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data)",
"tactic": "refine have ⟨_, _, h₁, _, eq⟩ := Bucket.exists_of_update ..; eq ▸ ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\ninst✝¹ : BEq α\ninst✝ : Hashable α\nm : Imp α β\nk : α\nh : m.size = Bucket.size m.buckets\nw✝¹ w✝ : List (AssocList α β)\nh₁ :\n m.buckets.val.data =\n w✝¹ ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w✝\nleft✝ : List.length w✝¹ = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\neq :\n (Bucket.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val\n (AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])\n (_ :\n USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <\n Array.size m.buckets.val)).val.data =\n w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝\n⊢ Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =\n Nat.sum\n (List.map (fun x => List.length (AssocList.toList x))\n (w✝¹ ++\n AssocList.modify k f\n m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::\n w✝))",
"tactic": "simp [h, h₁, Bucket.size_eq]"
}
] |
[
271,
31
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
265,
1
] |
Mathlib/Order/Bounded.lean
|
Set.bounded_le_inter_le
|
[
{
"state_after": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) s",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Bounded (fun x x_1 => x ≤ x_1) s",
"tactic": "refine' ⟨_, Bounded.mono (Set.inter_subset_left s _)⟩"
},
{
"state_after": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b})",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) s",
"tactic": "rw [← @bounded_le_inter_lt _ s _ a]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ns t : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) → Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b})",
"tactic": "exact Bounded.mono fun x ⟨hx, hx'⟩ => ⟨hx, le_of_lt hx'⟩"
}
] |
[
345,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.coe_prod
|
[] |
[
2009,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2006,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.le_zero_lf
|
[
{
"state_after": "x : PGame\n⊢ ((∀ (i : LeftMoves x), moveLeft x i ⧏ 0) ∧ ∀ (j : RightMoves 0), x ⧏ moveRight 0 j) ↔\n ∀ (i : LeftMoves x), moveLeft x i ⧏ 0",
"state_before": "x : PGame\n⊢ x ≤ 0 ↔ ∀ (i : LeftMoves x), moveLeft x i ⧏ 0",
"tactic": "rw [le_iff_forall_lf]"
},
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ ((∀ (i : LeftMoves x), moveLeft x i ⧏ 0) ∧ ∀ (j : RightMoves 0), x ⧏ moveRight 0 j) ↔\n ∀ (i : LeftMoves x), moveLeft x i ⧏ 0",
"tactic": "simp"
}
] |
[
651,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Order/BoundedOrder.lean
|
monotone_lt
|
[] |
[
568,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.hasBasis_nhds
|
[] |
[
694,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
11
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLinAlgEquiv_self
|
[] |
[
775,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
773,
1
] |
Mathlib/Algebra/Support.lean
|
Pi.mulSupport_mulSingle_subset
|
[] |
[
467,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
rootsOfUnity.coe_pow
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.368159\nN : Type ?u.368162\nG : Type ?u.368165\nR : Type u_1\nS : Type ?u.368171\nF : Type ?u.368174\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ+\ninst✝ : CommMonoid R\nζ : { x // x ∈ rootsOfUnity k R }\nm : ℕ\n⊢ ↑↑(ζ ^ m) = ↑↑ζ ^ m",
"tactic": "rw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]"
}
] |
[
136,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/List/Destutter.lean
|
List.destutter'_cons_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na b : α\nh : R b a\n⊢ destutter' R b (a :: l) = b :: destutter' R a l",
"tactic": "rw [destutter', if_pos h]"
}
] |
[
52,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.prod_eq_foldr
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.12132\nα : Type ?u.12135\nM : Type u_1\nN : Type ?u.12141\nP : Type ?u.12144\nM₀ : Type ?u.12147\nG : Type ?u.12150\nR : Type ?u.12153\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na✝ a : M\nl : List M\n⊢ prod (a :: l) = foldr (fun x x_1 => x * x_1) 1 (a :: l)",
"tactic": "rw [prod_cons, foldr_cons, prod_eq_foldr]"
}
] |
[
75,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
MonotoneOn.Ioc
|
[] |
[
151,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
11
] |
Mathlib/Data/Set/NAry.lean
|
Set.image_image2_antidistrib_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nα' : Type ?u.47893\nβ : Type u_4\nβ' : Type u_5\nγ : Type u_2\nγ' : Type ?u.47905\nδ : Type u_1\nδ' : Type ?u.47911\nε : Type ?u.47914\nε' : Type ?u.47917\nζ : Type ?u.47920\nζ' : Type ?u.47923\nν : Type ?u.47926\nf f'✝ : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\ng : γ → δ\nf' : β' → α → δ\ng' : β → β'\nh_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' (g' b) a\n⊢ image2 f' ((fun b => g' b) '' t) ((fun a => a) '' s) = image2 f' (g' '' t) s",
"tactic": "rw [image_id']"
}
] |
[
408,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.eq_iff
|
[
{
"state_after": "case mp\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : ↑f x = ↑f y\n⊢ y⁻¹ * x ∈ ker f\n\ncase mpr\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : y⁻¹ * x ∈ ker f\n⊢ ↑f x = ↑f y",
"state_before": "G : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\n⊢ ↑f x = ↑f y ↔ y⁻¹ * x ∈ ker f",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : ↑f x = ↑f y\n⊢ y⁻¹ * x ∈ ker f",
"tactic": "rw [mem_ker, map_mul, h, ← map_mul, inv_mul_self, map_one]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nG : Type u_1\nG' : Type ?u.513580\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.513589\ninst✝³ : AddGroup A\nN : Type ?u.513595\nP : Type ?u.513598\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_2\ninst✝ : MulOneClass M\nf : G →* M\nx y : G\nh : y⁻¹ * x ∈ ker f\n⊢ ↑f x = ↑f y",
"tactic": "rw [← one_mul x, ← mul_inv_self y, mul_assoc, map_mul, f.mem_ker.1 h, mul_one]"
}
] |
[
2788,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2785,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.limit.lift_extend
|
[
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasLimit F\nc : Cone F\nX : C\nf : X ⟶ c.pt\n⊢ lift F (Cone.extend c f) = f ≫ lift F c",
"tactic": "aesop_cat"
}
] |
[
312,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.Tendsto.mul_mul
|
[] |
[
635,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.image_eq_image
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni i' j : ι\nr : ComplexShape.Rel c i j\nr' : ComplexShape.Rel c i' j\n⊢ imageSubobject (eqToHom (_ : X C i = X C i') ≫ d C i' j) = imageSubobject (d C i' j)",
"state_before": "ι : Type u_1\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni i' j : ι\nr : ComplexShape.Rel c i j\nr' : ComplexShape.Rel c i' j\n⊢ imageSubobject (d C i j) = imageSubobject (d C i' j)",
"tactic": "rw [← eqToHom_comp_d C r r']"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni i' j : ι\nr : ComplexShape.Rel c i j\nr' : ComplexShape.Rel c i' j\n⊢ imageSubobject (eqToHom (_ : X C i = X C i') ≫ d C i' j) = imageSubobject (d C i' j)",
"tactic": "apply imageSubobject_iso_comp"
}
] |
[
344,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Analysis/Convex/Topology.lean
|
Convex.isPathConnected
|
[
{
"state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\n⊢ ∀ (x : E), x ∈ s → ∀ (y : E), y ∈ s → JoinedIn s x y",
"state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\n⊢ IsPathConnected s",
"tactic": "refine' isPathConnected_iff.mpr ⟨hne, _⟩"
},
{
"state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\n⊢ JoinedIn s x y",
"state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\n⊢ ∀ (x : E), x ∈ s → ∀ (y : E), y ∈ s → JoinedIn s x y",
"tactic": "intro x x_in y y_in"
},
{
"state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : [x-[ℝ]y] ⊆ s\n⊢ JoinedIn s x y",
"state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\n⊢ JoinedIn s x y",
"tactic": "have H := hconv.segment_subset x_in y_in"
},
{
"state_after": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : ↑(lineMap x y) '' Icc 0 1 ⊆ s\n⊢ JoinedIn s x y",
"state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : [x-[ℝ]y] ⊆ s\n⊢ JoinedIn s x y",
"tactic": "rw [segment_eq_image_lineMap] at H"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.191359\n𝕜 : Type ?u.191362\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhconv : Convex ℝ s\nhne : Set.Nonempty s\nx : E\nx_in : x ∈ s\ny : E\ny_in : y ∈ s\nH : ↑(lineMap x y) '' Icc 0 1 ⊆ s\n⊢ JoinedIn s x y",
"tactic": "exact\n JoinedIn.ofLine AffineMap.lineMap_continuous.continuousOn (lineMap_apply_zero _ _)\n (lineMap_apply_one _ _) H"
}
] |
[
350,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
11
] |
Mathlib/RingTheory/IntegralClosure.lean
|
FG_adjoin_of_finite
|
[
{
"state_after": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns : Set A\nhfs : Set.Finite s\nhis : ∀ (x : A), x ∈ s → IsIntegral R x\nx✝ : ∀ (x : A), x ∈ ∅ → IsIntegral R x\nx : A\n⊢ (∃ y, ↑(Algebra.linearMap R A) y = x) ↔ x ∈ Set.range ↑(algebraMap R A)",
"state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns : Set A\nhfs : Set.Finite s\nhis : ∀ (x : A), x ∈ s → IsIntegral R x\nx✝ : ∀ (x : A), x ∈ ∅ → IsIntegral R x\nx : A\n⊢ x ∈ span R ↑{1} ↔ x ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin R ∅)",
"tactic": "erw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, one_eq_range,\n LinearMap.mem_range, Algebra.mem_bot]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns : Set A\nhfs : Set.Finite s\nhis : ∀ (x : A), x ∈ s → IsIntegral R x\nx✝ : ∀ (x : A), x ∈ ∅ → IsIntegral R x\nx : A\n⊢ (∃ y, ↑(Algebra.linearMap R A) y = x) ↔ x ∈ Set.range ↑(algebraMap R A)",
"tactic": "rfl"
},
{
"state_after": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns✝ : Set A\nhfs : Set.Finite s✝\nhis✝ : ∀ (x : A), x ∈ s✝ → IsIntegral R x\na : A\ns : Set A\nx✝¹ : ¬a ∈ s\nx✝ : Set.Finite s\nih : (∀ (x : A), x ∈ s → IsIntegral R x) → FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : ∀ (x : A), x ∈ insert a s → IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s) * ↑Subalgebra.toSubmodule (Algebra.adjoin R {a}))",
"state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns✝ : Set A\nhfs : Set.Finite s✝\nhis✝ : ∀ (x : A), x ∈ s✝ → IsIntegral R x\na : A\ns : Set A\nx✝¹ : ¬a ∈ s\nx✝ : Set.Finite s\nih : (∀ (x : A), x ∈ s → IsIntegral R x) → FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : ∀ (x : A), x ∈ insert a s → IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R (insert a s)))",
"tactic": "rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.318888\nS : Type ?u.318891\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\ns✝ : Set A\nhfs : Set.Finite s✝\nhis✝ : ∀ (x : A), x ∈ s✝ → IsIntegral R x\na : A\ns : Set A\nx✝¹ : ¬a ∈ s\nx✝ : Set.Finite s\nih : (∀ (x : A), x ∈ s → IsIntegral R x) → FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : ∀ (x : A), x ∈ insert a s → IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R s) * ↑Subalgebra.toSubmodule (Algebra.adjoin R {a}))",
"tactic": "exact\n FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)\n (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s)"
}
] |
[
244,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
AddMonoidHom.mapMatrix_id
|
[] |
[
1413,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1412,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean
|
CategoryTheory.Limits.sigmaComparison_map_desc
|
[
{
"state_after": "case h\nβ : Type w\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\nj : β\n⊢ Sigma.ι (fun b => G.obj (f b)) j ≫ sigmaComparison G f ≫ G.map (Sigma.desc g) =\n Sigma.ι (fun b => G.obj (f b)) j ≫ Sigma.desc fun j => G.map (g j)",
"state_before": "β : Type w\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\n⊢ sigmaComparison G f ≫ G.map (Sigma.desc g) = Sigma.desc fun j => G.map (g j)",
"tactic": "ext j"
},
{
"state_after": "no goals",
"state_before": "case h\nβ : Type w\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nf : β → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : β) → f j ⟶ P\nj : β\n⊢ Sigma.ι (fun b => G.obj (f b)) j ≫ sigmaComparison G f ≫ G.map (Sigma.desc g) =\n Sigma.ι (fun b => G.obj (f b)) j ≫ Sigma.desc fun j => G.map (g j)",
"tactic": "simp only [Discrete.functor_obj, ι_comp_sigmaComparison_assoc, ← G.map_comp, colimit.ι_desc,\n Cofan.mk_pt, Cofan.mk_ι_app]"
}
] |
[
279,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integrable_prod_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.2396048\nβ : Type u_2\nβ' : Type ?u.2396054\nγ : Type ?u.2396057\nE : Type u_3\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nf : α × β → E\nh1f : AEStronglyMeasurable f (Measure.prod μ ν)\n⊢ Integrable f ↔ (∀ᵐ (x : α) ∂μ, Integrable fun y => f (x, y)) ∧ Integrable fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν",
"tactic": "simp [Integrable, h1f, hasFiniteIntegral_prod_iff', h1f.norm.integral_prod_right',\n h1f.prod_mk_left]"
}
] |
[
280,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.mapM'_eq_mapM
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → m β\nl : List α\n⊢ mapM' f l = mapM f l",
"tactic": "simp [go, mapM]"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → m β\nl : List α\nacc : List β\n⊢ mapM.loop f [] acc = do\n let __do_lift ← mapM' f []\n pure (reverse acc ++ __do_lift)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα : Type u_3\nβ : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → m β\nl✝ : List α\na : α\nl : List α\nacc : List β\n⊢ mapM.loop f (a :: l) acc = do\n let __do_lift ← mapM' f (a :: l)\n pure (reverse acc ++ __do_lift)",
"tactic": "simp [go l]"
}
] |
[
216,
34
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
212,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sup_eq_max
|
[] |
[
743,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
9
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.l_eq_bot
|
[] |
[
255,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_eq_preimage'
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' t → s = t\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ s = t → f ⁻¹' s = f ⁻¹' t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' t ↔ s = t",
"tactic": "constructor"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t s : Set α\nf : β → α\nhs ht : s ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' s",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ s = t → f ⁻¹' s = f ⁻¹' t",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t s : Set α\nf : β → α\nhs ht : s ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' s",
"tactic": "rfl"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s = t",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\n⊢ f ⁻¹' s = f ⁻¹' t → s = t",
"tactic": "intro h"
},
{
"state_after": "case mp.h₁\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s ⊆ t\n\ncase mp.h₂\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ t ⊆ s",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s = t",
"tactic": "apply Subset.antisymm"
},
{
"state_after": "no goals",
"state_before": "case mp.h₁\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s ⊆ t",
"tactic": "rw [← preimage_subset_preimage_iff hs, h]"
},
{
"state_after": "no goals",
"state_before": "case mp.h₂\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.77374\nι : Sort ?u.77377\nι' : Sort ?u.77380\nf✝ : ι → α\ns✝ t✝ s t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ t ⊆ s",
"tactic": "rw [← preimage_subset_preimage_iff ht, h]"
}
] |
[
836,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.exists_minimal_degree_vertex
|
[
{
"state_after": "case intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\n⊢ ∃ v, minDegree G = degree G v",
"state_before": "ι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\n⊢ ∃ v, minDegree G = degree G v",
"tactic": "obtain ⟨t, ht : _ = _⟩ := min_of_nonempty (univ_nonempty.image fun v => G.degree v)"
},
{
"state_after": "case intro.intro.intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nleft✝ : v ∈ univ\nht : Finset.min (image (fun v => degree G v) univ) = ↑(degree G v)\n⊢ ∃ v, minDegree G = degree G v",
"state_before": "case intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\n⊢ ∃ v, minDegree G = degree G v",
"tactic": "obtain ⟨v, _, rfl⟩ := mem_image.mp (mem_of_min ht)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nleft✝ : v ∈ univ\nht : Finset.min (image (fun v => degree G v) univ) = ↑(degree G v)\n⊢ ∃ v, minDegree G = degree G v",
"tactic": "refine' ⟨v, by simp [minDegree, ht]⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.289282\n𝕜 : Type ?u.289285\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty V\nv : V\nleft✝ : v ∈ univ\nht : Finset.min (image (fun v => degree G v) univ) = ↑(degree G v)\n⊢ minDegree G = degree G v",
"tactic": "simp [minDegree, ht]"
}
] |
[
1518,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1514,
1
] |
Mathlib/GroupTheory/Solvable.lean
|
not_solvable_of_mem_derivedSeries
|
[] |
[
213,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.X_mul_divMonomial
|
[] |
[
166,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.add_bind
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.11263\nδ : Type ?u.11266\na : α\ns t : Multiset α\nf g : α → Multiset β\n⊢ bind (s + t) f = bind s f + bind t f",
"tactic": "simp [bind]"
}
] |
[
119,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.ne_of_mem_erase
|
[] |
[
1879,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1879,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
hnot_anti
|
[] |
[
1083,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1083,
1
] |
Mathlib/CategoryTheory/Monad/Limits.lean
|
CategoryTheory.hasColimits_of_reflective
|
[] |
[
390,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.eval_one
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : Field K\nL : Type u_1\ninst✝ : Field L\nf : K →+* L\na : L\n⊢ eval f a 1 = 1",
"tactic": "simp [eval]"
}
] |
[
1503,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1503,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.prod_same_index_anti
|
[] |
[
932,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
IsCompact.div_closedBall
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx✝ y : E\nhs : IsCompact s\nhδ : 0 ≤ δ\nx : E\n⊢ s / closedBall x δ = x⁻¹ • cthickening δ s",
"tactic": "simp [div_eq_mul_inv, mul_comm, hs.mul_closedBall hδ]"
}
] |
[
293,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Data/Int/Cast/Basic.lean
|
Int.cast_zero
|
[] |
[
62,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Data/Polynomial/Expand.lean
|
Polynomial.expand_C
|
[] |
[
55,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.forall_mem_map_iff
|
[] |
[
1162,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1160,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.ModEq.eq
|
[] |
[
73,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
11
] |
Mathlib/Algebra/AddTorsor.lean
|
Equiv.pointReflection_fixed_iff_of_injective_bit0
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝¹ : AddGroup G\ninst✝ : AddTorsor G P\nx y : P\nh : Injective bit0\n⊢ ↑(pointReflection x) y = y ↔ y = x",
"tactic": "rw [pointReflection_apply, eq_comm, eq_vadd_iff_vsub_eq, ← neg_vsub_eq_vsub_rev,\n neg_eq_iff_add_eq_zero, ← bit0, ← bit0_zero, h.eq_iff, vsub_eq_zero_iff_eq, eq_comm]"
}
] |
[
471,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.pentagon
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nW : ModuleCat R\nX : ModuleCat R\nY : ModuleCat R\nZ : ModuleCat R\n⊢ tensorHom (associator W X Y).hom (𝟙 Z) ≫\n (associator W (tensorObj X Y) Z).hom ≫ tensorHom (𝟙 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom ≫ (associator W X (tensorObj Y Z)).hom",
"tactic": "convert pentagon_aux R W X Y Z using 1"
}
] |
[
135,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.coe_ereal_ennreal
|
[] |
[
2018,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2016,
1
] |
Mathlib/LinearAlgebra/CrossProduct.lean
|
jacobi_cross
|
[] |
[
166,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.mem_support_iff
|
[] |
[
97,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/Convex/StrictConvexSpace.lean
|
norm_add_lt_of_not_sameRay
|
[
{
"state_after": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : x ≠ 0 ∧ y ≠ 0 ∧ ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"state_before": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : ¬SameRay ℝ x y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"tactic": "simp only [sameRay_iff_inv_norm_smul_eq, not_or, ← Ne.def] at h"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"state_before": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : x ≠ 0 ∧ y ≠ 0 ∧ ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"tactic": "rcases h with ⟨hx, hy, hne⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"tactic": "rw [← norm_pos_iff] at hx hy"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"tactic": "have hxy : 0 < ‖x‖ + ‖y‖ := add_pos hx hy"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"tactic": "have :=\n combo_mem_ball_of_ne (inv_norm_smul_mem_closed_unit_ball x)\n (inv_norm_smul_mem_closed_unit_ball y) hne (div_pos hx hxy) (div_pos hy hxy)\n (by rw [← add_div, div_self hxy.ne'])"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖",
"tactic": "rwa [mem_ball_zero_iff, div_eq_inv_mul, div_eq_inv_mul, mul_smul, mul_smul, smul_inv_smul₀ hx.ne',\n smul_inv_smul₀ hy.ne', ← smul_add, norm_smul, Real.norm_of_nonneg (inv_pos.2 hxy).le, ←\n div_eq_inv_mul, div_lt_one hxy] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.54272\nE : Type u_1\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x‖ / (‖x‖ + ‖y‖) + ‖y‖ / (‖x‖ + ‖y‖) = 1",
"tactic": "rw [← add_div, div_self hxy.ne']"
}
] |
[
194,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.commute_sub_int
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nn : ℤ\n⊢ Function.Commute ↑f fun x => x - ↑n",
"tactic": "simpa only [sub_eq_add_neg] using\n (f.commute_add_int n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv"
}
] |
[
358,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.cyclotomic_prime_pow_eq_geom_sum
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ p ^ n) ^ i",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\n⊢ cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ p ^ n) ^ i",
"tactic": "have : ∀ m, (cyclotomic (p ^ (m + 1)) R = ∑ i in Finset.range p, (X ^ p ^ m) ^ i) ↔\n ((∑ i in Finset.range p, (X ^ p ^ m) ^ i) *\n ∏ x : ℕ in Finset.range (m + 1), cyclotomic (p ^ x) R) = X ^ p ^ (m + 1) - 1 := by\n intro m\n have := eq_cyclotomic_iff (R := R) (P := ∑ i in range p, (X ^ p ^ m) ^ i)\n (pow_pos hp.pos (m + 1))\n rw [eq_comm] at this\n rw [this, Nat.prod_properDivisors_prime_pow hp]"
},
{
"state_after": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i\n\ncase succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ cyclotomic (p ^ (Nat.succ n_n + 1)) R = ∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ p ^ n) ^ i",
"tactic": "induction' n with n_n n_ih"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) * ∏ i in Nat.properDivisors (p ^ (Nat.succ n_n + 1)), cyclotomic i R =\n X ^ p ^ (Nat.succ n_n + 1) - 1",
"state_before": "case succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ cyclotomic (p ^ (Nat.succ n_n + 1)) R = ∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i",
"tactic": "rw [((eq_cyclotomic_iff (pow_pos hp.pos (n_n.succ + 1)) _).mpr _).symm]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) * ∏ i in Nat.properDivisors (p ^ (Nat.succ n_n + 1)), cyclotomic i R =\n X ^ p ^ (Nat.succ n_n + 1) - 1",
"tactic": "rw [Nat.prod_properDivisors_prime_pow hp, Finset.prod_range_succ, n_ih]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : (∑ i in range p, (X ^ p ^ n_n) ^ i) * ∏ x in range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i in range p, (X ^ p ^ n_n) ^ i\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1",
"tactic": "rw [this] at n_ih"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : (∑ i in range p, (X ^ p ^ n_n) ^ i) * ∏ x in range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1\n⊢ (∑ i in range p, (X ^ p ^ Nat.succ n_n) ^ i) *\n ((∏ x in range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i in range p, (X ^ p ^ n_n) ^ i) =\n X ^ p ^ (Nat.succ n_n + 1) - 1",
"tactic": "rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\n⊢ ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"tactic": "intro m"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n ∑ i in range p, (X ^ p ^ m) ^ i = cyclotomic (p ^ (m + 1)) R ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"tactic": "have := eq_cyclotomic_iff (R := R) (P := ∑ i in range p, (X ^ p ^ m) ^ i)\n (pow_pos hp.pos (m + 1))"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n ∑ i in range p, (X ^ p ^ m) ^ i = cyclotomic (p ^ (m + 1)) R ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"tactic": "rw [eq_comm] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\np n : ℕ\nhp : Nat.Prime p\nm : ℕ\nthis :\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ i in Nat.properDivisors (p ^ (m + 1)), cyclotomic i R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1",
"tactic": "rw [this, Nat.prod_properDivisors_prime_pow hp]"
},
{
"state_after": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis✝ :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nthis : Fact (Nat.Prime p)\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i",
"state_before": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i",
"tactic": "haveI := Fact.mk hp"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis✝ :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i in range p, (X ^ p ^ m) ^ i ↔\n (∑ i in range p, (X ^ p ^ m) ^ i) * ∏ x in range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nthis : Fact (Nat.Prime p)\n⊢ cyclotomic (p ^ (Nat.zero + 1)) R = ∑ i in range p, (X ^ p ^ Nat.zero) ^ i",
"tactic": "simp [cyclotomic_prime]"
}
] |
[
576,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_congr
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ a✝ ∈ image f s ↔ a✝ ∈ image g s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\n⊢ image f s = image g s",
"tactic": "ext"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ (∃ x x_1, f x = a✝) ↔ ∃ x x_1, g x = a✝",
"state_before": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ a✝ ∈ image f s ↔ a✝ ∈ image g s",
"tactic": "simp_rw [mem_image, ← bex_def]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\n⊢ (∃ x x_1, f x = a✝) ↔ ∃ x x_1, g x = a✝",
"tactic": "exact bex_congr fun x hx => by rw [h hx]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.68613\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\nh : Set.EqOn f g ↑s\na✝ : β\nx : α\nhx : x ∈ s\n⊢ f x = a✝ ↔ g x = a✝",
"tactic": "rw [h hx]"
}
] |
[
361,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.toHomeomorph_inj
|
[] |
[
668,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.exists_subset_range_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.75384\nι : Sort ?u.75387\nι' : Sort ?u.75390\nf✝ : ι → α\ns t : Set α\nf : α → β\np : Set β → Prop\n⊢ (∃ s x, p s) ↔ ∃ s, p (f '' s)",
"tactic": "simp"
}
] |
[
813,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.ofReal_inv_of_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.816393\nβ : Type ?u.816396\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ\nhx : 0 < x\n⊢ (ENNReal.ofReal x)⁻¹ = ENNReal.ofReal x⁻¹",
"tactic": "rw [ENNReal.ofReal, ENNReal.ofReal, ← @coe_inv (Real.toNNReal x) (by simp [hx]), coe_eq_coe,\n ← Real.toNNReal_inv]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.816393\nβ : Type ?u.816396\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ\nhx : 0 < x\n⊢ Real.toNNReal x ≠ 0",
"tactic": "simp [hx]"
}
] |
[
2187,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2185,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
WfDvdMonoid.exists_irreducible_factor
|
[] |
[
93,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
StrictMonoOn.monotoneOn
|
[] |
[
458,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.zero_eq_ofReal
|
[] |
[
2123,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2122,
1
] |
Mathlib/Topology/ContinuousFunction/CocompactMap.lean
|
CocompactMap.coe_copy
|
[] |
[
112,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Topology/Separation.lean
|
separated_by_openEmbedding
|
[] |
[
1131,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1126,
1
] |
Mathlib/Data/Sum/Interval.lean
|
Sum.Ico_inl_inr
|
[] |
[
164,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_add_eq_iff_add_mul_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.576510\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nx : R\ny : { x // x ∈ M }\nz₁ z₂ : S\n⊢ mk' S x y + z₁ = z₂ ↔ ↑(algebraMap R S) x + z₁ * ↑(algebraMap R S) ↑y = z₂ * ↑(algebraMap R S) ↑y",
"tactic": "rw [← mk'_spec S x y, ← IsUnit.mul_left_inj (IsLocalization.map_units S y), right_distrib]"
}
] |
[
296,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.subset_image_symm_diff
|
[] |
[
438,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/LinearAlgebra/Matrix/Trace.lean
|
Matrix.trace_conjTranspose
|
[] |
[
75,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.leftDual_ρ
|
[
{
"state_after": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ ↑(ᘁX).ρ h = ᘁ↑X.ρ (inv h)\n\ncase x\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1",
"state_before": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ ↑(ᘁX).ρ h = ᘁ↑X.ρ h⁻¹",
"tactic": "rw [← SingleObj.inv_as_inv]"
},
{
"state_after": "no goals",
"state_before": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ ↑(ᘁX).ρ h = ᘁ↑X.ρ (inv h)\n\ncase x\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : LeftRigidCategory V\nh : ↑H\n⊢ SingleObj H.1",
"tactic": "rfl"
}
] |
[
754,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_ofDual
|
[] |
[
2315,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2314,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.range_coe_eq_Ioo
|
[
{
"state_after": "case h\nx : EReal\n⊢ x ∈ range Real.toEReal ↔ x ∈ Ioo ⊥ ⊤",
"state_before": "⊢ range Real.toEReal = Ioo ⊥ ⊤",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nx : EReal\n⊢ x ∈ range Real.toEReal ↔ x ∈ Ioo ⊥ ⊤",
"tactic": "induction x using EReal.rec <;> simp"
}
] |
[
351,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Data/Finsupp/Order.lean
|
Finsupp.coe_tsub
|
[] |
[
195,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.le_ceil
|
[] |
[
1126,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1125,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean
|
div_sq_cancel
|
[
{
"state_after": "case pos\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : a = 0\n⊢ a ^ 2 * b / a = a * b\n\ncase neg\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : ¬a = 0\n⊢ a ^ 2 * b / a = a * b",
"state_before": "G₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\n⊢ a ^ 2 * b / a = a * b",
"tactic": "by_cases ha : a = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : ¬a = 0\n⊢ a ^ 2 * b / a = a * b",
"tactic": "rw [sq, mul_assoc, mul_div_cancel_left _ ha]"
},
{
"state_after": "no goals",
"state_before": "case pos\nG₀ : Type u_1\ninst✝ : CommGroupWithZero G₀\na b : G₀\nha : a = 0\n⊢ a ^ 2 * b / a = a * b",
"tactic": "simp [ha]"
}
] |
[
198,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_eq_one_iff_inv_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.53359\nβ : Type ?u.53362\nG : Type u_1\ninst✝ : Group G\na b c d : G\n⊢ a * b = 1 ↔ a⁻¹ = b",
"tactic": "rw [mul_eq_one_iff_eq_inv, inv_eq_iff_eq_inv]"
}
] |
[
675,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.pair_apply
|
[] |
[
417,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Topology/Basic.lean
|
isOpen_singleton_iff_nhds_eq_pure
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ IsOpen {a} → 𝓝 a = pure a\n\ncase mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ 𝓝 a = pure a → IsOpen {a}",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ IsOpen {a} ↔ 𝓝 a = pure a",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a = pure a",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ IsOpen {a} → 𝓝 a = pure a",
"tactic": "intro h"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a ≤ pure a",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a = pure a",
"tactic": "apply le_antisymm _ (pure_le_nhds a)"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ {a} ∈ 𝓝 a",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ 𝓝 a ≤ pure a",
"tactic": "rw [le_pure_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : IsOpen {a}\n⊢ {a} ∈ 𝓝 a",
"tactic": "exact h.mem_nhds (mem_singleton a)"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : 𝓝 a = pure a\n⊢ IsOpen {a}",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ 𝓝 a = pure a → IsOpen {a}",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nh : 𝓝 a = pure a\n⊢ IsOpen {a}",
"tactic": "simp [isOpen_iff_nhds, h]"
}
] |
[
1257,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1250,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_hasStrictFDerivAt
|
[] |
[
633,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Mathlib/Logic/Hydra.lean
|
Relation.cutExpand_fibration
|
[
{
"state_after": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = (fun s => s.fst + s.snd) (s₁, s₂) + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ (fun s => s.fst + s.snd) a' = s",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\n⊢ Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s => s.fst + s.snd",
"tactic": "rintro ⟨s₁, s₂⟩ s ⟨t, a, hr, he⟩"
},
{
"state_after": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s",
"state_before": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = (fun s => s.fst + s.snd) (s₁, s₂) + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ (fun s => s.fst + s.snd) a' = s",
"tactic": "dsimp at he ⊢"
},
{
"state_after": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s",
"state_before": "case mk.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s",
"tactic": "obtain ⟨ha, hb⟩ := add_singleton_eq_iff.1 he"
},
{
"state_after": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α",
"state_before": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = s",
"tactic": "rw [hb]"
},
{
"state_after": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ ∨ a ∈ s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α",
"state_before": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α",
"tactic": "rw [add_assoc, mem_add] at ha"
},
{
"state_after": "case mk.intro.intro.intro.intro.inl\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\ncase mk.intro.intro.intro.intro.inr\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α",
"state_before": "case mk.intro.intro.intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nha : a ∈ s₁ ∨ a ∈ s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a\n\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\n⊢ DecidableEq α",
"tactic": "obtain h | h := ha"
},
{
"state_after": "case mk.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ erase s₁ a + t + {a} = s₁ + t\n\ncase mk.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ (erase s₁ a + t, s₂).fst + (erase s₁ a + t, s₂).snd = erase (s₁ + s₂ + t) a",
"state_before": "case mk.intro.intro.intro.intro.inl\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a",
"tactic": "refine' ⟨(s₁.erase a + t, s₂), GameAdd.fst ⟨t, a, hr, _⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ erase s₁ a + t + {a} = s₁ + t",
"tactic": "rw [add_comm, ← add_assoc, singleton_add, cons_erase h]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₁\n⊢ (erase s₁ a + t, s₂).fst + (erase s₁ a + t, s₂).snd = erase (s₁ + s₂ + t) a",
"tactic": "rw [add_assoc s₁, erase_add_left_pos _ h, add_right_comm, add_assoc]"
},
{
"state_after": "case mk.intro.intro.intro.intro.inr.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ erase (s₂ + t) a + {a} = s₂ + t\n\ncase mk.intro.intro.intro.intro.inr.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ (s₁, erase (s₂ + t) a).fst + (s₁, erase (s₂ + t) a).snd = erase (s₁ + s₂ + t) a",
"state_before": "case mk.intro.intro.intro.intro.inr\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.fst + a'.snd = erase (s₁ + s₂ + t) a",
"tactic": "refine' ⟨(s₁, (s₂ + t).erase a), GameAdd.snd ⟨t, a, hr, _⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.intro.inr.refine'_1\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ erase (s₂ + t) a + {a} = s₂ + t",
"tactic": "rw [add_comm, singleton_add, cons_erase h]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.intro.inr.refine'_2\nα : Type u_1\nr✝ r : α → α → Prop\ns₁ s₂ s t : Multiset α\na : α\nhr : ∀ (a' : α), a' ∈ t → r a' a\nhe : s + {a} = s₁ + s₂ + t\nhb : s = erase (s₁ + s₂ + t) a\nh : a ∈ s₂ + t\n⊢ (s₁, erase (s₂ + t) a).fst + (s₁, erase (s₂ + t) a).snd = erase (s₁ + s₂ + t) a",
"tactic": "rw [add_assoc, erase_add_right_pos _ h]"
}
] |
[
127,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.splits_one
|
[] |
[
137,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_sub
|
[] |
[
673,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.sum_mul_boole
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nb : β\nf✝ g : α → β\ninst✝¹ : NonAssocSemiring β\ninst✝ : DecidableEq α\ns : Finset α\nf : α → β\na : α\n⊢ (∑ x in s, f x * if a = x then 1 else 0) = if a ∈ s then f a else 0",
"tactic": "simp"
}
] |
[
77,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.ofEquiv_card
|
[] |
[
149,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_le_iff_right
|
[] |
[
165,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.sup_adj
|
[] |
[
255,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Multiset/Fold.lean
|
Multiset.fold_zero
|
[] |
[
59,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Commute.commutator_eq
|
[] |
[
41,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
List.stronglyMeasurable_prod'
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\nhl : ∀ (f : α → M), f ∈ [] → StronglyMeasurable f\n⊢ StronglyMeasurable (List.prod [])\n\ncase cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : ∀ (f_1 : α → M), f_1 ∈ f :: l → StronglyMeasurable f_1\n⊢ StronglyMeasurable (List.prod (f :: l))",
"state_before": "α : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\n⊢ StronglyMeasurable (List.prod l)",
"tactic": "induction' l with f l ihl"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (List.prod (f :: l))",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : ∀ (f_1 : α → M), f_1 ∈ f :: l → StronglyMeasurable f_1\n⊢ StronglyMeasurable (List.prod (f :: l))",
"tactic": "rw [List.forall_mem_cons] at hl"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (f * List.prod l)",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (List.prod (f :: l))",
"tactic": "rw [List.prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf✝ g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl✝ : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l✝ → StronglyMeasurable f\nf : α → M\nl : List (α → M)\nihl : (∀ (f : α → M), f ∈ l → StronglyMeasurable f) → StronglyMeasurable (List.prod l)\nhl : StronglyMeasurable f ∧ ∀ (x : α → M), x ∈ l → StronglyMeasurable x\n⊢ StronglyMeasurable (f * List.prod l)",
"tactic": "exact hl.1.mul (ihl hl.2)"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.109795\nγ : Type ?u.109798\nι : Type ?u.109801\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl✝ : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\nhl : ∀ (f : α → M), f ∈ [] → StronglyMeasurable f\n⊢ StronglyMeasurable (List.prod [])",
"tactic": "exact stronglyMeasurable_one"
}
] |
[
548,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.Nonempty.of_mul_right
|
[] |
[
393,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/Data/MvPolynomial/Supported.lean
|
MvPolynomial.supported_mono
|
[] |
[
116,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Finset/Interval.lean
|
Finset.Iic_eq_powerset
|
[] |
[
69,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
Rel.mem_interedges_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.1103\nι : Type ?u.1106\nκ : Type ?u.1109\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nx : α × β\n⊢ x ∈ interedges r s t ↔ x.fst ∈ s ∧ x.snd ∈ t ∧ r x.fst x.snd",
"tactic": "rw [interedges, mem_filter, Finset.mem_product, and_assoc]"
}
] |
[
61,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.im_le_abs
|
[] |
[
1047,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1046,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.emod_two_eq_zero_or_one
|
[
{
"state_after": "no goals",
"state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\n⊢ 0 ≤ 2",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\n⊢ abs 2 ≠ 0",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\n⊢ 2 ≠ 0",
"tactic": "decide"
},
{
"state_after": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\nk : ℕ\nh₁ : ↑(k + 2) < 2\nx✝ : 0 ≤ ↑(k + 2)\n⊢ 2 ≤ ↑k + ↑2",
"state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\nk : ℕ\nh₁ : ↑(k + 2) < 2\nx✝ : 0 ≤ ↑(k + 2)\n⊢ 2 ≤ ↑(k + 2)",
"tactic": "rw [Nat.cast_add]"
},
{
"state_after": "no goals",
"state_before": "a b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\nk : ℕ\nh₁ : ↑(k + 2) < 2\nx✝ : 0 ≤ ↑(k + 2)\n⊢ 2 ≤ ↑k + ↑2",
"tactic": "exact (le_add_iff_nonneg_left 2).2 (NonNeg.mk k)"
},
{
"state_after": "no goals",
"state_before": "a✝ b : ℤ\nn✝ : ℕ\nn : ℤ\nh : n % 2 < 2\nh₁✝ : 0 ≤ n % 2\na : ℕ\nx✝ : -[a+1] < 2\nh₁ : 0 ≤ -[a+1]\n⊢ -[a+1] = 0 ∨ -[a+1] = 1",
"tactic": "cases h₁"
}
] |
[
330,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.mem_preimage
|
[] |
[
68,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.right_le_of_le_left''
|
[] |
[
1107,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1105,
1
] |
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