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/Data/Set/Lattice.lean
|
Set.sUnion_subset
|
[] |
[
1053,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_iSup_of_mem
|
[] |
[
1046,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1044,
1
] |
Mathlib/Data/Finsupp/NeLocus.lean
|
Finsupp.coe_neLocus
|
[
{
"state_after": "case h\nα : Type u_1\nM : Type ?u.2198\nN : Type u_2\nP : Type ?u.2204\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq N\ninst✝ : Zero N\nf g : α →₀ N\nx✝ : α\n⊢ x✝ ∈ ↑(neLocus f g) ↔ x✝ ∈ {x | ↑f x ≠ ↑g x}",
"state_before": "α : Type u_1\nM : Type ?u.2198\nN : Type u_2\nP : Type ?u.2204\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq N\ninst✝ : Zero N\nf g : α →₀ N\n⊢ ↑(neLocus f g) = {x | ↑f x ≠ ↑g x}",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nM : Type ?u.2198\nN : Type u_2\nP : Type ?u.2204\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq N\ninst✝ : Zero N\nf g : α →₀ N\nx✝ : α\n⊢ x✝ ∈ ↑(neLocus f g) ↔ x✝ ∈ {x | ↑f x ≠ ↑g x}",
"tactic": "exact mem_neLocus"
}
] |
[
57,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean
|
ContinuousAffineMap.add_contLinear
|
[] |
[
138,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Algebra/Divisibility/Units.lean
|
IsUnit.dvd_mul_right
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝ : Monoid α\na b : α\nu : αˣ\n⊢ a ∣ b * ↑u ↔ a ∣ b",
"state_before": "α : Type u_1\ninst✝ : Monoid α\na b u : α\nhu : IsUnit u\n⊢ a ∣ b * u ↔ a ∣ b",
"tactic": "rcases hu with ⟨u, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : Monoid α\na b : α\nu : αˣ\n⊢ a ∣ b * ↑u ↔ a ∣ b",
"tactic": "apply Units.dvd_mul_right"
}
] |
[
87,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/Int/Parity.lean
|
Int.even_or_odd
|
[] |
[
68,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arccos_pos
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ 0 < arccos x ↔ x < 1",
"tactic": "simp [arccos]"
}
] |
[
358,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Analysis/Complex/Arg.lean
|
Complex.abs_add_eq
|
[] |
[
65,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.mkRat_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "d : Nat\nn : Int\nd0 : d ≠ 0\n⊢ mkRat n d = 0 ↔ n = 0",
"tactic": "simp [mkRat_def, d0]"
}
] |
[
101,
86
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
101,
1
] |
Mathlib/Logic/Embedding/Basic.lean
|
Function.Embedding.setValue_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\nf : α ↪ β\na : α\nb : β\ninst✝¹ : (a' : α) → Decidable (a' = a)\ninst✝ : (a' : α) → Decidable (↑f a' = b)\n⊢ ↑(setValue f a b) a = b",
"tactic": "simp [setValue]"
}
] |
[
208,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.fst_embedding_of_right_embedding
|
[
{
"state_after": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH : Embedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj X = ↑X\n⊢ (forget TopCat).map pullback.fst =\n ↑(homeoOfIso (asIso pullback.fst)) ∘\n (forget TopCat).map (pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S))",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH : Embedding ((forget TopCat).map g)\n⊢ Embedding ((forget TopCat).map pullback.fst)",
"tactic": "convert (homeoOfIso (asIso (pullback.fst : pullback f (𝟙 S) ⟶ _))).embedding.comp\n (pullback_map_embedding_of_embeddings (i₁ := 𝟙 X)\n f g f (𝟙 _) (homeoOfIso (Iso.refl _)).embedding H (𝟙 _) rfl (by simp))"
},
{
"state_after": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH : Embedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj X = ↑X\n⊢ (forget TopCat).map pullback.fst =\n (forget TopCat).map\n (pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S) ≫\n (asIso pullback.fst).hom)",
"state_before": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH : Embedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj X = ↑X\n⊢ (forget TopCat).map pullback.fst =\n ↑(homeoOfIso (asIso pullback.fst)) ∘\n (forget TopCat).map (pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S))",
"tactic": "erw [← coe_comp]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH : Embedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj X = ↑X\n⊢ (forget TopCat).map pullback.fst =\n (forget TopCat).map\n (pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S) ≫\n (asIso pullback.fst).hom)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH : Embedding ((forget TopCat).map g)\n⊢ g ≫ 𝟙 S = g ≫ 𝟙 S",
"tactic": "simp"
}
] |
[
305,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.ext_iff
|
[] |
[
159,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.coe_sup
|
[] |
[
416,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/RingTheory/Artinian.lean
|
isArtinian_of_linearEquiv
|
[] |
[
100,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Topology/PathConnected.lean
|
pathConnectedSpace_iff_connectedSpace
|
[
{
"state_after": "case mp\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\n⊢ PathConnectedSpace X → ConnectedSpace X\n\ncase mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\n⊢ ConnectedSpace X → PathConnectedSpace X",
"state_before": "X : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\n⊢ PathConnectedSpace X ↔ ConnectedSpace X",
"tactic": "constructor"
},
{
"state_after": "case mp\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nh : PathConnectedSpace X\n⊢ ConnectedSpace X",
"state_before": "case mp\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\n⊢ PathConnectedSpace X → ConnectedSpace X",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nh : PathConnectedSpace X\n⊢ ConnectedSpace X",
"tactic": "infer_instance"
},
{
"state_after": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ PathConnectedSpace X",
"state_before": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\n⊢ ConnectedSpace X → PathConnectedSpace X",
"tactic": "intro hX"
},
{
"state_after": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ ∃ x, pathComponent x = univ",
"state_before": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ PathConnectedSpace X",
"tactic": "rw [pathConnectedSpace_iff_eq]"
},
{
"state_after": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ pathComponent (Classical.arbitrary X) = univ",
"state_before": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ ∃ x, pathComponent x = univ",
"tactic": "use Classical.arbitrary X"
},
{
"state_after": "case mpr.refine'_1\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ IsOpen (pathComponent (Classical.arbitrary X))\n\ncase mpr.refine'_2\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ IsClosed (pathComponent (Classical.arbitrary X))",
"state_before": "case mpr\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ pathComponent (Classical.arbitrary X) = univ",
"tactic": "refine' IsClopen.eq_univ ⟨_, _⟩ (by simp)"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ Set.Nonempty (pathComponent (Classical.arbitrary X))",
"tactic": "simp"
},
{
"state_after": "case mpr.refine'_1\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ ∀ (a : X), a ∈ pathComponent (Classical.arbitrary X) → pathComponent (Classical.arbitrary X) ∈ 𝓝 a",
"state_before": "case mpr.refine'_1\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ IsOpen (pathComponent (Classical.arbitrary X))",
"tactic": "rw [isOpen_iff_mem_nhds]"
},
{
"state_after": "case mpr.refine'_1\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\n⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y",
"state_before": "case mpr.refine'_1\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ ∀ (a : X), a ∈ pathComponent (Classical.arbitrary X) → pathComponent (Classical.arbitrary X) ∈ 𝓝 a",
"tactic": "intro y y_in"
},
{
"state_after": "case mpr.refine'_1.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y",
"state_before": "case mpr.refine'_1\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\n⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y",
"tactic": "rcases(path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩"
},
{
"state_after": "case mpr.refine'_1.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ U ⊆ pathComponent (Classical.arbitrary X)",
"state_before": "case mpr.refine'_1.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ pathComponent (Classical.arbitrary X) ∈ 𝓝 y",
"tactic": "apply mem_of_superset U_in"
},
{
"state_after": "case mpr.refine'_1.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ U ⊆ pathComponent y",
"state_before": "case mpr.refine'_1.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ U ⊆ pathComponent (Classical.arbitrary X)",
"tactic": "rw [← pathComponent_congr y_in]"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_1.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\ny_in : y ∈ pathComponent (Classical.arbitrary X)\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ U ⊆ pathComponent y",
"tactic": "exact hU.subset_pathComponent (mem_of_mem_nhds U_in)"
},
{
"state_after": "case mpr.refine'_2\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ ∀ (x : X),\n (∀ (U : Set X), U ∈ 𝓝 x → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))) →\n x ∈ pathComponent (Classical.arbitrary X)",
"state_before": "case mpr.refine'_2\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ IsClosed (pathComponent (Classical.arbitrary X))",
"tactic": "rw [isClosed_iff_nhds]"
},
{
"state_after": "case mpr.refine'_2\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\nH : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))\n⊢ y ∈ pathComponent (Classical.arbitrary X)",
"state_before": "case mpr.refine'_2\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\n⊢ ∀ (x : X),\n (∀ (U : Set X), U ∈ 𝓝 x → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))) →\n x ∈ pathComponent (Classical.arbitrary X)",
"tactic": "intro y H"
},
{
"state_after": "case mpr.refine'_2.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\nH : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ y ∈ pathComponent (Classical.arbitrary X)",
"state_before": "case mpr.refine'_2\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\nH : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))\n⊢ y ∈ pathComponent (Classical.arbitrary X)",
"tactic": "rcases(path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩"
},
{
"state_after": "case mpr.refine'_2.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z✝ : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\nH : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\nz : X\nhz : z ∈ U\nhz' : z ∈ pathComponent (Classical.arbitrary X)\n⊢ y ∈ pathComponent (Classical.arbitrary X)",
"state_before": "case mpr.refine'_2.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\nH : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\n⊢ y ∈ pathComponent (Classical.arbitrary X)",
"tactic": "rcases H U U_in with ⟨z, hz, hz'⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_2.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.703689\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y✝ z✝ : X\nι : Type ?u.703704\nF : Set X\ninst✝ : LocPathConnectedSpace X\nhX : ConnectedSpace X\ny : X\nH : ∀ (U : Set X), U ∈ 𝓝 y → Set.Nonempty (U ∩ pathComponent (Classical.arbitrary X))\nU : Set X\nU_in : U ∈ 𝓝 y\nhU : IsPathConnected U\nz : X\nhz : z ∈ U\nhz' : z ∈ pathComponent (Classical.arbitrary X)\n⊢ y ∈ pathComponent (Classical.arbitrary X)",
"tactic": "exact (hU.joinedIn z hz y <| mem_of_mem_nhds U_in).joined.mem_pathComponent hz'"
}
] |
[
1232,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.mkPiField_apply
|
[] |
[
897,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
895,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.smul_eq_C_smul
|
[
{
"state_after": "case ofFractionRing\nK : Type u\ninst✝ : CommRing K\nR : Type ?u.178730\nr : K\nx : FractionRing K[X]\n⊢ r • { toFractionRing := x } = ↑Polynomial.C r • { toFractionRing := x }",
"state_before": "K : Type u\ninst✝ : CommRing K\nR : Type ?u.178730\nx : RatFunc K\nr : K\n⊢ r • x = ↑Polynomial.C r • x",
"tactic": "cases' x with x"
},
{
"state_after": "case ofFractionRing.H\nK : Type u\ninst✝ : CommRing K\nR : Type ?u.178730\nr : K\ny✝ : K[X] × { x // x ∈ K[X]⁰ }\n⊢ r • { toFractionRing := Localization.mk y✝.fst y✝.snd } =\n ↑Polynomial.C r • { toFractionRing := Localization.mk y✝.fst y✝.snd }",
"state_before": "case ofFractionRing\nK : Type u\ninst✝ : CommRing K\nR : Type ?u.178730\nr : K\nx : FractionRing K[X]\n⊢ r • { toFractionRing := x } = ↑Polynomial.C r • { toFractionRing := x }",
"tactic": "induction x using Localization.induction_on"
},
{
"state_after": "no goals",
"state_before": "case ofFractionRing.H\nK : Type u\ninst✝ : CommRing K\nR : Type ?u.178730\nr : K\ny✝ : K[X] × { x // x ∈ K[X]⁰ }\n⊢ r • { toFractionRing := Localization.mk y✝.fst y✝.snd } =\n ↑Polynomial.C r • { toFractionRing := Localization.mk y✝.fst y✝.snd }",
"tactic": "rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,\n Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]"
}
] |
[
463,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
FG_adjoin_singleton_of_integral
|
[
{
"state_after": "case intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nhx : IsIntegral R x\n⊢ FG (↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"tactic": "rcases hx with ⟨f, hfm, hfx⟩"
},
{
"state_after": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\n⊢ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))) ≤\n ↑Subalgebra.toSubmodule (Algebra.adjoin R {x})\n\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\n⊢ ↑Subalgebra.toSubmodule (Algebra.adjoin R {x}) ≤\n span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\n⊢ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))) =\n ↑Subalgebra.toSubmodule (Algebra.adjoin R {x})",
"tactic": "apply le_antisymm"
},
{
"state_after": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nr : A\nhr : r ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin R {x})\n⊢ r ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\n⊢ ↑Subalgebra.toSubmodule (Algebra.adjoin R {x}) ≤\n span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "intro r hr"
},
{
"state_after": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nr : A\nhr : r ∈ Algebra.adjoin R {x}\n⊢ r ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nr : A\nhr : r ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin R {x})\n⊢ r ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "change r ∈ Algebra.adjoin R ({x} : Set A) at hr"
},
{
"state_after": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nr : A\nhr : r ∈ AlgHom.range (aeval x)\n⊢ r ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nr : A\nhr : r ∈ Algebra.adjoin R {x}\n⊢ r ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "rw [Algebra.adjoin_singleton_eq_range_aeval] at hr"
},
{
"state_after": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) p ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nr : A\nhr : r ∈ AlgHom.range (aeval x)\n⊢ r ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "rcases(aeval x).mem_range.mp hr with ⟨p, rfl⟩"
},
{
"state_after": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) (p %ₘ f + f * (p /ₘ f)) ∈\n span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) p ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "rw [← modByMonic_add_div p hfm]"
},
{
"state_after": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : ↑(aeval x) f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) (p %ₘ f + f * (p /ₘ f)) ∈\n span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) (p %ₘ f + f * (p /ₘ f)) ∈\n span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "rw [← aeval_def] at hfx"
},
{
"state_after": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : ↑(aeval x) f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) (p %ₘ f) ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : ↑(aeval x) f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) (p %ₘ f + f * (p /ₘ f)) ∈\n span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "rw [AlgHom.map_add, AlgHom.map_mul, hfx, MulZeroClass.zero_mul, add_zero]"
},
{
"state_after": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : ↑(aeval x) f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\nthis : degree (p %ₘ f) ≤ degree f\n⊢ ↑(aeval x) (p %ₘ f) ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"state_before": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : ↑(aeval x) f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\n⊢ ↑(aeval x) (p %ₘ f) ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
"tactic": "have : degree (p %ₘ f) ≤ degree f := degree_modByMonic_le p hfm"
},
{
"state_after": "case intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : ↑(aeval x) f = 0\np : R[X]\nhr : ↑(aeval x) p ∈ AlgHom.range (aeval x)\nq : R[X]\nthis : degree q ≤ degree f\n⊢ ↑(aeval x) q ∈ span R ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))",
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{
"state_after": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\ns : A\nhs : s ∈ ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n⊢ s ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\n⊢ ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))) ⊆\n ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"tactic": "intro s hs"
},
{
"state_after": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\ns : A\nhs : s ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n⊢ s ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\ns : A\nhs : s ∈ ↑(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n⊢ s ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"tactic": "rw [Finset.mem_coe] at hs"
},
{
"state_after": "case intro.intro.a.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nk : ℕ\nhk : k ∈ Finset.range (natDegree f + 1)\nhs : (fun x x_1 => x ^ x_1) x k ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n⊢ (fun x x_1 => x ^ x_1) x k ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"state_before": "case intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\ns : A\nhs : s ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n⊢ s ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"tactic": "rcases Finset.mem_image.1 hs with ⟨k, hk, rfl⟩"
},
{
"state_after": "case intro.intro.a.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nk : ℕ\nhs : (fun x x_1 => x ^ x_1) x k ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n⊢ (fun x x_1 => x ^ x_1) x k ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"state_before": "case intro.intro.a.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nk : ℕ\nhk : k ∈ Finset.range (natDegree f + 1)\nhs : (fun x x_1 => x ^ x_1) x k ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n⊢ (fun x x_1 => x ^ x_1) x k ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"tactic": "clear hk"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.a.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.290022\nS : Type ?u.290025\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\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval₂ (algebraMap R A) x f = 0\nk : ℕ\nhs : (fun x x_1 => x ^ x_1) x k ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n⊢ (fun x x_1 => x ^ x_1) x k ∈ ↑(↑Subalgebra.toSubmodule (Algebra.adjoin R {x}))",
"tactic": "exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k"
}
] |
[
227,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.rightInverse_rangeSplitting
|
[] |
[
1176,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1173,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.MemRightTransversals.mul_inv_toFun_mem
|
[] |
[
426,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.empty_sdiff
|
[] |
[
2172,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2171,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycle.pow_iff
|
[
{
"state_after": "case intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ IsCycle (f ^ n) ↔ Nat.coprime n (orderOf f)",
"state_before": "ι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\n⊢ IsCycle (f ^ n) ↔ Nat.coprime n (orderOf f)",
"tactic": "cases nonempty_fintype β"
},
{
"state_after": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ IsCycle (f ^ n) → Nat.coprime n (orderOf f)\n\ncase intro.mpr\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ Nat.coprime n (orderOf f) → IsCycle (f ^ n)",
"state_before": "case intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ IsCycle (f ^ n) ↔ Nat.coprime n (orderOf f)",
"tactic": "constructor"
},
{
"state_after": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\n⊢ Nat.coprime n (orderOf f)",
"state_before": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ IsCycle (f ^ n) → Nat.coprime n (orderOf f)",
"tactic": "intro h"
},
{
"state_after": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\n⊢ Nat.coprime n (orderOf f)",
"state_before": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\n⊢ Nat.coprime n (orderOf f)",
"tactic": "have hr : support (f ^ n) = support f := by\n rw [hf.support_pow_eq_iff]\n rintro ⟨k, rfl⟩\n refine' h.ne_one _\n simp [pow_mul, pow_orderOf_eq_one]"
},
{
"state_after": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nthis : orderOf (f ^ n) = orderOf f\n⊢ Nat.coprime n (orderOf f)",
"state_before": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\n⊢ Nat.coprime n (orderOf f)",
"tactic": "have : orderOf (f ^ n) = orderOf f := by rw [h.orderOf, hr, hf.orderOf]"
},
{
"state_after": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nthis : orderOf f = 0 ∨ Nat.gcd (orderOf f) n = 1\n⊢ Nat.coprime n (orderOf f)",
"state_before": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nthis : orderOf (f ^ n) = orderOf f\n⊢ Nat.coprime n (orderOf f)",
"tactic": "rw [orderOf_pow, Nat.div_eq_self] at this"
},
{
"state_after": "case intro.mp.inl\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh✝ : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nh : orderOf f = 0\n⊢ Nat.coprime n (orderOf f)\n\ncase intro.mp.inr\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nh✝ : Nat.gcd (orderOf f) n = 1\n⊢ Nat.coprime n (orderOf f)",
"state_before": "case intro.mp\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nthis : orderOf f = 0 ∨ Nat.gcd (orderOf f) n = 1\n⊢ Nat.coprime n (orderOf f)",
"tactic": "cases' this with h"
},
{
"state_after": "ι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\n⊢ ¬orderOf f ∣ n",
"state_before": "ι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\n⊢ support (f ^ n) = support f",
"tactic": "rw [hf.support_pow_eq_iff]"
},
{
"state_after": "case intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nval✝ : Fintype β\nk : ℕ\nh : IsCycle (f ^ (orderOf f * k))\n⊢ False",
"state_before": "ι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\n⊢ ¬orderOf f ∣ n",
"tactic": "rintro ⟨k, rfl⟩"
},
{
"state_after": "case intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nval✝ : Fintype β\nk : ℕ\nh : IsCycle (f ^ (orderOf f * k))\n⊢ f ^ (orderOf f * k) = 1",
"state_before": "case intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nval✝ : Fintype β\nk : ℕ\nh : IsCycle (f ^ (orderOf f * k))\n⊢ False",
"tactic": "refine' h.ne_one _"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nval✝ : Fintype β\nk : ℕ\nh : IsCycle (f ^ (orderOf f * k))\n⊢ f ^ (orderOf f * k) = 1",
"tactic": "simp [pow_mul, pow_orderOf_eq_one]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\n⊢ orderOf (f ^ n) = orderOf f",
"tactic": "rw [h.orderOf, hr, hf.orderOf]"
},
{
"state_after": "no goals",
"state_before": "case intro.mp.inl\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh✝ : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nh : orderOf f = 0\n⊢ Nat.coprime n (orderOf f)",
"tactic": "exact absurd h (orderOf_pos _).ne'"
},
{
"state_after": "no goals",
"state_before": "case intro.mp.inr\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : IsCycle (f ^ n)\nhr : support (f ^ n) = support f\nh✝ : Nat.gcd (orderOf f) n = 1\n⊢ Nat.coprime n (orderOf f)",
"tactic": "rwa [Nat.coprime_iff_gcd_eq_one, Nat.gcd_comm]"
},
{
"state_after": "case intro.mpr\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\n⊢ IsCycle (f ^ n)",
"state_before": "case intro.mpr\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\n⊢ Nat.coprime n (orderOf f) → IsCycle (f ^ n)",
"tactic": "intro h"
},
{
"state_after": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\n⊢ IsCycle (f ^ n)",
"state_before": "case intro.mpr\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\n⊢ IsCycle (f ^ n)",
"tactic": "obtain ⟨m, hm⟩ := exists_pow_eq_self_of_coprime h"
},
{
"state_after": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\nhf' : IsCycle ((f ^ n) ^ m)\n⊢ IsCycle (f ^ n)",
"state_before": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\n⊢ IsCycle (f ^ n)",
"tactic": "have hf' : IsCycle ((f ^ n) ^ m) := by rwa [hm]"
},
{
"state_after": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\nhf' : IsCycle ((f ^ n) ^ m)\nx : β\nhx : x ∈ support (f ^ n)\n⊢ x ∈ support ((f ^ n) ^ m)",
"state_before": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\nhf' : IsCycle ((f ^ n) ^ m)\n⊢ IsCycle (f ^ n)",
"tactic": "refine' hf'.of_pow fun x hx => _"
},
{
"state_after": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\nhf' : IsCycle ((f ^ n) ^ m)\nx : β\nhx : x ∈ support (f ^ n)\n⊢ x ∈ support f",
"state_before": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\nhf' : IsCycle ((f ^ n) ^ m)\nx : β\nhx : x ∈ support (f ^ n)\n⊢ x ∈ support ((f ^ n) ^ m)",
"tactic": "rw [hm]"
},
{
"state_after": "no goals",
"state_before": "case intro.mpr.intro\nι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx✝ y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\nhf' : IsCycle ((f ^ n) ^ m)\nx : β\nhx : x ∈ support (f ^ n)\n⊢ x ∈ support f",
"tactic": "exact support_pow_le _ n hx"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1194880\nα : Type ?u.1194883\nβ : Type u_1\nf✝ g : Perm α\nx y : α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : Finite β\nf : Perm β\nhf : IsCycle f\nn : ℕ\nval✝ : Fintype β\nh : Nat.coprime n (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\n⊢ IsCycle ((f ^ n) ^ m)",
"tactic": "rwa [hm]"
}
] |
[
667,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
646,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.IsPath.transfer
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nhp : ∀ (e : Sym2 V), e ∈ edges p → e ∈ edgeSet H\npp : IsPath p\n⊢ IsPath (Walk.transfer p H hp)",
"tactic": "induction p with\n| nil => simp\n| cons _ _ ih =>\n simp only [Walk.transfer, cons_isPath_iff, support_transfer _ ] at pp ⊢\n exact ⟨ih _ pp.1, pp.2⟩"
},
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nu✝ : V\nhp : ∀ (e : Sym2 V), e ∈ edges Walk.nil → e ∈ edgeSet H\npp : IsPath Walk.nil\n⊢ IsPath (Walk.transfer Walk.nil H hp)",
"tactic": "simp"
},
{
"state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ (hp : ∀ (e : Sym2 V), e ∈ edges p✝ → e ∈ edgeSet H), IsPath p✝ → IsPath (Walk.transfer p✝ H hp)\nhp : ∀ (e : Sym2 V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H\npp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\n⊢ IsPath (Walk.transfer p✝ H (_ : ∀ (e : Sym2 V), e ∈ edges p✝ → e ∈ edgeSet H)) ∧ ¬u✝ ∈ support p✝",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ (hp : ∀ (e : Sym2 V), e ∈ edges p✝ → e ∈ edgeSet H), IsPath p✝ → IsPath (Walk.transfer p✝ H hp)\nhp : ∀ (e : Sym2 V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H\npp : IsPath (cons h✝ p✝)\n⊢ IsPath (Walk.transfer (cons h✝ p✝) H hp)",
"tactic": "simp only [Walk.transfer, cons_isPath_iff, support_transfer _ ] at pp ⊢"
},
{
"state_after": "no goals",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ (hp : ∀ (e : Sym2 V), e ∈ edges p✝ → e ∈ edgeSet H), IsPath p✝ → IsPath (Walk.transfer p✝ H hp)\nhp : ∀ (e : Sym2 V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H\npp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\n⊢ IsPath (Walk.transfer p✝ H (_ : ∀ (e : Sym2 V), e ∈ edges p✝ → e ∈ edgeSet H)) ∧ ¬u✝ ∈ support p✝",
"tactic": "exact ⟨ih _ pp.1, pp.2⟩"
}
] |
[
1719,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1713,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.exists_nat_pos_mul_gt
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.339781\nβ : Type ?u.339784\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ 0\nhb : b ≠ ⊤\nn : ℕ\nhn : b / a < ↑n\n⊢ b < ↑n * a",
"tactic": "rwa [← ENNReal.div_lt_iff (Or.inl ha) (Or.inr hb)]"
}
] |
[
1825,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1822,
1
] |
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.equivAddCircle_apply_mk
|
[] |
[
334,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_iUnion_eq_or_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.94012\nι : Sort ?u.94015\nι' : Sort ?u.94018\nι₂ : Sort ?u.94021\nκ : ι → Sort ?u.94026\nκ₁ : ι → Sort ?u.94031\nκ₂ : ι → Sort ?u.94036\nκ' : ι' → Sort ?u.94041\nb : β\np : β → Prop\ns : (x : β) → x = b ∨ p x → Set α\n⊢ (⋃ (x : β) (h : x = b ∨ p x), s x h) = s b (_ : b = b ∨ p b) ∪ ⋃ (x : β) (h : p x), s x (_ : x = b ∨ p x)",
"tactic": "simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]"
}
] |
[
838,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
836,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean
|
ContinuousMap.continuous_coe
|
[] |
[
179,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Analysis/NormedSpace/CompactOperator.lean
|
isCompactOperator_zero
|
[] |
[
71,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/CategoryTheory/Limits/ColimitLimit.lean
|
CategoryTheory.Limits.map_id_right_eq_curry_swap_map
|
[] |
[
52,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.const_max
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : PseudoEMetricSpace α\nf g : α → ℝ\nKf Kg : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x => max a (f x)",
"tactic": "simpa only [max_comm] using hf.max_const a"
}
] |
[
456,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Logic/Relation.lean
|
Acc.of_downward_closed
|
[] |
[
204,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Analysis/Convex/Independent.lean
|
Subsingleton.convexIndependent
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns✝ t : Set E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ ↑(convexHull 𝕜).toOrderHom (p '' s)\n⊢ x ∈ s",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns t : Set E\ninst✝ : Subsingleton ι\np : ι → E\n⊢ ConvexIndependent 𝕜 p",
"tactic": "intro s x hx"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns✝ t : Set E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ ↑(convexHull 𝕜).toOrderHom (p '' s)\nthis : Set.Nonempty (↑(convexHull 𝕜).toOrderHom (p '' s))\n⊢ x ∈ s",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns✝ t : Set E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ ↑(convexHull 𝕜).toOrderHom (p '' s)\n⊢ x ∈ s",
"tactic": "have : (convexHull 𝕜 (p '' s)).Nonempty := ⟨p x, hx⟩"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns✝ t : Set E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ ↑(convexHull 𝕜).toOrderHom (p '' s)\nthis : Set.Nonempty s\n⊢ x ∈ s",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns✝ t : Set E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ ↑(convexHull 𝕜).toOrderHom (p '' s)\nthis : Set.Nonempty (↑(convexHull 𝕜).toOrderHom (p '' s))\n⊢ x ∈ s",
"tactic": "rw [convexHull_nonempty_iff, Set.nonempty_image_iff] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nι : Type u_1\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns✝ t : Set E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ ↑(convexHull 𝕜).toOrderHom (p '' s)\nthis : Set.Nonempty s\n⊢ x ∈ s",
"tactic": "rwa [Subsingleton.mem_iff_nonempty]"
}
] |
[
71,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.cast_zneg
|
[] |
[
1064,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1061,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.one_mem_div_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.114914\nα : Type u_1\nβ : Type ?u.114920\nγ : Type ?u.114923\ninst✝ : Group α\ns t : Set α\na b : α\n⊢ 1 ∈ s / t ↔ ¬Disjoint s t",
"tactic": "simp [not_disjoint_iff_nonempty_inter, mem_div, div_eq_one, Set.Nonempty]"
}
] |
[
1163,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1162,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.aestronglyMeasurable
|
[] |
[
230,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
11
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.comp_equiv_symm_dotProduct
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.146320\nm : Type u_1\nn : Type u_2\no : Type ?u.146329\nm' : o → Type ?u.146334\nn' : o → Type ?u.146339\nR : Type ?u.146342\nS : Type ?u.146345\nα : Type v\nβ : Type w\nγ : Type ?u.146352\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v w : m → α\nx y : n → α\ne : m ≃ n\nx✝¹ : m\nx✝ : x✝¹ ∈ Finset.univ\n⊢ (u ∘ ↑e.symm) (↑e x✝¹) * x (↑e x✝¹) = u x✝¹ * (x ∘ ↑e) x✝¹",
"tactic": "simp only [Function.comp, Equiv.symm_apply_apply]"
}
] |
[
785,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
783,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean
|
AbstractCompletion.extend_def
|
[] |
[
136,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
Part.ωSup_eq_none
|
[] |
[
341,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Data/Nat/GCD/BigOperators.lean
|
Nat.coprime_prod_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nx : ℕ\ns : ι → ℕ\nt : Finset ι\n⊢ (fun y => coprime y x) 1",
"tactic": "simp"
}
] |
[
27,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
25,
1
] |
Mathlib/MeasureTheory/Constructions/Polish.lean
|
IsClosed.measurableSet_image_of_continuousOn_injOn
|
[
{
"state_after": "α : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n⊢ MeasurableSet (range fun x => f ↑x)",
"state_before": "α : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n⊢ MeasurableSet (f '' s)",
"tactic": "rw [image_eq_range]"
},
{
"state_after": "α : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace ↑s\n⊢ MeasurableSet (range fun x => f ↑x)",
"state_before": "α : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n⊢ MeasurableSet (range fun x => f ↑x)",
"tactic": "haveI : PolishSpace s := IsClosed.polishSpace hs"
},
{
"state_after": "case f_cont\nα : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace ↑s\n⊢ Continuous fun x => f ↑x\n\ncase f_inj\nα : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace ↑s\n⊢ Injective fun x => f ↑x",
"state_before": "α : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace ↑s\n⊢ MeasurableSet (range fun x => f ↑x)",
"tactic": "apply measurableSet_range_of_continuous_injective"
},
{
"state_after": "no goals",
"state_before": "case f_cont\nα : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace ↑s\n⊢ Continuous fun x => f ↑x",
"tactic": "rwa [continuousOn_iff_continuous_restrict] at f_cont"
},
{
"state_after": "no goals",
"state_before": "case f_inj\nα : Type ?u.320629\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.320635\nγ : Type u_2\ntγ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\nβ : Type u_1\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set γ\nhs : IsClosed s\nf : γ → β\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace ↑s\n⊢ Injective fun x => f ↑x",
"tactic": "rwa [injOn_iff_injective] at f_inj"
}
] |
[
581,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.nadd_one
|
[
{
"state_after": "case h\na✝ b c a : Ordinal\nIH : ∀ (k : Ordinal), k < a → k ♯ 1 = succ k\n⊢ a ♯ 1 = succ a",
"state_before": "a b c : Ordinal\n⊢ a ♯ 1 = succ a",
"tactic": "induction' a using Ordinal.induction with a IH"
},
{
"state_after": "case h\na✝ b c a : Ordinal\nIH : ∀ (k : Ordinal), k < a → k ♯ 1 = succ k\n⊢ ∀ (i : Ordinal), i < a → i ♯ 1 < succ a",
"state_before": "case h\na✝ b c a : Ordinal\nIH : ∀ (k : Ordinal), k < a → k ♯ 1 = succ k\n⊢ a ♯ 1 = succ a",
"tactic": "rw [nadd_def, blsub_one, nadd_zero, max_eq_right_iff, blsub_le_iff]"
},
{
"state_after": "case h\na✝ b c a : Ordinal\nIH : ∀ (k : Ordinal), k < a → k ♯ 1 = succ k\ni : Ordinal\nhi : i < a\n⊢ i ♯ 1 < succ a",
"state_before": "case h\na✝ b c a : Ordinal\nIH : ∀ (k : Ordinal), k < a → k ♯ 1 = succ k\n⊢ ∀ (i : Ordinal), i < a → i ♯ 1 < succ a",
"tactic": "intro i hi"
},
{
"state_after": "no goals",
"state_before": "case h\na✝ b c a : Ordinal\nIH : ∀ (k : Ordinal), k < a → k ♯ 1 = succ k\ni : Ordinal\nhi : i < a\n⊢ i ♯ 1 < succ a",
"tactic": "rwa [IH i hi, succ_lt_succ_iff]"
}
] |
[
302,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Filter.Tendsto.cexp
|
[] |
[
94,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
src/lean/Init/Control/ExceptCps.lean
|
ExceptCpsT.runCatch_pure
|
[] |
[
62,
95
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
62,
9
] |
Mathlib/Algebra/Order/WithZero.lean
|
one_le_inv₀
|
[] |
[
155,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.currySum_apply
|
[] |
[
1786,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1784,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
IsBoundedLinearMap.isBigO_sub
|
[] |
[
203,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.opow_lt_opow_iff_right
|
[] |
[
116,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
div_sub_one
|
[] |
[
169,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Order/Minimal.lean
|
eq_of_mem_minimals
|
[] |
[
93,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
GroupCat.SurjectiveOfEpiAuxs.one_smul
|
[
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nx : X'\ny : ↑(Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier))\n⊢ fromCoset\n { val := 1 *l ↑y,\n property :=\n (_ : 1 *l ↑y ∈ Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) } =\n fromCoset y",
"state_before": "A B : GroupCat\nf : A ⟶ B\nx : X'\ny : ↑(Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier))\n⊢ 1 • fromCoset y = fromCoset y",
"tactic": "change fromCoset _ = fromCoset _"
},
{
"state_after": "no goals",
"state_before": "A B : GroupCat\nf : A ⟶ B\nx : X'\ny : ↑(Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier))\n⊢ fromCoset\n { val := 1 *l ↑y,\n property :=\n (_ : 1 *l ↑y ∈ Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) } =\n fromCoset y",
"tactic": "simp only [one_leftCoset, Subtype.ext_iff_val]"
}
] |
[
144,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.zero_lf
|
[
{
"state_after": "x : PGame\n⊢ ((∃ i,\n (∀ (i' : LeftMoves 0), moveLeft 0 i' ⧏ moveLeft x i) ∧\n ∀ (j : RightMoves (moveLeft x i)), 0 ⧏ moveRight (moveLeft x i) j) ∨\n ∃ j,\n (∀ (i : LeftMoves (moveRight 0 j)), moveLeft (moveRight 0 j) i ⧏ x) ∧\n ∀ (j' : RightMoves x), moveRight 0 j ⧏ moveRight x j') ↔\n ∃ i, ∀ (j : RightMoves (moveLeft x i)), 0 ⧏ moveRight (moveLeft x i) j",
"state_before": "x : PGame\n⊢ 0 ⧏ x ↔ ∃ i, ∀ (j : RightMoves (moveLeft x i)), 0 ⧏ moveRight (moveLeft x i) j",
"tactic": "rw [lf_def]"
},
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ ((∃ i,\n (∀ (i' : LeftMoves 0), moveLeft 0 i' ⧏ moveLeft x i) ∧\n ∀ (j : RightMoves (moveLeft x i)), 0 ⧏ moveRight (moveLeft x i) j) ∨\n ∃ j,\n (∀ (i : LeftMoves (moveRight 0 j)), moveLeft (moveRight 0 j) i ⧏ x) ∧\n ∀ (j' : RightMoves x), moveRight 0 j ⧏ moveRight x j') ↔\n ∃ i, ∀ (j : RightMoves (moveLeft x i)), 0 ⧏ moveRight (moveLeft x i) j",
"tactic": "simp"
}
] |
[
681,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
ContDiffWithinAt.differentiableWithinAt_iteratedDerivWithin
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.53911\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nm : ℕ\nh : ContDiffWithinAt 𝕜 n f s x\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 (insert x s)\n⊢ DifferentiableWithinAt 𝕜 (iteratedDerivWithin m f s) s x",
"tactic": "simpa only [iteratedDerivWithin_eq_equiv_comp,\n LinearIsometryEquiv.comp_differentiableWithinAt_iff] using\n h.differentiableWithinAt_iteratedFDerivWithin hmn hs"
}
] |
[
167,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
MulLECancellable.mul_le_mul_iff_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.86235\ninst✝² : LE α\ninst✝¹ : CommSemigroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : MulLECancellable a\n⊢ b * a ≤ c * a ↔ b ≤ c",
"tactic": "rw [mul_comm b, mul_comm c, ha.mul_le_mul_iff_left]"
}
] |
[
1642,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1640,
11
] |
Mathlib/Data/Nat/Cast/WithTop.lean
|
Nat.cast_withTop
|
[] |
[
23,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
22,
1
] |
Mathlib/Analysis/Normed/Field/UnitBall.lean
|
coe_mul_unitBall
|
[] |
[
54,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
Mon_.comp_hom'
|
[] |
[
133,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.functorPushforward_comp
|
[
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nR : Sieve X\nY✝ : E\nf✝ : Y✝ ⟶ (F ⋙ G).obj X\n⊢ (functorPushforward (F ⋙ G) R).arrows f✝ ↔ (functorPushforward G (functorPushforward F R)).arrows f✝",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nR : Sieve X\n⊢ functorPushforward (F ⋙ G) R = functorPushforward G (functorPushforward F R)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nR : Sieve X\nY✝ : E\nf✝ : Y✝ ⟶ (F ⋙ G).obj X\n⊢ (functorPushforward (F ⋙ G) R).arrows f✝ ↔ (functorPushforward G (functorPushforward F R)).arrows f✝",
"tactic": "simp [R.arrows.functorPushforward_comp F G]"
}
] |
[
652,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.Valid.prevn
|
[] |
[
659,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
657,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.erase_union_singleton
|
[
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : x = a\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s\n\ncase neg\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : ¬x = a\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s",
"tactic": "by_cases h' : x = a"
},
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns : Finmap β\nx : α\nb : β x\nh : lookup x s = some b\n⊢ lookup x (erase x s ∪ singleton x b) = lookup x s",
"state_before": "case pos\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : x = a\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s",
"tactic": "subst a"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns : Finmap β\nx : α\nb : β x\nh : lookup x s = some b\n⊢ lookup x (erase x s ∪ singleton x b) = lookup x s",
"tactic": "rw [lookup_union_right not_mem_erase_self, lookup_singleton_eq, h]"
},
{
"state_after": "case neg\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : ¬x = a\nthis : ¬x ∈ singleton a b\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s",
"state_before": "case neg\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : ¬x = a\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s",
"tactic": "have : x ∉ singleton a b := by rwa [mem_singleton]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : ¬x = a\nthis : ¬x ∈ singleton a b\n⊢ lookup x (erase a s ∪ singleton a b) = lookup x s",
"tactic": "rw [lookup_union_left_of_not_in this, lookup_erase_ne h']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\nb : β a\ns : Finmap β\nh : lookup a s = some b\nx : α\nh' : ¬x = a\n⊢ ¬x ∈ singleton a b",
"tactic": "rwa [mem_singleton]"
}
] |
[
643,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Algebra/Homology/ShortExact/Preadditive.lean
|
CategoryTheory.Splitting.inr_iso_inv
|
[] |
[
238,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Combinatorics/Quiver/SingleObj.lean
|
Quiver.SingleObj.listToPath_pathToList
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx : SingleObj α\n⊢ listToPath (pathToList Path.nil) = Path.cast (_ : star α = star α) (_ : star α = star α) Path.nil\n\ncase cons\nα : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx y z : SingleObj α\np : Path (star α) y\na : y ⟶ z\nih : listToPath (pathToList p) = Path.cast (_ : star α = star α) (_ : y = star α) p\n⊢ listToPath (pathToList (Path.cons p a)) = Path.cast (_ : star α = star α) (_ : z = star α) (Path.cons p a)",
"state_before": "α : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx : SingleObj α\np : Path (star α) x\n⊢ listToPath (pathToList p) = Path.cast (_ : star α = star α) (_ : x = star α) p",
"tactic": "induction' p with y z p a ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx : SingleObj α\n⊢ listToPath (pathToList Path.nil) = Path.cast (_ : star α = star α) (_ : star α = star α) Path.nil",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx y z : SingleObj α\np : Path (star α) y\na : y ⟶ z\nih : listToPath (pathToList p) = p\n⊢ Path.cons (listToPath (pathToList p)) a = Path.cons p a",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx y z : SingleObj α\np : Path (star α) y\na : y ⟶ z\nih : listToPath (pathToList p) = Path.cast (_ : star α = star α) (_ : y = star α) p\n⊢ listToPath (pathToList (Path.cons p a)) = Path.cast (_ : star α = star α) (_ : z = star α) (Path.cons p a)",
"tactic": "dsimp at *"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.4805\nγ : Type ?u.4808\nx y z : SingleObj α\np : Path (star α) y\na : y ⟶ z\nih : listToPath (pathToList p) = p\n⊢ Path.cons (listToPath (pathToList p)) a = Path.cons p a",
"tactic": "rw [ih]"
}
] |
[
142,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Std/Data/Int/DivMod.lean
|
Int.fmod_self
|
[
{
"state_after": "a : Int\nthis : fmod (1 * a) a = 0\n⊢ fmod a a = 0",
"state_before": "a : Int\n⊢ fmod a a = 0",
"tactic": "have := mul_fmod_left 1 a"
},
{
"state_after": "no goals",
"state_before": "a : Int\nthis : fmod (1 * a) a = 0\n⊢ fmod a a = 0",
"tactic": "rwa [Int.one_mul] at this"
}
] |
[
483,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
482,
9
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.get_natCast
|
[] |
[
181,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeCommRing.isSupported_one
|
[] |
[
210,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.snd_list_prod
|
[
{
"state_after": "case nil\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\n⊢ snd (List.prod []) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst [])) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst []))) • snd x.snd)\n (List.enum []))\n\ncase cons\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n snd (List.prod xs) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs))\n⊢ snd (List.prod (x :: xs)) =\n List.sum\n (List.map\n (fun x_1 =>\n List.prod (List.take x_1.fst (List.map fst (x :: xs))) •\n op (List.prod (List.drop (Nat.succ x_1.fst) (List.map fst (x :: xs)))) • snd x_1.snd)\n (List.enum (x :: xs)))",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nl : List (tsze R M)\n⊢ snd (List.prod l) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst l)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst l))) • snd x.snd)\n (List.enum l))",
"tactic": "induction' l with x xs ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\n⊢ snd (List.prod []) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst [])) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst []))) • snd x.snd)\n (List.enum []))",
"tactic": "simp"
},
{
"state_after": "case cons\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n snd (List.prod xs) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs))\n⊢ snd (List.prod (x :: xs)) =\n List.sum\n (List.map\n (fun x_1 =>\n List.prod (List.take x_1.fst (List.map fst (x :: xs))) •\n op (List.prod (List.drop (Nat.succ x_1.fst) (List.map fst (x :: xs)))) • snd x_1.snd)\n ((0, x) :: List.map (Prod.map (fun x => x + 1) id) (List.enum xs)))",
"state_before": "case cons\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n snd (List.prod xs) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs))\n⊢ snd (List.prod (x :: xs)) =\n List.sum\n (List.map\n (fun x_1 =>\n List.prod (List.take x_1.fst (List.map fst (x :: xs))) •\n op (List.prod (List.drop (Nat.succ x_1.fst) (List.map fst (x :: xs)))) • snd x_1.snd)\n (List.enum (x :: xs)))",
"tactic": "rw [List.enum_cons, ← List.map_fst_add_enum_eq_enumFrom]"
},
{
"state_after": "case cons\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n snd (List.prod xs) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs))\n⊢ List.sum\n (List.map\n ((fun x_1 => fst x • x_1) ∘ fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs)) +\n op (List.prod (List.map fst xs)) • snd x =\n op (List.prod (List.map fst xs)) • snd x +\n List.sum\n (List.map\n (fun x_1 =>\n fst x •\n List.prod (List.take x_1.fst (List.map fst xs)) •\n op (List.prod (List.drop (x_1.fst + 1) (List.map fst xs))) • snd x_1.snd)\n (List.enum xs))",
"state_before": "case cons\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n snd (List.prod xs) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs))\n⊢ snd (List.prod (x :: xs)) =\n List.sum\n (List.map\n (fun x_1 =>\n List.prod (List.take x_1.fst (List.map fst (x :: xs))) •\n op (List.prod (List.drop (Nat.succ x_1.fst) (List.map fst (x :: xs)))) • snd x_1.snd)\n ((0, x) :: List.map (Prod.map (fun x => x + 1) id) (List.enum xs)))",
"tactic": "simp_rw [List.map_cons, List.map_map, Function.comp, Prod.map_snd, Prod.map_fst, id,\n List.take_zero, List.take_cons, List.prod_nil, List.prod_cons, snd_mul, one_smul, List.drop,\n mul_smul, List.sum_cons, fst_list_prod, ih, List.smul_sum, List.map_map]"
},
{
"state_after": "no goals",
"state_before": "case cons\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n snd (List.prod xs) =\n List.sum\n (List.map\n (fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs))\n⊢ List.sum\n (List.map\n ((fun x_1 => fst x • x_1) ∘ fun x =>\n List.prod (List.take x.fst (List.map fst xs)) •\n op (List.prod (List.drop (Nat.succ x.fst) (List.map fst xs))) • snd x.snd)\n (List.enum xs)) +\n op (List.prod (List.map fst xs)) • snd x =\n op (List.prod (List.map fst xs)) • snd x +\n List.sum\n (List.map\n (fun x_1 =>\n fst x •\n List.prod (List.take x_1.fst (List.map fst xs)) •\n op (List.prod (List.drop (x_1.fst + 1) (List.map fst xs))) • snd x_1.snd)\n (List.enum xs))",
"tactic": "exact add_comm _ _"
}
] |
[
691,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Homeomorph.isBigOWith_congr
|
[] |
[
2221,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2219,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Fin.isCycle_cycleRange
|
[
{
"state_after": "case mk\nn i : ℕ\nhi : i < n + 1\nh0 : { val := i, isLt := hi } ≠ 0\n⊢ IsCycle (cycleRange { val := i, isLt := hi })",
"state_before": "n : ℕ\ni : Fin (n + 1)\nh0 : i ≠ 0\n⊢ IsCycle (cycleRange i)",
"tactic": "cases' i with i hi"
},
{
"state_after": "case mk.zero\nn : ℕ\nhi : Nat.zero < n + 1\nh0 : { val := Nat.zero, isLt := hi } ≠ 0\n⊢ IsCycle (cycleRange { val := Nat.zero, isLt := hi })\n\ncase mk.succ\nn n✝ : ℕ\nhi : Nat.succ n✝ < n + 1\nh0 : { val := Nat.succ n✝, isLt := hi } ≠ 0\n⊢ IsCycle (cycleRange { val := Nat.succ n✝, isLt := hi })",
"state_before": "case mk\nn i : ℕ\nhi : i < n + 1\nh0 : { val := i, isLt := hi } ≠ 0\n⊢ IsCycle (cycleRange { val := i, isLt := hi })",
"tactic": "cases i"
},
{
"state_after": "no goals",
"state_before": "case mk.succ\nn n✝ : ℕ\nhi : Nat.succ n✝ < n + 1\nh0 : { val := Nat.succ n✝, isLt := hi } ≠ 0\n⊢ IsCycle (cycleRange { val := Nat.succ n✝, isLt := hi })",
"tactic": "exact isCycle_finRotate.extendDomain _"
},
{
"state_after": "no goals",
"state_before": "case mk.zero\nn : ℕ\nhi : Nat.zero < n + 1\nh0 : { val := Nat.zero, isLt := hi } ≠ 0\n⊢ IsCycle (cycleRange { val := Nat.zero, isLt := hi })",
"tactic": "exact (h0 rfl).elim"
}
] |
[
299,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
frobenius_def
|
[] |
[
349,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
add_nsmul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.9319\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : A\nm n : ℕ\n⊢ (m + n) • a = m • a + n • a",
"tactic": "induction m with\n| zero => rw [Nat.zero_add, zero_nsmul, zero_add]\n| succ m ih => rw [Nat.succ_add, Nat.succ_eq_add_one, succ_nsmul, ih, succ_nsmul, add_assoc]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type ?u.9319\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : A\nn : ℕ\n⊢ (Nat.zero + n) • a = Nat.zero • a + n • a",
"tactic": "rw [Nat.zero_add, zero_nsmul, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type ?u.9319\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : A\nn m : ℕ\nih : (m + n) • a = m • a + n • a\n⊢ (Nat.succ m + n) • a = Nat.succ m • a + n • a",
"tactic": "rw [Nat.succ_add, Nat.succ_eq_add_one, succ_nsmul, ih, succ_nsmul, add_assoc]"
}
] |
[
88,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.support_add
|
[] |
[
984,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
982,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.comp_mulIndicator
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.16343\nM : Type u_3\nN : Type ?u.16349\ninst✝² : One M\ninst✝¹ : One N\ns✝ t : Set α\nf✝ g : α → M\na : α\nh : M → β\nf : α → M\ns : Set α\nx : α\ninst✝ : DecidablePred fun x => x ∈ s\nthis : DecidablePred fun x => x ∈ s := Classical.decPred fun x => x ∈ s\n⊢ h (mulIndicator s f x) = piecewise s (h ∘ f) (const α (h 1)) x",
"tactic": "convert s.apply_piecewise f (const α 1) (fun _ => h) (x := x) using 2"
}
] |
[
255,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.castLE_rfl
|
[
{
"state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nh : n✝ ≤ n✝\n⊢ castLE h falsum = falsum\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nh : n✝ ≤ n✝\n⊢ castLE h (equal t₁✝ t₂✝) = equal t₁✝ t₂✝\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nh : n✝ ≤ n✝\n⊢ castLE h (rel R✝ ts✝) = rel R✝ ts✝\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ (h : n✝ ≤ n✝), castLE h f₁✝ = f₁✝\nih2 : ∀ (h : n✝ ≤ n✝), castLE h f₂✝ = f₂✝\nh : n✝ ≤ n✝\n⊢ castLE h (imp f₁✝ f₂✝) = imp f₁✝ f₂✝\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (h : n✝ + 1 ≤ n✝ + 1), castLE h f✝ = f✝\nh : n✝ ≤ n✝\n⊢ castLE h (all f✝) = all f✝",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝ n : ℕ\nh : n ≤ n\nφ : BoundedFormula L α n\n⊢ castLE h φ = φ",
"tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3"
},
{
"state_after": "no goals",
"state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nh : n✝ ≤ n✝\n⊢ castLE h falsum = falsum",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nh : n✝ ≤ n✝\n⊢ castLE h (equal t₁✝ t₂✝) = equal t₁✝ t₂✝",
"tactic": "simp [Fin.castLE_of_eq]"
},
{
"state_after": "no goals",
"state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nh : n✝ ≤ n✝\n⊢ castLE h (rel R✝ ts✝) = rel R✝ ts✝",
"tactic": "simp [Fin.castLE_of_eq]"
},
{
"state_after": "no goals",
"state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ (h : n✝ ≤ n✝), castLE h f₁✝ = f₁✝\nih2 : ∀ (h : n✝ ≤ n✝), castLE h f₂✝ = f₂✝\nh : n✝ ≤ n✝\n⊢ castLE h (imp f₁✝ f₂✝) = imp f₁✝ f₂✝",
"tactic": "simp [Fin.castLE_of_eq, ih1, ih2]"
},
{
"state_after": "no goals",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.48561\nP : Type ?u.48564\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.48592\nn✝¹ n : ℕ\nh✝ : n ≤ n\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ (h : n✝ + 1 ≤ n✝ + 1), castLE h f✝ = f✝\nh : n✝ ≤ n✝\n⊢ castLE h (all f✝) = all f✝",
"tactic": "simp [Fin.castLE_of_eq, ih3]"
}
] |
[
456,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.le_sup_lintegral
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\n⊢ (∑ x in SimpleFunc.range (pair f g), x.fst * ↑↑μ (↑(pair f g) ⁻¹' {x})) ⊔\n ∑ x in SimpleFunc.range (pair f g), x.snd * ↑↑μ (↑(pair f g) ⁻¹' {x}) ≤\n ∑ x in SimpleFunc.range (pair f g), (x.fst ⊔ x.snd) * ↑↑μ (↑(pair f g) ⁻¹' {x})",
"state_before": "α : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\n⊢ lintegral (map Prod.fst (pair f g)) μ ⊔ lintegral (map Prod.snd (pair f g)) μ ≤\n ∑ x in SimpleFunc.range (pair f g), (x.fst ⊔ x.snd) * ↑↑μ (↑(pair f g) ⁻¹' {x})",
"tactic": "rw [map_lintegral, map_lintegral]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\na : ℝ≥0∞ × ℝ≥0∞\nx✝ : a ∈ SimpleFunc.range (pair f g)\n⊢ a.fst ≤ a.fst ⊔ a.snd\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\na : ℝ≥0∞ × ℝ≥0∞\nx✝ : a ∈ SimpleFunc.range (pair f g)\n⊢ a.snd ≤ a.fst ⊔ a.snd",
"state_before": "α : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\n⊢ (∑ x in SimpleFunc.range (pair f g), x.fst * ↑↑μ (↑(pair f g) ⁻¹' {x})) ⊔\n ∑ x in SimpleFunc.range (pair f g), x.snd * ↑↑μ (↑(pair f g) ⁻¹' {x}) ≤\n ∑ x in SimpleFunc.range (pair f g), (x.fst ⊔ x.snd) * ↑↑μ (↑(pair f g) ⁻¹' {x})",
"tactic": "refine' sup_le _ _ <;> refine' Finset.sum_le_sum fun a _ => mul_le_mul_right' _ _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\na : ℝ≥0∞ × ℝ≥0∞\nx✝ : a ∈ SimpleFunc.range (pair f g)\n⊢ a.snd ≤ a.fst ⊔ a.snd",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\na : ℝ≥0∞ × ℝ≥0∞\nx✝ : a ∈ SimpleFunc.range (pair f g)\n⊢ a.fst ≤ a.fst ⊔ a.snd\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\na : ℝ≥0∞ × ℝ≥0∞\nx✝ : a ∈ SimpleFunc.range (pair f g)\n⊢ a.snd ≤ a.fst ⊔ a.snd",
"tactic": "exact le_sup_left"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\na : ℝ≥0∞ × ℝ≥0∞\nx✝ : a ∈ SimpleFunc.range (pair f g)\n⊢ a.snd ≤ a.fst ⊔ a.snd",
"tactic": "exact le_sup_right"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.897639\nγ : Type ?u.897642\nδ : Type ?u.897645\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α →ₛ ℝ≥0∞\n⊢ ∑ x in SimpleFunc.range (pair f g), (x.fst ⊔ x.snd) * ↑↑μ (↑(pair f g) ⁻¹' {x}) = lintegral (f ⊔ g) μ",
"tactic": "rw [sup_eq_map₂, map_lintegral]"
}
] |
[
1117,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean
|
aeSeq.measure_compl_aeSeqSet_eq_zero
|
[
{
"state_after": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ ↑↑μ ({x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) ∧ p x fun n => f n x}ᶜ) = 0",
"state_before": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ ↑↑μ (aeSeqSet hf pᶜ) = 0",
"tactic": "rw [aeSeqSet, compl_compl, measure_toMeasurable]"
},
{
"state_after": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nhf_eq : ∀ (i : ι), f i =ᵐ[μ] AEMeasurable.mk (f i) (_ : AEMeasurable (f i))\n⊢ ↑↑μ ({x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) ∧ p x fun n => f n x}ᶜ) = 0",
"state_before": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ ↑↑μ ({x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) ∧ p x fun n => f n x}ᶜ) = 0",
"tactic": "have hf_eq := fun i => (hf i).ae_eq_mk"
},
{
"state_after": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nhf_eq : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), f i a = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) a\n⊢ ↑↑μ ({x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) ∧ p x fun n => f n x}ᶜ) = 0",
"state_before": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nhf_eq : ∀ (i : ι), f i =ᵐ[μ] AEMeasurable.mk (f i) (_ : AEMeasurable (f i))\n⊢ ↑↑μ ({x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) ∧ p x fun n => f n x}ᶜ) = 0",
"tactic": "simp_rw [Filter.EventuallyEq, ← ae_all_iff] at hf_eq"
},
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type ?u.519115\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nhf_eq : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), f i a = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) a\n⊢ ↑↑μ ({x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) ∧ p x fun n => f n x}ᶜ) = 0",
"tactic": "exact Filter.Eventually.and hf_eq hp"
}
] |
[
108,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.comp_measurable
|
[] |
[
1556,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1553,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.mul_inv_rev
|
[
{
"state_after": "case pos\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹\n\ncase neg\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹",
"state_before": "σ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\n⊢ (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹",
"tactic": "by_cases h : constantCoeff σ k (φ * ψ) = 0"
},
{
"state_after": "case pos\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ 0 = ψ⁻¹ * φ⁻¹",
"state_before": "case pos\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹",
"tactic": "rw [inv_eq_zero.mpr h]"
},
{
"state_after": "case pos\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) φ = 0 ∨ ↑(constantCoeff σ k) ψ = 0\n⊢ 0 = ψ⁻¹ * φ⁻¹",
"state_before": "case pos\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ 0 = ψ⁻¹ * φ⁻¹",
"tactic": "simp only [map_mul, mul_eq_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case pos\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) φ = 0 ∨ ↑(constantCoeff σ k) ψ = 0\n⊢ 0 = ψ⁻¹ * φ⁻¹",
"tactic": "cases' h with h h <;> simp [inv_eq_zero.mpr h]"
},
{
"state_after": "case neg\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ ψ⁻¹ * φ⁻¹ * (φ * ψ) = 1",
"state_before": "case neg\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹",
"tactic": "rw [MvPowerSeries.inv_eq_iff_mul_eq_one h]"
},
{
"state_after": "case neg\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬↑(constantCoeff σ k) φ = 0 ∧ ¬↑(constantCoeff σ k) ψ = 0\n⊢ ψ⁻¹ * φ⁻¹ * (φ * ψ) = 1",
"state_before": "case neg\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬↑(constantCoeff σ k) (φ * ψ) = 0\n⊢ ψ⁻¹ * φ⁻¹ * (φ * ψ) = 1",
"tactic": "simp only [not_or, map_mul, mul_eq_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case neg\nσ : Type u_1\nR : Type ?u.1945318\nk : Type u_2\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬↑(constantCoeff σ k) φ = 0 ∧ ¬↑(constantCoeff σ k) ψ = 0\n⊢ ψ⁻¹ * φ⁻¹ * (φ * ψ) = 1",
"tactic": "rw [← mul_assoc, mul_assoc _⁻¹, MvPowerSeries.inv_mul_cancel _ h.left, mul_one,\n MvPowerSeries.inv_mul_cancel _ h.right]"
}
] |
[
1019,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1009,
11
] |
Mathlib/Algebra/Homology/Augment.lean
|
CochainComplex.augment_X_zero
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.adjugate_adjugate
|
[
{
"state_after": "case zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = Nat.zero\n⊢ adjugate (adjugate A) = det A ^ (Nat.zero - 2) • A\n\ncase succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn' : ℕ\nh_card : Fintype.card n = Nat.succ n'\n⊢ adjugate (adjugate A) = det A ^ (Nat.succ n' - 2) • A",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\n⊢ adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A",
"tactic": "cases' h_card : Fintype.card n with n'"
},
{
"state_after": "case succ.zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = Nat.succ Nat.zero\n⊢ adjugate (adjugate A) = det A ^ (Nat.succ Nat.zero - 2) • A\n\ncase succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\n⊢ adjugate (adjugate A) = det A ^ (Nat.succ (Nat.succ n✝) - 2) • A",
"state_before": "case succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn' : ℕ\nh_card : Fintype.card n = Nat.succ n'\n⊢ adjugate (adjugate A) = det A ^ (Nat.succ n' - 2) • A",
"tactic": "cases n'"
},
{
"state_after": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\n⊢ adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A",
"state_before": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\n⊢ adjugate (adjugate A) = det A ^ (Nat.succ (Nat.succ n✝) - 2) • A",
"tactic": "rw [← h_card]"
},
{
"state_after": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\n⊢ adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A",
"state_before": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\n⊢ adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A",
"tactic": "let A' := mvPolynomialX n n ℤ"
},
{
"state_after": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\n⊢ adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'",
"state_before": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\n⊢ adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A",
"tactic": "suffices adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A' by\n rw [← mvPolynomialX_mapMatrix_aeval ℤ A, ← AlgHom.map_adjugate, ← AlgHom.map_adjugate, this,\n ← AlgHom.map_det, ← AlgHom.map_pow, AlgHom.mapMatrix_apply, AlgHom.mapMatrix_apply,\n Matrix.map_smul' _ _ _ (_root_.map_mul _)]"
},
{
"state_after": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\n⊢ adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'",
"state_before": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\n⊢ adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'",
"tactic": "have h_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1 := by simp [h_card]"
},
{
"state_after": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\nis_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A')\n⊢ adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'",
"state_before": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\n⊢ adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'",
"tactic": "have is_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A') := fun x y =>\n mul_left_cancel₀ (det_mvPolynomialX_ne_zero n ℤ)"
},
{
"state_after": "case succ.succ.a\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\nis_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A')\n⊢ (fun x => det A' • x) (adjugate (adjugate A')) = (fun x => det A' • x) (det A' ^ (Fintype.card n - 2) • A')",
"state_before": "case succ.succ\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\nis_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A')\n⊢ adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'",
"tactic": "apply is_reg.matrix"
},
{
"state_after": "case succ.succ.a\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\nis_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A')\n⊢ det (mvPolynomialX n n ℤ) • adjugate (adjugate (mvPolynomialX n n ℤ)) =\n det (mvPolynomialX n n ℤ) • det (mvPolynomialX n n ℤ) ^ (Fintype.card n - 2) • mvPolynomialX n n ℤ",
"state_before": "case succ.succ.a\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\nis_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A')\n⊢ (fun x => det A' • x) (adjugate (adjugate A')) = (fun x => det A' • x) (det A' ^ (Fintype.card n - 2) • A')",
"tactic": "simp only"
},
{
"state_after": "no goals",
"state_before": "case succ.succ.a\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nh_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1\nis_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A')\n⊢ det (mvPolynomialX n n ℤ) • adjugate (adjugate (mvPolynomialX n n ℤ)) =\n det (mvPolynomialX n n ℤ) • det (mvPolynomialX n n ℤ) ^ (Fintype.card n - 2) • mvPolynomialX n n ℤ",
"tactic": "rw [smul_smul, ← pow_succ, h_card', det_smul_adjugate_adjugate]"
},
{
"state_after": "case zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = Nat.zero\nthis : IsEmpty n\n⊢ adjugate (adjugate A) = det A ^ (Nat.zero - 2) • A",
"state_before": "case zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = Nat.zero\n⊢ adjugate (adjugate A) = det A ^ (Nat.zero - 2) • A",
"tactic": "haveI : IsEmpty n := Fintype.card_eq_zero_iff.mp h_card"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = Nat.zero\nthis : IsEmpty n\n⊢ adjugate (adjugate A) = det A ^ (Nat.zero - 2) • A",
"tactic": "apply Subsingleton.elim"
},
{
"state_after": "no goals",
"state_before": "case succ.zero\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = Nat.succ Nat.zero\n⊢ adjugate (adjugate A) = det A ^ (Nat.succ Nat.zero - 2) • A",
"tactic": "exact (h h_card).elim"
},
{
"state_after": "no goals",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\nthis : adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A'\n⊢ adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A",
"tactic": "rw [← mvPolynomialX_mapMatrix_aeval ℤ A, ← AlgHom.map_adjugate, ← AlgHom.map_adjugate, this,\n ← AlgHom.map_det, ← AlgHom.map_pow, AlgHom.mapMatrix_apply, AlgHom.mapMatrix_apply,\n Matrix.map_smul' _ _ _ (_root_.map_mul _)]"
},
{
"state_after": "no goals",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = Nat.succ (Nat.succ n✝)\nA' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ\n⊢ Fintype.card n - 2 + 1 = Fintype.card n - 1",
"tactic": "simp [h_card]"
}
] |
[
548,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.reduce_isSpecial'
|
[] |
[
390,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.inv_prodComparison_map_fst
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nX Y : C\nD : Type u₂\ninst✝⁵ : Category D\nF : C ⥤ D\nA A' B B' : C\ninst✝⁴ : HasBinaryProduct A B\ninst✝³ : HasBinaryProduct A' B'\ninst✝² : HasBinaryProduct (F.obj A) (F.obj B)\ninst✝¹ : HasBinaryProduct (F.obj A') (F.obj B')\ninst✝ : IsIso (prodComparison F A B)\n⊢ inv (prodComparison F A B) ≫ F.map prod.fst = prod.fst",
"tactic": "simp [IsIso.inv_comp_eq]"
}
] |
[
1290,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1289,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_subset_Ici_self
|
[] |
[
413,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
AnalyticAt.neg
|
[] |
[
572,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.degree_basisDivisor_of_ne
|
[
{
"state_after": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ (x - y)⁻¹ ≠ 0",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ degree (basisDivisor x y) = 1",
"tactic": "rw [basisDivisor, degree_mul, degree_X_sub_C, degree_C, zero_add]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ (x - y)⁻¹ ≠ 0",
"tactic": "exact inv_ne_zero (sub_ne_zero_of_ne hxy)"
}
] |
[
153,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
|
SimpleGraph.ComponentCompl.hom_trans
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\nh : K ⊆ L\nh' : M ⊆ K\n⊢ ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Mᶜ)) C =\n ConnectedComponent.map (InduceHom Hom.id (_ : Kᶜ ⊆ Mᶜ)) (ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Kᶜ)) C)",
"state_before": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\nh : K ⊆ L\nh' : M ⊆ K\n⊢ hom (_ : M ⊆ L) C = hom h' (hom h C)",
"tactic": "change C.map _ = (C.map _).map _"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\nh : K ⊆ L\nh' : M ⊆ K\n⊢ ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Mᶜ)) C =\n ConnectedComponent.map\n (InduceHom (Hom.comp Hom.id Hom.id) (_ : Set.MapsTo (↑Hom.id ∘ fun x => ↑Hom.id x) (Lᶜ) (Mᶜ))) C",
"state_before": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\nh : K ⊆ L\nh' : M ⊆ K\n⊢ ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Mᶜ)) C =\n ConnectedComponent.map (InduceHom Hom.id (_ : Kᶜ ⊆ Mᶜ)) (ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Kᶜ)) C)",
"tactic": "erw [ConnectedComponent.map_comp, induceHom_comp]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\nh : K ⊆ L\nh' : M ⊆ K\n⊢ ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Mᶜ)) C =\n ConnectedComponent.map\n (InduceHom (Hom.comp Hom.id Hom.id) (_ : Set.MapsTo (↑Hom.id ∘ fun x => ↑Hom.id x) (Lᶜ) (Mᶜ))) C",
"tactic": "rfl"
}
] |
[
219,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.basisDivisor_inj
|
[
{
"state_after": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x = y\n⊢ x = y",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : basisDivisor x y = 0\n⊢ x = y",
"tactic": "simp_rw [basisDivisor, mul_eq_zero, X_sub_C_ne_zero, or_false_iff, C_eq_zero, inv_eq_zero,\n sub_eq_zero] at hxy"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x = y\n⊢ x = y",
"tactic": "exact hxy"
}
] |
[
139,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.transpose_submatrix
|
[] |
[
2403,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2401,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean
|
HasFDerivAt.clog
|
[] |
[
99,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_quadratic_le
|
[] |
[
1196,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1195,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.Valid.all
|
[
{
"state_after": "no goals",
"state_before": "f : Char → Bool\nx✝ : Substring\nh✝ : Valid x✝\nw✝² w✝¹ w✝ : List Char\nh : ValidFor w✝² w✝¹ w✝ x✝\n⊢ Substring.all x✝ f = List.all (toString x✝).data f",
"tactic": "simp [h.all, h.toString]"
}
] |
[
1063,
72
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1062,
1
] |
Mathlib/Data/Pi/Lex.lean
|
Pi.lex_lt_of_lt_of_preorder
|
[] |
[
70,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.ofDual_le_ofDual_iff
|
[] |
[
823,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
NonarchimedeanGroup.prod_self_subset
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : NonarchimedeanGroup G\nH : Type ?u.61803\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : TopologicalGroup H\nK : Type ?u.61824\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : NonarchimedeanGroup K\nU : Set (G × G)\nhU : U ∈ nhds 1\nV W : OpenSubgroup G\nh : ↑V ×ˢ ↑W ⊆ U\n⊢ ↑(V ⊓ W) ×ˢ ↑(V ⊓ W) ⊆ U",
"tactic": "refine' Set.Subset.trans (Set.prod_mono _ _) ‹_› <;> simp"
}
] |
[
110,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_top_iff_deriv
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3065764\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\n⊢ ContDiffOn 𝕜 ⊤ f₂ univ ↔ DifferentiableOn 𝕜 f₂ univ ∧ ContDiffOn 𝕜 ⊤ (derivWithin f₂ univ) univ",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3065764\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\n⊢ ContDiff 𝕜 ⊤ f₂ ↔ Differentiable 𝕜 f₂ ∧ ContDiff 𝕜 ⊤ (deriv f₂)",
"tactic": "simp only [← contDiffOn_univ, ← differentiableOn_univ, ← derivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3065764\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\n⊢ ContDiffOn 𝕜 ⊤ f₂ univ ↔ DifferentiableOn 𝕜 f₂ univ ∧ ContDiffOn 𝕜 ⊤ (derivWithin f₂ univ) univ",
"tactic": "rw [contDiffOn_top_iff_derivWithin uniqueDiffOn_univ]"
}
] |
[
2164,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2161,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean
|
skewAdjoint.conjugate
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type ?u.109551\ninst✝¹ : Ring R\ninst✝ : StarRing R\nx : R\nhx : x ∈ skewAdjoint R\nz : R\n⊢ z * x * star z ∈ skewAdjoint R",
"tactic": "simp only [mem_iff, star_mul, star_star, mem_iff.mp hx, neg_mul, mul_neg, mul_assoc]"
}
] |
[
491,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
|
Besicovitch.SatelliteConfig.exists_normalized
|
[
{
"state_after": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\n⊢ ∃ c', (∀ (n : Fin (Nat.succ N)), ‖c' n‖ ≤ 2) ∧ ∀ (i j : Fin (Nat.succ N)), i ≠ j → 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\n⊢ ∃ c', (∀ (n : Fin (Nat.succ N)), ‖c' n‖ ≤ 2) ∧ ∀ (i j : Fin (Nat.succ N)), i ≠ j → 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i"
},
{
"state_after": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\n⊢ ∃ c', (∀ (n : Fin (Nat.succ N)), ‖c' n‖ ≤ 2) ∧ ∀ (i j : Fin (Nat.succ N)), i ≠ j → 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "refine' ⟨c', fun n => norm_c'_le n, fun i j inej => _⟩"
},
{
"state_after": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nthis : ∀ (i j : Fin (Nat.succ N)), i ≠ j → ‖c a i‖ ≤ ‖c a j‖ → 1 - δ ≤ ‖c' i - c' j‖\nhij : ¬‖c a i‖ ≤ ‖c a j‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖\n\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j"
},
{
"state_after": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : ‖c a j‖ ≤ 2\n⊢ 1 - δ ≤ ‖c' i - c' j‖\n\ncase inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj)"
},
{
"state_after": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\n⊢ ‖c' i‖ ≤ 2",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\n⊢ ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2",
"tactic": "intro i"
},
{
"state_after": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\n⊢ ‖if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i‖ ≤ 2",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\n⊢ ‖c' i‖ ≤ 2",
"tactic": "simp only"
},
{
"state_after": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\nh : ‖c a i‖ ≤ 2\n⊢ ‖c a i‖ ≤ 2\n\ncase inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\nh : ¬‖c a i‖ ≤ 2\n⊢ ‖(2 / ‖c a i‖) • c a i‖ ≤ 2",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\n⊢ ‖if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i‖ ≤ 2",
"tactic": "split_ifs with h"
},
{
"state_after": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\nh : ¬‖c a i‖ ≤ 2\nhi : ‖c a i‖ = 0\n⊢ 0 ≤ 2",
"state_before": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\nh : ¬‖c a i‖ ≤ 2\n⊢ ‖(2 / ‖c a i‖) • c a i‖ ≤ 2",
"tactic": "by_cases hi : ‖a.c i‖ = 0 <;> field_simp [norm_smul, hi]"
},
{
"state_after": "no goals",
"state_before": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\nh : ¬‖c a i‖ ≤ 2\nhi : ‖c a i‖ = 0\n⊢ 0 ≤ 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\ni : Fin (Nat.succ N)\nh : ‖c a i‖ ≤ 2\n⊢ ‖c a i‖ ≤ 2",
"tactic": "exact h"
},
{
"state_after": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nthis : ∀ (i j : Fin (Nat.succ N)), i ≠ j → ‖c a i‖ ≤ ‖c a j‖ → 1 - δ ≤ ‖c' i - c' j‖\nhij : ¬‖c a i‖ ≤ ‖c a j‖\n⊢ 1 - δ ≤ ‖c' j - c' i‖",
"state_before": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nthis : ∀ (i j : Fin (Nat.succ N)), i ≠ j → ‖c a i‖ ≤ ‖c a j‖ → 1 - δ ≤ ‖c' i - c' j‖\nhij : ¬‖c a i‖ ≤ ‖c a j‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "rw [norm_sub_rev]"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nthis : ∀ (i j : Fin (Nat.succ N)), i ≠ j → ‖c a i‖ ≤ ‖c a j‖ → 1 - δ ≤ ‖c' i - c' j‖\nhij : ¬‖c a i‖ ≤ ‖c a j‖\n⊢ 1 - δ ≤ ‖c' j - c' i‖",
"tactic": "exact this j i inej.symm (le_of_not_le hij)"
},
{
"state_after": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : ‖c a j‖ ≤ 2\n⊢ 1 - δ ≤ ‖c a i - c a j‖",
"state_before": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : ‖c a j‖ ≤ 2\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "simp_rw [Hj, hij.trans Hj, if_true]"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : ‖c a j‖ ≤ 2\n⊢ 1 - δ ≤ ‖c a i - c a j‖",
"tactic": "exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej"
},
{
"state_after": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj"
},
{
"state_after": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : ‖c a i‖ ≤ 2\n⊢ 1 - δ ≤ ‖c' i - c' j‖\n\ncase inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\n⊢ ‖c a j‖ ≤ 2 ↔ False",
"tactic": "simpa only [not_le, iff_false_iff] using Hj"
},
{
"state_after": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : ‖c a i‖ ≤ 2\n⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖",
"state_before": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : ‖c a i‖ ≤ 2\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "simp_rw [Hi, if_true, H'j, if_false]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : ‖c a i‖ ≤ 2\n⊢ 1 - δ ≤ ‖c a i - (2 / ‖c a j‖) • c a j‖",
"tactic": "exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj"
},
{
"state_after": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\nH'i : ‖c a i‖ ≤ 2 ↔ False\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"state_before": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hi"
},
{
"state_after": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\nH'i : ‖c a i‖ ≤ 2 ↔ False\n⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖",
"state_before": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\nH'i : ‖c a i‖ ≤ 2 ↔ False\n⊢ 1 - δ ≤ ‖c' i - c' j‖",
"tactic": "simp_rw [H'i, if_false, H'j, if_false]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\nH'i : ‖c a i‖ ≤ 2 ↔ False\n⊢ 1 - δ ≤ ‖(2 / ‖c a i‖) • c a i - (2 / ‖c a j‖) • c a j‖",
"tactic": "exact exists_normalized_aux3 a lastc lastr hτ δ hδ1 i j inej Hi hij"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : c a (last N) = 0\nlastr : r a (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin (Nat.succ N) → E := fun i => if ‖c a i‖ ≤ 2 then c a i else (2 / ‖c a i‖) • c a i\nnorm_c'_le : ∀ (i : Fin (Nat.succ N)), ‖c' i‖ ≤ 2\ni j : Fin (Nat.succ N)\ninej : i ≠ j\nhij : ‖c a i‖ ≤ ‖c a j‖\nHj : 2 < ‖c a j‖\nH'j : ‖c a j‖ ≤ 2 ↔ False\nHi : 2 < ‖c a i‖\n⊢ ‖c a i‖ ≤ 2 ↔ False",
"tactic": "simpa only [not_le, iff_false_iff] using Hi"
}
] |
[
529,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.seq_mono
|
[] |
[
1967,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1966,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.support_single_subset
|
[] |
[
179,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
local_isCompact_isClosed_nhds_of_group
|
[
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"tactic": "obtain ⟨L, Lint, LU, Lcomp⟩ : ∃ (L : Set G), L ∈ 𝓝 (1 : G) ∧ L ⊆ U ∧ IsCompact L :=\n local_compact_nhds hU"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"tactic": "obtain ⟨V, Vnhds, hV⟩ : ∃ V ∈ 𝓝 (1 : G), ∀ v ∈ V, ∀ w ∈ V, v * w ∈ L := by\n have : (fun p : G × G => p.1 * p.2) ⁻¹' L ∈ 𝓝 ((1, 1) : G × G) := by\n refine' continuousAt_fst.mul continuousAt_snd _\n simpa only [mul_one] using Lint\n simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage]\n using this"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\nVL : closure V ⊆ L\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"tactic": "have VL : closure V ⊆ L :=\n calc\n closure V = {(1 : G)} * closure V := by simp only [singleton_mul, one_mul, image_id']\n _ ⊆ interior V * closure V :=\n mul_subset_mul_right\n (by simpa only [singleton_subset_iff] using mem_interior_iff_mem_nhds.2 Vnhds)\n _ = interior V * V := isOpen_interior.mul_closure _\n _ ⊆ V * V := mul_subset_mul_right interior_subset\n _ ⊆ L := by\n rintro x ⟨y, z, yv, zv, rfl⟩\n exact hV _ yv _ zv"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\nVL : closure V ⊆ L\n⊢ ∃ K, IsCompact K ∧ IsClosed K ∧ K ⊆ U ∧ 1 ∈ interior K",
"tactic": "exact\n ⟨closure V, isCompact_of_isClosed_subset Lcomp isClosed_closure VL, isClosed_closure,\n VL.trans LU, interior_mono subset_closure (mem_interior_iff_mem_nhds.2 Vnhds)⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nthis : (fun p => p.fst * p.snd) ⁻¹' L ∈ 𝓝 (1, 1)\n⊢ ∃ V, V ∈ 𝓝 1 ∧ ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\n⊢ ∃ V, V ∈ 𝓝 1 ∧ ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L",
"tactic": "have : (fun p : G × G => p.1 * p.2) ⁻¹' L ∈ 𝓝 ((1, 1) : G × G) := by\n refine' continuousAt_fst.mul continuousAt_snd _\n simpa only [mul_one] using Lint"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nthis : (fun p => p.fst * p.snd) ⁻¹' L ∈ 𝓝 (1, 1)\n⊢ ∃ V, V ∈ 𝓝 1 ∧ ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L",
"tactic": "simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage]\n using this"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\n⊢ L ∈ 𝓝 ((fun x => x.fst * x.snd) (1, 1))",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\n⊢ (fun p => p.fst * p.snd) ⁻¹' L ∈ 𝓝 (1, 1)",
"tactic": "refine' continuousAt_fst.mul continuousAt_snd _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\n⊢ L ∈ 𝓝 ((fun x => x.fst * x.snd) (1, 1))",
"tactic": "simpa only [mul_one] using Lint"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\n⊢ closure V = {1} * closure V",
"tactic": "simp only [singleton_mul, one_mul, image_id']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\n⊢ {1} ⊆ interior V",
"tactic": "simpa only [singleton_subset_iff] using mem_interior_iff_mem_nhds.2 Vnhds"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\ny z : G\nyv : y ∈ V\nzv : z ∈ V\n⊢ (fun x x_1 => x * x_1) y z ∈ L",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\n⊢ V * V ⊆ L",
"tactic": "rintro x ⟨y, z, yv, zv, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : LocallyCompactSpace G\nU : Set G\nhU : U ∈ 𝓝 1\nL : Set G\nLint : L ∈ 𝓝 1\nLU : L ⊆ U\nLcomp : IsCompact L\nV : Set G\nVnhds : V ∈ 𝓝 1\nhV : ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v * w ∈ L\ny z : G\nyv : y ∈ V\nzv : z ∈ V\n⊢ (fun x x_1 => x * x_1) y z ∈ L",
"tactic": "exact hV _ yv _ zv"
}
] |
[
1668,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1643,
1
] |
Mathlib/MeasureTheory/Group/Action.lean
|
MeasureTheory.isLocallyFiniteMeasure_of_smulInvariant
|
[
{
"state_after": "no goals",
"state_before": "G : Type ?u.57418\nM : Type ?u.57421\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁷ : Group G\ninst✝⁶ : MulAction G α\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousConstSMul G α\ninst✝ : MulAction.IsMinimal G α\nK U : Set α\nhU : IsOpen U\nhne : Set.Nonempty U\nhμU : ↑↑μ U ≠ ⊤\nx : α\ng : G\nhg : g • x ∈ U\n⊢ ↑↑μ ((fun x x_1 => x • x_1) g ⁻¹' U) ≠ ⊤",
"tactic": "rwa [measure_preimage_smul]"
}
] |
[
233,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Data/Nat/Multiplicity.lean
|
Nat.Prime.multiplicity_factorial_mul_succ
|
[
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have hp' := hp.prime"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have h0 : 2 ≤ p := hp.two_le"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have h1 : 1 ≤ p * n + 1 := Nat.le_add_left _ _"
},
{
"state_after": "case h2\nn p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\n⊢ p * n + 1 ≤ p * (n + 1)\n\nn p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have h2 : p * n + 1 ≤ p * (n + 1)"
},
{
"state_after": "case h3\nn p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\n⊢ p * n + 1 ≤ p * (n + 1) + 1\n\nn p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have h3 : p * n + 1 ≤ p * (n + 1) + 1"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nhm : multiplicity p (p * n)! ≠ ⊤\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have hm : multiplicity p (p * n)! ≠ ⊤ := by\n rw [Ne.def, eq_top_iff_not_finite, Classical.not_not, finite_nat_iff]\n exact ⟨hp.ne_one, factorial_pos _⟩"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ multiplicity p (p * n)! ≠ ⊤ → multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nhm : multiplicity p (p * n)! ≠ ⊤\n⊢ multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "revert hm"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nh4 : ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0\n⊢ multiplicity p (p * n)! ≠ ⊤ → multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ multiplicity p (p * n)! ≠ ⊤ → multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "have h4 : ∀ m ∈ Ico (p * n + 1) (p * (n + 1)), multiplicity p m = 0 := by\n intro m hm\n rw [multiplicity_eq_zero, ← not_dvd_iff_between_consec_multiples _ hp.pos]\n rw [mem_Ico] at hm\n exact ⟨n, lt_of_succ_le hm.1, hm.2⟩"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nh4 : ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0\n⊢ ∑ x in Ico 1 (p * n + 1), multiplicity p x ≠ ⊤ →\n ∑ x in Ico 1 (p * n + 1), multiplicity p x + ∑ x in Ico (p * n + 1) (p * (n + 1) + 1), multiplicity p x =\n ∑ x in Ico 1 (p * n + 1), multiplicity p x + (multiplicity p (n + 1) + 1)",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nh4 : ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0\n⊢ multiplicity p (p * n)! ≠ ⊤ → multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1",
"tactic": "simp_rw [← prod_Ico_id_eq_factorial, multiplicity.Finset.prod hp', ← sum_Ico_consecutive _ h1 h3,\n add_assoc]"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nh4 : ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0\nh : ∑ x in Ico 1 (p * n + 1), multiplicity p x ≠ ⊤\n⊢ ∑ x in Ico 1 (p * n + 1), multiplicity p x + ∑ x in Ico (p * n + 1) (p * (n + 1) + 1), multiplicity p x =\n ∑ x in Ico 1 (p * n + 1), multiplicity p x + (multiplicity p (n + 1) + 1)",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nh4 : ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0\n⊢ ∑ x in Ico 1 (p * n + 1), multiplicity p x ≠ ⊤ →\n ∑ x in Ico 1 (p * n + 1), multiplicity p x + ∑ x in Ico (p * n + 1) (p * (n + 1) + 1), multiplicity p x =\n ∑ x in Ico 1 (p * n + 1), multiplicity p x + (multiplicity p (n + 1) + 1)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nh4 : ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0\nh : ∑ x in Ico 1 (p * n + 1), multiplicity p x ≠ ⊤\n⊢ ∑ x in Ico 1 (p * n + 1), multiplicity p x + ∑ x in Ico (p * n + 1) (p * (n + 1) + 1), multiplicity p x =\n ∑ x in Ico 1 (p * n + 1), multiplicity p x + (multiplicity p (n + 1) + 1)",
"tactic": "rw [PartENat.add_left_cancel_iff h, sum_Ico_succ_top h2, multiplicity.mul hp',\n hp.multiplicity_self, sum_congr rfl h4, sum_const_zero, zero_add, add_comm (1 : PartENat)]"
},
{
"state_after": "no goals",
"state_before": "case h2\nn p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\n⊢ p * n + 1 ≤ p * (n + 1)",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "case h3\nn p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\n⊢ p * n + 1 ≤ p * (n + 1) + 1",
"tactic": "linarith"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ p ≠ 1 ∧ 0 < (p * n)!",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ multiplicity p (p * n)! ≠ ⊤",
"tactic": "rw [Ne.def, eq_top_iff_not_finite, Classical.not_not, finite_nat_iff]"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ p ≠ 1 ∧ 0 < (p * n)!",
"tactic": "exact ⟨hp.ne_one, factorial_pos _⟩"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : m ∈ Ico (p * n + 1) (p * (n + 1))\n⊢ multiplicity p m = 0",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ ∀ (m : ℕ), m ∈ Ico (p * n + 1) (p * (n + 1)) → multiplicity p m = 0",
"tactic": "intro m hm"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : m ∈ Ico (p * n + 1) (p * (n + 1))\n⊢ ∃ k, p * k < m ∧ m < p * (k + 1)",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : m ∈ Ico (p * n + 1) (p * (n + 1))\n⊢ multiplicity p m = 0",
"tactic": "rw [multiplicity_eq_zero, ← not_dvd_iff_between_consec_multiples _ hp.pos]"
},
{
"state_after": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : p * n + 1 ≤ m ∧ m < p * (n + 1)\n⊢ ∃ k, p * k < m ∧ m < p * (k + 1)",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : m ∈ Ico (p * n + 1) (p * (n + 1))\n⊢ ∃ k, p * k < m ∧ m < p * (k + 1)",
"tactic": "rw [mem_Ico] at hm"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : p * n + 1 ≤ m ∧ m < p * (n + 1)\n⊢ ∃ k, p * k < m ∧ m < p * (k + 1)",
"tactic": "exact ⟨n, lt_of_succ_le hm.1, hm.2⟩"
}
] |
[
148,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
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