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Mathlib/SetTheory/ZFC/Basic.lean | Class.mem_asymm | []
| [
1553,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1552,
1
]
|
Mathlib/Data/Complex/Exponential.lean | Complex.cos_add_cos | [
{
"state_after": "x y : ℂ\nh2 : 2 ≠ 0\n⊢ cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2)",
"state_before": "x y : ℂ\n⊢ cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2)",
"tactic": "have h2 : (2 : ℂ) ≠ 0 := by norm_num"
},
{
"state_after": "case calc_1\nx y : ℂ\nh2 : 2 ≠ 0\n⊢ cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2)\n\ncase calc_2\nx y : ℂ\nh2 : 2 ≠ 0\n⊢ cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) =\n cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +\n (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2))\n\ncase calc_3\nx y : ℂ\nh2 : 2 ≠ 0\n⊢ cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +\n (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) =\n 2 * cos ((x + y) / 2) * cos ((x - y) / 2)",
"state_before": "x y : ℂ\nh2 : 2 ≠ 0\n⊢ cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2)",
"tactic": "calc\n cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_\n _ =\n cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +\n (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) :=\n ?_\n _ = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := ?_"
},
{
"state_after": "no goals",
"state_before": "case calc_3\nx y : ℂ\nh2 : 2 ≠ 0\n⊢ cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +\n (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) =\n 2 * cos ((x + y) / 2) * cos ((x - y) / 2)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ 2 ≠ 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case calc_1\nx y : ℂ\nh2 : 2 ≠ 0\n⊢ cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2)",
"tactic": "congr <;> field_simp [h2]"
},
{
"state_after": "no goals",
"state_before": "case calc_2\nx y : ℂ\nh2 : 2 ≠ 0\n⊢ cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) =\n cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +\n (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2))",
"tactic": "rw [cos_add, cos_sub]"
}
]
| [
924,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
912,
1
]
|
Mathlib/Order/Filter/Basic.lean | Filter.Frequently.mono | []
| [
1278,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1276,
1
]
|
Mathlib/Topology/StoneCech.lean | continuous_stoneCechUnit | [
{
"state_after": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\nthis : ↑(Ultrafilter.map pure g) ≤ 𝓝 g\n⊢ Tendsto stoneCechUnit (↑g) (𝓝 (stoneCechUnit x))",
"state_before": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\n⊢ Tendsto stoneCechUnit (↑g) (𝓝 (stoneCechUnit x))",
"tactic": "have : (g.map pure).toFilter ≤ 𝓝 g := by\n rw [ultrafilter_converges_iff]\n exact (bind_pure _).symm"
},
{
"state_after": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\nthis✝ : ↑(Ultrafilter.map pure g) ≤ 𝓝 g\nthis : ↑(Ultrafilter.map stoneCechUnit g) ≤ 𝓝 (Quotient.mk (stoneCechSetoid α) g)\n⊢ Tendsto stoneCechUnit (↑g) (𝓝 (stoneCechUnit x))",
"state_before": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\nthis : ↑(Ultrafilter.map pure g) ≤ 𝓝 g\n⊢ Tendsto stoneCechUnit (↑g) (𝓝 (stoneCechUnit x))",
"tactic": "have : (g.map stoneCechUnit : Filter (StoneCech α)) ≤ 𝓝 ⟦g⟧ :=\n continuousAt_iff_ultrafilter.mp (continuous_quotient_mk'.tendsto g) _ this"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\nthis✝ : ↑(Ultrafilter.map pure g) ≤ 𝓝 g\nthis : ↑(Ultrafilter.map stoneCechUnit g) ≤ 𝓝 (Quotient.mk (stoneCechSetoid α) g)\n⊢ Tendsto stoneCechUnit (↑g) (𝓝 (stoneCechUnit x))",
"tactic": "rwa [show ⟦g⟧ = ⟦pure x⟧ from Quotient.sound <| convergent_eqv_pure gx] at this"
},
{
"state_after": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\n⊢ g = joinM (Ultrafilter.map pure g)",
"state_before": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\n⊢ ↑(Ultrafilter.map pure g) ≤ 𝓝 g",
"tactic": "rw [ultrafilter_converges_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\ng : Ultrafilter α\ngx : ↑g ≤ 𝓝 x\n⊢ g = joinM (Ultrafilter.map pure g)",
"tactic": "exact (bind_pure _).symm"
}
]
| [
312,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
]
|
Mathlib/Data/Prod/Basic.lean | Function.Injective.Prod_map | []
| [
315,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
]
|
Mathlib/RingTheory/DedekindDomain/Dvr.lean | IsDedekindDomain.isDedekindDomainDvr | []
| [
155,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
]
|
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ico_self | []
| [
283,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
]
|
Mathlib/Data/List/Range.lean | List.finRange_map_get | [
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\n⊢ length (map (get l) (finRange (length l))) = length l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\n⊢ ∀ (n : ℕ) (h₁ : n < length (map (get l) (finRange (length l)))) (h₂ : n < length l),\n get (map (get l) (finRange (length l))) { val := n, isLt := h₁ } = get l { val := n, isLt := h₂ }",
"tactic": "simp"
}
]
| [
215,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
]
|
Mathlib/Data/Polynomial/EraseLead.lean | Polynomial.eraseLead_add_monomial_natDegree_leadingCoeff | []
| [
68,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
]
|
Mathlib/Topology/PathConnected.lean | locPathConnected_of_bases | [
{
"state_after": "case path_connected_basis\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\n⊢ ∀ (x : X), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id",
"state_before": "X : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\n⊢ LocPathConnectedSpace X",
"tactic": "constructor"
},
{
"state_after": "case path_connected_basis\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\n⊢ HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id",
"state_before": "case path_connected_basis\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\n⊢ ∀ (x : X), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id",
"tactic": "intro x"
},
{
"state_after": "case path_connected_basis.h\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\n⊢ ∀ (i : ι), p i → ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i\n\ncase path_connected_basis.h'\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\n⊢ ∀ (i' : Set X), i' ∈ 𝓝 x ∧ IsPathConnected i' → ∃ i, p i ∧ s x i ⊆ id i'",
"state_before": "case path_connected_basis\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\n⊢ HasBasis (𝓝 x) (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id",
"tactic": "apply (h x).to_hasBasis"
},
{
"state_after": "case path_connected_basis.h\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\ni : ι\npi : p i\n⊢ ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i",
"state_before": "case path_connected_basis.h\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\n⊢ ∀ (i : ι), p i → ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i",
"tactic": "intro i pi"
},
{
"state_after": "no goals",
"state_before": "case path_connected_basis.h\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\ni : ι\npi : p i\n⊢ ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i",
"tactic": "exact ⟨s x i, ⟨(h x).mem_of_mem pi, h' x i pi⟩, by rfl⟩"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\ni : ι\npi : p i\n⊢ id (s x i) ⊆ s x i",
"tactic": "rfl"
},
{
"state_after": "case path_connected_basis.h'.intro\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\nU : Set X\nU_in : U ∈ 𝓝 x\n_hU : IsPathConnected U\n⊢ ∃ i, p i ∧ s x i ⊆ id U",
"state_before": "case path_connected_basis.h'\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\n⊢ ∀ (i' : Set X), i' ∈ 𝓝 x ∧ IsPathConnected i' → ∃ i, p i ∧ s x i ⊆ id i'",
"tactic": "rintro U ⟨U_in, _hU⟩"
},
{
"state_after": "case path_connected_basis.h'.intro.intro.intro\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\nU : Set X\nU_in : U ∈ 𝓝 x\n_hU : IsPathConnected U\ni : ι\npi : p i\nhi : s x i ⊆ U\n⊢ ∃ i, p i ∧ s x i ⊆ id U",
"state_before": "case path_connected_basis.h'.intro\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\nU : Set X\nU_in : U ∈ 𝓝 x\n_hU : IsPathConnected U\n⊢ ∃ i, p i ∧ s x i ⊆ id U",
"tactic": "rcases(h x).mem_iff.mp U_in with ⟨i, pi, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case path_connected_basis.h'.intro.intro.intro\nX : Type u_1\nY : Type ?u.702092\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_2\nF : Set X\np : ι → Prop\ns : X → ι → Set X\nh : ∀ (x : X), HasBasis (𝓝 x) p (s x)\nh' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)\nx : X\nU : Set X\nU_in : U ∈ 𝓝 x\n_hU : IsPathConnected U\ni : ι\npi : p i\nhi : s x i ⊆ U\n⊢ ∃ i, p i ∧ s x i ⊆ id U",
"tactic": "tauto"
}
]
| [
1210,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1200,
1
]
|
Mathlib/Data/Int/GCD.lean | Nat.xgcd_val | [
{
"state_after": "x y : ℕ\n⊢ xgcd x y = ((xgcd x y).fst, (xgcd x y).snd)",
"state_before": "x y : ℕ\n⊢ xgcd x y = (gcdA x y, gcdB x y)",
"tactic": "unfold gcdA gcdB"
},
{
"state_after": "case mk\nx y : ℕ\nfst✝ snd✝ : ℤ\n⊢ (fst✝, snd✝) = ((fst✝, snd✝).fst, (fst✝, snd✝).snd)",
"state_before": "x y : ℕ\n⊢ xgcd x y = ((xgcd x y).fst, (xgcd x y).snd)",
"tactic": "cases xgcd x y"
},
{
"state_after": "no goals",
"state_before": "case mk\nx y : ℕ\nfst✝ snd✝ : ℤ\n⊢ (fst✝, snd✝) = ((fst✝, snd✝).fst, (fst✝, snd✝).snd)",
"tactic": "rfl"
}
]
| [
125,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
]
|
Mathlib/Topology/Constructions.lean | exists_nhds_square | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.50921\nδ : Type ?u.50924\nε : Type ?u.50927\nζ : Type ?u.50930\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\ns : Set (α × α)\nx : α\nhx : s ∈ 𝓝 (x, x)\n⊢ ∃ U, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s",
"tactic": "simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and_assoc, and_left_comm] using hx"
}
]
| [
688,
98
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
686,
1
]
|
Mathlib/CategoryTheory/Monoidal/Free/Basic.lean | CategoryTheory.FreeMonoidalCategory.mk_α_inv | []
| [
211,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
]
|
Mathlib/Algebra/Group/Commute.lean | Commute.pow_self | []
| [
202,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
]
|
Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.HasBasis.cauchy_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nι : Sort u_1\np : ι → Prop\ns : ι → Set (α × α)\nh : HasBasis (𝓤 α) p s\nf : Filter α\n⊢ (∀ (i' : ι), p i' → ∃ i, i ∈ f ∧ id i ×ˢ id i ⊆ s i') ↔\n ∀ (i : ι), p i → ∃ t, t ∈ f ∧ ∀ (x : α), x ∈ t → ∀ (y : α), y ∈ t → (x, y) ∈ s i",
"tactic": "simp only [subset_def, Prod.forall, mem_prod_eq, and_imp, id, ball_mem_comm]"
}
]
| [
47,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
]
|
Std/Logic.lean | iff_congr | []
| [
78,
79
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
77,
1
]
|
Mathlib/Data/Set/Image.lean | Function.Surjective.preimage_subset_preimage_iff | [
{
"state_after": "case hs\nι : Sort ?u.104761\nα : Type u_2\nβ : Type u_1\nf : α → β\ns t : Set β\nhf : Surjective f\n⊢ s ⊆ range f",
"state_before": "ι : Sort ?u.104761\nα : Type u_2\nβ : Type u_1\nf : α → β\ns t : Set β\nhf : Surjective f\n⊢ f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t",
"tactic": "apply Set.preimage_subset_preimage_iff"
},
{
"state_after": "case hs\nι : Sort ?u.104761\nα : Type u_2\nβ : Type u_1\nf : α → β\ns t : Set β\nhf : Surjective f\n⊢ s ⊆ univ",
"state_before": "case hs\nι : Sort ?u.104761\nα : Type u_2\nβ : Type u_1\nf : α → β\ns t : Set β\nhf : Surjective f\n⊢ s ⊆ range f",
"tactic": "rw [Function.Surjective.range_eq hf]"
},
{
"state_after": "no goals",
"state_before": "case hs\nι : Sort ?u.104761\nα : Type u_2\nβ : Type u_1\nf : α → β\ns t : Set β\nhf : Surjective f\n⊢ s ⊆ univ",
"tactic": "apply subset_univ"
}
]
| [
1324,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1320,
1
]
|
Mathlib/Analysis/Convex/Gauge.lean | gauge_mono | []
| [
90,
99
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
]
|
Mathlib/Analysis/LocallyConvex/Basic.lean | balanced_iff_neg_mem | [
{
"state_after": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : ‖a‖ ≤ 1\nx : E\nhx : x ∈ s\n⊢ a • x ∈ s",
"state_before": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\n⊢ Balanced ℝ s ↔ ∀ ⦃x : E⦄, x ∈ s → -x ∈ s",
"tactic": "refine' ⟨fun h x => h.neg_mem_iff.2, fun h a ha => smul_set_subset_iff.2 fun x hx => _⟩"
},
{
"state_after": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : -1 ≤ a ∧ a ≤ 1\nx : E\nhx : x ∈ s\n⊢ a • x ∈ s",
"state_before": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : ‖a‖ ≤ 1\nx : E\nhx : x ∈ s\n⊢ a • x ∈ s",
"tactic": "rw [Real.norm_eq_abs, abs_le] at ha"
},
{
"state_after": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : -1 ≤ a ∧ a ≤ 1\nx : E\nhx : x ∈ s\n⊢ ((1 - a) / 2) • -x + ((a - -1) / 2) • x ∈ s",
"state_before": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : -1 ≤ a ∧ a ≤ 1\nx : E\nhx : x ∈ s\n⊢ a • x ∈ s",
"tactic": "rw [show a = -((1 - a) / 2) + (a - -1) / 2 by ring, add_smul, neg_smul, ← smul_neg]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : -1 ≤ a ∧ a ≤ 1\nx : E\nhx : x ∈ s\n⊢ ((1 - a) / 2) • -x + ((a - -1) / 2) • x ∈ s",
"tactic": "exact\n hs (h hx) hx (div_nonneg (sub_nonneg_of_le ha.2) zero_le_two)\n (div_nonneg (sub_nonneg_of_le ha.1) zero_le_two) (by ring)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : -1 ≤ a ∧ a ≤ 1\nx : E\nhx : x ∈ s\n⊢ a = -((1 - a) / 2) + (a - -1) / 2",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.232896\n𝕝 : Type ?u.232899\nE : Type u_1\nι : Sort ?u.232905\nκ : ι → Sort ?u.232910\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s\na : ℝ\nha : -1 ≤ a ∧ a ≤ 1\nx : E\nhx : x ∈ s\n⊢ (1 - a) / 2 + (a - -1) / 2 = 1",
"tactic": "ring"
}
]
| [
429,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
]
|
Mathlib/Analysis/InnerProductSpace/Basic.lean | InnerProductSpace.Core.inner_sub_right | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.603708\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y z : F\n⊢ inner x (y - z) = inner x y - inner x z",
"tactic": "simp [sub_eq_add_neg, inner_add_right, inner_neg_right]"
}
]
| [
297,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
1
]
|
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_C_mul_X_pow_le | []
| [
687,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
686,
1
]
|
Std/Logic.lean | not_forall_of_exists_not | []
| [
445,
27
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
444,
1
]
|
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval_bit1 | []
| [
394,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
393,
1
]
|
Mathlib/Data/Nat/Bits.lean | Nat.boddDiv2_eq | [
{
"state_after": "n✝ n : ℕ\n⊢ boddDiv2 n = ((boddDiv2 n).fst, (boddDiv2 n).snd)",
"state_before": "n✝ n : ℕ\n⊢ boddDiv2 n = (bodd n, div2 n)",
"tactic": "unfold bodd div2"
},
{
"state_after": "case mk\nn✝ n : ℕ\nfst✝ : Bool\nsnd✝ : ℕ\n⊢ (fst✝, snd✝) = ((fst✝, snd✝).fst, (fst✝, snd✝).snd)",
"state_before": "n✝ n : ℕ\n⊢ boddDiv2 n = ((boddDiv2 n).fst, (boddDiv2 n).snd)",
"tactic": "cases boddDiv2 n"
},
{
"state_after": "no goals",
"state_before": "case mk\nn✝ n : ℕ\nfst✝ : Bool\nsnd✝ : ℕ\n⊢ (fst✝, snd✝) = ((fst✝, snd✝).fst, (fst✝, snd✝).snd)",
"tactic": "rfl"
}
]
| [
42,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
]
|
Mathlib/Order/Filter/Ultrafilter.lean | Ultrafilter.map_id' | []
| [
234,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
233,
1
]
|
Mathlib/MeasureTheory/MeasurableSpace.lean | Measurable.of_uncurry_right | []
| [
701,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
699,
1
]
|
Mathlib/SetTheory/Ordinal/Exponential.lean | Ordinal.opow_dvd_opow_iff | []
| [
223,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
]
|
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.eq_of_eq | [
{
"state_after": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\nc : { x // x ∈ S }\nhc : ↑c * x = ↑c * y\n⊢ ↑g x = ↑g y",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\n⊢ ↑g x = ↑g y",
"tactic": "obtain ⟨c, hc⟩ := f.eq_iff_exists.1 h"
},
{
"state_after": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\nc : { x // x ∈ S }\nhc : ↑c * x = ↑c * y\n⊢ ↑(↑(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)⁻¹ * ↑(MonoidHom.restrict g S) c * ↑g x = ↑g y",
"state_before": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\nc : { x // x ∈ S }\nhc : ↑c * x = ↑c * y\n⊢ ↑g x = ↑g y",
"tactic": "rw [← one_mul (g x), ← IsUnit.liftRight_inv_mul (g.restrict S) hg c]"
},
{
"state_after": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\nc : { x // x ∈ S }\nhc : ↑c * x = ↑c * y\n⊢ ↑(↑(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)⁻¹ * ↑g ↑c * ↑g x = ↑g y",
"state_before": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\nc : { x // x ∈ S }\nhc : ↑c * x = ↑c * y\n⊢ ↑(↑(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)⁻¹ * ↑(MonoidHom.restrict g S) c * ↑g x = ↑g y",
"tactic": "show _ * g c * _ = _"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nx y : M\nh : ↑(toMap f) x = ↑(toMap f) y\nc : { x // x ∈ S }\nhc : ↑c * x = ↑c * y\n⊢ ↑(↑(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)⁻¹ * ↑g ↑c * ↑g x = ↑g y",
"tactic": "rw [mul_assoc, ← g.map_mul, hc, mul_comm, mul_inv_left hg, g.map_mul]"
}
]
| [
898,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
894,
1
]
|
Std/Data/Int/Lemmas.lean | Int.negSucc_eq' | [
{
"state_after": "m : Nat\n⊢ -↑m + -1 = -↑m - 1",
"state_before": "m : Nat\n⊢ -[m+1] = -↑m - 1",
"tactic": "simp only [negSucc_eq, Int.neg_add]"
},
{
"state_after": "no goals",
"state_before": "m : Nat\n⊢ -↑m + -1 = -↑m - 1",
"tactic": "rfl"
}
]
| [
1354,
95
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1354,
1
]
|
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.map_le_of_le_comap | []
| [
1423,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1422,
1
]
|
Mathlib/LinearAlgebra/Eigenspace/Basic.lean | Module.End.hasGeneralizedEigenvalue_of_hasGeneralizedEigenvalue_of_le | [
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : ↑(generalizedEigenspace f μ) k ≠ ⊥\n⊢ ↑(generalizedEigenspace f μ) m ≠ ⊥",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : HasGeneralizedEigenvalue f μ k\n⊢ HasGeneralizedEigenvalue f μ m",
"tactic": "unfold HasGeneralizedEigenvalue at *"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : ↑(generalizedEigenspace f μ) m = ⊥\n⊢ ↑(generalizedEigenspace f μ) k = ⊥",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : ↑(generalizedEigenspace f μ) k ≠ ⊥\n⊢ ↑(generalizedEigenspace f μ) m ≠ ⊥",
"tactic": "contrapose! hk"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : ↑(generalizedEigenspace f μ) m = ⊥\n⊢ ↑(generalizedEigenspace f μ) k ≤ ↑(generalizedEigenspace f μ) m",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : ↑(generalizedEigenspace f μ) m = ⊥\n⊢ ↑(generalizedEigenspace f μ) k = ⊥",
"tactic": "rw [← le_bot_iff, ← hk]"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : ↑(generalizedEigenspace f μ) m = ⊥\n⊢ ↑(generalizedEigenspace f μ) k ≤ ↑(generalizedEigenspace f μ) m",
"tactic": "exact (f.generalizedEigenspace μ).monotone hm"
}
]
| [
339,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
333,
1
]
|
Mathlib/MeasureTheory/Group/Prod.lean | MeasureTheory.measure_mul_right_null | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝³ : SigmaFinite ν\ninst✝² : SigmaFinite μ\ns : Set G\ninst✝¹ : MeasurableInv G\ninst✝ : IsMulLeftInvariant μ\ny : G\n⊢ ↑↑μ ((fun x => x * y) ⁻¹' s) = 0 ↔ ↑↑μ ((fun x => y⁻¹ * x) ⁻¹' s⁻¹)⁻¹ = 0",
"tactic": "simp_rw [← inv_preimage, preimage_preimage, mul_inv_rev, inv_inv]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝³ : SigmaFinite ν\ninst✝² : SigmaFinite μ\ns : Set G\ninst✝¹ : MeasurableInv G\ninst✝ : IsMulLeftInvariant μ\ny : G\n⊢ ↑↑μ ((fun x => y⁻¹ * x) ⁻¹' s⁻¹)⁻¹ = 0 ↔ ↑↑μ s = 0",
"tactic": "simp only [measure_inv_null μ, measure_preimage_mul]"
}
]
| [
219,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
]
|
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | BoxIntegral.IntegrationParams.hasBasis_toFilter | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\n⊢ HasBasis (toFilter l I) (fun r => ∀ (c : ℝ≥0), RCond l (r c)) fun r => {π | ∃ c, MemBaseSet l I c (r c) π}",
"tactic": "simpa only [setOf_exists] using hasBasis_iSup (l.hasBasis_toFilterDistortion I)"
}
]
| [
503,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
500,
1
]
|
Mathlib/CategoryTheory/Types.lean | CategoryTheory.Iso.toEquiv_id | []
| [
366,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
]
|
Mathlib/Topology/UniformSpace/UniformEmbedding.lean | UniformInducing.uniformContinuous | []
| [
95,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
]
|
Mathlib/RingTheory/Multiplicity.lean | multiplicity.dvd_iff_multiplicity_pos | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nhdvd : a ∣ b\nheq : 0 = multiplicity a b\n⊢ multiplicity a b < 1",
"tactic": "simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nhdvd : a ∣ b\nheq : 0 = multiplicity a b\n⊢ a ^ 1 ∣ b",
"tactic": "rwa [pow_one a]"
}
]
| [
280,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
]
|
Mathlib/LinearAlgebra/SymplecticGroup.lean | SymplecticGroup.neg_mem | [
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ (-A) ⬝ J l R ⬝ (-A)ᵀ = J l R",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ∈ symplecticGroup l R\n⊢ -A ∈ symplecticGroup l R",
"tactic": "rw [mem_iff] at h⊢"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ (-A) ⬝ J l R ⬝ (-A)ᵀ = J l R",
"tactic": "simp [h]"
}
]
| [
142,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
]
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | MeasureTheory.Lp.simpleFunc.coeFn_zero | []
| [
836,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
835,
1
]
|
Mathlib/Topology/Bases.lean | TopologicalSpace.IsTopologicalBasis.secondCountableTopology | []
| [
604,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
602,
11
]
|
Mathlib/Data/Sum/Basic.lean | Function.Injective.sum_map | []
| [
569,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
566,
1
]
|
Mathlib/Algebra/Category/ModuleCat/EpiMono.lean | ModuleCat.mono_iff_ker_eq_bot | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : Ring R\nX Y : ModuleCat R\nf : X ⟶ Y\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhf : LinearMap.ker f = ⊥\n⊢ Function.Injective ((forget (ModuleCat R)).map f)",
"tactic": "convert LinearMap.ker_eq_bot.1 hf"
}
]
| [
44,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
]
|
Mathlib/GroupTheory/CommutingProbability.lean | inv_card_commutator_le_commProb | []
| [
126,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
]
|
Mathlib/Topology/Separation.lean | Set.Subsingleton.closure | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\nhs : Set.Subsingleton s\n⊢ Set.Subsingleton (_root_.closure s)",
"tactic": "rcases hs.eq_empty_or_singleton with (rfl | ⟨x, rfl⟩) <;> simp"
}
]
| [
633,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
631,
1
]
|
Mathlib/Analysis/Calculus/Deriv/Linear.lean | LinearMap.hasDerivAtFilter | []
| [
87,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
11
]
|
Mathlib/Algebra/Associated.lean | Associates.exists_non_zero_rep | []
| [
979,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
978,
1
]
|
Mathlib/CategoryTheory/StructuredArrow.lean | CategoryTheory.CostructuredArrow.comp_left | []
| [
322,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
321,
1
]
|
Mathlib/Algebra/Ring/BooleanRing.lean | toBoolAlg_add | []
| [
353,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.biUnion_self | []
| [
926,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
925,
1
]
|
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.nontrivial | [
{
"state_after": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nh : degree f ≠ 0\n⊢ ∀ (x : R), IsUnit x → ¬↑C x = f",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nh : degree f ≠ 0\n⊢ span {f} ≠ ⊤",
"tactic": "simp_rw [Ne.def, span_singleton_eq_top, Polynomial.isUnit_iff, not_exists, not_and]"
},
{
"state_after": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R\nhx : IsUnit x\nh : degree (↑C x) ≠ 0\n⊢ False",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nh : degree f ≠ 0\n⊢ ∀ (x : R), IsUnit x → ¬↑C x = f",
"tactic": "rintro x hx rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R\nhx : IsUnit x\nh : degree (↑C x) ≠ 0\n⊢ False",
"tactic": "exact h (degree_C hx.ne_zero)"
}
]
| [
92,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
11
]
|
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.parallelPair_obj_zero | []
| [
227,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
]
|
Mathlib/ModelTheory/Definability.lean | Set.Definable.map_expansion | [
{
"state_after": "case intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\nL' : Language\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\nψ : Formula (L[[↑A]]) α\n⊢ Definable A L' (setOf (Formula.Realize ψ))",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\ns : Set (α → M)\nL' : Language\ninst✝¹ : Structure L' M\nh : Definable A L s\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\n⊢ Definable A L' s",
"tactic": "obtain ⟨ψ, rfl⟩ := h"
},
{
"state_after": "case intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\nL' : Language\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\nψ : Formula (L[[↑A]]) α\n⊢ setOf (Formula.Realize ψ) = setOf (Formula.Realize (LHom.onFormula (LHom.addConstants (↑A) φ) ψ))",
"state_before": "case intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\nL' : Language\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\nψ : Formula (L[[↑A]]) α\n⊢ Definable A L' (setOf (Formula.Realize ψ))",
"tactic": "refine' ⟨(φ.addConstants A).onFormula ψ, _⟩"
},
{
"state_after": "case intro.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\nL' : Language\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\nψ : Formula (L[[↑A]]) α\nx : α → M\n⊢ x ∈ setOf (Formula.Realize ψ) ↔ x ∈ setOf (Formula.Realize (LHom.onFormula (LHom.addConstants (↑A) φ) ψ))",
"state_before": "case intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\nL' : Language\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\nψ : Formula (L[[↑A]]) α\n⊢ setOf (Formula.Realize ψ) = setOf (Formula.Realize (LHom.onFormula (LHom.addConstants (↑A) φ) ψ))",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case intro.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type ?u.417\nB : Set M\nL' : Language\ninst✝¹ : Structure L' M\nφ : L →ᴸ L'\ninst✝ : LHom.IsExpansionOn φ M\nψ : Formula (L[[↑A]]) α\nx : α → M\n⊢ x ∈ setOf (Formula.Realize ψ) ↔ x ∈ setOf (Formula.Realize (LHom.onFormula (LHom.addConstants (↑A) φ) ψ))",
"tactic": "simp only [mem_setOf_eq, LHom.realize_onFormula]"
}
]
| [
59,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
]
|
src/lean/Init/SizeOf.lean | sizeOf_nat | []
| [
52,
59
]
| d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
52,
9
]
|
Mathlib/Data/Polynomial/EraseLead.lean | Polynomial.natDegree_not_mem_eraseLead_support | []
| [
109,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
]
|
Mathlib/Geometry/Manifold/ChartedSpace.lean | coe_achart | []
| [
568,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
567,
1
]
|
Mathlib/Analysis/Normed/Group/InfiniteSum.lean | summable_of_summable_nnnorm | []
| [
181,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
180,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Types.lean | CategoryTheory.Limits.Types.binaryProductIso_inv_comp_fst | []
| [
169,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
]
|
Mathlib/Data/Multiset/FinsetOps.lean | Multiset.ndunion_eq_union | []
| [
204,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
]
|
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.ghostFun_zsmul | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.801752\nT : Type ?u.801755\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.801770\nβ : Type ?u.801773\nx y : 𝕎 R\nm : ℤ\n⊢ WittVector.ghostFun (m • x) = m • WittVector.ghostFun x",
"tactic": "ghost_fun_tac m • (X 0 : MvPolynomial _ ℤ), ![x.coeff]"
}
]
| [
223,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
9
]
|
Mathlib/Data/List/Basic.lean | List.enum_map_snd | []
| [
3843,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3842,
1
]
|
Mathlib/Data/Polynomial/Basic.lean | Polynomial.toFinsupp_neg | [
{
"state_after": "case ofFinsupp\nR✝ : Type u\na b : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q : R✝[X]\nR : Type u\ninst✝ : Ring R\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp",
"state_before": "R✝ : Type u\na✝ b : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q : R✝[X]\nR : Type u\ninst✝ : Ring R\na : R[X]\n⊢ (-a).toFinsupp = -a.toFinsupp",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp\nR✝ : Type u\na b : R✝\nm n : ℕ\ninst✝¹ : Semiring R✝\np q : R✝[X]\nR : Type u\ninst✝ : Ring R\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp",
"tactic": "rw [← ofFinsupp_neg]"
}
]
| [
221,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
]
|
Mathlib/SetTheory/ZFC/Basic.lean | Class.coe_sUnion | [
{
"state_after": "no goals",
"state_before": "x y : ZFSet\n⊢ (∃ z, ↑x z ∧ y ∈ z) ↔ ∃ z, z ∈ x ∧ y ∈ z",
"tactic": "rfl"
}
]
| [
1688,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1686,
1
]
|
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.L1.ofReal_norm_eq_lintegral | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1305559\nδ : Type ?u.1305562\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : { x // x ∈ Lp β 1 }\n⊢ (∫⁻ (a : α), ↑‖↑↑f a‖₊ ∂μ) ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1305559\nδ : Type ?u.1305562\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : { x // x ∈ Lp β 1 }\n⊢ ENNReal.ofReal ‖f‖ = ∫⁻ (x : α), ↑‖↑↑f x‖₊ ∂μ",
"tactic": "rw [norm_def, ENNReal.ofReal_toReal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1305559\nδ : Type ?u.1305562\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : { x // x ∈ Lp β 1 }\n⊢ (∫⁻ (a : α), ↑‖↑↑f a‖₊ ∂μ) ≠ ⊤",
"tactic": "exact ne_of_lt (hasFiniteIntegral_coeFn f)"
}
]
| [
1327,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1324,
1
]
|
Mathlib/Topology/UniformSpace/UniformConvergence.lean | TendstoUniformlyOn.tendstoLocallyUniformlyOn | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝¹ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : TendstoUniformlyOn F f p s\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx : α\nx✝ : x ∈ s\n⊢ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ s → (f y, F n y) ∈ u",
"tactic": "simpa using h u hu"
}
]
| [
639,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
11
]
|
Mathlib/RingTheory/Ideal/Operations.lean | Submodule.top_smul | []
| [
200,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.prod.symmetry' | []
| [
1007,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1005,
1
]
|
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | Matrix.inv_adjugate | [
{
"state_after": "l : Type ?u.302074\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nh : IsUnit (det A)\n⊢ ((IsUnit.unit h)⁻¹ • A) ⬝ adjugate A = 1",
"state_before": "l : Type ?u.302074\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nh : IsUnit (det A)\n⊢ (adjugate A)⁻¹ = (IsUnit.unit h)⁻¹ • A",
"tactic": "refine' inv_eq_left_inv _"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.302074\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nh : IsUnit (det A)\n⊢ ((IsUnit.unit h)⁻¹ • A) ⬝ adjugate A = 1",
"tactic": "rw [smul_mul, mul_adjugate, Units.smul_def, smul_smul, h.val_inv_mul, one_smul]"
}
]
| [
514,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
512,
1
]
|
Mathlib/Topology/Algebra/Order/LeftRightLim.lean | Antitone.leftLim_le_rightLim | []
| [
332,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
]
|
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.Perm.ofSubtype_apply_coe | []
| [
455,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
]
|
Mathlib/Combinatorics/Additive/SalemSpencer.lean | AddSalemSpencer.le_rothNumberNat | []
| [
486,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
]
|
Mathlib/Algebra/Order/Monoid/WithTop.lean | WithTop.add_lt_top | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Add α\na✝ b✝ c d : WithTop α\nx y : α\ninst✝ : LT α\na b : WithTop α\n⊢ a + b < ⊤ ↔ a < ⊤ ∧ b < ⊤",
"tactic": "simp_rw [WithTop.lt_top_iff_ne_top, add_ne_top]"
}
]
| [
163,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | Polynomial.coe_pow | []
| [
2651,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2650,
1
]
|
Mathlib/CategoryTheory/Equivalence.lean | CategoryTheory.Equivalence.unitInv_app_inverse | [
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ e.inverse.map ((counit e).app Y) = (unitInv e).app (e.inverse.obj Y)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (unitInv e).app (e.inverse.obj Y) = e.inverse.map ((counit e).app Y)",
"tactic": "symm"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ 𝟙 (e.inverse.obj Y) = 𝟙 ((𝟭 C).obj (e.inverse.obj Y))",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ e.inverse.map ((counit e).app Y) = (unitInv e).app (e.inverse.obj Y)",
"tactic": "erw [← Iso.hom_comp_eq_id (e.unitIso.app _), unit_inverse_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ 𝟙 (e.inverse.obj Y) = 𝟙 ((𝟭 C).obj (e.inverse.obj Y))",
"tactic": "rfl"
}
]
| [
225,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
221,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_const_mul_left_iff | []
| [
1489,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1487,
1
]
|
Mathlib/ModelTheory/Syntax.lean | FirstOrder.Language.BoundedFormula.IsPrenex.induction_on_all_not | []
| [
772,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
767,
1
]
|
Mathlib/Algebra/Order/AbsoluteValue.lean | AbsoluteValue.map_neg | [
{
"state_after": "case pos\nR : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : a = 0\n⊢ ↑abv (-a) = ↑abv a\n\ncase neg\nR : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : ¬a = 0\n⊢ ↑abv (-a) = ↑abv a",
"state_before": "R : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\n⊢ ↑abv (-a) = ↑abv a",
"tactic": "by_cases ha : a = 0"
},
{
"state_after": "case neg\nR : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : ¬a = 0\n⊢ ¬↑abv (-a) = -↑abv a",
"state_before": "case neg\nR : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : ¬a = 0\n⊢ ↑abv (-a) = ↑abv a",
"tactic": "refine'\n (mul_self_eq_mul_self_iff.mp (by rw [← map_mul abv, neg_mul_neg, map_mul abv])).resolve_right _"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : ¬a = 0\n⊢ ¬↑abv (-a) = -↑abv a",
"tactic": "exact ((neg_lt_zero.mpr (abv.pos ha)).trans (abv.pos (neg_ne_zero.mpr ha))).ne'"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : a = 0\n⊢ ↑abv (-a) = ↑abv a",
"tactic": "simp [ha]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_1\ninst✝² : Ring R\ninst✝¹ : OrderedCommRing S\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na : R\nha : ¬a = 0\n⊢ ↑abv (-a) * ↑abv (-a) = ↑abv a * ↑abv a",
"tactic": "rw [← map_mul abv, neg_mul_neg, map_mul abv]"
}
]
| [
241,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
237,
11
]
|
Mathlib/Data/Multiset/Fintype.lean | Multiset.forall_coe | []
| [
98,
15
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
11
]
|
Mathlib/MeasureTheory/Measure/Regular.lean | MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable | []
| [
578,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
575,
1
]
|
Mathlib/CategoryTheory/EssentialImage.lean | CategoryTheory.Functor.essImage.ofIso | []
| [
61,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
]
|
Mathlib/Data/Dfinsupp/Lex.lean | Dfinsupp.lex_lt_of_lt | [
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt : x < y\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt : x < y\n⊢ Pi.Lex r (fun {i} x x_1 => x < x_1) ↑x ↑y",
"tactic": "simp_rw [Pi.Lex, le_antisymm_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\nr : ι → ι → Prop\ninst✝ : IsStrictOrder ι r\nx y : Π₀ (i : ι), α i\nhlt : x < y\n⊢ ∃ i, (∀ (j : ι), r j i → ↑x j ≤ ↑y j ∧ ↑y j ≤ ↑x j) ∧ ↑x i < ↑y i",
"tactic": "exact lex_lt_of_lt_of_preorder r hlt"
}
]
| [
66,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
]
|
Mathlib/Algebra/Associated.lean | Associates.isUnit_mk | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.295731\nγ : Type ?u.295734\nδ : Type ?u.295737\ninst✝ : CommMonoid α\na : α\n⊢ IsUnit (Associates.mk a) ↔ a ~ᵤ 1",
"tactic": "rw [isUnit_iff_eq_one, one_eq_mk_one, mk_eq_mk_iff_associated]"
}
]
| [
897,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
893,
1
]
|
Mathlib/Data/Set/Pointwise/Interval.lean | Set.image_const_sub_Iio | [
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iio b = Ioi (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a - x) '' Iio b = Ioi (a - b)",
"tactic": "have := image_comp (fun x => a + x) fun x => -x"
},
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iio b = Ioi (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iio b = Ioi (a - b)",
"tactic": "dsimp [Function.comp] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Iio b = Ioi (a - b)",
"tactic": "simp [sub_eq_add_neg, this, add_comm]"
}
]
| [
341,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
]
|
Std/Data/List/Lemmas.lean | List.get_set_ne | [
{
"state_after": "α : Type ?u.120711\nl : List α\ni j : Nat\nh : i ≠ j\na : α\nhj : j < length l\n⊢ j < length l",
"state_before": "α : Type ?u.120711\nl : List α\ni j : Nat\nh : i ≠ j\na : α\nhj : j < length (set l i a)\n⊢ j < length l",
"tactic": "simp at hj"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.120711\nl : List α\ni j : Nat\nh : i ≠ j\na : α\nhj : j < length l\n⊢ j < length l",
"tactic": "exact hj"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\ni j : Nat\nh : i ≠ j\na : α\nhj : j < length (set l i a)\n⊢ get (set l i a) { val := j, isLt := hj } = get l { val := j, isLt := (_ : j < length l) }",
"tactic": "rw [← Option.some_inj, ← get?_eq_get, get?_set_ne _ _ h, get?_eq_get]"
}
]
| [
842,
72
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
839,
9
]
|
Mathlib/Data/Num/Lemmas.lean | Num.toZNumNeg_succ | []
| [
1245,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1243,
1
]
|
Mathlib/Computability/Ackermann.lean | not_primrec₂_ack | []
| [
395,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
394,
1
]
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean | Multiset.all_one_of_le_one_le_of_prod_eq_one | [
{
"state_after": "ι : Type ?u.125685\nα : Type u_1\nβ : Type ?u.125691\nγ : Type ?u.125694\ninst✝ : OrderedCommMonoid α\ns t : Multiset α\na : α\n⊢ ∀ (a : List α), (∀ (x : α), x ∈ a → 1 ≤ x) → List.prod a = 1 → ∀ (x : α), x ∈ a → x = 1",
"state_before": "ι : Type ?u.125685\nα : Type u_1\nβ : Type ?u.125691\nγ : Type ?u.125694\ninst✝ : OrderedCommMonoid α\ns t : Multiset α\na : α\n⊢ ∀ (a : List α),\n (∀ (x : α), x ∈ Quotient.mk (List.isSetoid α) a → 1 ≤ x) →\n prod (Quotient.mk (List.isSetoid α) a) = 1 → ∀ (x : α), x ∈ Quotient.mk (List.isSetoid α) a → x = 1",
"tactic": "simp only [quot_mk_to_coe, coe_prod, mem_coe]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.125685\nα : Type u_1\nβ : Type ?u.125691\nγ : Type ?u.125694\ninst✝ : OrderedCommMonoid α\ns t : Multiset α\na : α\n⊢ ∀ (a : List α), (∀ (x : α), x ∈ a → 1 ≤ x) → List.prod a = 1 → ∀ (x : α), x ∈ a → x = 1",
"tactic": "exact fun l => List.all_one_of_le_one_le_of_prod_eq_one"
}
]
| [
391,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
]
|
Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | FreeAbelianGroup.support_nsmul | [
{
"state_after": "X : Type u_1\nk : ℕ\nh : k ≠ 0\na : FreeAbelianGroup X\n⊢ ↑k ≠ 0",
"state_before": "X : Type u_1\nk : ℕ\nh : k ≠ 0\na : FreeAbelianGroup X\n⊢ support (k • a) = support a",
"tactic": "apply support_zsmul k _ a"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nk : ℕ\nh : k ≠ 0\na : FreeAbelianGroup X\n⊢ ↑k ≠ 0",
"tactic": "exact_mod_cast h"
}
]
| [
193,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
1
]
|
Mathlib/FieldTheory/ChevalleyWarning.lean | MvPolynomial.sum_eval_eq_zero | [
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\n⊢ ∑ x : σ → K, ↑(eval x) f = 0",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\n⊢ ∑ x : σ → K, ↑(eval x) f = 0",
"tactic": "haveI : DecidableEq K := Classical.decEq K"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\n⊢ ∀ (x : σ →₀ ℕ), x ∈ support f → ∑ x_1 : σ → K, coeff x f * ∏ i : σ, x_1 i ^ ↑x i = 0",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\n⊢ ∑ x : σ → K, ↑(eval x) f = 0",
"tactic": "calc\n (∑ x, eval x f) = ∑ x : σ → K, ∑ d in f.support, f.coeff d * ∏ i, x i ^ d i := by\n simp only [eval_eq']\n _ = ∑ d in f.support, ∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i := sum_comm\n _ = 0 := sum_eq_zero ?_"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\n⊢ ∑ x : σ → K, coeff d f * ∏ i : σ, x i ^ ↑d i = 0",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\n⊢ ∀ (x : σ →₀ ℕ), x ∈ support f → ∑ x_1 : σ → K, coeff x f * ∏ i : σ, x_1 i ^ ↑x i = 0",
"tactic": "intro d hd"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\n⊢ ∃ i, ↑d i < q - 1\n\ncase intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∑ x : σ → K, coeff d f * ∏ i : σ, x i ^ ↑d i = 0",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\n⊢ ∑ x : σ → K, coeff d f * ∏ i : σ, x i ^ ↑d i = 0",
"tactic": "obtain ⟨i, hi⟩ : ∃ i, d i < q - 1"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∑ x : σ → K, coeff d f * ∏ i : σ, x i ^ ↑d i = 0",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\n⊢ ∃ i, ↑d i < q - 1\n\ncase intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∑ x : σ → K, coeff d f * ∏ i : σ, x i ^ ↑d i = 0",
"tactic": "exact f.exists_degree_lt (q - 1) h hd"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∑ x : σ → K, ∏ i : σ, x i ^ ↑d i = 0",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∑ x : σ → K, coeff d f * ∏ i : σ, x i ^ ↑d i = 0",
"tactic": "calc\n (∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i) = f.coeff d * ∑ x : σ → K, ∏ i, x i ^ d i :=\n mul_sum.symm\n _ = 0 := (mul_eq_zero.mpr ∘ Or.inr) ?_"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∀ (a : { j // j ≠ i } → K), ∑ x : { x // x ∘ Subtype.val = a }, ∏ j : σ, ↑x j ^ ↑d j = 0",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∑ x : σ → K, ∏ i : σ, x i ^ ↑d i = 0",
"tactic": "calc\n (∑ x : σ → K, ∏ i, x i ^ d i) =\n ∑ x₀ : { j // j ≠ i } → K, ∑ x : { x : σ → K // x ∘ (↑) = x₀ }, ∏ j, (x : σ → K) j ^ d j :=\n (Fintype.sum_fiberwise _ _).symm\n _ = 0 := Fintype.sum_eq_zero _ ?_"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\n⊢ ∑ x : { x // x ∘ Subtype.val = x₀ }, ∏ j : σ, ↑x j ^ ↑d j = 0",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\n⊢ ∀ (a : { j // j ≠ i } → K), ∑ x : { x // x ∘ Subtype.val = a }, ∏ j : σ, ↑x j ^ ↑d j = 0",
"tactic": "intro x₀"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\n⊢ ∑ x : { x // x ∘ Subtype.val = x₀ }, ∏ j : σ, ↑x j ^ ↑d j = 0",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\n⊢ ∑ x : { x // x ∘ Subtype.val = x₀ }, ∏ j : σ, ↑x j ^ ↑d j = 0",
"tactic": "let e : K ≃ { x // x ∘ ((↑) : _ → σ) = x₀ } := (Equiv.subtypeEquivCodomain _).symm"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\n⊢ ∀ (a : K), ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\n⊢ ∑ x : { x // x ∘ Subtype.val = x₀ }, ∏ j : σ, ↑x j ^ ↑d j = 0",
"tactic": "calc\n (∑ x : { x : σ → K // x ∘ (↑) = x₀ }, ∏ j, (x : σ → K) j ^ d j) =\n ∑ a : K, ∏ j : σ, (e a : σ → K) j ^ d j := (e.sum_comp _).symm\n _ = ∑ a : K, (∏ j, x₀ j ^ d j) * a ^ d i := (Fintype.sum_congr _ _ ?_)\n _ = (∏ j, x₀ j ^ d j) * ∑ a : K, a ^ d i := by rw [mul_sum]\n _ = 0 := by rw [sum_pow_lt_card_sub_one K _ hi, MulZeroClass.mul_zero]"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\n⊢ ∀ (a : K), ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"tactic": "intro a"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"tactic": "let e' : Sum { j // j = i } { j // j ≠ i } ≃ σ := Equiv.sumCompl _"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"tactic": "letI : Unique { j // j = i } :=\n { default := ⟨i, rfl⟩\n uniq := fun ⟨j, h⟩ => Subtype.val_injective h }"
},
{
"state_after": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ∀ (a_1 : { j // j ≠ i }), ↑(↑e a) ↑a_1 ^ ↑d ↑a_1 = x₀ a_1 ^ ↑d ↑a_1",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i",
"tactic": "calc\n (∏ j : σ, (e a : σ → K) j ^ d j) =\n (e a : σ → K) i ^ d i * ∏ j : { j // j ≠ i }, (e a : σ → K) j ^ d j :=\n by rw [← e'.prod_comp, Fintype.prod_sum_type, univ_unique, prod_singleton]; rfl\n _ = a ^ d i * ∏ j : { j // j ≠ i }, (e a : σ → K) j ^ d j := by\n rw [Equiv.subtypeEquivCodomain_symm_apply_eq]\n _ = a ^ d i * ∏ j, x₀ j ^ d j := (congr_arg _ (Fintype.prod_congr _ _ ?_))\n _ = (∏ j, x₀ j ^ d j) * a ^ d i := mul_comm _ _"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\n⊢ ∑ x : σ → K, ↑(eval x) f = ∑ x : σ → K, ∑ d in support f, coeff d f * ∏ i : σ, x i ^ ↑d i",
"tactic": "simp only [eval_eq']"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\n⊢ ∑ a : K, (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * a ^ ↑d i = (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * ∑ a : K, a ^ ↑d i",
"tactic": "rw [mul_sum]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\n⊢ (∏ j : { j // j ≠ i }, x₀ j ^ ↑d ↑j) * ∑ a : K, a ^ ↑d i = 0",
"tactic": "rw [sum_pow_lt_card_sub_one K _ hi, MulZeroClass.mul_zero]"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ↑(↑e a) (↑e' (Sum.inl default)) ^ ↑d (↑e' (Sum.inl default)) *\n ∏ a₂ : { j // j ≠ i }, ↑(↑e a) (↑e' (Sum.inr a₂)) ^ ↑d (↑e' (Sum.inr a₂)) =\n ↑(↑e a) i ^ ↑d i * ∏ j : { j // j ≠ i }, ↑(↑e a) ↑j ^ ↑d ↑j",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ∏ j : σ, ↑(↑e a) j ^ ↑d j = ↑(↑e a) i ^ ↑d i * ∏ j : { j // j ≠ i }, ↑(↑e a) ↑j ^ ↑d ↑j",
"tactic": "rw [← e'.prod_comp, Fintype.prod_sum_type, univ_unique, prod_singleton]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ↑(↑e a) (↑e' (Sum.inl default)) ^ ↑d (↑e' (Sum.inl default)) *\n ∏ a₂ : { j // j ≠ i }, ↑(↑e a) (↑e' (Sum.inr a₂)) ^ ↑d (↑e' (Sum.inr a₂)) =\n ↑(↑e a) i ^ ↑d i * ∏ j : { j // j ≠ i }, ↑(↑e a) ↑j ^ ↑d ↑j",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ↑(↑e a) i ^ ↑d i * ∏ j : { j // j ≠ i }, ↑(↑e a) ↑j ^ ↑d ↑j = a ^ ↑d i * ∏ j : { j // j ≠ i }, ↑(↑e a) ↑j ^ ↑d ↑j",
"tactic": "rw [Equiv.subtypeEquivCodomain_symm_apply_eq]"
},
{
"state_after": "case intro.mk\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\nj : σ\nhj : j ≠ i\n⊢ ↑(↑e a) ↑{ val := j, property := hj } ^ ↑d ↑{ val := j, property := hj } =\n x₀ { val := j, property := hj } ^ ↑d ↑{ val := j, property := hj }",
"state_before": "case intro\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\n⊢ ∀ (a_1 : { j // j ≠ i }), ↑(↑e a) ↑a_1 ^ ↑d ↑a_1 = x₀ a_1 ^ ↑d ↑a_1",
"tactic": "rintro ⟨j, hj⟩"
},
{
"state_after": "case intro.mk\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\nj : σ\nhj : j ≠ i\n⊢ ↑(↑e a) j ^ ↑d j = x₀ { val := j, property := hj } ^ ↑d j",
"state_before": "case intro.mk\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\nj : σ\nhj : j ≠ i\n⊢ ↑(↑e a) ↑{ val := j, property := hj } ^ ↑d ↑{ val := j, property := hj } =\n x₀ { val := j, property := hj } ^ ↑d ↑{ val := j, property := hj }",
"tactic": "show (e a : σ → K) j ^ d j = x₀ ⟨j, hj⟩ ^ d j"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nK : Type u_2\nσ : Type u_1\nι : Type ?u.9599\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : totalDegree f < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ support f\ni : σ\nhi : ↑d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = x₀ } := (Equiv.subtypeEquivCodomain x₀).symm\na : K\ne' : { j // j = i } ⊕ { j // j ≠ i } ≃ σ := Equiv.sumCompl fun j => j = i\nthis : Unique { j // j = i } :=\n { toInhabited := { default := { val := i, property := (_ : i = i) } },\n uniq := (_ : ∀ (x : { j // j = i }), x = default) }\nj : σ\nhj : j ≠ i\n⊢ ↑(↑e a) j ^ ↑d j = x₀ { val := j, property := hj } ^ ↑d j",
"tactic": "rw [Equiv.subtypeEquivCodomain_symm_apply_ne]"
}
]
| [
102,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
]
|
Mathlib/Data/MvPolynomial/Derivation.lean | MvPolynomial.derivation_C_mul | [
{
"state_after": "no goals",
"state_before": "σ : Type u_2\nR : Type u_1\nA : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Module (MvPolynomial σ R) A\nD : Derivation R (MvPolynomial σ R) A\na : R\nf : MvPolynomial σ R\n⊢ ↑D (↑C a * f) = a • ↑D f",
"tactic": "rw [C_mul', D.map_smul]"
}
]
| [
71,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
]
|
Mathlib/RingTheory/Subring/Basic.lean | Subring.map_equiv_eq_comap_symm | []
| [
1178,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1176,
1
]
|
Mathlib/CategoryTheory/Monad/Basic.lean | CategoryTheory.Comonad.coassoc | []
| [
163,
15
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
]
|
Mathlib/Topology/Sets/Closeds.lean | TopologicalSpace.Clopens.coe_inf | []
| [
329,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
9
]
|
Mathlib/Algebra/Order/Ring/Defs.lean | mul_lt_of_one_lt_right | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.89024\ninst✝ : StrictOrderedRing α\na b c : α\nha : a < 0\nh : 1 < b\n⊢ a * b < a",
"tactic": "simpa only [mul_one] using mul_lt_mul_of_neg_left h ha"
}
]
| [
719,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
718,
1
]
|
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.finitePresentation | []
| [
168,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
]
|
Mathlib/Computability/PartrecCode.lean | Nat.Partrec.Code.encode_lt_prec | [
{
"state_after": "cf cg : Code\nthis : ?m.360436\n⊢ encode cf < encode (prec cf cg) ∧ encode cg < encode (prec cf cg)\n\ncase this\ncf cg : Code\n⊢ ?m.360436",
"state_before": "cf cg : Code\n⊢ encode cf < encode (prec cf cg) ∧ encode cg < encode (prec cf cg)",
"tactic": "suffices"
},
{
"state_after": "case this\ncf cg : Code\n⊢ encode (pair cf cg) < encode (prec cf cg)",
"state_before": "cf cg : Code\nthis : ?m.360436\n⊢ encode cf < encode (prec cf cg) ∧ encode cg < encode (prec cf cg)\n\ncase this\ncf cg : Code\n⊢ ?m.360436",
"tactic": "exact (encode_lt_pair cf cg).imp (fun h => lt_trans h this) fun h => lt_trans h this"
},
{
"state_after": "case this\ncf cg : Code\n⊢ encode (pair cf cg) < encode (prec cf cg)",
"state_before": "case this\ncf cg : Code\n⊢ encode (pair cf cg) < encode (prec cf cg)",
"tactic": "change _"
},
{
"state_after": "no goals",
"state_before": "case this\ncf cg : Code\n⊢ encode (pair cf cg) < encode (prec cf cg)",
"tactic": "simp [encodeCode_eq, encodeCode]"
}
]
| [
223,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
1
]
|
Mathlib/Topology/Instances/Int.lean | Int.uniformEmbedding_coe_real | []
| [
47,
76
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
]
|
Mathlib/Order/UpperLower/Basic.lean | lowerClosure_univ | []
| [
1419,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1418,
1
]
|
Std/Data/Nat/Gcd.lean | Nat.coprime.gcd_mul_left_cancel_right | [
{
"state_after": "no goals",
"state_before": "k m n : Nat\nH : coprime k m\n⊢ gcd m (k * n) = gcd m n",
"tactic": "rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n]"
}
]
| [
267,
65
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
265,
1
]
|
Mathlib/Data/Set/Prod.lean | Set.mk_mem_prod | []
| [
69,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
]
|
Mathlib/Algebra/Order/Floor.lean | Int.ceil_add_ceil_le | [
{
"state_after": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌈b⌉ ≤ ↑⌈a + b⌉ + 1 - a",
"state_before": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ⌈a⌉ + ⌈b⌉ ≤ ⌈a + b⌉ + 1",
"tactic": "rw [← le_sub_iff_add_le, ceil_le, Int.cast_sub, Int.cast_add, Int.cast_one, le_sub_comm]"
},
{
"state_after": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ b + 1 ≤ ↑⌈a + b⌉ + 1 - a",
"state_before": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌈b⌉ ≤ ↑⌈a + b⌉ + 1 - a",
"tactic": "refine' (ceil_lt_add_one _).le.trans _"
},
{
"state_after": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ a + b ≤ ↑⌈a + b⌉",
"state_before": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ b + 1 ≤ ↑⌈a + b⌉ + 1 - a",
"tactic": "rw [le_sub_iff_add_le', ← add_assoc, add_le_add_iff_right]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.218257\nα : Type u_1\nβ : Type ?u.218263\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ a + b ≤ ↑⌈a + b⌉",
"tactic": "exact le_ceil _"
}
]
| [
1188,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1184,
1
]
|
Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | Orientation.measure_eq_volume | [
{
"state_after": "ι : Type ?u.6740\nF : Type u_1\ninst✝⁵ : Fintype ι\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : FiniteDimensional ℝ F\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\nm n : ℕ\n_i : Fact (finrank ℝ F = n)\no : Orientation ℝ F (Fin n)\nA :\n ↑↑(AlternatingMap.measure (volumeForm o))\n ↑(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F))) =\n 1\n⊢ AlternatingMap.measure (volumeForm o) = volume",
"state_before": "ι : Type ?u.6740\nF : Type u_1\ninst✝⁵ : Fintype ι\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : FiniteDimensional ℝ F\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\nm n : ℕ\n_i : Fact (finrank ℝ F = n)\no : Orientation ℝ F (Fin n)\n⊢ AlternatingMap.measure (volumeForm o) = volume",
"tactic": "have A : o.volumeForm.measure (stdOrthonormalBasis ℝ F).toBasis.parallelepiped = 1 :=\n Orientation.measure_orthonormalBasis o (stdOrthonormalBasis ℝ F)"
},
{
"state_after": "ι : Type ?u.6740\nF : Type u_1\ninst✝⁵ : Fintype ι\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : FiniteDimensional ℝ F\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\nm n : ℕ\n_i : Fact (finrank ℝ F = n)\no : Orientation ℝ F (Fin n)\nA :\n ↑↑(AlternatingMap.measure (volumeForm o))\n ↑(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F))) =\n 1\n⊢ addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F))) = volume",
"state_before": "ι : Type ?u.6740\nF : Type u_1\ninst✝⁵ : Fintype ι\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : FiniteDimensional ℝ F\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\nm n : ℕ\n_i : Fact (finrank ℝ F = n)\no : Orientation ℝ F (Fin n)\nA :\n ↑↑(AlternatingMap.measure (volumeForm o))\n ↑(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F))) =\n 1\n⊢ AlternatingMap.measure (volumeForm o) = volume",
"tactic": "rw [addHaarMeasure_unique o.volumeForm.measure\n (stdOrthonormalBasis ℝ F).toBasis.parallelepiped, A, one_smul]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6740\nF : Type u_1\ninst✝⁵ : Fintype ι\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : FiniteDimensional ℝ F\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\nm n : ℕ\n_i : Fact (finrank ℝ F = n)\no : Orientation ℝ F (Fin n)\nA :\n ↑↑(AlternatingMap.measure (volumeForm o))\n ↑(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F))) =\n 1\n⊢ addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F))) = volume",
"tactic": "simp only [volume, Basis.addHaar]"
}
]
| [
59,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
]
|
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