file_path
stringlengths 11
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stringclasses 4
values | start
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Mathlib/Data/Finset/Basic.lean
|
Finset.disjiUnion_disjiUnion
|
[
{
"state_after": "case intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\n⊢ False",
"state_before": "α : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\n⊢ False",
"tactic": "obtain ⟨xa, hfa, hga⟩ := mem_disjiUnion.mp hxa"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\nxb : β\nhfb : xb ∈ f ↑b\nhgb : x ∈ g xb\n⊢ False",
"state_before": "case intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\n⊢ False",
"tactic": "obtain ⟨xb, hfb, hgb⟩ := mem_disjiUnion.mp hxb"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\nxb : β\nhfb : xb ∈ f ↑b\nhgb : x ∈ g xb\n⊢ xa ≠ xb",
"state_before": "case intro.intro.intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\nxb : β\nhfb : xb ∈ f ↑b\nhgb : x ∈ g xb\n⊢ False",
"tactic": "refine'\n disjoint_left.mp\n (h2 (mem_disjiUnion.mpr ⟨_, a.prop, hfa⟩) (mem_disjiUnion.mpr ⟨_, b.prop, hfb⟩) _) hga\n hgb"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\nhfb : xa ∈ f ↑b\nhgb : x ∈ g xa\n⊢ False",
"state_before": "case intro.intro.intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\nxb : β\nhfb : xb ∈ f ↑b\nhgb : x ∈ g xb\n⊢ xa ≠ xb",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type ?u.491855\nβ : Type ?u.491858\nγ : Type ?u.491861\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ns : Finset α\nf : α → Finset β\ng : β → Finset γ\nh1 : Set.PairwiseDisjoint (↑s) f\nh2 : Set.PairwiseDisjoint (↑(disjiUnion s f h1)) g\na : { x // x ∈ s }\nx✝¹ : a ∈ ↑(attach s)\nb : { x // x ∈ s }\nx✝ : b ∈ ↑(attach s)\nhab : a ≠ b\nx : γ\nhxa :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n a\nhxb :\n x ∈\n (fun a =>\n disjiUnion (f ↑a) g (_ : ∀ (b : β), b ∈ ↑(f ↑a) → ∀ (c : β), c ∈ ↑(f ↑a) → b ≠ c → (_root_.Disjoint on g) b c))\n b\nxa : β\nhfa : xa ∈ f ↑a\nhga : x ∈ g xa\nhfb : xa ∈ f ↑b\nhgb : x ∈ g xa\n⊢ False",
"tactic": "exact disjoint_left.mp (h1 a.prop b.prop <| Subtype.coe_injective.ne hab) hfa hfb"
}
] |
[
3494,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3478,
1
] |
Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
|
Finset.card_pow_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : Finset.Nonempty A\nB : Finset α\nn : ℕ\n⊢ ↑(card (B ^ n)) ≤ (↑(card (A * B)) / ↑(card A)) ^ n * ↑(card A)",
"tactic": "simpa only [_root_.pow_zero, div_one] using card_pow_div_pow_le hA _ _ 0"
}
] |
[
252,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Data/Complex/ExponentialBounds.lean
|
Real.exp_one_lt_d9
|
[
{
"state_after": "no goals",
"state_before": "⊢ 1 / 10 ^ 10 + 2244083 / 825552 < 2.7182818286",
"tactic": "norm_num"
}
] |
[
44,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Order/Atoms.lean
|
Set.isSimpleOrder_Ici_iff_isCoatom
|
[] |
[
691,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/GroupTheory/GroupAction/ConjAct.lean
|
ConjAct.ofConjAct_inv
|
[] |
[
129,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
map_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.50695\nβ : Type ?u.50698\nM : Type ?u.50701\nN : Type ?u.50704\nP : Type ?u.50707\nG : Type u_1\nH : Type u_2\nF : Type u_3\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : Monoid G\ninst✝¹ : Monoid H\ninst✝ : MonoidHomClass F G H\nf : F\na : G\n⊢ ↑f (a ^ 0) = ↑f a ^ 0",
"tactic": "rw [pow_zero, pow_zero, map_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.50695\nβ : Type ?u.50698\nM : Type ?u.50701\nN : Type ?u.50704\nP : Type ?u.50707\nG : Type u_1\nH : Type u_2\nF : Type u_3\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : Monoid G\ninst✝¹ : Monoid H\ninst✝ : MonoidHomClass F G H\nf : F\na : G\nn : ℕ\n⊢ ↑f (a ^ (n + 1)) = ↑f a ^ (n + 1)",
"tactic": "rw [pow_succ, pow_succ, map_mul, map_pow f a n]"
}
] |
[
451,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.mul_self_norm
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.6573042\ninst✝ : IsROrC K\nz : K\n⊢ ‖z‖ * ‖z‖ = ↑normSq z",
"tactic": "rw [normSq_eq_def', sq]"
}
] |
[
705,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
705,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean
|
CategoryTheory.Limits.IsZero.hasZeroObject
|
[] |
[
224,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.TransvectionStruct.reverse_inv_prod_mul_prod
|
[
{
"state_after": "case nil\nn : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse [])) ⬝ List.prod (List.map toMatrix []) = 1\n\ncase cons\nn : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) = 1\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse (t :: L))) ⬝\n List.prod (List.map toMatrix (t :: L)) =\n 1",
"state_before": "n : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nL : List (TransvectionStruct n R)\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) = 1",
"tactic": "induction' L with t L IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nn : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse [])) ⬝ List.prod (List.map toMatrix []) = 1",
"tactic": "simp"
},
{
"state_after": "case cons\nn : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) = 1\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝\n (toMatrix (TransvectionStruct.inv t) ⬝ toMatrix t) ⬝\n List.prod (List.map toMatrix L) =\n 1",
"state_before": "case cons\nn : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) = 1\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse (t :: L))) ⬝\n List.prod (List.map toMatrix (t :: L)) =\n 1",
"tactic": "suffices\n (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod ⬝ (t.inv.toMatrix ⬝ t.toMatrix) ⬝\n (L.map toMatrix).prod = 1\n by simpa [Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case cons\nn : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) = 1\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝\n (toMatrix (TransvectionStruct.inv t) ⬝ toMatrix t) ⬝\n List.prod (List.map toMatrix L) =\n 1",
"tactic": "simpa [inv_mul] using IH"
},
{
"state_after": "no goals",
"state_before": "n : Type u_1\np : Type ?u.35992\nR : Type u₂\n𝕜 : Type ?u.35997\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) = 1\nthis :\n List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝\n (toMatrix (TransvectionStruct.inv t) ⬝ toMatrix t) ⬝\n List.prod (List.map toMatrix L) =\n 1\n⊢ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse (t :: L))) ⬝\n List.prod (List.map toMatrix (t :: L)) =\n 1",
"tactic": "simpa [Matrix.mul_assoc]"
}
] |
[
227,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.absolutelyContinuous_of_le
|
[] |
[
2358,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2357,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift_principal2
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.22457\nγ : Type ?u.22460\nι : Sort ?u.22463\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Set α → Filter β\nf : Filter α\ns : Set α\nhs : s ∈ f\n⊢ f ≤ 𝓟 s",
"tactic": "simp only [hs, le_principal_iff]"
}
] |
[
195,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_withDensity_iff_integrable_coe_smul
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ Integrable g ↔ Integrable fun x => ↑(f x) • g x\n\ncase neg\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : ¬AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ Integrable g ↔ Integrable fun x => ↑(f x) • g x",
"state_before": "α : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\n⊢ Integrable g ↔ Integrable fun x => ↑(f x) • g x",
"tactic": "by_cases H : AEStronglyMeasurable (fun x : α => (f x : ℝ) • g x) μ"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (∫⁻ (a : α), ↑‖g a‖₊ ∂Measure.withDensity μ fun x => ↑(f x)) < ⊤ ↔ (∫⁻ (a : α), ↑‖↑(f a) • g a‖₊ ∂μ) < ⊤",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ Integrable g ↔ Integrable fun x => ↑(f x) • g x",
"tactic": "simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, HasFiniteIntegral, H,\n true_and_iff]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (∫⁻ (a : α), ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) a ∂μ) < ⊤ ↔ (∫⁻ (a : α), ↑‖↑(f a) • g a‖₊ ∂μ) < ⊤\n\ncase pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ AEMeasurable fun a => ↑‖g a‖₊",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (∫⁻ (a : α), ↑‖g a‖₊ ∂Measure.withDensity μ fun x => ↑(f x)) < ⊤ ↔ (∫⁻ (a : α), ↑‖↑(f a) • g a‖₊ ∂μ) < ⊤",
"tactic": "rw [lintegral_withDensity_eq_lintegral_mul₀' hf.coe_nnreal_ennreal.aemeasurable]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ ((∫⁻ (a : α), ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) a ∂μ) < ⊤) = ((∫⁻ (a : α), ↑‖↑(f a) • g a‖₊ ∂μ) < ⊤)",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (∫⁻ (a : α), ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) a ∂μ) < ⊤ ↔ (∫⁻ (a : α), ↑‖↑(f a) • g a‖₊ ∂μ) < ⊤",
"tactic": "rw [iff_iff_eq]"
},
{
"state_after": "case pos.e_a.e_f\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (fun a => ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) a) = fun a => ↑‖↑(f a) • g a‖₊",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ ((∫⁻ (a : α), ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) a ∂μ) < ⊤) = ((∫⁻ (a : α), ↑‖↑(f a) • g a‖₊ ∂μ) < ⊤)",
"tactic": "congr"
},
{
"state_after": "case pos.e_a.e_f.h\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\nx : α\n⊢ ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) x = ↑‖↑(f x) • g x‖₊",
"state_before": "case pos.e_a.e_f\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (fun a => ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) a) = fun a => ↑‖↑(f a) • g a‖₊",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case pos.e_a.e_f.h\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\nx : α\n⊢ ((fun x => ↑(f x)) * fun a => ↑‖g a‖₊) x = ↑‖↑(f x) • g x‖₊",
"tactic": "simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul, Pi.mul_apply]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ AEMeasurable fun x => ↑(f x) * ↑‖g x‖₊",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ AEMeasurable fun a => ↑‖g a‖₊",
"tactic": "rw [aemeasurable_withDensity_ennreal_iff hf]"
},
{
"state_after": "case h.e'_5\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (fun x => ↑(f x) * ↑‖g x‖₊) = fun a => ↑‖↑(f a) • g a‖₊",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ AEMeasurable fun x => ↑(f x) * ↑‖g x‖₊",
"tactic": "convert H.ennnorm using 1"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\nx : α\n⊢ ↑(f x) * ↑‖g x‖₊ = ↑‖↑(f x) • g x‖₊",
"state_before": "case h.e'_5\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ (fun x => ↑(f x) * ↑‖g x‖₊) = fun a => ↑‖↑(f a) • g a‖₊",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\nx : α\n⊢ ↑(f x) * ↑‖g x‖₊ = ↑‖↑(f x) • g x‖₊",
"tactic": "simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.1006363\nγ : Type ?u.1006366\nδ : Type ?u.1006369\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : NormedAddCommGroup β\ninst✝² : NormedAddCommGroup γ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : ¬AEStronglyMeasurable (fun x => ↑(f x) • g x) μ\n⊢ Integrable g ↔ Integrable fun x => ↑(f x) • g x",
"tactic": "simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and_iff]"
}
] |
[
864,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
849,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
Bornology.isVonNBounded_covers
|
[] |
[
212,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.succ_eq_one_add
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ succ n = 1 + n",
"tactic": "rw [Nat.succ_eq_add_one, Nat.add_comm]"
}
] |
[
311,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
310,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.gc_homogeneousIdeal
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nI : HomogeneousIdeal 𝒜\nt : (Set (ProjectiveSpectrum 𝒜))ᵒᵈ\n⊢ (fun I => zeroLocus 𝒜 ↑I) I ≤ t ↔ I ≤ (fun t => vanishingIdeal t) t",
"tactic": "simpa [show I.toIdeal ≤ (vanishingIdeal t).toIdeal ↔ I ≤ vanishingIdeal t from Iff.rfl] using\n subset_zeroLocus_iff_le_vanishingIdeal t I.toIdeal"
}
] |
[
153,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
|
PiToModule.fromMatrix_apply
|
[] |
[
45,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.const_mul
|
[] |
[
99,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.abs_exp
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ ↑abs (exp z) = Real.exp z.re",
"tactic": "rw [exp_eq_exp_re_mul_sin_add_cos, map_mul, abs_exp_ofReal, abs_cos_add_sin_mul_I, mul_one]"
}
] |
[
2042,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2041,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.countable_iff_exists_surjective
|
[] |
[
109,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
11
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.cyclotomic'_splits
|
[
{
"state_after": "K : Type u_1\ninst✝ : Field K\nn : ℕ\n⊢ ∀ (j : K), j ∈ primitiveRoots n K → Splits (RingHom.id K) (X - ↑C j)",
"state_before": "K : Type u_1\ninst✝ : Field K\nn : ℕ\n⊢ Splits (RingHom.id K) (cyclotomic' n K)",
"tactic": "apply splits_prod (RingHom.id K)"
},
{
"state_after": "K : Type u_1\ninst✝ : Field K\nn : ℕ\nz : K\na✝ : z ∈ primitiveRoots n K\n⊢ Splits (RingHom.id K) (X - ↑C z)",
"state_before": "K : Type u_1\ninst✝ : Field K\nn : ℕ\n⊢ ∀ (j : K), j ∈ primitiveRoots n K → Splits (RingHom.id K) (X - ↑C j)",
"tactic": "intro z _"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝ : Field K\nn : ℕ\nz : K\na✝ : z ∈ primitiveRoots n K\n⊢ Splits (RingHom.id K) (X - ↑C z)",
"tactic": "simp only [splits_X_sub_C (RingHom.id K)]"
}
] |
[
159,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.one_sub_le_exp_minus_of_nonneg
|
[
{
"state_after": "case inl\nh : 0 ≤ 0\n⊢ 1 - 0 ≤ exp (-0)\n\ncase inr\ny : ℝ\nh✝ : 0 ≤ y\nh : 0 < y\n⊢ 1 - y ≤ exp (-y)",
"state_before": "y : ℝ\nh : 0 ≤ y\n⊢ 1 - y ≤ exp (-y)",
"tactic": "rcases eq_or_lt_of_le h with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nh : 0 ≤ 0\n⊢ 1 - 0 ≤ exp (-0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\ny : ℝ\nh✝ : 0 ≤ y\nh : 0 < y\n⊢ 1 - y ≤ exp (-y)",
"tactic": "exact (one_sub_lt_exp_minus_of_pos h).le"
}
] |
[
1973,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1970,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
Left.one_lt_inv_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : α\n⊢ 1 < a⁻¹ ↔ a < 1",
"tactic": "rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]"
}
] |
[
162,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_self
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ choose n n = 1",
"tactic": "induction n <;> simp [*, choose, choose_eq_zero_of_lt (lt_succ_self _)]"
}
] |
[
82,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Data/List/Permutation.lean
|
List.length_permutationsAux2
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ length (permutationsAux2 t ts [] ys f).snd = length ys",
"tactic": "induction ys generalizing f <;> simp [*]"
}
] |
[
172,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.zero_mul
|
[
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\nb : R\n⊢ 0 * b = 0",
"tactic": "simp"
}
] |
[
429,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_cons
|
[] |
[
314,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.coe_ofEq_apply
|
[] |
[
2032,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2031,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.filter_ne_fill
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ filter ((fun x x_1 => x ≠ x_1) a) (↑m.snd + ↑(replicate (↑m.fst) a)) = ↑m.snd",
"state_before": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ ↑(filterNe a (fill a m.fst m.snd)).snd = ↑m.snd",
"tactic": "rw [filterNe, ← val_eq_coe, Subtype.coe_mk, val_eq_coe, coe_fill]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ ∀ (a_1 : α), ¬a_1 ∈ filter ((fun x x_1 => x ≠ x_1) a) ↑(replicate (↑m.fst) a)\n\nα : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ ∀ (a_1 : α), a_1 ∈ ↑m.snd → (fun x x_1 => x ≠ x_1) a a_1",
"state_before": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ filter ((fun x x_1 => x ≠ x_1) a) (↑m.snd + ↑(replicate (↑m.fst) a)) = ↑m.snd",
"tactic": "rw [filter_add, filter_eq_self.2, add_right_eq_self, eq_zero_iff_forall_not_mem]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b✝ : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\nb : α\nhb : b ∈ filter ((fun x x_1 => x ≠ x_1) a) ↑(replicate (↑m.fst) a)\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ ∀ (a_1 : α), ¬a_1 ∈ filter ((fun x x_1 => x ≠ x_1) a) ↑(replicate (↑m.fst) a)",
"tactic": "intro b hb"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b✝ : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\nb : α\nhb : (↑m.fst ≠ 0 ∧ b = a) ∧ (fun x x_1 => x ≠ x_1) a b\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b✝ : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\nb : α\nhb : b ∈ filter ((fun x x_1 => x ≠ x_1) a) ↑(replicate (↑m.fst) a)\n⊢ False",
"tactic": "rw [mem_filter, Sym.mem_coe, mem_replicate] at hb"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b✝ : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\nb : α\nhb : (↑m.fst ≠ 0 ∧ b = a) ∧ (fun x x_1 => x ≠ x_1) a b\n⊢ False",
"tactic": "exact hb.2 hb.1.2.symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.55875\nn n' m✝ : ℕ\ns : Sym α n\na✝ b : α\ninst✝ : DecidableEq α\na : α\nm : (i : Fin (n + 1)) × Sym α (n - ↑i)\nh : ¬a ∈ m.snd\n⊢ ∀ (a_1 : α), a_1 ∈ ↑m.snd → (fun x x_1 => x ≠ x_1) a a_1",
"tactic": "exact fun a ha ha' => h <| ha'.symm ▸ ha"
}
] |
[
586,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Topology/Algebra/GroupWithZero.lean
|
Homeomorph.coe_mulLeft₀
|
[] |
[
259,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.single_lt_prod'
|
[] |
[
503,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.isOpen_iff_mem_balls
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.210790\n𝕝 : Type ?u.210793\n𝕝₂ : Type ?u.210796\nE : Type u_2\nF : Type ?u.210802\nG : Type ?u.210805\nι : Type u_3\nι' : Type ?u.210811\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nU : Set E\n⊢ IsOpen U ↔ ∀ (x : E), x ∈ U → ∃ s r, r > 0 ∧ ball (Finset.sup s p) x r ⊆ U",
"tactic": "simp_rw [← WithSeminorms.mem_nhds_iff hp _ U, isOpen_iff_mem_nhds]"
}
] |
[
333,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.mem_map_of_mem
|
[] |
[
1326,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1325,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage_emb
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1687696\nδ : Type ?u.1687699\nm : MeasurableSpace α\nμ ν✝ : Measure α\nmb : MeasurableSpace β\nν : Measure β\ng : α → β\nhg : MeasurePreserving g\nhge : MeasurableEmbedding g\nf : β → ℝ≥0∞\ns : Set β\n⊢ (∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ) = ∫⁻ (b : β) in s, f b ∂ν",
"tactic": "rw [← hg.map_eq, hge.restrict_map, hge.lintegral_map]"
}
] |
[
1351,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1348,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.smulRight
|
[] |
[
909,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
907,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
nndist_vadd_left
|
[] |
[
124,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
|
CategoryTheory.Limits.IsInitial.isSplitEpi_to
|
[] |
[
179,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ContinuousOrderHom.coe_toOrderHom
|
[] |
[
112,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
9
] |
Mathlib/Algebra/CharZero/Lemmas.lean
|
RingHom.injective_nat
|
[] |
[
202,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
InnerProductSpace.Core.inner_conj_symm
|
[] |
[
209,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.zigzag_isConnected
|
[
{
"state_after": "case h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ ∀ (p : Set J), ?j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\n\ncase j₀\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ J",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ IsConnected J",
"tactic": "apply IsConnected.of_induct"
},
{
"state_after": "case h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj : J\n⊢ j ∈ p\n\ncase j₀\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ J",
"state_before": "case h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ ∀ (p : Set J), ?j₀ ∈ p → (∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)) → ∀ (j : J), j ∈ p\n\ncase j₀\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ J",
"tactic": "intro p hp hjp j"
},
{
"state_after": "no goals",
"state_before": "case h\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj : J\nthis : ∀ (j₁ j₂ : J), Zigzag j₁ j₂ → (j₁ ∈ p ↔ j₂ ∈ p)\n⊢ j ∈ p\n\ncase j₀\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ J",
"tactic": "rwa [this j (Classical.arbitrary J) (h _ _)]"
},
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ : J\nk : Zigzag j₁ j₂\n⊢ j₁ ∈ p ↔ j₂ ∈ p",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj : J\n⊢ ∀ (j₁ j₂ : J), Zigzag j₁ j₂ → (j₁ ∈ p ↔ j₂ ∈ p)",
"tactic": "introv k"
},
{
"state_after": "case refl\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ : J\n⊢ j₁ ∈ p ↔ j₁ ∈ p\n\ncase tail\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nzag : Zag b✝ c✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\n⊢ j₁ ∈ p ↔ c✝ ∈ p",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ : J\nk : Zigzag j₁ j₂\n⊢ j₁ ∈ p ↔ j₂ ∈ p",
"tactic": "induction' k with _ _ _ zag k_ih"
},
{
"state_after": "no goals",
"state_before": "case refl\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ : J\n⊢ j₁ ∈ p ↔ j₁ ∈ p",
"tactic": "rfl"
},
{
"state_after": "case tail\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nzag : Zag b✝ c✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\n⊢ b✝ ∈ p ↔ c✝ ∈ p",
"state_before": "case tail\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nzag : Zag b✝ c✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\n⊢ j₁ ∈ p ↔ c✝ ∈ p",
"tactic": "rw [k_ih]"
},
{
"state_after": "case tail.inl\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\nzag : Nonempty (b✝ ⟶ c✝)\n⊢ b✝ ∈ p ↔ c✝ ∈ p\n\ncase tail.inr\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\nzag : Nonempty (c✝ ⟶ b✝)\n⊢ b✝ ∈ p ↔ c✝ ∈ p",
"state_before": "case tail\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nzag : Zag b✝ c✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\n⊢ b✝ ∈ p ↔ c✝ ∈ p",
"tactic": "cases' zag with zag zag"
},
{
"state_after": "case tail.inr\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\nzag : Nonempty (c✝ ⟶ b✝)\n⊢ b✝ ∈ p ↔ c✝ ∈ p",
"state_before": "case tail.inl\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\nzag : Nonempty (b✝ ⟶ c✝)\n⊢ b✝ ∈ p ↔ c✝ ∈ p\n\ncase tail.inr\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\nzag : Nonempty (c✝ ⟶ b✝)\n⊢ b✝ ∈ p ↔ c✝ ∈ p",
"tactic": "apply hjp (Nonempty.some zag)"
},
{
"state_after": "no goals",
"state_before": "case tail.inr\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : Nonempty J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\np : Set J\nhp : ?j₀ ∈ p\nhjp : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)\nj j₁ j₂ b✝ c✝ : J\na✝ : Relation.ReflTransGen Zag j₁ b✝\nk_ih : j₁ ∈ p ↔ b✝ ∈ p\nzag : Nonempty (c✝ ⟶ b✝)\n⊢ b✝ ∈ p ↔ c✝ ∈ p",
"tactic": "exact (hjp (Nonempty.some zag)).symm"
}
] |
[
328,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.cof_univ
|
[
{
"state_after": "α : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh : c < Cardinal.univ\n⊢ c < cof univ",
"state_before": "α : Type ?u.103379\nr : α → α → Prop\n⊢ Cardinal.univ ≤ cof univ",
"tactic": "refine' le_of_forall_lt fun c h => _"
},
{
"state_after": "case intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh : Cardinal.lift c < Cardinal.univ\n⊢ Cardinal.lift c < cof univ",
"state_before": "α : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh : c < Cardinal.univ\n⊢ c < cof univ",
"tactic": "rcases lt_univ'.1 h with ⟨c, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\n⊢ Cardinal.lift c < (#↑S)",
"state_before": "case intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\n⊢ Cardinal.lift c < cof univ",
"tactic": "rw [univ, ← lift_cof, ← Cardinal.lift_lift.{u+1, v, u}, Cardinal.lift_lt, ← Se]"
},
{
"state_after": "case intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\n⊢ False",
"state_before": "case intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\n⊢ Cardinal.lift c < (#↑S)",
"tactic": "refine' lt_of_not_ge fun h => _"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne : Cardinal.lift a = (#↑S)\n⊢ False",
"state_before": "case intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\n⊢ False",
"tactic": "cases' Cardinal.lift_down h with a e"
},
{
"state_after": "case intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\n⊢ False",
"state_before": "case intro.intro.intro.intro\nα : Type ?u.103379\nr : α → α → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne : Cardinal.lift a = (#↑S)\n⊢ False",
"tactic": "refine' Quotient.inductionOn a (fun α e => _) e"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf : ULift α ≃ ↑S\n⊢ False",
"state_before": "case intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\n⊢ False",
"tactic": "cases' Quotient.exact e with f"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf : ULift α ≃ ↑S\n⊢ False",
"tactic": "have f := Equiv.ulift.symm.trans f"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\n⊢ False",
"tactic": "let g a := (f a).1"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\n⊢ False",
"tactic": "let o := succ (sup.{u, u} g)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\n⊢ False",
"tactic": "rcases H o with ⟨b, h, l⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ b ≤ sup g",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ False",
"tactic": "refine' l (lt_succ_iff.2 _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ g (↑f.symm { val := b, property := h }) ≤ sup g",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ b ≤ sup g",
"tactic": "rw [← show g (f.symm ⟨b, h⟩) = b by simp]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ g (↑f.symm { val := b, property := h }) ≤ sup g",
"tactic": "apply le_sup"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.103379\nr : α✝ → α✝ → Prop\nc : Cardinal\nh✝¹ : Cardinal.lift c < Cardinal.univ\nS : Set Ordinal\nH : Unbounded (fun x x_1 => x < x_1) S\nSe : (#↑S) = cof (type fun x x_1 => x < x_1)\nh✝ : Cardinal.lift c ≥ (#↑S)\na : Cardinal\ne✝ : Cardinal.lift a = (#↑S)\nα : Type u\ne : Cardinal.lift (Quotient.mk Cardinal.isEquivalent α) = (#↑S)\nf✝ : ULift α ≃ ↑S\nf : α ≃ ↑S\ng : α → Ordinal := fun a => ↑(↑f a)\no : Ordinal := succ (sup g)\nb : Ordinal\nh : b ∈ S\nl : ¬(fun x x_1 => x < x_1) b o\n⊢ g (↑f.symm { val := b, property := h }) = b",
"tactic": "simp"
}
] |
[
775,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
758,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoLocallyUniformly.tendstoLocallyUniformlyOn
|
[] |
[
686,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
11
] |
Mathlib/RingTheory/Adjoin/Basic.lean
|
Algebra.adjoin_prod_le
|
[] |
[
235,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
mul_inv_lt_inv_mul_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝³ : Group α\ninst✝² : LT α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : α\n⊢ a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b",
"tactic": "rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,\n inv_mul_cancel_right]"
}
] |
[
418,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.QuasiMeasurePreserving.prod_of_left
|
[
{
"state_after": "α✝ : Type ?u.4535638\nα' : Type ?u.4535641\nβ✝ : Type ?u.4535644\nβ' : Type ?u.4535647\nγ✝ : Type ?u.4535650\nE : Type ?u.4535653\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : MeasurableSpace α'\ninst✝⁸ : MeasurableSpace β✝\ninst✝⁷ : MeasurableSpace β'\ninst✝⁶ : MeasurableSpace γ✝\nμ✝ μ' : Measure α✝\nν✝ ν' : Measure β✝\nτ✝ : Measure γ✝\ninst✝⁵ : NormedAddCommGroup E\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving fun x => f (x, y)\n⊢ QuasiMeasurePreserving f",
"state_before": "α✝ : Type ?u.4535638\nα' : Type ?u.4535641\nβ✝ : Type ?u.4535644\nβ' : Type ?u.4535647\nγ✝ : Type ?u.4535650\nE : Type ?u.4535653\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : MeasurableSpace α'\ninst✝⁸ : MeasurableSpace β✝\ninst✝⁷ : MeasurableSpace β'\ninst✝⁶ : MeasurableSpace γ✝\nμ✝ μ' : Measure α✝\nν✝ ν' : Measure β✝\nτ✝ : Measure γ✝\ninst✝⁵ : NormedAddCommGroup E\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving fun x => f (x, y)\n⊢ QuasiMeasurePreserving f",
"tactic": "rw [← prod_swap]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.4535638\nα' : Type ?u.4535641\nβ✝ : Type ?u.4535644\nβ' : Type ?u.4535647\nγ✝ : Type ?u.4535650\nE : Type ?u.4535653\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : MeasurableSpace α'\ninst✝⁸ : MeasurableSpace β✝\ninst✝⁷ : MeasurableSpace β'\ninst✝⁶ : MeasurableSpace γ✝\nμ✝ μ' : Measure α✝\nν✝ ν' : Measure β✝\nτ✝ : Measure γ✝\ninst✝⁵ : NormedAddCommGroup E\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving fun x => f (x, y)\n⊢ QuasiMeasurePreserving f",
"tactic": "convert (QuasiMeasurePreserving.prod_of_right (hf.comp measurable_swap) h2f).comp\n ((measurable_swap.measurePreserving (ν.prod μ)).symm\n MeasurableEquiv.prodComm).quasiMeasurePreserving"
}
] |
[
682,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.abs_exp_sub_one_le
|
[
{
"state_after": "no goals",
"state_before": "x : ℂ\nhx : ↑abs x ≤ 1\n⊢ ↑abs (exp x - 1) = ↑abs (exp x - ∑ m in range 1, x ^ m / ↑(Nat.factorial m))",
"tactic": "simp [sum_range_succ]"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\nhx : ↑abs x ≤ 1\n⊢ 0 < 1",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\nhx : ↑abs x ≤ 1\n⊢ ↑abs x ^ 1 * (↑(Nat.succ 1) * (↑(Nat.factorial 1) * ↑1)⁻¹) = 2 * ↑abs x",
"tactic": "simp [two_mul, mul_two, mul_add, mul_comm, add_mul]"
}
] |
[
1684,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1679,
1
] |
Std/Logic.lean
|
iff_self_and
|
[
{
"state_after": "no goals",
"state_before": "p q : Prop\n⊢ (p ↔ p ∧ q) ↔ p → q",
"tactic": "rw [@Iff.comm p, and_iff_left_iff_imp]"
}
] |
[
215,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
214,
9
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_union_inter_subset
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.51060\nβ : Type u_2\nβ' : Type ?u.51066\nγ : Type ?u.51069\nγ' : Type ?u.51072\nδ : Type ?u.51075\nδ' : Type ?u.51078\nε : Type ?u.51081\nε' : Type ?u.51084\nζ : Type ?u.51087\nζ' : Type ?u.51090\nν : Type ?u.51093\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Set α\nt✝ t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → α → β\ns t : Set α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 (fun b a => f b a) (s ∩ t) (s ∪ t) ⊆ image2 f s t",
"state_before": "α : Type u_1\nα' : Type ?u.51060\nβ : Type u_2\nβ' : Type ?u.51066\nγ : Type ?u.51069\nγ' : Type ?u.51072\nδ : Type ?u.51075\nδ' : Type ?u.51078\nε : Type ?u.51081\nε' : Type ?u.51084\nζ : Type ?u.51087\nζ' : Type ?u.51090\nν : Type ?u.51093\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Set α\nt✝ t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → α → β\ns t : Set α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 f (s ∪ t) (s ∩ t) ⊆ image2 f s t",
"tactic": "rw [image2_comm hf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.51060\nβ : Type u_2\nβ' : Type ?u.51066\nγ : Type ?u.51069\nγ' : Type ?u.51072\nδ : Type ?u.51075\nδ' : Type ?u.51078\nε : Type ?u.51081\nε' : Type ?u.51084\nζ : Type ?u.51087\nζ' : Type ?u.51090\nν : Type ?u.51093\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Set α\nt✝ t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → α → β\ns t : Set α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 (fun b a => f b a) (s ∩ t) (s ∪ t) ⊆ image2 f s t",
"tactic": "exact image2_inter_union_subset hf"
}
] |
[
471,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_zero_set
|
[] |
[
1681,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1680,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_sum
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nβ₂ : Type ?u.168938\nγ : Type u_2\nι : Sort ?u.168944\nι' : Sort ?u.168947\nκ : ι → Sort ?u.168952\nκ' : ι' → Sort ?u.168957\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : β ⊕ γ → α\nc : α\n⊢ (⨆ (x : β ⊕ γ), f x) ≤ c ↔ ((⨆ (i : β), f (Sum.inl i)) ⊔ ⨆ (j : γ), f (Sum.inr j)) ≤ c",
"tactic": "simp only [sup_le_iff, iSup_le_iff, Sum.forall]"
}
] |
[
1568,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1567,
1
] |
Mathlib/Data/List/Basic.lean
|
List.nthLe_of_eq
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.87083\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L L' : List α\nh : L = L'\ni : ℕ\nhi : i < length L\n⊢ nthLe L i hi = nthLe L' i (_ : i < length L')",
"tactic": "congr"
}
] |
[
1316,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1315,
1
] |
Mathlib/GroupTheory/Transfer.lean
|
Subgroup.leftTransversals.diff_self
|
[] |
[
72,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
StrictConcaveOn.translate_right
|
[] |
[
947,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
945,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.CostructuredArrow.map_id
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nf : CostructuredArrow S T\n⊢ (map (𝟙 T)).obj (mk f.hom) = mk f.hom",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nf : CostructuredArrow S T\n⊢ (map (𝟙 T)).obj f = f",
"tactic": "rw [eq_mk f]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nT T' T'' : D\nY Y' : C\nS : C ⥤ D\nf : CostructuredArrow S T\n⊢ (map (𝟙 T)).obj (mk f.hom) = mk f.hom",
"tactic": "simp"
}
] |
[
435,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.mk.inj_iff
|
[] |
[
211,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
11
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.lt_replicate_succ
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) ↔ m ≤ replicate n x",
"state_before": "α : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ m < replicate (n + 1) x ↔ m ≤ replicate n x",
"tactic": "rw [lt_iff_cons_le]"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) → m ≤ replicate n x\n\ncase mpr\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ m ≤ replicate n x → ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"state_before": "α : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) ↔ m ≤ replicate n x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nx' : α\nhx' : x' ::ₘ m ≤ replicate (n + 1) x\n⊢ m ≤ replicate n x",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) → m ≤ replicate n x",
"tactic": "rintro ⟨x', hx'⟩"
},
{
"state_after": "case mp.intro\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nx' : α\nhx' : x' ::ₘ m ≤ replicate (n + 1) x\nthis : x' = x\n⊢ m ≤ replicate n x",
"state_before": "case mp.intro\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nx' : α\nhx' : x' ::ₘ m ≤ replicate (n + 1) x\n⊢ m ≤ replicate n x",
"tactic": "have := eq_of_mem_replicate (mem_of_le hx' (mem_cons_self _ _))"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nx' : α\nhx' : x' ::ₘ m ≤ replicate (n + 1) x\nthis : x' = x\n⊢ m ≤ replicate n x",
"tactic": "rwa [this, replicate_succ, cons_le_cons_iff] at hx'"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nh : m ≤ replicate n x\n⊢ ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\n⊢ m ≤ replicate n x → ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"tactic": "intro h"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nh : m ≤ replicate n x\n⊢ ∃ a, a ::ₘ m ≤ x ::ₘ replicate n x",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nh : m ≤ replicate n x\n⊢ ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"tactic": "rw [replicate_succ]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.93086\nγ : Type ?u.93089\nm : Multiset α\nx : α\nn : ℕ\nh : m ≤ replicate n x\n⊢ ∃ a, a ::ₘ m ≤ x ::ₘ replicate n x",
"tactic": "exact ⟨x, cons_le_cons _ h⟩"
}
] |
[
988,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
979,
1
] |
Mathlib/Algebra/Order/Positive/Ring.lean
|
Positive.val_pow
|
[] |
[
116,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.deriv_log
|
[
{
"state_after": "no goals",
"state_before": "x✝ x : ℝ\nhx : x = 0\n⊢ deriv log x = x⁻¹",
"tactic": "rw [deriv_zero_of_not_differentiableAt (differentiableAt_log_iff.not_left.2 hx), hx, inv_zero]"
}
] |
[
73,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean
|
RightDerivMeasurableAux.differentiable_set_subset_d
|
[
{
"state_after": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\n⊢ x ∈ D f K",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\n⊢ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K} ⊆ D f K",
"tactic": "intro x hx"
},
{
"state_after": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\n⊢ ∀ (i : ℕ), x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\n⊢ x ∈ D f K",
"tactic": "rw [D, mem_iInter]"
},
{
"state_after": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\n⊢ ∀ (i : ℕ), x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)",
"tactic": "intro e"
},
{
"state_after": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "have : (0 : ℝ) < (1 / 2) ^ e := pow_pos (by norm_num) _"
},
{
"state_after": "case intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "rcases mem_a_of_differentiable this hx.1 with ⟨R, R_pos, hR⟩"
},
{
"state_after": "case intro.intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "case intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "obtain ⟨n, hn⟩ : ∃ n : ℕ, (1 / 2) ^ n < R :=\n exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ) / 2 < 1)"
},
{
"state_after": "case intro.intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\n⊢ ∃ i,\n ∀ (i_1 : ℕ),\n i_1 ≥ i →\n ∀ (i_3 : ℕ), i_3 ≥ i → ∃ i h, x ∈ A f i ((1 / 2) ^ i_1) ((1 / 2) ^ e) ∧ x ∈ A f i ((1 / 2) ^ i_3) ((1 / 2) ^ e)",
"state_before": "case intro.intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "simp only [mem_iUnion, mem_iInter, B, mem_inter_iff]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\n⊢ 0 < 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\n⊢ 1 / 2 < 1",
"tactic": "norm_num"
},
{
"state_after": "case intro.intro.intro.refine'_2\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ (1 / 2) ^ q ≤ (1 / 2) ^ n",
"state_before": "case intro.intro.intro.refine'_2\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ x ∈ A f (derivWithin f (Ici x) x) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "refine' hR _ ⟨pow_pos (by norm_num) _, lt_of_le_of_lt _ hn⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_2\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ (1 / 2) ^ q ≤ (1 / 2) ^ n",
"tactic": "exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (by assumption)"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ 0 < 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ 0 ≤ 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ 1 / 2 ≤ 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ n ≤ q",
"tactic": "assumption"
}
] |
[
588,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
576,
1
] |
Mathlib/Data/Multiset/Sum.lean
|
Multiset.zero_disjSum
|
[] |
[
38,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.mul_meas_ge_le_pow_snorm
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\nthis : 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n⊢ ε * ↑↑μ {x | ε ≤ ↑‖f x‖₊ ^ ENNReal.toReal p} ≤ snorm f p μ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\n⊢ ε * ↑↑μ {x | ε ≤ ↑‖f x‖₊ ^ ENNReal.toReal p} ≤ snorm f p μ ^ ENNReal.toReal p",
"tactic": "have : 1 / p.toReal * p.toReal = 1 := by\n refine' one_div_mul_cancel _\n rw [Ne, ENNReal.toReal_eq_zero_iff]\n exact not_or_of_not hp_ne_zero hp_ne_top"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\nthis : 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n⊢ ((ε * ↑↑μ {x | ε ≤ ↑‖f x‖₊ ^ ENNReal.toReal p}) ^ (1 / ENNReal.toReal p)) ^ ENNReal.toReal p ≤\n snorm f p μ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\nthis : 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n⊢ ε * ↑↑μ {x | ε ≤ ↑‖f x‖₊ ^ ENNReal.toReal p} ≤ snorm f p μ ^ ENNReal.toReal p",
"tactic": "rw [← ENNReal.rpow_one (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }), ← this, ENNReal.rpow_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\nthis : 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n⊢ ((ε * ↑↑μ {x | ε ≤ ↑‖f x‖₊ ^ ENNReal.toReal p}) ^ (1 / ENNReal.toReal p)) ^ ENNReal.toReal p ≤\n snorm f p μ ^ ENNReal.toReal p",
"tactic": "exact\n ENNReal.rpow_le_rpow (pow_mul_meas_ge_le_snorm μ hp_ne_zero hp_ne_top hf ε)\n ENNReal.toReal_nonneg"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\n⊢ ENNReal.toReal p ≠ 0",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\n⊢ 1 / ENNReal.toReal p * ENNReal.toReal p = 1",
"tactic": "refine' one_div_mul_cancel _"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\n⊢ ¬(p = 0 ∨ p = ⊤)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\n⊢ ENNReal.toReal p ≠ 0",
"tactic": "rw [Ne, ENNReal.toReal_eq_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.4415601\nG : Type ?u.4415604\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : AEStronglyMeasurable f μ\nε : ℝ≥0∞\n⊢ ¬(p = 0 ∨ p = ⊤)",
"tactic": "exact not_or_of_not hp_ne_zero hp_ne_top"
}
] |
[
1150,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1140,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.liminf_const
|
[] |
[
606,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_nat_mul_two_pi_add_pi
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ cos (↑n * (2 * π) + π) = -1",
"tactic": "simpa only [cos_zero] using (cos_periodic.nat_mul n).add_antiperiod_eq cos_antiperiodic"
}
] |
[
386,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Order/Disjoint.lean
|
codisjoint_inf_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderTop α\na b c : α\n⊢ Codisjoint (a ⊓ b) c ↔ Codisjoint a c ∧ Codisjoint b c",
"tactic": "simp only [codisjoint_iff, sup_inf_right, inf_eq_top_iff]"
}
] |
[
374,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.mem_iSup
|
[
{
"state_after": "R : Type u_2\nR₂ : Type ?u.237748\nK : Type ?u.237751\nM : Type u_3\nM₂ : Type ?u.237757\nV : Type ?u.237760\nS : Type ?u.237763\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\np : ι → Submodule R M\nm : M\n⊢ (∀ (b : Submodule R M), (∀ (i : ι), p i ≤ b) → span R {m} ≤ b) ↔ ∀ (N : Submodule R M), (∀ (i : ι), p i ≤ N) → m ∈ N",
"state_before": "R : Type u_2\nR₂ : Type ?u.237748\nK : Type ?u.237751\nM : Type u_3\nM₂ : Type ?u.237757\nV : Type ?u.237760\nS : Type ?u.237763\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\np : ι → Submodule R M\nm : M\n⊢ (m ∈ ⨆ (i : ι), p i) ↔ ∀ (N : Submodule R M), (∀ (i : ι), p i ≤ N) → m ∈ N",
"tactic": "rw [← span_singleton_le_iff_mem, le_iSup_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.237748\nK : Type ?u.237751\nM : Type u_3\nM₂ : Type ?u.237757\nV : Type ?u.237760\nS : Type ?u.237763\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nι : Sort u_1\np : ι → Submodule R M\nm : M\n⊢ (∀ (b : Submodule R M), (∀ (i : ι), p i ≤ b) → span R {m} ≤ b) ↔ ∀ (N : Submodule R M), (∀ (i : ι), p i ≤ N) → m ∈ N",
"tactic": "simp only [span_singleton_le_iff_mem]"
}
] |
[
731,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sInf_le_sInf
|
[] |
[
228,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.Ioc_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ Ioc a b = 0 ↔ ¬a < b",
"tactic": "rw [Ioc, Finset.val_eq_zero, Finset.Ioc_eq_empty_iff]"
}
] |
[
59,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
|
CategoryTheory.MonoidalOfChosenFiniteProducts.triangle
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y : C\n⊢ (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ BinaryFan.snd (ℬ 𝒯.cone.pt Y).cone)) =\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ BinaryFan.fst (ℬ X 𝒯.cone.pt).cone)\n (BinaryFan.snd (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ 𝟙 Y))",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y : C\n⊢ (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫\n tensorHom ℬ (𝟙 X) (BinaryFan.leftUnitor 𝒯.isLimit (ℬ 𝒯.cone.pt Y).isLimit).hom =\n tensorHom ℬ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X 𝒯.cone.pt).isLimit).hom (𝟙 Y)",
"tactic": "dsimp [tensorHom]"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y : C\n⊢ ∀ (j : Discrete WalkingPair),\n ((BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ BinaryFan.snd (ℬ 𝒯.cone.pt Y).cone))) ≫\n (ℬ X Y).cone.π.app j =\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ BinaryFan.fst (ℬ X 𝒯.cone.pt).cone)\n (BinaryFan.snd (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ 𝟙 Y)) ≫\n (ℬ X Y).cone.π.app j",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y : C\n⊢ (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ BinaryFan.snd (ℬ 𝒯.cone.pt Y).cone)) =\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ BinaryFan.fst (ℬ X 𝒯.cone.pt).cone)\n (BinaryFan.snd (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ 𝟙 Y))",
"tactic": "apply IsLimit.hom_ext (ℬ _ _).isLimit"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ : C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y : C\n⊢ ∀ (j : Discrete WalkingPair),\n ((BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ 𝟙 X)\n (BinaryFan.snd (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ BinaryFan.snd (ℬ 𝒯.cone.pt Y).cone))) ≫\n (ℬ X Y).cone.π.app j =\n IsLimit.lift (ℬ X Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ BinaryFan.fst (ℬ X 𝒯.cone.pt).cone)\n (BinaryFan.snd (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ 𝟙 Y)) ≫\n (ℬ X Y).cone.π.app j",
"tactic": "rintro ⟨⟨⟩⟩ <;> simp"
}
] |
[
305,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.lift_cof
|
[
{
"state_after": "α : Type ?u.17400\nr : α → α → Prop\no : Ordinal\n⊢ ∀ (α : Type v) (r : α → α → Prop) [inst : IsWellOrder α r], Cardinal.lift (cof (type r)) = cof (lift (type r))",
"state_before": "α : Type ?u.17400\nr : α → α → Prop\no : Ordinal\n⊢ Cardinal.lift (cof o) = cof (lift o)",
"tactic": "refine' inductionOn o _"
},
{
"state_after": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ Cardinal.lift (cof (type r)) = cof (lift (type r))",
"state_before": "α : Type ?u.17400\nr : α → α → Prop\no : Ordinal\n⊢ ∀ (α : Type v) (r : α → α → Prop) [inst : IsWellOrder α r], Cardinal.lift (cof (type r)) = cof (lift (type r))",
"tactic": "intro α r _"
},
{
"state_after": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ Cardinal.lift (cof (type r)) ≤ cof (lift (type r))\n\ncase a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"state_before": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ Cardinal.lift (cof (type r)) = cof (lift (type r))",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\n⊢ Cardinal.lift (cof (type r)) ≤ (#↑S)",
"state_before": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ Cardinal.lift (cof (type r)) ≤ cof (lift (type r))",
"tactic": "refine' le_cof_type.2 fun S H => _"
},
{
"state_after": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\nthis : Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ (#↑S)\n⊢ Cardinal.lift (cof (type r)) ≤ (#↑S)",
"state_before": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\n⊢ Cardinal.lift (cof (type r)) ≤ (#↑S)",
"tactic": "have : Cardinal.lift.{u, v} (#(ULift.up ⁻¹' S)) ≤ (#(S : Type (max u v))) := by\n rw [← Cardinal.lift_umax.{v, u}, ← Cardinal.lift_id'.{v, u} (#S)]\n refine mk_preimage_of_injective_lift.{v, max u v} ULift.up S (ULift.up_injective.{u, v})"
},
{
"state_after": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\nthis : Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ (#↑S)\n⊢ Unbounded r (ULift.up ⁻¹' S)",
"state_before": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\nthis : Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ (#↑S)\n⊢ Cardinal.lift (cof (type r)) ≤ (#↑S)",
"tactic": "refine' (Cardinal.lift_le.2 <| cof_type_le _).trans this"
},
{
"state_after": "no goals",
"state_before": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\nthis : Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ (#↑S)\n⊢ Unbounded r (ULift.up ⁻¹' S)",
"tactic": "exact fun a =>\n let ⟨⟨b⟩, bs, br⟩ := H ⟨a⟩\n ⟨b, bs, br⟩"
},
{
"state_after": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\n⊢ Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ Cardinal.lift (#↑S)",
"state_before": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\n⊢ Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ (#↑S)",
"tactic": "rw [← Cardinal.lift_umax.{v, u}, ← Cardinal.lift_id'.{v, u} (#S)]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set (ULift { α := α, r := r, wo := inst✝ }.α)\nH : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S\n⊢ Cardinal.lift (#↑(ULift.up ⁻¹' S)) ≤ Cardinal.lift (#↑S)",
"tactic": "refine mk_preimage_of_injective_lift.{v, max u v} ULift.up S (ULift.up_injective.{u, v})"
},
{
"state_after": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"state_before": "case a\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"tactic": "rcases cof_eq r with ⟨S, H, e'⟩"
},
{
"state_after": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nthis : (#↑(ULift.down ⁻¹' S)) ≤ Cardinal.lift (#↑S)\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"state_before": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"tactic": "have : (#ULift.down.{u, v} ⁻¹' S) ≤ Cardinal.lift.{u, v} (#S) :=\n ⟨⟨fun ⟨⟨x⟩, h⟩ => ⟨⟨x, h⟩⟩, fun ⟨⟨x⟩, h₁⟩ ⟨⟨y⟩, h₂⟩ e => by\n simp at e; congr⟩⟩"
},
{
"state_after": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nthis : (#↑(ULift.down ⁻¹' S)) ≤ Cardinal.lift (cof (type r))\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"state_before": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nthis : (#↑(ULift.down ⁻¹' S)) ≤ Cardinal.lift (#↑S)\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"tactic": "rw [e'] at this"
},
{
"state_after": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nthis : (#↑(ULift.down ⁻¹' S)) ≤ Cardinal.lift (cof (type r))\n⊢ Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) (ULift.down ⁻¹' S)",
"state_before": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nthis : (#↑(ULift.down ⁻¹' S)) ≤ Cardinal.lift (cof (type r))\n⊢ cof (lift (type r)) ≤ Cardinal.lift (cof (type r))",
"tactic": "refine' (cof_type_le _).trans this"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nα✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nthis : (#↑(ULift.down ⁻¹' S)) ≤ Cardinal.lift (cof (type r))\n⊢ Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) (ULift.down ⁻¹' S)",
"tactic": "exact fun ⟨a⟩ =>\n let ⟨b, bs, br⟩ := H a\n ⟨⟨b⟩, bs, br⟩"
},
{
"state_after": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nx✝¹ x✝ : ↑(ULift.down ⁻¹' S)\nx : α\nh₁ : { down := x } ∈ ULift.down ⁻¹' S\ny : α\nh₂ : { down := y } ∈ ULift.down ⁻¹' S\ne : x = y\n⊢ { val := { down := x }, property := h₁ } = { val := { down := y }, property := h₂ }",
"state_before": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nx✝¹ x✝ : ↑(ULift.down ⁻¹' S)\nx : α\nh₁ : { down := x } ∈ ULift.down ⁻¹' S\ny : α\nh₂ : { down := y } ∈ ULift.down ⁻¹' S\ne :\n (fun x =>\n match x with\n | { val := { down := x }, property := h } => { down := { val := x, property := h } })\n { val := { down := x }, property := h₁ } =\n (fun x =>\n match x with\n | { val := { down := x }, property := h } => { down := { val := x, property := h } })\n { val := { down := y }, property := h₂ }\n⊢ { val := { down := x }, property := h₁ } = { val := { down := y }, property := h₂ }",
"tactic": "simp at e"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.17400\nr✝ : α✝ → α✝ → Prop\no : Ordinal\nα : Type v\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nS : Set α\nH : Unbounded r S\ne' : (#↑S) = cof (type r)\nx✝¹ x✝ : ↑(ULift.down ⁻¹' S)\nx : α\nh₁ : { down := x } ∈ ULift.down ⁻¹' S\ny : α\nh₂ : { down := y } ∈ ULift.down ⁻¹' S\ne : x = y\n⊢ { val := { down := x }, property := h₁ } = { val := { down := y }, property := h₂ }",
"tactic": "congr"
}
] |
[
285,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_of_pos
|
[
{
"state_after": "x y : ℝ\nhx : 0 < x\n⊢ ↑(OrderIso.symm expOrderIso) { val := abs x, property := (_ : 0 < abs x) } =\n ↑(OrderIso.symm expOrderIso) { val := x, property := hx }",
"state_before": "x y : ℝ\nhx : 0 < x\n⊢ log x = ↑(OrderIso.symm expOrderIso) { val := x, property := hx }",
"tactic": "rw [log_of_ne_zero hx.ne']"
},
{
"state_after": "case h.e_6.h.e_val\nx y : ℝ\nhx : 0 < x\n⊢ abs x = x",
"state_before": "x y : ℝ\nhx : 0 < x\n⊢ ↑(OrderIso.symm expOrderIso) { val := abs x, property := (_ : 0 < abs x) } =\n ↑(OrderIso.symm expOrderIso) { val := x, property := hx }",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e_6.h.e_val\nx y : ℝ\nhx : 0 < x\n⊢ abs x = x",
"tactic": "exact abs_of_pos hx"
}
] |
[
55,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzOnWith.uniformContinuousOn
|
[] |
[
507,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
11
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.mk_smul
|
[
{
"state_after": "case pos\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nhq : q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q\n\ncase neg\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nhq : ¬q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q",
"state_before": "K : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q",
"tactic": "by_cases hq : q = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nhq : q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q",
"tactic": "rw [hq, mk_zero, mk_zero, ← ofFractionRing_smul, smul_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nhq : ¬q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q",
"tactic": "rw [mk_eq_localization_mk _ hq, mk_eq_localization_mk _ hq, ← Localization.smul_mk, ←\n ofFractionRing_smul]"
}
] |
[
479,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
LinearIsometry.coe_toAffineIsometry
|
[] |
[
113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Logic/Basic.lean
|
ite_and
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort ?u.40023\nσ : α → Sort ?u.40019\nf : α → β\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b c : α\nA : P → α\nB : ¬P → α\n⊢ (if P ∧ Q then a else b) = if P then if Q then a else b else b",
"tactic": "by_cases hp : P <;> by_cases hq : Q <;> simp [hp, hq]"
}
] |
[
1250,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1249,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_mul_of_one_lt_right'
|
[] |
[
441,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
uniformContinuous_uniformly_extend
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis : Prod.map f f ⁻¹' s ∈ 𝓤 β\n⊢ Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)",
"tactic": "rwa [← h_e.comap_uniformity] at this"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"tactic": "have : interior t ∈ 𝓝 (x₁, x₂) := isOpen_interior.mem_nhds hx_t"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"tactic": "let ⟨m₁, hm₁, m₂, hm₂, (hm : m₁ ×ˢ m₂ ⊆ interior t)⟩ := mem_nhds_prod_iff.mp this"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\na : β\nha₁ : a ∈ e ⁻¹' m₁\nleft✝ : (f a, DenseInducing.extend (_ : DenseInducing e) f x₁) ∈ s\nha₂ : (DenseInducing.extend (_ : DenseInducing e) f x₁, f a) ∈ s\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"tactic": "obtain ⟨_, ⟨a, ha₁, rfl⟩, _, ha₂⟩ := h_pnt hm₁"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\na : β\nha₁ : a ∈ e ⁻¹' m₁\nleft✝ : (f a, DenseInducing.extend (_ : DenseInducing e) f x₁) ∈ s\nha₂ : (DenseInducing.extend (_ : DenseInducing e) f x₁, f a) ∈ s\nb : β\nhb₁ : b ∈ e ⁻¹' m₂\nhb₂ : (f b, DenseInducing.extend (_ : DenseInducing e) f x₂) ∈ s\nright✝ : (DenseInducing.extend (_ : DenseInducing e) f x₂, f b) ∈ s\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\na : β\nha₁ : a ∈ e ⁻¹' m₁\nleft✝ : (f a, DenseInducing.extend (_ : DenseInducing e) f x₁) ∈ s\nha₂ : (DenseInducing.extend (_ : DenseInducing e) f x₁, f a) ∈ s\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"tactic": "obtain ⟨_, ⟨b, hb₁, rfl⟩, hb₂, _⟩ := h_pnt hm₂"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝² : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝¹ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis✝ : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\na : β\nha₁ : a ∈ e ⁻¹' m₁\nleft✝ : (f a, DenseInducing.extend (_ : DenseInducing e) f x₁) ∈ s\nha₂ : (DenseInducing.extend (_ : DenseInducing e) f x₁, f a) ∈ s\nb : β\nhb₁ : b ∈ e ⁻¹' m₂\nhb₂ : (f b, DenseInducing.extend (_ : DenseInducing e) f x₂) ∈ s\nright✝ : (DenseInducing.extend (_ : DenseInducing e) f x₂, f b) ∈ s\nthis : Prod.map f f (a, b) ∈ s\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝¹ : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\na : β\nha₁ : a ∈ e ⁻¹' m₁\nleft✝ : (f a, DenseInducing.extend (_ : DenseInducing e) f x₁) ∈ s\nha₂ : (DenseInducing.extend (_ : DenseInducing e) f x₁, f a) ∈ s\nb : β\nhb₁ : b ∈ e ⁻¹' m₂\nhb₂ : (f b, DenseInducing.extend (_ : DenseInducing e) f x₂) ∈ s\nright✝ : (DenseInducing.extend (_ : DenseInducing e) f x₂, f b) ∈ s\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"tactic": "have : Prod.map f f (a, b) ∈ s :=\n ts <| mem_preimage.2 <| interior_subset (@hm (e a, e b) ⟨ha₁, hb₁⟩)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp : s ○ (s ○ s) ⊆ d\nh_pnt :\n ∀ {a : α} {m : Set α},\n m ∈ 𝓝 a →\n ∃ c,\n c ∈ f '' (e ⁻¹' m) ∧\n (c, DenseInducing.extend (_ : DenseInducing e) f a) ∈ s ∧\n (DenseInducing.extend (_ : DenseInducing e) f a, c) ∈ s\nthis✝² : Prod.map f f ⁻¹' s ∈ 𝓤 β\nthis✝¹ : Prod.map f f ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α)\nt : Set (α × α)\nht : t ∈ 𝓤 α\nts : Prod.map e e ⁻¹' t ⊆ Prod.map f f ⁻¹' s\nx✝ : α × α\nx₁ x₂ : α\nhx_t : (x₁, x₂) ∈ interior t\nthis✝ : interior t ∈ 𝓝 (x₁, x₂)\nm₁ : Set α\nhm₁ : m₁ ∈ 𝓝 x₁\nm₂ : Set α\nhm₂ : m₂ ∈ 𝓝 x₂\nhm : m₁ ×ˢ m₂ ⊆ interior t\na : β\nha₁ : a ∈ e ⁻¹' m₁\nleft✝ : (f a, DenseInducing.extend (_ : DenseInducing e) f x₁) ∈ s\nha₂ : (DenseInducing.extend (_ : DenseInducing e) f x₁, f a) ∈ s\nb : β\nhb₁ : b ∈ e ⁻¹' m₂\nhb₂ : (f b, DenseInducing.extend (_ : DenseInducing e) f x₂) ∈ s\nright✝ : (DenseInducing.extend (_ : DenseInducing e) f x₂, f b) ∈ s\nthis : Prod.map f f (a, b) ∈ s\n⊢ (x₁, x₂) ∈\n Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) ⁻¹' d",
"tactic": "exact hs_comp ⟨f a, ha₂, ⟨f b, this, hb₂⟩⟩"
}
] |
[
490,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasurableEquiv.map_map_symm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2284080\nδ : Type ?u.2284083\nι : Type ?u.2284086\nR : Type ?u.2284089\nR' : Type ?u.2284092\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\ne : α ≃ᵐ β\n⊢ Measure.map (↑e) (Measure.map (↑(symm e)) ν) = ν",
"tactic": "simp [map_map e.measurable e.symm.measurable]"
}
] |
[
4303,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4302,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.of_injective
|
[] |
[
151,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Order/SymmDiff.lean
|
le_symmDiff_iff_left
|
[
{
"state_after": "ι : Type ?u.66703\nα : Type u_1\nβ : Type ?u.66709\nπ : ι → Type ?u.66714\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\nh : a ≤ a ∆ b\n⊢ Disjoint a b",
"state_before": "ι : Type ?u.66703\nα : Type u_1\nβ : Type ?u.66709\nπ : ι → Type ?u.66714\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a ≤ a ∆ b ↔ Disjoint a b",
"tactic": "refine' ⟨fun h => _, fun h => h.symmDiff_eq_sup.symm ▸ le_sup_left⟩"
},
{
"state_after": "ι : Type ?u.66703\nα : Type u_1\nβ : Type ?u.66709\nπ : ι → Type ?u.66714\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\nh : a ≤ (a ⊔ b) \\ (a ⊓ b)\n⊢ Disjoint a b",
"state_before": "ι : Type ?u.66703\nα : Type u_1\nβ : Type ?u.66709\nπ : ι → Type ?u.66714\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\nh : a ≤ a ∆ b\n⊢ Disjoint a b",
"tactic": "rw [symmDiff_eq_sup_sdiff_inf] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.66703\nα : Type u_1\nβ : Type ?u.66709\nπ : ι → Type ?u.66714\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\nh : a ≤ (a ⊔ b) \\ (a ⊓ b)\n⊢ Disjoint a b",
"tactic": "exact disjoint_iff_inf_le.mpr (le_sdiff_iff.1 <| inf_le_of_left_le h).le"
}
] |
[
451,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
List.aestronglyMeasurable_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.313618\nγ : Type ?u.313621\nι : Type ?u.313624\ninst✝⁵ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ (f : α → M), f ∈ l → AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x => List.prod (List.map (fun f => f x) l)) μ",
"tactic": "simpa only [← Pi.list_prod_apply] using l.aestronglyMeasurable_prod' hl"
}
] |
[
1389,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1386,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_liftAt_one_self
|
[
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.123113\nP : Type ?u.123116\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin (n + 1) → M\n⊢ Realize φ v (xs ∘ fun i => if ↑i < n then ↑castSucc i else succ i) = Realize φ v (xs ∘ ↑castSucc)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.123113\nP : Type ?u.123116\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin (n + 1) → M\n⊢ Realize (liftAt 1 n φ) v xs ↔ Realize φ v (xs ∘ ↑castSucc)",
"tactic": "rw [realize_liftAt_one (refl n), iff_eq_eq]"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.123113\nP : Type ?u.123116\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin (n + 1) → M\ni : Fin n\n⊢ (if ↑i < n then ↑castSucc i else succ i) = ↑castSucc i",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.123113\nP : Type ?u.123116\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin (n + 1) → M\n⊢ Realize φ v (xs ∘ fun i => if ↑i < n then ↑castSucc i else succ i) = Realize φ v (xs ∘ ↑castSucc)",
"tactic": "refine' congr rfl (congr rfl (funext fun i => _))"
},
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.123113\nP : Type ?u.123116\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin (n + 1) → M\ni : Fin n\n⊢ (if ↑i < n then ↑castSucc i else succ i) = ↑castSucc i",
"tactic": "rw [if_pos i.is_lt]"
}
] |
[
450,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/MeasureTheory/Group/Integration.lean
|
MeasureTheory.integral_div_right_eq_self
|
[
{
"state_after": "𝕜 : Type ?u.43478\nM : Type ?u.43481\nα : Type ?u.43484\nG : Type u_1\nE : Type u_2\nF : Type ?u.43493\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : IsMulRightInvariant μ\nf : G → E\ng : G\n⊢ (∫ (x : G), f (x * g⁻¹) ∂μ) = ∫ (x : G), f x ∂μ",
"state_before": "𝕜 : Type ?u.43478\nM : Type ?u.43481\nα : Type ?u.43484\nG : Type u_1\nE : Type u_2\nF : Type ?u.43493\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : IsMulRightInvariant μ\nf : G → E\ng : G\n⊢ (∫ (x : G), f (x / g) ∂μ) = ∫ (x : G), f x ∂μ",
"tactic": "simp_rw [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.43478\nM : Type ?u.43481\nα : Type ?u.43484\nG : Type u_1\nE : Type u_2\nF : Type ?u.43493\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : IsMulRightInvariant μ\nf : G → E\ng : G\n⊢ (∫ (x : G), f (x * g⁻¹) ∂μ) = ∫ (x : G), f x ∂μ",
"tactic": "rw [integral_mul_right_eq_self f g⁻¹]"
}
] |
[
123,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.nhdsWithin_basis_eball
|
[] |
[
636,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
635,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lt_bsup_of_ne_bsup
|
[] |
[
1519,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1517,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degrees_prod
|
[
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ degrees (∏ i in ∅, f i) ≤ ∑ i in ∅, degrees (f i)\n\ncase refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i) →\n degrees (∏ i in insert a s, f i) ≤ ∑ i in insert a s, degrees (f i)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i)",
"tactic": "refine' s.induction _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ degrees (∏ i in ∅, f i) ≤ ∑ i in ∅, degrees (f i)",
"tactic": "simp only [Finset.prod_empty, Finset.sum_empty, degrees_one, le_refl]"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i)\n⊢ degrees (∏ i in insert i s, f i) ≤ ∑ i in insert i s, degrees (f i)",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i) →\n degrees (∏ i in insert a s, f i) ≤ ∑ i in insert a s, degrees (f i)",
"tactic": "intro i s his ih"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i)\n⊢ degrees (f i * ∏ x in s, f x) ≤ degrees (f i) + ∑ x in s, degrees (f x)",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i)\n⊢ degrees (∏ i in insert i s, f i) ≤ ∑ i in insert i s, degrees (f i)",
"tactic": "rw [Finset.prod_insert his, Finset.sum_insert his]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.39868\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∏ i in s, f i) ≤ ∑ i in s, degrees (f i)\n⊢ degrees (f i * ∏ x in s, f x) ≤ degrees (f i) + ∑ x in s, degrees (f x)",
"tactic": "exact le_trans (degrees_mul _ _) (add_le_add_left ih _)"
}
] |
[
179,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PrimeMultiset.prod_ofPNatList
|
[
{
"state_after": "l : List ℕ+\nh : ∀ (p : ℕ+), p ∈ l → PNat.Prime p\nthis : prod (ofPNatMultiset (↑l) h) = Multiset.prod ↑l\n⊢ prod (ofPNatList l h) = List.prod l",
"state_before": "l : List ℕ+\nh : ∀ (p : ℕ+), p ∈ l → PNat.Prime p\n⊢ prod (ofPNatList l h) = List.prod l",
"tactic": "have := prod_ofPNatMultiset (l : Multiset ℕ+) h"
},
{
"state_after": "l : List ℕ+\nh : ∀ (p : ℕ+), p ∈ l → PNat.Prime p\nthis : prod (ofPNatMultiset (↑l) h) = List.prod l\n⊢ prod (ofPNatList l h) = List.prod l",
"state_before": "l : List ℕ+\nh : ∀ (p : ℕ+), p ∈ l → PNat.Prime p\nthis : prod (ofPNatMultiset (↑l) h) = Multiset.prod ↑l\n⊢ prod (ofPNatList l h) = List.prod l",
"tactic": "rw [Multiset.coe_prod] at this"
},
{
"state_after": "no goals",
"state_before": "l : List ℕ+\nh : ∀ (p : ℕ+), p ∈ l → PNat.Prime p\nthis : prod (ofPNatMultiset (↑l) h) = List.prod l\n⊢ prod (ofPNatList l h) = List.prod l",
"tactic": "exact this"
}
] |
[
213,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Algebra/BigOperators/Intervals.lean
|
Finset.prod_Ico_div_bot
|
[] |
[
246,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.tendsto_translation_number'
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nx : ℝ\n⊢ Tendsto (fun n => (↑(f ^ (n + 1)) x - x) / (↑n + 1)) atTop (𝓝 (τ f))",
"tactic": "exact_mod_cast (tendsto_add_atTop_iff_nat 1).2 (f.tendsto_translationNumber x)"
}
] |
[
803,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_hnot_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.56061\nα : Type u_1\nβ : Type ?u.56067\nπ : ι → Type ?u.56072\ninst✝ : CoheytingAlgebra α\na : α\n⊢ a ∆ (￢a) = ⊤",
"tactic": "rw [symmDiff_comm, hnot_symmDiff_self]"
}
] |
[
355,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.node4L_lemma₁
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.278351\ninst✝ : Preorder α\na b c d : ℕ\nlr₂ : 3 * (b + c + 1 + d) ≤ 16 * a + 9\nmr₂ : b + c + 1 ≤ 3 * d\nmm₁ : b ≤ 3 * c\n⊢ b < 3 * a + 1",
"tactic": "linarith"
}
] |
[
1143,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1142,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.real_le_real
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑x ≤ ↑y ↔ x ≤ y",
"tactic": "simp [le_def, ofReal']"
}
] |
[
1157,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1157,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.succ_pos'
|
[] |
[
170,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.hasEqualizer_of_hasKernel
|
[] |
[
355,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arcsin_projIcc
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ arcsin ↑(projIcc (-1) 1 (_ : -1 ≤ 1) x) = arcsin x",
"tactic": "rw [arcsin, Function.comp_apply, IccExtend_val, Function.comp_apply, IccExtend,\n Function.comp_apply]"
}
] |
[
61,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.le_mod_add_mod_of_dvd_add_of_not_dvd
|
[
{
"state_after": "m n a✝ b✝ c✝ d a b c : ℕ\nh : c ∣ a + b\nha : ¬c ∣ a\nhc : ¬c ≤ a % c + b % c\nthis : (a + b) % c = a % c + b % c\n⊢ False",
"state_before": "m n a✝ b✝ c✝ d a b c : ℕ\nh : c ∣ a + b\nha : ¬c ∣ a\nhc : ¬c ≤ a % c + b % c\n⊢ False",
"tactic": "have : (a + b) % c = a % c + b % c := add_mod_of_add_mod_lt (lt_of_not_ge hc)"
},
{
"state_after": "no goals",
"state_before": "m n a✝ b✝ c✝ d a b c : ℕ\nh : c ∣ a + b\nha : ¬c ∣ a\nhc : ¬c ≤ a % c + b % c\nthis : (a + b) % c = a % c + b % c\n⊢ False",
"tactic": "simp_all [dvd_iff_mod_eq_zero]"
}
] |
[
493,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
mem_span_finset
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\nN : Type ?u.761155\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ns : Finset M\nx : M\nhx : x ∈ span R ↑s\n⊢ x ∈ span R (_root_.id '' ↑s)",
"tactic": "rwa [Set.image_id]"
}
] |
[
1180,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1173,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
isCauSeq_of_decreasing_bounded
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "let ⟨k, hk⟩ := Archimedean.arch a ε0"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=\n ⟨k + k + 1, fun n hnm =>\n lt_of_lt_of_le\n (show a - (k + (k + 1)) • ε < -|f n| from\n lt_neg.1 <|\n lt_of_le_of_lt (ham n hnm)\n (by\n rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul]\n exact add_lt_add_of_le_of_lt hk (lt_of_le_of_lt hk (lt_add_of_pos_right _ ε0))))\n (neg_le.2 <| abs_neg (f n) ▸ le_abs_self _)⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "let l := Nat.find h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "have hl : ∀ n : ℕ, n ≥ m → f n > a - l • ε := Nat.find_spec h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "have hl0 : l ≠ 0 := fun hl0 =>\n not_lt_of_ge (ham m le_rfl)\n (lt_of_lt_of_le (by have := hl m (le_refl m); simpa [hl0] using this) (le_abs_self (f m)))"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : ¬(i ≥ m → a - Nat.pred (Nat.find h) • ε < f i)\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "cases' not_forall.1 (Nat.find_min h (Nat.pred_lt hl0)) with i hi"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : ¬(i ≥ m → a - Nat.pred (Nat.find h) • ε < f i)\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "rw [not_imp, not_lt] at hi"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\n⊢ ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "exists i"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\nj : ℕ\nhj : j ≥ i\n⊢ abs' (f j - f i) < ε",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\n⊢ ∀ (j : ℕ), j ≥ i → abs' (f j - f i) < ε",
"tactic": "intro j hj"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\nj : ℕ\nhj : j ≥ i\nhfij : f j ≤ f i\n⊢ f i < f j + ε",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\nj : ℕ\nhj : j ≥ i\nhfij : f j ≤ f i\n⊢ abs' (f j - f i) < ε",
"tactic": "rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add']"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\nj : ℕ\nhj : j ≥ i\nhfij : f j ≤ f i\n⊢ f i < f j + ε",
"tactic": "calc\n f i ≤ a - Nat.pred l • ε := hi.2\n _ = a - l • ε + ε := by\n conv =>\n rhs\n rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,\n add_sub_cancel]\n _ < f j + ε := add_lt_add_right (hl j (le_trans hi.1 hj)) _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm✝ : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nn : ℕ\nhnm : n ≥ m\n⊢ a + a < k • ε + (k • ε + ε)",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm✝ : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nn : ℕ\nhnm : n ≥ m\n⊢ a < -(a - (k + (k + 1)) • ε)",
"tactic": "rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm✝ : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nn : ℕ\nhnm : n ≥ m\n⊢ a + a < k • ε + (k • ε + ε)",
"tactic": "exact add_lt_add_of_le_of_lt hk (lt_of_le_of_lt hk (lt_add_of_pos_right _ ε0))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l = 0\nthis : f m > a - l • ε\n⊢ a < f m",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l = 0\n⊢ a < f m",
"tactic": "have := hl m (le_refl m)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l = 0\nthis : f m > a - l • ε\n⊢ a < f m",
"tactic": "simpa [hl0] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7193\ninst✝³ : Ring β\ninst✝² : LinearOrderedField α\ninst✝¹ : Archimedean α\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ (n : ℕ), n ≥ m → abs' (f n) ≤ a\nhnm : ∀ (n : ℕ), n ≥ m → f (Nat.succ n) ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ (n : ℕ), n ≥ m → a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε\nhl0 : l ≠ 0\ni : ℕ\nhi : i ≥ m ∧ f i ≤ a - Nat.pred (Nat.find h) • ε\nj : ℕ\nhj : j ≥ i\nhfij : f j ≤ f i\n⊢ a - Nat.pred l • ε = a - l • ε + ε",
"tactic": "conv =>\n rhs\n rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,\n add_sub_cancel]"
}
] |
[
71,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Std/Logic.lean
|
dite_not
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nP : Prop\ninst✝ : Decidable P\nx : ¬P → α\ny : ¬¬P → α\n⊢ dite (¬P) x y = dite P (fun h => y (_ : ¬¬P)) x",
"tactic": "by_cases h : P <;> simp [h]"
}
] |
[
704,
30
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
702,
9
] |
Mathlib/Data/Int/Parity.lean
|
Int.even_xor'_odd'
|
[
{
"state_after": "case inl.intro\nm n k : ℤ\n⊢ Xor' (k + k = 2 * k) (k + k = 2 * k + 1)\n\ncase inr.intro\nm n k : ℤ\n⊢ Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)",
"state_before": "m n✝ n : ℤ\n⊢ ∃ k, Xor' (n = 2 * k) (n = 2 * k + 1)",
"tactic": "rcases even_or_odd n with (⟨k, rfl⟩ | ⟨k, rfl⟩) <;> use k"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nm n k : ℤ\n⊢ Xor' (k + k = 2 * k) (k + k = 2 * k + 1)",
"tactic": "simpa only [← two_mul, Xor', true_and_iff, eq_self_iff_true, not_true, or_false_iff,\n and_false_iff] using (succ_ne_self (2 * k)).symm"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nm n k : ℤ\n⊢ Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)",
"tactic": "simp only [Xor', add_right_eq_self, false_or_iff, eq_self_iff_true, not_true, not_false_iff,\n one_ne_zero, and_self_iff]"
}
] |
[
86,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.Equiv.ge
|
[] |
[
749,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
748,
1
] |
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