file_path
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Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.continuous_iff_surjective
|
[] |
[
557,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean
|
SimplexCategory.δ_comp_σ_of_le
|
[
{
"state_after": "case mk\nn : ℕ\nj : Fin (n + 1)\ni : ℕ\nhi : i < n + 2\nH : { val := i, isLt := hi } ≤ ↑Fin.castSucc j\n⊢ δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ j) = σ j ≫ δ { val := i, isLt := hi }",
"state_before": "n : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i ≤ ↑Fin.castSucc j\n⊢ δ (↑Fin.castSucc i) ≫ σ (Fin.succ j) = σ j ≫ δ i",
"tactic": "rcases i with ⟨i, hi⟩"
},
{
"state_after": "case mk.mk\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nH : { val := i, isLt := hi } ≤ ↑Fin.castSucc { val := j, isLt := hj }\n⊢ δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ { val := j, isLt := hj }) =\n σ { val := j, isLt := hj } ≫ δ { val := i, isLt := hi }",
"state_before": "case mk\nn : ℕ\nj : Fin (n + 1)\ni : ℕ\nhi : i < n + 2\nH : { val := i, isLt := hi } ≤ ↑Fin.castSucc j\n⊢ δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ j) = σ j ≫ δ { val := i, isLt := hi }",
"tactic": "rcases j with ⟨j, hj⟩"
},
{
"state_after": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nH : { val := i, isLt := hi } ≤ ↑Fin.castSucc { val := j, isLt := hj }\nk : ℕ\nhk : k < len [n + 1] + 1\n⊢ ↑(↑(Hom.toOrderHom (δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ { val := j, isLt := hj })))\n { val := k, isLt := hk }) =\n ↑(↑(Hom.toOrderHom (σ { val := j, isLt := hj } ≫ δ { val := i, isLt := hi })) { val := k, isLt := hk })",
"state_before": "case mk.mk\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nH : { val := i, isLt := hi } ≤ ↑Fin.castSucc { val := j, isLt := hj }\n⊢ δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ { val := j, isLt := hj }) =\n σ { val := j, isLt := hj } ≫ δ { val := i, isLt := hi }",
"tactic": "ext ⟨k, hk⟩"
},
{
"state_after": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\n⊢ ↑(↑(Hom.toOrderHom (δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ { val := j, isLt := hj })))\n { val := k, isLt := hk }) =\n ↑(↑(Hom.toOrderHom (σ { val := j, isLt := hj } ≫ δ { val := i, isLt := hi })) { val := k, isLt := hk })",
"state_before": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nH : { val := i, isLt := hi } ≤ ↑Fin.castSucc { val := j, isLt := hj }\nk : ℕ\nhk : k < len [n + 1] + 1\n⊢ ↑(↑(Hom.toOrderHom (δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ { val := j, isLt := hj })))\n { val := k, isLt := hk }) =\n ↑(↑(Hom.toOrderHom (σ { val := j, isLt := hj } ≫ δ { val := i, isLt := hi })) { val := k, isLt := hk })",
"tactic": "simp at H hk"
},
{
"state_after": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\n⊢ ↑(if h :\n { val := j + 1, isLt := (_ : j + 1 < Nat.succ (n + 1 + 1)) } <\n if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) } then\n Fin.pred\n (if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) })\n (_ :\n (if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) ≠\n 0)\n else\n Fin.castLT\n (if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) })\n (_ :\n ↑(if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) <\n n + 1 + 1)) =\n ↑(if\n ↑Fin.castSucc\n (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then\n Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } ≠ 0)\n else { val := k, isLt := (_ : ↑{ val := k, isLt := hk } < n + 1) }) <\n { val := i, isLt := hi } then\n ↑Fin.castSucc\n (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then\n Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } ≠ 0)\n else { val := k, isLt := (_ : ↑{ val := k, isLt := hk } < n + 1) })\n else\n Fin.succ\n (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then\n Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } ≠ 0)\n else { val := k, isLt := (_ : ↑{ val := k, isLt := hk } < n + 1) }))",
"state_before": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\n⊢ ↑(↑(Hom.toOrderHom (δ (↑Fin.castSucc { val := i, isLt := hi }) ≫ σ (Fin.succ { val := j, isLt := hj })))\n { val := k, isLt := hk }) =\n ↑(↑(Hom.toOrderHom (σ { val := j, isLt := hj } ≫ δ { val := i, isLt := hi })) { val := k, isLt := hk })",
"tactic": "dsimp [σ, δ, Fin.predAbove, Fin.succAbove]"
},
{
"state_after": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\n⊢ (if j + 1 < if k < i then k else k + 1 then (if k < i then k else k + 1) - 1 else if k < i then k else k + 1) =\n if (if j < k then k - 1 else k) < i then if j < k then k - 1 else k else (if j < k then k - 1 else k) + 1",
"state_before": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\n⊢ ↑(if h :\n { val := j + 1, isLt := (_ : j + 1 < Nat.succ (n + 1 + 1)) } <\n if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) } then\n Fin.pred\n (if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) })\n (_ :\n (if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) ≠\n 0)\n else\n Fin.castLT\n (if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) })\n (_ :\n ↑(if\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } <\n { val := i, isLt := (_ : i < Nat.succ (n + 2)) } then\n { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) }\n else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) <\n n + 1 + 1)) =\n ↑(if\n ↑Fin.castSucc\n (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then\n Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } ≠ 0)\n else { val := k, isLt := (_ : ↑{ val := k, isLt := hk } < n + 1) }) <\n { val := i, isLt := hi } then\n ↑Fin.castSucc\n (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then\n Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } ≠ 0)\n else { val := k, isLt := (_ : ↑{ val := k, isLt := hk } < n + 1) })\n else\n Fin.succ\n (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then\n Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } ≠ 0)\n else { val := k, isLt := (_ : ↑{ val := k, isLt := hk } < n + 1) }))",
"tactic": "simp [Fin.lt_iff_val_lt_val, Fin.ite_val, Fin.dite_val]"
},
{
"state_after": "case mk.mk.a.h.h.mk.h.inl.inl.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : j + 1 < k\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k - 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inl.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : j + 1 < k\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k - 1 = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : k < i\nh✝¹ : j + 1 < k\nh✝ : ¬j < k\n⊢ k - 1 = k\n\ncase mk.mk.a.h.h.mk.h.inl.inr.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : ¬j + 1 < k\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k = k - 1\n\ncase mk.mk.a.h.h.mk.h.inl.inr.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : ¬j + 1 < k\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inl.inr.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : k < i\nh✝¹ : ¬j + 1 < k\nh✝ : ¬j < k\n⊢ k = k\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k + 1 - 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 - 1 = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : ¬k < i\nh✝¹ : j + 1 < k + 1\nh✝ : ¬j < k\n⊢ k + 1 - 1 = k + 1\n\ncase mk.mk.a.h.h.mk.h.inr.inr.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : ¬j + 1 < k + 1\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k + 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inr.inr.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : ¬j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inr.inr.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : ¬k < i\nh✝¹ : ¬j + 1 < k + 1\nh✝ : ¬j < k\n⊢ k + 1 = k + 1",
"state_before": "case mk.mk.a.h.h.mk.h\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\n⊢ (if j + 1 < if k < i then k else k + 1 then (if k < i then k else k + 1) - 1 else if k < i then k else k + 1) =\n if (if j < k then k - 1 else k) < i then if j < k then k - 1 else k else (if j < k then k - 1 else k) + 1",
"tactic": "split_ifs"
},
{
"state_after": "case mk.mk.a.h.h.mk.h.inr.inl.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k + 1 - 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 - 1 = k - 1 + 1",
"state_before": "case mk.mk.a.h.h.mk.h.inl.inl.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : j + 1 < k\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k - 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inl.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : j + 1 < k\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k - 1 = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : k < i\nh✝¹ : j + 1 < k\nh✝ : ¬j < k\n⊢ k - 1 = k\n\ncase mk.mk.a.h.h.mk.h.inl.inr.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : ¬j + 1 < k\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k = k - 1\n\ncase mk.mk.a.h.h.mk.h.inl.inr.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : k < i\nh✝² : ¬j + 1 < k\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inl.inr.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : k < i\nh✝¹ : ¬j + 1 < k\nh✝ : ¬j < k\n⊢ k = k\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k + 1 - 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 - 1 = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : ¬k < i\nh✝¹ : j + 1 < k + 1\nh✝ : ¬j < k\n⊢ k + 1 - 1 = k + 1\n\ncase mk.mk.a.h.h.mk.h.inr.inr.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : ¬j + 1 < k + 1\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k + 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inr.inr.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : ¬j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 = k - 1 + 1\n\ncase mk.mk.a.h.h.mk.h.inr.inr.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : ¬k < i\nh✝¹ : ¬j + 1 < k + 1\nh✝ : ¬j < k\n⊢ k + 1 = k + 1",
"tactic": "all_goals try simp <;> linarith"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.a.h.h.mk.h.inr.inl.inl.inl\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : k - 1 < i\n⊢ k + 1 - 1 = k - 1\n\ncase mk.mk.a.h.h.mk.h.inr.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 - 1 = k - 1 + 1",
"tactic": "all_goals cases k <;> simp at * <;> linarith"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.a.h.h.mk.h.inr.inr.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : ¬k < i\nh✝¹ : ¬j + 1 < k + 1\nh✝ : ¬j < k\n⊢ k + 1 = k + 1",
"tactic": "try simp <;> linarith"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.a.h.h.mk.h.inr.inr.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝² : ¬k < i\nh✝¹ : ¬j + 1 < k + 1\nh✝ : ¬j < k\n⊢ k + 1 = k + 1",
"tactic": "simp <;> linarith"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.a.h.h.mk.h.inr.inl.inl.inr\nn i : ℕ\nhi : i < n + 2\nj : ℕ\nhj : j < n + 1\nk : ℕ\nhk : k < n + 1 + 1\nH : i ≤ j\nh✝³ : ¬k < i\nh✝² : j + 1 < k + 1\nh✝¹ : j < k\nh✝ : ¬k - 1 < i\n⊢ k + 1 - 1 = k - 1 + 1",
"tactic": "cases k <;> simp at * <;> linarith"
}
] |
[
270,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.add_pf_add_gt
|
[
{
"state_after": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na b₂ b₁ : R\n⊢ a + (b₁ + b₂) = b₁ + (a + b₂)",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na b₂ c b₁ : R\nx✝ : a + b₂ = c\n⊢ a + (b₁ + b₂) = b₁ + c",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na b₂ b₁ : R\n⊢ a + (b₁ + b₂) = b₁ + (a + b₂)",
"tactic": "simp [add_left_comm]"
}
] |
[
324,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.single_smul
|
[
{
"state_after": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\n⊢ lp.single p i (c • a) = c • lp.single p i a",
"tactic": "refine' ext (funext (fun (j : α) => _))"
},
{
"state_after": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : j = i\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j\n\ncase neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j",
"state_before": "α : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j",
"tactic": "by_cases hi : j = i"
},
{
"state_after": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\nc : 𝕜\nj : α\na : E j\n⊢ ↑(lp.single p j (c • a)) j = ↑(c • lp.single p j a) j",
"state_before": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : j = i\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j",
"tactic": "subst hi"
},
{
"state_after": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\nc : 𝕜\nj : α\na : E j\n⊢ ↑(lp.single p j (c • a)) j = c • ↑(lp.single p j a) j",
"state_before": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\nc : 𝕜\nj : α\na : E j\n⊢ ↑(lp.single p j (c • a)) j = ↑(c • lp.single p j a) j",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\nc : 𝕜\nj : α\na : E j\n⊢ ↑(lp.single p j (c • a)) j = c • ↑(lp.single p j a) j",
"tactic": "simp [lp.single_apply_self]"
},
{
"state_after": "case neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (c • a)) j = c • ↑(lp.single p i a) j",
"state_before": "case neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (c • a)) j = ↑(c • lp.single p i a) j",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : α → Type u_2\np✝ q : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : DecidableEq α\np : ℝ≥0∞\ni : α\na : E i\nc : 𝕜\nj : α\nhi : ¬j = i\n⊢ ↑(lp.single p i (c • a)) j = c • ↑(lp.single p i a) j",
"tactic": "simp [lp.single_apply_ne p i _ hi]"
}
] |
[
1033,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1025,
11
] |
Mathlib/Topology/Algebra/GroupWithZero.lean
|
HasContinuousInv₀.of_nhds_one
|
[
{
"state_after": "α : Type ?u.61226\nβ : Type ?u.61229\nG₀ : Type u_1\ninst✝² : TopologicalSpace G₀\ninst✝¹ : GroupWithZero G₀\ninst✝ : ContinuousMul G₀\na : G₀\nh : Tendsto Inv.inv (𝓝 1) (𝓝 1)\nx : G₀\nhx : x ≠ 0\nhx' : x⁻¹ ≠ 0\n⊢ ContinuousAt Inv.inv x",
"state_before": "α : Type ?u.61226\nβ : Type ?u.61229\nG₀ : Type u_1\ninst✝² : TopologicalSpace G₀\ninst✝¹ : GroupWithZero G₀\ninst✝ : ContinuousMul G₀\na : G₀\nh : Tendsto Inv.inv (𝓝 1) (𝓝 1)\nx : G₀\nhx : x ≠ 0\n⊢ ContinuousAt Inv.inv x",
"tactic": "have hx' := inv_ne_zero hx"
},
{
"state_after": "α : Type ?u.61226\nβ : Type ?u.61229\nG₀ : Type u_1\ninst✝² : TopologicalSpace G₀\ninst✝¹ : GroupWithZero G₀\ninst✝ : ContinuousMul G₀\na : G₀\nh : Tendsto Inv.inv (𝓝 1) (𝓝 1)\nx : G₀\nhx : x ≠ 0\nhx' : x⁻¹ ≠ 0\n⊢ Tendsto ((fun x_1 => x_1 * x⁻¹⁻¹) ∘ Inv.inv ∘ fun x_1 => x * x_1) (𝓝 1) (𝓝 1)",
"state_before": "α : Type ?u.61226\nβ : Type ?u.61229\nG₀ : Type u_1\ninst✝² : TopologicalSpace G₀\ninst✝¹ : GroupWithZero G₀\ninst✝ : ContinuousMul G₀\na : G₀\nh : Tendsto Inv.inv (𝓝 1) (𝓝 1)\nx : G₀\nhx : x ≠ 0\nhx' : x⁻¹ ≠ 0\n⊢ ContinuousAt Inv.inv x",
"tactic": "rw [ContinuousAt, ← map_mul_left_nhds_one₀ hx, ← nhds_translation_mul_inv₀ hx',\n tendsto_map'_iff, tendsto_comap_iff]"
}
] |
[
308,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.single_mul_apply_aux
|
[] |
[
1642,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1640,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.restrictDom_apply
|
[] |
[
250,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.ofNat_mul_subNatNat
|
[
{
"state_after": "no goals",
"state_before": "m n k : Nat\n⊢ ↑m * subNatNat n k = subNatNat (m * n) (m * k)",
"tactic": "cases m with\n| zero => simp [ofNat_zero, Int.zero_mul, Nat.zero_mul]\n| succ m => cases n.lt_or_ge k with\n | inl h =>\n have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)\n simp [subNatNat_of_lt h, subNatNat_of_lt h']\n rw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h), ← neg_ofNat_succ, Nat.mul_sub_left_distrib,\n ← succ_pred_eq_of_pos (Nat.sub_pos_of_lt h')]; rfl\n | inr h =>\n have h' : succ m * k ≤ succ m * n := Nat.mul_le_mul_left _ h\n simp [subNatNat_of_le h, subNatNat_of_le h', Nat.mul_sub_left_distrib]"
},
{
"state_after": "no goals",
"state_before": "case zero\nn k : Nat\n⊢ ↑zero * subNatNat n k = subNatNat (zero * n) (zero * k)",
"tactic": "simp [ofNat_zero, Int.zero_mul, Nat.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case succ\nn k m : Nat\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"tactic": "cases n.lt_or_ge k with\n| inl h =>\nhave h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)\nsimp [subNatNat_of_lt h, subNatNat_of_lt h']\nrw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h), ← neg_ofNat_succ, Nat.mul_sub_left_distrib,\n← succ_pred_eq_of_pos (Nat.sub_pos_of_lt h')]; rfl\n| inr h =>\nhave h' : succ m * k ≤ succ m * n := Nat.mul_le_mul_left _ h\nsimp [subNatNat_of_le h, subNatNat_of_le h', Nat.mul_sub_left_distrib]"
},
{
"state_after": "case succ.inl\nn k m : Nat\nh : n < k\nh' : succ m * n < succ m * k\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"state_before": "case succ.inl\nn k m : Nat\nh : n < k\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"tactic": "have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)"
},
{
"state_after": "case succ.inl\nn k m : Nat\nh : n < k\nh' : succ m * n < succ m * k\n⊢ negOfNat (succ m * succ (pred (k - n))) = -[pred (succ m * k - succ m * n)+1]",
"state_before": "case succ.inl\nn k m : Nat\nh : n < k\nh' : succ m * n < succ m * k\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"tactic": "simp [subNatNat_of_lt h, subNatNat_of_lt h']"
},
{
"state_after": "case succ.inl\nn k m : Nat\nh : n < k\nh' : succ m * n < succ m * k\n⊢ negOfNat (succ (pred (succ m * k - succ m * n))) = -↑(succ (pred (succ (pred (succ m * k - succ m * n)))))",
"state_before": "case succ.inl\nn k m : Nat\nh : n < k\nh' : succ m * n < succ m * k\n⊢ negOfNat (succ m * succ (pred (k - n))) = -[pred (succ m * k - succ m * n)+1]",
"tactic": "rw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h), ← neg_ofNat_succ, Nat.mul_sub_left_distrib,\n ← succ_pred_eq_of_pos (Nat.sub_pos_of_lt h')]"
},
{
"state_after": "no goals",
"state_before": "case succ.inl\nn k m : Nat\nh : n < k\nh' : succ m * n < succ m * k\n⊢ negOfNat (succ (pred (succ m * k - succ m * n))) = -↑(succ (pred (succ (pred (succ m * k - succ m * n)))))",
"tactic": "rfl"
},
{
"state_after": "case succ.inr\nn k m : Nat\nh : n ≥ k\nh' : succ m * k ≤ succ m * n\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"state_before": "case succ.inr\nn k m : Nat\nh : n ≥ k\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"tactic": "have h' : succ m * k ≤ succ m * n := Nat.mul_le_mul_left _ h"
},
{
"state_after": "no goals",
"state_before": "case succ.inr\nn k m : Nat\nh : n ≥ k\nh' : succ m * k ≤ succ m * n\n⊢ ↑(succ m) * subNatNat n k = subNatNat (succ m * n) (succ m * k)",
"tactic": "simp [subNatNat_of_le h, subNatNat_of_le h', Nat.mul_sub_left_distrib]"
}
] |
[
424,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
412,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhdsWithin_Iio_self_neBot'
|
[] |
[
2416,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2415,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.tailings_disjoint_tailing
|
[] |
[
987,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
985,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
isOpenMap_quotient_mk'_mul
|
[
{
"state_after": "M : Type ?u.142323\nα : Type ?u.142326\nβ : Type ?u.142329\nΓ : Type u_1\ninst✝³ : Group Γ\nT : Type u_2\ninst✝² : TopologicalSpace T\ninst✝¹ : MulAction Γ T\ninst✝ : ContinuousConstSMul Γ T\nU : Set T\nhU : IsOpen U\n⊢ IsOpen (⋃ (a : Γ), (fun x x_1 => x • x_1) a '' U)",
"state_before": "M : Type ?u.142323\nα : Type ?u.142326\nβ : Type ?u.142329\nΓ : Type u_1\ninst✝³ : Group Γ\nT : Type u_2\ninst✝² : TopologicalSpace T\ninst✝¹ : MulAction Γ T\ninst✝ : ContinuousConstSMul Γ T\nU : Set T\nhU : IsOpen U\n⊢ IsOpen (Quotient.mk' '' U)",
"tactic": "rw [isOpen_coinduced, MulAction.quotient_preimage_image_eq_union_mul U]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.142323\nα : Type ?u.142326\nβ : Type ?u.142329\nΓ : Type u_1\ninst✝³ : Group Γ\nT : Type u_2\ninst✝² : TopologicalSpace T\ninst✝¹ : MulAction Γ T\ninst✝ : ContinuousConstSMul Γ T\nU : Set T\nhU : IsOpen U\n⊢ IsOpen (⋃ (a : Γ), (fun x x_1 => x • x_1) a '' U)",
"tactic": "exact isOpen_iUnion fun γ => isOpenMap_smul γ U hU"
}
] |
[
490,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean
|
isSelfAdjoint_intCast
|
[] |
[
224,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_getLast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.667113\ninst✝ : DecidableEq α\nl : List α\nx✝ x : α\nxs : List α\n⊢ ↑(formPerm (x :: xs)) (getLast (x :: xs) (_ : x :: xs ≠ [])) = x",
"tactic": "induction' xs using List.reverseRecOn with xs y _ generalizing x <;> simp"
}
] |
[
138,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Algebra/Category/Ring/Colimits.lean
|
CommRingCat.Colimits.quot_neg
|
[] |
[
223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.tensor_id
|
[
{
"state_after": "case H\nR : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nN : ModuleCat R\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑M ↑N) (tensorHom (𝟙 M) (𝟙 N)) =\n LinearMap.compr₂ (TensorProduct.mk R ↑M ↑N) (𝟙 (of R (↑M ⊗[R] ↑N)))",
"state_before": "R : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nN : ModuleCat R\n⊢ tensorHom (𝟙 M) (𝟙 N) = 𝟙 (of R (↑M ⊗[R] ↑N))",
"tactic": "apply TensorProduct.ext"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\ninst✝ : CommRing R\nM : ModuleCat R\nN : ModuleCat R\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑M ↑N) (tensorHom (𝟙 M) (𝟙 N)) =\n LinearMap.compr₂ (TensorProduct.mk R ↑M ↑N) (𝟙 (of R (↑M ⊗[R] ↑N)))",
"tactic": "rfl"
}
] |
[
70,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.strictMono_iff
|
[] |
[
1141,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1136,
1
] |
Mathlib/Algebra/Associated.lean
|
associated_one_of_mul_eq_one
|
[] |
[
445,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
Mathlib/CategoryTheory/Sites/Plus.lean
|
CategoryTheory.GrothendieckTopology.isoToPlus_inv
|
[
{
"state_after": "case hγ\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nhP : Presheaf.IsSheaf J P\n⊢ toPlus J P ≫ (isoToPlus J P hP).inv = 𝟙 P",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nhP : Presheaf.IsSheaf J P\n⊢ (isoToPlus J P hP).inv = plusLift J (𝟙 P) hP",
"tactic": "apply J.plusLift_unique"
},
{
"state_after": "case hγ\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nhP : Presheaf.IsSheaf J P\n⊢ toPlus J P = (isoToPlus J P hP).hom",
"state_before": "case hγ\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nhP : Presheaf.IsSheaf J P\n⊢ toPlus J P ≫ (isoToPlus J P hP).inv = 𝟙 P",
"tactic": "rw [Iso.comp_inv_eq, Category.id_comp]"
},
{
"state_after": "no goals",
"state_before": "case hγ\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nhP : Presheaf.IsSheaf J P\n⊢ toPlus J P = (isoToPlus J P hP).hom",
"tactic": "rfl"
}
] |
[
347,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Embedding.coe_injective
|
[
{
"state_after": "case mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ng : M ↪[L] N\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝) = ↑g\n⊢ mk toEmbedding✝ = g",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\nf g : M ↪[L] N\nh : ↑f = ↑g\n⊢ f = g",
"tactic": "cases f"
},
{
"state_after": "case mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ntoEmbedding✝¹ : M ↪ N\nmap_fun'✝¹ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝¹ (funMap f x) = funMap f (toEmbedding✝¹.toFun ∘ x)\nmap_rel'✝¹ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝¹.toFun ∘ x) ↔ RelMap r x\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝¹) = ↑(mk toEmbedding✝)\n⊢ mk toEmbedding✝¹ = mk toEmbedding✝",
"state_before": "case mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ng : M ↪[L] N\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝) = ↑g\n⊢ mk toEmbedding✝ = g",
"tactic": "cases g"
},
{
"state_after": "case mk.mk.e_toEmbedding\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ntoEmbedding✝¹ : M ↪ N\nmap_fun'✝¹ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝¹ (funMap f x) = funMap f (toEmbedding✝¹.toFun ∘ x)\nmap_rel'✝¹ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝¹.toFun ∘ x) ↔ RelMap r x\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝¹) = ↑(mk toEmbedding✝)\n⊢ toEmbedding✝¹ = toEmbedding✝",
"state_before": "case mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ntoEmbedding✝¹ : M ↪ N\nmap_fun'✝¹ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝¹ (funMap f x) = funMap f (toEmbedding✝¹.toFun ∘ x)\nmap_rel'✝¹ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝¹.toFun ∘ x) ↔ RelMap r x\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝¹) = ↑(mk toEmbedding✝)\n⊢ mk toEmbedding✝¹ = mk toEmbedding✝",
"tactic": "congr"
},
{
"state_after": "case mk.mk.e_toEmbedding.h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ntoEmbedding✝¹ : M ↪ N\nmap_fun'✝¹ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝¹ (funMap f x) = funMap f (toEmbedding✝¹.toFun ∘ x)\nmap_rel'✝¹ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝¹.toFun ∘ x) ↔ RelMap r x\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝¹) = ↑(mk toEmbedding✝)\nx : M\n⊢ ↑toEmbedding✝¹ x = ↑toEmbedding✝ x",
"state_before": "case mk.mk.e_toEmbedding\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ntoEmbedding✝¹ : M ↪ N\nmap_fun'✝¹ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝¹ (funMap f x) = funMap f (toEmbedding✝¹.toFun ∘ x)\nmap_rel'✝¹ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝¹.toFun ∘ x) ↔ RelMap r x\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝¹) = ↑(mk toEmbedding✝)\n⊢ toEmbedding✝¹ = toEmbedding✝",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.e_toEmbedding.h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.138427\ninst✝¹ : Structure L P\nQ : Type ?u.138435\ninst✝ : Structure L Q\ntoEmbedding✝¹ : M ↪ N\nmap_fun'✝¹ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝¹ (funMap f x) = funMap f (toEmbedding✝¹.toFun ∘ x)\nmap_rel'✝¹ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝¹.toFun ∘ x) ↔ RelMap r x\ntoEmbedding✝ : M ↪ N\nmap_fun'✝ :\n ∀ {n : ℕ} (f : Functions L n) (x : Fin n → M),\n Function.Embedding.toFun toEmbedding✝ (funMap f x) = funMap f (toEmbedding✝.toFun ∘ x)\nmap_rel'✝ : ∀ {n : ℕ} (r : Relations L n) (x : Fin n → M), RelMap r (toEmbedding✝.toFun ∘ x) ↔ RelMap r x\nh : ↑(mk toEmbedding✝¹) = ↑(mk toEmbedding✝)\nx : M\n⊢ ↑toEmbedding✝¹ x = ↑toEmbedding✝ x",
"tactic": "exact Function.funext_iff.1 h x"
}
] |
[
652,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
646,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.Parallel.vectorSpan_eq
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : Set P\nh : affineSpan k s₁ ∥ affineSpan k s₂\n⊢ direction (affineSpan k s₁) = direction (affineSpan k s₂)",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : Set P\nh : affineSpan k s₁ ∥ affineSpan k s₂\n⊢ vectorSpan k s₁ = vectorSpan k s₂",
"tactic": "simp_rw [← direction_affineSpan]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : Set P\nh : affineSpan k s₁ ∥ affineSpan k s₂\n⊢ direction (affineSpan k s₁) = direction (affineSpan k s₂)",
"tactic": "exact h.direction_eq"
}
] |
[
1789,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1786,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean
|
AEMeasurable.apply_continuousLinearMap
|
[] |
[
95,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
LinearEquiv.coe_ofInjectiveEndo
|
[] |
[
981,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
979,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
essSup_le_of_ae_le
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\n⊢ essSup (fun x => c) μ ≤ c",
"state_before": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\n⊢ essSup f μ ≤ c",
"tactic": "refine' (essSup_mono_ae hf).trans _"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\nhμ : μ = 0\n⊢ essSup (fun x => c) μ ≤ c\n\ncase neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\nhμ : ¬μ = 0\n⊢ essSup (fun x => c) μ ≤ c",
"state_before": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\n⊢ essSup (fun x => c) μ ≤ c",
"tactic": "by_cases hμ : μ = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\nhμ : μ = 0\n⊢ essSup (fun x => c) μ ≤ c",
"tactic": "simp [hμ]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nf : α → β\nc : β\nhf : f ≤ᵐ[μ] fun x => c\nhμ : ¬μ = 0\n⊢ essSup (fun x => c) μ ≤ c",
"tactic": "rwa [essSup_const]"
}
] |
[
171,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.div_eq_empty
|
[] |
[
589,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
588,
1
] |
Mathlib/Data/FinEnum.lean
|
FinEnum.pi.mem_enum
|
[
{
"state_after": "α : Type u\nβ✝ : α → Type v\nβ : α → Type (max u v)\ninst✝¹ : FinEnum α\ninst✝ : (a : α) → FinEnum (β a)\nf : (a : α) → β a\n⊢ ∃ a, (a ∈ pi (toList α) fun x => toList (β x)) ∧ (fun x => a x (_ : x ∈ toList α)) = f",
"state_before": "α : Type u\nβ✝ : α → Type v\nβ : α → Type (max u v)\ninst✝¹ : FinEnum α\ninst✝ : (a : α) → FinEnum (β a)\nf : (a : α) → β a\n⊢ f ∈ enum β",
"tactic": "simp [pi.enum]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ✝ : α → Type v\nβ : α → Type (max u v)\ninst✝¹ : FinEnum α\ninst✝ : (a : α) → FinEnum (β a)\nf : (a : α) → β a\n⊢ ∃ a, (a ∈ pi (toList α) fun x => toList (β x)) ∧ (fun x => a x (_ : x ∈ toList α)) = f",
"tactic": "refine' ⟨fun a _ => f a, mem_pi _ _, rfl⟩"
}
] |
[
257,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
edist_le_of_edist_le_geometric_two_of_tendsto
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n\n⊢ edist (f n) a ≤ 2 * C * 2⁻¹ ^ n",
"state_before": "α : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ edist (f n) a ≤ 2 * C / 2 ^ n",
"tactic": "simp only [div_eq_mul_inv, ENNReal.inv_pow] at *"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n\n⊢ edist (f n) a ≤ C * 2⁻¹ ^ n * 2",
"state_before": "α : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n\n⊢ edist (f n) a ≤ 2 * C * 2⁻¹ ^ n",
"tactic": "rw [mul_assoc, mul_comm]"
},
{
"state_after": "case h.e'_4\nα : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n\n⊢ C * 2⁻¹ ^ n * 2 = C * 2⁻¹ ^ n / (1 - 2⁻¹)",
"state_before": "α : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n\n⊢ edist (f n) a ≤ C * 2⁻¹ ^ n * 2",
"tactic": "convert edist_le_of_edist_le_geometric_of_tendsto 2⁻¹ C hu ha n using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\nβ : Type ?u.441500\nι : Type ?u.441503\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nhC : C ≠ ⊤\nf : ℕ → α\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n\n⊢ C * 2⁻¹ ^ n * 2 = C * 2⁻¹ ^ n / (1 - 2⁻¹)",
"tactic": "rw [ENNReal.one_sub_inv_two, div_eq_mul_inv, inv_inv]"
}
] |
[
355,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/Topology/MetricSpace/Baire.lean
|
dense_biUnion_interior_of_closed
|
[] |
[
335,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsTensorProduct.lift_eq
|
[
{
"state_after": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.87557\nN₂ : Type ?u.87560\nN : Type ?u.87563\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nf' : M₁ →ₗ[R] M₂ →ₗ[R] M'\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(LinearMap.comp (TensorProduct.lift f') ↑(LinearEquiv.symm (equiv h))) (↑(↑f x₁) x₂) = ↑(↑f' x₁) x₂",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.87557\nN₂ : Type ?u.87560\nN : Type ?u.87563\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nf' : M₁ →ₗ[R] M₂ →ₗ[R] M'\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(lift h f') (↑(↑f x₁) x₂) = ↑(↑f' x₁) x₂",
"tactic": "delta IsTensorProduct.lift"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.87557\nN₂ : Type ?u.87560\nN : Type ?u.87563\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nf' : M₁ →ₗ[R] M₂ →ₗ[R] M'\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(LinearMap.comp (TensorProduct.lift f') ↑(LinearEquiv.symm (equiv h))) (↑(↑f x₁) x₂) = ↑(↑f' x₁) x₂",
"tactic": "simp"
}
] |
[
110,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_symm_diff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.48465\nι : Sort ?u.48468\nι' : Sort ?u.48471\nf : α → β\ns✝ t✝ : Set α\nhf : Injective f\ns t : Set α\n⊢ f '' s ∆ t = (f '' s) ∆ (f '' t)",
"tactic": "simp_rw [Set.symmDiff_def, image_union, image_diff hf]"
}
] |
[
448,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.secondCountableTopology
|
[] |
[
258,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
11
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
AntitoneOn.intervalIntegrable
|
[] |
[
389,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Topology/Constructions.lean
|
nhds_swap
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.39611\nδ : Type ?u.39614\nε : Type ?u.39617\nζ : Type ?u.39620\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\na : α\nb : β\n⊢ map (fun p => (p.snd, p.fst)) (𝓝 b ×ˢ 𝓝 a) = map Prod.swap (𝓝 b ×ˢ 𝓝 a)",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.39611\nδ : Type ?u.39614\nε : Type ?u.39617\nζ : Type ?u.39620\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\na : α\nb : β\n⊢ 𝓝 (a, b) = map Prod.swap (𝓝 (b, a))",
"tactic": "rw [nhds_prod_eq, Filter.prod_comm, nhds_prod_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.39611\nδ : Type ?u.39614\nε : Type ?u.39617\nζ : Type ?u.39620\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\na : α\nb : β\n⊢ map (fun p => (p.snd, p.fst)) (𝓝 b ×ˢ 𝓝 a) = map Prod.swap (𝓝 b ×ˢ 𝓝 a)",
"tactic": "rfl"
}
] |
[
586,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
585,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
SimpleGraph.mk_mem_interedges_comm
|
[] |
[
426,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.empty_mulVec
|
[] |
[
302,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/CategoryTheory/Functor/FullyFaithful.lean
|
CategoryTheory.Functor.image_preimage
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nX✝ Y✝ : C\nF : C ⥤ D\ninst✝ : Full F\nX Y : C\nf : F.obj X ⟶ F.obj Y\n⊢ F.map (Full.preimage f) = f",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nX✝ Y✝ : C\nF : C ⥤ D\ninst✝ : Full F\nX Y : C\nf : F.obj X ⟶ F.obj Y\n⊢ F.map (F.preimage f) = f",
"tactic": "unfold preimage"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nX✝ Y✝ : C\nF : C ⥤ D\ninst✝ : Full F\nX Y : C\nf : F.obj X ⟶ F.obj Y\n⊢ F.map (Full.preimage f) = f",
"tactic": "aesop_cat"
}
] |
[
95,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_pi_add_coe_pi
|
[
{
"state_after": "no goals",
"state_before": "⊢ ↑π + ↑π = 0",
"tactic": "rw [← two_nsmul, two_nsmul_coe_pi]"
}
] |
[
173,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.IsPrime.prod_le
|
[] |
[
1066,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1064,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
ofAdd_add
|
[] |
[
152,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.nonempty_neLocus_iff
|
[] |
[
67,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Analysis/NormedSpace/ENorm.lean
|
ENorm.finite_dist_eq
|
[] |
[
222,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousOn.mono_dom
|
[] |
[
634,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.112523\nγ : Type ?u.112526\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\n⊢ insert a s ⊆ t ↔ a ∈ t ∧ s ⊆ t",
"tactic": "simp only [subset_iff, mem_insert, forall_eq, or_imp, forall_and]"
}
] |
[
1161,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1160,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.map_div_right_divisors
|
[
{
"state_after": "case a.mk\nn d nd : ℕ\n⊢ (d, nd) ∈\n map\n { toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n (divisors n) ↔\n (d, nd) ∈ divisorsAntidiagonal n",
"state_before": "n : ℕ\n⊢ map\n { toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n (divisors n) =\n divisorsAntidiagonal n",
"tactic": "ext ⟨d, nd⟩"
},
{
"state_after": "case a.mk\nn d nd : ℕ\n⊢ (d ∣ n ∧ n ≠ 0) ∧ n / d = nd ↔ d * nd = n ∧ n ≠ 0",
"state_before": "case a.mk\nn d nd : ℕ\n⊢ (d, nd) ∈\n map\n { toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n (divisors n) ↔\n (d, nd) ∈ divisorsAntidiagonal n",
"tactic": "simp only [mem_map, mem_divisorsAntidiagonal, Function.Embedding.coeFn_mk, mem_divisors,\n Prod.ext_iff, exists_prop, and_left_comm, exists_eq_left]"
},
{
"state_after": "case a.mk.mp\nn d nd : ℕ\n⊢ (d ∣ n ∧ n ≠ 0) ∧ n / d = nd → d * nd = n ∧ n ≠ 0\n\ncase a.mk.mpr\nn d nd : ℕ\n⊢ d * nd = n ∧ n ≠ 0 → (d ∣ n ∧ n ≠ 0) ∧ n / d = nd",
"state_before": "case a.mk\nn d nd : ℕ\n⊢ (d ∣ n ∧ n ≠ 0) ∧ n / d = nd ↔ d * nd = n ∧ n ≠ 0",
"tactic": "constructor"
},
{
"state_after": "case a.mk.mp.intro.intro.intro\nd k : ℕ\nhn : d * k ≠ 0\n⊢ d * (d * k / d) = d * k ∧ d * k ≠ 0",
"state_before": "case a.mk.mp\nn d nd : ℕ\n⊢ (d ∣ n ∧ n ≠ 0) ∧ n / d = nd → d * nd = n ∧ n ≠ 0",
"tactic": "rintro ⟨⟨⟨k, rfl⟩, hn⟩, rfl⟩"
},
{
"state_after": "case a.mk.mp.intro.intro.intro\nd k : ℕ\nhn : d * k ≠ 0\n⊢ d * k = d * k ∧ d * k ≠ 0",
"state_before": "case a.mk.mp.intro.intro.intro\nd k : ℕ\nhn : d * k ≠ 0\n⊢ d * (d * k / d) = d * k ∧ d * k ≠ 0",
"tactic": "rw [Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt]"
},
{
"state_after": "no goals",
"state_before": "case a.mk.mp.intro.intro.intro\nd k : ℕ\nhn : d * k ≠ 0\n⊢ d * k = d * k ∧ d * k ≠ 0",
"tactic": "exact ⟨rfl, hn⟩"
},
{
"state_after": "case a.mk.mpr.intro\nd nd : ℕ\nhn : d * nd ≠ 0\n⊢ (d ∣ d * nd ∧ d * nd ≠ 0) ∧ d * nd / d = nd",
"state_before": "case a.mk.mpr\nn d nd : ℕ\n⊢ d * nd = n ∧ n ≠ 0 → (d ∣ n ∧ n ≠ 0) ∧ n / d = nd",
"tactic": "rintro ⟨rfl, hn⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mk.mpr.intro\nd nd : ℕ\nhn : d * nd ≠ 0\n⊢ (d ∣ d * nd ∧ d * nd ≠ 0) ∧ d * nd / d = nd",
"tactic": "exact ⟨⟨dvd_mul_right _ _, hn⟩, Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt⟩"
}
] |
[
274,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.conj_conj
|
[] |
[
228,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/MeasureTheory/Covering/OneDim.lean
|
Real.Icc_mem_vitaliFamily_at_right
|
[
{
"state_after": "x y : ℝ\nhxy : x < y\n⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (vitaliFamily volume 1) x",
"state_before": "x y : ℝ\nhxy : x < y\n⊢ Icc x y ∈ VitaliFamily.setsAt (vitaliFamily volume 1) x",
"tactic": "rw [Icc_eq_closedBall]"
},
{
"state_after": "x y : ℝ\nhxy : x < y\n⊢ dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2)",
"state_before": "x y : ℝ\nhxy : x < y\n⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (vitaliFamily volume 1) x",
"tactic": "refine' closedBall_mem_vitaliFamily_of_dist_le_mul _ _ (by linarith)"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhxy : x < y\n⊢ dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2)",
"tactic": "rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhxy : x < y\n⊢ 0 < (y - x) / 2",
"tactic": "linarith"
}
] |
[
33,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.subset_pointwise_smul_iff
|
[] |
[
362,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_add_add_self
|
[
{
"state_after": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n⊢ inner (x + y) (x + y) = inner x x + 2 * inner x y + inner y y",
"state_before": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ inner (x + y) (x + y) = inner x x + 2 * inner x y + inner y y",
"tactic": "have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [← inner_conj_symm]; rfl"
},
{
"state_after": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n⊢ inner x x + inner x y + inner x y = inner x x + 2 * inner x y",
"state_before": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n⊢ inner (x + y) (x + y) = inner x x + 2 * inner x y + inner y y",
"tactic": "simp only [inner_add_add_self, this, add_left_inj]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n⊢ inner x x + inner x y + inner x y = inner x x + 2 * inner x y",
"tactic": "ring"
},
{
"state_after": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ ↑(starRingEnd ℝ) (inner x y) = inner x y",
"state_before": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ inner y x = inner x y",
"tactic": "rw [← inner_conj_symm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.1806744\nE : Type ?u.1806747\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ ↑(starRingEnd ℝ) (inner x y) = inner x y",
"tactic": "rfl"
}
] |
[
674,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
670,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_indicator_ge_of_bdd_below
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ (∫⁻ (a : α), Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) a ∂μ) ≤\n ∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ C • ↑↑μ s ^ (1 / ENNReal.toReal p) ≤ snorm (Set.indicator s f) p μ",
"tactic": "rw [ENNReal.smul_def, smul_eq_mul, snorm_eq_lintegral_rpow_nnnorm hp hp',\n ENNReal.le_rpow_one_div_iff (ENNReal.toReal_pos hp hp'),\n ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg, ← ENNReal.rpow_mul,\n one_div_mul_cancel (ENNReal.toReal_pos hp hp').ne.symm, ENNReal.rpow_one, ← set_lintegral_const,\n ← lintegral_indicator _ hs]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ ∀ᵐ (a : α) ∂μ, Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) a ≤ ↑‖Set.indicator s f a‖₊ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ (∫⁻ (a : α), Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) a ∂μ) ≤\n ∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ",
"tactic": "refine' lintegral_mono_ae _"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ ∀ᵐ (a : α) ∂μ, Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) a ≤ ↑‖Set.indicator s f a‖₊ ^ ENNReal.toReal p",
"tactic": "filter_upwards [hf] with x hx"
},
{
"state_after": "case h\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑(Set.indicator s (fun a => ‖f a‖₊) x) ^ ENNReal.toReal p",
"state_before": "case h\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p",
"tactic": "rw [nnnorm_indicator_eq_indicator_nnnorm]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nhxs : x ∈ s\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑(Set.indicator s (fun a => ‖f a‖₊) x) ^ ENNReal.toReal p\n\ncase neg\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nhxs : ¬x ∈ s\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑(Set.indicator s (fun a => ‖f a‖₊) x) ^ ENNReal.toReal p",
"state_before": "case h\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑(Set.indicator s (fun a => ‖f a‖₊) x) ^ ENNReal.toReal p",
"tactic": "by_cases hxs : x ∈ s"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhxs : x ∈ s\nhx : x ∈ s → C ≤ ‖f x‖₊\n⊢ ↑C ^ ENNReal.toReal p ≤ ↑‖f x‖₊ ^ ENNReal.toReal p",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nhxs : x ∈ s\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑(Set.indicator s (fun a => ‖f a‖₊) x) ^ ENNReal.toReal p",
"tactic": "simp only [Set.indicator_of_mem hxs] at hx⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhxs : x ∈ s\nhx : x ∈ s → C ≤ ‖f x‖₊\n⊢ ↑C ^ ENNReal.toReal p ≤ ↑‖f x‖₊ ^ ENNReal.toReal p",
"tactic": "exact ENNReal.rpow_le_rpow (ENNReal.coe_le_coe.2 (hx hxs)) ENNReal.toReal_nonneg"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.5996188\nF : Type u_2\nG : Type ?u.5996194\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nhp : p ≠ 0\nhp' : p ≠ ⊤\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nx : α\nhx : x ∈ s → C ≤ ‖Set.indicator s f x‖₊\nhxs : ¬x ∈ s\n⊢ Set.indicator s (fun x => ↑C ^ ENNReal.toReal p) x ≤ ↑(Set.indicator s (fun a => ‖f a‖₊) x) ^ ENNReal.toReal p",
"tactic": "simp [Set.indicator_of_not_mem hxs]"
}
] |
[
1593,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1579,
1
] |
Mathlib/GroupTheory/GroupAction/Opposite.lean
|
MulOpposite.smul_eq_mul_unop
|
[] |
[
105,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.snorm_indicator_const_le
|
[
{
"state_after": "case inl\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\n⊢ snorm (Set.indicator s fun x => c) 0 μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal 0)\n\ncase inr\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"state_before": "α : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "rcases eq_or_ne p 0 with (rfl | hp)"
},
{
"state_after": "case inr.inl\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhp : ⊤ ≠ 0\n⊢ snorm (Set.indicator s fun x => c) ⊤ μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal ⊤)\n\ncase inr.inr\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p ≠ ⊤\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"state_before": "case inr\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "rcases eq_or_ne p ∞ with (rfl | h'p)"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p ≠ ⊤\nt : Set α := toMeasurable μ s\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"state_before": "case inr.inr\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p ≠ ⊤\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "let t := toMeasurable μ s"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p ≠ ⊤\nt : Set α := toMeasurable μ s\n⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "calc\n snorm (s.indicator fun _ => c) p μ ≤ snorm (t.indicator fun _ => c) p μ :=\n snorm_mono (norm_indicator_le_of_subset (subset_toMeasurable _ _) _)\n _ = ‖c‖₊ * μ t ^ (1 / p.toReal) :=\n (snorm_indicator_const (measurableSet_toMeasurable _ _) hp h'p)\n _ = ‖c‖₊ * μ s ^ (1 / p.toReal) := by rw [measure_toMeasurable]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\n⊢ snorm (Set.indicator s fun x => c) 0 μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal 0)",
"tactic": "simp only [snorm_exponent_zero, zero_le']"
},
{
"state_after": "case inr.inl\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhp : ⊤ ≠ 0\n⊢ snormEssSup (Set.indicator s fun x => c) μ ≤ ↑‖c‖₊",
"state_before": "case inr.inl\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhp : ⊤ ≠ 0\n⊢ snorm (Set.indicator s fun x => c) ⊤ μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal ⊤)",
"tactic": "simp only [snorm_exponent_top, ENNReal.top_toReal, _root_.div_zero, ENNReal.rpow_zero, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhp : ⊤ ≠ 0\n⊢ snormEssSup (Set.indicator s fun x => c) μ ≤ ↑‖c‖₊",
"tactic": "exact snormEssSup_indicator_const_le _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.648257\nF : Type ?u.648260\nG : Type u_2\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p ≠ ⊤\nt : Set α := toMeasurable μ s\n⊢ ↑‖c‖₊ * ↑↑μ t ^ (1 / ENNReal.toReal p) = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "rw [measure_toMeasurable]"
}
] |
[
629,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
CompleteLatticeHom.toFun_eq_coe
|
[] |
[
689,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
688,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.range_comp
|
[] |
[
1231,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1229,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
Right.one_le_inv_iff
|
[
{
"state_after": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ 1 * a ≤ a⁻¹ * a ↔ a ≤ 1",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap 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_le_mul_iff_right a]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ 1 * a ≤ a⁻¹ * a ↔ a ≤ 1",
"tactic": "simp"
}
] |
[
228,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/Down.lean
|
Finset.memberSubfamily_nonMemberSubfamily
|
[
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\na✝ : Finset α\n⊢ a✝ ∈ memberSubfamily a (nonMemberSubfamily a 𝒜) ↔ a✝ ∈ ∅",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ memberSubfamily a (nonMemberSubfamily a 𝒜) = ∅",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\na✝ : Finset α\n⊢ a✝ ∈ memberSubfamily a (nonMemberSubfamily a 𝒜) ↔ a✝ ∈ ∅",
"tactic": "simp"
}
] |
[
125,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
PhragmenLindelof.eqOn_vertical_strip
|
[] |
[
346,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/List/BigOperators/Lemmas.lean
|
MonoidHom.unop_map_list_prod
|
[] |
[
174,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
11
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.le_pred_iff
|
[] |
[
706,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
705,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.cokernelIsoOfEq_inv_comp_desc
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\ne : Y ⟶ Z\nhe : f ≫ e = 0\n⊢ g ≫ e = 0",
"tactic": "simp [← h, he]"
},
{
"state_after": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf : X ⟶ Y\ninst✝¹ : HasCokernel f\ne : Y ⟶ Z\nhe : f ≫ e = 0\ninst✝ : HasCokernel f\n⊢ (cokernelIsoOfEq (_ : f = f)).inv ≫ cokernel.desc f e he = cokernel.desc f e (_ : f ≫ e = 0)",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\ne : Y ⟶ Z\nhe : f ≫ e = 0\n⊢ (cokernelIsoOfEq h).inv ≫ cokernel.desc f e he = cokernel.desc g e (_ : g ≫ e = 0)",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasCokernel f✝\nZ : C\nf : X ⟶ Y\ninst✝¹ : HasCokernel f\ne : Y ⟶ Z\nhe : f ≫ e = 0\ninst✝ : HasCokernel f\n⊢ (cokernelIsoOfEq (_ : f = f)).inv ≫ cokernel.desc f e he = cokernel.desc f e (_ : f ≫ e = 0)",
"tactic": "simp"
}
] |
[
849,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
846,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.update_preimage_pi
|
[
{
"state_after": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\n⊢ x ∈ update f i ⁻¹' pi s t ↔ x ∈ t i",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\n⊢ update f i ⁻¹' pi s t = t i",
"tactic": "ext x"
},
{
"state_after": "case h.refine'_1\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nh : x ∈ update f i ⁻¹' pi s t\n⊢ x ∈ t i\n\ncase h.refine'_2\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nhx : x ∈ t i\nj : ι\nhj : j ∈ s\n⊢ update f i x j ∈ t j",
"state_before": "case h\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\n⊢ x ∈ update f i ⁻¹' pi s t ↔ x ∈ t i",
"tactic": "refine' ⟨fun h => _, fun hx j hj => _⟩"
},
{
"state_after": "case h.e'_4\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nh : x ∈ update f i ⁻¹' pi s t\n⊢ x = update f i x i",
"state_before": "case h.refine'_1\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nh : x ∈ update f i ⁻¹' pi s t\n⊢ x ∈ t i",
"tactic": "convert h i hi"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nh : x ∈ update f i ⁻¹' pi s t\n⊢ x = update f i x i",
"tactic": "simp"
},
{
"state_after": "case h.refine'_2.inl\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nj : ι\nhj hi : j ∈ s\nhf : ∀ (j_1 : ι), j_1 ∈ s → j_1 ≠ j → f j_1 ∈ t j_1\nx : α j\nhx : x ∈ t j\n⊢ update f j x j ∈ t j\n\ncase h.refine'_2.inr\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nhx : x ∈ t i\nj : ι\nhj : j ∈ s\nh : j ≠ i\n⊢ update f i x j ∈ t j",
"state_before": "case h.refine'_2\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nhx : x ∈ t i\nj : ι\nhj : j ∈ s\n⊢ update f i x j ∈ t j",
"tactic": "obtain rfl | h := eq_or_ne j i"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.inl\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nj : ι\nhj hi : j ∈ s\nhf : ∀ (j_1 : ι), j_1 ∈ s → j_1 ≠ j → f j_1 ∈ t j_1\nx : α j\nhx : x ∈ t j\n⊢ update f j x j ∈ t j",
"tactic": "simpa"
},
{
"state_after": "case h.refine'_2.inr\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nhx : x ∈ t i\nj : ι\nhj : j ∈ s\nh : j ≠ i\n⊢ f j ∈ t j",
"state_before": "case h.refine'_2.inr\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nhx : x ∈ t i\nj : ι\nhj : j ∈ s\nh : j ≠ i\n⊢ update f i x j ∈ t j",
"tactic": "rw [update_noteq h]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.inr\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.163005\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\nhi : i ∈ s\nhf : ∀ (j : ι), j ∈ s → j ≠ i → f j ∈ t j\nx : α i\nhx : x ∈ t i\nj : ι\nhj : j ∈ s\nh : j ≠ i\n⊢ f j ∈ t j",
"tactic": "exact hf j hj h"
}
] |
[
857,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
|
ModuleCat.epi_iff_surjective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : Ring R\nX Y : ModuleCat R\nf : X ⟶ Y\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ Epi f ↔ Function.Surjective ↑f",
"tactic": "rw [epi_iff_range_eq_top, LinearMap.range_eq_top]"
}
] |
[
60,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.biUnion_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.505331\ninst✝¹ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\ninst✝ : DecidableEq α\na : α\nx : β\n⊢ x ∈ Finset.biUnion (insert a s) t ↔ x ∈ t a ∪ Finset.biUnion s t",
"tactic": "simp only [mem_biUnion, exists_prop, mem_union, mem_insert, or_and_right, exists_or,\n exists_eq_left]"
}
] |
[
3553,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3550,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_mul_cpow_one_add_sq
|
[
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ (∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t) =\n (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\n⊢ (∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t) =\n (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"tactic": "have : t + 1 ≠ 0 := by contrapose! ht; rwa [add_eq_zero_iff_eq_neg] at ht"
},
{
"state_after": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) (↑x * (1 + ↑x ^ 2) ^ t) x\n\ncase hint\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ IntervalIntegrable (fun y => ↑y * (1 + ↑y ^ 2) ^ t) MeasureTheory.volume a b",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ (∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t) =\n (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"tactic": "apply integral_eq_sub_of_hasDerivAt"
},
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t + 1 = 0\n⊢ t = -1",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\n⊢ t + 1 ≠ 0",
"tactic": "contrapose! ht"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t + 1 = 0\n⊢ t = -1",
"tactic": "rwa [add_eq_zero_iff_eq_neg] at ht"
},
{
"state_after": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) (↑x * (1 + ↑x ^ 2) ^ t) x",
"state_before": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) (↑x * (1 + ↑x ^ 2) ^ t) x",
"tactic": "intro x _"
},
{
"state_after": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ HasDerivAt (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) (↑x * (1 + ↑x ^ 2) ^ t) x",
"state_before": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\n⊢ HasDerivAt (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) (↑x * (1 + ↑x ^ 2) ^ t) x",
"tactic": "have g :\n ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z := by\n intro z hz\n convert (HasDerivAt.cpow_const (c := t + 1) (hasDerivAt_id _)\n (Or.inl hz)).div_const (2 * (t + 1)) using 1\n field_simp\n ring"
},
{
"state_after": "case h.e'_6\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) = fun y =>\n ((fun z => z ^ (t + 1) / (2 * (t + 1))) ∘ fun y => ↑1 + y ^ 2) ↑y\n\ncase h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ ↑x * (1 + ↑x ^ 2) ^ t = (↑1 + ↑x ^ 2) ^ t / 2 * (2 * ↑x)\n\ncase hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ 0 < (↑1 + ↑x ^ 2).re",
"state_before": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ HasDerivAt (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) (↑x * (1 + ↑x ^ 2) ^ t) x",
"tactic": "convert (HasDerivAt.comp (↑x) (g _) f).comp_ofReal using 1"
},
{
"state_after": "case h.e'_6.h.h.e'_5\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nx✝ : ℂ\n⊢ ↑1 = 1\n\ncase h.e'_7.h.e'_6\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\n⊢ ↑x = ↑x ^ (2 - 1)",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x",
"tactic": "convert (hasDerivAt_pow 2 (x : ℂ)).const_add 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6.h.h.e'_5\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nx✝ : ℂ\n⊢ ↑1 = 1",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h.e'_6\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\n⊢ ↑x = ↑x ^ (2 - 1)",
"tactic": "simp"
},
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\nz : ℂ\nhz : 0 < z.re\n⊢ HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\n⊢ ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z",
"tactic": "intro z hz"
},
{
"state_after": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\nz : ℂ\nhz : 0 < z.re\n⊢ z ^ t / 2 = (t + 1) * id z ^ (t + 1 - 1) * 1 / (2 * (t + 1))",
"state_before": "a b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\nz : ℂ\nhz : 0 < z.re\n⊢ HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z",
"tactic": "convert (HasDerivAt.cpow_const (c := t + 1) (hasDerivAt_id _)\n (Or.inl hz)).div_const (2 * (t + 1)) using 1"
},
{
"state_after": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\nz : ℂ\nhz : 0 < z.re\n⊢ z ^ t * (2 * (t + 1)) = (t + 1) * z ^ t * 2",
"state_before": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\nz : ℂ\nhz : 0 < z.re\n⊢ z ^ t / 2 = (t + 1) * id z ^ (t + 1 - 1) * 1 / (2 * (t + 1))",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\nz : ℂ\nhz : 0 < z.re\n⊢ z ^ t * (2 * (t + 1)) = (t + 1) * z ^ t * 2",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ (fun {b} => (1 + ↑b ^ 2) ^ (t + 1) / (↑2 * (t + 1))) = fun y =>\n ((fun z => z ^ (t + 1) / (2 * (t + 1))) ∘ fun y => ↑1 + y ^ 2) ↑y",
"tactic": "simp"
},
{
"state_after": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ ↑x * (1 + ↑x ^ 2) ^ t * 2 = (1 + ↑x ^ 2) ^ t * (2 * ↑x)",
"state_before": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ ↑x * (1 + ↑x ^ 2) ^ t = (↑1 + ↑x ^ 2) ^ t / 2 * (2 * ↑x)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ ↑x * (1 + ↑x ^ 2) ^ t * 2 = (1 + ↑x ^ 2) ^ t * (2 * ↑x)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "case hderiv\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\nx : ℝ\na✝ : x ∈ [[a, b]]\nf : HasDerivAt (fun y => ↑1 + y ^ 2) (2 * ↑x) ↑x\ng : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z => z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z\n⊢ 0 < (↑1 + ↑x ^ 2).re",
"tactic": "exact_mod_cast add_pos_of_pos_of_nonneg zero_lt_one (sq_nonneg x)"
},
{
"state_after": "case hint.hu\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun y => ↑y * (1 + ↑y ^ 2) ^ t",
"state_before": "case hint\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ IntervalIntegrable (fun y => ↑y * (1 + ↑y ^ 2) ^ t) MeasureTheory.volume a b",
"tactic": "apply Continuous.intervalIntegrable"
},
{
"state_after": "case hint.hu\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun y => (1 + ↑y ^ 2) ^ t",
"state_before": "case hint.hu\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun y => ↑y * (1 + ↑y ^ 2) ^ t",
"tactic": "refine' continuous_ofReal.mul _"
},
{
"state_after": "case hint.hu.hf\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun x => 1 + ↑x ^ 2\n\ncase hint.hu.hg\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun x => t\n\ncase hint.hu.h0\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ ∀ (a : ℝ), 0 < (1 + ↑a ^ 2).re ∨ (1 + ↑a ^ 2).im ≠ 0",
"state_before": "case hint.hu\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun y => (1 + ↑y ^ 2) ^ t",
"tactic": "apply Continuous.cpow"
},
{
"state_after": "no goals",
"state_before": "case hint.hu.hf\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun x => 1 + ↑x ^ 2",
"tactic": "exact continuous_const.add (continuous_ofReal.pow 2)"
},
{
"state_after": "no goals",
"state_before": "case hint.hu.hg\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ Continuous fun x => t",
"tactic": "exact continuous_const"
},
{
"state_after": "case hint.hu.h0\na✝ b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\na : ℝ\n⊢ 0 < (1 + ↑a ^ 2).re ∨ (1 + ↑a ^ 2).im ≠ 0",
"state_before": "case hint.hu.h0\na b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ ∀ (a : ℝ), 0 < (1 + ↑a ^ 2).re ∨ (1 + ↑a ^ 2).im ≠ 0",
"tactic": "intro a"
},
{
"state_after": "case hint.hu.h0\na✝ b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\na : ℝ\n⊢ 0 < 1 + a ^ 2 ∨ (1 + ↑(a ^ 2)).im ≠ 0",
"state_before": "case hint.hu.h0\na✝ b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\na : ℝ\n⊢ 0 < (1 + ↑a ^ 2).re ∨ (1 + ↑a ^ 2).im ≠ 0",
"tactic": "rw [add_re, one_re, ← ofReal_pow, ofReal_re]"
},
{
"state_after": "no goals",
"state_before": "case hint.hu.h0\na✝ b : ℝ\nn : ℕ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\na : ℝ\n⊢ 0 < 1 + a ^ 2 ∨ (1 + ↑(a ^ 2)).im ≠ 0",
"tactic": "exact Or.inl (add_pos_of_pos_of_nonneg zero_lt_one (sq_nonneg a))"
}
] |
[
589,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
PartitionOfUnity.exists_finset_nhd_support_subset
|
[] |
[
214,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.trans_refl
|
[
{
"state_after": "case mk\nα : Sort u\nβ : Sort v\nγ : Sort w\ntoFun✝ : α → β\ninvFun✝ : β → α\nleft_inv✝ : LeftInverse invFun✝ toFun✝\nright_inv✝ : Function.RightInverse invFun✝ toFun✝\n⊢ { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }.trans (Equiv.refl β) =\n { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }",
"state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\ne : α ≃ β\n⊢ e.trans (Equiv.refl β) = e",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Sort u\nβ : Sort v\nγ : Sort w\ntoFun✝ : α → β\ninvFun✝ : β → α\nleft_inv✝ : LeftInverse invFun✝ toFun✝\nright_inv✝ : Function.RightInverse invFun✝ toFun✝\n⊢ { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }.trans (Equiv.refl β) =\n { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }",
"tactic": "rfl"
}
] |
[
341,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
9
] |
Mathlib/Order/Max.lean
|
IsBot.isMin
|
[] |
[
225,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
11
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.conjTranspose_map
|
[] |
[
2252,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2250,
1
] |
Mathlib/Data/Fintype/Pi.lean
|
Fintype.piFinset_subset
|
[] |
[
57,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.Nodup.set
|
[] |
[
411,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
11
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.le_normalizer
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.383552\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.383561\ninst✝ : AddGroup A\nH K : Subgroup G\nx : G\nxH : x ∈ H\nn : G\n⊢ n ∈ H ↔ x * n * x⁻¹ ∈ H",
"tactic": "rw [H.mul_mem_cancel_right (H.inv_mem xH), H.mul_mem_cancel_left xH]"
}
] |
[
2173,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2172,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.pullbackDiagonalMapIso_inv_fst
|
[
{
"state_after": "C : Type u_2\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ (Iso.mk (lift (snd ≫ fst) (snd ≫ snd) (_ : (snd ≫ fst) ≫ i₁ = (snd ≫ snd) ≫ i₂))\n (lift (fst ≫ i₁ ≫ fst)\n (map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd))\n (_ :\n (fst ≫ i₁ ≫ fst) ≫ diagonal f =\n map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd) ≫\n map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i (_ : (i₁ ≫ snd) ≫ i = (i₁ ≫ fst) ≫ f)\n (_ : (i₂ ≫ snd) ≫ i = (i₂ ≫ fst) ≫ f)))).inv ≫\n fst =\n fst ≫ i₁ ≫ fst",
"state_before": "C : Type u_2\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ (pullbackDiagonalMapIso f i i₁ i₂).inv ≫ fst = fst ≫ i₁ ≫ fst",
"tactic": "delta pullbackDiagonalMapIso"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ (Iso.mk (lift (snd ≫ fst) (snd ≫ snd) (_ : (snd ≫ fst) ≫ i₁ = (snd ≫ snd) ≫ i₂))\n (lift (fst ≫ i₁ ≫ fst)\n (map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd))\n (_ :\n (fst ≫ i₁ ≫ fst) ≫ diagonal f =\n map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) (𝟙 V₁) (𝟙 V₂) snd (_ : i₁ ≫ snd = 𝟙 V₁ ≫ i₁ ≫ snd)\n (_ : i₂ ≫ snd = 𝟙 V₂ ≫ i₂ ≫ snd) ≫\n map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i (_ : (i₁ ≫ snd) ≫ i = (i₁ ≫ fst) ≫ f)\n (_ : (i₂ ≫ snd) ≫ i = (i₂ ≫ fst) ≫ f)))).inv ≫\n fst =\n fst ≫ i₁ ≫ fst",
"tactic": "simp"
}
] |
[
172,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.prodComm_symm
|
[] |
[
944,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
943,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesAt.analyticAt
|
[] |
[
420,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/ModelTheory/FinitelyGenerated.lean
|
FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family
|
[
{
"state_after": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) ↔ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ FG N ↔ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"tactic": "rw [fg_def]"
},
{
"state_after": "case mp\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) → ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N\n\ncase mpr\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ (∃ n s, LowerAdjoint.toFun (closure L) (range s) = N) → ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N",
"state_before": "L : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) ↔ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nS : Set M\nSfin : Set.Finite S\nhS : LowerAdjoint.toFun (closure L) S = N\n⊢ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"state_before": "case mp\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ (∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N) → ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"tactic": "rintro ⟨S, Sfin, hS⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nn : ℕ\nf : Fin n ↪ M\nSfin : Set.Finite (range ↑f)\nhS : LowerAdjoint.toFun (closure L) (range ↑f) = N\n⊢ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"state_before": "case mp.intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nS : Set M\nSfin : Set.Finite S\nhS : LowerAdjoint.toFun (closure L) S = N\n⊢ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"tactic": "obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nn : ℕ\nf : Fin n ↪ M\nSfin : Set.Finite (range ↑f)\nhS : LowerAdjoint.toFun (closure L) (range ↑f) = N\n⊢ ∃ n s, LowerAdjoint.toFun (closure L) (range s) = N",
"tactic": "exact ⟨n, f, hS⟩"
},
{
"state_after": "case mpr.intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nn : ℕ\ns : Fin n → M\nhs : LowerAdjoint.toFun (closure L) (range s) = N\n⊢ ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N",
"state_before": "case mpr\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\n⊢ (∃ n s, LowerAdjoint.toFun (closure L) (range s) = N) → ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N",
"tactic": "rintro ⟨n, s, hs⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nL : Language\nM : Type u_3\ninst✝ : Structure L M\nN : Substructure L M\nn : ℕ\ns : Fin n → M\nhs : LowerAdjoint.toFun (closure L) (range s) = N\n⊢ ∃ S, Set.Finite S ∧ LowerAdjoint.toFun (closure L) S = N",
"tactic": "refine' ⟨range s, finite_range s, hs⟩"
}
] |
[
63,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.zsmul_modEq_zsmul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ z • b - z • a = m • z • p ↔ b - a = m • p",
"tactic": "rw [← smul_sub, smul_comm, smul_right_inj hn]"
}
] |
[
172,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.coe_moebius_mul_coe_zeta
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Ring R\n⊢ ↑μ * ↑ζ = 1",
"tactic": "rw [← coe_coe, ← intCoe_mul, moebius_mul_coe_zeta, intCoe_one]"
}
] |
[
1051,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/Order/Minimal.lean
|
maximals_mono
|
[
{
"state_after": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\ninst✝ : IsAntisymm α r₂\nh : ∀ (a b : α), r₁ a b → r₂ a b\na : α\nha : a ∈ maximals r₂ s\nb : α\nhb : b ∈ s\nhab : r₁ a b\nthis : a = b\n⊢ r₁ b a",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\ninst✝ : IsAntisymm α r₂\nh : ∀ (a b : α), r₁ a b → r₂ a b\na : α\nha : a ∈ maximals r₂ s\nb : α\nhb : b ∈ s\nhab : r₁ a b\n⊢ r₁ b a",
"tactic": "have := eq_of_mem_maximals ha hb (h _ _ hab)"
},
{
"state_after": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b : α\ninst✝ : IsAntisymm α r₂\nh : ∀ (a b : α), r₁ a b → r₂ a b\na : α\nha : a ∈ maximals r₂ s\nhb : a ∈ s\nhab : r₁ a a\n⊢ r₁ a a",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\ninst✝ : IsAntisymm α r₂\nh : ∀ (a b : α), r₁ a b → r₂ a b\na : α\nha : a ∈ maximals r₂ s\nb : α\nhb : b ∈ s\nhab : r₁ a b\nthis : a = b\n⊢ r₁ b a",
"tactic": "subst this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b : α\ninst✝ : IsAntisymm α r₂\nh : ∀ (a b : α), r₁ a b → r₂ a b\na : α\nha : a ∈ maximals r₂ s\nhb : a ∈ s\nhab : r₁ a a\n⊢ r₁ a a",
"tactic": "exact hab"
}
] |
[
138,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Order/Minimal.lean
|
IsAntichain.max_maximals
|
[
{
"state_after": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b : α\nht : IsAntichain r t\nh : maximals r s ⊆ t\nhs : ∀ ⦃a : α⦄, a ∈ t → ∃ b, b ∈ maximals r s ∧ r b a\na : α\nha : a ∈ t\n⊢ a ∈ maximals r s",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na b : α\nht : IsAntichain r t\nh : maximals r s ⊆ t\nhs : ∀ ⦃a : α⦄, a ∈ t → ∃ b, b ∈ maximals r s ∧ r b a\n⊢ maximals r s = t",
"tactic": "refine' h.antisymm fun a ha => _"
},
{
"state_after": "case intro.intro\nα : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\nht : IsAntichain r t\nh : maximals r s ⊆ t\nhs : ∀ ⦃a : α⦄, a ∈ t → ∃ b, b ∈ maximals r s ∧ r b a\na : α\nha : a ∈ t\nb : α\nhb : b ∈ maximals r s\nhr : r b a\n⊢ a ∈ maximals r s",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b : α\nht : IsAntichain r t\nh : maximals r s ⊆ t\nhs : ∀ ⦃a : α⦄, a ∈ t → ∃ b, b ∈ maximals r s ∧ r b a\na : α\nha : a ∈ t\n⊢ a ∈ maximals r s",
"tactic": "obtain ⟨b, hb, hr⟩ := hs ha"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\nht : IsAntichain r t\nh : maximals r s ⊆ t\nhs : ∀ ⦃a : α⦄, a ∈ t → ∃ b, b ∈ maximals r s ∧ r b a\na : α\nha : a ∈ t\nb : α\nhb : b ∈ maximals r s\nhr : r b a\n⊢ a ∈ maximals r s",
"tactic": "rwa [of_not_not fun hab => ht (h hb) ha (Ne.symm hab) hr]"
}
] |
[
203,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.add_nat
|
[] |
[
241,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Matrix.toLinearMapₛₗ₂'_aux_eq
|
[] |
[
194,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Analysis/Convex/Hull.lean
|
Convex.convexHull_eq
|
[] |
[
89,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.mem_nhds_iff
|
[] |
[
945,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
944,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.card_add
|
[] |
[
735,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
734,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
hnot_top
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.176665\nα : Type u_1\nβ : Type ?u.176671\ninst✝ : CoheytingAlgebra α\na b c : α\n⊢ ￢⊤ = ⊥",
"tactic": "rw [← top_sdiff', sdiff_self]"
}
] |
[
1076,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1076,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ns t : Set (PrimeSpectrum R)\nhs : IsClosed s\nht : IsClosed t\n⊢ s ⊂ t ↔ vanishingIdeal t < vanishingIdeal s",
"tactic": "rw [Set.ssubset_def, vanishingIdeal_anti_mono_iff hs, vanishingIdeal_anti_mono_iff ht,\n lt_iff_le_not_le]"
}
] |
[
498,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Topology/Basic.lean
|
mem_closure_iff_nhds_neBot
|
[] |
[
1295,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1294,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.image
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → β\na✝ : Fintype ↑s\n⊢ Set.Finite (f '' s)",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → β\nhs : Set.Finite s\n⊢ Set.Finite (f '' s)",
"tactic": "cases hs"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nf : α → β\na✝ : Fintype ↑s\n⊢ Set.Finite (f '' s)",
"tactic": "apply toFinite"
}
] |
[
854,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/LinearAlgebra/Matrix/Block.lean
|
Matrix.blockTriangular_blockDiagonal
|
[
{
"state_after": "case mk.mk\nα : Type u_1\nβ : Type ?u.32691\nm : Type u_2\nn : Type ?u.32697\no : Type ?u.32700\nm' : α → Type ?u.32705\nn' : α → Type ?u.32710\nR : Type v\ninst✝² : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\nd : α → Matrix m m R\ni : m\ni' : α\nj : m\nj' : α\nh : (j, j').snd < (i, i').snd\n⊢ blockDiagonal d (i, i') (j, j') = 0",
"state_before": "α : Type u_1\nβ : Type ?u.32691\nm : Type u_2\nn : Type ?u.32697\no : Type ?u.32700\nm' : α → Type ?u.32705\nn' : α → Type ?u.32710\nR : Type v\ninst✝² : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\nd : α → Matrix m m R\n⊢ BlockTriangular (blockDiagonal d) Prod.snd",
"tactic": "rintro ⟨i, i'⟩ ⟨j, j'⟩ h"
},
{
"state_after": "case mk.mk.a\nα : Type u_1\nβ : Type ?u.32691\nm : Type u_2\nn : Type ?u.32697\no : Type ?u.32700\nm' : α → Type ?u.32705\nn' : α → Type ?u.32710\nR : Type v\ninst✝² : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\nd : α → Matrix m m R\ni : m\ni' : α\nj : m\nj' : α\nh : (j, j').snd < (i, i').snd\n⊢ { fst := j', snd := j }.fst < { fst := i', snd := i }.fst",
"state_before": "case mk.mk\nα : Type u_1\nβ : Type ?u.32691\nm : Type u_2\nn : Type ?u.32697\no : Type ?u.32700\nm' : α → Type ?u.32705\nn' : α → Type ?u.32710\nR : Type v\ninst✝² : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\nd : α → Matrix m m R\ni : m\ni' : α\nj : m\nj' : α\nh : (j, j').snd < (i, i').snd\n⊢ blockDiagonal d (i, i') (j, j') = 0",
"tactic": "rw [blockDiagonal'_eq_blockDiagonal, blockTriangular_blockDiagonal']"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.a\nα : Type u_1\nβ : Type ?u.32691\nm : Type u_2\nn : Type ?u.32697\no : Type ?u.32700\nm' : α → Type ?u.32705\nn' : α → Type ?u.32710\nR : Type v\ninst✝² : CommRing R\nM N : Matrix m m R\nb : m → α\ninst✝¹ : Preorder α\ninst✝ : DecidableEq α\nd : α → Matrix m m R\ni : m\ni' : α\nj : m\nj' : α\nh : (j, j').snd < (i, i').snd\n⊢ { fst := j', snd := j }.fst < { fst := i', snd := i }.fst",
"tactic": "exact h"
}
] |
[
123,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.402778\nι : Type ?u.402781\ninst✝⁴ : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\nf✝ g : α → β\ninst✝¹ : LinearOrder α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\na b : α\n⊢ AEStronglyMeasurable f (Measure.restrict μ (Ι a b)) ↔\n AEStronglyMeasurable f (Measure.restrict μ (Ioc a b)) ∧ AEStronglyMeasurable f (Measure.restrict μ (Ioc b a))",
"tactic": "rw [uIoc_eq_union, aestronglyMeasurable_union_iff]"
}
] |
[
1729,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1724,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
zpow_bit0_abs
|
[] |
[
221,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.join_map_join
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ bind μ join = join (join μ)",
"state_before": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ join (map join μ) = join (join μ)",
"tactic": "show bind μ join = join (join μ)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ bind μ join = bind μ fun a => bind (id a) id",
"state_before": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ bind μ join = join (join μ)",
"tactic": "rw [join_eq_bind, join_eq_bind, bind_bind measurable_id measurable_id]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ join = fun a => bind (id a) id",
"state_before": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ bind μ join = bind μ fun a => bind (id a) id",
"tactic": "apply congr_arg (bind μ)"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\nν : Measure (Measure α)\n⊢ join ν = bind (id ν) id",
"state_before": "α : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\n⊢ join = fun a => bind (id a) id",
"tactic": "funext ν"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.42082\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure (Measure α))\nν : Measure (Measure α)\n⊢ join ν = bind (id ν) id",
"tactic": "exact join_eq_bind ν"
}
] |
[
227,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.lt_inv_iff_lt_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.268007\nβ : Type ?u.268010\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ a < b⁻¹ ↔ b < a⁻¹",
"tactic": "simpa only [inv_inv] using @ENNReal.inv_lt_inv a⁻¹ b"
}
] |
[
1487,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1486,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.not_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.50009\nι : Sort x\nf g : Filter α\ns t : Set α\n⊢ ¬f ≤ g ↔ ∃ s, s ∈ g ∧ ¬s ∈ f",
"tactic": "simp_rw [le_def, not_forall, exists_prop]"
}
] |
[
336,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
11
] |
Mathlib/Analysis/Convex/Slope.lean
|
concaveOn_iff_slope_anti_adjacent
|
[] |
[
213,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_int_add
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.133790\nα : Type u_1\nβ : Type ?u.133796\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ : α\nz : ℤ\na : α\n⊢ ⌊↑z + a⌋ = z + ⌊a⌋",
"tactic": "simpa only [add_comm] using floor_add_int a z"
}
] |
[
748,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.lt_of_neg_lt_neg
|
[] |
[
885,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
884,
11
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Int.addRight_one_isCycle
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.3068304\nα : Type ?u.3068307\nβ : Type ?u.3068310\ninst✝ : DecidableEq α\nn : ℤ\nx✝ : ↑(Equiv.addRight 1) n ≠ n\n⊢ ↑(Equiv.addRight 1 ^ n) 0 = n",
"tactic": "simp"
}
] |
[
1864,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1863,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.aeval_modByMonic_eq_self_of_root
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\np✝ q✝ : R[X]\ninst✝¹ : Ring S\ninst✝ : Algebra R S\np q : R[X]\nhq : Monic q\nx : S\nhx : ↑(aeval x) q = 0\n⊢ ↑(aeval x) (p %ₘ q) = ↑(aeval x) p",
"tactic": "rw [modByMonic_eq_sub_mul_div p hq, _root_.map_sub, _root_.map_mul, hx, MulZeroClass.zero_mul,\n sub_zero]"
}
] |
[
119,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.op_smul_set_mul_eq_mul_smul_set
|
[] |
[
855,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
853,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.enumOrd_mem
|
[] |
[
2178,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2177,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.quotientMap_injective
|
[] |
[
411,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
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