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Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | CircleDeg1Lift.ext | [] | [
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Mathlib/MeasureTheory/Measure/Content.lean | MeasureTheory.Content.innerContent_iSup_nat | [
{
"state_after": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\n⊢ innerContent μ (⨆ (i : ℕ), U i) ≤ ∑' (i : ℕ), innerContent μ (U i)",
"tactic": "refine' iSup₂_le fun K hK => _"
},
{
"state_after": "case intro\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"tactic": "obtain ⟨t, ht⟩ :=\n K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])"
},
{
"state_after": "case intro.intro.intro.intro\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"state_before": "case intro\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"tactic": "rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with\n ⟨K', h1K', h2K', h3K'⟩"
},
{
"state_after": "case intro.intro.intro.intro\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"state_before": "case intro.intro.intro.intro\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"tactic": "let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩"
},
{
"state_after": "case h.e'_3.h.e'_1\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ K = Finset.sup t L\n\ncase intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ (Finset.sum t fun i => (fun s => ↑(toFun μ s)) (L i)) ≤ ∑' (i : ℕ), innerContent μ (U i)",
"state_before": "case intro.intro.intro.intro\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)",
"tactic": "convert le_trans (h3 t L) _"
},
{
"state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ ∀ (i : ℕ), i ∈ t → (fun s => ↑(toFun μ s)) (L i) ≤ innerContent μ (U i)",
"state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ (Finset.sum t fun i => (fun s => ↑(toFun μ s)) (L i)) ≤ ∑' (i : ℕ), innerContent μ (U i)",
"tactic": "refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t)"
},
{
"state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : ℕ\na✝ : i ∈ t\n⊢ (fun s => ↑(toFun μ s)) (L i) ≤ innerContent μ (U i)",
"state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ ∀ (i : ℕ), i ∈ t → (fun s => ↑(toFun μ s)) (L i) ≤ innerContent μ (U i)",
"tactic": "intro i _"
},
{
"state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : ℕ\na✝ : i ∈ t\n⊢ (fun s => ↑(toFun μ s)) (L i) ≤ ⨆ (_ : ↑(L i) ⊆ ↑(U i)), (fun s => ↑(toFun μ s)) (L i)",
"state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : ℕ\na✝ : i ∈ t\n⊢ (fun s => ↑(toFun μ s)) (L i) ≤ innerContent μ (U i)",
"tactic": "refine' le_trans _ (le_iSup _ (L i))"
},
{
"state_after": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : ℕ\na✝ : i ∈ t\n⊢ (fun s => ↑(toFun μ s)) (L i) ≤ (fun s => ↑(toFun μ s)) (L i)",
"state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : ℕ\na✝ : i ∈ t\n⊢ (fun s => ↑(toFun μ s)) (L i) ≤ ⨆ (_ : ↑(L i) ⊆ ↑(U i)), (fun s => ↑(toFun μ s)) (L i)",
"tactic": "refine' le_trans _ (le_iSup _ (h2K' i))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.convert_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\ni : ℕ\na✝ : i ∈ t\n⊢ (fun s => ↑(toFun μ s)) (L i) ≤ (fun s => ↑(toFun μ s)) (L i)",
"tactic": "rfl"
},
{
"state_after": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\n⊢ (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\n⊢ ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)",
"tactic": "intro t K"
},
{
"state_after": "case refine'_1\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\n⊢ (fun s => ↑(toFun μ s)) (Finset.sup ∅ K) ≤ Finset.sum ∅ fun i => (fun s => ↑(toFun μ s)) (K i)\n\ncase refine'_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\n⊢ ∀ ⦃a : ℕ⦄ {s : Finset ℕ},\n ¬a ∈ s →\n ((fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)) →\n (fun s => ↑(toFun μ s)) (Finset.sup (insert a s) K) ≤\n Finset.sum (insert a s) fun i => (fun s => ↑(toFun μ s)) (K i)",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\n⊢ (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)",
"tactic": "refine' Finset.induction_on t _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\n⊢ (fun s => ↑(toFun μ s)) (Finset.sup ∅ K) ≤ Finset.sum ∅ fun i => (fun s => ↑(toFun μ s)) (K i)",
"tactic": "simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]"
},
{
"state_after": "case refine'_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\nn : ℕ\ns : Finset ℕ\nhn : ¬n ∈ s\nih : (fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)\n⊢ (fun s => ↑(toFun μ s)) (Finset.sup (insert n s) K) ≤ Finset.sum (insert n s) fun i => (fun s => ↑(toFun μ s)) (K i)",
"state_before": "case refine'_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\n⊢ ∀ ⦃a : ℕ⦄ {s : Finset ℕ},\n ¬a ∈ s →\n ((fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)) →\n (fun s => ↑(toFun μ s)) (Finset.sup (insert a s) K) ≤\n Finset.sum (insert a s) fun i => (fun s => ↑(toFun μ s)) (K i)",
"tactic": "intro n s hn ih"
},
{
"state_after": "case refine'_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\nn : ℕ\ns : Finset ℕ\nhn : ¬n ∈ s\nih : (fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)\n⊢ (fun s => ↑(toFun μ s)) (K n ⊔ Finset.sup s K) ≤\n (fun s => ↑(toFun μ s)) (K n) + Finset.sum s fun x => (fun s => ↑(toFun μ s)) (K x)",
"state_before": "case refine'_2\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nt : Finset ℕ\nK : ℕ → Compacts G\nn : ℕ\ns : Finset ℕ\nhn : ¬n ∈ s\nih : (fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)\n⊢ (fun s => ↑(toFun μ s)) (Finset.sup (insert n s) K) ≤ Finset.sum (insert n s) fun i => (fun s => ↑(toFun μ s)) (K i)",
"tactic": "rw [Finset.sup_insert, Finset.sum_insert hn]"
},
{
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"tactic": "exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)"
},
{
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"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\n⊢ ↑K ⊆ ⋃ (i : ℕ), ↑(U i)",
"tactic": "rwa [← Opens.coe_iSup]"
},
{
"state_after": "case h.e'_3.h.e'_1.h\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ ↑K = ↑(Finset.sup t L)",
"state_before": "case h.e'_3.h.e'_1\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ K = Finset.sup t L",
"tactic": "ext1"
},
{
"state_after": "case h.e'_3.h.e'_1.h\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ ↑K = ⨆ (a : ℕ) (_ : a ∈ t), ↑(L a)",
"state_before": "case h.e'_3.h.e'_1.h\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ ↑K = ↑(Finset.sup t L)",
"tactic": "rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_1.h\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nU : ℕ → Opens G\nh3 :\n ∀ (t : Finset ℕ) (K : ℕ → Compacts G),\n (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ (i : ℕ), U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ (i : ℕ) (_ : i ∈ t), ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i\nh3K' : ↑K = ⋃ (i : ℕ) (_ : i ∈ t), K' i\nL : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }\n⊢ ↑K = ⨆ (a : ℕ) (_ : a ∈ t), ↑(L a)",
"tactic": "exact h3K'"
}
] | [
196,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
] |
Mathlib/Data/List/Zip.lean | List.mem_zip | [
{
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"tactic": "cases' h with _ _ _ h"
},
{
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"tactic": "simp"
},
{
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"state_before": "case tail\nα : Type u\nβ : Type u_1\nγ : Type ?u.74738\nδ : Type ?u.74741\nε : Type ?u.74744\na : α\nb : β\nhead✝¹ : α\nl₁ : List α\nhead✝ : β\nl₂ : List β\nh : Mem (a, b) (zipWith Prod.mk l₁ l₂)\n⊢ a ∈ head✝¹ :: l₁ ∧ b ∈ head✝ :: l₂",
"tactic": "have := mem_zip h"
},
{
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"tactic": "exact ⟨Mem.tail _ this.1, Mem.tail _ this.2⟩"
}
] | [
178,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
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Mathlib/Data/Matrix/Basic.lean | Matrix.row_mulVec | [
{
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"tactic": "ext"
},
{
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"tactic": "rfl"
}
] | [
2723,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2720,
1
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Mathlib/RingTheory/Localization/Basic.lean | Localization.algEquiv_mk' | [] | [
1070,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1069,
8
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Mathlib/RingTheory/Ideal/Operations.lean | Ideal.comap_isPrime | [
{
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"tactic": "rwa [mem_comap, map_mul] at h"
}
] | [
1497,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1496,
1
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Mathlib/Analysis/Convex/Basic.lean | convex_iff_pairwise_pos | [
{
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"tactic": "refine' convex_iff_forall_pos.trans ⟨fun h x hx y hy _ => h hx hy, _⟩"
},
{
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"tactic": "intro h x hx y hy a b ha hb hab"
},
{
"state_after": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.14989\nβ : Type ?u.14992\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhy : x ∈ s\n⊢ a • x + b • x ∈ s\n\ncase inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.14989\nβ : Type ?u.14992\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxy : x ≠ y\n⊢ a • x + b • y ∈ s",
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"tactic": "obtain rfl | hxy := eq_or_ne x y"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.14989\nβ : Type ?u.14992\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhy : x ∈ s\n⊢ a • x + b • x ∈ s",
"tactic": "rwa [Convex.combo_self hab]"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.14989\nβ : Type ?u.14992\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxy : x ≠ y\n⊢ a • x + b • y ∈ s",
"tactic": "exact h hx hy hxy ha hb hab"
}
] | [
163,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Topology/Connected.lean | IsConnected.iUnion_of_chain | [
{
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"tactic": "rw [inter_comm]"
},
{
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"tactic": "exact K i"
}
] | [
255,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/FieldTheory/RatFunc.lean | RatFunc.map_apply_ofFractionRing_mk | [
{
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"tactic": "refine (liftOn_ofFractionRing_mk n _ _ _).trans ?_"
},
{
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"tactic": "rw [dif_pos]"
}
] | [
657,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
651,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.mk_cons | [] | [
861,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
859,
1
] |
Mathlib/Algebra/Order/Pi.lean | Function.one_lt_const | [] | [
145,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/List/BigOperators/Lemmas.lean | unop_map_list_prod | [
{
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"tactic": "rw [map_list_prod f l, MulOpposite.unop_list_prod, List.map_map]"
}
] | [
165,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
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Mathlib/Algebra/Order/ToIntervalMod.lean | Ico_eq_locus_Ioc_eq_iUnion_Ioo | [
{
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"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c : α\nn : ℤ\n⊢ {b | toIcoMod hp a b = toIocMod hp a b} = ⋃ (z : ℤ), Set.Ioo (a + z • p) (a + p + z • p)",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c : α\nn : ℤ\nx✝ : α\n⊢ x✝ ∈ {b | toIcoMod hp a b = toIocMod hp a b} ↔ x✝ ∈ ⋃ (z : ℤ), Set.Ioo (a + z • p) (a + p + z • p)",
"tactic": "simp_rw [Set.mem_setOf, Set.mem_iUnion, ← Set.sub_mem_Ioo_iff_left, ←\n not_modEq_iff_toIcoMod_eq_toIocMod, modEq_iff_not_forall_mem_Ioo_mod hp, not_forall,\n Classical.not_not]"
}
] | [
693,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
688,
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Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoDiv_add_zsmul' | [
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ b - (toIcoDiv hp a b - m) • p ∈ Set.Ico (a + m • p) (a + m • p + p)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoDiv hp (a + m • p) b = toIcoDiv hp a b - m",
"tactic": "refine' toIcoDiv_eq_of_sub_zsmul_mem_Ico _ _"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ b - toIcoDiv hp a b • p + m • p ∈ Set.Ico (a + m • p) (a + p + m • p)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ b - (toIcoDiv hp a b - m) • p ∈ Set.Ico (a + m • p) (a + m • p + p)",
"tactic": "rw [sub_smul, ← sub_add, add_right_comm]"
},
{
"state_after": "no goals",
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"tactic": "simpa using sub_toIcoDiv_zsmul_mem_Ico hp a b"
}
] | [
245,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
241,
1
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Mathlib/MeasureTheory/Measure/GiryMonad.lean | MeasureTheory.Measure.measurable_lintegral | [
{
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"state_before": "α : Type u_1\nβ : Type ?u.4410\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ Measurable fun μ => ∫⁻ (x : α), f x ∂μ",
"tactic": "simp only [lintegral_eq_iSup_eapprox_lintegral, hf, SimpleFunc.lintegral]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.4410\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → ℝ≥0∞\nhf : Measurable f\nn : ℕ\ni : ℝ≥0∞\nx✝ : i ∈ SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n)\n⊢ Measurable fun μ => i * ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {i})",
"state_before": "α : Type u_1\nβ : Type ?u.4410\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ Measurable fun μ =>\n ⨆ (n : ℕ),\n ∑ x in SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n),\n x * ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {x})",
"tactic": "refine' measurable_iSup fun n => Finset.measurable_sum _ fun i _ => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.4410\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → ℝ≥0∞\nhf : Measurable f\nn : ℕ\ni : ℝ≥0∞\nx✝ : i ∈ SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n)\n⊢ Measurable fun μ => ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {i})",
"state_before": "α : Type u_1\nβ : Type ?u.4410\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → ℝ≥0∞\nhf : Measurable f\nn : ℕ\ni : ℝ≥0∞\nx✝ : i ∈ SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n)\n⊢ Measurable fun μ => i * ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {i})",
"tactic": "refine' Measurable.const_mul _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4410\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → ℝ≥0∞\nhf : Measurable f\nn : ℕ\ni : ℝ≥0∞\nx✝ : i ∈ SimpleFunc.range (SimpleFunc.eapprox (fun x => f x) n)\n⊢ Measurable fun μ => ↑↑μ (↑(SimpleFunc.eapprox (fun x => f x) n) ⁻¹' {i})",
"tactic": "exact measurable_coe ((SimpleFunc.eapprox f n).measurableSet_preimage _)"
}
] | [
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75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/Computability/TuringMachine.lean | Turing.TM2to1.tr_respects_aux₃ | [
{
"state_after": "case zero\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[Nat.zero]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n\ncase succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[Nat.succ n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }",
"state_before": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\n⊢ Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }",
"tactic": "induction' n with n IH"
},
{
"state_after": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) } ∈\n TM1.step (tr M)\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[Nat.succ n]) (Tape.mk' ∅ (addBottom L)) }",
"state_before": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[Nat.succ n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }",
"tactic": "refine' Reaches₀.head _ IH"
},
{
"state_after": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ some (TM1.stepAux (tr M (ret q)) v ((Tape.move Dir.right^[Nat.succ n]) (Tape.mk' ∅ (addBottom L)))) =\n some { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }",
"state_before": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) } ∈\n TM1.step (tr M)\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[Nat.succ n]) (Tape.mk' ∅ (addBottom L)) }",
"tactic": "simp only [Option.mem_def, TM1.step]"
},
{
"state_after": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ (bif false then\n TM1.stepAux (trNormal q) v (Tape.move Dir.right ((Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L))))\n else TM1.stepAux (goto fun x x => ret q) v ((Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)))) =\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }",
"state_before": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ some (TM1.stepAux (tr M (ret q)) v ((Tape.move Dir.right^[Nat.succ n]) (Tape.mk' ∅ (addBottom L)))) =\n some { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }",
"tactic": "rw [Option.some_inj, tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat,\n addBottom_nth_succ_fst, TM1.stepAux, iterate_succ', Function.comp_apply, Tape.move_right_left]"
},
{
"state_after": "no goals",
"state_before": "case succ\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\nn : ℕ\nIH :\n Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }\n⊢ (bif false then\n TM1.stepAux (trNormal q) v (Tape.move Dir.right ((Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L))))\n else TM1.stepAux (goto fun x x => ret q) v ((Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)))) =\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[n]) (Tape.mk' ∅ (addBottom L)) }",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case zero\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ Reaches₀ (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right^[Nat.zero]) (Tape.mk' ∅ (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' ∅ (addBottom L) }",
"tactic": "rfl"
}
] | [
2678,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2670,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | contDiff_const_smul | [] | [
1540,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1539,
1
] |
Mathlib/LinearAlgebra/Prod.lean | LinearMap.prodMap_comap_prod | [] | [
325,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | ContDiffOn.exp | [] | [
269,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
] |
Mathlib/Algebra/Algebra/Operations.lean | Submodule.bot_mul | [] | [
216,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.not_isBoundedUnder_of_tendsto_atBot | [] | [
139,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Std/Data/Int/Lemmas.lean | Int.zero_sub | [
{
"state_after": "no goals",
"state_before": "a : Int\n⊢ 0 - a = -a",
"tactic": "simp [Int.sub_eq_add_neg]"
}
] | [
347,
82
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
347,
11
] |
Mathlib/Algebra/Divisibility/Units.lean | IsUnit.dvd | [
{
"state_after": "case intro\nα : Type u_1\ninst✝ : Monoid α\na b : α\nu : αˣ\n⊢ ↑u ∣ a",
"state_before": "α : Type u_1\ninst✝ : Monoid α\na b u : α\nhu : IsUnit u\n⊢ u ∣ a",
"tactic": "rcases hu with ⟨u, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : Monoid α\na b : α\nu : αˣ\n⊢ ↑u ∣ a",
"tactic": "apply Units.coe_dvd"
}
] | [
81,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
Mathlib/Analysis/Convex/Function.lean | ConvexOn.add_strictConvexOn | [] | [
513,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
511,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean | CategoryTheory.StructuredArrow.ext_iff | [] | [
147,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Data/Set/Basic.lean | Set.union_univ | [] | [
878,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
877,
1
] |
Mathlib/Order/SuccPred/Basic.lean | WithTop.pred_untop | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.55953\ninst✝² : Preorder α\ninst✝¹ : OrderTop α\ninst✝ : PredOrder α\na : WithTop α\nha : a ≠ ⊤\n⊢ pred a ≠ ⊤",
"tactic": "induction a using WithTop.recTopCoe <;> simp"
}
] | [
1113,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1109,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean | BoxIntegral.Integrable.tendsto_integralSum_toFilteriUnion_single | [] | [
548,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
544,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.map_mul | [] | [
254,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
253,
11
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_cubic_le | [] | [
1234,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1233,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_right_pure_add_one | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12341\nγ : Type ?u.12344\ninst✝³ : LinearOrderedRing α\ninst✝² : FloorRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto ceil (𝓝[Ioi ↑n] ↑n) (pure (n + 1))",
"tactic": "simpa only [floor_intCast] using tendsto_ceil_right_pure_floor_add_one (n : α)"
}
] | [
112,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Logic/Equiv/List.lean | Denumerable.lower_raise' | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.56215\nβ : Type ?u.56218\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nm : ℕ\nl : List ℕ\nn : ℕ\n⊢ lower' (raise' (m :: l) n) n = m :: l",
"tactic": "simp [raise', lower', add_tsub_cancel_right, lower_raise']"
}
] | [
363,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | ZMod.χ₈'_eq_χ₄_mul_χ₈ | [
{
"state_after": "case head\n\n⊢ ↑χ₈' { val := 0, isLt := (_ : 0 < 7 + 1) } =\n ↑χ₄ ↑{ val := 0, isLt := (_ : 0 < 7 + 1) } * ↑χ₈ { val := 0, isLt := (_ : 0 < 7 + 1) }\n\ncase tail.head\n\n⊢ ↑χ₈' { val := 1, isLt := (_ : (fun a => a < 7 + 1) 1) } =\n ↑χ₄ ↑{ val := 1, isLt := (_ : (fun a => a < 7 + 1) 1) } * ↑χ₈ { val := 1, isLt := (_ : (fun a => a < 7 + 1) 1) }\n\ncase tail.tail.head\n\n⊢ ↑χ₈' { val := 2, isLt := (_ : (fun a => (fun a => a < 7 + 1) a) 2) } =\n ↑χ₄ ↑{ val := 2, isLt := (_ : (fun a => (fun a => a < 7 + 1) a) 2) } *\n ↑χ₈ { val := 2, isLt := (_ : (fun a => (fun a => a < 7 + 1) a) 2) }\n\ncase tail.tail.tail.head\n\n⊢ ↑χ₈' { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 7 + 1) a) a) 3) } =\n ↑χ₄ ↑{ val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 7 + 1) a) a) 3) } *\n ↑χ₈ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 7 + 1) a) a) 3) }\n\ncase tail.tail.tail.tail.head\n\n⊢ ↑χ₈' { val := 4, isLt := (_ : (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) 4) } =\n ↑χ₄ ↑{ val := 4, isLt := (_ : (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) 4) } *\n ↑χ₈ { val := 4, isLt := (_ : (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) 4) }\n\ncase tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈' { val := 5, isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) 5) } =\n ↑χ₄ ↑{ val := 5, isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) 5) } *\n ↑χ₈ { val := 5, isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) 5) }\n\ncase tail.tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈'\n { val := 6,\n isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) 6) } =\n ↑χ₄\n ↑{ val := 6,\n isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) 6) } *\n ↑χ₈\n { val := 6,\n isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) 6) }\n\ncase tail.tail.tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈'\n { val := 7,\n isLt :=\n (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a) 7) } =\n ↑χ₄\n ↑{ val := 7,\n isLt :=\n (_ :\n (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a)\n 7) } *\n ↑χ₈\n { val := 7,\n isLt :=\n (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a) 7) }",
"state_before": "a : ZMod 8\n⊢ ↑χ₈' a = ↑χ₄ ↑a * ↑χ₈ a",
"tactic": "fin_cases a"
},
{
"state_after": "no goals",
"state_before": "case head\n\n⊢ ↑χ₈' { val := 0, isLt := (_ : 0 < 7 + 1) } =\n ↑χ₄ ↑{ val := 0, isLt := (_ : 0 < 7 + 1) } * ↑χ₈ { val := 0, isLt := (_ : 0 < 7 + 1) }\n\ncase tail.head\n\n⊢ ↑χ₈' { val := 1, isLt := (_ : (fun a => a < 7 + 1) 1) } =\n ↑χ₄ ↑{ val := 1, isLt := (_ : (fun a => a < 7 + 1) 1) } * ↑χ₈ { val := 1, isLt := (_ : (fun a => a < 7 + 1) 1) }\n\ncase tail.tail.head\n\n⊢ ↑χ₈' { val := 2, isLt := (_ : (fun a => (fun a => a < 7 + 1) a) 2) } =\n ↑χ₄ ↑{ val := 2, isLt := (_ : (fun a => (fun a => a < 7 + 1) a) 2) } *\n ↑χ₈ { val := 2, isLt := (_ : (fun a => (fun a => a < 7 + 1) a) 2) }\n\ncase tail.tail.tail.head\n\n⊢ ↑χ₈' { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 7 + 1) a) a) 3) } =\n ↑χ₄ ↑{ val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 7 + 1) a) a) 3) } *\n ↑χ₈ { val := 3, isLt := (_ : (fun a => (fun a => (fun a => a < 7 + 1) a) a) 3) }\n\ncase tail.tail.tail.tail.head\n\n⊢ ↑χ₈' { val := 4, isLt := (_ : (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) 4) } =\n ↑χ₄ ↑{ val := 4, isLt := (_ : (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) 4) } *\n ↑χ₈ { val := 4, isLt := (_ : (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) 4) }\n\ncase tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈' { val := 5, isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) 5) } =\n ↑χ₄ ↑{ val := 5, isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) 5) } *\n ↑χ₈ { val := 5, isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) 5) }\n\ncase tail.tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈'\n { val := 6,\n isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) 6) } =\n ↑χ₄\n ↑{ val := 6,\n isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) 6) } *\n ↑χ₈\n { val := 6,\n isLt := (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) 6) }\n\ncase tail.tail.tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈'\n { val := 7,\n isLt :=\n (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a) 7) } =\n ↑χ₄\n ↑{ val := 7,\n isLt :=\n (_ :\n (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a)\n 7) } *\n ↑χ₈\n { val := 7,\n isLt :=\n (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a) 7) }",
"tactic": "all_goals decide"
},
{
"state_after": "no goals",
"state_before": "case tail.tail.tail.tail.tail.tail.tail.head\n\n⊢ ↑χ₈'\n { val := 7,\n isLt :=\n (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a) 7) } =\n ↑χ₄\n ↑{ val := 7,\n isLt :=\n (_ :\n (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a)\n 7) } *\n ↑χ₈\n { val := 7,\n isLt :=\n (_ : (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => (fun a => a < 7 + 1) a) a) a) a) a) a) 7) }",
"tactic": "decide"
}
] | [
216,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.isBigO_sub | [] | [
686,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
684,
1
] |
Mathlib/Combinatorics/DoubleCounting.lean | Finset.card_mul_le_card_mul | [] | [
98,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ici_prod_Ici | [] | [
1888,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1887,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | CategoryTheory.Limits.biprod.braiding'_eq_braiding | [
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nP✝ Q✝ : C\ninst✝ : HasBinaryBiproducts C\nP Q : C\n⊢ braiding' P Q = braiding P Q",
"tactic": "aesop_cat"
}
] | [
1831,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1830,
1
] |
Mathlib/Order/Bounds/Basic.lean | IsLUB.of_image | [] | [
1564,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1561,
1
] |
Mathlib/FieldTheory/SeparableDegree.lean | Polynomial.HasSeparableContraction.dvd_degree' | [] | [
84,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Analysis/Convex/Function.lean | ConvexOn.lt_right_of_left_lt | [
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.618434\nα : Type ?u.618437\nβ : Type u_3\nι : Type ?u.618443\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxz : f x < f (a • x + b • y)\n⊢ f (a • x + b • y) < f y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.618434\nα : Type ?u.618437\nβ : Type u_3\nι : Type ?u.618443\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y z : E\nhx : x ∈ s\nhy : y ∈ s\nhz : z ∈ openSegment 𝕜 x y\nhxz : f x < f z\n⊢ f z < f y",
"tactic": "obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.618434\nα : Type ?u.618437\nβ : Type u_3\nι : Type ?u.618443\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxz : f x < f (a • x + b • y)\n⊢ f (a • x + b • y) < f y",
"tactic": "exact hf.lt_right_of_left_lt' hx hy ha hb hab hxz"
}
] | [
817,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
814,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.restrictScalars_bot_eq_self | [
{
"state_after": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\nx : E\n⊢ x ∈ restrictScalars F ⊥ ↔ x ∈ K",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\n⊢ restrictScalars F ⊥ = K",
"tactic": "ext x"
},
{
"state_after": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap { x // x ∈ K } E) ↔ x ∈ K",
"state_before": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\nx : E\n⊢ x ∈ restrictScalars F ⊥ ↔ x ∈ K",
"tactic": "rw [mem_restrictScalars, mem_bot]"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap { x // x ∈ K } E) ↔ x ∈ K",
"tactic": "exact Set.ext_iff.mp Subtype.range_coe x"
}
] | [
265,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Mathlib/Analysis/Calculus/Deriv/Slope.lean | hasDerivAt_iff_tendsto_slope | [] | [
88,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.sign_toReal | [
{
"state_after": "case inl\nθ : Angle\nh : θ ≠ ↑π\nht : toReal θ < 0\n⊢ ↑SignType.sign (toReal θ) = sign θ\n\ncase inr.inl\nθ : Angle\nh : θ ≠ ↑π\nht : toReal θ = 0\n⊢ ↑SignType.sign (toReal θ) = sign θ\n\ncase inr.inr\nθ : Angle\nh : θ ≠ ↑π\nht : 0 < toReal θ\n⊢ ↑SignType.sign (toReal θ) = sign θ",
"state_before": "θ : Angle\nh : θ ≠ ↑π\n⊢ ↑SignType.sign (toReal θ) = sign θ",
"tactic": "rcases lt_trichotomy θ.toReal 0 with (ht | ht | ht)"
},
{
"state_after": "no goals",
"state_before": "case inl\nθ : Angle\nh : θ ≠ ↑π\nht : toReal θ < 0\n⊢ ↑SignType.sign (toReal θ) = sign θ",
"tactic": "simp [ht, toReal_neg_iff_sign_neg.1 ht]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nθ : Angle\nh : θ ≠ ↑π\nht : toReal θ = 0\n⊢ ↑SignType.sign (toReal θ) = sign θ",
"tactic": "simp [sign, ht, ← sin_toReal]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nθ : Angle\nh : θ ≠ ↑π\nht : 0 < toReal θ\n⊢ ↑SignType.sign (toReal θ) = sign θ",
"tactic": "rw [sign, ← sin_toReal, sign_pos ht,\n sign_pos\n (sin_pos_of_pos_of_lt_pi ht ((toReal_le_pi θ).lt_of_ne (toReal_eq_pi_iff.not.2 h)))]"
}
] | [
929,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
923,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean | Measurable.snd | [] | [
660,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
659,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean | Set.inter_vsub_subset | [] | [
693,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
692,
1
] |
Mathlib/Data/List/Chain.lean | List.chain'_map_of_chain' | [] | [
239,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
237,
1
] |
Mathlib/Algebra/Periodic.lean | Function.Periodic.zsmul | [
{
"state_after": "case ofNat\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.104199\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddGroup α\nh : Periodic f c\nn : ℕ\n⊢ Periodic f (Int.ofNat n • c)\n\ncase negSucc\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.104199\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddGroup α\nh : Periodic f c\nn : ℕ\n⊢ Periodic f (Int.negSucc n • c)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.104199\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddGroup α\nh : Periodic f c\nn : ℤ\n⊢ Periodic f (n • c)",
"tactic": "cases' n with n n"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.104199\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddGroup α\nh : Periodic f c\nn : ℕ\n⊢ Periodic f (Int.ofNat n • c)",
"tactic": "simpa only [Int.ofNat_eq_coe, coe_nat_zsmul] using h.nsmul n"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.104199\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddGroup α\nh : Periodic f c\nn : ℕ\n⊢ Periodic f (Int.negSucc n • c)",
"tactic": "simpa only [negSucc_zsmul] using (h.nsmul (n + 1)).neg"
}
] | [
237,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
234,
11
] |
Mathlib/Data/Polynomial/Splits.lean | Polynomial.roots_ne_zero_of_splits' | [
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\n⊢ False",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval₂ i x f = 0\nh : roots (map i f) = 0\n⊢ False",
"tactic": "rw [← eval_map] at hx"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\nthis : map i f ≠ 0\n⊢ False",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\n⊢ False",
"tactic": "have : f.map i ≠ 0 := by intro; simp_all"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\nthis : map i f ≠ 0\n⊢ False",
"tactic": "cases h.subst ((mem_roots this).2 hx)"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\na✝ : map i f = 0\n⊢ False",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\n⊢ map i f ≠ 0",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) ≠ 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\na✝ : map i f = 0\n⊢ False",
"tactic": "simp_all"
}
] | [
193,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigOWith.prod_rightl | [] | [
953,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
951,
1
] |
Mathlib/Order/Hom/Basic.lean | OrderHom.symm_dual_comp | [] | [
595,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
593,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean | MulHom.mem_srange | [] | [
766,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
765,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_neg_Icc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ -Icc a b = Icc (-b) (-a)",
"tactic": "simp [← Ici_inter_Iic, inter_comm]"
}
] | [
153,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Mathlib/Topology/DenseEmbedding.lean | DenseInducing.mk' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.27032\nδ : Type ?u.27035\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ni✝ : α → β\ndi : DenseInducing i✝\ninst✝¹ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\ninst✝ : TopologicalSpace γ\ni : α → β\nc : Continuous i\ndense : ∀ (x : β), x ∈ closure (range i)\nH : ∀ (a : α) (s : Set α), s ∈ 𝓝 a → ∃ t, t ∈ 𝓝 (i a) ∧ ∀ (b : α), i b ∈ t → b ∈ s\na : α\n⊢ comap i (𝓝 (i a)) ≤ 𝓝 a",
"tactic": "simpa [Filter.le_def] using H a"
}
] | [
228,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | BilinForm.toQuadraticForm_sum | [] | [
708,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
706,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean | MvPolynomial.vars_C_mul | [
{
"state_after": "case a\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\n⊢ i ∈ vars (↑C a * φ) ↔ i ∈ vars φ",
"state_before": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\n⊢ vars (↑C a * φ) = vars φ",
"tactic": "ext1 i"
},
{
"state_after": "case a\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\n⊢ (∃ d, coeff d (↑C a * φ) ≠ 0 ∧ i ∈ d.support) ↔ ∃ d, coeff d φ ≠ 0 ∧ i ∈ d.support",
"state_before": "case a\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\n⊢ i ∈ vars (↑C a * φ) ↔ i ∈ vars φ",
"tactic": "simp only [mem_vars, exists_prop, mem_support_iff]"
},
{
"state_after": "case a.h\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\n⊢ ∀ (a_1 : σ →₀ ℕ), coeff a_1 (↑C a * φ) ≠ 0 ∧ i ∈ a_1.support ↔ coeff a_1 φ ≠ 0 ∧ i ∈ a_1.support",
"state_before": "case a\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\n⊢ (∃ d, coeff d (↑C a * φ) ≠ 0 ∧ i ∈ d.support) ↔ ∃ d, coeff d φ ≠ 0 ∧ i ∈ d.support",
"tactic": "apply exists_congr"
},
{
"state_after": "case a.h\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\nd : σ →₀ ℕ\n⊢ coeff d (↑C a * φ) ≠ 0 ∧ i ∈ d.support ↔ coeff d φ ≠ 0 ∧ i ∈ d.support",
"state_before": "case a.h\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\n⊢ ∀ (a_1 : σ →₀ ℕ), coeff a_1 (↑C a * φ) ≠ 0 ∧ i ∈ a_1.support ↔ coeff a_1 φ ≠ 0 ∧ i ∈ a_1.support",
"tactic": "intro d"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\nd : σ →₀ ℕ\n⊢ coeff d (↑C a * φ) ≠ 0 ↔ coeff d φ ≠ 0",
"state_before": "case a.h\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\nd : σ →₀ ℕ\n⊢ coeff d (↑C a * φ) ≠ 0 ∧ i ∈ d.support ↔ coeff d φ ≠ 0 ∧ i ∈ d.support",
"tactic": "apply and_congr _ Iff.rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.169900\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\np q : MvPolynomial σ R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\na : A\nha : a ≠ 0\nφ : MvPolynomial σ A\ni : σ\nd : σ →₀ ℕ\n⊢ coeff d (↑C a * φ) ≠ 0 ↔ coeff d φ ≠ 0",
"tactic": "rw [coeff_C_mul, mul_ne_zero_iff, eq_true ha, true_and_iff]"
}
] | [
406,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
399,
1
] |
Mathlib/Order/OrdContinuous.lean | LeftOrdContinuous.id | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf : α → β\ns : Set α\nx : α\nh : IsLUB s x\n⊢ IsLUB (id '' s) (id x)",
"tactic": "simpa only [image_id] using h"
}
] | [
59,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
11
] |
Mathlib/Data/Fin/Basic.lean | Fin.mk_one | [] | [
643,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
642,
1
] |
Mathlib/GroupTheory/Submonoid/Inverses.lean | Submonoid.leftInv_leftInv_le | [
{
"state_after": "case intro.mk.intro\nM : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nx y : M\nz : { x // x ∈ S }\nh₁ : y * ↑z = 1\nh₂ : x * y = 1\n⊢ x ∈ S",
"state_before": "M : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\n⊢ leftInv (leftInv S) ≤ S",
"tactic": "rintro x ⟨⟨y, z, h₁⟩, h₂ : x * y = 1⟩"
},
{
"state_after": "case h.e'_4\nM : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nx y : M\nz : { x // x ∈ S }\nh₁ : y * ↑z = 1\nh₂ : x * y = 1\n⊢ x = ↑z",
"state_before": "case intro.mk.intro\nM : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nx y : M\nz : { x // x ∈ S }\nh₁ : y * ↑z = 1\nh₂ : x * y = 1\n⊢ x ∈ S",
"tactic": "convert z.prop"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nM : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nx y : M\nz : { x // x ∈ S }\nh₁ : y * ↑z = 1\nh₂ : x * y = 1\n⊢ x = ↑z",
"tactic": "rw [← mul_one x, ← h₁, ← mul_assoc, h₂, one_mul]"
}
] | [
75,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
] |
Mathlib/Algebra/DirectLimit.lean | Module.DirectLimit.of.zero_exact | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁶ : Preorder ι\nG : ι → Type w\ninst✝⁵ : (i : ι) → AddCommGroup (G i)\ninst✝⁴ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝¹ : DirectedSystem G fun i j h => ↑(f i j h)\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\ni : ι\nx : G i\nH : ↑(of R ι G f i) x = 0\nthis : Nonempty ι\nj : ι\nhj : ∀ (k : ι), k ∈ Dfinsupp.support (↑(DirectSum.lof R ι G i) x) → k ≤ j\nhxj : ↑(DirectSum.toModule R ι (G j) fun i => totalize G (fun i j H => f i j H) i j) (↑(DirectSum.lof R ι G i) x) = 0\nhx0 : x = 0\n⊢ ↑(f i i (_ : i ≤ i)) x = 0",
"tactic": "simp [hx0]"
},
{
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268,
39
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Mathlib/Topology/Algebra/UniformGroup.lean | uniformContinuous_div | [] | [
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37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/RingTheory/IsTensorProduct.lean | IsBaseChange.alg_hom_ext | [
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{
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215,
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Mathlib/Order/Filter/Basic.lean | Filter.disjoint_comap_iff | [
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Mathlib/FieldTheory/Adjoin.lean | IntermediateField.subsingleton_of_finrank_adjoin_eq_one | [] | [
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Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_iff_isBigOWith | [
{
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"state_before": "α : Type u_1\nβ : Type ?u.12161\nE : Type u_2\nF : Type u_3\nG : Type ?u.12170\nE' : Type ?u.12173\nF' : Type ?u.12176\nG' : Type ?u.12179\nE'' : Type ?u.12182\nF'' : Type ?u.12185\nG'' : Type ?u.12188\nR : Type ?u.12191\nR' : Type ?u.12194\n𝕜 : Type ?u.12197\n𝕜' : Type ?u.12200\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ f =O[l] g ↔ ∃ c, IsBigOWith c l f g",
"tactic": "rw [IsBigO_def]"
}
] | [
115,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.coequalizer.π_colimMap_desc | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\nX' Y' Z : C\nf' g' : X' ⟶ Y'\ninst✝ : HasCoequalizer f' g'\np : X ⟶ X'\nq : Y ⟶ Y'\nwf : f ≫ q = p ≫ f'\nwg : g ≫ q = p ≫ g'\nh : Y' ⟶ Z\nwh : f' ≫ h = g' ≫ h\n⊢ π f g ≫ colimMap (parallelPairHom f g f' g' p q wf wg) ≫ desc h wh = q ≫ h",
"tactic": "rw [ι_colimMap_assoc, parallelPairHom_app_one, coequalizer.π_desc]"
}
] | [
991,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
986,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | AlgHom.coe_codRestrict | [] | [
647,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
645,
1
] |
Mathlib/RingTheory/HahnSeries.lean | HahnSeries.embDomain_injective | [
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : embDomain f x = embDomain f y\ng : Γ\n⊢ coeff x g = coeff y g",
"state_before": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : embDomain f x = embDomain f y\n⊢ x = y",
"tactic": "ext g"
},
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : ∀ (a : Γ'), coeff (embDomain f x) a = coeff (embDomain f y) a\ng : Γ\n⊢ coeff x g = coeff y g",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : embDomain f x = embDomain f y\ng : Γ\n⊢ coeff x g = coeff y g",
"tactic": "rw [HahnSeries.ext_iff, Function.funext_iff] at xy"
},
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : ∀ (a : Γ'), coeff (embDomain f x) a = coeff (embDomain f y) a\ng : Γ\nxyg : coeff (embDomain f x) (↑f g) = coeff (embDomain f y) (↑f g)\n⊢ coeff x g = coeff y g",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : ∀ (a : Γ'), coeff (embDomain f x) a = coeff (embDomain f y) a\ng : Γ\n⊢ coeff x g = coeff y g",
"tactic": "have xyg := xy (f g)"
},
{
"state_after": "no goals",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : HahnSeries Γ R\nxy : ∀ (a : Γ'), coeff (embDomain f x) a = coeff (embDomain f y) a\ng : Γ\nxyg : coeff (embDomain f x) (↑f g) = coeff (embDomain f y) (↑f g)\n⊢ coeff x g = coeff y g",
"tactic": "rwa [embDomain_coeff, embDomain_coeff] at xyg"
}
] | [
338,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
333,
1
] |
Std/Logic.lean | or_iff_right_iff_imp | [
{
"state_after": "no goals",
"state_before": "a b : Prop\n⊢ (a ∨ b ↔ b) ↔ a → b",
"tactic": "rw [or_comm, or_iff_left_iff_imp]"
}
] | [
299,
36
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
298,
9
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.erase_add_left_neg | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.98762\nγ : Type ?u.98765\ninst✝ : DecidableEq α\ns✝ t✝ : Multiset α\na✝ b a : α\ns t : Multiset α\nh : ¬a ∈ t\n⊢ erase (s + t) a = erase s a + t",
"tactic": "rw [add_comm, erase_add_right_neg s h, add_comm]"
}
] | [
1065,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1064,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean | ContinuousAffineMap.toContinuousMap_coe | [] | [
105,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.attach_nonempty_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.288285\nγ : Type ?u.288288\ns : Finset α\n⊢ Finset.Nonempty (attach s) ↔ Finset.Nonempty s",
"tactic": "simp [Finset.Nonempty]"
}
] | [
2422,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2421,
1
] |
Std/Data/Rat/Lemmas.lean | Rat.inv_divInt | [
{
"state_after": "no goals",
"state_before": "n d : Int\n⊢ Rat.inv (n /. d) = d /. n",
"tactic": "if z : d = 0 then simp [z] else\ncases e : n /. d; rcases divInt_num_den z e with ⟨g, zg, rfl, rfl⟩\nsimp [inv_def, divInt_mul_right zg]"
},
{
"state_after": "no goals",
"state_before": "n d : Int\nz : d = 0\n⊢ Rat.inv (n /. d) = d /. n",
"tactic": "simp [z]"
},
{
"state_after": "case mk'\nn d : Int\nz : ¬d = 0\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ne : n /. d = mk' num✝ den✝\n⊢ Rat.inv (mk' num✝ den✝) = d /. n",
"state_before": "n d : Int\nz : ¬d = 0\n⊢ Rat.inv (n /. d) = d /. n",
"tactic": "cases e : n /. d"
},
{
"state_after": "case mk'.intro.intro.intro\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng : Int\nzg : g ≠ 0\nz : ¬↑den✝ * g = 0\ne : num✝ * g /. (↑den✝ * g) = mk' num✝ den✝\n⊢ Rat.inv (mk' num✝ den✝) = ↑den✝ * g /. (num✝ * g)",
"state_before": "case mk'\nn d : Int\nz : ¬d = 0\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ne : n /. d = mk' num✝ den✝\n⊢ Rat.inv (mk' num✝ den✝) = d /. n",
"tactic": "rcases divInt_num_den z e with ⟨g, zg, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk'.intro.intro.intro\nnum✝ : Int\nden✝ : Nat\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\ng : Int\nzg : g ≠ 0\nz : ¬↑den✝ * g = 0\ne : num✝ * g /. (↑den✝ * g) = mk' num✝ den✝\n⊢ Rat.inv (mk' num✝ den✝) = ↑den✝ * g /. (num✝ * g)",
"tactic": "simp [inv_def, divInt_mul_right zg]"
}
] | [
302,
38
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
299,
9
] |
Mathlib/FieldTheory/Separable.lean | Polynomial.separable_map | [
{
"state_after": "no goals",
"state_before": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\nf : F →+* K\np : F[X]\n⊢ Separable (map f p) ↔ Separable p",
"tactic": "simp_rw [separable_def, derivative_map, isCoprime_map]"
}
] | [
291,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/Topology/Algebra/StarSubalgebra.lean | StarSubalgebra.embedding_inclusion | [] | [
52,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Mathlib/Order/Heyting/Basic.lean | hnot_hnot_sdiff_distrib | [
{
"state_after": "case refine'_1\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢(a \\ b) ≤ ￢￢a \\ ￢￢b\n\ncase refine'_2\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢a \\ ￢￢b ≤ ￢￢(a \\ b)",
"state_before": "ι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢(a \\ b) = ￢￢a \\ ￢￢b",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢(￢￢a \\ ￢￢b) ≤ ￢(a ⊓ ￢b)",
"state_before": "case refine'_1\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢(a \\ b) ≤ ￢￢a \\ ￢￢b",
"tactic": "refine' hnot_le_comm.1 ((hnot_anti sdiff_le_inf_hnot).trans' _)"
},
{
"state_after": "case refine'_1\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢a ≤ ￢￢a \\ ￢￢b ⊔ ￢￢b",
"state_before": "case refine'_1\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢(￢￢a \\ ￢￢b) ≤ ￢(a ⊓ ￢b)",
"tactic": "rw [hnot_inf_distrib, hnot_le_iff_codisjoint_right, codisjoint_left_comm, ←\n hnot_le_iff_codisjoint_right]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢a ≤ ￢￢a \\ ￢￢b ⊔ ￢￢b",
"tactic": "exact le_sdiff_sup"
},
{
"state_after": "case refine'_2\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢a ≤ ￢￢(b ⊔ a \\ b)",
"state_before": "case refine'_2\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢a \\ ￢￢b ≤ ￢￢(a \\ b)",
"tactic": "rw [sdiff_le_iff, ← hnot_hnot_sup_distrib]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type ?u.179703\nα : Type u_1\nβ : Type ?u.179709\ninst✝ : CoheytingAlgebra α\na✝ b✝ c a b : α\n⊢ ￢￢a ≤ ￢￢(b ⊔ a \\ b)",
"tactic": "exact hnot_anti (hnot_anti le_sup_sdiff)"
}
] | [
1123,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1116,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean | HasDerivAt.lhopital_zero_nhds' | [
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, HasDerivAt f (f' x) x) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, HasDerivAt f (f' x) x\nhgg' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, HasDerivAt g (g' x) x) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, HasDerivAt g (g' x) x\nhg' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, g' x ≠ 0) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝[{a}ᶜ] a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝[{a}ᶜ] a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[{a}ᶜ] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[{a}ᶜ] a) (𝓝 0)\nhga : Tendsto g (𝓝[{a}ᶜ] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[{a}ᶜ] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[{a}ᶜ] a) l",
"tactic": "simp only [← Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, HasDerivAt f (f' x) x) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, HasDerivAt f (f' x) x\nhgg' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, HasDerivAt g (g' x) x) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, HasDerivAt g (g' x) x\nhg' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, g' x ≠ 0) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "exact ⟨lhopital_zero_nhds_left hff'.1 hgg'.1 hg'.1 hfa.1 hga.1 hdiv.1,\n lhopital_zero_nhds_right hff'.2 hgg'.2 hg'.2 hfa.2 hga.2 hdiv.2⟩"
}
] | [
329,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | dist_eq_zero | [] | [
2852,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2851,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | EMetric.cauchySeq_iff_le_tendsto_0 | [
{
"state_after": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\n⊢ (∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε) →\n ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)\n\ncase mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\n⊢ (∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)) →\n ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\n⊢ (∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε) ↔\n ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\n⊢ ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)",
"state_before": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\n⊢ (∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε) →\n ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)",
"tactic": "intro hs"
},
{
"state_after": "case mp.refine_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nn m N : β\nhn : N ≤ n\nhm : N ≤ m\n⊢ edist (s n) (s m) ≤ (fun N => diam (s '' Ici N)) N\n\ncase mp.refine_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\n⊢ Tendsto (fun N => diam (s '' Ici N)) atTop (𝓝 0)",
"state_before": "case mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\n⊢ ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)",
"tactic": "refine ⟨fun N => EMetric.diam (s '' Ici N), fun n m N hn hm => ?_, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.refine_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nn m N : β\nhn : N ≤ n\nhm : N ≤ m\n⊢ edist (s n) (s m) ≤ (fun N => diam (s '' Ici N)) N",
"tactic": "exact EMetric.edist_le_diam_of_mem (mem_image_of_mem _ hn) (mem_image_of_mem _ hm)"
},
{
"state_after": "case mp.refine_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\n⊢ ∀ᶠ (x : β) in atTop, diam (s '' Ici x) ≤ ε",
"state_before": "case mp.refine_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\n⊢ Tendsto (fun N => diam (s '' Ici N)) atTop (𝓝 0)",
"tactic": "refine ENNReal.tendsto_nhds_zero.2 fun ε ε0 => ?_"
},
{
"state_after": "case mp.refine_2.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\nN : β\nhN : ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\n⊢ ∀ᶠ (x : β) in atTop, diam (s '' Ici x) ≤ ε",
"state_before": "case mp.refine_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\n⊢ ∀ᶠ (x : β) in atTop, diam (s '' Ici x) ≤ ε",
"tactic": "rcases hs ε ε0 with ⟨N, hN⟩"
},
{
"state_after": "case mp.refine_2.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\nN : β\nhN : ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nn : β\nhn : N ≤ n\n⊢ ∀ (x : α), x ∈ s '' Ici n → ∀ (y : α), y ∈ s '' Ici n → edist x y ≤ ε",
"state_before": "case mp.refine_2.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\nN : β\nhN : ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\n⊢ ∀ᶠ (x : β) in atTop, diam (s '' Ici x) ≤ ε",
"tactic": "refine (eventually_ge_atTop N).mono fun n hn => EMetric.diam_le ?_"
},
{
"state_after": "case mp.refine_2.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\nN : β\nhN : ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nn : β\nhn : N ≤ n\nk : β\nhk : k ∈ Ici n\nl : β\nhl : l ∈ Ici n\n⊢ edist (s k) (s l) ≤ ε",
"state_before": "case mp.refine_2.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\nN : β\nhN : ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nn : β\nhn : N ≤ n\n⊢ ∀ (x : α), x ∈ s '' Ici n → ∀ (y : α), y ∈ s '' Ici n → edist x y ≤ ε",
"tactic": "rintro _ ⟨k, hk, rfl⟩ _ ⟨l, hl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.refine_2.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nε : ℝ≥0∞\nε0 : ε > 0\nN : β\nhN : ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε\nn : β\nhn : N ≤ n\nk : β\nhk : k ∈ Ici n\nl : β\nhl : l ∈ Ici n\n⊢ edist (s k) (s l) ≤ ε",
"tactic": "exact (hN _ (hn.trans hk) _ (hn.trans hl)).le"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"state_before": "case mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\n⊢ (∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)) →\n ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"tactic": "rintro ⟨b, ⟨b_bound, b_lim⟩⟩ ε εpos"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\nthis : ∀ᶠ (n : β) in atTop, b n < ε\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"state_before": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"tactic": "have : ∀ᶠ n in atTop, b n < ε := b_lim.eventually (gt_mem_nhds εpos)"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\nthis : ∀ᶠ (n : β) in atTop, b n < ε\nN : β\nhN : b N < ε\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"state_before": "case mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\nthis : ∀ᶠ (n : β) in atTop, b n < ε\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"tactic": "rcases this.exists with ⟨N, hN⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\nthis : ∀ᶠ (n : β) in atTop, b n < ε\nN : β\nhN : b N < ε\nm : β\nhm : N ≤ m\nn : β\nhn : N ≤ n\n⊢ edist (s m) (s n) < ε",
"state_before": "case mpr.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\nthis : ∀ᶠ (n : β) in atTop, b n < ε\nN : β\nhN : b N < ε\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε",
"tactic": "refine ⟨N, fun m hm n hn => ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.416936\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\ns : β → α\nb : β → ℝ≥0∞\nb_bound : ∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N\nb_lim : Tendsto b atTop (𝓝 0)\nε : ℝ≥0∞\nεpos : ε > 0\nthis : ∀ᶠ (n : β) in atTop, b n < ε\nN : β\nhN : b N < ε\nm : β\nhm : N ≤ m\nn : β\nhn : N ≤ n\n⊢ edist (s m) (s n) < ε",
"tactic": "calc edist (s m) (s n) ≤ b N := b_bound m n N hm hn\n_ < ε := hN"
}
] | [
1431,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1411,
1
] |
Std/Data/List/Lemmas.lean | List.forall_mem_append | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Prop\nl₁ l₂ : List α\n⊢ (∀ (x : α), x ∈ l₁ ++ l₂ → p x) ↔ (∀ (x : α), x ∈ l₁ → p x) ∧ ∀ (x : α), x ∈ l₂ → p x",
"tactic": "simp only [mem_append, or_imp, forall_and]"
}
] | [
225,
45
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
223,
1
] |
Mathlib/LinearAlgebra/Basic.lean | Submodule.map_id | [
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.474481\nR₂ : Type ?u.474484\nR₃ : Type ?u.474487\nR₄ : Type ?u.474490\nS : Type ?u.474493\nK : Type ?u.474496\nK₂ : Type ?u.474499\nM : Type u_1\nM' : Type ?u.474505\nM₁ : Type ?u.474508\nM₂ : Type ?u.474511\nM₃ : Type ?u.474514\nM₄ : Type ?u.474517\nN : Type ?u.474520\nN₂ : Type ?u.474523\nι : Type ?u.474526\nV : Type ?u.474529\nV₂ : Type ?u.474532\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\ninst✝ : RingHomSurjective σ₁₂\nF : Type ?u.475113\nsc : SemilinearMapClass F σ₁₂ M M₂\na : M\n⊢ a ∈ map LinearMap.id p ↔ a ∈ p",
"tactic": "simp"
}
] | [
726,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
725,
1
] |
Mathlib/Dynamics/Ergodic/Ergodic.lean | MeasureTheory.MeasurePreserving.ergodic_conjugate_iff | [
{
"state_after": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nβ : Type u_2\nm' : MeasurableSpace β\nμ' : Measure β\ns' : Set β\ng : α → β\ne : α ≃ᵐ β\nh : MeasurePreserving ↑e\nthis : MeasurePreserving (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ MeasurePreserving f\n⊢ Ergodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ Ergodic f",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nβ : Type u_2\nm' : MeasurableSpace β\nμ' : Measure β\ns' : Set β\ng : α → β\ne : α ≃ᵐ β\nh : MeasurePreserving ↑e\n⊢ Ergodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ Ergodic f",
"tactic": "have : MeasurePreserving (e ∘ f ∘ e.symm) μ' μ' ↔ MeasurePreserving f μ μ := by\n rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]"
},
{
"state_after": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nβ : Type u_2\nm' : MeasurableSpace β\nμ' : Measure β\ns' : Set β\ng : α → β\ne : α ≃ᵐ β\nthis : MeasurePreserving (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ MeasurePreserving f\nh : PreErgodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ PreErgodic f\n⊢ Ergodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ Ergodic f",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nβ : Type u_2\nm' : MeasurableSpace β\nμ' : Measure β\ns' : Set β\ng : α → β\ne : α ≃ᵐ β\nh : MeasurePreserving ↑e\nthis : MeasurePreserving (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ MeasurePreserving f\n⊢ Ergodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ Ergodic f",
"tactic": "replace h : PreErgodic (e ∘ f ∘ e.symm) μ' ↔ PreErgodic f μ := h.preErgodic_conjugate_iff"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nβ : Type u_2\nm' : MeasurableSpace β\nμ' : Measure β\ns' : Set β\ng : α → β\ne : α ≃ᵐ β\nthis : MeasurePreserving (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ MeasurePreserving f\nh : PreErgodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ PreErgodic f\n⊢ Ergodic (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ Ergodic f",
"tactic": "exact ⟨fun hf => { this.mp hf.toMeasurePreserving, h.mp hf.toPreErgodic with },\n fun hf => { this.mpr hf.toMeasurePreserving, h.mpr hf.toPreErgodic with }⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nβ : Type u_2\nm' : MeasurableSpace β\nμ' : Measure β\ns' : Set β\ng : α → β\ne : α ≃ᵐ β\nh : MeasurePreserving ↑e\n⊢ MeasurePreserving (↑e ∘ f ∘ ↑(MeasurableEquiv.symm e)) ↔ MeasurePreserving f",
"tactic": "rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]"
}
] | [
114,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean | summable_pow_mul_geometric_of_norm_lt_1 | [] | [
342,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean | OreLocalization.oreDiv_mul_oreDiv_comm | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommMonoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₁ * r₂ /ₒ (s₁ * s₂)",
"tactic": "rw [oreDiv_mul_char r₁ r₂ s₁ s₂ r₂ s₁ (by simp [mul_comm]), mul_comm s₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommMonoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ r₂ * ↑s₁ = ↑s₁ * r₂",
"tactic": "simp [mul_comm]"
}
] | [
446,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
444,
1
] |
Mathlib/CategoryTheory/Limits/Over.lean | CategoryTheory.Over.epi_iff_epi_left | [] | [
73,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean | MeasureTheory.eventually_mul_left_iff | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.490392\nG : Type u_1\nH : Type ?u.490398\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableSpace H\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\nμ : Measure G\ninst✝ : IsMulLeftInvariant μ\nt : G\np : G → Prop\n⊢ (∀ᵐ (x : G) ∂μ, p (t * x)) ↔ ∀ᵐ (x : G) ∂μ, p x",
"tactic": "conv_rhs => rw [Filter.Eventually, ← map_mul_left_ae μ t]; rfl"
}
] | [
322,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
320,
1
] |
Mathlib/Analysis/Convex/Basic.lean | convex_iff_openSegment_subset | [] | [
148,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.support_neg | [] | [
1300,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1295,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.sumElim_inr | [] | [
1329,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1327,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean | ContinuousLinearMap.isSelfAdjoint_iff' | [] | [
226,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
] |
Mathlib/LinearAlgebra/Pi.lean | LinearMap.coe_proj | [] | [
92,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/Algebra/Group/Opposite.lean | MulOpposite.opAddEquiv_toEquiv | [] | [
280,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
280,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.succ_ne_succ | [] | [
180,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
Mathlib/Data/Set/Basic.lean | Set.nmem_setOf_iff | [] | [
272,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
271,
1
] |
Mathlib/Topology/Order/Basic.lean | IsLUB.mem_upperBounds_of_tendsto | [
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nha : IsLUB s a\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\n⊢ f x ≤ b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nha : IsLUB s a\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\n⊢ b ∈ upperBounds (f '' s)",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\nha : IsLUB (s ∩ Ici x) a\n⊢ f x ≤ b",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nha : IsLUB s a\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\n⊢ f x ≤ b",
"tactic": "replace ha := ha.inter_Ici_of_mem hx"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\nha : IsLUB (s ∩ Ici x) a\nthis : NeBot (𝓝[s ∩ Ici x] a)\n⊢ f x ≤ b",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\nha : IsLUB (s ∩ Ici x) a\n⊢ f x ≤ b",
"tactic": "haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\nha : IsLUB (s ∩ Ici x) a\nthis : NeBot (𝓝[s ∩ Ici x] a)\n⊢ ∀ᶠ (c : α) in 𝓝[s ∩ Ici x] a, f x ≤ f c",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\nha : IsLUB (s ∩ Ici x) a\nthis : NeBot (𝓝[s ∩ Ici x] a)\n⊢ f x ≤ b",
"tactic": "refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : TopologicalSpace β\ninst✝⁶ : LinearOrder α\ninst✝⁵ : LinearOrder β\ninst✝⁴ : OrderTopology α\ninst✝³ : OrderTopology β\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderClosedTopology γ\nf : α → γ\ns : Set α\na : α\nb : γ\nhf : MonotoneOn f s\nhb : Tendsto f (𝓝[s] a) (𝓝 b)\nx : α\nhx : x ∈ s\nha : IsLUB (s ∩ Ici x) a\nthis : NeBot (𝓝[s ∩ Ici x] a)\n⊢ ∀ᶠ (c : α) in 𝓝[s ∩ Ici x] a, f x ≤ f c",
"tactic": "exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2"
}
] | [
2034,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2027,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean | sSupHom.dual_comp | [] | [
807,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
805,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left | [
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) s x\n⊢ ContinuousWithinAt f s x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\n⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt (↑e ∘ f) s x",
"tactic": "refine' ⟨(e.continuousAt hx).comp_continuousWithinAt, fun fe_cont => _⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x\n⊢ ContinuousWithinAt f (s ∩ f ⁻¹' e.source) x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) s x\n⊢ ContinuousWithinAt f s x",
"tactic": "rw [← continuousWithinAt_inter' h] at fe_cont⊢"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x\nthis : ContinuousWithinAt (↑(LocalHomeomorph.symm e) ∘ ↑e ∘ f) (s ∩ f ⁻¹' e.source) x\n⊢ ContinuousWithinAt f (s ∩ f ⁻¹' e.source) x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x\n⊢ ContinuousWithinAt f (s ∩ f ⁻¹' e.source) x",
"tactic": "have : ContinuousWithinAt (e.symm ∘ e ∘ f) (s ∩ f ⁻¹' e.source) x :=\n haveI : ContinuousWithinAt e.symm univ (e (f x)) :=\n (e.continuousAt_symm (e.map_source hx)).continuousWithinAt\n ContinuousWithinAt.comp this fe_cont (subset_univ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x\nthis : ContinuousWithinAt (↑(LocalHomeomorph.symm e) ∘ ↑e ∘ f) (s ∩ f ⁻¹' e.source) x\n⊢ ContinuousWithinAt f (s ∩ f ⁻¹' e.source) x",
"tactic": "exact this.congr (fun y hy => by simp [e.left_inv hy.2]) (by simp [e.left_inv hx])"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x\nthis : ContinuousWithinAt (↑(LocalHomeomorph.symm e) ∘ ↑e ∘ f) (s ∩ f ⁻¹' e.source) x\ny : γ\nhy : y ∈ s ∩ f ⁻¹' e.source\n⊢ f y = (↑(LocalHomeomorph.symm e) ∘ ↑e ∘ f) y",
"tactic": "simp [e.left_inv hy.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.100825\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\ns : Set γ\nx : γ\nhx : f x ∈ e.source\nh : f ⁻¹' e.source ∈ 𝓝[s] x\nfe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x\nthis : ContinuousWithinAt (↑(LocalHomeomorph.symm e) ∘ ↑e ∘ f) (s ∩ f ⁻¹' e.source) x\n⊢ f x = (↑(LocalHomeomorph.symm e) ∘ ↑e ∘ f) x",
"tactic": "simp [e.left_inv hx]"
}
] | [
1177,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1168,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | zero_pow_le_one | [
{
"state_after": "β : Type ?u.202613\nA : Type ?u.202616\nG : Type ?u.202619\nM : Type ?u.202622\nR : Type u_1\ninst✝ : OrderedSemiring R\na x y : R\nn✝ m n : ℕ\n⊢ 0 ≤ 1",
"state_before": "β : Type ?u.202613\nA : Type ?u.202616\nG : Type ?u.202619\nM : Type ?u.202622\nR : Type u_1\ninst✝ : OrderedSemiring R\na x y : R\nn✝ m n : ℕ\n⊢ 0 ^ (n + 1) ≤ 1",
"tactic": "rw [zero_pow n.succ_pos]"
},
{
"state_after": "no goals",
"state_before": "β : Type ?u.202613\nA : Type ?u.202616\nG : Type ?u.202619\nM : Type ?u.202622\nR : Type u_1\ninst✝ : OrderedSemiring R\na x y : R\nn✝ m n : ℕ\n⊢ 0 ≤ 1",
"tactic": "exact zero_le_one"
}
] | [
400,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
396,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.piecewise_le_piecewise | [] | [
2574,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2572,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Real.surjOn_cos | [
{
"state_after": "no goals",
"state_before": "⊢ SurjOn cos (Icc 0 π) (Icc (-1) 1)",
"tactic": "simpa only [cos_zero, cos_pi] using intermediate_value_Icc' pi_pos.le continuous_cos.continuousOn"
}
] | [
624,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
623,
1
] |
Mathlib/Order/Circular.lean | btw_cyclic | [] | [
183,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
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