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Mathlib/LinearAlgebra/Projection.lean
|
Submodule.coe_isComplEquivProj_apply
|
[] |
[
359,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.coe_ringEquiv_refl
|
[] |
[
447,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.toLocalizationMap_toMap
|
[] |
[
173,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/RingTheory/Bezout.lean
|
IsBezout.TFAE
|
[
{
"state_after": "case tfae_1_to_2\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\n⊢ IsNoetherianRing R → IsPrincipalIdealRing R\n\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"tactic": "tfae_have 1 → 2"
},
{
"state_after": "case tfae_2_to_3\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\n⊢ IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"tactic": "tfae_have 2 → 3"
},
{
"state_after": "case tfae_3_to_4\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n⊢ UniqueFactorizationMonoid R → WfDvdMonoid R\n\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"tactic": "tfae_have 3 → 4"
},
{
"state_after": "case tfae_4_to_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\n⊢ WfDvdMonoid R → IsNoetherianRing R\n\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\ntfae_4_to_1 : WfDvdMonoid R → IsNoetherianRing R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"tactic": "tfae_have 4 → 1"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\ntfae_4_to_1 : WfDvdMonoid R → IsNoetherianRing R\n⊢ List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]",
"tactic": "tfae_finish"
},
{
"state_after": "case tfae_1_to_2\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\nH : IsNoetherianRing R\n⊢ IsPrincipalIdealRing R",
"state_before": "case tfae_1_to_2\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\n⊢ IsNoetherianRing R → IsPrincipalIdealRing R",
"tactic": "intro H"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_to_2\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\nH : IsNoetherianRing R\n⊢ IsPrincipalIdealRing R",
"tactic": "exact ⟨fun I => isPrincipal_of_FG _ (IsNoetherian.noetherian _)⟩"
},
{
"state_after": "case tfae_2_to_3\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\n✝ : IsPrincipalIdealRing R\n⊢ UniqueFactorizationMonoid R",
"state_before": "case tfae_2_to_3\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\n⊢ IsPrincipalIdealRing R → UniqueFactorizationMonoid R",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case tfae_2_to_3\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\n✝ : IsPrincipalIdealRing R\n⊢ UniqueFactorizationMonoid R",
"tactic": "infer_instance"
},
{
"state_after": "case tfae_3_to_4\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n✝ : UniqueFactorizationMonoid R\n⊢ WfDvdMonoid R",
"state_before": "case tfae_3_to_4\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n⊢ UniqueFactorizationMonoid R → WfDvdMonoid R",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case tfae_3_to_4\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\n✝ : UniqueFactorizationMonoid R\n⊢ WfDvdMonoid R",
"tactic": "infer_instance"
},
{
"state_after": "case tfae_4_to_1.mk\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\n⊢ IsNoetherianRing R",
"state_before": "case tfae_4_to_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\n⊢ WfDvdMonoid R → IsNoetherianRing R",
"tactic": "rintro ⟨h⟩"
},
{
"state_after": "case tfae_4_to_1.mk\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\n⊢ WellFounded fun x x_1 => x > x_1",
"state_before": "case tfae_4_to_1.mk\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\n⊢ IsNoetherianRing R",
"tactic": "rw [isNoetherianRing_iff, isNoetherian_iff_fg_wellFounded]"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\n⊢ (fun x x_1 => x > x_1) ↪r DvdNotUnit",
"state_before": "case tfae_4_to_1.mk\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\n⊢ WellFounded fun x x_1 => x > x_1",
"tactic": "apply RelEmbedding.wellFounded _ h"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nthis : ∀ (I : { J // Ideal.FG J }), ∃ x, ↑I = Ideal.span {x}\n⊢ (fun x x_1 => x > x_1) ↪r DvdNotUnit",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\n⊢ (fun x x_1 => x > x_1) ↪r DvdNotUnit",
"tactic": "have : ∀ I : { J : Ideal R // J.FG }, ∃ x : R, (I : Ideal R) = Ideal.span {x} :=\n fun ⟨I, hI⟩ => (IsBezout.isPrincipal_of_FG I hI).1"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\n⊢ (fun x x_1 => x > x_1) ↪r DvdNotUnit",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nthis : ∀ (I : { J // Ideal.FG J }), ∃ x, ↑I = Ideal.span {x}\n⊢ (fun x x_1 => x > x_1) ↪r DvdNotUnit",
"tactic": "choose f hf using this"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\n⊢ (fun x x_1 => x > x_1) ↪r DvdNotUnit",
"tactic": "exact\n { toFun := f\n inj' := fun x y e => by ext1; rw [hf, hf, e]\n map_rel_iff' := by\n dsimp\n intro a b\n rw [← Ideal.span_singleton_lt_span_singleton, ← hf, ← hf]\n rfl }"
},
{
"state_after": "case a\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\nx y : { N // Submodule.FG N }\ne : f x = f y\n⊢ ↑x = ↑y",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\nx y : { N // Submodule.FG N }\ne : f x = f y\n⊢ x = y",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\nx y : { N // Submodule.FG N }\ne : f x = f y\n⊢ ↑x = ↑y",
"tactic": "rw [hf, hf, e]"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\n⊢ ∀ {a b : { N // Submodule.FG N }}, DvdNotUnit (f a) (f b) ↔ a > b",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\n⊢ ∀ {a b : { N // Submodule.FG N }},\n DvdNotUnit (↑{ toFun := f, inj' := (_ : ∀ (x y : { N // Submodule.FG N }), f x = f y → x = y) } a)\n (↑{ toFun := f, inj' := (_ : ∀ (x y : { N // Submodule.FG N }), f x = f y → x = y) } b) ↔\n a > b",
"tactic": "dsimp"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\na b : { N // Submodule.FG N }\n⊢ DvdNotUnit (f a) (f b) ↔ a > b",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\n⊢ ∀ {a b : { N // Submodule.FG N }}, DvdNotUnit (f a) (f b) ↔ a > b",
"tactic": "intro a b"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\na b : { N // Submodule.FG N }\n⊢ ↑b < ↑a ↔ a > b",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\na b : { N // Submodule.FG N }\n⊢ DvdNotUnit (f a) (f b) ↔ a > b",
"tactic": "rw [← Ideal.span_singleton_lt_span_singleton, ← hf, ← hf]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } → R\nhf : ∀ (I : { J // Ideal.FG J }), ↑I = Ideal.span {f I}\na b : { N // Submodule.FG N }\n⊢ ↑b < ↑a ↔ a > b",
"tactic": "rfl"
}
] |
[
146,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/MonoidAlgebra/Division.lean
|
AddMonoidAlgebra.of'_dvd_iff_modOf_eq_zero
|
[
{
"state_after": "case mp\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\n⊢ of' k G g ∣ x → x %ᵒᶠ g = 0\n\ncase mpr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\n⊢ x %ᵒᶠ g = 0 → of' k G g ∣ x",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\n⊢ of' k G g ∣ x ↔ x %ᵒᶠ g = 0",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\n⊢ of' k G g * x %ᵒᶠ g = 0",
"state_before": "case mp\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\n⊢ of' k G g ∣ x → x %ᵒᶠ g = 0",
"tactic": "rintro ⟨x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\ng : G\nx : AddMonoidAlgebra k G\n⊢ of' k G g * x %ᵒᶠ g = 0",
"tactic": "rw [of'_mul_modOf]"
},
{
"state_after": "case mpr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\nh : x %ᵒᶠ g = 0\n⊢ of' k G g ∣ x",
"state_before": "case mpr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\n⊢ x %ᵒᶠ g = 0 → of' k G g ∣ x",
"tactic": "intro h"
},
{
"state_after": "case mpr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\nh : x %ᵒᶠ g = 0\n⊢ of' k G g ∣ of' k G g * (x /ᵒᶠ g)",
"state_before": "case mpr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\nh : x %ᵒᶠ g = 0\n⊢ of' k G g ∣ x",
"tactic": "rw [← divOf_add_modOf x g, h, add_zero]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\nx : AddMonoidAlgebra k G\ng : G\nh : x %ᵒᶠ g = 0\n⊢ of' k G g ∣ of' k G g * (x /ᵒᶠ g)",
"tactic": "exact dvd_mul_right _ _"
}
] |
[
203,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.enumOrd_def'_nonempty
|
[] |
[
2169,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2167,
1
] |
Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
|
Geometry.SimplicialComplex.mem_vertices
|
[] |
[
154,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_finset_of_fintype
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\ninst✝ : Fintype α\n⊢ (#Finset α) = 2 ^ Fintype.card α",
"tactic": "simp [Pow.pow]"
}
] |
[
1323,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1322,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.natCast_inj
|
[] |
[
1368,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1367,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_insert'
|
[
{
"state_after": "ι✝ : Type u'\nι' : Type ?u.1222999\nR : Type ?u.1223002\nK : Type u_2\nM : Type ?u.1223008\nM' : Type ?u.1223011\nM'' : Type ?u.1223014\nV : Type u\nV' : Type ?u.1223019\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι✝ → V\ns✝ t : Set V\nx y z : V\nι : Type u_1\ns : Set ι\na : ι\nf : ι → V\nhas : ¬a ∈ s\n⊢ LinearIndependent K (((fun x => f ↑x) ∘ ↑((Equiv.optionEquivSumPUnit ↑s).trans (Equiv.Set.insert has).symm)) ∘ some) ∧\n ¬((fun x => f ↑x) ∘ ↑((Equiv.optionEquivSumPUnit ↑s).trans (Equiv.Set.insert has).symm)) none ∈\n span K\n (range (((fun x => f ↑x) ∘ ↑((Equiv.optionEquivSumPUnit ↑s).trans (Equiv.Set.insert has).symm)) ∘ some)) ↔\n (LinearIndependent K fun x => f ↑x) ∧ ¬f a ∈ span K (f '' s)",
"state_before": "ι✝ : Type u'\nι' : Type ?u.1222999\nR : Type ?u.1223002\nK : Type u_2\nM : Type ?u.1223008\nM' : Type ?u.1223011\nM'' : Type ?u.1223014\nV : Type u\nV' : Type ?u.1223019\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι✝ → V\ns✝ t : Set V\nx y z : V\nι : Type u_1\ns : Set ι\na : ι\nf : ι → V\nhas : ¬a ∈ s\n⊢ (LinearIndependent K fun x => f ↑x) ↔ (LinearIndependent K fun x => f ↑x) ∧ ¬f a ∈ span K (f '' s)",
"tactic": "rw [← linearIndependent_equiv ((Equiv.optionEquivSumPUnit _).trans (Equiv.Set.insert has).symm),\n linearIndependent_option]"
},
{
"state_after": "ι✝ : Type u'\nι' : Type ?u.1222999\nR : Type ?u.1223002\nK : Type u_2\nM : Type ?u.1223008\nM' : Type ?u.1223011\nM'' : Type ?u.1223014\nV : Type u\nV' : Type ?u.1223019\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι✝ → V\ns✝ t : Set V\nx y z : V\nι : Type u_1\ns : Set ι\na : ι\nf : ι → V\nhas : ¬a ∈ s\n⊢ (LinearIndependent K fun x => f ↑x) → (¬f a ∈ span K (f '' range Subtype.val) ↔ ¬f a ∈ span K (f '' s))",
"state_before": "ι✝ : Type u'\nι' : Type ?u.1222999\nR : Type ?u.1223002\nK : Type u_2\nM : Type ?u.1223008\nM' : Type ?u.1223011\nM'' : Type ?u.1223014\nV : Type u\nV' : Type ?u.1223019\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι✝ → V\ns✝ t : Set V\nx y z : V\nι : Type u_1\ns : Set ι\na : ι\nf : ι → V\nhas : ¬a ∈ s\n⊢ (LinearIndependent K fun x => f ↑x) → (¬f a ∈ span K (range fun x => f ↑x) ↔ ¬f a ∈ span K (f '' s))",
"tactic": "erw [range_comp f ((↑) : s → ι)]"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type u'\nι' : Type ?u.1222999\nR : Type ?u.1223002\nK : Type u_2\nM : Type ?u.1223008\nM' : Type ?u.1223011\nM'' : Type ?u.1223014\nV : Type u\nV' : Type ?u.1223019\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι✝ → V\ns✝ t : Set V\nx y z : V\nι : Type u_1\ns : Set ι\na : ι\nf : ι → V\nhas : ¬a ∈ s\n⊢ (LinearIndependent K fun x => f ↑x) → (¬f a ∈ span K (f '' range Subtype.val) ↔ ¬f a ∈ span K (f '' s))",
"tactic": "simp"
}
] |
[
1215,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1207,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_le_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.75125\nα : Type u_1\nβ : Type ?u.75131\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhb : 0 < b\n⊢ 1 ≤ a / b ↔ b ≤ a",
"tactic": "rw [le_div_iff hb, one_mul]"
}
] |
[
424,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Topology/Instances/Real.lean
|
Function.Periodic.compact_of_continuous'
|
[] |
[
213,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.denom_ne_zero
|
[] |
[
1207,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1206,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
hasDerivAt_neg'
|
[] |
[
255,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Rel.lean
|
Set.image_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.38153\nf : α → β\ns : Set α\n⊢ f '' s = Rel.image (Function.graph f) s",
"tactic": "simp [Set.image, Function.graph, Rel.image]"
}
] |
[
275,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.FiniteAtFilter.inf_ae_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf g : Filter α\n⊢ FiniteAtFilter μ (f ⊓ ae μ) → FiniteAtFilter μ f",
"state_before": "α : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf g : Filter α\n⊢ FiniteAtFilter μ (f ⊓ ae μ) ↔ FiniteAtFilter μ f",
"tactic": "refine' ⟨_, fun h => h.filter_mono inf_le_left⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\n⊢ FiniteAtFilter μ f",
"state_before": "α : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nf g : Filter α\n⊢ FiniteAtFilter μ (f ⊓ ae μ) → FiniteAtFilter μ f",
"tactic": "rintro ⟨s, ⟨t, ht, u, hu, rfl⟩, hμ⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\nthis : ↑↑μ t ≤ ↑↑μ (t ∩ u)\n⊢ FiniteAtFilter μ f\n\ncase this\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\n⊢ ↑↑μ t ≤ ↑↑μ (t ∩ u)",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\n⊢ FiniteAtFilter μ f",
"tactic": "suffices : μ t ≤ μ (t ∩ u)"
},
{
"state_after": "case this\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\n⊢ ↑↑μ t ≤ ↑↑μ (t ∩ u)",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\nthis : ↑↑μ t ≤ ↑↑μ (t ∩ u)\n⊢ FiniteAtFilter μ f\n\ncase this\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\n⊢ ↑↑μ t ≤ ↑↑μ (t ∩ u)",
"tactic": "exact ⟨t, ht, this.trans_lt hμ⟩"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u_1\nβ : Type ?u.852920\nγ : Type ?u.852923\nδ : Type ?u.852926\nι : Type ?u.852929\nR : Type ?u.852932\nR' : Type ?u.852935\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nf g : Filter α\nt : Set α\nht : t ∈ f\nu : Set α\nhu : u ∈ ae μ\nhμ : ↑↑μ (t ∩ u) < ⊤\n⊢ ↑↑μ t ≤ ↑↑μ (t ∩ u)",
"tactic": "exact measure_mono_ae (mem_of_superset hu fun x hu ht => ⟨ht, hu⟩)"
}
] |
[
4118,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4114,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_real_mul
|
[
{
"state_after": "case inl\nr : ℝ\nhr : 0 < r\n⊢ arg (↑r * 0) = arg 0\n\ncase inr\nx : ℂ\nr : ℝ\nhr : 0 < r\nhx : x ≠ 0\n⊢ arg (↑r * x) = arg x",
"state_before": "x : ℂ\nr : ℝ\nhr : 0 < r\n⊢ arg (↑r * x) = arg x",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx)"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℂ\nr : ℝ\nhr : 0 < r\nhx : x ≠ 0\n⊢ arg (↑r * x) = arg x",
"tactic": "conv_lhs =>\n rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,\n arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc]"
},
{
"state_after": "no goals",
"state_before": "case inl\nr : ℝ\nhr : 0 < r\n⊢ arg (↑r * 0) = arg 0",
"tactic": "rw [MulZeroClass.mul_zero]"
}
] |
[
185,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
iteratedFDerivWithin_eq_equiv_comp
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.9982\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ iteratedFDerivWithin 𝕜 n f s = ↑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F) ∘ iteratedDerivWithin n f s",
"tactic": "rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,\n Function.left_id]"
}
] |
[
100,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
padicNormE.norm_p_pow
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑p ^ n‖ = ↑p ^ (-↑n)",
"tactic": "rw [← norm_p_zpow, zpow_ofNat]"
}
] |
[
862,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
861,
1
] |
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.coe_zsmul
|
[] |
[
159,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Algebra/Invertible.lean
|
invOf_mul_self_assoc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Monoid α\na b : α\ninst✝ : Invertible a\n⊢ ⅟a * (a * b) = b",
"tactic": "rw [← mul_assoc, invOf_mul_self, one_mul]"
}
] |
[
123,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.denseRange
|
[] |
[
796,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
11
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_unique_iff
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\n⊢ (∀ (l : ι →₀ R), ↑l default = 0 ∨ v default = 0 → l = 0) ↔ v default ≠ 0",
"state_before": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\n⊢ LinearIndependent R v ↔ v default ≠ 0",
"tactic": "simp only [linearIndependent_iff, Finsupp.total_unique, smul_eq_zero]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\nh : ∀ (l : ι →₀ R), ↑l default = 0 ∨ v default = 0 → l = 0\nhv : v default = 0\n⊢ False",
"state_before": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\n⊢ (∀ (l : ι →₀ R), ↑l default = 0 ∨ v default = 0 → l = 0) ↔ v default ≠ 0",
"tactic": "refine' ⟨fun h hv => _, fun hv l hl => Finsupp.unique_ext <| hl.resolve_right hv⟩"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\nh : ∀ (l : ι →₀ R), ↑l default = 0 ∨ v default = 0 → l = 0\nhv : v default = 0\nthis : Finsupp.single default 1 = 0\n⊢ False",
"state_before": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\nh : ∀ (l : ι →₀ R), ↑l default = 0 ∨ v default = 0 → l = 0\nhv : v default = 0\n⊢ False",
"tactic": "have := h (Finsupp.single default 1) (Or.inr hv)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.1178902\nR : Type u_1\nK : Type ?u.1178908\nM : Type u_2\nM' : Type ?u.1178914\nM'' : Type ?u.1178917\nV : Type u\nV' : Type ?u.1178922\ninst✝⁷ : Ring R\ninst✝⁶ : Nontrivial R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M\ninst✝² : NoZeroSMulDivisors R M\ninst✝¹ : Module R M'\nv✝ : ι → M\ns t : Set M\nx y z : M\nv : ι → M\ninst✝ : Unique ι\nh : ∀ (l : ι →₀ R), ↑l default = 0 ∨ v default = 0 → l = 0\nhv : v default = 0\nthis : Finsupp.single default 1 = 0\n⊢ False",
"tactic": "exact one_ne_zero (Finsupp.single_eq_zero.1 this)"
}
] |
[
1121,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
Set.OrdConnected.isPreconnected
|
[] |
[
430,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Topology/LocalExtr.lean
|
Filter.EventuallyEq.isLocalExtrOn_iff
|
[] |
[
590,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
588,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cast_to_int
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : PosNum\n⊢ ↑↑n = ↑n",
"tactic": "rw [← to_nat_to_int, Int.cast_ofNat, cast_to_nat]"
}
] |
[
72,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
RingHom.ext_zmod
|
[
{
"state_after": "case a\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\na : ZMod n\n⊢ ↑f a = ↑g a",
"state_before": "n : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\n⊢ f = g",
"tactic": "ext a"
},
{
"state_after": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\n⊢ ↑f ↑k = ↑g ↑k",
"state_before": "case a\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\na : ZMod n\n⊢ ↑f a = ↑g a",
"tactic": "obtain ⟨k, rfl⟩ := ZMod.int_cast_surjective a"
},
{
"state_after": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\nφ : ℤ →+* R := comp f (Int.castRingHom (ZMod n))\n⊢ ↑f ↑k = ↑g ↑k",
"state_before": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\n⊢ ↑f ↑k = ↑g ↑k",
"tactic": "let φ : ℤ →+* R := f.comp (Int.castRingHom (ZMod n))"
},
{
"state_after": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\nφ : ℤ →+* R := comp f (Int.castRingHom (ZMod n))\nψ : ℤ →+* R := comp g (Int.castRingHom (ZMod n))\n⊢ ↑f ↑k = ↑g ↑k",
"state_before": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\nφ : ℤ →+* R := comp f (Int.castRingHom (ZMod n))\n⊢ ↑f ↑k = ↑g ↑k",
"tactic": "let ψ : ℤ →+* R := g.comp (Int.castRingHom (ZMod n))"
},
{
"state_after": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\nφ : ℤ →+* R := comp f (Int.castRingHom (ZMod n))\nψ : ℤ →+* R := comp g (Int.castRingHom (ZMod n))\n⊢ ↑φ k = ↑ψ k",
"state_before": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\nφ : ℤ →+* R := comp f (Int.castRingHom (ZMod n))\nψ : ℤ →+* R := comp g (Int.castRingHom (ZMod n))\n⊢ ↑f ↑k = ↑g ↑k",
"tactic": "show φ k = ψ k"
},
{
"state_after": "no goals",
"state_before": "case a.intro\nn : ℕ\nR : Type u_1\ninst✝ : Semiring R\nf g : ZMod n →+* R\nk : ℤ\nφ : ℤ →+* R := comp f (Int.castRingHom (ZMod n))\nψ : ℤ →+* R := comp g (Int.castRingHom (ZMod n))\n⊢ ↑φ k = ↑ψ k",
"tactic": "rw [φ.ext_int ψ]"
}
] |
[
1145,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1139,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sSup_eq_sUnion
|
[] |
[
120,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.isCauSeq_re
|
[
{
"state_after": "no goals",
"state_before": "f : CauSeq ℂ ↑abs\nε : ℝ\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → ↑abs (↑f j - ↑f i) < ε\nj : ℕ\nij : j ≥ i\n⊢ abs' ((fun n => (↑f n).re) j - (fun n => (↑f n).re) i) ≤ ↑abs (↑f j - ↑f i)",
"tactic": "simpa using abs_re_le_abs (f j - f i)"
}
] |
[
1240,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1238,
1
] |
Mathlib/LinearAlgebra/QuotientPi.lean
|
Submodule.quotientPi_aux.map_add
|
[] |
[
118,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.exists_mem_normalizedFactors
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\np' : α\nhp' : Irreducible p'\nhp'x : p' ∣ x\n⊢ ∃ p, p ∈ normalizedFactors x",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\n⊢ ∃ p, p ∈ normalizedFactors x",
"tactic": "obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\np' : α\nhp' : Irreducible p'\nhp'x : p' ∣ x\np : α\nhp : p ∈ normalizedFactors x\nright✝ : p' ~ᵤ p\n⊢ ∃ p, p ∈ normalizedFactors x",
"state_before": "case intro.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\np' : α\nhp' : Irreducible p'\nhp'x : p' ∣ x\n⊢ ∃ p, p ∈ normalizedFactors x",
"tactic": "obtain ⟨p, hp, _⟩ := exists_mem_normalizedFactors_of_dvd hx hp' hp'x"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\np' : α\nhp' : Irreducible p'\nhp'x : p' ∣ x\np : α\nhp : p ∈ normalizedFactors x\nright✝ : p' ~ᵤ p\n⊢ ∃ p, p ∈ normalizedFactors x",
"tactic": "exact ⟨p, hp⟩"
}
] |
[
646,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\n⊢ (∫⁻ (x : α), f x ∂μ) ≤ C",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\n⊢ (∫⁻ (x : α), f x ∂μ) ≤ C",
"tactic": "have : (∫⁻ x in univ, f x ∂μ) = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\n⊢ (∫⁻ (x : α) in univ, f x ∂μ) ≤ C",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\n⊢ (∫⁻ (x : α), f x ∂μ) ≤ C",
"tactic": "rw [← this]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (∫⁻ (x : α) in ⋃ (n : ℕ), S n, f x ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\n⊢ (∫⁻ (x : α) in univ, f x ∂μ) ≤ C",
"tactic": "refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ\n\ncase hs\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ MeasurableSet (⋃ (n : ℕ), S n)",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (∫⁻ (x : α) in ⋃ (n : ℕ), S n, f x ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ",
"tactic": "rw [← lintegral_indicator]"
},
{
"state_after": "case hs\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ MeasurableSet (⋃ (n : ℕ), S n)\n\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ\n\ncase hs\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ MeasurableSet (⋃ (n : ℕ), S n)",
"tactic": "swap"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ",
"tactic": "have h_integral_indicator : (⨆ n, ∫⁻ x in S n, f x ∂μ) = ⨆ n, ∫⁻ x, (S n).indicator f x ∂μ := by\n congr\n ext1 n\n rw [lintegral_indicator _ (hm _ (hS_meas n))]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ∫⁻ (a : α), ⨆ (n : ℕ), indicator (S n) f a ∂μ\n\ncase hf\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ ∀ (n : ℕ), Measurable fun x => indicator (S n) f x\n\ncase h_mono\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ Monotone fun n x => indicator (S n) f x",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ",
"tactic": "rw [h_integral_indicator, ← lintegral_iSup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\n⊢ (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ",
"tactic": "simp only [Measure.restrict_univ]"
},
{
"state_after": "no goals",
"state_before": "case hs\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ MeasurableSet (⋃ (n : ℕ), S n)",
"tactic": "exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)"
},
{
"state_after": "case e_s\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (fun n => ∫⁻ (x : α) in S n, f x ∂μ) = fun n => ∫⁻ (x : α), indicator (S n) f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ",
"tactic": "congr"
},
{
"state_after": "case e_s.h\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nn : ℕ\n⊢ (∫⁻ (x : α) in S n, f x ∂μ) = ∫⁻ (x : α), indicator (S n) f x ∂μ",
"state_before": "case e_s\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\n⊢ (fun n => ∫⁻ (x : α) in S n, f x ∂μ) = fun n => ∫⁻ (x : α), indicator (S n) f x ∂μ",
"tactic": "ext1 n"
},
{
"state_after": "no goals",
"state_before": "case e_s.h\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nn : ℕ\n⊢ (∫⁻ (x : α) in S n, f x ∂μ) = ∫⁻ (x : α), indicator (S n) f x ∂μ",
"tactic": "rw [lintegral_indicator _ (hm _ (hS_meas n))]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\n⊢ indicator (⋃ (n : ℕ), S n) (fun x => f x) x = ⨆ (n : ℕ), indicator (S n) f x",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ (∫⁻ (a : α), indicator (⋃ (n : ℕ), S n) (fun x => f x) a ∂μ) ≤ ∫⁻ (a : α), ⨆ (n : ℕ), indicator (S n) f a ∂μ",
"tactic": "refine' le_of_eq (lintegral_congr fun x => _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\n⊢ (if x ∈ ⋃ (n : ℕ), S n then f x else 0) = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\n⊢ indicator (⋃ (n : ℕ), S n) (fun x => f x) x = ⨆ (n : ℕ), indicator (S n) f x",
"tactic": "simp_rw [indicator_apply]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\n⊢ (if x ∈ ⋃ (n : ℕ), S n then f x else 0) = ⨆ (n : ℕ), if x ∈ S n then f x else 0\n\ncase neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ¬x ∈ iUnion S\n⊢ (if x ∈ ⋃ (n : ℕ), S n then f x else 0) = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "α : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\n⊢ (if x ∈ ⋃ (n : ℕ), S n then f x else 0) = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "by_cases hx_mem : x ∈ iUnion S"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\n⊢ f x = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\n⊢ (if x ∈ ⋃ (n : ℕ), S n then f x else 0) = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "simp only [hx_mem, if_true]"
},
{
"state_after": "case pos.intro\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\nn : ℕ\nhxn : x ∈ S n\n⊢ f x = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\n⊢ f x = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "obtain ⟨n, hxn⟩ := mem_iUnion.mp hx_mem"
},
{
"state_after": "case pos.intro.refine'_1\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\nn : ℕ\nhxn : x ∈ S n\n⊢ f x ≤ if x ∈ S n then f x else 0\n\ncase pos.intro.refine'_2\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\nn : ℕ\nhxn : x ∈ S n\ni : ℕ\n⊢ (if x ∈ S i then f x else 0) ≤ f x",
"state_before": "case pos.intro\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\nn : ℕ\nhxn : x ∈ S n\n⊢ f x = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "refine' le_antisymm (_root_.trans _ (le_iSup _ n)) (iSup_le fun i => _)"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.refine'_1\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\nn : ℕ\nhxn : x ∈ S n\n⊢ f x ≤ if x ∈ S n then f x else 0",
"tactic": "simp only [hxn, le_refl, if_true]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.refine'_2\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : x ∈ iUnion S\nn : ℕ\nhxn : x ∈ S n\ni : ℕ\n⊢ (if x ∈ S i then f x else 0) ≤ f x",
"tactic": "by_cases hxi : x ∈ S i <;> simp [hxi]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ¬x ∈ iUnion S\n⊢ 0 = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ¬x ∈ iUnion S\n⊢ (if x ∈ ⋃ (n : ℕ), S n then f x else 0) = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "simp only [hx_mem, if_false]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ¬∃ i, x ∈ S i\n⊢ 0 = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ¬x ∈ iUnion S\n⊢ 0 = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "rw [mem_iUnion] at hx_mem"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ∀ (i : ℕ), ¬x ∈ S i\n⊢ 0 = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ¬∃ i, x ∈ S i\n⊢ 0 = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "push_neg at hx_mem"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ∀ (i : ℕ), ¬x ∈ S i\nn : ℕ\n⊢ (if x ∈ S n then f x else 0) ≤ 0",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ∀ (i : ℕ), ¬x ∈ S i\n⊢ 0 = ⨆ (n : ℕ), if x ∈ S n then f x else 0",
"tactic": "refine' le_antisymm (zero_le _) (iSup_le fun n => _)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nx : α\nhx_mem : ∀ (i : ℕ), ¬x ∈ S i\nn : ℕ\n⊢ (if x ∈ S n then f x else 0) ≤ 0",
"tactic": "simp only [hx_mem n, if_false, nonpos_iff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ ∀ (n : ℕ), Measurable fun x => indicator (S n) f x",
"tactic": "exact fun n => hf_meas.indicator (hm _ (hS_meas n))"
},
{
"state_after": "case h_mono\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\n⊢ (fun n x => indicator (S n) f x) n₁ a ≤ (fun n x => indicator (S n) f x) n₂ a",
"state_before": "case h_mono\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\n⊢ Monotone fun n x => indicator (S n) f x",
"tactic": "intro n₁ n₂ hn₁₂ a"
},
{
"state_after": "case h_mono\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\n⊢ (if a ∈ S n₁ then f a else 0) ≤ if a ∈ S n₂ then f a else 0",
"state_before": "case h_mono\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\n⊢ (fun n x => indicator (S n) f x) n₁ a ≤ (fun n x => indicator (S n) f x) n₂ a",
"tactic": "simp_rw [indicator_apply]"
},
{
"state_after": "case h_mono.inl.inl\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : a ∈ S n₁\nh_1 : a ∈ S n₂\n⊢ f a ≤ f a\n\ncase h_mono.inl.inr\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : a ∈ S n₁\nh_1 : ¬a ∈ S n₂\n⊢ f a ≤ 0\n\ncase h_mono.inr.inl\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : ¬a ∈ S n₁\nh✝ : a ∈ S n₂\n⊢ 0 ≤ f a\n\ncase h_mono.inr.inr\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : ¬a ∈ S n₁\nh✝ : ¬a ∈ S n₂\n⊢ 0 ≤ 0",
"state_before": "case h_mono\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\n⊢ (if a ∈ S n₁ then f a else 0) ≤ if a ∈ S n₂ then f a else 0",
"tactic": "split_ifs with h h_1"
},
{
"state_after": "no goals",
"state_before": "case h_mono.inl.inl\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : a ∈ S n₁\nh_1 : a ∈ S n₂\n⊢ f a ≤ f a",
"tactic": "exact le_rfl"
},
{
"state_after": "no goals",
"state_before": "case h_mono.inl.inr\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : a ∈ S n₁\nh_1 : ¬a ∈ S n₂\n⊢ f a ≤ 0",
"tactic": "exact absurd (mem_of_mem_of_subset h (hS_mono hn₁₂)) h_1"
},
{
"state_after": "no goals",
"state_before": "case h_mono.inr.inl\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : ¬a ∈ S n₁\nh✝ : a ∈ S n₂\n⊢ 0 ≤ f a",
"tactic": "exact zero_le _"
},
{
"state_after": "no goals",
"state_before": "case h_mono.inr.inr\nα : Type u_1\nβ : Type ?u.1825244\nγ : Type ?u.1825247\nδ : Type ?u.1825250\nm m0 : MeasurableSpace α\nE : Type ?u.1825259\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (Measure.trim μ hm)\nC : ℝ≥0∞\nf : α → ℝ≥0∞\nhf_meas : Measurable f\nhf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → (∫⁻ (x : α) in s, f x ∂μ) ≤ C\nthis : (∫⁻ (x : α) in univ, f x ∂μ) = ∫⁻ (x : α), f x ∂μ\nS : ℕ → Set α\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : (⨆ (n : ℕ), ∫⁻ (x : α) in S n, f x ∂μ) = ⨆ (n : ℕ), ∫⁻ (x : α), indicator (S n) f x ∂μ\nn₁ n₂ : ℕ\nhn₁₂ : n₁ ≤ n₂\na : α\nh : ¬a ∈ S n₁\nh✝ : ¬a ∈ S n₂\n⊢ 0 ≤ 0",
"tactic": "exact le_rfl"
}
] |
[
2004,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1970,
1
] |
Mathlib/Data/List/Basic.lean
|
List.length_modifyNth
|
[] |
[
1556,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1555,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.surjective_decode_iget
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.385\nβ : Type ?u.388\nα : Type u_1\ninst✝¹ : Encodable α\ninst✝ : Inhabited α\nx : α\n⊢ (fun n => iget (decode n)) (encode x) = x",
"tactic": "simp_rw [Encodable.encodek]"
}
] |
[
83,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.ne_zero_of_ortho
|
[] |
[
613,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.finite
|
[] |
[
923,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Std/Data/List/Lemmas.lean
|
List.getLast?_eq_getLast
|
[] |
[
481,
19
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
479,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_row
|
[
{
"state_after": "n : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) (inl i) (inr ()) = 0",
"state_before": "n : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) (inl i) (inr ()) = 0",
"tactic": "have : listTransvecCol M = listTransvecCol (M ⬝ (listTransvecRow M).prod) := by\n simp [listTransvecCol, mul_listTransvecRow_last_col]"
},
{
"state_after": "n : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))\n⊢ (List.prod (listTransvecCol (M ⬝ List.prod (listTransvecRow M))) ⬝ (M ⬝ List.prod (listTransvecRow M))) (inl i)\n (inr ()) =\n 0",
"state_before": "n : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) (inl i) (inr ()) = 0",
"tactic": "rw [this, Matrix.mul_assoc]"
},
{
"state_after": "case hM\nn : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))\n⊢ (M ⬝ List.prod (listTransvecRow M)) (inr ()) (inr ()) ≠ 0",
"state_before": "n : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))\n⊢ (List.prod (listTransvecCol (M ⬝ List.prod (listTransvecRow M))) ⬝ (M ⬝ List.prod (listTransvecRow M))) (inl i)\n (inr ()) =\n 0",
"tactic": "apply listTransvecCol_mul_last_col"
},
{
"state_after": "no goals",
"state_before": "case hM\nn : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))\n⊢ (M ⬝ List.prod (listTransvecRow M)) (inr ()) (inr ()) ≠ 0",
"tactic": "simpa [mul_listTransvecRow_last_col] using hM"
},
{
"state_after": "no goals",
"state_before": "n : Type ?u.209504\np : Type ?u.209507\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\n⊢ listTransvecCol M = listTransvecCol (M ⬝ List.prod (listTransvecRow M))",
"tactic": "simp [listTransvecCol, mul_listTransvecRow_last_col]"
}
] |
[
519,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
aemeasurable_biSup
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.1526225\nγ : Type ?u.1526228\nγ₂ : Type ?u.1526231\nδ : Type u_2\nι✝ : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : CompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_1\nμ : MeasureTheory.Measure δ\ns : Set ι\nf : ι → δ → α\nhs : Set.Countable s\nhf : ∀ (i : ι), AEMeasurable (f i)\nthis : Encodable ↑s\n⊢ AEMeasurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b",
"state_before": "α : Type u_3\nβ : Type ?u.1526225\nγ : Type ?u.1526228\nγ₂ : Type ?u.1526231\nδ : Type u_2\nι✝ : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : CompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_1\nμ : MeasureTheory.Measure δ\ns : Set ι\nf : ι → δ → α\nhs : Set.Countable s\nhf : ∀ (i : ι), AEMeasurable (f i)\n⊢ AEMeasurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b",
"tactic": "haveI : Encodable s := hs.toEncodable"
},
{
"state_after": "α : Type u_3\nβ : Type ?u.1526225\nγ : Type ?u.1526228\nγ₂ : Type ?u.1526231\nδ : Type u_2\nι✝ : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : CompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_1\nμ : MeasureTheory.Measure δ\ns : Set ι\nf : ι → δ → α\nhs : Set.Countable s\nhf : ∀ (i : ι), AEMeasurable (f i)\nthis : Encodable ↑s\n⊢ AEMeasurable fun b => ⨆ (x : { i // i ∈ s }), f (↑x) b",
"state_before": "α : Type u_3\nβ : Type ?u.1526225\nγ : Type ?u.1526228\nγ₂ : Type ?u.1526231\nδ : Type u_2\nι✝ : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : CompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_1\nμ : MeasureTheory.Measure δ\ns : Set ι\nf : ι → δ → α\nhs : Set.Countable s\nhf : ∀ (i : ι), AEMeasurable (f i)\nthis : Encodable ↑s\n⊢ AEMeasurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b",
"tactic": "simp only [iSup_subtype']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.1526225\nγ : Type ?u.1526228\nγ₂ : Type ?u.1526231\nδ : Type u_2\nι✝ : Sort y\ns✝ t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : BorelSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : BorelSpace β\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : MeasurableSpace γ\ninst✝⁴ : BorelSpace γ\ninst✝³ : MeasurableSpace δ\ninst✝² : CompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_1\nμ : MeasureTheory.Measure δ\ns : Set ι\nf : ι → δ → α\nhs : Set.Countable s\nhf : ∀ (i : ι), AEMeasurable (f i)\nthis : Encodable ↑s\n⊢ AEMeasurable fun b => ⨆ (x : { i // i ∈ s }), f (↑x) b",
"tactic": "exact aemeasurable_iSup fun i => hf i"
}
] |
[
1291,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1287,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean
|
ContinuousLinearMap.fderiv_of_bilinear
|
[] |
[
155,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.clog_pos
|
[
{
"state_after": "b n : ℕ\nhb : 1 < b\nhn : 2 ≤ n\n⊢ 0 < clog b ((n + b - 1) / b) + 1",
"state_before": "b n : ℕ\nhb : 1 < b\nhn : 2 ≤ n\n⊢ 0 < clog b n",
"tactic": "rw [clog_of_two_le hb hn]"
},
{
"state_after": "no goals",
"state_before": "b n : ℕ\nhb : 1 < b\nhn : 2 ≤ n\n⊢ 0 < clog b ((n + b - 1) / b) + 1",
"tactic": "exact zero_lt_succ _"
}
] |
[
284,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.filter_apply_pos
|
[] |
[
914,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
914,
1
] |
Mathlib/Data/Nat/Choose/Sum.lean
|
Int.alternating_sum_range_choose_of_ne
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.188221\nn : ℕ\nh0 : n ≠ 0\n⊢ ∑ m in range (n + 1), (-1) ^ m * ↑(Nat.choose n m) = 0",
"tactic": "rw [Int.alternating_sum_range_choose, if_neg h0]"
}
] |
[
150,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/RingTheory/Coprime/Lemmas.lean
|
IsCoprime.prod_left_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\nx✝ : I\n⊢ x✝ ∈ ∅ → IsCoprime (s x✝) x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt✝ : Finset I\nb : I\nt : Finset I\nhbt : ¬b ∈ t\nih : IsCoprime (∏ i in t, s i) x ↔ ∀ (i : I), i ∈ t → IsCoprime (s i) x\n⊢ IsCoprime (∏ i in insert b t, s i) x ↔ ∀ (i : I), i ∈ insert b t → IsCoprime (s i) x",
"tactic": "rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]"
}
] |
[
66,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
Monovary.sum_mul_comp_perm_le_sum_mul
|
[] |
[
457,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Computability/Partrec.lean
|
Nat.Partrec.some
|
[] |
[
203,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
11
] |
Mathlib/Algebra/Module/Injective.lean
|
Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd'
|
[
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\nthis : r ∈ ideal i f y\n⊢ ↑(extendIdealTo i f h y) r = 0",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\n⊢ ↑(extendIdealTo i f h y) r = 0",
"tactic": "have : r ∈ ideal i f y := by\n change (r • y) ∈ (extensionOfMax i f).toLinearPMap.domain\n rw [eq1]\n apply Submodule.zero_mem _"
},
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\nthis : r ∈ ideal i f y\n⊢ ↑(idealTo i f y) { val := r, property := this } = 0",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\nthis : r ∈ ideal i f y\n⊢ ↑(extendIdealTo i f h y) r = 0",
"tactic": "rw [ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h y r this]"
},
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\nthis : r ∈ ideal i f y\n⊢ ↑(extensionOfMax i f).toLinearPMap { val := r • y, property := (_ : ↑{ val := r, property := this } ∈ ideal i f y) } =\n 0",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\nthis : r ∈ ideal i f y\n⊢ ↑(idealTo i f y) { val := r, property := this } = 0",
"tactic": "dsimp [ExtensionOfMaxAdjoin.idealTo]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\nthis : r ∈ ideal i f y\n⊢ ↑(extensionOfMax i f).toLinearPMap { val := r • y, property := (_ : ↑{ val := r, property := this } ∈ ideal i f y) } =\n 0",
"tactic": "simp only [LinearMap.coe_mk, eq1, Subtype.coe_mk, ← ZeroMemClass.zero_def,\n (extensionOfMax i f).toLinearPMap.map_zero]"
},
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\n⊢ r • y ∈ (extensionOfMax i f).toLinearPMap.domain",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\n⊢ r ∈ ideal i f y",
"tactic": "change (r • y) ∈ (extensionOfMax i f).toLinearPMap.domain"
},
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\n⊢ 0 ∈ (extensionOfMax i f).toLinearPMap.domain",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\n⊢ r • y ∈ (extensionOfMax i f).toLinearPMap.domain",
"tactic": "rw [eq1]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr : R\neq1 : r • y = 0\n⊢ 0 ∈ (extensionOfMax i f).toLinearPMap.domain",
"tactic": "apply Submodule.zero_mem _"
}
] |
[
340,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.left_lt_of_lt_right'
|
[] |
[
787,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.wittSub_zero
|
[
{
"state_after": "case a\np : ℕ\nR : Type ?u.60999\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ ↑(map (Int.castRingHom ℚ)) (wittSub p 0) = ↑(map (Int.castRingHom ℚ)) (X (0, 0) - X (1, 0))",
"state_before": "p : ℕ\nR : Type ?u.60999\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ wittSub p 0 = X (0, 0) - X (1, 0)",
"tactic": "apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective"
},
{
"state_after": "no goals",
"state_before": "case a\np : ℕ\nR : Type ?u.60999\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ ↑(map (Int.castRingHom ℚ)) (wittSub p 0) = ↑(map (Int.castRingHom ℚ)) (X (0, 0) - X (1, 0))",
"tactic": "simp only [wittSub, wittStructureRat, AlgHom.map_sub, RingHom.map_sub, rename_X,\n xInTermsOfW_zero, map_X, wittPolynomial_zero, bind₁_X_right, map_wittStructureInt]"
}
] |
[
276,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.inf_def
|
[] |
[
571,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
571,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.Nonempty.of_smul_right
|
[] |
[
147,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup₂_le
|
[] |
[
825,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
824,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_diff_cancel_left
|
[] |
[
2074,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2073,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.tr_eval
|
[
{
"state_after": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\n⊢ eval (TM2.step tr) (init c v) = halt <$> Code.eval c v",
"state_before": "c : Code\nv : List ℕ\n⊢ eval (TM2.step tr) (init c v) = halt <$> Code.eval c v",
"tactic": "obtain ⟨i, h₁, h₂⟩ := tr_init c v"
},
{
"state_after": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ x ∈ eval (TM2.step tr) (init c v) ↔ x ∈ halt <$> Code.eval c v",
"state_before": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\n⊢ eval (TM2.step tr) (init c v) = halt <$> Code.eval c v",
"tactic": "refine' Part.ext fun x => _"
},
{
"state_after": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ x ∈ eval (fun a => TM2.step tr a) i ↔ x ∈ halt <$> Code.eval c v",
"state_before": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ x ∈ eval (TM2.step tr) (init c v) ↔ x ∈ halt <$> Code.eval c v",
"tactic": "rw [reaches_eval h₂.to_reflTransGen]"
},
{
"state_after": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ x ∈ eval (fun a => TM2.step tr a) i ↔ ∃ a, a ∈ Code.eval c v ∧ halt a = x",
"state_before": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ x ∈ eval (fun a => TM2.step tr a) i ↔ x ∈ halt <$> Code.eval c v",
"tactic": "simp [-TM2.step]"
},
{
"state_after": "case intro.intro.refine'_1\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\n⊢ ∃ a, a ∈ Code.eval c v ∧ halt a = x\n\ncase intro.intro.refine'_2\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ (∃ a, a ∈ Code.eval c v ∧ halt a = x) → x ∈ eval (fun a => TM2.step tr a) i",
"state_before": "case intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ x ∈ eval (fun a => TM2.step tr a) i ↔ ∃ a, a ∈ Code.eval c v ∧ halt a = x",
"tactic": "refine' ⟨fun h => _, _⟩"
},
{
"state_after": "case intro.intro.refine'_1.intro.intro\nc✝ : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c✝ Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c✝ v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\nc : Cfg\nhc₁ : TrCfg c x\nhc₂ : c ∈ eval step (stepNormal c✝ Cont.halt v)\n⊢ ∃ a, a ∈ Code.eval c✝ v ∧ halt a = x",
"state_before": "case intro.intro.refine'_1\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\n⊢ ∃ a, a ∈ Code.eval c v ∧ halt a = x",
"tactic": "obtain ⟨c, hc₁, hc₂⟩ := tr_eval_rev tr_respects h₁ h"
},
{
"state_after": "case intro.intro.refine'_1.intro.intro\nc✝ : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c✝ Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c✝ v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\nc : Cfg\nhc₁ : TrCfg c x\nhc₂ : ∃ a, a ∈ Code.eval c✝ v ∧ Cfg.halt a = c\n⊢ ∃ a, a ∈ Code.eval c✝ v ∧ halt a = x",
"state_before": "case intro.intro.refine'_1.intro.intro\nc✝ : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c✝ Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c✝ v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\nc : Cfg\nhc₁ : TrCfg c x\nhc₂ : c ∈ eval step (stepNormal c✝ Cont.halt v)\n⊢ ∃ a, a ∈ Code.eval c✝ v ∧ halt a = x",
"tactic": "simp [stepNormal_eval] at hc₂"
},
{
"state_after": "case intro.intro.refine'_1.intro.intro.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nhc₁ : TrCfg (Cfg.halt v') x\n⊢ ∃ a, a ∈ Code.eval c v ∧ halt a = x",
"state_before": "case intro.intro.refine'_1.intro.intro\nc✝ : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c✝ Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c✝ v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\nc : Cfg\nhc₁ : TrCfg c x\nhc₂ : ∃ a, a ∈ Code.eval c✝ v ∧ Cfg.halt a = c\n⊢ ∃ a, a ∈ Code.eval c✝ v ∧ halt a = x",
"tactic": "obtain ⟨v', hv, rfl⟩ := hc₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1.intro.intro.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\nh : x ∈ eval (fun a => TM2.step tr a) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nhc₁ : TrCfg (Cfg.halt v') x\n⊢ ∃ a, a ∈ Code.eval c v ∧ halt a = x",
"tactic": "exact ⟨_, hv, hc₁.symm⟩"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"state_before": "case intro.intro.refine'_2\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nx : TM2.Cfg (fun x => Γ') Λ' (Option Γ')\n⊢ (∃ a, a ∈ Code.eval c v ∧ halt a = x) → x ∈ eval (fun a => TM2.step tr a) i",
"tactic": "rintro ⟨v', hv, rfl⟩"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nthis : Cfg.halt v' ∈ eval step (stepNormal c Cont.halt v) → ∃ b₂, TrCfg (Cfg.halt v') b₂ ∧ b₂ ∈ eval (TM2.step tr) i\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"state_before": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"tactic": "have := Turing.tr_eval (b₁ := Cfg.halt v') tr_respects h₁"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nthis : v' ∈ Code.eval c v → ∃ b₂, TrCfg (Cfg.halt v') b₂ ∧ b₂ ∈ eval (TM2.step tr) i\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"state_before": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nthis : Cfg.halt v' ∈ eval step (stepNormal c Cont.halt v) → ∃ b₂, TrCfg (Cfg.halt v') b₂ ∧ b₂ ∈ eval (TM2.step tr) i\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"tactic": "simp [stepNormal_eval, -TM2.step] at this"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro.intro.intro.refl\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nthis : v' ∈ Code.eval c v → ∃ b₂, TrCfg (Cfg.halt v') b₂ ∧ b₂ ∈ eval (TM2.step tr) i\nh : halt v' ∈ eval (TM2.step tr) i\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"state_before": "case intro.intro.refine'_2.intro.intro\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nthis : v' ∈ Code.eval c v → ∃ b₂, TrCfg (Cfg.halt v') b₂ ∧ b₂ ∈ eval (TM2.step tr) i\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"tactic": "obtain ⟨_, ⟨⟩, h⟩ := this hv"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2.intro.intro.intro.intro.refl\nc : Code\nv : List ℕ\ni : Cfg'\nh₁ : TrCfg (stepNormal c Cont.halt v) i\nh₂ : Reaches₁ (TM2.step tr) (init c v) i\nv' : List ℕ\nhv : v' ∈ Code.eval c v\nthis : v' ∈ Code.eval c v → ∃ b₂, TrCfg (Cfg.halt v') b₂ ∧ b₂ ∈ eval (TM2.step tr) i\nh : halt v' ∈ eval (TM2.step tr) i\n⊢ halt v' ∈ eval (fun a => TM2.step tr a) i",
"tactic": "exact h"
}
] |
[
1719,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1706,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lt_bsup_of_limit
|
[] |
[
1541,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1538,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhds_bind_nhdsWithin
|
[] |
[
43,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.neg_of_one_le
|
[] |
[
514,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
isCoprime_one_left
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\n⊢ 1 * 1 + 0 * x = 1",
"tactic": "rw [one_mul, zero_mul, add_zero]"
}
] |
[
83,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x (↑C a) = ↑f a",
"tactic": "simp [eval₂_eq_sum]"
}
] |
[
73,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isClopen_range_inl
|
[] |
[
1644,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1643,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat.one
|
[
{
"state_after": "p : ℕ\n⊢ (if h : p ≠ 1 ∧ 0 < 1 then Part.get (multiplicity p 1) (_ : multiplicity.Finite p 1) else 0) = 0",
"state_before": "p : ℕ\n⊢ padicValNat p 1 = 0",
"tactic": "unfold padicValNat"
},
{
"state_after": "case inl\np : ℕ\nh✝ : p ≠ 1 ∧ 0 < 1\n⊢ Part.get (multiplicity p 1) (_ : multiplicity.Finite p 1) = 0\n\ncase inr\np : ℕ\nh✝ : ¬(p ≠ 1 ∧ 0 < 1)\n⊢ 0 = 0",
"state_before": "p : ℕ\n⊢ (if h : p ≠ 1 ∧ 0 < 1 then Part.get (multiplicity p 1) (_ : multiplicity.Finite p 1) else 0) = 0",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ\nh✝ : p ≠ 1 ∧ 0 < 1\n⊢ Part.get (multiplicity p 1) (_ : multiplicity.Finite p 1) = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\np : ℕ\nh✝ : ¬(p ≠ 1 ∧ 0 < 1)\n⊢ 0 = 0",
"tactic": "rfl"
}
] |
[
77,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
11
] |
Mathlib/Algebra/Ring/Defs.lean
|
sub_one_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonAssocRing α\na b : α\n⊢ (a - 1) * b = a * b - b",
"tactic": "rw [sub_mul, one_mul]"
}
] |
[
410,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean
|
ModuleCat.MonoidalCategory.hexagon_reverse
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\n⊢ (α_ X Y Z).hom ≫ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =\n (α_ X Y Z).hom ≫ (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)",
"state_before": "R : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\n⊢ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =\n (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)",
"tactic": "apply (cancel_epi (α_ X Y Z).hom).1"
},
{
"state_after": "case H\nR : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\n⊢ ∀ (x : ↑X) (y : ↑Y) (z : ↑Z),\n ↑((α_ X Y Z).hom ≫ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv) ((x ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑((α_ X Y Z).hom ≫ (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)) ((x ⊗ₜ[R] y) ⊗ₜ[R] z)",
"state_before": "R : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\n⊢ (α_ X Y Z).hom ≫ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =\n (α_ X Y Z).hom ≫ (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)",
"tactic": "apply TensorProduct.ext_threefold"
},
{
"state_after": "case H\nR : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\nx : ↑X\ny : ↑Y\nz : ↑Z\n⊢ ↑((α_ X Y Z).hom ≫ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv) ((x ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑((α_ X Y Z).hom ≫ (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)) ((x ⊗ₜ[R] y) ⊗ₜ[R] z)",
"state_before": "case H\nR : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\n⊢ ∀ (x : ↑X) (y : ↑Y) (z : ↑Z),\n ↑((α_ X Y Z).hom ≫ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv) ((x ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑((α_ X Y Z).hom ≫ (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)) ((x ⊗ₜ[R] y) ⊗ₜ[R] z)",
"tactic": "intro x y z"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\nx : ↑X\ny : ↑Y\nz : ↑Z\n⊢ ↑((α_ X Y Z).hom ≫ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv) ((x ⊗ₜ[R] y) ⊗ₜ[R] z) =\n ↑((α_ X Y Z).hom ≫ (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y)) ((x ⊗ₜ[R] y) ⊗ₜ[R] z)",
"tactic": "rfl"
}
] |
[
61,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Order/Atoms.lean
|
IsSimpleOrder.bot_ne_top
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.23015\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\na b : α\nh : a ≠ b\n⊢ ⊥ ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.23015\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\n⊢ ⊥ ≠ ⊤",
"tactic": "obtain ⟨a, b, h⟩ := exists_pair_ne α"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.23015\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\na b : α\nh : a ≠ b\n⊢ ⊥ ≠ ⊤",
"tactic": "rcases eq_bot_or_eq_top a with (rfl | rfl) <;> rcases eq_bot_or_eq_top b with (rfl | rfl) <;>\n first |simpa|simpa using h.symm"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inr\nα : Type u_1\nβ : Type ?u.23015\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊤ ≠ ⊤\n⊢ ⊥ ≠ ⊤",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inr\nα : Type u_1\nβ : Type ?u.23015\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\nh : ⊤ ≠ ⊤\n⊢ ⊥ ≠ ⊤",
"tactic": "simpa using h.symm"
}
] |
[
448,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/CategoryTheory/Sites/Types.lean
|
CategoryTheory.eval_typesGlue
|
[
{
"state_after": "case h\nS : Type uᵒᵖ ⥤ Type u\nhs : IsSheaf typesGrothendieckTopology S\nα : Type u\nf : α → S.obj PUnit.op\nx : α\n⊢ eval S α (typesGlue S hs α f) x = f x",
"state_before": "S : Type uᵒᵖ ⥤ Type u\nhs : IsSheaf typesGrothendieckTopology S\nα : Type u\nf : α → S.obj PUnit.op\n⊢ eval S α (typesGlue S hs α f) = f",
"tactic": "funext x"
},
{
"state_after": "case h\nS : Type uᵒᵖ ⥤ Type u\nhs : IsSheaf typesGrothendieckTopology S\nα : Type u\nf : α → S.obj PUnit.op\nx : α\n⊢ S.map (↾fun x => PUnit.unit).op (f ((↾fun x_1 => x) (Classical.choose (_ : ∃ x, ∀ (y : PUnit), y = x)))) = f x",
"state_before": "case h\nS : Type uᵒᵖ ⥤ Type u\nhs : IsSheaf typesGrothendieckTopology S\nα : Type u\nf : α → S.obj PUnit.op\nx : α\n⊢ eval S α (typesGlue S hs α f) x = f x",
"tactic": "apply (IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans"
},
{
"state_after": "no goals",
"state_before": "case h\nS : Type uᵒᵖ ⥤ Type u\nhs : IsSheaf typesGrothendieckTopology S\nα : Type u\nf : α → S.obj PUnit.op\nx : α\n⊢ S.map (↾fun x => PUnit.unit).op (f ((↾fun x_1 => x) (Classical.choose (_ : ∃ x, ∀ (y : PUnit), y = x)))) = f x",
"tactic": "convert FunctorToTypes.map_id_apply S _"
}
] |
[
108,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMinOn.sup
|
[] |
[
461,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
8
] |
Mathlib/Data/LazyList/Basic.lean
|
LazyList.append_bind
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nxs : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\n⊢ LazyList.bind (append xs ys) f = append (LazyList.bind xs f) { fn := fun x => LazyList.bind (Thunk.get ys) f }",
"tactic": "match xs with\n| LazyList.nil => rfl\n| LazyList.cons x xs =>\n simp only [append, Thunk.get, LazyList.bind]\n have := append_bind xs.get ys f\n simp only [Thunk.get] at this\n rw [this, append_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nxs : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\n⊢ LazyList.bind (append nil ys) f = append (LazyList.bind nil f) { fn := fun x => LazyList.bind (Thunk.get ys) f }",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\n⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }",
"state_before": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\n⊢ LazyList.bind (append (cons x xs) ys) f =\n append (LazyList.bind (cons x xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }",
"tactic": "simp only [append, Thunk.get, LazyList.bind]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\nthis :\n LazyList.bind (append (Thunk.get xs) ys) f =\n append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }",
"state_before": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\n⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }",
"tactic": "have := append_bind xs.get ys f"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\nthis :\n LazyList.bind (append (Thunk.fn xs ()) ys) f =\n append (LazyList.bind (Thunk.fn xs ()) f) { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }",
"state_before": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\nthis :\n LazyList.bind (append (Thunk.get xs) ys) f =\n append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }",
"tactic": "simp only [Thunk.get] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nxs✝ : LazyList α\nys : Thunk (LazyList α)\nf : α → LazyList β\nx : α\nxs : Thunk (LazyList α)\nthis :\n LazyList.bind (append (Thunk.fn xs ()) ys) f =\n append (LazyList.bind (Thunk.fn xs ()) f) { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }",
"tactic": "rw [this, append_assoc]"
}
] |
[
190,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Order/Bounded.lean
|
Set.Unbounded.mono
|
[] |
[
34,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
32,
1
] |
Mathlib/Probability/Independence/ZeroOne.lean
|
ProbabilityTheory.indep_limsup_atTop_self
|
[
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\n⊢ Indep (limsup s atTop) (limsup s atTop)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\n⊢ Indep (limsup s atTop) (limsup s atTop)",
"tactic": "let ns : ι → Set ι := Set.Iic"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ Indep (limsup s atTop) (limsup s atTop)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\n⊢ Indep (limsup s atTop) (limsup s atTop)",
"tactic": "have hnsp : ∀ i, BddAbove (ns i) := fun i => bddAbove_Iic"
},
{
"state_after": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ ∀ (t : Set ι), BddAbove t → tᶜ ∈ atTop\n\ncase refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ Directed (fun x x_1 => x ≤ x_1) fun a => ns a\n\ncase refine'_3\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ ∀ (n : ι), ∃ a, n ∈ ns a",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ Indep (limsup s atTop) (limsup s atTop)",
"tactic": "refine' indep_limsup_self h_le h_indep _ _ hnsp _"
},
{
"state_after": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ ∀ (t : Set ι), (∃ x, x ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}) → ∃ a, ∀ (b : ι), a ≤ b → ¬b ∈ t",
"state_before": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ ∀ (t : Set ι), BddAbove t → tᶜ ∈ atTop",
"tactic": "simp only [mem_atTop_sets, ge_iff_le, Set.mem_compl_iff, BddAbove, upperBounds, Set.Nonempty]"
},
{
"state_after": "case refine'_1.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\n⊢ ∃ a, ∀ (b : ι), a ≤ b → ¬b ∈ t",
"state_before": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ ∀ (t : Set ι), (∃ x, x ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}) → ∃ a, ∀ (b : ι), a ≤ b → ¬b ∈ t",
"tactic": "rintro t ⟨a, ha⟩"
},
{
"state_after": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\n⊢ ∃ a, ∀ (b : ι), a ≤ b → ¬b ∈ t",
"state_before": "case refine'_1.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\n⊢ ∃ a, ∀ (b : ι), a ≤ b → ¬b ∈ t",
"tactic": "obtain ⟨b, hb⟩ : ∃ b, a < b := exists_gt a"
},
{
"state_after": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\n⊢ False",
"state_before": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\n⊢ ∃ a, ∀ (b : ι), a ≤ b → ¬b ∈ t",
"tactic": "refine' ⟨b, fun c hc hct => _⟩"
},
{
"state_after": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\nthis : ∀ (i : ι), i ∈ t → i < c\n⊢ False\n\ncase this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → i < c",
"state_before": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\n⊢ False",
"tactic": "suffices : ∀ i ∈ t, i < c"
},
{
"state_after": "case this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → i < c",
"state_before": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\nthis : ∀ (i : ι), i ∈ t → i < c\n⊢ False\n\ncase this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → i < c",
"tactic": "exact lt_irrefl c (this c hct)"
},
{
"state_after": "no goals",
"state_before": "case this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → a ≤ x}\nb : ι\nhb : a < b\nc : ι\nhc : b ≤ c\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → i < c",
"tactic": "exact fun i hi => (ha hi).trans_lt (hb.trans_le hc)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ Directed (fun x x_1 => x ≤ x_1) fun a => ns a",
"tactic": "exact Monotone.directed_le fun i j hij k hki => le_trans hki hij"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Iic\nhnsp : ∀ (i : ι), BddAbove (ns i)\n⊢ ∀ (n : ι), ∃ a, n ∈ ns a",
"tactic": "exact fun n => ⟨n, le_rfl⟩"
}
] |
[
145,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
AddCommGroup.DirectLimit.induction_on
|
[] |
[
316,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
11
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
derivWithin_inter
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nht : t ∈ 𝓝 x\n⊢ ↑(fderivWithin 𝕜 f (s ∩ t) x) 1 = ↑(fderivWithin 𝕜 f s x) 1",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nht : t ∈ 𝓝 x\n⊢ derivWithin f (s ∩ t) x = derivWithin f s x",
"tactic": "unfold derivWithin"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nht : t ∈ 𝓝 x\n⊢ ↑(fderivWithin 𝕜 f (s ∩ t) x) 1 = ↑(fderivWithin 𝕜 f s x) 1",
"tactic": "rw [fderivWithin_inter ht]"
}
] |
[
519,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_subset_closedBall
|
[] |
[
677,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/LinearAlgebra/UnitaryGroup.lean
|
Matrix.UnitaryGroup.ext
|
[] |
[
110,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
Left.inv_lt_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : α\n⊢ a⁻¹ < 1 ↔ 1 < a",
"tactic": "rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]"
}
] |
[
169,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_comm
|
[] |
[
128,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
tendsto_inv_nhdsWithin_Ici_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : TopologicalSpace α\nf : α → G\ns : Set α\nx : α\ninst✝² : TopologicalSpace H\ninst✝¹ : OrderedCommGroup H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[Ici a⁻¹] a⁻¹) (𝓝[Iic a] a)",
"tactic": "simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Ici _ _ _ _ a⁻¹"
}
] |
[
597,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
596,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.complementedLattice_iff
|
[
{
"state_after": "F : Type ?u.129007\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.129016\nδ : Type ?u.129019\ninst✝³ : Lattice α\ninst✝² : Lattice β\ninst✝¹ : BoundedOrder α\ninst✝ : BoundedOrder β\nf : α ≃o β\na✝ : ComplementedLattice α\n⊢ ComplementedLattice β",
"state_before": "F : Type ?u.129007\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.129016\nδ : Type ?u.129019\ninst✝³ : Lattice α\ninst✝² : Lattice β\ninst✝¹ : BoundedOrder α\ninst✝ : BoundedOrder β\nf : α ≃o β\n⊢ ComplementedLattice α → ComplementedLattice β",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.129007\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.129016\nδ : Type ?u.129019\ninst✝³ : Lattice α\ninst✝² : Lattice β\ninst✝¹ : BoundedOrder α\ninst✝ : BoundedOrder β\nf : α ≃o β\na✝ : ComplementedLattice α\n⊢ ComplementedLattice β",
"tactic": "exact f.complementedLattice"
},
{
"state_after": "F : Type ?u.129007\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.129016\nδ : Type ?u.129019\ninst✝³ : Lattice α\ninst✝² : Lattice β\ninst✝¹ : BoundedOrder α\ninst✝ : BoundedOrder β\nf : α ≃o β\na✝ : ComplementedLattice β\n⊢ ComplementedLattice α",
"state_before": "F : Type ?u.129007\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.129016\nδ : Type ?u.129019\ninst✝³ : Lattice α\ninst✝² : Lattice β\ninst✝¹ : BoundedOrder α\ninst✝ : BoundedOrder β\nf : α ≃o β\n⊢ ComplementedLattice β → ComplementedLattice α",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.129007\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.129016\nδ : Type ?u.129019\ninst✝³ : Lattice α\ninst✝² : Lattice β\ninst✝¹ : BoundedOrder α\ninst✝ : BoundedOrder β\nf : α ≃o β\na✝ : ComplementedLattice β\n⊢ ComplementedLattice α",
"tactic": "exact f.symm.complementedLattice"
}
] |
[
1385,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1383,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioo_filter_lt
|
[
{
"state_after": "case a\nι : Type ?u.141256\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝¹ b✝ a b c a✝ : α\n⊢ a✝ ∈ filter (fun x => x < c) (Ioo a b) ↔ a✝ ∈ Ioo a (min b c)",
"state_before": "ι : Type ?u.141256\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\n⊢ filter (fun x => x < c) (Ioo a b) = Ioo a (min b c)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type ?u.141256\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝¹ b✝ a b c a✝ : α\n⊢ a✝ ∈ filter (fun x => x < c) (Ioo a b) ↔ a✝ ∈ Ioo a (min b c)",
"tactic": "simp [and_assoc]"
}
] |
[
804,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
802,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean
|
CategoryTheory.unmop_inj
|
[] |
[
110,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Order/Filter/ENNReal.lean
|
ENNReal.eventually_le_limsup
|
[] |
[
28,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_plift_up
|
[] |
[
230,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.le_sSup
|
[
{
"state_after": "case f\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ underlying.obj f ⟶ underlying.obj (sSup s)\n\ncase w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ ?f ≫ arrow (sSup s) = arrow f",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ f ≤ sSup s",
"tactic": "fapply le_of_comm"
},
{
"state_after": "case f\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ underlying.obj f =\n underlying.obj\n (↑(equivShrink (Subobject A)).symm\n ↑{ val := ↑(equivShrink (Subobject A)) f,\n property := (_ : ↑(equivShrink (Subobject A)) f ∈ ↑(equivShrink (Subobject A)) '' s) })",
"state_before": "case f\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ underlying.obj f ⟶ underlying.obj (sSup s)",
"tactic": "refine' eqToHom _ ≫ Sigma.ι _ ⟨equivShrink (Subobject A) f, by simpa [Set.mem_image] using hf⟩\n ≫ factorThruImage _ ≫ (underlyingIso _).inv"
},
{
"state_after": "no goals",
"state_before": "case f\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ underlying.obj f =\n underlying.obj\n (↑(equivShrink (Subobject A)).symm\n ↑{ val := ↑(equivShrink (Subobject A)) f,\n property := (_ : ↑(equivShrink (Subobject A)) f ∈ ↑(equivShrink (Subobject A)) '' s) })",
"tactic": "exact (congr_arg (fun X : Subobject A => (X : C)) (Equiv.symm_apply_apply _ _).symm)"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ ↑(equivShrink (Subobject A)) f ∈ ↑(equivShrink (Subobject A)) '' s",
"tactic": "simpa [Set.mem_image] using hf"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ (eqToHom\n (_ :\n underlying.obj f =\n underlying.obj\n (↑(equivShrink (Subobject A)).symm\n ↑{ val := ↑(equivShrink (Subobject A)) f,\n property := (_ : ↑(equivShrink (Subobject A)) f ∈ ↑(equivShrink (Subobject A)) '' s) })) ≫\n Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j))\n { val := ↑(equivShrink (Subobject A)) f,\n property := (_ : ↑(equivShrink (Subobject A)) f ∈ ↑(equivShrink (Subobject A)) '' s) } ≫\n factorThruImage (smallCoproductDesc s) ≫ (underlyingIso (image.ι (smallCoproductDesc s))).inv) ≫\n arrow (sSup s) =\n arrow f",
"tactic": "simp [sSup, smallCoproductDesc]"
}
] |
[
693,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
688,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.isOption_neg_neg
|
[
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ IsOption (-x) (-y) ↔ IsOption x y",
"tactic": "rw [isOption_neg, neg_neg]"
}
] |
[
1238,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1237,
1
] |
Mathlib/Topology/Constructions.lean
|
nhds_prod_eq
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.32917\nδ : Type ?u.32920\nε : Type ?u.32923\nζ : Type ?u.32926\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\na : α\nb : β\n⊢ 𝓝 (a, b) = Filter.prod (𝓝 a) (𝓝 b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.32917\nδ : Type ?u.32920\nε : Type ?u.32923\nζ : Type ?u.32926\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\na : α\nb : β\n⊢ 𝓝 (a, b) = 𝓝 a ×ˢ 𝓝 b",
"tactic": "dsimp only [SProd.sprod]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.32917\nδ : Type ?u.32920\nε : Type ?u.32923\nζ : Type ?u.32926\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\na : α\nb : β\n⊢ 𝓝 (a, b) = Filter.prod (𝓝 a) (𝓝 b)",
"tactic": "rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _)\n (t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced]"
}
] |
[
514,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Topology/Constructions.lean
|
induced_to_pi
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.512017\nδ : Type ?u.512020\nε : Type ?u.512023\nζ : Type ?u.512026\nι : Type u_2\nπ : ι → Type u_3\nκ : Type ?u.512037\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf✝ : α → (i : ι) → π i\nX : Type u_1\nf : X → (i : ι) → π i\n⊢ (⨅ (i : ι), induced f (induced (fun f => f i) ((fun a => inst✝ a) i))) =\n ⨅ (i : ι), induced (fun x => f x i) inferInstance",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.512017\nδ : Type ?u.512020\nε : Type ?u.512023\nζ : Type ?u.512026\nι : Type u_2\nπ : ι → Type u_3\nκ : Type ?u.512037\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf✝ : α → (i : ι) → π i\nX : Type u_1\nf : X → (i : ι) → π i\n⊢ induced f Pi.topologicalSpace = ⨅ (i : ι), induced (fun x => f x i) inferInstance",
"tactic": "erw [induced_iInf]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.512017\nδ : Type ?u.512020\nε : Type ?u.512023\nζ : Type ?u.512026\nι : Type u_2\nπ : ι → Type u_3\nκ : Type ?u.512037\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf✝ : α → (i : ι) → π i\nX : Type u_1\nf : X → (i : ι) → π i\n⊢ (⨅ (i : ι), induced ((fun f => f i) ∘ f) (inst✝ i)) = ⨅ (i : ι), induced (fun x => f x i) inferInstance",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.512017\nδ : Type ?u.512020\nε : Type ?u.512023\nζ : Type ?u.512026\nι : Type u_2\nπ : ι → Type u_3\nκ : Type ?u.512037\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf✝ : α → (i : ι) → π i\nX : Type u_1\nf : X → (i : ι) → π i\n⊢ (⨅ (i : ι), induced f (induced (fun f => f i) ((fun a => inst✝ a) i))) =\n ⨅ (i : ι), induced (fun x => f x i) inferInstance",
"tactic": "simp only [induced_compose]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.512017\nδ : Type ?u.512020\nε : Type ?u.512023\nζ : Type ?u.512026\nι : Type u_2\nπ : ι → Type u_3\nκ : Type ?u.512037\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf✝ : α → (i : ι) → π i\nX : Type u_1\nf : X → (i : ι) → π i\n⊢ (⨅ (i : ι), induced ((fun f => f i) ∘ f) (inst✝ i)) = ⨅ (i : ι), induced (fun x => f x i) inferInstance",
"tactic": "rfl"
}
] |
[
1415,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1411,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.iUnion_cylinder_update
|
[
{
"state_after": "case h\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (y ∈ ⋃ (k : E n), cylinder (update x n k) (n + 1)) ↔ y ∈ cylinder x n",
"state_before": "E : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\n⊢ (⋃ (k : E n), cylinder (update x n k) (n + 1)) = cylinder x n",
"tactic": "ext y"
},
{
"state_after": "case h\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1) ↔ ∀ (i : ℕ), i < n → y i = x i",
"state_before": "case h\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (y ∈ ⋃ (k : E n), cylinder (update x n k) (n + 1)) ↔ y ∈ cylinder x n",
"tactic": "simp only [mem_cylinder_iff, mem_iUnion]"
},
{
"state_after": "case h.mp\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1) → ∀ (i : ℕ), i < n → y i = x i\n\ncase h.mpr\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (∀ (i : ℕ), i < n → y i = x i) → ∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1",
"state_before": "case h\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1) ↔ ∀ (i : ℕ), i < n → y i = x i",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nk : E n\nhk : ∀ (i : ℕ), i < n + 1 → y i = update x n k i\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"state_before": "case h.mp\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1) → ∀ (i : ℕ), i < n → y i = x i",
"tactic": "rintro ⟨k, hk⟩ i hi"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nk : E n\nhk : ∀ (i : ℕ), i < n + 1 → y i = update x n k i\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"tactic": "simpa [hi.ne] using hk i (Nat.lt_succ_of_lt hi)"
},
{
"state_after": "case h.mpr\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nH : ∀ (i : ℕ), i < n → y i = x i\n⊢ ∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1",
"state_before": "case h.mpr\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ (∀ (i : ℕ), i < n → y i = x i) → ∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1",
"tactic": "intro H"
},
{
"state_after": "case h.mpr\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nH : ∀ (i : ℕ), i < n → y i = x i\ni : ℕ\nhi : i < n + 1\n⊢ y i = update x n (y n) i",
"state_before": "case h.mpr\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nH : ∀ (i : ℕ), i < n → y i = x i\n⊢ ∃ i, ∀ (i_1 : ℕ), i_1 < n + 1 → y i_1 = update x n i i_1",
"tactic": "refine' ⟨y n, fun i hi => _⟩"
},
{
"state_after": "case h.mpr.inl\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nH : ∀ (i : ℕ), i < n → y i = x i\ni : ℕ\nhi : i < n + 1\nh'i : i < n\n⊢ y i = update x n (y n) i\n\ncase h.mpr.inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\ni : ℕ\nH : ∀ (i_1 : ℕ), i_1 < i → y i_1 = x i_1\nhi : i < i + 1\n⊢ y i = update x i (y i) i",
"state_before": "case h.mpr\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nH : ∀ (i : ℕ), i < n → y i = x i\ni : ℕ\nhi : i < n + 1\n⊢ y i = update x n (y n) i",
"tactic": "rcases Nat.lt_succ_iff_lt_or_eq.1 hi with (h'i | rfl)"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.inl\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\nH : ∀ (i : ℕ), i < n → y i = x i\ni : ℕ\nhi : i < n + 1\nh'i : i < n\n⊢ y i = update x n (y n) i",
"tactic": "simp [H i h'i, h'i.ne]"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\ni : ℕ\nH : ∀ (i_1 : ℕ), i_1 < i → y i_1 = x i_1\nhi : i < i + 1\n⊢ y i = update x i (y i) i",
"tactic": "simp"
}
] |
[
188,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.normSq_add_mul_I
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑normSq (↑x + ↑y * I) = x ^ 2 + y ^ 2",
"tactic": "rw [← mk_eq_add_mul_I, normSq_mk, sq, sq]"
}
] |
[
598,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.mem_closedBall_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.818815\nR' : Type ?u.818818\n𝕜 : Type u_2\n𝕜₂ : Type ?u.818824\n𝕜₃ : Type ?u.818827\n𝕝 : Type ?u.818830\nE : Type u_1\nE₂ : Type ?u.818836\nE₃ : Type ?u.818839\nF : Type ?u.818842\nG : Type ?u.818845\nι : Type ?u.818848\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np : Seminorm 𝕜 E\nx y : E\nr : ℝ\n⊢ y ∈ closedBall p 0 r ↔ ↑p y ≤ r",
"tactic": "rw [mem_closedBall, sub_zero]"
}
] |
[
665,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
DifferentiableWithinAt.smul
|
[] |
[
199,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Topology/Algebra/UniformRing.lean
|
UniformSpace.Completion.continuous_mul
|
[
{
"state_after": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => p.fst * p.snd",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "let m := (AddMonoidHom.mul : α →+ α →+ α).compr₂ toCompl"
},
{
"state_after": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\nthis : Continuous fun p => ↑(↑m p.fst) p.snd\n⊢ Continuous fun p => p.fst * p.snd",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "have : Continuous fun p : α × α => m p.1 p.2 := by\n apply (continuous_coe α).comp _\n simp only [AddMonoidHom.coe_mul, AddMonoidHom.coe_mulLeft]\n exact _root_.continuous_mul"
},
{
"state_after": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\nthis : Continuous fun p => ↑(↑m p.fst) p.snd\ndi : DenseInducing ↑toCompl\n⊢ Continuous fun p => p.fst * p.snd",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\nthis : Continuous fun p => ↑(↑m p.fst) p.snd\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "have di : DenseInducing (toCompl : α → Completion α) := denseInducing_coe"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\nthis : Continuous fun p => ↑(↑m p.fst) p.snd\ndi : DenseInducing ↑toCompl\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "convert di.extend_Z_bilin di this"
},
{
"state_after": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => ↑(↑AddMonoidHom.mul p.fst) p.snd",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => ↑(↑m p.fst) p.snd",
"tactic": "apply (continuous_coe α).comp _"
},
{
"state_after": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => p.fst * p.snd",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => ↑(↑AddMonoidHom.mul p.fst) p.snd",
"tactic": "simp only [AddMonoidHom.coe_mul, AddMonoidHom.coe_mulLeft]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalRing α\ninst✝ : UniformAddGroup α\nm : α →+ α →+ Completion α := AddMonoidHom.compr₂ AddMonoidHom.mul toCompl\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "exact _root_.continuous_mul"
}
] |
[
85,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Icc_mem_nhdsWithin_Ioi
|
[] |
[
425,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.ext_iff
|
[] |
[
86,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsBaseChange.lift_eq
|
[
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module R M\ninst✝⁸ : Module R N\ninst✝⁷ : Module S N\ninst✝⁶ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.200383\nQ : Type u_2\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module S Q\ninst✝¹ : Module R Q\ninst✝ : IsScalarTower R S Q\ng : M →ₗ[R] Q\nx : M\nhF : ∀ (s : S) (m : M), ↑(lift h g) (s • ↑f m) = s • ↑g m\n⊢ ↑(lift h g) (↑f x) = ↑g x",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module R M\ninst✝⁸ : Module R N\ninst✝⁷ : Module S N\ninst✝⁶ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.200383\nQ : Type u_2\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module S Q\ninst✝¹ : Module R Q\ninst✝ : IsScalarTower R S Q\ng : M →ₗ[R] Q\nx : M\n⊢ ↑(lift h g) (↑f x) = ↑g x",
"tactic": "have hF : ∀ (s : S) (m : M), h.lift g (s • f m) = s • g m := h.lift_eq _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module R M\ninst✝⁸ : Module R N\ninst✝⁷ : Module S N\ninst✝⁶ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.200383\nQ : Type u_2\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module S Q\ninst✝¹ : Module R Q\ninst✝ : IsScalarTower R S Q\ng : M →ₗ[R] Q\nx : M\nhF : ∀ (s : S) (m : M), ↑(lift h g) (s • ↑f m) = s • ↑g m\n⊢ ↑(lift h g) (↑f x) = ↑g x",
"tactic": "convert hF 1 x <;> rw [one_smul]"
}
] |
[
192,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
8
] |
Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean
|
ContinuousLinearMap.fderivWithin_of_bilinear
|
[] |
[
148,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
injective_toZ
|
[] |
[
363,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.associator_hom_hom
|
[
{
"state_after": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nX Y Z : Action V G\n⊢ (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) ≫ (α_ X.V Y.V Z.V).hom ≫ (𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) = (α_ X.V Y.V Z.V).hom",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nX Y Z : Action V G\n⊢ (α_ X Y Z).hom.hom = (α_ X.V Y.V Z.V).hom",
"tactic": "dsimp [Monoidal.transport_associator]"
},
{
"state_after": "no goals",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nX Y Z : Action V G\n⊢ (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) ≫ (α_ X.V Y.V Z.V).hom ≫ (𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) = (α_ X.V Y.V Z.V).hom",
"tactic": "simp"
}
] |
[
514,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.comap_cocompact
|
[] |
[
273,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Data/Nat/Interval.lean
|
Nat.image_Ico_mod
|
[
{
"state_after": "case inl\na b c n : ℕ\n⊢ image (fun x => x % 0) (Ico n (n + 0)) = range 0\n\ncase inr\na✝ b c n a : ℕ\nha : a ≠ 0\n⊢ image (fun x => x % a) (Ico n (n + a)) = range a",
"state_before": "a✝ b c n a : ℕ\n⊢ image (fun x => x % a) (Ico n (n + a)) = range a",
"tactic": "obtain rfl | ha := eq_or_ne a 0"
},
{
"state_after": "case inr.a\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ i ∈ image (fun x => x % a) (Ico n (n + a)) ↔ i ∈ range a",
"state_before": "case inr\na✝ b c n a : ℕ\nha : a ≠ 0\n⊢ image (fun x => x % a) (Ico n (n + a)) = range a",
"tactic": "ext i"
},
{
"state_after": "case inr.a\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ (∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i) ↔ i < a",
"state_before": "case inr.a\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ i ∈ image (fun x => x % a) (Ico n (n + a)) ↔ i ∈ range a",
"tactic": "simp only [mem_image, exists_prop, mem_range, mem_Ico]"
},
{
"state_after": "case inr.a.mp\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ (∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i) → i < a\n\ncase inr.a.mpr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ i < a → ∃ a_2, (n ≤ a_2 ∧ a_2 < n + a) ∧ a_2 % a = i",
"state_before": "case inr.a\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ (∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i) ↔ i < a",
"tactic": "constructor"
},
{
"state_after": "case inr.a.mpr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"state_before": "case inr.a.mpr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ i < a → ∃ a_2, (n ≤ a_2 ∧ a_2 < n + a) ∧ a_2 % a = i",
"tactic": "intro hia"
},
{
"state_after": "case inr.a.mpr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"state_before": "case inr.a.mpr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"tactic": "have hn := Nat.mod_add_div n a"
},
{
"state_after": "case inr.a.mpr.inl\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i\n\ncase inr.a.mpr.inr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"state_before": "case inr.a.mpr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"tactic": "obtain hi | hi := lt_or_le i (n % a)"
},
{
"state_after": "no goals",
"state_before": "case inl\na b c n : ℕ\n⊢ image (fun x => x % 0) (Ico n (n + 0)) = range 0",
"tactic": "rw [range_zero, add_zero, Ico_self, image_empty]"
},
{
"state_after": "case inr.a.mp.intro.intro\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nleft✝ : n ≤ i ∧ i < n + a\n⊢ i % a < a",
"state_before": "case inr.a.mp\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\n⊢ (∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i) → i < a",
"tactic": "rintro ⟨i, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mp.intro.intro\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nleft✝ : n ≤ i ∧ i < n + a\n⊢ i % a < a",
"tactic": "exact mod_lt i ha.bot_lt"
},
{
"state_after": "case inr.a.mpr.inl.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n ≤ i + a * (n / a + 1)\n\ncase inr.a.mpr.inl.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ i + a * (n / a + 1) < n + a\n\ncase inr.a.mpr.inl.refine'_3\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ (i + a * (n / a + 1)) % a = i",
"state_before": "case inr.a.mpr.inl\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"tactic": "refine' ⟨i + a * (n / a + 1), ⟨_, _⟩, _⟩"
},
{
"state_after": "case inr.a.mpr.inl.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n ≤ i + a + a * (n / a)",
"state_before": "case inr.a.mpr.inl.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n ≤ i + a * (n / a + 1)",
"tactic": "rw [add_comm (n / a), mul_add, mul_one, ← add_assoc]"
},
{
"state_after": "case inr.a.mpr.inl.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a ≤ i + a",
"state_before": "case inr.a.mpr.inl.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n ≤ i + a + a * (n / a)",
"tactic": "refine' hn.symm.le.trans (add_le_add_right _ _)"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mpr.inl.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a ≤ i + a",
"tactic": "simpa only [zero_add] using add_le_add (zero_le i) (Nat.mod_lt n ha.bot_lt).le"
},
{
"state_after": "case inr.a.mpr.inl.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a + a * (n / a + 1) ≤ n + a",
"state_before": "case inr.a.mpr.inl.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ i + a * (n / a + 1) < n + a",
"tactic": "refine' lt_of_lt_of_le (add_lt_add_right hi (a * (n / a + 1))) _"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mpr.inl.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a + a * (n / a + 1) ≤ n + a",
"tactic": "rw [mul_add, mul_one, ← add_assoc, hn]"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mpr.inl.refine'_3\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ (i + a * (n / a + 1)) % a = i",
"tactic": "rw [Nat.add_mul_mod_self_left, Nat.mod_eq_of_lt hia]"
},
{
"state_after": "case inr.a.mpr.inr.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ n ≤ i + a * (n / a)\n\ncase inr.a.mpr.inr.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ i + a * (n / a) < n + a\n\ncase inr.a.mpr.inr.refine'_3\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ (i + a * (n / a)) % a = i",
"state_before": "case inr.a.mpr.inr\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ ∃ a_1, (n ≤ a_1 ∧ a_1 < n + a) ∧ a_1 % a = i",
"tactic": "refine' ⟨i + a * (n / a), ⟨_, _⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mpr.inr.refine'_1\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ n ≤ i + a * (n / a)",
"tactic": "exact hn.symm.le.trans (add_le_add_right hi _)"
},
{
"state_after": "case inr.a.mpr.inr.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ i + a * (n / a) < a + n",
"state_before": "case inr.a.mpr.inr.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ i + a * (n / a) < n + a",
"tactic": "rw [add_comm n a]"
},
{
"state_after": "case inr.a.mpr.inr.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ a * (n / a) ≤ n % a + a * (n / a)",
"state_before": "case inr.a.mpr.inr.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ i + a * (n / a) < a + n",
"tactic": "refine' add_lt_add_of_lt_of_le hia (le_trans _ hn.le)"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mpr.inr.refine'_2\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ a * (n / a) ≤ n % a + a * (n / a)",
"tactic": "simp only [zero_le, le_add_iff_nonneg_left]"
},
{
"state_after": "no goals",
"state_before": "case inr.a.mpr.inr.refine'_3\na✝ b c n a : ℕ\nha : a ≠ 0\ni : ℕ\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : n % a ≤ i\n⊢ (i + a * (n / a)) % a = i",
"tactic": "rw [Nat.add_mul_mod_self_left, Nat.mod_eq_of_lt hia]"
}
] |
[
307,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Topology/MetricSpace/Kuratowski.lean
|
kuratowskiEmbedding.isometry
|
[] |
[
123,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
11
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.to_rightInvOn'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.8961\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\ns : Set α\nhf : AntilipschitzWith K (Set.restrict s f)\ng : β → α\nt : Set β\ng_maps : MapsTo g t s\ng_inv : RightInvOn g f t\nx y : ↑t\n⊢ edist (Set.restrict t g x) (Set.restrict t g y) ≤ ↑K * edist x y",
"tactic": "simpa only [restrict_apply, g_inv x.mem, g_inv y.mem, Subtype.edist_eq, Subtype.coe_mk] using\n hf ⟨g x, g_maps x.mem⟩ ⟨g y, g_maps y.mem⟩"
}
] |
[
151,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
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