mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
F : Presheaf C X
U V : Opens ↑X
x : ↑X
hxU : x ∈ U
hxV : x ∈ V
W : Opens ↑X
hxW : x ∈ W
iWU : W ⟶ U
iWV : W ⟶ V
sU : ToType (F.obj (op U))
sV : ToType (F.obj (op V))
ih : (ConcreteCategory.hom (F.map iWU.op)) sU = (ConcreteCategory.hom (F.map iWV.op)) sV
⊢ (ConcreteCategory.hom (F.germ U x hxU)) sU = (ConcreteCategory.hom (F.germ V x hxV)) sV

Tactic:
by
  rw [← F.germ_res iWU x hxW, ← F.germ_res iWV x hxW, ConcreteCategory.comp_apply,
    ConcreteCategory.comp_apply, ih]

Goal state after:
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
x : ↑X
hxU : x ∈ U
⊢ x ∈ iSup V

Tactic:
simp only [Opens.mem_iSup]

Goal state after:
case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
x : ↑X
hxU : x ∈ U
⊢ ∃ i, x ∈ V i
Module: Mathlib.Topology.Algebra.Module.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : ContinuousConstSMul R M
inst✝ : ContinuousAdd M
s : Set M
⊢ closure[inst✝⁴] s ⊆ ↑(span R s).topologicalClosure

Tactic:
rw [Submodule.topologicalClosure_coe]

Goal state after:
R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : ContinuousConstSMul R M
inst✝ : ContinuousAdd M
s : Set M
⊢ closure[inst✝⁴] s ⊆ closure[inst✝⁴] ↑(span R s)
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ x ∈ U, Function.Injective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map f.hom))
hsurj :
  ∀ (t : CC (G.obj.obj (op U))) (x : ↥U),
    ∃ V,
      ∃ (_ : ↑x ∈ V), ∃ iVU s, (ConcreteCategory.hom (f.hom.app (op V))) s = (ConcreteCategory.hom (G.obj.map iVU.op)) t
t : CC (G.obj.obj (op U))
V : ↥U → Opens ↑X
mV : ∀ (x : ↥U), ↑x ∈ V x
iVU : (x : ↥U) → V x ⟶ U
sf : (x : ↥U) → ToType (F.obj.obj (op (V x)))
heq : ∀ (x : ↥U), (ConcreteCategory.hom (f.hom.app (op (V x)))) (sf x) = (ConcreteCategory.hom (G.obj.map (iVU x).op)) t
x : ↑X
hxU : x ∈ U
⊢ x ∈ iSup V

Tactic:
simp only [Opens.mem_iSup]

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ x ∈ U, Function.Injective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map f.hom))
hsurj :
  ∀ (t : CC (G.obj.obj (op U))) (x : ↥U),
    ∃ V,
      ∃ (_ : ↑x ∈ V), ∃ iVU s, (ConcreteCategory.hom (f.hom.app (op V))) s = (ConcreteCategory.hom (G.obj.map iVU.op)) t
t : CC (G.obj.obj (op U))
V : ↥U → Opens ↑X
mV : ∀ (x : ↥U), ↑x ∈ V x
iVU : (x : ↥U) → V x ⟶ U
sf : (x : ↥U) → ToType (F.obj.obj (op (V x)))
heq : ∀ (x : ↥U), (ConcreteCategory.hom (f.hom.app (op (V x)))) (sf x) = (ConcreteCategory.hom (G.obj.map (iVU x).op)) t
x : ↑X
hxU : x ∈ U
⊢ ∃ i, x ∈ V i
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case property.refine_3.pair.right.e_a
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s✝ : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X :=
  fun x ↦
    match x with
    | { down := j } => WalkingPair.casesOn j U V
hι : U ⊔ V = iSup ι
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
m : s.pt ⟶ (F.interUnionPullbackCone U V).pt
h₁ : m ≫ (F.interUnionPullbackCone U V).fst = s.fst
h₂ : m ≫ (F.interUnionPullbackCone U V).snd = s.snd
a✝ : ULift.{w, 0} WalkingPair
⊢ m ≫ F.obj.map ((eqToHom ⋯).op ≫ (homOfLE ⋯).op) =
    F.interUnionPullbackConeLift U V s ≫ F.obj.map ((eqToHom ⋯).op ≫ (homOfLE ⋯).op)

Tactic:
· convert! h₂
      apply interUnionPullbackConeLift_right

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ModuleTopology

Goal state before:
pp: R : Type u_1
S : Type u_2
τR : TopologicalSpace R
τS : TopologicalSpace S
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
A : Type u_3
inst✝⁶ : AddCommMonoid A
inst✝⁵ : Module R A
aA : TopologicalSpace A
inst✝⁴ : IsModuleTopology R A
B' : Type u_5
inst✝³ : AddCommMonoid B'
inst✝² : Module S B'
aB' : TopologicalSpace B'
inst✝¹ : ContinuousAdd B'
inst✝ : ContinuousSMul S B'
σ : R →* S
hσ : Continuous ⇑σ
φ : A →ₑ+[σ] B'
⊢ Continuous ⇑φ

Tactic:
continuous_iff_le_induced

Goal state after:
R : Type u_1
S : Type u_2
τR : TopologicalSpace R
τS : TopologicalSpace S
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
A : Type u_3
inst✝⁶ : AddCommMonoid A
inst✝⁵ : Module R A
aA : TopologicalSpace A
inst✝⁴ : IsModuleTopology R A
B' : Type u_5
inst✝³ : AddCommMonoid B'
inst✝² : Module S B'
aB' : TopologicalSpace B'
inst✝¹ : ContinuousAdd B'
inst✝ : ContinuousSMul S B'
σ : R →* S
hσ : Continuous ⇑σ
φ : A →ₑ+[σ] B'
⊢ moduleTopology R A ≤ TopologicalSpace.induced (⇑φ) aB'
Module: Mathlib.Topology.Algebra.Module.ModuleTopology

Goal state before:
pp: R : Type u_1
S : Type u_2
τR : TopologicalSpace R
τS : TopologicalSpace S
inst✝⁷ : Ring R
inst✝⁶ : Ring S
A : Type u_3
inst✝⁵ : AddCommGroup A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : IsModuleTopology R A
B' : Type u_5
inst✝¹ : AddCommGroup B'
inst✝ : Module S B'
σ : R →+* S
hσ : IsOpenQuotientMap ⇑σ
φ : A →ₛₗ[σ] B'
hφ : Function.Surjective ⇑φ
this✝ : TopologicalSpace B' := moduleTopology S B'
this : IsModuleTopology S B'
⊢ moduleTopology S B' = TopologicalSpace.coinduced (⇑φ) inferInstance

Tactic:
exact (isQuotientMap_of_surjectiveₛₗ hσ φ hφ).eq_coinduced

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic

Goal state before:
pp: case mpr
R₁ : Type u_1
R₂ : Type u_2
inst✝⁷ : Semiring R₁
inst✝⁶ : Semiring R₂
σ₁₂ : R₁ →+* R₂
M₁ : Type u_4
inst✝⁵ : TopologicalSpace M₁
inst✝⁴ : AddCommMonoid M₁
M₂ : Type u_6
inst✝³ : TopologicalSpace M₂
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R₁ M₁
inst✝ : Module R₂ M₂
f : M₁ → M₂
⊢ f ∈ {f | Continuous[inst✝⁵, inst✝³] f} ∩ Set.range DFunLike.coe → f ∈ Set.range DFunLike.coe

Tactic:
rintro ⟨hfc, f, rfl⟩
    exact ⟨⟨f, hfc⟩, rfl⟩

Goal state after:
Module: Mathlib.Algebra.Category.Semigrp.Basic

Goal state before:
pp: X✝ Y✝ : Type u
X Y : MagmaCat
f : X ⟶ Y
x✝ : IsIso ((forget MagmaCat).map f)
i : (forget MagmaCat).obj X ≅ (forget MagmaCat).obj Y := asIso ((forget MagmaCat).map f)
e : ↑X ≃* ↑Y :=
  let __src := Hom.hom f;
  let __src_1 := i.toEquiv;
  { toFun := __src.toFun, invFun := __src_1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }
⊢ IsIso f

Tactic:
exact e.toMagmaCatIso.isIso_hom

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
F : J ⥤ C
i₀ : J
h : F.IsEventuallyConstantTo i₀
inst✝ : IsCofiltered J
j j' : J
φ : j ⟶ j'
i : J := IsCofiltered.min i₀ j
⊢ h.coneπApp j' = h.coneπApp j ≫ F.map φ

Tactic:
let α : i ⟶ i₀ := IsCofiltered.minToLeft _ _

Goal state after:
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
F : J ⥤ C
i₀ : J
h : F.IsEventuallyConstantTo i₀
inst✝ : IsCofiltered J
j j' : J
φ : j ⟶ j'
i : J := IsCofiltered.min i₀ j
α : i ⟶ i₀ := minToLeft i₀ j
⊢ h.coneπApp j' = h.coneπApp j ≫ F.map φ
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
M : ModuleCat R
I : Type u_1
X : I → TopModuleCat R
f : (i : I) → (X i).toModuleCat ⟶ M
J : Type u_2
inst✝ : Category.{v_1, u_2} J
F : J ⥤ TopModuleCat R
c : Cocone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))
hc : IsColimit c
s : Cocone F
⊢ Continuous (ModuleCat.Hom.hom (hc.desc ((forget₂ (TopModuleCat R) (ModuleCat R)).mapCocone s))).toFun

Tactic:
continuous_iff_le_induced

Goal state after:
R : Type u
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
M : ModuleCat R
I : Type u_1
X : I → TopModuleCat R
f : (i : I) → (X i).toModuleCat ⟶ M
J : Type u_2
inst✝ : Category.{v_1, u_2} J
F : J ⥤ TopModuleCat R
c : Cocone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))
hc : IsColimit c
s : Cocone F
⊢ instTopologicalSpaceCarrier R (ofCocone c).pt ≤
    TopologicalSpace.induced (ModuleCat.Hom.hom (hc.desc ((forget₂ (TopModuleCat R) (ModuleCat R)).mapCocone s))).toFun
      s.1.topologicalSpace
Module: Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝³ : Category.{v_1, u_1} J
inst✝² : Category.{v_2, u_2} C
F : J ⥤ C
i₀ : J
h : F.IsEventuallyConstantTo i₀
inst✝¹ : IsCofiltered J
c : Cone F
inst✝ : IsIso (c.π.app i₀)
j : J
i : J := IsCofiltered.min i₀ j
⊢ h.cone.π.app j = (asIso (c.π.app i₀)).symm.hom ≫ c.π.app j

Tactic:
let α : i ⟶ i₀ := IsCofiltered.minToLeft _ _

Goal state after:
J : Type u_1
C : Type u_2
inst✝³ : Category.{v_1, u_1} J
inst✝² : Category.{v_2, u_2} C
F : J ⥤ C
i₀ : J
h : F.IsEventuallyConstantTo i₀
inst✝¹ : IsCofiltered J
c : Cone F
inst✝ : IsIso (c.π.app i₀)
j : J
i : J := IsCofiltered.min i₀ j
α : i ⟶ i₀ := minToLeft i₀ j
⊢ h.cone.π.app j = (asIso (c.π.app i₀)).symm.hom ≫ c.π.app j
Module: Mathlib.Algebra.Category.ModuleCat.Topology.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X : TopModuleCat R
⊢ Continuous ((Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat) R ↑X.toModuleCat) id).toFun

Tactic:
continuous_iff_le_induced

Goal state after:
R : Type u
inst✝¹ : Ring R
inst✝ : TopologicalSpace R
X : TopModuleCat R
⊢ ((forget₂ (TopModuleCat R) TopCat ⋙ free R).obj X).topologicalSpace ≤
    TopologicalSpace.induced ((Finsupp.lift (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat) R ↑X.toModuleCat) id).toFun
      ((𝟭 (TopModuleCat R)).obj X).topologicalSpace
Module: Mathlib.CategoryTheory.Presentable.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
⊢ IsCardinalPresentable X κ ↔ (coyoneda.obj (op X) ⋙ uliftFunctor.{t, v₁}).IsCardinalAccessible κ

Tactic:
refine ⟨fun _ ↦ inferInstance, fun _ ↦ ⟨fun J _ _ ↦ ?_⟩⟩

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
x✝² : (coyoneda.obj (op X) ⋙ uliftFunctor.{t, v₁}).IsCardinalAccessible κ
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
⊢ PreservesColimitsOfShape J (coyoneda.obj (op X))
Module: Mathlib.CategoryTheory.Presentable.IsCardinalFiltered

Goal state before:
pp: κ : Cardinal.{w}
hκ : Fact κ.IsRegular
ι : Type w
f : ι → κ.ord.ToType
h : Function.Surjective fun i ↦ ⟨f i, ⋯⟩
hs : ∀ (j : κ.ord.ToType), ∃ k, j < f k
⊢ κ.ord.cof ≤ Cardinal.mk ι

Tactic:
← Ordinal.cof_toType

Goal state after:
κ : Cardinal.{w}
hκ : Fact κ.IsRegular
ι : Type w
f : ι → κ.ord.ToType
h : Function.Surjective fun i ↦ ⟨f i, ⋯⟩
hs : ∀ (j : κ.ord.ToType), ∃ k, j < f k
⊢ Order.cof κ.ord.ToType ≤ Cardinal.mk ι
Module: Mathlib.LinearAlgebra.TensorProduct.Pi

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_3
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
⊢ piScalarRightInv R S N ι ∘ₗ piScalarRightHom R S N ι = LinearMap.id

Tactic:
ext x i

Goal state after:
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_3
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : N
i : ι
⊢ ((AlgebraTensorModule.curry (piScalarRightInv R S N ι ∘ₗ piScalarRightHom R S N ι)) x ∘ₗ single R (fun i ↦ R) i) 1 =
    ((AlgebraTensorModule.curry LinearMap.id) x ∘ₗ single R (fun i ↦ R) i) 1
Module: Mathlib.Algebra.Module.SnakeLemma

Goal state before:
pp: R : Type u_1
inst✝¹⁸ : CommRing R
M₁ : Type u_2
M₂ : Type u_3
M₃ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
N₃ : Type u_7
inst✝¹⁷ : AddCommGroup M₁
inst✝¹⁶ : Module R M₁
inst✝¹⁵ : AddCommGroup M₂
inst✝¹⁴ : Module R M₂
inst✝¹³ : AddCommGroup M₃
inst✝¹² : Module R M₃
inst✝¹¹ : AddCommGroup N₁
inst✝¹⁰ : Module R N₁
inst✝⁹ : AddCommGroup N₂
inst✝⁸ : Module R N₂
inst✝⁷ : AddCommGroup N₃
inst✝⁶ : Module R N₃
i₁ : M₁ →ₗ[R] N₁
i₂ : M₂ →ₗ[R] N₂
i₃ : M₃ →ₗ[R] N₃
f₁ : M₁ →ₗ[R] M₂
f₂ : M₂ →ₗ[R] M₃
hf : Exact ⇑f₁ ⇑f₂
g₁ : N₁ →ₗ[R] N₂
g₂ : N₂ →ₗ[R] N₃
hg : Exact ⇑g₁ ⇑g₂
h₁ : g₁ ∘ₗ i₁ = i₂ ∘ₗ f₁
h₂ : g₂ ∘ₗ i₂ = i₃ ∘ₗ f₂
σ : M₃ → M₂
hσ : ⇑f₂ ∘ σ = id
ρ : N₂ → N₁
hρ : ρ ∘ ⇑g₁ = id
K₃ : Type u_9
C₁ : Type u_10
C₂ : Type u_11
inst✝⁵ : AddCommGroup K₃
inst✝⁴ : Module R K₃
inst✝³ : AddCommGroup C₁
inst✝² : Module R C₁
inst✝¹ : AddCommGroup C₂
inst✝ : Module R C₂
ι₃ : K₃ →ₗ[R] M₃
hι₃ : Exact ⇑ι₃ ⇑i₃
π₁ : N₁ →ₗ[R] C₁
hπ₁ : Exact ⇑i₁ ⇑π₁
π₂ : N₂ →ₗ[R] C₂
hπ₂ : Exact ⇑i₂ ⇑π₂
G : C₁ →ₗ[R] C₂
hF : G ∘ₗ π₁ = π₂ ∘ₗ g₁
h : Surjective ⇑π₁
H₂ : ∀ (x : K₃), g₁ (ρ (i₂ (σ (ι₃ x)))) = i₂ (σ (ι₃ x))
x : N₁
H : G (π₁ x) = 0
y : M₂
hy : i₂ y = g₁ x
⊢ g₂ (i₂ y) = 0

Tactic:
hy,

Goal state after:
R : Type u_1
inst✝¹⁸ : CommRing R
M₁ : Type u_2
M₂ : Type u_3
M₃ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
N₃ : Type u_7
inst✝¹⁷ : AddCommGroup M₁
inst✝¹⁶ : Module R M₁
inst✝¹⁵ : AddCommGroup M₂
inst✝¹⁴ : Module R M₂
inst✝¹³ : AddCommGroup M₃
inst✝¹² : Module R M₃
inst✝¹¹ : AddCommGroup N₁
inst✝¹⁰ : Module R N₁
inst✝⁹ : AddCommGroup N₂
inst✝⁸ : Module R N₂
inst✝⁷ : AddCommGroup N₃
inst✝⁶ : Module R N₃
i₁ : M₁ →ₗ[R] N₁
i₂ : M₂ →ₗ[R] N₂
i₃ : M₃ →ₗ[R] N₃
f₁ : M₁ →ₗ[R] M₂
f₂ : M₂ →ₗ[R] M₃
hf : Exact ⇑f₁ ⇑f₂
g₁ : N₁ →ₗ[R] N₂
g₂ : N₂ →ₗ[R] N₃
hg : Exact ⇑g₁ ⇑g₂
h₁ : g₁ ∘ₗ i₁ = i₂ ∘ₗ f₁
h₂ : g₂ ∘ₗ i₂ = i₃ ∘ₗ f₂
σ : M₃ → M₂
hσ : ⇑f₂ ∘ σ = id
ρ : N₂ → N₁
hρ : ρ ∘ ⇑g₁ = id
K₃ : Type u_9
C₁ : Type u_10
C₂ : Type u_11
inst✝⁵ : AddCommGroup K₃
inst✝⁴ : Module R K₃
inst✝³ : AddCommGroup C₁
inst✝² : Module R C₁
inst✝¹ : AddCommGroup C₂
inst✝ : Module R C₂
ι₃ : K₃ →ₗ[R] M₃
hι₃ : Exact ⇑ι₃ ⇑i₃
π₁ : N₁ →ₗ[R] C₁
hπ₁ : Exact ⇑i₁ ⇑π₁
π₂ : N₂ →ₗ[R] C₂
hπ₂ : Exact ⇑i₂ ⇑π₂
G : C₁ →ₗ[R] C₂
hF : G ∘ₗ π₁ = π₂ ∘ₗ g₁
h : Surjective ⇑π₁
H₂ : ∀ (x : K₃), g₁ (ρ (i₂ (σ (ι₃ x)))) = i₂ (σ (ι₃ x))
x : N₁
H : G (π₁ x) = 0
y : M₂
hy : i₂ y = g₁ x
⊢ g₂ (g₁ x) = 0
Module: Mathlib.FieldTheory.IntermediateField.Basic

Goal state before:
pp: K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
f : L →ₐ[K] L'
a₁✝ a₂✝ : IntermediateField K L
h : a₁✝.toSubalgebra = a₂✝.toSubalgebra
⊢ a₁✝ = a₂✝

Tactic:
toSubalgebra_inj

Goal state after:
K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
f : L →ₐ[K] L'
a₁✝ a₂✝ : IntermediateField K L
h : a₁✝ = a₂✝
⊢ a₁✝ = a₂✝
Module: Mathlib.Algebra.ContinuedFractions.Determinant

Goal state before:
pp: case succ
K : Type u_1
inst✝ : Field K
g : GenContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬g.TerminatedAt (n + 1 - 1)
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
IH :
  n = 0 ∨ ¬g.TerminatedAt (n - 1) →
    (g.contsAux n).a * pred_conts.b - (g.contsAux n).b * pred_conts.a =
      ∏ i ∈ Finset.range n, -(g.partNums.get? i).getD 0
pred_conts_eq : pred_conts = g.contsAux (n + 1)
⊢ pred_conts.a * (g.contsAux (n + 1 + 1)).b - pred_conts.b * (g.contsAux (n + 1 + 1)).a =
    ∏ i ∈ Finset.range (n + 1), -(g.partNums.get? i).getD 0

Tactic:
set ppred_conts := contsAux g n with ppred_conts_eq

Goal state after:
case succ
K : Type u_1
inst✝ : Field K
g : GenContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬g.TerminatedAt (n + 1 - 1)
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH :
  n = 0 ∨ ¬g.TerminatedAt (n - 1) →
    ppred_conts.a * pred_conts.b - ppred_conts.b * pred_conts.a = ∏ i ∈ Finset.range n, -(g.partNums.get? i).getD 0
ppred_conts_eq : ppred_conts = g.contsAux n
⊢ pred_conts.a * (g.contsAux (n + 1 + 1)).b - pred_conts.b * (g.contsAux (n + 1 + 1)).a =
    ∏ i ∈ Finset.range (n + 1), -(g.partNums.get? i).getD 0
Module: Mathlib.Algebra.ContinuedFractions.Determinant

Goal state before:
pp: K : Type u_1
inst✝ : Field K
g : GenContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬g.TerminatedAt (n + 1 - 1)
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH :
  n = 0 ∨ ¬g.TerminatedAt (n - 1) →
    ppred_conts.a * pred_conts.b - ppred_conts.b * pred_conts.a = ∏ i ∈ Finset.range n, -(g.partNums.get? i).getD 0
ppred_conts_eq : ppred_conts = g.contsAux n
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
not_terminated_at_n : ¬g.TerminatedAt n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
this : ppA * pB - ppB * pA = ∏ i ∈ Finset.range n, -(g.partNums.get? i).getD 0
⊢ pA * conts.b - pB * conts.a = ∏ i ∈ Finset.range (n + 1), -(g.partNums.get? i).getD 0

Tactic:
Finset.prod_range_succ,

Goal state after:
K : Type u_1
inst✝ : Field K
g : GenContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬g.TerminatedAt (n + 1 - 1)
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH :
  n = 0 ∨ ¬g.TerminatedAt (n - 1) →
    ppred_conts.a * pred_conts.b - ppred_conts.b * pred_conts.a = ∏ i ∈ Finset.range n, -(g.partNums.get? i).getD 0
ppred_conts_eq : ppred_conts = g.contsAux n
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
not_terminated_at_n : ¬g.TerminatedAt n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
this : ppA * pB - ppB * pA = ∏ i ∈ Finset.range n, -(g.partNums.get? i).getD 0
⊢ pA * conts.b - pB * conts.a = (∏ x ∈ Finset.range n, -(g.partNums.get? x).getD 0) * -(g.partNums.get? n).getD 0
Module: Mathlib.Algebra.ContinuedFractions.Determinant

Goal state before:
pp: case inl
K : Type u_1
inst✝ : Field K
g : GenContFract K
n : ℕ
terminatedAt_n : g.TerminatedAt n
⊢ g.nums n * g.dens n - g.dens n * g.nums n = ∏ i ∈ Finset.range (n + 1), -(g.partNums.get? i).getD 0

Tactic:
Finset.prod_range_succ,

Goal state after:
case inl
K : Type u_1
inst✝ : Field K
g : GenContFract K
n : ℕ
terminatedAt_n : g.TerminatedAt n
⊢ g.nums n * g.dens n - g.dens n * g.nums n =
    (∏ x ∈ Finset.range n, -(g.partNums.get? x).getD 0) * -(g.partNums.get? n).getD 0
Module: Mathlib.Data.Seq.Computation

Goal state before:
pp: α : Type u
c : Computation α
⊢ (empty α <|> c) = c

Tactic:
apply eq_of_bisim (fun c₁ c₂ => (empty α <|> c₂) = c₁) _ rfl

Goal state after:
α : Type u
c : Computation α
⊢ IsBisimulation fun c₁ c₂ ↦ (empty α <|> c₂) = c₁
Module: Mathlib.Data.Seq.Basic

Goal state before:
pp: α : Type u
R : α → α → Prop
inst✝ : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
n : ℕ
x y : α
hx : (s.drop n).head = some x
hy : s.get? (n + 1) = some y
this : ∀ (n : ℕ), motive (s.drop n)
⊢ R x y

Tactic:
exact (step x (s.drop (n + 1)) (head_eq_some hx ▸ this n)).left _ (by simpa)

Goal state after:
Module: Mathlib.Data.Nat.Fib.Basic

Goal state before:
pp: n : ℕ
h1 : fib (n + 1) = ∑ p ∈ Finset.antidiagonal n, p.1.choose p.2
h2 : fib (n + 1 + 1) = ∑ p ∈ Finset.antidiagonal (n + 1), p.1.choose p.2
⊢ ∑ p ∈ Finset.antidiagonal n, p.1.choose p.2 + ∑ p ∈ Finset.antidiagonal (n + 1), p.1.choose p.2 =
    ∑ p ∈ Finset.antidiagonal (n + 2), p.1.choose p.2

Tactic:
Finset.Nat.antidiagonal_succ_succ',

Goal state after:
n : ℕ
h1 : fib (n + 1) = ∑ p ∈ Finset.antidiagonal n, p.1.choose p.2
h2 : fib (n + 1 + 1) = ∑ p ∈ Finset.antidiagonal (n + 1), p.1.choose p.2
⊢ ∑ p ∈ Finset.antidiagonal n, p.1.choose p.2 + ∑ p ∈ Finset.antidiagonal (n + 1), p.1.choose p.2 =
    ∑
      p ∈
        Finset.cons (0, n + 2)
          (Finset.cons (n + 2, 0)
            (Finset.map ({ toFun := succ, inj' := succ_injective }.prodMap { toFun := succ, inj' := succ_injective })
              (Finset.antidiagonal n))
            ⋯)
          ⋯,
      p.1.choose p.2
Module: Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
⊢ Option.map (Pair.map Rat.cast)
      (Option.map (fun p ↦ { a := 1, b := ↑p.b }) (Stream'.Seq.get? ⟨IntFractPair.stream q, ⋯⟩ (n + 1))) =
    Option.map (fun p ↦ { a := 1, b := ↑p.b }) (Stream'.Seq.get? ⟨IntFractPair.stream v, ⋯⟩ (n + 1))

Tactic:
simp only [Stream'.Seq.get?]

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
⊢ Option.map (Pair.map Rat.cast) (Option.map (fun p ↦ { a := 1, b := ↑p.b }) (IntFractPair.stream q (n + 1))) =
    Option.map (fun p ↦ { a := 1, b := ↑p.b }) (IntFractPair.stream v (n + 1))