mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
P : E → Finset (Set X)
Pg : ∀ (i : E), ∀ t ∈ P i, t ⊆ ⇑g ⁻¹' {i}
Pdisj : ∀ (i : E), (↑(P i)).PairwiseDisjoint id
Pmeas : ∀ (i : E), ∀ t ∈ P i, MeasurableSet t
hP :
  ∀ (i : E),
    ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
      ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)
I3 :
  ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
    ∑ i ∈ g.range.sigma P, ‖i.fst‖ₑ * ‖((μ.transpose B).restrict s) i.snd‖ₑ + ↑δ
⊢ ∑ i ∈ g.range.sigma P, ‖∫ᵛ (x : X) in i.snd, g x - f x ∂[B; μ.restrict s]‖ₑ +
      ∑ i ∈ g.range.sigma P, ‖∫ᵛ (x : X) in i.snd, f x ∂[B; μ.restrict s]‖ₑ ≤
    ∑ i ∈ g.range.sigma P, ∫⁻ (x : X) in i.snd, ‖g x - f x‖ₑ ∂(μ.transpose B).variation +
      ∑ i ∈ g.range.sigma P, ‖∫ᵛ (x : X) in i.snd, f x ∂[B; μ.restrict s]‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
n✝ : ℕ
R : L.Relations n✝
x : Fin n✝ → DirectLimit G f
⊢ RelMap R (Quotient.lift (fun x ↦ (g x.fst) x.snd) ⋯ ∘ x) ↔ RelMap R x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
i : ι
x : G i
⊢ (lift L ι G f g Hg) ⟦Structure.Sigma.mk f i x⟧ = (g i) x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
i : ι
x : G i
⊢ (lift L ι G f g Hg) ((of L ι G f i) x) = (g i) x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
F : DirectLimit G f ↪[L] P
x✝ : DirectLimit G f
i j : ι
hij : i ≤ j
x : G i
⊢ (F.comp (of L ι G f j)) ((f i j hij) x) = (F.comp (of L ι G f i)) x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
F : DirectLimit G f ↪[L] P
x✝ : DirectLimit G f
i : ι
x : G i
⊢ F ((of L ι G f i) x) = (lift L ι G f (fun i ↦ F.comp (of L ι G f i)) ⋯) ((of L ι G f i) x)

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
⊢ toRatVec (hs.fract x) =
    toRatVec hs.base + ∑ i, Int.fract ((hs.basis.repr (toRatVec x - toRatVec hs.base)) i) • toRatVec ↑i

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
⊢ (lift L ι G f g Hg).toHom.range = ⨆ i, (g i).toHom.range

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
⊢ L.IsExtensionPair M N ↔
    ∀ (S : L.Substructure M), S.FG → ∀ (f : ↥S ↪[L] N) (m : M), ∃ g, f = g.comp (Substructure.inclusion ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
H : L.IsExtensionPair M N
X : Set M
left✝ : X.Countable
X_gen : (closure L).toFun X = ⊤
⊢ Countable ↑X

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x y : ι → ℕ
hy : y ∈ closure hs.basisSet
⊢ hs.fract (x + y) = hs.fract x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁷ : Preorder ι
G : ι → Type w
inst✝⁶ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝⁵ : IsDirectedOrder ι
inst✝⁴ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝³ : Nonempty ι
P : Type u₁
inst✝² : L.Structure P
g✝ : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) ((f i j hij) x) = (g✝ i) x
G' : ι → Type w'
inst✝¹ : (i : ι) → L.Structure (G' i)
f' : (i j : ι) → i ≤ j → G' i ↪[L] G' j
g : (i : ι) → G i ≃[L] G' i
inst✝ : DirectedSystem G' fun i j h ↦ ⇑(f' i j h)
H_commuting : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (f' i j hij) ((g i) x)
U : (i : ι) → G i ↪[L] DirectLimit G' f' := fun i ↦ (of L ι G' f' i).comp (g i).toEmbedding
⊢ ∀ (i j : ι) (hij : i ≤ j) (x : G i), (U j) ((f i j hij) x) = (U i) x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
H : L.IsExtensionPair M N
X : Set M
left✝ : X.Countable
X_gen : (closure L).toFun X = ⊤
x✝¹ : Countable ↑X
x✝ : Encodable ↑X
D : ↑X → Order.Cofinal (L.FGEquiv M N) := fun x ↦ H.definedAtLeft ↑x
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
⊢ X ⊆ ↑F.dom

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: case right
ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
i : ↑hs.basisSet
⊢ (hs.basis.repr (∑ i, Int.fract ((hs.basis.repr (toRatVec x - toRatVec hs.base)) i) • toRatVec ↑i)) i < 1

Tactic:
simp [← hs.basis_apply, Finsupp.single_apply, Int.fract_lt_one]

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
H : L.IsExtensionPair M N
X : Set M
left✝ : X.Countable
X_gen : (closure L).toFun X = ⊤
x✝² : Countable ↑X
x✝¹ : Encodable ↑X
D : ↑X → Order.Cofinal (L.FGEquiv M N) := fun x ↦ H.definedAtLeft ↑x
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
x✝ : X ⊆ ↑F.dom
⊢ F.dom = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
⊢ hs.fract x ∈ hs.fundamentalDomain

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
H : L.IsExtensionPair M N
X : Set M
left✝ : X.Countable
X_gen : (closure L).toFun X = ⊤
x✝² : Countable ↑X
x✝¹ : Encodable ↑X
D : ↑X → Order.Cofinal (L.FGEquiv M N) := fun x ↦ H.definedAtLeft ↑x
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
x✝ : X ⊆ ↑F.dom
isTop : F.dom = ⊤
⊢ ↑g ≤ (toEmbeddingOfEqTop isTop).toPartialEquiv

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁷ : Preorder ι
G : ι → Type w
inst✝⁶ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝⁵ : IsDirectedOrder ι
inst✝⁴ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝³ : Nonempty ι
P : Type u₁
inst✝² : L.Structure P
g✝ : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) ((f i j hij) x) = (g✝ i) x
G' : ι → Type w'
inst✝¹ : (i : ι) → L.Structure (G' i)
f' : (i j : ι) → i ≤ j → G' i ↪[L] G' j
g : (i : ι) → G i ≃[L] G' i
inst✝ : DirectedSystem G' fun i j h ↦ ⇑(f' i j h)
H_commuting : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (f' i j hij) ((g i) x)
U : (i : ι) → G i ↪[L] DirectLimit G' f' := fun i ↦ (of L ι G' f' i).comp (g i).toEmbedding
F : DirectLimit G f ↪[L] DirectLimit G' f' := lift L ι G f U ⋯
⊢ Function.Surjective ⇑F

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
M_cg : Structure.CG L M
g : L.FGEquiv M N
H : L.IsExtensionPair M N
⊢ ∃ f, ↑g ≤ f.toPartialEquiv

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
⊢ Countable ↑X

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁷ : Preorder ι
G : ι → Type w
inst✝⁶ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝⁵ : IsDirectedOrder ι
inst✝⁴ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝³ : Nonempty ι
P : Type u₁
inst✝² : L.Structure P
g✝ : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) ((f i j hij) x) = (g✝ i) x
G' : ι → Type w'
inst✝¹ : (i : ι) → L.Structure (G' i)
f' : (i j : ι) → i ≤ j → G' i ↪[L] G' j
g : (i : ι) → G i ≃[L] G' i
inst✝ : DirectedSystem G' fun i j h ↦ ⇑(f' i j h)
H_commuting : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (f' i j hij) ((g i) x)
⊢ DirectLimit G f ≃[L] DirectLimit G' f'

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝¹ : Countable ↑X
x✝ : Encodable ↑X
⊢ Countable ↑Y

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
hx : x ∈ hs.fundamentalDomain
⊢ hs.fract x = x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝³ : Countable ↑X
x✝² : Encodable ↑X
x✝¹ : Countable ↑Y
x✝ : Encodable ↑Y
D : ↑X ⊕ ↑Y → Order.Cofinal (L.FGEquiv M N) :=
  fun p ↦ Sum.recOn p (fun x ↦ ext_dom.definedAtLeft ↑x) fun y ↦ ext_cod.definedAtRight ↑y
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
⊢ X ⊆ ↑F.dom

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝⁴ : Countable ↑X
x✝³ : Encodable ↑X
x✝² : Countable ↑Y
x✝¹ : Encodable ↑Y
D : ↑X ⊕ ↑Y → Order.Cofinal (L.FGEquiv M N) :=
  fun p ↦ Sum.recOn p (fun x ↦ ext_dom.definedAtLeft ↑x) fun y ↦ ext_cod.definedAtRight ↑y
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
x✝ : X ⊆ ↑F.dom
⊢ Y ⊆ ↑F.cod

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
P : E → Finset (Set X)
Pg : ∀ (i : E), ∀ t ∈ P i, t ⊆ ⇑g ⁻¹' {i}
Pdisj : ∀ (i : E), (↑(P i)).PairwiseDisjoint id
Pmeas : ∀ (i : E), ∀ t ∈ P i, MeasurableSet t
hP :
  ∀ (i : E),
    ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
      ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)
I3 :
  ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
    ∑ i ∈ g.range.sigma P, ‖i.fst‖ₑ * ‖((μ.transpose B).restrict s) i.snd‖ₑ + ↑δ
i : E
p : Set X
j : E
q : Set X
hijpq : ⟨i, p⟩ ≠ ⟨j, q⟩
hi : (∃ y, g y = i) ∧ p ∈ P i
hj : (∃ y, g y = j) ∧ q ∈ P j
hij : i ≠ j
⊢ Disjoint (⇑g ⁻¹' {i}) (⇑g ⁻¹' {j})

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝⁵ : Countable ↑X
x✝⁴ : Encodable ↑X
x✝³ : Countable ↑Y
x✝² : Encodable ↑Y
D : ↑X ⊕ ↑Y → Order.Cofinal (L.FGEquiv M N) :=
  fun p ↦ Sum.recOn p (fun x ↦ ext_dom.definedAtLeft ↑x) fun y ↦ ext_cod.definedAtRight ↑y
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
x✝¹ : X ⊆ ↑F.dom
x✝ : Y ⊆ ↑F.cod
⊢ F.dom = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type u_1
inst✝⁵ : Countable ι
inst✝⁴ : Preorder ι
inst✝³ : IsDirectedOrder ι
inst✝² : Nonempty ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
h : ∀ (i : ι), CG L (G i)
inst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
⊢ CG L (DirectLimit G f)

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝³ : Preorder ι
inst✝² : Nonempty ι
inst✝¹ : IsDirectedOrder ι
M : Type u_1
inst✝ : L.Structure M
S : ι →o L.Substructure M
⊢ (liftInclusion S).toHom.range = ⨆ i, S i

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : Nonempty ι
inst✝¹ : IsDirectedOrder ι
M : Type u_1
inst✝ : L.Structure M
S : ι →o L.Substructure M
⊢ ∀ (x : DirectLimit (fun i ↦ ↥(S i)) fun x x_1 h ↦ Substructure.inclusion ⋯), (liftInclusion S) x ∈ ⨆ i, S i

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝⁵ : Countable ↑X
x✝⁴ : Encodable ↑X
x✝³ : Countable ↑Y
x✝² : Encodable ↑Y
D : ↑X ⊕ ↑Y → Order.Cofinal (L.FGEquiv M N) :=
  fun p ↦ Sum.recOn p (fun x ↦ ext_dom.definedAtLeft ↑x) fun y ↦ ext_cod.definedAtRight ↑y
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
x✝¹ : X ⊆ ↑F.dom
x✝ : Y ⊆ ↑F.cod
dom_top : F.dom = ⊤
⊢ F.cod = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
⊢ x ∈ s ↔ hs.fract x = hs.base ∧ ∀ (i : ↑hs.basisSet), 0 ≤ hs.floor x i ∧ (↑i ∉ hs.periods → hs.floor x i = 0)

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
M_cg : Structure.CG L M
N_cg : Structure.CG L N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
⊢ ∃ f, ↑g ≤ f.toEmbedding.toPartialEquiv

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : Nonempty ι
inst✝¹ : IsDirectedOrder ι
M : Type u_1
inst✝ : L.Structure M
S : ι →o L.Substructure M
liftInclusion_in_sup :
  ∀ (x : DirectLimit (fun i ↦ ↥(S i)) fun x x_1 h ↦ Substructure.inclusion ⋯), (liftInclusion S) x ∈ ⨆ i, S i
F : (DirectLimit (fun i ↦ ↥(S i)) fun x x_1 h ↦ Substructure.inclusion ⋯) ↪[L] ↥(⨆ i, S i) :=
  Embedding.codRestrict (⨆ i, S i) (liftInclusion S) liftInclusion_in_sup
⊢ Function.Surjective ⇑F

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Fraisse

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
S : L.Substructure M
fg : S.FG
⊢ { α := ↥S, str := inferInstance } ∈ L.age M

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Fraisse

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
⊢ ∀ C ∈ Quotient.mk' '' L.age M, ∃ V, ⟦{ α := ↥↑V, str := inferInstance }⟧ = C

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : Nonempty ι
inst✝¹ : IsDirectedOrder ι
M : Type u_1
inst✝ : L.Structure M
S : ι →o L.Substructure M
⊢ (DirectLimit (fun i ↦ ↥(S i)) fun x x_1 h ↦ Substructure.inclusion ⋯) ≃[L] ↥(iSup ⇑S)

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝³ : Preorder ι
inst✝² : Nonempty ι
inst✝¹ : IsDirectedOrder ι
M : Type u_1
inst✝ : L.Structure M
S : ι →o L.Substructure M
i : ι
x : ↥(S i)
⊢ (Equiv_iSup S).symm ((Substructure.inclusion ⋯) x) =
    (of L ι (fun x ↦ ↥(S x)) (fun x x_1 h ↦ Substructure.inclusion ⋯) i) x

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝³ : Preorder ι
inst✝² : Nonempty ι
inst✝¹ : IsDirectedOrder ι
M : Type u_1
inst✝ : L.Structure M
S : ι →o L.Substructure M
i : ι
⊢ (Equiv_iSup S).symm.toEmbedding.comp (Substructure.inclusion ⋯) =
    of L ι (fun x ↦ ↥(S x)) (fun x x_1 h ↦ Substructure.inclusion ⋯) i

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Fraisse

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
h : Countable M
⊢ (Quotient.mk' '' L.age M).Countable

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Graph

Goal state before:
pp: V : Type u
n : ℕ
⊢ ∀ (a b : Language.graph.Relations n), a = b

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Graph

Goal state before:
pp: V : Type u
inst✝ : Language.graph.Structure V
⊢ V ⊨ Theory.simpleGraph ↔ (Std.Irrefl fun x y ↦ RelMap adj ![x, y]) ∧ Std.Symm fun x y ↦ RelMap adj ![x, y]

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Graph

Goal state before:
pp: V : Type u
n : ℕ
G : SimpleGraph V
⊢ V ⊨ Theory.simpleGraph

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Graph

Goal state before:
pp: V : Type u
G : SimpleGraph V
⊢ Language.simpleGraphOfStructure V = G

Tactic:
by

Goal state after:
Module: Mathlib.ModelTheory.Graph

Goal state before:
pp: V : Type u
S : Language.graph.Structure V
inst✝ : V ⊨ Theory.simpleGraph
⊢ (simpleGraphOfStructure V).structure = S

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableDenseLinearOrder

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : DenselyOrdered β
inst✝¹ : NoMinOrder β
inst✝ : NoMaxOrder β
nonem : Nonempty β
S : Finset α
f : ↥S ↪o β
a : α
Slt : Finset β := Finset.image (⇑f) ({x ∈ S.attach | ↑x < a})
Sgt : Finset β := Finset.image (⇑f) ({x ∈ S.attach | a < ↑x})
x : β
hx : x ∈ Slt
y : β
hy : y ∈ Sgt
⊢ x < y

Tactic:
by

Goal state after: