mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Prod

Goal state before:
pp: case refine_2
R : Type u
M : Type v
M₂ : Type w
M₃ : Type y
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M
inst✝¹ : Module R M₂
inst✝ : Module R M₃
f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
h : f.ker ⊔ g.ker = ⊤
x y : M
this : y - x ∈ f.ker ⊔ g.ker
x' : M
hx' : x' ∈ f.ker
y' : M
hy' : y' ∈ g.ker
H : x' + y' = y - x
⊢ g (x' + x) = g y

Tactic:
·

Goal state after:
case refine_2
R : Type u
M : Type v
M₂ : Type w
M₃ : Type y
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M
inst✝¹ : Module R M₂
inst✝ : Module R M₃
f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
h : f.ker ⊔ g.ker = ⊤
x y : M
this : y - x ∈ f.ker ⊔ g.ker
x' : M
hx' : x' ∈ f.ker
y' : M
hy' : y' ∈ g.ker
H : x' + y' = y - x
⊢ g (x' + x) = g y
Module: Mathlib.LinearAlgebra.Prod

Goal state before:
pp: R : Type u_3
S : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
σ : R →+* S
inst✝⁶ : RingHomSurjective σ
inst✝⁵ : AddCommMonoid G
inst✝⁴ : Module R G
inst✝³ : AddCommMonoid H
inst✝² : Module S H
inst✝¹ : AddCommMonoid I
inst✝ : Module S I
f : G →ₛₗ[σ] H × I
hf₁ : Surjective (Prod.fst ∘ ⇑f)
hf : ∀ (g₁ g₂ : G), (f g₁).1 = (f g₂).1 → (f g₁).2 = (f g₂).2
f' : H →+ I
hf' : ∀ (x : H × I), (∃ x_1, f x_1 = x) ↔ f' x.1 = x.2
⊢ ∀ (m : S) (x : H), (↑f').toFun (m • x) = (RingHom.id S) m • (↑f').toFun x

Tactic:
intro s h

Goal state after:
R : Type u_3
S : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
σ : R →+* S
inst✝⁶ : RingHomSurjective σ
inst✝⁵ : AddCommMonoid G
inst✝⁴ : Module R G
inst✝³ : AddCommMonoid H
inst✝² : Module S H
inst✝¹ : AddCommMonoid I
inst✝ : Module S I
f : G →ₛₗ[σ] H × I
hf₁ : Surjective (Prod.fst ∘ ⇑f)
hf : ∀ (g₁ g₂ : G), (f g₁).1 = (f g₂).1 → (f g₁).2 = (f g₂).2
f' : H →+ I
hf' : ∀ (x : H × I), (∃ x_1, f x_1 = x) ↔ f' x.1 = x.2
s : S
h : H
⊢ (↑f').toFun (s • h) = (RingHom.id S) s • (↑f').toFun h
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case succ.hin.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : PredOrder ι
inst✝ : IsSuccArchimedean ι
i j : ι
hij : i ≤ j
h_exists : ∃ n, succ^[n] i = j
n : ℕ
hn_eq : succ^[n + 1] i = j
hn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j
⊢ succ^[n] i ≤ succ^[n.succ] i

pp: case succ.hin.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : PredOrder ι
inst✝ : IsSuccArchimedean ι
i j : ι
hij : i ≤ j
h_exists : ∃ n, succ^[n] i = j
n : ℕ
hn_eq : succ^[n + 1] i = j
hn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j
⊢ succ^[n] i ≠ succ^[n.succ] i

Tactic:
·

Goal state after:
case succ.hin.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : PredOrder ι
inst✝ : IsSuccArchimedean ι
i j : ι
hij : i ≤ j
h_exists : ∃ n, succ^[n] i = j
n : ℕ
hn_eq : succ^[n + 1] i = j
hn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j
⊢ succ^[n] i ≤ succ^[n.succ] i
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case succ.hin.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : PredOrder ι
inst✝ : IsSuccArchimedean ι
i j : ι
hij : i ≤ j
h_exists : ∃ n, succ^[n] i = j
n : ℕ
hn_eq : succ^[n + 1] i = j
hn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j
⊢ succ^[n] i ≠ succ^[n.succ] i

Tactic:
·

Goal state after:
case succ.hin.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : PredOrder ι
inst✝ : IsSuccArchimedean ι
i j : ι
hij : i ≤ j
h_exists : ∃ n, succ^[n] i = j
n : ℕ
hn_eq : succ^[n + 1] i = j
hn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j
⊢ succ^[n] i ≠ succ^[n.succ] i
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inl
ι : Type u_1
inst✝ : LinearOrder ι
i j k : ι
hij_lt : i < j
h : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k
x : ι
hx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1
y : ι
hy : y ∈ Set.Ioi i
h_le : y ≤ j
⊢ x ≤ y

pp: case inr
ι : Type u_1
inst✝ : LinearOrder ι
i j k : ι
hij_lt : i < j
h : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k
x : ι
hx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1
y : ι
hy : y ∈ Set.Ioi i
h_lt : j < y
⊢ x ≤ y

Tactic:
·

Goal state after:
case inl
ι : Type u_1
inst✝ : LinearOrder ι
i j k : ι
hij_lt : i < j
h : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k
x : ι
hx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1
y : ι
hy : y ∈ Set.Ioi i
h_le : y ≤ j
⊢ x ≤ y
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inr
ι : Type u_1
inst✝ : LinearOrder ι
i j k : ι
hij_lt : i < j
h : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k
x : ι
hx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1
y : ι
hy : y ∈ Set.Ioi i
h_lt : j < y
⊢ x ≤ y

Tactic:
·

Goal state after:
case inr
ι : Type u_1
inst✝ : LinearOrder ι
i j k : ι
hij_lt : i < j
h : (∀ x ∈ Set.Ioi i, k ≤ x) ∧ ∀ (x : ι), (∀ x_1 ∈ Set.Ioi i, x ≤ x_1) → x ≤ k
x : ι
hx : ∀ x_1 ∈ Set.Ioc i j, x ≤ x_1
y : ι
hy : y ∈ Set.Ioi i
h_lt : j < y
⊢ x ≤ y
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inr
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i = j
⊢ ∃ n, succ^[n] i = j

pp: case inl
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
⊢ ∃ n, succ^[n] i = j

Tactic:
·

Goal state after:
case inr
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i = j
⊢ ∃ n, succ^[n] i = j
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case succ.inl
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) < n + 1
⊢ (partialSups (disjointed f)) i = (partialSups f) i

pp: case succ.inr
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
⊢ (partialSups (disjointed f)) i = (partialSups f) i

Tactic:
·

Goal state after:
case succ.inl
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) < n + 1
⊢ (partialSups (disjointed f)) i = (partialSups f) i
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inl
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → ↥(Finset.Icc i j) := fun n ↦ ⟨succ^[n] i, ⋯⟩
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : n ≤ m
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

pp: case inr
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → ↥(Finset.Icc i j) := fun n ↦ ⟨succ^[n] i, ⋯⟩
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : m ≤ n
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Tactic:
·

Goal state after:
case inl
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → ↥(Finset.Icc i j) := fun n ↦ ⟨succ^[n] i, ⋯⟩
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : n ≤ m
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inr
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → ↥(Finset.Icc i j) := fun n ↦ ⟨succ^[n] i, ⋯⟩
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : m ≤ n
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Tactic:
·

Goal state after:
case inr
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → ↥(Finset.Icc i j) := fun n ↦ ⟨succ^[n] i, ⋯⟩
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : m ≤ n
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Tactic:
by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m :=
        Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [f, Subtype.mk_eq_mk] using hfnm
      rcases le_total n m with h_le | h_le
      · exact ⟨n, m, lt_of_le_of_ne h_le hnm_ne, hnm_eq⟩
      · exact ⟨m, n, lt_of_le_of_ne h_le hnm_ne.symm, hnm_eq.symm⟩

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case succ.inr
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
⊢ disjointed f i ⊔ (Iio i).sup ⇑(partialSups f) = (partialSups f) i

pp: case succ.inr
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
⊢ ∀ a ∈ Iio i, ((Iic a).sup fun x ↦ disjointed f x) = (partialSups f) a

Tactic:
·

Goal state after:
case succ.inr
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
⊢ disjointed f i ⊔ (Iio i).sup ⇑(partialSups f) = (partialSups f) i
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case succ.inr
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
⊢ ∀ a ∈ Iio i, ((Iic a).sup fun x ↦ disjointed f x) = (partialSups f) a

Tactic:
·

Goal state after:
case succ.inr
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
ih : ∀ (i : ι), #(Iio i) ≤ n → (partialSups (disjointed f)) i = (partialSups f) i
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
⊢ ∀ a ∈ Iio i, ((Iic a).sup fun x ↦ disjointed f x) = (partialSups f) a
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: case refine_1
α : Type u_3
β : Type u_4
inst✝¹ : DecidableEq α
inst✝ : CommMonoid β
s : ℕ → Finset α
hs : Monotone s
g : α → β
n : ℕ
⊢ disjointed s 0 = s 0

pp: case refine_2
α : Type u_3
β : Type u_4
inst✝¹ : DecidableEq α
inst✝ : CommMonoid β
s : ℕ → Finset α
hs : Monotone s
g : α → β
n k : ℕ
⊢ (Iio (k + 1)).sup s = s k

Tactic:
·

Goal state after:
case refine_1
α : Type u_3
β : Type u_4
inst✝¹ : DecidableEq α
inst✝ : CommMonoid β
s : ℕ → Finset α
hs : Monotone s
g : α → β
n : ℕ
⊢ disjointed s 0 = s 0
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hi : i < i0
⊢ toZ i0 i ≤ 0

pp: case refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hi : i < i0
⊢ toZ i0 i ≠ 0

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hi : i < i0
⊢ toZ i0 i ≤ 0
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: case refine_2
α : Type u_3
β : Type u_4
inst✝¹ : DecidableEq α
inst✝ : CommMonoid β
s : ℕ → Finset α
hs : Monotone s
g : α → β
n k : ℕ
⊢ (Iio (k + 1)).sup s = s k

Tactic:
·

Goal state after:
case refine_2
α : Type u_3
β : Type u_4
inst✝¹ : DecidableEq α
inst✝ : CommMonoid β
s : ℕ → Finset α
hs : Monotone s
g : α → β
n k : ℕ
⊢ (Iio (k + 1)).sup s = s k
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hi : i < i0
⊢ toZ i0 i ≠ 0

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hi : i < i0
⊢ toZ i0 i ≠ 0
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
h : i0 ≤ pred^[n] i0
⊢ -↑n ≤ toZ i0 (pred^[n] i0)

pp: case inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
h : pred^[n] i0 < i0
⊢ -↑n ≤ toZ i0 (pred^[n] i0)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
h : i0 ≤ pred^[n] i0
⊢ -↑n ≤ toZ i0 (pred^[n] i0)
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case succ.inr.hab
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
ih : ∀ (i : ι), #(Iio i) ≤ n → (Iic i).sup (disjointed f) = (Iic i).sup f
x : ι
hx : x ∈ Iio i
⊢ #(Iio x) < #(Iic x)

pp: case succ.inr.hbc
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
ih : ∀ (i : ι), #(Iio i) ≤ n → (Iic i).sup (disjointed f) = (Iic i).sup f
x : ι
hx : x ∈ Iio i
⊢ #(Iic x) ≤ #(Iio i)

Tactic:
·

Goal state after:
case succ.inr.hab
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
ih : ∀ (i : ι), #(Iio i) ≤ n → (Iic i).sup (disjointed f) = (Iic i).sup f
x : ι
hx : x ∈ Iio i
⊢ #(Iio x) < #(Iic x)
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
h : pred^[n] i0 < i0
⊢ -↑n ≤ toZ i0 (pred^[n] i0)

Tactic:
·

Goal state after:
case inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
h : pred^[n] i0 < i0
⊢ -↑n ≤ toZ i0 (pred^[n] i0)
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case succ.inr.hbc
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
ih : ∀ (i : ι), #(Iio i) ≤ n → (Iic i).sup (disjointed f) = (Iic i).sup f
x : ι
hx : x ∈ Iio i
⊢ #(Iic x) ≤ #(Iio i)

Tactic:
·

Goal state after:
case succ.inr.hbc
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
n : ℕ
i : ι
hi : #(Iio i) ≤ n + 1
hn : #(Iio i) = n + 1
hun : (Iio i).biUnion Iic = Iio i
ih : ∀ (i : ι), #(Iio i) ≤ n → (Iic i).sup (disjointed f) = (Iic i).sup f
x : ι
hx : x ∈ Iio i
⊢ #(Iic x) ≤ #(Iio i)
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMax (succ^[n] i0)
m : ℕ := (toZ i0 (succ^[n] i0)).toNat
h_eq : succ^[m] i0 = succ^[n] i0
hmn : m = n
⊢ toZ i0 (succ^[n] i0) = ↑n

pp: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMax (succ^[n] i0)
m : ℕ := (toZ i0 (succ^[n] i0)).toNat
h_eq : succ^[m] i0 = succ^[n] i0
hmn : ¬m = n
⊢ toZ i0 (succ^[n] i0) = ↑n

Tactic:
·

Goal state after:
case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMax (succ^[n] i0)
m : ℕ := (toZ i0 (succ^[n] i0)).toNat
h_eq : succ^[m] i0 = succ^[n] i0
hmn : m = n
⊢ toZ i0 (succ^[n] i0) = ↑n
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case zero
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
hn : ¬IsMin (pred^[0] i0)
⊢ toZ i0 (pred^[0] i0) = -↑0

pp: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

Tactic:
·

Goal state after:
case zero
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
hn : ¬IsMin (pred^[0] i0)
⊢ toZ i0 (pred^[0] i0) = -↑0
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

pp: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

Tactic:
·

Goal state after:
case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case e_a.e_s
α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
i : ι
step1 : f i \ (Iio i).sup f = (partialSups f) i \ (Iio i).sup f
r : ι
⊢ r ∈ (Iio i).biUnion Iic ↔ r ∈ Iio i

Tactic:
simpa only [mem_biUnion, mem_Iio, mem_Iic] using
    ⟨fun ⟨a, ha⟩ ↦ ha.2.trans_lt ha.1, fun hr ↦ ⟨r, hr, le_rfl⟩⟩

Goal state after:
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

Tactic:
·

Goal state after:
case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i0 ≤ i
hj : i0 ≤ j
⊢ i ≤ j

pp: case inl.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i0 ≤ i
hj : j < i0
⊢ i ≤ j

pp: case inr.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : i0 ≤ j
⊢ i ≤ j

pp: case inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : j < i0
⊢ i ≤ j

Tactic:
·

Goal state after:
case inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i0 ≤ i
hj : i0 ≤ j
⊢ i ≤ j
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inl.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i0 ≤ i
hj : j < i0
⊢ i ≤ j

pp: case inr.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : i0 ≤ j
⊢ i ≤ j

pp: case inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : j < i0
⊢ i ≤ j

Tactic:
·

Goal state after:
case inl.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i0 ≤ i
hj : j < i0
⊢ i ≤ j
Module: Mathlib.Order.Disjointed

Goal state before:
pp: case pos
α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
hf : Monotone f
i : ι
h : IsMax i
⊢ disjointed f (succ i) ⊔ f i = f (succ i)

pp: case neg
α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
hf : Monotone f
i : ι
h : ¬IsMax i
⊢ disjointed f (succ i) ⊔ f i = f (succ i)

Tactic:
·

Goal state after:
case pos
α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
hf : Monotone f
i : ι
h : IsMax i
⊢ disjointed f (succ i) ⊔ f i = f (succ i)
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: case inr.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : i0 ≤ j
⊢ i ≤ j

pp: case inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : j < i0
⊢ i ≤ j

Tactic:
·

Goal state after:
case inr.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 j i : ι
h_le : toZ i0 i ≤ toZ i0 j
hi : i < i0
hj : i0 ≤ j
⊢ i ≤ j