mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ (∏ x ∈ Ico a b, f x) * f b = ∏ x ∈ Icc a b, f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
a b c : α
⊢ map (addLeftEmbedding c) (Ioc a b) = Ioc (c + a) (c + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ f a * ∏ x ∈ Ioc a b, f x = ∏ x ∈ Icc a b, f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
a b c : α
⊢ map (addRightEmbedding c) (Ioc a b) = Ioc (a + c) (b + c)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ (∏ x ∈ Ioc a b, f x) * f a = ∏ x ∈ Icc a b, f x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
i : ι
hn : IsMin i
⊢ Iio i = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
i : ι
hn : IsMin i
⊢ disjointed f i = f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
a b c : α
⊢ map (addLeftEmbedding c) (Ioo a b) = Ioo (c + a) (c + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a < b
⊢ f a * ∏ x ∈ Ioo a b, f x = ∏ x ∈ Ico a b, f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
a b c : α
⊢ map (addRightEmbedding c) (Ioo a b) = Ioo (a + c) (b + c)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a < b
⊢ (∏ x ∈ Ioo a b, f x) * f a = ∏ x ∈ Ico a b, f x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : LinearOrder ι
i j k : ι
hij_lt : i < j
h : IsGLB (Set.Ioi i) k
⊢ IsGLB (Set.Ioc i j) k

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
i : ι
hf : ∀ j < i, Disjoint (f j) (f i)
⊢ disjointed f i = f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ c + x) (Icc a b) = Icc (c + a) (c + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ c + x) (Ico a b) = Ico (c + a) (c + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a < b
⊢ f b * ∏ x ∈ Ioo a b, f x = ∏ x ∈ Ioc a b, f x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i : ι
hi : succFn i ≤ i
j : ι
x✝ : i ≤ j
hij_lt : i < j
h_succFn_eq : succFn i = i
⊢ IsGLB (↑(Finset.Ioc i j)) i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ c + x) (Ioc a b) = Ioc (c + a) (c + b)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i : ι
hi : succFn i ≤ i
j : ι
x✝ : i ≤ j
hij_lt : i < j
h_succFn_eq : succFn i = i
h_glb : IsGLB (↑(Finset.Ioc i j)) i
⊢ i ∈ Finset.Ioc i j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ c + x) (Ioo a b) = Ioo (c + a) (c + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a < b
⊢ (∏ x ∈ Ioo a b, f x) * f b = ∏ x ∈ Ioc a b, f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
a b : α
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
h : a < b
⊢ (∏ x ∈ Ioo a b, f x) * f b = ∏ x ∈ Ioc a b, f x

Tactic:
by
  rw [mul_comm, mul_prod_Ioo_eq_prod_Ioc h]

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: M : Type u_2
inst✝ : CommMonoid M
a b : ℕ
hab : a < b
f : ℕ → M
⊢ a ∉ Ico (a + 1) b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ x + c) (Icc a b) = Icc (a + c) (b + c)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i : ι
hi : succFn i ≤ i
⊢ IsMax i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ x + c) (Ico a b) = Ico (a + c) (b + c)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ x + c) (Ioc a b) = Ioc (a + c) (b + c)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : LinearOrder ι
i j : ι
hij : i < j
⊢ succFn i ≤ j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: M : Type u_2
inst✝ : CommMonoid M
a b : ℕ
hab : a < b
f : ℕ → M
⊢ ∏ k ∈ Ico a b, f k = f a * ∏ k ∈ Ico (a + 1) b, f k

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Interval.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : IsOrderedCancelAddMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
⊢ image (fun x ↦ x + c) (Ioo a b) = Ioo (a + c) (b + c)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrderTop α
a : α
⊢ f a * ∏ x > a, f x = ∏ x ≥ a, f x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : LinearOrder ι
j i : ι
hij : j < succFn i
⊢ j ≤ i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrderTop α
a : α
⊢ (∏ x > a, f x) * f a = ∏ x ≥ a, f x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
p : α → Prop
hdiff : ∀ ⦃t : α⦄ ⦃i : ι⦄, p t → p (t \ f i)
⊢ ∀ ⦃i : ι⦄, p (f i) → p (disjointed f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : Preorder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
p : α → Prop
hdiff : ∀ ⦃t : α⦄ ⦃i : ι⦄, p t → p (t \ f i)
⊢ ∀ ⦃i : ι⦄, p (f i) → p (disjointed f i)

Tactic:
by
  classical
  intro i hpi
  rw [disjointed]
  suffices ∀ (s : Finset ι), p (f i \ s.sup f) from this _
  intro s
  induction s using Finset.induction with
  | empty => simpa only [sup_empty, sdiff_bot] using hpi
  | insert _ _ ht IH =>
    rw [sup_insert, sup_comm, ← sdiff_sdiff]
    exact hdiff IH

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrderBot α
a : α
⊢ f a * ∏ x < a, f x = ∏ x ≤ a, f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝² : CommMonoid M
f : α → M
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrderBot α
a : α
⊢ (∏ x < a, f x) * f a = ∏ x ≤ a, f x

Tactic:
by

Goal state after:
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
n : ℕ
⊢ succ^[n] i < j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝⁵ : CommMonoid M
a b : α
inst✝⁴ : LinearOrder α
inst✝³ : LocallyFiniteOrder α
inst✝² : AddMonoidWithOne α
inst✝¹ : SuccAddOrder α
inst✝ : NoMaxOrder α
hab : a ≤ b
f : α → M
⊢ (∏ x ∈ Ico a b, f x) * f b = ∏ x ∈ Ico a (b + 1), f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝⁶ : CommMonoid M
inst✝⁵ : LinearOrder α
inst✝⁴ : LocallyFiniteOrderBot α
inst✝³ : Add α
inst✝² : One α
inst✝¹ : SuccAddOrder α
inst✝ : NoMaxOrder α
a : α
f : α → M
⊢ ∏ i < a + 1, f i = f a * ∏ i < a, f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝⁶ : CommMonoid M
inst✝⁵ : LinearOrder α
inst✝⁴ : LocallyFiniteOrderBot α
inst✝³ : Add α
inst✝² : One α
inst✝¹ : SuccAddOrder α
inst✝ : NoMaxOrder α
a : α
f : α → M
⊢ ∏ i < a + 1, f i = (∏ i < a, f i) * f a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝⁶ : CommMonoid M
inst✝⁵ : LinearOrder α
inst✝⁴ : LocallyFiniteOrderBot α
inst✝³ : Add α
inst✝² : One α
inst✝¹ : SuccAddOrder α
inst✝ : NoMaxOrder α
a : α
f : α → M
⊢ ∏ i ≤ a + 1, f i = f (a + 1) * ∏ i ≤ a, f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝⁶ : CommMonoid M
inst✝⁵ : LinearOrder α
inst✝⁴ : LocallyFiniteOrderBot α
inst✝³ : Add α
inst✝² : One α
inst✝¹ : SuccAddOrder α
inst✝ : NoMaxOrder α
a : α
f : α → M
⊢ ∏ i ≤ a + 1, f i = (∏ i ≤ a, f i) * f (a + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: 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
⊢ (Iio i).biUnion Iic = Iio i

Tactic:
by

Goal state after:
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
f : ℕ → ↥(Finset.Icc i j) := fun n ↦ ⟨succ^[n] i, ⋯⟩
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
⊢ succ^[n] i = succ^[m] i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝⁴ : CommMonoid M
inst✝³ : LinearOrder α
inst✝² : Fintype α
inst✝¹ : LocallyFiniteOrderTop α
inst✝ : LocallyFiniteOrderBot α
f : α → α → M
⊢ ∏ i, ∏ j > i, f j i * f i j = ∏ i, ∏ j ∈ {i}ᶜ, f j i

Tactic:
by

Goal state after:
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

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

Goal state before:
pp: ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrder ι
inst✝ : SuccOrder ι
⊢ ∀ {a b : ι}, a ≤ b → ∃ n, succ^[n] a = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: 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
| ∏ i ∈ s n, g i

Tactic:
rw [← hs.partialSups_eq, ← disjiUnion_Iic_disjointed, Iic_eq_Icc,
    prod_disjiUnion, Nat.bot_eq_zero, ← Nat.range_succ_eq_Icc_zero, prod_range_succ', mul_comm]

Goal state after:
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
| (∏ x ∈ disjointed s 0, g x) * ∏ k ∈ range n, ∏ x ∈ disjointed s (k + 1), g x
Module: Mathlib.Order.SuccPred.LinearLocallyFinite

Goal state before:
pp: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 : ι
⊢ toZ i0 i0 = 0

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hi : i0 ≤ i
⊢ succ^[(toZ i0 i).toNat] i0 = i

Tactic:
by

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

Goal state before:
pp: 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
⊢ pred^[(-toZ i0 i).toNat] i0 = i

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: 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:
by

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite

Goal state before:
pp: 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
⊢ ∏ i ∈ s n, g i = (∏ i ∈ s 0, g i) * ∏ i ∈ range n, ∏ j ∈ s (i + 1) \ s i, g j

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
⊢ partialSups (disjointed f) = partialSups f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : PartialOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : Fintype ι
f : ι → α
⊢ univ.biUnion Iic = univ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: α : Type u_1
M : Type u_3
inst✝⁵ : CommMonoid M
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
f : α → M
a b c : α
⊢ ∏ x ∈ Ico a b, f (x + c) = ∏ x ∈ Ico (a + c) (b + c), f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: α : Type u_1
M : Type u_3
inst✝⁵ : CommMonoid M
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
f : α → M
a b c : α
⊢ ∏ x ∈ Ico a b, f (c + x) = ∏ x ∈ Ico (a + c) (b + c), f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: α : Type u_1
M : Type u_3
inst✝⁷ : CommMonoid M
f : α → M
inst✝⁶ : AddCommMonoid α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedCancelAddMonoid α
inst✝³ : ExistsAddOfLE α
inst✝² : LocallyFiniteOrder α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
⊢ ∏ x ∈ Ico (a + c) (b + c), f (x - c) = ∏ x ∈ Ico a b, f x

Tactic:
simp only [← map_add_right_Ico, prod_map, addRightEmbedding_apply, add_tsub_cancel_right]

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

Goal state before:
pp: 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)
⊢ toZ i0 (succ^[n] i0) = ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: α : Type u_1
M : Type u_3
inst✝⁷ : CommMonoid M
f : α → M
inst✝⁶ : AddCommMonoid α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedCancelAddMonoid α
inst✝³ : ExistsAddOfLE α
inst✝² : LocallyFiniteOrder α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
⊢ ∏ x ∈ Ico (a + c) (b + c), f (x - c) = ∏ x ∈ Ico a b, f x

Tactic:
by

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

Goal state before:
pp: 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)
h_pred_iterate_eq : pred^[n.succ] i0 = i0
⊢ pred^[n.succ] i0 = pred^[0] i0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: M : Type u_3
inst✝ : CommMonoid M
a b : ℕ
hab : a ≤ b
f : ℕ → M
⊢ ∏ k ∈ Ico a (b + 1), f k = (∏ k ∈ Ico a b, f k) * f b

Tactic:
by

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

Goal state before:
pp: 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)
⊢ pred^[n.succ] i0 < i0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : PartialOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : Fintype ι
f : ι → α
⊢ univ.sup (disjointed f) = univ.sup f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
i : ι
⊢ f i \ (Iio i).sup f = (partialSups f) i \ (Iio i).sup f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: M : Type u_3
inst✝ : CommMonoid M
f : ℕ → M
m n k : ℕ
hmn : m ≤ n
hnk : n ≤ k
⊢ (∏ i ∈ Ioc m n, f i) * ∏ i ∈ Ioc n k, f i = ∏ i ∈ Ioc m k, f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Prod

Goal state before:
pp: A : Type u_1
inst✝ : Monoid A
⊢ ∀ (a b : HasMulAntidiagonal A), a = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Prod

Goal state before:
pp: M : Type u_2
inst✝¹ : Monoid M
inst✝ : HasMulAntidiagonal M
a : M
⊢ (1, a) ∈ mulAntidiagonal a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Prod

Goal state before:
pp: A : Type u_2
inst✝ : Monoid A
H1 H2 : HasMulAntidiagonal A
⊢ mulAntidiagonal = mulAntidiagonal

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Prod

Goal state before:
pp: A : Type u_1
inst✝¹ : CommMonoid A
inst✝ : HasMulAntidiagonal A
n : A
xy : A × A
⊢ xy.swap ∈ mulAntidiagonal n ↔ xy ∈ mulAntidiagonal n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Prod

Goal state before:
pp: A : Type u_1
inst✝¹ : CommMonoid A
inst✝ : HasMulAntidiagonal A
n : A
x✝ : A × A
a b : A
⊢ (a, b) ∈ map (Equiv.prodComm A A).toEmbedding (mulAntidiagonal n) ↔ (a, b) ∈ mulAntidiagonal n

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → α
⊢ disjointed ⇑(partialSups f) = disjointed f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: M : Type u_3
inst✝ : CommMonoid M
a b : ℕ
hab : a ≤ b
f : ℕ → M
⊢ ∏ k ∈ Ioc a (b + 1), f k = (∏ k ∈ Ioc a b, f k) * f (b + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
f d : ι → α
hdisj : ∀ {i j : ι}, i < j → Disjoint (d i) (d j)
hsups : partialSups d = partialSups f
⊢ d = disjointed f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.BigOperators.Intervals

Goal state before:
pp: M : Type u_3
inst✝ : CommMonoid M
a b : ℕ
hab : a ≤ b + 1
f : ℕ → M
⊢ ∏ k ∈ Icc a (b + 1), f k = (∏ k ∈ Icc a b, f k) * f (b + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Prod

Goal state before:
pp: A : Type u_1
inst✝¹ : CancelMonoid A
inst✝ : HasMulAntidiagonal A
p q : A × A
n : A
hp : p ∈ mulAntidiagonal n
hq : q ∈ mulAntidiagonal n
⊢ p = q ↔ p.1 = q.1

Tactic:
by

Goal state after:
Module: Mathlib.Order.Disjointed

Goal state before:
pp: ι : Type u_2
inst✝¹ : PartialOrder ι
inst✝ : LocallyFiniteOrderBot ι
α : Type u_3
f : ι → Set α
n : ι
⊢ ⋃ i ∈ Iic n, disjointed f i = (partialSups f) n

Tactic:
by

Goal state after: