mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: case inr.inr
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
b : α
inst✝ : IsStrictOrderedRing α
s : Set α
hs : IsLUB s b
ha : 0 ≤ 0
ne : s.Nonempty
⊢ IsLUB {0} 0

Tactic:
]

Goal state after:
case inr.inr
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
b : α
inst✝ : IsStrictOrderedRing α
s : Set α
hs : IsLUB s b
ha : 0 ≤ 0
ne : s.Nonempty
⊢ IsLUB {0} 0
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_4
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
h : 0 < a
b : α
this : 0 < a / max (b + 1) 1
⊢ b < max (b + 1) 1

Tactic:
]

Goal state after:
α : Type u_4
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
h : 0 < a
b : α
this : 0 < a / max (b + 1) 1
⊢ b < max (b + 1) 1
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_4
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
h : 0 < a
b c : α
hc₀ : 0 < c
hc : b * c < a
⊢ b * c < a

Tactic:
]

Goal state after:
α : Type u_4
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
h : 0 < a
b c : α
hc₀ : 0 < c
hc : b * c < a
⊢ b * c < a
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
⊢ a < 0 ↔ a < 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ a * c ≤ b ↔ a * c ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Ring α
na nb nc : ℤ
da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = ↑k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ↑da * ↑db * ⅟↑db * ⅟↑da * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑k * ↑nc * ⅟↑da * ⅟↑db = ↑nc * ⅟↑dc

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Ring α
na nb nc : ℤ
da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = ↑k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ↑da * ↑db * ⅟↑db * ⅟↑da * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑k * ↑nc * ⅟↑da * ⅟↑db = ↑nc * ⅟↑dc
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ b ≤ -(a * -c) ↔ b ≤ -(a * -c)

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Ring α
na nb nc : ℤ
da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = ↑k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑da * ⅟↑db = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Ring α
na nb nc : ℤ
da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = ↑k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑da * ⅟↑db = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ b ≤ a * c ↔ b ≤ a * c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ a * c < b ↔ a * c < b

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: x y : ℕ
⊢ ¬x ≤ y ↔ ¬x ≤ y

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ b < -(a * -c) ↔ b < -(a * -c)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ b < a * c ↔ b < a * c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
ha : a < 0
hb : b < 0
⊢ b ≤ a ↔ b ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
ha : a < 0
hb : b < 0
⊢ b⁻¹ ≤ a ↔ b⁻¹ ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
ha : a < 0
hb : b < 0
⊢ b ≤ a⁻¹ ↔ b ≤ a⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
ha : a < 0
hb : b < 0
this : a⁻¹ < b⁻¹ ↔ b < a
⊢ a⁻¹ < b⁻¹ ↔ b < a

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
ha : a < 0
hb : b < 0
this : a⁻¹ < b⁻¹ ↔ b < a
⊢ a⁻¹ < b⁻¹ ↔ b < a
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: α : Type u
l : List α
n : ℕ
h : n < l.length
⊢ [l[n]] = [l.get ⟨n, h⟩]

Tactic:
]

Goal state after:
α : Type u
l : List α
n : ℕ
h : n < l.length
⊢ [l[n]] = [l.get ⟨n, h⟩]
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: α : Type u
x : List α
n : ℕ
⊢ x.length ≤ n ↔ x.length ≤ n

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: α : Type u
x y : List α
n : ℕ
⊢ take n (x ++ y) = take n x ↔ y = [] ∨ n ≤ x.length

Tactic:
]

Goal state after:
α : Type u
x y : List α
n : ℕ
⊢ take n (x ++ y) = take n x ↔ y = [] ∨ n ≤ x.length
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
⊢ AntitoneOn (fun x ↦ x⁻¹) (Set.Ioi 0)

Tactic:
by
  convert! sub_inv_antitoneOn_Ioi (α := α)
  exact (sub_zero _).symm

Goal state after:
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: α : Type u
x : List α
m n : ℕ
⊢ drop m x = drop m x

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c ≤ 0
h : b ≤ a
⊢ a * c⁻¹ ≤ b * c⁻¹

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c ≤ 0
h : b ≤ a
⊢ a * c⁻¹ ≤ b * c⁻¹
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
h : b < a
⊢ a * c⁻¹ < b * c⁻¹

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
h : b < a
⊢ a * c⁻¹ < b * c⁻¹
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: case neg
α : Type u
a : α
l : List α
ih : ∀ (h : l ≠ []), drop (l.length - 1) l = [l.getLast h]
h : a :: l ≠ []
hl : ¬l = []
⊢ [l.getLast hl] = [(a :: l).getLast h]

Tactic:
]

Goal state after:
case neg
α : Type u
a : α
l : List α
ih : ∀ (h : l ≠ []), drop (l.length - 1) l = [l.getLast h]
h : a :: l ≠ []
hl : ¬l = []
⊢ [l.getLast hl] = [(a :: l).getLast h]
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ b ≤ a ↔ b ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b c : α
hc : c < 0
⊢ b < a ↔ b < a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
⊢ a ≤ b ↔ a ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
⊢ b ≤ a ↔ b ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
⊢ a < b ↔ a < b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
⊢ b < a ↔ b < a

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: α : Type u
inst✝ : Inhabited α
l₁ l₂ : List α
n : ℕ
h : l₁.length = n
⊢ takeI l₁.length (l₁ ++ l₂) = l₁

Tactic:
]

Goal state after:
α : Type u
inst✝ : Inhabited α
l₁ l₂ : List α
n : ℕ
h : l₁.length = n
⊢ takeI l₁.length (l₁ ++ l₂) = l₁
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
h : a ≤ b
⊢ a ≤ b

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
h : a ≤ b
⊢ a ≤ b
Module: Mathlib.Data.List.TakeDrop

Goal state before:
pp: α : Type u
l₁ l₂ : List α
n : ℕ
a : α
h : l₁.length = n
⊢ takeD l₁.length (l₁ ++ l₂) a = l₁

Tactic:
]

Goal state after:
α : Type u
l₁ l₂ : List α
n : ℕ
a : α
h : l₁.length = n
⊢ takeD l₁.length (l₁ ++ l₂) a = l₁
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
h : a < b
⊢ a < b

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
hb : b < 0
h : a < b
⊢ a < b
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
h1 : a < 0
h2 : -1 < a
this : 1 / a < -1
⊢ 1 / a < -1

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
h1 : a < 0
h2 : -1 < a
this : 1 / a < -1
⊢ 1 / a < -1
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
h1 : a < 0
h2 : -1 ≤ a
this : 1 / a ≤ -1
⊢ 1 / a ≤ -1

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
h1 : a < 0
h2 : -1 ≤ a
this : 1 / a ≤ -1
⊢ 1 / a ≤ -1
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
h : a < b
⊢ a < b

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a b : α
h : a < b
⊢ a < b
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
a2 : 2 ≤ a
⊢ 1 / a ≤ 1 - 2⁻¹

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
a2 : 2 ≤ a
⊢ 1 / a ≤ 1 - 2⁻¹
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Field α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
inst✝ : IsStrictOrderedRing α
a : α
a2 : 2 ≤ a
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Field α
inst✝¹ : PartialOrder α
inst✝ : IsStrictOrderedRing α
a b c d : α
hc : c ≠ 0
hd : d ≠ 0
⊢ a / c < b / d ↔ a / c < b / d

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Field α
inst✝¹ : PartialOrder α
inst✝ : IsStrictOrderedRing α
a b c d : α
hc : c ≠ 0
hd : d ≠ 0
⊢ a / c ≤ b / d ↔ a / c ≤ b / d

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
R : α → β → Prop
a✝ : α
b✝ : β
l₁✝ : List α
l₂✝ : List β
h₁ : R a✝ b✝
h₂ : Forall₂ R l₁✝ l₂✝
x : α
y : β
hx : (x, y) = (a✝, b✝) ∨ (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝
⊢ R x y

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
R : α → β → Prop
a✝ : α
b✝ : β
l₁✝ : List α
l₂✝ : List β
h₁ : R a✝ b✝
h₂ : Forall₂ R l₁✝ l₂✝
x : α
y : β
hx : (x, y) = (a✝, b✝) ∨ (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝
⊢ R x y
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
R : α → β → Prop
l : List α
l₁ l₂ : List β
h : Forall₂ R l (l₁ ++ l₂)
h' : Forall₂ R (take l₁.length l) l₁
⊢ Forall₂ R (take l₁.length l) l₁

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
R : α → β → Prop
l : List α
l₁ l₂ : List β
h : Forall₂ R l (l₁ ++ l₂)
h' : Forall₂ R (take l₁.length l) l₁
⊢ Forall₂ R (take l₁.length l) l₁
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
R : α → β → Prop
l : List α
l₁ l₂ : List β
h : Forall₂ R l (l₁ ++ l₂)
h' : Forall₂ R (drop l₁.length l) l₂
⊢ Forall₂ R (drop l₁.length l) l₂

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
R : α → β → Prop
l : List α
l₁ l₂ : List β
h : Forall₂ R l (l₁ ++ l₂)
h' : Forall₂ R (drop l₁.length l) l₂
⊢ Forall₂ R (drop l₁.length l) l₂
Module: Mathlib.Data.List.InsertIdx

Goal state before:
pp: case inl
α : Type u
l : List α
n : ℕ
a b : α
h : a ∈ l
hn : l.length < n
⊢ a = b ∨ a ∈ l

Tactic:
]

Goal state after:
case inl
α : Type u
l : List α
n : ℕ
a b : α
h : a ∈ l
hn : l.length < n
⊢ a = b ∨ a ∈ l
Module: Mathlib.Data.List.InsertIdx

Goal state before:
pp: case inl
α : Type u
l : List α
n : ℕ
a b : α
h : a ∈ l.insertIdx n b
hn : l.length < n
⊢ a = b ∨ a ∈ l

Tactic:
| inl hn =>

Goal state after:
Module: Mathlib.Data.List.InsertIdx

Goal state before:
pp: case inr
α : Type u
l : List α
n : ℕ
a b : α
h : a = b ∨ a ∈ l
hn : l.length ≥ n
⊢ a = b ∨ a ∈ l

Tactic:
]

Goal state after:
case inr
α : Type u
l : List α
n : ℕ
a b : α
h : a = b ∨ a ∈ l
hn : l.length ≥ n
⊢ a = b ∨ a ∈ l
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
R : α → β → Prop
l₁ : List α
l₂ : List β
h : Forall₂ R l₁.reverse l₂.reverse
⊢ Forall₂ R l₁.reverse.reverse l₂.reverse.reverse

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
R : α → β → Prop
l₁ : List α
l₂ : List β
h : Forall₂ R l₁.reverse l₂.reverse
⊢ Forall₂ R l₁.reverse.reverse l₂.reverse.reverse
Module: Mathlib.Data.List.InsertIdx

Goal state before:
pp: α : Type u
β : Type v
p : α → Prop
f : (a : α) → p a → β
l : List α
hl : ∀ (a : α), a ∈ l → p a
n✝ : ℕ
a : α
as : List α
h✝ : ∀ (a_1 : α), a_1 ∈ a :: as → p a_1
h : p a ∧ ∀ (x : α), x ∈ as → p x
n : ℕ
⊢ (pmap f (a :: as) h✝).eraseIdx (n + 1) = pmap f ((a :: as).eraseIdx (n + 1)) ⋯

Tactic:
]

Goal state after:
α : Type u
β : Type v
p : α → Prop
f : (a : α) → p a → β
l : List α
hl : ∀ (a : α), a ∈ l → p a
n✝ : ℕ
a : α
as : List α
h✝ : ∀ (a_1 : α), a_1 ∈ a :: as → p a_1
h : p a ∧ ∀ (x : α), x ∈ as → p x
n : ℕ
⊢ (pmap f (a :: as) h✝).eraseIdx (n + 1) = pmap f ((a :: as).eraseIdx (n + 1)) ⋯
Module: Mathlib.Data.List.InsertIdx

Goal state before:
pp: α : Type u
β : Type v
a : α
l : List α
x : α
n : ℕ
hn : n ≤ l.length
⊢ n ≤ l.length

Tactic:
]

Goal state after:
α : Type u
β : Type v
a : α
l : List α
x : α
n : ℕ
hn : n ≤ l.length
⊢ n ≤ l.length
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
R : α → β → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
⊢ p a = true

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
R : α → β → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : p a = true
⊢ p a = true
Module: Mathlib.Data.List.InsertIdx

Goal state before:
pp: α : Type u
β : Type v
a : α
l : List α
x : α
n k : ℕ
hk' : n + k < l.length
⊢ n + k < l.length

Tactic:
]

Goal state after:
α : Type u
β : Type v
a : α
l : List α
x : α
n k : ℕ
hk' : n + k < l.length
⊢ n + k < l.length
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
⊢ 1 / |a| = 1 / |a|

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
R : α → β → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
⊢ ¬p a = true

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
R : α → β → Prop
p : α → Bool
q : β → Bool
hpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
h : ¬p a = true
⊢ ¬p a = true
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R : α → β → Prop
P : γ → δ → Prop
f : α → Option γ
g : β → Option δ
hfg : (R ⇒ Option.Rel P) f g
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
⊢ Forall₂ P
    (match f a with
    | none => filterMap f as
    | some b => b :: filterMap f as)
    (match g b with
    | none => filterMap g bs
    | some b => b :: filterMap g bs)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R : α → β → Prop
P : γ → δ → Prop
f : α → Option γ
g : β → Option δ
hfg : (R ⇒ Option.Rel P) f g
a : α
as : List α
b : β
bs : List β
h₁ : R a b
h₂ : Forall₂ R as bs
⊢ Forall₂ P
    (match f a with
    | none => filterMap f as
    | some b => b :: filterMap f as)
    (match g b with
    | none => filterMap g bs
    | some b => b :: filterMap g bs)
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
ε : α
hε : 0 < ε
n : ℕ
B : α
B_pos : 0 < B
pos : 0 < 1 + ↑n * (B + 1) ^ (n - 1)
q r : α
hr : |r| ≤ B
hqr : |q - r| ≤ 1 ∧ |q - r| + |q - r| * ↑n * (B + 1) ^ (n - 1) ≤ ε
⊢ |q - r| * ↑n * max |q| |r| ^ (n - 1) < ε

Tactic:
]

Goal state after:
α : Type u_2
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
ε : α
hε : 0 < ε
n : ℕ
B : α
B_pos : 0 < B
pos : 0 < 1 + ↑n * (B + 1) ^ (n - 1)
q r : α
hr : |r| ≤ B
hqr : |q - r| ≤ 1 ∧ |q - r| + |q - r| * ↑n * (B + 1) ^ (n - 1) ≤ ε
⊢ |q - r| * ↑n * max |q| |r| ^ (n - 1) < ε
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: case mpr.slnil
α : Type u_1
β : Type u_2
R : α → β → Prop
l₂ l : List β
l₁ : List α
hl1 : Forall₂ R l₁ []
⊢ SublistForall₂ R [] []

Tactic:
]

Goal state after:
case mpr.slnil
α : Type u_1
β : Type u_2
R : α → β → Prop
l₂ l : List β
l₁ : List α
hl1 : Forall₂ R l₁ []
⊢ SublistForall₂ R [] []
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a b ε : α
hε : 0 < ε
h : 2 * (ε * a) * b ≤ (ε * a) ^ 2 + b ^ 2
⊢ 2 * a * b = 2 * a * b

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: α : Type u_1
l : List α
n : ℕ
h : n < l.length
a : α
⊢ (l.eraseIdx n).insertIdx n a ~ a :: l.eraseIdx n

Tactic:
]

Goal state after:
α : Type u_1
l : List α
n : ℕ
h : n < l.length
a : α
⊢ (l.eraseIdx n).insertIdx n a ~ a :: l.eraseIdx n
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a b ε : α
hε : 0 < ε
h : 2 * (ε * a) * b ≤ (ε * a) ^ 2 + b ^ 2
⊢ ε * a ^ 2 + ε⁻¹ * b ^ 2 = ε * a ^ 2 + ε⁻¹ * b ^ 2

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: α : Type u_1
l : List α
n : ℕ
h : n < l.length
a : α
⊢ n ≤ l.length - 1

Tactic:
]

Goal state after:
α : Type u_1
l : List α
n : ℕ
h : n < l.length
a : α
⊢ n ≤ l.length - 1
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: α : Type u_1
l₁ l₂ : List α
m n : ℕ
hm : m ≤ l₁.length
hn : n ≤ l₂.length
a : α
⊢ l₁ ~ l₂ ↔ l₁ ~ l₂

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Forall2

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R S : α → β → Prop
P : γ → δ → Prop
Rₐ : α → α → Prop
inst✝ : IsTrans α Rₐ
c : List α
⊢ ∀ (a b : List α), SublistForall₂ Rₐ a b → SublistForall₂ Rₐ b c → SublistForall₂ Rₐ a c

Tactic:
induction c with
    | nil =>
      rintro _ _ h1 h2
      cases h2
      exact h1
    | cons _ _ ih =>
      rintro a b h1 h2
      obtain - | ⟨hbc, tbc⟩ | btc := h2
      · cases h1
        exact SublistForall₂.nil
      · obtain - | ⟨hab, tab⟩ | atb := h1
        · exact SublistForall₂.nil
        · exact SublistForall₂.cons (_root_.trans hab hbc) (ih _ _ tab tbc)
        · exact SublistForall₂.cons_right (ih _ _ atb tbc)
      · exact SublistForall₂.cons_right (ih _ _ h1 btc)

Goal state after:
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: case inr
α : Type u_1
l₁ l₂ : List α
n : ℕ
a : α
this : ∀ {l₁ l₂ : List α}, l₁.length ≤ l₂.length → (l₁.insertIdx n a ~ l₂.insertIdx n a ↔ l₁ ~ l₂)
hle : ¬l₁.length ≤ l₂.length
⊢ l₁ ~ l₂ ↔ l₁ ~ l₂

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: case inl
α : Type u_1
n : ℕ
a : α
l₁ l₂ : List α
hle : l₁.length ≤ l₂.length
hn₁ : l₁.length < n
⊢ l₁ ~ l₂.insertIdx n a ↔ l₁ ~ l₂

Tactic:
]

Goal state after:
case inl
α : Type u_1
n : ℕ
a : α
l₁ l₂ : List α
hle : l₁.length ≤ l₂.length
hn₁ : l₁.length < n
⊢ l₁ ~ l₂.insertIdx n a ↔ l₁ ~ l₂
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: case inl.inl
α : Type u_1
n : ℕ
a : α
l₁ l₂ : List α
hle : l₁.length ≤ l₂.length
hn₁ : l₁.length < n
hn₂ : l₂.length < n
⊢ l₁ ~ l₂ ↔ l₁ ~ l₂

Tactic:
]

Goal state after:
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: case inl.inr.ha
α : Type u_1
n : ℕ
a : α
l₁ l₂ : List α
hle : l₁.length ≤ l₂.length
hn₁ : (l₂.insertIdx n a).length < n
hn₂ : l₂.length ≥ n
h : l₁ ~ l₂.insertIdx n a
⊢ False

Tactic:
]

Goal state after:
case inl.inr.ha
α : Type u_1
n : ℕ
a : α
l₁ l₂ : List α
hle : l₁.length ≤ l₂.length
hn₁ : (l₂.insertIdx n a).length < n
hn₂ : l₂.length ≥ n
h : l₁ ~ l₂.insertIdx n a
⊢ False
Module: Mathlib.Data.List.Perm.Basic

Goal state before:
pp: case inl
α : Type u_1
l₁ l₂ : List α
m n : ℕ
hm : m < l₁.length
h : ∃ h, l₂[n] = l₁[m]
⊢ l₁.eraseIdx m ~ l₂.eraseIdx n ↔ l₁ ~ l₂

Tactic:
]

Goal state after:
case inl
α : Type u_1
l₁ l₂ : List α
m n : ℕ
hm : m < l₁.length
h : ∃ h, l₂[n] = l₁[m]
⊢ l₁.eraseIdx m ~ l₂.eraseIdx n ↔ l₁ ~ l₂