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

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
h : a ≤ b
⊢ a + (b - a) = b

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
h : a ≤ b
⊢ a + (b - a) = b
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
h : b ≤ c
h2 : a ≤ c - b
⊢ c ≤ c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
h : a ≤ c
h2 : b ≤ c - a
⊢ c ≤ c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
h : c ≤ b
⊢ a ≤ b ↔ a ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hab : b ≤ a
hcb : c ≤ b
⊢ a - b = a - b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
h : c ≤ b
⊢ a - b = a - b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hb : AddLECancellable b
h : b ≤ a
⊢ a = c + b ↔ a = c + b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hb : AddLECancellable b
h : b ≤ a
⊢ c + (a - b) = c + (a - b)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hbc : AddLECancellable (b - c)
h₁ : b ≤ a
h₂ : c ≤ b
⊢ a = a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
hd : AddLECancellable d
hba : b ≤ a
hdc : d ≤ c
⊢ a + c - (b + d) = a + c - (b + d)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
ha : AddLECancellable a
h : a ≤ c
⊢ b ≤ c - a ↔ a + b ≤ c

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
ha : AddLECancellable a
h : a ≤ c
⊢ b ≤ c - a ↔ a + b ≤ c
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hb : AddLECancellable b
hba : b ≤ a
⊢ a - b < c ↔ a < b + c

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hb : AddLECancellable b
hba : b ≤ a
⊢ a - b < c ↔ a < b + c
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hb : AddLECancellable b
hc : AddLECancellable c
h₁ : b ≤ a
h₂ : c ≤ a
⊢ a < b + c ↔ a < b + c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
ha : AddLECancellable a
hc : AddLECancellable c
h₁ : a ≤ b
h₂ : c ≤ b
⊢ a + c ≤ b ↔ a + c ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hc : AddLECancellable c
h : c ≤ b
⊢ a < b - c ↔ a + c < b

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hc : AddLECancellable c
h : c ≤ b
⊢ a < b - c ↔ a + c < b
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hab : AddLECancellable (a - b)
h₁ : b ≤ a
h₂ : c ≤ a
h₃ : a - b = a - c
⊢ a - b + b = a - b + c

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hab : AddLECancellable (a - b)
h₁ : b ≤ a
h₂ : c ≤ a
h₃ : a - b = a - c
⊢ a - b + b = a - b + c
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hab : AddLECancellable (a - b)
h₁ : b ≤ a
h₂ : c ≤ a
h₃ : a - b = a - c
⊢ a = a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hc : AddLECancellable c
h : c ≤ a
h2 : a < b
⊢ a < b

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hc : AddLECancellable c
h : c ≤ a
h2 : a < b
⊢ a < b
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hc : AddLECancellable c
hcb : c ≤ b
⊢ a + b = a + b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hac : AddLECancellable (a - c)
h : c ≤ a
⊢ b + a = b + a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hab : AddLECancellable (a - b)
h : b ≤ a
⊢ b - c = b - c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝⁶ : AddCommSemigroup α
inst✝⁵ : PartialOrder α
inst✝⁴ : ExistsAddOfLE α
inst✝³ : AddLeftMono α
inst✝² : Sub α
inst✝¹ : OrderedSub α
a c : α
inst✝ : AddLeftReflectLE α
d : α
h : c ≤ c + d
⊢ a - d = a - d

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ a + (b - a) = b ↔ a ≤ b

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ a + (b - a) = b ↔ a ≤ b
Module: Mathlib.Algebra.Order.Sub.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ a ≤ b ↔ a ≤ b

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ ¬a ≤ b ↔ ¬a ≤ b

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
ha : AddLECancellable a
hc : AddLECancellable c
h : c ≤ a
⊢ a + c ≤ a + b → c ≤ b

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
ha : AddLECancellable a
hc : AddLECancellable c
h : c ≤ a
⊢ a + c ≤ a + b → c ≤ b
Module: Mathlib.Algebra.Order.Sub.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ b < a ↔ b < a

Tactic:
]

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

Goal state before:
pp: case inl
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
h : a ≤ b
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case inr
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
h : b ≤ a
⊢ a - b = a - b

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
hc : AddLECancellable c
h : c ≤ a
⊢ a < b ↔ a < b

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
ha : AddLECancellable a
h₁ : b < a
h₂ : 0 < b
h : a - b = a
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
ha : AddLECancellable a
h₁ : b < a
h₂ : 0 < b
h : a - b = a
⊢ False
Module: Mathlib.Algebra.Order.Sub.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a : α
ha : AddLECancellable a
h : a < a
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a : α
ha : AddLECancellable a
h : a < a
⊢ False
Module: Mathlib.Algebra.Order.Sub.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : AddCommMonoid α
inst✝⁴ : LinearOrder α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : AddLeftReflectLE α
a b : α
⊢ min a b = min a b

Tactic:
]

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

Goal state before:
pp: case inl
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
h : a ≤ b
⊢ b = b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Goal state before:
pp: case coe.coe
α : Type u
β : Type v
inst✝³ : Add α
F : Type u_1
inst✝² : Add β
inst✝¹ : FunLike F α β
inst✝ : AddHomClass F α β
f : F
a✝¹ a✝ : α
⊢ map ⇑f ↑(a✝¹ + a✝) = ↑(f (a✝¹ + a✝))

Tactic:
]

Goal state after:
case coe.coe
α : Type u
β : Type v
inst✝³ : Add α
F : Type u_1
inst✝² : Add β
inst✝¹ : FunLike F α β
inst✝ : AddHomClass F α β
f : F
a✝¹ a✝ : α
⊢ map ⇑f ↑(a✝¹ + a✝) = ↑(f (a✝¹ + a✝))
Module: Mathlib.Algebra.Order.Sub.Basic

Goal state before:
pp: case inr
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
h : b ≤ a
⊢ a = a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ max b a = max b a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ a = a

pp: case h
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ min a b ≤ a

Tactic:
]

Goal state after:
case h
α : Type u_1
inst✝⁴ : AddCommMonoid α
inst✝³ : LinearOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b : α
⊢ min a b ≤ a
Module: Mathlib.Algebra.Order.Sub.Basic

Goal state before:
pp: case inl
α : Type u_1
inst✝⁵ : AddCommMonoid α
inst✝⁴ : LinearOrder α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : AddLeftReflectLE α
a b : α
h : a ≤ b
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Goal state before:
pp: case top.zero
α : Type u
β : Type v
inst✝ : AddMonoid α
⊢ (match ⊤, 0 + 1 with
    | Option.some a, n => ↑(n • a)
    | none, 0 => 0
    | none, _n.succ => ⊤) =
    (match ⊤, 0 with
      | Option.some a, n => ↑(n • a)
      | none, 0 => 0
      | none, _n.succ => ⊤) +
      ⊤

Tactic:
simp [succ_nsmul, coe_add]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Goal state before:
pp: case top.succ
α : Type u
β : Type v
inst✝ : AddMonoid α
n✝ : ℕ
⊢ (match ⊤, n✝ + 1 + 1 with
    | Option.some a, n => ↑(n • a)
    | none, 0 => 0
    | none, _n.succ => ⊤) =
    (match ⊤, n✝ + 1 with
      | Option.some a, n => ↑(n • a)
      | none, 0 => 0
      | none, _n.succ => ⊤) +
      ⊤

Tactic:
simp [succ_nsmul, coe_add]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Goal state before:
pp: case coe.zero
α : Type u
β : Type v
inst✝ : AddMonoid α
a✝ : α
⊢ (match ↑a✝, 0 + 1 with
    | Option.some a, n => ↑(n • a)
    | none, 0 => 0
    | none, _n.succ => ⊤) =
    (match ↑a✝, 0 with
      | Option.some a, n => ↑(n • a)
      | none, 0 => 0
      | none, _n.succ => ⊤) +
      ↑a✝

Tactic:
simp [succ_nsmul, coe_add]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Goal state before:
pp: case coe.succ
α : Type u
β : Type v
inst✝ : AddMonoid α
a✝ : α
n✝ : ℕ
⊢ (match ↑a✝, n✝ + 1 + 1 with
    | Option.some a, n => ↑(n • a)
    | none, 0 => 0
    | none, _n.succ => ⊤) =
    (match ↑a✝, n✝ + 1 with
      | Option.some a, n => ↑(n • a)
      | none, 0 => 0
      | none, _n.succ => ⊤) +
      ↑a✝

Tactic:
simp [succ_nsmul, coe_add]

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

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : AddCommMonoid α
inst✝² : LinearOrder α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : AddRightReflectLE α
a b c : α
h✝ : b ≤ a
h : b ≤ b + ⋯.choose
this : a = b + ⋯.choose
⊢ ⋯.choose ≤ c ↔ a ≤ c + b

Tactic:
]

Goal state after:
case pos
α : Type u_1
inst✝³ : AddCommMonoid α
inst✝² : LinearOrder α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : AddRightReflectLE α
a b c : α
h✝ : b ≤ a
h : b ≤ b + ⋯.choose
this : a = b + ⋯.choose
⊢ ⋯.choose ≤ c ↔ a ≤ c + b
Module: Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Goal state before:
pp: α : Type u
β : Type v
M : Type u_1
N : Type u_2
inst✝¹ : One M
inst✝ : One N
f : OneHom M N
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : AddCommMonoid α
inst✝² : LinearOrder α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : AddRightReflectLE α
a b c : α
h✝ : b ≤ a
h : b ≤ b + ⋯.choose
this : a = b + ⋯.choose
⊢ ⋯.choose ≤ c ↔ ⋯.choose ≤ c

Tactic:
]

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

Goal state before:
pp: case neg
α : Type u_1
inst✝³ : AddCommMonoid α
inst✝² : LinearOrder α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : AddRightReflectLE α
a b c : α
h : a < b
⊢ 0 ≤ c ↔ a ≤ c + b

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝³ : AddCommMonoid α
inst✝² : LinearOrder α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : AddRightReflectLE α
a b c : α
h : a < b
⊢ 0 ≤ c ↔ a ≤ c + b
Module: Mathlib.Data.Nat.Cast.Order.Ring

Goal state before:
pp: case inl
α : Type u_2
inst✝⁶ : CommSemiring α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedRing α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : AddLeftReflectLE α
m n : ℕ
h : m ≤ n
⊢ 0 = 0

pp: case inl
α : Type u_2
inst✝⁶ : CommSemiring α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedRing α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : AddLeftReflectLE α
m n : ℕ
h : m ≤ n
⊢ ↑m ≤ ↑n

Tactic:
]

Goal state after:
case inl
α : Type u_2
inst✝⁶ : CommSemiring α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedRing α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : AddLeftReflectLE α
m n : ℕ
h : m ≤ n
⊢ ↑m ≤ ↑n
Module: Mathlib.Data.Nat.Cast.Order.Ring

Goal state before:
pp: case inr
α : Type u_2
inst✝⁶ : CommSemiring α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedRing α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : AddLeftReflectLE α
n m : ℕ
h : n ≤ m + n
⊢ ↑m = ↑m

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝³ : AddCommGroupWithOne R
inst✝² : PartialOrder R
inst✝¹ : AddLeftMono R
inst✝ : ZeroLEOneClass R
m n : ℤ
h : 0 ≤ n - m
⊢ ↑m ≤ ↑n

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : AddCommGroupWithOne R
inst✝² : PartialOrder R
inst✝¹ : AddLeftMono R
inst✝ : ZeroLEOneClass R
m n : ℤ
h : 0 ≤ n - m
⊢ ↑m ≤ ↑n
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝³ : AddCommGroupWithOne R
inst✝² : PartialOrder R
inst✝¹ : AddLeftMono R
inst✝ : ZeroLEOneClass R
m n : ℤ
k : ℕ
hk : ↑k = n - m
⊢ 0 ≤ ↑k

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : AddCommGroupWithOne R
inst✝² : PartialOrder R
inst✝¹ : AddLeftMono R
inst✝ : ZeroLEOneClass R
m n : ℤ
k : ℕ
hk : ↑k = n - m
⊢ 0 ≤ ↑k
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝⁴ : AddCommGroupWithOne R
inst✝³ : PartialOrder R
inst✝² : AddLeftMono R
inst✝¹ : ZeroLEOneClass R
inst✝ : NeZero 1
m n : ℤ
⊢ m ≤ n ↔ m ≤ n

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝⁴ : AddCommGroupWithOne R
inst✝³ : PartialOrder R
inst✝² : AddLeftMono R
inst✝¹ : ZeroLEOneClass R
inst✝ : NeZero 1
n : ℤ
⊢ n ≤ 0 ↔ n ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝⁴ : AddCommGroupWithOne R
inst✝³ : PartialOrder R
inst✝² : AddLeftMono R
inst✝¹ : ZeroLEOneClass R
inst✝ : NeZero 1
n : ℤ
⊢ 0 < n ↔ 0 < n

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝⁴ : AddCommGroupWithOne R
inst✝³ : PartialOrder R
inst✝² : AddLeftMono R
inst✝¹ : ZeroLEOneClass R
inst✝ : NeZero 1
n : ℤ
⊢ n < 0 ↔ n < 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
a : ℤ
h : a < 0
⊢ a ≤ -1

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
a : ℤ
h : a < 0
⊢ a ≤ -1
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hn : 0 < n
this : 0 ≤ x + ↑n
⊢ 0 ≤ x + ↑n

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hn : 0 < n
this : 0 ≤ x + ↑n
⊢ 0 ≤ x + ↑n
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hn : n < 0
this : x + ↑n ≤ 0
⊢ x + ↑n ≤ 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hn : n < 0
this : x + ↑n ≤ 0
⊢ x + ↑n ≤ 0
Module: Mathlib.Algebra.Order.Ring.Cast

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hnx' : n < 0 → x + ↑n ≤ 0
⊢ 0 ≤ ↑n ∧ 0 ≤ x + ↑n ∨ ↑n ≤ 0 ∧ x + ↑n ≤ 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hnx' : n < 0 → x + ↑n ≤ 0
⊢ 0 ≤ ↑n ∧ 0 ≤ x + ↑n ∨ ↑n ≤ 0 ∧ x + ↑n ≤ 0
Module: Mathlib.Order.WithBot

Goal state before:
pp: α : Type u_1
a b : WithBot α
ha : a ≠ ⊥
hb : b ≠ ⊥
⊢ a = b ↔ a = b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Canonical

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c : α
h : a < c * b
⊢ a * b⁻¹ < c

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c : α
h : a < c * b
⊢ a * b⁻¹ < c
Module: Mathlib.Algebra.Order.GroupWithZero.Canonical

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
h : a * b < c * d
hc : 0 < c
hh : a⁻¹ ≤ c⁻¹
ha : a ≠ 0
⊢ b < d

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
h : a * b < c * d
hc : 0 < c
hh : a⁻¹ ≤ c⁻¹
ha : a ≠ 0
⊢ b < d
Module: Mathlib.Data.Nat.Find

Goal state before:
pp: m✝¹ n✝ k✝ : ℕ
p q : ℕ → Prop
inst✝ : DecidablePred p
H : ∃ n, p n
n : ℕ
pn : p n
m✝ m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc lbp k
k : ℕ
kn : n ≤ k + m.succ
y : ℕ
r : y.lbp k
_a : ∀ (k_1 : ℕ), k_1 ≤ k → ¬p k_1
⊢ n ≤ k + m + 1

Tactic:
]

Goal state after:
m✝¹ n✝ k✝ : ℕ
p q : ℕ → Prop
inst✝ : DecidablePred p
H : ∃ n, p n
n : ℕ
pn : p n
m✝ m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc lbp k
k : ℕ
kn : n ≤ k + m.succ
y : ℕ
r : y.lbp k
_a : ∀ (k_1 : ℕ), k_1 ≤ k → ¬p k_1
⊢ n ≤ k + m + 1
Module: Mathlib.Data.Nat.Find

Goal state before:
pp: m✝ n✝ k : ℕ
p q : ℕ → Prop
inst✝ : DecidablePred p
H : ∃ n, p n
m : ℕ
IH : (y : ℕ) → y.lbp m → (∀ (n : ℕ), n < y → ¬p n) → { n // p n ∧ ∀ (m : ℕ), m < n → ¬p m }
al : ∀ (n : ℕ), n < m → ¬p n
pm : ¬p m
n : ℕ
h : n ≤ m
e : n = m
⊢ ¬p m

Tactic:
]

Goal state after:
m✝ n✝ k : ℕ
p q : ℕ → Prop
inst✝ : DecidablePred p
H : ∃ n, p n
m : ℕ
IH : (y : ℕ) → y.lbp m → (∀ (n : ℕ), n < y → ¬p n) → { n // p n ∧ ∀ (m : ℕ), m < n → ¬p m }
al : ∀ (n : ℕ), n < m → ¬p n
pm : ¬p m
n : ℕ
h : n ≤ m
e : n = m
⊢ ¬p m
Module: Mathlib.Data.Nat.Find

Goal state before:
pp: case refine_1
n : ℕ
p : ℕ → Prop
inst✝ : DecidablePred p
hₘ : ∃ m, p (m + n)
hₙ : ∃ n, p n
hn : n ≤ Nat.find hₙ
m : ℕ
hm : m < Nat.find hₘ + n
hpm : p m
hnm : n ≤ m
⊢ p m

Tactic:
]

Goal state after:
case refine_1
n : ℕ
p : ℕ → Prop
inst✝ : DecidablePred p
hₘ : ∃ m, p (m + n)
hₙ : ∃ n, p n
hn : n ≤ Nat.find hₙ
m : ℕ
hm : m < Nat.find hₘ + n
hpm : p m
hnm : n ≤ m
⊢ p m