Theorem: WithTop.isLUB_sSup'
File: Mathlib/Order/ConditionallyCompleteLattice/Basic.lean

tactic: constructor
state_before: α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ IsLUB s (sSup s)
state_after: case left
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ sSup s ∈ upperBounds s

case right
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ sSup s ∈ lowerBounds (upperBounds s)

tactic: show ite _ _ _ ∈ _
state_before: case left
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ sSup s ∈ upperBounds s
state_after: case left
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ (if ⊤ ∈ s then ⊤ else if BddAbove ((fun a => ↑a) ⁻¹' s) then ↑(sSup ((fun a => ↑a) ⁻¹' s)) else ⊤) ∈ upperBounds s

tactic: split_ifs with h₁ h₂
state_before: case left
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ (if ⊤ ∈ s then ⊤ else if BddAbove ((fun a => ↑a) ⁻¹' s) then ↑(sSup ((fun a => ↑a) ⁻¹' s)) else ⊤) ∈ upperBounds s
state_after: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
⊢ ⊤ ∈ upperBounds s

case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ∈ upperBounds s

case neg
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ⊤ ∈ upperBounds s

tactic: intro _ _
state_before: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
⊢ ⊤ ∈ upperBounds s
state_after: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
a✝¹ : WithTop β
a✝ : a✝¹ ∈ s
⊢ a✝¹ ≤ ⊤

tactic: exact le_top
state_before: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
a✝¹ : WithTop β
a✝ : a✝¹ ∈ s
⊢ a✝¹ ≤ ⊤
state_after: no goals

tactic: rintro (⟨⟩ | a) ha
state_before: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ∈ upperBounds s
state_after: case pos.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
ha : none ∈ s
⊢ none ≤ ↑(sSup ((fun a => ↑a) ⁻¹' s))

case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
a : β
ha : Option.some a ∈ s
⊢ Option.some a ≤ ↑(sSup ((fun a => ↑a) ⁻¹' s))

tactic: apply coe_le_coe.2
state_before: case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
a : β
ha : Option.some a ∈ s
⊢ Option.some a ≤ ↑(sSup ((fun a => ↑a) ⁻¹' s))
state_after: case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
a : β
ha : Option.some a ∈ s
⊢ a ≤ sSup ((fun a => ↑a) ⁻¹' s)

tactic: exact le_csSup h₂ ha
state_before: case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
a : β
ha : Option.some a ∈ s
⊢ a ≤ sSup ((fun a => ↑a) ⁻¹' s)
state_after: no goals

tactic: contradiction
state_before: case pos.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
ha : none ∈ s
⊢ none ≤ ↑(sSup ((fun a => ↑a) ⁻¹' s))
state_after: no goals

tactic: intro _ _
state_before: case neg
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ⊤ ∈ upperBounds s
state_after: case neg
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
a✝¹ : WithTop β
a✝ : a✝¹ ∈ s
⊢ a✝¹ ≤ ⊤

tactic: exact le_top
state_before: case neg
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
a✝¹ : WithTop β
a✝ : a✝¹ ∈ s
⊢ a✝¹ ≤ ⊤
state_after: no goals

tactic: show ite _ _ _ ∈ _
state_before: case right
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ sSup s ∈ lowerBounds (upperBounds s)
state_after: case right
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ (if ⊤ ∈ s then ⊤ else if BddAbove ((fun a => ↑a) ⁻¹' s) then ↑(sSup ((fun a => ↑a) ⁻¹' s)) else ⊤) ∈
    lowerBounds (upperBounds s)

tactic: split_ifs with h₁ h₂
state_before: case right
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
⊢ (if ⊤ ∈ s then ⊤ else if BddAbove ((fun a => ↑a) ⁻¹' s) then ↑(sSup ((fun a => ↑a) ⁻¹' s)) else ⊤) ∈
    lowerBounds (upperBounds s)
state_after: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
⊢ ⊤ ∈ lowerBounds (upperBounds s)

case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds (upperBounds s)

case neg
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ⊤ ∈ lowerBounds (upperBounds s)

tactic: rintro (⟨⟩ | a) ha
state_before: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
⊢ ⊤ ∈ lowerBounds (upperBounds s)
state_after: case pos.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
ha : none ∈ upperBounds s
⊢ ⊤ ≤ none

case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
a : β
ha : Option.some a ∈ upperBounds s
⊢ ⊤ ≤ Option.some a

tactic: exact le_rfl
state_before: case pos.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
ha : none ∈ upperBounds s
⊢ ⊤ ≤ none
state_after: no goals

tactic: exact False.elim (not_top_le_coe a (ha h₁))
state_before: case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
a : β
ha : Option.some a ∈ upperBounds s
⊢ ⊤ ≤ Option.some a
state_after: no goals

tactic: rintro (⟨⟩ | b) hb
state_before: case pos
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds (upperBounds s)
state_after: case pos.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
hb : none ∈ upperBounds s
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ≤ none

case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ≤ Option.some b

tactic: refine coe_le_coe.2 (csSup_le ?_ ?_)
state_before: case pos.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ≤ Option.some b
state_after: case pos.some.refine_1
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ((fun a => ↑a) ⁻¹' s).Nonempty

case pos.some.refine_2
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ∀ b_1 ∈ (fun a => ↑a) ⁻¹' s, b_1 ≤ b

tactic: exact le_top
state_before: case pos.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
hb : none ∈ upperBounds s
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ≤ none
state_after: no goals

tactic: rcases hs with ⟨⟨⟩ | b, hb⟩
state_before: case pos.some.refine_1
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ((fun a => ↑a) ⁻¹' s).Nonempty
state_after: case pos.some.refine_1.intro.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb✝ : Option.some b ∈ upperBounds s
hb : none ∈ s
⊢ ((fun a => ↑a) ⁻¹' s).Nonempty

case pos.some.refine_1.intro.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b✝ : β
hb✝ : Option.some b✝ ∈ upperBounds s
b : β
hb : Option.some b ∈ s
⊢ ((fun a => ↑a) ⁻¹' s).Nonempty

tactic: exact absurd hb h₁
state_before: case pos.some.refine_1.intro.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb✝ : Option.some b ∈ upperBounds s
hb : none ∈ s
⊢ ((fun a => ↑a) ⁻¹' s).Nonempty
state_after: no goals

tactic: exact ⟨b, hb⟩
state_before: case pos.some.refine_1.intro.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b✝ : β
hb✝ : Option.some b✝ ∈ upperBounds s
b : β
hb : Option.some b ∈ s
⊢ ((fun a => ↑a) ⁻¹' s).Nonempty
state_after: no goals

tactic: intro a ha
state_before: case pos.some.refine_2
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ∀ b_1 ∈ (fun a => ↑a) ⁻¹' s, b_1 ≤ b
state_after: case pos.some.refine_2
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
a : β
ha : a ∈ (fun a => ↑a) ⁻¹' s
⊢ a ≤ b

tactic: exact coe_le_coe.1 (hb ha)
state_before: case pos.some.refine_2
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
a : β
ha : a ∈ (fun a => ↑a) ⁻¹' s
⊢ a ≤ b
state_after: no goals

tactic: rintro (⟨⟩ | b) hb
state_before: case neg
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ⊤ ∈ lowerBounds (upperBounds s)
state_after: case neg.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
hb : none ∈ upperBounds s
⊢ ⊤ ≤ none

case neg.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ⊤ ≤ Option.some b

tactic: exact le_rfl
state_before: case neg.none
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
hb : none ∈ upperBounds s
⊢ ⊤ ≤ none
state_after: no goals

tactic: exfalso
state_before: case neg.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ ⊤ ≤ Option.some b
state_after: case neg.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ False

tactic: apply h₂
state_before: case neg.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ False
state_after: case neg.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ BddAbove ((fun a => ↑a) ⁻¹' s)

tactic: use b
state_before: case neg.some
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ BddAbove ((fun a => ↑a) ⁻¹' s)
state_after: case h
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ b ∈ upperBounds ((fun a => ↑a) ⁻¹' s)

tactic: intro a ha
state_before: case h
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
⊢ b ∈ upperBounds ((fun a => ↑a) ⁻¹' s)
state_after: case h
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
a : β
ha : a ∈ (fun a => ↑a) ⁻¹' s
⊢ a ≤ b

tactic: exact coe_le_coe.1 (hb ha)
state_before: case h
α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLinearOrderBot α
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : ¬BddAbove ((fun a => ↑a) ⁻¹' s)
b : β
hb : Option.some b ∈ upperBounds s
a : β
ha : a ∈ (fun a => ↑a) ⁻¹' s
⊢ a ≤ b
state_after: no goals