-- Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
theorem isGLB_sInf' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)}
    (hs : BddBelow s) : IsGLB s (sInf s)

Proof state:
case neg
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ upperBounds (lowerBounds s)

Tactic:
rintro (⟨⟩ | a) ha

Result:
case neg.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
ha : none ∈ lowerBounds s
⊢ none ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))

case neg.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ lowerBounds s
⊢ Option.some a ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))

Proof state:
case neg
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds s

Tactic:
rintro (⟨⟩ | a) ha

Result:
case neg.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
ha : none ∈ s
⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ≤ none

case neg.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ≤ Option.some a

Proof state:
case neg.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
ha : none ∈ lowerBounds s
⊢ False

Tactic:
apply h

Result:
case neg.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
ha : none ∈ lowerBounds s
⊢ s ⊆ {⊤}

Proof state:
case left
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ lowerBounds s

Tactic:
split_ifs with h

Result:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
⊢ ⊤ ∈ lowerBounds s

case neg
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds s

Proof state:
case h
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
b : β
hb : Option.some b ∈ lowerBounds s
c : β
hc : c ∈ (fun a => ↑a) ⁻¹' s
⊢ b ≤ c

Tactic:
exact coe_le_coe.1 (hb hc)

Result:
no goals

Proof state:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
a✝¹ : WithTop β
a✝ : a✝¹ ∈ lowerBounds s
⊢ a✝¹ ≤ ⊤

Tactic:
exact le_top

Result:
no goals

Proof state:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
⊢ ⊤ ∈ lowerBounds s

Tactic:
intro a ha

Result:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
a : WithTop β
ha : a ∈ s
⊢ ⊤ ≤ a

Proof state:
case left
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ (if s ⊆ {⊤} ∨ ¬BddBelow s then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ lowerBounds s

Tactic:
simp only [hs, not_true_eq_false, or_false]

Result:
case left
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ lowerBounds s

Proof state:
case right
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ (if s ⊆ {⊤} then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ upperBounds (lowerBounds s)

Tactic:
split_ifs with h

Result:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
⊢ ⊤ ∈ upperBounds (lowerBounds s)

case neg
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
⊢ ↑(sInf ((fun a => ↑a) ⁻¹' s)) ∈ upperBounds (lowerBounds s)

Proof state:
case neg.some.refine_2
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ lowerBounds s
b : β
hb : b ∈ (fun a => ↑a) ⁻¹' s
⊢ ↑a ≤ ↑b

Tactic:
exact ha hb

Result:
no goals

Proof state:
case neg.some.intro.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
b : β
hb : Option.some b ∈ lowerBounds s
⊢ BddBelow ((fun a => ↑a) ⁻¹' s)

Tactic:
use b

Result:
case h
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
b : β
hb : Option.some b ∈ lowerBounds s
⊢ b ∈ lowerBounds ((fun a => ↑a) ⁻¹' s)

Proof state:
case neg.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
ha : none ∈ lowerBounds s
⊢ none ≤ ↑(sInf ((fun a => ↑a) ⁻¹' s))

Tactic:
exfalso

Result:
case neg.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
ha : none ∈ lowerBounds s
⊢ False

Proof state:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
⊢ ⊤ ∈ upperBounds (lowerBounds s)

Tactic:
intro _ _

Result:
case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : s ⊆ {⊤}
a✝¹ : WithTop β
a✝ : a✝¹ ∈ lowerBounds s
⊢ a✝¹ ≤ ⊤

Proof state:
case left
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ sInf s ∈ lowerBounds s

Tactic:
show ite _ _ _ ∈ _

Result:
case left
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ (if s ⊆ {⊤} ∨ ¬BddBelow s then ⊤ else ↑(sInf ((fun a => ↑a) ⁻¹' s))) ∈ lowerBounds s

Proof state:
case neg.some.refine_2
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ lowerBounds s
b : β
hb : b ∈ (fun a => ↑a) ⁻¹' s
⊢ a ≤ b

Tactic:
rw [← coe_le_coe]

Result:
case neg.some.refine_2
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ lowerBounds s
b : β
hb : b ∈ (fun a => ↑a) ⁻¹' s
⊢ ↑a ≤ ↑b

Proof state:
case neg.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
⊢ BddBelow ((fun a => ↑a) ⁻¹' s)

Tactic:
rcases hs with ⟨⟨⟩ | b, hb⟩

Result:
case neg.some.intro.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
hb : none ∈ lowerBounds s
⊢ BddBelow ((fun a => ↑a) ⁻¹' s)

case neg.some.intro.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
b : β
hb : Option.some b ∈ lowerBounds s
⊢ BddBelow ((fun a => ↑a) ⁻¹' s)

Proof state:
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ IsGLB s (sInf s)

Tactic:
constructor

Result:
case left
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ sInf s ∈ lowerBounds s

case right
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : BddBelow s
⊢ sInf s ∈ upperBounds (lowerBounds s)