Theorem: CompleteLattice.wellFounded_characterisations
File: Mathlib/Order/CompactlyGenerated/Basic.lean

tactic: tfae_have 1 → 2
state_before: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE
state_after: case tfae_1_to_2
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
⊢ (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α

ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE

tactic: tfae_have 2 → 3
state_before: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE
state_after: case tfae_2_to_3
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
⊢ IsSupFiniteCompact α → IsSupClosedCompact α

ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE

tactic: tfae_have 3 → 1
state_before: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE
state_after: case tfae_3_to_1
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
⊢ IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1

ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
tfae_3_to_1 : IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE

tactic: tfae_have 2 ↔ 4
state_before: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
tfae_3_to_1 : IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE
state_after: case tfae_2_iff_4
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
tfae_3_to_1 : IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
⊢ IsSupFiniteCompact α ↔ ∀ (k : α), IsCompactElement k

ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
tfae_3_to_1 : IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
tfae_2_iff_4 : IsSupFiniteCompact α ↔ ∀ (k : α), IsCompactElement k
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE

tactic: tfae_finish
state_before: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
tfae_3_to_1 : IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
tfae_2_iff_4 : IsSupFiniteCompact α ↔ ∀ (k : α), IsCompactElement k
⊢ [WellFounded fun x x_1 => x > x_1, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ (k : α), IsCompactElement k].TFAE
state_after: no goals

tactic: exact WellFounded.isSupFiniteCompact α
state_before: case tfae_1_to_2
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
⊢ (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
state_after: no goals

tactic: exact IsSupFiniteCompact.isSupClosedCompact α
state_before: case tfae_2_to_3
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
⊢ IsSupFiniteCompact α → IsSupClosedCompact α
state_after: no goals

tactic: exact IsSupClosedCompact.wellFounded α
state_before: case tfae_3_to_1
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
⊢ IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
state_after: no goals

tactic: exact isSupFiniteCompact_iff_all_elements_compact α
state_before: case tfae_2_iff_4
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι → α
tfae_1_to_2 : (WellFounded fun x x_1 => x > x_1) → IsSupFiniteCompact α
tfae_2_to_3 : IsSupFiniteCompact α → IsSupClosedCompact α
tfae_3_to_1 : IsSupClosedCompact α → WellFounded fun x x_1 => x > x_1
⊢ IsSupFiniteCompact α ↔ ∀ (k : α), IsCompactElement k
state_after: no goals