Theorem: ContractingWith.efixedPoint_eq_of_edist_lt_top
File: Mathlib/Topology/MetricSpace/Contracting.lean

tactic: refine' (hf.eq_or_edist_eq_top_of_fixedPoints _ _).elim id fun h' ↦ False.elim (ne_of_lt _ h')
  <;> try apply efixedPoint_isFixedPt
state_before: α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
⊢ efixedPoint f hf x hx = efixedPoint f hf y hy

tactic: change edistLtTopSetoid.Rel _ _
state_before: case refine'_3
α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) < ⊤

tactic: trans x
state_before: case refine'_3
α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ Setoid.Rel edistLtTopSetoid (efixedPoint f hf x hx) (efixedPoint f hf y hy)

tactic: trans y
state_before: α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ Setoid.Rel edistLtTopSetoid x (efixedPoint f hf y hy)

tactic: exacts [lt_top_iff_ne_top.2 h, hf.edist_efixedPoint_lt_top hy]
state_before: α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ Setoid.Rel edistLtTopSetoid x y

α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ Setoid.Rel edistLtTopSetoid y (efixedPoint f hf y hy)

tactic: apply efixedPoint_isFixedPt
state_before: case refine'_3
α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) < ⊤

tactic: apply Setoid.symm'
state_before: α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ Setoid.Rel edistLtTopSetoid (efixedPoint f hf x hx) x

tactic: exact hf.edist_efixedPoint_lt_top hx
state_before: case a
α : Type u_1
inst✝ : EMetricSpace α
cs : CompleteSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x : α
hx : edist x (f x) ≠ ⊤
y : α
hy : edist y (f y) ≠ ⊤
h : edist x y ≠ ⊤
h' : edist (efixedPoint f hf x hx) (efixedPoint f hf y hy) = ⊤
⊢ Setoid.Rel edistLtTopSetoid x (efixedPoint f hf x hx)