-- Mathlib/Dynamics/PeriodicPts/Defs.lean
theorem periodicOrbit_chain (r : α → α → Prop) {f : α → α} {x : α} :
    (periodicOrbit f x).Chain r ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Tactic:
have hM := Nat.sub_add_cancel (succ_le_iff.2 hx')

Result:
case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Tactic:
have hx' := minimalPeriod_pos_of_mem_periodicPts hx

Result:
case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos.refine_1.intro
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
hr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)
H : ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)
n : ℕ
hn : n < minimalPeriod f x - succ 0 + succ 0
⊢ r (f^[n] x) (f^[n + 1] x)

Tactic:
rcases eq_or_lt_of_le (Nat.lt_succ_iff.1 hn) with hM' | hM'

Result:
case pos.refine_1.intro.inl
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
hr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)
H : ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)
n : ℕ
hn : n < minimalPeriod f x - succ 0 + succ 0
hM' : n = minimalPeriod f x - succ 0
⊢ r (f^[n] x) (f^[n + 1] x)

case pos.refine_1.intro.inr
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
hr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)
H : ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)
n : ℕ
hn : n < minimalPeriod f x - succ 0 + succ 0
hM' : n < minimalPeriod f x - succ 0
⊢ r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos.refine_2
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
H : ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)
⊢ r (f^[minimalPeriod f x - succ 0] x) (f^[minimalPeriod f x] x)

Tactic:
nth_rw 2 [← hM]

Result:
case pos.refine_2
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
H : ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)
⊢ r (f^[minimalPeriod f x - succ 0] x) (f^[minimalPeriod f x - succ 0 + succ 0] x)

Proof state:
case neg
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∉ periodicPts f
⊢ Cycle.Chain r Cycle.nil ↔ ∀ n < 0, r (f^[n] x) (f^[n + 1] x)

Tactic:
simp

Result:
no goals

Proof state:
α : Type u_1
r : α → α → Prop
f : α → α
x : α
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Tactic:
by_cases hx : x ∈ periodicPts f

Result:
case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

case neg
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∉ periodicPts f
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos.refine_1
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
⊢ (r (f^[minimalPeriod f x - succ 0] x) (f^[0] x) ∧ ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)) →
    ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)

Tactic:
rintro ⟨hr, H⟩ n hn

Result:
case pos.refine_1.intro
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
hr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)
H : ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)
n : ℕ
hn : n < minimalPeriod f x - succ 0 + succ 0
⊢ r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos.refine_2
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
H : ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)
⊢ r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)

Tactic:
rw [iterate_zero_apply]

Result:
case pos.refine_2
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
H : ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)
⊢ r (f^[minimalPeriod f x - succ 0] x) x

Proof state:
case neg
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∉ periodicPts f
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)

Tactic:
rw [periodicOrbit_eq_nil_of_not_periodic_pt hx, minimalPeriod_eq_zero_of_nmem_periodicPts hx]

Result:
case neg
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∉ periodicPts f
⊢ Cycle.Chain r Cycle.nil ↔ ∀ n < 0, r (f^[n] x) (f^[n + 1] x)

Proof state:
case pos.refine_2
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
H : ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)
⊢ r (f^[minimalPeriod f x - succ 0] x) (f^[minimalPeriod f x - succ 0 + succ 0] x)

Tactic:
exact H _ (Nat.lt_succ_self _)

Result:
no goals

Proof state:
case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
⊢ (r (f^[minimalPeriod f x - succ 0] x) (f^[0] x) ∧ ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)) ↔
    ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)

Tactic:
refine ⟨?_, fun H => ⟨?_, fun m hm => H _ (hm.trans (Nat.lt_succ_self _))⟩⟩

Result:
case pos.refine_1
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
⊢ (r (f^[minimalPeriod f x - succ 0] x) (f^[0] x) ∧ ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)) →
    ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)

case pos.refine_2
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
H : ∀ n < minimalPeriod f x - succ 0 + succ 0, r (f^[n] x) (f^[n + 1] x)
⊢ r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)

Proof state:
case pos.refine_1.intro.inr
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
hr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)
H : ∀ m < minimalPeriod f x - succ 0, r (f^[m] x) (f^[m.succ] x)
n : ℕ
hn : n < minimalPeriod f x - succ 0 + succ 0
hM' : n < minimalPeriod f x - succ 0
⊢ r (f^[n] x) (f^[n + 1] x)

Tactic:
exact H _ hM'

Result:
no goals