chore(Dynamics/PeriodicPts): don't import MonoidWithZero (#20765)

For this, split the file into `.Defs` and `.Lemmas`.

Author: YaelDillies

Archive/Imo/Imo2006Q5.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
 import Mathlib.Algebra.Polynomial.Roots
-import Mathlib.Dynamics.PeriodicPts
+import Mathlib.Dynamics.PeriodicPts.Lemmas
 
 /-!
 # IMO 2006 Q5

Mathlib.lean

@@ -3071,7 +3071,8 @@ import Mathlib.Dynamics.Flow
 import Mathlib.Dynamics.Minimal
 import Mathlib.Dynamics.Newton
 import Mathlib.Dynamics.OmegaLimit
-import Mathlib.Dynamics.PeriodicPts
+import Mathlib.Dynamics.PeriodicPts.Defs
+import Mathlib.Dynamics.PeriodicPts.Lemmas
 import Mathlib.Dynamics.TopologicalEntropy.CoverEntropy
 import Mathlib.Dynamics.TopologicalEntropy.DynamicalEntourage
 import Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Mathlib/Dynamics/PeriodicPts.lean renamed to Mathlib/Dynamics/PeriodicPts/Defs.lean

@@ -3,13 +3,11 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Algebra.GroupPower.IterateHom
-import Mathlib.Algebra.Ring.Divisibility.Basic
-import Mathlib.Algebra.Ring.Int.Defs
+import Batteries.Data.Nat.Gcd
+import Mathlib.Algebra.Order.Group.Nat
+import Mathlib.Algebra.Order.Sub.Basic
 import Mathlib.Data.List.Cycle
-import Mathlib.Data.Nat.GCD.Basic
-import Mathlib.Data.Nat.Prime.Basic
-import Mathlib.Data.PNat.Basic
+import Mathlib.Data.PNat.Notation
 import Mathlib.Dynamics.FixedPoints.Basic
 
 /-!
@@ -43,6 +41,8 @@ is a periodic point of `f` of period `n` if and only if `minimalPeriod f x | n`.
 
 -/
 
+assert_not_exists MonoidWithZero
+
 
 open Set
 
@@ -164,7 +164,7 @@ theorem eq_of_apply_eq_same (hx : IsPeriodicPt f n x) (hy : IsPeriodicPt f n y)
 then `x = y`. -/
 theorem eq_of_apply_eq (hx : IsPeriodicPt f m x) (hy : IsPeriodicPt f n y) (hm : 0 < m) (hn : 0 < n)
     (h : f x = f y) : x = y :=
-  (hx.mul_const n).eq_of_apply_eq_same (hy.const_mul m) (mul_pos hm hn) h
+  (hx.mul_const n).eq_of_apply_eq_same (hy.const_mul m) (Nat.mul_pos hm hn) h
 
 end IsPeriodicPt
 
@@ -187,9 +187,6 @@ theorem bijOn_ptsOfPeriod (f : α → α) {n : ℕ} (hn : 0 < n) :
     ⟨f^[n.pred] x, hx.apply_iterate _, by
       rw [← comp_apply (f := f), comp_iterate_pred_of_pos f hn, hx.eq]⟩⟩
 
-theorem directed_ptsOfPeriod_pNat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=
-  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩
-
 /-- The set of periodic points of a map `f : α → α`. -/
 def periodicPts (f : α → α) : Set α :=
   { x : α | ∃ n > 0, IsPeriodicPt f n x }
@@ -208,7 +205,7 @@ theorem isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate (hx : x ∈ peri
     change _ = _
     convert (hm.apply_iterate ((n / r + 1) * r - n)).eq <;>
       rw [← iterate_add_apply, Nat.sub_add_cancel this, iterate_mul, (hr'.iterate _).eq]
-  rw [add_mul, one_mul]
+  rw [Nat.add_mul, one_mul]
   exact (Nat.lt_div_mul_add hr).le
 
 variable (f)
@@ -219,10 +216,6 @@ theorem bUnion_ptsOfPeriod : ⋃ n > 0, ptsOfPeriod f n = periodicPts f :=
 theorem iUnion_pNat_ptsOfPeriod : ⋃ n : ℕ+, ptsOfPeriod f n = periodicPts f :=
   iSup_subtype.trans <| bUnion_ptsOfPeriod f
 
-theorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=
-  iUnion_pNat_ptsOfPeriod f ▸
-    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pNat f) fun i => bijOn_ptsOfPeriod f i.pos
-
 variable {f}
 
 theorem Semiconj.mapsTo_periodicPts {g : α → β} (h : Semiconj g fa fb) :
@@ -339,7 +332,7 @@ theorem not_isPeriodicPt_of_pos_of_lt_minimalPeriod :
   | _ + 1, _, hn => fun hp => Nat.succ_ne_zero _ (hp.eq_zero_of_lt_minimalPeriod hn)
 
 theorem IsPeriodicPt.minimalPeriod_dvd (hx : IsPeriodicPt f n x) : minimalPeriod f x ∣ n :=
-  (eq_or_lt_of_le <| n.zero_le).elim (fun hn0 => hn0 ▸ dvd_zero _) fun hn0 =>
+  (eq_or_lt_of_le <| n.zero_le).elim (fun hn0 => hn0 ▸ Nat.dvd_zero _) fun hn0 =>
     -- Porting note: `Nat.dvd_iff_mod_eq_zero` gained explicit arguments
     Nat.dvd_iff_mod_eq_zero.2 <|
       (hx.mod <| isPeriodicPt_minimalPeriod f x).eq_zero_of_lt_minimalPeriod <|
@@ -354,40 +347,11 @@ theorem minimalPeriod_eq_minimalPeriod_iff {g : β → β} {y : β} :
     minimalPeriod f x = minimalPeriod g y ↔ ∀ n, IsPeriodicPt f n x ↔ IsPeriodicPt g n y := by
   simp_rw [isPeriodicPt_iff_minimalPeriod_dvd, dvd_right_iff_eq]
 
-theorem minimalPeriod_eq_prime {p : ℕ} [hp : Fact p.Prime] (hper : IsPeriodicPt f p x)
-    (hfix : ¬IsFixedPt f x) : minimalPeriod f x = p :=
-  (hp.out.eq_one_or_self_of_dvd _ hper.minimalPeriod_dvd).resolve_left
-    (mt minimalPeriod_eq_one_iff_isFixedPt.1 hfix)
-
-theorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)
-    (hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by
-  apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>
-    rwa [← isPeriodicPt_iff_minimalPeriod_dvd]
-
 theorem Commute.minimalPeriod_of_comp_dvd_lcm {g : α → α} (h : Commute f g) :
     minimalPeriod (f ∘ g) x ∣ Nat.lcm (minimalPeriod f x) (minimalPeriod g x) := by
   rw [← isPeriodicPt_iff_minimalPeriod_dvd]
   exact (isPeriodicPt_minimalPeriod f x).comp_lcm h (isPeriodicPt_minimalPeriod g x)
 
-theorem Commute.minimalPeriod_of_comp_dvd_mul {g : α → α} (h : Commute f g) :
-    minimalPeriod (f ∘ g) x ∣ minimalPeriod f x * minimalPeriod g x :=
-  dvd_trans h.minimalPeriod_of_comp_dvd_lcm (lcm_dvd_mul _ _)
-
-theorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : α → α} (h : Commute f g)
-    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :
-    minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x := by
-  apply h.minimalPeriod_of_comp_dvd_mul.antisymm
-  suffices
-    ∀ {f g : α → α},
-      Commute f g →
-        Coprime (minimalPeriod f x) (minimalPeriod g x) →
-          minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from
-    hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)
-  intro f g h hco
-  refine hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h ?_ ?_).minimalPeriod_dvd
-  · exact (isPeriodicPt_minimalPeriod _ _).const_mul _
-  · exact (isPeriodicPt_minimalPeriod _ _).mul_const _
-
 private theorem minimalPeriod_iterate_eq_div_gcd_aux (h : 0 < gcd (minimalPeriod f x) n) :
     minimalPeriod f^[n] x = minimalPeriod f x / Nat.gcd (minimalPeriod f x) n := by
   apply Nat.dvd_antisymm
@@ -521,16 +485,6 @@ theorem isPeriodicPt_prod_map (x : α × β) :
     IsPeriodicPt (Prod.map f g) n x ↔ IsPeriodicPt f n x.1 ∧ IsPeriodicPt g n x.2 := by
   simp [IsPeriodicPt]
 
-theorem minimalPeriod_prod_map (f : α → α) (g : β → β) (x : α × β) :
-    minimalPeriod (Prod.map f g) x = (minimalPeriod f x.1).lcm (minimalPeriod g x.2) :=
-  eq_of_forall_dvd <| by cases x; simp [← isPeriodicPt_iff_minimalPeriod_dvd, Nat.lcm_dvd_iff]
-
-theorem minimalPeriod_fst_dvd : minimalPeriod f x.1 ∣ minimalPeriod (Prod.map f g) x := by
-  rw [minimalPeriod_prod_map]; exact Nat.dvd_lcm_left _ _
-
-theorem minimalPeriod_snd_dvd : minimalPeriod g x.2 ∣ minimalPeriod (Prod.map f g) x := by
-  rw [minimalPeriod_prod_map]; exact Nat.dvd_lcm_right _ _
-
 end Function
 
 namespace MulAction
@@ -584,7 +538,7 @@ theorem zpow_smul_eq_iff_period_dvd {j : ℤ} {g : G} {a : α} :
     g ^ j • a = a ↔ (period g a : ℤ) ∣ j := by
   rcases j with n | n
   · rw [Int.ofNat_eq_coe, zpow_natCast, Int.natCast_dvd_natCast, pow_smul_eq_iff_period_dvd]
-  · rw [Int.negSucc_coe, zpow_neg, zpow_natCast, inv_smul_eq_iff, eq_comm, dvd_neg,
+  · rw [Int.negSucc_coe, zpow_neg, zpow_natCast, inv_smul_eq_iff, eq_comm, Int.dvd_neg,
       Int.natCast_dvd_natCast, pow_smul_eq_iff_period_dvd]
 
 @[to_additive (attr := simp)]

Mathlib/Dynamics/PeriodicPts/Lemmas.lean

@@ -0,0 +1,75 @@
+/-
+Copyright (c) 2020 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Data.Nat.GCD.Basic
+import Mathlib.Data.Nat.Prime.Basic
+import Mathlib.Data.PNat.Basic
+import Mathlib.Dynamics.PeriodicPts.Defs
+
+/-!
+# Extra lemmas about periodic points
+-/
+
+open Nat Set
+
+namespace Function
+variable {α : Type*} {f : α → α} {x y : α}
+
+open Function (Commute)
+
+theorem directed_ptsOfPeriod_pNat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=
+  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩
+
+variable (f) in
+theorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=
+  iUnion_pNat_ptsOfPeriod f ▸
+    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pNat f) fun i => bijOn_ptsOfPeriod f i.pos
+
+theorem minimalPeriod_eq_prime {p : ℕ} [hp : Fact p.Prime] (hper : IsPeriodicPt f p x)
+    (hfix : ¬IsFixedPt f x) : minimalPeriod f x = p :=
+  (hp.out.eq_one_or_self_of_dvd _ hper.minimalPeriod_dvd).resolve_left
+    (mt minimalPeriod_eq_one_iff_isFixedPt.1 hfix)
+
+theorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)
+    (hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by
+  apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>
+    rwa [← isPeriodicPt_iff_minimalPeriod_dvd]
+
+theorem Commute.minimalPeriod_of_comp_dvd_mul {g : α → α} (h : Commute f g) :
+    minimalPeriod (f ∘ g) x ∣ minimalPeriod f x * minimalPeriod g x :=
+  dvd_trans h.minimalPeriod_of_comp_dvd_lcm (lcm_dvd_mul _ _)
+
+theorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : α → α} (h : Commute f g)
+    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :
+    minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x := by
+  apply h.minimalPeriod_of_comp_dvd_mul.antisymm
+  suffices
+    ∀ {f g : α → α},
+      Commute f g →
+        Coprime (minimalPeriod f x) (minimalPeriod g x) →
+          minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from
+    hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)
+  intro f g h hco
+  refine hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h ?_ ?_).minimalPeriod_dvd
+  · exact (isPeriodicPt_minimalPeriod _ _).const_mul _
+  · exact (isPeriodicPt_minimalPeriod _ _).mul_const _
+
+end Function
+
+namespace Function
+
+variable {α β : Type*} {f : α → α} {g : β → β} {x : α × β} {a : α} {b : β} {m n : ℕ}
+
+theorem minimalPeriod_prod_map (f : α → α) (g : β → β) (x : α × β) :
+    minimalPeriod (Prod.map f g) x = (minimalPeriod f x.1).lcm (minimalPeriod g x.2) :=
+  eq_of_forall_dvd <| by cases x; simp [← isPeriodicPt_iff_minimalPeriod_dvd, Nat.lcm_dvd_iff]
+
+theorem minimalPeriod_fst_dvd : minimalPeriod f x.1 ∣ minimalPeriod (Prod.map f g) x := by
+  rw [minimalPeriod_prod_map]; exact Nat.dvd_lcm_left _ _
+
+theorem minimalPeriod_snd_dvd : minimalPeriod g x.2 ∣ minimalPeriod (Prod.map f g) x := by
+  rw [minimalPeriod_prod_map]; exact Nat.dvd_lcm_right _ _
+
+end Function

Mathlib/GroupTheory/GroupAction/FixedPoints.lean

@@ -5,7 +5,7 @@ Authors: Emilie Burgun
 -/
 import Mathlib.Algebra.Group.Commute.Basic
 import Mathlib.Data.Set.Pointwise.SMul
-import Mathlib.Dynamics.PeriodicPts
+import Mathlib.Dynamics.PeriodicPts.Defs
 import Mathlib.GroupTheory.GroupAction.Defs
 
 /-!
@@ -83,7 +83,7 @@ theorem fixedBy_subset_fixedBy_zpow (g : G) (j : ℤ) :
     fixedBy α g ⊆ fixedBy α (g ^ j) := by
   intro a a_in_fixedBy
   rw [mem_fixedBy, zpow_smul_eq_iff_minimalPeriod_dvd,
-    minimalPeriod_eq_one_iff_fixedBy.mpr a_in_fixedBy, Nat.cast_one]
+    minimalPeriod_eq_one_iff_fixedBy.mpr a_in_fixedBy, Int.natCast_one]
   exact one_dvd j
 
 variable (M α) in

Mathlib/GroupTheory/GroupAction/Period.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Emilie Burgun
 -/
 
-import Mathlib.Dynamics.PeriodicPts
+import Mathlib.Dynamics.PeriodicPts.Lemmas
 import Mathlib.GroupTheory.Exponent
 import Mathlib.GroupTheory.GroupAction.Basic
 

Mathlib/GroupTheory/GroupAction/Quotient.lean

@@ -6,7 +6,7 @@ Authors: Chris Hughes, Thomas Browning
 import Mathlib.Algebra.Group.Subgroup.Actions
 import Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas
 import Mathlib.Data.Fintype.BigOperators
-import Mathlib.Dynamics.PeriodicPts
+import Mathlib.Dynamics.PeriodicPts.Defs
 import Mathlib.GroupTheory.Commutator.Basic
 import Mathlib.GroupTheory.Coset.Basic
 import Mathlib.GroupTheory.GroupAction.Basic

Mathlib/GroupTheory/OrderOfElement.lean

@@ -8,6 +8,7 @@ import Mathlib.Algebra.Group.Subgroup.Finite
 import Mathlib.Algebra.Module.NatInt
 import Mathlib.Algebra.Order.Group.Action
 import Mathlib.Algebra.Order.Ring.Abs
+import Mathlib.Dynamics.PeriodicPts.Lemmas
 import Mathlib.GroupTheory.Index
 import Mathlib.Order.Interval.Set.Infinite
 import Mathlib.Tactic.Positivity