imo.imo2006_q5 ⟷ Archive.Imo.Imo2006Q5

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,7 +3,7 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
-import Data.Polynomial.RingDivision
+import Algebra.Polynomial.RingDivision
 import Dynamics.PeriodicPts
 
 #align_import imo.imo2006_q5 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -53,7 +53,7 @@ absolute value. -/
 theorem Int.natAbs_eq_of_chain_dvd {l : Cycle ℤ} {x y : ℤ} (hl : l.Chain (· ∣ ·)) (hx : x ∈ l)
     (hy : y ∈ l) : x.natAbs = y.natAbs :=
   by
-  rw [Cycle.chain_iff_pairwise] at hl 
+  rw [Cycle.chain_iff_pairwise] at hl
   exact Int.natAbs_eq_of_dvd_dvd (hl x hx y hy) (hl y hy x hx)
 #align imo2006_q5.int.nat_abs_eq_of_chain_dvd Imo2006Q5.Int.natAbs_eq_of_chain_dvd
 
@@ -115,19 +115,19 @@ theorem Polynomial.isPeriodicPt_eval_two {P : Polynomial ℤ} {t : ℤ}
     · exact (irrefl 0 hk).elim
     · have H := IH k
       rw [hk'.is_fixed_pt.eq, sub_self, Int.sign_zero, eq_comm, Int.sign_eq_zero_iff_zero,
-        sub_eq_zero] at H 
+        sub_eq_zero] at H
       simp [is_periodic_pt, is_fixed_pt, H]
   · -- We take two nonequal consecutive entries.
-    rw [Cycle.chain_map, periodic_orbit_chain' _ ht] at HC' 
-    push_neg at HC' 
+    rw [Cycle.chain_map, periodic_orbit_chain' _ ht] at HC'
+    push_neg at HC'
     cases' HC' with n hn
     -- They must have opposite sign, so that P^{k + 1}(t) - P^k(t) = P^{k + 2}(t) - P^{k + 1}(t).
     cases' Int.natAbs_eq_natAbs_iff.1 (Habs n n.succ) with hn' hn'
     · apply (hn _).elim
       convert hn' <;> simp only [Function.iterate_succ_apply']
     -- We deduce P^{k + 2}(t) = P^k(t) and hence P(P(t)) = t.
-    · rw [neg_sub, sub_right_inj] at hn' 
-      simp only [Function.iterate_succ_apply'] at hn' 
+    · rw [neg_sub, sub_right_inj] at hn'
+      simp only [Function.iterate_succ_apply'] at hn'
       exact @is_periodic_pt_of_mem_periodic_pts_of_is_periodic_pt_iterate _ _ t 2 n ht hn'.symm
 #align imo2006_q5.polynomial.is_periodic_pt_eval_two Imo2006Q5.Polynomial.isPeriodicPt_eval_two
 
@@ -147,8 +147,8 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     simpa using hP
   have hPX' : P - X ≠ 0 := by
     intro h
-    rw [h, nat_degree_zero] at hPX 
-    rw [← hPX] at hP 
+    rw [h, nat_degree_zero] at hPX
+    rw [← hPX] at hP
     exact (zero_le_one.not_lt hP).elim
   -- If every root of P(P(t)) - t is also a root of P(t) - t, then we're done.
   by_cases H : (P.comp P - X).roots.toFinset ⊆ (P - X).roots.toFinset
@@ -159,10 +159,10 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
   -- Otherwise, take a, b with P(a) = b, P(b) = a, a ≠ b.
   · rcases Finset.not_subset.1 H with ⟨a, ha, hab⟩
     replace ha := is_root_of_mem_roots (Multiset.mem_toFinset.1 ha)
-    simp [sub_eq_zero] at ha 
-    simp [mem_roots hPX'] at hab 
+    simp [sub_eq_zero] at ha
+    simp [mem_roots hPX'] at hab
     set b := P.eval a
-    rw [sub_eq_zero] at hab 
+    rw [sub_eq_zero] at hab
     -- More auxiliary lemmas on degrees.
     have hPab : (P + X - a - b).natDegree = P.nat_degree :=
       by
@@ -176,8 +176,8 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
       exact zero_lt_one.trans hP
     have hPab' : P + X - a - b ≠ 0 := by
       intro h
-      rw [h, nat_degree_zero] at hPab 
-      rw [← hPab] at hP 
+      rw [h, nat_degree_zero] at hPab
+      rw [← hPab] at hP
       exact (zero_le_one.not_lt hP).elim
     -- We claim that every root of P(P(t)) - t is a root of P(t) + t - a - b. This allows us to
     -- conclude the problem.
@@ -189,7 +189,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     · -- Let t be a root of P(P(t)) - t, define u = P(t).
       intro t ht
       replace ht := is_root_of_mem_roots (Multiset.mem_toFinset.1 ht)
-      simp [sub_eq_zero] at ht 
+      simp [sub_eq_zero] at ht
       simp only [mem_roots hPab', sub_eq_iff_eq_add, Multiset.mem_toFinset, is_root.def, eval_sub,
         eval_add, eval_X, eval_C, eval_int_cast, Int.cast_id, zero_add]
       -- An auxiliary lemma proved earlier implies we only need to show |t - a| = |u - b| and
@@ -215,7 +215,7 @@ theorem imo2006_q5 {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ} (hk : 0
   apply (Finset.card_le_card fun t ht => _).trans (imo2006_q5' hP)
   have hP' : P.comp P - X ≠ 0 := by simpa using polynomial.iterate_comp_sub_X_ne hP zero_lt_two
   replace ht := is_root_of_mem_roots (Multiset.mem_toFinset.1 ht)
-  simp only [sub_eq_zero, is_root.def, eval_sub, iterate_comp_eval, eval_X] at ht 
+  simp only [sub_eq_zero, is_root.def, eval_sub, iterate_comp_eval, eval_X] at ht
   simpa [mem_roots hP', sub_eq_zero] using polynomial.is_periodic_pt_eval_two ⟨k, hk, ht⟩
 #align imo2006_q5 imo2006_q5
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -154,7 +154,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
   by_cases H : (P.comp P - X).roots.toFinset ⊆ (P - X).roots.toFinset
   ·
     exact
-      (Finset.card_le_of_subset H).trans
+      (Finset.card_le_card H).trans
         ((Multiset.toFinset_card_le _).trans ((card_roots' _).trans_eq hPX))
   -- Otherwise, take a, b with P(a) = b, P(b) = a, a ≠ b.
   · rcases Finset.not_subset.1 H with ⟨a, ha, hab⟩
@@ -184,7 +184,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     suffices H' : (P.comp P - X).roots.toFinset ⊆ (P + X - a - b).roots.toFinset
     ·
       exact
-        (Finset.card_le_of_subset H').trans
+        (Finset.card_le_card H').trans
           ((Multiset.toFinset_card_le _).trans <| (card_roots' _).trans_eq hPab)
     · -- Let t be a root of P(P(t)) - t, define u = P(t).
       intro t ht
@@ -212,7 +212,7 @@ open imo2006_q5
 theorem imo2006_q5 {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ} (hk : 0 < k) :
     ((P.comp^[k]) X - X).roots.toFinset.card ≤ P.natDegree :=
   by
-  apply (Finset.card_le_of_subset fun t ht => _).trans (imo2006_q5' hP)
+  apply (Finset.card_le_card fun t ht => _).trans (imo2006_q5' hP)
   have hP' : P.comp P - X ≠ 0 := by simpa using polynomial.iterate_comp_sub_X_ne hP zero_lt_two
   replace ht := is_root_of_mem_roots (Multiset.mem_toFinset.1 ht)
   simp only [sub_eq_zero, is_root.def, eval_sub, iterate_comp_eval, eval_X] at ht 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,8 +3,8 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
-import Mathbin.Data.Polynomial.RingDivision
-import Mathbin.Dynamics.PeriodicPts
+import Data.Polynomial.RingDivision
+import Dynamics.PeriodicPts
 
 #align_import imo.imo2006_q5 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
-
-! This file was ported from Lean 3 source module imo.imo2006_q5
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Polynomial.RingDivision
 import Mathbin.Dynamics.PeriodicPts
 
+#align_import imo.imo2006_q5 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # IMO 2006 Q5
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 
 ! This file was ported from Lean 3 source module imo.imo2006_q5
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
+! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Dynamics.PeriodicPts
 /-!
 # IMO 2006 Q5
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Let $P(x)$ be a polynomial of degree $n>1$ with integer coefficients, and let $k$ be a positive
 integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x))\ldots))$, where $P$ occurs $k$ times.
 Prove that there are at most $n$ integers $t$ such that $Q(t)=t$.

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -161,7 +161,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
         rw [natDegree_add_eq_left_of_natDegree_lt]
         simpa using hP
       rwa [natDegree_sub_eq_left_of_natDegree_lt]
-      rw [h₁, natDegree_int_cast]
+      rw [h₁, natDegree_intCast]
       exact zero_lt_one.trans hP
     have hPab' : P + (X : ℤ[X]) - a - b ≠ 0 := by
       intro h
@@ -178,7 +178,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
       replace ht := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ht)
       rw [IsRoot.def, eval_sub, eval_comp, eval_X, sub_eq_zero] at ht
       simp only [mem_roots hPab', sub_eq_iff_eq_add, Multiset.mem_toFinset, IsRoot.def,
-        eval_sub, eval_add, eval_X, eval_C, eval_int_cast, Int.cast_id, zero_add]
+        eval_sub, eval_add, eval_X, eval_C, eval_intCast, Int.cast_id, zero_add]
       -- An auxiliary lemma proved earlier implies we only need to show |t - a| = |u - b| and
       -- |t - b| = |u - a|. We prove this by establishing that each side of either equation divides
       -- the other.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -150,8 +150,8 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
   -- Otherwise, take a, b with P(a) = b, P(b) = a, a ≠ b.
   · rcases Finset.not_subset.1 H with ⟨a, ha, hab⟩
     replace ha := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ha)
-    rw [IsRoot.definition, eval_sub, eval_comp, eval_X, sub_eq_zero] at ha
-    rw [Multiset.mem_toFinset, mem_roots hPX', IsRoot.definition, eval_sub, eval_X, sub_eq_zero]
+    rw [IsRoot.def, eval_sub, eval_comp, eval_X, sub_eq_zero] at ha
+    rw [Multiset.mem_toFinset, mem_roots hPX', IsRoot.def, eval_sub, eval_X, sub_eq_zero]
       at hab
     set b := P.eval a
     -- More auxiliary lemmas on degrees.
@@ -176,8 +176,8 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     · -- Let t be a root of P(P(t)) - t, define u = P(t).
       intro t ht
       replace ht := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ht)
-      rw [IsRoot.definition, eval_sub, eval_comp, eval_X, sub_eq_zero] at ht
-      simp only [mem_roots hPab', sub_eq_iff_eq_add, Multiset.mem_toFinset, IsRoot.definition,
+      rw [IsRoot.def, eval_sub, eval_comp, eval_X, sub_eq_zero] at ht
+      simp only [mem_roots hPab', sub_eq_iff_eq_add, Multiset.mem_toFinset, IsRoot.def,
         eval_sub, eval_add, eval_X, eval_C, eval_int_cast, Int.cast_id, zero_add]
       -- An auxiliary lemma proved earlier implies we only need to show |t - a| = |u - b| and
       -- |t - b| = |u - a|. We prove this by establishing that each side of either equation divides
@@ -201,8 +201,8 @@ theorem imo2006_q5 {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ} (hk : 0
   have hP' : P.comp P - X ≠ 0 := by
     simpa [Nat.iterate] using Polynomial.iterate_comp_sub_X_ne hP zero_lt_two
   replace ht := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ht)
-  rw [IsRoot.definition, eval_sub, iterate_comp_eval, eval_X, sub_eq_zero] at ht
-  rw [Multiset.mem_toFinset, mem_roots hP', IsRoot.definition, eval_sub, eval_comp, eval_X,
+  rw [IsRoot.def, eval_sub, iterate_comp_eval, eval_X, sub_eq_zero] at ht
+  rw [Multiset.mem_toFinset, mem_roots hP', IsRoot.def, eval_sub, eval_comp, eval_X,
     sub_eq_zero]
   exact Polynomial.isPeriodicPt_eval_two ⟨k, hk, ht⟩
 #align imo2006_q5 imo2006_q5

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

@@ -3,7 +3,7 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
-import Mathlib.Data.Polynomial.RingDivision
+import Mathlib.Algebra.Polynomial.RingDivision
 import Mathlib.Dynamics.PeriodicPts
 
 #align_import imo.imo2006_q5 from "leanprover-community/mathlib"@"308826471968962c6b59c7ff82a22757386603e3"

This will become an error in 2024-03-16 nightly, possibly not permanently.

Co-authored-by: Scott Morrison <scott@tqft.net>

@@ -150,8 +150,9 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
   -- Otherwise, take a, b with P(a) = b, P(b) = a, a ≠ b.
   · rcases Finset.not_subset.1 H with ⟨a, ha, hab⟩
     replace ha := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ha)
-    rw [IsRoot.def, eval_sub, eval_comp, eval_X, sub_eq_zero] at ha
-    rw [Multiset.mem_toFinset, mem_roots hPX', IsRoot.def, eval_sub, eval_X, sub_eq_zero] at hab
+    rw [IsRoot.definition, eval_sub, eval_comp, eval_X, sub_eq_zero] at ha
+    rw [Multiset.mem_toFinset, mem_roots hPX', IsRoot.definition, eval_sub, eval_X, sub_eq_zero]
+      at hab
     set b := P.eval a
     -- More auxiliary lemmas on degrees.
     have hPab : (P + (X : ℤ[X]) - a - b).natDegree = P.natDegree := by
@@ -175,9 +176,9 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     · -- Let t be a root of P(P(t)) - t, define u = P(t).
       intro t ht
       replace ht := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ht)
-      rw [IsRoot.def, eval_sub, eval_comp, eval_X, sub_eq_zero] at ht
-      simp only [mem_roots hPab', sub_eq_iff_eq_add, Multiset.mem_toFinset, IsRoot.def, eval_sub,
-        eval_add, eval_X, eval_C, eval_int_cast, Int.cast_id, zero_add]
+      rw [IsRoot.definition, eval_sub, eval_comp, eval_X, sub_eq_zero] at ht
+      simp only [mem_roots hPab', sub_eq_iff_eq_add, Multiset.mem_toFinset, IsRoot.definition,
+        eval_sub, eval_add, eval_X, eval_C, eval_int_cast, Int.cast_id, zero_add]
       -- An auxiliary lemma proved earlier implies we only need to show |t - a| = |u - b| and
       -- |t - b| = |u - a|. We prove this by establishing that each side of either equation divides
       -- the other.
@@ -200,7 +201,8 @@ theorem imo2006_q5 {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ} (hk : 0
   have hP' : P.comp P - X ≠ 0 := by
     simpa [Nat.iterate] using Polynomial.iterate_comp_sub_X_ne hP zero_lt_two
   replace ht := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ht)
-  rw [IsRoot.def, eval_sub, iterate_comp_eval, eval_X, sub_eq_zero] at ht
-  rw [Multiset.mem_toFinset, mem_roots hP', IsRoot.def, eval_sub, eval_comp, eval_X, sub_eq_zero]
+  rw [IsRoot.definition, eval_sub, iterate_comp_eval, eval_X, sub_eq_zero] at ht
+  rw [Multiset.mem_toFinset, mem_roots hP', IsRoot.definition, eval_sub, eval_comp, eval_X,
+    sub_eq_zero]
   exact Polynomial.isPeriodicPt_eval_two ⟨k, hk, ht⟩
 #align imo2006_q5 imo2006_q5

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -169,8 +169,8 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
       exact (zero_le_one.not_lt hP).elim
     -- We claim that every root of P(P(t)) - t is a root of P(t) + t - a - b. This allows us to
     -- conclude the problem.
-    suffices H' : (P.comp P - X).roots.toFinset ⊆ (P + (X : ℤ[X]) - a - b).roots.toFinset
-    · exact (Finset.card_le_card H').trans
+    suffices H' : (P.comp P - X).roots.toFinset ⊆ (P + (X : ℤ[X]) - a - b).roots.toFinset from
+      (Finset.card_le_card H').trans
         ((Multiset.toFinset_card_le _).trans <| (card_roots' _).trans_eq hPab)
     · -- Let t be a root of P(P(t)) - t, define u = P(t).
       intro t ht

Change a few lemma names that have historically bothered me.

• Finset.card_le_of_subset → Finset.card_le_card
• Multiset.card_le_of_le → Multiset.card_le_card
• Multiset.card_lt_of_lt → Multiset.card_lt_card
• Set.ncard_le_of_subset → Set.ncard_le_ncard
• Finset.image_filter → Finset.filter_image
• CompleteLattice.finset_sup_compact_of_compact → CompleteLattice.isCompactElement_finset_sup

@@ -145,7 +145,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     exact (zero_le_one.not_lt hP).elim
   -- If every root of P(P(t)) - t is also a root of P(t) - t, then we're done.
   by_cases H : (P.comp P - X).roots.toFinset ⊆ (P - X).roots.toFinset
-  · exact (Finset.card_le_of_subset H).trans
+  · exact (Finset.card_le_card H).trans
       ((Multiset.toFinset_card_le _).trans ((card_roots' _).trans_eq hPX))
   -- Otherwise, take a, b with P(a) = b, P(b) = a, a ≠ b.
   · rcases Finset.not_subset.1 H with ⟨a, ha, hab⟩
@@ -170,7 +170,7 @@ theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
     -- We claim that every root of P(P(t)) - t is a root of P(t) + t - a - b. This allows us to
     -- conclude the problem.
     suffices H' : (P.comp P - X).roots.toFinset ⊆ (P + (X : ℤ[X]) - a - b).roots.toFinset
-    · exact (Finset.card_le_of_subset H').trans
+    · exact (Finset.card_le_card H').trans
         ((Multiset.toFinset_card_le _).trans <| (card_roots' _).trans_eq hPab)
     · -- Let t be a root of P(P(t)) - t, define u = P(t).
       intro t ht
@@ -196,7 +196,7 @@ open Imo2006Q5
 /-- The general problem follows easily from the k = 2 case. -/
 theorem imo2006_q5 {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ} (hk : 0 < k) :
     (P.comp^[k] X - X).roots.toFinset.card ≤ P.natDegree := by
-  refine' (Finset.card_le_of_subset fun t ht => _).trans (imo2006_q5' hP)
+  refine' (Finset.card_le_card fun t ht => _).trans (imo2006_q5' hP)
   have hP' : P.comp P - X ≠ 0 := by
     simpa [Nat.iterate] using Polynomial.iterate_comp_sub_X_ne hP zero_lt_two
   replace ht := isRoot_of_mem_roots (Multiset.mem_toFinset.1 ht)

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
-
-! This file was ported from Lean 3 source module imo.imo2006_q5
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Polynomial.RingDivision
 import Mathlib.Dynamics.PeriodicPts
 
+#align_import imo.imo2006_q5 from "leanprover-community/mathlib"@"308826471968962c6b59c7ff82a22757386603e3"
+
 /-!
 # IMO 2006 Q5
 

@@ -71,7 +71,7 @@ theorem Polynomial.isPeriodicPt_eval_two {P : Polynomial ℤ} {t : ℤ}
     (ht : t ∈ periodicPts fun x => P.eval x) : IsPeriodicPt (fun x => P.eval x) 2 t := by
   -- The cycle [P(t) - t, P(P(t)) - P(t), ...]
   let C : Cycle ℤ := (periodicOrbit (fun x => P.eval x) t).map fun x => P.eval x - x
-  have HC : ∀ {n : ℕ}, ((fun x => P.eval x)^[n + 1]) t - ((fun x => P.eval x)^[n]) t ∈ C := by
+  have HC : ∀ {n : ℕ}, (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t ∈ C := by
     intro n
     rw [Cycle.mem_map, Function.iterate_succ_apply']
     exact ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩
@@ -79,24 +79,24 @@ theorem Polynomial.isPeriodicPt_eval_two {P : Polynomial ℤ} {t : ℤ}
   have Hdvd : C.Chain (· ∣ ·) := by
     rw [Cycle.chain_map, periodicOrbit_chain' _ ht]
     intro n
-    convert sub_dvd_eval_sub (((fun x => P.eval x)^[n + 1]) t) (((fun x => P.eval x)^[n]) t) P <;>
+    convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P <;>
       rw [Function.iterate_succ_apply']
   -- Any two entries in C have the same absolute value.
   have Habs :
     ∀ m n : ℕ,
-      (((fun x => P.eval x)^[m + 1]) t - ((fun x => P.eval x)^[m]) t).natAbs =
-        (((fun x => P.eval x)^[n + 1]) t - ((fun x => P.eval x)^[n]) t).natAbs :=
+      ((fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t).natAbs =
+        ((fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t).natAbs :=
     fun m n => Int.natAbs_eq_of_chain_dvd Hdvd HC HC
   -- We case on whether the elements on C are pairwise equal.
   by_cases HC' : C.Chain (· = ·)
   · -- Any two entries in C are equal.
     have Heq :
       ∀ m n : ℕ,
-        ((fun x => P.eval x)^[m + 1]) t - ((fun x => P.eval x)^[m]) t =
-          ((fun x => P.eval x)^[n + 1]) t - ((fun x => P.eval x)^[n]) t :=
+        (fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t =
+          (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t :=
       fun m n => Cycle.chain_iff_pairwise.1 HC' _ HC _ HC
     -- The sign of P^n(t) - t is the same as P(t) - t for positive n. Proven by induction on n.
-    have IH : ∀ n : ℕ, (((fun x => P.eval x)^[n + 1]) t - t).sign = (P.eval t - t).sign := by
+    have IH : ∀ n : ℕ, ((fun x => P.eval x)^[n + 1] t - t).sign = (P.eval t - t).sign := by
       intro n
       induction' n with n IH
       · rfl
@@ -127,7 +127,7 @@ theorem Polynomial.isPeriodicPt_eval_two {P : Polynomial ℤ} {t : ℤ}
 #align imo2006_q5.polynomial.is_periodic_pt_eval_two Imo2006Q5.Polynomial.isPeriodicPt_eval_two
 
 theorem Polynomial.iterate_comp_sub_X_ne {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ}
-    (hk : 0 < k) : (P.comp^[k]) X - X ≠ 0 := by
+    (hk : 0 < k) : P.comp^[k] X - X ≠ 0 := by
   rw [sub_ne_zero]
   apply_fun natDegree
   simpa using (one_lt_pow hP hk.ne').ne'
@@ -198,7 +198,7 @@ open Imo2006Q5
 
 /-- The general problem follows easily from the k = 2 case. -/
 theorem imo2006_q5 {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ} (hk : 0 < k) :
-    ((P.comp^[k]) X - X).roots.toFinset.card ≤ P.natDegree := by
+    (P.comp^[k] X - X).roots.toFinset.card ≤ P.natDegree := by
   refine' (Finset.card_le_of_subset fun t ht => _).trans (imo2006_q5' hP)
   have hP' : P.comp P - X ≠ 0 := by
     simpa [Nat.iterate] using Polynomial.iterate_comp_sub_X_ne hP zero_lt_two