ring_theory.polynomial.eisenstein.is_integral ⟷ Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

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

@@ -101,7 +101,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       rw [zero_add, pow_one] at hi ⊢
       exact (cyclotomic_comp_X_add_one_isEisensteinAt p).Mem hi
     · intro i hi
-      rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd, ←
+      rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.intCast_zmod_eq_zero_iff_dvd, ←
         Int.coe_castRingHom, ← coeff_map, map_comp, map_cyclotomic, Polynomial.map_add, map_X,
         Polynomial.map_one, pow_add, pow_one, cyclotomic_mul_prime_dvd_eq_pow, pow_comp, ←
         ZMod.expand_card, coeff_expand hp.out.pos]
@@ -113,7 +113,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rw [hk, pow_succ', mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
-        rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
+        rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.intCast_zmod_eq_zero_iff_dvd,
           ← Int.coe_castRingHom, ← coeff_map] at hn
         simpa [map_comp] using hn
       · exact ⟨p ^ n, by rw [pow_succ']⟩

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

@@ -68,13 +68,13 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
       nat_degree_cyclotomic, Nat.totient_prime hp.out] at hi
     simp only [hi.trans_le (Nat.sub_le _ _), sum_ite_eq', mem_range, if_true,
-      Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.coe_nat_dvd]
+      Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.natCast_dvd_natCast]
     exact hp.out.dvd_choose_self i.succ_ne_zero (lt_tsub_iff_right.1 hi)
   · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime, eval_add, eval_X, eval_one, zero_add,
       eval_geom_sum, one_geom_sum, Ideal.submodule_span_eq, Ideal.span_singleton_pow,
       Ideal.mem_span_singleton]
     intro h
-    obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
+    obtain ⟨k, hk⟩ := Int.natCast_dvd_natCast.1 h
     rw [← mul_assoc, mul_one, mul_assoc] at hk
     nth_rw 1 [← Nat.mul_one p] at hk
     rw [mul_right_inj' hp.out.ne_zero] at hk
@@ -110,20 +110,20 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
           nat_degree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _),
           Nat.add_one_sub_one] at hn hi
-        rw [hk, pow_succ, mul_assoc] at hi
+        rw [hk, pow_succ', mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
         rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
           ← Int.coe_castRingHom, ← coeff_map] at hn
         simpa [map_comp] using hn
-      · exact ⟨p ^ n, by rw [pow_succ]⟩
+      · exact ⟨p ^ n, by rw [pow_succ']⟩
   · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime_pow_eq_geom_sum hp.out, eval_add,
       eval_X, eval_one, zero_add, eval_finset_sum]
     simp only [eval_pow, eval_X, one_pow, sum_const, card_range, Nat.smul_one_eq_coe,
       submodule_span_eq, Ideal.submodule_span_eq, Ideal.span_singleton_pow,
       Ideal.mem_span_singleton]
     intro h
-    obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
+    obtain ⟨k, hk⟩ := Int.natCast_dvd_natCast.1 h
     rw [← mul_assoc, mul_one, mul_assoc] at hk
     nth_rw 1 [← Nat.mul_one p] at hk
     rw [mul_right_inj' hp.out.ne_zero] at hk
@@ -424,7 +424,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBa
     z ∈ adjoin R ({B.gen} : Set L) := by
   induction' n with n hn
   · simpa using hz
-  · rw [pow_succ, mul_smul] at hz
+  · rw [pow_succ', mul_smul] at hz
     exact
       hn
         (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (IsIntegral.smul _ hzint)

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

@@ -56,7 +56,7 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
       (fun i hi => _) _
   · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
     refine' (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => _
-    rw [nat_degree_X_add_C] at h 
+    rw [nat_degree_X_add_C] at h
     exact zero_ne_one h.symm
   · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, map_sum]
     conv =>
@@ -66,7 +66,7 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
       ext
       rw [lcoeff_apply, ← C_eq_nat_cast, C_mul_X_pow_eq_monomial, coeff_monomial]
     rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
-      nat_degree_cyclotomic, Nat.totient_prime hp.out] at hi 
+      nat_degree_cyclotomic, Nat.totient_prime hp.out] at hi
     simp only [hi.trans_le (Nat.sub_le _ _), sum_ite_eq', mem_range, if_true,
       Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.coe_nat_dvd]
     exact hp.out.dvd_choose_self i.succ_ne_zero (lt_tsub_iff_right.1 hi)
@@ -75,9 +75,9 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
       Ideal.mem_span_singleton]
     intro h
     obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
-    rw [← mul_assoc, mul_one, mul_assoc] at hk 
-    nth_rw 1 [← Nat.mul_one p] at hk 
-    rw [mul_right_inj' hp.out.ne_zero] at hk 
+    rw [← mul_assoc, mul_one, mul_assoc] at hk
+    nth_rw 1 [← Nat.mul_one p] at hk
+    rw [mul_right_inj' hp.out.ne_zero] at hk
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 #align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_X_add_one_isEisensteinAt
 -/
@@ -94,7 +94,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       _ _
   · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
     refine' (cyclotomic.monic _ ℤ).comp (monic_X_add_C 1) fun h => _
-    rw [nat_degree_X_add_C] at h 
+    rw [nat_degree_X_add_C] at h
     exact zero_ne_one h.symm
   · induction' n with n hn
     · intro i hi
@@ -109,12 +109,12 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rintro ⟨k, hk⟩
         rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
           nat_degree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _),
-          Nat.add_one_sub_one] at hn hi 
-        rw [hk, pow_succ, mul_assoc] at hi 
+          Nat.add_one_sub_one] at hn hi
+        rw [hk, pow_succ, mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
         rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
-          ← Int.coe_castRingHom, ← coeff_map] at hn 
+          ← Int.coe_castRingHom, ← coeff_map] at hn
         simpa [map_comp] using hn
       · exact ⟨p ^ n, by rw [pow_succ]⟩
   · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime_pow_eq_geom_sum hp.out, eval_add,
@@ -124,9 +124,9 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       Ideal.mem_span_singleton]
     intro h
     obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
-    rw [← mul_assoc, mul_one, mul_assoc] at hk 
-    nth_rw 1 [← Nat.mul_one p] at hk 
-    rw [mul_right_inj' hp.out.ne_zero] at hk 
+    rw [← mul_assoc, mul_one, mul_assoc] at hk
+    nth_rw 1 [← Nat.mul_one p] at hk
+    rw [mul_right_inj' hp.out.ne_zero] at hk
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 #align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt
 -/
@@ -166,7 +166,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
   choose! f hf using
     hei.is_weakly_eisenstein_at.exists_mem_adjoin_mul_eq_pow_nat_degree_le (minpoly.aeval R B.gen)
       (minpoly.monic hBint)
-  simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf 
+  simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf
   -- The Eisenstein condition shows that `p` divides `Q.coeff 0`
   -- if `p^n.succ` divides the following multiple of `Q.coeff 0^n.succ`:
   suffices p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
@@ -181,7 +181,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
   have aux : ∀ i ∈ (range (Q.nat_degree + 1)).eraseₓ 0, B.dim ≤ i + n :=
     by
     intro i hi
-    simp only [mem_range, mem_erase] at hi 
+    simp only [mem_range, mem_erase] at hi
     rw [hn]
     exact le_add_pred_of_pos _ hi.1
   have hintsum :
@@ -245,13 +245,13 @@ theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [Co
     (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
     z ∈ adjoin A ({x} : Set B) := by
   choose! f hf using hQ
-  rw [aeval_eq_sum_range, sum_range] at hz 
+  rw [aeval_eq_sum_range, sum_range] at hz
   conv_lhs at hz =>
     congr
     skip
     ext
     rw [hf i (mem_range.2 (Fin.is_lt i)), ← smul_smul]
-  rw [← smul_sum] at hz 
+  rw [← smul_sum] at hz
   rw [← smul_right_injective _ hp hz]
   exact
     Subalgebra.sum_mem _ fun _ _ =>
@@ -279,14 +279,14 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
   let P₁ := P.map (algebraMap R L)
   -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
   -- `Q.degree < P.degree` and `Q ≠ 0`.
-  rw [adjoin_singleton_eq_range_aeval] at hz 
+  rw [adjoin_singleton_eq_range_aeval] at hz
   obtain ⟨Q₁, hQ⟩ := hz
   set Q := Q₁ %ₘ P with hQ₁
   replace hQ : aeval B.gen Q = p • z
-  · rw [← mod_by_monic_add_div Q₁ (minpoly.monic hBint)] at hQ 
+  · rw [← mod_by_monic_add_div Q₁ (minpoly.monic hBint)] at hQ
     simpa using hQ
   by_cases hQzero : Q = 0
-  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ 
+  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
     cases' hQ with H H₁
     · have : Function.Injective (algebraMap R L) :=
         by
@@ -307,14 +307,14 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
     exact n
     -- Two technical results we will need about `P.nat_degree` and `Q.nat_degree`.
     have H := degree_mod_by_monic_lt Q₁ (minpoly.monic hBint)
-    rw [← hQ₁, ← hP] at H 
+    rw [← hQ₁, ← hP] at H
     replace H :=
       Nat.lt_iff_add_one_le.1
         (lt_of_lt_of_le
           (lt_of_le_of_lt (Nat.lt_iff_add_one_le.1 (Nat.lt_of_succ_lt_succ (mem_range.1 hj)))
             (lt_succ_self _))
           (Nat.lt_iff_add_one_le.1 ((nat_degree_lt_nat_degree_iff hQzero).2 H)))
-    rw [add_assoc] at H 
+    rw [add_assoc] at H
     have Hj : Q.nat_degree + 1 = j + 1 + (Q.nat_degree - j) := by
       rw [← add_comm 1, ← add_comm 1, add_assoc, add_right_inj, ←
         Nat.add_sub_assoc (Nat.lt_of_succ_lt_succ (mem_range.1 hj)).le, add_comm,
@@ -347,7 +347,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
       rw [(minpoly.monic hBint).natDegree_map, add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
       · exact Nat.zero_le _
       · refine' one_le_iff_ne_zero.2 fun h => _
-        rw [h] at hk 
+        rw [h] at hk
         simpa using hk
       · infer_instance
     -- The Eisenstein condition shows that `p` divides `Q.coeff j`
@@ -366,15 +366,15 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
     -- Using `hQ : aeval B.gen Q = p • z`, we write `p • z` as a sum of terms of degree less than
     -- `j+1`, that are multiples of `p` by induction, and terms of degree at least `j+1`.
     rw [aeval_eq_sum_range, Hj, range_add, sum_union (disjoint_range_add_left_embedding _ _),
-      sum_congr rfl hg, add_comm] at hQ 
+      sum_congr rfl hg, add_comm] at hQ
     -- We multiply this equality by `B.gen ^ (P.nat_degree-(j+2))`, so we can use `hf₁` on the terms
     -- we didn't know were multiples of `p`, and we take the norm on both sides.
     replace hQ := congr_arg (fun x => x * B.gen ^ (P.nat_degree - (j + 2))) hQ
-    simp_rw [sum_map, addLeftEmbedding_apply, add_mul, sum_mul, mul_assoc] at hQ 
+    simp_rw [sum_map, addLeftEmbedding_apply, add_mul, sum_mul, mul_assoc] at hQ
     rw [←
       insert_erase (mem_range.2 (tsub_pos_iff_lt.2 <| Nat.lt_of_succ_lt_succ <| mem_range.1 hj)),
       sum_insert (not_mem_erase 0 _), add_zero, sum_congr rfl hf₁, ← mul_sum, ← mul_sum, add_assoc,
-      ← mul_add, smul_mul_assoc, ← pow_add, Algebra.smul_def] at hQ 
+      ← mul_add, smul_mul_assoc, ← pow_add, Algebra.smul_def] at hQ
     replace hQ := congr_arg (norm K) (eq_sub_of_add_eq hQ)
     -- We obtain an equality of elements of `K`, but everything is integral, so we can move to `R`
     -- and simplify `hQ`.
@@ -395,7 +395,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
       rw [add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
       · exact zero_le _
       · refine' one_le_iff_ne_zero.2 fun h => _
-        rw [h] at hk 
+        rw [h] at hk
         simpa using hk
     obtain ⟨r, hr⟩ := is_integral_iff.1 (is_integral_norm K hintsum)
     rw [Algebra.smul_def, mul_assoc, ← mul_sub, _root_.map_mul, algebraMap_apply R K L, map_pow,
@@ -403,12 +403,12 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
       finrank B, ← hr, power_basis.norm_gen_eq_coeff_zero_minpoly,
       minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map,
       show (-1 : K) = algebraMap R K (-1) by simp, ← map_pow, ← map_pow, ← _root_.map_mul, ←
-      map_pow, ← _root_.map_mul, ← map_pow, ← _root_.map_mul] at hQ 
+      map_pow, ← _root_.map_mul, ← map_pow, ← _root_.map_mul] at hQ
     -- We can now finish the proof.
     have hppdiv : p ^ B.dim ∣ p ^ B.dim * r := dvd_mul_of_dvd_left dvd_rfl _
     rwa [← IsFractionRing.injective R K hQ, mul_comm, ← Units.coe_neg_one, mul_pow, ←
       Units.val_pow_eq_pow_val, ← Units.val_pow_eq_pow_val, mul_assoc,
-      IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv 
+      IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv
 #align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt
 -/
 
@@ -424,7 +424,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBa
     z ∈ adjoin R ({B.gen} : Set L) := by
   induction' n with n hn
   · simpa using hz
-  · rw [pow_succ, mul_smul] at hz 
+  · rw [pow_succ, mul_smul] at hz
     exact
       hn
         (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (IsIntegral.smul _ hzint)

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

@@ -108,8 +108,8 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       · simp only [ite_eq_right_iff]
         rintro ⟨k, hk⟩
         rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
-          nat_degree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _), Nat.succ_sub_one] at
-          hn hi 
+          nat_degree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _),
+          Nat.add_one_sub_one] at hn hi 
         rw [hk, pow_succ, mul_assoc] at hi 
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
@@ -189,8 +189,8 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
       (z * B.gen ^ n - ∑ x : ℕ in (range (Q.nat_degree + 1)).eraseₓ 0, Q.coeff x • f (x + n)) :=
     by
     refine'
-      isIntegral_sub (isIntegral_mul hzint (IsIntegral.pow hBint _))
-        (IsIntegral.sum _ fun i hi => isIntegral_smul _ _)
+      IsIntegral.sub (IsIntegral.mul hzint (IsIntegral.pow hBint _))
+        (IsIntegral.sum _ fun i hi => IsIntegral.smul _ _)
     exact adjoin_le_integralClosure hBint (hf _ (aux i hi)).1
   obtain ⟨r, hr⟩ := is_integral_iff.1 (is_integral_norm K hintsum)
   use r
@@ -386,10 +386,10 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
             ∑ x : ℕ in range (j + 1), g x • B.gen ^ x * B.gen ^ (P.nat_degree - (j + 2)))) :=
       by
       refine'
-        isIntegral_sub (isIntegral_mul hzint (IsIntegral.pow hBint _))
-          (isIntegral_add (IsIntegral.sum _ fun k hk => isIntegral_smul _ _)
+        IsIntegral.sub (IsIntegral.mul hzint (IsIntegral.pow hBint _))
+          (IsIntegral.add (IsIntegral.sum _ fun k hk => IsIntegral.smul _ _)
             (IsIntegral.sum _ fun k hk =>
-              isIntegral_mul (isIntegral_smul _ (IsIntegral.pow hBint _)) (IsIntegral.pow hBint _)))
+              IsIntegral.mul (IsIntegral.smul _ (IsIntegral.pow hBint _)) (IsIntegral.pow hBint _)))
       refine' adjoin_le_integralClosure hBint (hf _ _).1
       rw [(minpoly.monic hBint).natDegree_map (algebraMap R L)]
       rw [add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
@@ -427,7 +427,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBa
   · rw [pow_succ, mul_smul] at hz 
     exact
       hn
-        (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (isIntegral_smul _ hzint)
+        (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (IsIntegral.smul _ hzint)
           hz hei)
 #align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt
 -/

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

@@ -58,7 +58,7 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     refine' (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => _
     rw [nat_degree_X_add_C] at h 
     exact zero_ne_one h.symm
-  · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, LinearMap.map_sum]
+  · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, map_sum]
     conv =>
       congr
       congr

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

@@ -3,10 +3,10 @@ Copyright (c) 2022 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 -/
-import Mathbin.Data.Nat.Choose.Dvd
-import Mathbin.RingTheory.IntegrallyClosed
-import Mathbin.RingTheory.Norm
-import Mathbin.RingTheory.Polynomial.Cyclotomic.Expand
+import Data.Nat.Choose.Dvd
+import RingTheory.IntegrallyClosed
+import RingTheory.Norm
+import RingTheory.Polynomial.Cyclotomic.Expand
 
 #align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.eisenstein.is_integral
-! leanprover-community/mathlib commit e8e130de9dba4ed6897183c3193c752ffadbcc77
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Choose.Dvd
 import Mathbin.RingTheory.IntegrallyClosed
 import Mathbin.RingTheory.Norm
 import Mathbin.RingTheory.Polynomial.Cyclotomic.Expand
 
+#align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"
+
 /-!
 # Eisenstein polynomials
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/fdc286cc6967a012f41b87f76dcd2797b53152af

@@ -47,8 +47,8 @@ local notation "𝓟" => Submodule.span ℤ {p}
 
 open Polynomial
 
-#print cyclotomic_comp_x_add_one_isEisensteinAt /-
-theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
+#print cyclotomic_comp_X_add_one_isEisensteinAt /-
+theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 :=
   by
   refine'
@@ -82,11 +82,11 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
     nth_rw 1 [← Nat.mul_one p] at hk 
     rw [mul_right_inj' hp.out.ne_zero] at hk 
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
-#align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_x_add_one_isEisensteinAt
+#align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_X_add_one_isEisensteinAt
 -/
 
-#print cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt /-
-theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
+#print cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt /-
+theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
     ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 :=
   by
   refine'
@@ -102,7 +102,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
   · induction' n with n hn
     · intro i hi
       rw [zero_add, pow_one] at hi ⊢
-      exact (cyclotomic_comp_x_add_one_isEisensteinAt p).Mem hi
+      exact (cyclotomic_comp_X_add_one_isEisensteinAt p).Mem hi
     · intro i hi
       rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd, ←
         Int.coe_castRingHom, ← coeff_map, map_comp, map_cyclotomic, Polynomial.map_add, map_X,
@@ -131,7 +131,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
     nth_rw 1 [← Nat.mul_one p] at hk 
     rw [mul_right_inj' hp.out.ne_zero] at hk 
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
-#align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt
+#align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt
 -/
 
 end Cyclotomic
@@ -148,12 +148,12 @@ local notation "𝓟" => Submodule.span R {p}
 
 open IsIntegrallyClosed PowerBasis Nat Polynomial IsScalarTower
 
-#print dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at /-
+#print dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt /-
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `Q : R[X]` is such that
 `aeval B.gen Q = p • z`, then `p ∣ Q.coeff 0`. -/
-theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
+theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
     (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 :=
   by
@@ -239,7 +239,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
   · rw [aeval_eq_sum_range,
       Finset.add_sum_erase (range (Q.nat_degree + 1)) fun i => Q.coeff i • B.gen ^ i]
     simp
-#align dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at
+#align dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt
 -/
 
 #print mem_adjoin_of_dvd_coeff_of_dvd_aeval /-
@@ -262,13 +262,13 @@ theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [Co
 #align mem_adjoin_of_dvd_coeff_of_dvd_aeval mem_adjoin_of_dvd_coeff_of_dvd_aeval
 -/
 
-#print mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at /-
+#print mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt /-
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
 `z ∈ adjoin R {B.gen}`. -/
-theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
-    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
+theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L} (hp : Prime p)
+    (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
     (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
     z ∈ adjoin R ({B.gen} : Set L) :=
   by
@@ -304,7 +304,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
   refine' mem_adjoin_of_dvd_coeff_of_dvd_aeval hp.ne_zero (fun i => _) hQ
   refine' Nat.case_strong_induction_on i _ fun j hind => _
   · intro H
-    exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at hp hBint hQ hzint hei
+    exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt hp hBint hQ hzint hei
   · intro hj
     refine' hp.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd _ hndiv
     exact n
@@ -412,16 +412,16 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     rwa [← IsFractionRing.injective R K hQ, mul_comm, ← Units.coe_neg_one, mul_pow, ←
       Units.val_pow_eq_pow_val, ← Units.val_pow_eq_pow_val, mul_assoc,
       IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv 
-#align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at
+#align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt
 -/
 
-#print mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at /-
+#print mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt /-
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p ^ n • z ∈ adjoin R {B.gen}`,
 then `z ∈ adjoin R {B.gen}`. Together with `algebra.discr_mul_is_integral_mem_adjoin` this result
 often allows to compute the ring of integers of `L`. -/
-theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
+theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {n : ℕ} {z : L} (hzint : IsIntegral R z)
     (hz : p ^ n • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
     z ∈ adjoin R ({B.gen} : Set L) := by
@@ -430,9 +430,9 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : Powe
   · rw [pow_succ, mul_smul] at hz 
     exact
       hn
-        (mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at hp hBint
-          (isIntegral_smul _ hzint) hz hei)
-#align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at
+        (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (isIntegral_smul _ hzint)
+          hz hei)
+#align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt
 -/
 
 end IsIntegral

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.eisenstein.is_integral
-! leanprover-community/mathlib commit 5bfbcca0a7ffdd21cf1682e59106d6c942434a32
+! leanprover-community/mathlib commit e8e130de9dba4ed6897183c3193c752ffadbcc77
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.Polynomial.Cyclotomic.Expand
 
 /-!
 # Eisenstein polynomials
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
 In this file we gather more miscellaneous results about Eisenstein polynomials
 
 ## Main results

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -44,6 +44,7 @@ local notation "𝓟" => Submodule.span ℤ {p}
 
 open Polynomial
 
+#print cyclotomic_comp_x_add_one_isEisensteinAt /-
 theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
     ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 :=
   by
@@ -79,7 +80,9 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
     rw [mul_right_inj' hp.out.ne_zero] at hk 
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 #align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_x_add_one_isEisensteinAt
+-/
 
+#print cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt /-
 theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
     ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 :=
   by
@@ -126,6 +129,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
     rw [mul_right_inj' hp.out.ne_zero] at hk 
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 #align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt
+-/
 
 end Cyclotomic
 
@@ -141,6 +145,7 @@ local notation "𝓟" => Submodule.span R {p}
 
 open IsIntegrallyClosed PowerBasis Nat Polynomial IsScalarTower
 
+#print dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at /-
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `Q : R[X]` is such that
@@ -232,7 +237,9 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
       Finset.add_sum_erase (range (Q.nat_degree + 1)) fun i => Q.coeff i • B.gen ^ i]
     simp
 #align dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at
+-/
 
+#print mem_adjoin_of_dvd_coeff_of_dvd_aeval /-
 theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [CommRing B]
     [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
     (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
@@ -250,7 +257,9 @@ theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [Co
     Subalgebra.sum_mem _ fun _ _ =>
       Subalgebra.smul_mem _ (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton _)) _) _
 #align mem_adjoin_of_dvd_coeff_of_dvd_aeval mem_adjoin_of_dvd_coeff_of_dvd_aeval
+-/
 
+#print mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at /-
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
@@ -401,7 +410,9 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
       Units.val_pow_eq_pow_val, ← Units.val_pow_eq_pow_val, mul_assoc,
       IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv 
 #align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at
+-/
 
+#print mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at /-
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p ^ n • z ∈ adjoin R {B.gen}`,
@@ -419,6 +430,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : Powe
         (mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at hp hBint
           (isIntegral_smul _ hzint) hz hei)
 #align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at
+-/
 
 end IsIntegral
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -40,7 +40,6 @@ section Cyclotomic
 
 variable (p : ℕ)
 
--- mathport name: expr𝓟
 local notation "𝓟" => Submodule.span ℤ {p}
 
 open Polynomial
@@ -138,7 +137,6 @@ variable [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [IsSepa
 
 variable [IsDomain R] [IsFractionRing R K] [IsIntegrallyClosed R]
 
--- mathport name: expr𝓟
 local notation "𝓟" => Submodule.span R {p}
 
 open IsIntegrallyClosed PowerBasis Nat Polynomial IsScalarTower

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -374,8 +374,8 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     have hintsum :
       IsIntegral R
         (z * B.gen ^ (P.nat_degree - (j + 2)) -
-          ((∑ x : ℕ in (range (Q.nat_degree - j)).eraseₓ 0,
-              Q.coeff (j + 1 + x) • f (x + P.nat_degree - 1)) +
+          (∑ x : ℕ in (range (Q.nat_degree - j)).eraseₓ 0,
+              Q.coeff (j + 1 + x) • f (x + P.nat_degree - 1) +
             ∑ x : ℕ in range (j + 1), g x • B.gen ^ x * B.gen ^ (P.nat_degree - (j + 2)))) :=
       by
       refine'

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -203,7 +203,6 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
             ∑ x : ℕ in (range (Q.nat_degree + 1)).eraseₓ 0, p • Q.coeff x • f (x + n)) :=
       (congr_arg (norm K) (eq_sub_of_add_eq _))
     _ = _ := _
-    
   · simp only [Algebra.smul_def, algebraMap_apply R K L, Algebra.norm_algebraMap, _root_.map_mul,
       _root_.map_pow, finrank_K_L, power_basis.norm_gen_eq_coeff_zero_minpoly,
       minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map, ← hn]
@@ -224,7 +223,6 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
       by rw [pow_zero]
     _ = aeval B.gen Q * B.gen ^ n := _
     _ = _ := by rw [hQ, Algebra.smul_mul_assoc]
-    
   · have :
       ∀ i ∈ (range (Q.nat_degree + 1)).eraseₓ 0,
         Q.coeff i • (B.gen ^ i * B.gen ^ n) = p • Q.coeff i • f (i + n) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -171,7 +171,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
     have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
       hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
     refine' Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd hp (_ : _ ^ n.succ ∣ _) hndiv
-    convert(IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
+    convert (IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
     push_cast
     ring_nf; simp [pow_right_comm _ _ 2]
   -- We claim the quotient of `Q^n * _` by `p^n` is the following `r`:

mathlib commit https://github.com/leanprover-community/mathlib/commit/34ebaffc1d1e8e783fc05438ec2e70af87275ac9

@@ -149,7 +149,7 @@ extension of `K`, generated by an integral power basis `B` such that the minimal
 `aeval B.gen Q = p • z`, then `p ∣ Q.coeff 0`. -/
 theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
-    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.Coeff 0 :=
+    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 :=
   by
   -- First define some abbreviations.
   letI := B.finite_dimensional
@@ -166,9 +166,9 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
   simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf 
   -- The Eisenstein condition shows that `p` divides `Q.coeff 0`
   -- if `p^n.succ` divides the following multiple of `Q.coeff 0^n.succ`:
-  suffices p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).Coeff 0 ^ n)
+  suffices p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
     by
-    have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).Coeff 0 := fun h =>
+    have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
       hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
     refine' Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd hp (_ : _ ^ n.succ ∣ _) hndiv
     convert(IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
@@ -239,7 +239,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
 
 theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [CommRing B]
     [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
-    (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.Coeff i) (hz : aeval x Q = p • z) :
+    (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
     z ∈ adjoin A ({x} : Set B) := by
   choose! f hf using hQ
   rw [aeval_eq_sum_range, sum_range] at hz 
@@ -265,7 +265,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     z ∈ adjoin R ({B.gen} : Set L) :=
   by
   -- First define some abbreviations.
-  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).Coeff 0 := fun h =>
+  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
     hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
   letI := FiniteDimensional B
   set P := minpoly R B.gen with hP
@@ -349,7 +349,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     -- if `p^n.succ` divides the following multiple of `Q.coeff (succ j)^n.succ`:
     suffices
       p ^ n.succ ∣
-        Q.coeff (succ j) ^ n.succ * (minpoly R B.gen).Coeff 0 ^ (succ j + (P.nat_degree - (j + 2)))
+        Q.coeff (succ j) ^ n.succ * (minpoly R B.gen).coeff 0 ^ (succ j + (P.nat_degree - (j + 2)))
       by
       convert this
       rw [Nat.succ_eq_add_one, add_assoc, ← Nat.add_sub_assoc H, ← add_assoc, add_comm (j + 1),

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -56,7 +56,7 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
       (fun i hi => _) _
   · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
     refine' (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => _
-    rw [nat_degree_X_add_C] at h
+    rw [nat_degree_X_add_C] at h 
     exact zero_ne_one h.symm
   · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, LinearMap.map_sum]
     conv =>
@@ -66,7 +66,7 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
       ext
       rw [lcoeff_apply, ← C_eq_nat_cast, C_mul_X_pow_eq_monomial, coeff_monomial]
     rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
-      nat_degree_cyclotomic, Nat.totient_prime hp.out] at hi
+      nat_degree_cyclotomic, Nat.totient_prime hp.out] at hi 
     simp only [hi.trans_le (Nat.sub_le _ _), sum_ite_eq', mem_range, if_true,
       Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.coe_nat_dvd]
     exact hp.out.dvd_choose_self i.succ_ne_zero (lt_tsub_iff_right.1 hi)
@@ -75,9 +75,9 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
       Ideal.mem_span_singleton]
     intro h
     obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
-    rw [← mul_assoc, mul_one, mul_assoc] at hk
-    nth_rw 1 [← Nat.mul_one p] at hk
-    rw [mul_right_inj' hp.out.ne_zero] at hk
+    rw [← mul_assoc, mul_one, mul_assoc] at hk 
+    nth_rw 1 [← Nat.mul_one p] at hk 
+    rw [mul_right_inj' hp.out.ne_zero] at hk 
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 #align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_x_add_one_isEisensteinAt
 
@@ -92,11 +92,11 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
       _ _
   · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
     refine' (cyclotomic.monic _ ℤ).comp (monic_X_add_C 1) fun h => _
-    rw [nat_degree_X_add_C] at h
+    rw [nat_degree_X_add_C] at h 
     exact zero_ne_one h.symm
   · induction' n with n hn
     · intro i hi
-      rw [zero_add, pow_one] at hi⊢
+      rw [zero_add, pow_one] at hi ⊢
       exact (cyclotomic_comp_x_add_one_isEisensteinAt p).Mem hi
     · intro i hi
       rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd, ←
@@ -107,12 +107,12 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rintro ⟨k, hk⟩
         rw [nat_degree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, nat_degree_X_add_C, mul_one,
           nat_degree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _), Nat.succ_sub_one] at
-          hn hi
-        rw [hk, pow_succ, mul_assoc] at hi
+          hn hi 
+        rw [hk, pow_succ, mul_assoc] at hi 
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
         rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
-          ← Int.coe_castRingHom, ← coeff_map] at hn
+          ← Int.coe_castRingHom, ← coeff_map] at hn 
         simpa [map_comp] using hn
       · exact ⟨p ^ n, by rw [pow_succ]⟩
   · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime_pow_eq_geom_sum hp.out, eval_add,
@@ -122,9 +122,9 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
       Ideal.mem_span_singleton]
     intro h
     obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
-    rw [← mul_assoc, mul_one, mul_assoc] at hk
-    nth_rw 1 [← Nat.mul_one p] at hk
-    rw [mul_right_inj' hp.out.ne_zero] at hk
+    rw [← mul_assoc, mul_one, mul_assoc] at hk 
+    nth_rw 1 [← Nat.mul_one p] at hk 
+    rw [mul_right_inj' hp.out.ne_zero] at hk 
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 #align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt
 
@@ -163,7 +163,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
   choose! f hf using
     hei.is_weakly_eisenstein_at.exists_mem_adjoin_mul_eq_pow_nat_degree_le (minpoly.aeval R B.gen)
       (minpoly.monic hBint)
-  simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf
+  simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf 
   -- The Eisenstein condition shows that `p` divides `Q.coeff 0`
   -- if `p^n.succ` divides the following multiple of `Q.coeff 0^n.succ`:
   suffices p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).Coeff 0 ^ n)
@@ -178,7 +178,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
   have aux : ∀ i ∈ (range (Q.nat_degree + 1)).eraseₓ 0, B.dim ≤ i + n :=
     by
     intro i hi
-    simp only [mem_range, mem_erase] at hi
+    simp only [mem_range, mem_erase] at hi 
     rw [hn]
     exact le_add_pred_of_pos _ hi.1
   have hintsum :
@@ -242,13 +242,13 @@ theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [Co
     (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.Coeff i) (hz : aeval x Q = p • z) :
     z ∈ adjoin A ({x} : Set B) := by
   choose! f hf using hQ
-  rw [aeval_eq_sum_range, sum_range] at hz
+  rw [aeval_eq_sum_range, sum_range] at hz 
   conv_lhs at hz =>
     congr
     skip
     ext
     rw [hf i (mem_range.2 (Fin.is_lt i)), ← smul_smul]
-  rw [← smul_sum] at hz
+  rw [← smul_sum] at hz 
   rw [← smul_right_injective _ hp hz]
   exact
     Subalgebra.sum_mem _ fun _ _ =>
@@ -274,14 +274,14 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
   let P₁ := P.map (algebraMap R L)
   -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
   -- `Q.degree < P.degree` and `Q ≠ 0`.
-  rw [adjoin_singleton_eq_range_aeval] at hz
+  rw [adjoin_singleton_eq_range_aeval] at hz 
   obtain ⟨Q₁, hQ⟩ := hz
   set Q := Q₁ %ₘ P with hQ₁
   replace hQ : aeval B.gen Q = p • z
-  · rw [← mod_by_monic_add_div Q₁ (minpoly.monic hBint)] at hQ
+  · rw [← mod_by_monic_add_div Q₁ (minpoly.monic hBint)] at hQ 
     simpa using hQ
   by_cases hQzero : Q = 0
-  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
+  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ 
     cases' hQ with H H₁
     · have : Function.Injective (algebraMap R L) :=
         by
@@ -302,14 +302,14 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     exact n
     -- Two technical results we will need about `P.nat_degree` and `Q.nat_degree`.
     have H := degree_mod_by_monic_lt Q₁ (minpoly.monic hBint)
-    rw [← hQ₁, ← hP] at H
+    rw [← hQ₁, ← hP] at H 
     replace H :=
       Nat.lt_iff_add_one_le.1
         (lt_of_lt_of_le
           (lt_of_le_of_lt (Nat.lt_iff_add_one_le.1 (Nat.lt_of_succ_lt_succ (mem_range.1 hj)))
             (lt_succ_self _))
           (Nat.lt_iff_add_one_le.1 ((nat_degree_lt_nat_degree_iff hQzero).2 H)))
-    rw [add_assoc] at H
+    rw [add_assoc] at H 
     have Hj : Q.nat_degree + 1 = j + 1 + (Q.nat_degree - j) := by
       rw [← add_comm 1, ← add_comm 1, add_assoc, add_right_inj, ←
         Nat.add_sub_assoc (Nat.lt_of_succ_lt_succ (mem_range.1 hj)).le, add_comm,
@@ -342,7 +342,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
       rw [(minpoly.monic hBint).natDegree_map, add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
       · exact Nat.zero_le _
       · refine' one_le_iff_ne_zero.2 fun h => _
-        rw [h] at hk
+        rw [h] at hk 
         simpa using hk
       · infer_instance
     -- The Eisenstein condition shows that `p` divides `Q.coeff j`
@@ -361,15 +361,15 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     -- Using `hQ : aeval B.gen Q = p • z`, we write `p • z` as a sum of terms of degree less than
     -- `j+1`, that are multiples of `p` by induction, and terms of degree at least `j+1`.
     rw [aeval_eq_sum_range, Hj, range_add, sum_union (disjoint_range_add_left_embedding _ _),
-      sum_congr rfl hg, add_comm] at hQ
+      sum_congr rfl hg, add_comm] at hQ 
     -- We multiply this equality by `B.gen ^ (P.nat_degree-(j+2))`, so we can use `hf₁` on the terms
     -- we didn't know were multiples of `p`, and we take the norm on both sides.
     replace hQ := congr_arg (fun x => x * B.gen ^ (P.nat_degree - (j + 2))) hQ
-    simp_rw [sum_map, addLeftEmbedding_apply, add_mul, sum_mul, mul_assoc] at hQ
+    simp_rw [sum_map, addLeftEmbedding_apply, add_mul, sum_mul, mul_assoc] at hQ 
     rw [←
       insert_erase (mem_range.2 (tsub_pos_iff_lt.2 <| Nat.lt_of_succ_lt_succ <| mem_range.1 hj)),
       sum_insert (not_mem_erase 0 _), add_zero, sum_congr rfl hf₁, ← mul_sum, ← mul_sum, add_assoc,
-      ← mul_add, smul_mul_assoc, ← pow_add, Algebra.smul_def] at hQ
+      ← mul_add, smul_mul_assoc, ← pow_add, Algebra.smul_def] at hQ 
     replace hQ := congr_arg (norm K) (eq_sub_of_add_eq hQ)
     -- We obtain an equality of elements of `K`, but everything is integral, so we can move to `R`
     -- and simplify `hQ`.
@@ -390,7 +390,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
       rw [add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
       · exact zero_le _
       · refine' one_le_iff_ne_zero.2 fun h => _
-        rw [h] at hk
+        rw [h] at hk 
         simpa using hk
     obtain ⟨r, hr⟩ := is_integral_iff.1 (is_integral_norm K hintsum)
     rw [Algebra.smul_def, mul_assoc, ← mul_sub, _root_.map_mul, algebraMap_apply R K L, map_pow,
@@ -398,12 +398,12 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
       finrank B, ← hr, power_basis.norm_gen_eq_coeff_zero_minpoly,
       minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map,
       show (-1 : K) = algebraMap R K (-1) by simp, ← map_pow, ← map_pow, ← _root_.map_mul, ←
-      map_pow, ← _root_.map_mul, ← map_pow, ← _root_.map_mul] at hQ
+      map_pow, ← _root_.map_mul, ← map_pow, ← _root_.map_mul] at hQ 
     -- We can now finish the proof.
     have hppdiv : p ^ B.dim ∣ p ^ B.dim * r := dvd_mul_of_dvd_left dvd_rfl _
     rwa [← IsFractionRing.injective R K hQ, mul_comm, ← Units.coe_neg_one, mul_pow, ←
       Units.val_pow_eq_pow_val, ← Units.val_pow_eq_pow_val, mul_assoc,
-      IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv
+      IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv 
 #align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at
 
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
@@ -417,7 +417,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : Powe
     z ∈ adjoin R ({B.gen} : Set L) := by
   induction' n with n hn
   · simpa using hz
-  · rw [pow_succ, mul_smul] at hz
+  · rw [pow_succ, mul_smul] at hz 
     exact
       hn
         (mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at hp hBint

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -34,7 +34,7 @@ variable {R : Type u}
 
 open Ideal Algebra Finset
 
-open BigOperators Polynomial
+open scoped BigOperators Polynomial
 
 section Cyclotomic
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -173,8 +173,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
     refine' Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd hp (_ : _ ^ n.succ ∣ _) hndiv
     convert(IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
     push_cast
-    ring_nf
-    simp [pow_right_comm _ _ 2]
+    ring_nf; simp [pow_right_comm _ _ 2]
   -- We claim the quotient of `Q^n * _` by `p^n` is the following `r`:
   have aux : ∀ i ∈ (range (Q.nat_degree + 1)).eraseₓ 0, B.dim ≤ i + n :=
     by
@@ -209,7 +208,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
       _root_.map_pow, finrank_K_L, power_basis.norm_gen_eq_coeff_zero_minpoly,
       minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map, ← hn]
     ring
-  swap
+  swap;
   ·
     simp_rw [← smul_sum, ← smul_sub, Algebra.smul_def p, algebraMap_apply R K L, _root_.map_mul,
       Algebra.norm_algebraMap, finrank_K_L, hr, ← hn]

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -4,14 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.eisenstein.is_integral
-! leanprover-community/mathlib commit ceb887ddf3344dab425292e497fa2af91498437c
+! leanprover-community/mathlib commit 5bfbcca0a7ffdd21cf1682e59106d6c942434a32
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Choose.Dvd
 import Mathbin.RingTheory.IntegrallyClosed
 import Mathbin.RingTheory.Norm
-import Mathbin.RingTheory.Polynomial.Cyclotomic.Basic
+import Mathbin.RingTheory.Polynomial.Cyclotomic.Expand
 
 /-!
 # Eisenstein polynomials

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -149,7 +149,7 @@ extension of `K`, generated by an integral power basis `B` such that the minimal
 `aeval B.gen Q = p • z`, then `p ∣ Q.coeff 0`. -/
 theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
-    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 :=
+    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.Coeff 0 :=
   by
   -- First define some abbreviations.
   letI := B.finite_dimensional
@@ -166,9 +166,9 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
   simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf
   -- The Eisenstein condition shows that `p` divides `Q.coeff 0`
   -- if `p^n.succ` divides the following multiple of `Q.coeff 0^n.succ`:
-  suffices p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
+  suffices p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).Coeff 0 ^ n)
     by
-    have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
+    have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).Coeff 0 := fun h =>
       hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
     refine' Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd hp (_ : _ ^ n.succ ∣ _) hndiv
     convert(IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
@@ -240,7 +240,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
 
 theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [CommRing B]
     [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
-    (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
+    (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.Coeff i) (hz : aeval x Q = p • z) :
     z ∈ adjoin A ({x} : Set B) := by
   choose! f hf using hQ
   rw [aeval_eq_sum_range, sum_range] at hz
@@ -266,7 +266,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     z ∈ adjoin R ({B.gen} : Set L) :=
   by
   -- First define some abbreviations.
-  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
+  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).Coeff 0 := fun h =>
     hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
   letI := FiniteDimensional B
   set P := minpoly R B.gen with hP
@@ -350,7 +350,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     -- if `p^n.succ` divides the following multiple of `Q.coeff (succ j)^n.succ`:
     suffices
       p ^ n.succ ∣
-        Q.coeff (succ j) ^ n.succ * (minpoly R B.gen).coeff 0 ^ (succ j + (P.nat_degree - (j + 2)))
+        Q.coeff (succ j) ^ n.succ * (minpoly R B.gen).Coeff 0 ^ (succ j + (P.nat_degree - (j + 2)))
       by
       convert this
       rw [Nat.succ_eq_add_one, add_assoc, ← Nat.add_sub_assoc H, ← add_assoc, add_comm (j + 1),

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

@@ -205,13 +205,13 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
       (congr_arg (norm K) (eq_sub_of_add_eq _))
     _ = _ := _
     
-  · simp only [Algebra.smul_def, algebra_map_apply R K L, Algebra.norm_algebraMap, _root_.map_mul,
+  · simp only [Algebra.smul_def, algebraMap_apply R K L, Algebra.norm_algebraMap, _root_.map_mul,
       _root_.map_pow, finrank_K_L, power_basis.norm_gen_eq_coeff_zero_minpoly,
       minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map, ← hn]
     ring
   swap
   ·
-    simp_rw [← smul_sum, ← smul_sub, Algebra.smul_def p, algebra_map_apply R K L, _root_.map_mul,
+    simp_rw [← smul_sum, ← smul_sub, Algebra.smul_def p, algebraMap_apply R K L, _root_.map_mul,
       Algebra.norm_algebraMap, finrank_K_L, hr, ← hn]
   calc
     _ =
@@ -394,8 +394,8 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
         rw [h] at hk
         simpa using hk
     obtain ⟨r, hr⟩ := is_integral_iff.1 (is_integral_norm K hintsum)
-    rw [Algebra.smul_def, mul_assoc, ← mul_sub, _root_.map_mul, algebra_map_apply R K L, map_pow,
-      Algebra.norm_algebraMap, _root_.map_mul, algebra_map_apply R K L, Algebra.norm_algebraMap,
+    rw [Algebra.smul_def, mul_assoc, ← mul_sub, _root_.map_mul, algebraMap_apply R K L, map_pow,
+      Algebra.norm_algebraMap, _root_.map_mul, algebraMap_apply R K L, Algebra.norm_algebraMap,
       finrank B, ← hr, power_basis.norm_gen_eq_coeff_zero_minpoly,
       minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map,
       show (-1 : K) = algebraMap R K (-1) by simp, ← map_pow, ← map_pow, ← _root_.map_mul, ←

mathlib commit https://github.com/leanprover-community/mathlib/commit/02ba8949f486ebecf93fe7460f1ed0564b5e442c

@@ -99,7 +99,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
       rw [zero_add, pow_one] at hi⊢
       exact (cyclotomic_comp_x_add_one_isEisensteinAt p).Mem hi
     · intro i hi
-      rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_coe_zMod_eq_zero_iff_dvd, ←
+      rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd, ←
         Int.coe_castRingHom, ← coeff_map, map_comp, map_cyclotomic, Polynomial.map_add, map_X,
         Polynomial.map_one, pow_add, pow_one, cyclotomic_mul_prime_dvd_eq_pow, pow_comp, ←
         ZMod.expand_card, coeff_expand hp.out.pos]
@@ -111,7 +111,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rw [hk, pow_succ, mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
-        rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_coe_zMod_eq_zero_iff_dvd,
+        rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
           ← Int.coe_castRingHom, ← coeff_map] at hn
         simpa [map_comp] using hn
       · exact ⟨p ^ n, by rw [pow_succ]⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -171,7 +171,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
     have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
       hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
     refine' Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd hp (_ : _ ^ n.succ ∣ _) hndiv
-    convert (IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
+    convert(IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
     push_cast
     ring_nf
     simp [pow_right_comm _ _ 2]

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.eisenstein.is_integral
-! leanprover-community/mathlib commit 966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2
+! leanprover-community/mathlib commit ceb887ddf3344dab425292e497fa2af91498437c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Data.Nat.Choose.Dvd
 import Mathbin.RingTheory.IntegrallyClosed
 import Mathbin.RingTheory.Norm
 import Mathbin.RingTheory.Polynomial.Cyclotomic.Basic

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -45,7 +45,7 @@ local notation "𝓟" => Submodule.span ℤ {p}
 open Polynomial
 
 theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
-    ((cyclotomic p ℤ).comp (x + 1)).IsEisensteinAt 𝓟 :=
+    ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 :=
   by
   refine'
     monic.is_eisenstein_at_of_mem_of_not_mem _
@@ -81,7 +81,7 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
 #align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_x_add_one_isEisensteinAt
 
 theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
-    ((cyclotomic (p ^ (n + 1)) ℤ).comp (x + 1)).IsEisensteinAt 𝓟 :=
+    ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 :=
   by
   refine'
     monic.is_eisenstein_at_of_mem_of_not_mem _

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -201,7 +201,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
         norm K
           (p • (z * B.gen ^ n) -
             ∑ x : ℕ in (range (Q.nat_degree + 1)).eraseₓ 0, p • Q.coeff x • f (x + n)) :=
-      congr_arg (norm K) (eq_sub_of_add_eq _)
+      (congr_arg (norm K) (eq_sub_of_add_eq _))
     _ = _ := _
     
   · simp only [Algebra.smul_def, algebra_map_apply R K L, Algebra.norm_algebraMap, _root_.map_mul,

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -272,7 +272,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
     exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt hp hBint hQ hzint hei
   · intro hj
     convert hp.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd _ hndiv
-    exact n
+    · exact n
     -- Two technical results we will need about `P.natDegree` and `Q.natDegree`.
     have H := degree_modByMonic_lt Q₁ (minpoly.monic hBint)
     rw [← hQ₁, ← hP] at H

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.

@@ -56,7 +56,7 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
       congr
       next => skip
       ext
-      rw [lcoeff_apply, ← C_eq_nat_cast, C_mul_X_pow_eq_monomial, coeff_monomial]
+      rw [lcoeff_apply, ← C_eq_natCast, C_mul_X_pow_eq_monomial, coeff_monomial]
     rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
       natDegree_cyclotomic, Nat.totient_prime hp.out] at hi
     simp only [hi.trans_le (Nat.sub_le _ _), sum_ite_eq', mem_range, if_true,
@@ -88,7 +88,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       rw [Nat.zero_add, pow_one] at hi ⊢
       exact (cyclotomic_comp_X_add_one_isEisensteinAt p).mem hi
     · intro i hi
-      rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
+      rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.intCast_zmod_eq_zero_iff_dvd,
         show ↑(_  : ℤ) = Int.castRingHom (ZMod p) _ by rfl, ← coeff_map, map_comp, map_cyclotomic,
         Polynomial.map_add, map_X, Polynomial.map_one, pow_add, pow_one,
         cyclotomic_mul_prime_dvd_eq_pow, pow_comp, ← ZMod.expand_card, coeff_expand hp.out.pos]
@@ -100,7 +100,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rw [hk, pow_succ', mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
-        rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
+        rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.intCast_zmod_eq_zero_iff_dvd,
            show ↑(_  : ℤ) = Int.castRingHom (ZMod p) _ by rfl, ← coeff_map] at hn
         simpa [map_comp] using hn
       · exact ⟨p ^ n, by rw [pow_succ']⟩

Reduce the diff of #11499

Renames

All in the Int namespace:

• ofNat_eq_cast → ofNat_eq_natCast
• cast_eq_cast_iff_Nat → natCast_inj
• natCast_eq_ofNat → ofNat_eq_natCast
• coe_nat_sub → natCast_sub
• coe_nat_nonneg → natCast_nonneg
• sign_coe_add_one → sign_natCast_add_one
• nat_succ_eq_int_succ → natCast_succ
• succ_neg_nat_succ → succ_neg_natCast_succ
• coe_pred_of_pos → natCast_pred_of_pos
• coe_nat_div → natCast_div
• coe_nat_ediv → natCast_ediv
• sign_coe_nat_of_nonzero → sign_natCast_of_ne_zero
• toNat_coe_nat → toNat_natCast
• toNat_coe_nat_add_one → toNat_natCast_add_one
• coe_nat_dvd → natCast_dvd_natCast
• coe_nat_dvd_left → natCast_dvd
• coe_nat_dvd_right → dvd_natCast
• le_coe_nat_sub → le_natCast_sub
• succ_coe_nat_pos → succ_natCast_pos
• coe_nat_modEq_iff → natCast_modEq_iff
• coe_natAbs → natCast_natAbs
• coe_nat_eq_zero → natCast_eq_zero
• coe_nat_ne_zero → natCast_ne_zero
• coe_nat_ne_zero_iff_pos → natCast_ne_zero_iff_pos
• abs_coe_nat → abs_natCast
• coe_nat_nonpos_iff → natCast_nonpos_iff

Also rename Nat.coe_nat_dvd to Nat.cast_dvd_cast

@@ -60,13 +60,13 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
       natDegree_cyclotomic, Nat.totient_prime hp.out] at hi
     simp only [hi.trans_le (Nat.sub_le _ _), sum_ite_eq', mem_range, if_true,
-      Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.coe_nat_dvd]
+      Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.natCast_dvd_natCast]
     exact hp.out.dvd_choose_self i.succ_ne_zero (lt_tsub_iff_right.1 hi)
   · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime, eval_add, eval_X, eval_one, zero_add,
       eval_geom_sum, one_geom_sum, Ideal.submodule_span_eq, Ideal.span_singleton_pow,
       Ideal.mem_span_singleton]
     intro h
-    obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
+    obtain ⟨k, hk⟩ := Int.natCast_dvd_natCast.1 h
     rw [mul_assoc, mul_comm 1, mul_one] at hk
     nth_rw 1 [← Nat.mul_one p] at hk
     rw [mul_right_inj' hp.out.ne_zero] at hk
@@ -110,7 +110,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       submodule_span_eq, Ideal.submodule_span_eq, Ideal.span_singleton_pow,
       Ideal.mem_span_singleton]
     intro h
-    obtain ⟨k, hk⟩ := Int.coe_nat_dvd.1 h
+    obtain ⟨k, hk⟩ := Int.natCast_dvd_natCast.1 h
     rw [mul_assoc, mul_comm 1, mul_one] at hk
     nth_rw 1 [← Nat.mul_one p] at hk
     rw [mul_right_inj' hp.out.ne_zero] at hk

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -97,13 +97,13 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
         rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
           natDegree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _), Nat.add_one_sub_one]
           at hn hi
-        rw [hk, pow_succ, mul_assoc] at hi
+        rw [hk, pow_succ', mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]
         replace hn := hn (lt_of_mul_lt_mul_left' hi)
         rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
            show ↑(_  : ℤ) = Int.castRingHom (ZMod p) _ by rfl, ← coeff_map] at hn
         simpa [map_comp] using hn
-      · exact ⟨p ^ n, by rw [pow_succ]⟩
+      · exact ⟨p ^ n, by rw [pow_succ']⟩
   · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime_pow_eq_geom_sum hp.out, eval_add,
       eval_X, eval_one, zero_add, eval_finset_sum]
     simp only [eval_pow, eval_X, one_pow, sum_const, card_range, Nat.smul_one_eq_coe,
@@ -383,7 +383,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBa
     z ∈ adjoin R ({B.gen} : Set L) := by
   induction' n with n hn
   · simpa using hz
-  · rw [_root_.pow_succ, mul_smul] at hz
+  · rw [_root_.pow_succ', mul_smul] at hz
     exact
       hn (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (hzint.smul _) hz hei)
 #align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -123,9 +123,7 @@ end Cyclotomic
 section IsIntegral
 
 variable {K : Type v} {L : Type z} {p : R} [CommRing R] [Field K] [Field L]
-
 variable [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [IsSeparable K L]
-
 variable [IsDomain R] [IsFractionRing R K] [IsIntegrallyClosed R]
 
 local notation "𝓟" => Submodule.span R {(p : R)}

• Prove isSemisimple_of_mem_adjoin: if two commuting endomorphisms of a finite-dimensional vector space over a perfect field are both semisimple, then every endomorphism in the algebra generated by them (in particular their product and sum) is semisimple.

• In the same file LinearAlgebra/Semisimple.lean, eq_zero_of_isNilpotent_isSemisimple and isSemisimple_of_squarefree_aeval_eq_zero are golfed, and IsSemisimple.minpoly_squarefree is proved

RingTheory/SimpleModule.lean:

• Define IsSemisimpleRing R to mean that R is a semisimple R-module. add properties of simple modules and a characterization (they are exactly the quotients of the ring by maximal left ideals).

• The annihilator of a semisimple module is a radical ideal.

• Any module over a semisimple ring is semisimple.

• A finite product of semisimple rings is semisimple.

• Any quotient of a semisimple ring is semisimple.

• Add Artin--Wedderburn as a TODO (proof_wanted).

• Order/Atoms.lean: add the instance from IsSimpleOrder to ComplementedLattice, so that IsSimpleModule → IsSemisimpleModule is automatically inferred.

Prerequisites for showing a product of semisimple rings is semisimple:

• Algebra/Module/Submodule/Map.lean: generalize orderIsoMapComap so that it only requires RingHomSurjective rather than RingHomInvPair

• Algebra/Ring/CompTypeclasses.lean, Mathlib/Algebra/Ring/Pi.lean, Algebra/Ring/Prod.lean: add RingHomSurjective instances

RingTheory/Artinian.lean:

• quotNilradicalEquivPi: the quotient of a commutative Artinian ring R by its nilradical is isomorphic to the (finite) product of its quotients by maximal ideals (therefore a product of fields). equivPi: if the ring is moreover reduced, then the ring itself is a product of fields. Deduce that R is a semisimple ring and both R and R[X] are decomposition monoids. Requires RingEquiv.quotientBot in RingTheory/Ideal/QuotientOperations.lean.

• Data/Polynomial/Eval.lean: the polynomial ring over a finite product of rings is isomorphic to the product of polynomial rings over individual rings. (Used to show R[X] is a decomposition monoid.)

Other necessary results:

• FieldTheory/Minpoly/Field.lean: the minimal polynomial of an element in a reduced algebra over a field is radical.

• RingTheory/PowerBasis.lean: generalize PowerBasis.finiteDimensional and rename it to .finite.

Annihilator stuff, some of which do not end up being used:

• RingTheory/Ideal/Operations.lean: define Module.annihilator and redefine Submodule.annihilator in terms of it; add lemmas, including one that says an arbitrary intersection of radical ideals is radical. The new lemma Ideal.isRadical_iff_pow_one_lt depends on pow_imp_self_of_one_lt in Mathlib/Data/Nat/Interval.lean, which is also used to golf the proof of isRadical_iff_pow_one_lt.

• Algebra/Module/Torsion.lean: add a lemma and an instance (unused)

• Data/Polynomial/Module/Basic.lean: add a def (unused) and a lemma

• LinearAlgebra/AnnihilatingPolynomial.lean: add lemma span_minpoly_eq_annihilator

Some results about idempotent linear maps (projections) and idempotent elements, used to show that any (left) ideal in a semisimple ring is spanned by an idempotent element (unused):

• LinearAlgebra/Projection.lean: add def isIdempotentElemEquiv

• LinearAlgebra/Span.lean: add two lemmas

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -140,7 +140,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
     (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
     (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 := by
   -- First define some abbreviations.
-  letI := B.finiteDimensional
+  letI := B.finite
   let P := minpoly R B.gen
   obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
   have finrank_K_L : FiniteDimensional.finrank K L = B.dim := B.finrank
@@ -243,7 +243,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
   -- First define some abbreviations.
   have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
     hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
-  letI := finiteDimensional B
+  have := B.finite
   set P := minpoly R B.gen with hP
   obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
   haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L

I ran tryAtEachStep on all files under Mathlib to find all locations where omega succeeds. For each that was a linarith without an only, I tried replacing it with omega, and I verified that elaboration time got smaller. (In almost all cases, there was a noticeable speedup.) I also replaced some slow aesops along the way.

@@ -328,7 +328,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
         (minpoly.monic hBint).natDegree_map (algebraMap R K), ←
         minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, natDegree_minpoly, hn, Nat.sub_one,
         Nat.pred_succ]
-      linarith
+      omega
 
     -- Using `hQ : aeval B.gen Q = p • z`, we write `p • z` as a sum of terms of degree less than
     -- `j+1`, that are multiples of `p` by induction, and terms of degree at least `j+1`.

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>

@@ -253,8 +253,8 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
   rw [adjoin_singleton_eq_range_aeval] at hz
   obtain ⟨Q₁, hQ⟩ := hz
   set Q := Q₁ %ₘ P with hQ₁
-  replace hQ : aeval B.gen Q = p • z
-  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
+  replace hQ : aeval B.gen Q = p • z := by
+    rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
     simpa using hQ
   by_cases hQzero : Q = 0
   · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
@@ -290,8 +290,9 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
     -- By induction hypothesis we can find `g : ℕ → R` such that
     -- `k ∈ range (j + 1) → Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k`-
     choose! g hg using hind
-    replace hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k = algebraMap R L p * g k • B.gen ^ k
-    · intro k hk
+    replace hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k =
+        algebraMap R L p * g k • B.gen ^ k := by
+      intro k hk
       rw [hg k (mem_range_succ_iff.1 hk)
         (mem_range_succ_iff.2
           (le_trans (mem_range_succ_iff.1 hk) (succ_le_iff.1 (mem_range_succ_iff.1 hj)).le)),

• Introduce typeclass DecompositionMonoid, which says every element in the monoid is primal, i.e., whenever an element divides a product b * c, it can be factored into a product such that the factors divides b and c respectively. A domain is called pre-Schreier if its multiplicative monoid is a decomposition monoid, and these are more general than GCD domains.

• Show that any GCDMonoid is a DecompositionMonoid. In order for lemmas about DecompositionMonoids to automatically apply to UniqueFactorizationMonoids, we add instances from UniqueFactorizationMonoid α to Nonempty (NormalizedGCDMonoid α) to Nonempty (GCDMonoid α) to DecompositionMonoid α. (Zulip) See the bottom of message for an updated diagram of classes and instances.

• Introduce binary predicate IsRelPrime which says that the only common divisors of the two elements are units. Replace previous occurrences in mathlib by this predicate.

• Duplicate all lemmas about IsCoprime in Coprime/Basic (except three lemmas about smul) to IsRelPrime. Due to import constraints, they are spread into three files Algebra/Divisibility/Units (including key lemmas assuming DecompositionMonoid), GroupWithZero/Divisibility, and Coprime/Basic.

• Show IsCoprime always imply IsRelPrime and is equivalent to it in Bezout rings. To reduce duplication, the definition of Bezout rings and the GCDMonoid instance are moved from RingTheory/Bezout to RingTheory/PrincipalIdealDomain, and some results in PrincipalIdealDomain are generalized to Bezout rings.

• Remove the recently added file Squarefree/UniqueFactorizationMonoid and place the results appropriately within Squarefree/Basic. All results are generalized to DecompositionMonoid or weaker except the last one.

Zulip

With this PR, all the following instances (indicated by arrows) now work; this PR fills the central part.

                                                                          EuclideanDomain (bundled)
                                                                              ↙          ↖
                                                                 IsPrincipalIdealRing ← Field (bundled)
                                                                            ↓             ↓
         NormalizationMonoid ←          NormalizedGCDMonoid → GCDMonoid  IsBezout ← ValuationRing ← DiscreteValuationRing
                   ↓                             ↓                 ↘       ↙
Nonempty NormalizationMonoid ← Nonempty NormalizedGCDMonoid →  Nonempty GCDMonoid → IsIntegrallyClosed
                                                 ↑                    ↓
                    WfDvdMonoid ← UniqueFactorizationMonoid → DecompositionMonoid
                                                 ↑
                                       IsPrincipalIdealRing

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Oliver Nash <github@olivernash.org>

@@ -370,7 +370,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
     have hppdiv : p ^ B.dim ∣ p ^ B.dim * r := dvd_mul_of_dvd_left dvd_rfl _
     rwa [← IsFractionRing.injective R K hQ, mul_comm, ← Units.coe_neg_one, mul_pow, ←
       Units.val_pow_eq_pow_val, ← Units.val_pow_eq_pow_val, mul_assoc,
-      IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv
+      Units.dvd_mul_left, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv
 #align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt
 
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable

@@ -95,7 +95,7 @@ theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (
       · simp only [ite_eq_right_iff]
         rintro ⟨k, hk⟩
         rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
-          natDegree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _), Nat.succ_sub_one]
+          natDegree_cyclotomic, Nat.totient_prime_pow hp.out (Nat.succ_pos _), Nat.add_one_sub_one]
           at hn hi
         rw [hk, pow_succ, mul_assoc] at hi
         rw [hk, mul_comm, Nat.mul_div_cancel _ hp.out.pos]

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

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

@@ -44,7 +44,7 @@ open Polynomial
 theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 := by
   refine Monic.isEisensteinAt_of_mem_of_not_mem ?_
-      (Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (by exact_mod_cast hp.out.ne_zero)).2 <|
+      (Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (mod_cast hp.out.ne_zero)).2 <|
         Nat.prime_iff_prime_int.1 hp.out) (fun {i hi} => ?_) ?_
   · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
     refine (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => ?_
@@ -77,7 +77,7 @@ set_option linter.uppercaseLean3 false in
 theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
     ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 := by
   refine Monic.isEisensteinAt_of_mem_of_not_mem ?_
-      (Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (by exact_mod_cast hp.out.ne_zero)).2 <|
+      (Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (mod_cast hp.out.ne_zero)).2 <|
         Nat.prime_iff_prime_int.1 hp.out) ?_ ?_
   · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp]
     refine (cyclotomic.monic _ ℤ).comp (monic_X_add_C 1) fun h => ?_

Zulip

Initially I just wanted to add more dot notations for IsIntegral and IsAlgebraic (done in #8437); then I noticed near-duplicates Algebra.isIntegral_of_finite [Field R] [Ring A] and RingHom.IsIntegral.of_finite [CommRing R] [CommRing A] so I went on to generalize the latter to cover the former, and generalized everything in the IntegralClosure file to the noncommutative case whenever possible.

In the process I noticed more golfs, which result in this PR. Most notably, isIntegral_of_mem_of_FG is now proven using Cayley-Hamilton and doesn't depend on the Noetherian case isIntegral_of_noetherian; the latter is now proven using the former. In total the golfs makes mathlib 227 lines leaner (+487 -714).

The main changes are in the single file RingTheory/IntegralClosure:

• Change the definition of Algebra.IsIntegral which makes it unfold to IsIntegral rather than RingHom.IsIntegralElem because the former has much more APIs.

• Fix lemma names involving is_integral which are actually about IsIntegralElem: RingHom.is_integral_map → RingHom.isIntegralElem_map RingHom.is_integral_of_mem_closure → RingHom.IsIntegralElem.of_mem_closure RingHom.is_integral_zero/one → RingHom.isIntegralElem_zero/one RingHom.is_integral_add/neg/sub/mul/of_mul_unit → RingHom.IsIntegralElem.add/neg/sub/mul/of_mul_unit

• Add a lemma Algebra.IsIntegral.of_injective.

• Move isIntegral_of_(submodule_)noetherian down and golf them.

• Remove (Algebra.)isIntegral_of_finite that work only over fields, in favor of the more general (Algebra.)isIntegral.of_finite.

• Merge duplicate lemmas isIntegral_of_isScalarTower and isIntegral_tower_top_of_isIntegral into IsIntegral.tower_top.

• Golf IsIntegral.of_mem_of_fg by first proving IsIntegral.of_finite using Cayley-Hamilton.

• Add a docstring mentioning the Kurosh problem at Algebra.IsIntegral.finite. The negative solution to the problem means the theorem doesn't generalize to noncommutative algebras.

• Golf IsIntegral.tmul and isField_of_isIntegral_of_isField(').

• Combine isIntegral_trans_aux into isIntegral_trans and golf.

• Add Algebra namespace to isIntegral_sup.

• rename lemmas for dot notation: RingHom.isIntegral_trans → RingHom.IsIntegral.trans RingHom.isIntegral_quotient/tower_bot/top_of_isIntegral → RingHom.IsIntegral.quotient/tower_bot/top isIntegral_of_mem_closure' → IsIntegral.of_mem_closure' (and the '' version) isIntegral_of_surjective → Algebra.isIntegral_of_surjective

The next changed file is RingTheory/Algebraic:

• Rename: of_larger_base → tower_top (for consistency with IsIntegral) Algebra.isAlgebraic_of_finite → Algebra.IsAlgebraic.of_finite Algebra.isAlgebraic_trans → Algebra.IsAlgebraic.trans

• Add new lemmasAlgebra.IsIntegral.isAlgebraic, isAlgebraic_algHom_iff, and Algebra.IsAlgebraic.of_injective to streamline some proofs.

The generalization from CommRing to Ring requires an additional lemma scaleRoots_eval₂_mul_of_commute in Polynomial/ScaleRoots.

A lemma Algebra.lmul_injective is added to Algebra/Bilinear (in order to golf the proof of IsIntegral.of_mem_of_fg).

In all other files, I merely fix the changed names, or use newly available dot notations.

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -174,9 +174,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
   have hintsum :
     IsIntegral R
       (z * B.gen ^ n - ∑ x : ℕ in (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n)) := by
-    refine
-      IsIntegral.sub (IsIntegral.mul hzint (IsIntegral.pow hBint _))
-        (IsIntegral.sum _ fun i hi => IsIntegral.smul _ ?_)
+    refine (hzint.mul (hBint.pow _)).sub (.sum _ fun i hi => .smul _ ?_)
     exact adjoin_le_integralClosure hBint (hf _ (aux i hi)).1
   obtain ⟨r, hr⟩ := isIntegral_iff.1 (isIntegral_norm K hintsum)
   use r
@@ -351,10 +349,9 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
         (∑ x : ℕ in (range (Q.natDegree - j)).erase 0,
           Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
             ∑ x : ℕ in range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2)))) := by
-      refine IsIntegral.sub (IsIntegral.mul hzint (IsIntegral.pow hBint _))
-          (IsIntegral.add (IsIntegral.sum _ fun k hk => IsIntegral.smul _ ?_)
-            (IsIntegral.sum _ fun k _ =>
-              IsIntegral.mul (IsIntegral.smul _ (IsIntegral.pow hBint _)) (IsIntegral.pow hBint _)))
+      refine (hzint.mul (hBint.pow _)).sub
+        (.add (.sum _ fun k hk => .smul _ ?_)
+          (.sum _ fun k _ => .mul (.smul _ (.pow hBint _)) (hBint.pow _)))
       refine adjoin_le_integralClosure hBint (hf _ ?_).1
       rw [(minpoly.monic hBint).natDegree_map (algebraMap R L)]
       rw [add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
@@ -389,9 +386,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBa
   · simpa using hz
   · rw [_root_.pow_succ, mul_smul] at hz
     exact
-      hn
-        (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint
-          (IsIntegral.smul _ hzint) hz hei)
+      hn (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (hzint.smul _) hz hei)
 #align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt
 
 end IsIntegral

This PR tests a string-based tool for renaming declarations.

Inspired by this Zulip thread, I am trying to reduce the diff of #8406.

This PR makes the following renames:

| From | To |

@@ -175,8 +175,8 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
     IsIntegral R
       (z * B.gen ^ n - ∑ x : ℕ in (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n)) := by
     refine
-      isIntegral_sub (isIntegral_mul hzint (IsIntegral.pow hBint _))
-        (IsIntegral.sum _ fun i hi => isIntegral_smul _ ?_)
+      IsIntegral.sub (IsIntegral.mul hzint (IsIntegral.pow hBint _))
+        (IsIntegral.sum _ fun i hi => IsIntegral.smul _ ?_)
     exact adjoin_le_integralClosure hBint (hf _ (aux i hi)).1
   obtain ⟨r, hr⟩ := isIntegral_iff.1 (isIntegral_norm K hintsum)
   use r
@@ -351,10 +351,10 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis
         (∑ x : ℕ in (range (Q.natDegree - j)).erase 0,
           Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
             ∑ x : ℕ in range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2)))) := by
-      refine isIntegral_sub (isIntegral_mul hzint (IsIntegral.pow hBint _))
-          (isIntegral_add (IsIntegral.sum _ fun k hk => isIntegral_smul _ ?_)
+      refine IsIntegral.sub (IsIntegral.mul hzint (IsIntegral.pow hBint _))
+          (IsIntegral.add (IsIntegral.sum _ fun k hk => IsIntegral.smul _ ?_)
             (IsIntegral.sum _ fun k _ =>
-              isIntegral_mul (isIntegral_smul _ (IsIntegral.pow hBint _)) (IsIntegral.pow hBint _)))
+              IsIntegral.mul (IsIntegral.smul _ (IsIntegral.pow hBint _)) (IsIntegral.pow hBint _)))
       refine adjoin_le_integralClosure hBint (hf _ ?_).1
       rw [(minpoly.monic hBint).natDegree_map (algebraMap R L)]
       rw [add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
@@ -391,7 +391,7 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBa
     exact
       hn
         (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint
-          (isIntegral_smul _ hzint) hz hei)
+          (IsIntegral.smul _ hzint) hz hei)
 #align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt
 
 end IsIntegral

Also _root_.map_smul when in the neighbourhood.

@@ -50,7 +50,7 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     refine (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => ?_
     rw [natDegree_X_add_C] at h
     exact zero_ne_one h.symm
-  · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, LinearMap.map_sum]
+  · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, map_sum]
     conv =>
       congr
       congr

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -216,7 +216,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
     simp
 #align dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt
 
-theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [CommRing B]
+theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type*} [CommSemiring A] [CommRing B]
     [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
     (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
     z ∈ adjoin A ({x} : Set B) := by

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.eisenstein.is_integral
-! leanprover-community/mathlib commit 5bfbcca0a7ffdd21cf1682e59106d6c942434a32
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Nat.Choose.Dvd
 import Mathlib.RingTheory.IntegrallyClosed
 import Mathlib.RingTheory.Norm
 import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
 
+#align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf1682e59106d6c942434a32"
+
 /-!
 # Eisenstein polynomials
 In this file we gather more miscellaneous results about Eisenstein polynomials

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

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

@@ -162,7 +162,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
       p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) by
     have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
       hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
-    refine @Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd R _ _ _ _ n hp (?_ : _ ∣ _)  hndiv
+    refine @Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd R _ _ _ _ n hp (?_ : _ ∣ _) hndiv
     convert (IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
     push_cast
     ring_nf

@@ -18,11 +18,11 @@ import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
 In this file we gather more miscellaneous results about Eisenstein polynomials
 
 ## Main results
-* `mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at`: let `K` be the field of fraction
+* `mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt`: let `K` be the field of fraction
   of an integrally closed domain `R` and let `L` be a separable extension of `K`, generated by an
   integral power basis `B` such that the minimal polynomial of `B.gen` is Eisenstein at `p`. Given
   `z : L` integral over `R`, if `p ^ n • z ∈ adjoin R {B.gen}`, then `z ∈ adjoin R {B.gen}`.
-  Together with `algebra.discr_mul_is_integral_mem_adjoin` this result often allows to compute the
+  Together with `Algebra.discr_mul_isIntegral_mem_adjoin` this result often allows to compute the
   ring of integers of `L`.
 
 -/
@@ -40,13 +40,11 @@ section Cyclotomic
 
 variable (p : ℕ)
 
--- Porting note: to prevent Lean complaigning about this notation
-set_option quotPrecheck false
-local notation "𝓟" => Submodule.span ℤ {↑p}
+local notation "𝓟" => Submodule.span ℤ {(p : ℤ)}
 
 open Polynomial
 
-theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
+theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
     ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 := by
   refine Monic.isEisensteinAt_of_mem_of_not_mem ?_
       (Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (by exact_mod_cast hp.out.ne_zero)).2 <|
@@ -77,9 +75,9 @@ theorem cyclotomic_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] :
     rw [mul_right_inj' hp.out.ne_zero] at hk
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 set_option linter.uppercaseLean3 false in
-#align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_x_add_one_isEisensteinAt
+#align cyclotomic_comp_X_add_one_is_eisenstein_at cyclotomic_comp_X_add_one_isEisensteinAt
 
-theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
+theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
     ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 := by
   refine Monic.isEisensteinAt_of_mem_of_not_mem ?_
       (Ideal.IsPrime.ne_top <| (Ideal.span_singleton_prime (by exact_mod_cast hp.out.ne_zero)).2 <|
@@ -91,7 +89,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
   · induction' n with n hn
     · intro i hi
       rw [Nat.zero_add, pow_one] at hi ⊢
-      exact (cyclotomic_comp_x_add_one_isEisensteinAt p).mem hi
+      exact (cyclotomic_comp_X_add_one_isEisensteinAt p).mem hi
     · intro i hi
       rw [Ideal.submodule_span_eq, Ideal.mem_span_singleton, ← ZMod.int_cast_zmod_eq_zero_iff_dvd,
         show ↑(_  : ℤ) = Int.castRingHom (ZMod p) _ by rfl, ← coeff_map, map_comp, map_cyclotomic,
@@ -121,7 +119,7 @@ theorem cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt [hp : Fact p.Prime] (
     rw [mul_right_inj' hp.out.ne_zero] at hk
     exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
 set_option linter.uppercaseLean3 false in
-#align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_x_add_one_isEisensteinAt
+#align cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt
 
 end Cyclotomic
 
@@ -133,9 +131,7 @@ variable [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [IsSepa
 
 variable [IsDomain R] [IsFractionRing R K] [IsIntegrallyClosed R]
 
--- Porting note: to prevent Lean complaigning about this notation
-set_option quotPrecheck false
-local notation "𝓟" => Submodule.span R {p}
+local notation "𝓟" => Submodule.span R {(p : R)}
 
 open IsIntegrallyClosed PowerBasis Nat Polynomial IsScalarTower
 
@@ -143,7 +139,7 @@ open IsIntegrallyClosed PowerBasis Nat Polynomial IsScalarTower
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `Q : R[X]` is such that
 `aeval B.gen Q = p • z`, then `p ∣ Q.coeff 0`. -/
-theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
+theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
     (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 := by
   -- First define some abbreviations.
@@ -221,7 +217,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B :
   · rw [aeval_eq_sum_range,
       Finset.add_sum_erase (range (Q.natDegree + 1)) fun i => Q.coeff i • B.gen ^ i]
     simp
-#align dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at
+#align dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt
 
 theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [CommRing B]
     [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
@@ -245,7 +241,7 @@ theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type _} [CommSemiring A] [Co
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
 `z ∈ adjoin R {B.gen}`. -/
-theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
+theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
     (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
     z ∈ adjoin R ({B.gen} : Set L) := by
@@ -276,15 +272,15 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     · rw [H₁]
       exact Subalgebra.zero_mem _
   -- It is enough to prove that all coefficients of `Q` are divisible by `p`, by induction.
-  -- The base case is `dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at`.
+  -- The base case is `dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt`.
   refine mem_adjoin_of_dvd_coeff_of_dvd_aeval hp.ne_zero (fun i => ?_) hQ
   induction' i using Nat.case_strong_induction_on with j hind
   · intro _
-    exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at hp hBint hQ hzint hei
+    exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt hp hBint hQ hzint hei
   · intro hj
     convert hp.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd _ hndiv
     exact n
-    -- Two technical results we will need about `P.nat_degree` and `Q.nat_degree`.
+    -- Two technical results we will need about `P.natDegree` and `Q.natDegree`.
     have H := degree_modByMonic_lt Q₁ (minpoly.monic hBint)
     rw [← hQ₁, ← hP] at H
     replace H := Nat.lt_iff_add_one_le.1
@@ -342,7 +338,7 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     -- `j+1`, that are multiples of `p` by induction, and terms of degree at least `j+1`.
     rw [aeval_eq_sum_range, Hj, range_add, sum_union (disjoint_range_addLeftEmbedding _ _),
       sum_congr rfl hg, add_comm] at hQ
-    -- We multiply this equality by `B.gen ^ (P.nat_degree-(j+2))`, so we can use `hf₁` on the terms
+    -- We multiply this equality by `B.gen ^ (P.natDegree-(j+2))`, so we can use `hf₁` on the terms
     -- we didn't know were multiples of `p`, and we take the norm on both sides.
     replace hQ := congr_arg (fun x => x * B.gen ^ (P.natDegree - (j + 2))) hQ
     simp_rw [sum_map, addLeftEmbedding_apply, add_mul, sum_mul, mul_assoc] at hQ
@@ -381,14 +377,14 @@ theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : PowerBas
     rwa [← IsFractionRing.injective R K hQ, mul_comm, ← Units.coe_neg_one, mul_pow, ←
       Units.val_pow_eq_pow_val, ← Units.val_pow_eq_pow_val, mul_assoc,
       IsUnit.dvd_mul_left _ _ _ ⟨_, rfl⟩, mul_comm, ← Nat.succ_eq_add_one, hn] at hppdiv
-#align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at
+#align mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt
 
 /-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
 extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
 `B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p ^ n • z ∈ adjoin R {B.gen}`,
-then `z ∈ adjoin R {B.gen}`. Together with `algebra.discr_mul_is_integral_mem_adjoin` this result
+then `z ∈ adjoin R {B.gen}`. Together with `Algebra.discr_mul_isIntegral_mem_adjoin` this result
 often allows to compute the ring of integers of `L`. -/
-theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : PowerBasis K L}
+theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
     (hp : Prime p) (hBint : IsIntegral R B.gen) {n : ℕ} {z : L} (hzint : IsIntegral R z)
     (hz : p ^ n • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
     z ∈ adjoin R ({B.gen} : Set L) := by
@@ -397,8 +393,8 @@ theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : Powe
   · rw [_root_.pow_succ, mul_smul] at hz
     exact
       hn
-        (mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at hp hBint
+        (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint
           (isIntegral_smul _ hzint) hz hei)
-#align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at
+#align mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt
 
 end IsIntegral

Fix also some names in RingTheory.Polynomial.Eisenstein.Basic that are in the wrong namespace.