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

@@ -51,7 +51,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     by
     rcases hlim 2 one_lt_two with ⟨c, cgrowth, ctop, clim⟩
     have : tendsto (fun n => u 0 / c n) at_top (𝓝 0) :=
-      tendsto_const_nhds.div_at_top (tendsto_nat_cast_atTop_iff.2 Ctop)
+      tendsto_const_nhds.div_at_top (tendsto_natCast_atTop_iff.2 Ctop)
     apply le_of_tendsto_of_tendsto' this clim fun n => _
     simp_rw [div_eq_inv_mul]
     exact mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))
@@ -313,7 +313,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       by
       apply div_le_div _ _ _ le_rfl
       · apply Real.rpow_nonneg (sq_nonneg _)
-      · rw [← Real.rpow_nat_cast]
+      · rw [← Real.rpow_natCast]
         apply Real.rpow_le_rpow_of_exponent_ge A
         · exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)
         · exact (Nat.sub_one_lt_floor _).le

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

@@ -62,7 +62,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     have L : ∀ᶠ n in at_top, u (c n) - c n * l ≤ ε * c n :=
       by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
-        Asymptotics.isLittleO_iff] at clim 
+        Asymptotics.isLittleO_iff] at clim
       filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)] with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
@@ -70,7 +70,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
         _ ≤ ε * c n := by
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
-          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
+          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
           exact le_trans (le_abs_self _) hn
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
@@ -100,7 +100,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       have A : a ≤ N - 1 := by linarith only [aN, Npos]
       have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
       have := (ha _ A).1
-      rwa [B] at this 
+      rwa [B] at this
     calc
       u n - n * l ≤ u (c N) - c (N - 1) * l :=
         by
@@ -128,7 +128,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     have L : ∀ᶠ n : ℕ in at_top, (c n : ℝ) * l - u (c n) ≤ ε * c n :=
       by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
-        Asymptotics.isLittleO_iff] at clim 
+        Asymptotics.isLittleO_iff] at clim
       filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)] with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
@@ -136,7 +136,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
         _ ≤ ε * c n := by
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
-          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
+          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
           exact le_trans (neg_le_abs _) hn
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ,
@@ -173,7 +173,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
         have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
         have := (ha _ aN').1
-        rwa [B] at this 
+        rwa [B] at this
       _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
       _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
       _ = ε * (1 + l) * c (N - 1) := by ring
@@ -188,7 +188,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         by
         apply tendsto.mono_left _ nhdsWithin_le_nhds
         exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
-      simp only [MulZeroClass.zero_mul, add_zero] at L 
+      simp only [MulZeroClass.zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 l hd).And self_mem_nhdsWithin).exists
     filter_upwards [B ε εpos, Ioi_mem_at_top 0] with n hn npos
     simp_rw [div_eq_inv_mul]
@@ -211,7 +211,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         exact
           tendsto_const_nhds.add
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
-      simp only [MulZeroClass.zero_mul, add_zero] at L 
+      simp only [MulZeroClass.zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 d hd).And self_mem_nhdsWithin).exists
     filter_upwards [A ε εpos, Ioi_mem_at_top 0] with n hn npos
     simp_rw [div_eq_inv_mul]
@@ -258,7 +258,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
       exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
   have B : tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / ⌊c k ^ n⌋₊) at_top (𝓝 (c k)) :=
     by
-    simp only [one_mul, div_one] at A 
+    simp only [one_mul, div_one] at A
     convert A
     ext1 n
     simp (disch :=
@@ -294,7 +294,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       refine'
         sum_le_sum_of_subset_of_nonneg _ fun i hi hident => div_nonneg zero_le_one (sq_nonneg _)
       intro i hi
-      simp only [mem_filter, mem_range] at hi 
+      simp only [mem_filter, mem_range] at hi
       simp only [hi.1, mem_Ico, and_true_iff]
       apply Nat.floor_le_of_le
       apply le_of_lt
@@ -371,7 +371,7 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
       by
       apply sum_le_sum_of_subset_of_nonneg
       · intro i hi
-        simp only [mem_filter, mem_range] at hi 
+        simp only [mem_filter, mem_range] at hi
         simpa only [hi.1, mem_filter, mem_range, true_and_iff] using
           hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))
       · intro i hi hident

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

@@ -137,7 +137,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         _ ≤ ε * c n := by
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
           simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
-          exact le_trans (neg_le_abs_self _) hn
+          exact le_trans (neg_le_abs _) hn
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ,
         ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=

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

@@ -312,7 +312,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     _ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c⁻¹ ^ 2) :=
       by
       apply div_le_div _ _ _ le_rfl
-      · apply Real.rpow_nonneg_of_nonneg (sq_nonneg _)
+      · apply Real.rpow_nonneg (sq_nonneg _)
       · rw [← Real.rpow_nat_cast]
         apply Real.rpow_le_rpow_of_exponent_ge A
         · exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)

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

@@ -286,7 +286,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     have : c ^ 3 = c ^ 2 * c := by ring
     simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
     rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
-    simpa using pow_le_pow hc.le one_le_two
+    simpa using pow_le_pow_right hc.le one_le_two
   calc
     ∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2 ≤
         ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, 1 / (c ^ i) ^ 2 :=
@@ -351,7 +351,7 @@ theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹)
     (1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
     _ ≤ c ^ i - 1 := by
       simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using
-        pow_le_pow hc.le hident
+        pow_le_pow_right hc.le hident
     _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
 #align mul_pow_le_nat_floor_pow mul_pow_le_nat_floor_pow
 -/
@@ -385,7 +385,7 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
         simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
       · exact sq_pos_of_pos (pow_pos cpos _)
       rw [one_mul, ← mul_pow]
-      apply pow_le_pow_of_le_left (pow_nonneg cpos.le _)
+      apply pow_le_pow_left (pow_nonneg cpos.le _)
       rw [← div_eq_inv_mul, le_div_iff A, mul_comm]
       exact mul_pow_le_nat_floor_pow hc i
     _ ≤ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2 :=

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

@@ -84,7 +84,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     let N := Nat.find exN
     have ncN : n < c N := Nat.find_spec exN
     have aN : a + 1 ≤ N := by
-      by_contra' h
+      by_contra! h
       have cNM : c N ≤ M := by
         apply le_max'
         apply mem_image_of_mem
@@ -151,7 +151,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     let N := Nat.find exN
     have ncN : n < c N := Nat.find_spec exN
     have aN : a + 1 ≤ N := by
-      by_contra' h
+      by_contra! h
       have cNM : c N ≤ M := by
         apply le_max'
         apply mem_image_of_mem

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Analysis.SpecificLimits.Basic
-import Mathbin.Analysis.SpecialFunctions.Pow.Real
+import Analysis.SpecificLimits.Basic
+import Analysis.SpecialFunctions.Pow.Real
 
 #align_import analysis.specific_limits.floor_pow from "leanprover-community/mathlib"@"599fffe78f0e11eb6a034e834ec51882167b9688"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.specific_limits.floor_pow
-! leanprover-community/mathlib commit 599fffe78f0e11eb6a034e834ec51882167b9688
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecificLimits.Basic
 import Mathbin.Analysis.SpecialFunctions.Pow.Real
 
+#align_import analysis.specific_limits.floor_pow from "leanprover-community/mathlib"@"599fffe78f0e11eb6a034e834ec51882167b9688"
+
 /-!
 # Results on discretized exponentials
 

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

@@ -30,6 +30,7 @@ open Filter Finset
 
 open scoped Topology BigOperators
 
+#print tendsto_div_of_monotone_of_exists_subseq_tendsto_div /-
 /-- If a monotone sequence `u` is such that `u n / n` tends to a limit `l` along subsequences with
 exponential growth rate arbitrarily close to `1`, then `u n / n` tends to `l`. -/
 theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
@@ -227,8 +228,10 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         rwa [inv_mul_cancel, one_mul]
         exact Nat.cast_ne_zero.2 (ne_of_gt npos)
 #align tendsto_div_of_monotone_of_exists_subseq_tendsto_div tendsto_div_of_monotone_of_exists_subseq_tendsto_div
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic tactic.field_simp.ne_zero -/
+#print tendsto_div_of_monotone_of_tendsto_div_floor_pow /-
 /-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
 `c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
 `c`s converging to `1`. -/
@@ -269,7 +272,9 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
   filter_upwards [(tendsto_order.1 B).2 a hk] with n hn
   exact (div_le_iff (H n)).1 hn.le
 #align tendsto_div_of_monotone_of_tendsto_div_floor_pow tendsto_div_of_monotone_of_tendsto_div_floor_pow
+-/
 
+#print sum_div_pow_sq_le_div_sq /-
 /-- The sum of `1/(c^i)^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
 constant. -/
 theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
@@ -336,7 +341,9 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       apply div_le_div _ B (sq_pos_of_pos hj) le_rfl
       exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)
 #align sum_div_pow_sq_le_div_sq sum_div_pow_sq_le_div_sq
+-/
 
+#print mul_pow_le_nat_floor_pow /-
 theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ :=
   by
   have cpos : 0 < c := zero_lt_one.trans hc
@@ -350,7 +357,9 @@ theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹)
         pow_le_pow hc.le hident
     _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
 #align mul_pow_le_nat_floor_pow mul_pow_le_nat_floor_pow
+-/
 
+#print sum_div_nat_floor_pow_sq_le_div_sq /-
 /-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
 constant. -/
 theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
@@ -392,4 +401,5 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
       field_simp [cpos.ne', (sub_pos.2 hc).ne']
       ring
 #align sum_div_nat_floor_pow_sq_le_div_sq sum_div_nat_floor_pow_sq_le_div_sq
+-/
 

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

@@ -273,7 +273,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
 /-- The sum of `1/(c^i)^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
 constant. -/
 theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
-    (∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2) ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 :=
+    ∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 :=
   by
   have cpos : 0 < c := zero_lt_one.trans hc
   have A : 0 < c⁻¹ ^ 2 := sq_pos_of_pos (inv_pos.2 cpos)
@@ -286,7 +286,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
     simpa using pow_le_pow hc.le one_le_two
   calc
-    (∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2) ≤
+    ∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2 ≤
         ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, 1 / (c ^ i) ^ 2 :=
       by
       refine'
@@ -354,13 +354,13 @@ theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹)
 /-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
 constant. -/
 theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
-    (∑ i in (range N).filterₓ fun i => j < ⌊c ^ i⌋₊, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2) ≤
+    ∑ i in (range N).filterₓ fun i => j < ⌊c ^ i⌋₊, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2 ≤
       c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 :=
   by
   have cpos : 0 < c := zero_lt_one.trans hc
   have A : 0 < 1 - c⁻¹ := sub_pos.2 (inv_lt_one hc)
   calc
-    (∑ i in (range N).filterₓ fun i => j < ⌊c ^ i⌋₊, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2) ≤
+    ∑ i in (range N).filterₓ fun i => j < ⌊c ^ i⌋₊, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2 ≤
         ∑ i in (range N).filterₓ fun i => j < c ^ i, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2 :=
       by
       apply sum_le_sum_of_subset_of_nonneg

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

@@ -74,7 +74,6 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
           simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
           exact le_trans (le_abs_self _) hn
-        
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
       eventually_at_top.1 (cgrowth.and L)
@@ -124,7 +123,6 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
         apply mul_nonneg εpos.le
         linarith only [εpos, lnonneg]
-      
   have B : ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in at_top, (n : ℝ) * l - u n ≤ ε * (1 + l) * n :=
     by
     intro ε εpos
@@ -142,7 +140,6 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
           simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
           exact le_trans (neg_le_abs_self _) hn
-        
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ,
         ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
@@ -186,7 +183,6 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         by
         refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
         exact mul_nonneg εpos.le (add_nonneg zero_le_one lnonneg)
-      
   refine' tendsto_order.2 ⟨fun d hd => _, fun d hd => _⟩
   · obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, d + ε * (1 + l) < l ∧ 0 < ε :=
       by
@@ -209,7 +205,6 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         by
         refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
         linarith only [hn]
-      
   · obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, l + ε * (1 + ε + l) < d ∧ 0 < ε :=
       by
       have L : tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) :=
@@ -231,7 +226,6 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       _ < d := by
         rwa [inv_mul_cancel, one_mul]
         exact Nat.cast_ne_zero.2 (ne_of_gt npos)
-      
 #align tendsto_div_of_monotone_of_exists_subseq_tendsto_div tendsto_div_of_monotone_of_exists_subseq_tendsto_div
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic tactic.field_simp.ne_zero -/
@@ -341,7 +335,6 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       by
       apply div_le_div _ B (sq_pos_of_pos hj) le_rfl
       exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)
-    
 #align sum_div_pow_sq_le_div_sq sum_div_pow_sq_le_div_sq
 
 theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ :=
@@ -356,7 +349,6 @@ theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹)
       simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using
         pow_le_pow hc.le hident
     _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
-    
 #align mul_pow_le_nat_floor_pow mul_pow_le_nat_floor_pow
 
 /-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
@@ -399,6 +391,5 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
       congr 1
       field_simp [cpos.ne', (sub_pos.2 hc).ne']
       ring
-    
 #align sum_div_nat_floor_pow_sq_le_div_sq sum_div_nat_floor_pow_sq_le_div_sq
 

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

@@ -65,7 +65,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
         Asymptotics.isLittleO_iff] at clim 
-      filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
+      filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)] with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
         u (c n) - c n * l = (u (c n) / c n - l) * c n := by
@@ -79,7 +79,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
       eventually_at_top.1 (cgrowth.and L)
     let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
-    filter_upwards [Ici_mem_at_top M]with n hn
+    filter_upwards [Ici_mem_at_top M] with n hn
     have exN : ∃ N, n < c N :=
       by
       rcases(tendsto_at_top.1 Ctop (n + 1)).exists with ⟨N, hN⟩
@@ -133,7 +133,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
         Asymptotics.isLittleO_iff] at clim 
-      filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
+      filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)] with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
         (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
@@ -148,7 +148,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
       eventually_at_top.1 (cgrowth.and L)
     let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
-    filter_upwards [Ici_mem_at_top M]with n hn
+    filter_upwards [Ici_mem_at_top M] with n hn
     have exN : ∃ N, n < c N :=
       by
       rcases(tendsto_at_top.1 Ctop (n + 1)).exists with ⟨N, hN⟩
@@ -196,7 +196,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
       simp only [MulZeroClass.zero_mul, add_zero] at L 
       exact (((tendsto_order.1 L).2 l hd).And self_mem_nhdsWithin).exists
-    filter_upwards [B ε εpos, Ioi_mem_at_top 0]with n hn npos
+    filter_upwards [B ε εpos, Ioi_mem_at_top 0] with n hn npos
     simp_rw [div_eq_inv_mul]
     calc
       d < n⁻¹ * n * (l - ε * (1 + l)) :=
@@ -220,7 +220,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
       simp only [MulZeroClass.zero_mul, add_zero] at L 
       exact (((tendsto_order.1 L).2 d hd).And self_mem_nhdsWithin).exists
-    filter_upwards [A ε εpos, Ioi_mem_at_top 0]with n hn npos
+    filter_upwards [A ε εpos, Ioi_mem_at_top 0] with n hn npos
     simp_rw [div_eq_inv_mul]
     calc
       (n : ℝ)⁻¹ * u n ≤ (n : ℝ)⁻¹ * (n * l + ε * (1 + ε + l) * n) :=
@@ -272,7 +272,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
         tactic.field_simp.ne_zero) only [(zero_lt_one.trans (cone k)).ne',
       Ne.def, not_false_iff, (H n).ne', field_simps]
     ring
-  filter_upwards [(tendsto_order.1 B).2 a hk]with n hn
+  filter_upwards [(tendsto_order.1 B).2 a hk] with n hn
   exact (div_le_iff (H n)).1 hn.le
 #align tendsto_div_of_monotone_of_tendsto_div_floor_pow tendsto_div_of_monotone_of_tendsto_div_floor_pow
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.specific_limits.floor_pow
-! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
+! leanprover-community/mathlib commit 599fffe78f0e11eb6a034e834ec51882167b9688
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.SpecialFunctions.Pow.Real
 /-!
 # Results on discretized exponentials
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We state several auxiliary results pertaining to sequences of the form `⌊c^n⌋₊`.
 
 * `tendsto_div_of_monotone_of_tendsto_div_floor_pow`: If a monotone sequence `u` is such that

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

@@ -61,7 +61,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     have L : ∀ᶠ n in at_top, u (c n) - c n * l ≤ ε * c n :=
       by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
-        Asymptotics.isLittleO_iff] at clim
+        Asymptotics.isLittleO_iff] at clim 
       filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
@@ -69,7 +69,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
         _ ≤ ε * c n := by
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
-          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
+          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
           exact le_trans (le_abs_self _) hn
         
     obtain ⟨a, ha⟩ :
@@ -100,7 +100,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       have A : a ≤ N - 1 := by linarith only [aN, Npos]
       have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
       have := (ha _ A).1
-      rwa [B] at this
+      rwa [B] at this 
     calc
       u n - n * l ≤ u (c N) - c (N - 1) * l :=
         by
@@ -129,7 +129,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     have L : ∀ᶠ n : ℕ in at_top, (c n : ℝ) * l - u (c n) ≤ ε * c n :=
       by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
-        Asymptotics.isLittleO_iff] at clim
+        Asymptotics.isLittleO_iff] at clim 
       filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
@@ -137,7 +137,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
         _ ≤ ε * c n := by
           refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
-          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
+          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn 
           exact le_trans (neg_le_abs_self _) hn
         
     obtain ⟨a, ha⟩ :
@@ -175,7 +175,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
         have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
         have := (ha _ aN').1
-        rwa [B] at this
+        rwa [B] at this 
       _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
       _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
       _ = ε * (1 + l) * c (N - 1) := by ring
@@ -191,7 +191,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         by
         apply tendsto.mono_left _ nhdsWithin_le_nhds
         exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
-      simp only [MulZeroClass.zero_mul, add_zero] at L
+      simp only [MulZeroClass.zero_mul, add_zero] at L 
       exact (((tendsto_order.1 L).2 l hd).And self_mem_nhdsWithin).exists
     filter_upwards [B ε εpos, Ioi_mem_at_top 0]with n hn npos
     simp_rw [div_eq_inv_mul]
@@ -215,7 +215,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         exact
           tendsto_const_nhds.add
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
-      simp only [MulZeroClass.zero_mul, add_zero] at L
+      simp only [MulZeroClass.zero_mul, add_zero] at L 
       exact (((tendsto_order.1 L).2 d hd).And self_mem_nhdsWithin).exists
     filter_upwards [A ε εpos, Ioi_mem_at_top 0]with n hn npos
     simp_rw [div_eq_inv_mul]
@@ -261,7 +261,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
       exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
   have B : tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / ⌊c k ^ n⌋₊) at_top (𝓝 (c k)) :=
     by
-    simp only [one_mul, div_one] at A
+    simp only [one_mul, div_one] at A 
     convert A
     ext1 n
     simp (disch :=
@@ -295,7 +295,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       refine'
         sum_le_sum_of_subset_of_nonneg _ fun i hi hident => div_nonneg zero_le_one (sq_nonneg _)
       intro i hi
-      simp only [mem_filter, mem_range] at hi
+      simp only [mem_filter, mem_range] at hi 
       simp only [hi.1, mem_Ico, and_true_iff]
       apply Nat.floor_le_of_le
       apply le_of_lt
@@ -370,7 +370,7 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
       by
       apply sum_le_sum_of_subset_of_nonneg
       · intro i hi
-        simp only [mem_filter, mem_range] at hi
+        simp only [mem_filter, mem_range] at hi 
         simpa only [hi.1, mem_filter, mem_range, true_and_iff] using
           hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))
       · intro i hi hident

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

@@ -25,7 +25,7 @@ We state several auxiliary results pertaining to sequences of the form `⌊c^n
 
 open Filter Finset
 
-open Topology BigOperators
+open scoped Topology BigOperators
 
 /-- If a monotone sequence `u` is such that `u n / n` tends to a limit `l` along subsequences with
 exponential growth rate arbitrarily close to `1`, then `u n / n` tends to `l`. -/

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

@@ -282,8 +282,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
   have A : 0 < c⁻¹ ^ 2 := sq_pos_of_pos (inv_pos.2 cpos)
   have B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹ :=
     by
-    rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]
-    swap
+    rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]; swap
     · exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)
     have : c ^ 3 = c ^ 2 * c := by ring
     simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
@@ -325,8 +324,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       have I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2 :=
         by
         apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)
-        · rw [one_div]
-          exact inv_pos.2 (sq_pos_of_pos hj)
+        · rw [one_div]; exact inv_pos.2 (sq_pos_of_pos hj)
         rw [Real.log_rpow A]
         simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_bit0, Nat.cast_one, mul_neg,
           neg_inj]

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -4,12 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.specific_limits.floor_pow
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecificLimits.Basic
-import Mathbin.Analysis.SpecialFunctions.Pow
+import Mathbin.Analysis.SpecialFunctions.Pow.Real
 
 /-!
 # Results on discretized exponentials

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -60,8 +60,8 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
     have L : ∀ᶠ n in at_top, u (c n) - c n * l ≤ ε * c n :=
       by
-      rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isOCat_one_iff ℝ, Asymptotics.isOCat_iff] at
-        clim
+      rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
+        Asymptotics.isLittleO_iff] at clim
       filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
@@ -128,8 +128,8 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
     have L : ∀ᶠ n : ℕ in at_top, (c n : ℝ) * l - u (c n) ≤ ε * c n :=
       by
-      rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isOCat_one_iff ℝ, Asymptotics.isOCat_iff] at
-        clim
+      rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
+        Asymptotics.isLittleO_iff] at clim
       filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -191,7 +191,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         by
         apply tendsto.mono_left _ nhdsWithin_le_nhds
         exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
-      simp only [zero_mul, add_zero] at L
+      simp only [MulZeroClass.zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 l hd).And self_mem_nhdsWithin).exists
     filter_upwards [B ε εpos, Ioi_mem_at_top 0]with n hn npos
     simp_rw [div_eq_inv_mul]
@@ -215,7 +215,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         exact
           tendsto_const_nhds.add
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
-      simp only [zero_mul, add_zero] at L
+      simp only [MulZeroClass.zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 d hd).And self_mem_nhdsWithin).exists
     filter_upwards [A ε εpos, Ioi_mem_at_top 0]with n hn npos
     simp_rw [div_eq_inv_mul]

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

@@ -114,7 +114,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         · apply mul_le_mul_of_nonneg_right _ lnonneg
           linarith only [IcN]
       _ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l :=
-        add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl
+        (add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl)
       _ = ε * (1 + ε + l) * c (N - 1) := by ring
       _ ≤ ε * (1 + ε + l) * n :=
         by
@@ -177,7 +177,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         have := (ha _ aN').1
         rwa [B] at this
       _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
-      _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := add_le_add (ha _ aN').2 le_rfl
+      _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
       _ = ε * (1 + l) * c (N - 1) := by ring
       _ ≤ ε * (1 + l) * n :=
         by
@@ -231,7 +231,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       
 #align tendsto_div_of_monotone_of_exists_subseq_tendsto_div tendsto_div_of_monotone_of_exists_subseq_tendsto_div
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic tactic.field_simp.ne_zero -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic tactic.field_simp.ne_zero -/
 /-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
 `c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
 `c`s converging to `1`. -/

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

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.

@@ -40,7 +40,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
   have lnonneg : 0 ≤ l := by
     rcases hlim 2 one_lt_two with ⟨c, _, ctop, clim⟩
     have : Tendsto (fun n => u 0 / c n) atTop (𝓝 0) :=
-      tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)
+      tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_iff.2 ctop)
     apply le_of_tendsto_of_tendsto' this clim fun n => ?_
     gcongr
     exact hmono (zero_le _)
@@ -198,7 +198,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
   have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
     intro n
     refine' zero_lt_one.trans_le _
-    simp only [Real.rpow_nat_cast, Nat.one_le_cast, Nat.one_le_floor_iff,
+    simp only [Real.rpow_natCast, Nat.one_le_cast, Nat.one_le_floor_iff,
       one_le_pow_of_one_le (cone k).le n]
   have A :
     Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / c k ^ (n + 1) * c k / (⌊c k ^ n⌋₊ / c k ^ n))
@@ -251,7 +251,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     _ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / ((1 : ℝ) - c⁻¹ ^ 2) := by
       gcongr
       · exact sub_nonneg.2 C.le
-      · rw [← Real.rpow_nat_cast]
+      · rw [← Real.rpow_natCast]
         exact Real.rpow_le_rpow_of_exponent_ge A C.le (Nat.sub_one_lt_floor _).le
     _ = c ^ 2 * ((1 : ℝ) - c⁻¹ ^ 2)⁻¹ / j ^ 2 := by
       have I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = (1 : ℝ) / j ^ 2 := by

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -148,7 +148,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
         simpa [B] using (ha _ aN').1
       _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
-      _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
+      _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := add_le_add (ha _ aN').2 le_rfl
       _ = ε * (1 + l) * c (N - 1) := by ring
       _ ≤ ε * (1 + l) * n := by gcongr
   refine' tendsto_order.2 ⟨fun d hd => _, fun d hd => _⟩

@@ -54,7 +54,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       have cnpos : 0 < c n := cnpos'
       calc
         u (c n) - c n * l = (u (c n) / c n - l) * c n := by
-          simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
+          simp only [cnpos.ne', Ne, Nat.cast_eq_zero, not_false_iff, field_simps]
         _ ≤ ε * c n := by
           gcongr
           refine (le_abs_self _).trans ?_
@@ -108,7 +108,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       have cnpos : 0 < c n := cnpos'
       calc
         (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
-          simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
+          simp only [cnpos.ne', Ne, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
         _ ≤ ε * c n := by
           gcongr
           refine le_trans (neg_le_abs _) ?_

The current design for abs is flawed:

• The Abs notation typeclass has exactly two instances: one for [Neg α] [Sup α], one for [Inv α] [Sup α]. This means that:
  • We can't write a meaningful hover for Abs.abs
  • Fields have two Abs instances!
• We have the multiplicative definition but:
  • All the lemmas in Algebra.Order.Group.Abs are about the additive version.
  • The only lemmas about the multiplicative version are in Algebra.Order.Group.PosPart, and they get additivised to duplicates of the lemmas in Algebra.Order.Group.Abs!

This PR changes the notation typeclass with two new definitions (related through to_additive): mabs and abs. abs inherits the |a| notation and mabs gets |a|ₘ instead.

The first half of Algebra.Order.Group.Abs gets multiplicativised. A later PR will multiplicativise the second half, and another one will deduplicate the lemmas in Algebra.Order.Group.PosPart.

Part of #9411.

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -111,7 +111,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
         _ ≤ ε * c n := by
           gcongr
-          refine le_trans (neg_le_abs_self _) ?_
+          refine le_trans (neg_le_abs _) ?_
           simpa using hn
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ,

@@ -41,9 +41,9 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     rcases hlim 2 one_lt_two with ⟨c, _, ctop, clim⟩
     have : Tendsto (fun n => u 0 / c n) atTop (𝓝 0) :=
       tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)
-    apply le_of_tendsto_of_tendsto' this clim fun n => _
-    simp_rw [div_eq_inv_mul]
-    exact fun n => mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))
+    apply le_of_tendsto_of_tendsto' this clim fun n => ?_
+    gcongr
+    exact hmono (zero_le _)
   have A : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, u n - n * l ≤ ε * (1 + ε + l) * n := by
     intro ε εpos
     rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
@@ -56,9 +56,9 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         u (c n) - c n * l = (u (c n) / c n - l) * c n := by
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
         _ ≤ ε * c n := by
-          refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
-          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
-          exact le_trans (le_abs_self _) hn
+          gcongr
+          refine (le_abs_self _).trans ?_
+          simpa using hn
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
       eventually_atTop.1 (cgrowth.and L)
@@ -89,23 +89,15 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       have := (ha _ A).1
       rwa [B] at this
     calc
-      u n - n * l ≤ u (c N) - c (N - 1) * l := by
-        apply sub_le_sub (hmono ncN.le)
-        apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg
+      u n - n * l ≤ u (c N) - c (N - 1) * l := by gcongr; exact hmono ncN.le
       _ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring
       _ ≤ ε * c N + ε * c (N - 1) * l := by
-        apply add_le_add
-        · apply (ha _ _).2
-          exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN
-        · apply mul_le_mul_of_nonneg_right _ lnonneg
-          linarith only [IcN]
-      _ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l :=
-        (add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl)
+        gcongr
+        · exact (ha N (a.le_succ.trans aN)).2
+        · linarith only [IcN]
+      _ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l := by gcongr
       _ = ε * (1 + ε + l) * c (N - 1) := by ring
-      _ ≤ ε * (1 + ε + l) * n := by
-        refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
-        apply mul_nonneg εpos.le
-        linarith only [εpos, lnonneg]
+      _ ≤ ε * (1 + ε + l) * n := by gcongr
   have B : ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (n : ℝ) * l - u n ≤ ε * (1 + l) * n := by
     intro ε εpos
     rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
@@ -118,9 +110,9 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
           simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
         _ ≤ ε * c n := by
-          refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
-          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
-          exact le_trans (neg_le_abs_self _) hn
+          gcongr
+          refine le_trans (neg_le_abs_self _) ?_
+          simpa using hn
     obtain ⟨a, ha⟩ :
       ∃ a : ℕ,
         ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
@@ -149,19 +141,16 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       simpa only [not_lt] using Nat.find_min exN this
     calc
       (n : ℝ) * l - u n ≤ c N * l - u (c (N - 1)) := by
-        refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _
-        exact neg_le_neg (hmono cNn)
+        gcongr
+        exact hmono cNn
       _ ≤ (1 + ε) * c (N - 1) * l - u (c (N - 1)) := by
-        refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
+        gcongr
         have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
-        have := (ha _ aN').1
-        rwa [B] at this
+        simpa [B] using (ha _ aN').1
       _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
       _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
       _ = ε * (1 + l) * c (N - 1) := by ring
-      _ ≤ ε * (1 + l) * n := by
-        refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
-        exact mul_nonneg εpos.le (add_nonneg zero_le_one lnonneg)
+      _ ≤ ε * (1 + l) * n := by gcongr
   refine' tendsto_order.2 ⟨fun d hd => _, fun d hd => _⟩
   · obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, d + ε * (1 + l) < l ∧ 0 < ε := by
       have L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[>] 0) (𝓝 (d + 0 * (1 + l))) := by
@@ -177,9 +166,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         · linarith only [hε]
         · exact Nat.cast_ne_zero.2 (ne_of_gt npos)
       _ = (n : ℝ)⁻¹ * (n * l - ε * (1 + l) * n) := by ring
-      _ ≤ (n : ℝ)⁻¹ * u n := by
-        refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
-        linarith only [hn]
+      _ ≤ (n : ℝ)⁻¹ * u n := by gcongr; linarith only [hn]
   · obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, l + ε * (1 + ε + l) < d ∧ 0 < ε := by
       have L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) := by
         apply Tendsto.mono_left _ nhdsWithin_le_nhds
@@ -188,17 +175,11 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
       simp only [zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists
-    filter_upwards [A ε εpos, Ioi_mem_atTop 0] with n hn npos
-    simp_rw [div_eq_inv_mul]
+    filter_upwards [A ε εpos, Ioi_mem_atTop 0] with n hn (npos : 0 < n)
     calc
-      (n : ℝ)⁻¹ * u n ≤ (n : ℝ)⁻¹ * (n * l + ε * (1 + ε + l) * n) := by
-        refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
-        linarith only [hn]
-      _ = (n : ℝ)⁻¹ * n * (l + ε * (1 + ε + l)) := by ring
-      _ < d := by
-        rwa [inv_mul_cancel, one_mul]
-        exact Nat.cast_ne_zero.2 (ne_of_gt npos)
-
+      u n / n ≤ (n * l + ε * (1 + ε + l) * n) / n := by gcongr; linarith only [hn]
+      _ = (l + ε * (1 + ε + l)) := by field_simp; ring
+      _ < d := hε
 #align tendsto_div_of_monotone_of_exists_subseq_tendsto_div tendsto_div_of_monotone_of_exists_subseq_tendsto_div
 
 /-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
@@ -251,12 +232,11 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
     rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
     simpa using pow_le_pow_right hc.le one_le_two
+  have C : c⁻¹ ^ 2 < 1 := pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero
   calc
-    (∑ i in (range N).filter fun i => j < c ^ i, (1 : ℝ) / (c ^ i) ^ 2) ≤
+    (∑ i in (range N).filter (j < c ^ ·), (1 : ℝ) / (c ^ i) ^ 2) ≤
         ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (1 : ℝ) / (c ^ i) ^ 2 := by
-      refine'
-        sum_le_sum_of_subset_of_nonneg _ fun i _hi _hident => div_nonneg zero_le_one (sq_nonneg _)
-      intro i hi
+      refine sum_le_sum_of_subset_of_nonneg (fun i hi ↦ ?_) (by intros; positivity)
       simp only [mem_filter, mem_range] at hi
       simp only [hi.1, mem_Ico, and_true_iff]
       apply Nat.floor_le_of_le
@@ -266,50 +246,38 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     _ = ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (c⁻¹ ^ 2) ^ i := by
       congr 1 with i
       simp [← pow_mul, mul_comm]
-    _ ≤ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ / ((1 : ℝ) - c⁻¹ ^ 2) := by
-      apply geom_sum_Ico_le_of_lt_one (sq_nonneg _)
-      rw [sq_lt_one_iff (inv_nonneg.2 (zero_le_one.trans hc.le))]
-      exact inv_lt_one hc
+    _ ≤ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ / ((1 : ℝ) - c⁻¹ ^ 2) :=
+      geom_sum_Ico_le_of_lt_one (sq_nonneg _) C
     _ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / ((1 : ℝ) - c⁻¹ ^ 2) := by
-      apply div_le_div _ _ _ le_rfl
-      · apply Real.rpow_nonneg (sq_nonneg _)
+      gcongr
+      · exact sub_nonneg.2 C.le
       · rw [← Real.rpow_nat_cast]
-        apply Real.rpow_le_rpow_of_exponent_ge A
-        · exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)
-        · exact (Nat.sub_one_lt_floor _).le
-      · simpa only [inv_pow, sub_pos] using inv_lt_one (one_lt_pow hc two_ne_zero)
+        exact Real.rpow_le_rpow_of_exponent_ge A C.le (Nat.sub_one_lt_floor _).le
     _ = c ^ 2 * ((1 : ℝ) - c⁻¹ ^ 2)⁻¹ / j ^ 2 := by
       have I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = (1 : ℝ) / j ^ 2 := by
         apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)
-        · rw [one_div]
-          exact inv_pos.2 (sq_pos_of_pos hj)
+        · rw [Set.mem_Ioi]; positivity
         rw [Real.log_rpow A]
-        simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg,
-          neg_inj]
+        simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]
         field_simp [(Real.log_pos hc).ne']
         ring
       rw [Real.rpow_sub A, I]
       have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'
       field_simp [hj.ne', (zero_lt_one.trans hc).ne']
       ring
-    _ ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 := by
-      apply div_le_div _ B (sq_pos_of_pos hj) le_rfl
-      exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)
-
+    _ ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 := by gcongr
 #align sum_div_pow_sq_le_div_sq sum_div_pow_sq_le_div_sq
 
 theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ := by
   have cpos : 0 < c := zero_lt_one.trans hc
-  rcases Nat.eq_zero_or_pos i with (rfl | hi)
+  rcases eq_or_ne i 0 with (rfl | hi)
   · simp only [pow_zero, Nat.floor_one, Nat.cast_one, mul_one, sub_le_self_iff, inv_nonneg, cpos.le]
-  have hident : 1 ≤ i := hi
   calc
     (1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
     _ ≤ c ^ i - 1 := by
-      simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using
-        pow_le_pow_right hc.le hident
+      gcongr
+      simpa only [← div_eq_mul_inv, one_le_div cpos, pow_one] using le_self_pow hc.le hi
     _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
-
 #align mul_pow_le_nat_floor_pow mul_pow_le_nat_floor_pow
 
 /-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
@@ -323,12 +291,8 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
     (∑ i in (range N).filter (j < ⌊c ^ ·⌋₊), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
         ∑ i in (range N).filter (j < c ^ ·), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2 := by
       apply sum_le_sum_of_subset_of_nonneg
-      · intro i hi
-        simp only [mem_filter, mem_range] at hi
-        simpa only [hi.1, mem_filter, mem_range, true_and_iff] using
-          hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))
-      · intro i _hi _hident
-        exact div_nonneg zero_le_one (sq_nonneg _)
+      · exact monotone_filter_right _ fun k hk ↦ hk.trans_le <| Nat.floor_le (by positivity)
+      · intros; positivity
     _ ≤ ∑ i in (range N).filter (j < c ^ ·), (1 - c⁻¹)⁻¹ ^ 2 * ((1 : ℝ) / (c ^ i) ^ 2) := by
       refine' sum_le_sum fun i _hi => _
       rw [mul_div_assoc', mul_one, div_le_div_iff]; rotate_left
@@ -337,16 +301,15 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
         simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
       · exact sq_pos_of_pos (pow_pos cpos _)
       rw [one_mul, ← mul_pow]
-      apply pow_le_pow_left (pow_nonneg cpos.le _)
+      gcongr
       rw [← div_eq_inv_mul, le_div_iff A, mul_comm]
       exact mul_pow_le_nat_floor_pow hc i
     _ ≤ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2 := by
       rw [← mul_sum, ← mul_div_assoc']
-      refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)
+      gcongr
       exact sum_div_pow_sq_le_div_sq N hj hc
     _ = c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 := by
       congr 1
-      field_simp [cpos.ne', (sub_pos.2 hc).ne']
+      field_simp [(sub_pos.2 hc).ne']
       ring!
-
 #align sum_div_nat_floor_pow_sq_le_div_sq sum_div_nat_floor_pow_sq_le_div_sq

This better matches other lemma names.

From LeanAPAP

@@ -272,7 +272,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
       exact inv_lt_one hc
     _ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / ((1 : ℝ) - c⁻¹ ^ 2) := by
       apply div_le_div _ _ _ le_rfl
-      · apply Real.rpow_nonneg_of_nonneg (sq_nonneg _)
+      · apply Real.rpow_nonneg (sq_nonneg _)
       · rw [← Real.rpow_nat_cast]
         apply Real.rpow_le_rpow_of_exponent_ge A
         · exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)

The names for lemmas about monotonicity of (a ^ ·) and (· ^ n) were a mess. This PR tidies up everything related by following the naming convention for (a * ·) and (· * b). Namely, (a ^ ·) is pow_right and (· ^ n) is pow_left in lemma names. All lemma renames follow the corresponding multiplication lemma names closely.

Renames

Algebra.GroupPower.Order

• pow_mono → pow_right_mono
• pow_le_pow → pow_le_pow_right
• pow_le_pow_of_le_left → pow_le_pow_left
• pow_lt_pow_of_lt_left → pow_lt_pow_left
• strictMonoOn_pow → pow_left_strictMonoOn
• pow_strictMono_right → pow_right_strictMono
• pow_lt_pow → pow_lt_pow_right
• pow_lt_pow_iff → pow_lt_pow_iff_right
• pow_le_pow_iff → pow_le_pow_iff_right
• self_lt_pow → lt_self_pow
• strictAnti_pow → pow_right_strictAnti
• pow_lt_pow_iff_of_lt_one → pow_lt_pow_iff_right_of_lt_one
• pow_lt_pow_of_lt_one → pow_lt_pow_right_of_lt_one
• lt_of_pow_lt_pow → lt_of_pow_lt_pow_left
• le_of_pow_le_pow → le_of_pow_le_pow_left
• pow_lt_pow₀ → pow_lt_pow_right₀

Algebra.GroupPower.CovariantClass

• pow_le_pow_of_le_left' → pow_le_pow_left'
• nsmul_le_nsmul_of_le_right → nsmul_le_nsmul_right
• pow_lt_pow' → pow_lt_pow_right'
• nsmul_lt_nsmul → nsmul_lt_nsmul_left
• pow_strictMono_left → pow_right_strictMono'
• nsmul_strictMono_right → nsmul_left_strictMono
• StrictMono.pow_right' → StrictMono.pow_const
• StrictMono.nsmul_left → StrictMono.const_nsmul
• pow_strictMono_right' → pow_left_strictMono
• nsmul_strictMono_left → nsmul_right_strictMono
• Monotone.pow_right → Monotone.pow_const
• Monotone.nsmul_left → Monotone.const_nsmul
• lt_of_pow_lt_pow' → lt_of_pow_lt_pow_left'
• lt_of_nsmul_lt_nsmul → lt_of_nsmul_lt_nsmul_right
• pow_le_pow' → pow_le_pow_right'
• nsmul_le_nsmul → nsmul_le_nsmul_left
• pow_le_pow_of_le_one' → pow_le_pow_right_of_le_one'
• nsmul_le_nsmul_of_nonpos → nsmul_le_nsmul_left_of_nonpos
• le_of_pow_le_pow' → le_of_pow_le_pow_left'
• le_of_nsmul_le_nsmul' → le_of_nsmul_le_nsmul_right'
• pow_le_pow_iff' → pow_le_pow_iff_right'
• nsmul_le_nsmul_iff → nsmul_le_nsmul_iff_left
• pow_lt_pow_iff' → pow_lt_pow_iff_right'
• nsmul_lt_nsmul_iff → nsmul_lt_nsmul_iff_left

Data.Nat.Pow

• Nat.pow_lt_pow_of_lt_left → Nat.pow_lt_pow_left
• Nat.pow_le_iff_le_left → Nat.pow_le_pow_iff_left
• Nat.pow_lt_iff_lt_left → Nat.pow_lt_pow_iff_left

Lemmas added

• pow_le_pow_iff_left
• pow_lt_pow_iff_left
• pow_right_injective
• pow_right_inj
• Nat.pow_le_pow_left to have the correct name since Nat.pow_le_pow_of_le_left is in Std.
• Nat.pow_le_pow_right to have the correct name since Nat.pow_le_pow_of_le_right is in Std.

Lemmas removed

• self_le_pow was a duplicate of le_self_pow.
• Nat.pow_lt_pow_of_lt_right is defeq to pow_lt_pow_right.
• Nat.pow_right_strictMono is defeq to pow_right_strictMono.
• Nat.pow_le_iff_le_right is defeq to pow_le_pow_iff_right.
• Nat.pow_lt_iff_lt_right is defeq to pow_lt_pow_iff_right.

Other changes

• A bunch of proofs have been golfed.
• Some lemma assumptions have been turned from 0 < n or 1 ≤ n to n ≠ 0.
• A few Nat lemmas have been protected.
• One docstring has been fixed.

@@ -250,7 +250,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
     have : c ^ 3 = c ^ 2 * c := by ring
     simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
     rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
-    simpa using pow_le_pow hc.le one_le_two
+    simpa using pow_le_pow_right hc.le one_le_two
   calc
     (∑ i in (range N).filter fun i => j < c ^ i, (1 : ℝ) / (c ^ i) ^ 2) ≤
         ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (1 : ℝ) / (c ^ i) ^ 2 := by
@@ -307,7 +307,7 @@ theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹)
     (1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
     _ ≤ c ^ i - 1 := by
       simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using
-        pow_le_pow hc.le hident
+        pow_le_pow_right hc.le hident
     _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
 
 #align mul_pow_le_nat_floor_pow mul_pow_le_nat_floor_pow
@@ -337,7 +337,7 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
         simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
       · exact sq_pos_of_pos (pow_pos cpos _)
       rw [one_mul, ← mul_pow]
-      apply pow_le_pow_of_le_left (pow_nonneg cpos.le _)
+      apply pow_le_pow_left (pow_nonneg cpos.le _)
       rw [← div_eq_inv_mul, le_div_iff A, mul_comm]
       exact mul_pow_le_nat_floor_pow hc i
     _ ≤ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2 := by

To fit with the "please try harder" convention of ! tactics.

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

@@ -70,7 +70,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     let N := Nat.find exN
     have ncN : n < c N := Nat.find_spec exN
     have aN : a + 1 ≤ N := by
-      by_contra' h
+      by_contra! h
       have cNM : c N ≤ M := by
         apply le_max'
         apply mem_image_of_mem
@@ -133,7 +133,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     let N := Nat.find exN
     have ncN : n < c N := Nat.find_spec exN
     have aN : a + 1 ≤ N := by
-      by_contra' h
+      by_contra! h
       have cNM : c N ≤ M := by
         apply le_max'
         apply mem_image_of_mem

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -19,8 +19,6 @@ We state several auxiliary results pertaining to sequences of the form `⌊c^n
   to `1/j^2`, up to a multiplicative constant.
 -/
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
-
 open Filter Finset
 
 open Topology BigOperators

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -67,7 +67,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
     filter_upwards [Ici_mem_atTop M] with n hn
     have exN : ∃ N, n < c N := by
-      rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
+      rcases (tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
       exact ⟨N, by linarith only [hN]⟩
     let N := Nat.find exN
     have ncN : n < c N := Nat.find_spec exN
@@ -130,7 +130,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
     filter_upwards [Ici_mem_atTop M] with n hn
     have exN : ∃ N, n < c N := by
-      rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
+      rcases (tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
       exact ⟨N, by linarith only [hN]⟩
     let N := Nat.find exN
     have ncN : n < c N := Nat.find_spec exN

mathport was forgetting a space in filter_upwards [...]with instead of filter_upwards [...] with.

@@ -65,7 +65,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
       eventually_atTop.1 (cgrowth.and L)
     let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
-    filter_upwards [Ici_mem_atTop M]with n hn
+    filter_upwards [Ici_mem_atTop M] with n hn
     have exN : ∃ N, n < c N := by
       rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
       exact ⟨N, by linarith only [hN]⟩
@@ -114,7 +114,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
     have L : ∀ᶠ n : ℕ in atTop, (c n : ℝ) * l - u (c n) ≤ ε * c n := by
       rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
         Asymptotics.isLittleO_iff] at clim
-      filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)]with n hn cnpos'
+      filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
       have cnpos : 0 < c n := cnpos'
       calc
         (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
@@ -128,7 +128,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
       eventually_atTop.1 (cgrowth.and L)
     let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
-    filter_upwards [Ici_mem_atTop M]with n hn
+    filter_upwards [Ici_mem_atTop M] with n hn
     have exN : ∃ N, n < c N := by
       rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
       exact ⟨N, by linarith only [hN]⟩
@@ -171,7 +171,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
       simp only [zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists
-    filter_upwards [B ε εpos, Ioi_mem_atTop 0]with n hn npos
+    filter_upwards [B ε εpos, Ioi_mem_atTop 0] with n hn npos
     simp_rw [div_eq_inv_mul]
     calc
       d < (n : ℝ)⁻¹ * n * (l - ε * (1 + l)) := by
@@ -190,7 +190,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
       simp only [zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists
-    filter_upwards [A ε εpos, Ioi_mem_atTop 0]with n hn npos
+    filter_upwards [A ε εpos, Ioi_mem_atTop 0] with n hn npos
     simp_rw [div_eq_inv_mul]
     calc
       (n : ℝ)⁻¹ * u n ≤ (n : ℝ)⁻¹ * (n * l + ε * (1 + ε + l) * n) := by
@@ -235,7 +235,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
     ext1 n
     field_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne']
     ring
-  filter_upwards [(tendsto_order.1 B).2 a hk]with n hn
+  filter_upwards [(tendsto_order.1 B).2 a hk] with n hn
   exact (div_le_iff (H n)).1 hn.le
 #align tendsto_div_of_monotone_of_tendsto_div_floor_pow tendsto_div_of_monotone_of_tendsto_div_floor_pow
 

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -169,7 +169,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
       have L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[>] 0) (𝓝 (d + 0 * (1 + l))) := by
         apply Tendsto.mono_left _ nhdsWithin_le_nhds
         exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
-      simp only [MulZeroClass.zero_mul, add_zero] at L
+      simp only [zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists
     filter_upwards [B ε εpos, Ioi_mem_atTop 0]with n hn npos
     simp_rw [div_eq_inv_mul]
@@ -188,7 +188,7 @@ theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (
         exact
           tendsto_const_nhds.add
             (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
-      simp only [MulZeroClass.zero_mul, add_zero] at L
+      simp only [zero_mul, add_zero] at L
       exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists
     filter_upwards [A ε εpos, Ioi_mem_atTop 0]with n hn npos
     simp_rw [div_eq_inv_mul]

@@ -19,9 +19,7 @@ We state several auxiliary results pertaining to sequences of the form `⌊c^n
   to `1/j^2`, up to a multiplicative constant.
 -/
 
--- porting note: elaboration of some occurrences of `^` were incorrect.
--- see: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/!4.234085
-local macro_rules | `($x ^ $y)   => `(HPow.hPow $x $y)
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
 
 open Filter Finset
 

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.specific_limits.floor_pow
-! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.Analysis.SpecialFunctions.Pow.Real
 
+#align_import analysis.specific_limits.floor_pow from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
+
 /-!
 # Results on discretized exponentials
 

Notes:

1. This PR also contains a fix to the filter_upwards tactic, see Zulip
2. Several situations required extra type annotations in order to get the expressions to elaborate the same way (because of heterogeneous operations in Lean 4).
3. ^ n was not being elaborated correctly for n : ℕ and a coercion was being inserted. This seems to be the same issue as this one and I used Floris' fix there.

Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: qawbecrdtey <qawbecrdtey@naver.com>