imo.imo2006_q3 ⟷ Archive.Imo.Imo2006Q3

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

@@ -40,7 +40,7 @@ namespace imo2006_q3
 theorem lhs_ineq {x y : ℝ} (hxy : 0 ≤ x * y) :
     16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3 :=
   by
-  conv_rhs => rw [pow_succ']
+  conv_rhs => rw [pow_succ]
   refine' mul_le_mul_of_nonneg_right _ (sq_nonneg _)
   apply le_of_sub_nonneg
   calc

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

@@ -143,7 +143,7 @@ theorem proof₂ (M : ℝ)
     by intro; ring
   have h₂ : ∀ x : ℝ, (2 - 3 * x) ^ 2 + 2 ^ 2 + (2 + 3 * x) ^ 2 = 18 * x ^ 2 + 12 := by intro; ring
   have := h (2 - 3 * sqrt 2) 2 (2 + 3 * sqrt 2)
-  rw [lhs_identity, h₁, h₂, sq_sqrt zero_le_two, abs_neg, abs_eq_self.mpr, ← div_le_iff] at this 
+  rw [lhs_identity, h₁, h₂, sq_sqrt zero_le_two, abs_neg, abs_eq_self.mpr, ← div_le_iff] at this
   · convert this using 1; ring
   · apply pow_pos; norm_num
   · exact mul_nonneg (mul_nonneg (sq_nonneg _) zero_le_two) (sqrt_nonneg _)

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

@@ -84,9 +84,9 @@ theorem subst_wlog {x y z s : ℝ} (hxy : 0 ≤ x * y) (hxyz : x + y + z = 0) :
         le_trans (mul_le_mul_of_nonneg_left (lhs_ineq hxy) hs) mid_ineq
       _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=
         div_le_div_of_le four_pow_four_pos.le <|
-          pow_le_pow_of_le_left (add_nonneg (mul_nonneg zero_lt_three.le (sq_nonneg _)) hs)
+          pow_le_pow_left (add_nonneg (mul_nonneg zero_lt_three.le (sq_nonneg _)) hs)
             (add_le_add_right rhs_ineq _) _
-  le_of_pow_le_pow _ (mul_nonneg (sqrt_nonneg _) (sq_nonneg _)) Nat.succ_pos' <|
+  le_of_pow_le_pow_left _ (mul_nonneg (sqrt_nonneg _) (sq_nonneg _)) Nat.succ_pos' <|
     calc
       (32 * |x * y * z * s|) ^ 2 = 32 * (2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)) := by
         rw [mul_pow, sq_abs, hz] <;> ring

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

@@ -3,7 +3,7 @@ Copyright (c) 2021 Tian Chen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Tian Chen
 -/
-import Mathbin.Analysis.SpecialFunctions.Sqrt
+import Analysis.SpecialFunctions.Sqrt
 
 #align_import imo.imo2006_q3 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 

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

@@ -2,14 +2,11 @@
 Copyright (c) 2021 Tian Chen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Tian Chen
-
-! This file was ported from Lean 3 source module imo.imo2006_q3
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecialFunctions.Sqrt
 
+#align_import imo.imo2006_q3 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # IMO 2006 Q3
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Tian Chen
 
 ! This file was ported from Lean 3 source module imo.imo2006_q3
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
+! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Analysis.SpecialFunctions.Sqrt
 /-!
 # IMO 2006 Q3
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Determine the least real number $M$ such that
 $$
 \left| ab(a^2 - b^2) + bc(b^2 - c^2) + ca(c^2 - a^2) \right|

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

@@ -46,7 +46,6 @@ theorem lhs_ineq {x y : ℝ} (hxy : 0 ≤ x * y) :
   calc
     ((x + y) ^ 2) ^ 2 - 16 * x ^ 2 * y ^ 2 = (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) := by ring
     _ ≥ 0 := mul_nonneg (sq_nonneg _) <| add_nonneg (sq_nonneg _) <| mul_nonneg zero_lt_four.le hxy
-    
 #align imo2006_q3.lhs_ineq Imo2006Q3.lhs_ineq
 
 theorem four_pow_four_pos : (0 : ℝ) < 4 ^ 4 :=
@@ -62,7 +61,6 @@ theorem mid_ineq {s t : ℝ} : s * t ^ 3 ≤ (3 * t + s) ^ 4 / 4 ^ 4 :=
         _ ≥ 0 :=
           mul_nonneg (sq_nonneg _) <|
             add_nonneg (sq_nonneg _) <| mul_nonneg zero_le_two (sq_nonneg _)
-        
 #align imo2006_q3.mid_ineq Imo2006Q3.mid_ineq
 
 /-- Replacing `x` and `y` with their average decreases the right side. -/
@@ -71,7 +69,6 @@ theorem rhs_ineq {x y : ℝ} : 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y)
     calc
       _ = (x - y) ^ 2 := by ring
       _ ≥ 0 := sq_nonneg _
-      
 #align imo2006_q3.rhs_ineq Imo2006Q3.rhs_ineq
 
 theorem zero_lt_32 : (0 : ℝ) < 32 := by norm_num
@@ -89,7 +86,6 @@ theorem subst_wlog {x y z s : ℝ} (hxy : 0 ≤ x * y) (hxyz : x + y + z = 0) :
         div_le_div_of_le four_pow_four_pos.le <|
           pow_le_pow_of_le_left (add_nonneg (mul_nonneg zero_lt_three.le (sq_nonneg _)) hs)
             (add_le_add_right rhs_ineq _) _
-      
   le_of_pow_le_pow _ (mul_nonneg (sqrt_nonneg _) (sq_nonneg _)) Nat.succ_pos' <|
     calc
       (32 * |x * y * z * s|) ^ 2 = 32 * (2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)) := by
@@ -99,7 +95,6 @@ theorem subst_wlog {x y z s : ℝ} (hxy : 0 ≤ x * y) (hxyz : x + y + z = 0) :
       _ = (sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2) ^ 2 := by
         rw [mul_pow, sq_sqrt zero_le_two, hz, ← pow_mul, ← mul_add, mul_pow, ← mul_comm_div, ←
           mul_assoc, show 32 / 4 ^ 4 * 2 ^ 4 = (2 : ℝ) by norm_num, show 2 * 2 = 4 by rfl]
-      
 #align imo2006_q3.subst_wlog Imo2006Q3.subst_wlog
 
 /-- Proof that `M = 9 * sqrt 2 / 32` works with the substitution. -/
@@ -131,7 +126,6 @@ theorem proof₁ {a b c : ℝ} :
     _ = _ := congr_arg _ <| lhs_identity a b c
     _ ≤ _ := (subst_proof₁ (a - b) (b - c) (c - a) (-(a + b + c)) (by ring))
     _ = _ := by ring
-    
 #align imo2006_q3.proof₁ Imo2006Q3.proof₁
 
 theorem proof₂ (M : ℝ)

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

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.

@@ -71,7 +71,7 @@ theorem subst_wlog {x y z s : ℝ} (hxy : 0 ≤ x * y) (hxyz : x + y + z = 0) :
       _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 := by
           gcongr (?_ + _) ^ 4 / _
           apply rhs_ineq
-  refine le_of_pow_le_pow 2 (by positivity) (by positivity) ?_
+  refine le_of_pow_le_pow_left two_ne_zero (by positivity) ?_
   calc
     (32 * |x * y * z * s|) ^ 2 = 32 * (2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)) := by
       rw [mul_pow, sq_abs, hz]; ring

• 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) 2021 Tian Chen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Tian Chen
-
-! This file was ported from Lean 3 source module imo.imo2006_q3
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecialFunctions.Sqrt
 import Mathlib.Tactic.Polyrith
 
+#align_import imo.imo2006_q3 from "leanprover-community/mathlib"@"308826471968962c6b59c7ff82a22757386603e3"
+
 /-!
 # IMO 2006 Q3
 

@@ -9,6 +9,7 @@ Authors: Tian Chen
 ! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecialFunctions.Sqrt
+import Mathlib.Tactic.Polyrith
 
 /-!
 # IMO 2006 Q3
@@ -39,61 +40,49 @@ namespace Imo2006Q3
 /-- Replacing `x` and `y` with their average increases the left side. -/
 theorem lhs_ineq {x y : ℝ} (hxy : 0 ≤ x * y) :
     16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3 := by
-  conv_rhs => rw [pow_succ']
-  refine' mul_le_mul_of_nonneg_right _ (sq_nonneg _)
-  apply le_of_sub_nonneg
-  calc
-    ((x + y) ^ 2) ^ 2 - 16 * x ^ 2 * y ^ 2 = (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) := by ring
-    _ ≥ 0 := mul_nonneg (sq_nonneg _) <| add_nonneg (sq_nonneg _) <| mul_nonneg zero_lt_four.le hxy
+  have : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0 := by positivity
+  calc 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2 := by gcongr; linarith
+    _ = ((x + y) ^ 2) ^ 3 := by ring
 #align imo2006_q3.lhs_ineq Imo2006Q3.lhs_ineq
 
-theorem four_pow_four_pos : (0 : ℝ) < 4 ^ 4 :=
-  pow_pos zero_lt_four _
+theorem four_pow_four_pos : (0 : ℝ) < 4 ^ 4 := by norm_num
 #align imo2006_q3.four_pow_four_pos Imo2006Q3.four_pow_four_pos
 
-theorem mid_ineq {s t : ℝ} : s * t ^ 3 ≤ (3 * t + s) ^ 4 / 4 ^ 4 :=
-  (le_div_iff four_pow_four_pos).mpr <|
-    le_of_sub_nonneg <|
-      calc
-        (3 * t + s) ^ 4 - s * t ^ 3 * 4 ^ 4 = (s - t) ^ 2 * ((s + 7 * t) ^ 2 + 2 * (4 * t) ^ 2) :=
-          by ring
-        _ ≥ 0 :=
-          mul_nonneg (sq_nonneg _) <|
-            add_nonneg (sq_nonneg _) <| mul_nonneg zero_le_two (sq_nonneg _)
+theorem mid_ineq {s t : ℝ} : s * t ^ 3 ≤ (3 * t + s) ^ 4 / 4 ^ 4 := by
+  rw [le_div_iff four_pow_four_pos]
+  have : 0 ≤ (s - t) ^ 2 * ((s + 7 * t) ^ 2 + 2 * (4 * t) ^ 2) := by positivity
+  linarith
 #align imo2006_q3.mid_ineq Imo2006Q3.mid_ineq
 
 /-- Replacing `x` and `y` with their average decreases the right side. -/
-theorem rhs_ineq {x y : ℝ} : 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) :=
-  le_of_sub_nonneg <|
-    calc
-      _ = (x - y) ^ 2 := by ring
-      _ ≥ 0 := sq_nonneg _
+theorem rhs_ineq {x y : ℝ} : 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) := by
+  have : 0 ≤ (x - y) ^ 2 := by positivity
+  linarith
 #align imo2006_q3.rhs_ineq Imo2006Q3.rhs_ineq
 
 theorem zero_lt_32 : (0 : ℝ) < 32 := by norm_num
 #align imo2006_q3.zero_lt_32 Imo2006Q3.zero_lt_32
 
 theorem subst_wlog {x y z s : ℝ} (hxy : 0 ≤ x * y) (hxyz : x + y + z = 0) :
-    32 * |x * y * z * s| ≤ sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 :=
-  have hz : (x + y) ^ 2 = z ^ 2 := neg_eq_of_add_eq_zero_right hxyz ▸ (neg_sq _).symm
-  have hs : 0 ≤ 2 * s ^ 2 := mul_nonneg zero_le_two (sq_nonneg s)
+    32 * |x * y * z * s| ≤ sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 := by
+  have hz : (x + y) ^ 2 = z ^ 2 := by linear_combination (x + y - z) * hxyz
   have this :=
     calc
-      2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2) ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=
-        le_trans (mul_le_mul_of_nonneg_left (lhs_ineq hxy) hs) mid_ineq
-      _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=
-        div_le_div_of_le four_pow_four_pos.le <|
-          pow_le_pow_of_le_left (add_nonneg (mul_nonneg zero_lt_three.le (sq_nonneg _)) hs)
-            (add_le_add_right rhs_ineq _) _
-  le_of_pow_le_pow _ (mul_nonneg (sqrt_nonneg _) (sq_nonneg _)) Nat.succ_pos' <|
-    calc
-      (32 * |x * y * z * s|) ^ 2 = 32 * (2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)) := by
-        rw [mul_pow, sq_abs, hz]; ring
-      _ ≤ 32 * ((2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4) :=
-        (mul_le_mul_of_nonneg_left this zero_lt_32.le)
-      _ = (sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2) ^ 2 := by
-        rw [mul_pow, sq_sqrt zero_le_two, hz, ← pow_mul, ← mul_add, mul_pow, ← mul_comm_div,
-          ← mul_assoc, show 32 / 4 ^ 4 * 2 ^ 4 = (2 : ℝ) by norm_num, show 2 * 2 = 4 by rfl]
+      2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)
+        ≤ _ * _ ^ 3 := by gcongr; exact lhs_ineq hxy
+      _ ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 := mid_ineq
+      _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 := by
+          gcongr (?_ + _) ^ 4 / _
+          apply rhs_ineq
+  refine le_of_pow_le_pow 2 (by positivity) (by positivity) ?_
+  calc
+    (32 * |x * y * z * s|) ^ 2 = 32 * (2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)) := by
+      rw [mul_pow, sq_abs, hz]; ring
+    _ ≤ 32 * ((2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4) := by gcongr
+    _ = (sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2) ^ 2 := by
+      field_simp
+      rw [mul_pow, sq_sqrt zero_le_two, hz]
+      ring
 #align imo2006_q3.subst_wlog Imo2006Q3.subst_wlog
 
 /-- Proof that `M = 9 * sqrt 2 / 32` works with the substitution. -/
@@ -103,26 +92,16 @@ theorem subst_proof₁ (x y z s : ℝ) (hxyz : x + y + z = 0) :
   · rw [div_mul_eq_mul_div, le_div_iff' zero_lt_32]
     exact subst_wlog h' hxyz
   cases' (mul_nonneg_of_three x y z).resolve_left h' with h h
-  · specialize this y z x _ h
-    · rw [← hxyz]; ring
-    · convert this using 2 <;> ring
-  · specialize this z x y _ h
-    · rw [← hxyz]; ring
-    · convert this using 2 <;> ring
+  · convert this y z x _ h using 2 <;> linarith
+  · convert this z x y _ h using 2 <;> linarith
 #align imo2006_q3.subst_proof₁ Imo2006Q3.subst_proof₁
 
-theorem lhs_identity (a b c : ℝ) :
-    a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) =
-      (a - b) * (b - c) * (c - a) * -(a + b + c) :=
-  by ring
-#align imo2006_q3.lhs_identity Imo2006Q3.lhs_identity
-
 theorem proof₁ {a b c : ℝ} :
     |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤
       9 * sqrt 2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 :=
   calc
-    _ = _ := congr_arg _ <| lhs_identity a b c
-    _ ≤ _ := (subst_proof₁ (a - b) (b - c) (c - a) (-(a + b + c)) (by ring))
+    _ = |(a - b) * (b - c) * (c - a) * -(a + b + c)| := by ring_nf
+    _ ≤ _ := subst_proof₁ (a - b) (b - c) (c - a) (-(a + b + c)) (by ring)
     _ = _ := by ring
 #align imo2006_q3.proof₁ Imo2006Q3.proof₁
 
@@ -131,18 +110,19 @@ theorem proof₂ (M : ℝ)
       |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤
         M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2) :
     9 * sqrt 2 / 32 ≤ M := by
-  have h₁ :
-    ∀ x : ℝ,
-      (2 - 3 * x - 2) * (2 - (2 + 3 * x)) * (2 + 3 * x - (2 - 3 * x)) *
-          -(2 - 3 * x + 2 + (2 + 3 * x)) =
-        -(18 ^ 2 * x ^ 2 * x) :=
-    by intro; ring
-  have h₂ : ∀ x : ℝ, (2 - 3 * x) ^ 2 + 2 ^ 2 + (2 + 3 * x) ^ 2 = 18 * x ^ 2 + 12 := by intro; ring
-  have := h (2 - 3 * sqrt 2) 2 (2 + 3 * sqrt 2)
-  rw [lhs_identity, h₁, h₂, sq_sqrt zero_le_two, abs_neg, abs_eq_self.mpr, ← div_le_iff] at this
-  · convert this using 1; ring
-  · apply pow_pos; norm_num
-  · exact mul_nonneg (mul_nonneg (sq_nonneg _) zero_le_two) (sqrt_nonneg _)
+  set α := sqrt (2:ℝ)
+  have hα : α ^ 2 = 2 := sq_sqrt (by norm_num)
+  let a := 2 - 3 * α
+  let c := 2 + 3 * α
+  calc _ = 18 ^ 2 * 2 * α / 48 ^ 2 := by ring
+    _ ≤ M := ?_
+  rw [div_le_iff (by positivity)]
+  calc 18 ^ 2 * 2 * α
+      = 18 ^ 2 * α ^ 2 * α := by linear_combination -324 * α * hα
+    _ = abs (-(18 ^ 2 * α ^ 2 * α)) := by rw [abs_neg, abs_of_nonneg]; positivity
+    _ = |a * 2 * (a ^ 2 - 2 ^ 2) + 2 * c * (2 ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| := by ring_nf
+    _ ≤ M * (a ^ 2 + 2 ^ 2 + c ^ 2) ^ 2 := by apply h
+    _ = M * 48 ^ 2 := by linear_combination (324 * α ^ 2 + 1080) * M * hα
 #align imo2006_q3.proof₂ Imo2006Q3.proof₂
 
 end Imo2006Q3
@@ -154,5 +134,5 @@ theorem imo2006_q3 (M : ℝ) :
         |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤
           M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2) ↔
       9 * sqrt 2 / 32 ≤ M :=
-  ⟨proof₂ M, fun h _ _ _ => le_trans proof₁ <| mul_le_mul_of_nonneg_right h <| sq_nonneg _⟩
+  ⟨proof₂ M, fun h _ _ _ => proof₁.trans (by gcongr)⟩
 #align imo2006_q3 imo2006_q3