imo.imo2008_q4 ⟷ Archive.Imo.Imo2008Q4

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

@@ -5,7 +5,7 @@ Authors: Manuel Candales
 -/
 import Data.Real.Basic
 import Data.Real.Sqrt
-import Data.Real.Nnreal
+import Data.Real.NNReal
 import Tactic.LinearCombination
 
 #align_import imo.imo2008_q4 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"

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

@@ -69,15 +69,15 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
   have h₁ : f 1 = 1 :=
     by
     specialize H₂ 1 1 1 1 zero_lt_one zero_lt_one zero_lt_one zero_lt_one rfl
-    norm_num [← two_mul] at H₂ 
-    rw [mul_div_mul_left (f 1 ^ 2) (f 1) two_ne_zero] at H₂ 
+    norm_num [← two_mul] at H₂
+    rw [mul_div_mul_left (f 1 ^ 2) (f 1) two_ne_zero] at H₂
     rwa [← (div_eq_iff h₀).mpr (sq (f 1))]
   have h₂ : ∀ x > 0, (f x - x) * (f x - 1 / x) = 0 :=
     by
     intro x hx
     have h1xss : 1 * x = sqrt x * sqrt x := by rw [one_mul, mul_self_sqrt (le_of_lt hx)]
     specialize H₂ 1 x (sqrt x) (sqrt x) zero_lt_one hx (sqrt_pos.mpr hx) (sqrt_pos.mpr hx) h1xss
-    rw [h₁, one_pow 2, sq_sqrt (le_of_lt hx), ← two_mul (f x), ← two_mul x] at H₂ 
+    rw [h₁, one_pow 2, sq_sqrt (le_of_lt hx), ← two_mul (f x), ← two_mul x] at H₂
     have hx_ne_0 : x ≠ 0 := ne_of_gt hx
     have hfx_ne_0 : f x ≠ 0 := by specialize H₁ x hx; exact ne_of_gt H₁
     field_simp at H₂ ⊢
@@ -92,24 +92,24 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
   have hab : a * b > 0 := mul_pos ha hb
   have habss : a * b = sqrt (a * b) * sqrt (a * b) := (mul_self_sqrt (le_of_lt hab)).symm
   specialize H₂ a b (sqrt (a * b)) (sqrt (a * b)) ha hb (sqrt_pos.mpr hab) (sqrt_pos.mpr hab) habss
-  rw [sq_sqrt (le_of_lt hab), ← two_mul (f (a * b)), ← two_mul (a * b)] at H₂ 
-  rw [hfa₂, hfb₂] at H₂ 
+  rw [sq_sqrt (le_of_lt hab), ← two_mul (f (a * b)), ← two_mul (a * b)] at H₂
+  rw [hfa₂, hfb₂] at H₂
   have h2ab_ne_0 : 2 * (a * b) ≠ 0 := mul_ne_zero two_ne_zero (ne_of_gt hab)
   specialize h₃ (a * b) hab
   cases' h₃ with hab₁ hab₂
   -- f(ab) = ab → b^4 = 1 → b = 1 → f(b) = b → false
-  · field_simp [hab₁] at H₂ 
-    field_simp [ne_of_gt hb] at H₂ 
+  · field_simp [hab₁] at H₂
+    field_simp [ne_of_gt hb] at H₂
     have hb₁ : b ^ 4 = 1 := by linear_combination -H₂
     obtain hb₂ := abs_eq_one_of_pow_eq_one b 4 (show 4 ≠ 0 by norm_num) hb₁
-    rw [abs_of_pos hb] at hb₂ ; rw [hb₂] at hfb₁ ; exact hfb₁ h₁
+    rw [abs_of_pos hb] at hb₂; rw [hb₂] at hfb₁; exact hfb₁ h₁
   -- f(ab) = 1/ab → a^4 = 1 → a = 1 → f(a) = 1/a → false
   · have hb_ne_0 : b ≠ 0 := ne_of_gt hb
-    field_simp [hab₂] at H₂ 
+    field_simp [hab₂] at H₂
     have H₃ : 2 * b ^ 4 * (a ^ 4 - 1) = 0 := by linear_combination H₂
     have h2b4_ne_0 : 2 * b ^ 4 ≠ 0 := mul_ne_zero two_ne_zero (pow_ne_zero 4 hb_ne_0)
     have ha₁ : a ^ 4 = 1 := by simpa [sub_eq_zero, h2b4_ne_0] using H₃
     obtain ha₂ := abs_eq_one_of_pow_eq_one a 4 (show 4 ≠ 0 by norm_num) ha₁
-    rw [abs_of_pos ha] at ha₂ ; rw [ha₂] at hfa₁ ; norm_num at hfa₁ 
+    rw [abs_of_pos ha] at ha₂; rw [ha₂] at hfa₁; norm_num at hfa₁
 #align imo2008_q4 imo2008_q4
 

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

@@ -83,7 +83,7 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
     field_simp at H₂ ⊢
     linear_combination 1 / 2 * H₂
   have h₃ : ∀ x > 0, f x = x ∨ f x = 1 / x := by simpa [sub_eq_zero] using h₂
-  by_contra' h
+  by_contra! h
   rcases h with ⟨⟨b, hb, hfb₁⟩, ⟨a, ha, hfa₁⟩⟩
   obtain hfa₂ := Or.resolve_right (h₃ a ha) hfa₁
   -- f(a) ≠ 1/a, f(a) = a

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

@@ -3,10 +3,10 @@ Copyright (c) 2021 Manuel Candales. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Manuel Candales
 -/
-import Mathbin.Data.Real.Basic
-import Mathbin.Data.Real.Sqrt
-import Mathbin.Data.Real.Nnreal
-import Mathbin.Tactic.LinearCombination
+import Data.Real.Basic
+import Data.Real.Sqrt
+import Data.Real.Nnreal
+import Tactic.LinearCombination
 
 #align_import imo.imo2008_q4 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Manuel Candales. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Manuel Candales
-
-! This file was ported from Lean 3 source module imo.imo2008_q4
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Real.Basic
 import Mathbin.Data.Real.Sqrt
 import Mathbin.Data.Real.Nnreal
 import Mathbin.Tactic.LinearCombination
 
+#align_import imo.imo2008_q4 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # IMO 2008 Q4
 

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: Manuel Candales
 
 ! This file was ported from Lean 3 source module imo.imo2008_q4
-! 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.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Tactic.LinearCombination
 
 /-!
 # IMO 2008 Q4
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
 Find all functions `f : (0,∞) → (0,∞)` (so, `f` is a function from the positive real
 numbers to the positive real numbers) such that
       ```

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.

@@ -29,8 +29,7 @@ open Real
 namespace Imo2008Q4
 
 theorem abs_eq_one_of_pow_eq_one (x : ℝ) (n : ℕ) (hn : n ≠ 0) (h : x ^ n = 1) : |x| = 1 := by
-  rw [← pow_left_inj (abs_nonneg x) zero_le_one (pos_iff_ne_zero.2 hn), one_pow, pow_abs, h,
-    abs_one]
+  rw [← pow_left_inj (abs_nonneg x) zero_le_one hn, one_pow, pow_abs, h, abs_one]
 #align imo2008_q4.abs_eq_one_of_pow_eq_one Imo2008Q4.abs_eq_one_of_pow_eq_one
 
 end Imo2008Q4

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

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

@@ -68,7 +68,7 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
     field_simp at H₂ ⊢
     linear_combination 1 / 2 * H₂
   have h₃ : ∀ x > 0, f x = x ∨ f x = 1 / x := by simpa [sub_eq_zero] using h₂
-  by_contra' h
+  by_contra! h
   rcases h with ⟨⟨b, hb, hfb₁⟩, ⟨a, ha, hfa₁⟩⟩
   obtain hfa₂ := Or.resolve_right (h₃ a ha) hfa₁
   -- f(a) ≠ 1/a, f(a) = a

@@ -83,7 +83,7 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
   specialize h₃ (a * b) hab
   cases' h₃ with hab₁ hab₂
   -- f(ab) = ab → b^4 = 1 → b = 1 → f(b) = b → false
-  · rw [ hab₁, div_left_inj' h2ab_ne_0 ] at H₂
+  · rw [hab₁, div_left_inj' h2ab_ne_0] at H₂
     field_simp at H₂
     have hb₁ : b ^ 4 = 1 := by linear_combination -H₂
     obtain hb₂ := abs_eq_one_of_pow_eq_one b 4 (show 4 ≠ 0 by norm_num) hb₁

norm_num was passing the wrong syntax node to elabSimpArgs when elaborating, which essentially had the effect of ignoring all arguments it was passed, i.e. norm_num [add_comm] would not try to commute addition in the simp step. The fix itself is very simple (though not obvious to debug!), probably using TSyntax more would help avoid such issues in future.

Due to this bug many norm_num [blah] became rw [blah]; norm_num or similar, sometimes with porting notes, sometimes not, we fix these porting notes and other regressions during the port also.

Interestingly cancel_denoms uses norm_num [<- mul_assoc] internally, so cancel_denoms also got stronger with this change.

@@ -56,8 +56,8 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
   have h₀ : f 1 ≠ 0 := by specialize H₁ 1 zero_lt_one; exact ne_of_gt H₁
   have h₁ : f 1 = 1 := by
     specialize H₂ 1 1 1 1 zero_lt_one zero_lt_one zero_lt_one zero_lt_one rfl
-    norm_num at H₂
-    rw [← two_mul, ← two_mul, mul_div_mul_left (f 1 ^ 2) (f 1) two_ne_zero] at H₂
+    norm_num [← two_mul] at H₂
+    rw [mul_div_mul_left (f 1 ^ 2) (f 1) two_ne_zero] at H₂
     rwa [← (div_eq_iff h₀).mpr (sq (f 1))]
   have h₂ : ∀ x > 0, (f x - x) * (f x - 1 / x) = 0 := by
     intro x hx

The main reasons is that having h : 0 < denom in the context should suffice for field_simp to do its job, without the need to manually pass h.ne or similar.

Quite a few have := … ≠ 0 could be dropped, and some field_simp calls no longer need explicit arguments; this is promising.

This does break some proofs where field_simp was not used as a closing tactic, and it now shuffles terms around a bit different. These were fixed. Using field_simp in the middle of a proof seems rather fragile anyways.

As a drive-by contribution, positivity now knows about π > 0.

fixes: #4835

Co-authored-by: Matthew Ballard <matt@mrb.email>

@@ -48,10 +48,8 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
       rw [h w hw, h x hx, h (y ^ 2) (pow_pos hy 2), h (z ^ 2) (pow_pos hz 2)]
     · intro w x y z hw hx hy hz hprod
       rw [h w hw, h x hx, h (y ^ 2) (pow_pos hy 2), h (z ^ 2) (pow_pos hz 2)]
-      have hy2z2 : y ^ 2 + z ^ 2 ≠ 0 := ne_of_gt (add_pos (pow_pos hy 2) (pow_pos hz 2))
-      have hz2y2 : z ^ 2 + y ^ 2 ≠ 0 := ne_of_gt (add_pos (pow_pos hz 2) (pow_pos hy 2))
       have hp2 : w ^ 2 * x ^ 2 = y ^ 2 * z ^ 2 := by linear_combination (w * x + y * z) * hprod
-      field_simp [ne_of_gt hw, ne_of_gt hx, ne_of_gt hy, ne_of_gt hz, hy2z2, hz2y2, hp2]
+      field_simp [hp2]
       ring
   -- proof that the only solutions are f(x) = x or f(x) = 1/x
   intro H₂
@@ -66,7 +64,6 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
     have h1xss : 1 * x = sqrt x * sqrt x := by rw [one_mul, mul_self_sqrt (le_of_lt hx)]
     specialize H₂ 1 x (sqrt x) (sqrt x) zero_lt_one hx (sqrt_pos.mpr hx) (sqrt_pos.mpr hx) h1xss
     rw [h₁, one_pow 2, sq_sqrt (le_of_lt hx), ← two_mul (f x), ← two_mul x] at H₂
-    have hx_ne_0 : x ≠ 0 := ne_of_gt hx
     have hfx_ne_0 : f x ≠ 0 := by specialize H₁ x hx; exact ne_of_gt H₁
     field_simp at H₂ ⊢
     linear_combination 1 / 2 * H₂
@@ -82,12 +79,12 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
   specialize H₂ a b (sqrt (a * b)) (sqrt (a * b)) ha hb (sqrt_pos.mpr hab) (sqrt_pos.mpr hab) habss
   rw [sq_sqrt (le_of_lt hab), ← two_mul (f (a * b)), ← two_mul (a * b)] at H₂
   rw [hfa₂, hfb₂] at H₂
-  have h2ab_ne_0 : 2 * (a * b) ≠ 0 := mul_ne_zero two_ne_zero (ne_of_gt hab)
+  have h2ab_ne_0 : 2 * (a * b) ≠ 0 := by positivity
   specialize h₃ (a * b) hab
   cases' h₃ with hab₁ hab₂
   -- f(ab) = ab → b^4 = 1 → b = 1 → f(b) = b → false
-  · field_simp [hab₁] at H₂
-    field_simp [ne_of_gt hb] at H₂
+  · rw [ hab₁, div_left_inj' h2ab_ne_0 ] at H₂
+    field_simp at H₂
     have hb₁ : b ^ 4 = 1 := by linear_combination -H₂
     obtain hb₂ := abs_eq_one_of_pow_eq_one b 4 (show 4 ≠ 0 by norm_num) hb₁
     rw [abs_of_pos hb] at hb₂; rw [hb₂] at hfb₁; exact hfb₁ h₁

• 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) 2021 Manuel Candales. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Manuel Candales
-
-! This file was ported from Lean 3 source module imo.imo2008_q4
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Real.Basic
 import Mathlib.Data.Real.Sqrt
 import Mathlib.Data.Real.NNReal
 import Mathlib.Tactic.LinearCombination
 
+#align_import imo.imo2008_q4 from "leanprover-community/mathlib"@"308826471968962c6b59c7ff82a22757386603e3"
+
 /-!
 # IMO 2008 Q4
 Find all functions `f : (0,∞) → (0,∞)` (so, `f` is a function from the positive real

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

@@ -93,7 +93,7 @@ theorem imo2008_q4 (f : ℝ → ℝ) (H₁ : ∀ x > 0, f x > 0) :
     field_simp [ne_of_gt hb] at H₂
     have hb₁ : b ^ 4 = 1 := by linear_combination -H₂
     obtain hb₂ := abs_eq_one_of_pow_eq_one b 4 (show 4 ≠ 0 by norm_num) hb₁
-    rw [abs_of_pos hb] at hb₂ ; rw [hb₂] at hfb₁ ; exact hfb₁ h₁
+    rw [abs_of_pos hb] at hb₂; rw [hb₂] at hfb₁; exact hfb₁ h₁
   -- f(ab) = 1/ab → a^4 = 1 → a = 1 → f(a) = 1/a → false
   · have hb_ne_0 : b ≠ 0 := ne_of_gt hb
     field_simp [hab₂] at H₂