imo.imo1972_q5 ⟷ Archive.Imo.Imo1972Q5

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

@@ -53,7 +53,7 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
       _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))
       _ = 2 * k := (two_mul _).symm
   -- Suppose the conclusion does not hold.
-  by_contra' hneg
+  by_contra! hneg
   set k' := k / ‖g y‖
   -- Demonstrate that `k' < k` using `hneg`.
   have H₁ : k' < k :=
@@ -98,7 +98,7 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
   obtain ⟨x, hx⟩ := hf3
   set k := ⨆ x, ‖f x‖
   have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2
-  by_contra' H
+  by_contra! H
   have hgy : 0 < ‖g y‖ := by linarith
   have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x)
   have : k / ‖g y‖ < k := (div_lt_iff hgy).mpr (lt_mul_of_one_lt_right k_pos H)

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

@@ -3,8 +3,8 @@ Copyright (c) 2020 Ruben Van de Velde, Stanislas Polu. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Ruben Van de Velde, Stanislas Polu
 -/
-import Mathbin.Data.Real.Basic
-import Mathbin.Analysis.NormedSpace.Basic
+import Data.Real.Basic
+import Analysis.NormedSpace.Basic
 
 #align_import imo.imo1972_q5 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -69,7 +69,7 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
   -- Demonstrate that `k ≤ k'` using `hk₂`.
   have H₂ : k ≤ k' :=
     by
-    have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0
+    have h₁ : ∃ x : ℝ, x ∈ S := by use‖f 0‖; exact Set.mem_range_self 0
     have h₂ : ∀ x, ‖f x‖ ≤ k' := by
       intro x
       rw [le_div_iff]

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Ruben Van de Velde, Stanislas Polu. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Ruben Van de Velde, Stanislas Polu
-
-! This file was ported from Lean 3 source module imo.imo1972_q5
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Real.Basic
 import Mathbin.Analysis.NormedSpace.Basic
 
+#align_import imo.imo1972_q5 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # IMO 1972 Q5
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Ruben Van de Velde, Stanislas Polu
 
 ! This file was ported from Lean 3 source module imo.imo1972_q5
-! leanprover-community/mathlib commit 5f25c089cb34db4db112556f23c50d12da81b297
+! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.NormedSpace.Basic
 /-!
 # IMO 1972 Q5
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Problem: `f` and `g` are real-valued functions defined on the real line. For all `x` and `y`,
 `f(x + y) + f(x - y) = 2f(x)g(y)`. `f` is not identically zero and `|f(x)| ≤ 1` for all `x`.
 Prove that `|g(x)| ≤ 1` for all `x`.

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

@@ -52,7 +52,6 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
       _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (norm_add_le _ _)
       _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))
       _ = 2 * k := (two_mul _).symm
-      
   -- Suppose the conclusion does not hold.
   by_contra' hneg
   set k' := k / ‖g y‖
@@ -64,7 +63,6 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
       calc
         0 < ‖f x‖ := norm_pos_iff.mpr hx
         _ ≤ k := hk₁ x
-        
     rw [div_lt_iff]
     apply lt_mul_of_one_lt_right h₁ hneg
     exact trans zero_lt_one hneg
@@ -85,7 +83,6 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
   calc
     k' < k := H₁
     _ ≤ k' := H₂
-    
 
 /-- IMO 1972 Q5
 
@@ -115,7 +112,6 @@ example (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x
       _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]
       _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)
       _ ≤ 2 * k := by linarith [h (x + y), h (x - y)]
-      
   linarith
 
 end Imo1972Q5

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

A PR accompanying #12339.

Zulip discussion

@@ -53,8 +53,8 @@ theorem imo1972_q5 (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y
         0 < ‖f x‖ := norm_pos_iff.mpr hx
         _ ≤ k := hk₁ x
     rw [div_lt_iff]
-    apply lt_mul_of_one_lt_right h₁ hneg
-    exact zero_lt_one.trans hneg
+    · apply lt_mul_of_one_lt_right h₁ hneg
+    · exact zero_lt_one.trans hneg
   -- Demonstrate that `k ≤ k'` using `hk₂`.
   have H₂ : k ≤ k' := by
     have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0

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

@@ -41,8 +41,8 @@ theorem imo1972_q5 (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y
     calc
       2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]
       _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]
-      _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (norm_add_le _ _)
-      _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))
+      _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := norm_add_le _ _
+      _ ≤ k + k := add_le_add (hk₁ _) (hk₁ _)
       _ = 2 * k := (two_mul _).symm
   set k' := k / ‖g y‖
   -- Demonstrate that `k' < k` using `hneg`.
@@ -98,6 +98,6 @@ theorem imo1972_q5' (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x -
     calc
       2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]
       _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]
-      _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)
+      _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := abs_add _ _
       _ ≤ 2 * k := by linarith [h (x + y), h (x - y)]
   linarith

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -33,7 +33,7 @@ theorem imo1972_q5 (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y
   let k : ℝ := sSup S
   -- Show that `‖f x‖ ≤ k`.
   have hk₁ : ∀ x, ‖f x‖ ≤ k := by
-    have h : BddAbove S := ⟨1, Set.forall_range_iff.mpr hf2⟩
+    have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩
     intro x
     exact le_csSup h (Set.mem_range_self x)
   -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -82,7 +82,7 @@ This is a more concise version of the proof proposed by Ruben Van de Velde.
 -/
 theorem imo1972_q5' (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x * g y)
     (hf2 : BddAbove (Set.range fun x => ‖f x‖)) (hf3 : ∃ x, f x ≠ 0) (y : ℝ) : ‖g y‖ ≤ 1 := by
-  -- porting note: moved `by_contra!` up to avoid a bug
+  -- Porting note: moved `by_contra!` up to avoid a bug
   by_contra! H
   obtain ⟨x, hx⟩ := hf3
   set k := ⨆ x, ‖f x‖

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

This follows on from #6964.

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

@@ -91,7 +91,7 @@ theorem imo1972_q5' (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x -
   have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x)
   have : k / ‖g y‖ < k := (div_lt_iff hgy).mpr (lt_mul_of_one_lt_right k_pos H)
   have : k ≤ k / ‖g y‖ := by
-    suffices : ∀ x, ‖f x‖ ≤ k / ‖g y‖; exact ciSup_le this
+    suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this
     intro x
     suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by
       rwa [le_div_iff hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]

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

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

@@ -27,7 +27,7 @@ from `hneg` directly), finally raising a contradiction with `k' < k'`.
 theorem imo1972_q5 (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x * g y)
     (hf2 : ∀ y, ‖f y‖ ≤ 1) (hf3 : ∃ x, f x ≠ 0) (y : ℝ) : ‖g y‖ ≤ 1 := by
   -- Suppose the conclusion does not hold.
-  by_contra' hneg
+  by_contra! hneg
   set S := Set.range fun x => ‖f x‖
   -- Introduce `k`, the supremum of `f`.
   let k : ℝ := sSup S
@@ -82,8 +82,8 @@ This is a more concise version of the proof proposed by Ruben Van de Velde.
 -/
 theorem imo1972_q5' (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x * g y)
     (hf2 : BddAbove (Set.range fun x => ‖f x‖)) (hf3 : ∃ x, f x ≠ 0) (y : ℝ) : ‖g y‖ ≤ 1 := by
-  -- porting note: moved `by_contra'` up to avoid a bug
-  by_contra' H
+  -- porting note: moved `by_contra!` up to avoid a bug
+  by_contra! H
   obtain ⟨x, hx⟩ := hf3
   set k := ⨆ x, ‖f x‖
   have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2

• 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) 2020 Ruben Van de Velde, Stanislas Polu. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Ruben Van de Velde, Stanislas Polu
-
-! This file was ported from Lean 3 source module imo.imo1972_q5
-! leanprover-community/mathlib commit 5f25c089cb34db4db112556f23c50d12da81b297
-! 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.Analysis.NormedSpace.Basic
 
+#align_import imo.imo1972_q5 from "leanprover-community/mathlib"@"5f25c089cb34db4db112556f23c50d12da81b297"
+
 /-!
 # IMO 1972 Q5
 

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>