imo.imo2020_q2
⟷
Archive.Imo.Imo2020Q2
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -47,7 +47,7 @@ theorem imo2020_q2 (a b c d : ℝ) (hd0 : 0 < d) (hdc : d ≤ c) (hcb : c ≤ b)
(3 * a + 3 * b + c + 3 * d) * (c * c) +
(3 * a + 3 * b + 3 * c + d) * (d * d) :=
by
- apply_rules [add_le_add] <;> refine' mul_le_mul_of_nonneg_right _ (mul_self_nonneg _) <;>
+ apply_rules [add_le_add] <;> refine' mul_le_mul_of_nonneg_right _ (hMul_self_nonneg _) <;>
linarith
_ <
(a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b) +
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
-import Mathbin.Analysis.MeanInequalities
+import Analysis.MeanInequalities
#align_import imo.imo2020_q2 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-
-! This file was ported from Lean 3 source module imo.imo2020_q2
-! 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.MeanInequalities
+#align_import imo.imo2020_q2 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
/-!
# IMO 2020 Q2
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: Joseph Myers, Yury Kudryashov
! This file was ported from Lean 3 source module imo.imo2020_q2
-! 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.
-/
@@ -13,6 +13,9 @@ import Mathbin.Analysis.MeanInequalities
/-!
# IMO 2020 Q2
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
The real numbers `a`, `b`, `c`, `d` are such that `a ≥ b ≥ c ≥ d > 0` and `a + b + c + d = 1`.
Prove that `(a + 2b + 3c + 4d) a^a b^b c^c d^d < 1`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -57,6 +57,5 @@ theorem imo2020_q2 (a b c d : ℝ) (hd0 : 0 < d) (hdc : d ≤ c) (hcb : c ≤ b)
(lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))
_ = (a + b + c + d) ^ 3 := by ring
_ = 1 := by simp [h1]
-
#align imo2020_q2 imo2020_q2
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090
@@ -31,17 +31,15 @@ theorem imo2020_q2 (a b c d : ℝ) (hd0 : 0 < d) (hdc : d ≤ c) (hcb : c ≤ b)
(a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d =
(a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) := by ac_rfl
_ ≤ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) := by gcongr; linarith
- _ = (a + 2 * b + 3 * c + 4 * d) * (a * a) + (a + 2 * b + 3 * c + 4 * d) * (b * b)
- + (a + 2 * b + 3 * c + 4 * d) * (c * c) + (a + 2 * b + 3 * c + 4 * d) * (d * d) := by ring
- _ ≤ (a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b)
- + (3 * a + 3 * b + c + 3 * d) * (c * c) + (3 * a + 3 * b + 3 * c + d) * (d * d) := by
- -- Porting note: after #2220 is fixed we can change all `a * a` to `a ^ 2` and automation
- -- will work better
- gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _ <;> (try linarith) <;> apply mul_self_nonneg
- _ < (a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b)
- + (3 * a + 3 * b + c + 3 * d) * (c * c) + (3 * a + 3 * b + 3 * c + d) * (d * d)
+ _ = (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2
+ + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 + (a + 2 * b + 3 * c + 4 * d) * d ^ 2 := by ring
+ _ ≤ (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2
+ + (3 * a + 3 * b + c + 3 * d) * c ^ 2 + (3 * a + 3 * b + 3 * c + d) * d ^ 2 := by
+ gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _ <;> linarith
+ _ < (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2
+ + (3 * a + 3 * b + c + 3 * d) * c ^ 2 + (3 * a + 3 * b + 3 * c + d) * d ^ 2
+ (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) :=
(lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))
- _ = HPow.hPow (a + b + c + d) 3 := by ring -- Porting note: See issue #2220
+ _ = (a + b + c + d) ^ 3 := by ring
_ = 1 := by simp [h1]
#align imo2020_q2 imo2020_q2
@@ -2,14 +2,11 @@
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-
-! This file was ported from Lean 3 source module imo.imo2020_q2
-! 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.Analysis.MeanInequalities
+#align_import imo.imo2020_q2 from "leanprover-community/mathlib"@"5f25c089cb34db4db112556f23c50d12da81b297"
+
/-!
# IMO 2020 Q2
This PR completes the port of the entire Probability
folder to mathlib4.
@@ -32,27 +32,19 @@ theorem imo2020_q2 (a b c d : ℝ) (hd0 : 0 < d) (hdc : d ≤ c) (hcb : c ≤ b)
refine' geom_mean_le_arith_mean4_weighted _ _ _ _ _ _ _ _ h1 <;> linarith
calc
(a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d =
- (a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) :=
- by ac_rfl
- _ ≤ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) :=
- (mul_le_mul_of_nonneg_left hp (by linarith))
- _ =
- (a + 2 * b + 3 * c + 4 * d) * (a * a) + (a + 2 * b + 3 * c + 4 * d) * (b * b) +
- (a + 2 * b + 3 * c + 4 * d) * (c * c) +
- (a + 2 * b + 3 * c + 4 * d) * (d * d) :=
- by simp only [mul_add]
- _ ≤
- (a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b) +
- (3 * a + 3 * b + c + 3 * d) * (c * c) +
- (3 * a + 3 * b + 3 * c + d) * (d * d) := by
- apply_rules [add_le_add] <;> refine' mul_le_mul_of_nonneg_right _ (mul_self_nonneg _) <;>
- linarith
- _ <
- (a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b) +
- (3 * a + 3 * b + c + 3 * d) * (c * c) +
- (3 * a + 3 * b + 3 * c + d) * (d * d) +
- (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) :=
- (lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))
+ (a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) := by ac_rfl
+ _ ≤ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) := by gcongr; linarith
+ _ = (a + 2 * b + 3 * c + 4 * d) * (a * a) + (a + 2 * b + 3 * c + 4 * d) * (b * b)
+ + (a + 2 * b + 3 * c + 4 * d) * (c * c) + (a + 2 * b + 3 * c + 4 * d) * (d * d) := by ring
+ _ ≤ (a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b)
+ + (3 * a + 3 * b + c + 3 * d) * (c * c) + (3 * a + 3 * b + 3 * c + d) * (d * d) := by
+ -- Porting note: after #2220 is fixed we can change all `a * a` to `a ^ 2` and automation
+ -- will work better
+ gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _ <;> (try linarith) <;> apply mul_self_nonneg
+ _ < (a + 3 * b + 3 * c + 3 * d) * (a * a) + (3 * a + b + 3 * c + 3 * d) * (b * b)
+ + (3 * a + 3 * b + c + 3 * d) * (c * c) + (3 * a + 3 * b + 3 * c + d) * (d * d)
+ + (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) :=
+ (lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))
_ = HPow.hPow (a + b + c + d) 3 := by ring -- Porting note: See issue #2220
_ = 1 := by simp [h1]
#align imo2020_q2 imo2020_q2
The unported dependencies are
algebra.order.module
init.core
linear_algebra.free_module.finite.rank
algebra.order.monoid.cancel.defs
algebra.abs
algebra.group_power.lemmas
init.data.list.basic
linear_algebra.free_module.rank
algebra.order.monoid.cancel.basic
init.data.list.default
topology.subset_properties
init.logic
The following 1 dependencies have changed in mathlib3 since they were ported, which may complicate porting this file