state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case mk.intro.intro.mk.intro.intro
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpa... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | ext v | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refi... | Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.mk.intro.intro.h
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalS... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | show ((e.pullback f).coordChangeL 𝕜 (e'.pullback f) b) v = (e.coordChangeL 𝕜 e' (f b)) v | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refi... | Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.mk.intro.intro.h
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalS... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [e.coordChangeL_apply e' hb, (e.pullback f).coordChangeL_apply' _] | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refi... | Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.mk.intro.intro.h
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalS... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | exacts [rfl, hb] | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refi... | Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finset ι
hs_nonempty : Finset.... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _ | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib.Algebra.BigOperators.Order.33_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib_Algebra_BigOperators_Order |
case refine'_1
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finset ι
hs_non... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp [hs_nonempty.ne_empty] | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib.Algebra.BigOperators.Order.33_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib_Algebra_BigOperators_Order |
case refine'_2
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finset ι
hs_non... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact Multiset.forall_mem_map_iff.mpr hs | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib.Algebra.BigOperators.Order.33_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib_Algebra_BigOperators_Order |
case refine'_3
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finset ι
hs_non... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [Multiset.map_map] | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib.Algebra.BigOperators.Order.33_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib_Algebra_BigOperators_Order |
case refine'_3
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finset ι
hs_non... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rfl | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib.Algebra.BigOperators.Order.33_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
submultiplicative on `{x | p x}`, i.e., `p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be
a nonempty finite family of elements of `M` such that `∀ i ∈ s, p (g i)`. Then
`f (∏ x in s, g x) ≤ ∏ x in s, f (g x)`. -/
@[to_... | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_one : f 1 = 1
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finset ι
hs : ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rcases eq_empty_or_nonempty s with (rfl | hs_nonempty) | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be a finite family of elements of `M` such
that `∀ i ∈ s, p (g i)`. Then `f (∏ i in s, g i) ≤ ∏ i in... | Mathlib.Algebra.BigOperators.Order.70_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be a finite family of elements of `M` such
that `∀ i ∈ s, p (g i)`. Then `f (∏ i in s, g i) ≤ ∏ i in... | Mathlib_Algebra_BigOperators_Order |
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_one : f 1 = 1
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
hs : ∀ i ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp [h_one] | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be a finite family of elements of `M` such
that `∀ i ∈ s, p (g i)`. Then `f (∏ i in s, g i) ≤ ∏ i in... | Mathlib.Algebra.BigOperators.Order.70_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be a finite family of elements of `M` such
that `∀ i ∈ s, p (g i)`. Then `f (∏ i in s, g i) ≤ ∏ i in... | Mathlib_Algebra_BigOperators_Order |
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
p : M → Prop
h_one : f 1 = 1
h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y
hp_mul : ∀ (x y : M), p x → p y → p (x * y)
g : ι → M
s : Finse... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be a finite family of elements of `M` such
that `∀ i ∈ s, p (g i)`. Then `f (∏ i in s, g i) ≤ ∏ i in... | Mathlib.Algebra.BigOperators.Order.70_0.ewL52iF1Dz3xeLh | /-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M → N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x → p y → f (x * y) ≤ f x * f y`. Let `g i`, `i ∈ s`, be a finite family of elements of `M` such
that `∀ i ∈ s, p (g i)`. Then `f (∏ i in s, g i) ≤ ∏ i in... | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
h_one : f 1 = 1
h_mul : ∀ (x y : M), f (x * y) ≤ f x * f y
s : Finset ι
g : ι → M
⊢ f (∏ i in s, g i) ≤ ∏ i in s, f (g i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _ | /-- If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, ... | Mathlib.Algebra.BigOperators.Order.90_0.ewL52iF1Dz3xeLh | /-- If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, ... | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
h_one : f 1 = 1
h_mul : ∀ (x y : M), f (x * y) ≤ f x * f y
s : Finset ι
g : ι → M
⊢ Multiset.prod (Multiset.map f (Multiset.map (fun i => g i) s.val)) ≤ ∏ i... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [Multiset.map_map] | /-- If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, ... | Mathlib.Algebra.BigOperators.Order.90_0.ewL52iF1Dz3xeLh | /-- If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, ... | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f : M → N
h_one : f 1 = 1
h_mul : ∀ (x y : M), f (x * y) ≤ f x * f y
s : Finset ι
g : ι → M
⊢ Multiset.prod (Multiset.map (f ∘ fun i => g i) s.val) ≤ ∏ i in s, f (g i... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rfl | /-- If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, ... | Mathlib.Algebra.BigOperators.Order.90_0.ewL52iF1Dz3xeLh | /-- If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, ... | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
h : ∀ i ∈ s, 1 ≤ f i
⊢ 1 ≤ ∏ i in s, 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_const_one] | @[to_additive sum_nonneg]
theorem one_le_prod' (h : ∀ i ∈ s, 1 ≤ f i) : 1 ≤ ∏ i in s, f i :=
le_trans (by | Mathlib.Algebra.BigOperators.Order.138_0.ewL52iF1Dz3xeLh | @[to_additive sum_nonneg]
theorem one_le_prod' (h : ∀ i ∈ s, 1 ≤ f i) : 1 ≤ ∏ i in s, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
h : ∀ i ∈ s, f i ≤ 1
⊢ ∏ i in s, 1 = 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_const_one] | @[to_additive sum_nonpos]
theorem prod_le_one' (h : ∀ i ∈ s, f i ≤ 1) : ∏ i in s, f i ≤ 1 :=
(prod_le_prod' h).trans_eq (by | Mathlib.Algebra.BigOperators.Order.150_0.ewL52iF1Dz3xeLh | @[to_additive sum_nonpos]
theorem prod_le_one' (h : ∀ i ∈ s, f i ≤ 1) : ∏ i in s, f i ≤ 1 | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
h : s ⊆ t
hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i
⊢ ∏ i in s, f i ≤ ∏ i in t, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | classical calc
∏ i in s, f i ≤ (∏ i in t \ s, f i) * ∏ i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = ∏ i in t \ s ∪ s, f i := (prod_union sdiff_disjoint).symm
_ = ∏ i in t, f i := by rw [sdiff_union_of_subset h] | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i := by
| Mathlib.Algebra.BigOperators.Order.156_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
h : s ⊆ t
hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i
⊢ ∏ i in s, f i ≤ ∏ i in t, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | calc
∏ i in s, f i ≤ (∏ i in t \ s, f i) * ∏ i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = ∏ i in t \ s ∪ s, f i := (prod_union sdiff_disjoint).symm
_ = ∏ i in t, f i := by rw [sdiff_union_of_subset h] | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i := by
classical | Mathlib.Algebra.BigOperators.Order.156_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
h : s ⊆ t
hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i
⊢ ∀ i ∈ t \ s, 1 ≤ f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simpa only [mem_sdiff, and_imp] | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i := by
classical calc
∏ i in s, f i ≤ (∏ i in t \ s, f i) * ∏ i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by | Mathlib.Algebra.BigOperators.Order.156_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
h : s ⊆ t
hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i
⊢ ∏ i in t \ s ∪ s, f i = ∏ i in t, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [sdiff_union_of_subset h] | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i := by
classical calc
∏ i in s, f i ≤ (∏ i in t \ s, f i) * ∏ i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem... | Mathlib.Algebra.BigOperators.Order.156_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
⊢ (∀ i ∈ s, 1 ≤ f i) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | classical
refine Finset.induction_on s
(fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h), fun _ ↦ rfl⟩) ?_
intro a s ha ih H
have : ∀ i ∈ s, 1 ≤ f i := fun _ ↦ H _ ∘ mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_i... | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) := by
| Mathlib.Algebra.BigOperators.Order.181_0.ewL52iF1Dz3xeLh | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
⊢ (∀ i ∈ s, 1 ≤ f i) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine Finset.induction_on s
(fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h), fun _ ↦ rfl⟩) ?_ | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) := by
classical
| Mathlib.Algebra.BigOperators.Order.181_0.ewL52iF1Dz3xeLh | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
a ∉ s →
((∀ i ∈ s, 1 ≤ f i) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1)) →
(∀ i ∈ insert a s, 1 ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | intro a s ha ih H | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h), fun _ ↦ rfl⟩) ?_
| Mathlib.Algebra.BigOperators.Order.181_0.ewL52iF1Dz3xeLh | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s✝ t : Finset ι
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, 1 ≤ f i) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1)
H : ∀ i ∈ insert a s, 1 ≤ f i
⊢ ∏ i in inse... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | have : ∀ i ∈ s, 1 ≤ f i := fun _ ↦ H _ ∘ mem_insert_of_mem | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h), fun _ ↦ rfl⟩) ?_
intro a s ha ih H
| Mathlib.Algebra.BigOperators.Order.181_0.ewL52iF1Dz3xeLh | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s✝ t : Finset ι
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, 1 ≤ f i) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1)
H : ∀ i ∈ insert a s, 1 ≤ f i
this : ∀ i ∈ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this] | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h), fun _ ↦ rfl⟩) ?_
intro a s ha ih H
have : ∀ i ∈ s,... | Mathlib.Algebra.BigOperators.Order.181_0.ewL52iF1Dz3xeLh | @[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(∀ i ∈ s, 1 ≤ f i) → ((∏ i in s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
i j : ι
hf : ∀ i ∈ s, 1 ≤ f i
hi : i ∈ s
hj : j ∈ s
hne : i ≠ j
⊢ i ∉ {j} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simpa | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k :=
calc
f i * f j = ∏ k in .cons i {j} (by | Mathlib.Algebra.BigOperators.Order.211_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
i j : ι
hf : ∀ i ∈ s, 1 ≤ f i
hi : i ∈ s
hj : j ∈ s
hne : i ≠ j
⊢ f i * f j = ∏ k in cons i {j} (_ : i ∉ {j}), f k | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_cons, prod_singleton] | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k :=
calc
f i * f j = ∏ k in .cons i {j} (by simpa), f k := by | Mathlib.Algebra.BigOperators.Order.211_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
i j : ι
hf : ∀ i ∈ s, 1 ≤ f i
hi : i ∈ s
hj : j ∈ s
hne : i ≠ j
⊢ ∏ k in cons i {j} (_ : i ∉ {j}), f k ≤ ∏ k in s, f k | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ ↦ hf k hk | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k :=
calc
f i * f j = ∏ k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ ≤ ∏ k in s, f k := by
| Mathlib.Algebra.BigOperators.Order.211_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f g : ι → N
s t : Finset ι
i j : ι
hf : ∀ i ∈ s, 1 ≤ f i
hi : i ∈ s
hj : j ∈ s
hne : i ≠ j
⊢ cons i {j} (_ : i ∉ {j}) ⊆ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp [cons_subset, *] | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k :=
calc
f i * f j = ∏ k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ ≤ ∏ k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k... | Mathlib.Algebra.BigOperators.Order.211_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
f i * f j ≤ ∏ k in s, f k | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f✝ g : ι → N
s✝ t s : Finset ι
f : ι → N
n : N
h : ∀ x ∈ s, f x ≤ n
⊢ Finset.prod s f ≤ n ^ card s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _ | @[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card := by
| Mathlib.Algebra.BigOperators.Order.220_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card | Mathlib_Algebra_BigOperators_Order |
case refine'_1
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f✝ g : ι → N
s✝ t s : Finset ι
f : ι → N
n : N
h : ∀ x ∈ s, f x ≤ n
⊢ ∀ x ∈ Multiset.map f s.val, x ≤ n | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simpa using h | @[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
· | Mathlib.Algebra.BigOperators.Order.220_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card | Mathlib_Algebra_BigOperators_Order |
case refine'_2
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : CommMonoid M
inst✝ : OrderedCommMonoid N
f✝ g : ι → N
s✝ t s : Finset ι
f : ι → N
n : N
h : ∀ x ∈ s, f x ≤ n
⊢ n ^ Multiset.card (Multiset.map f s.val) ≤ n ^ card s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp | @[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
· simpa using h
· | Mathlib.Algebra.BigOperators.Order.220_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G✝ : Type u_6
k : Type u_7
R : Type u_8
G : Type u_9
inst✝ : LinearOrderedAddCommGroup G
f : ι → G
s : Finset ι
hf : ∀ i ∈ s, 0 ≤ f i
⊢ |∑ i in s, f i| = ∑ i in s, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [abs_of_nonneg (Finset.sum_nonneg hf)] | theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
(hf : ∀ i ∈ s, 0 ≤ f i) : |∑ i : ι in s, f i| = ∑ i : ι in s, f i := by
| Mathlib.Algebra.BigOperators.Order.271_0.ewL52iF1Dz3xeLh | theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
(hf : ∀ i ∈ s, 0 ≤ f i) : |∑ i : ι in s, f i| = ∑ i : ι in s, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G✝ : Type u_6
k : Type u_7
R : Type u_8
G : Type u_9
inst✝ : LinearOrderedAddCommGroup G
f : ι → G
s : Finset ι
hf : ∀ (i : ι), 0 ≤ f i
⊢ |∑ i in s, f i| = ∑ i in s, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [abs_of_nonneg (Finset.sum_nonneg' hf)] | theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
(hf : ∀ i, 0 ≤ f i) : |∑ i : ι in s, f i| = ∑ i : ι in s, f i := by
| Mathlib.Algebra.BigOperators.Order.276_0.ewL52iF1Dz3xeLh | theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
(hf : ∀ i, 0 ≤ f i) : |∑ i : ι in s, f i| = ∑ i : ι in s, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq β
f : α → β
s : Finset α
t : Finset β
Hf : ∀ a ∈ s, f a ∈ t
n : ℕ
hn : ∀ a ∈ t, card (filter (fun x => f x = a) s) ≤ n
⊢ ∑ _a in t, n = n * card t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp [mul_comm] | theorem card_le_mul_card_image_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, (s.filter fun x ↦ f x = a).card ≤ n) :
s.card ≤ n * t.card :=
calc
s.card = ∑ a in t, (s.filter fun x ↦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ ≤ ∑ _a in t, n := ... | Mathlib.Algebra.BigOperators.Order.290_0.ewL52iF1Dz3xeLh | theorem card_le_mul_card_image_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, (s.filter fun x ↦ f x = a).card ≤ n) :
s.card ≤ n * t.card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq β
f : α → β
s : Finset α
t : Finset β
Hf : ∀ a ∈ s, f a ∈ t
n : ℕ
hn : ∀ a ∈ t, n ≤ card (filter (fun x => f x = a) s)
⊢ n * card t = ∑ _a in t, n | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp [mul_comm] | theorem mul_card_image_le_card_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (s.filter fun x ↦ f x = a).card) :
n * t.card ≤ s.card :=
calc
n * t.card = ∑ _a in t, n := by | Mathlib.Algebra.BigOperators.Order.304_0.ewL52iF1Dz3xeLh | theorem mul_card_image_le_card_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (s.filter fun x ↦ f x = a).card) :
n * t.card ≤ s.card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq β
f : α → β
s : Finset α
t : Finset β
Hf : ∀ a ∈ s, f a ∈ t
n : ℕ
hn : ∀ a ∈ t, n ≤ card (filter (fun x => f x = a) s)
⊢ ∑ a in t, card (filter (fun x => f x = a) s) = card s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [← card_eq_sum_card_fiberwise Hf] | theorem mul_card_image_le_card_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (s.filter fun x ↦ f x = a).card) :
n * t.card ≤ s.card :=
calc
n * t.card = ∑ _a in t, n := by simp [mul_comm]
_ ≤ ∑ a in t, (s.filter fun x ↦ f x = a).card := sum_le_... | Mathlib.Algebra.BigOperators.Order.304_0.ewL52iF1Dz3xeLh | theorem mul_card_image_le_card_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (s.filter fun x ↦ f x = a).card) :
n * t.card ≤ s.card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, card (filter (fun x => a ∈ x) B) ≤ n
⊢ ∑ t in B, card (s ∩ t) ≤ card s * n | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine' le_trans _ (s.sum_le_card_nsmul _ _ h) | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
(∑ t in B, (s ∩ t).card) ≤ s.card * n := by
| Mathlib.Algebra.BigOperators.Order.324_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
(∑ t in B, (s ∩ t).card) ≤ s.card * n | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, card (filter (fun x => a ∈ x) B) ≤ n
⊢ ∑ t in B, card (s ∩ t) ≤ ∑ x in s, card (filter (fun x_1 => x ∈ x_1) B) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp_rw [← filter_mem_eq_inter, card_eq_sum_ones, sum_filter] | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
(∑ t in B, (s ∩ t).card) ≤ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
| Mathlib.Algebra.BigOperators.Order.324_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
(∑ t in B, (s ∩ t).card) ≤ s.card * n | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, card (filter (fun x => a ∈ x) B) ≤ n
⊢ (∑ x in B, ∑ a in s, if a ∈ x then 1 else 0) ≤ ∑ x in s, ∑ a in B, if x ∈ a then 1 else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact sum_comm.le | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
(∑ t in B, (s ∩ t).card) ≤ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [← filter_mem_eq_i... | Mathlib.Algebra.BigOperators.Order.324_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
(∑ t in B, (s ∩ t).card) ≤ s.card * n | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
inst✝ : Fintype α
h : ∀ (a : α), card (filter (fun x => a ∈ x) B) ≤ n
⊢ ∑ s in B, card s = ∑ s in B, card (univ ∩ s) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp_rw [univ_inter] | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card ≤ n) :
∑ s in B, s.card ≤ Fintype.card α * n :=
calc
∑ s in B, s.card = ∑ s in B, (univ ∩ s).card := by | Mathlib.Algebra.BigOperators.Order.333_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card ≤ n) :
∑ s in B, s.card ≤ Fintype.card α * n | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, n ≤ card (filter (fun x => a ∈ x) B)
⊢ card s * n ≤ ∑ t in B, card (s ∩ t) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | apply (s.card_nsmul_le_sum _ _ h).trans | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card := by
| Mathlib.Algebra.BigOperators.Order.342_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, n ≤ card (filter (fun x => a ∈ x) B)
⊢ ∑ x in s, card (filter (fun x_1 => x ∈ x_1) B) ≤ ∑ t in B, card (s ∩ t) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp_rw [← filter_mem_eq_inter, card_eq_sum_ones, sum_filter] | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
| Mathlib.Algebra.BigOperators.Order.342_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, n ≤ card (filter (fun x => a ∈ x) B)
⊢ (∑ x in s, ∑ a in B, if x ∈ a then 1 else 0) ≤ ∑ x in B, ∑ a in s, if a ∈ x then 1 else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact sum_comm.le | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [← filter_mem_eq_inter, c... | Mathlib.Algebra.BigOperators.Order.342_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
inst✝ : Fintype α
h : ∀ (a : α), n ≤ card (filter (fun x => a ∈ x) B)
⊢ ∑ s in B, card (univ ∩ s) = ∑ s in B, card s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp_rw [univ_inter] | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype α] (h : ∀ a, n ≤ (B.filter (a ∈ ·)).card) :
Fintype.card α * n ≤ ∑ s in B, s.card :=
calc
Fintype.card α * n ≤ ∑ s in B, (univ ∩ s).card := le_sum_card_int... | Mathlib.Algebra.BigOperators.Order.351_0.ewL52iF1Dz3xeLh | /-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype α] (h : ∀ a, n ≤ (B.filter (a ∈ ·)).card) :
Fintype.card α * n ≤ ∑ s in B, s.card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
inst✝ : Fintype α
h : ∀ (a : α), card (filter (fun x => a ∈ x) B) = n
⊢ ∑ s in B, card s = Fintype.card α * n | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp_rw [Fintype.card, ← sum_card_inter fun a _ ↦ h a, univ_inter] | /-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card = n) :
∑ s in B, s.card = Fintype.card α * n := by
| Mathlib.Algebra.BigOperators.Order.367_0.ewL52iF1Dz3xeLh | /-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card = n) :
∑ s in B, s.card = Fintype.card α * n | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s✝ : Finset α
B : Finset (Finset α)
n : ℕ
s : Finset ι
f : ι → Finset α
hs : Set.PairwiseDisjoint (↑s) f
hf : ∀ i ∈ s, Finset.Nonempty (f i)
⊢ card s ≤ card (Finset.biUnion s f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [card_biUnion hs, card_eq_sum_ones] | theorem card_le_card_biUnion {s : Finset ι} {f : ι → Finset α} (hs : (s : Set ι).PairwiseDisjoint f)
(hf : ∀ i ∈ s, (f i).Nonempty) : s.card ≤ (s.biUnion f).card := by
| Mathlib.Algebra.BigOperators.Order.374_0.ewL52iF1Dz3xeLh | theorem card_le_card_biUnion {s : Finset ι} {f : ι → Finset α} (hs : (s : Set ι).PairwiseDisjoint f)
(hf : ∀ i ∈ s, (f i).Nonempty) : s.card ≤ (s.biUnion f).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s✝ : Finset α
B : Finset (Finset α)
n : ℕ
s : Finset ι
f : ι → Finset α
hs : Set.PairwiseDisjoint (↑s) f
hf : ∀ i ∈ s, Finset.Nonempty (f i)
⊢ ∑ x in s, 1 ≤ ∑ u in s, card (f u) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact sum_le_sum fun i hi ↦ (hf i hi).card_pos | theorem card_le_card_biUnion {s : Finset ι} {f : ι → Finset α} (hs : (s : Set ι).PairwiseDisjoint f)
(hf : ∀ i ∈ s, (f i).Nonempty) : s.card ≤ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
| Mathlib.Algebra.BigOperators.Order.374_0.ewL52iF1Dz3xeLh | theorem card_le_card_biUnion {s : Finset ι} {f : ι → Finset α} (hs : (s : Set ι).PairwiseDisjoint f)
(hf : ∀ i ∈ s, (f i).Nonempty) : s.card ≤ (s.biUnion f).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s✝ : Finset α
B : Finset (Finset α)
n : ℕ
s : Finset ι
f : ι → Finset α
hs : Set.PairwiseDisjoint (↑s) f
⊢ card s ≤ card (Finset.biUnion s f) + card (filter (fun i => f i = ∅) s) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [← Finset.filter_card_add_filter_neg_card_eq_card fun i ↦ f i = ∅, add_comm] | theorem card_le_card_biUnion_add_card_fiber {s : Finset ι} {f : ι → Finset α}
(hs : (s : Set ι).PairwiseDisjoint f) :
s.card ≤ (s.biUnion f).card + (s.filter fun i ↦ f i = ∅).card := by
| Mathlib.Algebra.BigOperators.Order.380_0.ewL52iF1Dz3xeLh | theorem card_le_card_biUnion_add_card_fiber {s : Finset ι} {f : ι → Finset α}
(hs : (s : Set ι).PairwiseDisjoint f) :
s.card ≤ (s.biUnion f).card + (s.filter fun i ↦ f i = ∅).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : DecidableEq α
s✝ : Finset α
B : Finset (Finset α)
n : ℕ
s : Finset ι
f : ι → Finset α
hs : Set.PairwiseDisjoint (↑s) f
⊢ card (filter (fun a => ¬f a = ∅) s) + card (filter (fun i => f i = ∅) s) ≤
card (Fi... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi ↦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_ | theorem card_le_card_biUnion_add_card_fiber {s : Finset ι} {f : ι → Finset α}
(hs : (s : Set ι).PairwiseDisjoint f) :
s.card ≤ (s.biUnion f).card + (s.filter fun i ↦ f i = ∅).card := by
rw [← Finset.filter_card_add_filter_neg_card_eq_card fun i ↦ f i = ∅, add_comm]
| Mathlib.Algebra.BigOperators.Order.380_0.ewL52iF1Dz3xeLh | theorem card_le_card_biUnion_add_card_fiber {s : Finset ι} {f : ι → Finset α}
(hs : (s : Set ι).PairwiseDisjoint f) :
s.card ≤ (s.biUnion f).card + (s.filter fun i ↦ f i = ∅).card | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ ∏ x in s, f x ≤ ∏ x in t, f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | classical calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
rw [← prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h ↦ n_h h
_ ≤ ∏ x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp on... | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
| Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ ∏ x in s, f x ≤ ∏ x in t, f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
rw [← prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h ↦ n_h h
_ ≤ ∏ x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_fi... | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
classical | Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ ∏ x in s, f x = (∏ x in filter (fun x => f x = 1) s, f x) * ∏ x in filter (fun x => f x ≠ 1) s, f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [← prod_union, filter_union_filter_neg_eq] | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
classical calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
| Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ Disjoint (filter (fun x => f x = 1) s) (filter (fun x => f x ≠ 1) s) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact disjoint_filter.2 fun _ _ h n_h ↦ n_h h | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
classical calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
rw [← prod_union, filter_union_filter_neg_e... | Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ ∀ i ∈ filter (fun x => f x = 1) s, f i ≤ 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp only [mem_filter, and_imp] | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
classical calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
rw [← prod_union, filter_union_filter_neg_e... | Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ ∀ i ∈ s, f i = 1 → f i ≤ 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact fun _ _ ↦ le_of_eq | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
classical calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
rw [← prod_union, filter_union_filter_neg_e... | Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : CanonicallyOrderedCommMonoid M
f : ι → M
s t : Finset ι
h : ∀ x ∈ s, f x ≠ 1 → x ∈ t
⊢ filter (fun x => f x ≠ 1) s ⊆ t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simpa only [subset_iff, mem_filter, and_imp] | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x := by
classical calc
∏ x in s, f x = (∏ x in s.filter fun x ↦ f x = 1, f x) *
∏ x in s.filter fun x ↦ f x ≠ 1, f x := by
rw [← prod_union, filter_union_filter_neg_e... | Mathlib.Algebra.BigOperators.Order.423_0.ewL52iF1Dz3xeLh | @[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
hs : Finset.Nonempty s
hlt : ∀ i ∈ s, f i < g i
⊢ s.val ≠ ∅ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | aesop | @[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : ∀ i ∈ s, f i < g i) :
∏ i in s, f i < ∏ i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by | Mathlib.Algebra.BigOperators.Order.451_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : ∀ i ∈ s, f i < g i) :
∏ i in s, f i < ∏ i in s, g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ ∏ j in s, f j < ∏ j in t, f j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (∏ j in s, f j) hlt
_ ≤ ∏ j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
· simp [Finset.insert_subset_iff, h, ht]
· intro x hx h'x
simp only [mem_insert, ... | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
| Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ ∏ j in s, f j < ∏ j in t, f j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (∏ j in s, f j) hlt
_ ≤ ∏ j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
· simp [Finset.insert_subset_iff, h, ht]
· intro x hx h'x
simp only [mem_insert, not_or] at... | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical | Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ ∏ j in s, f j < ∏ j in insert i s, f j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_insert hs] | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
| Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ ∏ j in s, f j < f i * ∏ x in s, f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact lt_mul_of_one_lt_left' (∏ j in s, f j) hlt | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
| Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ ∏ j in insert i s, f j ≤ ∏ j in t, f j | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | apply prod_le_prod_of_subset_of_one_le' | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_o... | Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
case h
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ insert i s ⊆ t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp [Finset.insert_subset_iff, h, ht] | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_o... | Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
case hf
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
⊢ ∀ i_1 ∈ t, i_1 ∉ insert i s → 1 ≤ f i_1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | intro x hx h'x | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_o... | Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
case hf
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
x : ι
hx : x ∈ t
h'x : x ∉ insert i s
⊢ 1 ≤ f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp only [mem_insert, not_or] at h'x | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_o... | Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
case hf
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : s ⊆ t
i : ι
ht : i ∈ t
hs : i ∉ s
hlt : 1 < f i
hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j
x : ι
hx : x ∈ t
h'x : ¬x = i ∧ x ∉ s
⊢ 1 ≤ f x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact hle x hx h'x.2 | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j := by
classical calc
∏ j in s, f j < ∏ j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_o... | Mathlib.Algebra.BigOperators.Order.478_0.ewL52iF1Dz3xeLh | @[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) : ∏ j in s, f j < ∏ j in t, f j | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
i j : ι
hij : j ≠ i
hi : i ∈ s
hj : j ∈ s
hlt : 1 < f j
hle : ∀ k ∈ s, k ≠ i → 1 ≤ f k
⊢ f i = ∏ k in {i}, f k | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_singleton] | @[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ι} (hij : j ≠ i) (hi : i ∈ s) (hj : j ∈ s) (hlt : 1 < f j)
(hle : ∀ k ∈ s, k ≠ i → 1 ≤ f k) : f i < ∏ k in s, f k :=
calc
f i = ∏ k in {i}, f k := by | Mathlib.Algebra.BigOperators.Order.494_0.ewL52iF1Dz3xeLh | @[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ι} (hij : j ≠ i) (hi : i ∈ s) (hj : j ∈ s) (hlt : 1 < f j)
(hle : ∀ k ∈ s, k ≠ i → 1 ≤ f k) : f i < ∏ k in s, f k | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : ∀ i ∈ s, 1 < f i
hs : Finset.Nonempty s
⊢ 1 ≤ ∏ i in s, 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_const_one] | @[to_additive sum_pos]
theorem one_lt_prod (h : ∀ i ∈ s, 1 < f i) (hs : s.Nonempty) : 1 < ∏ i in s, f i :=
lt_of_le_of_lt (by | Mathlib.Algebra.BigOperators.Order.505_0.ewL52iF1Dz3xeLh | @[to_additive sum_pos]
theorem one_lt_prod (h : ∀ i ∈ s, 1 < f i) (hs : s.Nonempty) : 1 < ∏ i in s, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
h : ∀ i ∈ s, f i < 1
hs : Finset.Nonempty s
⊢ ∏ i in s, 1 ≤ 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [prod_const_one] | @[to_additive]
theorem prod_lt_one (h : ∀ i ∈ s, f i < 1) (hs : s.Nonempty) : ∏ i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by | Mathlib.Algebra.BigOperators.Order.511_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_lt_one (h : ∀ i ∈ s, f i < 1) (hs : s.Nonempty) : ∏ i in s, f i < 1 | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f✝ g✝ : ι → M
s t : Finset ι
f g : ι → M
h : ∀ i ∈ s, f i ≤ g i
⊢ ∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | classical
revert h
refine Finset.induction_on s (fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h),
fun _ ↦ rfl⟩) fun a s ha ih H ↦ ?_
specialize ih fun i ↦ H i ∘ Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, ← ih]
exact
mul... | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i := by
| Mathlib.Algebra.BigOperators.Order.529_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f✝ g✝ : ι → M
s t : Finset ι
f g : ι → M
h : ∀ i ∈ s, f i ≤ g i
⊢ ∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | revert h | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i := by
classical
| Mathlib.Algebra.BigOperators.Order.529_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f✝ g✝ : ι → M
s t : Finset ι
f g : ι → M
⊢ (∀ i ∈ s, f i ≤ g i) → (∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine Finset.induction_on s (fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h),
fun _ ↦ rfl⟩) fun a s ha ih H ↦ ?_ | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i := by
classical
revert h
| Mathlib.Algebra.BigOperators.Order.529_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f✝ g✝ : ι → M
s✝ t : Finset ι
f g : ι → M
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, f i ≤ g i) → (∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i)
H : ∀ i ∈ insert a s, f i ≤ g... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | specialize ih fun i ↦ H i ∘ Finset.mem_insert_of_mem | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h),
fun _ ↦ rfl⟩) fun a s ha ih H ↦ ?_
| Mathlib.Algebra.BigOperators.Order.529_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f✝ g✝ : ι → M
s✝ t : Finset ι
f g : ι → M
a : ι
s : Finset ι
ha : a ∉ s
H : ∀ i ∈ insert a s, f i ≤ g i
ih : ∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i
⊢ ∏ i in insert a s, f... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, ← ih] | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h),
fun _ ↦ rfl⟩) fun a s ha ih H ↦ ?_
spec... | Mathlib.Algebra.BigOperators.Order.529_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCancelCommMonoid M
f✝ g✝ : ι → M
s✝ t : Finset ι
f g : ι → M
a : ι
s : Finset ι
ha : a ∉ s
H : ∀ i ∈ insert a s, f i ≤ g i
ih : ∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i
⊢ f a * ∏ x in s, f x ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i ↦ H i ∘ Finset.mem_insert_of_mem) | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h),
fun _ ↦ rfl⟩) fun a s ha ih H ↦ ?_
spec... | Mathlib.Algebra.BigOperators.Order.529_0.ewL52iF1Dz3xeLh | @[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i in s, f i) = ∏ i in s, g i) ↔ ∀ i ∈ s, f i = g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
inst✝ : DecidableEq ι
⊢ ∏ i in s \ t, f i ≤ ∏ i in t \ s, f i ↔ ∏ i in s, f i ≤ ∏ i in t, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [← mul_le_mul_iff_right, ← prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
← prod_union, inter_comm, sdiff_union_inter] | @[to_additive] lemma prod_sdiff_le_prod_sdiff :
∏ i in s \ t, f i ≤ ∏ i in t \ s, f i ↔ ∏ i in s, f i ≤ ∏ i in t, f i := by
| Mathlib.Algebra.BigOperators.Order.546_0.ewL52iF1Dz3xeLh | @[to_additive] lemma prod_sdiff_le_prod_sdiff :
∏ i in s \ t, f i ≤ ∏ i in t \ s, f i ↔ ∏ i in s, f i ≤ ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
inst✝ : DecidableEq ι
⊢ Disjoint (t \ s) (s ∩ t) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simpa only [inter_comm] using disjoint_sdiff_inter t s | @[to_additive] lemma prod_sdiff_le_prod_sdiff :
∏ i in s \ t, f i ≤ ∏ i in t \ s, f i ↔ ∏ i in s, f i ≤ ∏ i in t, f i := by
rw [← mul_le_mul_iff_right, ← prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
← prod_union, inter_comm, sdiff_union_inter];
| Mathlib.Algebra.BigOperators.Order.546_0.ewL52iF1Dz3xeLh | @[to_additive] lemma prod_sdiff_le_prod_sdiff :
∏ i in s \ t, f i ≤ ∏ i in t \ s, f i ↔ ∏ i in s, f i ≤ ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
inst✝ : DecidableEq ι
⊢ ∏ i in s \ t, f i < ∏ i in t \ s, f i ↔ ∏ i in s, f i < ∏ i in t, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | rw [← mul_lt_mul_iff_right, ← prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
← prod_union, inter_comm, sdiff_union_inter] | @[to_additive] lemma prod_sdiff_lt_prod_sdiff :
∏ i in s \ t, f i < ∏ i in t \ s, f i ↔ ∏ i in s, f i < ∏ i in t, f i := by
| Mathlib.Algebra.BigOperators.Order.552_0.ewL52iF1Dz3xeLh | @[to_additive] lemma prod_sdiff_lt_prod_sdiff :
∏ i in s \ t, f i < ∏ i in t \ s, f i ↔ ∏ i in s, f i < ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : OrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
inst✝ : DecidableEq ι
⊢ Disjoint (t \ s) (s ∩ t) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simpa only [inter_comm] using disjoint_sdiff_inter t s | @[to_additive] lemma prod_sdiff_lt_prod_sdiff :
∏ i in s \ t, f i < ∏ i in t \ s, f i ↔ ∏ i in s, f i < ∏ i in t, f i := by
rw [← mul_lt_mul_iff_right, ← prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
← prod_union, inter_comm, sdiff_union_inter];
| Mathlib.Algebra.BigOperators.Order.552_0.ewL52iF1Dz3xeLh | @[to_additive] lemma prod_sdiff_lt_prod_sdiff :
∏ i in s \ t, f i < ∏ i in t \ s, f i ↔ ∏ i in s, f i < ∏ i in t, f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
Hlt : ∏ i in s, f i < ∏ i in s, g i
⊢ ∃ i ∈ s, f i < g i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | contrapose! Hlt with Hle | @[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : ∏ i in s, f i < ∏ i in s, g i) : ∃ i ∈ s, f i < g i := by
| Mathlib.Algebra.BigOperators.Order.564_0.ewL52iF1Dz3xeLh | @[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : ∏ i in s, f i < ∏ i in s, g i) : ∃ i ∈ s, f i < g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
Hle : ∀ i ∈ s, g i ≤ f i
⊢ ∏ i in s, g i ≤ ∏ i in s, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact prod_le_prod' Hle | @[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : ∏ i in s, f i < ∏ i in s, g i) : ∃ i ∈ s, f i < g i := by
contrapose! Hlt with Hle
| Mathlib.Algebra.BigOperators.Order.564_0.ewL52iF1Dz3xeLh | @[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : ∏ i in s, f i < ∏ i in s, g i) : ∃ i ∈ s, f i < g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
hs : Finset.Nonempty s
Hle : ∏ i in s, f i ≤ ∏ i in s, g i
⊢ ∃ i ∈ s, f i ≤ g i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | contrapose! Hle with Hlt | @[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : ∏ i in s, f i ≤ ∏ i in s, g i) :
∃ i ∈ s, f i ≤ g i := by
| Mathlib.Algebra.BigOperators.Order.571_0.ewL52iF1Dz3xeLh | @[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : ∏ i in s, f i ≤ ∏ i in s, g i) :
∃ i ∈ s, f i ≤ g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f g : ι → M
s t : Finset ι
hs : Finset.Nonempty s
Hlt : ∀ i ∈ s, g i < f i
⊢ ∏ i in s, g i < ∏ i in s, f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact prod_lt_prod_of_nonempty' hs Hlt | @[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : ∏ i in s, f i ≤ ∏ i in s, g i) :
∃ i ∈ s, f i ≤ g i := by
contrapose! Hle with Hlt
| Mathlib.Algebra.BigOperators.Order.571_0.ewL52iF1Dz3xeLh | @[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : ∏ i in s, f i ≤ ∏ i in s, g i) :
∃ i ∈ s, f i ≤ g i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f✝ g : ι → M
s t : Finset ι
f : ι → M
h₁ : ∏ i in s, f i = 1
h₂ : ∃ i ∈ s, f i ≠ 1
⊢ ∃ i ∈ s, 1 < f i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | contrapose! h₁ | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i := by
| Mathlib.Algebra.BigOperators.Order.579_0.ewL52iF1Dz3xeLh | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f✝ g : ι → M
s t : Finset ι
f : ι → M
h₂ : ∃ i ∈ s, f i ≠ 1
h₁ : ∀ i ∈ s, f i ≤ 1
⊢ ∏ i in s, f i ≠ 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂ | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i := by
contrapose! h₁
| Mathlib.Algebra.BigOperators.Order.579_0.ewL52iF1Dz3xeLh | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i | Mathlib_Algebra_BigOperators_Order |
case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f✝ g : ι → M
s t : Finset ι
f : ι → M
h₁ : ∀ i ∈ s, f i ≤ 1
i : ι
m : i ∈ s
i_ne : f i ≠ 1
⊢ ∏ i in s, f i ≠ 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | apply ne_of_lt | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i := by
contrapose! h₁
obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂
| Mathlib.Algebra.BigOperators.Order.579_0.ewL52iF1Dz3xeLh | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i | Mathlib_Algebra_BigOperators_Order |
case intro.intro.h
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : LinearOrderedCancelCommMonoid M
f✝ g : ι → M
s t : Finset ι
f : ι → M
h₁ : ∀ i ∈ s, f i ≤ 1
i : ι
m : i ∈ s
i_ne : f i ≠ 1
⊢ ∏ i in s, f i < 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | calc
∏ j in s, f j < ∏ j in s, 1 := prod_lt_prod' h₁ ⟨i, m, (h₁ i m).lt_of_ne i_ne⟩
_ = 1 := prod_const_one | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i := by
contrapose! h₁
obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂
apply ne_of_lt
| Mathlib.Algebra.BigOperators.Order.579_0.ewL52iF1Dz3xeLh | @[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i in s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s t : Finset ι
h0 : ∀ i ∈ s, 0 ≤ f i
h1 : ∀ i ∈ s, f i ≤ g i
⊢ ∏ i in s, f i ≤ ∏ i in s, g i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | induction' s using Finset.induction with a s has ih h | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case empty
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s t : Finset ι
h0 : ∀ i ∈ ∅, 0 ≤ f i
h1 : ∀ i ∈ ∅, f i ≤ g i
⊢ ∏ i in ∅, f i ≤ ∏ i in ∅, g i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0 ≤ f i
h1 : ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | simp only [prod_insert has] | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0 ≤ f i
h1 : ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | apply mul_le_mul | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert.h₁
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0 ≤ f i
h1... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact h1 a (mem_insert_self a s) | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert.h₂
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0 ≤ f i
h1... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | refine ih (fun x H ↦ h0 _ ?_) (fun x H ↦ h1 _ ?_) | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert.h₂.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact mem_insert_of_mem H | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert.h₂.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | exact mem_insert_of_mem H | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert.c0
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0 ≤ f i
h1... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | apply prod_nonneg fun x H ↦ h0 x (mem_insert_of_mem H) | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
case insert.b0
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s✝ t : Finset ι
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, 0 ≤ f i) → (∀ i ∈ s, f i ≤ g i) → ∏ i in s, f i ≤ ∏ i in s, g i
h0 : ∀ i ∈ insert a s, 0 ≤ f i
h1... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s)) | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib.Algebra.BigOperators.Order.604_0.ewL52iF1Dz3xeLh | /-- If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, ... | Mathlib_Algebra_BigOperators_Order |
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝ : OrderedCommSemiring R
f g : ι → R
s t : Finset ι
h0 : ∀ i ∈ s, 0 ≤ f i
h1 : ∀ i ∈ s, f i ≤ 1
⊢ ∏ i in s, f i ≤ 1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import... | convert ← prod_le_prod h0 h1 | /-- If each `f i`, `i ∈ s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ 1) : ∏ i in s, f i ≤ 1 := by
| Mathlib.Algebra.BigOperators.Order.629_0.ewL52iF1Dz3xeLh | /-- If each `f i`, `i ∈ s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ 1) : ∏ i in s, f i ≤ 1 | Mathlib_Algebra_BigOperators_Order |
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