statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
smul_upper_bounds_subset_lower_bounds_smul (hc : c ≤ 0) :
c • upper_bounds s ⊆ lower_bounds (c • s) | (antitone_smul_left hc).image_upper_bounds_subset_lower_bounds_image | lemma | smul_upper_bounds_subset_lower_bounds_smul | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone_smul_left",
"lower_bounds",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below.smul_of_nonpos (hc : c ≤ 0) (hs : bdd_below s) : bdd_above (c • s) | (antitone_smul_left hc).map_bdd_below hs | lemma | bdd_below.smul_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone_smul_left",
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.smul_of_nonpos (hc : c ≤ 0) (hs : bdd_above s) : bdd_below (c • s) | (antitone_smul_left hc).map_bdd_above hs | lemma | bdd_above.smul_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone_smul_left",
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_max_of_nonpos (ha : a ≤ 0) (b₁ b₂ : M) : a • max b₁ b₂ = min (a • b₁) (a • b₂) | (antitone_smul_left ha : antitone (_ : M → M)).map_max | lemma | smul_max_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone",
"antitone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_min_of_nonpos (ha : a ≤ 0) (b₁ b₂ : M) : a • min b₁ b₂ = max (a • b₁) (a • b₂) | (antitone_smul_left ha : antitone (_ : M → M)).map_min | lemma | smul_min_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone",
"antitone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_bounds_smul_of_neg (hc : c < 0) : lower_bounds (c • s) = c • upper_bounds s | (order_iso.smul_left_dual M hc).upper_bounds_image | lemma | lower_bounds_smul_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"lower_bounds",
"order_iso.smul_left_dual",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_bounds_smul_of_neg (hc : c < 0) : upper_bounds (c • s) = c • lower_bounds s | (order_iso.smul_left_dual M hc).lower_bounds_image | lemma | upper_bounds_smul_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"lower_bounds",
"order_iso.smul_left_dual",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_smul_iff_of_neg (hc : c < 0) : bdd_below (c • s) ↔ bdd_above s | (order_iso.smul_left_dual M hc).bdd_above_image | lemma | bdd_below_smul_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"bdd_above",
"bdd_below",
"order_iso.smul_left_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_smul_iff_of_neg (hc : c < 0) : bdd_above (c • s) ↔ bdd_below s | (order_iso.smul_left_dual M hc).bdd_below_image | lemma | bdd_above_smul_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"bdd_above",
"bdd_below",
"order_iso.smul_left_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_monoid {ι : Type*} {Z : ι → Type*} [∀ i, ordered_comm_monoid (Z i)] :
ordered_comm_monoid (Π i, Z i) | { mul_le_mul_left := λ f g w h i, mul_le_mul_left' (w i) _,
..pi.partial_order,
..pi.comm_monoid, } | instance | pi.ordered_comm_monoid | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [
"mul_le_mul_left",
"mul_le_mul_left'",
"ordered_comm_monoid",
"pi.comm_monoid",
"pi.partial_order"
] | The product of a family of ordered commutative monoids is an ordered commutative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_cancel_comm_monoid [∀ i, ordered_cancel_comm_monoid $ f i] :
ordered_cancel_comm_monoid (Π i : I, f i) | by refine_struct { mul := (*), one := (1 : Π i, f i), le := (≤), lt := (<),
npow := monoid.npow, .. pi.partial_order, .. pi.monoid };
tactic.pi_instance_derive_field | instance | pi.ordered_cancel_comm_monoid | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [
"ordered_cancel_comm_monoid",
"pi.monoid",
"pi.partial_order",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_group [∀ i, ordered_comm_group $ f i] :
ordered_comm_group (Π i : I, f i) | { mul := (*), one := (1 : Π i, f i), le := (≤), lt := (<),
npow := monoid.npow,
..pi.comm_group,
..pi.ordered_comm_monoid, } | instance | pi.ordered_comm_group | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [
"ordered_comm_group",
"pi.comm_group",
"pi.ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_const_of_one_le (ha : 1 ≤ a) : 1 ≤ const β a | λ _, ha | lemma | function.one_le_const_of_one_le | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_le_one_of_le_one (ha : a ≤ 1) : const β a ≤ 1 | λ _, ha | lemma | function.const_le_one_of_le_one | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_const : 1 ≤ const β a ↔ 1 ≤ a | @const_le_const _ _ _ _ 1 _ | lemma | function.one_le_const | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_const : 1 < const β a ↔ 1 < a | @const_lt_const _ _ _ _ 1 a | lemma | function.one_lt_const | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_le_one : const β a ≤ 1 ↔ a ≤ 1 | @const_le_const _ _ _ _ _ 1 | lemma | function.const_le_one | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_lt_one : const β a < 1 ↔ a < 1 | @const_lt_const _ _ _ _ _ 1 | lemma | function.const_lt_one | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function_const_nonneg_of_pos [preorder α] (ha : 0 < a) : 0 ≤ const ι a | const_nonneg_of_nonneg _ ha.le | lemma | tactic.function_const_nonneg_of_pos | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function_const_ne_zero : a ≠ 0 → const ι a ≠ 0 | const_ne_zero.2 | lemma | tactic.function_const_ne_zero | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function_const_pos [preorder α] : 0 < a → 0 < const ι a | const_pos.2 | lemma | tactic.function_const_pos | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_const : expr → tactic strictness | | `(function.const %%ι %%a) := do
strict_a ← core a,
match strict_a with
| positive p := positive <$> to_expr ``(function_const_pos %%ι %%p)
<|> nonnegative <$> to_expr ``(function_const_nonneg_of_pos %%ι %%p)
| nonnegative p := nonnegative <$> to_expr ``(const_nonneg_of_nonneg %%ι %%p)
| no... | def | tactic.positivity_const | algebra.order | src/algebra/order/pi.lean | [
"algebra.order.ring.defs",
"algebra.ring.pi",
"tactic.positivity"
] | [] | Extension for the `positivity` tactic: `function.const` is positive/nonnegative/nonzero if its
input is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cSup_one : Sup (1 : set α) = 1 | cSup_singleton _ | lemma | cSup_one | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"cSup_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cInf_one : Inf (1 : set α) = 1 | cInf_singleton _ | lemma | cInf_one | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"cInf_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cSup_inv (hs₀ : s.nonempty) (hs₁ : bdd_below s) : Sup s⁻¹ = (Inf s)⁻¹ | by { rw ←image_inv, exact ((order_iso.inv α).map_cInf' hs₀ hs₁).symm } | lemma | cSup_inv | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"bdd_below",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cInf_inv (hs₀ : s.nonempty) (hs₁ : bdd_above s) : Inf s⁻¹ = (Sup s)⁻¹ | by { rw ←image_inv, exact ((order_iso.inv α).map_cSup' hs₀ hs₁).symm } | lemma | cInf_inv | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"bdd_above",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cSup_mul (hs₀ : s.nonempty) (hs₁ : bdd_above s) (ht₀ : t.nonempty)
(ht₁ : bdd_above t) :
Sup (s * t) = Sup s * Sup t | cSup_image2_eq_cSup_cSup (λ _, (order_iso.mul_right _).to_galois_connection)
(λ _, (order_iso.mul_left _).to_galois_connection) hs₀ hs₁ ht₀ ht₁ | lemma | cSup_mul | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"bdd_above",
"cSup_image2_eq_cSup_cSup",
"order_iso.mul_left",
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cInf_mul (hs₀ : s.nonempty) (hs₁ : bdd_below s) (ht₀ : t.nonempty)
(ht₁ : bdd_below t) :
Inf (s * t) = Inf s * Inf t | cInf_image2_eq_cInf_cInf (λ _, (order_iso.mul_right _).symm.to_galois_connection)
(λ _, (order_iso.mul_left _).symm.to_galois_connection) hs₀ hs₁ ht₀ ht₁ | lemma | cInf_mul | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"bdd_below",
"cInf_image2_eq_cInf_cInf",
"order_iso.mul_left",
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cSup_div (hs₀ : s.nonempty) (hs₁ : bdd_above s) (ht₀ : t.nonempty)
(ht₁ : bdd_below t) :
Sup (s / t) = Sup s / Inf t | by rw [div_eq_mul_inv, cSup_mul hs₀ hs₁ ht₀.inv ht₁.inv, cSup_inv ht₀ ht₁, div_eq_mul_inv] | lemma | cSup_div | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"bdd_above",
"bdd_below",
"cSup_inv",
"cSup_mul",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cInf_div (hs₀ : s.nonempty) (hs₁ : bdd_below s) (ht₀ : t.nonempty)
(ht₁ : bdd_above t) :
Inf (s / t) = Inf s / Sup t | by rw [div_eq_mul_inv, cInf_mul hs₀ hs₁ ht₀.inv ht₁.inv, cInf_inv ht₀ ht₁, div_eq_mul_inv] | lemma | cInf_div | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"bdd_above",
"bdd_below",
"cInf_inv",
"cInf_mul",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_one : Sup (1 : set α) = 1 | Sup_singleton | lemma | Sup_one | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Sup_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_one : Inf (1 : set α) = 1 | Inf_singleton | lemma | Inf_one | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Inf_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_inv (s : set α) : Sup s⁻¹ = (Inf s)⁻¹ | by { rw [←image_inv, Sup_image], exact ((order_iso.inv α).map_Inf _).symm } | lemma | Sup_inv | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Sup_image",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_inv (s : set α) : Inf s⁻¹ = (Sup s)⁻¹ | by { rw [←image_inv, Inf_image], exact ((order_iso.inv α).map_Sup _).symm } | lemma | Inf_inv | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Inf_image",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_mul : Sup (s * t) = Sup s * Sup t | Sup_image2_eq_Sup_Sup (λ _, (order_iso.mul_right _).to_galois_connection) $
λ _, (order_iso.mul_left _).to_galois_connection | lemma | Sup_mul | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Sup_image2_eq_Sup_Sup",
"order_iso.mul_left",
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_mul : Inf (s * t) = Inf s * Inf t | Inf_image2_eq_Inf_Inf (λ _, (order_iso.mul_right _).symm.to_galois_connection) $
λ _, (order_iso.mul_left _).symm.to_galois_connection | lemma | Inf_mul | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Inf_image2_eq_Inf_Inf",
"order_iso.mul_left",
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_div : Sup (s / t) = Sup s / Inf t | by simp_rw [div_eq_mul_inv, Sup_mul, Sup_inv] | lemma | Sup_div | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Sup_inv",
"Sup_mul",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_div : Inf (s / t) = Inf s / Sup t | by simp_rw [div_eq_mul_inv, Inf_mul, Inf_inv] | lemma | Inf_div | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"Inf_inv",
"Inf_mul",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Ioo],
split,
{ rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩, split,
exact (mul_lt_mul_left hr).mpr a_h_left_left,
exact (mul_lt_mul_left hr).mpr a_h_left_right, },
{ rintro ⟨a_left, a_right⟩,
use x / r,
refine ⟨⟨(lt_div_iff' hr).mpr a_... | lemma | linear_ordered_field.smul_Ioo | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"div_lt_iff'",
"lt_div_iff'",
"mul_div_cancel'",
"mul_lt_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Icc : r • Icc a b = Icc (r • a) (r • b) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Icc],
split,
{ rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩, split,
exact (mul_le_mul_left hr).mpr a_h_left_left,
exact (mul_le_mul_left hr).mpr a_h_left_right, },
{ rintro ⟨a_left, a_right⟩,
use x / r,
refine ⟨⟨(le_div_iff' hr).mpr a_... | lemma | linear_ordered_field.smul_Icc | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"div_le_iff'",
"le_div_iff'",
"mul_div_cancel'",
"mul_le_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Ico : r • Ico a b = Ico (r • a) (r • b) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Ico],
split,
{ rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩, split,
exact (mul_le_mul_left hr).mpr a_h_left_left,
exact (mul_lt_mul_left hr).mpr a_h_left_right, },
{ rintro ⟨a_left, a_right⟩,
use x / r,
refine ⟨⟨(le_div_iff' hr).mpr a_... | lemma | linear_ordered_field.smul_Ico | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"div_lt_iff'",
"le_div_iff'",
"mul_div_cancel'",
"mul_le_mul_left",
"mul_lt_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Ioc],
split,
{ rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩, split,
exact (mul_lt_mul_left hr).mpr a_h_left_left,
exact (mul_le_mul_left hr).mpr a_h_left_right, },
{ rintro ⟨a_left, a_right⟩,
use x / r,
refine ⟨⟨(lt_div_iff' hr).mpr a_... | lemma | linear_ordered_field.smul_Ioc | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"div_le_iff'",
"lt_div_iff'",
"mul_div_cancel'",
"mul_le_mul_left",
"mul_lt_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Ioi : r • Ioi a = Ioi (r • a) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Ioi],
split,
{ rintro ⟨a_w, a_h_left, rfl⟩,
exact (mul_lt_mul_left hr).mpr a_h_left, },
{ rintro h,
use x / r,
split,
exact (lt_div_iff' hr).mpr h,
exact mul_div_cancel' _ (ne_of_gt hr), }
end | lemma | linear_ordered_field.smul_Ioi | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"lt_div_iff'",
"mul_div_cancel'",
"mul_lt_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Iio : r • Iio a = Iio (r • a) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Iio],
split,
{ rintro ⟨a_w, a_h_left, rfl⟩,
exact (mul_lt_mul_left hr).mpr a_h_left, },
{ rintro h,
use x / r,
split,
exact (div_lt_iff' hr).mpr h,
exact mul_div_cancel' _ (ne_of_gt hr), }
end | lemma | linear_ordered_field.smul_Iio | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"div_lt_iff'",
"mul_div_cancel'",
"mul_lt_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Ici : r • Ici a = Ici (r • a) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Ioi],
split,
{ rintro ⟨a_w, a_h_left, rfl⟩,
exact (mul_le_mul_left hr).mpr a_h_left, },
{ rintro h,
use x / r,
split,
exact (le_div_iff' hr).mpr h,
exact mul_div_cancel' _ (ne_of_gt hr), }
end | lemma | linear_ordered_field.smul_Ici | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"le_div_iff'",
"mul_div_cancel'",
"mul_le_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Iic : r • Iic a = Iic (r • a) | begin
ext x,
simp only [mem_smul_set, smul_eq_mul, mem_Iio],
split,
{ rintro ⟨a_w, a_h_left, rfl⟩,
exact (mul_le_mul_left hr).mpr a_h_left, },
{ rintro h,
use x / r,
split,
exact (div_le_iff' hr).mpr h,
exact mul_div_cancel' _ (ne_of_gt hr), }
end | lemma | linear_ordered_field.smul_Iic | algebra.order | src/algebra/order/pointwise.lean | [
"algebra.bounds",
"data.set.pointwise.smul"
] | [
"div_le_iff'",
"mul_div_cancel'",
"mul_le_mul_left",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.sum_smul_comp_perm_le_sum_smul (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g (σ i) ≤ ∑ i in s, f i • g i | begin
classical,
revert hσ σ hfg,
apply finset.induction_on_max_value (λ i, to_lex (g i, f i)) s,
{ simp only [le_rfl, finset.sum_empty, implies_true_iff] },
intros a s has hamax hind σ hfg hσ,
set τ : perm ι := σ.trans (swap a (σ a)) with hτ,
have hτs : {x | τ x ≠ x} ⊆ s,
{ intros x hx,
simp only [... | lemma | monovary_on.sum_smul_comp_perm_le_sum_smul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"eq_or_ne",
"equiv.coe_trans",
"equiv.swap_comp_apply",
"finset.induction_on_max_value",
"le_rfl",
"monovary_on",
"ne_of_apply_ne",
"prod.lex.le_iff",
"smul_add_smul_le_smul_add_smul'",
"to_lex"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_smul_comp_perm_eq_sum_smul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g (σ i) = ∑ i in s, f i • g i ↔ monovary_on f (g ∘ σ) s | begin
classical,
refine ⟨not_imp_not.1 $ λ h, _, λ h, (hfg.sum_smul_comp_perm_le_sum_smul hσ).antisymm _⟩,
{ rw monovary_on at h,
push_neg at h,
obtain ⟨x, hx, y, hy, hgxy, hfxy⟩ := h,
set τ : perm ι := (swap x y).trans σ,
have hτs : {x | τ x ≠ x} ⊆ s,
{ refine (set_support_mul_subset σ $ swap... | lemma | monovary_on.sum_smul_comp_perm_eq_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"eq_or_ne",
"equiv.coe_trans",
"monovary_on",
"set.union_subset",
"smul_add_smul_lt_smul_add_smul"
] | **Equality case of Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_smul_comp_perm_lt_sum_smul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g (σ i) < ∑ i in s, f i • g i ↔ ¬ monovary_on f (g ∘ σ) s | by simp [←hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ,
lt_iff_le_and_ne, hfg.sum_smul_comp_perm_le_sum_smul hσ] | lemma | monovary_on.sum_smul_comp_perm_lt_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"lt_iff_le_and_ne",
"monovary_on"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_comp_perm_smul_le_sum_smul (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) • g i ≤ ∑ i in s, f i • g i | begin
convert hfg.sum_smul_comp_perm_le_sum_smul
(show {x | σ⁻¹ x ≠ x} ⊆ s, by simp only [set_support_inv_eq, hσ]) using 1,
exact σ.sum_comp' s (λ i j, f i • g j) hσ,
end | lemma | monovary_on.sum_comp_perm_smul_le_sum_smul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_comp_perm_smul_eq_sum_smul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) • g i = ∑ i in s, f i • g i ↔ monovary_on (f ∘ σ) g s | begin
have hσinv : {x | σ⁻¹ x ≠ x} ⊆ s := (set_support_inv_eq _).subset.trans hσ,
refine (iff.trans _ $ hfg.sum_smul_comp_perm_eq_sum_smul_iff hσinv).trans ⟨λ h, _, λ h, _⟩,
{ simpa only [σ.sum_comp' s (λ i j, f i • g j) hσ] },
{ convert h.comp_right σ,
{ rw [comp.assoc, inv_def, symm_comp_self, comp.right_... | lemma | monovary_on.sum_comp_perm_smul_eq_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on",
"set.image_perm"
] | **Equality case of Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_comp_perm_smul_lt_sum_smul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) • g i < ∑ i in s, f i • g i ↔ ¬ monovary_on (f ∘ σ) g s | by simp [←hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ,
lt_iff_le_and_ne, hfg.sum_comp_perm_smul_le_sum_smul hσ] | lemma | monovary_on.sum_comp_perm_smul_lt_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"lt_iff_le_and_ne",
"monovary_on"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not monovary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_smul_le_sum_smul_comp_perm (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g i ≤ ∑ i in s, f i • g (σ i) | hfg.dual_right.sum_smul_comp_perm_le_sum_smul hσ | lemma | antivary_on.sum_smul_le_sum_smul_comp_perm | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_smul_eq_sum_smul_comp_perm_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g (σ i) = ∑ i in s, f i • g i ↔ antivary_on f (g ∘ σ) s | (hfg.dual_right.sum_smul_comp_perm_eq_sum_smul_iff hσ).trans monovary_on_to_dual_right | lemma | antivary_on.sum_smul_eq_sum_smul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on",
"monovary_on_to_dual_right"
] | **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
`g`, which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_smul_lt_sum_smul_comp_perm_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g i < ∑ i in s, f i • g (σ i) ↔ ¬ antivary_on f (g ∘ σ) s | by simp [←hfg.sum_smul_eq_sum_smul_comp_perm_iff hσ, lt_iff_le_and_ne, eq_comm,
hfg.sum_smul_le_sum_smul_comp_perm hσ] | lemma | antivary_on.sum_smul_lt_sum_smul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on",
"lt_iff_le_and_ne"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_smul_le_sum_comp_perm_smul (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g i ≤ ∑ i in s, f (σ i) • g i | hfg.dual_right.sum_comp_perm_smul_le_sum_smul hσ | lemma | antivary_on.sum_smul_le_sum_comp_perm_smul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_smul_eq_sum_comp_perm_smul_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) • g i = ∑ i in s, f i • g i ↔ antivary_on (f ∘ σ) g s | (hfg.dual_right.sum_comp_perm_smul_eq_sum_smul_iff hσ).trans monovary_on_to_dual_right | lemma | antivary_on.sum_smul_eq_sum_comp_perm_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on",
"monovary_on_to_dual_right"
] | **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
`g`, which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_smul_lt_sum_comp_perm_smul_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g i < ∑ i in s, f (σ i) • g i ↔ ¬ antivary_on (f ∘ σ) g s | by simp [←hfg.sum_smul_eq_sum_comp_perm_smul_iff hσ, eq_comm, lt_iff_le_and_ne,
hfg.sum_smul_le_sum_comp_perm_smul hσ] | lemma | antivary_on.sum_smul_lt_sum_comp_perm_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on",
"lt_iff_le_and_ne"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_smul_comp_perm_le_sum_smul (hfg : monovary f g) :
∑ i, f i • g (σ i) ≤ ∑ i, f i • g i | (hfg.monovary_on _).sum_smul_comp_perm_le_sum_smul $ λ i _, mem_univ _ | lemma | monovary.sum_smul_comp_perm_le_sum_smul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_smul_comp_perm_eq_sum_smul_iff (hfg : monovary f g) :
∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ monovary f (g ∘ σ) | by simp [(hfg.monovary_on _).sum_smul_comp_perm_eq_sum_smul_iff (λ i _, mem_univ _)] | lemma | monovary.sum_smul_comp_perm_eq_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Equality case of Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_smul_comp_perm_lt_sum_smul_iff (hfg : monovary f g) :
∑ i, f i • g (σ i) < ∑ i, f i • g i ↔ ¬ monovary f (g ∘ σ) | by simp [(hfg.monovary_on _).sum_smul_comp_perm_lt_sum_smul_iff (λ i _, mem_univ _)] | lemma | monovary.sum_smul_comp_perm_lt_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_comp_perm_smul_le_sum_smul (hfg : monovary f g) :
∑ i, f (σ i) • g i ≤ ∑ i, f i • g i | (hfg.monovary_on _).sum_comp_perm_smul_le_sum_smul $ λ i _, mem_univ _ | lemma | monovary.sum_comp_perm_smul_le_sum_smul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : monovary f g) :
∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ monovary (f ∘ σ) g | by simp [(hfg.monovary_on _).sum_comp_perm_smul_eq_sum_smul_iff (λ i _, mem_univ _)] | lemma | monovary.sum_comp_perm_smul_eq_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Equality case of Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_comp_perm_smul_lt_sum_smul_iff (hfg : monovary f g) :
∑ i, f (σ i) • g i < ∑ i, f i • g i ↔ ¬ monovary (f ∘ σ) g | by simp [(hfg.monovary_on _).sum_comp_perm_smul_lt_sum_smul_iff (λ i _, mem_univ _)] | lemma | monovary.sum_comp_perm_smul_lt_sum_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_smul_le_sum_smul_comp_perm (hfg : antivary f g) :
∑ i, f i • g i ≤ ∑ i, f i • g (σ i) | (hfg.antivary_on _).sum_smul_le_sum_smul_comp_perm $ λ i _, mem_univ _ | lemma | antivary.sum_smul_le_sum_smul_comp_perm | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_smul_eq_sum_smul_comp_perm_iff (hfg : antivary f g) :
∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ antivary f (g ∘ σ) | by simp [(hfg.antivary_on _).sum_smul_eq_sum_smul_comp_perm_iff (λ i _, mem_univ _)] | lemma | antivary.sum_smul_eq_sum_smul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
`g`, which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_smul_lt_sum_smul_comp_perm_iff (hfg : antivary f g) :
∑ i, f i • g i < ∑ i, f i • g (σ i) ↔ ¬ antivary f (g ∘ σ) | by simp [(hfg.antivary_on _).sum_smul_lt_sum_smul_comp_perm_iff (λ i _, mem_univ _)] | lemma | antivary.sum_smul_lt_sum_smul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_smul_le_sum_comp_perm_smul (hfg : antivary f g) :
∑ i, f i • g i ≤ ∑ i, f (σ i) • g i | (hfg.antivary_on _).sum_smul_le_sum_comp_perm_smul $ λ i _, mem_univ _ | lemma | antivary.sum_smul_le_sum_comp_perm_smul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_smul_eq_sum_comp_perm_smul_iff (hfg : antivary f g) :
∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ antivary (f ∘ σ) g | by simp [(hfg.antivary_on _).sum_smul_eq_sum_comp_perm_smul_iff (λ i _, mem_univ _)] | lemma | antivary.sum_smul_eq_sum_comp_perm_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
`g`, which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_smul_lt_sum_comp_perm_smul_iff (hfg : antivary f g) :
∑ i, f i • g i < ∑ i, f (σ i) • g i ↔ ¬ antivary (f ∘ σ) g | by simp [(hfg.antivary_on _).sum_smul_lt_sum_comp_perm_smul_iff (λ i _, mem_univ _)] | lemma | antivary.sum_smul_lt_sum_comp_perm_smul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_mul_comp_perm_le_sum_mul (hfg : monovary_on f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g (σ i) ≤ ∑ i in s, f i * g i | hfg.sum_smul_comp_perm_le_sum_smul hσ | lemma | monovary_on.sum_mul_comp_perm_le_sum_mul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_mul_comp_perm_eq_sum_mul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g (σ i) = ∑ i in s, f i * g i ↔ monovary_on f (g ∘ σ) s | hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ | lemma | monovary_on.sum_mul_comp_perm_eq_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Equality case of Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_mul_comp_perm_lt_sum_mul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i • g (σ i) < ∑ i in s, f i • g i ↔ ¬ monovary_on f (g ∘ σ) s | hfg.sum_smul_comp_perm_lt_sum_smul_iff hσ | lemma | monovary_on.sum_mul_comp_perm_lt_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_comp_perm_mul_le_sum_mul (hfg : monovary_on f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) * g i ≤ ∑ i in s, f i * g i | hfg.sum_comp_perm_smul_le_sum_smul hσ | lemma | monovary_on.sum_comp_perm_mul_le_sum_mul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_comp_perm_mul_eq_sum_mul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) * g i = ∑ i in s, f i * g i ↔ monovary_on (f ∘ σ) g s | hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ | lemma | monovary_on.sum_comp_perm_mul_eq_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Equality case of Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_comp_perm_mul_lt_sum_mul_iff (hfg : monovary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) * g i < ∑ i in s, f i * g i ↔ ¬ monovary_on (f ∘ σ) g s | hfg.sum_comp_perm_smul_lt_sum_smul_iff hσ | lemma | monovary_on.sum_comp_perm_mul_lt_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary_on"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not monovary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_mul_le_sum_mul_comp_perm (hfg : antivary_on f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g i ≤ ∑ i in s, f i * g (σ i) | hfg.sum_smul_le_sum_smul_comp_perm hσ | lemma | antivary_on.sum_mul_le_sum_mul_comp_perm | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_mul_eq_sum_mul_comp_perm_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g (σ i) = ∑ i in s, f i * g i ↔ antivary_on f (g ∘ σ) s | hfg.sum_smul_eq_sum_smul_comp_perm_iff hσ | lemma | antivary_on.sum_mul_eq_sum_mul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_mul_lt_sum_mul_comp_perm_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g i < ∑ i in s, f i * g (σ i) ↔ ¬ antivary_on f (g ∘ σ) s | hfg.sum_smul_lt_sum_smul_comp_perm_iff hσ | lemma | antivary_on.sum_mul_lt_sum_mul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_mul_le_sum_comp_perm_mul (hfg : antivary_on f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g i ≤ ∑ i in s, f (σ i) * g i | hfg.sum_smul_le_sum_comp_perm_smul hσ | lemma | antivary_on.sum_mul_le_sum_comp_perm_mul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_mul_eq_sum_comp_perm_mul_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f (σ i) * g i = ∑ i in s, f i * g i ↔ antivary_on (f ∘ σ) g s | hfg.sum_smul_eq_sum_comp_perm_smul_iff hσ | lemma | antivary_on.sum_mul_eq_sum_comp_perm_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.sum_mul_lt_sum_comp_perm_mul_iff (hfg : antivary_on f g s)
(hσ : {x | σ x ≠ x} ⊆ s) :
∑ i in s, f i * g i < ∑ i in s, f (σ i) * g i ↔ ¬ antivary_on (f ∘ σ) g s | hfg.sum_smul_lt_sum_comp_perm_smul_iff hσ | lemma | antivary_on.sum_mul_lt_sum_comp_perm_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary_on"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_mul_comp_perm_le_sum_mul (hfg : monovary f g) :
∑ i, f i * g (σ i) ≤ ∑ i, f i * g i | hfg.sum_smul_comp_perm_le_sum_smul | lemma | monovary.sum_mul_comp_perm_le_sum_mul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_mul_comp_perm_eq_sum_mul_iff (hfg : monovary f g) :
∑ i, f i * g (σ i) = ∑ i, f i * g i ↔ monovary f (g ∘ σ) | hfg.sum_smul_comp_perm_eq_sum_smul_iff | lemma | monovary.sum_mul_comp_perm_eq_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Equality case of Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_mul_comp_perm_lt_sum_mul_iff (hfg : monovary f g) :
∑ i, f i * g (σ i) < ∑ i, f i * g i ↔ ¬ monovary f (g ∘ σ) | hfg.sum_smul_comp_perm_lt_sum_smul_iff | lemma | monovary.sum_mul_comp_perm_lt_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_comp_perm_mul_le_sum_mul (hfg : monovary f g) :
∑ i, f (σ i) * g i ≤ ∑ i, f i * g i | hfg.sum_comp_perm_smul_le_sum_smul | lemma | monovary.sum_comp_perm_mul_le_sum_mul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_comp_perm_mul_eq_sum_mul_iff (hfg : monovary f g) :
∑ i, f (σ i) * g i = ∑ i, f i * g i ↔ monovary (f ∘ σ) g | hfg.sum_comp_perm_smul_eq_sum_smul_iff | lemma | monovary.sum_comp_perm_mul_eq_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Equality case of Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_comp_perm_mul_lt_sum_mul_iff (hfg : monovary f g) :
∑ i, f (σ i) * g i < ∑ i, f i * g i ↔ ¬ monovary (f ∘ σ) g | hfg.sum_comp_perm_smul_lt_sum_smul_iff | lemma | monovary.sum_comp_perm_mul_lt_sum_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"monovary"
] | **Strict inequality case of Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_mul_le_sum_mul_comp_perm (hfg : antivary f g) :
∑ i, f i * g i ≤ ∑ i, f i * g (σ i) | hfg.sum_smul_le_sum_smul_comp_perm | lemma | antivary.sum_mul_le_sum_mul_comp_perm | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_mul_eq_sum_mul_comp_perm_iff (hfg : antivary f g) :
∑ i, f i * g (σ i) = ∑ i, f i * g i ↔ antivary f (g ∘ σ) | hfg.sum_smul_eq_sum_smul_comp_perm_iff | lemma | antivary.sum_mul_eq_sum_mul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_mul_lt_sum_mul_comp_perm_iff (hfg : antivary f g) :
∑ i, f i • g i < ∑ i, f i • g (σ i) ↔ ¬ antivary f (g ∘ σ) | hfg.sum_smul_lt_sum_smul_comp_perm_iff | lemma | antivary.sum_mul_lt_sum_mul_comp_perm_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_mul_le_sum_comp_perm_mul (hfg : antivary f g) :
∑ i, f i * g i ≤ ∑ i, f (σ i) * g i | hfg.sum_smul_le_sum_comp_perm_smul | lemma | antivary.sum_mul_le_sum_comp_perm_mul | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_mul_eq_sum_comp_perm_mul_iff (hfg : antivary f g) :
∑ i, f (σ i) * g i = ∑ i, f i * g i ↔ antivary (f ∘ σ) g | hfg.sum_smul_eq_sum_comp_perm_smul_iff | lemma | antivary.sum_mul_eq_sum_comp_perm_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.sum_mul_lt_sum_comp_perm_mul_iff (hfg : antivary f g) :
∑ i, f i * g i < ∑ i, f (σ i) * g i ↔ ¬ antivary (f ∘ σ) g | hfg.sum_smul_lt_sum_comp_perm_smul_iff | lemma | antivary.sum_mul_lt_sum_comp_perm_mul_iff | algebra.order | src/algebra/order/rearrangement.lean | [
"algebra.big_operators.basic",
"algebra.order.module",
"data.prod.lex",
"group_theory.perm.support",
"order.monotone.monovary",
"tactic.abel"
] | [
"antivary"
] | **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_smul (R M : Type*)
[ordered_semiring R] [ordered_add_comm_monoid M] [smul_with_zero R M] : Prop | (smul_lt_smul_of_pos : ∀ {a b : M}, ∀ {c : R}, a < b → 0 < c → c • a < c • b)
(lt_of_smul_lt_smul_of_pos : ∀ {a b : M}, ∀ {c : R}, c • a < c • b → 0 < c → a < b) | class | ordered_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"ordered_add_comm_monoid",
"ordered_semiring",
"smul_lt_smul_of_pos",
"smul_with_zero"
] | The ordered scalar product property is when an ordered additive commutative monoid
with a partial order has a scalar multiplication which is compatible with the order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_lt_smul_of_pos : a < b → 0 < c → c • a < c • b | ordered_smul.smul_lt_smul_of_pos | lemma | smul_lt_smul_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul_of_nonneg (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c • a ≤ c • b | begin
rcases h₁.eq_or_lt with rfl|hab,
{ refl },
{ rcases h₂.eq_or_lt with rfl|hc,
{ rw [zero_smul, zero_smul] },
{ exact (smul_lt_smul_of_pos hab hc).le } }
end | lemma | smul_le_smul_of_nonneg | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_lt_smul_of_pos",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonneg (hc : 0 ≤ c) (ha : 0 ≤ a) : 0 ≤ c • a | calc (0 : M) = c • (0 : M) : (smul_zero c).symm
... ≤ c • a : smul_le_smul_of_nonneg ha hc | lemma | smul_nonneg | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_le_smul_of_nonneg",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonpos_of_nonneg_of_nonpos (hc : 0 ≤ c) (ha : a ≤ 0) : c • a ≤ 0 | @smul_nonneg R Mᵒᵈ _ _ _ _ _ _ hc ha | lemma | smul_nonpos_of_nonneg_of_nonpos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_smul_eq_smul_of_pos_of_le (h₁ : c • a = c • b) (hc : 0 < c) (hle : a ≤ b) :
a = b | hle.lt_or_eq.resolve_left $ λ hlt, (smul_lt_smul_of_pos hlt hc).ne h₁ | lemma | eq_of_smul_eq_smul_of_pos_of_le | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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