File: Mathlib/Algebra/Order/Rearrangement.lean

full_name: MonovaryOn.sum_smul_comp_perm_le_sum_smul
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `g`. -/
theorem MonovaryOn.sum_smul_comp_perm_le_sum_smul (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i 
goal: ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i

full_name: MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff
code: /-- **Equality case of the 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`. -/
theorem MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ MonovaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ MonovaryOn f (g ∘ ⇑σ) ↑s

full_name: MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ ⇑σ) ↑s

full_name: MonovaryOn.sum_comp_perm_smul_le_sum_smul
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `f`. -/
theorem MonovaryOn.sum_comp_perm_smul_le_sum_smul (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f (σ i) • g i ≤ ∑ i ∈ s, f i • g i 
goal: ∑ i ∈ s, f (σ i) • g i ≤ ∑ i ∈ s, f i • g i

full_name: MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff
code: /-- **Equality case of the 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`. -/
theorem MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s

full_name: MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ ⇑σ) g ↑s

full_name: AntivaryOn.sum_smul_le_sum_smul_comp_perm
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `g`. -/
theorem AntivaryOn.sum_smul_le_sum_smul_comp_perm (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f i • g (σ i) 
goal: ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f i • g (σ i)

full_name: AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ AntivaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ AntivaryOn f (g ∘ ⇑σ) ↑s

full_name: AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i • g i < ∑ i ∈ s, f i • g (σ i) ↔ ¬AntivaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i • g i < ∑ i ∈ s, f i • g (σ i) ↔ ¬AntivaryOn f (g ∘ ⇑σ) ↑s

full_name: AntivaryOn.sum_smul_le_sum_comp_perm_smul
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `f`. -/
theorem AntivaryOn.sum_smul_le_sum_comp_perm_smul (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f (σ i) • g i 
goal: ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f (σ i) • g i

full_name: AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ AntivaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ AntivaryOn (f ∘ ⇑σ) g ↑s

full_name: AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i • g i < ∑ i ∈ s, f (σ i) • g i ↔ ¬AntivaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f i • g i < ∑ i ∈ s, f (σ i) • g i ↔ ¬AntivaryOn (f ∘ ⇑σ) g ↑s

full_name: Monovary.sum_smul_comp_perm_le_sum_smul
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `g`. -/
theorem Monovary.sum_smul_comp_perm_le_sum_smul (hfg : Monovary f g) :
    ∑ i, f i • g (σ i) ≤ ∑ i, f i • g i 
goal: ∑ i : ι, f i • g (σ i) ≤ ∑ i : ι, f i • g i

full_name: Monovary.sum_smul_comp_perm_eq_sum_smul_iff
code: /-- **Equality case of the 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`. -/
theorem 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) 
goal: ∑ i : ι, f i • g (σ i) = ∑ i : ι, f i • g i ↔ Monovary f (g ∘ ⇑σ)

full_name: Monovary.sum_smul_comp_perm_lt_sum_smul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem 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) 
goal: ∑ i : ι, f i • g (σ i) < ∑ i : ι, f i • g i ↔ ¬Monovary f (g ∘ ⇑σ)

full_name: Monovary.sum_comp_perm_smul_le_sum_smul
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together. Stated by permuting the entries of `f`. -/
theorem Monovary.sum_comp_perm_smul_le_sum_smul (hfg : Monovary f g) :
    ∑ i, f (σ i) • g i ≤ ∑ i, f i • g i 
goal: ∑ i : ι, f (σ i) • g i ≤ ∑ i : ι, f i • g i

full_name: Monovary.sum_comp_perm_smul_eq_sum_smul_iff
code: /-- **Equality case of the 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`. -/
theorem 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 
goal: ∑ i : ι, f (σ i) • g i = ∑ i : ι, f i • g i ↔ Monovary (f ∘ ⇑σ) g

full_name: Monovary.sum_comp_perm_smul_lt_sum_smul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem 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 
goal: ∑ i : ι, f (σ i) • g i < ∑ i : ι, f i • g i ↔ ¬Monovary (f ∘ ⇑σ) g

full_name: Antivary.sum_smul_le_sum_smul_comp_perm
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `g`. -/
theorem Antivary.sum_smul_le_sum_smul_comp_perm (hfg : Antivary f g) :
    ∑ i, f i • g i ≤ ∑ i, f i • g (σ i) 
goal: ∑ i : ι, f i • g i ≤ ∑ i : ι, f i • g (σ i)

full_name: Antivary.sum_smul_comp_perm_eq_sum_smul_iff
code: /-- **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`. -/
theorem Antivary.sum_smul_comp_perm_eq_sum_smul_iff (hfg : Antivary f g) :
    ∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ Antivary f (g ∘ σ) 
goal: ∑ i : ι, f i • g (σ i) = ∑ i : ι, f i • g i ↔ Antivary f (g ∘ ⇑σ)

full_name: Antivary.sum_smul_lt_sum_smul_comp_perm_iff
code: /-- **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`. -/
theorem 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) 
goal: ∑ i : ι, f i • g i < ∑ i : ι, f i • g (σ i) ↔ ¬Antivary f (g ∘ ⇑σ)

full_name: Antivary.sum_smul_le_sum_comp_perm_smul
code: /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `f`. -/
theorem Antivary.sum_smul_le_sum_comp_perm_smul (hfg : Antivary f g) :
    ∑ i, f i • g i ≤ ∑ i, f (σ i) • g i 
goal: ∑ i : ι, f i • g i ≤ ∑ i : ι, f (σ i) • g i

full_name: Antivary.sum_comp_perm_smul_eq_sum_smul_iff
code: /-- **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`. -/
theorem Antivary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : Antivary f g) :
    ∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ Antivary (f ∘ σ) g 
goal: ∑ i : ι, f (σ i) • g i = ∑ i : ι, f i • g i ↔ Antivary (f ∘ ⇑σ) g

full_name: Antivary.sum_smul_lt_sum_comp_perm_smul_iff
code: /-- **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`. -/
theorem 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 
goal: ∑ i : ι, f i • g i < ∑ i : ι, f (σ i) • g i ↔ ¬Antivary (f ∘ ⇑σ) g

full_name: MonovaryOn.sum_mul_comp_perm_le_sum_mul
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `g`. -/
theorem MonovaryOn.sum_mul_comp_perm_le_sum_mul (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i * g (σ i) ≤ ∑ i ∈ s, f i * g i 
goal: ∑ i ∈ s, f i * g (σ i) ≤ ∑ i ∈ s, f i * g i

full_name: MonovaryOn.sum_mul_comp_perm_eq_sum_mul_iff
code: /-- **Equality case of the 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`. -/
theorem MonovaryOn.sum_mul_comp_perm_eq_sum_mul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ MonovaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ MonovaryOn f (g ∘ ⇑σ) ↑s

full_name: MonovaryOn.sum_mul_comp_perm_lt_sum_mul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem MonovaryOn.sum_mul_comp_perm_lt_sum_mul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ ⇑σ) ↑s

full_name: MonovaryOn.sum_comp_perm_mul_le_sum_mul
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `f`. -/
theorem MonovaryOn.sum_comp_perm_mul_le_sum_mul (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f (σ i) * g i ≤ ∑ i ∈ s, f i * g i 
goal: ∑ i ∈ s, f (σ i) * g i ≤ ∑ i ∈ s, f i * g i

full_name: MonovaryOn.sum_comp_perm_mul_eq_sum_mul_iff
code: /-- **Equality case of the 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`. -/
theorem MonovaryOn.sum_comp_perm_mul_eq_sum_mul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ MonovaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s

full_name: MonovaryOn.sum_comp_perm_mul_lt_sum_mul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem MonovaryOn.sum_comp_perm_mul_lt_sum_mul_iff (hfg : MonovaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f (σ i) * g i < ∑ i ∈ s, f i * g i ↔ ¬MonovaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f (σ i) * g i < ∑ i ∈ s, f i * g i ↔ ¬MonovaryOn (f ∘ ⇑σ) g ↑s

full_name: AntivaryOn.sum_mul_le_sum_mul_comp_perm
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `g`. -/
theorem AntivaryOn.sum_mul_le_sum_mul_comp_perm (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i * g i ≤ ∑ i ∈ s, f i * g (σ i) 
goal: ∑ i ∈ s, f i * g i ≤ ∑ i ∈ s, f i * g (σ i)

full_name: AntivaryOn.sum_mul_eq_sum_mul_comp_perm_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_mul_eq_sum_mul_comp_perm_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ AntivaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ AntivaryOn f (g ∘ ⇑σ) ↑s

full_name: AntivaryOn.sum_mul_lt_sum_mul_comp_perm_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_mul_lt_sum_mul_comp_perm_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i * g i < ∑ i ∈ s, f i * g (σ i) ↔ ¬AntivaryOn f (g ∘ σ) s 
goal: ∑ i ∈ s, f i * g i < ∑ i ∈ s, f i * g (σ i) ↔ ¬AntivaryOn f (g ∘ ⇑σ) ↑s

full_name: AntivaryOn.sum_mul_le_sum_comp_perm_mul
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `f`. -/
theorem AntivaryOn.sum_mul_le_sum_comp_perm_mul (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i * g i ≤ ∑ i ∈ s, f (σ i) * g i 
goal: ∑ i ∈ s, f i * g i ≤ ∑ i ∈ s, f (σ i) * g i

full_name: AntivaryOn.sum_comp_perm_mul_eq_sum_mul_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_comp_perm_mul_eq_sum_mul_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ AntivaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ AntivaryOn (f ∘ ⇑σ) g ↑s

full_name: AntivaryOn.sum_mul_lt_sum_comp_perm_mul_iff
code: /-- **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`. -/
theorem AntivaryOn.sum_mul_lt_sum_comp_perm_mul_iff (hfg : AntivaryOn f g s)
    (hσ : {x | σ x ≠ x} ⊆ s) :
    ∑ i ∈ s, f i * g i < ∑ i ∈ s, f (σ i) * g i ↔ ¬AntivaryOn (f ∘ σ) g s 
goal: ∑ i ∈ s, f i * g i < ∑ i ∈ s, f (σ i) * g i ↔ ¬AntivaryOn (f ∘ ⇑σ) g ↑s

full_name: Monovary.sum_mul_comp_perm_le_sum_mul
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `g`. -/
theorem Monovary.sum_mul_comp_perm_le_sum_mul (hfg : Monovary f g) :
    ∑ i, f i * g (σ i) ≤ ∑ i, f i * g i 
goal: ∑ i : ι, f i * g (σ i) ≤ ∑ i : ι, f i * g i

full_name: Monovary.sum_mul_comp_perm_eq_sum_mul_iff
code: /-- **Equality case of the 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`. -/
theorem 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) 
goal: ∑ i : ι, f i * g (σ i) = ∑ i : ι, f i * g i ↔ Monovary f (g ∘ ⇑σ)

full_name: Monovary.sum_mul_comp_perm_lt_sum_mul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem 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) 
goal: ∑ i : ι, f i * g (σ i) < ∑ i : ι, f i * g i ↔ ¬Monovary f (g ∘ ⇑σ)

full_name: Monovary.sum_comp_perm_mul_le_sum_mul
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
`g` monovary together. Stated by permuting the entries of `f`. -/
theorem Monovary.sum_comp_perm_mul_le_sum_mul (hfg : Monovary f g) :
    ∑ i, f (σ i) * g i ≤ ∑ i, f i * g i 
goal: ∑ i : ι, f (σ i) * g i ≤ ∑ i : ι, f i * g i

full_name: Monovary.sum_comp_perm_mul_eq_sum_mul_iff
code: /-- **Equality case of the 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`. -/
theorem 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 
goal: ∑ i : ι, f (σ i) * g i = ∑ i : ι, f i * g i ↔ Monovary (f ∘ ⇑σ) g

full_name: Monovary.sum_comp_perm_mul_lt_sum_mul_iff
code: /-- **Strict inequality case of the 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`. -/
theorem 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 
goal: ∑ i : ι, f (σ i) * g i < ∑ i : ι, f i * g i ↔ ¬Monovary (f ∘ ⇑σ) g

full_name: Antivary.sum_mul_le_sum_mul_comp_perm
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `g`. -/
theorem Antivary.sum_mul_le_sum_mul_comp_perm (hfg : Antivary f g) :
    ∑ i, f i * g i ≤ ∑ i, f i * g (σ i) 
goal: ∑ i : ι, f i * g i ≤ ∑ i : ι, f i * g (σ i)

full_name: Antivary.sum_mul_eq_sum_mul_comp_perm_iff
code: /-- **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`. -/
theorem 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) 
goal: ∑ i : ι, f i * g (σ i) = ∑ i : ι, f i * g i ↔ Antivary f (g ∘ ⇑σ)

full_name: Antivary.sum_mul_lt_sum_mul_comp_perm_iff
code: /-- **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`. -/
theorem 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) 
goal: ∑ i : ι, f i • g i < ∑ i : ι, f i • g (σ i) ↔ ¬Antivary f (g ∘ ⇑σ)

full_name: Antivary.sum_mul_le_sum_comp_perm_mul
code: /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
`g` antivary together. Stated by permuting the entries of `f`. -/
theorem Antivary.sum_mul_le_sum_comp_perm_mul (hfg : Antivary f g) :
    ∑ i, f i * g i ≤ ∑ i, f (σ i) * g i 
goal: ∑ i : ι, f i * g i ≤ ∑ i : ι, f (σ i) * g i

full_name: Antivary.sum_comp_perm_mul_eq_sum_mul_iff
code: /-- **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`. -/
theorem Antivary.sum_comp_perm_mul_eq_sum_mul_iff (hfg : Antivary f g) :
    ∑ i, f (σ i) * g i = ∑ i, f i * g i ↔ Antivary (f ∘ σ) g 
goal: ∑ i : ι, f (σ i) * g i = ∑ i : ι, f i * g i ↔ Antivary (f ∘ ⇑σ) g

full_name: Antivary.sum_mul_lt_sum_comp_perm_mul_iff
code: /-- **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`. -/
theorem 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 
goal: ∑ i : ι, f i * g i < ∑ i : ι, f (σ i) * g i ↔ ¬Antivary (f ∘ ⇑σ) g