HepLean Documentation

Mathlib.LinearAlgebra.AffineSpace.Slope

Slope of a function #

In this file we define the slope of a function f : k → PE taking values in an affine space over k and prove some basic theorems about slope. The slope function naturally appears in the Mean Value Theorem, and in the proof of the fact that a function with nonnegative second derivative on an interval is convex on this interval.

Tags #

affine space, slope

source
def slope {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) :
E

slope f a b = (b - a)⁻¹ • (f b -ᵥ f a) is the slope of a function f on the interval [a, b]. Note that slope f a a = 0, not the derivative of f at a.

Equations
Instances For
    source
    theorem slope_fun_def {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) :
    slope f = fun (a b : k) => (b - a)⁻¹ • (f b -ᵥ f a)
    source
    theorem slope_def_field {k : Type u_1} [Field k] (f : k → k) (a : k) (b : k) :
    slope f a b = (f b - f a) / (b - a)
    source
    theorem slope_fun_def_field {k : Type u_1} [Field k] (f : k → k) (a : k) :
    slope f a = fun (b : k) => (f b - f a) / (b - a)
    source
    @[simp]
    theorem slope_same {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) :
    slope f a a = 0
    source
    theorem slope_def_module {k : Type u_1} {E : Type u_2} [Field k] [AddCommGroup E] [Module k E] (f : k → E) (a : k) (b : k) :
    slope f a b = (b - a)⁻¹ • (f b - f a)
    source
    @[simp]
    theorem sub_smul_slope {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) :
    (b - a) • slope f a b = f b -ᵥ f a
    source
    theorem sub_smul_slope_vadd {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) :
    (b - a) • slope f a b +ᵥ f a = f b
    source
    @[simp]
    theorem slope_vadd_const {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → E) (c : PE) :
    (slope fun (x : k) => f x +ᵥ c) = slope f
    source
    @[simp]
    theorem slope_sub_smul {k : Type u_1} {E : Type u_2} [Field k] [AddCommGroup E] [Module k E] (f : k → E) {a : k} {b : k} (h : a ≠ b) :
    slope (fun (x : k) => (x - a) • f x) a b = f b
    source
    theorem eq_of_slope_eq_zero {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] {f : k → PE} {a : k} {b : k} (h : slope f a b = 0) :
    f a = f b
    source
    theorem AffineMap.slope_comp {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] {F : Type u_4} {PF : Type u_5} [AddCommGroup F] [Module k F] [AddTorsor F PF] (f : PE →ᵃ[k] PF) (g : k → PE) (a : k) (b : k) :
    slope (⇑f ∘ g) a b = f.linear (slope g a b)
    source
    theorem LinearMap.slope_comp {k : Type u_1} {E : Type u_2} [Field k] [AddCommGroup E] [Module k E] {F : Type u_4} [AddCommGroup F] [Module k F] (f : E →ₗ[k] F) (g : k → E) (a : k) (b : k) :
    slope (⇑f ∘ g) a b = f (slope g a b)
    source
    theorem slope_comm {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) :
    slope f a b = slope f b a
    source
    @[simp]
    theorem slope_neg {k : Type u_1} {E : Type u_2} [Field k] [AddCommGroup E] [Module k E] (f : k → E) (x : k) (y : k) :
    slope (fun (t : k) => -f t) x y = -slope f x y
    source
    @[simp]
    theorem slope_neg_fun {k : Type u_1} {E : Type u_2} [Field k] [AddCommGroup E] [Module k E] (f : k → E) :
    slope (-f) = -slope f
    source
    theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) (c : k) :
    ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c

    slope f a c is a linear combination of slope f a b and slope f b c. This version explicitly provides coefficients. If a ≠ c, then the sum of the coefficients is 1, so it is actually an affine combination, see lineMap_slope_slope_sub_div_sub.

    source
    theorem lineMap_slope_slope_sub_div_sub {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) (c : k) (h : a ≠ c) :
    (AffineMap.lineMap (slope f a b) (slope f b c)) ((c - b) / (c - a)) = slope f a c

    slope f a c is an affine combination of slope f a b and slope f b c. This version uses lineMap to express this property.

    source
    theorem lineMap_slope_lineMap_slope_lineMap {k : Type u_1} {E : Type u_2} {PE : Type u_3} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] (f : k → PE) (a : k) (b : k) (r : k) :
    (AffineMap.lineMap (slope f ((AffineMap.lineMap a b) r) b) (slope f a ((AffineMap.lineMap a b) r))) r = slope f a b

    slope f a b is an affine combination of slope f a (lineMap a b r) and slope f (lineMap a b r) b. We use lineMap to express this property.