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refactor(linear_algebra/affine_space): move def of
slope to a new f…
…ile (#11361) Also add a few trivial lemmas.
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| Original file line number | Diff line number | Diff line change |
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| /- | ||
| Copyright (c) 2020 Yury G. Kudryashov. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Yury G. Kudryashov | ||
| -/ | ||
| import algebra.order.module | ||
| import linear_algebra.affine_space.affine_map | ||
| import tactic.field_simp | ||
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| /-! | ||
| # 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 | ||
| -/ | ||
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| open affine_map | ||
| variables {k E PE : Type*} [field k] [add_comm_group E] [module k E] [add_torsor E PE] | ||
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| include E | ||
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| /-- `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`. -/ | ||
| def slope (f : k → PE) (a b : k) : E := (b - a)⁻¹ • (f b -ᵥ f a) | ||
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| omit E | ||
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| lemma slope_def_field (f : k → k) (a b : k) : slope f a b = (f b - f a) / (b - a) := | ||
| div_eq_inv_mul.symm | ||
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| @[simp] lemma slope_same (f : k → PE) (a : k) : (slope f a a : E) = 0 := | ||
| by rw [slope, sub_self, inv_zero, zero_smul] | ||
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| include E | ||
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| lemma slope_def_module (f : k → E) (a b : k) : slope f a b = (b - a)⁻¹ • (f b - f a) := rfl | ||
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| @[simp] lemma sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a := | ||
| begin | ||
| rcases eq_or_ne a b with rfl | hne, | ||
| { rw [sub_self, zero_smul, vsub_self] }, | ||
| { rw [slope, smul_inv_smul₀ (sub_ne_zero.2 hne.symm)] } | ||
| end | ||
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| lemma sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b := | ||
| by rw [sub_smul_slope, vsub_vadd] | ||
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| @[simp] lemma slope_vadd_const (f : k → E) (c : PE) : | ||
| slope (λ x, f x +ᵥ c) = slope f := | ||
| begin | ||
| ext a b, | ||
| simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub] | ||
| end | ||
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| @[simp] lemma slope_sub_smul (f : k → E) {a b : k} (h : a ≠ b): | ||
| slope (λ x, (x - a) • f x) a b = f b := | ||
| by simp [slope, inv_smul_smul₀ (sub_ne_zero.2 h.symm)] | ||
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| lemma eq_of_slope_eq_zero {f : k → PE} {a b : k} (h : slope f a b = (0:E)) : f a = f b := | ||
| by rw [← sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd] | ||
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| lemma slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := | ||
| by rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub] | ||
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| /-- `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 `line_map_slope_slope_sub_div_sub`. -/ | ||
| lemma sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (f : k → PE) (a b c : k) : | ||
| ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c := | ||
| begin | ||
| by_cases hab : a = b, | ||
| { subst hab, | ||
| rw [sub_self, zero_div, zero_smul, zero_add], | ||
| by_cases hac : a = c, | ||
| { simp [hac] }, | ||
| { rw [div_self (sub_ne_zero.2 $ ne.symm hac), one_smul] } }, | ||
| by_cases hbc : b = c, { subst hbc, simp [sub_ne_zero.2 (ne.symm hab)] }, | ||
| rw [add_comm], | ||
| simp_rw [slope, div_eq_inv_mul, mul_smul, ← smul_add, | ||
| smul_inv_smul₀ (sub_ne_zero.2 $ ne.symm hab), smul_inv_smul₀ (sub_ne_zero.2 $ ne.symm hbc), | ||
| vsub_add_vsub_cancel], | ||
| end | ||
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| /-- `slope f a c` is an affine combination of `slope f a b` and `slope f b c`. This version uses | ||
| `line_map` to express this property. -/ | ||
| lemma line_map_slope_slope_sub_div_sub (f : k → PE) (a b c : k) (h : a ≠ c) : | ||
| line_map (slope f a b) (slope f b c) ((c - b) / (c - a)) = slope f a c := | ||
| by field_simp [sub_ne_zero.2 h.symm, ← sub_div_sub_smul_slope_add_sub_div_sub_smul_slope f a b c, | ||
| line_map_apply_module] | ||
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| /-- `slope f a b` is an affine combination of `slope f a (line_map a b r)` and | ||
| `slope f (line_map a b r) b`. We use `line_map` to express this property. -/ | ||
| lemma line_map_slope_line_map_slope_line_map (f : k → PE) (a b r : k) : | ||
| line_map (slope f (line_map a b r) b) (slope f a (line_map a b r)) r = slope f a b := | ||
| begin | ||
| obtain (rfl|hab) : a = b ∨ a ≠ b := classical.em _, { simp }, | ||
| rw [slope_comm _ a, slope_comm _ a, slope_comm _ _ b], | ||
| convert line_map_slope_slope_sub_div_sub f b (line_map a b r) a hab.symm using 2, | ||
| rw [line_map_apply_ring, eq_div_iff (sub_ne_zero.2 hab), sub_mul, one_mul, mul_sub, ← sub_sub, | ||
| sub_sub_cancel] | ||
| end |