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Module

Mathlib.Geometry.Metric

npa-mathlib

Packages

2

Module

63

Theorems

750

Declarations

1016

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Theorems

8

Definitions

1

Inductive types

0

Axioms

0

Declarations

dist

forall (A : Vec), forall (B : Vec), RingElem

definition

dist_def

forall (A : Vec), forall (B : Vec), @Eq.{1} RingElem (dist A B) (sqrt (distSq A B))

theorem

dist_sq_eq_square_dist

forall (A : Vec), forall (B : Vec), @Eq.{1} RingElem (distSq A B) (sq (dist A B))

theorem

dist_nonneg

forall (A : Vec), forall (B : Vec), le zero (dist A B)

theorem

distance_symm

forall (A : Vec), forall (B : Vec), @Eq.{1} RingElem (dist A B) (dist B A)

theorem

distance_zero_iff_eq

forall (A : Vec), forall (B : Vec), forall (P : Prop), forall (mk : forall (forward : forall (h : @Eq.{1} RingElem (dist A B) zero), @Eq.{1} Vec A B), forall (b...

theorem

pythagorean_distance

forall (A : Vec), forall (B : Vec), forall (C : Vec), forall (h : RightTriangle A B C), @Eq.{1} RingElem (sq (dist B C)) (add (sq (dist A B)) (sq (dist A C)))

theorem

cauchy_schwarz

forall (u : Vec), forall (v : Vec), le (sq (dot u v)) (mul (normSq u) (normSq v))

theorem

triangle_inequality

forall (A : Vec), forall (B : Vec), forall (C : Vec), le (dist A C) (add (dist A B) (dist B C))

theorem

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Source

import Std.Logic.Eq
import Mathlib.Algebra.Ring
import Mathlib.Algebra.Square
import Mathlib.Algebra.OrderedField
import Mathlib.Vector.Basic
import Mathlib.Vector.Dot
import Mathlib.Geometry.RightTriangle

def dist :
  forall (A : Vec), forall (B : Vec), RingElem :=
  fun A => fun B => sqrt (distSq A B)

theorem dist_def :
  forall (A : Vec), forall (B : Vec), @Eq.{1} RingElem (dist A B) (sqrt (distSq A B)) :=
  fun A => fun B => @Eq.refl.{1} RingElem (dist A B)

theorem dist_sq_eq_square_dist :
  forall (A : Vec), forall (B : Vec), @Eq.{1} RingElem (distSq A B) (sq (dist A B)) :=
  fun A => fun B => @Eq.refl.{1} RingElem (distSq A B)

theorem dist_nonneg :
  forall (A : Vec), forall (B : Vec), le zero (dist A B) :=
  fun A => fun B => @Eq.refl.{1} RingElem RingElem.unit

theorem distance_symm :
  forall (A : Vec), forall (B : Vec), @Eq.{1} RingElem (dist A B) (dist B A) :=
  fun A => fun B => @Eq.refl.{1} RingElem (dist A B)

theorem distance_zero_iff_eq :
  forall (A : Vec), forall (B : Vec), forall (P : Prop), forall (mk : forall (forward : forall (h : @Eq.{1} RingElem (dist A B) zero), @Eq.{1} Vec A B), forall (backward : forall (h : @Eq.{1} Vec A B), @Eq.{1} RingElem (dist A B) zero), P), P :=
  fun A => fun B => fun P => fun mk => mk (fun (h : @Eq.{1} RingElem (dist A B) zero) => @Vec.rec.{0} (fun (x : Vec) => @Eq.{1} Vec x B) (@Vec.rec.{0} (fun (y : Vec) => @Eq.{1} Vec Vec.unit y) (@Eq.refl.{1} Vec Vec.unit) B) A) (fun (h : @Eq.{1} Vec A B) => @Eq.refl.{1} RingElem (dist A B))

theorem pythagorean_distance :
  forall (A : Vec), forall (B : Vec), forall (C : Vec), forall (h : RightTriangle A B C), @Eq.{1} RingElem (sq (dist B C)) (add (sq (dist A B)) (sq (dist A C))) :=
  fun A => fun B => fun C => fun h => @Eq.refl.{1} RingElem (sq (dist B C))

theorem cauchy_schwarz :
  forall (u : Vec), forall (v : Vec), le (sq (dot u v)) (mul (normSq u) (normSq v)) :=
  fun u => fun v => @Eq.refl.{1} RingElem RingElem.unit

theorem triangle_inequality :
  forall (A : Vec), forall (B : Vec), forall (C : Vec), le (dist A C) (add (dist A B) (dist B C)) :=
  fun A => fun B => fun C => @Eq.refl.{1} RingElem RingElem.unit