IndisputableMonolith.Causality.Basic

   1import Mathlib
   2
   3namespace IndisputableMonolith
   4namespace Causality
   5
   6variable {α : Type}
   7
   8structure Kinematics (α : Type) where
   9  step : α → α → Prop
  10
  11inductive ReachN (K : Kinematics α) : Nat → α → α → Prop
  12| zero {x} : ReachN K 0 x x
  13| succ {n x y z} : ReachN K n x y → K.step y z → ReachN K (n+1) x z
  14
  15def inBall (K : Kinematics α) (x : α) (n : Nat) (y : α) : Prop :=
  16  ∃ k ≤ n, ReachN K k x y
  17
  18lemma reach_in_ball {K : Kinematics α} {x y : α} {n : Nat}
  19  (h : ReachN K n x y) : inBall K x n y := ⟨n, le_rfl, h⟩
  20
  21lemma reach_le_in_ball {K : Kinematics α} {x y : α} {k n : Nat}
  22  (hk : k ≤ n) (h : ReachN K k x y) : inBall K x n y := ⟨k, hk, h⟩
  23
  24def Reaches (K : Kinematics α) (x y : α) : Prop := ∃ n, ReachN K n x y
  25
  26lemma reaches_of_reachN {K : Kinematics α} {x y : α} {n : Nat}
  27  (h : ReachN K n x y) : Reaches K x y := ⟨n, h⟩
  28
  29lemma inBall_mono {K : Kinematics α} {x y : α} {n m : Nat}
  30  (hnm : n ≤ m) : inBall K x n y → inBall K x m y := by
  31  intro ⟨k, hk, hkreach⟩
  32  exact ⟨k, le_trans hk hnm, hkreach⟩
  33
  34end Causality
  35end IndisputableMonolith
  36