`Chain`

definition

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module: IndisputableMonolith.Recognition
domain: Recognition
line: 35 · github
papers citing: none yet

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IndisputableMonolith.Recognition on GitHub at line 35.

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All declarations in this module, on Recognition.

explainer page

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depends on

used by

formal source

  32  U : Type
  33  R : U → U → Prop
  34
  35structure Chain (M : RecognitionStructure) where
  36  n : Nat
  37  f : Fin (n+1) → M.U
  38  ok : ∀ i : Fin n, M.R (f i.castSucc) (f i.succ)
  39
  40namespace Chain
  41variable {M : RecognitionStructure} (ch : Chain M)
  42def head : M.U := by
  43  have hpos : 0 < ch.n + 1 := Nat.succ_pos _
  44  exact ch.f ⟨0, hpos⟩
  45def last : M.U := by
  46  have hlt : ch.n < ch.n + 1 := Nat.lt_succ_self _
  47  exact ch.f ⟨ch.n, hlt⟩
  48end Chain
  49
  50structure Ledger (M : RecognitionStructure) where
  51  debit : M.U → ℤ
  52  credit : M.U → ℤ
  53
  54@[simp] def Ledger.Carrier {M : RecognitionStructure} (_ : Ledger M) : Type :=
  55  M.U
  56
  57def phi {M} (L : Ledger M) : M.U → ℤ := fun u => L.debit u - L.credit u
  58
  59def chainFlux {M} (L : Ledger M) (ch : Chain M) : ℤ :=
  60  phi L (Chain.last ch) - phi L (Chain.head ch)
  61
  62class Conserves {M} (L : Ledger M) : Prop where
  63  conserve : ∀ ch : Chain M, ch.head = ch.last → chainFlux L ch = 0
  64
  65lemma chainFlux_zero_of_loop {M} (L : Ledger M) [Conserves L] (ch : Chain M) (h : ch.head = ch.last) :

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