`mp_holds`

proved

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module: IndisputableMonolith.Recognition
domain: Recognition
line: 19 · github
papers citing: none yet

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IndisputableMonolith.Recognition on GitHub at line 19.

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All declarations in this module, on Recognition.

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formal source

  16/-- MP: It is impossible for Nothing to recognize itself. -/
  17def MP : Prop := ¬ ∃ _ : Recognize Nothing Nothing, True
  18
  19theorem mp_holds : MP := by
  20  intro h
  21  rcases h with ⟨⟨r, _⟩, _⟩
  22  cases r
  23
  24/-- Alias used in the manuscript: "Nonexistence cannot recognize itself." -/
  25abbrev NothingCannotRecognizeItself : Prop := MP
  26
  27/-- Direct alias proof of MP for the manuscript phrasing. -/
  28theorem nothing_cannot_recognize_itself : NothingCannotRecognizeItself :=
  29  mp_holds
  30
  31structure RecognitionStructure where
  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

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