reverse_involution

formal source

  35def reverse : Fin 8 → Fin 8 := fun k => ⟨7 - k.val, by omega⟩
  36
  37/-- Reversal is an involution: reverse ∘ reverse = id. -/
  38theorem reverse_involution (k : Fin 8) : reverse (reverse k) = k := by
  39  apply Fin.ext
  40  simp only [reverse]
  41  omega
  42
  43/-- Reversal swaps first and last. -/
  44theorem reverse_swaps_endpoints :
  45    reverse ⟨0, by decide⟩ = ⟨7, by decide⟩ ∧
  46    reverse ⟨7, by decide⟩ = ⟨0, by decide⟩ := by
  47  refine ⟨?_, ?_⟩
  48  · apply Fin.ext; simp [reverse]
  49  · apply Fin.ext; simp [reverse]
  50
  51/-- Mid-life stages invert before early ones:
  52    reverse of stage 6 (generativity) = stage 1 (autonomy),
  53    reverse of stage 0 (trust) = stage 7 (integrity).
  54    So going in reverse from old-age, we pass stage 6's inverse at
  55    position 1 before stage 0's inverse at position 7. -/
  56theorem midlife_inverts_first :
  57    (reverse ⟨6, by decide⟩).val < (reverse ⟨0, by decide⟩).val := by
  58  simp [reverse]
  59
  60structure DevelopmentReversalCert where
  61  stage_count : Fintype.card EriksonStage = 8
  62  two_cube : Fintype.card EriksonStage = 2 ^ 3
  63  involution : ∀ k : Fin 8, reverse (reverse k) = k
  64  endpoints_swap : reverse ⟨0, by decide⟩ = ⟨7, by decide⟩
  65  midlife_first : (reverse ⟨6, by decide⟩).val < (reverse ⟨0, by decide⟩).val
  66
  67def developmentReversalCert : DevelopmentReversalCert where
  68  stage_count := eriksonCount