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jonaprieto avatar jonaprieto commented on June 2, 2024

The number calls following the recursive definition may seems more likely to this one:

contraction₀ : Prop  Prop
contraction₀ (suc n) φ with contra-view φ
... | impl φ₁ φ₂ φ₃ = contraction₀ n ((φ₁ ∧ φ₂) ⇒ φ₃)
... | other _       = φ
contraction₀ zero φ = φ

steps-contraction : Prop  ℕ
steps-contraction φ with contra-view φ
steps-contraction .(φ₁ ⇒ (φ₂ ⇒ φ₃))      | impl φ₁ φ₂ φ₃ with impl-view φ₃
steps-contraction .(φ₁ ⇒ (φ₂ ⇒ (φ ⇒ ψ))) | impl φ₁ φ₂ .(φ ⇒ ψ) | (impl φ ψ) =
  2 + steps-contraction (φ ⇒ ψ)
steps-contraction .(φ₁ ⇒ (φ₂ ⇒ φ₃))      | impl φ₁ φ₂ φ₃ | (other .φ₃) = 2
steps-contraction φ                      | other .φ      = 1

Something like counting the number of chains implications.

from agda-metis.

jonaprieto avatar jonaprieto commented on June 2, 2024

I renamed the steps-contraction to ♯contractions and simplify the counting.

♯contractions : Prop  ℕ
♯contractions φ with contra-view φ
♯contractions .(φ₁ ⇒ (φ₂ ⇒ φ₃)) | impl φ₁ φ₂ φ₃ = 2 + ♯contractions φ₃
♯contractions φ                 | other .φ      = 1

from agda-metis.

jonaprieto avatar jonaprieto commented on June 2, 2024
p ==> (q ==> r) 
-------------
p /\ q  ==> r
  • Consider this case
p ==> (q /\ r) 
-------------
p ==> q   and  p ==> r

from agda-metis.

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